Subgroups — #11776847

Order 1152.

431 conjugacy classes of subgroups.

The subgroup lattice has 366 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#3743	C₂	1	1	yes	maximal, characteristic
2	#10590331	C₂	1	1	yes	maximal, characteristic
3	#10590256	C₂	1	1	yes	maximal, characteristic
4	C₃×#4458	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×#4457	C₂	1	1	yes	maximal, characteristic, complemented
6	#10589100	C₂	1	1	yes	maximal, characteristic, complemented
7	#10589098	C₂	1	1	yes	maximal, characteristic, complemented
8	#68568	—	3	1	no	maximal
9	#68528	—	3	1	no	maximal
10	C₂×#3743	C₄	1	1	yes	characteristic
11	C₂×#3743	C₄	1	1	yes	characteristic
12	C₄²×C₃⋊D₃	C₂²	1	1	yes	characteristic
13	C₂×#3710	—	2	1	no	—
14	C₃²×C₄×C₈	C₂²	1	1	yes	characteristic
15	#63642	C₄	1	1	yes	characteristic
16	#63642	C₄	1	1	yes	characteristic
17	C₄×C₃⋊(C₃⋊C₈)	C₂²	1	1	yes	characteristic
18	C₃×C₄⋊(C₃⋊C₈)	C₂²	1	1	yes	characteristic
19	C₃×C₄⋊(C₃⋊C₈)	C₂²	1	1	yes	characteristic
20	C₃×C₄⋊(C₃⋊C₈)	C₂²	1	1	yes	characteristic
21	C₃×C₄⋊(C₃⋊C₈)	C₂²	1	1	yes	characteristic
22	#63572	—	2	1	no	—
23	#63557	—	2	1	no	—
24	#63553	—	2	1	no	—
25	#4833	—	3	1	no	—
26	#4833	—	3	1	no	—
27	D₃×C₄⋊C₈	—	3	1	no	—
28	D₃×C₄⋊C₈	—	3	1	no	—
29	C₄×ΓL(1,𝔽₂₅)	—	3	1	no	—
30	C₄×ΓL(1,𝔽₂₅)	—	3	1	no	—
31	C₄×ΓL(1,𝔽₂₅)	—	6	1	no	—
32	C₃×#335	—	3	1	no	—
33	C₃×#334	—	3	1	no	—
34	#4480	—	3	1	no	—
35	#4480	—	3	1	no	—
36	#4479	—	3	1	no	—
37	#4479	—	3	1	no	—
38	#4458	D₃	1	1	yes	characteristic, complemented
39	#4457	D₃	1	1	yes	characteristic, complemented
40	C₂×C₄×C₃⋊D₃	C₂×C₄	1	1	yes	characteristic
41	C₂×C₄×C₃⋊D₃	D₄	1	1	yes	characteristic
42	C₂×C₄×C₃⋊D₃	Q₈	1	1	yes	characteristic
43	C₂×C₃²×C₈	C₂×C₄	1	1	yes	characteristic
44	C₂×C₃²×C₈	C₂×C₄	1	1	yes	characteristic
45	C₃²×C₄²	C₂³	1	1	yes	characteristic
46	C₄×C₃⋊Dic₃	C₂×C₄	1	1	yes	characteristic
47	C₄×C₃⋊Dic₃	D₄	1	1	yes	characteristic
48	C₄×C₃⋊Dic₃	Q₈	1	1	yes	characteristic
