Subgroups — C₃×#10596244 #13179524

Order 1728.

1844 conjugacy classes of subgroups.

The subgroup lattice has 666 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#41×C₃⋊D₃	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₃×#3830	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃³×AGL(1,ℤ/8)	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#64160	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×#64141	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃×#64125	C₂	1	2	yes	maximal, complemented
7	C₃×#64124	C₂	1	2	yes	maximal, complemented
8	C₃×#64121	C₂	1	2	yes	maximal, complemented
9	C₃×#64120	C₂	1	2	yes	maximal, complemented
10	C₃×#64118	C₂	1	1	yes	maximal, characteristic, complemented
11	C₃×M₁₆×C₃⋊D₃	C₂	1	1	yes	maximal, characteristic, complemented
12	#10596244	C₃	1	1	yes	maximal, characteristic, direct product factor
13	C₃×#5773	—	3	4	no	maximal
14	C₃³×#41	C₂²	1	1	yes	characteristic
15	C₂×C₃³×D₄	C₂²	1	1	yes	characteristic, complemented
16	C₂×C₃×#638	—	2	1	no	—
17	C₂²×C₃×C₃⋊Dic₃	—	2	1	no	—
18	C₃×#3832	C₂²	1	1	yes	characteristic, complemented
19	C₃×Q₈×C₃⋊D₃	C₂²	1	1	yes	characteristic, complemented
20	C₃×#3830	—	2	1	no	—
21	C₃×#3830	—	2	1	no	—
22	C₃×#3830	—	2	1	no	—
23	C₃×#3830	C₂²	1	2	yes	—
24	C₃×#3830	C₂²	1	1	yes	characteristic, complemented
25	C₃×#3830	C₂²	1	2	yes	complemented
26	C₃×D₄×C₃⋊D₃	—	2	1	no	—
27	C₃×D₄×C₃⋊D₃	C₂²	1	1	yes	characteristic, complemented
28	C₃×#3828	—	2	1	no	—
29	C₃×#3828	C₂²	1	1	yes	characteristic
30	C₂×C₃×C₄×C₃⋊D₃	—	2	1	no	—
31	C₂×C₃×C₄×C₃⋊D₃	C₂²	1	1	yes	characteristic
32	C₂×C₃×#634	C₂²	1	1	yes	characteristic, complemented
33	C₃³×SD₁₆	C₂²	1	2	yes	complemented
34	C₃³×D₈	C₂²	1	2	yes	complemented
35	C₃³×M₁₆	C₂²	1	1	yes	characteristic, complemented
36	C₃×#3756	C₂²	1	2	yes	complemented
37	C₃×#3754	C₂²	1	2	yes	complemented
38	C₃×#3754	C₂²	1	2	yes	complemented
39	C₃×#3753	C₂²	1	2	yes	complemented
40	C₃×#3748	C₂²	1	1	yes	characteristic, complemented
41	C₃×#3746	C₂²	1	2	yes	complemented
42	C₃×#3744	C₂²	1	2	yes	complemented
43	C₃×#3743	C₂²	1	2	yes	complemented
44	C₃×C₈×C₃⋊D₃	C₂²	1	2	yes	complemented
45	#41×C₃⋊D₃	C₂×C₃	1	1	yes	characteristic, complemented
46	C₂×#3830	C₂×C₃	1	1	yes	characteristic, complemented
47	C₃×D₃×#41	—	3	4	no	—
48	C₂×C₃×#237	—	3	4	no	—
49	C₃²×AGL(1,ℤ/8)	—	2	4	no	—
50	C₃²×AGL(1,ℤ/8)	D₃	1	4	yes	complemented
51	C₃²×AGL(1,ℤ/8)	C₂×C₃	1	1	yes	characteristic, complemented
52	#64160	C₂×C₃	1	1	yes	characteristic, complemented
53	#64141	C₂×C₃	1	1	yes	characteristic, complemented
54	#64125	C₂×C₃	1	2	yes	complemented
55	#64124	C₂×C₃	1	2	yes	complemented
56	#64121	C₂×C₃	1	2	yes	complemented
57	#64120	C₂×C₃	1	2	yes	complemented
58	#64118	C₂×C₃	1	1	yes	characteristic, complemented
59	M₁₆×C₃⋊D₃	C₂×C₃	1	1	yes	characteristic, complemented
