Subgroups — #11777082

Order 1152.

379 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#10592384	C₂	1	1	yes	maximal, characteristic
2	#10590331	C₂	1	1	yes	maximal, characteristic
3	#10590260	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	#68574	—	3	1	no	maximal
9	#68534	—	3	1	no	maximal
10	C₄²×C₃⋊D₃	C₂²	1	1	yes	characteristic
11	C₂×#3710	—	2	1	no	—
12	C₃²×C₄×C₈	C₂²	1	1	yes	characteristic
13	#63647	C₄	1	1	yes	characteristic
14	#63647	C₄	1	1	yes	characteristic
15	#63641	C₄	1	1	yes	characteristic
16	#63641	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	#63555	—	2	1	no	—
25	#4835	—	3	1	no	—
26	#4835	—	3	1	no	—
27	D₃×C₄⋊C₈	—	3	1	no	—
28	D₃×C₄⋊C₈	—	3	1	no	—
29	#4685	—	3	1	no	—
30	#4685	—	3	1	no	—
31	#4685	—	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	C₂×C₃×C₃⋊C₈	—	2	1	no	—
52	C₂×C₃×C₃⋊C₈	—	2	1	no	—
53	C₂×C₃×C₃⋊C₈	—	2	1	no	—
54	C₂×C₃×C₃⋊C₈	—	2	1	no	—
55	#3710	—	4	2	no	—
56	#1573	—	9	1	no	Sylow
57	C₂×D₃×C₈	—	6	1	no	—
58	C₂×D₃×C₈	—	6	1	no	—
59	C₄²×D₃	—	3	1	no	—
60	C₄²×D₃	—	3	1	no	—
61	C₄²×D₃	—	6	1	no	—
62	C₃×C₄⋊C₈	—	3	1	no	—
63	C₃×C₄⋊C₈	—	3	1	no	—
64	C₃×C₄⋊C₈	—	3	1	no	—
65	C₃×C₄⋊C₈	—	3	1	no	—
66	C₃×C₄×C₈	—	2	1	no	—
67	C₃×C₄×C₈	C₂×D₃	1	1	yes	characteristic
68	C₃×C₄×C₈	C₂×D₃	1	1	yes	characteristic
69	(C₂×D₃)⋊C₈	—	3	1	no	—
70	(C₂×D₃)⋊C₈	—	3	1	no	—
71	(C₂×D₃)⋊C₈	—	3	1	no	—
72	(C₂×D₃)⋊C₈	—	3	1	no	—
73	(C₂×D₃)⋊C₈	—	6	1	no	—
74	(C₂×D₃)⋊C₈	—	6	1	no	—
75	(C₂×D₃)⋊C₈	—	6	1	no	—
76	(C₂×D₃)⋊C₈	—	6	1	no	—
77	#815	—	6	1	no	—
78	#815	—	6	1	no	—
79	Dic₃⋊C₈	—	3	1	no	—
80	Dic₃⋊C₈	—	3	1	no	—
81	Dic₃⋊C₈	—	3	1	no	—
82	Dic₃⋊C₈	—	3	1	no	—
83	Dic₃⋊C₈	—	6	1	no	—
84	Dic₃⋊C₈	—	6	1	no	—
85	Dic₃⋊C₈	—	6	1	no	—
86	Dic₃⋊C₈	—	6	1	no	—
87	C₄⋊(C₃⋊C₈)	C₂×D₃	1	1	yes	characteristic
88	C₄⋊(C₃⋊C₈)	C₂×D₃	1	1	yes	characteristic
89	C₄⋊(C₃⋊C₈)	C₂×D₃	1	1	yes	characteristic
90	C₄⋊(C₃⋊C₈)	C₂×D₃	1	1	yes	characteristic
91	C₄×C₃⋊C₈	—	3	1	no	—
92	C₄×C₃⋊C₈	—	3	1	no	—
93	C₄×C₃⋊C₈	—	6	1	no	—
94	C₂²×C₃⋊D₃	C₄⋊C₄	1	1	yes	characteristic
95	C₂×C₃²×C₄	C₂²×C₄	1	1	yes	characteristic
96	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
97	C₂×C₃²×C₄	C₂×Q₈	1	1	yes	characteristic
98	C₂×C₃⋊Dic₃	C₄⋊C₄	1	1	yes	characteristic
99	C₂×C₃⋊Dic₃	C₄⋊C₄	1	1	yes	characteristic
100	C₂×C₃⋊Dic₃	C₄⋊C₄	1	1	yes	characteristic
101	C₄×C₃⋊D₃	—	2	2	no	—
102	C₄×C₃⋊D₃	—	2	1	no	—
103	C₄×C₃⋊D₃	—	2	1	no	—
