Subgroups — #3287

Order 128.

252 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#135	C₂	1	2	yes	maximal, complemented
2	#550	C₂	1	2	yes	maximal, complemented
3	#493	C₂	1	1	yes	maximal, characteristic, complemented
4	#466	C₂	1	2	yes	maximal, complemented
5	#460	C₂	1	2	yes	maximal, complemented
6	#449	C₂	1	4	yes	maximal, complemented
7	C₄×Q₁₆	C₂	1	1	yes	maximal, characteristic, complemented
8	C₈×D₄	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#41	C₂²	1	4	yes	—
10	#135	—	2	8	no	—
11	C₂×Q₁₆	C₂²	1	1	yes	characteristic
12	C₂×SD₁₆	C₂²	1	4	yes	—
13	C₂×D₈	C₂²	1	2	yes	—
14	C₂×D₈	C₂²	1	2	yes	—
15	C₂²×C₈	C₂²	1	2	yes	—
16	#127	—	2	2	no	—
17	#127	C₂²	1	2	yes	—
18	#121	—	2	4	no	—
19	#121	C₂²	1	4	yes	—
20	C₄×Q₈	C₂²	1	2	yes	—
21	C₄×D₄	—	2	2	no	—
22	C₄×D₄	C₂²	1	1	yes	characteristic
23	#107	C₂²	1	1	yes	characteristic
24	C₄⋊C₈	C₂²	1	1	yes	characteristic
25	#103	C₂²	1	2	yes	—
26	#102	C₂²	1	4	yes	—
27	C₂²⋊C₈	C₂²	1	2	yes	—
28	C₄×C₈	C₂²	1	1	yes	characteristic
29	#41	—	2	8	no	—
30	#41	—	4	4	no	—
31	C₂×Q₈	C₂³	1	2	yes	—
32	C₂×D₄	—	2	4	no	—
33	C₂×D₄	—	2	4	no	—
34	C₂×D₄	—	2	2	no	—
35	C₂×D₄	—	2	2	no	—
36	C₂×D₄	D₄	1	1	yes	characteristic
37	C₂×D₄	C₂³	1	4	yes	—
38	C₂²×C₄	—	2	4	no	—
39	C₂²×C₄	C₂³	1	2	yes	—
40	Q₁₆	D₄	1	4	yes	complemented
41	SD₁₆	—	2	8	no	—
42	D₈	—	2	4	no	—
43	D₈	—	4	2	no	—
44	C₂×C₈	—	2	4	no	—
45	C₂×C₈	C₂³	1	1	yes	characteristic
46	C₂×C₈	C₂³	1	2	yes	—
47	C₂×C₈	C₂³	1	1	yes	characteristic
48	C₄⋊C₄	—	2	2	no	—
49	C₄⋊C₄	D₄	1	1	yes	characteristic
50	C₄⋊C₄	C₂³	1	2	yes	—
51	C₂²⋊C₄	—	2	4	no	—
52	C₂²⋊C₄	D₄	1	2	yes	—
53	C₄²	—	2	2	no	—
54	C₄²	C₂³	1	1	yes	characteristic
55	C₂³	—	2	4	no	—
56	C₂³	C₂×D₄	1	2	yes	—
57	Q₈	—	2	2	no	—
58	Q₈	C₂×D₄	1	4	yes	—
59	D₄	—	2	2	no	—
60	D₄	—	2	4	no	—
61	D₄	—	2	4	no	—
62	D₄	—	4	8	no	—
63	D₄	—	4	4	no	—
64	D₄	—	8	2	no	—
65	D₄	—	8	4	no	—
66	C₂×C₄	—	2	2	no	—
67	C₂×C₄	—	2	2	no	—
68	C₂×C₄	—	2	2	no	—
69	C₂×C₄	—	2	2	no	—
70	C₂×C₄	—	4	8	no	—
71	C₂×C₄	C₂×D₄	1	1	yes	characteristic
72	C₂×C₄	C₂×D₄	1	2	yes	—
73	C₂×C₄	C₂×D₄	1	1	yes	characteristic
74	C₂×C₄	C₂⁴	1	1	yes	characteristic
75	C₈	—	2	2	no	—
76	C₈	—	2	1	no	—
77	C₈	C₂×D₄	1	2	yes	—
78	C₂²	—	2	2	no	—
79	C₂²	—	2	2	no	—
80	C₂²	—	2	2	no	—
81	C₂²	—	4	4	no	—
82	C₂²	—	4	4	no	—
83	C₂²	—	4	4	no	—
84	C₂²	C₂²×D₄	1	1	yes	characteristic
85	C₄	—	2	2	no	—
86	C₄	—	2	4	no	—
87	C₄	—	2	1	no	—
88	C₄	—	4	2	no	—
89	C₄	#135	1	2	yes	complemented
90	C₄	C₂²×D₄	1	1	yes	characteristic
91	C₄	H(5,𝔽₂)	1	1	yes	characteristic
92	C₂	—	4	2	no	—
93	C₂	—	8	4	no	—
94	C₂	D₄²	1	1	yes	characteristic
95	C₂	C₂×#135	1	1	yes	characteristic
96	C₂	#576	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×D₄ ◁ G
LECS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
1. C₂×#135	C₂	semidirect
2. #550	C₂	semidirect
3. #493	C₂	semidirect
4. #466	C₂	semidirect
5. #460	C₂	semidirect
6. #449	C₂	semidirect
7. C₄×Q₁₆	C₂	semidirect
8. C₈×D₄	C₂	semidirect
40. Q₁₆	D₄	semidirect
89. C₄	#135	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
#550	2
#493	2
#466	2
#460	2
#449	2
C₄×Q₁₆	2
C₈×D₄	2