Subgroups — #3352

Order 128.

168 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#552	C₂	1	2	yes	maximal
2	#499	C₂	1	1	yes	maximal, characteristic, complemented
3	#485	C₂	1	2	yes	maximal, complemented
4	#484	C₂	1	2	yes	maximal, complemented
5	#479	C₂	1	2	yes	maximal, complemented
6	#470	C₂	1	2	yes	maximal, complemented
7	C₄×Q₁₆	C₂	1	1	yes	maximal, characteristic, complemented
8	#436	C₂	1	1	yes	maximal, characteristic
9	#428	C₂	1	2	yes	maximal, complemented
10	C₂×Q₁₆	C₂²	1	1	yes	characteristic
11	C₂×M₁₆	C₂²	1	2	yes	—
12	#126	—	2	2	no	—
13	#125	—	2	2	no	—
14	#125	C₂²	1	2	yes	—
15	#123	—	2	2	no	—
16	C₂²⋊Q₈	C₂²	1	2	yes	—
17	C₂²⋊Q₈	C₂²	1	2	yes	—
18	C₄×Q₈	C₂²	1	2	yes	—
19	C₄×D₄	C₂²	1	1	yes	characteristic
20	#117	—	2	2	no	—
21	#117	C₂²	1	2	yes	—
22	C₂×C₄⋊C₄	C₂²	1	2	yes	—
23	#107	C₂²	1	1	yes	characteristic
24	#107	C₂²	1	2	yes	—
25	#107	C₂²	1	1	yes	characteristic
26	#107	C₂²	1	1	yes	characteristic
27	#106	C₂²	1	2	yes	—
28	#106	C₂²	1	2	yes	—
29	C₄⋊C₈	C₂²	1	1	yes	characteristic
30	#103	C₂²	1	2	yes	—
31	#103	C₂²	1	2	yes	—
32	#103	C₂²	1	2	yes	—
33	C₂²⋊C₈	C₂²	1	2	yes	—
34	C₄×C₈	C₂²	1	1	yes	characteristic
35	C₂×Q₈	C₂³	1	2	yes	—
36	C₂×D₄	D₄	1	1	yes	characteristic
37	C₂²×C₄	—	2	2	no	—
38	C₂²×C₄	C₂³	1	2	yes	—
39	Q₁₆	—	2	2	no	—
40	M₁₆	—	2	4	no	—
41	C₂×C₈	C₂³	1	1	yes	characteristic
42	C₂×C₈	C₂³	1	2	yes	—
43	C₂×C₈	C₂³	1	1	yes	characteristic
44	C₄⋊C₄	—	2	2	no	—
45	C₄⋊C₄	—	2	2	no	—
46	C₄⋊C₄	—	2	2	no	—
47	C₄⋊C₄	—	2	2	no	—
48	C₄⋊C₄	—	2	2	no	—
49	C₄⋊C₄	—	2	2	no	—
50	C₄⋊C₄	D₄	1	1	yes	characteristic
51	C₄⋊C₄	C₂³	1	2	yes	—
52	C₄⋊C₄	C₂³	1	2	yes	—
53	C₄⋊C₄	C₂³	1	2	yes	—
54	C₂²⋊C₄	—	2	2	no	—
55	C₂²⋊C₄	—	2	2	no	—
56	C₂²⋊C₄	—	2	2	no	—
57	C₂²⋊C₄	D₄	1	2	yes	—
58	C₄²	—	2	2	no	—
59	C₄²	—	2	2	no	—
60	C₄²	C₂³	1	1	yes	characteristic
61	C₂³	C₂×D₄	1	2	yes	—
62	Q₈	—	2	2	no	—
63	Q₈	—	2	2	no	—
64	D₄	—	4	1	no	—
65	C₂×C₄	—	2	2	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	—	2	2	no	—
71	C₂×C₄	—	4	4	no	—
72	C₂×C₄	C₂×D₄	1	1	yes	characteristic
73	C₂×C₄	C₂×D₄	1	2	yes	—
74	C₂×C₄	C₂×D₄	1	1	yes	characteristic
75	C₂×C₄	C₂⁴	1	1	yes	characteristic
76	C₈	—	2	2	no	—
77	C₈	—	2	1	no	—
78	C₈	#41	1	2	yes	—
79	C₂²	—	2	2	no	—
80	C₂²	—	2	2	no	—
81	C₂²	—	2	2	no	—
82	C₂²	C₂²×D₄	1	1	yes	characteristic
83	C₄	—	2	2	no	—
84	C₄	—	2	1	no	—
85	C₄	—	2	1	no	—
86	C₄	—	4	2	no	—
87	C₄	—	4	2	no	—
88	C₄	—	4	2	no	—
89	C₄	—	4	2	no	—
90	C₄	C₂×#41	1	1	yes	characteristic
91	C₄	2₋¹⁺⁴	1	1	yes	characteristic
92	C₂	—	4	2	no	—
93	C₂	#547	1	1	yes	characteristic
94	C₂	#575	1	1	yes	characteristic
95	C₂	#578	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴
prime radical	C₂×D₄	D₄
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
2. #499	C₂	semidirect
3. #485	C₂	semidirect
4. #484	C₂	semidirect
5. #479	C₂	semidirect
6. #470	C₂	semidirect
7. C₄×Q₁₆	C₂	semidirect
9. #428	C₂	semidirect

Maximal subgroups

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

subgroup	index
#552	2
#499	2
#485	2
#484	2
#479	2
#470	2
C₄×Q₁₆	2
#436	2
#428	2