Subgroups — #10592930

Order 576.

606 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃²×Q₈	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#3752	C₂	1	3	yes	maximal
3	#63658	C₂	1	3	yes	maximal, complemented
4	#5231	—	3	4	no	maximal
5	C₂³×C₃⋊D₃	C₂²	1	1	yes	characteristic
6	C₂×C₃²×Q₈	—	2	6	no	—
7	C₂²×C₃²×C₄	C₂²	1	3	yes	—
8	C₂²×C₃⋊Dic₃	C₂²	1	3	yes	—
9	#3752	—	2	6	no	—
10	C₂²×C₃×Q₈	D₃	1	4	yes	complemented
11	C₂×(C₂×D₃)⋊C₄	—	3	12	no	—
12	C₃⋊#95	—	3	12	no	—
13	C₂³×C₃²	C₂³	1	1	yes	characteristic
14	C₂²×C₃⋊D₃	—	2	3	no	—
15	C₂²×C₃⋊D₃	—	2	3	no	—
16	C₂²×C₃⋊D₃	Q₈	1	2	yes	complemented
17	C₃²×Q₈	—	2	8	no	—
18	C₂×C₃²×C₄	—	2	6	no	—
19	C₂×C₃²×C₄	D₄	1	6	yes	—
20	C₂×C₃⋊Dic₃	—	2	3	no	—
21	C₂×C₃⋊Dic₃	—	2	6	no	—
22	#393	—	9	1	no	Sylow
23	C₂³×D₃	—	3	4	no	—
24	C₂×C₃×Q₈	—	2	24	no	—
25	C₂²×C₃×C₄	C₂×D₃	1	12	yes	—
26	C₂²×Dic₃	—	3	12	no	—
27	(C₂×D₃)⋊C₄	—	6	24	no	—
28	C₂²×C₃²	C₂×D₄	1	3	yes	—
29	C₂²×C₃²	C₂×Q₈	1	1	yes	characteristic
30	C₂²×C₃²	#41	1	3	yes	—
31	C₂×C₃⋊D₃	—	2	6	no	—
32	C₂×C₃⋊D₃	—	4	1	no	—
33	C₂×C₃⋊D₃	—	4	3	no	—
34	C₃²×C₄	—	2	6	no	—
35	C₃⋊Dic₃	—	4	3	no	—
36	C₂²×Q₈	C₃⋊D₃	1	1	yes	characteristic, complemented
37	C₂×C₂²⋊C₄	—	9	3	no	—
38	#95	—	9	3	no	—
39	C₂³×C₃	C₂²×D₃	1	4	yes	—
40	C₂²×D₃	—	3	8	no	—
41	C₂²×D₃	—	6	12	no	—
42	C₂²×D₃	—	6	12	no	—
43	C₃×Q₈	—	2	32	no	—
44	C₂×C₃×C₄	—	2	24	no	—
45	C₂×C₃×C₄	#67	1	24	yes	—
46	C₂×Dic₃	—	6	24	no	—
47	C₂×Dic₃	—	6	12	no	—
48	C₂×C₃²	#120	1	1	yes	characteristic
49	C₂×C₃²	C₂²⋊Q₈	1	3	yes	—
50	C₂×C₃²	#124	1	3	yes	—
51	C₃⋊D₃	—	4	2	no	—
52	C₂⁴	—	9	1	no	—
53	C₂×Q₈	—	2	6	no	—
54	C₂²×C₄	—	9	3	no	—
55	C₂²×C₄	C₂×C₃⋊D₃	1	3	yes	—
56	C₂²⋊C₄	—	18	6	no	—
57	C₂²×C₃	D₃×Q₈	1	4	yes	—
58	C₂²×C₃	#239	1	12	yes	—
59	C₂²×C₃	C₂×#67	1	12	yes	—
60	C₂×D₃	—	6	24	no	—
61	C₂×D₃	—	12	12	no	—
62	C₂×D₃	—	12	4	no	—
63	C₃×C₄	—	2	24	no	—
64	Dic₃	—	12	12	no	—
65	C₃²	#393	1	1	yes	characteristic, complemented, Sylow
66	C₂³	—	9	2	no	—
67	C₂³	—	18	3	no	—
68	C₂³	—	18	3	no	—
69	C₂³	C₂²×C₃⋊D₃	1	1	yes	characteristic
70	Q₈	—	2	8	no	—
71	C₂×C₄	—	2	6	no	—
72	C₂×C₄	—	18	6	no	—
73	C₂×C₄	—	18	3	no	—
74	C₂×C₄	#638	1	6	yes	—
75	C₂×C₃	#946	1	12	yes	—
76	C₂×C₃	#947	1	12	yes	—
77	C₂×C₃	#953	1	4	yes	—
78	D₃	—	12	8	no	—
79	C₂²	—	18	6	no	—
80	C₂²	—	36	3	no	—
81	C₂²	—	36	1	no	—
82	C₂²	Q₈×C₃⋊D₃	1	1	yes	characteristic
83	C₂²	#3832	1	3	yes	—
84	C₂²	C₂×#638	1	3	yes	—
85	C₄	—	2	6	no	—
86	C₄	—	36	3	no	—
87	C₃	#5231	1	4	yes	complemented
88	C₂	—	36	2	no	—
89	C₂	#64155	1	3	yes	—
90	C₂	#64156	1	3	yes	—
91	C₂	#64162	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂²×C₃⋊D₃
commutator, socle, cosocle	C₂³×C₃²	C₂³
Fitting	C₂²×C₃²×Q₈	C₂
prime radical	C₂³×C₃⋊D₃	C₂²
hypercenter	C₂²×Q₈	C₃⋊D₃
hypocenter	C₃²	#393
LCS	G ▷ C₂³×C₃² ▷ C₃²
UCS	C₁ ◁ C₂³ ◁ C₂²×Q₈

Complements

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

N	H ≅ G/N	type
1. C₂²×C₃²×Q₈	C₂	semidirect
3. #63658	C₂	semidirect
10. C₂²×C₃×Q₈	D₃	semidirect
16. C₂²×C₃⋊D₃	Q₈	semidirect
36. C₂²×Q₈	C₃⋊D₃	semidirect
65. C₃²	#393	semidirect
87. C₃	#5231	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₃²×Q₈	2
C₂×#3752	2
#63658	2
#5231	3