Subgroups — #11777043

Order 1152.

327 conjugacy classes of subgroups.

The subgroup lattice has 326 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#4833	C₂	1	1	yes	maximal, characteristic
2	#10590263	C₂	1	1	yes	maximal, characteristic
3	#10590261	C₂	1	1	yes	maximal, characteristic
4	#10589346	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×#4479	C₂	1	1	yes	maximal, characteristic, complemented
6	#10589099	C₂	1	1	yes	maximal, characteristic, complemented
7	#10589098	C₂	1	1	yes	maximal, characteristic, complemented
8	#70747	—	3	1	no	maximal
9	#68573	—	3	1	no	maximal
10	C₃×C₄²×D₃	C₂²	1	1	yes	characteristic
11	C₃²×C₄⋊C₈	C₂²	1	1	yes	characteristic
12	C₄⋊(C₃⋊(C₃⋊C₈))	C₂²	1	1	yes	characteristic
13	C₄×C₃⋊(C₃⋊C₈)	C₂²	1	1	yes	characteristic
14	C₃×(C₂×D₃)⋊C₈	—	2	1	no	—
15	C₃×C₈×Dic₃	—	2	1	no	—
16	C₃×C₄⋊(C₃⋊C₈)	C₂²	1	1	yes	characteristic
17	C₃×C₄⋊(C₃⋊C₈)	C₂²	1	1	yes	characteristic
18	C₃×C₄×C₃⋊C₈	C₂²	1	1	yes	characteristic
19	Dic₃⋊(C₃⋊C₈)	C₄	1	1	yes	characteristic
20	Dic₃⋊(C₃⋊C₈)	C₄	1	1	yes	characteristic
21	(C₂×D₃)⋊(C₃⋊C₈)	—	2	1	no	—
22	(C₂×D₃)⋊(C₃⋊C₈)	C₄	1	1	yes	characteristic
23	(C₂×D₃)⋊(C₃⋊C₈)	C₄	1	1	yes	characteristic
24	#63554	—	2	1	no	—
25	C₃×#431	—	3	1	no	—
26	#4926	—	3	1	no	—
27	#4925	—	3	1	no	—
28	#4835	—	3	1	no	—
29	#4833	D₃	1	1	yes	characteristic
30	#4685	—	3	1	no	—
31	C₃×#335	—	3	1	no	—
32	#4480	—	3	1	no	—
33	#4480	—	3	1	no	—
34	#4480	—	3	1	no	—
35	#4480	—	3	1	no	—
36	#4480	—	6	1	no	—
37	#4479	—	3	1	no	—
38	#4479	D₃	1	1	yes	characteristic, complemented
39	#4458	—	3	1	no	—
40	C₂×C₃×C₄×D₃	C₂×C₄	1	1	yes	characteristic
41	C₂×C₃×C₄×D₃	D₄	1	1	yes	characteristic
42	C₂×C₃×C₄×D₃	Q₈	1	1	yes	characteristic
43	C₂×C₃²×C₈	—	2	1	no	—
44	C₃²×C₄²	C₂³	1	1	yes	characteristic
45	C₂×C₃⋊(C₃⋊C₈)	—	2	1	no	—
46	C₂×C₃⋊(C₃⋊C₈)	C₂×C₄	1	1	yes	characteristic
47	C₂×C₃⋊(C₃⋊C₈)	C₂×C₄	1	1	yes	characteristic
48	C₃×C₄×Dic₃	C₂×C₄	1	1	yes	characteristic
49	C₃×C₄×Dic₃	D₄	1	1	yes	characteristic
50	C₃×C₄×Dic₃	Q₈	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	C₂×C₄	1	1	yes	characteristic
