Finite Group Databasebeta

← #23307

Subgroups — #23307

Order 256.

526 conjugacy classes of subgroups; only the 157 normal subgroups are shown.

normal subgroups only

This group has 526 conjugacy classes of subgroups — too many to draw. Only its 157 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×#120C211yesmaximal, characteristic
2#1955C212yesmaximal, complemented
3#1954C211yesmaximal, characteristic, complemented
4#1924C212yesmaximal, complemented
5#1924C212yesmaximal, complemented
6#1923C212yesmaximal, complemented
7#1869C211yesmaximal, characteristic, complemented
8#1869C212yesmaximal, complemented
9#1779C212yesmaximal, complemented
10C24×C4C2211yescharacteristic
11C23×C8C2211yescharacteristic, complemented
12C2×#120C411yescharacteristic
13C2×C4×D4C2211yescharacteristic
14C2×C4×D4C2212yes—
15C2×C4⋊C8C2212yescomplemented
16C2×C4⋊C8C2211yescharacteristic, complemented
17C2×C22⋊C8C2212yes—
18C2×C22⋊C8C2212yes—
19C2×C22⋊C8C2212yes—
20C2×C22⋊C8C2212yes—
21C2×C22⋊C8C2212yes—
22C2×C22⋊C8C2212yes—
23C2×C22⋊C8C2212yescomplemented
24C2×C22⋊C8C2211yescharacteristic, complemented
25C2×#97C2212yescomplemented
26C2×C4×C8C2211yescharacteristic, complemented
27#386C412yes—
28#386C411yescharacteristic
29#385C411yescharacteristic
30#385C412yes—
31#379C411yescharacteristic
32C4×C22⋊C4C2212yes—
33C4×C22⋊C4C2211yescharacteristic
34#336C2212yescomplemented
35#336C2212yescomplemented
36#336C2212yescomplemented
37C25C2×C411yescharacteristic
38C22×D4C2×C412yes—
39C22×D4C2×C411yescharacteristic
40C23×C4—41no—
41C23×C4—42no—
42C23×C4—41no—
43C23×C4—42no—
44C23×C4C2×C411yescharacteristic
45C23×C4C2×C412yes—
46C23×C4C2311yescharacteristic
47C23×C4C2311yescharacteristic
48C23×C4C2312yes—
49C22×C8—41no—
50C22×C8—42no—
51C22×C8C2312yes—
52C22×C8C2312yes—
53C22×C8C2312yes—
54C22×C8C2312yes—
55#120—24no—
56C4×D4—41no—
57C4×D4—42no—
58C4×D4—41no—
59C4×D4—42no—
60C2×C4⋊C4C2×C412yes—
61C2×C4⋊C4C2×C411yescharacteristic
62C2×C22⋊C4C2×C411yescharacteristic
63C2×C22⋊C4C2×C412yes—
64C2×C22⋊C4C2×C412yes—
65C2×C22⋊C4C2×C411yescharacteristic
66C2×C22⋊C4C2×C412yes—
67C2×C22⋊C4C2×C411yescharacteristic
68C2×C22⋊C4C2×C412yes—
69C2×C22⋊C4C2×C411yescharacteristic
70C2×C42C2311yescharacteristic
71C2×C42C2312yes—
72C4⋊C8—42no—
73C4⋊C8—41no—
74C22⋊C8—42no—
75C22⋊C8—42no—
76C22⋊C8—42no—
77C22⋊C8—42no—
78C22⋊C8—42no—
79C22⋊C8—42no—
80C22⋊C8—42no—
81C22⋊C8—41no—
82#97—42no—
83C4×C8—41no—
84#95C2×C412yes—
85#95C2×C412yes—
86#95C2×C412yes—
87C24—22no—
88C24—24no—
89C24—41no—
90C24—42no—
91C24—41no—
92C24C22×C411yescharacteristic
93C24C22×C411yescharacteristic
94C24C22×C412yes—
95C2×D4—24no—
96C2×D4—24no—
97C2×D4—41no—
98C2×D4—42no—
99C2×D4—41no—
100C2×D4M1614yes—
101C22×C4—21no—
102C22×C4—22no—
103C22×C4—22no—
104C22×C4—22no—
105C22×C4—22no—
106C22×C4—22no—
107C22×C4—22no—
108C22×C4—22no—
109C22×C4—21no—
110C22×C4—22no—
111C22×C4—21no—
112C22×C4—22no—
113C22×C4—21no—
114C22×C4—22no—
115C22×C4—42no—
116C22×C4—42no—
117C22×C4—42no—
118C22×C4—41no—
119C22×C4—42no—
120C22×C4—44no—
121C22×C4—48no—
122C22×C4—44no—
123C22×C4C22×C411yescharacteristic
