Subgroups — C₃²×#5757 #13179134

Order 1728.

1912 conjugacy classes of subgroups.

The subgroup lattice has 418 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×#1004	C₂	1	2	yes	maximal, complemented
2	C₂×C₃³×D₈	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃²×#932	C₂	1	2	yes	maximal, complemented
4	C₃²×#912	C₂	1	2	yes	maximal, complemented
5	C₃²×#911	C₂	1	4	yes	maximal, complemented
6	C₃²×D₃×D₈	C₂	1	2	yes	maximal, complemented
7	C₃²×#904	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃²×#901	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃²×#576	—	3	1	no	maximal
10	C₃×#5757	C₃	1	4	yes	maximal, direct product factor
11	C₂×C₃³×D₄	C₂²	1	2	yes	complemented
12	C₂×C₃²×#67	—	2	4	no	—
13	C₃²×#237	—	2	2	no	—
14	C₃²×#237	C₂²	1	4	yes	complemented
15	C₃²×D₃×D₄	—	2	2	no	—
16	C₃²×D₃×D₄	C₂²	1	4	yes	complemented
17	C₃²×#235	C₂²	1	1	yes	characteristic
18	C₃²×#235	C₂²	1	2	yes	complemented
19	C₃³×D₈	C₂²	1	2	yes	complemented
20	C₃³×D₈	C₂²	1	2	yes	complemented
21	C₂×C₃³×C₈	C₂²	1	1	yes	characteristic, complemented
22	C₃²×#214	C₂²	1	4	yes	complemented
23	C₃²×#213	C₂²	1	4	yes	complemented
24	C₃²×#208	C₂²	1	1	yes	characteristic, complemented
25	C₃²×Dic₁₂	C₂²	1	1	yes	characteristic, complemented
26	C₃²×D₂₄	C₂²	1	1	yes	characteristic, complemented
27	C₃²×#204	C₂²	1	2	yes	complemented
28	C₃²×ΓL(1,𝔽₂₅)	C₂²	1	2	yes	complemented
29	C₃²×D₃×C₈	C₂²	1	2	yes	complemented
30	C₃²×H(5,𝔽₂)	—	3	2	no	—
31	C₃×#1004	C₂×C₃	1	8	yes	complemented
32	C₃²×AGL(1,ℤ/8)	—	3	2	no	—
33	C₃²×AGL(1,ℤ/8)	—	3	4	no	—
34	C₃²×#135	—	3	2	no	—
35	C₃²×#135	—	3	1	no	—
36	C₂×C₃²×D₈	—	2	4	no	—
37	C₂×C₃²×D₈	—	3	2	no	—
38	C₂×C₃²×D₈	D₃	1	1	yes	characteristic, complemented
39	C₂×C₃²×D₈	C₂×C₃	1	4	yes	complemented
40	C₃²×#131	—	3	1	no	—
41	C₃×#932	C₂×C₃	1	8	yes	complemented
42	C₃×#912	C₂×C₃	1	8	yes	complemented
43	C₃×#911	C₂×C₃	1	16	yes	complemented
44	C₃×D₃×D₈	C₂×C₃	1	8	yes	complemented
45	C₃×#904	C₂×C₃	1	4	yes	complemented
46	C₃×#901	C₂×C₃	1	4	yes	complemented