54	C₂×C₃×C₃⋊C₈	C₂×C₄	1	1	yes	characteristic
55	#1572	—	9	1	no	Sylow
56	C₂×C₃×C₄²	—	3	1	no	—
57	C₄²×D₃	C₂×D₃	1	1	yes	characteristic
58	C₃×C₄⋊C₈	—	2	1	no	—
59	C₃×C₄⋊C₈	—	3	1	no	—
60	C₃×C₄⋊C₈	—	3	1	no	—
61	C₃×C₄⋊C₈	C₂×D₃	1	1	yes	characteristic
62	C₃×C₄⋊C₈	C₂×D₃	1	1	yes	characteristic
63	C₃×C₂²⋊C₈	—	6	1	no	—
64	C₃×C₄×C₈	—	3	1	no	—
65	C₃×C₄×C₈	—	6	1	no	—
66	#830	—	3	1	no	—
67	#830	—	3	1	no	—
68	#830	—	6	1	no	—
69	(C₂×D₃)⋊C₈	—	2	1	no	—
70	(C₂×D₃)⋊C₈	—	3	1	no	—
71	(C₂×D₃)⋊C₈	—	3	1	no	—
72	(C₂×D₃)⋊C₈	—	6	1	no	—
73	#815	—	6	1	no	—
74	Dic₃⋊C₈	—	3	1	no	—
75	Dic₃⋊C₈	—	3	1	no	—
76	C₈×Dic₃	—	2	1	no	—
77	C₄⋊(C₃⋊C₈)	—	3	1	no	—
78	C₄⋊(C₃⋊C₈)	—	3	1	no	—
79	C₄⋊(C₃⋊C₈)	—	3	1	no	—
80	C₄⋊(C₃⋊C₈)	—	3	1	no	—
81	C₄⋊(C₃⋊C₈)	—	6	1	no	—
82	C₄⋊(C₃⋊C₈)	C₂×D₃	1	1	yes	characteristic
83	C₄⋊(C₃⋊C₈)	C₂×D₃	1	1	yes	characteristic
84	#803	—	6	1	no	—
85	C₄×C₃⋊C₈	—	3	1	no	—
86	C₄×C₃⋊C₈	—	3	1	no	—
87	C₄×C₃⋊C₈	—	6	1	no	—
88	C₄×C₃⋊C₈	C₂×D₃	1	1	yes	characteristic
89	C₂²×C₃×D₃	C₄⋊C₄	1	1	yes	characteristic
90	C₂×C₃²×C₄	C₂²×C₄	1	1	yes	characteristic
91	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
92	C₂×C₃²×C₄	C₂×Q₈	1	1	yes	characteristic
93	C₂×C₃×Dic₃	C₄⋊C₄	1	1	yes	characteristic
94	C₂×C₃×Dic₃	C₄⋊C₄	1	1	yes	characteristic
95	C₂×C₃×Dic₃	C₄⋊C₄	1	1	yes	characteristic
96	C₃×C₄×D₃	—	2	1	no	—
97	C₃×C₄×D₃	—	2	1	no	—
98	C₃×C₄×D₃	—	2	1	no	—
99	C₃×C₄×D₃	—	2	1	no	—
100	C₃×C₄×D₃	—	4	1	no	—
101	C₃²×C₈	—	4	1	no	—
102	C₃⋊(C₃⋊C₈)	—	2	1	no	—
103	C₃⋊(C₃⋊C₈)	—	2	1	no	—
104	C₃⋊(C₃⋊C₈)	—	4	1	no	—
105	C₃×C₃⋊C₈	—	2	1	no	—
106	C₃×C₃⋊C₈	—	2	1	no	—
107	C₃×C₃⋊C₈	—	4	1	no	—
108	C₃×C₃⋊C₈	—	4	1	no	—
109	#432	—	9	1	no	—
110	#431	—	3	1	no	—
111	#431	—	9	1	no	—
112	#335	—	3	1	no	—
113	#335	—	9	1	no	—
114	#334	—	9	1	no	—
115	#334	—	9	1	no	—
116	C₂²×C₃×C₄	—	3	1	no	—
117	C₂²×C₃×C₄	—	3	1	no	—
118	C₂²×C₃×C₄	—	3	1	no	—
119	C₂×C₄×D₃	Dic₆	1	1	yes	characteristic
120	C₂×C₄×D₃	C₄×D₃	1	1	yes	characteristic
121	C₂×C₄×D₃	#67	1	1	yes	characteristic
122	C₂×C₃×C₈	—	2	1	no	—
123	C₂×C₃×C₈	—	2	1	no	—
124	C₂×C₃×C₈	—	3	1	no	—
