Subgroups — #9778

Order 256.

195 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#500	C₂	1	1	yes	maximal, characteristic
2	#1837	C₂	1	1	yes	maximal, characteristic
3	#1549	C₂	1	2	yes	maximal
4	#1548	C₂	1	2	yes	maximal
5	#1452	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊Q₈	C₂²	1	2	yes	—
7	#500	—	2	2	no	—
8	#500	C₄	1	4	yes	—
9	#431	C₂²	1	1	yes	characteristic
10	#431	C₂²	1	1	yes	characteristic
11	C₂×#107	—	2	2	no	—
12	#423	C₂²	1	2	yes	—
13	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
14	#341	—	2	2	no	—
15	#333	—	2	4	no	—
16	#332	—	2	4	no	—
17	#322	—	2	2	no	—
18	C₂²×Q₈	—	2	2	no	—
19	C₂×M₁₆	—	2	2	no	—
20	C₂²×C₈	D₄	1	2	yes	—
21	C₄⋊Q₈	—	2	2	no	—
22	C₄⋊Q₈	—	2	2	no	—
23	C₄⋊Q₈	—	2	2	no	—
24	C₄⋊Q₈	C₂×C₄	1	4	yes	—
25	C₂×C₄⋊C₄	—	2	2	no	—
26	C₂×C₄⋊C₄	—	2	2	no	—
27	C₂×C₄²	C₂³	1	1	yes	characteristic
28	#107	—	2	8	no	—
29	C₄⋊C₈	—	2	2	no	—
30	C₄⋊C₈	—	2	2	no	—
31	C₄⋊C₈	D₄	1	2	yes	—
32	C₄⋊C₈	D₄	1	2	yes	—
33	C₂²⋊C₈	—	2	1	no	—
34	C₂²⋊C₈	—	2	1	no	—
35	C₄×C₈	—	2	1	no	—
36	C₄×C₈	—	2	1	no	—
37	C₄×C₈	C₂×C₄	1	2	yes	—
38	C₄×C₈	D₄	1	2	yes	—
39	C₂×Q₈	—	2	4	no	—
40	C₂×Q₈	—	4	2	no	—
41	C₂×Q₈	—	8	4	no	—
42	C₂²×C₄	—	4	2	no	—
43	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
44	C₂²×C₄	#41	1	1	yes	characteristic
45	C₂²×C₄	#41	1	1	yes	characteristic
46	M₁₆	—	4	4	no	—
47	C₂×C₈	—	2	4	no	—
48	C₂×C₈	—	2	2	no	—
49	C₂×C₈	—	2	2	no	—
50	C₂×C₈	—	4	2	no	—
51	C₂×C₈	C₂²⋊C₄	1	4	yes	—
52	C₄⋊C₄	—	2	4	no	—
53	C₄⋊C₄	—	2	4	no	—
54	C₄⋊C₄	—	4	2	no	—
55	C₄⋊C₄	—	4	2	no	—
56	C₄²	C₂²×C₄	1	1	yes	characteristic
57	C₄²	C₂×D₄	1	1	yes	characteristic
58	C₄²	C₂×D₄	1	1	yes	characteristic
59	C₄²	C₂×D₄	1	1	yes	characteristic
60	C₂³	#124	1	1	yes	characteristic
61	Q₈	—	8	4	no	—
62	Q₈	—	8	4	no	—
63	C₂×C₄	—	2	2	no	—
64	C₂×C₄	—	2	2	no	—
65	C₂×C₄	—	4	4	no	—
66	C₂×C₄	—	8	4	no	—
67	C₂×C₄	C₂×C₂²⋊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	#121	1	1	yes	characteristic
71	C₂×C₄	#121	1	1	yes	characteristic
72	C₂×C₄	#127	1	1	yes	characteristic
73	C₂×C₄	#137	1	4	yes	—
74	C₈	—	2	4	no	—
75	C₈	—	4	2	no	—
76	C₈	—	4	2	no	—
77	C₈	—	8	2	no	—
78	C₂²	—	2	1	no	—
79	C₂²	—	2	1	no	—
80	C₂²	#390	1	1	yes	characteristic
81	C₂²	#491	1	2	yes	—
82	C₄	—	2	1	no	—
83	C₄	—	2	1	no	—
84	C₄	—	8	4	no	—
85	C₄	#419	1	2	yes	—
86	C₄	#472	1	2	yes	—
87	C₄	#473	1	2	yes	—
88	C₄	#497	1	2	yes	—
89	C₂	—	2	2	no	—
90	C₂	#1960	1	1	yes	characteristic
91	C₂	#1978	1	1	yes	characteristic
92	C₂	#1979	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	#390
commutator	C₄²	C₂²×C₄
Frattini, cosocle	C₂×C₄²	C₂³
prime radical	C₂³	#124
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

Maximal subgroups

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

subgroup	index
C₂×#500	2
#1837	2
#1549	2
#1548	2
#1452	2