Subgroups — #67661

Order 384.

152 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×#97	C₂	1	1	yes	maximal, characteristic
2	#4925	C₂	1	2	yes	maximal
3	#1281	—	3	1	no	maximal, Sylow
4	C₂²×C₃×C₈	C₄	1	2	yes	—
5	C₂×C₃×C₄²	C₂²	1	1	yes	characteristic
6	C₃×#97	—	2	2	no	—
7	#830	C₄	1	4	yes	—
8	C₄⋊(C₃⋊C₈)	—	2	2	no	—
9	C₄×C₃⋊C₈	—	2	2	no	—
10	#431	—	3	2	no	—
11	C₂×#97	D₃	1	1	yes	characteristic
12	C₂²×C₃×C₄	C₂×C₄	1	2	yes	—
13	C₂²×C₃×C₄	C₂×C₄	1	1	yes	characteristic
14	C₂×C₃×C₈	—	2	2	no	—
15	C₂×C₃×C₈	—	2	2	no	—
16	C₂×C₃×C₈	C₈	1	4	yes	complemented
17	C₃×C₄²	D₄	1	1	yes	characteristic
18	C₃×C₄²	D₄	1	2	yes	—
19	C₃×C₄²	Q₈	1	1	yes	characteristic
20	C₂×C₃⋊C₈	—	2	4	no	—
21	C₂²×C₈	Dic₃	1	2	yes	—
22	C₂×C₄²	C₂×D₃	1	1	yes	characteristic
23	C₄⋊C₈	—	6	2	no	—
24	C₂²⋊C₈	—	3	4	no	—
25	#97	—	2	2	no	—
26	C₄×C₈	—	6	2	no	—
27	C₂³×C₃	C₄²	1	1	yes	characteristic
28	C₂×C₃×C₄	—	4	2	no	—
29	C₂×C₃×C₄	C₂²⋊C₄	1	2	yes	—
30	C₂×C₃×C₄	C₂²⋊C₄	1	1	yes	characteristic
31	C₂×C₃×C₄	C₄⋊C₄	1	2	yes	—
32	C₂×C₃×C₄	C₄⋊C₄	1	1	yes	characteristic
33	C₂×C₃×C₄	C₂×C₈	1	2	yes	—
34	C₂×C₃×C₄	M₁₆	1	2	yes	—
35	C₃×C₈	—	2	4	no	—
36	C₃⋊C₈	—	4	4	no	—
37	C₂²×C₄	C₄×D₃	1	2	yes	—
38	C₂²×C₄	C₂×Dic₃	1	1	yes	characteristic
39	C₂×C₈	—	2	2	no	—
40	C₂×C₈	—	2	2	no	—
41	C₂×C₈	—	6	4	no	—
42	C₂×C₈	C₃⋊C₈	1	4	yes	complemented
43	C₄²	Dic₆	1	1	yes	characteristic
44	C₄²	D₁₂	1	1	yes	characteristic
45	C₄²	#67	1	2	yes	—
46	C₂²×C₃	—	2	2	no	—
47	C₂²×C₃	#95	1	1	yes	characteristic
48	C₂²×C₃	C₄×C₈	1	1	yes	characteristic
49	C₂²×C₃	#97	1	1	yes	characteristic
50	C₃×C₄	—	2	2	no	—
51	C₃×C₄	—	2	2	no	—
52	C₃×C₄	C₂²⋊C₈	1	2	yes	—
53	C₃×C₄	C₄⋊C₈	1	2	yes	—
54	C₂³	C₄×Dic₃	1	1	yes	characteristic
55	C₂×C₄	—	4	2	no	—
56	C₂×C₄	C₂×C₃⋊C₈	1	2	yes	—
57	C₂×C₄	#208	1	2	yes	—
58	C₂×C₄	Dic₃⋊C₄	1	2	yes	—
59	C₂×C₄	C₄⋊Dic₃	1	1	yes	characteristic
60	C₂×C₄	(C₂×D₃)⋊C₄	1	2	yes	—
61	C₂×C₄	#217	1	1	yes	characteristic
62	C₈	—	2	4	no	—
63	C₈	—	12	4	no	—
64	C₂×C₃	—	2	1	no	—
65	C₂×C₃	—	2	1	no	—
66	C₂×C₃	#336	1	1	yes	characteristic
67	C₂×C₃	#344	1	2	yes	—
68	C₂²	—	2	2	no	—
69	C₂²	C₄×C₃⋊C₈	1	1	yes	characteristic
70	C₂²	#803	1	1	yes	characteristic
71	C₂²	C₃⋊#95	1	1	yes	characteristic
72	C₄	—	2	2	no	—
73	C₄	—	2	2	no	—
74	C₄	C₄⋊(C₃⋊C₈)	1	2	yes	—
75	C₄	#830	1	2	yes	—
76	C₃	#1281	1	1	yes	characteristic, complemented, Sylow
77	C₂	—	2	1	no	—
78	C₂	—	2	1	no	—
79	C₂	#4524	1	1	yes	characteristic
80	C₂	#4556	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	#217
commutator	C₂²×C₃	C₄×C₈
Frattini	C₂×C₄²	C₂×D₃
Fitting	C₂×C₃×#97	C₂
socle	C₂²×C₃	#95
prime radical	C₂³×C₃	C₄²
cosocle	C₂×C₃×C₄²	C₂²
hypercenter	C₂×#97	D₃
hypocenter	C₃	#1281
LCS	G ▷ C₂²×C₃ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×#97

Complements

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

N	H ≅ G/N	type
16. C₂×C₃×C₈	C₈	semidirect
42. C₂×C₈	C₃⋊C₈	semidirect
76. C₃	#1281	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃×#97	2
#4925	2
#1281	3