Subgroups — C₃×#10596240 #13179520

Order 1728.

1856 conjugacy classes of subgroups.

The subgroup lattice has 422 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#64366	C₂	1	2	yes	maximal, complemented
2	C₃³×#135	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×#64158	C₂	1	2	yes	maximal, complemented
4	C₃×#64128	C₂	1	2	yes	maximal, complemented
5	C₃×#64123	C₂	1	4	yes	maximal, complemented
6	C₃×D₈×C₃⋊D₃	C₂	1	2	yes	maximal, complemented
7	C₂×C₃×#3745	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃×#64110	C₂	1	1	yes	maximal, characteristic, complemented
9	#10596240	C₃	1	1	yes	maximal, characteristic, direct product factor
10	C₃×#5769	—	3	4	no	maximal
11	C₃³×#41	C₂²	1	2	yes	complemented
12	C₃×#3832	C₂²	1	4	yes	complemented
13	C₃×D₄×C₃⋊D₃	—	2	4	no	—
14	C₃×D₄×C₃⋊D₃	C₂²	1	4	yes	complemented
15	C₃×#3828	—	2	2	no	—
16	C₃×#3828	C₂²	1	1	yes	characteristic
17	C₂×C₃×C₄⋊(C₃⋊D₃)	—	2	2	no	—
18	C₂×C₃×C₄⋊(C₃⋊D₃)	C₂²	1	2	yes	complemented
19	C₃³×Q₁₆	C₂²	1	1	yes	characteristic, complemented
20	C₃³×SD₁₆	C₂²	1	2	yes	complemented
21	C₃³×D₈	C₂²	1	1	yes	characteristic, complemented
22	C₂×C₃³×C₈	C₂²	1	1	yes	characteristic, complemented
23	C₃×#3755	C₂²	1	4	yes	complemented
24	C₃×#3753	C₂²	1	4	yes	complemented
25	C₃×#3748	C₂²	1	1	yes	characteristic, complemented
26	C₃×#3745	C₂²	1	2	yes	complemented
27	C₃×#3745	C₂²	1	2	yes	complemented
28	C₃×#3743	C₂²	1	2	yes	complemented
29	C₃×C₈×C₃⋊D₃	C₂²	1	2	yes	complemented
30	#64366	C₂×C₃	1	2	yes	complemented
31	C₃×#1009	—	3	8	no	—
32	C₃²×#135	—	2	4	no	—
33	C₃²×#135	D₃	1	4	yes	complemented
34	C₃²×#135	C₂×C₃	1	1	yes	characteristic, complemented
35	#64158	C₂×C₃	1	2	yes	complemented
36	#64128	C₂×C₃	1	2	yes	complemented
37	#64123	C₂×C₃	1	4	yes	complemented
38	D₈×C₃⋊D₃	C₂×C₃	1	2	yes	complemented
39	C₂×#3745	C₂×C₃	1	1	yes	characteristic, complemented
40	#64110	C₂×C₃	1	1	yes	characteristic, complemented
41	C₃×#949	—	3	8	no	—
