Finite Group Databasebeta

Subgroups — #10619

Order 256.

209 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 209 conjugacy classes of subgroups — too many to draw. Only its 71 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	#1954	C₂	1	1	yes	maximal, characteristic
2	#1844	C₂	1	1	yes	maximal, characteristic, complemented
3	#1802	C₂	1	1	yes	maximal, characteristic
4	#1802	C₂	1	1	yes	maximal, characteristic
5	#1296	C₂	1	1	yes	maximal, characteristic, complemented
6	#1296	C₂	1	1	yes	maximal, characteristic, complemented
7	#1277	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₄×D₄	C₂²	1	1	yes	characteristic
9	C₂×#106	—	2	1	no	—
10	C₂×#106	—	2	1	no	—
11	C₂×C₄⋊C₈	C₂²	1	1	yes	characteristic
12	C₂×C₄⋊C₈	C₂²	1	1	yes	characteristic
13	C₂×C₄⋊C₈	C₂²	1	1	yes	characteristic
14	C₂×#102	—	2	1	no	—
15	C₂×#102	—	2	1	no	—
16	C₂×C₂²⋊C₈	—	2	1	no	—
17	C₂×C₂²⋊C₈	—	2	1	no	—
18	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
19	#384	C₂²	1	1	yes	characteristic
20	#384	C₂²	1	1	yes	characteristic
21	#340	—	2	1	no	—
22	#340	—	2	1	no	—
23	C₂²×D₄	D₄	1	1	yes	characteristic
24	C₂³×C₄	—	2	1	no	—
25	C₂²×C₈	—	2	1	no	—
26	C₂²×C₈	—	2	1	no	—
27	C₂²×C₈	—	2	1	no	—
28	C₂²×C₈	D₄	1	1	yes	characteristic
29	C₂²×C₈	D₄	1	1	yes	characteristic
30	C₄×D₄	—	2	1	no	—
31	C₄×D₄	—	2	1	no	—
32	C₄×D₄	—	2	1	no	—
33	C₄×D₄	—	2	1	no	—
34	C₂×C₄⋊C₄	—	2	1	no	—
35	C₂×C₄⋊C₄	—	2	1	no	—
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	—	2	1	no	—
40	C₂×C₄⋊C₄	D₄	1	1	yes	characteristic
41	C₂×C₂²⋊C₄	—	2	1	no	—
42	C₂×C₄²	C₂³	1	1	yes	characteristic
43	#106	—	4	2	no	—
44	#106	—	4	2	no	—
45	C₄⋊C₈	—	2	1	no	—
46	C₄⋊C₈	—	2	1	no	—
47	C₄⋊C₈	—	2	1	no	—
48	C₄⋊C₈	—	2	1	no	—
49	C₄⋊C₈	—	2	1	no	—
50	C₄⋊C₈	—	2	1	no	—
51	#102	—	4	2	no	—
52	#102	—	4	2	no	—
53	C₂²⋊C₈	—	4	2	no	—
54	C₂²⋊C₈	—	4	2	no	—
55	C₄×C₈	—	2	1	no	—
56	C₄×C₈	—	2	1	no	—
57	C₂⁴	—	2	1	no	—
58	C₂×D₄	—	2	1	no	—
59	C₂×D₄	—	2	1	no	—
60	C₂×D₄	—	2	1	no	—
61	C₂×D₄	—	2	1	no	—
62	C₂×D₄	—	2	1	no	—
63	C₂×D₄	SD₁₆	1	2	yes	—
64	C₂²×C₄	—	2	1	no	—
65	C₂²×C₄	—	4	1	no	—
66	C₂²×C₄	—	4	1	no	—
67	C₂²×C₄	—	4	1	no	—
68	C₂²×C₄	—	4	1	no	—
69	C₂²×C₄	—	4	1	no	—
70	C₂²×C₄	—	4	1	no	—
71	C₂²×C₄	—	4	1	no	—
72	C₂²×C₄	—	4	1	no	—
73	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
74	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
75	C₂²×C₄	#41	1	1	yes	characteristic
76	C₂×C₈	—	2	1	no	—
77	C₂×C₈	—	2	1	no	—
78	C₂×C₈	—	2	1	no	—
79	C₂×C₈	—	2	1	no	—
80	C₂×C₈	—	4	1	no	—
81	C₂×C₈	—	4	1	no	—
82	C₂×C₈	—	4	1	no	—
83	C₂×C₈	—	4	1	no	—
84	C₂×C₈	—	4	1	no	—
85	C₂×C₈	—	4	1	no	—
86	C₂×C₈	—	4	1	no	—
87	C₂×C₈	—	4	1	no	—
88	C₂×C₈	—	4	1	no	—
89	C₂×C₈	D₈	1	2	yes	—
90	C₂×C₈	SD₁₆	1	2	yes	—
91	C₄⋊C₄	—	2	1	no	—
92	C₄⋊C₄	—	2	1	no	—
93	C₄⋊C₄	—	4	1	no	—
94	C₄⋊C₄	—	4	1	no	—
95	C₄⋊C₄	—	4	1	no	—
96	C₄⋊C₄	—	4	1	no	—
97	C₄⋊C₄	—	4	1	no	—
98	C₄⋊C₄	—	4	1	no	—
99	C₄⋊C₄	—	4	1	no	—
100	C₄⋊C₄	—	4	1	no	—
101	C₄⋊C₄	—	8	1	no	—