104	C₄×C₃⋊D₃	—	2	1	no	—
105	C₄×C₃⋊D₃	—	2	1	no	—
106	C₃²×C₈	—	2	1	no	—
107	C₃²×C₈	—	2	1	no	—
108	C₃⋊(C₃⋊C₈)	—	2	1	no	—
109	C₃⋊(C₃⋊C₈)	—	2	1	no	—
110	C₃×C₃⋊C₈	—	4	1	no	—
111	C₃×C₃⋊C₈	—	4	1	no	—
112	C₃×C₃⋊C₈	—	4	1	no	—
113	C₃×C₃⋊C₈	—	4	1	no	—
114	#432	—	9	1	no	—
115	#431	—	9	1	no	—
116	C₂×C₄⋊C₈	—	9	1	no	—
117	#335	—	3	1	no	—
118	#335	—	9	1	no	—
119	#334	—	3	1	no	—
120	#334	—	9	1	no	—
121	C₂×C₄×D₃	—	3	1	no	—
122	C₂×C₄×D₃	—	3	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	—	6	1	no	—
128	C₂×C₄×D₃	—	6	1	no	—
129	C₂×C₄×D₃	—	6	1	no	—
130	C₂×C₃×C₈	—	2	1	no	—
131	C₂×C₃×C₈	—	2	1	no	—
132	C₂×C₃×C₈	—	6	1	no	—
133	C₂×C₃×C₈	—	6	1	no	—
134	C₂×C₃×C₈	—	6	1	no	—
135	C₂×C₃×C₈	—	6	1	no	—
136	C₂×C₃×C₈	C₄×D₃	1	1	yes	characteristic
137	C₂×C₃×C₈	C₄×D₃	1	1	yes	characteristic
138	C₂×C₃×C₈	C₄×D₃	1	1	yes	characteristic
139	C₂×C₃×C₈	C₄×D₃	1	1	yes	characteristic
140	C₃×C₄²	—	2	1	no	—
141	C₃×C₄²	C₂²×D₃	1	1	yes	characteristic
142	C₃×C₄²	C₂²×D₃	1	1	yes	characteristic
143	C₄×Dic₃	—	3	1	no	—
144	C₄×Dic₃	—	3	1	no	—
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	—	6	1	no	—
150	C₄×Dic₃	—	6	1	no	—
151	C₄×Dic₃	—	6	1	no	—
152	C₂×C₃⋊C₈	—	2	1	no	—
153	C₂×C₃⋊C₈	—	2	1	no	—
154	C₂×C₃⋊C₈	—	2	1	no	—
155	C₂×C₃⋊C₈	—	2	1	no	—
156	C₂×C₃⋊C₈	—	3	1	no	—
157	C₂×C₃⋊C₈	—	3	1	no	—
158	C₂×C₃⋊C₈	—	3	1	no	—
159	C₂×C₃⋊C₈	—	3	1	no	—
160	C₂×C₃⋊C₈	—	6	1	no	—
161	C₂×C₃⋊C₈	—	6	1	no	—
162	D₃×C₈	—	12	2	no	—
163	D₃×C₈	—	12	2	no	—
164	C₂²×C₃²	C₂×C₄⋊C₄	1	1	yes	characteristic
165	C₂×C₃⋊D₃	—	2	1	no	—
166	C₂×C₃⋊D₃	—	2	1	no	—
167	C₂×C₃⋊D₃	#108	1	2	yes	—
168	C₃²×C₄	—	2	1	no	—
169	C₃²×C₄	#131	1	1	yes	characteristic
170	C₃²×C₄	#131	1	1	yes	characteristic
171	C₃²×C₄	AGL(1,ℤ/8)	1	1	yes	characteristic
172	C₃²×C₄	#137	1	1	yes	characteristic
173	C₃⋊Dic₃	—	2	1	no	—
174	C₃⋊Dic₃	—	2	1	no	—
175	C₃⋊Dic₃	—	2	1	no	—
176	C₂²×C₈	—	18	1	no	—
177	C₂×C₄²	—	9	1	no	—
178	C₄⋊C₈	—	3	1	no	—
179	C₄⋊C₈	—	3	1	no	—
180	C₄⋊C₈	—	3	1	no	—
181	C₄⋊C₈	—	3	1	no	—
182	C₄⋊C₈	—	9	1	no	—
183	C₄⋊C₈	—	9	1	no	—
184	C₄⋊C₈	—	18	1	no	—
185	C₂²⋊C₈	—	9	1	no	—
186	C₂²⋊C₈	—	9	1	no	—
187	C₂²⋊C₈	—	18	1	no	—
188	#97	—	18	1	no	—
189	C₄×C₈	—	9	1	no	—
190	C₄×C₈	D₃²	1	1	yes	characteristic
191	C₂²×D₃	—	3	1	no	—
192	C₂²×D₃	—	3	1	no	—
193	C₂²×D₃	—	6	1	no	—
194	C₂×C₃×C₄	—	2	1	no	—
195	C₂×C₃×C₄	—	2	1	no	—
196	C₂×C₃×C₄	—	2	1	no	—
197	C₂×C₃×C₄	C₂×C₄×D₃	1	1	yes	characteristic