125	C₂×C₃×C₈	—	3	1	no	—
126	C₂×C₃×C₈	—	4	1	no	—
127	C₂×C₃×C₈	—	6	1	no	—
128	C₂×C₃×C₈	—	6	1	no	—
129	C₃×C₄²	—	2	1	no	—
130	C₃×C₄²	—	3	1	no	—
131	C₃×C₄²	—	3	1	no	—
132	C₃×C₄²	—	3	1	no	—
133	C₃×C₄²	C₂²×D₃	1	1	yes	characteristic
134	C₃×C₄²	C₂²×D₃	1	1	yes	characteristic
135	C₄×Dic₃	Dic₆	1	1	yes	characteristic
136	C₄×Dic₃	C₄×D₃	1	1	yes	characteristic
137	C₄×Dic₃	#67	1	1	yes	characteristic
138	C₂×C₃⋊C₈	—	2	1	no	—
139	C₂×C₃⋊C₈	—	2	1	no	—
140	C₂×C₃⋊C₈	—	3	1	no	—
141	C₂×C₃⋊C₈	—	3	1	no	—
142	C₂×C₃⋊C₈	—	3	1	no	—
143	C₂×C₃⋊C₈	—	3	1	no	—
144	C₂×C₃⋊C₈	—	6	1	no	—
145	C₂×C₃⋊C₈	—	6	1	no	—
146	C₂×C₃⋊C₈	—	6	1	no	—
147	C₂×C₃⋊C₈	—	6	1	no	—
148	C₂×C₃⋊C₈	—	12	1	no	—
149	C₂×C₃⋊C₈	C₄×D₃	1	1	yes	characteristic
150	C₂×C₃⋊C₈	C₄×D₃	1	1	yes	characteristic
151	C₂²×C₃²	C₂×C₄⋊C₄	1	1	yes	characteristic
152	C₂×C₃×D₃	—	2	1	no	—
153	C₂×C₃×D₃	—	2	1	no	—
154	C₂×C₃×D₃	—	2	1	no	—
155	C₃²×C₄	—	2	1	no	—
156	C₃²×C₄	#131	1	1	yes	characteristic
157	C₃²×C₄	#131	1	1	yes	characteristic
158	C₃²×C₄	#135	1	1	yes	characteristic
159	C₃²×C₄	#135	1	1	yes	characteristic
160	C₃×Dic₃	—	2	1	no	—
161	C₃×Dic₃	—	2	1	no	—
162	C₃×Dic₃	—	2	1	no	—
163	C₂×C₄²	—	3	1	no	—
164	C₄⋊C₈	—	3	1	no	—
165	C₄⋊C₈	—	3	1	no	—
166	C₄⋊C₈	—	9	1	no	—
167	C₄⋊C₈	—	9	1	no	—
168	C₄⋊C₈	—	9	1	no	—
169	C₄⋊C₈	D₃²	1	1	yes	characteristic
170	C₂²⋊C₈	—	6	1	no	—
171	C₂²⋊C₈	—	9	1	no	—
172	C₂²⋊C₈	—	9	1	no	—
173	C₂²⋊C₈	—	18	1	no	—
174	#97	—	18	1	no	—
175	C₄×C₈	—	3	1	no	—
176	C₄×C₈	—	6	1	no	—
177	C₄×C₈	—	9	1	no	—
178	C₂³×C₃	—	3	1	no	—
179	C₂²×D₃	Dic₃⋊C₄	1	1	yes	characteristic
180	C₂×C₃×C₄	—	2	1	no	—
181	C₂×C₃×C₄	—	2	1	no	—
182	C₂×C₃×C₄	—	2	1	no	—
183	C₂×C₃×C₄	—	3	1	no	—
184	C₂×C₃×C₄	—	3	1	no	—
185	C₂×C₃×C₄	—	3	1	no	—
186	C₂×C₃×C₄	—	6	1	no	—
187	C₂×C₃×C₄	—	6	1	no	—
188	C₂×C₃×C₄	—	6	1	no	—
189	C₂×C₃×C₄	—	6	1	no	—
190	C₂×C₃×C₄	—	12	1	no	—
191	C₂×C₃×C₄	C₂×Dic₆	1	1	yes	characteristic
192	C₂×C₃×C₄	C₂×C₄×D₃	1	1	yes	characteristic
193	C₂×C₃×C₄	C₂×C₄×D₃	1	1	yes	characteristic
194	C₂×C₃×C₄	D₃×D₄	1	1	yes	characteristic
195	C₂×C₃×C₄	D₃×Q₈	1	1	yes	characteristic