42	C₃×#919	—	3	8	no	—
43	C₃×#914	—	3	16	no	—
44	C₃×D₃×D₈	—	3	8	no	—
45	C₂×C₃×D₂₄	—	3	4	no	—
46	C₃×#901	—	3	4	no	—
47	C₂²×C₃×C₃⋊D₃	—	2	2	no	—
48	C₂²×C₃×C₃⋊D₃	—	2	4	no	—
49	C₃³×Q₈	C₂³	1	2	yes	—
50	C₃³×D₄	—	2	2	no	—
51	C₃³×D₄	C₂³	1	2	yes	—
52	C₂×C₃³×C₄	—	2	2	no	—
53	C₂×C₃³×C₄	C₂³	1	1	yes	characteristic
54	C₃×#638	—	2	4	no	—
55	C₃×#638	D₄	1	2	yes	—
56	C₃×C₄⋊(C₃⋊D₃)	—	2	4	no	—
57	C₃×C₄⋊(C₃⋊D₃)	—	2	2	no	—
58	C₃×C₄⋊(C₃⋊D₃)	D₄	1	1	yes	characteristic
59	C₃×C₄⋊(C₃⋊D₃)	C₂³	1	4	yes	—
60	C₃×C₄×C₃⋊D₃	—	2	4	no	—
61	C₃×C₄×C₃⋊D₃	C₂³	1	2	yes	—
62	C₃×#634	D₄	1	1	yes	characteristic
63	C₃³×C₈	C₂³	1	1	yes	characteristic
64	C₃³×C₈	C₂³	1	1	yes	characteristic
65	C₃×C₃⋊(C₃⋊C₈)	C₂³	1	2	yes	—
66	C₃×#576	—	9	1	no	—
67	#5769	—	3	4	no	—
68	C₃²×#41	—	2	8	no	—
69	C₃²×#41	C₂×D₃	1	8	yes	complemented
70	C₃²×#41	C₂²×C₃	1	2	yes	complemented
71	#3832	C₂²×C₃	1	4	yes	complemented
72	D₄×C₃⋊D₃	—	2	4	no	—
73	D₄×C₃⋊D₃	C₂²×C₃	1	4	yes	complemented
74	#3828	—	2	2	no	—
75	#3828	C₂²×C₃	1	1	yes	characteristic
76	C₂×C₄⋊(C₃⋊D₃)	—	2	2	no	—
77	C₂×C₄⋊(C₃⋊D₃)	C₂²×C₃	1	2	yes	complemented
78	C₃×#239	—	3	16	no	—
79	C₃×D₃×D₄	—	3	16	no	—
80	C₃×D₃×D₄	—	6	16	no	—
81	C₃×#235	—	3	4	no	—
82	C₃×#235	—	6	8	no	—
83	C₂×C₃×D₁₂	—	3	8	no	—
84	C₂×C₃×D₁₂	—	6	8	no	—
85	C₃²×Q₁₆	—	2	4	no	—
86	C₃²×Q₁₆	C₂×D₃	1	4	yes	complemented
87	C₃²×Q₁₆	C₂²×C₃	1	1	yes	characteristic, complemented
88	C₃²×SD₁₆	—	2	8	no	—
89	C₃²×SD₁₆	C₂×D₃	1	8	yes	complemented
90	C₃²×SD₁₆	C₂²×C₃	1	2	yes	complemented
91	C₃²×D₈	—	2	4	no	—
92	C₃²×D₈	C₂×D₃	1	4	yes	complemented
93	C₃²×D₈	C₂²×C₃	1	1	yes	characteristic, complemented
94	C₂×C₃²×C₈	—	2	4	no	—
95	C₂×C₃²×C₈	C₂×D₃	1	4	yes	complemented
96	C₂×C₃²×C₈	C₂²×C₃	1	1	yes	characteristic, complemented
97	#3755	C₂²×C₃	1	4	yes	complemented
98	#3753	C₂²×C₃	1	4	yes	complemented
99	#3748	C₂²×C₃	1	1	yes	characteristic, complemented
100	#3745	C₂²×C₃	1	2	yes	complemented
101	#3745	C₂²×C₃	1	2	yes	complemented
