Subgroups — #11772496

Order 1152.

337 conjugacy classes of subgroups.

The subgroup lattice has 276 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#63645	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×#819	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃×#812	C₂	1	1	yes	maximal, characteristic
4	#67817	—	3	1	no	maximal
5	#67732	—	3	1	no	maximal
6	C₂²×C₃²×C₈	C₂²	1	1	yes	characteristic
7	C₂×C₄⋊(C₃⋊Dic₃)	—	2	1	no	—
8	C₂×C₃×#208	—	2	1	no	—
9	#63645	—	2	1	no	—
10	#63645	C₄	1	2	yes	—
11	C₃×#819	—	2	1	no	—
12	C₃×#819	C₄	1	2	yes	complemented
13	C₃×#812	—	2	1	no	—
14	C₃×#812	C₄	1	2	yes	complemented
15	C₂×C₃×C₃⋊C₁₆	—	2	1	no	—
16	C₂×C₃×M₃₂	—	3	1	no	—
17	C₂×C₃×#108	—	3	1	no	—
18	C₂×#819	D₃	1	1	yes	characteristic
19	C₂×#818	—	3	1	no	—
20	C₂×#818	—	3	1	no	—
21	C₂×#818	—	6	1	no	—
22	C₂×#812	D₃	1	1	yes	characteristic
23	C₂²×C₃²×C₄	D₄	1	1	yes	characteristic
24	C₂²×C₃⋊Dic₃	—	4	1	no	—
25	C₂×C₃²×C₈	C₂×C₄	1	1	yes	characteristic
26	C₂×C₃²×C₈	C₂×C₄	1	1	yes	characteristic
27	C₂×C₃²×C₈	C₂×C₄	1	1	yes	characteristic
28	C₂×C₃²×C₈	D₄	1	1	yes	characteristic
29	C₂×C₃²×C₈	D₄	1	1	yes	characteristic
30	C₂×C₃²×C₈	Q₈	1	1	yes	characteristic
31	C₄⋊(C₃⋊Dic₃)	—	2	2	no	—
32	C₄⋊(C₃⋊Dic₃)	—	4	1	no	—
33	C₃×#208	—	2	2	no	—
34	C₃×#208	—	4	1	no	—
35	C₂×C₃×C₃⋊C₈	—	4	1	no	—
36	C₃×C₃⋊C₁₆	—	2	2	no	—
37	#1387	—	9	1	no	Sylow
38	C₂×C₃×M₁₆	—	6	1	no	—
39	C₂²×C₃×C₈	—	2	1	no	—
40	C₂²×C₃×C₈	C₂×D₃	1	1	yes	characteristic
41	C₂²×C₃×C₈	C₂×D₃	1	1	yes	characteristic
42	C₂×C₄⋊Dic₃	—	6	2	no	—
43	C₂×C₄⋊Dic₃	—	6	1	no	—
44	C₂×C₄⋊Dic₃	—	6	1	no	—
45	C₂×#208	—	2	1	no	—
46	C₃×M₃₂	—	3	2	no	—
47	C₃×M₃₂	—	6	1	no	—
48	C₂×C₃×C₁₆	—	6	1	no	—
49	C₃×#108	—	3	2	no	—
50	C₃×#108	—	6	1	no	—
51	#819	—	2	1	no	—
52	#819	Dic₃	1	2	yes	complemented
53	#818	—	3	2	no	—
54	#818	—	3	2	no	—
55	#818	—	6	2	no	—
56	#818	—	6	2	no	—
57	#818	—	6	1	no	—
58	#818	—	6	1	no	—
59	#812	—	2	1	no	—
60	#812	Dic₃	1	2	yes	complemented
61	C₂×C₃⋊C₁₆	—	2	1	no	—
62	C₂³×C₃²	SD₁₆	1	1	yes	characteristic
63	C₂×C₃²×C₄	C₂²⋊C₄	1	1	yes	characteristic
64	C₂×C₃²×C₄	C₂²⋊C₄	1	1	yes	characteristic
65	C₂×C₃²×C₄	C₄⋊C₄	1	1	yes	characteristic
66	C₂×C₃²×C₄	D₈	1	1	yes	characteristic
67	C₂×C₃²×C₄	SD₁₆	1	1	yes	characteristic
68	C₂×C₃²×C₄	Q₁₆	1	1	yes	characteristic
69	C₂×C₃⋊Dic₃	—	4	2	no	—
70	C₂×C₃⋊Dic₃	—	8	2	no	—
71	C₃²×C₈	C₄²	1	1	yes	characteristic
72	C₃²×C₈	C₂²⋊C₄	1	1	yes	characteristic
