Subgroups — #11771167

Order 1152.

355 conjugacy classes of subgroups.

The subgroup lattice has 298 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#10592624	C₂	1	1	yes	maximal, characteristic
2	C₃×#4924	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃×C₄⋊(C₃⋊C₈)	C₂	1	1	yes	maximal, characteristic
4	#67714	—	3	1	no	maximal
5	#67707	—	3	1	no	maximal
6	C₂×C₃²×C₄²	C₂²	1	1	yes	characteristic
7	C₂×C₃×#208	—	2	1	no	—
8	C₂²×C₃×C₃⋊C₈	—	2	1	no	—
9	#63657	C₄	1	2	yes	—
10	C₄⋊(C₃⋊(C₃⋊C₈))	—	2	1	no	—
11	C₄×C₃⋊(C₃⋊C₈)	—	2	1	no	—
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)	C₄	1	2	yes	—
16	C₃×#423	—	3	1	no	—
17	C₂×C₃×C₄⋊C₈	—	3	1	no	—
18	#4925	—	3	1	no	—
19	#4925	—	3	1	no	—
20	#4925	—	6	1	no	—
21	#4924	D₃	1	1	yes	characteristic
22	C₂×C₄⋊(C₃⋊C₈)	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	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	C₃×#208	—	4	1	no	—
32	C₃×#208	—	4	1	no	—
33	C₂×C₃×C₃⋊C₈	—	2	2	no	—
34	C₂×C₃×C₃⋊C₈	—	2	2	no	—
35	C₂×C₃×C₃⋊C₈	—	4	1	no	—
36	C₂×C₃×C₃⋊C₈	—	4	1	no	—
37	#1301	—	9	1	no	Sylow
38	C₂×C₃×M₁₆	—	6	1	no	—
39	C₂²×C₃×C₈	—	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×#208	—	2	1	no	—
44	C₂²×C₃⋊C₈	—	2	1	no	—
45	C₃×C₄⋊C₈	—	3	2	no	—
46	C₃×C₄⋊C₈	—	3	2	no	—
47	C₃×C₄⋊C₈	—	6	1	no	—
48	C₃×C₄⋊C₈	—	6	1	no	—
49	#830	—	3	2	no	—
50	#830	—	3	2	no	—
51	#830	—	6	2	no	—
52	C₄⋊(C₃⋊C₈)	—	2	1	no	—
53	C₄⋊(C₃⋊C₈)	—	2	1	no	—
54	C₄⋊(C₃⋊C₈)	—	6	2	no	—
55	C₄⋊(C₃⋊C₈)	—	6	1	no	—
56	C₄⋊(C₃⋊C₈)	—	6	1	no	—
57	C₄⋊(C₃⋊C₈)	Dic₃	1	2	yes	—
58	C₄⋊(C₃⋊C₈)	Dic₃	1	2	yes	—
59	C₄×C₃⋊C₈	—	6	2	no	—
60	C₄×C₃⋊C₈	—	6	1	no	—
61	C₄×C₃⋊C₈	—	6	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	—	9	1	no	—
79	#423	—	3	1	no	—
80	C₂×C₄⋊C₈	—	3	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	C₄×D₃	1	1	yes	characteristic
88	C₂²×C₃×C₄	#67	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	C₂×C₃×C₈	—	12	1	no	—
95	C₂×C₃×C₈	—	12	1	no	—
96	C₃×C₄²	—	2	1	no	—
97	C₃×C₄²	—	2	1	no	—
98	C₃×C₄²	—	2	1	no	—
99	C₃×C₄²	—	2	1	no	—
100	C₃×C₄²	C₄×D₃	1	1	yes	characteristic
101	C₃×C₄²	C₄×D₃	1	1	yes	characteristic
102	C₃×C₄²	D₁₂	1	1	yes	characteristic
103	C₃×C₄²	D₁₂	1	1	yes	characteristic
104	C₃×C₄²	C₂×Dic₃	1	1	yes	characteristic
105	C₃×C₄²	C₂×Dic₃	1	1	yes	characteristic
106	C₃×C₄²	#67	1	1	yes	characteristic
107	C₃×C₄²	#67	1	1	yes	characteristic
108	#208	—	4	1	no	—
109	#208	—	4	1	no	—
110	C₂×C₃⋊C₈	—	2	2	no	—
111	C₂×C₃⋊C₈	—	2	2	no	—
112	C₂×C₃⋊C₈	—	4	1	no	—
113	C₂×C₃⋊C₈	—	4	1	no	—
114	C₂×C₃⋊C₈	—	6	4	no	—
115	C₂×C₃⋊C₈	—	6	2	no	—
116	C₂×C₃⋊C₈	—	6	2	no	—
117	C₂²×C₃²	—	2	1	no	—
118	C₂²×C₃²	—	2	1	no	—
119	C₂²×C₃²	#95	1	1	yes	characteristic
120	C₂²×C₃²	#106	1	1	yes	characteristic
121	C₂²×C₃²	#107	1	1	yes	characteristic
122	C₃²×C₄	—	2	1	no	—