102	#3743	C₂²×C₃	1	2	yes	complemented
103	C₈×C₃⋊D₃	C₂²×C₃	1	2	yes	complemented
104	C₃×#215	—	3	16	no	—
105	C₃×#213	—	3	16	no	—
106	C₃×#208	—	3	4	no	—
107	C₃×D₂₄	—	3	8	no	—
108	C₃×D₂₄	—	3	8	no	—
109	C₃×ΓL(1,𝔽₂₅)	—	3	8	no	—
110	C₃×D₃×C₈	—	3	8	no	—
111	C₂²×C₃³	—	2	2	no	—
112	C₂²×C₃³	C₂×D₄	1	1	yes	characteristic
113	C₂×C₃×C₃⋊D₃	—	2	4	no	—
114	C₂×C₃×C₃⋊D₃	—	8	4	no	—
115	C₂×C₃×C₃⋊D₃	—	8	2	no	—
116	C₂×C₃×C₃⋊D₃	C₂×D₄	1	2	yes	—
117	C₃³×C₄	—	2	2	no	—
118	C₃³×C₄	C₂×D₄	1	1	yes	characteristic
119	C₃³×C₄	C₂⁴	1	1	yes	characteristic
120	C₃×C₃⋊Dic₃	C₂×D₄	1	2	yes	—
121	C₃×H(5,𝔽₂)	—	9	2	no	—
122	#1009	—	3	8	no	—
123	C₃×AGL(1,ℤ/8)	—	9	2	no	—
124	C₃×AGL(1,ℤ/8)	—	9	4	no	—
125	C₃×#135	—	2	4	no	—
126	C₃×#135	—	9	2	no	—
127	C₃×#135	C₃×D₃	1	4	yes	complemented
128	C₃×#135	C₃⋊D₃	1	1	yes	characteristic, complemented
129	C₂×C₃×D₈	—	9	2	no	—
130	C₂×C₃×D₈	—	9	1	no	—
131	C₃×#131	—	9	1	no	—
132	#949	—	3	8	no	—
133	#919	—	3	8	no	—
134	#914	—	3	16	no	—
135	D₃×D₈	—	3	8	no	—
136	C₂×D₂₄	—	3	4	no	—
137	#901	—	3	4	no	—
138	C₂²×C₃⋊D₃	—	2	2	no	—
139	C₂²×C₃⋊D₃	—	2	4	no	—
140	C₂²×C₃×D₃	—	6	16	no	—
141	C₂²×C₃×D₃	—	6	8	no	—
142	C₃²×Q₈	—	2	8	no	—
143	C₃²×Q₈	C₂²×D₃	1	8	yes	—
144	C₃²×Q₈	C₂³×C₃	1	2	yes	—
145	C₃²×D₄	—	2	2	no	—
146	C₃²×D₄	—	2	8	no	—
147	C₃²×D₄	—	2	8	no	—
148	C₃²×D₄	—	4	8	no	—
149	C₃²×D₄	C₂²×D₃	1	8	yes	—
150	C₃²×D₄	C₂³×C₃	1	2	yes	—
151	C₂×C₃²×C₄	—	2	2	no	—
152	C₂×C₃²×C₄	—	2	4	no	—
153	C₂×C₃²×C₄	—	2	8	no	—
154	C₂×C₃²×C₄	—	4	8	no	—
155	C₂×C₃²×C₄	C₂²×D₃	1	4	yes	—
156	C₂×C₃²×C₄	C₂³×C₃	1	1	yes	characteristic
157	#638	—	2	4	no	—
158	#638	C₃×D₄	1	2	yes	—
159	C₄⋊(C₃⋊D₃)	—	2	4	no	—
160	C₄⋊(C₃⋊D₃)	—	2	2	no	—
161	C₄⋊(C₃⋊D₃)	C₃×D₄	1	1	yes	characteristic
162	C₄⋊(C₃⋊D₃)	C₂³×C₃	1	4	yes	—
163	C₄×C₃⋊D₃	—	2	4	no	—
164	C₄×C₃⋊D₃	C₂³×C₃	1	2	yes	—
165	#634	C₃×D₄	1	1	yes	characteristic
166	C₃×#67	—	3	8	no	—
167	C₃×#67	—	6	16	no	—
168	C₃×D₁₂	—	3	4	no	—
169	C₃×D₁₂	—	3	16	no	—
170	C₃×D₁₂	—	6	16	no	—
