Subgroups — #73097

Order 384.

104 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Q₈×SL(2,𝔽₃)	C₂	1	1	yes	maximal, characteristic
2	C₄⋊#226	C₂	1	1	yes	maximal, characteristic
3	SL(2,𝔽₃)⋊C₈	C₂	1	1	yes	maximal, characteristic
4	#1645	—	3	1	no	maximal, Sylow
5	C₂×#226	—	2	1	no	—
6	C₄×SL(2,𝔽₃)	—	2	1	no	—
7	C₄×SL(2,𝔽₃)	C₂²	1	1	yes	characteristic
8	#835	—	4	1	no	maximal
9	Q₈²	D₃	1	1	yes	characteristic
10	#487	—	3	1	no	—
11	#475	—	3	1	no	—
12	#462	—	3	1	no	—
13	#333	—	3	1	no	—
14	Q₈⋊C₈	—	3	1	no	—
15	Q₈⋊C₈	—	3	1	no	—
16	C₂×C₃×Q₈	—	4	1	no	—
17	C₂×SL(2,𝔽₃)	D₄	1	1	yes	characteristic
18	#226	—	4	1	no	—
19	C₄⋊Dic₃	—	4	1	no	—
20	C₂×C₃⋊C₈	—	4	1	no	—
21	C₂×Q₁₆	—	6	1	no	—
22	C₄⋊Q₈	—	3	1	no	—
23	C₄⋊Q₈	—	3	1	no	—
24	C₄⋊Q₈	—	6	1	no	—
25	C₄×Q₈	—	2	1	no	—
26	C₄×Q₈	—	3	1	no	—
27	C₄×Q₈	C₂×D₃	1	1	yes	characteristic
28	#106	—	6	1	no	—
29	C₄⋊C₈	—	3	1	no	—
30	C₄⋊C₈	—	3	1	no	—
31	#103	—	6	1	no	—
32	#103	—	6	2	no	—
33	#103	—	6	1	no	—
34	C₄×C₈	—	3	1	no	—
35	C₃×Q₈	—	4	2	no	—
36	C₃×Q₈	—	8	1	no	—
37	C₂×C₃×C₄	—	4	1	no	—
38	C₂×C₃×C₄	—	8	1	no	—
39	C₂×Dic₃	—	8	1	no	—
40	SL(2,𝔽₃)	SD₁₆	1	1	yes	characteristic
41	C₃⋊C₈	—	8	1	no	—
42	C₂×Q₈	—	3	2	no	—
43	C₂×Q₈	—	6	1	no	—
44	C₂×Q₈	#67	1	1	yes	characteristic
45	C₂×Q₈	S₄	1	1	yes	characteristic
46	Q₁₆	—	12	1	no	—
47	Q₁₆	—	12	1	no	—
48	C₂×C₈	—	6	1	no	—
49	C₂×C₈	—	6	1	no	—
50	C₂×C₈	—	6	1	no	—
51	C₄⋊C₄	—	3	1	no	—
52	C₄⋊C₄	—	3	1	no	—
53	C₄⋊C₄	—	6	1	no	—
54	C₄⋊C₄	—	6	1	no	—
55	C₄⋊C₄	—	6	1	no	—
56	C₄⋊C₄	—	6	1	no	—
57	C₄²	—	3	1	no	—
58	C₄²	—	6	1	no	—
59	C₂²×C₃	—	4	1	no	—
60	C₃×C₄	—	4	1	no	—
61	C₃×C₄	—	4	1	no	—
62	C₃×C₄	—	8	2	no	—
63	Dic₃	—	16	1	no	—
64	Q₈	—	2	1	no	—
65	Q₈	—	3	1	no	—
66	Q₈	—	12	1	no	—
67	Q₈	—	12	1	no	—
68	Q₈	—	12	2	no	—
69	Q₈	#215	1	1	yes	characteristic
70	Q₈	#226	1	2	yes	complemented
71	C₂×C₄	—	2	1	no	—
72	C₂×C₄	—	3	1	no	—
73	C₂×C₄	—	3	1	no	—
74	C₂×C₄	—	6	1	no	—
75	C₂×C₄	—	12	1	no	—
76	C₂×C₄	C₂×S₄	1	1	yes	characteristic
77	C₈	—	12	1	no	—
78	C₈	—	12	1	no	—
79	C₈	—	12	1	no	—
80	C₂×C₃	—	4	1	no	—
81	C₂×C₃	—	4	1	no	—
82	C₂×C₃	—	4	1	no	—
83	C₂²	GL(2,ℤ/4)	1	1	yes	characteristic
84	C₄	—	2	2	no	—
85	C₄	—	3	1	no	—
86	C₄	—	3	1	no	—
87	C₄	—	6	1	no	—
88	C₄	—	12	1	no	—
89	C₄	—	24	1	no	—
90	C₄	C₂×#226	1	1	yes	characteristic
91	C₄	#983	1	1	yes	characteristic
92	C₃	—	4	1	no	Sylow
93	C₂	#5417	1	1	yes	characteristic
94	C₂	#5419	1	1	yes	characteristic
95	C₂	#5428	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	GL(2,ℤ/4)
commutator, cosocle	C₄×SL(2,𝔽₃)	C₂²
Frattini	C₂×C₄	C₂×S₄
Fitting	Q₈²	D₃
prime radical	C₂×SL(2,𝔽₃)	D₄
hypercenter	C₂×Q₈	S₄
hypocenter	SL(2,𝔽₃)	SD₁₆
LCS	G ▷ C₄×SL(2,𝔽₃) ▷ C₂×SL(2,𝔽₃) ▷ SL(2,𝔽₃)
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×Q₈

Complements

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

N	H ≅ G/N	type
70. Q₈	#226	semidirect

Maximal subgroups

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

subgroup	index
Q₈×SL(2,𝔽₃)	2
C₄⋊#226	2
SL(2,𝔽₃)⋊C₈	2
#1645	3
#835	4