Finite Group Databasebeta

← #24010

Subgroups — #24010

Order 256.

714 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 714 conjugacy classes of subgroups — too many to draw. Only its 145 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
2C2×#485C212yesmaximal, complemented
3#2280C211yesmaximal, characteristic
4#2036C212yesmaximal, complemented
5#2035C212yesmaximal, complemented
6#1873C212yesmaximal, complemented
7#1853C211yesmaximal, characteristic, complemented
8#1852C211yesmaximal, characteristic, complemented
9#1787C211yesmaximal, characteristic, complemented
10#1787C211yesmaximal, characteristic, complemented
11#1779C211yesmaximal, characteristic
12C23×Q8C2211yescharacteristic
13C24×C4C2211yescharacteristic
14C2×C22⋊Q8—21no—
15C2×C22⋊Q8—21no—
16C2×C22⋊Q8—28no—
17C2×C22⋊Q8C2212yes—
18C2×C22⋊Q8C2212yes—
19C2×#117C2211yescharacteristic
20C2×#117C2211yescharacteristic
21C22×C4⋊C4—21no—
22C22×C4⋊C4C2211yescharacteristic
23C22×C22⋊C4—21no—
24#485—28no—
25C2×#107C2212yescomplemented
26C2×#106C2212yescomplemented
27C2×#103C2212yescomplemented
28C2×#103C2212yescomplemented
29C2×#103C2212yescomplemented
30C2×#103C2212yescomplemented
31C2×C22⋊C8C2212yes—
32C2×C22⋊C8C2211yescharacteristic
33C2×C22⋊C8C2211yescharacteristic
34C2×C22⋊C8C2212yes—
35#381—21no—
36#380C2211yescharacteristic
37#380C2211yescharacteristic
38#380C2211yescharacteristic
39#380C2211yescharacteristic
40#379—21no—
41C4×C22⋊C4—21no—
42C4×C22⋊C4—21no—
43#340C2212yescomplemented
44#340C2212yescomplemented
45C25D411yescharacteristic
46C22×Q8—24no—
47C22×Q8—21no—
48C22×Q8—22no—
49C22×Q8—22no—
50C22×Q8—42no—
51C22×Q8C2312yes—
52C23×C4—21no—
53C23×C4—21no—
54C23×C4—24no—
55C23×C4—22no—
56C23×C4—22no—
57C23×C4—42no—
58C23×C4D411yescharacteristic
59C23×C4D411yescharacteristic
60C23×C4D411yescharacteristic
61C23×C4C2311yescharacteristic
62C23×C4C2311yescharacteristic
63C23×C4C2311yescharacteristic
64C22×C8C2312yes—
65C22×C8C2312yes—
66C22⋊Q8—28no—
67C22⋊Q8—28no—
68C22⋊Q8—24no—
69C22⋊Q8—24no—
70C22⋊Q8—24no—
71C22⋊Q8D418yes—
72#117—24no—
73#117—42no—
74C2×C4⋊C4—28no—
75C2×C4⋊C4—22no—
76C2×C4⋊C4—22no—
77C2×C4⋊C4—21no—
78C2×C4⋊C4—24no—
79C2×C4⋊C4C2311yescharacteristic
80C2×C4⋊C4C2311yescharacteristic
81C2×C4⋊C4C2311yescharacteristic
82C2×C4⋊C4C2311yescharacteristic
83C2×C4⋊C4C2312yes—
84C2×C22⋊C4—28no—
85C2×C22⋊C4—22no—
86C2×C22⋊C4—22no—
87C2×C22⋊C4—21no—
88C2×C22⋊C4—21no—
89C2×C22⋊C4—21no—
90C2×C22⋊C4—21no—
91C2×C22⋊C4—21no—
92C2×C22⋊C4—21no—
93C2×C42—21no—
94C2×C42—21no—
95#107—24no—
96#106—24no—
97#103—24no—
98#103—24no—
99#103—42no—
100#103—42no—
101C22⋊C8—24no—
102C22⋊C8—22no—
103C22⋊C8—22no—
104C22⋊C8—42no—
105#95—21no—
106#95—21no—
107#95—21no—
108#95—21no—
109C24—22no—
110C24—22no—
111C24—22no—
112C24—22no—
113C24—22no—
114C24—41no—
115C24—41no—
116C24C2×D411yescharacteristic
117C24C2×D411yescharacteristic
118C24C2×D411yescharacteristic
119C2×Q8—24no—
120C2×Q8—22no—
121C2×Q8—22no—
122C2×Q8—22no—
123C2×Q8—22no—
