Finite Group Databasebeta

Subgroups — #9571

Order 256.

227 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 227 conjugacy classes of subgroups — too many to draw. Only its 77 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	#1982	C₂	1	1	yes	maximal, characteristic
2	#1957	C₂	1	1	yes	maximal, characteristic
3	#1947	C₂	1	1	yes	maximal, characteristic, complemented
4	#1770	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#331	C₂	1	1	yes	maximal, characteristic
6	#1296	C₂	1	1	yes	maximal, characteristic, complemented
7	#1277	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#127	C₂²	1	1	yes	characteristic
9	C₂×#107	—	2	1	no	—
10	C₂×C₄⋊C₈	C₂²	1	1	yes	characteristic
11	C₂×C₄⋊C₈	C₂²	1	1	yes	characteristic
12	C₂×C₄⋊C₈	C₂²	1	1	yes	characteristic
13	C₂×#102	—	2	1	no	—
14	C₂×#102	—	2	1	no	—
15	C₂×#102	—	2	1	no	—
16	C₂×#102	—	2	1	no	—
17	C₂×#97	—	2	1	no	—
18	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
19	#384	C₂²	1	1	yes	characteristic
20	C₄×C₄⋊C₄	C₂²	1	1	yes	characteristic
21	#340	—	2	1	no	—
22	#336	—	2	1	no	—
23	#331	C₄	1	4	yes	complemented
24	C₂²×D₄	—	2	1	no	—
25	C₂²×D₄	—	2	1	no	—
26	C₂²×D₄	—	2	1	no	—
27	C₂²×C₈	—	2	1	no	—
28	C₂²×C₈	—	2	1	no	—
29	C₂²×C₈	—	2	1	no	—
30	C₂²×C₈	D₄	1	1	yes	characteristic
31	C₂²×C₈	D₄	1	1	yes	characteristic
32	#127	—	2	1	no	—
33	#127	—	2	1	no	—
34	#127	—	2	1	no	—
35	#127	C₂×C₄	1	2	yes	—
36	C₂×C₄⋊C₄	—	2	1	no	—
37	C₂×C₄⋊C₄	—	2	1	no	—
38	C₂×C₄⋊C₄	—	2	1	no	—
39	C₂×C₄⋊C₄	D₄	1	1	yes	characteristic
40	C₂×C₄⋊C₄	D₄	1	1	yes	characteristic
41	C₂×C₄²	—	2	1	no	—
42	C₂×C₄²	C₂³	1	1	yes	characteristic
43	#107	—	4	2	no	—
44	C₄⋊C₈	—	2	1	no	—
45	C₄⋊C₈	—	2	1	no	—
46	C₄⋊C₈	—	2	1	no	—
47	C₄⋊C₈	—	2	1	no	—
48	C₄⋊C₈	C₂×C₄	1	2	yes	—
49	C₄⋊C₈	C₂×C₄	1	2	yes	—
50	#102	—	4	2	no	—
51	#102	—	4	2	no	—
52	#102	—	4	2	no	—
53	#102	—	4	2	no	—
54	#97	—	4	2	no	—
55	C₄×C₈	—	2	1	no	—
56	C₄×C₈	—	2	1	no	—
57	#95	—	2	1	no	—
58	C₂⁴	—	4	1	no	—
59	C₂×D₄	—	2	2	no	—
60	C₂×D₄	—	2	2	no	—
61	C₂×D₄	—	2	2	no	—
62	C₂×D₄	—	4	1	no	—
63	C₂×D₄	—	4	1	no	—
64	C₂×D₄	—	4	1	no	—
65	C₂×D₄	—	4	1	no	—
66	C₂×D₄	—	4	1	no	—
67	C₂×D₄	—	4	1	no	—
68	C₂×D₄	—	4	1	no	—
69	C₂×D₄	—	4	1	no	—
70	C₂×D₄	—	4	1	no	—
71	C₂×D₄	—	4	1	no	—
72	C₂×D₄	—	4	1	no	—
73	C₂×D₄	—	8	1	no	—
74	C₂×D₄	—	8	1	no	—
75	C₂²×C₄	—	2	1	no	—
76	C₂²×C₄	—	2	1	no	—
77	C₂²×C₄	—	4	1	no	—
78	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
79	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
80	C₂²×C₄	#41	1	1	yes	characteristic
81	C₂×C₈	—	2	1	no	—
82	C₂×C₈	—	2	1	no	—
83	C₂×C₈	—	2	2	no	—
84	C₂×C₈	—	2	2	no	—
85	C₂×C₈	—	2	1	no	—
86	C₂×C₈	—	2	1	no	—
87	C₂×C₈	—	2	1	no	—
88	C₂×C₈	—	4	1	no	—
89	C₂×C₈	—	4	1	no	—
90	C₂×C₈	—	4	1	no	—
91	C₂×C₈	—	4	1	no	—
92	C₂×C₈	—	4	1	no	—
93	C₂×C₈	—	4	1	no	—
94	C₂×C₈	—	4	1	no	—
95	C₂×C₈	D₈	1	2	yes	—
96	C₄⋊C₄	—	2	1	no	—
97	C₄⋊C₄	—	2	1	no	—
98	C₄⋊C₄	—	2	1	no	—
99	C₄⋊C₄	—	2	1	no	—
100	C₄⋊C₄	—	4	1	no	—
101	C₄⋊C₄	—	4	1	no	—
102	C₄⋊C₄	—	4	1	no	—
103	C₄⋊C₄	—	4	1	no	—
