Finite Group Databasebeta

← #23923

Subgroups — #23923

Order 256.

974 conjugacy classes of subgroups; only the 153 normal subgroups are shown.

normal subgroups only

This group has 974 conjugacy classes of subgroups — too many to draw. Only its 153 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×#525C212yesmaximal, complemented
2C2×#452C212yesmaximal, complemented
3C2×#449C212yesmaximal, complemented
4#2001C211yesmaximal, characteristic, complemented
5#2000C212yesmaximal, complemented
6#1999C211yesmaximal, characteristic, complemented
7#1875C212yesmaximal, complemented
8#1873C212yesmaximal, complemented
9#1779C211yesmaximal, characteristic, complemented
10C22×#41C2212yes—
11C24×C4C2211yescharacteristic
12C22×Q16C2211yescharacteristic, complemented
13C22×SD16C2212yescomplemented
14C22×D8C2211yescharacteristic, complemented
15#525—28no—
16#525—28no—
17C2×#123—22no—
18C2×C22⋊Q8C2212yes—
19C2×C22⋊Q8C2212yes—
20C2×#121C2212yes—
21C2×#121C2212yes—
22C2×#120—22no—
23C2×C4×D4—22no—
24C2×C4×D4—22no—
25C2×#117C2212yes—
26#452—28no—
27#449—28no—
28C2×#103C2212yescomplemented
29C2×#103C2212yescomplemented
30C2×#102C2212yescomplemented
31C2×#102C2212yescomplemented
32C2×C22⋊C8C2212yes—
33C2×C22⋊C8C2212yes—
34C2×C22⋊C8C2211yescharacteristic
35C2×C22⋊C8C2211yescharacteristic
36#340C2211yescharacteristic, complemented
37#340C2212yescomplemented
38#340C2211yescharacteristic, complemented
39C25D411yescharacteristic
40C2×#41—24no—
41C2×#41—24no—
42C2×#41—24no—
43C2×#41—24no—
44C2×#41—42no—
45C2×#41—42no—
46C22×Q8C2312yes—
47C22×D4—22no—
48C22×D4—22no—
49C22×D4—22no—
50C22×D4C2312yes—
51C23×C4—22no—
52C23×C4—24no—
53C23×C4—24no—
54C23×C4—41no—
55C23×C4—41no—
56C23×C4D411yescharacteristic
57C23×C4D412yes—
58C23×C4C2311yescharacteristic
59C23×C4C2312yes—
60C2×Q16—24no—
61C2×Q16—41no—
62C2×SD16—24no—
63C2×SD16—24no—
64C2×SD16—42no—
65C2×D8—24no—
66C2×D8—41no—
67C22×C8C2311yescharacteristic
68C22×C8C2312yes—
69C22×C8C2311yescharacteristic
70#123—216no—
71C22⋊Q8—24no—
72C22⋊Q8—24no—
73C22⋊Q8—24no—
74C22⋊Q8D418yescomplemented
75#121—24no—
76#121—24no—
77#121—24no—
78#121D418yescomplemented
79#120—216no—
80C4×D4—216no—
81C4×D4—216no—
82#117—24no—
83#117—24no—
84C2×C4⋊C4—22no—
85C2×C4⋊C4—22no—
86C2×C4⋊C4C2312yes—
87C2×C4⋊C4C2312yes—
88C2×C22⋊C4—22no—
89C2×C22⋊C4—22no—
90C2×C22⋊C4—22no—
91C2×C22⋊C4—22no—
92C2×C22⋊C4—22no—
93C2×C42—22no—
94#103—24no—
95#103—24no—
96#102—24no—
97#102—24no—
98C22⋊C8—24no—
99C22⋊C8—24no—
100C22⋊C8—41no—
101C22⋊C8—41no—
102C24—24no—
103C24—22no—
104C24—22no—
105C24—41no—
106C24—42no—
107C24—41no—
108C24C2×D412yes—
109C24C2×D411yescharacteristic
110#41—48no—
111#41—48no—
112#41—48no—
113#41—48no—
114C2×Q8—22no—
115C2×Q8—22no—
116C2×Q8—22no—
117C2×Q8—22no—
118C2×Q8—22no—
119C2×Q8C2×D414yes—
120C2×D4—28no—
121C2×D4—28no—
122C2×D4—28no—
123C2×D4—22no—
124C2×D4—22no—
125C2×D4—22no—
126C2×D4—22no—
127C2×D4—22no—
128C2×D4—44no—
129C2×D4—44no—
130C2×D4—44no—
131C2×D4—44no—
132C2×D4—44no—
133C2×D4—44no—
134C2×D4C2×D414yes—
135C22×C4—22no—
136C22×C4—22no—
137C22×C4—22no—
138C22×C4—22no—
139C22×C4—28no—
140C22×C4—24no—
