Finite Group Databasebeta

← #8359

Subgroups — #8359

Order 256.

604 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 604 conjugacy classes of subgroups — too many to draw. Only its 79 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
1C2×#546C211yesmaximal, characteristic
2#1789C212yesmaximal, complemented
3C4×#99C211yesmaximal, characteristic, complemented
4#1507C212yesmaximal, complemented
5#1465C211yesmaximal, characteristic, complemented
6C22×#41C2211yescharacteristic
7C23×D4C411yescharacteristic, complemented
8#546—24no—
9#546—24no—
10C2×#124C411yescharacteristic, complemented
11C2×#123—21no—
12C2×C22⋊Q8C411yescharacteristic, complemented
13C2×#121—21no—
14C2×#121C2211yescharacteristic
15C2×#120—21no—
16C2×C4×D4C411yescharacteristic, complemented
17C2×C4×D4C2211yescharacteristic
18C22×C22⋊C4—21no—
19C2×#104—24no—
20C2×#99—21no—
21#407C2212yescomplemented
22C4×C22⋊C4C2212yescomplemented
23#342—21no—
24#327—24no—
25#324—21no—
26#323—21no—
27C25—21no—
28C2×#41—22no—
29C2×#41—44no—
30C2×#41D414yes—
31C22×Q8C2×C411yescharacteristic
32C22×D4—21no—
33C22×D4—21no—
34C22×D4—22no—
35C22×D4—22no—
36C22×D4—21no—
37C22×D4—44no—
38C22×D4C2×C411yescharacteristic
39C22×D4C2×C411yescharacteristic
40C22×D4D411yescharacteristic
41C23×C4—21no—
42C23×C4—21no—
43C23×C4C2×C411yescharacteristic
44C23×C4C2311yescharacteristic
45C2×M16—22no—
46#124—22no—
47#124—22no—
48#123—28no—
49C22⋊Q8—22no—
50C22⋊Q8—22no—
51#121—28no—
52#121—22no—
53#121D414yes—
54#120—28no—
55C4×D4—22no—
56C4×D4—22no—
57C4×D4—22no—
58C4×D4—22no—
59C2×C4⋊C4—21no—
60C2×C4⋊C4C2×C411yescharacteristic
61C2×C4⋊C4D411yescharacteristic
62C2×C22⋊C4—28no—
63C2×C22⋊C4—21no—
64C2×C22⋊C4—21no—
65C2×C22⋊C4—21no—
66C2×C22⋊C4—21no—
67C2×C22⋊C4—21no—
68C2×C22⋊C4—22no—
69C2×C22⋊C4D411yescharacteristic
70C2×C22⋊C4D411yescharacteristic
71C2×C42—22no—
72C2×C42C2×C411yescharacteristic
73#104—48no—
74#99—24no—
75C22⋊C8—22no—
76C22⋊C8—22no—
77#95—22no—
78C24—21no—
79C24—24no—
80C24—21no—
81C24—44no—
82C24—42no—
83C24—42no—
84C24—82no—
85C24C22⋊C411yescharacteristic
86C24C2×D411yescharacteristic
87#41—28no—
88#41—44no—
89#41—48no—
90C2×Q8—21no—
91C2×Q8—42no—
92C2×Q8C22⋊C412yes—
93C2×D4—24no—
94C2×D4—24no—
95C2×D4—22no—
96C2×D4—22no—
97C2×D4—24no—
98C2×D4—24no—
99C2×D4—21no—
100C2×D4—21no—
101C2×D4—41no—
102C2×D4—42no—
103C2×D4—48no—
104C2×D4—44no—
105C2×D4—44no—
106C2×D4—44no—
107C2×D4—44no—
108C2×D4—82no—
109C2×D4C22⋊C412yes—
110C2×D4C22⋊C412yes—
111C22×C4—21no—
112C22×C4—21no—
113C22×C4—24no—
114C22×C4—21no—
115C22×C4—21no—
116C22×C4—24no—
117C22×C4—42no—
118C22×C4—42no—
119C22×C4—44no—
120C22×C4—42no—
121C22×C4—82no—
122C22×C4C22⋊C411yescharacteristic
