Finite Group Databasebeta

Subgroups — #10547

Order 256.

193 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 193 conjugacy classes of subgroups — too many to draw. Only its 53 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	#3135	C₂	1	1	yes	maximal, characteristic
2	#1940	C₂	1	1	yes	maximal, characteristic, complemented
3	#1542	C₂	1	1	yes	maximal, characteristic, complemented
4	#1540	C₂	1	1	yes	maximal, characteristic
5	#1503	C₂	1	1	yes	maximal, characteristic, complemented
6	#1502	C₂	1	1	yes	maximal, characteristic
7	#1275	C₂	1	1	yes	maximal, characteristic, complemented
8	#533	C₂²	1	1	yes	characteristic
9	#532	C₂²	1	1	yes	characteristic
10	#491	—	2	1	no	—
11	#490	—	2	1	no	—
12	#489	—	2	1	no	—
13	#489	—	2	1	no	—
14	#431	C₂²	1	1	yes	characteristic
15	#431	C₂²	1	1	yes	characteristic
16	C₂×C₄⋊C₈	—	2	1	no	—
17	C₂×C₄⋊C₈	C₂²	1	1	yes	characteristic
18	#416	—	2	1	no	—
19	#416	—	2	1	no	—
20	C₂×#97	C₂²	1	1	yes	characteristic
21	C₄×C₄⋊C₄	C₂²	1	1	yes	characteristic
22	#336	—	2	1	no	—
23	#333	—	2	1	no	—
24	#332	—	2	1	no	—
25	#332	—	2	1	no	—
26	#331	—	2	1	no	—
27	#328	—	2	1	no	—
28	#328	—	2	1	no	—
29	#327	—	2	1	no	—
30	#327	—	2	1	no	—
31	C₂×#41	—	2	1	no	—
32	C₂²×C₈	—	2	1	no	—
33	C₂²×C₈	D₄	1	1	yes	characteristic
34	C₂²×C₈	D₄	1	1	yes	characteristic
35	C₄⋊Q₈	D₄	1	1	yes	characteristic
36	C₄⋊Q₈	D₄	1	1	yes	characteristic
37	C₄⋊Q₈	D₄	1	1	yes	characteristic
38	#127	D₄	1	1	yes	characteristic
39	#125	—	2	1	no	—
40	#124	—	2	1	no	—
41	C₂²⋊Q₈	—	2	1	no	—
42	C₂²⋊Q₈	—	2	1	no	—
43	#121	—	2	1	no	—
44	#121	—	2	1	no	—
45	C₄×Q₈	—	2	1	no	—
46	C₄×Q₈	—	2	1	no	—
47	C₄×D₄	—	2	1	no	—
48	C₄×D₄	—	2	1	no	—
49	#117	—	2	1	no	—
50	C₂×C₄⋊C₄	—	2	1	no	—
51	C₂×C₄²	—	2	1	no	—
52	C₂×C₄²	C₂³	1	1	yes	characteristic
53	C₄⋊C₈	—	2	1	no	—
54	C₄⋊C₈	—	2	1	no	—
55	C₄⋊C₈	—	2	1	no	—
56	C₄⋊C₈	—	2	1	no	—
57	C₄⋊C₈	—	4	2	no	—
58	#103	—	2	1	no	—
59	#103	—	2	1	no	—
60	#103	—	2	1	no	—
61	#103	—	2	1	no	—
62	#102	—	2	1	no	—
63	#102	—	2	1	no	—
64	#102	—	2	1	no	—
65	#102	—	2	1	no	—
66	C₂²⋊C₈	—	2	1	no	—
67	C₂²⋊C₈	—	2	1	no	—
68	#97	—	2	1	no	—
69	#97	—	2	1	no	—
70	C₄×C₈	—	2	1	no	—
71	C₄×C₈	—	2	1	no	—
72	#95	D₄	1	1	yes	characteristic
73	#95	D₄	1	1	yes	characteristic
74	#41	—	4	1	no	—
75	#41	—	4	1	no	—
76	#41	—	4	1	no	—
77	#41	—	4	1	no	—
78	C₂×Q₈	—	2	1	no	—
79	C₂×Q₈	—	2	1	no	—
80	C₂×Q₈	—	2	1	no	—
81	C₂×D₄	—	2	1	no	—
82	C₂×D₄	—	2	1	no	—
83	C₂×D₄	—	2	1	no	—
84	C₂×D₄	—	4	1	no	—
85	C₂²×C₄	—	2	1	no	—
86	C₂²×C₄	—	2	1	no	—
87	C₂²×C₄	—	4	1	no	—
88	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
89	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
90	C₂²×C₄	#41	1	1	yes	characteristic
91	C₂×C₈	—	2	1	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	—	4	1	no	—