49	C₂×C₃⋊(C₃⋊C₈)	C₂×C₄	1	1	yes	characteristic
50	C₂×C₃⋊(C₃⋊C₈)	C₂×C₄	1	1	yes	characteristic
51	#3743	—	2	2	no	—
52	#3743	C₈	1	4	yes	complemented
53	C₂×C₃×C₃⋊C₈	—	2	1	no	—
54	C₂×C₃×C₃⋊C₈	—	2	1	no	—
55	C₂×C₃×C₃⋊C₈	—	2	1	no	—
56	C₂×C₃×C₃⋊C₈	—	2	1	no	—
57	#3710	—	4	2	no	—
58	#1565	—	9	1	no	Sylow
59	C₂×ΓL(1,𝔽₂₅)	—	3	1	no	—
60	C₂×ΓL(1,𝔽₂₅)	—	3	1	no	—
61	C₂×ΓL(1,𝔽₂₅)	—	3	1	no	—
62	C₂×ΓL(1,𝔽₂₅)	—	3	1	no	—
63	C₂×ΓL(1,𝔽₂₅)	—	6	1	no	—
64	C₂×ΓL(1,𝔽₂₅)	—	6	1	no	—
65	C₂×D₃×C₈	—	6	1	no	—
66	C₂×D₃×C₈	—	6	1	no	—
67	C₄²×D₃	—	3	1	no	—
68	C₄²×D₃	—	3	1	no	—
69	C₄²×D₃	—	6	1	no	—
70	C₃×C₄⋊C₈	—	3	1	no	—
71	C₃×C₄⋊C₈	—	3	1	no	—
72	C₃×C₄⋊C₈	—	3	1	no	—
73	C₃×C₄⋊C₈	—	3	1	no	—
74	C₃×C₄×C₈	—	2	1	no	—
75	C₃×C₄×C₈	C₂×D₃	1	1	yes	characteristic
76	C₃×C₄×C₈	C₂×D₃	1	1	yes	characteristic
77	(C₂×D₃)⋊C₈	—	6	1	no	—
78	(C₂×D₃)⋊C₈	—	6	1	no	—
79	#815	—	3	1	no	—
80	#815	—	3	1	no	—
81	#815	—	3	1	no	—
82	#815	—	3	1	no	—
83	#815	—	6	1	no	—
84	#815	—	6	1	no	—
85	Dic₃⋊C₈	—	6	1	no	—
86	Dic₃⋊C₈	—	6	1	no	—
87	C₈×Dic₃	—	6	1	no	—
88	C₈×Dic₃	—	6	1	no	—
89	C₄⋊(C₃⋊C₈)	C₂×D₃	1	1	yes	characteristic
90	C₄⋊(C₃⋊C₈)	C₂×D₃	1	1	yes	characteristic
91	C₄⋊(C₃⋊C₈)	C₂×D₃	1	1	yes	characteristic
92	C₄⋊(C₃⋊C₈)	C₂×D₃	1	1	yes	characteristic
93	C₄×C₃⋊C₈	—	3	1	no	—
94	C₄×C₃⋊C₈	—	3	1	no	—
95	C₄×C₃⋊C₈	—	6	1	no	—
96	C₂²×C₃⋊D₃	C₄⋊C₄	1	1	yes	characteristic
97	C₂×C₃²×C₄	C₂²×C₄	1	1	yes	characteristic
98	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
99	C₂×C₃²×C₄	C₂×Q₈	1	1	yes	characteristic
100	C₂×C₃⋊Dic₃	C₄⋊C₄	1	1	yes	characteristic
101	C₂×C₃⋊Dic₃	C₄⋊C₄	1	1	yes	characteristic
102	C₂×C₃⋊Dic₃	C₄⋊C₄	1	1	yes	characteristic
103	C₄×C₃⋊D₃	—	2	1	no	—
104	C₄×C₃⋊D₃	—	2	1	no	—
105	C₄×C₃⋊D₃	—	4	1	no	—
106	C₄×C₃⋊D₃	C₂×C₈	1	2	yes	—
107	C₄×C₃⋊D₃	M₁₆	1	2	yes	—
108	C₃²×C₈	—	2	1	no	—
109	C₃²×C₈	C₂×C₈	1	2	yes	complemented
110	C₃⋊(C₃⋊C₈)	—	2	1	no	—
111	C₃⋊(C₃⋊C₈)	C₂×C₈	1	2	yes	complemented
112	C₃×C₃⋊C₈	—	4	1	no	—
113	C₃×C₃⋊C₈	—	4	1	no	—
