Subgroups — #11771208

Order 1152.

449 conjugacy classes of subgroups.

The subgroup lattice has 320 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₄⋊(C₃⋊Dic₃)	C₂	1	1	yes	maximal, characteristic
2	C₃×#4925	C₂	1	1	yes	maximal, characteristic
3	C₃×#4924	C₂	1	1	yes	maximal, characteristic
4	#67746	—	3	1	no	maximal
5	#67715	—	3	1	no	maximal
6	C₂×C₃²×C₄²	C₂²	1	1	yes	characteristic
7	C₂×C₄⋊(C₃⋊Dic₃)	—	2	1	no	—
8	C₂×C₄×C₃⋊Dic₃	—	2	1	no	—
9	C₂×C₃×#208	—	2	1	no	—
10	#63658	C₄	1	2	yes	—
11	C₃×#830	—	2	1	no	—
12	C₃×C₄⋊(C₃⋊C₈)	—	2	1	no	—
13	C₃×C₄⋊(C₃⋊C₈)	C₄	1	2	yes	complemented
14	C₃×C₄⋊(C₃⋊C₈)	C₄	1	2	yes	complemented
15	C₃×C₄×C₃⋊C₈	—	2	1	no	—
16	C₃×#431	—	3	1	no	—
17	C₃×#423	—	3	1	no	—
18	C₄×C₄⋊Dic₃	—	3	1	no	—
19	C₄×C₄⋊Dic₃	—	3	1	no	—
20	C₄×C₄⋊Dic₃	—	6	1	no	—
21	#4925	D₃	1	1	yes	characteristic
22	#4924	D₃	1	1	yes	characteristic
23	C₂²×C₃²×C₄	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₃²×C₄	D₄	1	1	yes	characteristic
25	C₂²×C₃²×C₄	D₄	1	1	yes	characteristic
26	C₂²×C₃⋊Dic₃	—	2	2	no	—
27	C₃²×C₄²	C₂×C₄	1	1	yes	characteristic
28	C₃²×C₄²	C₂×C₄	1	1	yes	characteristic
29	C₃²×C₄²	D₄	1	1	yes	characteristic
30	C₃²×C₄²	Q₈	1	1	yes	characteristic
31	C₄⋊(C₃⋊Dic₃)	—	2	4	no	—
32	C₄×C₃⋊Dic₃	—	4	1	no	—
33	C₄×C₃⋊Dic₃	—	4	1	no	—
34	C₃×#208	—	4	1	no	—
35	C₃×#208	—	4	1	no	—
36	C₂×C₃×C₃⋊C₈	—	2	2	no	—
37	C₂×C₃×C₃⋊C₈	—	2	2	no	—
38	#1303	—	9	1	no	Sylow
39	C₂×C₃×M₁₆	—	6	1	no	—
40	C₂×C₃×C₄²	—	2	1	no	—
41	C₂×C₃×C₄²	C₂×D₃	1	1	yes	characteristic
42	C₂×C₃×C₄²	C₂×D₃	1	1	yes	characteristic
43	C₂×C₄⋊Dic₃	—	6	2	no	—
44	C₂×C₄⋊Dic₃	—	6	1	no	—
45	C₂×C₄⋊Dic₃	—	6	1	no	—
46	C₂×C₄×Dic₃	—	6	2	no	—
47	C₂×C₄×Dic₃	—	6	1	no	—
48	C₂×C₄×Dic₃	—	6	1	no	—
49	C₂×#208	—	2	1	no	—
50	C₃×C₄⋊C₈	—	3	2	no	—
51	C₃×C₄⋊C₈	—	3	2	no	—
52	C₃×C₄⋊C₈	—	6	1	no	—
53	C₃×C₂²⋊C₈	—	6	1	no	—
54	C₃×C₄×C₈	—	6	1	no	—
55	C₃⋊#95	—	3	2	no	—
56	C₃⋊#95	—	3	2	no	—
57	C₃⋊#95	—	6	2	no	—
58	#830	—	2	1	no	—
59	C₄⋊(C₃⋊C₈)	—	2	1	no	—
60	C₄⋊(C₃⋊C₈)	Dic₃	1	2	yes	complemented
61	C₄⋊(C₃⋊C₈)	Dic₃	1	2	yes	complemented
62	C₄×C₃⋊C₈	—	2	1	no	—
63	C₂³×C₃²	C₂²⋊C₄	1	1	yes	characteristic
64	C₂×C₃²×C₄	—	2	2	no	—
