Finite Group Databasebeta

← #14821

Subgroups — #14821

Order 256.

1095 conjugacy classes of subgroups; only the 251 normal subgroups are shown.

normal subgroups only

This group has 1095 conjugacy classes of subgroups — too many to draw. Only its 251 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×C4⋊C4C211yesmaximal, characteristic
2C2×#398C212yesmaximal, complemented
3C2×#385C212yesmaximal
4C2×#380C211yesmaximal, characteristic
5#1441C214yesmaximal, complemented
6#1441C212yesmaximal, complemented
7C22⋊#95C212yesmaximal
8C22⋊#95C211yesmaximal, characteristic
9C24×C4C2211yescharacteristic
10C24×C4C2211yescharacteristic
11C24×C4C2211yescharacteristic
12C22×C4⋊C4—24no—
13C22×C4⋊C4—28no—
14C22×C4⋊C4C2211yescharacteristic
15C22×C4⋊C4C2214yes—
16C22×C4⋊C4C2212yes—
17C22×C4⋊C4C2212yes—
18C22×C4⋊C4C2211yescharacteristic
19C22×C22⋊C4C2212yes—
20C22×C22⋊C4C2212yes—
21#398—28no—
22#385—28no—
23#380—24no—
24C2×#95C2214yes—
25C2×#95C2212yes—
26C2×#95C2212yes—
27C2×#95C2214yes—
28C2×#95C2214yes—
29C2×#95C2212yes—
30C25C2311yescharacteristic
31C23×C4—22no—
32C23×C4—22no—
33C23×C4—24no—
34C23×C4—24no—
35C23×C4—22no—
36C23×C4—24no—
37C23×C4—24no—
38C23×C4—24no—
39C23×C4—24no—
40C23×C4—42no—
41C23×C4—42no—
42C23×C4D414yes—
43C23×C4D414yes—
44C23×C4C2312yes—
45C23×C4C2312yes—
46C23×C4C2311yescharacteristic
47C23×C4C2312yes—
48C23×C4C2314yes—
49C23×C4C2311yescharacteristic
50C23×C4C2312yes—
51C2×C4⋊C4—232no—
52C2×C4⋊C4—24no—
53C2×C4⋊C4—24no—
54C2×C4⋊C4—24no—
55C2×C4⋊C4—24no—
56C2×C4⋊C4—24no—
57C2×C4⋊C4—22no—
58C2×C4⋊C4—22no—
59C2×C4⋊C4—28no—
60C2×C4⋊C4—24no—
61C2×C4⋊C4—28no—
62C2×C4⋊C4—28no—
63C2×C4⋊C4—42no—
64C2×C4⋊C4D418yes—
65C2×C22⋊C4—24no—
66C2×C22⋊C4—24no—
67C2×C22⋊C4—24no—
68C2×C22⋊C4—24no—
69C2×C22⋊C4—24no—
70C2×C22⋊C4Q818yes—
71#95—24no—
72#95—28no—
73#95—24no—
74#95—28no—
75#95—42no—
76#95—44no—
77C24—22no—
78C24—22no—
79C24—22no—
80C24—22no—
81C24C2×D412yes—
82C24C2×D412yes—
83C24C2×Q814yes—
84C24#4112yes—
85C24#4114yes—
86C24C2411yescharacteristic
87C22×C4—28no—
88C22×C4—216no—
89C22×C4—28no—
90C22×C4—28no—
91C22×C4—22no—
92C22×C4—22no—
93C22×C4—22no—
94C22×C4—22no—
95C22×C4—22no—
96C22×C4—22no—
97C22×C4—216no—
98C22×C4—28no—
99C22×C4—28no—
100C22×C4—24no—
101C22×C4—24no—
102C22×C4—22no—
103C22×C4—22no—
104C22×C4—22no—
105C22×C4—24no—
106C22×C4—24no—
107C22×C4—22no—
108C22×C4—24no—
109C22×C4—24no—
110C22×C4—22no—
111C22×C4—24no—
112C22×C4—24no—
113C22×C4—28no—
114C22×C4—22no—
115C22×C4—22no—
116C22×C4—22no—
117C22×C4—24no—
118C22×C4—22no—
119C22×C4—22no—
120C22×C4—22no—
121C22×C4—24no—
122C22×C4—24no—
123C22×C4—24no—
124C22×C4—44no—
125C22×C4—416no—