196	C₂×C₃×C₄	C₂×#67	1	1	yes	characteristic
197	C₂×Dic₃	Dic₃⋊C₄	1	1	yes	characteristic
198	C₂×Dic₃	Dic₃⋊C₄	1	1	yes	characteristic
199	C₂×Dic₃	Dic₃⋊C₄	1	1	yes	characteristic
200	C₄×D₃	—	2	1	no	—
201	C₄×D₃	—	2	1	no	—
202	C₄×D₃	—	2	1	no	—
203	C₄×D₃	—	2	1	no	—
204	C₄×D₃	—	4	1	no	—
205	C₃×C₈	—	4	1	no	—
206	C₃×C₈	—	4	1	no	—
207	C₃×C₈	—	6	1	no	—
208	C₃×C₈	—	6	1	no	—
209	C₃×C₈	—	8	1	no	—
210	C₃×C₈	—	12	1	no	—
211	C₃×C₈	—	12	1	no	—
212	C₃⋊C₈	—	2	1	no	—
213	C₃⋊C₈	—	2	1	no	—
214	C₃⋊C₈	—	4	1	no	—
215	C₃⋊C₈	—	4	1	no	—
216	C₃⋊C₈	—	6	2	no	—
217	C₃⋊C₈	—	6	1	no	—
218	C₃⋊C₈	—	6	1	no	—
219	C₃⋊C₈	—	6	1	no	—
220	C₃⋊C₈	—	6	1	no	—
221	C₃⋊C₈	—	12	1	no	—
222	C₃⋊C₈	—	12	1	no	—
223	C₃⋊C₈	—	12	1	no	—
224	C₃⋊C₈	—	24	1	no	—
225	C₂×C₃²	#424	1	1	yes	characteristic
226	C₂×C₃²	#427	1	1	yes	characteristic
227	C₂×C₃²	#430	1	1	yes	characteristic
228	C₃×D₃	—	4	1	no	—
229	C₂²×C₄	—	3	1	no	—
230	C₂²×C₄	—	3	1	no	—
231	C₂²×C₄	—	3	1	no	—
232	C₂×C₈	—	2	1	no	—
233	C₂×C₈	—	3	1	no	—
234	C₂×C₈	—	3	1	no	—
235	C₂×C₈	—	6	1	no	—
236	C₂×C₈	—	6	1	no	—
237	C₂×C₈	—	9	1	no	—
238	C₂×C₈	—	9	1	no	—
239	C₂×C₈	—	18	1	no	—
240	C₄²	—	3	1	no	—
241	C₄²	—	3	1	no	—
242	C₄²	—	3	1	no	—
243	C₄²	C₂×D₃²	1	1	yes	characteristic
244	C₂²×C₃	—	2	1	no	—
245	C₂²×C₃	—	6	1	no	—
246	C₂²×C₃	—	6	1	no	—
247	C₂²×C₃	—	6	1	no	—
248	C₂²×C₃	D₃×C₄⋊C₄	1	1	yes	characteristic
249	C₂²×C₃	C₂×Dic₃⋊C₄	1	1	yes	characteristic
250	C₂×D₃	—	2	1	no	—
251	C₂×D₃	—	2	1	no	—
252	C₂×D₃	—	2	1	no	—
253	C₃×C₄	—	2	1	no	—
254	C₃×C₄	—	2	1	no	—
255	C₃×C₄	—	2	1	no	—
256	C₃×C₄	—	2	1	no	—
257	C₃×C₄	—	2	1	no	—
258	C₃×C₄	—	2	1	no	—
259	C₃×C₄	—	4	1	no	—
260	C₃×C₄	—	6	1	no	—
261	C₃×C₄	—	6	1	no	—
262	C₃×C₄	—	6	1	no	—
263	C₃×C₄	#901	1	1	yes	characteristic
264	C₃×C₄	#901	1	1	yes	characteristic
265	C₃×C₄	#901	1	1	yes	characteristic
266	C₃×C₄	#907	1	1	yes	characteristic
267	C₃×C₄	#912	1	1	yes	characteristic
268	C₃×C₄	#919	1	1	yes	characteristic
269	C₃×C₄	#950	1	1	yes	characteristic
270	C₃×C₄	#950	1	1	yes	characteristic
271	Dic₃	—	2	1	no	—
272	Dic₃	—	2	1	no	—
273	Dic₃	—	2	1	no	—