123	C₃²×C₄	—	2	1	no	—
124	C₃²×C₄	—	2	1	no	—
125	C₃²×C₄	#102	1	1	yes	characteristic
126	C₃²×C₄	#102	1	1	yes	characteristic
127	C₃²×C₄	#103	1	1	yes	characteristic
128	C₃²×C₄	#103	1	1	yes	characteristic
129	C₃²×C₄	#108	1	2	yes	—
130	C₂×M₁₆	—	6	1	no	—
131	C₂²×C₈	—	6	1	no	—
132	C₂×C₄²	D₃²	1	1	yes	characteristic
133	C₄⋊C₈	—	3	2	no	—
134	C₄⋊C₈	—	3	2	no	—
135	C₄⋊C₈	—	6	1	no	—
136	C₄⋊C₈	—	6	1	no	—
137	C₄⋊C₈	—	18	1	no	—
138	C₂²⋊C₈	—	9	2	no	—
139	C₄×C₈	—	18	1	no	—
140	C₂³×C₃	—	2	1	no	—
141	C₂³×C₃	Dic₃⋊C₄	1	1	yes	characteristic
142	C₂³×C₃	Dic₃⋊C₄	1	1	yes	characteristic
143	C₂×C₃×C₄	—	2	2	no	—
144	C₂×C₃×C₄	—	2	2	no	—
145	C₂×C₃×C₄	—	2	1	no	—
146	C₂×C₃×C₄	—	2	1	no	—
147	C₂×C₃×C₄	—	2	1	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	—	4	2	no	—
156	C₂×C₃×C₄	—	4	1	no	—
157	C₂×C₃×C₄	—	4	1	no	—
158	C₂×C₃×C₄	—	4	2	no	—
159	C₂×C₃×C₄	C₄×Dic₃	1	1	yes	characteristic
160	C₂×C₃×C₄	C₄×Dic₃	1	1	yes	characteristic
161	C₂×C₃×C₄	Dic₃⋊C₄	1	1	yes	characteristic
162	C₂×C₃×C₄	Dic₃⋊C₄	1	1	yes	characteristic
163	C₂×C₃×C₄	C₄⋊Dic₃	1	1	yes	characteristic
164	C₂×C₃×C₄	C₄⋊Dic₃	1	1	yes	characteristic
165	C₂×C₃×C₄	(C₂×D₃)⋊C₄	1	1	yes	characteristic
166	C₂×C₃×C₄	(C₂×D₃)⋊C₄	1	1	yes	characteristic
167	C₂×C₃×C₄	(C₂×D₃)⋊C₄	1	1	yes	characteristic
168	C₂×C₃×C₄	(C₂×D₃)⋊C₄	1	1	yes	characteristic
169	C₂×C₃×C₄	#213	1	1	yes	characteristic
170	C₂×C₃×C₄	#213	1	1	yes	characteristic
171	C₂×C₃×C₄	#214	1	1	yes	characteristic
172	C₂×C₃×C₄	#214	1	1	yes	characteristic
173	C₂×C₃×C₄	#215	1	1	yes	characteristic
174	C₂×C₃×C₄	#215	1	1	yes	characteristic
175	C₂×C₃×C₄	#216	1	1	yes	characteristic
176	C₂×C₃×C₄	#216	1	1	yes	characteristic
177	C₂×C₃×C₄	#217	1	1	yes	characteristic
178	C₂×C₃×C₄	#217	1	1	yes	characteristic
179	C₃×C₈	—	12	2	no	—
180	C₃×C₈	—	12	2	no	—
181	C₃⋊C₈	—	4	2	no	—
182	C₃⋊C₈	—	4	2	no	—
183	C₃⋊C₈	—	12	4	no	—
184	C₃⋊C₈	—	12	2	no	—
185	C₃⋊C₈	—	12	2	no	—
186	C₂×C₃²	—	2	1	no	—
187	C₂×C₃²	—	2	1	no	—
188	C₂×C₃²	#340	1	1	yes	characteristic
189	C₂×C₃²	#341	1	1	yes	characteristic
190	C₂×C₃²	#344	1	1	yes	characteristic
191	C₂²×C₄	#624	1	1	yes	characteristic
192	C₂²×C₄	#625	1	1	yes	characteristic
193	C₂²×C₄	#627	1	1	yes	characteristic
194	M₁₆	—	12	1	no	—
195	M₁₆	—	12	1	no	—
196	C₂×C₈	—	6	2	no	—
197	C₂×C₈	—	6	2	no	—
198	C₂×C₈	—	12	1	no	—
199	C₂×C₈	—	12	1	no	—
200	C₂×C₈	—	18	2	no	—
201	C₄²	D₃×Dic₃	1	1	yes	characteristic
202	C₄²	D₃×Dic₃	1	1	yes	characteristic
203	C₄²	#626	1	1	yes	characteristic
204	C₄²	#626	1	1	yes	characteristic
205	C₂²×C₃	—	2	1	no	—
206	C₂²×C₃	—	2	1	no	—
207	C₂²×C₃	—	2	1	no	—
208	C₂²×C₃	—	2	1	no	—
209	C₂²×C₃	—	2	1	no	—
210	C₂²×C₃	—	2	1	no	—
211	C₂²×C₃	—	2	1	no	—
212	C₂²×C₃	—	4	1	no	—
213	C₂²×C₃	—	4	1	no	—
214	C₂²×C₃	#807	1	1	yes	characteristic
215	C₂²×C₃	#807	1	1	yes	characteristic