47	C₂³×C₃³	—	2	2	no	—
48	C₂²×C₃²×D₃	—	2	4	no	—
49	C₃³×D₄	—	2	2	no	—
50	C₃³×D₄	C₂³	1	4	yes	—
51	C₂×C₃³×C₄	C₂³	1	1	yes	characteristic
52	C₃²×#67	—	2	2	no	—
53	C₃²×#67	—	2	4	no	—
54	C₃²×#67	—	2	4	no	—
55	C₃²×#67	D₄	1	2	yes	—
56	C₂×C₃²×Dic₃	—	2	4	no	—
57	C₃²×D₁₂	D₄	1	1	yes	characteristic
58	C₃²×D₁₂	C₂³	1	2	yes	—
59	C₃²×C₄×D₃	—	2	2	no	—
60	C₃²×C₄×D₃	C₂³	1	2	yes	—
61	C₃²×Dic₆	D₄	1	1	yes	characteristic
62	C₃²×Dic₆	C₂³	1	2	yes	—
63	C₃³×C₈	C₂³	1	1	yes	characteristic
64	C₃³×C₈	C₂³	1	1	yes	characteristic
65	C₃²×C₃⋊C₈	C₂³	1	2	yes	—
66	C₃×#576	—	3	4	no	—
67	#5757	C₃²	1	1	yes	characteristic, direct product factor
68	C₃²×#41	—	3	1	no	—
69	C₃²×#41	—	3	4	no	—
70	C₃²×#41	—	3	2	no	—
71	C₃²×#41	—	6	2	no	—
72	C₂×C₃²×D₄	—	2	8	no	—
73	C₂×C₃²×D₄	—	3	4	no	—
74	C₂×C₃²×D₄	—	6	2	no	—
75	C₂×C₃²×D₄	—	6	4	no	—
76	C₂×C₃²×D₄	C₂×D₃	1	2	yes	complemented
77	C₂×C₃²×D₄	C₂²×C₃	1	8	yes	complemented
78	C₂×C₃×#67	—	2	16	no	—
79	C₃×#237	—	2	8	no	—
80	C₃×#237	C₂²×C₃	1	16	yes	complemented
81	C₃×D₃×D₄	—	2	8	no	—
82	C₃×D₃×D₄	C₂²×C₃	1	16	yes	complemented
83	C₃×#235	C₂²×C₃	1	4	yes	—
84	C₃×#235	C₂²×C₃	1	8	yes	complemented
85	C₃²×Q₁₆	—	3	1	no	—
86	C₃²×SD₁₆	—	3	4	no	—
87	C₃²×SD₁₆	—	3	2	no	—
88	C₃²×D₈	—	2	8	no	—
89	C₃²×D₈	—	2	8	no	—
90	C₃²×D₈	—	3	4	no	—
91	C₃²×D₈	—	3	1	no	—
92	C₃²×D₈	C₂×D₃	1	2	yes	complemented
93	C₃²×D₈	C₂×D₃	1	2	yes	complemented
94	C₃²×D₈	C₂²×C₃	1	8	yes	complemented
95	C₃²×D₈	C₂²×C₃	1	8	yes	complemented
96	C₃²×M₁₆	—	3	1	no	—
97	C₃²×M₁₆	—	3	2	no	—
98	C₂×C₃²×C₈	—	2	4	no	—
99	C₂×C₃²×C₈	—	3	2	no	—
100	C₂×C₃²×C₈	C₂×D₃	1	1	yes	characteristic, complemented
101	C₂×C₃²×C₈	C₂²×C₃	1	4	yes	complemented
102	C₃×#214	C₂²×C₃	1	16	yes	complemented
103	C₃×#213	C₂²×C₃	1	16	yes	complemented
104	C₃×#208	C₂²×C₃	1	4	yes	complemented
105	C₃×Dic₁₂	C₂²×C₃	1	4	yes	complemented
106	C₃×D₂₄	C₂²×C₃	1	4	yes	complemented
107	C₃×#204	C₂²×C₃	1	8	yes	complemented
108	C₃×ΓL(1,𝔽₂₅)	C₂²×C₃	1	8	yes	complemented
109	C₃×D₃×C₈	C₂²×C₃	1	8	yes	complemented
110	C₂²×C₃³	—	2	4	no	—
