Finite Group Databasebeta

Subgroups — #8839

Order 256.

233 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 233 conjugacy classes of subgroups — too many to draw. Only its 85 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	#2988	C₂	1	1	yes	maximal, characteristic
2	#1842	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄×#104	C₂	1	1	yes	maximal, characteristic
4	#1565	C₂	1	1	yes	maximal, characteristic, complemented
5	#1477	C₂	1	1	yes	maximal, characteristic
6	#1283	C₂	1	1	yes	maximal, characteristic, complemented
7	#1276	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄×#41	C₂²	1	1	yes	characteristic
9	C₈×Q₈	—	2	1	no	—
10	C₈×D₄	—	2	2	no	—
11	C₈×D₄	—	2	1	no	—
12	#431	—	2	1	no	—
13	#431	C₂²	1	1	yes	characteristic
14	#431	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊C₈	—	2	1	no	—
16	C₂×C₄⋊C₈	—	2	2	no	—
17	C₂×C₄⋊C₈	C₂²	1	1	yes	characteristic
18	C₂×C₄⋊C₈	C₂²	1	1	yes	characteristic
19	C₂×#104	C₄	1	1	yes	characteristic
20	C₂×#104	C₄	1	1	yes	characteristic
21	C₄×M₁₆	C₂²	1	1	yes	characteristic
22	C₄³	C₂²	1	1	yes	characteristic
23	#339	C₄	1	1	yes	characteristic
24	#339	C₄	1	1	yes	characteristic
25	#335	—	2	1	no	—
26	#334	—	2	1	no	—
27	Q₈⋊C₈	—	2	1	no	—
28	D₄⋊C₈	—	2	1	no	—
29	C₂×#41	C₂×C₄	1	1	yes	characteristic
30	C₂×M₁₆	C₂×C₄	1	1	yes	characteristic
31	C₂×M₁₆	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₈	—	2	2	no	—
33	C₂²×C₈	—	2	1	no	—
34	C₂²×C₈	D₄	1	2	yes	—
35	C₄×Q₈	D₄	1	1	yes	characteristic
36	C₄×Q₈	Q₈	1	1	yes	characteristic
37	C₄×D₄	—	2	1	no	—
38	C₄×D₄	—	2	1	no	—
39	C₄×D₄	D₄	1	1	yes	characteristic
40	C₄×D₄	Q₈	1	1	yes	characteristic
41	#117	—	2	1	no	—
42	#117	C₂×C₄	1	1	yes	characteristic
43	C₂×C₄²	—	2	1	no	—
44	C₂×C₄²	—	2	1	no	—
45	C₂×C₄²	—	2	1	no	—
46	C₂×C₄²	C₂×C₄	1	1	yes	characteristic
47	C₂×C₄²	C₂×C₄	1	1	yes	characteristic
48	C₂×C₄²	C₂³	1	1	yes	characteristic
49	C₄⋊C₈	—	2	2	no	—
50	C₄⋊C₈	—	2	1	no	—
51	C₄⋊C₈	—	2	1	no	—
52	C₄⋊C₈	—	2	1	no	—
53	C₄⋊C₈	—	2	1	no	—
54	C₄⋊C₈	—	2	1	no	—
55	C₄⋊C₈	—	4	4	no	—
56	C₄⋊C₈	D₄	1	1	yes	characteristic
57	C₄⋊C₈	D₄	1	1	yes	characteristic
58	#104	—	2	2	no	—
59	#104	C₈	1	4	yes	complemented
60	C₂²⋊C₈	—	2	2	no	—
61	C₂²⋊C₈	—	2	1	no	—
62	C₂²⋊C₈	—	2	1	no	—
63	#97	—	2	1	no	—
64	C₄×C₈	—	2	2	no	—
65	C₄×C₈	—	2	1	no	—
66	C₄×C₈	—	2	1	no	—
67	C₄×C₈	—	2	1	no	—
68	#41	—	2	1	no	—
69	#41	—	2	1	no	—
70	#41	C₂×C₈	1	2	yes	—
71	#41	M₁₆	1	2	yes	—
72	C₂×Q₈	C₄⋊C₄	1	1	yes	characteristic
73	C₂×D₄	—	2	1	no	—
74	C₂×D₄	C₄⋊C₄	1	1	yes	characteristic
75	C₂²×C₄	—	2	1	no	—
76	C₂²×C₄	—	2	1	no	—
77	C₂²×C₄	—	2	1	no	—
78	C₂²×C₄	—	2	1	no	—
79	C₂²×C₄	—	2	1	no	—
80	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
81	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
82	C₂²×C₄	#41	1	1	yes	characteristic
83	M₁₆	—	2	1	no	—
84	M₁₆	—	2	1	no	—
85	M₁₆	—	2	1	no	—
86	M₁₆	C₂×C₈	1	2	yes	complemented
87	C₂×C₈	—	2	1	no	—
88	C₂×C₈	—	2	1	no	—
89	C₂×C₈	—	2	1	no	—
90	C₂×C₈	—	2	2	no	—
91	C₂×C₈	—	2	1	no	—
92	C₂×C₈	—	4	2	no	—
93	C₂×C₈	—	4	2	no	—
94	C₂×C₈	—	4	1	no	—
