Subgroups — C₂²×#994 #87531

Order 384.

1335 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#994	C₂	1	12	yes	maximal, direct product factor
2	C₂²×#41⋊C₃	C₂	1	3	yes	maximal, complemented
3	C₂²×H(5,𝔽₂)	C₃	1	1	yes	maximal, characteristic, complemented, Sylow
4	C₂²×C₃×Q₈	—	4	1	no	maximal
5	#994	C₂²	1	16	yes	direct product factor
6	C₂×#41⋊C₃	C₂²	1	18	yes	complemented
7	C₂²×SL(2,𝔽₃)	C₂²	1	1	yes	characteristic, complemented
8	C₂×H(5,𝔽₂)	—	3	12	no	—
9	C₂×H(5,𝔽₂)	C₂×C₃	1	12	yes	complemented
10	C₂²×#41	—	3	1	no	—
11	C₂²×#41	C₂×C₃	1	3	yes	—
12	C₂³×D₄	—	3	3	no	—
13	C₂×C₃×Q₈	—	4	12	no	—
14	C₂²×C₃×C₄	—	4	3	no	—
15	#41⋊C₃	C₂³	1	12	yes	complemented
16	C₂×SL(2,𝔽₃)	C₂³	1	3	yes	complemented
17	C₂⁵	—	3	2	no	—
18	H(5,𝔽₂)	—	3	48	no	—
19	H(5,𝔽₂)	—	3	32	no	—
20	H(5,𝔽₂)	C₂²×C₃	1	16	yes	complemented
21	C₂×#41	—	3	12	no	—
22	C₂×#41	—	3	12	no	—
23	C₂×#41	—	3	18	no	—
24	C₂×#41	C₂²×C₃	1	18	yes	—
25	C₂²×Q₈	A₄	1	1	yes	characteristic
26	C₂²×Q₈	C₂²×C₃	1	1	yes	characteristic
27	C₂²×D₄	—	3	3	no	—
28	C₂²×D₄	—	3	36	no	—
29	C₂²×D₄	—	3	36	no	—
30	C₂²×D₄	—	3	3	no	—
31	C₂³×C₄	—	3	3	no	—
32	C₂³×C₃	—	4	1	no	—
33	C₃×Q₈	—	4	16	no	—
34	C₂×C₃×C₄	—	4	18	no	—
35	SL(2,𝔽₃)	C₂⁴	1	1	yes	characteristic, complemented
36	C₂⁴	—	3	24	no	—
37	C₂⁴	—	3	3	no	—
38	C₂⁴	—	12	8	no	—
39	#41	—	3	24	no	—
40	#41	—	3	48	no	—
41	#41	—	3	16	no	—
42	#41	—	3	36	no	—
43	#41	C₂³×C₃	1	12	yes	—
44	C₂×Q₈	—	3	3	no	—
45	C₂×Q₈	C₂×A₄	1	12	yes	—
46	C₂×Q₈	C₂³×C₃	1	3	yes	—
47	C₂×D₄	—	3	18	no	—
48	C₂×D₄	—	3	18	no	—
49	C₂×D₄	—	3	144	no	—
50	C₂×D₄	—	3	48	no	—
51	C₂×D₄	—	3	36	no	—
52	C₂²×C₄	—	3	1	no	—
53	C₂²×C₄	—	3	18	no	—
54	C₂²×C₄	—	3	18	no	—
55	C₂²×C₄	C₂×A₄	1	3	yes	—
56	C₂²×C₃	—	4	4	no	—
57	C₂²×C₃	—	4	3	no	—
58	C₃×C₄	—	4	12	no	—
59	C₂³	—	3	32	no	—
60	C₂³	—	3	18	no	—
61	C₂³	—	6	12	no	—
62	C₂³	—	12	48	no	—
63	C₂³	C₂²×A₄	1	1	yes	characteristic
64	Q₈	—	3	3	no	—
65	Q₈	—	3	2	no	—
66	Q₈	C₂²×A₄	1	16	yes	—
67	Q₈	C₂⁴×C₃	1	1	yes	characteristic
68	D₄	—	3	12	no	—
69	D₄	—	3	36	no	—
70	D₄	—	3	48	no	—
71	C₂×C₄	—	3	3	no	—
72	C₂×C₄	—	3	3	no	—
73	C₂×C₄	—	3	36	no	—
74	C₂×C₄	—	3	12	no	—
75	C₂×C₄	C₂²×A₄	1	18	yes	—
76	C₂×C₃	—	4	6	no	—
77	C₂×C₃	—	4	1	no	—
78	C₂²	—	3	12	no	—
79	C₂²	—	6	36	no	—
80	C₂²	—	12	32	no	—
81	C₂²	#994	1	4	yes	direct product factor
82	C₂²	C₂³×A₄	1	3	yes	—
83	C₄	—	3	3	no	—
84	C₄	—	3	1	no	—
85	C₄	C₂³×A₄	1	12	yes	—
86	C₃	—	4	1	no	Sylow
87	C₂	—	6	12	no	—
88	C₂	C₂×#994	1	6	yes	direct product factor
89	C₂	C₂⁴×A₄	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³	C₂²×A₄
commutator, cosocle, hypocenter	Q₈	C₂⁴×C₃
Frattini	C₂	C₂⁴×A₄
Fitting	C₂²×H(5,𝔽₂)	C₃
hypercenter	C₂²×Q₈	A₄
LCS	G ▷ Q₈
UCS	C₁ ◁ C₂³ ◁ C₂²×Q₈

Complements

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

N	H ≅ G/N	type
1. C₂×#994	C₂	direct (N × H)
2. C₂²×#41⋊C₃	C₂	semidirect
3. C₂²×H(5,𝔽₂)	C₃	semidirect
5. #994	C₂²	direct (N × H)
6. C₂×#41⋊C₃	C₂²	semidirect
7. C₂²×SL(2,𝔽₃)	C₂²	semidirect
9. C₂×H(5,𝔽₂)	C₂×C₃	semidirect
15. #41⋊C₃	C₂³	semidirect
16. C₂×SL(2,𝔽₃)	C₂³	semidirect
20. H(5,𝔽₂)	C₂²×C₃	semidirect
35. SL(2,𝔽₃)	C₂⁴	semidirect
81. C₂²	#994	direct (N × H)
88. C₂	C₂×#994	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#994	2
C₂²×#41⋊C₃	2
C₂²×H(5,𝔽₂)	3
C₂²×C₃×Q₈	4

Subgroups — C22×#994 #87531

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#994 #87531