111	C₂²×C₃³	—	8	2	no	—
112	C₂²×C₃³	C₂×D₄	1	1	yes	characteristic
113	C₂×C₃²×D₃	—	2	2	no	—
114	C₂×C₃²×D₃	—	8	4	no	—
115	C₂×C₃²×D₃	C₂×D₄	1	2	yes	—
116	C₃³×C₄	C₂×D₄	1	1	yes	characteristic
117	C₃³×C₄	C₂⁴	1	1	yes	characteristic
118	C₃²×Dic₃	—	2	2	no	—
119	C₃²×Dic₃	C₂×D₄	1	2	yes	—
120	C₃×H(5,𝔽₂)	—	3	8	no	—
121	#1004	C₂×C₃²	1	2	yes	complemented
122	C₃×AGL(1,ℤ/8)	—	3	8	no	—
123	C₃×AGL(1,ℤ/8)	—	3	16	no	—
124	C₃×#135	—	3	8	no	—
125	C₃×#135	—	3	4	no	—
126	C₂×C₃×D₈	—	2	4	no	—
127	C₂×C₃×D₈	—	3	8	no	—
128	C₂×C₃×D₈	C₃×D₃	1	4	yes	complemented
129	C₂×C₃×D₈	C₂×C₃²	1	1	yes	characteristic, complemented
130	C₃×#131	—	3	4	no	—
131	#932	C₂×C₃²	1	2	yes	complemented
132	#912	C₂×C₃²	1	2	yes	complemented
133	#911	C₂×C₃²	1	4	yes	complemented
134	D₃×D₈	C₂×C₃²	1	2	yes	complemented
135	#904	C₂×C₃²	1	1	yes	characteristic, complemented
136	#901	C₂×C₃²	1	1	yes	characteristic, complemented
137	C₂³×C₃²	—	2	2	no	—
138	C₂³×C₃²	—	2	8	no	—
139	C₂³×C₃²	—	4	8	no	—
140	C₂³×C₃²	—	6	4	no	—
141	C₂²×C₃×D₃	—	2	16	no	—
142	C₃²×Q₈	—	3	1	no	—
143	C₃²×Q₈	—	3	2	no	—
144	C₃²×D₄	—	2	2	no	—
145	C₃²×D₄	—	2	16	no	—
146	C₃²×D₄	—	2	8	no	—
147	C₃²×D₄	—	3	2	no	—
148	C₃²×D₄	—	3	1	no	—
149	C₃²×D₄	—	3	2	no	—
150	C₃²×D₄	—	4	8	no	—
151	C₃²×D₄	—	6	4	no	—
152	C₃²×D₄	—	6	4	no	—
153	C₃²×D₄	—	6	2	no	—
154	C₃²×D₄	C₂²×D₃	1	4	yes	—
155	C₃²×D₄	C₂³×C₃	1	16	yes	—
156	C₂×C₃²×C₄	—	2	4	no	—
157	C₂×C₃²×C₄	—	3	2	no	—
158	C₂×C₃²×C₄	—	6	4	no	—
159	C₂×C₃²×C₄	—	6	2	no	—
160	C₂×C₃²×C₄	C₂²×D₃	1	1	yes	characteristic
161	C₂×C₃²×C₄	C₂³×C₃	1	4	yes	—
162	C₃×#67	—	2	8	no	—
163	C₃×#67	—	2	16	no	—
164	C₃×#67	—	2	16	no	—
165	C₃×#67	C₃×D₄	1	8	yes	—
166	C₂×C₃×Dic₃	—	2	16	no	—
167	C₃×D₁₂	C₃×D₄	1	4	yes	—
168	C₃×D₁₂	C₂³×C₃	1	8	yes	—
169	C₃×C₄×D₃	—	2	8	no	—
170	C₃×C₄×D₃	C₂³×C₃	1	8	yes	—
171	C₃×Dic₆	C₃×D₄	1	4	yes	—
172	C₃×Dic₆	C₂³×C₃	1	8	yes	—
173	C₃²×C₈	—	2	4	no	—
174	C₃²×C₈	—	2	4	no	—
175	C₃²×C₈	—	3	2	no	—
176	C₃²×C₈	C₂²×D₃	1	1	yes	characteristic
177	C₃²×C₈	C₂²×D₃	1	1	yes	characteristic
178	C₃²×C₈	C₂³×C₃	1	4	yes	—