60	C₃×#951	—	3	4	no	—
61	C₃×#932	—	3	4	no	—
62	C₃×#916	—	3	8	no	—
63	C₃×#915	—	3	8	no	—
64	C₃×#912	—	3	8	no	—
65	C₃×#911	—	3	8	no	—
66	C₃×#909	—	3	4	no	—
67	C₃×D₃×M₁₆	—	3	4	no	—
68	C₂³×C₃³	—	2	1	no	—
69	C₂²×C₃×C₃⋊D₃	—	2	1	no	—
70	C₂²×C₃×C₃⋊D₃	D₄	1	1	yes	characteristic
71	C₃³×Q₈	C₂³	1	1	yes	characteristic
72	C₃³×D₄	—	2	1	no	—
73	C₃³×D₄	—	2	1	no	—
74	C₃³×D₄	C₂³	1	2	yes	—
75	C₃³×D₄	C₂³	1	1	yes	characteristic
76	C₂×C₃³×C₄	—	2	1	no	—
77	C₂×C₃³×C₄	C₂³	1	1	yes	characteristic
78	C₃×#638	—	2	1	no	—
79	C₃×#638	—	2	1	no	—
80	C₃×#638	—	2	2	no	—
81	C₃×#638	—	2	2	no	—
82	C₃×#638	—	2	1	no	—
83	C₂×C₃×C₃⋊Dic₃	—	2	1	no	—
84	C₂×C₃×C₃⋊Dic₃	—	2	1	no	—
85	C₂×C₃×C₃⋊Dic₃	—	2	2	no	—
86	C₂×C₃×C₃⋊Dic₃	—	2	2	no	—
87	C₂×C₃×C₃⋊Dic₃	D₄	1	1	yes	characteristic
88	C₃×C₄⋊(C₃⋊D₃)	—	2	1	no	—
89	C₃×C₄⋊(C₃⋊D₃)	C₂³	1	1	yes	characteristic
90	C₃×C₄×C₃⋊D₃	—	2	1	no	—
91	C₃×C₄×C₃⋊D₃	—	2	1	no	—
92	C₃×C₄×C₃⋊D₃	—	2	1	no	—
93	C₃×C₄×C₃⋊D₃	D₄	1	1	yes	characteristic
94	C₃×C₄×C₃⋊D₃	D₄	1	1	yes	characteristic
95	C₃×C₄×C₃⋊D₃	C₂³	1	1	yes	characteristic
96	C₃×C₄×C₃⋊D₃	C₂³	1	1	yes	characteristic
97	C₃×#634	—	2	1	no	—
98	C₃×#634	—	2	1	no	—
99	C₃×#634	C₂³	1	2	yes	—
100	C₃×#634	C₂³	1	1	yes	characteristic
101	C₃³×C₈	C₂³	1	2	yes	—
102	C₃×C₃⋊(C₃⋊C₈)	C₂³	1	2	yes	—
103	C₃×#575	—	9	1	no	—
104	#5773	—	3	4	no	—
105	C₃²×#41	—	2	4	no	—
106	C₃²×#41	C₂×D₃	1	4	yes	—
107	C₃²×#41	C₂²×C₃	1	1	yes	characteristic
108	C₂×C₃²×D₄	—	2	4	no	—
109	C₂×C₃²×D₄	C₂×D₃	1	4	yes	complemented
110	C₂×C₃²×D₄	C₂²×C₃	1	1	yes	characteristic, complemented
111	C₂×#638	—	2	1	no	—
112	C₂²×C₃⋊Dic₃	—	2	1	no	—
113	#3832	C₂²×C₃	1	1	yes	characteristic, complemented
114	Q₈×C₃⋊D₃	C₂²×C₃	1	1	yes	characteristic, complemented
115	#3830	—	2	1	no	—
116	#3830	—	2	1	no	—
117	#3830	—	2	1	no	—
118	#3830	C₂²×C₃	1	2	yes	—
119	#3830	C₂²×C₃	1	2	yes	complemented
120	#3830	C₂²×C₃	1	1	yes	characteristic, complemented
121	D₄×C₃⋊D₃	—	2	1	no	—
122	D₄×C₃⋊D₃	C₂²×C₃	1	1	yes	characteristic, complemented
123	#3828	—	2	1	no	—
124	#3828	C₂²×C₃	1	1	yes	characteristic
125	C₂×C₄×C₃⋊D₃	—	2	1	no	—
126	C₂×C₄×C₃⋊D₃	C₂²×C₃	1	1	yes	characteristic
127	C₂×#634	C₂²×C₃	1	1	yes	characteristic, complemented
128	C₂×C₃×#67	—	6	4	no	—
129	C₂²×C₃×Dic₃	—	6	4	no	—
130	C₃×#239	—	3	4	no	—
131	C₃×D₃×Q₈	—	3	4	no	—
132	C₃×#237	—	3	8	no	—
133	C₃×#237	—	3	4	no	—
134	C₃×#237	—	3	8	no	—
135	C₃×#237	—	6	4	no	—
136	C₃×#237	—	6	4	no	—
137	C₃×#237	—	6	4	no	—
138	C₃×D₃×D₄	—	3	4	no	—
139	C₃×D₃×D₄	—	6	4	no	—
140	C₃×#235	—	3	4	no	—
141	C₃×#235	—	6	4	no	—
142	C₂×C₃×C₄×D₃	—	3	4	no	—
143	C₂×C₃×C₄×D₃	—	6	4	no	—
144	C₂×C₃×Dic₆	—	3	4	no	—
145	C₃²×SD₁₆	—	2	8	no	—
146	C₃²×SD₁₆	C₂×D₃	1	8	yes	complemented
147	C₃²×SD₁₆	C₂²×C₃	1	2	yes	complemented
148	C₃²×D₈	—	2	8	no	—
149	C₃²×D₈	C₂×D₃	1	8	yes	complemented
150	C₃²×D₈	C₂²×C₃	1	2	yes	complemented
151	C₃²×M₁₆	—	2	4	no	—
