Finite Group Databasebeta

Subgroups — #11782774

Order 1152.

492 conjugacy classes of subgroups; only the 161 normal subgroups are shown.

normal subgroups only

This group has 492 conjugacy classes of subgroups — too many to draw. Only its 161 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#10592381	C₂	1	1	yes	maximal, characteristic
2	#10592380	C₂	1	1	yes	maximal, characteristic
3	#10592380	C₂	1	1	yes	maximal, characteristic
4	#10592380	C₂	1	1	yes	maximal, characteristic
5	#10592380	C₂	1	1	yes	maximal, characteristic
6	#10592379	C₂	1	1	yes	maximal, characteristic
7	C₃²×#322	C₂	1	1	yes	maximal, characteristic
8	#68096	—	3	4	no	maximal
9	#64078	C₂²	1	1	yes	characteristic
10	#64078	C₂²	1	1	yes	characteristic
11	#64078	C₂²	1	1	yes	characteristic
12	#64078	C₂²	1	1	yes	characteristic
13	C₃²×C₄×C₈	C₂²	1	1	yes	characteristic
14	C₃²×C₄×C₈	C₂²	1	1	yes	characteristic
15	C₃²×C₄×C₈	C₂²	1	1	yes	characteristic
16	#63645	—	2	2	no	—
17	#63644	—	2	2	no	—
18	#63643	—	2	2	no	—
19	#63643	—	2	2	no	—
20	#63643	—	2	2	no	—
21	#63643	—	2	2	no	—
22	#4682	—	3	4	no	—
23	#4681	—	3	4	no	—
24	#4681	—	3	4	no	—
25	#4681	—	3	4	no	—
26	#4681	—	3	4	no	—
27	#4680	—	3	4	no	—
28	C₃×#322	D₃	1	4	yes	—
29	C₂×#634	—	2	1	no	—
30	C₂×#634	—	2	1	no	—
31	C₂×#634	—	2	1	no	—
32	C₂×#634	—	2	1	no	—
33	C₂×C₃²×C₈	D₄	1	2	yes	—
34	C₂×C₃²×C₈	D₄	1	2	yes	—
35	C₂×C₃²×C₈	D₄	1	2	yes	—
36	C₃²×C₄²	C₂³	1	1	yes	characteristic
37	C₄⋊(C₃⋊Dic₃)	—	2	1	no	—
38	C₄⋊(C₃⋊Dic₃)	—	2	1	no	—
39	C₄⋊(C₃⋊Dic₃)	—	2	1	no	—
40	C₄⋊(C₃⋊Dic₃)	—	2	1	no	—
41	C₄⋊(C₃⋊Dic₃)	—	2	1	no	—
42	C₄⋊(C₃⋊Dic₃)	—	2	1	no	—
43	C₄⋊(C₃⋊Dic₃)	—	2	1	no	—
44	C₄⋊(C₃⋊Dic₃)	—	2	1	no	—
45	#1718	—	9	1	no	Sylow
46	#869	—	3	4	no	—
47	#869	—	3	4	no	—
48	#869	—	3	4	no	—
49	#869	—	3	4	no	—
50	C₃×C₄×C₈	C₂×D₃	1	4	yes	—
51	C₃×C₄×C₈	C₂×D₃	1	4	yes	—
52	C₃×C₄×C₈	C₂×D₃	1	4	yes	—
53	#818	—	6	8	no	—
54	#817	—	6	8	no	—
55	#816	—	6	8	no	—
56	#816	—	6	8	no	—
57	#816	—	6	8	no	—
58	#816	—	6	8	no	—
59	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
60	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
61	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
62	C₂×C₃⋊Dic₃	—	4	1	no	—
63	C₂×C₃⋊Dic₃	—	4	1	no	—
64	C₂×C₃⋊Dic₃	—	4	1	no	—
65	C₂×C₃⋊Dic₃	—	4	1	no	—
66	#634	—	4	2	no	—
67	#634	—	4	2	no	—
68	#634	—	4	2	no	—
69	#634	—	4	2	no	—
70	C₃²×C₈	—	2	2	no	—
71	C₃²×C₈	—	2	2	no	—
72	C₃²×C₈	—	2	2	no	—
73	#500	—	9	1	no	—
74	#498	—	9	1	no	—
75	#487	—	9	1	no	—
76	#487	—	9	1	no	—
77	#487	—	9	1	no	—
78	#487	—	9	1	no	—
79	#322	C₃⋊D₃	1	1	yes	characteristic
80	C₂×Dic₆	—	6	4	no	—
81	C₂×Dic₆	—	6	4	no	—
82	C₂×Dic₆	—	6	4	no	—
83	C₂×Dic₆	—	6	4	no	—
84	C₂×C₃×C₈	D₁₂	1	8	yes	—
85	C₂×C₃×C₈	D₁₂	1	8	yes	—
86	C₂×C₃×C₈	D₁₂	1	8	yes	—
87	C₃×C₄²	C₂²×D₃	1	4	yes	—
88	C₄⋊Dic₃	—	6	4	no	—
89	C₄⋊Dic₃	—	6	4	no	—
90	C₄⋊Dic₃	—	6	4	no	—
91	C₄⋊Dic₃	—	6	4	no	—
92	C₄⋊Dic₃	—	6	4	no	—
93	C₄⋊Dic₃	—	6	4	no	—
94	C₄⋊Dic₃	—	6	4	no	—
95	C₄⋊Dic₃	—	6	4	no	—
96	C₂²×C₃²	#127	1	1	yes	characteristic
97	C₃²×C₄	#135	1	2	yes	—