179	C₃²×C₈	C₂³×C₃	1	4	yes	—
180	C₃×C₃⋊C₈	C₂³×C₃	1	8	yes	—
181	#576	—	3	1	no	Sylow
182	C₂×C₃³	—	2	1	no	—
183	C₂×C₃³	—	4	4	no	—
184	C₂×C₃³	C₂²×D₄	1	1	yes	characteristic
185	C₃²×D₃	—	2	2	no	—
186	C₃²×D₃	—	4	2	no	—
187	C₃×#41	—	3	4	no	—
188	C₃×#41	—	3	16	no	—
189	C₃×#41	—	3	8	no	—
190	C₃×#41	—	6	8	no	—
191	C₂×C₃×D₄	—	2	8	no	—
192	C₂×C₃×D₄	—	3	16	no	—
193	C₂×C₃×D₄	—	6	8	no	—
194	C₂×C₃×D₄	—	6	16	no	—
195	C₂×C₃×D₄	C₂×C₃×D₃	1	8	yes	complemented
196	C₂×C₃×D₄	C₂²×C₃²	1	2	yes	complemented
197	C₂×#67	—	2	4	no	—
198	#237	—	2	2	no	—
199	#237	C₂²×C₃²	1	4	yes	complemented
200	D₃×D₄	—	2	2	no	—
201	D₃×D₄	C₂²×C₃²	1	4	yes	complemented
202	#235	C₂²×C₃²	1	1	yes	characteristic
203	#235	C₂²×C₃²	1	2	yes	complemented
204	C₃×Q₁₆	—	3	4	no	—
205	C₃×SD₁₆	—	3	16	no	—
206	C₃×SD₁₆	—	3	8	no	—
207	C₃×D₈	—	2	8	no	—
208	C₃×D₈	—	2	8	no	—
209	C₃×D₈	—	3	16	no	—
210	C₃×D₈	—	3	4	no	—
211	C₃×D₈	C₂×C₃×D₃	1	8	yes	complemented
212	C₃×D₈	C₂×C₃×D₃	1	8	yes	complemented
213	C₃×D₈	C₂²×C₃²	1	2	yes	complemented
214	C₃×D₈	C₂²×C₃²	1	2	yes	complemented
215	C₃×M₁₆	—	3	4	no	—
216	C₃×M₁₆	—	3	8	no	—
217	C₂×C₃×C₈	—	2	4	no	—
218	C₂×C₃×C₈	—	3	8	no	—
219	C₂×C₃×C₈	C₂×C₃×D₃	1	4	yes	complemented
220	C₂×C₃×C₈	C₂²×C₃²	1	1	yes	characteristic, complemented
221	#214	C₂²×C₃²	1	4	yes	complemented
222	#213	C₂²×C₃²	1	4	yes	complemented
223	#208	C₂²×C₃²	1	1	yes	characteristic, complemented
224	Dic₁₂	C₂²×C₃²	1	1	yes	characteristic, complemented
225	D₂₄	C₂²×C₃²	1	1	yes	characteristic, complemented
226	#204	C₂²×C₃²	1	2	yes	complemented
227	ΓL(1,𝔽₂₅)	C₂²×C₃²	1	2	yes	complemented
228	D₃×C₈	C₂²×C₃²	1	2	yes	complemented
229	C₂²×C₃²	—	2	4	no	—
230	C₂²×C₃²	—	2	4	no	—
231	C₂²×C₃²	—	2	16	no	—
232	C₂²×C₃²	—	3	2	no	—
233	C₂²×C₃²	—	4	16	no	—
234	C₂²×C₃²	—	6	2	no	—
235	C₂²×C₃²	—	8	2	no	—
236	C₂²×C₃²	—	8	8	no	—
237	C₂²×C₃²	—	8	16	no	—
238	C₂²×C₃²	—	24	4	no	—
239	C₂²×C₃²	D₃×D₄	1	1	yes	characteristic
240	C₂²×C₃²	C₂×C₃×D₄	1	4	yes	—
241	C₂×C₃×D₃	—	2	8	no	—
242	C₂×C₃×D₃	—	8	16	no	—
243	C₂×C₃×D₃	C₂×C₃×D₄	1	8	yes	—
