Subgroups — #11791632

Order 1152.

653 conjugacy classes of subgroups.

The subgroup lattice has 318 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×#869	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊(C₃⋊(C₃⋊C₈))	C₂	1	1	yes	maximal, characteristic
3	C₃×C₄×C₄⋊Dic₃	C₂	1	1	yes	maximal, characteristic
4	#10589163	C₂	1	4	yes	maximal
5	#71780	—	3	1	no	maximal
6	#69490	—	3	1	no	maximal
7	C₂²×C₃×Dic₆	—	2	1	no	—
8	C₂×C₃²×C₄²	C₂²	1	1	yes	characteristic
9	C₂²×C₃⋊(C₃⋊C₈)	C₂²	1	2	yes	—
10	C₂×C₃×C₄⋊Dic₃	—	2	1	no	—
11	C₂×C₃×C₄⋊Dic₃	C₂²	1	2	yes	—
12	C₂×C₃×C₄⋊Dic₃	C₂²	1	2	yes	—
13	C₂×C₃×C₄×Dic₃	—	2	1	no	—
14	C₃×#869	—	2	2	no	—
15	C₃×#869	C₄	1	4	yes	complemented
16	C₄⋊(C₃⋊(C₃⋊C₈))	—	2	2	no	—
17	C₃×C₃⋊#95	—	2	1	no	—
18	C₂×C₃×C₄⋊Q₈	—	3	1	no	—
19	C₂×#869	D₃	1	1	yes	characteristic
20	C₃×C₄×C₄⋊C₄	—	3	1	no	—
21	C₄×C₄⋊Dic₃	D₃	1	1	yes	characteristic
22	C₂×C₄⋊(C₃⋊C₈)	—	3	1	no	—
23	C₂×C₄⋊(C₃⋊C₈)	—	3	1	no	—
24	C₂×C₄⋊(C₃⋊C₈)	—	6	1	no	—
25	#4529	—	3	4	no	—
26	#4529	—	3	4	no	—
27	C₂²×C₃²×C₄	D₄	1	1	yes	characteristic
28	C₂²×C₃²×C₄	D₄	1	1	yes	characteristic
29	C₂²×C₃²×C₄	C₂³	1	1	yes	characteristic
30	C₂²×C₃×Dic₃	—	2	2	no	—
31	C₂²×C₃×Dic₃	—	2	2	no	—
32	C₂×C₃×Dic₆	—	2	4	no	—
33	C₂×C₃×Dic₆	—	4	2	no	—
34	C₂×C₃×Dic₆	—	4	2	no	—
35	C₃²×C₄²	C₂×C₄	1	2	yes	—
36	C₃²×C₄²	D₄	1	2	yes	—
37	C₂×C₃⋊(C₃⋊C₈)	—	2	2	no	—
38	C₂×C₃⋊(C₃⋊C₈)	—	2	4	no	—
39	C₃×C₄⋊Dic₃	—	2	4	no	—
40	C₃×C₄⋊Dic₃	—	2	2	no	—
41	C₃×C₄⋊Dic₃	—	2	2	no	—
42	C₃×C₄⋊Dic₃	C₂×C₄	1	4	yes	—
43	C₃×C₄⋊Dic₃	Q₈	1	4	yes	complemented
44	C₃×C₄×Dic₃	—	4	2	no	—
45	#1988	—	9	1	no	Sylow
46	C₂²×C₃×Q₈	—	6	1	no	—
47	C₂²×Dic₆	—	2	1	no	—
48	C₃×C₄⋊Q₈	—	3	4	no	—
49	C₃×C₄⋊Q₈	—	6	2	no	—
50	C₂×C₃×C₄⋊C₄	—	3	2	no	—
51	C₂×C₃×C₄⋊C₄	—	3	2	no	—
52	C₂×C₃×C₄⋊C₄	—	6	1	no	—
53	C₂×C₃×C₄²	—	2	1	no	—
54	C₂×C₃×C₄²	—	6	1	no	—
55	C₂×C₃×C₄²	C₂×D₃	1	1	yes	characteristic
56	C₂×C₃×C₄²	C₂×D₃	1	1	yes	characteristic
57	C₂×C₄⋊Dic₃	—	2	1	no	—
58	C₂×C₄⋊Dic₃	C₂×D₃	1	2	yes	—
59	C₂×C₄⋊Dic₃	C₂×D₃	1	2	yes	—
60	C₂×C₄×Dic₃	—	2	1	no	—
61	C₂²×C₃⋊C₈	—	3	2	no	—
