Subgroups — C₂²×#859 #85500

Order 384.

439 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×SL(2,𝔽₃)	C₂	1	1	yes	maximal, characteristic
2	C₂×#859	C₂	1	6	yes	maximal, direct product factor
3	C₂²×#103	—	3	1	no	maximal, Sylow
4	C₂³×Dic₃	—	4	1	no	maximal
5	C₂²×SL(2,𝔽₃)	C₄	1	4	yes	complemented
6	C₂²×SL(2,𝔽₃)	C₂²	1	3	yes	—
7	#859	C₂²	1	4	yes	direct product factor
8	C₂³×Q₈	D₃	1	1	yes	characteristic
9	C₂³×C₈	—	3	1	no	—
10	C₂²×C₄⋊C₄	—	3	1	no	—
11	C₂×#103	—	3	6	no	—
12	C₂×#103	—	3	6	no	—
13	C₂⁴×C₃	—	4	1	no	—
14	C₂²×Dic₃	—	4	6	no	—
15	C₂²×Dic₃	—	4	8	no	—
16	C₂×SL(2,𝔽₃)	C₂×C₄	1	6	yes	complemented
17	C₂×SL(2,𝔽₃)	C₂³	1	1	yes	characteristic
18	C₂²×Q₈	—	3	4	no	—
19	C₂²×Q₈	—	3	3	no	—
20	C₂²×Q₈	Dic₃	1	4	yes	complemented
21	C₂²×Q₈	C₂×D₃	1	3	yes	—
22	C₂³×C₄	—	3	1	no	—
23	C₂³×C₄	—	6	1	no	—
24	C₂²×C₈	—	3	6	no	—
25	C₂²×C₈	—	6	4	no	—
26	C₂×C₄⋊C₄	—	3	6	no	—
27	C₂×C₄⋊C₄	—	6	3	no	—
28	#103	—	3	12	no	—
29	#103	—	3	4	no	—
30	C₂³×C₃	—	4	4	no	—
31	C₂³×C₃	—	4	4	no	—
32	C₂³×C₃	—	4	4	no	—
33	C₂³×C₃	—	4	3	no	—
34	C₂×Dic₃	—	4	4	no	—
35	C₂×Dic₃	—	4	24	no	—
36	SL(2,𝔽₃)	C₂²×C₄	1	1	yes	characteristic, complemented
37	C₂⁴	S₄	1	1	yes	characteristic
38	C₂×Q₈	—	3	3	no	—
39	C₂×Q₈	—	3	12	no	—
40	C₂×Q₈	—	3	6	no	—
41	C₂×Q₈	—	6	6	no	—
42	C₂×Q₈	—	6	1	no	—
43	C₂×Q₈	C₂×Dic₃	1	6	yes	complemented
44	C₂×Q₈	C₂²×D₃	1	1	yes	characteristic
45	C₂²×C₄	—	3	4	no	—
46	C₂²×C₄	—	3	4	no	—
47	C₂²×C₄	—	3	3	no	—
48	C₂²×C₄	—	3	3	no	—
49	C₂²×C₄	—	6	6	no	—
50	C₂²×C₄	—	12	4	no	—
51	C₂×C₈	—	3	4	no	—
52	C₂×C₈	—	6	12	no	—
53	C₄⋊C₄	—	3	4	no	—
54	C₄⋊C₄	—	6	6	no	—
55	C₂²×C₃	—	4	16	no	—
56	C₂²×C₃	—	4	6	no	—
57	C₂²×C₃	—	4	6	no	—
58	C₂²×C₃	—	4	6	no	—
59	C₂²×C₃	—	4	1	no	—
60	Dic₃	—	4	8	no	—
61	C₂³	#226	1	4	yes	—
62	C₂³	GL(2,𝔽₃)	1	4	yes	—
63	C₂³	SL(2,ℤ/4)	1	4	yes	—
64	C₂³	C₂×S₄	1	3	yes	—
65	Q₈	—	3	6	no	—
66	Q₈	—	3	1	no	—
67	Q₈	—	6	3	no	—
68	Q₈	—	6	4	no	—
69	Q₈	C₂²×Dic₃	1	1	yes	characteristic, complemented
70	C₂×C₄	—	3	6	no	—
71	C₂×C₄	—	3	6	no	—
72	C₂×C₄	—	3	12	no	—
73	C₂×C₄	—	3	1	no	—
74	C₂×C₄	—	3	3	no	—
75	C₂×C₄	—	6	4	no	—
76	C₂×C₄	—	12	12	no	—
77	C₈	—	6	4	no	—
78	C₂×C₃	—	4	1	no	—
79	C₂×C₃	—	4	1	no	—
80	C₂×C₃	—	4	1	no	—
81	C₂×C₃	—	4	12	no	—
82	C₂²	#859	1	16	yes	direct product factor
83	C₂²	C₂×#226	1	6	yes	—
84	C₂²	C₂×GL(2,𝔽₃)	1	6	yes	—
85	C₂²	C₂×SL(2,ℤ/4)	1	6	yes	—
86	C₂²	C₂²×S₄	1	1	yes	characteristic
87	C₄	—	3	1	no	—
88	C₄	—	3	1	no	—
89	C₄	—	3	6	no	—
90	C₄	—	12	4	no	—
91	C₃	—	4	1	no	Sylow
92	C₂	C₂×#859	1	12	yes	direct product factor
93	C₂	C₂²×#226	1	1	yes	characteristic
94	C₂	C₂²×GL(2,𝔽₃)	1	1	yes	characteristic
95	C₂	C₂²×SL(2,ℤ/4)	1	1	yes	characteristic

	subgroup	G/H
center, socle, hypercenter	C₂⁴	S₄
commutator, hypocenter	SL(2,𝔽₃)	C₂²×C₄
Frattini	C₂²	C₂²×S₄
Fitting	C₂³×Q₈	D₃
prime radical	C₂³×SL(2,𝔽₃)	C₂
cosocle	C₂×SL(2,𝔽₃)	C₂³
LCS	G ▷ SL(2,𝔽₃)
UCS	C₁ ◁ C₂⁴

Complements

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

N	H ≅ G/N	type
2. C₂×#859	C₂	direct (N × H)
5. C₂²×SL(2,𝔽₃)	C₄	semidirect
7. #859	C₂²	direct (N × H)
16. C₂×SL(2,𝔽₃)	C₂×C₄	semidirect
20. C₂²×Q₈	Dic₃	semidirect
36. SL(2,𝔽₃)	C₂²×C₄	semidirect
43. C₂×Q₈	C₂×Dic₃	semidirect
69. Q₈	C₂²×Dic₃	semidirect
82. C₂²	#859	direct (N × H)
92. C₂	C₂×#859	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×SL(2,𝔽₃)	2
C₂×#859	2
C₂²×#103	3
C₂³×Dic₃	4

Subgroups — C22×#859 #85500

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#859 #85500