104	C₄⋊C₄	—	8	1	no	—
105	C₄²	—	4	1	no	—
106	C₄²	—	4	1	no	—
107	C₄²	C₂²×C₄	1	1	yes	characteristic
108	C₄²	#41	1	1	yes	characteristic
109	C₄²	#41	1	1	yes	characteristic
110	C₄²	#41	1	1	yes	characteristic
111	C₂³	—	4	2	no	—
112	C₂³	—	8	1	no	—
113	C₂³	—	8	1	no	—
114	C₂³	—	8	1	no	—
115	C₂³	—	8	1	no	—
116	C₂³	—	8	1	no	—
117	C₂³	—	8	1	no	—
118	C₂³	#121	1	1	yes	characteristic
119	D₄	—	4	2	no	—
120	D₄	—	4	2	no	—
121	D₄	—	4	2	no	—
122	D₄	—	4	2	no	—
123	D₄	—	8	2	no	—
124	D₄	—	8	2	no	—
125	D₄	—	8	1	no	—
126	D₄	—	8	1	no	—
127	D₄	—	8	1	no	—
128	D₄	—	8	1	no	—
129	C₂×C₄	—	2	1	no	—
130	C₂×C₄	—	2	1	no	—
131	C₂×C₄	—	4	1	no	—
132	C₂×C₄	—	4	1	no	—
133	C₂×C₄	—	4	1	no	—
134	C₂×C₄	—	4	1	no	—
135	C₂×C₄	—	4	1	no	—
136	C₂×C₄	—	4	1	no	—
137	C₂×C₄	—	8	1	no	—
138	C₂×C₄	—	8	1	no	—
139	C₂×C₄	—	8	1	no	—
140	C₂×C₄	#117	1	1	yes	characteristic
141	C₂×C₄	C₄×D₄	1	1	yes	characteristic
142	C₂×C₄	C₄×D₄	1	1	yes	characteristic
143	C₂×C₄	#123	1	1	yes	characteristic
144	C₂×C₄	#124	1	1	yes	characteristic
145	C₂×C₄	#126	1	1	yes	characteristic
146	C₂×C₄	C₂×D₈	1	1	yes	characteristic
147	C₂×C₄	#135	1	1	yes	characteristic
148	C₂×C₄	AGL(1,ℤ/8)	1	1	yes	characteristic
149	C₂×C₄	AGL(1,ℤ/8)	1	1	yes	characteristic
150	C₂×C₄	AGL(1,ℤ/8)	1	1	yes	characteristic
151	C₂×C₄	AGL(1,ℤ/8)	1	1	yes	characteristic
152	C₂×C₄	#137	1	1	yes	characteristic
153	C₂×C₄	#137	1	1	yes	characteristic
154	C₈	—	2	2	no	—
155	C₈	—	4	2	no	—
156	C₈	—	4	2	no	—
157	C₈	—	4	1	no	—
158	C₈	—	8	1	no	—
159	C₂²	—	8	2	no	—
160	C₂²	—	8	2	no	—
161	C₂²	—	8	2	no	—
162	C₂²	—	16	1	no	—
163	C₂²	—	16	1	no	—
164	C₂²	—	16	1	no	—
165	C₂²	—	16	1	no	—
166	C₂²	#388	1	1	yes	characteristic
167	C₂²	#459	1	1	yes	characteristic
168	C₂²	#464	1	1	yes	characteristic
169	C₂²	#468	1	1	yes	characteristic
170	C₂²	#469	1	1	yes	characteristic
171	C₂²	#471	1	1	yes	characteristic
172	C₂²	#472	1	1	yes	characteristic
173	C₄	—	2	1	no	—
174	C₄	—	2	1	no	—
175	C₄	—	8	1	no	—
176	C₄	—	8	1	no	—
177	C₄	—	16	1	no	—
178	C₄	C₄×D₈	1	1	yes	characteristic
179	C₄	#440	1	1	yes	characteristic
180	C₄	#440	1	1	yes	characteristic
181	C₄	#442	1	1	yes	characteristic
182	C₄	#480	1	1	yes	characteristic
183	C₄	#482	1	1	yes	characteristic
184	C₄	#489	1	1	yes	characteristic
185	C₄	#490	1	1	yes	characteristic
186	C₂	—	16	2	no	—
187	C₂	#1685	1	1	yes	characteristic
188	C₂	#1687	1	1	yes	characteristic
189	C₂	#1698	1	1	yes	characteristic
190	C₂	#1702	1	1	yes	characteristic
191	C₂	#1925	1	1	yes	characteristic
192	C₂	#1931	1	1	yes	characteristic
193	C₂	#1936	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³	#121
commutator	C₄²	C₂²×C₄
Frattini, cosocle	C₂×C₄²	C₂³
prime radical	C₂×#127	C₂²
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
3. #1947	C₂	semidirect
4. #1770	C₂	semidirect
6. #1296	C₂	semidirect
7. #1277	C₂	semidirect
23. #331	C₄	semidirect

Maximal subgroups

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

subgroup	index
#1982	2
#1957	2
#1947	2
#1770	2
C₂×#331	2
#1296	2
#1277	2