152	C₃²×M₁₆	C₂×D₃	1	4	yes	complemented
153	C₃²×M₁₆	C₂²×C₃	1	1	yes	characteristic, complemented
154	#3756	C₂²×C₃	1	2	yes	complemented
155	#3754	C₂²×C₃	1	2	yes	complemented
156	#3754	C₂²×C₃	1	2	yes	complemented
157	#3753	C₂²×C₃	1	2	yes	complemented
158	#3748	C₂²×C₃	1	1	yes	characteristic, complemented
159	#3746	C₂²×C₃	1	2	yes	complemented
160	#3744	C₂²×C₃	1	2	yes	complemented
161	#3743	C₂²×C₃	1	2	yes	complemented
162	C₈×C₃⋊D₃	C₂²×C₃	1	2	yes	complemented
163	C₃×#216	—	3	8	no	—
164	C₃×#214	—	3	8	no	—
165	C₃×#214	—	3	8	no	—
166	C₃×#213	—	3	8	no	—
167	C₃×#208	—	3	4	no	—
168	C₃×Dic₁₂	—	3	8	no	—
169	C₃×#204	—	3	8	no	—
170	C₃×ΓL(1,𝔽₂₅)	—	3	8	no	—
171	C₃×D₃×C₈	—	3	8	no	—
172	C₂²×C₃³	—	2	2	no	—
173	C₂²×C₃³	—	2	1	no	—
174	C₂²×C₃³	—	4	2	no	—
175	C₂²×C₃³	C₂×D₄	1	1	yes	characteristic
176	C₂×C₃×C₃⋊D₃	—	2	1	no	—
177	C₂×C₃×C₃⋊D₃	—	4	1	no	—
178	C₂×C₃×C₃⋊D₃	—	4	2	no	—
179	C₂×C₃×C₃⋊D₃	C₂×D₄	1	1	yes	characteristic
180	C₂×C₃×C₃⋊D₃	C₂×D₄	1	1	yes	characteristic
181	C₃³×C₄	—	2	1	no	—
182	C₃³×C₄	C₂×D₄	1	1	yes	characteristic
183	C₃³×C₄	C₂⁴	1	1	yes	characteristic
184	C₃×C₃⋊Dic₃	—	2	1	no	—
185	C₃×C₃⋊Dic₃	—	2	2	no	—
186	C₃×C₃⋊Dic₃	C₂×D₄	1	1	yes	characteristic
187	C₃×C₃⋊Dic₃	C₂×D₄	1	1	yes	characteristic
188	C₂×C₃×#41	—	9	1	no	—
189	C₂×C₃×#41	—	9	1	no	—
190	D₃×#41	—	3	4	no	—
191	C₂×#237	—	3	4	no	—
192	C₃×#137	—	9	1	no	—
193	C₃×#137	—	9	2	no	—
194	C₃×#137	—	9	1	no	—
195	C₃×AGL(1,ℤ/8)	—	2	4	no	—
196	C₃×AGL(1,ℤ/8)	—	9	1	no	—
197	C₃×AGL(1,ℤ/8)	—	9	2	no	—
198	C₃×AGL(1,ℤ/8)	C₃×D₃	1	4	yes	complemented
199	C₃×AGL(1,ℤ/8)	C₃⋊D₃	1	1	yes	characteristic, complemented
200	C₃×#135	—	9	2	no	—
201	C₃×#135	—	9	2	no	—
202	C₂×C₃×M₁₆	—	9	1	no	—
203	#951	—	3	4	no	—
204	#932	—	3	4	no	—
205	#916	—	3	8	no	—
206	#915	—	3	8	no	—
207	#912	—	3	8	no	—
208	#911	—	3	8	no	—
209	#909	—	3	4	no	—
210	D₃×M₁₆	—	3	4	no	—
211	C₂³×C₃²	—	2	1	no	—
212	C₂³×C₃²	—	2	4	no	—
213	C₂³×C₃²	—	4	4	no	—
214	C₂²×C₃⋊D₃	—	2	1	no	—
215	C₂²×C₃⋊D₃	C₃×D₄	1	1	yes	characteristic
216	C₂²×C₃×D₃	—	3	4	no	—
217	C₂²×C₃×D₃	—	6	4	no	—
218	C₃²×Q₈	—	2	4	no	—
219	C₃²×Q₈	C₂²×D₃	1	4	yes	—
220	C₃²×Q₈	C₂³×C₃	1	1	yes	characteristic
221	C₃²×D₄	—	2	1	no	—
222	C₃²×D₄	—	2	1	no	—
223	C₃²×D₄	—	2	4	no	—
224	C₃²×D₄	—	2	8	no	—
225	C₃²×D₄	—	2	4	no	—
226	C₃²×D₄	—	2	4	no	—
227	C₃²×D₄	—	4	4	no	—
228	C₃²×D₄	—	4	4	no	—
229	C₃²×D₄	C₂²×D₃	1	8	yes	—
230	C₃²×D₄	C₂²×D₃	1	4	yes	—
231	C₃²×D₄	C₂³×C₃	1	2	yes	—
232	C₃²×D₄	C₂³×C₃	1	1	yes	characteristic
233	C₂×C₃²×C₄	—	2	1	no	—
234	C₂×C₃²×C₄	—	2	4	no	—
235	C₂×C₃²×C₄	—	2	4	no	—
236	C₂×C₃²×C₄	—	4	4	no	—
237	C₂×C₃²×C₄	C₂²×D₃	1	4	yes	—
238	C₂×C₃²×C₄	C₂³×C₃	1	1	yes	characteristic
239	#638	—	2	1	no	—
240	#638	—	2	1	no	—
241	#638	—	2	2	no	—
242	#638	—	2	2	no	—
243	#638	—	2	1	no	—
244	C₂×C₃⋊Dic₃	—	2	1	no	—
245	C₂×C₃⋊Dic₃	—	2	1	no	—
246	C₂×C₃⋊Dic₃	—	2	2	no	—