171	C₃×D₁₂	—	6	8	no	—
172	C₃×C₄×D₃	—	3	8	no	—
173	C₃×C₄×D₃	—	6	16	no	—
174	C₃×Dic₆	—	3	4	no	—
175	C₃²×C₈	—	2	4	no	—
176	C₃²×C₈	—	2	4	no	—
177	C₃²×C₈	C₂²×D₃	1	4	yes	—
178	C₃²×C₈	C₂²×D₃	1	4	yes	—
179	C₃²×C₈	C₂³×C₃	1	1	yes	characteristic
180	C₃²×C₈	C₂³×C₃	1	1	yes	characteristic
181	C₃⋊(C₃⋊C₈)	C₂³×C₃	1	2	yes	—
182	C₃×C₃⋊C₈	—	3	8	no	—
183	#576	—	9	1	no	Sylow
184	C₂×C₃³	—	2	1	no	—
185	C₂×C₃³	—	4	2	no	—
186	C₂×C₃³	C₂²×D₄	1	1	yes	characteristic
187	C₃×C₃⋊D₃	—	2	2	no	—
188	C₃×C₃⋊D₃	—	4	4	no	—
189	C₃×#41	—	2	8	no	—
190	C₃×#41	—	9	1	no	—
191	C₃×#41	—	9	4	no	—
192	C₃×#41	—	18	2	no	—
193	C₃×#41	C₂×C₃×D₃	1	8	yes	complemented
194	C₃×#41	C₂×C₃⋊D₃	1	2	yes	complemented
195	C₂×C₃×D₄	—	9	4	no	—
196	C₂×C₃×D₄	—	9	2	no	—
197	C₂×C₃×D₄	—	18	2	no	—
198	C₂×C₃×D₄	—	18	4	no	—
199	#239	—	3	16	no	—
200	D₃×D₄	—	3	16	no	—
201	D₃×D₄	—	6	16	no	—
202	#235	—	3	4	no	—
203	#235	—	6	8	no	—
204	C₂×D₁₂	—	3	8	no	—
205	C₂×D₁₂	—	6	8	no	—
206	C₃×Q₁₆	—	2	4	no	—
207	C₃×Q₁₆	C₂×C₃×D₃	1	4	yes	complemented
208	C₃×Q₁₆	C₂×C₃⋊D₃	1	1	yes	characteristic, complemented
209	C₃×SD₁₆	—	2	8	no	—
210	C₃×SD₁₆	—	9	4	no	—
211	C₃×SD₁₆	C₂×C₃×D₃	1	8	yes	complemented
212	C₃×SD₁₆	C₂×C₃⋊D₃	1	2	yes	complemented
213	C₃×D₈	—	2	4	no	—
214	C₃×D₈	—	9	4	no	—
215	C₃×D₈	—	9	2	no	—
216	C₃×D₈	—	9	2	no	—
217	C₃×D₈	C₂×C₃×D₃	1	4	yes	complemented
218	C₃×D₈	C₂×C₃⋊D₃	1	1	yes	characteristic, complemented
219	C₃×M₁₆	—	9	1	no	—
220	C₃×M₁₆	—	9	2	no	—
221	C₂×C₃×C₈	—	2	4	no	—
222	C₂×C₃×C₈	—	9	2	no	—
223	C₂×C₃×C₈	C₂×C₃×D₃	1	4	yes	complemented
224	C₂×C₃×C₈	C₂×C₃⋊D₃	1	1	yes	characteristic, complemented
225	#215	—	3	16	no	—
226	#213	—	3	16	no	—
227	#208	—	3	4	no	—
228	D₂₄	—	3	8	no	—
229	D₂₄	—	3	8	no	—
230	ΓL(1,𝔽₂₅)	—	3	8	no	—
231	D₃×C₈	—	3	8	no	—
232	C₂²×C₃²	—	2	2	no	—
233	C₂²×C₃²	—	2	4	no	—
234	C₂²×C₃²	—	2	8	no	—
235	C₂²×C₃²	—	4	8	no	—
236	C₂²×C₃²	D₃×D₄	1	4	yes	—
237	C₂²×C₃²	C₂×C₃×D₄	1	1	yes	characteristic
238	C₂×C₃⋊D₃	—	2	4	no	—
