Finite Group Databasebeta

Subgroups — #9754

Order 256.

208 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 208 conjugacy classes of subgroups — too many to draw. Only its 69 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	#3131	C₂	1	1	yes	maximal, characteristic
2	#1770	C₂	1	1	yes	maximal, characteristic, complemented
3	#1564	C₂	1	1	yes	maximal, characteristic, complemented
4	#1503	C₂	1	1	yes	maximal, characteristic, complemented
5	#1502	C₂	1	1	yes	maximal, characteristic
6	#1481	C₂	1	1	yes	maximal, characteristic, complemented
7	#1480	C₂	1	1	yes	maximal, characteristic
8	#533	C₂²	1	1	yes	characteristic
9	#532	C₂²	1	1	yes	characteristic
10	#488	—	2	1	no	—
11	#488	—	2	1	no	—
12	#487	C₄	1	1	yes	characteristic, complemented
13	#487	C₄	1	1	yes	characteristic, complemented
14	#486	C₄	1	1	yes	characteristic, complemented
15	#486	C₄	1	1	yes	characteristic, complemented
16	#431	C₂²	1	1	yes	characteristic
17	#431	C₂²	1	1	yes	characteristic
18	C₂×C₄⋊C₈	C₂²	1	1	yes	characteristic
19	#416	—	2	1	no	—
20	#416	—	2	1	no	—
21	C₂×#97	—	2	1	no	—
22	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
23	C₄×C₄⋊C₄	C₂²	1	1	yes	characteristic
24	#336	—	2	1	no	—
25	#335	—	2	1	no	—
26	#334	—	2	1	no	—
27	#328	—	2	2	no	—
28	#327	—	2	2	no	—
29	Q₈⋊C₈	—	2	2	no	—
30	D₄⋊C₈	—	2	2	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	C₂×C₄	1	1	yes	characteristic
36	C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic
37	C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic
38	#127	C₂×C₄	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	D₄	1	2	yes	—
57	#103	—	2	1	no	—
58	#103	—	2	1	no	—
59	#103	—	2	1	no	—
60	#103	—	2	1	no	—
61	#102	—	2	1	no	—
62	#102	—	2	1	no	—
63	#102	—	2	1	no	—
64	#102	—	2	1	no	—
65	C₂²⋊C₈	—	2	1	no	—
66	C₂²⋊C₈	—	2	1	no	—
67	#97	—	4	2	no	—
68	C₄×C₈	—	2	1	no	—
69	C₄×C₈	—	2	1	no	—
70	C₄×C₈	C₂×C₄	1	1	yes	characteristic
71	C₄×C₈	C₂×C₄	1	1	yes	characteristic
72	C₄×C₈	D₄	1	1	yes	characteristic
73	C₄×C₈	D₄	1	1	yes	characteristic
74	#95	D₄	1	2	yes	—
75	#41	—	4	1	no	—
76	#41	—	4	1	no	—
77	#41	—	4	1	no	—
78	#41	—	4	1	no	—
79	C₂×Q₈	—	2	1	no	—
80	C₂×Q₈	—	2	1	no	—
81	C₂×Q₈	—	2	1	no	—
82	C₂×D₄	—	2	1	no	—
83	C₂×D₄	—	2	1	no	—
84	C₂×D₄	—	2	1	no	—
85	C₂×D₄	—	4	1	no	—
86	C₂²×C₄	—	2	2	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	2	no	—
96	C₂×C₈	—	4	1	no	—
97	C₂×C₈	—	4	1	no	—
98	C₂×C₈	—	4	1	no	—
99	C₂×C₈	—	4	1	no	—
100	C₂×C₈	C₂²⋊C₄	1	1	yes	characteristic
101	C₂×C₈	C₂²⋊C₄	1	1	yes	characteristic
102	C₂×C₈	C₂²⋊C₄	1	1	yes	characteristic
103	C₂×C₈	C₂²⋊C₄	1	1	yes	characteristic