73	C₃²×C₈	C₄⋊C₄	1	1	yes	characteristic
74	C₃²×C₈	C₄⋊C₄	1	1	yes	characteristic
75	C₃×C₃⋊C₈	—	4	2	no	—
76	C₂×M₃₂	—	3	1	no	—
77	C₂×#108	—	3	1	no	—
78	C₂×#107	—	9	1	no	—
79	C₂²×C₃×C₄	—	2	1	no	—
80	C₂²×C₃×C₄	D₁₂	1	1	yes	characteristic
81	C₂²×C₃×C₄	#67	1	1	yes	characteristic
82	C₂²×Dic₃	—	12	2	no	—
83	C₂²×Dic₃	—	12	1	no	—
84	C₂²×Dic₃	—	12	1	no	—
85	C₃×M₁₆	—	6	2	no	—
86	C₃×M₁₆	—	12	1	no	—
87	C₂×C₃×C₈	—	2	1	no	—
88	C₂×C₃×C₈	—	2	1	no	—
89	C₂×C₃×C₈	—	2	1	no	—
90	C₂×C₃×C₈	—	2	1	no	—
91	C₂×C₃×C₈	—	2	1	no	—
92	C₂×C₃×C₈	—	2	1	no	—
93	C₂×C₃×C₈	—	12	1	no	—
94	C₂×C₃×C₈	Dic₆	1	1	yes	characteristic
95	C₂×C₃×C₈	Dic₆	1	1	yes	characteristic
96	C₂×C₃×C₈	C₄×D₃	1	1	yes	characteristic
97	C₂×C₃×C₈	C₄×D₃	1	1	yes	characteristic
98	C₂×C₃×C₈	C₄×D₃	1	1	yes	characteristic
99	C₂×C₃×C₈	C₄×D₃	1	1	yes	characteristic
100	C₂×C₃×C₈	D₁₂	1	1	yes	characteristic
101	C₂×C₃×C₈	C₂×Dic₃	1	1	yes	characteristic
102	C₂×C₃×C₈	C₂×Dic₃	1	1	yes	characteristic
103	C₂×C₃×C₈	#67	1	1	yes	characteristic
104	C₂×C₃×C₈	#67	1	1	yes	characteristic
105	C₂×C₃×C₈	#67	1	1	yes	characteristic
106	C₄⋊Dic₃	—	6	4	no	—
107	C₄⋊Dic₃	—	6	2	no	—
108	C₄⋊Dic₃	—	6	2	no	—
109	C₄⋊Dic₃	—	12	2	no	—
110	C₄⋊Dic₃	—	12	1	no	—
111	C₄⋊Dic₃	—	12	1	no	—
112	#208	—	2	2	no	—
113	#208	—	4	1	no	—
114	C₂×C₃⋊C₈	—	4	1	no	—
115	C₃×C₁₆	—	6	2	no	—
116	C₃⋊C₁₆	—	2	2	no	—
117	C₂²×C₃²	—	2	1	no	—
118	C₂²×C₃²	—	2	1	no	—
119	C₂²×C₃²	#102	1	1	yes	characteristic
120	C₂²×C₃²	#103	1	1	yes	characteristic
121	C₂²×C₃²	#106	1	1	yes	characteristic
122	C₃²×C₄	#95	1	1	yes	characteristic
123	C₃²×C₄	#102	1	1	yes	characteristic
124	C₃²×C₄	#103	1	1	yes	characteristic
125	C₃²×C₄	#107	1	1	yes	characteristic
126	C₃⋊Dic₃	—	8	2	no	—
127	C₂×M₁₆	—	6	1	no	—
128	C₂²×C₈	D₃²	1	1	yes	characteristic
129	C₂×C₄⋊C₄	—	18	1	no	—
130	M₃₂	—	3	2	no	—
131	M₃₂	—	6	1	no	—
132	C₂×C₁₆	—	6	1	no	—
133	#108	—	3	2	no	—
134	#108	—	6	1	no	—
135	#107	—	9	2	no	—
136	#107	—	18	1	no	—
137	C₂³×C₃	—	2	1	no	—
138	C₂³×C₃	#204	1	1	yes	characteristic
139	C₂³×C₃	#215	1	1	yes	characteristic
140	C₂×C₃×C₄	—	2	1	no	—
141	C₂×C₃×C₄	—	2	1	no	—
142	C₂×C₃×C₄	—	2	1	no	—
143	C₂×C₃×C₄	—	2	1	no	—
144	C₂×C₃×C₄	—	2	1	no	—
145	C₂×C₃×C₄	—	2	1	no	—
146	C₂×C₃×C₄	#204	1	1	yes	characteristic
147	C₂×C₃×C₄	D₂₄	1	1	yes	characteristic
148	C₂×C₃×C₄	Dic₁₂	1	1	yes	characteristic
149	C₂×C₃×C₄	Dic₃⋊C₄	1	1	yes	characteristic
150	C₂×C₃×C₄	C₄⋊Dic₃	1	1	yes	characteristic
