Finite Group Databasebeta

Subgroups — #11858604

Order 1152.

558 conjugacy classes of subgroups; only the 189 normal subgroups are shown.

normal subgroups only

This group has 558 conjugacy classes of subgroups — too many to draw. Only its 189 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	#10596650	C₂	1	2	yes	maximal
2	#10596647	C₂	1	1	yes	maximal, characteristic
3	#10593544	C₂	1	6	yes	maximal, complemented
4	#10593543	C₂	1	3	yes	maximal, complemented
5	C₂×#63746	C₂	1	3	yes	maximal, complemented
6	#41×C₃⋊D₃	C₂²	1	1	yes	characteristic
7	Q₈×#157	—	2	2	no	—
8	#64289	C₄	1	2	yes	complemented
9	D₄×#157	—	2	6	no	—
10	#64287	C₄	1	6	yes	complemented
11	#64286	C₂²	1	6	yes	—
12	C₂×C₄⋊#157	C₂²	1	6	yes	—
13	#64283	C₂²	1	3	yes	—
14	C₂×#3787	C₂²	1	3	yes	—
15	#63746	C₂²	1	3	yes	—
16	#63746	C₂²	1	6	yes	—
17	#63746	C₂²	1	1	yes	characteristic
18	#63745	C₂²	1	6	yes	—
19	C₂²×#157	—	2	6	no	—
20	C₃²×#41	Q₈	1	1	yes	characteristic
21	#3832	C₂×C₄	1	1	yes	characteristic
22	Q₈×C₃⋊D₃	D₄	1	1	yes	characteristic
23	#3830	C₂×C₄	1	3	yes	—
24	D₄×C₃⋊D₃	D₄	1	3	yes	—
25	#3828	Q₈	1	3	yes	—
26	C₂×C₄×C₃⋊D₃	C₂³	1	3	yes	—
27	C₂²⋊#157	—	2	6	no	—
28	#3792	C₂×C₄	1	6	yes	complemented
29	C₂×#622	C₂×C₄	1	6	yes	complemented
30	C₄⋊#157	—	2	6	no	—
31	C₄⋊#157	C₂³	1	2	yes	—
32	C₄⋊#157	C₂³	1	6	yes	—
33	C₄×#157	—	2	6	no	—
34	#3788	C₂×C₄	1	6	yes	complemented
35	#3787	C₂×C₄	1	6	yes	complemented
36	#3787	C₂³	1	3	yes	—
37	#3787	C₂³	1	1	yes	characteristic
38	#2913	—	9	1	no	maximal, Sylow
39	D₃×#41	—	12	1	no	—
40	C₂²×C₃⋊D₃	C₂×D₄	1	3	yes	—
41	C₂×#157	—	2	6	no	—
42	C₂×#157	—	2	2	no	—
43	C₂×#157	—	8	6	no	—
44	C₃²×Q₈	C₄⋊C₄	1	1	yes	characteristic
45	C₃²×D₄	C₄⋊C₄	1	3	yes	—
46	C₂×C₃²×C₄	C₂×Q₈	1	3	yes	—
47	#638	C₄⋊C₄	1	6	yes	—
48	C₂×C₃⋊Dic₃	C₂²×C₄	1	3	yes	—
49	C₄⋊(C₃⋊D₃)	C₄⋊C₄	1	3	yes	—
50	C₄×C₃⋊D₃	C₂²×C₄	1	3	yes	—
51	C₄×C₃⋊D₃	C₂×D₄	1	3	yes	—
52	C₄×C₃⋊D₃	C₂×Q₈	1	3	yes	—
53	C₄×C₃⋊D₃	C₂⁴	1	1	yes	characteristic
54	#634	C₄⋊C₄	1	3	yes	—
55	#622	C₂²×C₄	1	2	yes	—
56	#622	C₂²×C₄	1	6	yes	—
57	C₂×#131	—	9	1	no	—
58	#520	—	9	2	no	—
59	#428	—	9	6	no	—
60	#427	—	9	3	no	—
61	C₂×#107	—	9	3	no	—
62	C₃×#41	—	4	1	no	—
63	#239	—	12	1	no	—
64	D₃×Q₈	—	12	1	no	—
65	#237	—	12	3	no	—
66	D₃×D₄	—	12	3	no	—
67	#235	—	12	3	no	—
68	C₂×C₄×D₃	—	12	3	no	—
69	C₂²×C₃²	C₂×C₄⋊C₄	1	3	yes	—
70	C₂×C₃⋊D₃	—	2	3	no	—
71	C₂×C₃⋊D₃	—	2	3	no	—
72	C₂×C₃⋊D₃	C₂×C₄⋊C₄	1	3	yes	—
73	C₂×C₃⋊D₃	C₂²×D₄	1	1	yes	characteristic
74	#157	—	4	6	no	—
75	#157	—	4	2	no	—
76	C₃²×C₄	C₂×C₄⋊C₄	1	3	yes	—
77	C₃²×C₄	C₂²×Q₈	1	1	yes	characteristic
78	C₃⋊Dic₃	C₂×C₄⋊C₄	1	3	yes	—
79	C₃⋊Dic₃	C₂³×C₄	1	1	yes	characteristic
80	C₂×#41	—	9	1	no	—
81	#131	—	9	6	no	—
82	#131	—	9	2	no	—
83	C₂×M₁₆	—	9	3	no	—
84	C₂²×C₈	—	9	3	no	—
85	C₄×Q₈	—	18	2	no	—
86	C₄×D₄	—	18	6	no	—
87	#117	—	9	6	no	—
88	C₂×C₄⋊C₄	—	9	6	no	—
89	#107	—	9	3	no	—
90	#107	—	9	6	no	—
91	#107	—	9	1	no	—