65	C₂×C₃²×C₄	—	4	1	no	—
66	C₂×C₃²×C₄	C₄²	1	1	yes	characteristic
67	C₂×C₃²×C₄	C₂²⋊C₄	1	1	yes	characteristic
68	C₂×C₃²×C₄	C₂²⋊C₄	1	1	yes	characteristic
69	C₂×C₃²×C₄	C₄⋊C₄	1	1	yes	characteristic
70	C₂×C₃²×C₄	C₄⋊C₄	1	1	yes	characteristic
71	C₂×C₃²×C₄	C₄⋊C₄	1	1	yes	characteristic
72	C₂×C₃²×C₄	D₈	1	1	yes	characteristic
73	C₂×C₃²×C₄	SD₁₆	1	1	yes	characteristic
74	C₂×C₃²×C₄	SD₁₆	1	1	yes	characteristic
75	C₂×C₃²×C₄	Q₁₆	1	1	yes	characteristic
76	C₂×C₃⋊Dic₃	—	2	4	no	—
77	C₂×C₃⋊Dic₃	—	4	2	no	—
78	C₂×C₃⋊Dic₃	—	4	2	no	—
79	C₃×C₃⋊C₈	—	4	2	no	—
80	C₃×C₃⋊C₈	—	4	2	no	—
81	#431	—	3	1	no	—
82	#423	—	3	1	no	—
83	C₄×C₄⋊C₄	—	9	1	no	—
84	C₂²×C₃×C₄	—	2	1	no	—
85	C₂²×C₃×C₄	—	2	1	no	—
86	C₂²×C₃×C₄	—	2	1	no	—
87	C₂²×C₃×C₄	C₄×D₃	1	1	yes	characteristic
88	C₂²×C₃×C₄	C₄×D₃	1	1	yes	characteristic
89	C₂²×C₃×C₄	D₁₂	1	1	yes	characteristic
90	C₂²×C₃×C₄	D₁₂	1	1	yes	characteristic
91	C₂²×C₃×C₄	#67	1	1	yes	characteristic
92	C₂²×C₃×C₄	#67	1	1	yes	characteristic
93	C₂²×Dic₃	—	6	4	no	—
94	C₂²×Dic₃	—	6	2	no	—
95	C₂²×Dic₃	—	6	2	no	—
96	C₃×M₁₆	—	12	1	no	—
97	C₃×M₁₆	—	12	1	no	—
98	C₂×C₃×C₈	—	6	2	no	—
99	C₂×C₃×C₈	—	6	2	no	—
100	C₃×C₄²	—	2	1	no	—
101	C₃×C₄²	—	2	1	no	—
102	C₃×C₄²	—	2	1	no	—
103	C₃×C₄²	—	2	1	no	—
104	C₃×C₄²	Dic₆	1	1	yes	characteristic
105	C₃×C₄²	Dic₆	1	1	yes	characteristic
106	C₃×C₄²	C₄×D₃	1	1	yes	characteristic
107	C₃×C₄²	C₄×D₃	1	1	yes	characteristic
108	C₃×C₄²	C₂×Dic₃	1	1	yes	characteristic
109	C₃×C₄²	C₂×Dic₃	1	1	yes	characteristic
110	C₃×C₄²	#67	1	1	yes	characteristic
111	C₃×C₄²	#67	1	1	yes	characteristic
112	C₄⋊Dic₃	—	6	8	no	—
113	C₄⋊Dic₃	—	6	4	no	—
114	C₄⋊Dic₃	—	6	4	no	—
115	C₄×Dic₃	—	12	2	no	—
116	C₄×Dic₃	—	12	2	no	—
117	C₄×Dic₃	—	12	1	no	—
118	C₄×Dic₃	—	12	1	no	—
119	C₄×Dic₃	—	12	1	no	—
120	C₄×Dic₃	—	12	1	no	—
121	#208	—	4	1	no	—
122	#208	—	4	1	no	—
123	C₂×C₃⋊C₈	—	2	2	no	—
124	C₂×C₃⋊C₈	—	2	2	no	—
125	C₂²×C₃²	—	2	1	no	—
126	C₂²×C₃²	—	2	1	no	—
127	C₂²×C₃²	#95	1	1	yes	characteristic
128	C₂²×C₃²	#102	1	1	yes	characteristic
129	C₂²×C₃²	#103	1	1	yes	characteristic
130	C₃²×C₄	—	2	1	no	—
131	C₃²×C₄	—	2	1	no	—
132	C₃²×C₄	—	2	1	no	—