62	C₂²×C₃⋊C₈	—	3	2	no	—
63	C₂²×C₃⋊C₈	—	6	2	no	—
64	#869	—	2	2	no	—
65	#869	Dic₃	1	4	yes	complemented
66	C₃×#95	—	6	1	no	—
67	C₃⋊#95	—	2	1	no	—
68	C₄⋊(C₃⋊C₈)	—	6	4	no	—
69	C₄⋊(C₃⋊C₈)	—	6	2	no	—
70	C₄⋊(C₃⋊C₈)	—	6	2	no	—
71	C₂³×C₃²	C₂×D₄	1	1	yes	characteristic
72	C₂×C₃²×C₄	—	2	2	no	—
73	C₂×C₃²×C₄	C₂²⋊C₄	1	2	yes	—
74	C₂×C₃²×C₄	C₂²⋊C₄	1	2	yes	—
75	C₂×C₃²×C₄	D₈	1	2	yes	—
76	C₂×C₃²×C₄	SD₁₆	1	2	yes	—
77	C₂×C₃²×C₄	C₂²×C₄	1	1	yes	characteristic
78	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
79	C₂×C₃²×C₄	C₂×Q₈	1	2	yes	—
80	C₂×C₃²×C₄	#41	1	2	yes	—
81	C₂×C₃×Dic₃	—	2	4	no	—
82	C₂×C₃×Dic₃	—	2	4	no	—
83	C₂×C₃×Dic₃	—	4	2	no	—
84	C₂×C₃×Dic₃	—	4	2	no	—
85	C₂×C₃×Dic₃	—	4	2	no	—
86	C₂×C₃×Dic₃	—	4	2	no	—
87	C₃×Dic₆	—	4	8	no	—
88	C₃⋊(C₃⋊C₈)	—	4	2	no	—
89	C₂×C₄⋊Q₈	—	3	1	no	—
90	C₂×C₄⋊C₈	—	9	1	no	—
91	C₄×C₄⋊C₄	—	3	1	no	—
92	#340	—	9	4	no	—
93	C₂×C₃×Q₈	—	6	4	no	—
94	C₂×C₃×Q₈	—	12	2	no	—
95	C₂×C₃×Q₈	—	12	2	no	—
96	C₂²×C₃×C₄	—	2	1	no	—
97	C₂²×C₃×C₄	—	2	1	no	—
98	C₂²×C₃×C₄	—	2	1	no	—
99	C₂²×C₃×C₄	—	6	2	no	—
100	C₂²×C₃×C₄	—	6	2	no	—
101	C₂²×C₃×C₄	#67	1	1	yes	characteristic
102	C₂²×C₃×C₄	#67	1	1	yes	characteristic
103	C₂²×C₃×C₄	#67	1	1	yes	characteristic
104	C₂²×C₃×C₄	#67	1	1	yes	characteristic
105	C₂²×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
106	C₂²×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
107	C₂²×Dic₃	—	2	2	no	—
108	C₂²×Dic₃	—	2	2	no	—
109	C₂×Dic₆	—	2	4	no	—
110	C₂×Dic₆	—	4	2	no	—
111	C₂×Dic₆	—	4	2	no	—
112	C₃×C₄⋊C₄	—	3	4	no	—
113	C₃×C₄⋊C₄	—	3	4	no	—
114	C₃×C₄⋊C₄	—	6	4	no	—
115	C₃×C₄⋊C₄	—	6	2	no	—
116	C₃×C₄⋊C₄	—	6	2	no	—
117	C₃×C₄²	—	2	2	no	—
118	C₃×C₄²	—	2	2	no	—
119	C₃×C₄²	—	12	2	no	—
120	C₃×C₄²	C₄×D₃	1	2	yes	—
121	C₃×C₄²	D₁₂	1	2	yes	—
122	C₃×C₄²	C₂×Dic₃	1	2	yes	—
123	C₃×C₄²	#67	1	2	yes	—
124	C₄⋊Dic₃	—	2	4	no	—
125	C₄⋊Dic₃	—	2	2	no	—
126	C₄⋊Dic₃	—	2	2	no	—
127	C₄⋊Dic₃	Dic₆	1	4	yes	complemented
128	C₄⋊Dic₃	C₂×Dic₃	1	4	yes	—
129	C₄×Dic₃	—	4	2	no	—
130	C₂×C₃⋊C₈	—	6	4	no	—
131	C₂×C₃⋊C₈	—	6	2	no	—
132	C₂×C₃⋊C₈	—	6	4	no	—
133	C₂×C₃⋊C₈	—	6	4	no	—
