Subgroups — #69066

Order 384.

176 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Dic₂₄	C₂	1	4	yes	maximal
2	#4699	C₂	1	2	yes	maximal
3	C₃×C₄×C₁₆	C₂	1	1	yes	maximal, characteristic
4	#2247	—	3	1	no	maximal, Sylow
5	C₂×Dic₁₂	—	2	2	no	—
6	C₂×Dic₁₂	C₂²	1	4	yes	—
7	#869	—	2	2	no	—
8	C₂×C₃×C₁₆	C₂²	1	2	yes	—
9	C₃×C₄×C₈	C₂²	1	1	yes	characteristic
10	Dic₂₄	—	2	8	no	—
11	C₂×Q₃₂	—	3	4	no	—
12	#494	—	3	2	no	—
13	C₄×C₁₆	D₃	1	1	yes	characteristic
14	C₂×Dic₆	—	2	4	no	—
15	C₂×C₃×C₈	D₄	1	1	yes	characteristic
16	C₂×C₃×C₈	C₂³	1	1	yes	characteristic
17	C₃×C₄²	D₄	1	1	yes	characteristic
18	C₄⋊Dic₃	—	4	4	no	—
19	Dic₁₂	—	2	8	no	—
20	Dic₁₂	—	4	4	no	—
21	C₃×C₁₆	D₄	1	4	yes	—
22	C₂×Q₁₆	—	3	4	no	—
23	C₂×Q₁₆	—	6	2	no	—
24	C₄⋊Q₈	—	6	2	no	—
25	Q₃₂	—	6	8	no	—
26	C₂×C₁₆	C₂×D₃	1	2	yes	—
27	C₄×C₈	C₂×D₃	1	1	yes	characteristic
28	C₂×C₃×C₄	D₈	1	2	yes	—
29	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
30	C₂×Dic₃	—	4	4	no	—
31	Dic₆	—	4	4	no	—
32	Dic₆	—	4	4	no	—
33	C₃×C₈	D₈	1	2	yes	—
34	C₃×C₈	C₂×D₄	1	2	yes	—
35	C₂×Q₈	—	6	4	no	—
36	Q₁₆	—	6	8	no	—
37	Q₁₆	—	12	4	no	—
38	C₂×C₈	D₁₂	1	1	yes	characteristic
39	C₂×C₈	C₂²×D₃	1	1	yes	characteristic
40	C₄⋊C₄	—	12	4	no	—
41	C₄²	D₁₂	1	1	yes	characteristic
42	C₁₆	D₁₂	1	4	yes	—
43	C₂²×C₃	C₂×D₈	1	1	yes	characteristic
44	C₃×C₄	Q₃₂	1	4	yes	complemented
45	C₃×C₄	#127	1	1	yes	characteristic
46	C₃×C₄	C₂×D₈	1	1	yes	characteristic
47	Dic₃	—	8	4	no	—
48	Q₈	—	12	4	no	—
49	Q₈	—	12	4	no	—
50	C₂×C₄	—	12	4	no	—
51	C₂×C₄	D₂₄	1	2	yes	—
52	C₂×C₄	C₂×D₁₂	1	1	yes	characteristic
53	C₈	D₂₄	1	2	yes	—
54	C₈	C₂×D₁₂	1	2	yes	—
55	C₂×C₃	#493	1	1	yes	characteristic
56	C₂×C₃	C₂×Q₃₂	1	2	yes	—
57	C₂²	C₂×D₂₄	1	1	yes	characteristic
58	C₄	—	24	4	no	—
59	C₄	Dic₂₄	1	4	yes	complemented
60	C₄	#874	1	1	yes	characteristic
61	C₄	C₂×D₂₄	1	1	yes	characteristic
62	C₃	#2247	1	1	yes	characteristic, complemented, Sylow
63	C₂	#4695	1	1	yes	characteristic
64	C₂	C₂×Dic₂₄	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×D₂₄
commutator, cosocle	C₂×C₃×C₈	C₂³
Frattini	C₂×C₈	C₂²×D₃
Fitting	C₃×C₄×C₁₆	C₂
socle, prime radical	C₂²×C₃	C₂×D₈
hypercenter	C₄×C₁₆	D₃
hypocenter	C₃	#2247
LCS	G ▷ C₂×C₃×C₈ ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×C₁₆

Complements

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

N	H ≅ G/N	type
44. C₃×C₄	Q₃₂	semidirect
59. C₄	Dic₂₄	semidirect
62. C₃	#2247	semidirect

Maximal subgroups

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

subgroup	index
C₂×Dic₂₄	2
#4699	2
C₃×C₄×C₁₆	2
#2247	3