244	C₃²×C₄	—	2	4	no	—
245	C₃²×C₄	—	2	4	no	—
246	C₃²×C₄	—	3	2	no	—
247	C₃²×C₄	—	6	2	no	—
248	C₃²×C₄	D₃×D₄	1	1	yes	characteristic
249	C₃²×C₄	C₂×C₃×D₄	1	4	yes	—
250	C₃²×C₄	C₂³×D₃	1	1	yes	characteristic
251	C₃²×C₄	C₂⁴×C₃	1	4	yes	—
252	C₃×Dic₃	—	2	8	no	—
253	C₃×Dic₃	C₂×C₃×D₄	1	8	yes	—
254	H(5,𝔽₂)	—	3	2	no	—
255	AGL(1,ℤ/8)	—	3	2	no	—
256	AGL(1,ℤ/8)	—	3	4	no	—
257	#135	—	3	2	no	—
258	#135	—	3	1	no	—
259	C₂×D₈	—	3	2	no	—
260	C₂×D₈	C₃²×D₃	1	1	yes	characteristic, complemented
261	#131	—	3	1	no	—
262	C₃³	#576	1	1	yes	characteristic, complemented, Sylow
263	C₂³×C₃	—	2	8	no	—
264	C₂³×C₃	—	2	2	no	—
265	C₂³×C₃	—	4	8	no	—
266	C₂³×C₃	—	6	16	no	—
267	C₂²×D₃	—	2	4	no	—
268	C₃×Q₈	—	3	4	no	—
269	C₃×Q₈	—	3	8	no	—
270	C₃×D₄	—	2	8	no	—
271	C₃×D₄	—	2	2	no	—
272	C₃×D₄	—	2	16	no	—
273	C₃×D₄	—	3	8	no	—
274	C₃×D₄	—	3	8	no	—
275	C₃×D₄	—	3	4	no	—
276	C₃×D₄	—	4	8	no	—
277	C₃×D₄	—	6	16	no	—
278	C₃×D₄	—	6	16	no	—
279	C₃×D₄	—	6	8	no	—
280	C₃×D₄	C₂²×C₃×D₃	1	16	yes	—
281	C₃×D₄	C₂³×C₃²	1	4	yes	—
282	C₂×C₃×C₄	—	2	4	no	—
283	C₂×C₃×C₄	—	3	8	no	—
284	C₂×C₃×C₄	—	6	16	no	—
285	C₂×C₃×C₄	—	6	8	no	—
286	C₂×C₃×C₄	C₂²×C₃×D₃	1	4	yes	—
287	C₂×C₃×C₄	C₂³×C₃²	1	1	yes	characteristic
288	#67	—	2	2	no	—
289	#67	—	2	4	no	—
290	#67	—	2	4	no	—
291	#67	C₃²×D₄	1	2	yes	—
292	C₂×Dic₃	—	2	4	no	—
293	D₁₂	C₃²×D₄	1	1	yes	characteristic
294	D₁₂	C₂³×C₃²	1	2	yes	—
295	C₄×D₃	—	2	2	no	—
296	C₄×D₃	C₂³×C₃²	1	2	yes	—
297	Dic₆	C₃²×D₄	1	1	yes	characteristic
298	Dic₆	C₂³×C₃²	1	2	yes	—
299	C₃×C₈	—	2	4	no	—
300	C₃×C₈	—	2	4	no	—
301	C₃×C₈	—	3	8	no	—
302	C₃×C₈	C₂²×C₃×D₃	1	4	yes	—
303	C₃×C₈	C₂²×C₃×D₃	1	4	yes	—
304	C₃×C₈	C₂³×C₃²	1	1	yes	characteristic
305	C₃×C₈	C₂³×C₃²	1	1	yes	characteristic
306	C₃⋊C₈	C₂³×C₃²	1	2	yes	—
307	C₂×C₃²	—	2	8	no	—
308	C₂×C₃²	—	2	4	no	—
309	C₂×C₃²	—	2	1	no	—
310	C₂×C₃²	—	2	4	no	—
311	C₂×C₃²	—	4	4	no	—
312	C₂×C₃²	—	4	16	no	—
313	C₂×C₃²	—	6	2	no	—
314	C₂×C₃²	—	8	16	no	—
315	C₂×C₃²	—	12	2	no	—
316	C₂×C₃²	C₂×D₃×D₄	1	1	yes	characteristic
317	C₂×C₃²	C₂²×C₃×D₄	1	4	yes	—
318	C₃×D₃	—	2	8	no	—