114	C₃×C₃⋊C₈	—	4	1	no	—
115	C₃×C₃⋊C₈	—	4	1	no	—
116	#431	—	9	1	no	—
117	C₂×C₄⋊C₈	—	9	1	no	—
118	C₄×M₁₆	—	9	1	no	—
119	#335	—	3	1	no	—
120	#335	—	9	1	no	—
121	#334	—	3	1	no	—
122	#334	—	9	1	no	—
123	C₂×C₄×D₃	—	3	1	no	—
124	C₂×C₄×D₃	—	3	1	no	—
125	C₂×C₄×D₃	—	3	1	no	—
126	C₂×C₄×D₃	—	3	1	no	—
127	C₂×C₄×D₃	—	3	1	no	—
128	C₂×C₄×D₃	—	3	1	no	—
129	C₂×C₄×D₃	—	6	1	no	—
130	C₂×C₄×D₃	—	6	1	no	—
131	C₂×C₄×D₃	—	6	1	no	—
132	C₂×C₃×C₈	—	2	1	no	—
133	C₂×C₃×C₈	—	2	1	no	—
134	C₂×C₃×C₈	—	6	1	no	—
135	C₂×C₃×C₈	—	6	1	no	—
136	C₂×C₃×C₈	—	6	1	no	—
137	C₂×C₃×C₈	—	6	1	no	—
138	C₂×C₃×C₈	C₄×D₃	1	1	yes	characteristic
139	C₂×C₃×C₈	C₄×D₃	1	1	yes	characteristic
140	C₂×C₃×C₈	C₄×D₃	1	1	yes	characteristic
141	C₂×C₃×C₈	C₄×D₃	1	1	yes	characteristic
142	C₃×C₄²	—	2	1	no	—
143	C₃×C₄²	C₂²×D₃	1	1	yes	characteristic
144	C₃×C₄²	C₂²×D₃	1	1	yes	characteristic
145	C₄×Dic₃	—	3	1	no	—
146	C₄×Dic₃	—	3	1	no	—
147	C₄×Dic₃	—	3	1	no	—
148	C₄×Dic₃	—	3	1	no	—
149	C₄×Dic₃	—	3	1	no	—
150	C₄×Dic₃	—	3	1	no	—
151	C₄×Dic₃	—	6	1	no	—
152	C₄×Dic₃	—	6	1	no	—
153	C₄×Dic₃	—	6	1	no	—
154	C₂×C₃⋊C₈	—	2	1	no	—
155	C₂×C₃⋊C₈	—	2	1	no	—
156	C₂×C₃⋊C₈	—	2	1	no	—
157	C₂×C₃⋊C₈	—	2	1	no	—
158	C₂×C₃⋊C₈	—	3	1	no	—
159	C₂×C₃⋊C₈	—	3	1	no	—
160	C₂×C₃⋊C₈	—	3	1	no	—
161	C₂×C₃⋊C₈	—	3	1	no	—
162	C₂×C₃⋊C₈	—	6	1	no	—
163	C₂×C₃⋊C₈	—	6	1	no	—
164	ΓL(1,𝔽₂₅)	—	3	4	no	—
165	ΓL(1,𝔽₂₅)	—	3	4	no	—
166	ΓL(1,𝔽₂₅)	—	6	4	no	—
167	ΓL(1,𝔽₂₅)	—	6	2	no	—
168	ΓL(1,𝔽₂₅)	—	6	2	no	—
169	ΓL(1,𝔽₂₅)	—	6	4	no	—
170	D₃×C₈	—	12	2	no	—
171	D₃×C₈	—	12	2	no	—
172	C₂²×C₃²	C₂×C₄⋊C₄	1	1	yes	characteristic
173	C₂×C₃⋊D₃	—	2	1	no	—
174	C₂×C₃⋊D₃	—	2	1	no	—
175	C₂×C₃⋊D₃	C₄⋊C₈	1	2	yes	—
176	C₃²×C₄	—	2	1	no	—
177	C₃²×C₄	C₂²×C₈	1	1	yes	characteristic
178	C₃²×C₄	C₂×M₁₆	1	1	yes	characteristic
179	C₃²×C₄	AGL(1,ℤ/8)	1	1	yes	characteristic
180	C₃²×C₄	#137	1	1	yes	characteristic
181	C₃⋊Dic₃	—	2	1	no	—
182	C₃⋊Dic₃	—	2	1	no	—
183	C₃⋊Dic₃	C₄⋊C₈	1	2	yes	—
184	C₂×M₁₆	—	9	1	no	—
