Subgroups — #73167

Order 384.

193 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×SL(2,𝔽₃)	C₂	1	3	yes	maximal
2	#2276	C₃	1	1	yes	maximal, characteristic, complemented, Sylow
3	C₂²×SL(2,𝔽₃)	C₂²	1	1	yes	characteristic
4	C₄×SL(2,𝔽₃)	C₄	1	6	yes	complemented
5	C₃×#95	—	4	1	no	maximal
6	C₂×C₄×Q₈	C₂×C₃	1	3	yes	—
7	C₄×C₄⋊C₄	—	3	3	no	—
8	#376	—	3	1	no	—
9	C₂²×C₃×C₄	—	4	3	no	—
10	C₂×SL(2,𝔽₃)	C₂×C₄	1	3	yes	—
11	C₂²×Q₈	C₂²×C₃	1	1	yes	characteristic
12	C₄×Q₈	—	3	6	no	—
13	C₄×Q₈	C₃×C₄	1	6	yes	complemented
14	C₂×C₄⋊C₄	—	3	3	no	—
15	C₂×C₄²	—	3	3	no	—
16	C₂×C₄²	—	3	2	no	—
17	#95	—	3	3	no	—
18	#95	A₄	1	1	yes	characteristic
19	C₂³×C₃	—	4	1	no	—
20	C₂×C₃×C₄	—	4	6	no	—
21	C₂×C₃×C₄	—	8	6	no	—
22	SL(2,𝔽₃)	C₄²	1	1	yes	characteristic, complemented
23	C₂×Q₈	—	3	3	no	—
24	C₂×Q₈	C₂×C₃×C₄	1	3	yes	—
25	C₂²×C₄	—	3	1	no	—
26	C₂²×C₄	—	3	3	no	—
27	C₂²×C₄	C₂×A₄	1	3	yes	—
28	C₄⋊C₄	—	3	6	no	—
29	C₄⋊C₄	—	3	6	no	—
30	C₄²	—	3	6	no	—
31	C₄²	—	3	6	no	—
32	C₄²	—	12	2	no	—
33	C₂²×C₃	—	4	3	no	—
34	C₂²×C₃	—	4	1	no	—
35	C₂²×C₃	—	4	3	no	—
36	C₃×C₄	—	8	6	no	—
37	C₂³	C₂²×A₄	1	1	yes	characteristic
38	Q₈	—	3	3	no	—
39	Q₈	—	3	2	no	—
40	Q₈	C₃×C₄²	1	1	yes	characteristic
41	C₂×C₄	—	2	6	no	—
42	C₂×C₄	—	3	3	no	—
43	C₂×C₄	—	3	3	no	—
44	C₂×C₄	—	3	6	no	—
45	C₂×C₄	—	6	3	no	—
46	C₂×C₄	—	6	3	no	—
47	C₂×C₄	C₄×A₄	1	6	yes	—
48	C₂×C₃	—	4	1	no	—
49	C₂×C₃	—	4	3	no	—
50	C₂×C₃	—	4	3	no	—
51	C₂²	C₂×C₄×A₄	1	3	yes	—
52	C₂²	#994	1	1	yes	characteristic
53	C₂²	2₋¹⁺⁴⋊C₃	1	3	yes	—
54	C₄	—	2	6	no	—
55	C₄	—	3	3	no	—
56	C₄	—	3	1	no	—
57	C₄	—	6	6	no	—
58	C₃	—	4	1	no	Sylow
59	C₂	C₄²×A₄	1	1	yes	characteristic
60	C₂	#519⋊C₃	1	3	yes	—
61	C₂	#520⋊C₃	1	3	yes	—

	subgroup	G/H
center, Frattini, socle	C₂³	C₂²×A₄
commutator, hypocenter	Q₈	C₃×C₄²
Fitting	#2276	C₃
prime radical	C₂²×SL(2,𝔽₃)	C₂²
cosocle	C₂²×Q₈	C₂²×C₃
hypercenter	#95	A₄
LCS	G ▷ Q₈
UCS	C₁ ◁ C₂³ ◁ #95

Complements

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

N	H ≅ G/N	type
2. #2276	C₃	semidirect
4. C₄×SL(2,𝔽₃)	C₄	semidirect
13. C₄×Q₈	C₃×C₄	semidirect
22. SL(2,𝔽₃)	C₄²	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×SL(2,𝔽₃)	2
#2276	3
C₃×#95	4