124C22×C4C22×C412yes—
125C22×C4C22×C412yes—
126C22×C4C22×C411yescharacteristic
127C22×C4C22×C412yes—
128C22×C4C22×C411yescharacteristic
129C22×C4C22×C411yescharacteristic
130C22×C4C2411yescharacteristic
131C2×C8—22no—
132C2×C8—22no—
133C2×C8—22no—
134C2×C8—22no—
135C2×C8—22no—
136C2×C8—22no—
137C2×C8—22no—
138C2×C8—22no—
139C2×C8—22no—
140C2×C8—22no—
141C2×C8—22no—
142C2×C8—22no—
143C2×C8—44no—
144C4⋊C4—41no—
145C4⋊C4—42no—
146C22⋊C4—22no—
147C22⋊C4—24no—
148C22⋊C4—24no—
149C22⋊C4—41no—
150C22⋊C4—42no—
151C22⋊C4—41no—
152C22⋊C4—42no—
153C22⋊C4—41no—
154C42—21no—
155C42—22no—
156C42—21no—
157C42—22no—
158C23—21no—
159C23—22no—
160C23—21no—
161C23—22no—
162C23—22no—
163C23—22no—
164C23—22no—
165C23—22no—
166C23—22no—
167C23—22no—
168C23—22no—
169C23—21no—
170C23—21no—
171C23—22no—
172C23—48no—
173C23—44no—
174C23—44no—
175C23—48no—
176C23—44no—
177C23C2×M1612yes—
178C23C2×M1612yes—
179C23#13114yes—
180C23C23×C411yescharacteristic
181D4—44no—
182D4—48no—
183C2×C4—21no—
184C2×C4—21no—
185C2×C4—22no—
186C2×C4—21no—
187C2×C4—22no—
188C2×C4—21no—
189C2×C4—22no—
190C2×C4—21no—
191C2×C4—22no—
192C2×C4—22no—
193C2×C4—22no—
194C2×C4—22no—
195C2×C4—22no—
196C2×C4—22no—
197C2×C4—22no—
198C2×C4—22no—
199C2×C4—22no—
200C2×C4—48no—
201C2×C4—44no—
202C2×C4—42no—
203C2×C4—41no—
204C2×C4—41no—
205C2×C4—42no—
206C2×C4—42no—
207C2×C4—42no—
208C2×C4—42no—
209C2×C4—41no—
210C2×C4—41no—
211C2×C4—42no—
212C2×C4C2×M1612yes—
213C2×C4H(5,𝔽2)11yescharacteristic
214C2×C4H(5,𝔽2)12yes—
215C2×C4H(5,𝔽2)11yescharacteristic
216C2×C4H(5,𝔽2)12yes—
217C8—42no—
218C8—42no—
219C8—42no—
220C8—42no—
221C22—22no—
222C22—22no—
223C22—24no—
224C22—24no—
225C22—24no—
226C22—24no—
227C22—24no—
228C22—416no—
229C22—41no—
230C22—42no—
231C22—41no—
232C22—42no—
233C22—42no—
234C22—42no—
235C22—42no—
236C22—42no—
237C22—41no—
238C22—41no—
239C22#51811yescharacteristic
240C22#51812yes—
241C22C22×M1611yescharacteristic
242C22C2×#13112yes—
243C22#56811yescharacteristic
244C4—22no—
245C4—41no—
246C4—41no—
247C4—42no—
248C4—42no—
249C4—42no—
250C4—41no—
251C4#56012yes—
252C4#56011yescharacteristic
253C4H(3,𝔽4)11yescharacteristic
254C2—42no—
255C2—42no—
256C2—44no—
257C2#237511yescharacteristic
258C2#297311yescharacteristic
259C2#297412yes—
260C2#297612yes—
261C2#297711yescharacteristic
subgroupG/H
center, Frattini, cosocleC22×C4C24
commutator, socleC23C23×C4
prime radicalC2×#120C4
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C22×C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #1955C2semidirect
3. #1954C2semidirect
4. #1924C2semidirect
5. #1924C2semidirect
6. #1923C2semidirect
7. #1869C2semidirect
8. #1869C2semidirect
9. #1779C2semidirect
11. C23×C8C22semidirect
15. C2×C4⋊C8C22semidirect
16. C2×C4⋊C8C22semidirect
23. C2×C22⋊C8C22semidirect
24. C2×C22⋊C8C22semidirect
25. C2×#97C22semidirect
26. C2×C4×C8C22semidirect
34. #336C22semidirect
35. #336C22semidirect
36. #336C22semidirect

Maximal subgroups

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

subgroupindex
C4×#1202
#19552
#19542
#19242
#19232
#18692
#17792