274	C₃²	#1572	1	1	yes	characteristic, complemented, Sylow
275	C₂³	—	3	1	no	—
276	C₂×C₄	—	3	1	no	—
277	C₂×C₄	—	3	1	no	—
278	C₂×C₄	—	3	1	no	—
279	C₂×C₄	—	6	1	no	—
280	C₂×C₄	—	6	1	no	—
281	C₂×C₄	—	6	1	no	—
282	C₂×C₄	—	6	1	no	—
283	C₂×C₄	—	12	1	no	—
284	C₂×C₄	D₃×Dic₆	1	1	yes	characteristic
285	C₂×C₄	C₄×D₃²	1	1	yes	characteristic
286	C₂×C₄	D₃×#67	1	1	yes	characteristic
287	C₈	—	4	1	no	—
288	C₈	—	6	1	no	—
289	C₈	—	6	1	no	—
290	C₈	—	12	1	no	—
291	C₈	—	12	1	no	—
292	C₈	—	18	1	no	—
293	C₈	—	18	1	no	—
294	C₈	—	36	1	no	—
295	C₂×C₃	—	2	1	no	—
296	C₂×C₃	—	2	1	no	—
297	C₂×C₃	—	2	1	no	—
298	C₂×C₃	—	12	1	no	—
299	C₂×C₃	#4839	1	1	yes	characteristic
300	C₂×C₃	#4880	1	1	yes	characteristic
301	C₂×C₃	#4893	1	1	yes	characteristic
302	C₂×C₃	#4998	1	1	yes	characteristic
303	C₂×C₃	#5102	1	1	yes	characteristic
304	C₂×C₃	#5124	1	1	yes	characteristic
305	D₃	—	4	1	no	—
306	C₂²	—	6	1	no	—
307	C₂²	—	6	1	no	—
308	C₂²	—	6	1	no	—
309	C₂²	D₃×Dic₃⋊C₄	1	1	yes	characteristic
310	C₄	—	2	1	no	—
311	C₄	—	6	1	no	—
312	C₄	—	6	1	no	—
313	C₄	—	6	1	no	—
314	C₄	#63803	1	1	yes	characteristic
315	C₄	#63804	1	1	yes	characteristic
316	C₄	#63933	1	1	yes	characteristic
317	C₄	#63949	1	1	yes	characteristic
318	C₃	—	2	1	no	—
319	C₃	#68573	1	1	yes	characteristic, complemented
320	C₃	#70747	1	1	yes	characteristic, complemented
321	C₂	—	12	1	no	—
322	C₂	#10590571	1	1	yes	characteristic
323	C₂	#10590694	1	1	yes	characteristic
324	C₂	#10590973	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	D₃×#67
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	#1572
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. #10589346	C₂	semidirect
5. C₃×#4479	C₂	semidirect
6. #10589099	C₂	semidirect
7. #10589098	C₂	semidirect
38. #4479	D₃	semidirect
274. C₃²	#1572	semidirect
319. C₃	#68573	semidirect
320. C₃	#70747	semidirect

Maximal subgroups

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

subgroup	index
C₃×#4833	2
#10590263	2
#10590261	2
#10589346	2
C₃×#4479	2
#10589099	2
#10589098	2
#70747	3
#68573	3