Subgroups — #2189

Order 128.

110 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#135	C₂	1	1	yes	maximal, characteristic
2	C₂×Q₃₂	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×SD₃₂	C₂	1	1	yes	maximal, characteristic, complemented
4	#467	C₂	1	1	yes	maximal, characteristic, complemented
5	#358	C₂	1	1	yes	maximal, characteristic, complemented
6	#357	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#41	—	2	1	no	—
9	#135	—	2	2	no	—
10	#135	—	2	2	no	—
11	C₂×Q₁₆	C₂²	1	1	yes	characteristic
12	C₂×Q₁₆	C₂²	1	1	yes	characteristic
13	C₂×SD₁₆	—	2	1	no	—
14	C₂×D₈	C₂²	1	1	yes	characteristic
15	C₂²×C₈	C₂²	1	1	yes	characteristic
16	C₂²⋊Q₈	—	2	1	no	—
17	Q₃₂	—	2	2	no	—
18	SD₃₂	—	2	2	no	—
19	C₂×C₁₆	C₂²	1	1	yes	characteristic
20	C₂×C₁₆	C₂²	1	1	yes	characteristic
21	#107	C₂²	1	1	yes	characteristic
22	#103	—	2	1	no	—
23	#41	—	2	4	no	—
24	#41	—	4	2	no	—
25	C₂×Q₈	—	2	1	no	—
26	C₂×Q₈	—	2	1	no	—
27	C₂×D₄	—	2	1	no	—
28	C₂×D₄	—	4	1	no	—
29	C₂²×C₄	—	4	1	no	—
30	C₂²×C₄	D₄	1	1	yes	characteristic
31	Q₁₆	—	2	1	no	—
32	Q₁₆	—	2	1	no	—
33	Q₁₆	—	2	1	no	—
34	Q₁₆	D₄	1	2	yes	—
35	SD₁₆	—	2	4	no	—
36	D₈	—	2	1	no	—
37	D₈	D₄	1	2	yes	—
38	C₂×C₈	—	2	1	no	—
39	C₂×C₈	—	2	1	no	—
40	C₂×C₈	D₄	1	1	yes	characteristic
41	C₂×C₈	C₂³	1	1	yes	characteristic
42	C₄⋊C₄	—	2	1	no	—
43	C₄⋊C₄	—	4	1	no	—
44	C₂²⋊C₄	—	4	1	no	—
45	C₁₆	—	2	1	no	—
46	C₁₆	—	2	1	no	—
47	C₂³	—	4	1	no	—
48	C₂³	D₈	1	1	yes	characteristic
49	Q₈	—	2	2	no	—
50	Q₈	—	4	1	no	—
51	Q₈	—	4	1	no	—
52	Q₈	—	4	1	no	—
53	D₄	—	2	2	no	—
54	D₄	—	4	4	no	—
55	D₄	—	4	1	no	—
56	C₂×C₄	—	2	1	no	—
57	C₂×C₄	—	2	1	no	—
58	C₂×C₄	—	4	4	no	—
59	C₂×C₄	—	4	1	no	—
60	C₂×C₄	—	4	1	no	—
61	C₂×C₄	—	4	1	no	—
62	C₂×C₄	D₈	1	1	yes	characteristic
63	C₂×C₄	C₂×D₄	1	1	yes	characteristic
64	C₈	—	2	1	no	—
65	C₈	C₂×D₄	1	1	yes	characteristic
66	C₈	C₂×D₄	1	1	yes	characteristic
67	C₂²	—	2	1	no	—
68	C₂²	—	2	1	no	—
69	C₂²	—	2	1	no	—
70	C₂²	—	4	2	no	—
71	C₂²	—	8	1	no	—
72	C₂²	—	8	1	no	—
73	C₂²	C₂×D₈	1	1	yes	characteristic
74	C₄	—	2	1	no	—
75	C₄	—	4	2	no	—
76	C₄	—	8	1	no	—
77	C₄	—	8	1	no	—
78	C₄	#120	1	1	yes	characteristic
79	C₄	AGL(1,ℤ/8)	1	1	yes	characteristic
80	C₂	—	4	1	no	—
81	C₂	—	8	2	no	—
82	C₂	#447	1	1	yes	characteristic
83	C₂	#508	1	1	yes	characteristic
84	C₂	#510	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	C₂×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂³
prime radical	C₂×#135	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
2. C₂×Q₃₂	C₂	semidirect
3. C₂×SD₃₂	C₂	semidirect
4. #467	C₂	semidirect
5. #358	C₂	semidirect
6. #357	C₂	semidirect
7. C₂²⋊C₁₆	C₂	semidirect

Maximal subgroups

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

subgroup	index
C₂×#135	2
C₂×Q₃₂	2
C₂×SD₃₂	2
#467	2
#358	2
#357	2
C₂²⋊C₁₆	2