Subgroups — #64319

Order 288.

347 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Q₈×C₃⋊D₃	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃×#239	C₂	1	2	yes	maximal, complemented
3	#3802	C₂	1	3	yes	maximal, complemented
4	C₄×D₃²	C₂	1	3	yes	maximal, complemented
5	#3795	C₂	1	6	yes	maximal, complemented
6	D₃×#41	—	3	2	no	maximal
7	C₂×D₃²	C₂²	1	3	yes	—
8	C₃²×Q₈	C₂²	1	1	yes	characteristic, complemented
9	C₄×C₃⋊D₃	C₂²	1	3	yes	—
10	#634	C₂²	1	3	yes	complemented
11	C₃×D₁₂	C₂²	1	6	yes	complemented
12	C₃×C₄×D₃	C₂²	1	6	yes	complemented
13	#625	C₂²	1	6	yes	complemented
14	#624	C₂²	1	1	yes	characteristic
15	D₃×Dic₃	C₂²	1	6	yes	complemented
16	C₃×#41	—	3	2	no	—
17	#239	D₃	1	2	yes	complemented
18	D₃×Q₈	—	3	2	no	—
19	D₃×Q₈	—	6	1	no	—
20	#237	—	3	6	no	—
21	D₃×D₄	—	3	6	no	—
22	#235	—	3	6	no	—
23	C₂×C₄×D₃	—	3	6	no	—
24	C₂×C₃⋊D₃	C₂³	1	1	yes	characteristic
25	C₂×C₃×D₃	C₂³	1	6	yes	—
26	D₃²	—	2	6	no	—
27	C₃²×C₄	C₂³	1	3	yes	—
28	C₃⋊Dic₃	C₂³	1	3	yes	—
29	C₃×Dic₃	C₂³	1	2	yes	—
30	C₂×#41	—	9	1	no	Sylow
31	C₂²×D₃	—	3	6	no	—
32	C₃×Q₈	—	2	1	no	—
33	C₃×Q₈	C₂×D₃	1	2	yes	complemented
34	C₃×D₄	—	3	6	no	—
35	C₂×C₃×C₄	—	3	6	no	—
36	#67	—	3	12	no	—
37	C₂×Dic₃	—	3	6	no	—
38	D₁₂	C₂×D₃	1	6	yes	complemented
39	C₄×D₃	—	3	6	no	—
40	C₄×D₃	—	3	6	no	—
41	C₄×D₃	—	3	2	no	—
42	C₄×D₃	—	6	3	no	—
43	C₄×D₃	C₂×D₃	1	6	yes	complemented
44	Dic₆	—	3	6	no	—
45	Dic₆	—	6	3	no	—
46	C₂×C₃²	C₂⁴	1	1	yes	characteristic
47	C₃⋊D₃	#41	1	2	yes	complemented
48	C₃×D₃	—	2	6	no	—
49	#41	—	3	2	no	—
50	#41	—	9	6	no	—
51	C₂×Q₈	—	9	1	no	—
52	C₂×D₄	—	9	3	no	—
53	C₂²×C₄	—	9	3	no	—
54	C₂²×C₃	—	3	6	no	—
55	C₂×D₃	—	3	2	no	—
56	C₂×D₃	—	6	1	no	—
57	C₂×D₃	—	6	12	no	—
58	C₂×D₃	C₂²×D₃	1	6	yes	—
59	C₃×C₄	—	2	3	no	—
60	C₃×C₄	—	3	2	no	—
61	C₃×C₄	C₂²×D₃	1	6	yes	—
62	Dic₃	—	3	6	no	—
63	Dic₃	—	6	3	no	—
64	Dic₃	C₂²×D₃	1	2	yes	—
65	C₃²	C₂×#41	1	1	yes	characteristic, complemented, Sylow
66	C₂³	—	9	3	no	—
67	Q₈	—	9	3	no	—
68	Q₈	D₃²	1	1	yes	characteristic, complemented
69	D₄	—	3	6	no	—
70	D₄	—	9	6	no	—
71	C₂×C₄	—	3	6	no	—
72	C₂×C₄	—	9	1	no	—
73	C₂×C₄	—	9	6	no	—
74	C₂×C₄	—	9	3	no	—
75	C₂×C₃	—	2	1	no	—
76	C₂×C₃	—	6	6	no	—
77	C₂×C₃	C₂³×D₃	1	2	yes	—
78	D₃	—	2	6	no	—
79	D₃	—	3	4	no	—
80	D₃	—	6	2	no	—
81	C₂²	—	3	6	no	—
82	C₂²	—	9	1	no	—
83	C₂²	—	18	6	no	—
84	C₄	—	3	2	no	—
85	C₄	—	9	3	no	—
86	C₄	C₂×D₃²	1	3	yes	—
87	C₃	—	2	1	no	—
88	C₃	D₃×#41	1	2	yes	complemented
89	C₂	—	6	6	no	—
90	C₂	—	9	2	no	—
91	C₂	C₂²×D₃²	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂	C₂²×D₃²
commutator, socle, cosocle	C₂×C₃²	C₂⁴
Fitting	C₃²×Q₈	C₂²
hypercenter	Q₈	D₃²
hypocenter	C₃²	C₂×#41
LCS	G ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂ ◁ Q₈

Complements

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

N	H ≅ G/N	type
1. Q₈×C₃⋊D₃	C₂	semidirect
2. C₃×#239	C₂	semidirect
3. #3802	C₂	semidirect
4. C₄×D₃²	C₂	semidirect
5. #3795	C₂	semidirect
8. C₃²×Q₈	C₂²	semidirect
10. #634	C₂²	semidirect
11. C₃×D₁₂	C₂²	semidirect
12. C₃×C₄×D₃	C₂²	semidirect
13. #625	C₂²	semidirect
15. D₃×Dic₃	C₂²	semidirect
17. #239	D₃	semidirect
33. C₃×Q₈	C₂×D₃	semidirect
38. D₁₂	C₂×D₃	semidirect
43. C₄×D₃	C₂×D₃	semidirect
47. C₃⋊D₃	#41	semidirect
65. C₃²	C₂×#41	semidirect
68. Q₈	D₃²	semidirect
88. C₃	D₃×#41	semidirect

Maximal subgroups

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

subgroup	index
Q₈×C₃⋊D₃	2
C₃×#239	2
#3802	2
C₄×D₃²	2
#3795	2
D₃×#41	3