247	C₂×C₃⋊Dic₃	—	2	2	no	—
248	C₂×C₃⋊Dic₃	C₃×D₄	1	1	yes	characteristic
249	C₄⋊(C₃⋊D₃)	—	2	1	no	—
250	C₄⋊(C₃⋊D₃)	C₂³×C₃	1	1	yes	characteristic
251	C₄×C₃⋊D₃	—	2	1	no	—
252	C₄×C₃⋊D₃	—	2	1	no	—
253	C₄×C₃⋊D₃	—	2	1	no	—
254	C₄×C₃⋊D₃	C₃×D₄	1	1	yes	characteristic
255	C₄×C₃⋊D₃	C₃×D₄	1	1	yes	characteristic
256	C₄×C₃⋊D₃	C₂³×C₃	1	1	yes	characteristic
257	C₄×C₃⋊D₃	C₂³×C₃	1	1	yes	characteristic
258	#634	—	2	1	no	—
259	#634	—	2	1	no	—
260	#634	C₂³×C₃	1	2	yes	—
261	#634	C₂³×C₃	1	1	yes	characteristic
262	C₃×#67	—	6	4	no	—
263	C₃×#67	—	6	4	no	—
264	C₃×#67	—	6	8	no	—
265	C₃×#67	—	6	8	no	—
266	C₃×#67	—	6	4	no	—
267	C₂×C₃×Dic₃	—	3	4	no	—
268	C₂×C₃×Dic₃	—	6	4	no	—
269	C₂×C₃×Dic₃	—	6	8	no	—
270	C₂×C₃×Dic₃	—	6	8	no	—
271	C₂×C₃×Dic₃	—	6	4	no	—
272	C₃×D₁₂	—	3	4	no	—
273	C₃×D₁₂	—	6	4	no	—
274	C₃×C₄×D₃	—	3	4	no	—
275	C₃×C₄×D₃	—	3	4	no	—
276	C₃×C₄×D₃	—	3	4	no	—
277	C₃×C₄×D₃	—	3	4	no	—
278	C₃×C₄×D₃	—	6	4	no	—
279	C₃×C₄×D₃	—	6	4	no	—
280	C₃×C₄×D₃	—	6	4	no	—
281	C₃×Dic₆	—	3	8	no	—
282	C₃×Dic₆	—	3	4	no	—
283	C₃×Dic₆	—	6	4	no	—
284	C₃×Dic₆	—	6	4	no	—
285	C₃²×C₈	—	2	8	no	—
286	C₃²×C₈	C₂²×D₃	1	8	yes	—
287	C₃²×C₈	C₂³×C₃	1	2	yes	—
288	C₃⋊(C₃⋊C₈)	C₂³×C₃	1	2	yes	—
289	C₃×C₃⋊C₈	—	3	8	no	—
290	#575	—	9	1	no	Sylow
291	C₂×C₃³	—	2	1	no	—
292	C₂×C₃³	—	4	1	no	—
293	C₂×C₃³	—	4	2	no	—
294	C₂×C₃³	C₂²×D₄	1	1	yes	characteristic
295	C₃×C₃⋊D₃	—	2	1	no	—
296	C₃×C₃⋊D₃	—	2	1	no	—
297	C₃×C₃⋊D₃	—	4	1	no	—
298	C₃×#41	—	2	4	no	—
299	C₃×#41	—	9	1	no	—
300	C₃×#41	—	9	2	no	—
301	C₃×#41	—	9	1	no	—
302	C₃×#41	—	9	2	no	—
303	C₃×#41	—	9	1	no	—
304	C₃×#41	—	18	1	no	—
305	C₃×#41	—	18	1	no	—
306	C₃×#41	—	18	1	no	—
307	C₃×#41	—	18	1	no	—
308	C₃×#41	C₂×C₃×D₃	1	4	yes	—
309	C₃×#41	C₂×C₃⋊D₃	1	1	yes	characteristic
310	C₂×C₃×Q₈	—	9	1	no	—
311	C₂×C₃×Q₈	—	9	1	no	—
312	C₂×C₃×D₄	—	2	4	no	—
313	C₂×C₃×D₄	—	9	1	no	—
314	C₂×C₃×D₄	—	18	1	no	—
315	C₂×C₃×D₄	—	18	1	no	—
316	C₂×C₃×D₄	C₂×C₃×D₃	1	4	yes	complemented
317	C₂×C₃×D₄	C₂×C₃⋊D₃	1	1	yes	characteristic, complemented
318	C₂²×C₃×C₄	—	9	1	no	—
319	C₂²×C₃×C₄	—	18	1	no	—
320	C₂²×C₃×C₄	—	18	1	no	—
321	C₂×#67	—	6	4	no	—
322	C₂²×Dic₃	—	6	4	no	—
323	#239	—	3	4	no	—
324	D₃×Q₈	—	3	4	no	—
325	#237	—	3	8	no	—
326	#237	—	3	4	no	—
327	#237	—	3	8	no	—
328	#237	—	6	4	no	—
329	#237	—	6	4	no	—
330	#237	—	6	4	no	—
331	D₃×D₄	—	3	4	no	—
332	D₃×D₄	—	6	4	no	—
333	#235	—	3	4	no	—
334	#235	—	6	4	no	—
335	C₂×C₄×D₃	—	3	4	no	—
336	C₂×C₄×D₃	—	6	4	no	—
337	C₂×Dic₆	—	3	4	no	—
338	C₃×Q₁₆	—	9	2	no	—
339	C₃×Q₁₆	—	9	2	no	—
340	C₃×SD₁₆	—	2	8	no	—
341	C₃×SD₁₆	—	9	2	no	—
342	C₃×SD₁₆	—	9	2	no	—
343	C₃×SD₁₆	—	9	2	no	—
344	C₃×SD₁₆	C₂×C₃×D₃	1	8	yes	complemented
345	C₃×SD₁₆	C₂×C₃⋊D₃	1	2	yes	complemented
346	C₃×D₈	—	2	8	no	—
347	C₃×D₈	—	9	2	no	—
348	C₃×D₈	C₂×C₃×D₃	1	8	yes	complemented
349	C₃×D₈	C₂×C₃⋊D₃	1	2	yes	complemented
350	C₃×M₁₆	—	2	4	no	—
351	C₃×M₁₆	—	9	1	no	—
