Finite Group Databasebeta

← #32922

Subgroups — #32922

Order 256.

709 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 709 conjugacy classes of subgroups — too many to draw. Only its 149 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
1C4×H(5,𝔽2)C211yesmaximal, characteristic
2#3039C212yesmaximal, complemented
3C2×#455C212yesmaximal
4#2547C211yesmaximal, characteristic, complemented
5#1885C212yesmaximal, complemented
6#1882C214yesmaximal, complemented
7#1806C212yesmaximal, complemented
8#1800C211yesmaximal, characteristic, complemented
9C2×H(5,𝔽2)C2211yescharacteristic
10C2×#137C2214yes—
11C2×C4⋊Q8C2211yescharacteristic, complemented
12C2×#124C2212yes—
13#520—22no—
14#520C2212yes—
15#518—22no—
16#518—22no—
17#518C2211yescharacteristic
18C4×#41—22no—
19C4×#41C2212yes—
20C2×C4×D4—22no—
21C2×C4×D4—22no—
22C2×C4×D4C2211yescharacteristic
23#462—22no—
24#462—22no—
25#461—22no—
26#461—22no—
27#455—28no—
28#423C2212yescomplemented
29C2×#104C2214yes—
30#419C2214yescomplemented
31C2×#100C2212yes—
32#397—22no—
33#391C2212yescomplemented
34#380—22no—
35C4×C4⋊C4C2211yescharacteristic, complemented
36#343C2212yescomplemented
37#339C2214yescomplemented
38H(5,𝔽2)—24no—
39H(5,𝔽2)—22no—
40H(5,𝔽2)D414yes—
41C2×#41—22no—
42C2×#41C2312yes—
43C22×Q8C2312yes—
44C22×D4—22no—
45C22×D4—22no—
46C22×D4C2311yescharacteristic
47C23×C4—22no—
48C23×C4D412yes—
49#137—28no—
50#137—28no—
51C2×Q16—24no—
52C2×SD16—24no—
53C2×M16C2314yes—
54C4⋊Q8—22no—
55C4⋊Q8—22no—
56#124—24no—
57#124—24no—
58C4×Q8—22no—
59C4×Q8D412yes—
60C4×Q8D412yes—
61C4×D4—24no—
62C4×D4—24no—
63C4×D4—24no—
64C4×D4—24no—
65C4×D4—22no—
66C4×D4—22no—
67C4×D4—24no—
68C4×D4—22no—
69C4×D4—22no—
70C4×D4—22no—
71C4×D4—22no—
72C4×D4—22no—
73C4×D4D412yes—
74C4×D4D412yes—
75#117—24no—
76#117—22no—
77#117—22no—
78#117C2312yes—
79C2×C4⋊C4—22no—
80C2×C4⋊C4—22no—
81C2×C4⋊C4—21no—
82C2×C4⋊C4—21no—
83C2×C4⋊C4—21no—
84C2×C4⋊C4C2311yescharacteristic
85C2×C22⋊C4—22no—
86C2×C22⋊C4—24no—
87C2×C22⋊C4—22no—
88C2×C22⋊C4—24no—
89C2×C22⋊C4D412yes—
90C2×C42—22no—
91C2×C42—22no—
92C2×C42C2312yes—
93C2×C42C2311yescharacteristic
94C4⋊C8—22no—
95C4⋊C8—22no—
96#104—28no—
97#103—24no—
98#103—24no—
99#100—24no—
100#95—21no—
101#95—24no—
102C24—22no—
103C24C2×D412yes—
104#41—28no—
105#41—28no—
106#41—22no—
107#41—22no—
108#41C2×D414yes—
109#41#4114yes—
110C2×Q8—24no—
111C2×Q8—22no—
112C2×Q8—22no—
113C2×Q8—24no—
114C2×Q8C2×D412yes—
115C2×D4—24no—
116C2×D4—28no—
117C2×D4—28no—
118C2×D4—28no—
119C2×D4—22no—
120C2×D4—28no—
121C2×D4—22no—
122C2×D4—22no—
123C2×D4C2×D412yes—
124C2×D4C2×D412yes—
125C2×D4#4112yes—
126C22×C4—22no—
127C22×C4—22no—
128C22×C4—22no—
129C22×C4—24no—
130C22×C4—22no—
131C22×C4—22no—
132C22×C4—22no—
