Subgroups — #12674

Order 256.

179 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#367	C₂	1	1	yes	maximal, characteristic
2	C₂×#366	C₂	1	1	yes	maximal, characteristic
3	C₂×#346	C₂	1	1	yes	maximal, characteristic
4	C₄×#107	C₂	1	2	yes	maximal
5	#1380	C₂	1	2	yes	maximal
6	C₂²×C₁₆	C₂²	1	1	yes	characteristic
7	C₂²×C₁₆	C₂²	1	1	yes	characteristic
8	C₂×#107	C₂²	1	2	yes	—
9	C₂×#107	C₂²	1	2	yes	—
10	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
11	C₄×C₄⋊C₄	—	2	2	no	—
12	#367	C₄	1	4	yes	complemented
13	#366	C₄	1	4	yes	complemented
14	#346	C₄	1	4	yes	complemented
15	#340	—	2	2	no	—
16	C₂²×C₈	D₄	1	1	yes	characteristic
17	C₂²×C₈	C₂³	1	1	yes	characteristic
18	C₂×C₄⋊C₄	—	2	2	no	—
19	C₂×C₄⋊C₄	—	2	2	no	—
20	C₂×C₄²	—	4	2	no	—
21	C₂×C₄²	D₄	1	1	yes	characteristic
22	C₂×C₁₆	—	2	1	no	—
23	C₂×C₁₆	—	2	1	no	—
24	C₂×C₁₆	C₂×C₄	1	2	yes	—
25	C₂×C₁₆	C₂×C₄	1	2	yes	—
26	C₂×C₁₆	C₂×C₄	1	2	yes	—
27	C₂×C₁₆	C₂×C₄	1	2	yes	—
28	#107	—	2	2	no	—
29	#107	—	2	2	no	—
30	#107	C₂×C₄	1	4	yes	—
31	#107	C₂×C₄	1	4	yes	—
32	C₄×C₈	C₂×C₄	1	2	yes	—
33	C₄×C₈	Q₈	1	2	yes	—
34	#95	—	4	2	no	—
35	C₂²×C₄	—	4	2	no	—
36	C₂²×C₄	—	4	2	no	—
37	C₂²×C₄	D₈	1	2	yes	—
38	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
39	C₂×C₈	—	2	1	no	—
40	C₂×C₈	—	2	1	no	—
41	C₂×C₈	C₄⋊C₄	1	2	yes	—
42	C₂×C₈	C₂²×C₄	1	1	yes	characteristic
43	C₂×C₈	C₂²×C₄	1	1	yes	characteristic
44	C₂×C₈	C₂²×C₄	1	1	yes	characteristic
45	C₂×C₈	C₂×Q₈	1	1	yes	characteristic
46	C₂×C₈	#41	1	1	yes	characteristic
47	C₂×C₈	#41	1	1	yes	characteristic
48	C₄⋊C₄	—	2	4	no	—
49	C₄⋊C₄	—	2	4	no	—
50	C₄⋊C₄	—	4	2	no	—
51	C₄⋊C₄	—	4	2	no	—
52	C₄²	—	8	4	no	—
53	C₄²	C₄⋊C₄	1	2	yes	—
54	C₄²	Q₁₆	1	2	yes	—
55	C₁₆	—	2	2	no	—
56	C₁₆	C₄²	1	4	yes	complemented
57	C₂³	C₂×D₈	1	1	yes	characteristic
58	C₂×C₄	—	2	2	no	—
59	C₂×C₄	—	2	2	no	—
60	C₂×C₄	—	4	4	no	—
61	C₂×C₄	—	4	4	no	—
62	C₂×C₄	—	8	2	no	—
63	C₂×C₄	—	8	2	no	—
64	C₂×C₄	—	8	2	no	—
65	C₂×C₄	—	8	2	no	—
66	C₂×C₄	#107	1	4	yes	—
67	C₂×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
68	C₂×C₄	C₄×D₄	1	1	yes	characteristic
69	C₂×C₄	C₄×D₄	1	1	yes	characteristic
70	C₂×C₄	C₂×Q₁₆	1	1	yes	characteristic
71	C₂×C₄	#135	1	1	yes	characteristic
72	C₂×C₄	#135	1	1	yes	characteristic
73	C₈	—	2	2	no	—
74	C₈	C₂×C₄²	1	1	yes	characteristic
75	C₈	#117	1	1	yes	characteristic
76	C₈	C₄×Q₈	1	1	yes	characteristic
77	C₈	C₄×Q₈	1	1	yes	characteristic
78	C₂²	C₂×#107	1	1	yes	characteristic
79	C₂²	C₄×D₈	1	1	yes	characteristic
80	C₂²	C₄×D₈	1	1	yes	characteristic
81	C₂²	#509	1	2	yes	—
82	C₂²	#510	1	2	yes	—
83	C₄	—	2	4	no	—
84	C₄	—	8	4	no	—
85	C₄	—	8	4	no	—
86	C₄	C₄×C₄⋊C₄	1	1	yes	characteristic
87	C₄	#427	1	1	yes	characteristic
88	C₄	C₄×Q₁₆	1	1	yes	characteristic
89	C₄	C₄×Q₁₆	1	1	yes	characteristic
90	C₂	C₄×#107	1	1	yes	characteristic
91	C₂	#2159	1	2	yes	—
92	C₂	#2175	1	2	yes	—
93	C₂	#2176	1	1	yes	characteristic
94	C₂	#2177	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³	C₂×D₈
commutator	C₈	C₂×C₄²
Frattini, cosocle	C₂²×C₈	C₂³
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
12. #367	C₄	semidirect
13. #366	C₄	semidirect
14. #346	C₄	semidirect
56. C₁₆	C₄²	semidirect

Maximal subgroups

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

subgroup	index
C₂×#367	2
C₂×#366	2
C₂×#346	2
C₄×#107	2
#1380	2