352	C₃×M₁₆	—	9	2	no	—
353	C₃×M₁₆	C₂×C₃×D₃	1	4	yes	complemented
354	C₃×M₁₆	C₂×C₃⋊D₃	1	1	yes	characteristic, complemented
355	C₂×C₃×C₈	—	9	2	no	—
356	#216	—	3	8	no	—
357	#214	—	3	8	no	—
358	#214	—	3	8	no	—
359	#213	—	3	8	no	—
360	#208	—	3	4	no	—
361	Dic₁₂	—	3	8	no	—
362	#204	—	3	8	no	—
363	ΓL(1,𝔽₂₅)	—	3	8	no	—
364	D₃×C₈	—	3	8	no	—
365	C₂²×C₃²	—	2	2	no	—
366	C₂²×C₃²	—	2	1	no	—
367	C₂²×C₃²	—	2	4	no	—
368	C₂²×C₃²	—	2	8	no	—
369	C₂²×C₃²	—	2	4	no	—
370	C₂²×C₃²	—	4	4	no	—
371	C₂²×C₃²	—	4	8	no	—
372	C₂²×C₃²	—	4	2	no	—
373	C₂²×C₃²	—	4	8	no	—
374	C₂²×C₃²	—	8	8	no	—
375	C₂²×C₃²	D₃×D₄	1	4	yes	—
376	C₂²×C₃²	C₂×C₃×D₄	1	1	yes	characteristic
377	C₂×C₃⋊D₃	—	2	1	no	—
378	C₂×C₃⋊D₃	—	4	1	no	—
379	C₂×C₃⋊D₃	—	4	2	no	—
380	C₂×C₃⋊D₃	C₂×C₃×D₄	1	1	yes	characteristic
381	C₂×C₃⋊D₃	C₂×C₃×D₄	1	1	yes	characteristic
382	C₂×C₃×D₃	—	3	4	no	—
383	C₂×C₃×D₃	—	3	4	no	—
384	C₂×C₃×D₃	—	6	4	no	—
385	C₂×C₃×D₃	—	12	4	no	—
386	C₂×C₃×D₃	—	12	8	no	—
387	C₃²×C₄	—	2	1	no	—
388	C₃²×C₄	—	2	4	no	—
389	C₃²×C₄	—	2	4	no	—
390	C₃²×C₄	—	2	4	no	—
391	C₃²×C₄	—	4	4	no	—
392	C₃²×C₄	D₃×D₄	1	4	yes	—
393	C₃²×C₄	C₂×C₃×D₄	1	1	yes	characteristic
394	C₃²×C₄	C₂³×D₃	1	4	yes	—
395	C₃²×C₄	C₂⁴×C₃	1	1	yes	characteristic
396	C₃⋊Dic₃	—	2	1	no	—
397	C₃⋊Dic₃	—	2	2	no	—
398	C₃⋊Dic₃	C₂×C₃×D₄	1	1	yes	characteristic
399	C₃⋊Dic₃	C₂×C₃×D₄	1	1	yes	characteristic
400	C₃×Dic₃	—	3	4	no	—
401	C₃×Dic₃	—	3	4	no	—
402	C₃×Dic₃	—	6	4	no	—
403	C₃×Dic₃	—	6	8	no	—
404	C₂×#41	—	9	1	no	—
405	C₂×#41	—	9	1	no	—
406	#137	—	9	1	no	—
407	#137	—	9	2	no	—
408	#137	—	9	1	no	—
409	AGL(1,ℤ/8)	—	9	1	no	—
410	AGL(1,ℤ/8)	—	9	2	no	—
411	AGL(1,ℤ/8)	C₃×C₃⋊D₃	1	1	yes	characteristic, complemented
412	#135	—	9	2	no	—
413	#135	—	9	2	no	—
414	C₂×M₁₆	—	9	1	no	—
415	C₃³	#575	1	1	yes	characteristic, complemented, Sylow
416	C₂³×C₃	—	2	4	no	—
417	C₂³×C₃	—	2	1	no	—
418	C₂³×C₃	—	4	4	no	—
419	C₂³×C₃	—	9	1	no	—
420	C₂³×C₃	—	18	1	no	—
421	C₂²×D₃	—	3	4	no	—
422	C₂²×D₃	—	6	4	no	—
423	C₃×Q₈	—	2	4	no	—
424	C₃×Q₈	—	9	2	no	—
425	C₃×Q₈	—	9	1	no	—
426	C₃×Q₈	—	18	1	no	—
427	C₃×Q₈	—	18	1	no	—
428	C₃×Q₈	C₂²×C₃×D₃	1	4	yes	—
429	C₃×Q₈	C₂²×C₃⋊D₃	1	1	yes	characteristic
430	C₃×D₄	—	2	4	no	—
431	C₃×D₄	—	2	1	no	—
432	C₃×D₄	—	2	4	no	—
433	C₃×D₄	—	2	1	no	—
434	C₃×D₄	—	2	4	no	—
435	C₃×D₄	—	2	8	no	—
436	C₃×D₄	—	4	4	no	—
437	C₃×D₄	—	4	4	no	—
438	C₃×D₄	—	9	1	no	—
439	C₃×D₄	—	18	1	no	—
440	C₃×D₄	—	18	1	no	—
441	C₃×D₄	—	18	1	no	—
442	C₃×D₄	—	18	2	no	—
443	C₃×D₄	—	18	2	no	—
444	C₃×D₄	—	18	1	no	—
445	C₃×D₄	C₂²×C₃×D₃	1	8	yes	—
446	C₃×D₄	C₂²×C₃×D₃	1	4	yes	—
447	C₃×D₄	C₂²×C₃⋊D₃	1	2	yes	—
448	C₃×D₄	C₂²×C₃⋊D₃	1	1	yes	characteristic
449	C₂×C₃×C₄	—	2	4	no	—
450	C₂×C₃×C₄	—	2	1	no	—
451	C₂×C₃×C₄	—	2	4	no	—
452	C₂×C₃×C₄	—	4	4	no	—
453	C₂×C₃×C₄	—	9	1	no	—
454	C₂×C₃×C₄	—	9	1	no	—
455	C₂×C₃×C₄	—	9	1	no	—
456	C₂×C₃×C₄	—	9	1	no	—
