Finite Group Databasebeta

Subgroups — #28527

Order 256.

515 conjugacy classes of subgroups; only the 167 normal subgroups are shown.

normal subgroups only

This group has 515 conjugacy classes of subgroups — too many to draw. Only its 167 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×#500	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#474	C₂	1	2	yes	maximal, complemented
3	C₂×C₄×D₈	C₂	1	1	yes	maximal, characteristic
4	#2542	C₂	1	2	yes	maximal, complemented
5	#2093	C₂	1	4	yes	maximal, complemented
6	#1970	C₂	1	2	yes	maximal
7	#1845	C₂	1	2	yes	maximal, complemented
8	C₈×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×D₈	C₂²	1	1	yes	characteristic
10	C₂×C₄⋊Q₈	C₂²	1	2	yes	—
11	C₂×C₂²⋊Q₈	—	2	4	no	—
12	C₂×C₄×D₄	C₂²	1	2	yes	—
13	#500	—	2	4	no	—
14	#474	—	2	8	no	—
15	C₄×D₈	—	2	4	no	—
16	C₂×#107	C₂²	1	2	yes	—
17	C₂×#107	C₂²	1	4	yes	—
18	C₂×#107	C₂²	1	1	yes	characteristic
19	C₂×#107	C₂²	1	2	yes	—
20	C₂×C₄⋊C₈	C₂²	1	1	yes	characteristic
21	C₂×#102	C₂²	1	2	yes	—
22	C₂×#102	C₂²	1	4	yes	—
23	C₂×C₄×C₈	C₂²	1	2	yes	—
24	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
25	#399	—	2	4	no	—
26	#390	C₂²	1	4	yes	—
27	#384	C₂²	1	4	yes	—
28	C₄×C₄⋊C₄	C₂²	1	1	yes	characteristic
29	#336	C₂²	1	2	yes	—
30	C₂²×Q₈	—	2	2	no	—
31	C₂²×D₄	C₂³	1	2	yes	—
32	C₂³×C₄	—	2	2	no	—
33	C₂×D₈	—	2	2	no	—
34	C₂×D₈	—	2	2	no	—
35	C₂×D₈	Q₈	1	4	yes	complemented
36	C₂²×C₈	C₂³	1	1	yes	characteristic
37	C₂²×C₈	C₂³	1	2	yes	—
38	C₂²×C₈	C₂³	1	1	yes	characteristic
39	C₄⋊Q₈	—	2	4	no	—
40	C₄⋊Q₈	—	2	4	no	—
41	C₂²⋊Q₈	—	4	16	no	—
42	C₄×D₄	—	2	4	no	—
43	C₄×D₄	—	2	4	no	—
44	C₂×C₄⋊C₄	—	2	8	no	—
45	C₂×C₄⋊C₄	—	2	2	no	—
46	C₂×C₄⋊C₄	D₄	1	2	yes	—
47	C₂×C₄⋊C₄	C₂³	1	2	yes	—
48	C₂×C₄⋊C₄	C₂³	1	4	yes	—
49	C₂×C₂²⋊C₄	—	2	2	no	—
50	C₂×C₂²⋊C₄	—	2	8	no	—
51	C₂×C₄²	C₂³	1	2	yes	—
52	C₂×C₄²	C₂³	1	1	yes	characteristic
53	#107	—	2	2	no	—
54	#107	—	2	8	no	—
55	#107	—	2	4	no	—
56	#107	—	2	4	no	—
57	C₄⋊C₈	D₄	1	4	yes	—
58	#102	—	2	4	no	—
59	#102	—	2	8	no	—
60	C₄×C₈	—	2	4	no	—
61	C₄×C₈	D₄	1	4	yes	—
62	#95	D₄	1	2	yes	—
63	C₂⁴	—	2	2	no	—
64	C₂×Q₈	—	4	4	no	—
65	C₂×Q₈	—	4	8	no	—
66	C₂×D₄	—	2	2	no	—
67	C₂×D₄	—	2	2	no	—
68	C₂×D₄	—	2	2	no	—
69	C₂×D₄	—	2	2	no	—
70	C₂×D₄	—	2	2	no	—
71	C₂×D₄	C₂×Q₈	1	4	yes	—
72	C₂²×C₄	—	2	4	no	—
73	C₂²×C₄	—	2	2	no	—
74	C₂²×C₄	—	4	4	no	—
75	C₂²×C₄	—	4	8	no	—
76	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
77	C₂²×C₄	C₂×D₄	1	4	yes	—
78	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
79	C₂²×C₄	C₂⁴	1	1	yes	characteristic