185	C₂×M₁₆	—	9	1	no	—
186	C₂²×C₈	—	18	1	no	—
187	C₂×C₄²	—	9	1	no	—
188	C₄⋊C₈	—	3	1	no	—
189	C₄⋊C₈	—	3	1	no	—
190	C₄⋊C₈	—	3	1	no	—
191	C₄⋊C₈	—	3	1	no	—
192	C₄⋊C₈	—	18	1	no	—
193	C₂²⋊C₈	—	18	1	no	—
194	#97	—	9	1	no	—
195	#97	—	9	1	no	—
196	C₄×C₈	—	9	1	no	—
197	C₄×C₈	—	18	1	no	—
198	C₄×C₈	D₃²	1	1	yes	characteristic
199	C₂²×D₃	—	3	1	no	—
200	C₂²×D₃	—	3	1	no	—
201	C₂²×D₃	—	6	1	no	—
202	C₂×C₃×C₄	—	2	1	no	—
203	C₂×C₃×C₄	—	2	1	no	—
204	C₂×C₃×C₄	—	2	1	no	—
205	C₂×C₃×C₄	C₂×C₄×D₃	1	1	yes	characteristic
206	C₂×C₃×C₄	C₂×C₄×D₃	1	1	yes	characteristic
207	C₂×C₃×C₄	D₃×D₄	1	1	yes	characteristic
208	C₂×C₃×C₄	D₃×D₄	1	1	yes	characteristic
209	C₂×C₃×C₄	D₃×Q₈	1	1	yes	characteristic
210	C₂×C₃×C₄	D₃×Q₈	1	1	yes	characteristic
211	C₂×Dic₃	—	3	1	no	—
212	C₂×Dic₃	—	3	1	no	—
213	C₂×Dic₃	—	3	1	no	—
214	C₂×Dic₃	—	3	1	no	—
215	C₂×Dic₃	—	3	1	no	—
216	C₂×Dic₃	—	3	1	no	—
217	C₂×Dic₃	—	6	1	no	—
218	C₂×Dic₃	—	6	1	no	—
219	C₂×Dic₃	—	6	1	no	—
220	C₄×D₃	—	3	2	no	—
221	C₄×D₃	—	3	2	no	—
222	C₄×D₃	—	3	2	no	—
223	C₄×D₃	—	3	2	no	—
224	C₄×D₃	—	6	2	no	—
225	C₄×D₃	—	6	2	no	—
226	C₄×D₃	—	6	1	no	—
227	C₄×D₃	—	6	1	no	—
228	C₄×D₃	—	6	1	no	—
229	C₄×D₃	—	6	1	no	—
230	C₄×D₃	—	12	1	no	—
231	C₄×D₃	—	12	1	no	—
232	C₄×D₃	—	12	1	no	—
233	C₄×D₃	—	12	1	no	—
234	C₄×D₃	—	12	2	no	—
235	C₃×C₈	—	2	2	no	—
236	C₃×C₈	—	2	1	no	—
237	C₃×C₈	—	2	1	no	—
238	C₃×C₈	—	2	2	no	—
239	C₃×C₈	—	12	1	no	—
240	C₃×C₈	—	12	1	no	—
241	C₃×C₈	—	12	1	no	—
242	C₃×C₈	—	12	1	no	—
243	C₃×C₈	D₃×C₈	1	2	yes	complemented
244	C₃×C₈	D₃×C₈	1	2	yes	complemented
245	C₃⋊C₈	—	3	2	no	—
246	C₃⋊C₈	—	3	2	no	—
247	C₃⋊C₈	—	4	1	no	—
248	C₃⋊C₈	—	4	1	no	—
249	C₃⋊C₈	—	4	1	no	—
250	C₃⋊C₈	—	4	1	no	—
251	C₃⋊C₈	—	6	2	no	—
252	C₃⋊C₈	—	6	2	no	—
253	C₃⋊C₈	—	6	1	no	—
254	C₃⋊C₈	—	6	1	no	—
255	C₂×C₃²	C₂×C₄⋊C₈	1	1	yes	characteristic
256	C₂×C₃²	#428	1	1	yes	characteristic
257	C₂×C₃²	#430	1	1	yes	characteristic
258	C₃⋊D₃	—	2	2	no	—
259	C₂²×C₄	—	9	1	no	—
260	C₂²×C₄	—	9	1	no	—
