Finite Group Databasebeta

← #14720

Subgroups — #14720

Order 256.

1247 conjugacy classes of subgroups; only the 267 normal subgroups are shown.

normal subgroups only

This group has 1247 conjugacy classes of subgroups — too many to draw. Only its 267 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
1C22×C22⋊Q8C211yesmaximal, characteristic
2C23×C4⋊C4C211yesmaximal, characteristic
3C2×#395C212yesmaximal, complemented
4C2×#385C212yesmaximal
5#1444C214yesmaximal, complemented
6#1441C212yesmaximal, complemented
7C22⋊#95C212yesmaximal
8C22⋊#95C211yesmaximal, characteristic
9C23×Q8C2211yescharacteristic
10C24×C4C2211yescharacteristic
11C24×C4C2211yescharacteristic
12C24×C4C2211yescharacteristic
13C2×C22⋊Q8—24no—
14C2×C22⋊Q8—28no—
15C22×C4⋊C4—24no—
16C22×C4⋊C4—28no—
17C22×C4⋊C4C2212yes—
18C22×C4⋊C4C2211yescharacteristic
19C22×C4⋊C4C2212yes—
20C22×C4⋊C4C2214yes—
21C22×C4⋊C4C2212yes—
22C22×C22⋊C4C2212yes—
23C22×C22⋊C4C2212yes—
24#395—28no—
25#385—28no—
26C2×#95C2212yes—
27C2×#95C2214yes—
28C2×#95C2214yes—
29C2×#95C2214yes—
30C2×#95C2212yes—
31C25C2311yescharacteristic
32C22×Q8—24no—
33C22×Q8—22no—
34C22×Q8—24no—
35C22×Q8—42no—
36C23×C4—22no—
37C23×C4—22no—
38C23×C4—24no—
39C23×C4—24no—
40C23×C4—22no—
41C23×C4—24no—
42C23×C4—24no—
43C23×C4—24no—
44C23×C4—24no—
45C23×C4—42no—
46C23×C4—42no—
47C23×C4D414yes—
48C23×C4Q814yescomplemented
49C23×C4C2312yes—
50C23×C4C2312yes—
51C23×C4C2312yes—
52C23×C4C2311yescharacteristic
53C23×C4C2314yes—
54C23×C4C2311yescharacteristic
55C23×C4C2312yes—
56C22⋊Q8—232no—
57C2×C4⋊C4—232no—
58C2×C4⋊C4—24no—
59C2×C4⋊C4—24no—
60C2×C4⋊C4—24no—
61C2×C4⋊C4—24no—
62C2×C4⋊C4—24no—
63C2×C4⋊C4—24no—
64C2×C4⋊C4—22no—
65C2×C4⋊C4—24no—
66C2×C4⋊C4—24no—
67C2×C4⋊C4—24no—
68C2×C4⋊C4—28no—
69C2×C4⋊C4—28no—
70C2×C4⋊C4—44no—
71C2×C4⋊C4D418yes—
72C2×C22⋊C4—24no—
73C2×C22⋊C4—24no—
74C2×C22⋊C4—24no—
75C2×C22⋊C4—24no—
76C2×C22⋊C4D418yes—
77C2×C22⋊C4Q818yescomplemented
78#95—28no—
79#95—24no—
80#95—28no—
81#95—28no—
82#95—42no—
83C24—22no—
84C24—22no—
85C24—22no—
86C24—22no—
87C24C2×D414yes—
88C24C2×D412yes—
89C24C2×Q814yes—
90C24C2×Q812yes—
91C24#4112yes—
92C24C2411yescharacteristic
93C2×Q8—28no—
94C2×Q8—416no—
95C2×Q8—44no—
96C2×Q8—44no—
97C22×C4—28no—
98C22×C4—216no—
99C22×C4—28no—
100C22×C4—28no—
101C22×C4—22no—
102C22×C4—22no—
103C22×C4—22no—
104C22×C4—22no—
105C22×C4—22no—
106C22×C4—22no—
107C22×C4—216no—
108C22×C4—28no—
109C22×C4—28no—
110C22×C4—24no—
111C22×C4—24no—
112C22×C4—24no—
113C22×C4—24no—
114C22×C4—22no—
115C22×C4—22no—
116C22×C4—22no—
117C22×C4—22no—
118C22×C4—22no—
119C22×C4—22no—
120C22×C4—22no—
121C22×C4—22no—
122C22×C4—24no—
123C22×C4—24no—
124C22×C4—24no—
125C22×C4—24no—
126C22×C4—22no—
127C22×C4—24no—
128C22×C4—24no—
129C22×C4—24no—
130C22×C4—24no—
131C22×C4—28no—
