Subgroups — #73125

Order 384.

125 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#131⋊C₃	C₂	1	1	yes	maximal, characteristic
2	#5426	C₂	1	2	yes	maximal
3	#1815	—	3	1	no	maximal, Sylow
4	C₂×#41⋊C₃	C₂²	1	1	yes	characteristic
5	#131⋊C₃	C₄	1	2	yes	complemented
6	#859	—	2	2	no	—
7	#817	—	4	1	no	maximal
8	C₂×#131	D₃	1	1	yes	characteristic
9	C₂×#108	—	3	1	no	—
10	C₂×#106	—	3	1	no	—
11	#417	—	3	2	no	—
12	#343	—	3	2	no	—
13	#41⋊C₃	C₂×C₄	1	1	yes	characteristic
14	#41⋊C₃	Q₈	1	1	yes	characteristic
15	C₂×SL(2,𝔽₃)	D₄	1	1	yes	characteristic
16	C₂×C₃×C₈	—	4	1	no	—
17	C₄⋊Dic₃	—	4	2	no	—
18	C₂×#41	C₂×D₃	1	1	yes	characteristic
19	#131	—	3	2	no	—
20	#131	Dic₃	1	2	yes	complemented
21	C₂×M₁₆	—	3	1	no	—
22	C₂×M₁₆	—	3	2	no	—
23	C₂²×C₈	—	3	1	no	—
24	C₂×C₄⋊C₄	—	3	2	no	—
25	#108	—	6	2	no	—
26	#106	—	6	1	no	—
27	#106	—	6	1	no	—
28	#103	—	6	2	no	—
29	#102	—	6	2	no	—
30	C₂×C₃×C₄	—	4	1	no	—
31	C₂×Dic₃	—	8	2	no	—
32	SL(2,𝔽₃)	C₄⋊C₄	1	1	yes	characteristic
33	C₃×C₈	—	4	2	no	—
34	#41	—	3	1	no	—
35	#41	—	3	1	no	—
36	#41	Dic₆	1	1	yes	characteristic
37	#41	C₂×Dic₃	1	1	yes	characteristic
38	C₂×Q₈	D₁₂	1	1	yes	characteristic
39	C₂×D₄	—	3	1	no	—
40	C₂²×C₄	—	3	1	no	—
41	C₂²×C₄	—	6	2	no	—
42	M₁₆	—	3	2	no	—
43	M₁₆	—	6	1	no	—
44	M₁₆	—	6	2	no	—
45	M₁₆	—	6	2	no	—
46	C₂×C₈	—	3	1	no	—
47	C₂×C₈	—	3	2	no	—
48	C₂×C₈	—	3	2	no	—
49	C₂×C₈	—	6	2	no	—
50	C₂×C₈	S₄	1	1	yes	characteristic
51	C₄⋊C₄	—	6	2	no	—
52	C₄⋊C₄	—	6	2	no	—
53	C₂²×C₃	—	4	1	no	—
54	C₃×C₄	—	4	1	no	—
55	C₃×C₄	—	4	1	no	—
56	Dic₃	—	16	2	no	—
57	C₂³	—	3	1	no	—
58	Q₈	—	3	1	no	—
59	Q₈	C₄⋊Dic₃	1	1	yes	characteristic
60	D₄	—	3	1	no	—
61	D₄	—	3	1	no	—
62	D₄	—	6	1	no	—
63	C₂×C₄	—	3	1	no	—
64	C₂×C₄	—	3	1	no	—
65	C₂×C₄	—	3	1	no	—
66	C₂×C₄	—	3	1	no	—
67	C₂×C₄	—	3	1	no	—
68	C₂×C₄	—	12	2	no	—
69	C₂×C₄	—	24	2	no	—
70	C₂×C₄	C₂×S₄	1	1	yes	characteristic
71	C₈	—	3	2	no	—
72	C₈	—	12	2	no	—
73	C₈	SL(2,ℤ/4)	1	2	yes	—
74	C₂×C₃	—	4	2	no	—
75	C₂×C₃	—	4	1	no	—
76	C₂²	—	3	1	no	—
77	C₂²	—	3	1	no	—
78	C₂²	—	6	2	no	—
79	C₂²	C₄⋊S₄	1	1	yes	characteristic
80	C₄	—	3	1	no	—
81	C₄	—	3	1	no	—
82	C₄	—	24	2	no	—
83	C₄	A₄⋊Q₈	1	1	yes	characteristic
84	C₄	C₂×SL(2,ℤ/4)	1	1	yes	characteristic
85	C₃	—	4	1	no	Sylow
86	C₂	—	6	1	no	—
87	C₂	—	6	1	no	—
88	C₂	#5406	1	2	yes	—
89	C₂	C₄⋊SL(2,ℤ/4)	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	C₄⋊S₄
commutator	#41⋊C₃	C₂×C₄
Frattini	C₂×C₄	C₂×S₄
Fitting	C₂×#131	D₃
prime radical, cosocle	C₂×#41⋊C₃	C₂²
hypercenter	C₂×C₈	S₄
hypocenter	SL(2,𝔽₃)	C₄⋊C₄
LCS	G ▷ #41⋊C₃ ▷ SL(2,𝔽₃)
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈

Complements

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

N	H ≅ G/N	type
5. #131⋊C₃	C₄	semidirect
20. #131	Dic₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×#131⋊C₃	2
#5426	2
#1815	3
#817	4