457	C₂×C₃×C₄	—	9	1	no	—
458	C₂×C₃×C₄	—	18	1	no	—
459	C₂×C₃×C₄	—	18	1	no	—
460	C₂×C₃×C₄	—	18	1	no	—
461	C₂×C₃×C₄	—	18	2	no	—
462	C₂×C₃×C₄	—	18	2	no	—
463	C₂×C₃×C₄	—	18	1	no	—
464	C₂×C₃×C₄	—	18	1	no	—
465	C₂×C₃×C₄	C₂²×C₃×D₃	1	4	yes	—
466	C₂×C₃×C₄	C₂²×C₃⋊D₃	1	1	yes	characteristic
467	#67	—	6	4	no	—
468	#67	—	6	4	no	—
469	#67	—	6	8	no	—
470	#67	—	6	8	no	—
471	#67	—	6	4	no	—
472	C₂×Dic₃	—	3	4	no	—
473	C₂×Dic₃	—	6	4	no	—
474	C₂×Dic₃	—	6	8	no	—
475	C₂×Dic₃	—	6	8	no	—
476	C₂×Dic₃	—	6	4	no	—
477	D₁₂	—	3	4	no	—
478	D₁₂	—	6	4	no	—
479	C₄×D₃	—	3	4	no	—
480	C₄×D₃	—	3	4	no	—
481	C₄×D₃	—	3	4	no	—
482	C₄×D₃	—	3	4	no	—
483	C₄×D₃	—	6	4	no	—
484	C₄×D₃	—	6	4	no	—
485	C₄×D₃	—	6	4	no	—
486	Dic₆	—	3	8	no	—
487	Dic₆	—	3	4	no	—
488	Dic₆	—	6	4	no	—
489	Dic₆	—	6	4	no	—
490	C₃×C₈	—	2	8	no	—
491	C₃×C₈	—	9	2	no	—
492	C₃×C₈	C₂²×C₃×D₃	1	8	yes	—
493	C₃×C₈	C₂²×C₃⋊D₃	1	2	yes	—
494	C₃⋊C₈	—	3	8	no	—
495	C₂×C₃²	—	2	1	no	—
496	C₂×C₃²	—	2	4	no	—
497	C₂×C₃²	—	2	4	no	—
498	C₂×C₃²	—	4	4	no	—
499	C₂×C₃²	—	4	1	no	—
500	C₂×C₃²	—	4	2	no	—
501	C₂×C₃²	—	4	4	no	—
502	C₂×C₃²	—	4	8	no	—
503	C₂×C₃²	—	8	4	no	—
504	C₂×C₃²	—	8	8	no	—
505	C₂×C₃²	C₂×D₃×D₄	1	4	yes	—
506	C₂×C₃²	C₂²×C₃×D₄	1	1	yes	characteristic
507	C₃⋊D₃	—	2	1	no	—
508	C₃⋊D₃	—	2	1	no	—
509	C₃⋊D₃	—	4	1	no	—
510	C₃×D₃	—	6	4	no	—
511	C₃×D₃	—	6	4	no	—
512	C₃×D₃	—	12	4	no	—
513	#41	—	9	1	no	—
514	#41	—	9	2	no	—
515	#41	—	9	1	no	—
516	#41	—	9	2	no	—
517	#41	—	9	1	no	—
518	#41	—	18	1	no	—
519	#41	—	18	1	no	—
520	#41	—	18	1	no	—
521	#41	—	18	1	no	—
522	#41	C₂×C₃×C₃⋊D₃	1	1	yes	characteristic
523	C₂×Q₈	—	9	1	no	—
524	C₂×Q₈	—	9	1	no	—
525	C₂×D₄	—	9	1	no	—
526	C₂×D₄	—	18	1	no	—
527	C₂×D₄	—	18	1	no	—
528	C₂×D₄	C₂×C₃×C₃⋊D₃	1	1	yes	characteristic, complemented
529	C₂²×C₄	—	9	1	no	—
530	C₂²×C₄	—	18	1	no	—
531	C₂²×C₄	—	18	1	no	—
532	Q₁₆	—	9	2	no	—
533	Q₁₆	—	9	2	no	—
534	SD₁₆	—	9	2	no	—
535	SD₁₆	—	9	2	no	—
536	SD₁₆	—	9	2	no	—
537	SD₁₆	C₂×C₃×C₃⋊D₃	1	2	yes	complemented
538	D₈	—	9	2	no	—
539	D₈	C₂×C₃×C₃⋊D₃	1	2	yes	complemented
540	M₁₆	—	9	1	no	—
541	M₁₆	—	9	2	no	—
542	M₁₆	C₂×C₃×C₃⋊D₃	1	1	yes	characteristic, complemented
543	C₂×C₈	—	9	2	no	—
544	C₂²×C₃	—	2	8	no	—
545	C₂²×C₃	—	2	4	no	—
546	C₂²×C₃	—	2	1	no	—
547	C₂²×C₃	—	2	2	no	—
548	C₂²×C₃	—	2	4	no	—
549	C₂²×C₃	—	4	4	no	—
550	C₂²×C₃	—	4	8	no	—
551	C₂²×C₃	—	4	2	no	—
552	C₂²×C₃	—	4	8	no	—
553	C₂²×C₃	—	8	8	no	—
554	C₂²×C₃	—	9	1	no	—
555	C₂²×C₃	—	9	1	no	—
556	C₂²×C₃	—	18	1	no	—
557	C₂²×C₃	—	36	1	no	—
558	C₂²×C₃	—	36	2	no	—
559	C₂²×C₃	C₃×D₃×D₄	1	4	yes	—
560	C₂²×C₃	D₄×C₃⋊D₃	1	1	yes	characteristic
561	C₂×D₃	—	3	4	no	—
562	C₂×D₃	—	3	4	no	—
563	C₂×D₃	—	6	4	no	—
564	C₂×D₃	—	12	4	no	—
565	C₂×D₃	—	12	8	no	—
566	C₃×C₄	—	2	4	no	—
567	C₃×C₄	—	2	1	no	—
568	C₃×C₄	—	2	4	no	—
569	C₃×C₄	—	2	4	no	—
570	C₃×C₄	—	4	4	no	—
571	C₃×C₄	—	9	1	no	—
572	C₃×C₄	—	9	1	no	—
573	C₃×C₄	—	18	1	no	—
