Subgroups — #2222

Order 128.

70 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Q₃₂	C₂	1	1	yes	maximal, characteristic, complemented
2	#504	C₂	1	1	yes	maximal, characteristic
3	#473	C₂	1	2	yes	maximal
4	#368	C₂	1	1	yes	maximal, characteristic, complemented
5	#362	C₂	1	2	yes	maximal, complemented
6	#137	—	2	2	no	—
7	C₂×Q₁₆	C₂²	1	2	yes	—
8	#131	C₂²	1	1	yes	characteristic
9	Q₃₂	—	2	1	no	—
10	Q₃₂	—	2	1	no	—
11	M₃₂	—	2	1	no	—
12	M₃₂	C₂²	1	1	yes	characteristic
13	C₂×C₁₆	—	2	1	no	—
14	C₂×C₁₆	C₂²	1	1	yes	characteristic
15	#108	C₂²	1	2	yes	—
16	#101	—	2	2	no	—
17	#41	D₄	1	1	yes	characteristic
18	C₂×Q₈	—	2	2	no	—
19	Q₁₆	—	2	2	no	—
20	Q₁₆	—	2	2	no	—
21	Q₁₆	—	4	2	no	—
22	SD₁₆	—	4	2	no	—
23	M₁₆	—	2	1	no	—
24	M₁₆	—	2	2	no	—
25	M₁₆	D₄	1	1	yes	characteristic
26	C₂×C₈	—	2	1	no	—
27	C₂×C₈	C₂³	1	1	yes	characteristic
28	C₁₆	—	2	1	no	—
29	C₁₆	D₄	1	1	yes	characteristic
30	C₁₆	D₄	1	1	yes	characteristic
31	Q₈	—	4	2	no	—
32	Q₈	—	4	2	no	—
33	Q₈	D₈	1	1	yes	characteristic
34	D₄	—	2	1	no	—
35	D₄	D₈	1	1	yes	characteristic
36	C₂×C₄	—	2	1	no	—
37	C₂×C₄	—	4	2	no	—
38	C₂×C₄	C₂×D₄	1	1	yes	characteristic
39	C₈	—	2	1	no	—
40	C₈	—	4	2	no	—
41	C₈	C₂×D₄	1	1	yes	characteristic
42	C₈	#41	1	1	yes	characteristic
43	C₂²	—	2	1	no	—
44	C₂²	#135	1	1	yes	characteristic
45	C₄	—	2	1	no	—
46	C₄	—	8	2	no	—
47	C₄	#121	1	1	yes	characteristic
48	C₄	C₂×D₈	1	1	yes	characteristic
49	C₂	—	2	1	no	—
50	C₂	—	4	1	no	—
51	C₂	#466	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂	#466
commutator, Frattini, cosocle	C₂×C₈	C₂³
prime radical	#41	D₄
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #131 ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

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

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