Subgroups — #1521

Order 128.

109 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#529	C₂	1	1	yes	maximal, characteristic
2	#431	C₂	1	1	yes	maximal, characteristic, complemented
3	#383	C₂	1	1	yes	maximal, characteristic, complemented
4	#328	C₂	1	2	yes	maximal, complemented
5	#327	C₂	1	2	yes	maximal
6	#126	—	2	1	no	—
7	#125	C₄	1	1	yes	characteristic, complemented
8	#124	C₄	1	1	yes	characteristic, complemented
9	#123	—	2	1	no	—
10	C₂²⋊Q₈	C₂²	1	1	yes	characteristic
11	#121	C₂²	1	1	yes	characteristic
12	C₄×Q₈	C₄	1	1	yes	characteristic, complemented
13	C₄×D₄	—	2	1	no	—
14	C₄×D₄	C₄	1	1	yes	characteristic, complemented
15	#117	—	2	1	no	—
16	C₂×C₄²	C₂²	1	1	yes	characteristic
17	C₄⋊C₈	—	2	1	no	—
18	C₂²⋊C₈	C₂²	1	2	yes	—
19	C₄×C₈	—	2	1	no	—
20	#95	—	2	1	no	—
21	#95	—	2	1	no	—
22	#95	C₂²	1	2	yes	—
23	C₂×Q₈	C₂×C₄	1	1	yes	characteristic
24	C₂×D₄	—	2	1	no	—
25	C₂×D₄	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₄	—	2	1	no	—
27	C₂²×C₄	—	2	2	no	—
28	C₂²×C₄	D₄	1	1	yes	characteristic
29	C₂²×C₄	D₄	1	1	yes	characteristic
30	C₂²×C₄	C₂³	1	1	yes	characteristic
31	C₂×C₈	—	2	2	no	—
32	C₄⋊C₄	—	2	1	no	—
33	C₄⋊C₄	—	2	1	no	—
34	C₄⋊C₄	—	2	1	no	—
35	C₄⋊C₄	—	2	1	no	—
36	C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
37	C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
38	C₂²⋊C₄	—	2	1	no	—
39	C₂²⋊C₄	—	2	1	no	—
40	C₂²⋊C₄	—	2	1	no	—
41	C₂²⋊C₄	—	2	1	no	—
42	C₂²⋊C₄	—	2	1	no	—
43	C₄²	—	2	1	no	—
44	C₄²	C₂×C₄	1	1	yes	characteristic
45	C₄²	C₂×C₄	1	1	yes	characteristic
46	C₄²	D₄	1	1	yes	characteristic
47	C₄²	D₄	1	1	yes	characteristic
48	C₂³	—	2	1	no	—
49	C₂³	C₂×D₄	1	1	yes	characteristic
50	Q₈	—	4	1	no	—
51	D₄	—	4	1	no	—
52	D₄	—	4	2	no	—
53	C₂×C₄	—	2	2	no	—
54	C₂×C₄	—	2	1	no	—
55	C₂×C₄	—	2	1	no	—
56	C₂×C₄	—	2	1	no	—
57	C₂×C₄	—	4	1	no	—
58	C₂×C₄	—	4	1	no	—
59	C₂×C₄	—	4	2	no	—
60	C₂×C₄	—	4	2	no	—
61	C₂×C₄	—	4	2	no	—
62	C₂×C₄	—	4	2	no	—
63	C₂×C₄	C₂²⋊C₄	1	1	yes	characteristic
64	C₂×C₄	C₂²⋊C₄	1	1	yes	characteristic
65	C₂×C₄	C₂²⋊C₄	1	1	yes	characteristic
66	C₂×C₄	C₂²⋊C₄	1	1	yes	characteristic
67	C₂×C₄	C₂²×C₄	1	1	yes	characteristic
68	C₂×C₄	C₂×D₄	1	1	yes	characteristic
69	C₈	—	4	2	no	—
70	C₂²	—	2	1	no	—
71	C₂²	—	2	1	no	—
72	C₂²	—	4	1	no	—
73	C₂²	—	4	1	no	—
74	C₂²	—	4	1	no	—
75	C₂²	C₂×C₂²⋊C₄	1	1	yes	characteristic
76	C₂²	#135	1	2	yes	—
77	C₄	—	2	1	no	—
78	C₄	—	2	1	no	—
79	C₄	—	2	1	no	—
80	C₄	—	2	1	no	—
81	C₄	—	2	1	no	—
82	C₄	—	2	1	no	—
83	C₄	—	4	1	no	—
84	C₄	—	4	1	no	—
85	C₄	—	4	1	no	—
86	C₄	—	8	2	no	—
87	C₂	—	2	2	no	—
88	C₂	—	8	1	no	—
89	C₂	#410	1	1	yes	characteristic
90	C₂	#416	1	1	yes	characteristic
91	C₂	#421	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	C₂×C₂²⋊C₄
commutator	C₂×C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₄	C₂³
prime radical	#121	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁

Complements

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

N	H ≅ G/N	type
2. #431	C₂	semidirect
3. #383	C₂	semidirect
4. #328	C₂	semidirect
7. #125	C₄	semidirect
8. #124	C₄	semidirect
12. C₄×Q₈	C₄	semidirect
14. C₄×D₄	C₄	semidirect

Maximal subgroups

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

subgroup	index
#529	2
#431	2
#383	2
#328	2
#327	2