239	C₂×C₃⋊D₃	—	8	4	no	—
240	C₂×C₃⋊D₃	—	8	2	no	—
241	C₂×C₃⋊D₃	C₂×C₃×D₄	1	2	yes	—
242	C₂×C₃×D₃	—	3	8	no	—
243	C₂×C₃×D₃	—	6	16	no	—
244	C₂×C₃×D₃	—	24	16	no	—
245	C₂×C₃×D₃	—	24	8	no	—
246	C₃²×C₄	—	2	2	no	—
247	C₃²×C₄	—	2	4	no	—
248	C₃²×C₄	—	2	4	no	—
249	C₃²×C₄	—	2	8	no	—
250	C₃²×C₄	—	4	8	no	—
251	C₃²×C₄	D₃×D₄	1	4	yes	—
252	C₃²×C₄	C₂×C₃×D₄	1	1	yes	characteristic
253	C₃²×C₄	C₂³×D₃	1	4	yes	—
254	C₃²×C₄	C₂⁴×C₃	1	1	yes	characteristic
255	C₃⋊Dic₃	C₂×C₃×D₄	1	2	yes	—
256	C₃×Dic₃	—	3	8	no	—
257	H(5,𝔽₂)	—	9	2	no	—
258	AGL(1,ℤ/8)	—	9	2	no	—
259	AGL(1,ℤ/8)	—	9	4	no	—
260	#135	—	9	2	no	—
261	#135	C₃×C₃⋊D₃	1	1	yes	characteristic, complemented
262	C₂×D₈	—	9	2	no	—
263	C₂×D₈	—	9	1	no	—
264	#131	—	9	1	no	—
265	C₃³	#576	1	1	yes	characteristic, complemented, Sylow
266	C₂³×C₃	—	18	4	no	—
267	C₂³×C₃	—	18	2	no	—
268	C₂²×D₃	—	6	16	no	—
269	C₂²×D₃	—	6	8	no	—
270	C₃×Q₈	—	2	8	no	—
271	C₃×Q₈	—	9	1	no	—
272	C₃×Q₈	C₂²×C₃×D₃	1	8	yes	—
273	C₃×Q₈	C₂²×C₃⋊D₃	1	2	yes	—
274	C₃×D₄	—	2	8	no	—
275	C₃×D₄	—	2	2	no	—
276	C₃×D₄	—	2	8	no	—
277	C₃×D₄	—	4	8	no	—
278	C₃×D₄	—	9	2	no	—
279	C₃×D₄	—	9	4	no	—
280	C₃×D₄	—	9	1	no	—
281	C₃×D₄	—	18	4	no	—
282	C₃×D₄	—	18	4	no	—
283	C₃×D₄	—	18	2	no	—
284	C₃×D₄	C₂²×C₃×D₃	1	8	yes	—
285	C₃×D₄	C₂²×C₃⋊D₃	1	2	yes	—
286	C₂×C₃×C₄	—	2	8	no	—
287	C₂×C₃×C₄	—	2	2	no	—
288	C₂×C₃×C₄	—	2	4	no	—
289	C₂×C₃×C₄	—	4	8	no	—
290	C₂×C₃×C₄	—	9	2	no	—
291	C₂×C₃×C₄	—	18	4	no	—
292	C₂×C₃×C₄	C₂²×C₃×D₃	1	4	yes	—
293	C₂×C₃×C₄	C₂²×C₃⋊D₃	1	1	yes	characteristic
294	#67	—	3	8	no	—
295	#67	—	6	16	no	—
296	D₁₂	—	3	16	no	—
297	D₁₂	—	3	4	no	—
298	D₁₂	—	6	16	no	—
299	D₁₂	—	6	8	no	—
300	C₄×D₃	—	3	8	no	—
301	C₄×D₃	—	6	16	no	—
302	Dic₆	—	3	4	no	—
303	C₃×C₈	—	2	4	no	—
304	C₃×C₈	—	2	4	no	—
305	C₃×C₈	—	9	2	no	—
306	C₃×C₈	C₂²×C₃×D₃	1	4	yes	—
307	C₃×C₈	C₂²×C₃×D₃	1	4	yes	—
308	C₃×C₈	C₂²×C₃⋊D₃	1	1	yes	characteristic
309	C₃×C₈	C₂²×C₃⋊D₃	1	1	yes	characteristic
310	C₃⋊C₈	—	3	8	no	—
311	C₂×C₃²	—	2	1	no	—
