Finite Group Databasebeta

← #23919

Subgroups — #23919

Order 256.

1126 conjugacy classes of subgroups; only the 173 normal subgroups are shown.

normal subgroups only

This group has 1126 conjugacy classes of subgroups — too many to draw. Only its 173 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, complemented
2C22×#121C211yesmaximal, characteristic
3C2×#448C212yesmaximal, complemented
4C2×#450C212yesmaximal, complemented
5#2000C214yesmaximal, complemented
6#1874C212yesmaximal, complemented
7#1872C212yesmaximal, complemented
8#1779C211yesmaximal, characteristic, complemented
9C23×Q8C2211yescharacteristic, complemented
10C23×D4—21no—
11C23×D4C2211yescharacteristic
12C24×C4C2211yescharacteristic
13C22×SD16C2214yescomplemented
14C2×C22⋊Q8—21no—
15C2×C22⋊Q8—21no—
16C2×C22⋊Q8—28no—
17C2×C22⋊Q8C2212yes—
18C2×C22⋊Q8C2212yescomplemented
19C2×#121—21no—
20C2×#121—21no—
21C2×#121—28no—
22C2×#121C2212yes—
23C2×#121C2212yes—
24C22×C4⋊C4—21no—
25C22×C4⋊C4C2211yescharacteristic
26C22×C4⋊C4C2211yescharacteristic, complemented
27C22×C22⋊C4—21no—
28C22×C22⋊C4—21no—
29#450—28no—
30#448—28no—
31C2×#103C2214yescomplemented
32C2×#102C2214yescomplemented
33C2×C22⋊C8C2212yes—
34C2×C22⋊C8C2212yescomplemented
35C2×C22⋊C8C2212yescomplemented
36#340C2214yescomplemented
37C25—21no—
38C25D411yescharacteristic
39C22×Q8—24no—
40C22×Q8—21no—
41C22×Q8—24no—
42C22×Q8—42no—
43C22×Q8C2312yes—
44C22×D4—28no—
45C22×D4—22no—
46C22×D4—22no—
47C22×D4—24no—
48C22×D4—21no—
49C22×D4—24no—
50C22×D4—42no—
51C22×D4—44no—
52C22×D4—44no—
53C22×D4C2312yes—
54C23×C4—21no—
55C23×C4—21no—
56C23×C4—21no—
57C23×C4—24no—
58C23×C4—24no—
59C23×C4—42no—
60C23×C4D411yescharacteristic
61C23×C4D411yescharacteristic
62C23×C4D411yescharacteristic
63C23×C4C2311yescharacteristic
64C23×C4C2311yescharacteristic
65C23×C4C2311yescharacteristic
66C2×SD16—28no—
67C2×SD16—28no—
68C2×SD16—44no—
69C22×C8C2314yes—
70C22⋊Q8—28no—
71C22⋊Q8—28no—
72C22⋊Q8—24no—
73C22⋊Q8—24no—
74C22⋊Q8—24no—
75C22⋊Q8D418yescomplemented
76#121—28no—
77#121—28no—
78#121—24no—
79#121—24no—
80#121—24no—
81#121D418yes—
82C2×C4⋊C4—28no—
83C2×C4⋊C4—22no—
84C2×C4⋊C4—22no—
85C2×C4⋊C4—21no—
86C2×C4⋊C4—21no—
87C2×C4⋊C4—24no—
88C2×C4⋊C4—24no—
89C2×C4⋊C4C2312yes—
90C2×C4⋊C4C2312yes—
91C2×C22⋊C4—28no—
92C2×C22⋊C4—28no—
93C2×C22⋊C4—22no—
94C2×C22⋊C4—22no—
95C2×C22⋊C4—22no—
96C2×C22⋊C4—22no—
97#103—28no—
98#102—28no—
99C22⋊C8—24no—
100C22⋊C8—24no—
101C22⋊C8—42no—
102C24—22no—
103C24—22no—
104C24—22no—
105C24—22no—
106C24—24no—
107C24—22no—
108C24—41no—
109C24—41no—
110C24—42no—
111C24—42no—
112C24—42no—
113C24—42no—
114C24—42no—
115C24—42no—
116C24SD1614yescomplemented
117C24C2×D411yescharacteristic
118C24C2×D411yescharacteristic
119C24C2×D411yescharacteristic
120C2×Q8—24no—
121C2×Q8—22no—
122C2×Q8—24no—
123C2×Q8—24no—
124C2×Q8—416no—
125C2×Q8—42no—
126C2×Q8—42no—
127C2×Q8C2×D414yescomplemented
128C2×D4—24no—
129C2×D4—28no—
130C2×D4—28no—
131C2×D4—24no—
132C2×D4—22no—
133C2×D4—24no—
134C2×D4—416no—
135C2×D4—42no—
136C2×D4—432no—
137C2×D4—44no—
138C2×D4—44no—
139C2×D4—44no—
140C2×D4—44no—
141C2×D4—42no—
142C2×D4C2×D414yes—
143C22×C4—22no—
144C22×C4—22no—
145C22×C4—22no—
146C22×C4—22no—
147C22×C4—24no—
148C22×C4—24no—
149C22×C4—24no—
150C22×C4—22no—
151C22×C4—22no—
152C22×C4—22no—
153C22×C4—22no—
154C22×C4—22no—
155C22×C4—22no—
156C22×C4—22no—
157C22×C4—22no—
158C22×C4—22no—
159C22×C4—22no—
160C22×C4—22no—
161C22×C4—22no—
162C22×C4—41no—
163C22×C4—41no—
164C22×C4—41no—
165C22×C4—416no—
166C22×C4—42no—
167C22×C4—42no—
168C22×C4—42no—
