Subgroups — #11771168

Order 1152.

365 conjugacy classes of subgroups.

The subgroup lattice has 300 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄⋊(C₃⋊(C₃⋊C₈))	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	#67745	—	3	1	no	maximal
5	#67714	—	3	1	no	maximal
6	C₂×C₃²×C₄²	C₂²	1	1	yes	characteristic
7	C₂²×C₃⋊(C₃⋊C₈)	—	2	1	no	—
8	C₂×C₃×#208	—	2	1	no	—
9	C₄⋊(C₃⋊(C₃⋊C₈))	—	2	1	no	—
10	C₄⋊(C₃⋊(C₃⋊C₈))	C₄	1	2	yes	—
11	C₃×#830	C₄	1	2	yes	—
12	C₃×C₄⋊(C₃⋊C₈)	—	2	1	no	—
13	C₃×C₄⋊(C₃⋊C₈)	—	2	1	no	—
14	C₃×C₄⋊(C₃⋊C₈)	C₄	1	2	yes	—
15	C₃×C₄×C₃⋊C₈	—	2	1	no	—
16	C₃×#431	—	3	1	no	—
17	C₃×#423	—	3	1	no	—
18	#4925	D₃	1	1	yes	characteristic
19	#4924	D₃	1	1	yes	characteristic
20	C₂×C₄⋊(C₃⋊C₈)	—	3	1	no	—
21	C₂×C₄⋊(C₃⋊C₈)	—	3	1	no	—
22	C₂×C₄⋊(C₃⋊C₈)	—	6	1	no	—
23	C₂²×C₃²×C₄	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₃²×C₄	D₄	1	1	yes	characteristic
25	C₂²×C₃²×C₄	Q₈	1	1	yes	characteristic
26	C₃²×C₄²	C₂×C₄	1	1	yes	characteristic
27	C₃²×C₄²	C₂×C₄	1	1	yes	characteristic
28	C₃²×C₄²	D₄	1	1	yes	characteristic
29	C₃²×C₄²	D₄	1	1	yes	characteristic
30	C₂×C₃⋊(C₃⋊C₈)	—	2	2	no	—
31	C₂×C₃⋊(C₃⋊C₈)	—	4	1	no	—
32	C₂×C₃⋊(C₃⋊C₈)	—	4	1	no	—
33	C₃×#208	—	4	1	no	—
34	C₃×#208	—	4	1	no	—
35	C₂×C₃×C₃⋊C₈	—	2	2	no	—
36	C₂×C₃×C₃⋊C₈	—	2	2	no	—
37	#1301	—	9	1	no	Sylow
38	C₂×C₃×M₁₆	—	6	1	no	—
39	C₂×C₃×C₄²	—	2	1	no	—
40	C₂×C₃×C₄²	C₂×D₃	1	1	yes	characteristic
41	C₂×C₃×C₄²	C₂×D₃	1	1	yes	characteristic
42	C₂×#208	—	2	1	no	—
43	C₂²×C₃⋊C₈	—	6	2	no	—
44	C₂²×C₃⋊C₈	—	6	1	no	—
45	C₂²×C₃⋊C₈	—	6	1	no	—
46	C₃×C₄⋊C₈	—	3	2	no	—
47	C₃×C₄⋊C₈	—	6	1	no	—
48	C₃×C₄⋊C₈	—	6	1	no	—
49	C₃×C₂²⋊C₈	—	3	2	no	—
50	C₃×C₄×C₈	—	6	1	no	—
51	#830	Dic₃	1	2	yes	—
52	C₄⋊(C₃⋊C₈)	—	2	1	no	—
53	C₄⋊(C₃⋊C₈)	—	2	1	no	—
54	C₄⋊(C₃⋊C₈)	—	3	2	no	—
55	C₄⋊(C₃⋊C₈)	—	3	2	no	—
56	C₄⋊(C₃⋊C₈)	—	6	2	no	—
57	C₄⋊(C₃⋊C₈)	—	6	2	no	—
58	C₄⋊(C₃⋊C₈)	—	6	1	no	—
59	C₄⋊(C₃⋊C₈)	—	6	1	no	—
60	C₄⋊(C₃⋊C₈)	Dic₃	1	2	yes	—
61	C₄×C₃⋊C₈	—	2	1	no	—
62	C₂³×C₃²	C₄⋊C₄	1	1	yes	characteristic
63	C₂×C₃²×C₄	—	2	2	no	—
64	C₂×C₃²×C₄	—	4	1	no	—
65	C₂×C₃²×C₄	C₄²	1	1	yes	characteristic
66	C₂×C₃²×C₄	C₂²⋊C₄	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	D₈	1	1	yes	characteristic
72	C₂×C₃²×C₄	SD₁₆	1	1	yes	characteristic
73	C₂×C₃²×C₄	SD₁₆	1	1	yes	characteristic
74	C₂×C₃²×C₄	Q₁₆	1	1	yes	characteristic
75	C₃⋊(C₃⋊C₈)	—	4	2	no	—
76	C₃×C₃⋊C₈	—	4	2	no	—
77	C₃×C₃⋊C₈	—	4	2	no	—
