Finite Group Databasebeta

Subgroups — #20821

Order 256.

405 conjugacy classes of subgroups; only the 207 normal subgroups are shown.

normal subgroups only

This group has 405 conjugacy classes of subgroups — too many to draw. Only its 207 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×#405	C₂	1	1	yes	maximal, characteristic
2	#2314	C₂	1	2	yes	maximal
3	#1846	C₂	1	1	yes	maximal, characteristic, complemented
4	#1845	C₂	1	1	yes	maximal, characteristic, complemented
5	#1836	C₂	1	2	yes	maximal, complemented
6	#1809	C₂	1	2	yes	maximal
7	#1808	C₂	1	2	yes	maximal
8	#1776	C₂	1	2	yes	maximal, complemented
9	C₄×#107	C₂	1	2	yes	maximal, complemented
10	C₂²×M₁₆	C₂²	1	1	yes	characteristic
11	C₂³×C₈	C₂²	1	1	yes	characteristic
12	C₂×#117	C₂²	1	1	yes	characteristic
13	C₂×#107	C₂²	1	2	yes	—
14	C₂×#107	C₂²	1	2	yes	—
15	C₂×#107	C₂²	1	2	yes	—
16	C₂×#106	C₂²	1	2	yes	—
17	#405	C₄	1	8	yes	complemented
18	C₂×#97	C₂²	1	1	yes	characteristic
19	C₂×#97	C₂²	1	1	yes	characteristic
20	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
21	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
22	#388	—	2	4	no	—
23	#387	—	2	4	no	—
24	#384	C₂²	1	2	yes	—
25	#382	C₂²	1	2	yes	—
26	#380	C₂²	1	4	yes	—
27	C₄×C₄⋊C₄	C₂²	1	4	yes	—
28	#340	C₂²	1	4	yes	—
29	#340	C₂²	1	4	yes	—
30	C₂³×C₄	C₂³	1	1	yes	characteristic
31	C₂×M₁₆	—	2	2	no	—
32	C₂×M₁₆	—	2	2	no	—
33	C₂×M₁₆	C₂×C₄	1	4	yes	—
34	C₂²×C₈	—	2	2	no	—
35	C₂²×C₈	—	2	2	no	—
36	C₂²×C₈	C₂×C₄	1	4	yes	—
37	C₂²×C₈	C₂³	1	1	yes	characteristic
38	C₂²×C₈	C₂³	1	1	yes	characteristic
39	C₂²×C₈	C₂³	1	1	yes	characteristic
40	C₂²×C₈	C₂³	1	1	yes	characteristic
41	#117	C₂×C₄	1	4	yes	—
42	#117	Q₈	1	4	yes	—
43	C₂×C₄⋊C₄	—	2	4	no	—
44	C₂×C₄⋊C₄	D₄	1	2	yes	—
45	C₂×C₄⋊C₄	C₂³	1	4	yes	—
46	C₂×C₄⋊C₄	C₂³	1	4	yes	—
47	C₂×C₂²⋊C₄	—	2	4	no	—
48	C₂×C₂²⋊C₄	D₄	1	2	yes	—
49	C₂×C₄²	—	2	4	no	—
50	C₂×C₄²	C₂³	1	1	yes	characteristic
51	C₂×C₄²	C₂³	1	1	yes	characteristic
52	#107	—	2	4	no	—
53	#107	—	2	4	no	—
54	#107	—	2	4	no	—
55	#106	—	2	4	no	—
56	#97	C₂×C₄	1	4	yes	—
57	#97	C₂×C₄	1	4	yes	—
58	C₄×C₈	C₂×C₄	1	4	yes	—
59	C₄×C₈	C₂×C₄	1	4	yes	—
60	#95	—	2	4	no	—
61	#95	—	2	4	no	—
62	#95	—	2	4	no	—
63	C₂⁴	C₂×D₄	1	1	yes	characteristic
64	C₂²×C₄	—	2	1	no	—
65	C₂²×C₄	—	2	1	no	—
66	C₂²×C₄	—	2	1	no	—
67	C₂²×C₄	—	2	1	no	—
68	C₂²×C₄	—	2	4	no	—
69	C₂²×C₄	—	2	4	no	—
70	C₂²×C₄	C₂²×C₄	1	2	yes	—
71	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
72	C₂²×C₄	C₂×D₄	1	2	yes	—
73	C₂²×C₄	C₂×D₄	1	2	yes	—
74	C₂²×C₄	C₂×Q₈	1	2	yes	—
75	C₂²×C₄	C₂⁴	1	1	yes	characteristic
76	M₁₆	—	2	8	no	—
77	C₂×C₈	—	2	1	no	—
78	C₂×C₈	—	2	1	no	—
