Subgroups — #2381

Order 128.

212 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic
2	#391	C₂	1	4	yes	maximal
3	#387	C₂	1	8	yes	maximal
4	#383	C₂	1	2	yes	maximal
5	C₂²×Q₈	C₂²	1	2	yes	—
6	C₄⋊Q₈	C₄	1	8	yes	complemented
7	C₂×C₄⋊C₄	C₂²	1	4	yes	—
8	C₂×C₄⋊C₄	C₂²	1	4	yes	—
9	C₂×C₄²	C₂²	1	4	yes	—
10	C₂×C₄²	C₂²	1	1	yes	characteristic
11	#95	C₂²	1	8	yes	—
12	#95	C₂²	1	4	yes	—
13	#95	C₂²	1	8	yes	—
14	C₂×Q₈	—	4	2	no	—
15	C₂×Q₈	—	4	2	no	—
16	C₂×Q₈	C₂×C₄	1	8	yes	—
17	C₂²×C₄	C₂³	1	8	yes	—
18	C₂²×C₄	C₂³	1	2	yes	—
19	C₂²×C₄	C₂³	1	1	yes	characteristic
20	C₂²×C₄	C₂³	1	4	yes	—
21	C₄⋊C₄	—	4	4	no	—
22	C₄⋊C₄	C₂×C₄	1	16	yes	—
23	C₄²	—	4	4	no	—
24	C₄²	C₂×C₄	1	4	yes	—
25	C₂³	C₂⁴	1	1	yes	characteristic
26	Q₈	—	4	8	no	—
27	C₂×C₄	—	2	4	no	—
28	C₂×C₄	—	2	4	no	—
29	C₂×C₄	—	2	2	no	—
30	C₂×C₄	—	2	4	no	—
31	C₂×C₄	—	2	8	no	—
32	C₂×C₄	—	2	16	no	—
33	C₂×C₄	C₂²×C₄	1	4	yes	—
34	C₂×C₄	C₂²×C₄	1	2	yes	—
35	C₂×C₄	C₂²×C₄	1	8	yes	—
36	C₂²	C₂³×C₄	1	1	yes	characteristic
37	C₂²	H(5,𝔽₂)	1	2	yes	—
38	C₂²	2₋¹⁺⁴	1	4	yes	—
39	C₄	—	2	8	no	—
40	C₄	—	2	2	no	—
41	C₄	—	2	4	no	—
42	C₄	—	4	8	no	—
43	C₂	#518	1	1	yes	characteristic
44	C₂	#519	1	2	yes	—
45	C₂	#563	1	4	yes	—

	subgroup	G/H
center, Frattini, socle, prime radical, cosocle	C₂³	C₂⁴
commutator	C₂²	C₂³×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
6. C₄⋊Q₈	C₄	semidirect

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