151	C₂×C₃×C₄	(C₂×D₃)⋊C₄	1	1	yes	characteristic
152	C₂×C₃×C₄	(C₂×D₃)⋊C₄	1	1	yes	characteristic
153	C₂×C₃×C₄	(C₂×D₃)⋊C₄	1	1	yes	characteristic
154	C₂×C₃×C₄	#213	1	1	yes	characteristic
155	C₂×C₃×C₄	#214	1	1	yes	characteristic
156	C₂×C₃×C₄	#216	1	1	yes	characteristic
157	C₂×C₃×C₄	#217	1	1	yes	characteristic
158	C₂×Dic₃	—	12	4	no	—
159	C₂×Dic₃	—	12	2	no	—
160	C₂×Dic₃	—	12	2	no	—
161	C₂×Dic₃	—	24	4	no	—
162	C₂×Dic₃	—	24	2	no	—
163	C₂×Dic₃	—	24	2	no	—
164	C₃×C₈	—	2	1	no	—
165	C₃×C₈	—	2	1	no	—
166	C₃×C₈	—	2	1	no	—
167	C₃×C₈	—	2	1	no	—
168	C₃×C₈	—	12	2	no	—
169	C₃×C₈	C₄×Dic₃	1	1	yes	characteristic
170	C₃×C₈	C₄×Dic₃	1	1	yes	characteristic
171	C₃×C₈	Dic₃⋊C₄	1	1	yes	characteristic
172	C₃×C₈	Dic₃⋊C₄	1	1	yes	characteristic
173	C₃×C₈	Dic₃⋊C₄	1	1	yes	characteristic
174	C₃×C₈	C₄⋊Dic₃	1	1	yes	characteristic
175	C₃×C₈	(C₂×D₃)⋊C₄	1	1	yes	characteristic
176	C₃×C₈	#217	1	1	yes	characteristic
177	C₃⋊C₈	—	4	2	no	—
178	C₂×C₃²	—	2	1	no	—
179	C₂×C₃²	—	2	1	no	—
180	C₂×C₃²	#340	1	1	yes	characteristic
181	C₂×C₃²	#361	1	1	yes	characteristic
182	C₂×C₃²	#362	1	1	yes	characteristic
183	C₂²×C₄	—	36	1	no	—
184	C₂²×C₄	#626	1	1	yes	characteristic
185	M₁₆	—	6	2	no	—
186	M₁₆	—	12	1	no	—
187	C₂×C₈	—	12	1	no	—
188	C₂×C₈	D₃×Dic₃	1	1	yes	characteristic
189	C₂×C₈	D₃×Dic₃	1	1	yes	characteristic
190	C₂×C₈	#624	1	1	yes	characteristic
191	C₂×C₈	#625	1	1	yes	characteristic
192	C₂×C₈	#626	1	1	yes	characteristic
193	C₂×C₈	#627	1	1	yes	characteristic
194	C₄⋊C₄	—	18	2	no	—
195	C₄⋊C₄	—	36	1	no	—
196	C₁₆	—	6	2	no	—
197	C₂²×C₃	—	2	2	no	—
198	C₂²×C₃	—	2	2	no	—
199	C₂²×C₃	—	2	1	no	—
200	C₂²×C₃	—	2	1	no	—
201	C₂²×C₃	—	2	1	no	—
202	C₂²×C₃	—	2	1	no	—
203	C₂²×C₃	—	2	1	no	—
204	C₂²×C₃	—	2	1	no	—
205	C₂²×C₃	—	2	1	no	—
206	C₂²×C₃	#808	1	1	yes	characteristic
207	C₂²×C₃	#809	1	1	yes	characteristic
208	C₂²×C₃	#816	1	1	yes	characteristic
209	C₂²×C₃	#817	1	1	yes	characteristic
210	C₂²×C₃	#821	1	1	yes	characteristic
211	C₂²×C₃	#835	1	1	yes	characteristic
212	C₃×C₄	—	2	1	no	—
213	C₃×C₄	—	2	1	no	—
214	C₃×C₄	—	2	1	no	—
215	C₃×C₄	—	2	1	no	—
216	C₃×C₄	#807	1	1	yes	characteristic
217	C₃×C₄	#810	1	1	yes	characteristic
218	C₃×C₄	#816	1	1	yes	characteristic
219	C₃×C₄	#818	1	1	yes	characteristic
220	C₃×C₄	#821	1	1	yes	characteristic
221	C₃×C₄	C₃⋊#95	1	1	yes	characteristic
222	C₃×C₄	C₃⋊#95	1	1	yes	characteristic
223	C₃×C₄	#832	1	1	yes	characteristic
