Subgroups — #11786212

Order 1152.

625 conjugacy classes of subgroups.

The subgroup lattice has 291 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#63644	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×#817	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃×C₈×Dic₃	C₂	1	1	yes	maximal, characteristic
4	#10591349	C₂	1	2	yes	maximal
5	C₂×#63573	C₂	1	2	yes	maximal
6	#70554	—	3	1	no	maximal
7	#69764	—	3	1	no	maximal
8	C₂²×C₃²×C₈	C₂²	1	1	yes	characteristic
9	C₂×C₄⋊(C₃⋊Dic₃)	C₂²	1	2	yes	—
10	C₂×C₃×C₄⋊Dic₃	C₂²	1	2	yes	—
11	C₂×C₃×C₄×Dic₃	C₂²	1	1	yes	characteristic
12	C₂²×C₃×C₃⋊C₈	C₂²	1	1	yes	characteristic
13	C₂×Dic₃⋊Dic₃	—	2	4	no	—
14	#63644	—	2	2	no	—
15	C₃×#817	—	2	2	no	—
16	C₃×C₈×Dic₃	C₄	1	4	yes	complemented
17	#63573	—	2	4	no	—
18	C₂×C₃×#106	—	3	1	no	—
19	C₂×C₃×C₄×C₈	—	3	1	no	—
20	C₂×#817	—	3	1	no	—
21	C₂×#817	—	3	1	no	—
22	C₂×#817	—	3	2	no	—
23	C₂×#817	—	6	1	no	—
24	C₂×#817	D₃	1	1	yes	characteristic
25	C₂×C₈×Dic₃	D₃	1	1	yes	characteristic
26	#4975	—	3	2	no	—
27	C₂×#808	—	3	2	no	—
28	#4935	—	3	2	no	—
29	C₂²×C₃²×C₄	C₂³	1	1	yes	characteristic
30	C₂²×C₃⋊Dic₃	—	2	2	no	—
31	C₂²×C₃×Dic₃	—	2	2	no	—
32	C₂²×C₃×Dic₃	D₄	1	2	yes	—
33	C₂×C₃²×C₈	C₂×C₄	1	2	yes	—
34	C₂×C₃²×C₈	D₄	1	2	yes	—
35	C₂×C₃²×C₈	Q₈	1	2	yes	—
36	C₄⋊(C₃⋊Dic₃)	—	2	2	no	—
37	C₄⋊(C₃⋊Dic₃)	—	2	2	no	—
38	C₃×C₄⋊Dic₃	—	2	2	no	—
39	C₃×C₄⋊Dic₃	—	2	2	no	—
40	C₃×C₄×Dic₃	C₂×C₄	1	2	yes	—
41	C₃×C₄×Dic₃	Q₈	1	2	yes	—
42	C₂×C₃×C₃⋊C₈	C₂×C₄	1	2	yes	—
43	C₂×C₃×C₃⋊C₈	D₄	1	2	yes	—
44	C₂×C₃×C₃⋊C₈	Q₈	1	2	yes	—
45	Dic₃⋊Dic₃	—	4	8	no	—
46	#1844	—	9	1	no	Sylow
47	C₂²×C₃×C₈	—	2	1	no	—
48	C₂²×C₃×C₈	—	3	1	no	—
49	C₂²×C₃×C₈	C₂×D₃	1	1	yes	characteristic
50	C₂²×C₃×C₈	C₂×D₃	1	1	yes	characteristic
51	C₂×C₃×C₄⋊C₄	—	3	2	no	—
52	C₂×C₃×C₄²	—	3	1	no	—
53	C₂×C₄⋊Dic₃	—	3	2	no	—
54	C₂×C₄⋊Dic₃	—	3	2	no	—
55	C₂×C₄⋊Dic₃	—	6	2	no	—
56	C₂×C₄⋊Dic₃	—	6	4	no	—
57	C₂×C₄⋊Dic₃	C₂×D₃	1	2	yes	—
58	C₂×Dic₃⋊C₄	—	6	4	no	—
59	C₂×C₄×Dic₃	C₂×D₃	1	1	yes	characteristic
60	C₂²×C₃⋊C₈	C₂×D₃	1	1	yes	characteristic
