Finite Group Databasebeta

Subgroups — #27852

Order 256.

336 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 336 conjugacy classes of subgroups — too many to draw. Only its 119 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	#2496	C₂	1	2	yes	maximal
2	#2099	C₂	1	2	yes	maximal, complemented
3	#2076	C₂	1	2	yes	maximal
4	#1943	C₂	1	1	yes	maximal, characteristic, complemented
5	#1941	C₂	1	1	yes	maximal, characteristic, complemented
6	#1921	C₂	1	2	yes	maximal, complemented
7	#1845	C₂	1	1	yes	maximal, characteristic, complemented
8	#1809	C₂	1	1	yes	maximal, characteristic
9	#1809	C₂	1	1	yes	maximal, characteristic
10	#1776	C₂	1	1	yes	maximal, characteristic, complemented
11	#1753	C₂	1	1	yes	maximal, characteristic
12	C₂²×M₁₆	C₂²	1	1	yes	characteristic
13	C₂×#126	—	2	2	no	—
14	C₂×#107	C₂²	1	1	yes	characteristic
15	C₂×#107	C₂²	1	1	yes	characteristic
16	C₂×#107	C₂²	1	2	yes	—
17	C₂×#106	C₂²	1	1	yes	characteristic
18	C₂×#106	C₂²	1	1	yes	characteristic
19	C₂×C₂²⋊C₈	C₂²	1	1	yes	characteristic
20	C₂×#97	C₂²	1	1	yes	characteristic
21	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
22	#400	—	2	2	no	—
23	#397	—	2	2	no	—
24	#389	C₂²	1	2	yes	—
25	#389	C₂²	1	2	yes	—
26	#387	—	2	2	no	—
27	#384	C₂²	1	2	yes	—
28	#380	C₂²	1	2	yes	—
29	#380	C₂²	1	2	yes	—
30	C₄×C₄⋊C₄	C₂²	1	2	yes	—
31	C₄×C₂²⋊C₄	C₂²	1	1	yes	characteristic
32	#340	C₂²	1	2	yes	—
33	#340	C₂²	1	2	yes	—
34	#340	C₂²	1	2	yes	—
35	#340	C₂²	1	2	yes	—
36	#340	C₂²	1	2	yes	—
37	#336	C₂²	1	1	yes	characteristic
38	#336	C₂²	1	1	yes	characteristic
39	C₂³×C₄	C₂³	1	1	yes	characteristic
40	C₂×M₁₆	—	2	2	no	—
41	C₂×M₁₆	—	2	2	no	—
42	C₂×M₁₆	—	2	2	no	—
43	C₂²×C₈	C₂³	1	1	yes	characteristic
44	C₂²×C₈	C₂³	1	1	yes	characteristic
45	C₂²×C₈	C₂³	1	1	yes	characteristic
46	C₂²×C₈	C₂³	1	1	yes	characteristic
47	#126	—	4	8	no	—
48	C₂×C₄⋊C₄	—	2	2	no	—
49	C₂×C₄⋊C₄	—	2	2	no	—
50	C₂×C₄⋊C₄	—	2	2	no	—
51	C₂×C₄⋊C₄	—	2	2	no	—
52	C₂×C₄⋊C₄	C₂³	1	2	yes	—
53	C₂×C₄⋊C₄	C₂³	1	2	yes	—
54	C₂×C₄⋊C₄	C₂³	1	2	yes	—
55	C₂×C₄⋊C₄	C₂³	1	2	yes	—
56	C₂×C₂²⋊C₄	—	2	2	no	—
57	C₂×C₂²⋊C₄	—	2	2	no	—
58	C₂×C₂²⋊C₄	D₄	1	2	yes	—
59	C₂×C₄²	—	2	2	no	—
60	C₂×C₄²	C₂³	1	1	yes	characteristic
61	C₂×C₄²	C₂³	1	1	yes	characteristic
62	#107	—	2	4	no	—
63	#107	—	2	2	no	—
64	#107	—	2	2	no	—
65	#106	—	2	2	no	—
66	#106	—	2	2	no	—
67	C₂²⋊C₈	—	2	2	no	—
68	#97	—	2	2	no	—
69	C₄×C₈	—	2	2	no	—
70	#95	—	2	2	no	—
71	#95	—	2	2	no	—
72	#95	—	2	2	no	—
73	#95	—	2	2	no	—
74	#95	—	2	2	no	—
75	#95	D₄	1	2	yes	—
76	C₂⁴	C₂×D₄	1	1	yes	characteristic
77	C₂²×C₄	—	2	1	no	—
78	C₂²×C₄	—	2	1	no	—
79	C₂²×C₄	—	2	1	no	—
80	C₂²×C₄	—	2	1	no	—
81	C₂²×C₄	—	2	1	no	—
82	C₂²×C₄	—	2	1	no	—
83	C₂²×C₄	—	2	2	no	—
84	C₂²×C₄	—	2	2	no	—
85	C₂²×C₄	—	2	2	no	—
86	C₂²×C₄	—	2	2	no	—
87	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
88	C₂²×C₄	C₂×D₄	1	2	yes	—
89	C₂²×C₄	C₂×D₄	1	2	yes	—
