Subgroups — #2091

Order 128.

111 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#430	C₂	1	3	yes	maximal, complemented
2	#407	C₂	1	3	yes	maximal
3	#338	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂³×C₄	C₂²	1	1	yes	characteristic
5	C₂×M₁₆	C₂²	1	3	yes	—
6	C₂×M₁₆	C₂²	1	3	yes	—
7	#108	—	2	6	no	—
8	C₂²⋊C₈	—	2	6	no	—
9	C₂⁴	Q₈	1	1	yes	characteristic
10	C₂²×C₄	—	2	3	no	—
11	C₂²×C₄	—	2	3	no	—
12	C₂²×C₄	Q₈	1	1	yes	characteristic
13	C₂²×C₄	C₂³	1	1	yes	characteristic
14	M₁₆	—	2	6	no	—
15	M₁₆	—	2	6	no	—
16	C₂×C₈	—	2	3	no	—
17	C₂×C₈	D₄	1	6	yes	—
18	C₂³	—	2	3	no	—
19	C₂³	—	4	2	no	—
20	C₂³	C₂×Q₈	1	1	yes	characteristic
21	C₂×C₄	—	2	3	no	—
22	C₂×C₄	—	4	3	no	—
23	C₂×C₄	—	4	1	no	—
24	C₂×C₄	C₂×D₄	1	3	yes	—
25	C₂×C₄	#41	1	3	yes	—
26	C₈	—	2	6	no	—
27	C₈	—	4	3	no	—
28	C₂²	—	2	6	no	—
29	C₂²	—	4	1	no	—
30	C₂²	—	4	3	no	—
31	C₂²	—	4	1	no	—
32	C₂²	C₂²⋊Q₈	1	3	yes	—
33	C₄	—	4	1	no	—
34	C₄	#123	1	3	yes	—
35	C₄	#127	1	1	yes	characteristic
36	C₂	—	2	3	no	—
37	C₂	—	4	2	no	—
38	C₂	#399	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#127
commutator, Frattini, cosocle	C₂²×C₄	C₂³
socle	C₂	#399
prime radical	C₂⁴	Q₈
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

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

N	H ≅ G/N	type
1. #430	C₂	semidirect
3. #338	C₂	semidirect

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