133	C₃²×C₄	#102	1	1	yes	characteristic
134	C₃²×C₄	#103	1	1	yes	characteristic
135	C₃²×C₄	#104	1	2	yes	—
136	C₃²×C₄	#106	1	1	yes	characteristic
137	C₃²×C₄	#107	1	1	yes	characteristic
138	C₃⋊Dic₃	—	4	4	no	—
139	C₂×M₁₆	—	6	1	no	—
140	C₂×C₄⋊C₄	—	18	1	no	—
141	C₂×C₄²	—	18	1	no	—
142	C₂×C₄²	D₃²	1	1	yes	characteristic
143	C₄⋊C₈	—	3	2	no	—
144	C₄⋊C₈	—	3	2	no	—
145	C₄⋊C₈	—	6	1	no	—
146	C₂²⋊C₈	—	6	1	no	—
147	C₄×C₈	—	6	1	no	—
148	#95	—	9	2	no	—
149	C₂³×C₃	—	2	1	no	—
150	C₂³×C₃	(C₂×D₃)⋊C₄	1	1	yes	characteristic
151	C₂³×C₃	(C₂×D₃)⋊C₄	1	1	yes	characteristic
152	C₂×C₃×C₄	—	2	4	no	—
153	C₂×C₃×C₄	—	2	2	no	—
154	C₂×C₃×C₄	—	2	2	no	—
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	—	2	1	no	—
162	C₂×C₃×C₄	—	2	1	no	—
163	C₂×C₃×C₄	—	2	1	no	—
164	C₂×C₃×C₄	—	2	1	no	—
165	C₂×C₃×C₄	—	4	1	no	—
166	C₂×C₃×C₄	—	4	1	no	—
167	C₂×C₃×C₄	—	4	2	no	—
168	C₂×C₃×C₄	#204	1	1	yes	characteristic
169	C₂×C₃×C₄	#204	1	1	yes	characteristic
170	C₂×C₃×C₄	D₂₄	1	1	yes	characteristic
171	C₂×C₃×C₄	Dic₁₂	1	1	yes	characteristic
172	C₂×C₃×C₄	C₄×Dic₃	1	1	yes	characteristic
173	C₂×C₃×C₄	C₄×Dic₃	1	1	yes	characteristic
174	C₂×C₃×C₄	Dic₃⋊C₄	1	1	yes	characteristic
175	C₂×C₃×C₄	Dic₃⋊C₄	1	1	yes	characteristic
176	C₂×C₃×C₄	Dic₃⋊C₄	1	1	yes	characteristic
177	C₂×C₃×C₄	Dic₃⋊C₄	1	1	yes	characteristic
178	C₂×C₃×C₄	C₄⋊Dic₃	1	1	yes	characteristic
179	C₂×C₃×C₄	C₄⋊Dic₃	1	1	yes	characteristic
180	C₂×C₃×C₄	(C₂×D₃)⋊C₄	1	1	yes	characteristic
181	C₂×C₃×C₄	(C₂×D₃)⋊C₄	1	1	yes	characteristic
182	C₂×C₃×C₄	#213	1	1	yes	characteristic
183	C₂×C₃×C₄	#214	1	1	yes	characteristic
184	C₂×C₃×C₄	#215	1	1	yes	characteristic
185	C₂×C₃×C₄	#216	1	1	yes	characteristic
186	C₂×C₃×C₄	#217	1	1	yes	characteristic
187	C₂×C₃×C₄	#217	1	1	yes	characteristic
188	C₂×Dic₃	—	6	8	no	—
189	C₂×Dic₃	—	6	4	no	—
190	C₂×Dic₃	—	6	4	no	—
191	C₂×Dic₃	—	12	4	no	—
192	C₂×Dic₃	—	12	4	no	—
193	C₂×Dic₃	—	12	2	no	—
194	C₂×Dic₃	—	12	2	no	—
195	C₂×Dic₃	—	12	2	no	—
196	C₂×Dic₃	—	12	2	no	—
197	C₃×C₈	—	12	2	no	—
198	C₃×C₈	—	12	2	no	—
199	C₃⋊C₈	—	4	2	no	—
200	C₃⋊C₈	—	4	2	no	—
201	C₂×C₃²	—	2	1	no	—
202	C₂×C₃²	—	2	1	no	—
203	C₂×C₃²	#339	1	1	yes	characteristic