78	#431	—	3	1	no	—
79	#423	—	3	1	no	—
80	C₂×C₄⋊C₈	—	9	1	no	—
81	C₂²×C₃×C₄	—	2	1	no	—
82	C₂²×C₃×C₄	—	2	1	no	—
83	C₂²×C₃×C₄	—	2	1	no	—
84	C₂²×C₃×C₄	Dic₆	1	1	yes	characteristic
85	C₂²×C₃×C₄	Dic₆	1	1	yes	characteristic
86	C₂²×C₃×C₄	C₄×D₃	1	1	yes	characteristic
87	C₂²×C₃×C₄	D₁₂	1	1	yes	characteristic
88	C₂²×C₃×C₄	C₂×Dic₃	1	1	yes	characteristic
89	C₂²×C₃×C₄	#67	1	1	yes	characteristic
90	C₃×M₁₆	—	12	1	no	—
91	C₃×M₁₆	—	12	1	no	—
92	C₂×C₃×C₈	—	6	2	no	—
93	C₂×C₃×C₈	—	6	2	no	—
94	C₃×C₄²	—	2	1	no	—
95	C₃×C₄²	—	2	1	no	—
96	C₃×C₄²	—	2	1	no	—
97	C₃×C₄²	—	2	1	no	—
98	C₃×C₄²	C₄×D₃	1	1	yes	characteristic
99	C₃×C₄²	C₄×D₃	1	1	yes	characteristic
100	C₃×C₄²	C₄×D₃	1	1	yes	characteristic
101	C₃×C₄²	D₁₂	1	1	yes	characteristic
102	C₃×C₄²	C₂×Dic₃	1	1	yes	characteristic
103	C₃×C₄²	#67	1	1	yes	characteristic
104	C₃×C₄²	#67	1	1	yes	characteristic
105	C₃×C₄²	#67	1	1	yes	characteristic
106	#208	—	4	1	no	—
107	#208	—	4	1	no	—
108	C₂×C₃⋊C₈	—	2	2	no	—
109	C₂×C₃⋊C₈	—	2	2	no	—
110	C₂×C₃⋊C₈	—	6	4	no	—
111	C₂×C₃⋊C₈	—	6	2	no	—
112	C₂×C₃⋊C₈	—	6	2	no	—
113	C₂×C₃⋊C₈	—	12	2	no	—
114	C₂×C₃⋊C₈	—	12	2	no	—
115	C₂×C₃⋊C₈	—	12	1	no	—
116	C₂×C₃⋊C₈	—	12	1	no	—
117	C₂×C₃⋊C₈	—	12	1	no	—
118	C₂×C₃⋊C₈	—	12	1	no	—
119	C₂²×C₃²	—	2	1	no	—
120	C₂²×C₃²	—	2	1	no	—
121	C₂²×C₃²	#95	1	1	yes	characteristic
122	C₂²×C₃²	#106	1	1	yes	characteristic
123	C₂²×C₃²	#107	1	1	yes	characteristic
124	C₃²×C₄	—	2	1	no	—
125	C₃²×C₄	—	2	1	no	—
126	C₃²×C₄	—	2	1	no	—
127	C₃²×C₄	#102	1	1	yes	characteristic
128	C₃²×C₄	#102	1	1	yes	characteristic
129	C₃²×C₄	#103	1	1	yes	characteristic
130	C₃²×C₄	#103	1	1	yes	characteristic
131	C₃²×C₄	#108	1	2	yes	—
132	C₂×M₁₆	—	6	1	no	—
133	C₂²×C₈	—	18	1	no	—
134	C₂×C₄²	D₃²	1	1	yes	characteristic
135	C₄⋊C₈	—	3	2	no	—
136	C₄⋊C₈	—	6	1	no	—
137	C₄⋊C₈	—	6	1	no	—
138	C₄⋊C₈	—	9	2	no	—
139	C₄⋊C₈	—	18	1	no	—
140	C₂²⋊C₈	—	3	2	no	—
141	C₄×C₈	—	6	1	no	—
142	C₂³×C₃	—	2	1	no	—
143	C₂³×C₃	Dic₃⋊C₄	1	1	yes	characteristic
144	C₂³×C₃	C₄⋊Dic₃	1	1	yes	characteristic
145	C₂×C₃×C₄	—	2	4	no	—
146	C₂×C₃×C₄	—	2	2	no	—
147	C₂×C₃×C₄	—	2	2	no	—
148	C₂×C₃×C₄	—	2	1	no	—
149	C₂×C₃×C₄	—	2	1	no	—
150	C₂×C₃×C₄	—	2	1	no	—
151	C₂×C₃×C₄	—	2	1	no	—
152	C₂×C₃×C₄	—	2	1	no	—
153	C₂×C₃×C₄	—	2	1	no	—
154	C₂×C₃×C₄	—	2	1	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	—	4	1	no	—
159	C₂×C₃×C₄	—	4	1	no	—
160	C₂×C₃×C₄	—	4	2	no	—