134	C₂×C₃⋊C₈	—	6	2	no	—
135	C₂×C₃⋊C₈	—	12	4	no	—
136	C₂²×C₃²	C₂×C₂²⋊C₄	1	1	yes	characteristic
137	C₂²×C₃²	C₂²⋊Q₈	1	2	yes	—
138	C₂²×C₃²	C₂×D₈	1	1	yes	characteristic
139	C₂²×C₃²	C₂×SD₁₆	1	1	yes	characteristic
140	C₂²×C₃²	#137	1	2	yes	—
141	C₃²×C₄	—	2	2	no	—
142	C₃²×C₄	#102	1	4	yes	—
143	C₃²×C₄	C₄×Q₈	1	2	yes	—
144	C₃²×C₄	#124	1	1	yes	characteristic
145	C₃²×C₄	C₄⋊Q₈	1	1	yes	characteristic
146	C₃×Dic₃	—	4	4	no	—
147	C₃×Dic₃	—	4	4	no	—
148	C₂²×Q₈	—	6	1	no	—
149	C₂²×C₈	—	9	2	no	—
150	C₄⋊Q₈	—	3	4	no	—
151	C₄⋊Q₈	—	6	2	no	—
152	C₂×C₄⋊C₄	—	3	2	no	—
153	C₂×C₄⋊C₄	—	3	2	no	—
154	C₂×C₄⋊C₄	—	6	1	no	—
155	C₂×C₄²	—	6	1	no	—
156	C₂×C₄²	D₃²	1	1	yes	characteristic
157	C₄⋊C₈	—	18	2	no	—
158	#95	—	6	1	no	—
159	C₂³×C₃	—	2	1	no	—
160	C₂³×C₃	C₂×#67	1	1	yes	characteristic
161	C₂³×C₃	C₂×#67	1	1	yes	characteristic
162	C₃×Q₈	—	12	8	no	—
163	C₂×C₃×C₄	—	2	2	no	—
164	C₂×C₃×C₄	—	2	2	no	—
165	C₂×C₃×C₄	—	2	2	no	—
166	C₂×C₃×C₄	—	2	2	no	—
167	C₂×C₃×C₄	—	2	2	no	—
168	C₂×C₃×C₄	—	2	2	no	—
169	C₂×C₃×C₄	—	2	2	no	—
170	C₂×C₃×C₄	—	2	2	no	—
171	C₂×C₃×C₄	—	2	1	no	—
172	C₂×C₃×C₄	—	2	1	no	—
173	C₂×C₃×C₄	—	4	2	no	—
174	C₂×C₃×C₄	—	6	4	no	—
175	C₂×C₃×C₄	—	6	4	no	—
176	C₂×C₃×C₄	—	12	2	no	—
177	C₂×C₃×C₄	—	12	2	no	—
178	C₂×C₃×C₄	—	12	2	no	—
179	C₂×C₃×C₄	—	12	2	no	—
180	C₂×C₃×C₄	(C₂×D₃)⋊C₄	1	2	yes	—
181	C₂×C₃×C₄	(C₂×D₃)⋊C₄	1	2	yes	—
182	C₂×C₃×C₄	#213	1	2	yes	—
183	C₂×C₃×C₄	#213	1	2	yes	—
184	C₂×C₃×C₄	#214	1	2	yes	—
185	C₂×C₃×C₄	#215	1	2	yes	—
186	C₂×C₃×C₄	#217	1	2	yes	—
187	C₂×C₃×C₄	#217	1	2	yes	—
188	C₂×C₃×C₄	C₂×Dic₆	1	2	yes	—
189	C₂×C₃×C₄	C₂×C₄×D₃	1	1	yes	characteristic
190	C₂×C₃×C₄	C₂×D₁₂	1	1	yes	characteristic
191	C₂×C₃×C₄	#235	1	2	yes	—
192	C₂×C₃×C₄	D₃×Q₈	1	2	yes	—
193	C₂×C₃×C₄	#239	1	2	yes	—
194	C₂×C₃×C₄	C₂²×Dic₃	1	1	yes	characteristic
195	C₂×C₃×C₄	C₂×#67	1	1	yes	characteristic
196	C₂×Dic₃	—	2	4	no	—
197	C₂×Dic₃	—	2	4	no	—
198	C₂×Dic₃	—	4	2	no	—
199	C₂×Dic₃	—	4	2	no	—
200	C₂×Dic₃	—	4	2	no	—
201	C₂×Dic₃	—	4	2	no	—
202	Dic₆	—	4	8	no	—
203	C₃⋊C₈	—	12	4	no	—
204	C₃⋊C₈	—	12	2	no	—
