Subgroups — #71297

Order 384.

212 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#494	C₂	1	1	yes	maximal, characteristic, complemented
2	#5088	C₂	1	4	yes	maximal, complemented
3	#4695	C₂	1	1	yes	maximal, characteristic
4	C₈×C₃⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
5	#1709	—	3	1	no	maximal, Sylow
6	C₂×C₃×Q₁₆	—	2	2	no	—
7	C₃×C₄⋊Q₈	C₂²	1	2	yes	—
8	C₂×D₂₄	—	2	2	no	—
9	#874	C₂²	1	2	yes	—
10	C₃×C₄×C₈	C₂²	1	1	yes	characteristic
11	#809	—	2	8	no	—
12	C₄×C₃⋊C₈	C₂²	1	2	yes	—
13	#494	D₃	1	1	yes	characteristic, complemented
14	#493	—	3	1	no	—
15	#486	—	3	4	no	—
16	C₈²	—	3	1	no	—
17	C₂×C₃×Q₈	—	2	2	no	—
18	C₂×D₁₂	—	2	4	no	—
19	C₂×D₁₂	—	2	2	no	—
20	C₃×Q₁₆	—	4	4	no	—
21	C₂×C₃×C₈	D₄	1	2	yes	—
22	C₃×C₄⋊C₄	—	2	4	no	—
23	C₃×C₄²	C₂³	1	1	yes	characteristic
24	C₂×C₃⋊C₈	D₄	1	4	yes	—
25	D₂₄	—	4	4	no	—
26	C₂×Q₁₆	—	2	2	no	—
27	C₂×D₈	—	6	2	no	—
28	C₄⋊Q₈	C₂×D₃	1	2	yes	—
29	#127	—	3	2	no	—
30	#102	—	6	8	no	—
31	C₄×C₈	—	3	2	no	—
32	C₄×C₈	C₂×D₃	1	1	yes	characteristic
33	C₂²×D₃	—	4	2	no	—
34	C₃×Q₈	—	4	4	no	—
35	C₂×C₃×C₄	—	4	2	no	—
36	C₂×C₃×C₄	C₂×D₄	1	2	yes	—
37	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
38	D₁₂	—	4	4	no	—
39	D₁₂	—	4	8	no	—
40	C₃×C₈	D₈	1	4	yes	—
41	C₃⋊C₈	—	2	4	no	—
42	C₂×Q₈	—	2	2	no	—
43	C₂×D₄	—	6	4	no	—
44	C₂×D₄	—	6	2	no	—
45	Q₁₆	—	4	4	no	—
46	D₈	—	12	4	no	—
47	C₂×C₈	—	3	4	no	—
48	C₂×C₈	#67	1	2	yes	—
49	C₄⋊C₄	—	2	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	—	8	2	no	—
55	C₃×C₄	C₂×D₈	1	2	yes	—
56	C₃×C₄	#135	1	4	yes	—
57	C₂³	—	12	2	no	—
58	Q₈	—	4	4	no	—
59	D₄	—	12	8	no	—
60	D₄	—	12	4	no	—
61	C₂×C₄	—	4	2	no	—
62	C₂×C₄	D₃×D₄	1	2	yes	—
63	C₂×C₄	C₂×#67	1	1	yes	characteristic
64	C₈	—	6	4	no	—
65	C₈	#213	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	4	no	—
70	C₂²	—	24	2	no	—
71	C₂²	#940	1	1	yes	characteristic
72	C₄	—	8	2	no	—
73	C₄	#919	1	4	yes	—
74	C₄	C₂×#213	1	2	yes	—
75	C₃	#1709	1	1	yes	characteristic, complemented, Sylow
76	C₂	—	48	2	no	—
77	C₂	#5073	1	1	yes	characteristic
78	C₂	#5191	1	2	yes	—

	subgroup	G/H
center	C₂²	#940
commutator, cosocle	C₃×C₄²	C₂³
Frattini	C₄²	C₂²×D₃
Fitting	C₃×#494	C₂
socle	C₂²×C₃	#127
prime radical	#4695	C₂
hypercenter	#494	D₃
hypocenter	C₃	#1709
LCS	G ▷ C₃×C₄² ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #494

Complements

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

N	H ≅ G/N	type
1. C₃×#494	C₂	semidirect
2. #5088	C₂	semidirect
4. C₈×C₃⋊C₈	C₂	semidirect
13. #494	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
C₃×#494	2
#5088	2
#4695	2
C₈×C₃⋊C₈	2
#1709	3