204	C₂×C₃²	#340	1	1	yes	characteristic
205	C₂×C₃²	#344	1	1	yes	characteristic
206	C₂²×C₄	—	18	2	no	—
207	C₂²×C₄	#624	1	1	yes	characteristic
208	C₂²×C₄	#626	1	1	yes	characteristic
209	C₂²×C₄	#626	1	1	yes	characteristic
210	M₁₆	—	12	1	no	—
211	M₁₆	—	12	1	no	—
212	C₂×C₈	—	6	2	no	—
213	C₂×C₈	—	6	2	no	—
214	C₄⋊C₄	—	18	4	no	—
215	C₄²	—	36	1	no	—
216	C₄²	—	36	1	no	—
217	C₄²	D₃×Dic₃	1	1	yes	characteristic
218	C₄²	D₃×Dic₃	1	1	yes	characteristic
219	C₄²	#625	1	1	yes	characteristic
220	C₄²	#627	1	1	yes	characteristic
221	C₂²×C₃	—	2	2	no	—
222	C₂²×C₃	—	2	2	no	—
223	C₂²×C₃	—	2	1	no	—
224	C₂²×C₃	—	2	1	no	—
225	C₂²×C₃	—	2	1	no	—
226	C₂²×C₃	—	2	1	no	—
227	C₂²×C₃	—	2	1	no	—
228	C₂²×C₃	—	2	1	no	—
229	C₂²×C₃	—	2	1	no	—
230	C₂²×C₃	#809	1	1	yes	characteristic
231	C₂²×C₃	#810	1	1	yes	characteristic
232	C₂²×C₃	#816	1	1	yes	characteristic
233	C₂²×C₃	#821	1	1	yes	characteristic
234	C₂²×C₃	C₃⋊#95	1	1	yes	characteristic
235	C₂²×C₃	C₃⋊#95	1	1	yes	characteristic
236	C₃×C₄	—	2	2	no	—
237	C₃×C₄	—	2	1	no	—
238	C₃×C₄	—	2	1	no	—
239	C₃×C₄	—	2	1	no	—
240	C₃×C₄	—	2	1	no	—
241	C₃×C₄	—	2	1	no	—
242	C₃×C₄	—	2	1	no	—
243	C₃×C₄	—	2	1	no	—
244	C₃×C₄	—	2	1	no	—
245	C₃×C₄	—	2	2	no	—
246	C₃×C₄	—	2	1	no	—
247	C₃×C₄	—	2	1	no	—
248	C₃×C₄	—	4	1	no	—
249	C₃×C₄	—	4	1	no	—
250	C₃×C₄	#807	1	1	yes	characteristic
251	C₃×C₄	#808	1	1	yes	characteristic
252	C₃×C₄	#816	1	1	yes	characteristic
253	C₃×C₄	#817	1	1	yes	characteristic
254	C₃×C₄	#818	1	1	yes	characteristic
255	C₃×C₄	#821	1	1	yes	characteristic
256	C₃×C₄	#825	1	2	yes	—
257	C₃×C₄	#832	1	1	yes	characteristic
258	C₃×C₄	#835	1	1	yes	characteristic
259	C₃×C₄	#837	1	2	yes	—
260	Dic₃	—	12	8	no	—
261	Dic₃	—	12	4	no	—
262	Dic₃	—	12	4	no	—
263	C₃²	#1303	1	1	yes	characteristic, complemented, Sylow
264	C₂³	#3722	1	1	yes	characteristic
265	C₂×C₄	—	2	2	no	—
266	C₂×C₄	—	4	1	no	—
267	C₂×C₄	—	18	4	no	—
268	C₂×C₄	—	36	2	no	—
269	C₂×C₄	—	36	2	no	—
270	C₂×C₄	#3714	1	1	yes	characteristic
271	C₂×C₄	#3716	1	1	yes	characteristic
272	C₂×C₄	#3717	1	1	yes	characteristic
273	C₂×C₄	#3719	1	1	yes	characteristic