79	C₂×C₈	—	2	1	no	—
80	C₂×C₈	—	2	1	no	—
81	C₂×C₈	—	2	8	no	—
82	C₂×C₈	C₂²×C₄	1	2	yes	—
83	C₂×C₈	C₂²×C₄	1	2	yes	—
84	C₂×C₈	C₂²×C₄	1	2	yes	—
85	C₂×C₈	C₂²×C₄	1	2	yes	—
86	C₂×C₈	#41	1	2	yes	—
87	C₂×C₈	#41	1	2	yes	—
88	C₂×C₈	#41	1	2	yes	—
89	C₂×C₈	#41	1	2	yes	—
90	C₄⋊C₄	—	2	4	no	—
91	C₄⋊C₄	—	2	4	no	—
92	C₄⋊C₄	—	2	4	no	—
93	C₄⋊C₄	—	2	4	no	—
94	C₄⋊C₄	—	8	4	no	—
95	C₄⋊C₄	C₄⋊C₄	1	8	yes	—
96	C₂²⋊C₄	—	8	4	no	—
97	C₂²⋊C₄	C₄⋊C₄	1	8	yes	—
98	C₄²	—	8	4	no	—
99	C₄²	C₂²×C₄	1	2	yes	—
100	C₄²	C₂²×C₄	1	2	yes	—
101	C₄²	C₂×Q₈	1	2	yes	—
102	C₄²	C₂×Q₈	1	2	yes	—
103	C₂³	—	2	1	no	—
104	C₂³	—	2	1	no	—
105	C₂³	—	2	2	no	—
106	C₂³	—	2	2	no	—
107	C₂³	C₂×C₄⋊C₄	1	2	yes	—
108	C₂³	C₂²×D₄	1	1	yes	characteristic
109	C₂×C₄	—	2	1	no	—
110	C₂×C₄	—	2	1	no	—
111	C₂×C₄	—	2	2	no	—
112	C₂×C₄	—	2	2	no	—
113	C₂×C₄	—	2	2	no	—
114	C₂×C₄	—	2	2	no	—
115	C₂×C₄	—	2	2	no	—
116	C₂×C₄	—	2	2	no	—
117	C₂×C₄	—	2	2	no	—
118	C₂×C₄	—	2	2	no	—
119	C₂×C₄	—	4	4	no	—
120	C₂×C₄	—	4	4	no	—
121	C₂×C₄	—	4	4	no	—
122	C₂×C₄	—	4	4	no	—
123	C₂×C₄	—	4	4	no	—
124	C₂×C₄	—	4	4	no	—
125	C₂×C₄	C₂×C₄⋊C₄	1	4	yes	—
126	C₂×C₄	C₂×C₄⋊C₄	1	4	yes	—
127	C₂×C₄	C₂×C₄⋊C₄	1	2	yes	—
128	C₂×C₄	C₂³×C₄	1	1	yes	characteristic
129	C₂×C₄	C₂²×Q₈	1	1	yes	characteristic
130	C₂×C₄	C₂×#41	1	1	yes	characteristic
131	C₂×C₄	C₂×#41	1	1	yes	characteristic
132	C₂×C₄	C₂×#41	1	1	yes	characteristic
133	C₂×C₄	C₂×#41	1	1	yes	characteristic
134	C₈	—	2	2	no	—
135	C₈	—	2	2	no	—
136	C₈	#117	1	4	yes	—
137	C₈	#117	1	4	yes	—
138	C₂²	—	2	4	no	—
139	C₂²	—	2	2	no	—
140	C₂²	—	4	2	no	—
141	C₂²	—	4	1	no	—
142	C₂²	—	4	1	no	—
143	C₂²	C₂²×C₄⋊C₄	1	1	yes	characteristic
144	C₂²	#532	1	1	yes	characteristic
145	C₂²	#532	1	1	yes	characteristic
146	C₂²	#576	1	2	yes	—
147	C₂²	#578	1	2	yes	—
148	C₄	—	2	4	no	—
149	C₄	—	2	4	no	—
150	C₄	—	2	2	no	—
151	C₄	—	8	4	no	—
152	C₄	—	8	4	no	—
153	C₄	C₂×#117	1	1	yes	characteristic
154	C₄	C₂×#117	1	1	yes	characteristic
155	C₄	#533	1	1	yes	characteristic
156	C₄	#533	1	1	yes	characteristic
157	C₂	—	4	2	no	—
158	C₂	#2285	1	1	yes	characteristic
159	C₂	#2913	1	2	yes	—
160	C₂	#3175	1	2	yes	—
161	C₂	#3176	1	1	yes	characteristic
162	C₂	#3177	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄
Frattini, cosocle	C₂²×C₄	C₂⁴
prime radical	C₂⁴	C₂×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
3. #1846	C₂	semidirect
4. #1845	C₂	semidirect
5. #1836	C₂	semidirect
8. #1776	C₂	semidirect
9. C₄×#107	C₂	semidirect
17. #405	C₄	semidirect

Maximal subgroups

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

subgroup	index
C₂×#405	2
#2314	2
#1846	2
#1845	2
#1836	2
#1809	2
#1808	2
#1776	2
C₄×#107	2