216	C₂²×C₃	#808	1	1	yes	characteristic
217	C₂²×C₃	#808	1	1	yes	characteristic
218	C₂²×C₃	C₃⋊#95	1	1	yes	characteristic
219	C₂²×C₃	C₃⋊#95	1	1	yes	characteristic
220	C₃×C₄	—	2	2	no	—
221	C₃×C₄	—	2	1	no	—
222	C₃×C₄	—	2	1	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	2	no	—
232	C₃×C₄	—	4	1	no	—
233	C₃×C₄	—	4	1	no	—
234	C₃×C₄	#809	1	1	yes	characteristic
235	C₃×C₄	#809	1	1	yes	characteristic
236	C₃×C₄	#810	1	1	yes	characteristic
237	C₃×C₄	#810	1	1	yes	characteristic
238	C₃×C₄	#819	1	2	yes	—
239	C₃×C₄	#822	1	2	yes	—
240	C₃×C₄	#832	1	1	yes	characteristic
241	C₃×C₄	#832	1	1	yes	characteristic
242	C₃×C₄	#835	1	1	yes	characteristic
243	C₃×C₄	#835	1	1	yes	characteristic
244	C₃²	#1301	1	1	yes	characteristic, complemented, Sylow
245	C₂³	#3724	1	1	yes	characteristic
246	C₂×C₄	—	2	2	no	—
247	C₂×C₄	—	4	1	no	—
248	C₂×C₄	#3713	1	1	yes	characteristic
249	C₂×C₄	#3715	1	1	yes	characteristic
250	C₂×C₄	#3715	1	1	yes	characteristic
251	C₂×C₄	#3718	1	1	yes	characteristic
252	C₂×C₄	Dic₃²	1	1	yes	characteristic
253	C₂×C₄	(C₂×D₃)⋊Dic₃	1	1	yes	characteristic
254	C₂×C₄	(C₂×D₃)⋊Dic₃	1	1	yes	characteristic
255	C₂×C₄	#3722	1	1	yes	characteristic
256	C₂×C₄	Dic₃⋊Dic₃	1	1	yes	characteristic
257	C₂×C₄	Dic₃⋊Dic₃	1	1	yes	characteristic
258	C₈	—	12	2	no	—
259	C₈	—	12	2	no	—
260	C₈	—	36	2	no	—
261	C₂×C₃	—	2	1	no	—
262	C₂×C₃	—	2	1	no	—
263	C₂×C₃	—	2	1	no	—
264	C₂×C₃	—	2	1	no	—
265	C₂×C₃	—	2	1	no	—
266	C₂×C₃	—	2	1	no	—
267	C₂×C₃	—	2	1	no	—
268	C₂×C₃	—	4	1	no	—
269	C₂×C₃	—	4	1	no	—
270	C₂×C₃	#4529	1	1	yes	characteristic
271	C₂×C₃	#4529	1	1	yes	characteristic
272	C₂×C₃	#4552	1	1	yes	characteristic
273	C₂×C₃	#4556	1	1	yes	characteristic
274	C₂×C₃	#4558	1	1	yes	characteristic
275	C₂×C₃	#4559	1	1	yes	characteristic
276	C₂²	—	2	1	no	—
277	C₂²	—	2	1	no	—
278	C₂²	#63574	1	1	yes	characteristic
279	C₂²	#63576	1	1	yes	characteristic
280	C₂²	#63579	1	1	yes	characteristic
281	C₄	—	2	1	no	—
282	C₄	—	2	1	no	—
283	C₄	—	2	1	no	—
284	C₄	#63562	1	1	yes	characteristic
285	C₄	#63562	1	1	yes	characteristic
286	C₄	#63565	1	1	yes	characteristic
287	C₄	#63565	1	1	yes	characteristic
288	C₄	#63577	1	2	yes	—
289	C₃	—	2	1	no	—
290	C₃	#67707	1	1	yes	characteristic, complemented
291	C₃	#67714	1	1	yes	characteristic, complemented
292	C₂	—	2	1	no	—
293	C₂	—	2	1	no	—
294	C₂	#10589163	1	1	yes	characteristic
295	C₂	#10589166	1	1	yes	characteristic
296	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	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
244. C₃²	#1301	semidirect
290. C₃	#67707	semidirect
291. C₃	#67714	semidirect

Maximal subgroups

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

subgroup	index
#10592624	2
C₃×#4924	2
C₂×C₃×C₄⋊(C₃⋊C₈)	2
#67714	3
#67707	3