Subgroups — #73311

Order 384.

648 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#869	C₂	1	6	yes	maximal
2	C₃×C₄³	C₂	1	1	yes	maximal, characteristic
3	#4935	C₂	1	8	yes	maximal
4	#2868	—	3	1	no	maximal, Sylow
5	C₂²×Dic₆	C₂²	1	4	yes	—
6	C₂×C₃×C₄²	C₂²	1	3	yes	—
7	C₂×C₃×C₄²	C₂²	1	4	yes	—
8	C₂×C₄⋊Dic₃	C₂²	1	24	yes	—
9	#869	—	2	24	no	—
10	C₂×C₄⋊Q₈	—	3	6	no	—
11	#384	—	3	8	no	—
12	C₄³	D₃	1	1	yes	characteristic
13	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
14	C₂²×C₃×C₄	C₂³	1	6	yes	—
15	C₂²×Dic₃	C₂³	1	8	yes	—
16	C₂×Dic₆	—	2	24	no	—
17	C₃×C₄²	D₄	1	12	yes	—
18	C₃×C₄²	Q₈	1	16	yes	complemented
19	C₄⋊Dic₃	—	2	48	no	—
20	C₂²×Q₈	—	3	4	no	—
21	C₄⋊Q₈	—	6	24	no	—
22	C₂×C₄⋊C₄	—	3	24	no	—
23	C₂×C₄²	C₂×D₃	1	4	yes	—
24	C₂×C₄²	C₂×D₃	1	3	yes	—
25	C₂³×C₃	C₂⁴	1	1	yes	characteristic
26	C₂×C₃×C₄	C₂×D₄	1	6	yes	—
27	C₂×C₃×C₄	C₂×D₄	1	12	yes	—
28	C₂×C₃×C₄	C₂×Q₈	1	24	yes	—
29	C₂×Dic₃	—	2	24	no	—
30	Dic₆	—	4	16	no	—
31	C₂×Q₈	—	6	24	no	—
32	C₂²×C₄	—	3	8	no	—
33	C₂²×C₄	C₂²×D₃	1	6	yes	—
34	C₂²×C₄	C₂²×D₃	1	1	yes	characteristic
35	C₄⋊C₄	—	6	48	no	—
36	C₄²	Dic₆	1	16	yes	complemented
37	C₄²	D₁₂	1	12	yes	—
38	C₂²×C₃	C₂²×D₄	1	3	yes	—
39	C₂²×C₃	C₂²×Q₈	1	4	yes	—
40	C₃×C₄	#127	1	4	yes	—
41	C₃×C₄	C₄⋊Q₈	1	24	yes	complemented
42	Dic₃	—	4	8	no	—
43	C₂³	C₂³×D₃	1	1	yes	characteristic
44	Q₈	—	12	16	no	—
45	C₂×C₄	—	6	24	no	—
46	C₂×C₄	C₂×Dic₆	1	24	yes	—
47	C₂×C₄	C₂×D₁₂	1	12	yes	—
48	C₂×C₄	C₂×D₁₂	1	6	yes	—
49	C₂×C₃	C₂×#127	1	1	yes	characteristic
50	C₂×C₃	C₂×C₄⋊Q₈	1	6	yes	—
51	C₂²	C₂²×Dic₆	1	4	yes	—
52	C₂²	C₂²×D₁₂	1	3	yes	—
53	C₄	—	12	8	no	—
54	C₄	#869	1	24	yes	complemented
55	C₄	#874	1	4	yes	—
56	C₃	#2868	1	1	yes	characteristic, complemented, Sylow
57	C₂	C₂×#869	1	6	yes	—
58	C₂	C₂×#874	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₃
commutator, socle, prime radical, cosocle	C₂³×C₃	C₂⁴
Fitting	C₃×C₄³	C₂
hypercenter	C₄³	D₃
hypocenter	C₃	#2868
LCS	G ▷ C₂³×C₃ ▷ C₃
UCS	C₁ ◁ C₂³ ◁ C₄³

Complements

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

N	H ≅ G/N	type
18. C₃×C₄²	Q₈	semidirect
36. C₄²	Dic₆	semidirect
41. C₃×C₄	C₄⋊Q₈	semidirect
54. C₄	#869	semidirect
56. C₃	#2868	semidirect

Maximal subgroups

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

subgroup	index
C₂×#869	2
C₃×C₄³	2
#4935	2
#2868	3