95	C₂×C₈	—	4	1	no	—
96	C₂×C₈	—	8	2	no	—
97	C₄⋊C₄	—	2	1	no	—
98	C₄⋊C₄	—	2	1	no	—
99	C₄⋊C₄	C₄⋊C₄	1	1	yes	characteristic
100	C₄⋊C₄	C₄⋊C₄	1	1	yes	characteristic
101	C₂²⋊C₄	—	2	1	no	—
102	C₂²⋊C₄	—	2	1	no	—
103	C₂²⋊C₄	—	2	1	no	—
104	C₄²	—	2	2	no	—
105	C₄²	—	2	2	no	—
106	C₄²	—	2	1	no	—
107	C₄²	—	2	1	no	—
108	C₄²	—	2	1	no	—
109	C₄²	—	2	1	no	—
110	C₄²	—	2	1	no	—
111	C₄²	—	2	1	no	—
112	C₄²	—	4	2	no	—
113	C₄²	C₂×C₈	1	2	yes	complemented
114	C₄²	C₂×D₄	1	1	yes	characteristic
115	C₄²	C₂×D₄	1	1	yes	characteristic
116	C₄²	C₂×Q₈	1	1	yes	characteristic
117	C₄²	#41	1	1	yes	characteristic
118	C₂³	—	2	1	no	—
119	C₂³	C₄×D₄	1	1	yes	characteristic
120	Q₈	—	2	1	no	—
121	Q₈	C₄⋊C₈	1	2	yes	complemented
122	D₄	—	2	2	no	—
123	D₄	—	2	2	no	—
124	D₄	—	2	1	no	—
125	D₄	C₄⋊C₈	1	2	yes	complemented
126	C₂×C₄	—	2	2	no	—
127	C₂×C₄	—	2	2	no	—
128	C₂×C₄	—	2	1	no	—
129	C₂×C₄	—	2	2	no	—
130	C₂×C₄	—	2	2	no	—
131	C₂×C₄	—	2	2	no	—
132	C₂×C₄	—	2	1	no	—
133	C₂×C₄	—	2	1	no	—
134	C₂×C₄	—	2	2	no	—
135	C₂×C₄	—	2	1	no	—
136	C₂×C₄	—	2	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	—	8	1	no	—
143	C₂×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
144	C₂×C₄	C₄×D₄	1	1	yes	characteristic
145	C₂×C₄	#120	1	1	yes	characteristic
146	C₂×C₄	C₂²⋊Q₈	1	1	yes	characteristic
147	C₂×C₄	C₂²⋊Q₈	1	1	yes	characteristic
148	C₂×C₄	#123	1	1	yes	characteristic
149	C₂×C₄	C₂²×C₈	1	1	yes	characteristic
150	C₂×C₄	C₂×M₁₆	1	1	yes	characteristic
151	C₂×C₄	#131	1	1	yes	characteristic
152	C₂×C₄	#131	1	1	yes	characteristic
153	C₈	—	2	2	no	—
154	C₈	—	4	1	no	—
155	C₈	—	4	2	no	—
156	C₈	—	4	1	no	—
157	C₈	—	4	1	no	—
158	C₈	—	8	1	no	—
159	C₈	—	8	1	no	—
160	C₂²	—	2	1	no	—
161	C₂²	—	2	1	no	—
162	C₂²	—	2	2	no	—
163	C₂²	—	4	1	no	—
164	C₂²	—	4	1	no	—
165	C₂²	#385	1	1	yes	characteristic
166	C₂²	C₈×D₄	1	1	yes	characteristic
167	C₂²	#435	1	1	yes	characteristic
168	C₄	—	2	4	no	—
169	C₄	—	2	2	no	—
170	C₄	—	2	1	no	—
171	C₄	—	2	1	no	—
172	C₄	—	2	1	no	—
173	C₄	—	4	2	no	—
174	C₄	—	4	1	no	—
175	C₄	—	4	1	no	—
176	C₄	C₂×C₄⋊C₈	1	1	yes	characteristic
177	C₄	#424	1	1	yes	characteristic
178	C₄	C₈×D₄	1	1	yes	characteristic
179	C₄	#435	1	1	yes	characteristic
180	C₄	#453	1	1	yes	characteristic
181	C₄	#456	1	1	yes	characteristic
182	C₂	—	2	1	no	—
183	C₂	—	2	1	no	—
184	C₂	—	4	2	no	—
185	C₂	#1850	1	1	yes	characteristic
186	C₂	#1859	1	1	yes	characteristic
187	C₂	#1861	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#120
commutator	C₂×C₄	C₂²×C₈
Frattini, cosocle	C₂×C₄²	C₂³
socle	C₂²	#385
prime radical	C₂×#41	C₂×C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ 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. #1842	C₂	semidirect
4. #1565	C₂	semidirect
6. #1283	C₂	semidirect
7. #1276	C₂	semidirect
59. #104	C₈	semidirect
86. M₁₆	C₂×C₈	semidirect
113. C₄²	C₂×C₈	semidirect
121. Q₈	C₄⋊C₈	semidirect
125. D₄	C₄⋊C₈	semidirect

Maximal subgroups

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

subgroup	index
#2988	2
#1842	2
C₄×#104	2
#1565	2
#1477	2
#1283	2
#1276	2