224	Dic₃	—	24	4	no	—
225	Dic₃	—	24	2	no	—
226	Dic₃	—	24	2	no	—
227	C₃²	#1387	1	1	yes	characteristic, complemented, Sylow
228	C₂³	#3717	1	1	yes	characteristic
229	C₂×C₄	—	36	2	no	—
230	C₂×C₄	—	72	2	no	—
231	C₂×C₄	#3714	1	1	yes	characteristic
232	C₂×C₄	#3716	1	1	yes	characteristic
233	C₂×C₄	#3719	1	1	yes	characteristic
234	C₂×C₄	(C₂×D₃)⋊Dic₃	1	1	yes	characteristic
235	C₂×C₄	#3722	1	1	yes	characteristic
236	C₂×C₄	Dic₃⋊Dic₃	1	1	yes	characteristic
237	C₈	—	12	2	no	—
238	C₈	Dic₃²	1	1	yes	characteristic
239	C₈	(C₂×D₃)⋊Dic₃	1	1	yes	characteristic
240	C₈	Dic₃⋊Dic₃	1	1	yes	characteristic
241	C₈	#3724	1	1	yes	characteristic
242	C₂×C₃	—	2	2	no	—
243	C₂×C₃	—	2	2	no	—
244	C₂×C₃	—	2	1	no	—
245	C₂×C₃	—	2	1	no	—
246	C₂×C₃	—	2	1	no	—
247	C₂×C₃	—	2	1	no	—
248	C₂×C₃	—	2	1	no	—
249	C₂×C₃	—	2	1	no	—
250	C₂×C₃	—	2	1	no	—
251	C₂×C₃	#4495	1	1	yes	characteristic
252	C₂×C₃	#4496	1	1	yes	characteristic
253	C₂×C₃	#4516	1	1	yes	characteristic
254	C₂×C₃	#4517	1	1	yes	characteristic
255	C₂×C₃	#4529	1	1	yes	characteristic
256	C₂×C₃	#4551	1	1	yes	characteristic
257	C₂²	—	2	1	no	—
258	C₂²	—	2	1	no	—
259	C₂²	#63564	1	1	yes	characteristic
260	C₂²	#63566	1	1	yes	characteristic
261	C₂²	#63573	1	1	yes	characteristic
262	C₄	—	72	2	no	—
263	C₄	#63563	1	1	yes	characteristic
264	C₄	#63567	1	1	yes	characteristic
265	C₄	#63575	1	1	yes	characteristic
266	C₄	#63579	1	1	yes	characteristic
267	C₃	—	2	1	no	—
268	C₃	#67732	1	1	yes	characteristic, complemented
269	C₃	#67817	1	1	yes	characteristic, complemented
270	C₂	—	2	1	no	—
271	C₂	—	2	1	no	—
272	C₂	#10589128	1	1	yes	characteristic
273	C₂	#10589131	1	1	yes	characteristic
274	C₂	#10589164	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#63564
commutator	C₃²×C₈	C₄²
Frattini, hypercenter	C₂²×C₈	D₃²
Fitting, cosocle	C₂²×C₃²×C₈	C₂²
socle	C₂²×C₃²	#102
prime radical	C₂³×C₃²	SD₁₆
hypocenter	C₃²	#1387
LCS	G ▷ C₃²×C₈ ▷ C₃²×C₄ ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈

Complements

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

N	H ≅ G/N	type
12. C₃×#819	C₄	semidirect
14. C₃×#812	C₄	semidirect
52. #819	Dic₃	semidirect
60. #812	Dic₃	semidirect
227. C₃²	#1387	semidirect
268. C₃	#67732	semidirect
269. C₃	#67817	semidirect

Maximal subgroups

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

subgroup	index
C₂×#63645	2
C₂×C₃×#819	2
C₂×C₃×#812	2
#67817	3
#67732	3