61	C₃×#106	—	6	2	no	—
62	C₃×C₄×C₈	—	3	4	no	—
63	#817	—	2	2	no	—
64	#817	—	6	4	no	—
65	#817	—	6	4	no	—
66	#817	—	6	2	no	—
67	#817	—	6	2	no	—
68	C₈×Dic₃	Dic₃	1	4	yes	complemented
69	#808	—	6	4	no	—
70	C₂³×C₃²	C₂×D₄	1	1	yes	characteristic
71	C₂×C₃²×C₄	C₂²×C₄	1	1	yes	characteristic
72	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
73	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
74	C₂×C₃²×C₄	C₂×Q₈	1	1	yes	characteristic
75	C₂×C₃²×C₄	C₂×Q₈	1	1	yes	characteristic
76	C₂×C₃²×C₄	C₂×Q₈	1	1	yes	characteristic
77	C₂×C₃⋊Dic₃	—	4	2	no	—
78	C₂×C₃⋊Dic₃	—	4	4	no	—
79	C₂×C₃×Dic₃	—	4	2	no	—
80	C₂×C₃×Dic₃	—	4	4	no	—
81	C₂×C₃×Dic₃	C₄⋊C₄	1	4	yes	—
82	C₂×C₃×Dic₃	SD₁₆	1	8	yes	—
83	C₃²×C₈	C₄⋊C₄	1	4	yes	complemented
84	C₃×C₃⋊C₈	C₄⋊C₄	1	4	yes	complemented
85	C₂×#106	—	3	1	no	—
86	C₂×#106	—	9	2	no	—
87	C₂×#106	—	9	1	no	—
88	C₂×C₄×C₈	—	3	1	no	—
89	#384	—	9	2	no	—
90	C₂²×C₃×C₄	—	2	1	no	—
91	C₂²×C₃×C₄	—	3	2	no	—
92	C₂²×C₃×C₄	—	6	2	no	—
93	C₂²×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
94	C₂²×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
95	C₂²×Dic₃	—	2	2	no	—
96	C₂²×Dic₃	—	6	2	no	—
97	C₂²×Dic₃	—	6	2	no	—
98	C₂²×Dic₃	—	12	2	no	—
99	C₂²×Dic₃	D₁₂	1	2	yes	—
100	C₂×C₃×C₈	—	2	2	no	—
101	C₂×C₃×C₈	—	2	2	no	—
102	C₂×C₃×C₈	—	2	2	no	—
103	C₂×C₃×C₈	—	3	2	no	—
104	C₂×C₃×C₈	—	3	2	no	—
105	C₂×C₃×C₈	—	3	2	no	—
106	C₂×C₃×C₈	Dic₆	1	2	yes	—
107	C₂×C₃×C₈	Dic₆	1	2	yes	—
108	C₂×C₃×C₈	C₄×D₃	1	2	yes	—
109	C₂×C₃×C₈	D₁₂	1	2	yes	—
110	C₂×C₃×C₈	C₂×Dic₃	1	2	yes	—
111	C₂×C₃×C₈	#67	1	2	yes	—
112	C₃×C₄⋊C₄	—	6	2	no	—
113	C₃×C₄⋊C₄	—	6	2	no	—
114	C₃×C₄²	—	3	2	no	—
115	C₃×C₄²	—	3	2	no	—
116	C₄⋊Dic₃	—	2	2	no	—
117	C₄⋊Dic₃	—	2	2	no	—
118	C₄⋊Dic₃	—	6	4	no	—
119	C₄⋊Dic₃	—	6	2	no	—
120	C₄⋊Dic₃	—	6	2	no	—
121	C₄⋊Dic₃	—	6	2	no	—
122	C₄⋊Dic₃	—	6	2	no	—
123	C₄⋊Dic₃	—	12	2	no	—
124	C₄⋊Dic₃	—	12	8	no	—
125	Dic₃⋊C₄	—	12	8	no	—
126	C₄×Dic₃	Dic₆	1	2	yes	—
127	C₄×Dic₃	C₂×Dic₃	1	2	yes	—
128	C₂×C₃⋊C₈	Dic₆	1	2	yes	—
129	C₂×C₃⋊C₈	D₁₂	1	2	yes	—
