Subgroups — #73120

Order 384.

118 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	#5423	C₂	1	2	yes	maximal
3	#1814	—	3	1	no	maximal, Sylow
4	C₂×#41⋊C₃	C₂²	1	1	yes	characteristic
5	#131⋊C₃	C₄	1	1	yes	characteristic, complemented
6	#131⋊C₃	C₄	1	1	yes	characteristic, complemented
7	GU(2,𝔽₃)	—	2	2	no	—
8	#819	—	4	1	no	maximal
9	C₂×#131	D₃	1	1	yes	characteristic
10	C₂×#108	—	3	1	no	—
11	#427	—	3	1	no	—
12	#421	—	3	2	no	—
13	#344	—	3	2	no	—
14	#41⋊C₃	C₂×C₄	1	1	yes	characteristic
15	#41⋊C₃	D₄	1	1	yes	characteristic
16	C₂×SL(2,𝔽₃)	Q₈	1	1	yes	characteristic
17	C₂×C₃×C₈	—	4	1	no	—
18	#208	—	4	2	no	—
19	C₂×#41	C₂×D₃	1	1	yes	characteristic
20	#131	—	3	1	no	—
21	#131	—	3	1	no	—
22	#131	Dic₃	1	1	yes	characteristic
23	#131	Dic₃	1	1	yes	characteristic
24	C₂×M₁₆	—	3	1	no	—
25	C₂×M₁₆	—	3	2	no	—
26	C₂²×C₈	—	3	1	no	—
27	#117	—	3	2	no	—
28	#108	—	6	1	no	—
29	#108	—	6	1	no	—
30	#107	—	6	1	no	—
31	#106	—	6	1	no	—
32	#104	—	6	2	no	—
33	#104	—	6	2	no	—
34	C₂×C₃×C₄	—	4	1	no	—
35	SL(2,𝔽₃)	C₄⋊C₄	1	1	yes	characteristic
36	C₃×C₈	—	4	1	no	—
37	C₃×C₈	—	4	1	no	—
38	C₃⋊C₈	—	8	2	no	—
39	#41	—	3	1	no	—
40	#41	—	3	1	no	—
41	#41	D₁₂	1	1	yes	characteristic
42	#41	C₂×Dic₃	1	1	yes	characteristic
43	C₂×Q₈	Dic₆	1	1	yes	characteristic
44	C₂×D₄	—	3	1	no	—
45	C₂²×C₄	—	3	1	no	—
46	M₁₆	—	3	1	no	—
47	M₁₆	—	3	1	no	—
48	M₁₆	—	6	1	no	—
49	M₁₆	—	6	2	no	—
50	M₁₆	—	6	2	no	—
51	C₂×C₈	—	3	1	no	—
52	C₂×C₈	—	3	1	no	—
53	C₂×C₈	—	3	1	no	—
54	C₂×C₈	—	3	1	no	—
55	C₂×C₈	—	3	1	no	—
56	C₂×C₈	—	6	2	no	—
57	C₂×C₈	S₄	1	1	yes	characteristic
58	C₄⋊C₄	—	6	2	no	—
59	C₂²⋊C₄	—	6	2	no	—
60	C₄²	—	6	2	no	—
61	C₂²×C₃	—	4	1	no	—
62	C₃×C₄	—	4	1	no	—
63	C₃×C₄	—	4	1	no	—
64	C₂³	—	3	1	no	—
65	Q₈	—	3	1	no	—
66	Q₈	C₄⋊Dic₃	1	1	yes	characteristic
67	D₄	—	3	1	no	—
68	D₄	—	3	1	no	—
69	D₄	—	6	1	no	—
70	C₂×C₄	—	3	1	no	—
71	C₂×C₄	—	3	1	no	—
72	C₂×C₄	—	3	1	no	—
73	C₂×C₄	—	3	1	no	—
74	C₂×C₄	—	3	1	no	—
75	C₂×C₄	—	12	2	no	—
76	C₂×C₄	C₂×S₄	1	1	yes	characteristic
77	C₈	—	3	1	no	—
78	C₈	—	3	1	no	—
79	C₈	—	12	2	no	—
80	C₈	SL(2,ℤ/4)	1	1	yes	characteristic
81	C₈	SL(2,ℤ/4)	1	1	yes	characteristic
82	C₂×C₃	—	4	1	no	—
83	C₂×C₃	—	8	1	no	—
84	C₂²	—	3	1	no	—
85	C₂²	—	3	1	no	—
86	C₂²	—	12	1	no	—
87	C₂²	A₄⋊Q₈	1	1	yes	characteristic
88	C₄	—	3	1	no	—
89	C₄	—	3	1	no	—
90	C₄	—	24	2	no	—
91	C₄	C₄⋊S₄	1	1	yes	characteristic
92	C₄	C₂×SL(2,ℤ/4)	1	1	yes	characteristic
93	C₃	—	4	1	no	Sylow
94	C₂	—	2	1	no	—
95	C₂	—	6	1	no	—
96	C₂	—	6	1	no	—
97	C₂	C₄⋊SL(2,ℤ/4)	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₄⋊S₄
commutator	#41⋊C₃	C₂×C₄
Frattini	C₂×C₄	C₂×S₄
Fitting	C₂×#131	D₃
socle	C₂	C₄⋊SL(2,ℤ/4)
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 ◁ C₄ ◁ 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
6. #131⋊C₃	C₄	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
#5423	2
#1814	3
#819	4