261	C₂²×C₄	—	9	1	no	—
262	M₁₆	—	9	4	no	—
263	M₁₆	—	18	2	no	—
264	C₂×C₈	—	6	1	no	—
265	C₂×C₈	—	6	1	no	—
266	C₂×C₈	—	6	1	no	—
267	C₂×C₈	—	6	1	no	—
268	C₂×C₈	—	9	1	no	—
269	C₂×C₈	—	9	1	no	—
270	C₂×C₈	—	36	2	no	—
271	C₂×C₈	#624	1	1	yes	characteristic
272	C₂×C₈	#624	1	1	yes	characteristic
273	C₄²	—	9	1	no	—
274	C₄²	—	9	1	no	—
275	C₄²	—	9	1	no	—
276	C₄²	C₂×D₃²	1	1	yes	characteristic
277	C₂²×C₃	—	2	1	no	—
278	C₂²×C₃	D₃×C₄⋊C₄	1	1	yes	characteristic
279	C₂²×C₃	D₃×C₄⋊C₄	1	1	yes	characteristic
280	C₂×D₃	—	3	2	no	—
281	C₂×D₃	—	3	2	no	—
282	C₂×D₃	—	6	2	no	—
283	C₂×D₃	—	6	1	no	—
284	C₂×D₃	—	6	1	no	—
285	C₂×D₃	—	6	1	no	—
286	C₂×D₃	—	6	1	no	—
287	C₂×D₃	—	12	1	no	—
288	C₂×D₃	—	12	1	no	—
289	C₃×C₄	—	2	2	no	—
290	C₃×C₄	—	2	1	no	—
291	C₃×C₄	—	2	1	no	—
292	C₃×C₄	—	2	1	no	—
293	C₃×C₄	—	2	1	no	—
294	C₃×C₄	—	2	1	no	—
295	C₃×C₄	—	2	1	no	—
296	C₃×C₄	C₂×D₃×C₈	1	1	yes	characteristic
297	C₃×C₄	C₂×D₃×C₈	1	1	yes	characteristic
298	C₃×C₄	D₃×M₁₆	1	1	yes	characteristic
299	C₃×C₄	D₃×M₁₆	1	1	yes	characteristic
300	C₃×C₄	#911	1	1	yes	characteristic
301	C₃×C₄	#914	1	1	yes	characteristic
302	C₃×C₄	#915	1	1	yes	characteristic
303	C₃×C₄	#918	1	1	yes	characteristic
304	Dic₃	—	3	2	no	—
305	Dic₃	—	3	2	no	—
306	Dic₃	—	6	2	no	—
307	Dic₃	—	6	1	no	—
308	Dic₃	—	6	1	no	—
309	Dic₃	—	6	1	no	—
310	Dic₃	—	6	1	no	—
311	Dic₃	—	12	1	no	—
312	Dic₃	—	12	1	no	—
313	C₃²	#1565	1	1	yes	characteristic, complemented, Sylow
314	C₂³	—	9	1	no	—
315	C₂×C₄	—	9	1	no	—
316	C₂×C₄	—	9	2	no	—
317	C₂×C₄	—	9	2	no	—
318	C₂×C₄	—	9	1	no	—
319	C₂×C₄	—	9	1	no	—
320	C₂×C₄	—	18	1	no	—
321	C₂×C₄	—	18	1	no	—
322	C₂×C₄	—	36	1	no	—
323	C₂×C₄	#3797	1	1	yes	characteristic
324	C₂×C₄	#3802	1	1	yes	characteristic
325	C₂×C₄	C₂×#624	1	1	yes	characteristic
326	C₈	—	2	1	no	—
327	C₈	—	9	2	no	—
328	C₈	—	12	1	no	—
329	C₈	—	12	1	no	—
330	C₈	—	12	1	no	—
331	C₈	—	12	1	no	—
332	C₈	—	18	1	no	—
333	C₈	#3710	1	2	yes	complemented
334	C₂×C₃	—	2	1	no	—