130	C₂×C₃⋊C₈	C₂×Dic₃	1	2	yes	—
131	C₂²×C₃²	C₂×C₄⋊C₄	1	1	yes	characteristic
132	C₂²×C₃²	#127	1	1	yes	characteristic
133	C₂²×C₃²	C₄⋊Q₈	1	1	yes	characteristic
134	C₂²×C₃²	C₂×SD₁₆	1	4	yes	—
135	C₃²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
136	C₃²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
137	C₃²×C₄	C₄⋊Q₈	1	1	yes	characteristic
138	C₃²×C₄	C₄⋊Q₈	1	1	yes	characteristic
139	C₃⋊Dic₃	—	8	2	no	—
140	C₃×Dic₃	—	8	2	no	—
141	C₃×Dic₃	#106	1	8	yes	complemented
142	C₂²×C₈	—	3	1	no	—
143	C₂²×C₈	D₃²	1	1	yes	characteristic
144	C₂×C₄⋊C₄	—	3	2	no	—
145	C₂×C₄⋊C₄	—	9	2	no	—
146	C₂×C₄⋊C₄	—	18	4	no	—
147	C₂×C₄²	—	3	1	no	—
148	#106	—	6	2	no	—
149	#106	—	18	4	no	—
150	#106	—	18	2	no	—
151	C₄×C₈	—	3	4	no	—
152	C₂³×C₃	—	2	1	no	—
153	C₂³×C₃	C₂×D₁₂	1	1	yes	characteristic
154	C₂³×C₃	D₃×D₄	1	1	yes	characteristic
155	C₂×C₃×C₄	—	2	1	no	—
156	C₂×C₃×C₄	—	2	1	no	—
157	C₂×C₃×C₄	—	2	1	no	—
158	C₂×C₃×C₄	—	2	1	no	—
159	C₂×C₃×C₄	—	2	1	no	—
160	C₂×C₃×C₄	—	2	1	no	—
161	C₂×C₃×C₄	—	3	4	no	—
162	C₂×C₃×C₄	—	3	8	no	—
163	C₂×C₃×C₄	—	12	2	no	—
164	C₂×C₃×C₄	—	12	4	no	—
165	C₂×C₃×C₄	C₂×Dic₆	1	1	yes	characteristic
166	C₂×C₃×C₄	C₂×Dic₆	1	1	yes	characteristic
167	C₂×C₃×C₄	C₂×Dic₆	1	1	yes	characteristic
168	C₂×C₃×C₄	C₂×Dic₆	1	1	yes	characteristic
169	C₂×C₃×C₄	C₂×C₄×D₃	1	1	yes	characteristic
170	C₂×C₃×C₄	C₂×D₁₂	1	1	yes	characteristic
171	C₂×C₃×C₄	C₂×D₁₂	1	1	yes	characteristic
172	C₂×C₃×C₄	D₃×D₄	1	1	yes	characteristic
173	C₂×C₃×C₄	D₃×Q₈	1	1	yes	characteristic
174	C₂×C₃×C₄	D₃×Q₈	1	1	yes	characteristic
175	C₂×C₃×C₄	C₂²×Dic₃	1	1	yes	characteristic
176	C₂×C₃×C₄	C₂×#67	1	1	yes	characteristic
177	C₂×Dic₃	—	4	4	no	—
178	C₂×Dic₃	—	4	2	no	—
179	C₂×Dic₃	—	12	4	no	—
180	C₂×Dic₃	—	12	2	no	—
181	C₂×Dic₃	—	12	4	no	—
182	C₂×Dic₃	—	12	4	no	—
183	C₂×Dic₃	—	12	2	no	—
184	C₂×Dic₃	—	24	4	no	—
185	C₂×Dic₃	#204	1	8	yes	—
186	C₂×Dic₃	C₄⋊Dic₃	1	4	yes	—
187	C₃×C₈	—	2	4	no	—
188	C₃×C₈	—	3	4	no	—
189	C₃×C₈	Dic₃⋊C₄	1	4	yes	complemented
190	C₃×C₈	C₄⋊Dic₃	1	4	yes	complemented