161	C₂×C₃×C₄	#204	1	1	yes	characteristic
162	C₂×C₃×C₄	#204	1	1	yes	characteristic
163	C₂×C₃×C₄	D₂₄	1	1	yes	characteristic
164	C₂×C₃×C₄	Dic₁₂	1	1	yes	characteristic
165	C₂×C₃×C₄	C₄×Dic₃	1	1	yes	characteristic
166	C₂×C₃×C₄	C₄×Dic₃	1	1	yes	characteristic
167	C₂×C₃×C₄	Dic₃⋊C₄	1	1	yes	characteristic
168	C₂×C₃×C₄	Dic₃⋊C₄	1	1	yes	characteristic
169	C₂×C₃×C₄	Dic₃⋊C₄	1	1	yes	characteristic
170	C₂×C₃×C₄	C₄⋊Dic₃	1	1	yes	characteristic
171	C₂×C₃×C₄	(C₂×D₃)⋊C₄	1	1	yes	characteristic
172	C₂×C₃×C₄	(C₂×D₃)⋊C₄	1	1	yes	characteristic
173	C₂×C₃×C₄	(C₂×D₃)⋊C₄	1	1	yes	characteristic
174	C₂×C₃×C₄	(C₂×D₃)⋊C₄	1	1	yes	characteristic
175	C₂×C₃×C₄	#213	1	1	yes	characteristic
176	C₂×C₃×C₄	#214	1	1	yes	characteristic
177	C₂×C₃×C₄	#215	1	1	yes	characteristic
178	C₂×C₃×C₄	#216	1	1	yes	characteristic
179	C₂×C₃×C₄	#217	1	1	yes	characteristic
180	C₂×C₃×C₄	#217	1	1	yes	characteristic
181	C₃×C₈	—	12	2	no	—
182	C₃×C₈	—	12	2	no	—
183	C₃⋊C₈	—	4	2	no	—
184	C₃⋊C₈	—	4	2	no	—
185	C₃⋊C₈	—	12	4	no	—
186	C₃⋊C₈	—	12	2	no	—
187	C₃⋊C₈	—	12	2	no	—
188	C₂×C₃²	—	2	1	no	—
189	C₂×C₃²	—	2	1	no	—
190	C₂×C₃²	#340	1	1	yes	characteristic
191	C₂×C₃²	#341	1	1	yes	characteristic
192	C₂×C₃²	#344	1	1	yes	characteristic
193	C₂²×C₄	D₃×Dic₃	1	1	yes	characteristic
194	C₂²×C₄	#626	1	1	yes	characteristic
195	C₂²×C₄	#627	1	1	yes	characteristic
196	M₁₆	—	12	1	no	—
197	M₁₆	—	12	1	no	—
198	C₂×C₈	—	6	2	no	—
199	C₂×C₈	—	6	2	no	—
200	C₂×C₈	—	18	2	no	—
201	C₂×C₈	—	36	1	no	—
202	C₂×C₈	—	36	1	no	—
203	C₄²	D₃×Dic₃	1	1	yes	characteristic
204	C₄²	#624	1	1	yes	characteristic
205	C₄²	#625	1	1	yes	characteristic
206	C₄²	#626	1	1	yes	characteristic
207	C₂²×C₃	—	2	2	no	—
208	C₂²×C₃	—	2	2	no	—
209	C₂²×C₃	—	2	1	no	—
210	C₂²×C₃	—	2	1	no	—
211	C₂²×C₃	—	2	1	no	—
212	C₂²×C₃	—	2	1	no	—
213	C₂²×C₃	—	2	1	no	—
214	C₂²×C₃	—	2	1	no	—
215	C₂²×C₃	—	2	1	no	—
216	C₂²×C₃	#807	1	1	yes	characteristic
217	C₂²×C₃	#808	1	1	yes	characteristic
218	C₂²×C₃	#817	1	1	yes	characteristic
219	C₂²×C₃	#818	1	1	yes	characteristic
220	C₂²×C₃	C₃⋊#95	1	1	yes	characteristic
221	C₂²×C₃	C₃⋊#95	1	1	yes	characteristic
222	C₃×C₄	—	2	2	no	—
223	C₃×C₄	—	2	1	no	—
224	C₃×C₄	—	2	1	no	—
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	—	2	1	no	—
230	C₃×C₄	—	2	1	no	—
231	C₃×C₄	—	2	1	no	—
232	C₃×C₄	—	2	1	no	—
233	C₃×C₄	—	2	2	no	—
234	C₃×C₄	—	4	1	no	—
235	C₃×C₄	—	4	1	no	—
236	C₃×C₄	#809	1	1	yes	characteristic
