Finite Group Databasebeta

Subgroups — #20503

Order 256.

603 conjugacy classes of subgroups; only the 299 normal subgroups are shown.

normal subgroups only

This group has 603 conjugacy classes of subgroups — too many to draw. Only its 299 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#442	C₂	1	2	yes	maximal
2	C₂×C₄×D₈	C₂	1	1	yes	maximal, characteristic
3	#2275	C₂	1	2	yes	maximal, complemented
4	#1776	C₂	1	2	yes	maximal, complemented
5	C₄×#106	C₂	1	1	yes	maximal, characteristic, complemented
6	#1762	C₂	1	2	yes	maximal, complemented
7	C₄×#102	C₂	1	4	yes	maximal, complemented
8	#1726	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×D₈	C₂²	1	1	yes	characteristic
10	C₂×C₄×D₄	C₂²	1	2	yes	—
11	C₂×C₄×D₄	C₂²	1	4	yes	—
12	#442	C₄	1	16	yes	complemented
13	C₄×D₈	C₄	1	8	yes	complemented
14	C₂×#107	C₂²	1	1	yes	characteristic
15	C₂×#106	C₂²	1	2	yes	—
16	C₂×#102	C₂²	1	2	yes	—
17	C₂×#102	C₂²	1	4	yes	—
18	C₂×#97	C₂²	1	1	yes	characteristic
19	C₂×#97	C₂²	1	2	yes	—
20	C₂×C₄×C₈	C₂²	1	2	yes	—
21	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
22	C₄×C₄⋊C₄	C₂²	1	4	yes	—
23	C₄×C₄⋊C₄	C₂²	1	2	yes	—
24	C₄×C₂²⋊C₄	—	2	2	no	—
25	C₄×C₂²⋊C₄	—	2	4	no	—
26	#376	C₂²	1	1	yes	characteristic
27	C₂×#95	—	2	2	no	—
28	#340	C₂²	1	2	yes	—
29	#340	C₂²	1	4	yes	—
30	C₂²×D₄	C₂³	1	2	yes	—
31	C₂³×C₄	—	2	2	no	—
32	C₂³×C₄	—	2	4	no	—
33	C₂×D₈	C₂×C₄	1	4	yes	—
34	C₂×D₈	C₂×C₄	1	8	yes	—
35	C₂²×C₈	C₂³	1	1	yes	characteristic
36	C₂²×C₈	C₂³	1	2	yes	—
37	C₂²×C₈	C₂³	1	1	yes	characteristic
38	C₄×D₄	—	2	8	no	—
39	C₄×D₄	—	2	4	no	—
40	C₄×D₄	C₂×C₄	1	8	yes	complemented
41	C₄×D₄	C₂×C₄	1	16	yes	complemented
42	C₂×C₄⋊C₄	C₂³	1	4	yes	—
43	C₂×C₄⋊C₄	C₂³	1	2	yes	—
44	C₂×C₂²⋊C₄	—	2	4	no	—
45	C₂×C₂²⋊C₄	—	2	2	no	—
46	C₂×C₄²	—	2	4	no	—
47	C₂×C₄²	—	2	2	no	—
48	C₂×C₄²	C₂³	1	1	yes	characteristic
49	C₂×C₄²	C₂³	1	2	yes	—
50	#107	C₂×C₄	1	4	yes	complemented
51	#106	C₂×C₄	1	8	yes	complemented
52	#102	C₂×C₄	1	8	yes	complemented
53	#102	C₂×C₄	1	16	yes	complemented
54	#97	—	2	2	no	—
55	#97	C₂×C₄	1	8	yes	complemented
56	C₄×C₈	—	2	4	no	—
57	C₄×C₈	C₂×C₄	1	4	yes	complemented
58	#95	—	2	2	no	—
59	#95	—	2	4	no	—
60	#95	D₄	1	2	yes	—
61	#95	D₄	1	2	yes	—
62	C₂⁴	—	2	2	no	—
63	C₂×D₄	—	2	4	no	—
64	C₂×D₄	—	2	2	no	—
65	C₂×D₄	C₂²×C₄	1	4	yes	—
66	C₂×D₄	C₂²×C₄	1	8	yes	—
67	C₂²×C₄	—	2	2	no	—
68	C₂²×C₄	—	2	4	no	—
69	C₂²×C₄	—	2	16	no	—
70	C₂²×C₄	—	2	8	no	—
71	C₂²×C₄	—	8	4	no	—
72	C₂²×C₄	—	8	4	no	—
73	C₂²×C₄	—	8	4	no	—
74	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
75	C₂²×C₄	C₂×D₄	1	4	yes	—
76	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