123C22×C4C22⋊C411yescharacteristic
124C22×C4C22⋊C411yescharacteristic
125C22×C4C22×C411yescharacteristic
126C22×C4C2×D412yes—
127C22×C4C2×D412yes—
128C22×C4C2×D411yescharacteristic
129M16—44no—
130C2×C8—42no—
131C4⋊C4—24no—
132C4⋊C4—21no—
133C4⋊C4—22no—
134C4⋊C4C22⋊C412yes—
135C22⋊C4—24no—
136C22⋊C4—24no—
137C22⋊C4—24no—
138C22⋊C4—24no—
139C22⋊C4—22no—
140C22⋊C4—22no—
141C22⋊C4—24no—
142C22⋊C4—24no—
143C22⋊C4—42no—
144C42—21no—
145C42—21no—
146C42—44no—
147C23—24no—
148C23—21no—
149C23—22no—
150C23—21no—
151C23—21no—
152C23—21no—
153C23—22no—
154C23—41no—
155C23—41no—
156C23—41no—
157C23—41no—
158C23—42no—
159C23—42no—
160C23—48no—
161C23—42no—
162C23—48no—
163C23—41no—
164C23—41no—
165C23—41no—
166C23—41no—
167C23—84no—
168C23—88no—
169C23—88no—
170C23#10412yescomplemented
171C23C2×C22⋊C411yescharacteristic
172C23#12012yes—
173Q8—24no—
174Q8—42no—
175D4—24no—
176D4—22no—
177D4—42no—
178D4—44no—
179D4—48no—
180D4—48no—
181D4—42no—
182D4—84no—
183D4#9914yescomplemented
184C2×C4—24no—
185C2×C4—22no—
186C2×C4—21no—
187C2×C4—21no—
188C2×C4—22no—
189C2×C4—22no—
190C2×C4—21no—
191C2×C4—21no—
192C2×C4—21no—
193C2×C4—41no—
194C2×C4—42no—
195C2×C4—48no—
196C2×C4—42no—
197C2×C4—42no—
198C2×C4—42no—
199C2×C4—42no—
200C2×C4—41no—
201C2×C4—41no—
202C2×C4—44no—
203C2×C4—84no—
204C2×C4—82no—
205C2×C4—82no—
206C2×C4C2×C22⋊C411yescharacteristic
207C2×C4C2×C22⋊C411yescharacteristic
208C2×C4#12012yes—
209C8—82no—
210C22—21no—
211C22—21no—
212C22—22no—
213C22—22no—
214C22—24no—
215C22—22no—
216C22—22no—
217C22—42no—
218C22—44no—
219C22—44no—
220C22—42no—
221C22—42no—
222C22—41no—
223C22—41no—
224C22—42no—
225C22—42no—
226C22—42no—
227C22—42no—
228C22—81no—
229C22—81no—
230C22—82no—
231C22—82no—
232C22—816no—
233C22#37911yescharacteristic
234C22C2×#10411yescharacteristic
235C22#42111yescharacteristic
236C4—22no—
237C4—21no—
238C4—42no—
239C4—41no—
240C4—42no—
241C4—41no—
242C4—41no—
243C4—84no—
244C4C2×#9912yes—
245C2—21no—
246C2—21no—
247C2—42no—
248C2—44no—
249C2—42no—
250C2—82no—
251C2#178111yescharacteristic
252C2#178911yescharacteristic
253C2#179511yescharacteristic
subgroupG/H
center, socleC22#379
commutatorC22×C4C22×C4
Frattini, cosocleC23×C4C23
prime radicalC2×#546C2
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #1789C2semidirect
3. C4×#99C2semidirect
4. #1507C2semidirect
5. #1465C2semidirect
7. C23×D4C4semidirect
10. C2×#124C4semidirect
12. C2×C22⋊Q8C4semidirect
16. C2×C4×D4C4semidirect
21. #407C22semidirect
22. C4×C22⋊C4C22semidirect
170. C23#104semidirect
183. D4#99semidirect

Maximal subgroups

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

subgroupindex
C2×#5462
#17892
C4×#992
#15072
#14652