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	—	2	1	no	—
109	C₄⋊C₄	—	2	1	no	—
110	C₄⋊C₄	—	4	2	no	—
111	C₄⋊C₄	—	4	1	no	—
112	C₄⋊C₄	—	4	1	no	—
113	C₂²⋊C₄	—	4	1	no	—
114	C₂²⋊C₄	—	4	1	no	—
115	C₂²⋊C₄	—	4	1	no	—
116	C₄²	—	4	1	no	—
117	C₄²	—	4	1	no	—
118	C₄²	—	4	1	no	—
119	C₄²	C₂²×C₄	1	1	yes	characteristic
120	C₄²	C₂×D₄	1	1	yes	characteristic
121	C₄²	C₂×D₄	1	1	yes	characteristic
122	C₄²	#41	1	1	yes	characteristic
123	C₂³	—	4	1	no	—
124	C₂³	#121	1	1	yes	characteristic
125	Q₈	—	4	1	no	—
126	Q₈	—	4	1	no	—
127	Q₈	—	4	2	no	—
128	Q₈	—	8	1	no	—
129	D₄	—	4	1	no	—
130	D₄	—	4	1	no	—
131	D₄	—	4	2	no	—
132	D₄	—	8	1	no	—
133	D₄	—	8	1	no	—
134	D₄	—	8	1	no	—
135	C₂×C₄	—	2	2	no	—
136	C₂×C₄	—	4	1	no	—
137	C₂×C₄	—	4	2	no	—
138	C₂×C₄	—	4	2	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	—	8	1	no	—
144	C₂×C₄	—	8	1	no	—
145	C₂×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
146	C₂×C₄	C₄×D₄	1	1	yes	characteristic
147	C₂×C₄	C₄×D₄	1	1	yes	characteristic
148	C₂×C₄	#121	1	1	yes	characteristic
149	C₂×C₄	#124	1	1	yes	characteristic
150	C₂×C₄	#127	1	1	yes	characteristic
151	C₂×C₄	#135	1	1	yes	characteristic
152	C₂×C₄	#135	1	1	yes	characteristic
153	C₂×C₄	AGL(1,ℤ/8)	1	1	yes	characteristic
154	C₂×C₄	#137	1	1	yes	characteristic
155	C₈	—	2	1	no	—
156	C₈	—	2	1	no	—
157	C₈	—	2	1	no	—
158	C₈	—	2	1	no	—
159	C₈	—	4	2	no	—
160	C₈	—	8	1	no	—
161	C₈	—	8	1	no	—
162	C₂²	—	2	1	no	—
163	C₂²	—	2	1	no	—
164	C₂²	—	8	1	no	—
165	C₂²	—	8	1	no	—
166	C₂²	—	8	1	no	—
167	C₂²	#390	1	1	yes	characteristic
168	C₂²	#463	1	1	yes	characteristic
169	C₂²	#464	1	1	yes	characteristic
170	C₄	—	2	1	no	—
171	C₄	—	2	1	no	—
172	C₄	—	2	1	no	—
173	C₄	—	8	2	no	—
174	C₄	—	8	1	no	—
175	C₄	—	8	1	no	—
176	C₄	—	8	1	no	—
177	C₄	#416	1	1	yes	characteristic
178	C₄	#417	1	1	yes	characteristic
179	C₄	#444	1	2	yes	—
180	C₄	#463	1	1	yes	characteristic
181	C₄	#464	1	1	yes	characteristic
182	C₂	—	2	1	no	—
183	C₂	—	2	1	no	—
184	C₂	—	16	1	no	—
185	C₂	#1975	1	1	yes	characteristic
186	C₂	#1976	1	1	yes	characteristic
187	C₂	#1983	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#121
commutator	C₄²	C₂²×C₄
Frattini, cosocle	C₂×C₄²	C₂³
socle	C₂²	#390
prime radical	#532	C₂²
LCS	G ▷ 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. #1770	C₂	semidirect
3. #1564	C₂	semidirect
4. #1503	C₂	semidirect
6. #1481	C₂	semidirect
12. #487	C₄	semidirect
13. #487	C₄	semidirect
14. #486	C₄	semidirect
15. #486	C₄	semidirect

Maximal subgroups

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

subgroup	index
#3131	2
#1770	2
#1564	2
#1503	2
#1502	2
#1481	2
#1480	2