77	C₂²×C₄	C₂⁴	1	1	yes	characteristic
78	D₈	C₄²	1	16	yes	complemented
79	C₂×C₈	—	2	2	no	—
80	C₂×C₈	—	2	2	no	—
81	C₂×C₈	—	2	2	no	—
82	C₂×C₈	C₂²×C₄	1	2	yes	—
83	C₂×C₈	C₂²×C₄	1	4	yes	—
84	C₂×C₈	C₂²×C₄	1	2	yes	—
85	C₂×C₈	C₂²×C₄	1	4	yes	—
86	C₄⋊C₄	—	2	4	no	—
87	C₄⋊C₄	—	2	2	no	—
88	C₄⋊C₄	C₂²×C₄	1	8	yes	—
89	C₄⋊C₄	C₂²×C₄	1	4	yes	—
90	C₂²⋊C₄	—	2	8	no	—
91	C₂²⋊C₄	—	2	16	no	—
92	C₄²	—	8	4	no	—
93	C₄²	—	8	2	no	—
94	C₄²	C₂²×C₄	1	2	yes	—
95	C₄²	C₂²×C₄	1	4	yes	—
96	C₄²	#41	1	4	yes	—
97	C₄²	#41	1	2	yes	—
98	C₂³	—	2	8	no	—
99	C₂³	—	2	4	no	—
100	C₂³	—	4	4	no	—
101	C₂³	—	4	2	no	—
102	C₂³	—	4	2	no	—
103	C₂³	C₂²×D₄	1	1	yes	characteristic
104	D₄	—	2	4	no	—
105	D₄	—	2	8	no	—
106	D₄	C₂×C₄²	1	8	yes	—
107	C₂×C₄	—	2	4	no	—
108	C₂×C₄	—	2	2	no	—
109	C₂×C₄	—	2	2	no	—
110	C₂×C₄	—	2	4	no	—
111	C₂×C₄	—	2	8	no	—
112	C₂×C₄	—	2	4	no	—
113	C₂×C₄	—	4	4	no	—
114	C₂×C₄	—	4	4	no	—
115	C₂×C₄	—	4	4	no	—
116	C₂×C₄	—	8	16	no	—
117	C₂×C₄	—	8	8	no	—
118	C₂×C₄	C₄×D₄	1	2	yes	—
119	C₂×C₄	C₄×D₄	1	8	yes	—
120	C₂×C₄	C₄×D₄	1	2	yes	—
121	C₂×C₄	C₂³×C₄	1	2	yes	—
122	C₂×C₄	C₂³×C₄	1	1	yes	characteristic
123	C₂×C₄	C₂×#41	1	2	yes	—
124	C₂×C₄	C₂×#41	1	1	yes	characteristic
125	C₈	—	2	2	no	—
126	C₈	—	2	4	no	—
127	C₈	C₂×C₄²	1	4	yes	—
128	C₂²	—	2	8	no	—
129	C₂²	—	4	8	no	—
130	C₂²	—	4	8	no	—
131	C₂²	—	4	4	no	—
132	C₂²	—	4	4	no	—
133	C₂²	C₂×C₄×D₄	1	2	yes	—
134	C₂²	C₂×C₄×D₄	1	1	yes	characteristic
135	C₂²	#576	1	2	yes	—
136	C₂²	#577	1	2	yes	—
137	C₄	—	2	8	no	—
138	C₄	—	2	2	no	—
139	C₄	—	2	2	no	—
140	C₄	—	4	8	no	—
141	C₄	—	4	4	no	—
142	C₄	C₂²×C₄²	1	1	yes	characteristic
143	C₄	C₄×#41	1	1	yes	characteristic
144	C₄	C₄×#41	1	2	yes	—
145	C₂	—	4	8	no	—
146	C₂	C₄²×D₄	1	1	yes	characteristic
147	C₂	#2946	1	4	yes	—
148	C₂	#2948	1	1	yes	characteristic
149	C₂	#2952	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³	C₂²×D₄
commutator	C₄	C₂²×C₄²
Frattini, cosocle	C₂²×C₄	C₂⁴
prime radical	C₂²×D₈	C₂²
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #376 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
3. #2275	C₂	semidirect
4. #1776	C₂	semidirect
5. C₄×#106	C₂	semidirect
6. #1762	C₂	semidirect
7. C₄×#102	C₂	semidirect
8. #1726	C₂	semidirect
12. #442	C₄	semidirect
13. C₄×D₈	C₄	semidirect
40. C₄×D₄	C₂×C₄	semidirect
41. C₄×D₄	C₂×C₄	semidirect
50. #107	C₂×C₄	semidirect
51. #106	C₂×C₄	semidirect
52. #102	C₂×C₄	semidirect
53. #102	C₂×C₄	semidirect
55. #97	C₂×C₄	semidirect
57. C₄×C₈	C₂×C₄	semidirect
78. D₈	C₄²	semidirect

Maximal subgroups

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

subgroup	index
C₂×#442	2
C₂×C₄×D₈	2
#2275	2
#1776	2
C₄×#106	2
#1762	2
C₄×#102	2
#1726	2