205	C₃⋊C₈	—	12	2	no	—
206	C₂×C₃²	#391	1	1	yes	characteristic
207	C₂×C₃²	C₂×#102	1	1	yes	characteristic
208	C₂×C₃²	#419	1	1	yes	characteristic
209	C₂×C₃²	#474	1	2	yes	—
210	C₂×C₃²	#475	1	2	yes	—
211	C₂×Q₈	—	6	4	no	—
212	C₂×Q₈	—	12	2	no	—
213	C₂×Q₈	—	12	2	no	—
214	C₂²×C₄	—	6	2	no	—
215	C₂²×C₄	—	6	2	no	—
216	C₂²×C₄	#625	1	1	yes	characteristic
217	C₂²×C₄	#625	1	1	yes	characteristic
218	C₂²×C₄	C₂×D₃²	1	1	yes	characteristic
219	C₂×C₈	—	18	4	no	—
220	C₂×C₈	—	18	2	no	—
221	C₄⋊C₄	—	3	4	no	—
222	C₄⋊C₄	—	3	4	no	—
223	C₄⋊C₄	—	6	4	no	—
224	C₄⋊C₄	—	6	2	no	—
225	C₄⋊C₄	—	6	2	no	—
226	C₄²	—	12	2	no	—
227	C₄²	D₃×Dic₃	1	2	yes	—
228	C₄²	#626	1	2	yes	—
229	C₂²×C₃	—	2	1	no	—
230	C₂²×C₃	—	2	1	no	—
231	C₂²×C₃	—	2	2	no	—
232	C₂²×C₃	—	2	2	no	—
233	C₂²×C₃	—	2	1	no	—
234	C₂²×C₃	#924	1	2	yes	—
235	C₂²×C₃	C₂×(C₂×D₃)⋊C₄	1	1	yes	characteristic
236	C₂²×C₃	C₂×#213	1	1	yes	characteristic
237	C₂²×C₃	C₂×#213	1	1	yes	characteristic
238	C₂²×C₃	C₂×#214	1	1	yes	characteristic
239	C₂²×C₃	C₂×#215	1	1	yes	characteristic
240	C₂²×C₃	#942	1	2	yes	—
241	C₂²×C₃	#946	1	2	yes	—
242	C₂²×C₃	#951	1	2	yes	—
243	C₂²×C₃	C₂×#217	1	1	yes	characteristic
244	C₃×C₄	—	2	2	no	—
245	C₃×C₄	—	2	2	no	—
246	C₃×C₄	—	2	4	no	—
247	C₃×C₄	—	2	2	no	—
248	C₃×C₄	—	2	1	no	—
249	C₃×C₄	—	2	1	no	—
250	C₃×C₄	—	4	2	no	—
251	C₃×C₄	—	12	4	no	—
252	C₃×C₄	—	12	4	no	—
253	C₃×C₄	#809	1	4	yes	—
254	C₃×C₄	#832	1	4	yes	—
255	C₃×C₄	C₄×Dic₆	1	2	yes	—
256	C₃×C₄	#869	1	1	yes	characteristic
257	C₃×C₄	#875	1	1	yes	characteristic
258	C₃×C₄	#944	1	1	yes	characteristic
259	C₃×C₄	Q₈×Dic₃	1	2	yes	—
260	C₃×C₄	#947	1	1	yes	characteristic
261	Dic₃	—	4	4	no	—
262	Dic₃	—	4	4	no	—
263	C₃²	#1988	1	1	yes	characteristic, complemented, Sylow
264	C₂³	C₂×#625	1	1	yes	characteristic
265	Q₈	—	12	8	no	—
266	C₂×C₄	—	2	2	no	—
267	C₂×C₄	—	6	4	no	—
268	C₂×C₄	—	6	4	no	—
269	C₂×C₄	—	12	2	no	—
270	C₂×C₄	—	12	2	no	—
271	C₂×C₄	—	12	2	no	—
272	C₂×C₄	—	12	2	no	—
273	C₂×C₄	#3713	1	2	yes	—
274	C₂×C₄	#3715	1	2	yes	—
275	C₂×C₄	(C₂×D₃)⋊Dic₃	1	2	yes	—