141C22×C4—22no—
142C22×C4—22no—
143C22×C4—22no—
144C22×C4—24no—
145C22×C4—24no—
146C22×C4—22no—
147C22×C4—22no—
148C22×C4—21no—
149C22×C4—21no—
150C22×C4—41no—
151C22×C4—42no—
152C22×C4—44no—
153C22×C4—48no—
154C22×C4—44no—
155C22×C4—44no—
156C22×C4—44no—
157C22×C4C2×D411yescharacteristic
158C22×C4C2×D412yes—
159C22×C4C2×D414yes—
160C22×C4C2×D414yes—
161C22×C4C2411yescharacteristic
162Q16—44no—
163SD16—48no—
164D8—44no—
165C2×C8—22no—
166C2×C8—22no—
167C2×C8—22no—
168C2×C8—22no—
169C2×C8—21no—
170C2×C8—22no—
171C2×C8—21no—
172C4⋊C4—28no—
173C4⋊C4—28no—
174C4⋊C4—22no—
175C4⋊C4—22no—
176C4⋊C4C2×D414yes—
177C4⋊C4C2×D414yes—
178C22⋊C4—28no—
179C22⋊C4—28no—
180C22⋊C4—28no—
181C22⋊C4—28no—
182C22⋊C4—28no—
183C42—28no—
184C23—22no—
185C23—22no—
186C23—22no—
187C23—21no—
188C23—22no—
189C23—22no—
190C23—22no—
191C23—22no—
192C23—21no—
193C23—22no—
194C23—24no—
195C23—44no—
196C23—48no—
197C23—44no—
198C23—48no—
199C23—44no—
200C23—42no—
201C23—42no—
202C23—42no—
203C23—42no—
204C23—42no—
205C23—42no—
206C23#12014yes—
207C23#13512yes—
208C23C22×D411yescharacteristic
209Q8—24no—
210Q8—24no—
211Q8—42no—
212Q8—42no—
213D4—24no—
214D4—24no—
215D4—416no—
216D4—416no—
217D4—416no—
218D4—42no—
219D4—42no—
220C2×C4—22no—
221C2×C4—22no—
222C2×C4—24no—
223C2×C4—24no—
224C2×C4—24no—
225C2×C4—24no—
226C2×C4—24no—
227C2×C4—24no—
228C2×C4—22no—
229C2×C4—24no—
230C2×C4—21no—
231C2×C4—44no—
232C2×C4—48no—
233C2×C4—416no—
234C2×C4—42no—
235C2×C4—42no—
236C2×C4—42no—
237C2×C4—42no—
238C2×C4—42no—
239C2×C4—42no—
240C2×C4—41no—
241C2×C4—42no—
242C2×C4—41no—
243C2×C4#12014yes—
244C2×C4C22×D412yes—
245C2×C4C22×D412yes—
246C2×C4H(5,𝔽2)11yescharacteristic
247C2×C4H(5,𝔽2)11yescharacteristic
248C8—41no—
249C8—42no—
250C8—41no—
251C22—24no—
252C22—22no—
253C22—24no—
254C22—24no—
255C22—24no—
256C22—416no—
257C22—41no—
258C22—42no—
259C22—42no—
260C22—41no—
261C22—42no—
262C22—42no—
263C22—42no—
264C22—44no—
265C22—44no—
266C22—44no—
267C22—82no—
268C22—82no—
269C22—82no—
270C22—82no—
271C22C2×#12012yes—
272C22#53411yescharacteristic
273C22C2×#13511yescharacteristic
274C22#57511yescharacteristic
275C22#57512yes—
276C4—24no—
277C4—41no—
278C4—44no—
279C4—44no—
280C4—44no—
281C4D4211yescharacteristic
282C4D4212yes—
283C4D4211yescharacteristic
284C2—44no—
285C2—42no—
286C2—84no—
287C2#240311yescharacteristic
288C2#300212yes—
289C2#300512yes—
290C2#319111yescharacteristic
291C2#319711yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#525C2semidirect
2. C2×#452C2semidirect
3. C2×#449C2semidirect
4. #2001C2semidirect
5. #2000C2semidirect
6. #1999C2semidirect
7. #1875C2semidirect
8. #1873C2semidirect
9. #1779C2semidirect
12. C22×Q16C22semidirect
13. C22×SD16C22semidirect
14. C22×D8C22semidirect
28. C2×#103C22semidirect
29. C2×#103C22semidirect
30. C2×#102C22semidirect
31. C2×#102C22semidirect
36. #340C22semidirect
37. #340C22semidirect
38. #340C22semidirect
74. C22⋊Q8D4semidirect
78. #121D4semidirect

Maximal subgroups

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

subgroupindex
C2×#5252
C2×#4522
C2×#4492
#20012
#20002
#19992
#18752
#18732
#17792