Subgroups — #389

Order 64.

85 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊C₄	C₂	1	2	yes	maximal
3	C₂×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄²	C₂	1	1	yes	maximal, characteristic
5	#95	C₂	1	2	yes	maximal
6	C₂²×C₄	C₂²	1	2	yes	—
7	C₂²×C₄	C₂²	1	2	yes	—
8	C₂²×C₄	C₂²	1	1	yes	characteristic
9	C₂²×C₄	C₂²	1	1	yes	characteristic
10	C₂²×C₄	C₂²	1	1	yes	characteristic
11	C₄⋊C₄	—	2	4	no	—
12	C₄⋊C₄	—	2	2	no	—
13	C₄⋊C₄	C₄	1	4	yes	complemented
14	C₄²	—	2	2	no	—
15	C₂³	C₂³	1	1	yes	characteristic
16	C₂×C₄	—	2	1	no	—
17	C₂×C₄	—	2	1	no	—
18	C₂×C₄	—	2	2	no	—
19	C₂×C₄	—	2	2	no	—
20	C₂×C₄	—	2	1	no	—
21	C₂×C₄	—	2	1	no	—
22	C₂×C₄	—	2	2	no	—
23	C₂×C₄	—	2	2	no	—
24	C₂×C₄	—	2	2	no	—
25	C₂×C₄	C₂×C₄	1	4	yes	—
26	C₂×C₄	C₂×C₄	1	2	yes	—
27	C₂×C₄	D₄	1	2	yes	—
28	C₂×C₄	D₄	1	2	yes	—
29	C₂×C₄	Q₈	1	2	yes	—
30	C₂×C₄	Q₈	1	2	yes	—
31	C₂²	C₂²×C₄	1	1	yes	characteristic
32	C₂²	C₂×D₄	1	1	yes	characteristic
33	C₂²	C₂×D₄	1	1	yes	characteristic
34	C₂²	C₂×Q₈	1	1	yes	characteristic
35	C₂²	C₂×Q₈	1	1	yes	characteristic
36	C₂²	#41	1	1	yes	characteristic
37	C₂²	#41	1	1	yes	characteristic
38	C₄	—	2	4	no	—
39	C₄	—	2	2	no	—
40	C₄	—	2	2	no	—
41	C₄	—	4	2	no	—
42	C₄	C₄⋊C₄	1	4	yes	complemented
43	C₂	C₂×C₄⋊C₄	1	1	yes	characteristic
44	C₂	C₄×D₄	1	1	yes	characteristic
45	C₂	C₄×Q₈	1	1	yes	characteristic
46	C₂	#121	1	1	yes	characteristic
47	C₂	C₂²⋊Q₈	1	1	yes	characteristic
48	C₂	#125	1	1	yes	characteristic
49	C₂	C₄⋊Q₈	1	1	yes	characteristic

	subgroup	G/H
center, Frattini, socle, prime radical, cosocle	C₂³	C₂³
commutator	C₂²	C₂²×C₄
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

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

N	H ≅ G/N	type
13. C₄⋊C₄	C₄	semidirect
42. C₄	C₄⋊C₄	semidirect

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