80	D₈	—	2	8	no	—
81	C₂×C₈	—	2	2	no	—
82	C₂×C₈	—	2	1	no	—
83	C₂×C₈	—	2	1	no	—
84	C₂×C₈	—	2	2	no	—
85	C₂×C₈	C₂×D₄	1	2	yes	—
86	C₂×C₈	C₂×D₄	1	4	yes	—
87	C₂×C₈	C₂×D₄	1	2	yes	—
88	C₂×C₈	C₂×Q₈	1	2	yes	—
89	C₂×C₈	#41	1	2	yes	—
90	C₄⋊C₄	—	2	2	no	—
91	C₄⋊C₄	—	2	4	no	—
92	C₄⋊C₄	—	2	2	no	—
93	C₄⋊C₄	—	2	4	no	—
94	C₄⋊C₄	—	4	4	no	—
95	C₄⋊C₄	—	4	16	no	—
96	C₄⋊C₄	D₈	1	8	yes	complemented
97	C₂²⋊C₄	—	4	4	no	—
98	C₂²⋊C₄	—	4	16	no	—
99	C₄²	—	2	2	no	—
100	C₄²	—	2	2	no	—
101	C₄²	C₂×D₄	1	2	yes	—
102	C₄²	C₂×D₄	1	2	yes	—
103	C₂³	—	2	4	no	—
104	C₂³	—	4	2	no	—
105	C₂³	—	4	2	no	—
106	C₂³	—	4	4	no	—
107	C₂³	—	4	4	no	—
108	C₂³	C₂²×D₄	1	1	yes	characteristic
109	Q₈	—	8	8	no	—
110	D₄	—	2	4	no	—
111	D₄	—	2	4	no	—
112	D₄	—	4	2	no	—
113	D₄	—	4	2	no	—
114	C₂×C₄	—	2	4	no	—
115	C₂×C₄	—	2	4	no	—
116	C₂×C₄	—	2	2	no	—
117	C₂×C₄	—	4	2	no	—
118	C₂×C₄	—	4	4	no	—
119	C₂×C₄	—	4	8	no	—
120	C₂×C₄	—	4	4	no	—
121	C₂×C₄	—	8	8	no	—
122	C₂×C₄	#127	1	2	yes	—
123	C₂×C₄	#127	1	2	yes	—
124	C₂×C₄	C₂×D₈	1	8	yes	—
125	C₂×C₄	C₂×D₈	1	4	yes	—
126	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
127	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
128	C₂×C₄	C₂²×Q₈	1	1	yes	characteristic
129	C₂×C₄	C₂×#41	1	1	yes	characteristic
130	C₂×C₄	2₋¹⁺⁴	1	1	yes	characteristic
131	C₂×C₄	2₋¹⁺⁴	1	1	yes	characteristic
132	C₈	—	2	2	no	—
133	C₈	—	2	4	no	—
134	C₈	C₂²⋊Q₈	1	4	yes	complemented
135	C₂²	—	4	8	no	—
136	C₂²	—	4	4	no	—
137	C₂²	—	8	4	no	—
138	C₂²	—	8	2	no	—
139	C₂²	—	8	2	no	—
140	C₂²	C₂×#127	1	1	yes	characteristic
141	C₂²	#547	1	1	yes	characteristic
142	C₂²	D₄×Q₈	1	1	yes	characteristic
143	C₂²	C₂²×D₈	1	2	yes	—
144	C₂²	#578	1	2	yes	—
145	C₄	—	2	8	no	—
146	C₄	—	2	2	no	—
147	C₄	—	8	2	no	—
148	C₄	—	8	4	no	—
149	C₄	#493	1	4	yes	—
150	C₄	C₂×C₂²⋊Q₈	1	1	yes	characteristic
151	C₄	#536	1	1	yes	characteristic
152	C₄	#547	1	1	yes	characteristic
153	C₄	D₄×Q₈	1	1	yes	characteristic
154	C₂	—	8	4	no	—
155	C₂	#2542	1	1	yes	characteristic
156	C₂	C₂×#493	1	1	yes	characteristic
157	C₂	#3156	1	1	yes	characteristic
158	C₂	#3334	1	2	yes	—
159	C₂	Q₈×D₈	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
1. C₂×#500	C₂	semidirect
2. C₂×#474	C₂	semidirect
4. #2542	C₂	semidirect
5. #2093	C₂	semidirect
7. #1845	C₂	semidirect
8. C₈×C₄⋊C₄	C₂	semidirect
35. C₂×D₈	Q₈	semidirect
96. C₄⋊C₄	D₈	semidirect
134. C₈	C₂²⋊Q₈	semidirect

Maximal subgroups

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

subgroup	index
C₂×#500	2
C₂×#474	2
C₂×C₄×D₈	2
#2542	2
#2093	2
#1970	2
#1845	2
C₈×C₄⋊C₄	2