312	C₂×C₃²	—	2	4	no	—
313	C₂×C₃²	—	2	4	no	—
314	C₂×C₃²	—	4	4	no	—
315	C₂×C₃²	—	4	2	no	—
316	C₂×C₃²	—	4	8	no	—
317	C₂×C₃²	—	8	8	no	—
318	C₂×C₃²	C₂×D₃×D₄	1	4	yes	—
319	C₂×C₃²	C₂²×C₃×D₄	1	1	yes	characteristic
320	C₃⋊D₃	—	2	2	no	—
321	C₃⋊D₃	—	4	4	no	—
322	C₃×D₃	—	6	8	no	—
323	C₃×D₃	—	12	16	no	—
324	#41	—	9	1	no	—
325	#41	—	9	4	no	—
326	#41	—	18	2	no	—
327	#41	C₂×C₃×C₃⋊D₃	1	2	yes	complemented
328	C₂×D₄	—	9	4	no	—
329	C₂×D₄	—	9	2	no	—
330	C₂×D₄	—	18	2	no	—
331	C₂×D₄	—	18	4	no	—
332	Q₁₆	C₂×C₃×C₃⋊D₃	1	1	yes	characteristic, complemented
333	SD₁₆	—	9	4	no	—
334	SD₁₆	C₂×C₃×C₃⋊D₃	1	2	yes	complemented
335	D₈	—	9	4	no	—
336	D₈	—	9	2	no	—
337	D₈	—	9	2	no	—
338	D₈	C₂×C₃×C₃⋊D₃	1	1	yes	characteristic, complemented
339	M₁₆	—	9	1	no	—
340	M₁₆	—	9	2	no	—
341	C₂×C₈	—	9	2	no	—
342	C₂×C₈	C₂×C₃×C₃⋊D₃	1	1	yes	characteristic, complemented
343	C₂²×C₃	—	2	8	no	—
344	C₂²×C₃	—	2	2	no	—
345	C₂²×C₃	—	2	4	no	—
346	C₂²×C₃	—	4	8	no	—
347	C₂²×C₃	—	9	2	no	—
348	C₂²×C₃	—	18	4	no	—
349	C₂²×C₃	—	72	2	no	—
350	C₂²×C₃	—	72	4	no	—
351	C₂²×C₃	C₃×D₃×D₄	1	4	yes	—
352	C₂²×C₃	D₄×C₃⋊D₃	1	1	yes	characteristic
353	C₂×D₃	—	3	8	no	—
354	C₂×D₃	—	6	16	no	—
355	C₂×D₃	—	24	16	no	—
356	C₂×D₃	—	24	8	no	—
357	C₃×C₄	—	2	8	no	—
358	C₃×C₄	—	2	2	no	—
359	C₃×C₄	—	2	4	no	—
360	C₃×C₄	—	2	4	no	—
361	C₃×C₄	—	4	8	no	—
362	C₃×C₄	—	9	2	no	—
363	C₃×C₄	C₃×D₃×D₄	1	4	yes	—
364	C₃×C₄	D₄×C₃⋊D₃	1	1	yes	characteristic
365	C₃×C₄	C₂³×C₃×D₃	1	4	yes	—
366	C₃×C₄	C₂³×C₃⋊D₃	1	1	yes	characteristic
367	Dic₃	—	3	8	no	—
368	C₃²	—	2	4	no	—
369	C₃²	#5769	1	4	yes	complemented
370	C₃²	C₃×#576	1	1	yes	characteristic, complemented
371	C₂³	—	18	4	no	—
372	C₂³	—	18	2	no	—
373	Q₈	—	9	1	no	—
374	Q₈	C₂²×C₃×C₃⋊D₃	1	2	yes	—
375	D₄	—	2	2	no	—
376	D₄	—	9	2	no	—
377	D₄	—	9	4	no	—
378	D₄	—	9	1	no	—
379	D₄	—	18	4	no	—
380	D₄	—	18	4	no	—
381	D₄	—	18	2	no	—
382	D₄	C₂²×C₃×C₃⋊D₃	1	2	yes	—
383	C₂×C₄	—	2	2	no	—
384	C₂×C₄	—	9	2	no	—
385	C₂×C₄	—	18	4	no	—
