Subgroups — #64316

Order 288.

312 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#3832	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃×D₃×Q₈	C₂	1	2	yes	maximal, complemented
3	#3799	C₂	1	6	yes	maximal, complemented
4	#3798	C₂	1	3	yes	maximal, complemented
5	#3797	C₂	1	3	yes	maximal, complemented
6	#1007	—	3	2	no	maximal
7	C₃²×Q₈	C₂²	1	1	yes	characteristic
8	C₄⋊(C₃⋊D₃)	C₂²	1	3	yes	—
9	C₄×C₃⋊D₃	C₂²	1	3	yes	—
10	C₃×C₄×D₃	C₂²	1	6	yes	—
11	C₃×Dic₆	C₂²	1	6	yes	—
12	#627	C₂²	1	3	yes	—
13	#626	C₂²	1	6	yes	—
14	#625	C₂²	1	1	yes	characteristic
15	#624	C₂²	1	6	yes	—
16	C₂×C₃×Q₈	—	3	2	no	—
17	#239	—	3	2	no	—
18	#239	—	3	6	no	—
19	#239	—	6	1	no	—
20	D₃×Q₈	—	3	6	no	—
21	D₃×Q₈	D₃	1	2	yes	complemented
22	#235	—	3	6	no	—
23	#235	—	3	6	no	—
24	C₂×C₃⋊D₃	C₂³	1	3	yes	—
25	C₂×C₃×D₃	C₂³	1	2	yes	—
26	C₃²×C₄	C₂³	1	3	yes	—
27	C₃⋊Dic₃	C₂³	1	1	yes	characteristic
28	C₃×Dic₃	C₂³	1	6	yes	—
29	2₋¹⁺⁴	—	9	1	no	Sylow
30	C₃×Q₈	—	2	1	no	—
31	C₃×Q₈	—	3	6	no	—
32	C₃×Q₈	C₂×D₃	1	2	yes	—
33	C₂×C₃×C₄	—	3	6	no	—
34	#67	—	3	2	no	—
35	#67	—	3	6	no	—
36	D₁₂	—	3	6	no	—
37	D₁₂	—	3	6	no	—
38	D₁₂	—	6	3	no	—
39	C₄×D₃	—	3	6	no	—
40	C₄×D₃	—	3	12	no	—
41	C₄×D₃	—	6	3	no	—
42	C₄×D₃	C₂×D₃	1	6	yes	—
43	Dic₆	—	3	6	no	—
44	Dic₆	C₂×D₃	1	6	yes	—
45	C₂×C₃²	C₂⁴	1	1	yes	characteristic
46	C₃⋊D₃	—	2	3	no	—
47	C₃×D₃	—	2	2	no	—
48	#41	—	9	3	no	—
49	#41	—	9	6	no	—
50	#41	—	9	1	no	—
51	C₂×Q₈	—	3	2	no	—
52	C₂×Q₈	—	9	3	no	—
53	C₂²×C₃	—	3	2	no	—
54	C₂×D₃	—	3	6	no	—
55	C₂×D₃	—	6	3	no	—
56	C₂×D₃	C₂²×D₃	1	2	yes	—
57	C₃×C₄	—	2	3	no	—
58	C₃×C₄	—	3	6	no	—
59	C₃×C₄	C₂²×D₃	1	6	yes	—
60	Dic₃	—	3	2	no	—
61	Dic₃	—	6	1	no	—
62	Dic₃	C₂²×D₃	1	6	yes	—
63	C₃²	2₋¹⁺⁴	1	1	yes	characteristic, complemented, Sylow
64	Q₈	—	3	6	no	—
65	Q₈	—	9	3	no	—
66	Q₈	D₃²	1	1	yes	characteristic
67	D₄	—	9	6	no	—
68	D₄	—	9	1	no	—
69	D₄	—	9	3	no	—
70	C₂×C₄	—	3	6	no	—
71	C₂×C₄	—	9	6	no	—
72	C₂×C₄	—	9	3	no	—
73	C₂×C₃	—	2	1	no	—
74	C₂×C₃	—	6	2	no	—
75	C₂×C₃	C₂³×D₃	1	2	yes	—
76	D₃	—	2	2	no	—
77	D₃	—	6	6	no	—
78	D₃	—	12	3	no	—
79	C₂²	—	3	2	no	—
80	C₂²	—	9	3	no	—
81	C₄	—	3	6	no	—
82	C₄	—	9	1	no	—
83	C₄	C₂×D₃²	1	3	yes	—
84	C₃	—	2	1	no	—
85	C₃	#1007	1	2	yes	complemented
86	C₂	—	6	2	no	—
87	C₂	—	18	3	no	—
88	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	2₋¹⁺⁴
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. #3832	C₂	semidirect
2. C₃×D₃×Q₈	C₂	semidirect
3. #3799	C₂	semidirect
4. #3798	C₂	semidirect
5. #3797	C₂	semidirect
21. D₃×Q₈	D₃	semidirect
63. C₃²	2₋¹⁺⁴	semidirect
85. C₃	#1007	semidirect

Maximal subgroups

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

subgroup	index
#3832	2
C₃×D₃×Q₈	2
#3799	2
#3798	2
#3797	2
#1007	3