Subgroups — #9839

Order 256.

271 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#494	C₂	1	1	yes	maximal, characteristic
2	#1990	C₂	1	4	yes	maximal
3	#1844	C₂	1	1	yes	maximal, characteristic
4	C₂×#322	C₂	1	1	yes	maximal, characteristic
5	C₂²×Q₁₆	—	2	2	no	—
6	C₂×C₄⋊Q₈	C₂²	1	2	yes	—
7	#494	—	2	2	no	—
8	#494	C₄	1	4	yes	complemented
9	C₂×#106	—	2	2	no	—
10	C₂×C₄×C₈	C₂²	1	2	yes	—
11	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
12	#384	C₂²	1	2	yes	—
13	#340	—	2	8	no	—
14	#322	—	2	2	no	—
15	C₂²×Q₈	—	2	2	no	—
16	C₂×Q₁₆	—	2	8	no	—
17	C₂×Q₁₆	—	4	4	no	—
18	C₂×Q₁₆	—	4	4	no	—
19	C₂²×C₈	D₄	1	4	yes	—
20	C₂²×C₈	D₄	1	2	yes	—
21	C₄⋊Q₈	—	2	2	no	—
22	C₄⋊Q₈	—	2	4	no	—
23	C₄⋊Q₈	C₂×C₄	1	4	yes	—
24	C₂×C₄⋊C₄	—	2	4	no	—
25	C₂×C₄⋊C₄	—	2	4	no	—
26	C₂×C₄⋊C₄	—	2	2	no	—
27	C₂×C₄²	C₂³	1	1	yes	characteristic
28	#106	—	4	4	no	—
29	C₄×C₈	—	2	2	no	—
30	C₄×C₈	—	2	2	no	—
31	C₄×C₈	C₂×C₄	1	2	yes	—
32	C₄×C₈	D₄	1	2	yes	—
33	C₂×Q₈	—	2	4	no	—
34	C₂×Q₈	—	4	2	no	—
35	C₂×Q₈	—	4	4	no	—
36	C₂×Q₈	—	4	4	no	—
37	C₂²×C₄	—	4	2	no	—
38	C₂²×C₄	—	4	2	no	—
39	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
40	C₂²×C₄	C₂×D₄	1	2	yes	—
41	Q₁₆	—	4	16	no	—
42	C₂×C₈	—	2	4	no	—
43	C₂×C₈	—	2	8	no	—
44	C₂×C₈	C₂²⋊C₄	1	4	yes	—
45	C₂×C₈	D₈	1	4	yes	—
46	C₂×C₈	SD₁₆	1	4	yes	—
47	C₄⋊C₄	—	2	8	no	—
48	C₄⋊C₄	—	4	4	no	—
49	C₄⋊C₄	—	4	4	no	—
50	C₄⋊C₄	—	4	8	no	—
51	C₄²	C₂²×C₄	1	1	yes	characteristic
52	C₄²	C₂×D₄	1	1	yes	characteristic
53	C₄²	#41	1	2	yes	—
54	C₂³	#127	1	1	yes	characteristic
55	Q₈	—	4	8	no	—
56	Q₈	—	8	4	no	—
57	C₂×C₄	—	4	4	no	—
58	C₂×C₄	—	8	2	no	—
59	C₂×C₄	—	8	2	no	—
60	C₂×C₄	—	8	2	no	—
61	C₂×C₄	—	8	4	no	—
62	C₂×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
63	C₂×C₄	C₄×D₄	1	2	yes	—
64	C₂×C₄	#121	1	2	yes	—
65	C₂×C₄	#124	1	1	yes	characteristic
66	C₂×C₄	C₂×D₈	1	2	yes	—
67	C₂×C₄	C₂×SD₁₆	1	2	yes	—
68	C₂×C₄	#137	1	8	yes	—
69	C₈	—	4	4	no	—
70	C₈	#102	1	8	yes	—
71	C₂²	#390	1	1	yes	characteristic
72	C₂²	#492	1	1	yes	characteristic
73	C₂²	#493	1	1	yes	characteristic
74	C₂²	#497	1	4	yes	—
75	C₄	—	8	4	no	—
76	C₄	—	16	2	no	—
77	C₄	C₂×#102	1	2	yes	—
78	C₄	#441	1	4	yes	—
79	C₄	#470	1	4	yes	—
80	C₄	#486	1	2	yes	—
81	C₂	#1716	1	2	yes	—
82	C₂	#1722	1	2	yes	—
83	C₂	#1957	1	1	yes	characteristic
84	C₂	#1971	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂³	#127
commutator	C₄²	C₂²×C₄
Frattini, cosocle	C₂×C₄²	C₂³
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

Complements

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

N	H ≅ G/N	type
8. #494	C₄	semidirect

Maximal subgroups

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

subgroup	index
C₂×#494	2
#1990	2
#1844	2
C₂×#322	2