386	C₂×C₄	C₂²×C₃×C₃⋊D₃	1	1	yes	characteristic
387	C₈	—	9	2	no	—
388	C₈	C₂²×C₃×C₃⋊D₃	1	1	yes	characteristic
389	C₈	C₂²×C₃×C₃⋊D₃	1	1	yes	characteristic
390	C₂×C₃	—	2	4	no	—
391	C₂×C₃	—	2	1	no	—
392	C₂×C₃	—	2	4	no	—
393	C₂×C₃	—	4	4	no	—
394	C₂×C₃	—	4	8	no	—
395	C₂×C₃	—	4	2	no	—
396	C₂×C₃	—	8	8	no	—
397	C₂×C₃	—	18	2	no	—
398	C₂×C₃	—	36	4	no	—
399	C₂×C₃	C₂×C₃×D₃×D₄	1	4	yes	—
400	C₂×C₃	C₂×D₄×C₃⋊D₃	1	1	yes	characteristic
401	D₃	—	6	8	no	—
402	D₃	—	12	16	no	—
403	C₂²	—	2	2	no	—
404	C₂²	—	9	2	no	—
405	C₂²	—	18	4	no	—
406	C₂²	—	72	2	no	—
407	C₂²	—	72	4	no	—
408	C₂²	C₃×D₄×C₃⋊D₃	1	1	yes	characteristic
409	C₄	—	2	2	no	—
410	C₄	—	9	2	no	—
411	C₄	C₃×D₄×C₃⋊D₃	1	1	yes	characteristic
412	C₄	C₂³×C₃×C₃⋊D₃	1	1	yes	characteristic
413	C₃	—	2	4	no	—
414	C₃	C₃×#5769	1	4	yes	complemented
415	C₃	#10596240	1	1	yes	characteristic, direct product factor
416	C₂	—	2	1	no	—
417	C₂	—	4	2	no	—
418	C₂	—	18	2	no	—
419	C₂	—	36	4	no	—
420	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×#135	C₂
socle	C₂×C₃³	C₂²×D₄
hypercenter	C₃×#135	C₃⋊D₃
hypocenter	C₃²	C₃×#576
LCS	G ▷ C₃²×C₄ ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂×C₃ ◁ C₂×C₃×C₄ ◁ C₃×#135

Complements

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

N	H ≅ G/N	type
1. C₃×#64366	C₂	semidirect
2. C₃³×#135	C₂	semidirect
3. C₃×#64158	C₂	semidirect
4. C₃×#64128	C₂	semidirect
5. C₃×#64123	C₂	semidirect
6. C₃×D₈×C₃⋊D₃	C₂	semidirect
7. C₂×C₃×#3745	C₂	semidirect
8. C₃×#64110	C₂	semidirect
9. #10596240	C₃	direct (N × H)
11. C₃³×#41	C₂²	semidirect
12. C₃×#3832	C₂²	semidirect
14. C₃×D₄×C₃⋊D₃	C₂²	semidirect
18. C₂×C₃×C₄⋊(C₃⋊D₃)	C₂²	semidirect
19. C₃³×Q₁₆	C₂²	semidirect
20. C₃³×SD₁₆	C₂²	semidirect
21. C₃³×D₈	C₂²	semidirect
22. C₂×C₃³×C₈	C₂²	semidirect
23. C₃×#3755	C₂²	semidirect
24. C₃×#3753	C₂²	semidirect
25. C₃×#3748	C₂²	semidirect
26. C₃×#3745	C₂²	semidirect
27. C₃×#3745	C₂²	semidirect
28. C₃×#3743	C₂²	semidirect