574	C₃×C₄	—	18	2	no	—
575	C₃×C₄	C₃×D₃×D₄	1	4	yes	—
576	C₃×C₄	D₄×C₃⋊D₃	1	1	yes	characteristic
577	C₃×C₄	C₂³×C₃×D₃	1	4	yes	—
578	C₃×C₄	C₂³×C₃⋊D₃	1	1	yes	characteristic
579	Dic₃	—	3	4	no	—
580	Dic₃	—	3	4	no	—
581	Dic₃	—	6	4	no	—
582	Dic₃	—	6	8	no	—
583	C₃²	—	2	4	no	—
584	C₃²	#5773	1	4	yes	complemented
585	C₃²	C₃×#575	1	1	yes	characteristic, complemented
586	C₂³	—	2	1	no	—
587	C₂³	—	9	1	no	—
588	C₂³	—	18	1	no	—
589	Q₈	—	9	1	no	—
590	Q₈	—	9	2	no	—
591	Q₈	—	18	1	no	—
592	Q₈	—	18	1	no	—
593	Q₈	C₂²×C₃×C₃⋊D₃	1	1	yes	characteristic
594	D₄	—	2	1	no	—
595	D₄	—	2	1	no	—
596	D₄	—	9	1	no	—
597	D₄	—	18	1	no	—
598	D₄	—	18	1	no	—
599	D₄	—	18	1	no	—
600	D₄	—	18	2	no	—
601	D₄	—	18	2	no	—
602	D₄	—	18	1	no	—
603	D₄	C₂²×C₃×C₃⋊D₃	1	2	yes	—
604	D₄	C₂²×C₃×C₃⋊D₃	1	1	yes	characteristic
605	C₂×C₄	—	2	1	no	—
606	C₂×C₄	—	9	1	no	—
607	C₂×C₄	—	9	1	no	—
608	C₂×C₄	—	9	1	no	—
609	C₂×C₄	—	9	1	no	—
610	C₂×C₄	—	9	1	no	—
611	C₂×C₄	—	18	1	no	—
612	C₂×C₄	—	18	1	no	—
613	C₂×C₄	—	18	1	no	—
614	C₂×C₄	—	18	2	no	—
615	C₂×C₄	—	18	2	no	—
616	C₂×C₄	—	18	1	no	—
617	C₂×C₄	—	18	1	no	—
618	C₂×C₄	C₂²×C₃×C₃⋊D₃	1	1	yes	characteristic
619	C₈	—	9	2	no	—
620	C₈	C₂²×C₃×C₃⋊D₃	1	2	yes	—
621	C₂×C₃	—	2	4	no	—
622	C₂×C₃	—	2	1	no	—
623	C₂×C₃	—	2	4	no	—
624	C₂×C₃	—	4	4	no	—
625	C₂×C₃	—	4	4	no	—
626	C₂×C₃	—	4	8	no	—
627	C₂×C₃	—	4	2	no	—
628	C₂×C₃	—	4	1	no	—
629	C₂×C₃	—	8	8	no	—
630	C₂×C₃	—	8	4	no	—
631	C₂×C₃	—	18	1	no	—
632	C₂×C₃	—	18	1	no	—
633	C₂×C₃	—	36	1	no	—
634	C₂×C₃	C₂×C₃×D₃×D₄	1	4	yes	—
635	C₂×C₃	C₂×D₄×C₃⋊D₃	1	1	yes	characteristic
636	D₃	—	6	4	no	—
637	D₃	—	6	4	no	—
638	D₃	—	12	4	no	—
639	C₂²	—	2	1	no	—
640	C₂²	—	2	2	no	—
641	C₂²	—	4	2	no	—
642	C₂²	—	9	1	no	—
643	C₂²	—	9	1	no	—
644	C₂²	—	18	1	no	—
645	C₂²	—	36	1	no	—
646	C₂²	—	36	2	no	—
647	C₂²	C₃×D₄×C₃⋊D₃	1	1	yes	characteristic
648	C₄	—	2	1	no	—
649	C₄	—	9	1	no	—
650	C₄	—	9	1	no	—
651	C₄	—	18	1	no	—
652	C₄	—	18	2	no	—
653	C₄	C₃×D₄×C₃⋊D₃	1	1	yes	characteristic
654	C₄	C₂³×C₃×C₃⋊D₃	1	1	yes	characteristic
655	C₃	—	2	4	no	—
656	C₃	C₃×#5773	1	4	yes	complemented
657	C₃	#10596244	1	1	yes	characteristic, direct product factor
658	C₂	—	2	1	no	—
659	C₂	—	4	2	no	—
660	C₂	—	4	1	no	—
661	C₂	—	18	1	no	—
662	C₂	—	18	1	no	—
663	C₂	—	36	1	no	—
664	C₂	C₂×C₃×D₄×C₃⋊D₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₃	C₂×D₄×C₃⋊D₃
commutator, cosocle	C₃²×C₄	C₂⁴×C₃
Frattini	C₄	C₂³×C₃×C₃⋊D₃
Fitting	C₃³×AGL(1,ℤ/8)	C₂
socle	C₂×C₃³	C₂²×D₄
hypercenter	C₃×AGL(1,ℤ/8)	C₃⋊D₃
hypocenter	C₃²	C₃×#575
LCS	G ▷ C₃²×C₄ ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂×C₃ ◁ C₂×C₃×C₄ ◁ C₃×AGL(1,ℤ/8)

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. C₃×#41×C₃⋊D₃	C₂	semidirect
2. C₂×C₃×#3830	C₂	semidirect
3. C₃³×AGL(1,ℤ/8)	C₂	semidirect
4. C₃×#64160	C₂	semidirect
5. C₃×#64141	C₂	semidirect
6. C₃×#64125	C₂	semidirect