198	C₂×C₃×C₄	C₂×C₄×D₃	1	1	yes	characteristic
199	C₂×C₃×C₄	D₃×D₄	1	1	yes	characteristic
200	C₂×C₃×C₄	D₃×D₄	1	1	yes	characteristic
201	C₂×C₃×C₄	D₃×Q₈	1	1	yes	characteristic
202	C₂×C₃×C₄	D₃×Q₈	1	1	yes	characteristic
203	C₂×Dic₃	—	3	1	no	—
204	C₂×Dic₃	—	3	1	no	—
205	C₂×Dic₃	—	3	1	no	—
206	C₂×Dic₃	—	3	1	no	—
207	C₂×Dic₃	—	3	1	no	—
208	C₂×Dic₃	—	3	1	no	—
209	C₂×Dic₃	—	6	1	no	—
210	C₂×Dic₃	—	6	1	no	—
211	C₂×Dic₃	—	6	1	no	—
212	C₄×D₃	—	6	2	no	—
213	C₄×D₃	—	6	2	no	—
214	C₄×D₃	—	6	4	no	—
215	C₄×D₃	—	6	1	no	—
216	C₄×D₃	—	6	1	no	—
217	C₄×D₃	—	6	1	no	—
218	C₄×D₃	—	6	1	no	—
219	C₄×D₃	—	6	1	no	—
220	C₄×D₃	—	6	1	no	—
221	C₄×D₃	—	6	1	no	—
222	C₄×D₃	—	6	1	no	—
223	C₄×D₃	—	12	1	no	—
224	C₄×D₃	—	12	1	no	—
225	C₄×D₃	—	12	1	no	—
226	C₄×D₃	—	12	1	no	—
227	C₃×C₈	—	2	1	no	—
228	C₃×C₈	—	2	1	no	—
229	C₃×C₈	—	2	1	no	—
230	C₃×C₈	—	2	1	no	—
231	C₃×C₈	—	4	1	no	—
232	C₃×C₈	—	4	1	no	—
233	C₃×C₈	—	12	1	no	—
234	C₃×C₈	—	12	1	no	—
235	C₃×C₈	—	12	1	no	—
236	C₃×C₈	—	12	1	no	—
237	C₃⋊C₈	—	4	1	no	—
238	C₃⋊C₈	—	4	1	no	—
239	C₃⋊C₈	—	4	1	no	—
240	C₃⋊C₈	—	4	1	no	—
241	C₃⋊C₈	—	6	1	no	—
242	C₃⋊C₈	—	6	1	no	—
243	C₃⋊C₈	—	6	1	no	—
244	C₃⋊C₈	—	6	1	no	—
245	C₃⋊C₈	—	12	1	no	—
246	C₃⋊C₈	—	12	1	no	—
247	C₂×C₃²	#424	1	1	yes	characteristic
248	C₂×C₃²	#428	1	1	yes	characteristic
249	C₂×C₃²	C₂×#108	1	1	yes	characteristic
250	C₃⋊D₃	—	2	2	no	—
251	C₂²×C₄	—	9	1	no	—
252	C₂²×C₄	—	9	1	no	—
253	C₂²×C₄	—	9	1	no	—
254	C₂×C₈	—	6	1	no	—
255	C₂×C₈	—	6	1	no	—
256	C₂×C₈	—	6	1	no	—
257	C₂×C₈	—	6	1	no	—
258	C₂×C₈	—	9	1	no	—
259	C₂×C₈	—	9	1	no	—
260	C₂×C₈	—	36	2	no	—
261	C₂×C₈	#624	1	1	yes	characteristic
262	C₂×C₈	#624	1	1	yes	characteristic
263	C₄²	—	9	1	no	—
264	C₄²	—	9	1	no	—
265	C₄²	—	9	1	no	—
266	C₄²	C₂×D₃²	1	1	yes	characteristic
267	C₂²×C₃	—	2	1	no	—
268	C₂²×C₃	D₃×C₄⋊C₄	1	1	yes	characteristic
269	C₂²×C₃	D₃×C₄⋊C₄	1	1	yes	characteristic
270	C₂×D₃	—	3	2	no	—
271	C₂×D₃	—	3	2	no	—
272	C₂×D₃	—	6	2	no	—
273	C₂×D₃	—	6	1	no	—
274	C₂×D₃	—	6	1	no	—
275	C₂×D₃	—	6	1	no	—
276	C₂×D₃	—	6	1	no	—
277	C₂×D₃	—	12	1	no	—
278	C₂×D₃	—	12	1	no	—
279	C₃×C₄	—	2	2	no	—
280	C₃×C₄	—	2	1	no	—
281	C₃×C₄	—	2	1	no	—
282	C₃×C₄	—	2	1	no	—
283	C₃×C₄	—	2	1	no	—
284	C₃×C₄	—	2	1	no	—
285	C₃×C₄	—	2	1	no	—
286	C₃×C₄	#901	1	1	yes	characteristic
287	C₃×C₄	#901	1	1	yes	characteristic
288	C₃×C₄	#907	1	1	yes	characteristic