29. C₃×C₈×C₃⋊D₃	C₂²	semidirect
30. #64366	C₂×C₃	semidirect
33. C₃²×#135	D₃	semidirect
34. C₃²×#135	C₂×C₃	semidirect
35. #64158	C₂×C₃	semidirect
36. #64128	C₂×C₃	semidirect
37. #64123	C₂×C₃	semidirect
38. D₈×C₃⋊D₃	C₂×C₃	semidirect
39. C₂×#3745	C₂×C₃	semidirect
40. #64110	C₂×C₃	semidirect
69. C₃²×#41	C₂×D₃	semidirect
70. C₃²×#41	C₂²×C₃	semidirect
71. #3832	C₂²×C₃	semidirect
73. D₄×C₃⋊D₃	C₂²×C₃	semidirect
77. C₂×C₄⋊(C₃⋊D₃)	C₂²×C₃	semidirect
86. C₃²×Q₁₆	C₂×D₃	semidirect
87. C₃²×Q₁₆	C₂²×C₃	semidirect
89. C₃²×SD₁₆	C₂×D₃	semidirect
90. C₃²×SD₁₆	C₂²×C₃	semidirect
92. C₃²×D₈	C₂×D₃	semidirect
93. C₃²×D₈	C₂²×C₃	semidirect
95. C₂×C₃²×C₈	C₂×D₃	semidirect
96. C₂×C₃²×C₈	C₂²×C₃	semidirect
97. #3755	C₂²×C₃	semidirect
98. #3753	C₂²×C₃	semidirect
99. #3748	C₂²×C₃	semidirect
100. #3745	C₂²×C₃	semidirect
101. #3745	C₂²×C₃	semidirect
102. #3743	C₂²×C₃	semidirect
103. C₈×C₃⋊D₃	C₂²×C₃	semidirect
127. C₃×#135	C₃×D₃	semidirect
128. C₃×#135	C₃⋊D₃	semidirect
193. C₃×#41	C₂×C₃×D₃	semidirect
194. C₃×#41	C₂×C₃⋊D₃	semidirect
207. C₃×Q₁₆	C₂×C₃×D₃	semidirect
208. C₃×Q₁₆	C₂×C₃⋊D₃	semidirect
211. C₃×SD₁₆	C₂×C₃×D₃	semidirect
212. C₃×SD₁₆	C₂×C₃⋊D₃	semidirect
217. C₃×D₈	C₂×C₃×D₃	semidirect
218. C₃×D₈	C₂×C₃⋊D₃	semidirect
223. C₂×C₃×C₈	C₂×C₃×D₃	semidirect
224. C₂×C₃×C₈	C₂×C₃⋊D₃	semidirect
261. #135	C₃×C₃⋊D₃	semidirect
265. C₃³	#576	semidirect
327. #41	C₂×C₃×C₃⋊D₃	semidirect
332. Q₁₆	C₂×C₃×C₃⋊D₃	semidirect
334. SD₁₆	C₂×C₃×C₃⋊D₃	semidirect
338. D₈	C₂×C₃×C₃⋊D₃	semidirect
342. C₂×C₈	C₂×C₃×C₃⋊D₃	semidirect
369. C₃²	#5769	semidirect
370. C₃²	C₃×#576	semidirect
414. C₃	C₃×#5769	semidirect
415. C₃	#10596240	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×#64366	2
C₃³×#135	2
C₃×#64158	2
C₃×#64128	2
C₃×#64123	2
C₃×D₈×C₃⋊D₃	2
C₂×C₃×#3745	2
C₃×#64110	2
#10596240	3
C₃×#5769	3

Subgroups — C3×#10596240 #13179520

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#10596240 #13179520