319	C₃×D₃	—	4	8	no	—
320	#41	—	3	1	no	—
321	#41	—	3	4	no	—
322	#41	—	3	2	no	—
323	#41	—	6	2	no	—
324	C₂×D₄	—	3	4	no	—
325	C₂×D₄	—	6	2	no	—
326	C₂×D₄	—	6	4	no	—
327	C₂×D₄	C₂×C₃²×D₃	1	2	yes	complemented
328	Q₁₆	—	3	1	no	—
329	SD₁₆	—	3	4	no	—
330	SD₁₆	—	3	2	no	—
331	D₈	—	3	4	no	—
332	D₈	—	3	1	no	—
333	D₈	C₂×C₃²×D₃	1	2	yes	complemented
334	D₈	C₂×C₃²×D₃	1	2	yes	complemented
335	M₁₆	—	3	1	no	—
336	M₁₆	—	3	2	no	—
337	C₂×C₈	—	3	2	no	—
338	C₂×C₈	C₂×C₃²×D₃	1	1	yes	characteristic, complemented
339	C₂²×C₃	—	2	16	no	—
340	C₂²×C₃	—	2	4	no	—
341	C₂²×C₃	—	2	4	no	—
342	C₂²×C₃	—	3	8	no	—
343	C₂²×C₃	—	4	16	no	—
344	C₂²×C₃	—	6	8	no	—
345	C₂²×C₃	—	8	8	no	—
346	C₂²×C₃	—	8	2	no	—
347	C₂²×C₃	—	8	16	no	—
348	C₂²×C₃	—	24	16	no	—
349	C₂²×C₃	C₃×D₃×D₄	1	4	yes	—
350	C₂²×C₃	C₂×C₃²×D₄	1	1	yes	characteristic
351	C₂×D₃	—	2	2	no	—
352	C₂×D₃	—	8	4	no	—
353	C₂×D₃	C₂×C₃²×D₄	1	2	yes	—
354	C₃×C₄	—	2	4	no	—
355	C₃×C₄	—	2	4	no	—
356	C₃×C₄	—	3	8	no	—
357	C₃×C₄	—	6	8	no	—
358	C₃×C₄	C₃×D₃×D₄	1	4	yes	—
359	C₃×C₄	C₂×C₃²×D₄	1	1	yes	characteristic
360	C₃×C₄	C₂³×C₃×D₃	1	4	yes	—
361	C₃×C₄	C₂⁴×C₃²	1	1	yes	characteristic
362	Dic₃	—	2	2	no	—
363	Dic₃	C₂×C₃²×D₄	1	2	yes	—
364	C₃²	—	2	4	no	—
365	C₃²	#5757	1	1	yes	characteristic, direct product factor
366	C₃²	C₃×#576	1	4	yes	complemented
367	C₂³	—	2	2	no	—
368	C₂³	—	6	4	no	—
369	Q₈	—	3	1	no	—
370	Q₈	—	3	2	no	—
371	D₄	—	2	2	no	—
372	D₄	—	3	2	no	—
373	D₄	—	3	2	no	—
374	D₄	—	3	1	no	—
375	D₄	—	6	4	no	—
376	D₄	—	6	4	no	—
377	D₄	—	6	2	no	—
378	D₄	C₂²×C₃²×D₃	1	4	yes	—
379	C₂×C₄	—	3	2	no	—
380	C₂×C₄	—	6	4	no	—
381	C₂×C₄	—	6	2	no	—
382	C₂×C₄	C₂²×C₃²×D₃	1	1	yes	characteristic
383	C₈	—	3	2	no	—
384	C₈	C₂²×C₃²×D₃	1	1	yes	characteristic
385	C₈	C₂²×C₃²×D₃	1	1	yes	characteristic
386	C₂×C₃	—	2	8	no	—
387	C₂×C₃	—	2	4	no	—
388	C₂×C₃	—	2	1	no	—
389	C₂×C₃	—	2	4	no	—
390	C₂×C₃	—	4	16	no	—
391	C₂×C₃	—	4	4	no	—
392	C₂×C₃	—	6	8	no	—
393	C₂×C₃	—	8	16	no	—
394	C₂×C₃	—	12	8	no	—
395	C₂×C₃	C₂×C₃×D₃×D₄	1	4	yes	—
396	C₂×C₃	C₂²×C₃²×D₄	1	1	yes	characteristic