92	#106	—	9	6	no	—
93	C₂²×D₃	—	12	3	no	—
94	C₃×Q₈	—	4	1	no	—
95	C₃×D₄	—	4	3	no	—
96	C₂×C₃×C₄	—	4	3	no	—
97	#67	—	12	6	no	—
98	C₂×Dic₃	—	12	3	no	—
99	D₁₂	—	12	3	no	—
100	C₄×D₃	—	12	3	no	—
101	C₄×D₃	—	12	3	no	—
102	C₄×D₃	—	12	3	no	—
103	C₄×D₃	—	12	1	no	—
104	Dic₆	—	12	3	no	—
105	C₂×C₃²	—	2	3	no	—
106	C₂×C₃²	C₂²×C₄⋊C₄	1	1	yes	characteristic
107	C₃⋊D₃	—	2	3	no	—
108	C₃⋊D₃	#576	1	1	yes	characteristic
109	C₃⋊D₃	#578	1	1	yes	characteristic
110	#41	—	9	1	no	—
111	#41	—	9	3	no	—
112	#41	—	9	3	no	—
113	#41	PSU(3,𝔽₂)	1	1	yes	characteristic
114	C₂×Q₈	—	9	1	no	—
115	C₂×D₄	—	9	3	no	—
116	C₂²×C₄	—	9	3	no	—
117	C₂²×C₄	—	18	6	no	—
118	M₁₆	—	9	6	no	—
119	M₁₆	—	9	6	no	—
120	C₂×C₈	—	9	3	no	—
121	C₂×C₈	—	9	1	no	—
122	C₂×C₈	—	9	6	no	—
123	C₂×C₈	—	9	6	no	—
124	C₄⋊C₄	—	9	2	no	—
125	C₄⋊C₄	—	9	6	no	—
126	C₄⋊C₄	—	18	6	no	—
127	C₂²⋊C₄	—	18	6	no	—
128	C₄²	—	18	6	no	—
129	C₂²×C₃	—	4	3	no	—
130	C₂×D₃	—	12	1	no	—
131	C₂×D₃	—	12	3	no	—
132	C₂×D₃	—	24	3	no	—
133	C₂×D₃	—	24	3	no	—
134	C₃×C₄	—	4	1	no	—
135	C₃×C₄	—	4	3	no	—
136	Dic₃	—	12	3	no	—
137	Dic₃	—	12	1	no	—
138	C₃²	#2913	1	1	yes	characteristic, complemented, Sylow
139	C₂³	—	9	3	no	—
140	Q₈	—	9	3	no	—
141	Q₈	#3777	1	1	yes	characteristic
142	D₄	—	9	6	no	—
143	D₄	—	9	3	no	—
144	D₄	#3777	1	3	yes	—
145	C₂×C₄	—	9	3	no	—
146	C₂×C₄	—	9	1	no	—
147	C₂×C₄	—	9	3	no	—
148	C₂×C₄	—	9	3	no	—
149	C₂×C₄	—	9	3	no	—
150	C₂×C₄	—	18	2	no	—
151	C₂×C₄	—	18	6	no	—
152	C₂×C₄	—	72	6	no	—
153	C₂×C₄	C₂×PSU(3,𝔽₂)	1	3	yes	—
154	C₈	—	9	2	no	—
155	C₈	—	9	6	no	—
156	C₂×C₃	—	4	1	no	—
157	C₂×C₃	—	8	3	no	—
158	D₃	—	12	1	no	—
159	D₃	—	12	1	no	—
160	D₃	—	24	3	no	—
161	C₂²	—	9	1	no	—
162	C₂²	—	9	3	no	—
163	C₂²	—	18	3	no	—
164	C₂²	—	18	3	no	—
165	C₂²	C₂×#3777	1	3	yes	—
166	C₄	—	9	1	no	—
167	C₄	—	9	3	no	—
168	C₄	—	36	6	no	—
169	C₄	—	36	2	no	—
170	C₄	C₂×#3777	1	3	yes	—
171	C₄	C₂²×PSU(3,𝔽₂)	1	1	yes	characteristic
172	C₃	—	4	1	no	—
173	C₂	—	2	3	no	—
174	C₂	—	9	1	no	—
175	C₂	—	9	1	no	—
176	C₂	—	18	3	no	—
177	C₂	C₂²×#3777	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂²×#3777
commutator	C₃⋊Dic₃	C₂³×C₄
Frattini	C₄	C₂²×PSU(3,𝔽₂)
Fitting	C₃²×#41	Q₈
socle	C₂×C₃²	C₂²×C₄⋊C₄
prime radical	#41×C₃⋊D₃	C₂²
cosocle	C₄×C₃⋊D₃	C₂⁴
hypercenter	#41	PSU(3,𝔽₂)
hypocenter	C₃²	#2913
LCS	G ▷ C₃⋊Dic₃ ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂ ◁ #41

Complements

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

N	H ≅ G/N	type
3. #10593544	C₂	semidirect
4. #10593543	C₂	semidirect
5. C₂×#63746	C₂	semidirect
8. #64289	C₄	semidirect
10. #64287	C₄	semidirect
28. #3792	C₂×C₄	semidirect
29. C₂×#622	C₂×C₄	semidirect
34. #3788	C₂×C₄	semidirect
35. #3787	C₂×C₄	semidirect
138. C₃²	#2913	semidirect

Maximal subgroups

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

subgroup	index
#10596650	2
#10596647	2
#10593544	2
#10593543	2
C₂×#63746	2
#2913	9