90	C₂²×C₄	C₂⁴	1	1	yes	characteristic
91	M₁₆	—	4	4	no	—
92	C₂×C₈	—	2	1	no	—
93	C₂×C₈	—	2	1	no	—
94	C₂×C₈	—	2	1	no	—
95	C₂×C₈	—	2	1	no	—
96	C₂×C₈	—	2	1	no	—
97	C₂×C₈	—	2	1	no	—
98	C₂×C₈	—	2	1	no	—
99	C₂×C₈	—	2	1	no	—
100	C₂×C₈	#41	1	2	yes	—
101	C₂×C₈	#41	1	2	yes	—
102	C₂×C₈	#41	1	2	yes	—
103	C₂×C₈	#41	1	2	yes	—
104	C₄⋊C₄	—	2	2	no	—
105	C₄⋊C₄	—	2	2	no	—
106	C₄⋊C₄	—	2	2	no	—
107	C₄⋊C₄	—	2	2	no	—
108	C₄⋊C₄	—	2	2	no	—
109	C₄⋊C₄	—	2	2	no	—
110	C₄⋊C₄	—	2	2	no	—
111	C₄⋊C₄	—	2	2	no	—
112	C₄⋊C₄	—	4	4	no	—
113	C₄⋊C₄	—	4	4	no	—
114	C₄⋊C₄	—	4	4	no	—
115	C₄⋊C₄	—	4	4	no	—
116	C₂²⋊C₄	—	2	4	no	—
117	C₂²⋊C₄	—	4	4	no	—
118	C₂²⋊C₄	—	4	4	no	—
119	C₄²	—	2	1	no	—
120	C₄²	—	2	1	no	—
121	C₄²	—	2	1	no	—
122	C₄²	—	2	1	no	—
123	C₄²	—	4	4	no	—
124	C₂³	—	2	1	no	—
125	C₂³	—	2	1	no	—
126	C₂³	—	2	1	no	—
127	C₂³	—	2	2	no	—
128	C₂³	—	2	2	no	—
129	C₂³	C₂²×D₄	1	1	yes	characteristic
130	C₂×C₄	—	2	2	no	—
131	C₂×C₄	—	2	2	no	—
132	C₂×C₄	—	2	1	no	—
133	C₂×C₄	—	2	2	no	—
134	C₂×C₄	—	2	1	no	—
135	C₂×C₄	—	2	1	no	—
136	C₂×C₄	—	2	2	no	—
137	C₂×C₄	—	2	2	no	—
138	C₂×C₄	—	4	1	no	—
139	C₂×C₄	—	4	1	no	—
140	C₂×C₄	—	4	1	no	—
141	C₂×C₄	—	4	1	no	—
142	C₂×C₄	—	4	2	no	—
143	C₂×C₄	—	4	2	no	—
144	C₂×C₄	—	4	2	no	—
145	C₂×C₄	—	4	2	no	—
146	C₂×C₄	—	4	2	no	—
147	C₂×C₄	—	4	2	no	—
148	C₂×C₄	—	4	2	no	—
149	C₂×C₄	—	4	2	no	—
150	C₂×C₄	—	4	2	no	—
151	C₂×C₄	—	4	2	no	—
152	C₂×C₄	—	4	2	no	—
153	C₂×C₄	—	4	2	no	—
154	C₂×C₄	AGL(1,ℤ/8)	1	4	yes	—
155	C₂×C₄	C₂×#41	1	1	yes	characteristic
156	C₂×C₄	C₂×#41	1	1	yes	characteristic
157	C₂×C₄	C₂×#41	1	1	yes	characteristic
158	C₂×C₄	C₂×#41	1	1	yes	characteristic
159	C₂×C₄	2₋¹⁺⁴	1	1	yes	characteristic
160	C₂×C₄	2₋¹⁺⁴	1	1	yes	characteristic
161	C₈	—	2	2	no	—
162	C₈	—	4	1	no	—
163	C₈	—	4	1	no	—
164	C₈	#126	1	4	yes	complemented
165	C₂²	—	4	2	no	—
166	C₂²	—	4	2	no	—
167	C₂²	—	4	1	no	—
168	C₂²	—	4	1	no	—
169	C₂²	—	4	1	no	—
170	C₂²	#532	1	1	yes	characteristic
171	C₂²	#547	1	1	yes	characteristic
172	C₂²	#547	1	1	yes	characteristic
173	C₂²	C₂×AGL(1,ℤ/8)	1	2	yes	—
174	C₂²	#578	1	2	yes	—
175	C₄	—	2	4	no	—
176	C₄	—	4	2	no	—
177	C₄	—	4	1	no	—
178	C₄	—	8	2	no	—
179	C₄	—	8	2	no	—
180	C₄	—	8	2	no	—
181	C₄	—	8	2	no	—
182	C₄	C₂×#126	1	1	yes	characteristic
183	C₄	#541	1	1	yes	characteristic
184	C₄	#552	1	1	yes	characteristic
185	C₄	#556	1	1	yes	characteristic
186	C₂	—	8	1	no	—
187	C₂	#2496	1	1	yes	characteristic
188	C₂	#3177	1	1	yes	characteristic
189	C₂	#3184	1	1	yes	characteristic
190	C₂	#3343	1	2	yes	—
191	C₂	#3353	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
2. #2099	C₂	semidirect
4. #1943	C₂	semidirect
5. #1941	C₂	semidirect
6. #1921	C₂	semidirect
7. #1845	C₂	semidirect
10. #1776	C₂	semidirect
164. C₈	#126	semidirect

Maximal subgroups

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

subgroup	index
#2496	2
#2099	2
#2076	2
#1943	2
#1941	2
#1921	2
#1845	2
#1809	2
#1776	2
#1753	2