102	C₄⋊C₄	—	8	1	no	—
103	C₂²⋊C₄	—	4	1	no	—
104	C₂²⋊C₄	—	4	1	no	—
105	C₄²	#41	1	1	yes	characteristic
106	C₄²	#41	1	1	yes	characteristic
107	C₄²	#41	1	1	yes	characteristic
108	C₄²	#41	1	1	yes	characteristic
109	C₂³	—	2	2	no	—
110	C₂³	—	4	1	no	—
111	C₂³	—	4	1	no	—
112	C₂³	—	4	1	no	—
113	C₂³	—	4	1	no	—
114	C₂³	—	4	1	no	—
115	C₂³	—	4	1	no	—
116	C₂³	#121	1	1	yes	characteristic
117	D₄	—	2	2	no	—
118	D₄	—	2	2	no	—
119	D₄	—	4	1	no	—
120	D₄	—	4	1	no	—
121	C₂×C₄	—	2	1	no	—
122	C₂×C₄	—	2	1	no	—
123	C₂×C₄	—	4	1	no	—
124	C₂×C₄	—	4	1	no	—
125	C₂×C₄	—	4	1	no	—
126	C₂×C₄	—	8	1	no	—
127	C₂×C₄	—	8	1	no	—
128	C₂×C₄	—	8	1	no	—
129	C₂×C₄	—	8	1	no	—
130	C₂×C₄	—	8	1	no	—
131	C₂×C₄	—	8	1	no	—
132	C₂×C₄	—	8	1	no	—
133	C₂×C₄	—	8	1	no	—
134	C₂×C₄	—	8	1	no	—
135	C₂×C₄	—	8	1	no	—
136	C₂×C₄	#123	1	1	yes	characteristic
137	C₂×C₄	#123	1	1	yes	characteristic
138	C₂×C₄	#123	1	1	yes	characteristic
139	C₂×C₄	#124	1	1	yes	characteristic
140	C₂×C₄	#126	1	1	yes	characteristic
141	C₂×C₄	#126	1	1	yes	characteristic
142	C₂×C₄	C₂×D₈	1	1	yes	characteristic
143	C₂×C₄	C₂×SD₁₆	1	1	yes	characteristic
144	C₂×C₄	C₂×SD₁₆	1	1	yes	characteristic
145	C₂×C₄	#135	1	1	yes	characteristic
146	C₂×C₄	AGL(1,ℤ/8)	1	1	yes	characteristic
147	C₂×C₄	AGL(1,ℤ/8)	1	1	yes	characteristic
148	C₂×C₄	#137	1	1	yes	characteristic
149	C₂×C₄	#137	1	1	yes	characteristic
150	C₈	—	2	2	no	—
151	C₈	—	2	2	no	—
152	C₈	—	8	1	no	—
153	C₈	—	8	1	no	—
154	C₈	—	8	1	no	—
155	C₂²	—	4	2	no	—
156	C₂²	—	4	2	no	—
157	C₂²	—	4	2	no	—
158	C₂²	—	8	1	no	—
159	C₂²	—	8	1	no	—
160	C₂²	—	8	1	no	—
161	C₂²	—	8	1	no	—
162	C₂²	#397	1	1	yes	characteristic
163	C₂²	#459	1	1	yes	characteristic
164	C₂²	#461	1	1	yes	characteristic
165	C₂²	#465	1	1	yes	characteristic
166	C₂²	#468	1	1	yes	characteristic
167	C₂²	#471	1	1	yes	characteristic
168	C₂²	#471	1	1	yes	characteristic
169	C₄	—	2	1	no	—
170	C₄	—	2	1	no	—
171	C₄	—	8	1	no	—
172	C₄	—	16	1	no	—
173	C₄	—	16	1	no	—
174	C₄	#480	1	1	yes	characteristic
175	C₄	#481	1	1	yes	characteristic
176	C₄	#481	1	1	yes	characteristic
177	C₄	#482	1	1	yes	characteristic
178	C₄	#483	1	1	yes	characteristic
179	C₄	#485	1	1	yes	characteristic
180	C₄	#486	1	1	yes	characteristic
181	C₄	#489	1	1	yes	characteristic
182	C₂	—	8	2	no	—
183	C₂	#1665	1	1	yes	characteristic
184	C₂	#1671	1	1	yes	characteristic
185	C₂	#1685	1	1	yes	characteristic
186	C₂	#1691	1	1	yes	characteristic
187	C₂	#2071	1	1	yes	characteristic
188	C₂	#2077	1	1	yes	characteristic
189	C₂	#2079	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³	#121
commutator, Frattini, cosocle	C₂×C₄²	C₂³
prime radical	C₂²×D₄	D₄
LCS	G ▷ C₂×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
2. #1844	C₂	semidirect
5. #1296	C₂	semidirect
6. #1296	C₂	semidirect
7. #1277	C₂	semidirect

Maximal subgroups

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

subgroup	index
#1954	2
#1844	2
#1802	2
#1296	2
#1277	2