Subgroups — #10592792

Order 576.

294 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic
2	#64078	C₂	1	1	yes	maximal, characteristic
3	#63637	C₂	1	4	yes	maximal
4	C₄×C₃⋊(C₃⋊C₈)	C₂	1	1	yes	maximal, characteristic
5	#5093	—	3	4	no	maximal
6	C₂×C₃²×Q₈	—	2	1	no	—
7	C₂×#634	C₂²	1	2	yes	—
8	C₃²×C₄⋊C₄	—	2	1	no	—
9	C₃²×C₄⋊C₄	C₂²	1	2	yes	—
10	C₃²×C₄²	C₂²	1	1	yes	characteristic
11	C₄⋊(C₃⋊Dic₃)	—	2	1	no	—
12	C₄⋊(C₃⋊Dic₃)	—	2	1	no	—
13	C₂×C₃⋊(C₃⋊C₈)	C₂²	1	2	yes	—
14	C₃×C₄⋊Q₈	D₃	1	4	yes	—
15	#869	—	3	4	no	—
16	#810	—	3	16	no	—
17	C₄×C₃⋊C₈	—	3	4	no	—
18	C₃²×Q₈	—	4	2	no	—
19	C₂×C₃²×C₄	—	2	2	no	—
20	C₂×C₃²×C₄	D₄	1	1	yes	characteristic
21	C₂×C₃²×C₄	D₄	1	1	yes	characteristic
22	C₂×C₃²×C₄	C₂³	1	1	yes	characteristic
23	C₂×C₃⋊Dic₃	—	2	2	no	—
24	#634	—	2	4	no	—
25	C₃⋊(C₃⋊C₈)	—	2	2	no	—
26	#487	—	9	1	no	Sylow
27	C₂×C₃×Q₈	—	2	4	no	—
28	C₂×Dic₆	—	3	8	no	—
29	C₃×C₄⋊C₄	—	2	4	no	—
30	C₃×C₄⋊C₄	C₂×D₃	1	8	yes	—
31	C₃×C₄²	C₂×D₃	1	4	yes	—
32	C₄⋊Dic₃	—	6	4	no	—
33	C₄⋊Dic₃	—	6	4	no	—
34	C₂×C₃⋊C₈	—	3	8	no	—
35	C₂²×C₃²	C₂×D₄	1	1	yes	characteristic
36	C₃²×C₄	—	4	2	no	—
37	C₃²×C₄	SD₁₆	1	2	yes	—
38	C₃²×C₄	Q₁₆	1	2	yes	—
39	C₃²×C₄	#41	1	2	yes	—
40	C₃⋊Dic₃	—	4	2	no	—
41	C₄⋊Q₈	—	9	1	no	—
42	C₄⋊Q₈	C₃⋊D₃	1	1	yes	characteristic
43	#103	—	9	4	no	—
44	C₄×C₈	—	9	1	no	—
45	C₃×Q₈	—	4	8	no	—
46	C₂×C₃×C₄	—	2	8	no	—
47	C₂×C₃×C₄	#67	1	4	yes	—
48	C₂×C₃×C₄	#67	1	4	yes	—
49	C₂×C₃×C₄	C₂²×D₃	1	4	yes	—
50	C₂×Dic₃	—	6	8	no	—
51	Dic₆	—	6	16	no	—
52	C₃⋊C₈	—	6	8	no	—
53	C₂×C₃²	#124	1	1	yes	characteristic
54	C₂×C₃²	C₂×SD₁₆	1	1	yes	characteristic
55	C₂×C₃²	C₂×Q₁₆	1	1	yes	characteristic
56	C₂×Q₈	—	2	1	no	—
57	C₂×Q₈	—	9	2	no	—
58	C₂×C₈	—	9	2	no	—
59	C₄⋊C₄	—	2	1	no	—
60	C₄⋊C₄	—	18	1	no	—
61	C₄⋊C₄	—	18	1	no	—
62	C₄⋊C₄	C₂×C₃⋊D₃	1	2	yes	—
63	C₄²	C₂×C₃⋊D₃	1	1	yes	characteristic
64	C₂²×C₃	C₂×#67	1	4	yes	—
65	C₃×C₄	—	4	8	no	—
66	C₃×C₄	#214	1	8	yes	—
67	C₃×C₄	#216	1	8	yes	—
68	C₃×C₄	#239	1	8	yes	—
69	Dic₃	—	12	8	no	—
70	C₃²	#487	1	1	yes	characteristic, complemented, Sylow
71	Q₈	—	4	2	no	—
72	Q₈	—	18	4	no	—
73	C₂×C₄	—	2	2	no	—
74	C₂×C₄	—	18	2	no	—
75	C₂×C₄	#638	1	1	yes	characteristic
76	C₂×C₄	#638	1	1	yes	characteristic
77	C₂×C₄	C₂²×C₃⋊D₃	1	1	yes	characteristic
78	C₈	—	18	2	no	—
79	C₂×C₃	C₂×#214	1	4	yes	—
80	C₂×C₃	C₂×#216	1	4	yes	—
81	C₂×C₃	#947	1	4	yes	—
82	C₂²	C₂×#638	1	1	yes	characteristic
83	C₄	—	4	2	no	—
84	C₄	—	36	2	no	—
85	C₄	#3754	1	2	yes	—
86	C₄	#3756	1	2	yes	—
87	C₄	#3832	1	2	yes	—
88	C₃	#5093	1	4	yes	complemented
89	C₂	C₂×#3754	1	1	yes	characteristic
90	C₂	C₂×#3756	1	1	yes	characteristic
91	C₂	#64156	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#638
commutator, cosocle	C₂×C₃²×C₄	C₂³
Frattini	C₂×C₄	C₂²×C₃⋊D₃
Fitting	C₃²×C₄⋊Q₈	C₂
socle, prime radical	C₂²×C₃²	C₂×D₄
hypercenter	C₄⋊Q₈	C₃⋊D₃
hypocenter	C₃²	#487
LCS	G ▷ C₂×C₃²×C₄ ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
70. C₃²	#487	semidirect
88. C₃	#5093	semidirect

Maximal subgroups

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

subgroup	index
C₃²×C₄⋊Q₈	2
#64078	2
#63637	2
C₄×C₃⋊(C₃⋊C₈)	2
#5093	3