Subgroups — #64367

Order 288.

438 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×#41	C₂	1	1	yes	maximal, characteristic, complemented
2	Q₈×C₃⋊D₃	C₂	1	2	yes	maximal, complemented
3	#3830	C₂	1	6	yes	maximal, complemented
4	#3828	C₂	1	3	yes	maximal, complemented
5	C₂×#634	C₂	1	3	yes	maximal, complemented
6	#1010	—	3	4	no	maximal
7	C₃²×Q₈	C₂²	1	1	yes	characteristic
8	C₃²×D₄	C₂²	1	3	yes	—
9	C₂×C₃²×C₄	C₂²	1	3	yes	—
10	#638	C₂²	1	6	yes	—
11	C₂×C₃⋊Dic₃	C₂²	1	6	yes	—
12	C₄⋊(C₃⋊D₃)	C₂²	1	1	yes	characteristic
13	C₄×C₃⋊D₃	C₂²	1	6	yes	—
14	#634	C₂²	1	3	yes	—
15	#634	C₂²	1	6	yes	—
16	C₃×#41	D₃	1	4	yes	complemented
17	D₃×Q₈	—	3	8	no	—
18	#237	—	3	24	no	—
19	#235	—	3	12	no	—
20	C₂×Dic₆	—	3	12	no	—
21	C₂²×C₃²	C₂³	1	3	yes	—
22	C₂×C₃⋊D₃	C₂³	1	2	yes	—
23	C₃²×C₄	C₂³	1	1	yes	characteristic
24	C₃²×C₄	C₂³	1	3	yes	—
25	C₃⋊Dic₃	C₂³	1	6	yes	—
26	2₋¹⁺⁴	—	9	1	no	Sylow
27	C₃×Q₈	C₂×D₃	1	4	yes	—
28	C₃×D₄	C₂×D₃	1	12	yes	—
29	C₂×C₃×C₄	C₂×D₃	1	12	yes	—
30	#67	—	3	24	no	—
31	C₂×Dic₃	—	3	24	no	—
32	D₁₂	—	3	4	no	—
33	C₄×D₃	—	3	24	no	—
34	Dic₆	—	3	12	no	—
35	Dic₆	—	3	24	no	—
36	C₂×C₃²	—	2	3	no	—
37	C₂×C₃²	C₂⁴	1	1	yes	characteristic
38	C₃⋊D₃	—	2	2	no	—
39	#41	—	9	6	no	—
40	#41	—	9	3	no	—
41	#41	C₃⋊D₃	1	1	yes	characteristic, complemented
42	C₂×Q₈	—	9	2	no	—
43	C₂×Q₈	—	9	3	no	—
44	C₂²×C₃	C₂²×D₃	1	12	yes	—
45	C₂×D₃	—	3	8	no	—
46	C₃×C₄	C₂²×D₃	1	4	yes	—
47	C₃×C₄	C₂²×D₃	1	12	yes	—
48	Dic₃	—	3	24	no	—
49	C₃²	2₋¹⁺⁴	1	1	yes	characteristic, complemented, Sylow
50	Q₈	—	9	6	no	—
51	Q₈	—	9	3	no	—
52	Q₈	C₂×C₃⋊D₃	1	1	yes	characteristic
53	D₄	—	9	6	no	—
54	D₄	—	9	1	no	—
55	D₄	C₂×C₃⋊D₃	1	3	yes	—
56	C₂×C₄	—	9	6	no	—
57	C₂×C₄	—	9	6	no	—
58	C₂×C₄	C₂×C₃⋊D₃	1	3	yes	—
59	C₂×C₃	—	2	12	no	—
60	C₂×C₃	C₂³×D₃	1	4	yes	—
61	D₃	—	6	8	no	—
62	C₂²	—	9	2	no	—
63	C₂²	C₂²×C₃⋊D₃	1	3	yes	—
64	C₄	—	9	6	no	—
65	C₄	C₂²×C₃⋊D₃	1	3	yes	—
66	C₄	C₂²×C₃⋊D₃	1	1	yes	characteristic
67	C₃	#1010	1	4	yes	complemented
68	C₂	—	2	3	no	—
69	C₂	—	18	2	no	—
70	C₂	C₂³×C₃⋊D₃	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂	C₂³×C₃⋊D₃
commutator, socle, cosocle	C₂×C₃²	C₂⁴
Fitting	C₃²×#41	C₂
hypercenter	#41	C₃⋊D₃
hypocenter	C₃²	2₋¹⁺⁴
LCS	G ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂ ◁ #41

Complements

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

N	H ≅ G/N	type
1. C₃²×#41	C₂	semidirect
2. Q₈×C₃⋊D₃	C₂	semidirect
3. #3830	C₂	semidirect
4. #3828	C₂	semidirect
5. C₂×#634	C₂	semidirect
16. C₃×#41	D₃	semidirect
41. #41	C₃⋊D₃	semidirect
49. C₃²	2₋¹⁺⁴	semidirect
67. C₃	#1010	semidirect

Maximal subgroups

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

subgroup	index
C₃²×#41	2
Q₈×C₃⋊D₃	2
#3830	2
#3828	2
C₂×#634	2
#1010	3