237	C₃×C₄	#810	1	1	yes	characteristic
238	C₃×C₄	#816	1	1	yes	characteristic
239	C₃×C₄	#816	1	1	yes	characteristic
240	C₃×C₄	#821	1	1	yes	characteristic
241	C₃×C₄	#821	1	1	yes	characteristic
242	C₃×C₄	#822	1	2	yes	—
243	C₃×C₄	#822	1	2	yes	—
244	C₃×C₄	#832	1	1	yes	characteristic
245	C₃×C₄	#835	1	1	yes	characteristic
246	C₃²	#1301	1	1	yes	characteristic, complemented, Sylow
247	C₂³	Dic₃⋊Dic₃	1	1	yes	characteristic
248	C₂×C₄	—	2	2	no	—
249	C₂×C₄	—	4	1	no	—
250	C₂×C₄	#3714	1	1	yes	characteristic
251	C₂×C₄	#3716	1	1	yes	characteristic
252	C₂×C₄	#3717	1	1	yes	characteristic
253	C₂×C₄	#3719	1	1	yes	characteristic
254	C₂×C₄	Dic₃²	1	1	yes	characteristic
255	C₂×C₄	(C₂×D₃)⋊Dic₃	1	1	yes	characteristic
256	C₂×C₄	(C₂×D₃)⋊Dic₃	1	1	yes	characteristic
257	C₂×C₄	#3722	1	1	yes	characteristic
258	C₂×C₄	Dic₃⋊Dic₃	1	1	yes	characteristic
259	C₂×C₄	#3724	1	1	yes	characteristic
260	C₈	—	12	2	no	—
261	C₈	—	12	2	no	—
262	C₈	—	36	2	no	—
263	C₂×C₃	—	2	2	no	—
264	C₂×C₃	—	2	2	no	—
265	C₂×C₃	—	2	1	no	—
266	C₂×C₃	—	2	1	no	—
267	C₂×C₃	—	2	1	no	—
268	C₂×C₃	—	2	1	no	—
269	C₂×C₃	—	2	1	no	—
270	C₂×C₃	—	2	1	no	—
271	C₂×C₃	—	2	1	no	—
272	C₂×C₃	#4529	1	1	yes	characteristic
273	C₂×C₃	#4533	1	1	yes	characteristic
274	C₂×C₃	#4551	1	1	yes	characteristic
275	C₂×C₃	#4556	1	1	yes	characteristic
276	C₂×C₃	#4558	1	1	yes	characteristic
277	C₂×C₃	#4558	1	1	yes	characteristic
278	C₂²	—	2	1	no	—
279	C₂²	—	2	1	no	—
280	C₂²	#63573	1	1	yes	characteristic
281	C₂²	#63575	1	1	yes	characteristic
282	C₂²	#63579	1	1	yes	characteristic
283	C₄	—	2	1	no	—
284	C₄	—	2	1	no	—
285	C₄	—	2	1	no	—
286	C₄	#63563	1	1	yes	characteristic
287	C₄	#63564	1	1	yes	characteristic
288	C₄	#63566	1	1	yes	characteristic
289	C₄	#63567	1	1	yes	characteristic
290	C₄	#63578	1	2	yes	—
291	C₃	—	2	1	no	—
292	C₃	#67714	1	1	yes	characteristic, complemented
293	C₃	#67745	1	1	yes	characteristic, complemented
294	C₂	—	2	1	no	—
295	C₂	—	2	1	no	—
296	C₂	#10589164	1	1	yes	characteristic
297	C₂	#10589165	1	1	yes	characteristic
298	C₂	#10589169	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	C₄⋊C₄
hypocenter	C₃²	#1301
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
246. C₃²	#1301	semidirect
292. C₃	#67714	semidirect
293. C₃	#67745	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄⋊(C₃⋊(C₃⋊C₈))	2
C₃×#4925	2
C₃×#4924	2
#67745	3
#67714	3