126C22×C4—416no—
127C22×C4—44no—
128C22×C4—44no—
129C22×C4C2×D412yes—
130C22×C4C2×D414yes—
131C22×C4C2×D414yes—
132C22×C4C2×D412yes—
133C22×C4C2×D414yes—
134C22×C4C2×D414yes—
135C22×C4C2×Q818yes—
136C22×C4#4114yes—
137C4⋊C4—216no—
138C4⋊C4—216no—
139C4⋊C4—44no—
140C4⋊C4—44no—
141C4⋊C4—416no—
142C22⋊C4—216no—
143C22⋊C4—48no—
144C23—22no—
145C23—24no—
146C23—22no—
147C23—24no—
148C23—22no—
149C23—22no—
150C23—24no—
151C23—24no—
152C23—28no—
153C23—28no—
154C23—28no—
155C23—28no—
156C23#12114yes—
157C23C22⋊Q818yes—
158C23#12314yes—
159C23#12518yes—
160C23C4⋊Q814yes—
161C23C22×D411yescharacteristic
162C23C22×D412yes—
163C23C22×D411yescharacteristic
164C23C22×Q812yes—
165C23C2×#4111yescharacteristic
166C23C2×#4112yes—
167C23C2×#4112yes—
168C23H(5,𝔽2)12yes—
169C232−1+412yes—
170C2×C4—232no—
171C2×C4—24no—
172C2×C4—24no—
173C2×C4—24no—
174C2×C4—22no—
175C2×C4—24no—
176C2×C4—24no—
177C2×C4—24no—
178C2×C4—22no—
179C2×C4—28no—
180C2×C4—22no—
181C2×C4—24no—
182C2×C4—28no—
183C2×C4—22no—
184C2×C4—22no—
185C2×C4—28no—
186C2×C4—28no—
187C2×C4—28no—
188C2×C4—416no—
189C2×C4—416no—
190C2×C4—44no—
191C2×C4—42no—
192C2×C4—42no—
193C2×C4—44no—
194C2×C4—44no—
195C2×C4—44no—
196C2×C4—44no—
197C2×C4—44no—
198C2×C4—42no—
199C2×C4—42no—
200C2×C4—42no—
201C2×C4—44no—
202C2×C4—42no—
203C2×C4—44no—
204C2×C4—42no—
205C2×C4—42no—
206C2×C4—44no—
207C2×C4—44no—
208C2×C4#12014yes—
209C2×C4#12118yes—
210C22—24no—
211C22—24no—
212C22—24no—
213C22—24no—
214C22—216no—
215C22—24no—
216C22—216no—
217C22—24no—
218C22#39818yescomplemented
219C22C2×#12011yescharacteristic
220C22C2×#12111yescharacteristic
221C22C2×#12112yes—
222C22C2×C22⋊Q812yes—
223C22C2×#12311yescharacteristic
224C22#52511yescharacteristic
225C22#52512yes—
226C22C2×#12512yes—
227C22C2×C4⋊Q811yescharacteristic
228C22#53411yescharacteristic
229C22#53611yescharacteristic
230C22#53712yes—
231C22#53912yes—
232C22#54614yes—
233C22D4×Q814yes—
234C22#55214yes—
235C22#55414yes—
236C4—28no—
237C4—24no—
238C4—42no—
239C4—44no—
240C4—48no—
241C4—82no—
242C4—82no—
243C2—28no—
244C2C2×#39811yescharacteristic
245C2#240511yescharacteristic
246C2#241714yes—
247C2#243411yescharacteristic
248C2#247314yes—
249C2#248411yescharacteristic
250C2#258311yescharacteristic
251C2#261811yescharacteristic
252C2#284711yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C24
prime radicalC25C23
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. C2×#398C2semidirect
5. #1441C2semidirect
6. #1441C2semidirect
218. C22#398semidirect

Maximal subgroups

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

subgroupindex
C23×C4⋊C42
C2×#3982
C2×#3852
C2×#3802
#14412
C22⋊#952