335	C₂×C₃	—	2	1	no	—
336	C₂×C₃	—	2	1	no	—
337	C₂×C₃	D₃×C₄⋊C₈	1	1	yes	characteristic
338	C₂×C₃	D₃×C₄⋊C₈	1	1	yes	characteristic
339	C₂×C₃	#4861	1	1	yes	characteristic
340	C₂×C₃	#4881	1	1	yes	characteristic
341	C₂×C₃	#4893	1	1	yes	characteristic
342	C₂×C₃	#4893	1	1	yes	characteristic
343	D₃	—	6	2	no	—
344	D₃	—	6	2	no	—
345	D₃	—	12	2	no	—
346	C₂²	—	9	2	no	—
347	C₂²	—	18	1	no	—
348	C₂²	—	18	1	no	—
349	C₂²	#63900	1	1	yes	characteristic
350	C₄	—	2	1	no	—
351	C₄	—	9	2	no	—
352	C₄	—	18	1	no	—
353	C₄	—	18	1	no	—
354	C₄	#63798	1	1	yes	characteristic
355	C₄	#63811	1	1	yes	characteristic
356	C₄	C₂×#3710	1	1	yes	characteristic
357	C₄	#63818	1	1	yes	characteristic
358	C₃	—	2	1	no	—
359	C₃	#68528	1	1	yes	characteristic, complemented
360	C₃	#68568	1	1	yes	characteristic, complemented
361	C₂	—	18	2	no	—
362	C₂	#10590331	1	1	yes	characteristic
363	C₂	#10590688	1	1	yes	characteristic
364	C₂	#10590914	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#3802
commutator	C₃²×C₄	C₂²×C₈
Frattini	C₄²	C₂×D₃²
Fitting	C₃²×C₄×C₈	C₂²
socle	C₂²×C₃²	C₂×C₄⋊C₄
prime radical	C₂²×C₃⋊D₃	C₄⋊C₄
cosocle	C₃²×C₄²	C₂³
hypercenter	C₄×C₈	D₃²
hypocenter	C₃²	#1565
LCS	G ▷ C₃²×C₄ ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×C₈

Complements

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

N	H ≅ G/N	type
4. C₃×#4458	C₂	semidirect
5. C₃×#4457	C₂	semidirect
6. #10589100	C₂	semidirect
7. #10589098	C₂	semidirect
38. #4458	D₃	semidirect
39. #4457	D₃	semidirect
52. #3743	C₈	semidirect
109. C₃²×C₈	C₂×C₈	semidirect
111. C₃⋊(C₃⋊C₈)	C₂×C₈	semidirect
243. C₃×C₈	D₃×C₈	semidirect
244. C₃×C₈	D₃×C₈	semidirect
313. C₃²	#1565	semidirect
333. C₈	#3710	semidirect
359. C₃	#68528	semidirect
360. C₃	#68568	semidirect

Maximal subgroups

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

subgroup	index
C₄×#3743	2
#10590331	2
#10590256	2
C₃×#4458	2
C₃×#4457	2
#10589100	2
#10589098	2
#68568	3
#68528	3