191	C₃⋊C₈	C₄⋊Dic₃	1	4	yes	complemented
192	C₂×C₃²	#384	1	1	yes	characteristic
193	C₂×C₃²	C₂×#106	1	2	yes	—
194	C₂×C₃²	#492	1	2	yes	—
195	C₂×C₃²	#498	1	2	yes	—
196	C₂²×C₄	—	3	2	no	—
197	C₂²×C₄	—	6	2	no	—
198	C₂²×C₄	—	18	2	no	—
199	C₂²×C₄	C₂×D₃²	1	1	yes	characteristic
200	C₂×C₈	—	3	2	no	—
201	C₂×C₈	—	3	2	no	—
202	C₂×C₈	—	3	2	no	—
203	C₂×C₈	D₃×Dic₃	1	2	yes	—
204	C₂×C₈	#626	1	2	yes	—
205	C₂×C₈	#627	1	2	yes	—
206	C₄⋊C₄	—	6	2	no	—
207	C₄⋊C₄	—	6	2	no	—
208	C₄⋊C₄	—	18	2	no	—
209	C₄⋊C₄	—	18	2	no	—
210	C₄⋊C₄	—	36	8	no	—
211	C₄²	—	3	2	no	—
212	C₄²	—	3	2	no	—
213	C₂²×C₃	—	2	4	no	—
214	C₂²×C₃	—	2	1	no	—
215	C₂²×C₃	—	2	1	no	—
216	C₂²×C₃	—	2	1	no	—
217	C₂²×C₃	#869	1	1	yes	characteristic
218	C₂²×C₃	#874	1	1	yes	characteristic
219	C₂²×C₃	#888	1	1	yes	characteristic
220	C₂²×C₃	D₃×C₄⋊C₄	1	1	yes	characteristic
221	C₂²×C₃	C₂×#204	1	4	yes	—
222	C₂²×C₃	D₃×SD₁₆	1	4	yes	—
223	C₂²×C₃	C₂×C₄⋊Dic₃	1	1	yes	characteristic
224	C₂²×C₃	#940	1	1	yes	characteristic
225	C₃×C₄	—	2	1	no	—
226	C₃×C₄	—	2	1	no	—
227	C₃×C₄	—	2	1	no	—
228	C₃×C₄	—	2	1	no	—
229	C₃×C₄	—	3	8	no	—
230	C₃×C₄	—	24	2	no	—
231	C₃×C₄	#869	1	1	yes	characteristic
232	C₃×C₄	#869	1	1	yes	characteristic
233	C₃×C₄	#888	1	1	yes	characteristic
234	C₃×C₄	D₃×C₄⋊C₄	1	1	yes	characteristic
235	C₃×C₄	C₂×Dic₃⋊C₄	1	1	yes	characteristic
236	C₃×C₄	C₂×C₄⋊Dic₃	1	1	yes	characteristic
237	C₃×C₄	C₂×C₄⋊Dic₃	1	1	yes	characteristic
238	C₃×C₄	#944	1	1	yes	characteristic
239	Dic₃	—	8	2	no	—
240	Dic₃	—	24	4	no	—
241	Dic₃	—	24	2	no	—
242	Dic₃	—	24	2	no	—
243	Dic₃	#817	1	8	yes	complemented
244	C₃²	#1844	1	1	yes	characteristic, complemented, Sylow
245	C₂³	D₃×D₁₂	1	1	yes	characteristic
246	C₂×C₄	—	3	8	no	—
247	C₂×C₄	—	3	4	no	—
248	C₂×C₄	—	12	4	no	—
249	C₂×C₄	—	12	2	no	—
250	C₂×C₄	—	36	4	no	—
251	C₂×C₄	—	36	2	no	—
252	C₂×C₄	D₃×Dic₆	1	1	yes	characteristic
253	C₂×C₄	D₃×Dic₆	1	1	yes	characteristic
254	C₂×C₄	D₃×D₁₂	1	1	yes	characteristic
255	C₂×C₄	C₂×D₃×Dic₃	1	1	yes	characteristic
256	C₂×C₄	C₂×#626	1	1	yes	characteristic