274	C₂×C₄	Dic₃²	1	1	yes	characteristic
275	C₂×C₄	(C₂×D₃)⋊Dic₃	1	1	yes	characteristic
276	C₂×C₄	(C₂×D₃)⋊Dic₃	1	1	yes	characteristic
277	C₂×C₄	Dic₃⋊Dic₃	1	1	yes	characteristic
278	C₂×C₄	Dic₃⋊Dic₃	1	1	yes	characteristic
279	C₂×C₄	#3724	1	1	yes	characteristic
280	C₈	—	12	2	no	—
281	C₈	—	12	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	—	2	1	no	—
288	C₂×C₃	—	2	1	no	—
289	C₂×C₃	—	2	1	no	—
290	C₂×C₃	—	2	1	no	—
291	C₂×C₃	#4529	1	1	yes	characteristic
292	C₂×C₃	#4532	1	1	yes	characteristic
293	C₂×C₃	#4551	1	1	yes	characteristic
294	C₂×C₃	#4555	1	1	yes	characteristic
295	C₂×C₃	#4556	1	1	yes	characteristic
296	C₂×C₃	#4559	1	1	yes	characteristic
297	C₂²	—	2	1	no	—
298	C₂²	—	2	1	no	—
299	C₂²	#63564	1	1	yes	characteristic
300	C₂²	#63567	1	1	yes	characteristic
301	C₂²	#63579	1	1	yes	characteristic
302	C₄	—	2	1	no	—
303	C₄	—	2	1	no	—
304	C₄	—	2	1	no	—
305	C₄	—	36	4	no	—
306	C₄	#63563	1	1	yes	characteristic
307	C₄	#63566	1	1	yes	characteristic
308	C₄	#63568	1	2	yes	—
309	C₄	#63573	1	1	yes	characteristic
310	C₄	#63575	1	1	yes	characteristic
311	C₃	—	2	1	no	—
312	C₃	#67715	1	1	yes	characteristic, complemented
313	C₃	#67746	1	1	yes	characteristic, complemented
314	C₂	—	2	1	no	—
315	C₂	—	2	1	no	—
316	C₂	#10589160	1	1	yes	characteristic
317	C₂	#10589164	1	1	yes	characteristic
318	C₂	#10589171	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	(C₂×D₃)⋊Dic₃
commutator	C₂×C₃²×C₄	C₄²
Frattini, hypercenter	C₂×C₄²	D₃²
Fitting, cosocle	C₂×C₃²×C₄²	C₂²
socle	C₂²×C₃²	#95
prime radical	C₂³×C₃²	C₂²⋊C₄
hypocenter	C₃²	#1303
LCS	G ▷ C₂×C₃²×C₄ ▷ C₂²×C₃² ▷ C₃²
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄²

Complements

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

N	H ≅ G/N	type
13. C₃×C₄⋊(C₃⋊C₈)	C₄	semidirect
14. C₃×C₄⋊(C₃⋊C₈)	C₄	semidirect
60. C₄⋊(C₃⋊C₈)	Dic₃	semidirect
61. C₄⋊(C₃⋊C₈)	Dic₃	semidirect
263. C₃²	#1303	semidirect
312. C₃	#67715	semidirect
313. C₃	#67746	semidirect

Maximal subgroups

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

subgroup	index
C₄×C₄⋊(C₃⋊Dic₃)	2
C₃×#4925	2
C₃×#4924	2
#67746	3
#67715	3