397	D₃	—	2	2	no	—
398	D₃	—	4	2	no	—
399	C₂²	—	2	4	no	—
400	C₂²	—	3	2	no	—
401	C₂²	—	6	2	no	—
402	C₂²	—	8	2	no	—
403	C₂²	—	24	4	no	—
404	C₂²	C₃²×D₃×D₄	1	1	yes	characteristic
405	C₄	—	3	2	no	—
406	C₄	—	6	2	no	—
407	C₄	C₃²×D₃×D₄	1	1	yes	characteristic
408	C₄	C₂³×C₃²×D₃	1	1	yes	characteristic
409	C₃	—	2	4	no	—
410	C₃	C₃×#5757	1	4	yes	direct product factor
411	C₃	C₃²×#576	1	1	yes	characteristic, complemented
412	C₂	—	2	1	no	—
413	C₂	—	4	4	no	—
414	C₂	—	6	2	no	—
415	C₂	—	12	2	no	—
416	C₂	C₂×C₃²×D₃×D₄	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₃²	C₂×D₃×D₄
commutator, cosocle	C₃×C₄	C₂⁴×C₃²
Frattini	C₄	C₂³×C₃²×D₃
Fitting	C₂×C₃³×D₈	C₂
socle	C₂×C₃³	C₂²×D₄
hypercenter	C₂×C₃²×D₈	D₃
hypocenter	C₃	C₃²×#576
LCS	G ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₃² ◁ C₂×C₃²×C₄ ◁ C₂×C₃²×D₈

Complements

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

N	H ≅ G/N	type
1. C₃²×#1004	C₂	semidirect
2. C₂×C₃³×D₈	C₂	semidirect
3. C₃²×#932	C₂	semidirect
4. C₃²×#912	C₂	semidirect
5. C₃²×#911	C₂	semidirect
6. C₃²×D₃×D₈	C₂	semidirect
7. C₃²×#904	C₂	semidirect
8. C₃²×#901	C₂	semidirect
10. C₃×#5757	C₃	direct (N × H)
11. C₂×C₃³×D₄	C₂²	semidirect
14. C₃²×#237	C₂²	semidirect
16. C₃²×D₃×D₄	C₂²	semidirect
18. C₃²×#235	C₂²	semidirect
19. C₃³×D₈	C₂²	semidirect
20. C₃³×D₈	C₂²	semidirect
21. C₂×C₃³×C₈	C₂²	semidirect
22. C₃²×#214	C₂²	semidirect
23. C₃²×#213	C₂²	semidirect
24. C₃²×#208	C₂²	semidirect
25. C₃²×Dic₁₂	C₂²	semidirect
26. C₃²×D₂₄	C₂²	semidirect
27. C₃²×#204	C₂²	semidirect
28. C₃²×ΓL(1,𝔽₂₅)	C₂²	semidirect
29. C₃²×D₃×C₈	C₂²	semidirect
31. C₃×#1004	C₂×C₃	semidirect
38. C₂×C₃²×D₈	D₃	semidirect
39. C₂×C₃²×D₈	C₂×C₃	semidirect
41. C₃×#932	C₂×C₃	semidirect
42. C₃×#912	C₂×C₃	semidirect
43. C₃×#911	C₂×C₃	semidirect
44. C₃×D₃×D₈	C₂×C₃	semidirect
45. C₃×#904	C₂×C₃	semidirect
46. C₃×#901	C₂×C₃	semidirect
67. #5757	C₃²	direct (N × H)
76. C₂×C₃²×D₄	C₂×D₃	semidirect
77. C₂×C₃²×D₄	C₂²×C₃	semidirect
80. C₃×#237	C₂²×C₃	semidirect