257	C₂×C₄	C₂×#627	1	1	yes	characteristic
258	C₈	—	3	4	no	—
259	C₈	Dic₃⋊Dic₃	1	4	yes	complemented
260	C₂×C₃	—	2	1	no	—
261	C₂×C₃	—	2	2	no	—
262	C₂×C₃	—	2	2	no	—
263	C₂×C₃	—	2	2	no	—
264	C₂×C₃	#4680	1	2	yes	—
265	C₂×C₃	#4693	1	2	yes	—
266	C₂×C₃	#4855	1	2	yes	—
267	C₂×C₃	D₃×#106	1	2	yes	—
268	C₂×C₃	#4935	1	1	yes	characteristic
269	C₂×C₃	#4975	1	1	yes	characteristic
270	C₂×C₃	C₂×#817	1	2	yes	—
271	C₂×C₃	#5175	1	2	yes	—
272	C₂²	D₃×#204	1	4	yes	—
273	C₂²	D₃×C₄⋊Dic₃	1	1	yes	characteristic
274	C₂²	#63911	1	1	yes	characteristic
275	C₂²	#63918	1	1	yes	characteristic
276	C₄	—	3	8	no	—
277	C₄	—	24	2	no	—
278	C₄	—	72	2	no	—
279	C₄	#63866	1	1	yes	characteristic
280	C₄	D₃×C₄⋊Dic₃	1	1	yes	characteristic
281	C₄	#63918	1	1	yes	characteristic
282	C₄	C₂×Dic₃⋊Dic₃	1	1	yes	characteristic
283	C₃	—	2	1	no	—
284	C₃	#69764	1	1	yes	characteristic, complemented
285	C₃	#70554	1	1	yes	characteristic, complemented
286	C₂	D₃×#817	1	2	yes	—
287	C₂	#10590877	1	2	yes	—
288	C₂	#10590889	1	2	yes	—
289	C₂	#10591349	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	D₃×D₁₂
commutator	C₂×C₃²×C₄	C₂²×C₄
Frattini	C₂²×C₄	C₂×D₃²
Fitting	C₂²×C₃²×C₈	C₂²
socle, prime radical	C₂³×C₃²	C₂×D₄
cosocle	C₂²×C₃²×C₄	C₂³
hypercenter	C₂²×C₈	D₃²
hypocenter	C₃²	#1844
LCS	G ▷ C₂×C₃²×C₄ ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×C₈

Complements

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

N	H ≅ G/N	type
16. C₃×C₈×Dic₃	C₄	semidirect
68. C₈×Dic₃	Dic₃	semidirect
83. C₃²×C₈	C₄⋊C₄	semidirect
84. C₃×C₃⋊C₈	C₄⋊C₄	semidirect
141. C₃×Dic₃	#106	semidirect
189. C₃×C₈	Dic₃⋊C₄	semidirect
190. C₃×C₈	C₄⋊Dic₃	semidirect
191. C₃⋊C₈	C₄⋊Dic₃	semidirect
243. Dic₃	#817	semidirect
244. C₃²	#1844	semidirect
259. C₈	Dic₃⋊Dic₃	semidirect
284. C₃	#69764	semidirect
285. C₃	#70554	semidirect

Maximal subgroups

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

subgroup	index
C₂×#63644	2
C₂×C₃×#817	2
C₂×C₃×C₈×Dic₃	2
#10591349	2
C₂×#63573	2
#70554	3
#69764	3