276	C₂×C₄	(C₂×D₃)⋊Dic₃	1	2	yes	—
277	C₂×C₄	D₃×Dic₆	1	2	yes	—
278	C₂×C₄	#3799	1	2	yes	—
279	C₂×C₄	C₂×D₃×Dic₃	1	1	yes	characteristic
280	C₂×C₄	C₂×#626	1	1	yes	characteristic
281	C₈	—	36	2	no	—
282	C₂×C₃	—	2	2	no	—
283	C₂×C₃	—	2	2	no	—
284	C₂×C₃	—	2	1	no	—
285	C₂×C₃	—	2	1	no	—
286	C₂×C₃	—	2	1	no	—
287	C₂×C₃	#4929	1	1	yes	characteristic
288	C₂×C₃	C₂×#809	1	1	yes	characteristic
289	C₂×C₃	#5004	1	1	yes	characteristic
290	C₂×C₃	#5007	1	2	yes	—
291	C₂×C₃	#5020	1	2	yes	—
292	C₂×C₃	#5087	1	2	yes	—
293	C₂×C₃	#5094	1	2	yes	—
294	C₂×C₃	C₂×#832	1	1	yes	characteristic
295	C₂×C₃	#5225	1	1	yes	characteristic
296	C₂×C₃	#5229	1	1	yes	characteristic
297	C₂²	C₂×#3713	1	1	yes	characteristic
298	C₂²	C₂×#3715	1	1	yes	characteristic
299	C₂²	#63827	1	2	yes	—
300	C₂²	#63856	1	2	yes	—
301	C₂²	C₂×(C₂×D₃)⋊Dic₃	1	1	yes	characteristic
302	C₄	—	2	2	no	—
303	C₄	—	12	4	no	—
304	C₄	—	12	4	no	—
305	C₄	#63562	1	4	yes	—
306	C₄	Dic₃×Dic₆	1	2	yes	—
307	C₄	#63864	1	1	yes	characteristic
308	C₄	#63866	1	1	yes	characteristic
309	C₃	—	2	1	no	—
310	C₃	#69490	1	1	yes	characteristic, complemented
311	C₃	#71780	1	1	yes	characteristic, complemented
312	C₂	#10590269	1	2	yes	—
313	C₂	#10590285	1	2	yes	—
314	C₂	C₂×#63562	1	1	yes	characteristic
315	C₂	#10591099	1	1	yes	characteristic
316	C₂	#10591220	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×#625
commutator	C₂×C₃²×C₄	C₂²×C₄
Frattini	C₂²×C₄	C₂×D₃²
Fitting	C₂×C₃²×C₄²	C₂²
socle, prime radical	C₂³×C₃²	C₂×D₄
cosocle	C₂²×C₃²×C₄	C₂³
hypercenter	C₂×C₄²	D₃²
hypocenter	C₃²	#1988
LCS	G ▷ C₂×C₃²×C₄ ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂³ ◁ C₂×C₄²

Complements

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

N	H ≅ G/N	type
15. C₃×#869	C₄	semidirect
43. C₃×C₄⋊Dic₃	Q₈	semidirect
65. #869	Dic₃	semidirect
127. C₄⋊Dic₃	Dic₆	semidirect
263. C₃²	#1988	semidirect
310. C₃	#69490	semidirect
311. C₃	#71780	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃×#869	2
C₂×C₄⋊(C₃⋊(C₃⋊C₈))	2
C₃×C₄×C₄⋊Dic₃	2
#10589163	2
#71780	3
#69490	3