98	C₃²×C₄	#137	1	2	yes	—
99	C₃²×C₄	#137	1	2	yes	—
100	C₃⋊Dic₃	—	8	1	no	—
101	C₃⋊Dic₃	—	8	1	no	—
102	C₃⋊Dic₃	—	8	1	no	—
103	C₃⋊Dic₃	—	8	1	no	—
104	C₄⋊Q₈	—	9	1	no	—
105	C₄⋊Q₈	—	9	1	no	—
106	C₄⋊Q₈	—	9	1	no	—
107	C₄⋊Q₈	—	9	1	no	—
108	#107	—	18	2	no	—
109	#106	—	18	2	no	—
110	#103	—	18	2	no	—
111	#103	—	18	2	no	—
112	#103	—	18	2	no	—
113	#103	—	18	2	no	—
114	C₄×C₈	C₂×C₃⋊D₃	1	1	yes	characteristic
115	C₄×C₈	C₂×C₃⋊D₃	1	1	yes	characteristic
116	C₄×C₈	C₂×C₃⋊D₃	1	1	yes	characteristic
117	C₂×C₃×C₄	C₂×D₁₂	1	4	yes	—
118	C₂×C₃×C₄	C₂×D₁₂	1	4	yes	—
119	C₂×C₃×C₄	C₂×D₁₂	1	4	yes	—
120	C₂×Dic₃	—	12	4	no	—
121	C₂×Dic₃	—	12	4	no	—
122	C₂×Dic₃	—	12	4	no	—
123	C₂×Dic₃	—	12	4	no	—
124	Dic₆	—	12	8	no	—
125	Dic₆	—	12	8	no	—
126	Dic₆	—	12	8	no	—
127	Dic₆	—	12	8	no	—
128	C₃×C₈	—	2	8	no	—
129	C₃×C₈	—	2	8	no	—
130	C₃×C₈	—	2	8	no	—
131	C₂×C₃²	#495	1	1	yes	characteristic
132	C₂×C₃²	#497	1	1	yes	characteristic
133	C₂×C₃²	#497	1	1	yes	characteristic
134	C₂×Q₈	—	18	1	no	—
135	C₂×Q₈	—	18	1	no	—
136	C₂×Q₈	—	18	1	no	—
137	C₂×Q₈	—	18	1	no	—
138	C₂×C₈	C₄⋊(C₃⋊D₃)	1	2	yes	—
139	C₂×C₈	C₄⋊(C₃⋊D₃)	1	2	yes	—
140	C₂×C₈	C₄⋊(C₃⋊D₃)	1	2	yes	—
141	C₄⋊C₄	—	18	1	no	—
142	C₄⋊C₄	—	18	1	no	—
143	C₄⋊C₄	—	18	1	no	—
144	C₄⋊C₄	—	18	1	no	—
145	C₄⋊C₄	—	18	1	no	—
146	C₄⋊C₄	—	18	1	no	—
147	C₄⋊C₄	—	18	1	no	—
148	C₄⋊C₄	—	18	1	no	—
149	C₄²	C₂²×C₃⋊D₃	1	1	yes	characteristic
150	C₂²×C₃	#874	1	4	yes	—
151	C₃×C₄	#904	1	8	yes	—
152	C₃×C₄	#909	1	8	yes	—
153	C₃×C₄	#909	1	8	yes	—
154	Dic₃	—	24	4	no	—
155	Dic₃	—	24	4	no	—
156	Dic₃	—	24	4	no	—
157	Dic₃	—	24	4	no	—
158	C₃²	#1718	1	1	yes	characteristic, complemented, Sylow
159	Q₈	—	36	2	no	—
160	Q₈	—	36	2	no	—
161	Q₈	—	36	2	no	—
162	Q₈	—	36	2	no	—
163	C₂×C₄	—	36	1	no	—
164	C₂×C₄	—	36	1	no	—
165	C₂×C₄	—	36	1	no	—
166	C₂×C₄	—	36	1	no	—
167	C₂×C₄	C₂×C₄⋊(C₃⋊D₃)	1	1	yes	characteristic
168	C₂×C₄	C₂×C₄⋊(C₃⋊D₃)	1	1	yes	characteristic
169	C₂×C₄	C₂×C₄⋊(C₃⋊D₃)	1	1	yes	characteristic
170	C₈	—	2	2	no	—
171	C₈	—	2	2	no	—
172	C₈	—	2	2	no	—
173	C₂×C₃	#4696	1	4	yes	—
174	C₂×C₃	#4715	1	4	yes	—
175	C₂×C₃	#4715	1	4	yes	—
176	C₂²	#64083	1	1	yes	characteristic
177	C₄	—	72	1	no	—
178	C₄	—	72	1	no	—
179	C₄	—	72	1	no	—
180	C₄	—	72	1	no	—
181	C₄	#64113	1	2	yes	—
182	C₄	#64118	1	2	yes	—
183	C₄	#64118	1	2	yes	—
184	C₃	#68096	1	4	yes	complemented
185	C₂	#10592395	1	1	yes	characteristic
186	C₂	#10592414	1	1	yes	characteristic
187	C₂	#10592414	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#64083
commutator, cosocle	C₃²×C₄²	C₂³
Frattini	C₄²	C₂²×C₃⋊D₃
Fitting	C₃²×#322	C₂
socle, prime radical	C₂²×C₃²	#127
hypercenter	#322	C₃⋊D₃
hypocenter	C₃²	#1718
LCS	G ▷ C₃²×C₄² ▷ C₂²×C₃² ▷ C₃²
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #322

Complements

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

N	H ≅ G/N	type
158. C₃²	#1718	semidirect
184. C₃	#68096	semidirect

Maximal subgroups

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

subgroup	index
#10592381	2
#10592380	2
#10592379	2
C₃²×#322	2
#68096	3