Subgroups — #8278

Order 256.

197 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#1767	C₂	1	1	yes	maximal, characteristic
2	C₄×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic
3	#1744	C₂	1	1	yes	maximal, characteristic
4	C₂×#335	C₂	1	2	yes	maximal
5	#1277	C₂	1	2	yes	maximal
6	C₂×C₄⋊C₈	C₂²	1	2	yes	—
7	C₂×C₄⋊C₈	C₂²	1	2	yes	—
8	C₂×#97	—	2	1	no	—
9	C₂×C₄×C₈	—	2	1	no	—
10	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
11	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
12	C₄³	C₂²	1	1	yes	characteristic
13	#336	—	2	1	no	—
14	#336	—	2	1	no	—
15	#336	C₄	1	4	yes	—
16	#335	C₄	1	8	yes	complemented
17	C₂²×C₈	—	2	2	no	—
18	C₂²×C₈	—	2	2	no	—
19	C₂²×C₈	C₂×C₄	1	2	yes	—
20	C₂²×C₈	C₂×C₄	1	2	yes	—
21	C₂×C₄²	C₂×C₄	1	2	yes	—
22	C₂×C₄²	D₄	1	1	yes	characteristic
23	C₂×C₄²	D₄	1	1	yes	characteristic
24	C₂×C₄²	Q₈	1	2	yes	—
25	C₂×C₄²	C₂³	1	1	yes	characteristic
26	C₄⋊C₈	—	2	2	no	—
27	C₄⋊C₈	—	2	2	no	—
28	C₄⋊C₈	C₂×C₄	1	4	yes	—
29	C₄⋊C₈	C₂×C₄	1	4	yes	—
30	#97	—	4	2	no	—
31	C₄×C₈	—	2	2	no	—
32	C₄×C₈	—	4	2	no	—
33	C₄×C₈	C₂×C₄	1	4	yes	—
34	C₂²×C₄	C₄⋊C₄	1	2	yes	—
35	C₂²×C₄	C₄⋊C₄	1	2	yes	—
36	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
37	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
38	C₂²×C₄	C₂×Q₈	1	1	yes	characteristic
39	C₂×C₈	—	2	4	no	—
40	C₂×C₈	—	2	2	no	—
41	C₂×C₈	—	2	4	no	—
42	C₂×C₈	—	2	4	no	—
43	C₂×C₈	—	2	4	no	—
44	C₂×C₈	—	4	2	no	—
45	C₂×C₈	—	4	2	no	—
46	C₂×C₈	—	4	2	no	—
47	C₂×C₈	—	4	2	no	—
48	C₂×C₈	C₄²	1	4	yes	—
49	C₄²	—	2	4	no	—
50	C₄²	—	2	2	no	—
51	C₄²	—	4	2	no	—
52	C₄²	D₈	1	2	yes	—
53	C₄²	Q₁₆	1	2	yes	—
54	C₄²	C₂²×C₄	1	2	yes	—
55	C₄²	#41	1	2	yes	—
56	C₂³	C₂×C₄⋊C₄	1	1	yes	characteristic
57	C₂×C₄	—	2	2	no	—
58	C₂×C₄	—	2	2	no	—
59	C₂×C₄	—	2	2	no	—
60	C₂×C₄	—	2	2	no	—
61	C₂×C₄	—	2	2	no	—
62	C₂×C₄	—	2	2	no	—
63	C₂×C₄	#107	1	4	yes	—
64	C₂×C₄	C₂×C₄²	1	1	yes	characteristic
65	C₂×C₄	#117	1	1	yes	characteristic
66	C₂×C₄	C₄×D₄	1	2	yes	—
67	C₂×C₄	C₄×Q₈	1	2	yes	—
68	C₂×C₄	#131	1	4	yes	—
69	C₂×C₄	C₂×D₈	1	1	yes	characteristic
70	C₂×C₄	C₂×Q₁₆	1	1	yes	characteristic
71	C₂×C₄	#135	1	2	yes	—
72	C₈	—	2	4	no	—
73	C₈	—	4	2	no	—
74	C₈	—	4	4	no	—
75	C₈	—	4	4	no	—
76	C₂²	C₄×C₄⋊C₄	1	1	yes	characteristic
77	C₂²	#424	1	2	yes	—
78	C₂²	C₂×#107	1	1	yes	characteristic
79	C₂²	#427	1	1	yes	characteristic
80	C₂²	#430	1	2	yes	—
81	C₄	—	2	4	no	—
82	C₄	—	2	2	no	—
83	C₄	—	4	2	no	—
84	C₄	#405	1	2	yes	—
85	C₄	#433	1	2	yes	—
86	C₄	C₄×D₈	1	2	yes	—
87	C₄	C₄×Q₁₆	1	2	yes	—
88	C₂	#1571	1	2	yes	—
89	C₂	#1572	1	2	yes	—
90	C₂	#1768	1	1	yes	characteristic
91	C₂	C₄×#107	1	1	yes	characteristic
92	C₂	#1778	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×D₄
commutator	C₂×C₄	C₂×C₄²
Frattini, cosocle	C₂×C₄²	C₂³
socle, prime radical	C₂³	C₂×C₄⋊C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

Complements

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

N	H ≅ G/N	type
16. #335	C₄	semidirect

Maximal subgroups

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

subgroup	index
#1767	2
C₄×C₄⋊C₈	2
#1744	2
C₂×#335	2
#1277	2