Subgroups — #1347

Order 128.

63 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#466	C₂	1	1	yes	maximal, characteristic
2	C₂²⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
3	#340	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×D₈	C₄	1	1	yes	characteristic, complemented
5	C₂²×C₈	C₂²	1	1	yes	characteristic
6	#121	—	2	1	no	—
7	C₂×C₄⋊C₄	—	2	1	no	—
8	C₂×C₁₆	—	2	1	no	—
9	#107	C₄	1	1	yes	characteristic, complemented
10	#102	—	2	1	no	—
11	C₂×D₄	—	2	1	no	—
12	C₂×D₄	—	4	1	no	—
13	C₂²×C₄	—	4	1	no	—
14	C₂²×C₄	D₄	1	1	yes	characteristic
15	D₈	—	4	1	no	—
16	C₂×C₈	—	4	1	no	—
17	C₂×C₈	C₂×C₄	1	1	yes	characteristic
18	C₂×C₈	D₄	1	1	yes	characteristic
19	C₄⋊C₄	—	2	2	no	—
20	C₄⋊C₄	—	2	1	no	—
21	C₄⋊C₄	—	4	1	no	—
22	C₂²⋊C₄	—	4	1	no	—
23	C₁₆	—	4	1	no	—
24	C₂³	—	4	1	no	—
25	C₂³	D₈	1	1	yes	characteristic
26	D₄	—	4	1	no	—
27	D₄	—	4	1	no	—
28	D₄	—	8	1	no	—
29	D₄	—	8	1	no	—
30	C₂×C₄	—	2	1	no	—
31	C₂×C₄	—	2	1	no	—
32	C₂×C₄	—	4	2	no	—
33	C₂×C₄	—	4	1	no	—
34	C₂×C₄	—	8	1	no	—
35	C₂×C₄	—	8	1	no	—
36	C₂×C₄	C₂²⋊C₄	1	1	yes	characteristic
37	C₂×C₄	SD₁₆	1	1	yes	characteristic
38	C₈	—	2	1	no	—
39	C₈	—	2	1	no	—
40	C₂²	—	2	1	no	—
41	C₂²	—	2	1	no	—
42	C₂²	—	8	1	no	—
43	C₂²	—	8	1	no	—
44	C₂²	—	8	1	no	—
45	C₂²	#102	1	1	yes	characteristic
46	C₂²	D₁₆	1	1	yes	characteristic
47	C₂²	SD₃₂	1	1	yes	characteristic
48	C₄	—	2	1	no	—
49	C₄	—	8	2	no	—
50	C₄	—	8	1	no	—
51	C₄	#99	1	1	yes	characteristic
52	C₄	#104	1	1	yes	characteristic
53	C₂	—	2	1	no	—
54	C₂	—	2	1	no	—
55	C₂	—	16	1	no	—
56	C₂	#327	1	1	yes	characteristic
57	C₂	#357	1	1	yes	characteristic
58	C₂	#360	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	#102
commutator	C₂×C₈	C₂×C₄
Frattini, cosocle	C₂²×C₈	C₂²
prime radical	#466	C₂
LCS	G ▷ C₂×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
2. C₂²⋊C₁₆	C₂	semidirect
3. #340	C₂	semidirect
4. C₂×D₈	C₄	semidirect
9. #107	C₄	semidirect

Maximal subgroups

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

subgroup	index
#466	2
C₂²⋊C₁₆	2
#340	2