7. C₃×#64124	C₂	semidirect
8. C₃×#64121	C₂	semidirect
9. C₃×#64120	C₂	semidirect
10. C₃×#64118	C₂	semidirect
11. C₃×M₁₆×C₃⋊D₃	C₂	semidirect
12. #10596244	C₃	direct (N × H)
15. C₂×C₃³×D₄	C₂²	semidirect
18. C₃×#3832	C₂²	semidirect
19. C₃×Q₈×C₃⋊D₃	C₂²	semidirect
24. C₃×#3830	C₂²	semidirect
25. C₃×#3830	C₂²	semidirect
27. C₃×D₄×C₃⋊D₃	C₂²	semidirect
32. C₂×C₃×#634	C₂²	semidirect
33. C₃³×SD₁₆	C₂²	semidirect
34. C₃³×D₈	C₂²	semidirect
35. C₃³×M₁₆	C₂²	semidirect
36. C₃×#3756	C₂²	semidirect
37. C₃×#3754	C₂²	semidirect
38. C₃×#3754	C₂²	semidirect
39. C₃×#3753	C₂²	semidirect
40. C₃×#3748	C₂²	semidirect
41. C₃×#3746	C₂²	semidirect
42. C₃×#3744	C₂²	semidirect
43. C₃×#3743	C₂²	semidirect
44. C₃×C₈×C₃⋊D₃	C₂²	semidirect
45. #41×C₃⋊D₃	C₂×C₃	semidirect
46. C₂×#3830	C₂×C₃	semidirect
50. C₃²×AGL(1,ℤ/8)	D₃	semidirect
51. C₃²×AGL(1,ℤ/8)	C₂×C₃	semidirect
52. #64160	C₂×C₃	semidirect
53. #64141	C₂×C₃	semidirect
54. #64125	C₂×C₃	semidirect
55. #64124	C₂×C₃	semidirect
56. #64121	C₂×C₃	semidirect
57. #64120	C₂×C₃	semidirect
58. #64118	C₂×C₃	semidirect
59. M₁₆×C₃⋊D₃	C₂×C₃	semidirect
109. C₂×C₃²×D₄	C₂×D₃	semidirect
110. C₂×C₃²×D₄	C₂²×C₃	semidirect
113. #3832	C₂²×C₃	semidirect
114. Q₈×C₃⋊D₃	C₂²×C₃	semidirect
119. #3830	C₂²×C₃	semidirect
120. #3830	C₂²×C₃	semidirect
122. D₄×C₃⋊D₃	C₂²×C₃	semidirect
127. C₂×#634	C₂²×C₃	semidirect
146. C₃²×SD₁₆	C₂×D₃	semidirect
147. C₃²×SD₁₆	C₂²×C₃	semidirect
149. C₃²×D₈	C₂×D₃	semidirect
150. C₃²×D₈	C₂²×C₃	semidirect
152. C₃²×M₁₆	C₂×D₃	semidirect
153. C₃²×M₁₆	C₂²×C₃	semidirect
154. #3756	C₂²×C₃	semidirect
155. #3754	C₂²×C₃	semidirect
156. #3754	C₂²×C₃	semidirect
157. #3753	C₂²×C₃	semidirect
158. #3748	C₂²×C₃	semidirect
159. #3746	C₂²×C₃	semidirect
160. #3744	C₂²×C₃	semidirect
161. #3743	C₂²×C₃	semidirect
162. C₈×C₃⋊D₃	C₂²×C₃	semidirect
198. C₃×AGL(1,ℤ/8)	C₃×D₃	semidirect
199. C₃×AGL(1,ℤ/8)	C₃⋊D₃	semidirect
316. C₂×C₃×D₄	C₂×C₃×D₃	semidirect
317. C₂×C₃×D₄	C₂×C₃⋊D₃	semidirect
344. C₃×SD₁₆	C₂×C₃×D₃	semidirect
345. C₃×SD₁₆	C₂×C₃⋊D₃	semidirect
348. C₃×D₈	C₂×C₃×D₃	semidirect
349. C₃×D₈	C₂×C₃⋊D₃	semidirect
353. C₃×M₁₆	C₂×C₃×D₃	semidirect
354. C₃×M₁₆	C₂×C₃⋊D₃	semidirect
411. AGL(1,ℤ/8)	C₃×C₃⋊D₃	semidirect
415. C₃³	#575	semidirect
528. C₂×D₄	C₂×C₃×C₃⋊D₃	semidirect
537. SD₁₆	C₂×C₃×C₃⋊D₃	semidirect
539. D₈	C₂×C₃×C₃⋊D₃	semidirect
542. M₁₆	C₂×C₃×C₃⋊D₃	semidirect
584. C₃²	#5773	semidirect
585. C₃²	C₃×#575	semidirect
656. C₃	C₃×#5773	semidirect
657. C₃	#10596244	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₃×#41×C₃⋊D₃	2
C₂×C₃×#3830	2
C₃³×AGL(1,ℤ/8)	2
C₃×#64160	2
C₃×#64141	2
C₃×#64125	2
C₃×#64124	2
C₃×#64121	2
C₃×#64120	2
C₃×#64118	2
C₃×M₁₆×C₃⋊D₃	2
#10596244	3
C₃×#5773	3

Subgroups — C3×#10596244 #13179524

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#10596244 #13179524