82. C₃×D₃×D₄	C₂²×C₃	semidirect
84. C₃×#235	C₂²×C₃	semidirect
92. C₃²×D₈	C₂×D₃	semidirect
93. C₃²×D₈	C₂×D₃	semidirect
94. C₃²×D₈	C₂²×C₃	semidirect
95. C₃²×D₈	C₂²×C₃	semidirect
100. C₂×C₃²×C₈	C₂×D₃	semidirect
101. C₂×C₃²×C₈	C₂²×C₃	semidirect
102. C₃×#214	C₂²×C₃	semidirect
103. C₃×#213	C₂²×C₃	semidirect
104. C₃×#208	C₂²×C₃	semidirect
105. C₃×Dic₁₂	C₂²×C₃	semidirect
106. C₃×D₂₄	C₂²×C₃	semidirect
107. C₃×#204	C₂²×C₃	semidirect
108. C₃×ΓL(1,𝔽₂₅)	C₂²×C₃	semidirect
109. C₃×D₃×C₈	C₂²×C₃	semidirect
121. #1004	C₂×C₃²	semidirect
128. C₂×C₃×D₈	C₃×D₃	semidirect
129. C₂×C₃×D₈	C₂×C₃²	semidirect
131. #932	C₂×C₃²	semidirect
132. #912	C₂×C₃²	semidirect
133. #911	C₂×C₃²	semidirect
134. D₃×D₈	C₂×C₃²	semidirect
135. #904	C₂×C₃²	semidirect
136. #901	C₂×C₃²	semidirect
195. C₂×C₃×D₄	C₂×C₃×D₃	semidirect
196. C₂×C₃×D₄	C₂²×C₃²	semidirect
199. #237	C₂²×C₃²	semidirect
201. D₃×D₄	C₂²×C₃²	semidirect
203. #235	C₂²×C₃²	semidirect
211. C₃×D₈	C₂×C₃×D₃	semidirect
212. C₃×D₈	C₂×C₃×D₃	semidirect
213. C₃×D₈	C₂²×C₃²	semidirect
214. C₃×D₈	C₂²×C₃²	semidirect
219. C₂×C₃×C₈	C₂×C₃×D₃	semidirect
220. C₂×C₃×C₈	C₂²×C₃²	semidirect
221. #214	C₂²×C₃²	semidirect
222. #213	C₂²×C₃²	semidirect
223. #208	C₂²×C₃²	semidirect
224. Dic₁₂	C₂²×C₃²	semidirect
225. D₂₄	C₂²×C₃²	semidirect
226. #204	C₂²×C₃²	semidirect
227. ΓL(1,𝔽₂₅)	C₂²×C₃²	semidirect
228. D₃×C₈	C₂²×C₃²	semidirect
260. C₂×D₈	C₃²×D₃	semidirect
262. C₃³	#576	semidirect
327. C₂×D₄	C₂×C₃²×D₃	semidirect
333. D₈	C₂×C₃²×D₃	semidirect
334. D₈	C₂×C₃²×D₃	semidirect
338. C₂×C₈	C₂×C₃²×D₃	semidirect
365. C₃²	#5757	direct (N × H)
366. C₃²	C₃×#576	semidirect
410. C₃	C₃×#5757	direct (N × H)
411. C₃	C₃²×#576	semidirect

Maximal subgroups

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

subgroup	index
C₃²×#1004	2
C₂×C₃³×D₈	2
C₃²×#932	2
C₃²×#912	2
C₃²×#911	2
C₃²×D₃×D₈	2
C₃²×#904	2
C₃²×#901	2
C₃²×#576	3
C₃×#5757	3

Subgroups — C32×#5757 #13179134

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃²×#5757 #13179134