289	C₃×C₄	#907	1	1	yes	characteristic
290	C₃×C₄	#911	1	1	yes	characteristic
291	C₃×C₄	#914	1	1	yes	characteristic
292	C₃×C₄	#915	1	1	yes	characteristic
293	C₃×C₄	#918	1	1	yes	characteristic
294	Dic₃	—	6	2	no	—
295	Dic₃	—	6	1	no	—
296	Dic₃	—	6	1	no	—
297	Dic₃	—	6	1	no	—
298	Dic₃	—	6	1	no	—
299	Dic₃	—	6	1	no	—
300	Dic₃	—	6	1	no	—
301	Dic₃	—	12	1	no	—
302	Dic₃	—	12	1	no	—
303	C₃²	#1573	1	1	yes	characteristic, complemented, Sylow
304	C₂³	—	9	1	no	—
305	C₂×C₄	—	9	1	no	—
306	C₂×C₄	—	9	1	no	—
307	C₂×C₄	—	9	1	no	—
308	C₂×C₄	—	18	2	no	—
309	C₂×C₄	—	18	1	no	—
310	C₂×C₄	—	18	1	no	—
311	C₂×C₄	—	18	1	no	—
312	C₂×C₄	—	18	1	no	—
313	C₂×C₄	#3797	1	1	yes	characteristic
314	C₂×C₄	#3802	1	1	yes	characteristic
315	C₂×C₄	C₂×#624	1	1	yes	characteristic
316	C₈	—	2	1	no	—
317	C₈	—	2	1	no	—
318	C₈	—	12	1	no	—
319	C₈	—	12	1	no	—
320	C₈	—	12	1	no	—
321	C₈	—	12	1	no	—
322	C₈	—	18	1	no	—
323	C₈	—	18	1	no	—
324	C₂×C₃	—	2	1	no	—
325	C₂×C₃	—	2	1	no	—
326	C₂×C₃	—	2	1	no	—
327	C₂×C₃	#4839	1	1	yes	characteristic
328	C₂×C₃	#4839	1	1	yes	characteristic
329	C₂×C₃	#4861	1	1	yes	characteristic
330	C₂×C₃	#4881	1	1	yes	characteristic
331	C₂×C₃	D₃×#108	1	1	yes	characteristic
332	C₂×C₃	D₃×#108	1	1	yes	characteristic
333	D₃	—	6	2	no	—
334	D₃	—	6	2	no	—
335	D₃	—	12	2	no	—
336	C₂²	—	9	2	no	—
337	C₂²	—	18	1	no	—
338	C₂²	—	18	1	no	—
339	C₂²	#63900	1	1	yes	characteristic
340	C₄	—	2	1	no	—
341	C₄	—	18	1	no	—
342	C₄	—	18	1	no	—
343	C₄	—	18	1	no	—
344	C₄	#63798	1	1	yes	characteristic
345	C₄	#63811	1	1	yes	characteristic
346	C₄	#63817	1	1	yes	characteristic
347	C₄	#63817	1	1	yes	characteristic
348	C₃	—	2	1	no	—
349	C₃	#68534	1	1	yes	characteristic, complemented
350	C₃	#68574	1	1	yes	characteristic, complemented
351	C₂	—	18	2	no	—
352	C₂	#10590332	1	1	yes	characteristic
353	C₂	#10590688	1	1	yes	characteristic
354	C₂	#10590913	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#3802
commutator	C₂×C₃²×C₄	C₂²×C₄
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	#1573
LCS	G ▷ C₂×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
303. C₃²	#1573	semidirect
349. C₃	#68534	semidirect
350. C₃	#68574	semidirect

Maximal subgroups

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

subgroup	index
#10592384	2
#10590331	2
#10590260	2
C₃×#4458	2
C₃×#4457	2
#10589100	2
#10589098	2
#68574	3
#68534	3