133C22×C4—22no—
134C22×C4—22no—
135C22×C4—22no—
136C22×C4—22no—
137C22×C4—44no—
138C22×C4—84no—
139C22×C4C2×D411yescharacteristic
140C22×C4C2×D412yes—
141C22×C4C2×D411yescharacteristic
142C22×C4C2411yescharacteristic
143Q16—48no—
144SD16—48no—
145M16—24no—
146M16—24no—
147C2×C8—24no—
148C4⋊C4—24no—
149C4⋊C4—22no—
150C4⋊C4—22no—
151C4⋊C4—22no—
152C4⋊C4—42no—
153C4⋊C4—42no—
154C4⋊C4—41no—
155C4⋊C4—41no—
156C4⋊C4C2×D412yes—
157C4⋊C4C2×D412yes—
158C22⋊C4—24no—
159C22⋊C4—24no—
160C22⋊C4—22no—
161C22⋊C4—22no—
162C22⋊C4—22no—
163C22⋊C4—24no—
164C22⋊C4—48no—
165C42—24no—
166C42—22no—
167C42—22no—
168C42—22no—
169C42—22no—
170C42—22no—
171C42C2×D412yes—
172C42C2×D412yes—
173C23—28no—
174C23—22no—
175C23—22no—
176C23—22no—
177C23—42no—
178C23—42no—
179C23—82no—
180C23—82no—
181C23#12114yes—
182C23C22×D411yescharacteristic
183Q8—22no—
184Q8—44no—
185Q8—44no—
186Q8#12114yes—
187D4—216no—
188D4—24no—
189D4—24no—
190D4—24no—
191D4—22no—
192D4—24no—
193D4#12114yes—
194C2×C4—28no—
195C2×C4—22no—
196C2×C4—24no—
197C2×C4—24no—
198C2×C4—22no—
199C2×C4—22no—
200C2×C4—22no—
201C2×C4—22no—
202C2×C4—41no—
203C2×C4—44no—
204C2×C4—44no—
205C2×C4—42no—
206C2×C4—42no—
207C2×C4—42no—
208C2×C4—42no—
209C2×C4—42no—
210C2×C4—42no—
211C2×C4—42no—
212C2×C4—82no—
213C2×C4—82no—
214C2×C4—82no—
215C2×C4—84no—
216C2×C4—82no—
217C2×C4#12012yes—
218C2×C4#12011yescharacteristic
219C2×C4#12011yescharacteristic
220C2×C4C22×D411yescharacteristic
221C2×C4C22×D412yes—
222C2×C4C2×#4112yes—
223C2×C4C2×#4111yescharacteristic
224C8—44no—
225C22—21no—
226C22—21no—
227C22—24no—
228C22—24no—
229C22—44no—
230C22—42no—
231C22—42no—
232C22—42no—
233C22—42no—
234C22—82no—
235C22—88no—
236C22C2×#12011yescharacteristic
237C22C2×#12111yescharacteristic
238C22#52511yescharacteristic
239C4—24no—
240C4—21no—
241C4—22no—
242C4—42no—
243C4—42no—
244C4—42no—
245C4—82no—
246C4—82no—
247C4#45512yescomplemented
248C4C2×#12112yes—
249C4#52512yes—
250C2—21no—
251C2—21no—
252C2—44no—
253C2—44no—
254C2#284711yescharacteristic
255C2C2×#45511yescharacteristic
256C2#302211yescharacteristic
subgroupG/H
center, socleC22C2×#120
commutator, Frattini, cosocleC22×C4C24
prime radicalC2×H(5,𝔽2)C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #3039C2semidirect
4. #2547C2semidirect
5. #1885C2semidirect
6. #1882C2semidirect
7. #1806C2semidirect
8. #1800C2semidirect
11. C2×C4⋊Q8C22semidirect
28. #423C22semidirect
30. #419C22semidirect
33. #391C22semidirect
35. C4×C4⋊C4C22semidirect
36. #343C22semidirect
37. #339C22semidirect
247. C4#455semidirect

Maximal subgroups

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

subgroupindex
C4×H(5,𝔽2)2
#30392
C2×#4552
#25472
#18852
#18822
#18062
#18002