Subgroups — #68060

Order 384.

212 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#4699	C₂	1	1	yes	maximal, characteristic, complemented
2	#4695	C₂	1	1	yes	maximal, characteristic
3	#4694	C₂	1	4	yes	maximal, complemented
4	C₃×C₈²	C₂	1	1	yes	maximal, characteristic, complemented
5	#1709	—	3	1	no	maximal, Sylow
6	C₂×Dic₁₂	—	2	2	no	—
7	C₂×D₂₄	—	2	2	no	—
8	#874	C₂²	1	2	yes	—
9	#869	C₂²	1	2	yes	—
10	C₃×C₄×C₈	C₂²	1	2	yes	—
11	C₃×C₄×C₈	C₂²	1	1	yes	characteristic
12	#821	—	2	8	no	—
13	#494	—	3	1	no	—
14	#493	—	3	1	no	—
15	#486	—	3	4	no	—
16	C₈²	D₃	1	1	yes	characteristic, complemented
17	C₂×D₁₂	—	2	4	no	—
18	C₂×D₁₂	—	2	2	no	—
19	C₂×Dic₆	—	2	2	no	—
20	C₂×C₃×C₈	D₄	1	4	yes	—
21	C₂×C₃×C₈	D₄	1	2	yes	—
22	C₃×C₄²	C₂³	1	1	yes	characteristic
23	C₄⋊Dic₃	—	2	4	no	—
24	Dic₁₂	—	4	4	no	—
25	D₂₄	—	4	4	no	—
26	C₂×Q₁₆	—	6	2	no	—
27	C₂×D₈	—	6	2	no	—
28	C₄⋊Q₈	—	3	2	no	—
29	#127	—	3	2	no	—
30	#102	—	6	8	no	—
31	C₄×C₈	C₂×D₃	1	2	yes	—
32	C₄×C₈	C₂×D₃	1	1	yes	characteristic
33	C₂²×D₃	—	4	2	no	—
34	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
35	C₂×C₃×C₄	C₂×D₄	1	2	yes	—
36	C₂×Dic₃	—	4	2	no	—
37	D₁₂	—	4	4	no	—
38	D₁₂	—	4	8	no	—
39	Dic₆	—	4	4	no	—
40	C₃×C₈	—	2	4	no	—
41	C₃×C₈	D₈	1	4	yes	—
42	C₂×Q₈	—	6	2	no	—
43	C₂×D₄	—	6	4	no	—
44	C₂×D₄	—	6	2	no	—
45	Q₁₆	—	12	4	no	—
46	D₈	—	12	4	no	—
47	C₂×C₈	D₁₂	1	4	yes	—
48	C₂×C₈	D₁₂	1	2	yes	—
49	C₄⋊C₄	—	6	4	no	—
50	C₄²	C₂²×D₃	1	1	yes	characteristic
51	C₂²×C₃	#127	1	1	yes	characteristic
52	C₂×D₃	—	8	2	no	—
53	C₂×D₃	—	8	4	no	—
54	C₃×C₄	C₂×D₈	1	2	yes	—
55	C₃×C₄	#135	1	4	yes	—
56	Dic₃	—	8	2	no	—
57	C₂³	—	12	2	no	—
58	Q₈	—	12	4	no	—
59	D₄	—	12	4	no	—
60	D₄	—	12	8	no	—
61	C₂×C₄	—	12	2	no	—
62	C₂×C₄	C₂×D₁₂	1	1	yes	characteristic
63	C₂×C₄	C₂×D₁₂	1	2	yes	—
64	C₈	—	2	4	no	—
65	C₈	D₂₄	1	4	yes	—
66	C₂×C₃	#493	1	1	yes	characteristic
67	C₂×C₃	#495	1	2	yes	—
68	D₃	—	16	2	no	—
69	C₂²	—	24	2	no	—
70	C₂²	—	24	4	no	—
71	C₂²	#874	1	1	yes	characteristic
72	C₄	—	24	2	no	—
73	C₄	C₂×D₂₄	1	2	yes	—
74	C₄	#904	1	4	yes	—
75	C₃	#1709	1	1	yes	characteristic, complemented, Sylow
76	C₂	—	48	2	no	—
77	C₂	#4695	1	1	yes	characteristic
78	C₂	#4696	1	2	yes	—

	subgroup	G/H
center	C₂²	#874
commutator, cosocle	C₃×C₄²	C₂³
Frattini	C₄²	C₂²×D₃
Fitting	C₃×C₈²	C₂
socle	C₂²×C₃	#127
prime radical	#4695	C₂
hypercenter	C₈²	D₃
hypocenter	C₃	#1709
LCS	G ▷ C₃×C₄² ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈²

Complements

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

N	H ≅ G/N	type
1. #4699	C₂	semidirect
3. #4694	C₂	semidirect
4. C₃×C₈²	C₂	semidirect
16. C₈²	D₃	semidirect
75. C₃	#1709	semidirect

Maximal subgroups

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

subgroup	index
#4699	2
#4695	2
#4694	2
C₃×C₈²	2
#1709	3