98	C₂×C₈	—	4	1	no	—
99	C₂×C₈	—	4	1	no	—
100	C₂×C₈	—	4	1	no	—
101	C₂×C₈	—	4	1	no	—
102	C₄⋊C₄	—	2	1	no	—
103	C₄⋊C₄	—	2	1	no	—
104	C₄⋊C₄	—	2	1	no	—
105	C₄⋊C₄	—	2	1	no	—
106	C₄⋊C₄	—	2	1	no	—
107	C₄⋊C₄	—	2	1	no	—
108	C₄⋊C₄	—	4	2	no	—
109	C₄⋊C₄	—	4	1	no	—
110	C₄⋊C₄	—	4	1	no	—
111	C₂²⋊C₄	—	4	1	no	—
112	C₂²⋊C₄	—	4	1	no	—
113	C₂²⋊C₄	—	4	1	no	—
114	C₄²	—	4	1	no	—
115	C₄²	—	4	1	no	—
116	C₄²	—	4	1	no	—
117	C₄²	C₂×D₄	1	1	yes	characteristic
118	C₄²	C₂×D₄	1	1	yes	characteristic
119	C₄²	#41	1	1	yes	characteristic
120	C₄²	#41	1	1	yes	characteristic
121	C₂³	—	4	1	no	—
122	C₂³	#121	1	1	yes	characteristic
123	Q₈	—	4	1	no	—
124	Q₈	—	4	1	no	—
125	Q₈	—	8	1	no	—
126	Q₈	—	8	1	no	—
127	D₄	—	4	1	no	—
128	D₄	—	4	1	no	—
129	D₄	—	8	1	no	—
130	D₄	—	8	1	no	—
131	D₄	—	8	1	no	—
132	D₄	—	8	1	no	—
133	C₂×C₄	—	2	2	no	—
134	C₂×C₄	—	4	1	no	—
135	C₂×C₄	—	4	1	no	—
136	C₂×C₄	—	4	1	no	—
137	C₂×C₄	—	4	1	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	1	no	—
143	C₂×C₄	—	4	1	no	—
144	C₂×C₄	—	8	1	no	—
145	C₂×C₄	—	8	1	no	—
146	C₂×C₄	#120	1	1	yes	characteristic
147	C₂×C₄	#121	1	1	yes	characteristic
148	C₂×C₄	#121	1	1	yes	characteristic
149	C₂×C₄	#123	1	1	yes	characteristic
150	C₂×C₄	#123	1	1	yes	characteristic
151	C₂×C₄	#124	1	1	yes	characteristic
152	C₂×C₄	#135	1	1	yes	characteristic
153	C₂×C₄	#135	1	1	yes	characteristic
154	C₂×C₄	AGL(1,ℤ/8)	1	1	yes	characteristic
155	C₂×C₄	#137	1	1	yes	characteristic
156	C₈	—	4	1	no	—
157	C₈	—	4	1	no	—
158	C₈	—	8	1	no	—
159	C₈	—	8	1	no	—
160	C₈	—	8	1	no	—
161	C₂²	—	2	1	no	—
162	C₂²	—	2	1	no	—
163	C₂²	—	8	1	no	—
164	C₂²	—	8	1	no	—
165	C₂²	—	8	1	no	—
166	C₂²	#394	1	1	yes	characteristic
167	C₂²	#463	1	1	yes	characteristic
168	C₂²	#464	1	1	yes	characteristic
169	C₄	—	2	1	no	—
170	C₄	—	2	1	no	—
171	C₄	—	2	1	no	—
172	C₄	—	2	1	no	—
173	C₄	—	4	2	no	—
174	C₄	—	8	1	no	—
175	C₄	—	8	1	no	—
176	C₄	—	8	1	no	—
177	C₄	—	8	1	no	—
178	C₄	#449	1	1	yes	characteristic
179	C₄	#452	1	1	yes	characteristic
180	C₄	#488	1	1	yes	characteristic
181	C₄	#489	1	1	yes	characteristic
182	C₂	—	2	1	no	—
183	C₂	—	2	1	no	—
184	C₂	—	16	1	no	—
185	C₂	#2042	1	1	yes	characteristic
186	C₂	#2050	1	1	yes	characteristic
187	C₂	#2053	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#124
commutator, Frattini, cosocle	C₂×C₄²	C₂³
socle	C₂²	#394
prime radical	#532	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ 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
2. #1940	C₂	semidirect
3. #1542	C₂	semidirect
5. #1503	C₂	semidirect
7. #1275	C₂	semidirect

Maximal subgroups

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

subgroup	index
#3135	2
#1940	2
#1542	2
#1540	2
#1503	2
#1502	2
#1275	2