132C22×C4—44no—
133C22×C4—416no—
134C22×C4—416no—
135C22×C4—44no—
136C22×C4—44no—
137C22×C4C2×D414yes—
138C22×C4C2×D414yes—
139C22×C4C2×D412yes—
140C22×C4C2×D414yes—
141C22×C4C2×D414yes—
142C22×C4C2×D412yes—
143C22×C4C2×D414yes—
144C22×C4C2×Q818yes—
145C22×C4C2×Q814yes—
146C22×C4#4114yes—
147C4⋊C4—216no—
148C4⋊C4—28no—
149C4⋊C4—216no—
150C4⋊C4—216no—
151C4⋊C4—416no—
152C22⋊C4—216no—
153C22⋊C4—216no—
154C23—22no—
155C23—24no—
156C23—22no—
157C23—24no—
158C23—22no—
159C23—22no—
160C23—24no—
161C23—24no—
162C23—28no—
163C23—28no—
164C23—28no—
165C23—28no—
166C23#12014yes—
167C23C22⋊Q818yes—
168C23C22⋊Q814yes—
169C23C4⋊Q814yes—
170C23C4⋊Q818yes—
171C23C22×D411yescharacteristic
172C23C22×D412yes—
173C23C22×D412yes—
174C23C22×Q811yescharacteristic
175C23C22×Q812yes—
176C23C2×#4111yescharacteristic
177C23C2×#4112yes—
178C23H(5,𝔽2)12yes—
179C232−1+412yes—
180Q8—416no—
181C2×C4—232no—
182C2×C4—24no—
183C2×C4—22no—
184C2×C4—24no—
185C2×C4—24no—
186C2×C4—24no—
187C2×C4—24no—
188C2×C4—24no—
189C2×C4—22no—
190C2×C4—28no—
191C2×C4—24no—
192C2×C4—24no—
193C2×C4—22no—
194C2×C4—24no—
195C2×C4—22no—
196C2×C4—28no—
197C2×C4—22no—
198C2×C4—28no—
199C2×C4—28no—
200C2×C4—28no—
201C2×C4—28no—
202C2×C4—28no—
203C2×C4—416no—
204C2×C4—416no—
205C2×C4—42no—
206C2×C4—42no—
207C2×C4—42no—
208C2×C4—44no—
209C2×C4—42no—
210C2×C4—42no—
211C2×C4—42no—
212C2×C4—44no—
213C2×C4—44no—
214C2×C4—44no—
215C2×C4—44no—
216C2×C4—44no—
217C2×C4—44no—
218C2×C4—44no—
219C2×C4#12014yes—
220C2×C4C22⋊Q818yescomplemented
221C22—24no—
222C22—24no—
223C22—24no—
224C22—24no—
225C22—24no—
226C22—24no—
227C22—216no—
228C22—216no—
229C22#39518yescomplemented
230C22C2×#12011yescharacteristic
231C22C2×#12011yescharacteristic
232C22C2×C22⋊Q811yescharacteristic
233C22C2×C22⋊Q812yes—
234C22C2×C22⋊Q812yes—
235C22#52511yescharacteristic
236C22#52512yes—
237C22C2×C4⋊Q811yescharacteristic
238C22C2×C4⋊Q812yes—
239C22#53411yescharacteristic
240C22#53611yescharacteristic
241C22#53912yes—
242C22#54412yes—
243C22D4214yes—
244C22#54714yes—
245C22D4×Q814yes—
246C22#55414yes—
247C4—28no—
248C4—24no—
249C4—42no—
250C4—44no—
251C4—44no—
252C4—44no—
253C4—48no—
254C2—28no—
255C2C2×#39511yescharacteristic
256C2#240311yescharacteristic
257C2#240611yescharacteristic
258C2#243411yescharacteristic
259C2#244014yes—
260C2#244914yes—
261C2#258311yescharacteristic
262C2#262711yescharacteristic
263C2#284811yescharacteristic
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
3. C2×#395C2semidirect
5. #1444C2semidirect
6. #1441C2semidirect
48. C23×C4Q8semidirect
77. C2×C22⋊C4Q8semidirect
220. C2×C4C22⋊Q8semidirect
229. C22#395semidirect

Maximal subgroups

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

subgroupindex
C22×C22⋊Q82
C23×C4⋊C42
C2×#3952
C2×#3852
#14442
#14412
C22⋊#952