Subgroups — #2239

Order 128.

62 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#470	C₂	1	1	yes	maximal, characteristic
2	#430	C₂	1	1	yes	maximal, characteristic
3	#365	C₂	1	2	yes	maximal, complemented
4	#362	C₂	1	2	yes	maximal, complemented
5	#349	C₂	1	1	yes	maximal, characteristic
6	C₂×Q₁₆	C₂²	1	1	yes	characteristic
7	C₂×M₁₆	—	2	1	no	—
8	C₂×M₁₆	C₂²	1	1	yes	characteristic
9	C₂²⋊Q₈	—	2	1	no	—
10	M₃₂	C₂²	1	2	yes	—
11	#108	—	2	1	no	—
12	#108	C₂²	1	2	yes	—
13	#106	C₂²	1	1	yes	characteristic
14	#103	—	2	1	no	—
15	C₂×Q₈	—	2	1	no	—
16	C₂²×C₄	D₄	1	1	yes	characteristic
17	Q₁₆	—	2	2	no	—
18	M₁₆	—	2	2	no	—
19	M₁₆	—	2	2	no	—
20	M₁₆	—	4	1	no	—
21	C₂×C₈	—	4	1	no	—
22	C₂×C₈	D₄	1	1	yes	characteristic
23	C₂×C₈	C₂³	1	1	yes	characteristic
24	C₄⋊C₄	—	2	1	no	—
25	C₄⋊C₄	—	4	1	no	—
26	C₂²⋊C₄	—	4	1	no	—
27	C₁₆	—	2	2	no	—
28	C₂³	SD₁₆	1	1	yes	characteristic
29	Q₈	—	4	1	no	—
30	Q₈	—	4	1	no	—
31	C₂×C₄	—	2	1	no	—
32	C₂×C₄	—	2	1	no	—
33	C₂×C₄	—	4	1	no	—
34	C₂×C₄	—	4	1	no	—
35	C₂×C₄	SD₁₆	1	1	yes	characteristic
36	C₂×C₄	C₂×D₄	1	1	yes	characteristic
37	C₈	—	2	1	no	—
38	C₈	—	4	2	no	—
39	C₈	#41	1	2	yes	—
40	C₂²	—	2	1	no	—
41	C₂²	—	4	1	no	—
42	C₂²	C₂×SD₁₆	1	1	yes	characteristic
43	C₄	—	2	1	no	—
44	C₄	—	8	1	no	—
45	C₄	—	8	1	no	—
46	C₄	#123	1	1	yes	characteristic
47	C₄	AGL(1,ℤ/8)	1	1	yes	characteristic
48	C₂	—	2	1	no	—
49	C₂	—	4	1	no	—
50	C₂	#481	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂	#481
commutator, Frattini, cosocle	C₂×C₈	C₂³
prime radical	C₂³	SD₁₆
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×M₁₆ ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

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

N	H ≅ G/N	type
3. #365	C₂	semidirect
4. #362	C₂	semidirect

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