169C22×C4—42no—
170C22×C4—42no—
171C22×C4—42no—
172C22×C4C2×D411yescharacteristic
173C22×C4C2×D411yescharacteristic
174C22×C4C2×D411yescharacteristic
175C22×C4C2×D414yes—
176C22×C4C2×D414yes—
177C22×C4C2411yescharacteristic
178SD16—416no—
179C2×C8—24no—
180C2×C8—24no—
181C2×C8—24no—
182C4⋊C4—24no—
183C4⋊C4—24no—
184C4⋊C4—28no—
185C4⋊C4—28no—
186C4⋊C4—22no—
187C4⋊C4—22no—
188C4⋊C4C2×D414yes—
189C4⋊C4C2×D414yescomplemented
190C22⋊C4—28no—
191C22⋊C4—28no—
192C22⋊C4—28no—
193C22⋊C4—28no—
194C23—21no—
195C23—21no—
196C23—21no—
197C23—21no—
198C23—21no—
199C23—21no—
200C23—21no—
201C23—21no—
202C23—21no—
203C23—21no—
204C23—21no—
205C23—21no—
206C23—21no—
207C23—21no—
208C23—28no—
209C23—28no—
210C23—24no—
211C23—44no—
212C23—44no—
213C23—44no—
214C23—44no—
215C23—44no—
216C23—48no—
217C23—48no—
218C23—48no—
219C23—42no—
220C23—42no—
221C23—42no—
222C23—42no—
223C23—42no—
224C23—42no—
225C23—84no—
226C23—84no—
227C23—84no—
228C23—84no—
229C23#12012yes—
230C23#12012yes—
231C23C2×SD1612yes—
232C23C2×SD1612yes—
233C23C2×SD1612yes—
234C23AGL(1,ℤ/8)12yes—
235C23#13712yes—
236C23C22×D411yescharacteristic
237Q8—28no—
238Q8—48no—
239Q8—44no—
240D4—28no—
241D4—48no—
242D4—416no—
243D4—416no—
244D4—44no—
245C2×C4—21no—
246C2×C4—21no—
247C2×C4—21no—
248C2×C4—21no—
249C2×C4—28no—
250C2×C4—28no—
251C2×C4—21no—
252C2×C4—24no—
253C2×C4—24no—
254C2×C4—24no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—44no—
258C2×C4—44no—
259C2×C4—44no—
260C2×C4—48no—
261C2×C4—48no—
262C2×C4—48no—
263C2×C4—42no—
264C2×C4—42no—
265C2×C4—42no—
266C2×C4—42no—
267C2×C4—42no—
268C2×C4—42no—
269C2×C4—44no—
270C2×C4#12012yes—
271C2×C4#12012yes—
272C2×C4C22×D412yes—
273C2×C4C22×D412yes—
274C2×C4H(5,𝔽2)12yes—
275C8—44no—
276C22—22no—
277C22—22no—
278C22—22no—
279C22—22no—
280C22—22no—
281C22—22no—
282C22—22no—
283C22—22no—
284C22—22no—
285C22—22no—
286C22—22no—
287C22—22no—
288C22—22no—
289C22—22no—
290C22—22no—
291C22—416no—
292C22—41no—
293C22—41no—
294C22—41no—
295C22—41no—
296C22—44no—
297C22—44no—
298C22—44no—
299C22—816no—
300C22—82no—
301C22—82no—
302C22—82no—
303C22—82no—
304C22#44814yescomplemented
305C22#45014yescomplemented
306C22C2×#12011yescharacteristic
307C22C2×#12011yescharacteristic
308C22#53411yescharacteristic
309C22C22×SD1611yescharacteristic
310C22C2×AGL(1,ℤ/8)11yescharacteristic
311C22C2×#13711yescharacteristic
312C22#57511yescharacteristic
313C4—22no—
314C4—22no—
315C4—41no—
316C4—44no—
317C4—44no—
318C4—44no—
319C4D4214yes—
320C2—24no—
321C2—24no—
322C2—42no—
323C2—84no—
324C2#240311yescharacteristic
325C2C2×#45011yescharacteristic
326C2C2×#44811yescharacteristic
327C2#300611yescharacteristic
328C2#300711yescharacteristic
329C2#318711yescharacteristic
330C2#319311yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC22×#121C2
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. C22×C22⋊Q8C2semidirect
3. C2×#448C2semidirect
4. C2×#450C2semidirect
5. #2000C2semidirect
6. #1874C2semidirect
7. #1872C2semidirect
8. #1779C2semidirect
9. C23×Q8C22semidirect
13. C22×SD16C22semidirect
18. C2×C22⋊Q8C22semidirect
26. C22×C4⋊C4C22semidirect
31. C2×#103C22semidirect
32. C2×#102C22semidirect
34. C2×C22⋊C8C22semidirect
35. C2×C22⋊C8C22semidirect
36. #340C22semidirect
75. C22⋊Q8D4semidirect
116. C24SD16semidirect
127. C2×Q8C2×D4semidirect
189. C4⋊C4C2×D4semidirect
304. C22#448semidirect
305. C22#450semidirect

Maximal subgroups

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

subgroupindex
C22×C22⋊Q82
C22×#1212
C2×#4482
C2×#4502
#20002
#18742
#18722
#17792