124C2×Q8—22no—
125C2×Q8—48no—
126C2×Q8—48no—
127C2×Q8—42no—
128C2×Q8—42no—
129C2×Q8C2×D414yes—
130C22×C4—22no—
131C22×C4—21no—
132C22×C4—21no—
133C22×C4—22no—
134C22×C4—22no—
135C22×C4—24no—
136C22×C4—24no—
137C22×C4—22no—
138C22×C4—22no—
139C22×C4—22no—
140C22×C4—22no—
141C22×C4—22no—
142C22×C4—22no—
143C22×C4—22no—
144C22×C4—22no—
145C22×C4—22no—
146C22×C4—21no—
147C22×C4—21no—
148C22×C4—22no—
149C22×C4—21no—
150C22×C4—21no—
151C22×C4—21no—
152C22×C4—21no—
153C22×C4—41no—
154C22×C4—41no—
155C22×C4—41no—
156C22×C4—48no—
157C22×C4—48no—
158C22×C4—42no—
159C22×C4—42no—
160C22×C4—42no—
161C22×C4—42no—
162C22×C4C2×D411yescharacteristic
163C22×C4C2×D411yescharacteristic
164C22×C4C2×D414yes—
165C22×C4C2×D411yescharacteristic
166C22×C4#4112yes—
167C22×C4#4112yes—
168C22×C4C2411yescharacteristic
169C2×C8—22no—
170C2×C8—22no—
171C2×C8—22no—
172C2×C8—22no—
173C2×C8—22no—
174C2×C8—22no—
175C4⋊C4—24no—
176C4⋊C4—28no—
177C4⋊C4—28no—
178C4⋊C4—22no—
179C4⋊C4—22no—
180C4⋊C4—22no—
181C4⋊C4—22no—
182C4⋊C4—22no—
183C4⋊C4C2×D414yes—
184C22⋊C4—28no—
185C22⋊C4—28no—
186C22⋊C4—42no—
187C22⋊C4—42no—
188C22⋊C4—42no—
189C22⋊C4—42no—
190C22⋊C4—42no—
191C22⋊C4—42no—
192C42—42no—
193C42—42no—
194C23—21no—
195C23—21no—
196C23—21no—
197C23—21no—
198C23—28no—
199C23—21no—
200C23—21no—
201C23—21no—
202C23—21no—
203C23—21no—
204C23—21no—
205C23—21no—
206C23—21no—
207C23—21no—
208C23—21no—
209C23—21no—
210C23—21no—
211C23—44no—
212C23—44no—
213C23—44no—
214C23—44no—
215C23—44no—
216C23—44no—
217C23#12012yes—
218C23#12312yes—
219C23#13512yes—
220C23#13712yes—
221C23#13712yes—
222C23C22×D411yescharacteristic
223Q8—24no—
224Q8—24no—
225Q8—48no—
226Q8—42no—
227Q8—42no—
228C2×C4—21no—
229C2×C4—21no—
230C2×C4—21no—
231C2×C4—21no—
232C2×C4—24no—
233C2×C4—24no—
234C2×C4—24no—
235C2×C4—24no—
236C2×C4—24no—
237C2×C4—24no—
238C2×C4—21no—
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—44no—
242C2×C4—44no—
243C2×C4—44no—
244C2×C4—48no—
245C2×C4—48no—
246C2×C4—42no—
247C2×C4—42no—
248C2×C4—42no—
249C2×C4—42no—
250C2×C4—42no—
251C2×C4—42no—
252C2×C4—41no—
253C2×C4—41no—
254C2×C4—41no—
255C2×C4—41no—
256C2×C4—41no—
257C2×C4—41no—
258C2×C4—41no—
259C2×C4—41no—
260C2×C4—41no—
261C2×C4—41no—
262C2×C4—41no—
263C2×C4—41no—
264C2×C4#12012yes—
265C2×C4#12312yes—
266C2×C4C22×D412yes—
267C2×C4C2×#4111yescharacteristic
268C2×C4C2×#4111yescharacteristic
269C2×C4H(5,𝔽2)12yes—
270C8—42no—
271C8—42no—
272C22—22no—
273C22—22no—
274C22—22no—
275C22—22no—
276C22—22no—
277C22—22no—
278C22—22no—
279C22—22no—
280C22—22no—
281C22—22no—
282C22—22no—
283C22—22no—
284C22—416no—
285C22—41no—
286C22—41no—
287C22—41no—
288C22—41no—
289C22—41no—
290C22—41no—
291C22—41no—
292C22—41no—
293C22#48514yescomplemented
294C22C2×#12011yescharacteristic
295C22C2×#12311yescharacteristic
296C22#53411yescharacteristic
297C22C2×#13511yescharacteristic
298C22C2×#13711yescharacteristic
299C22C2×#13711yescharacteristic
300C22#57511yescharacteristic
301C4—22no—
302C4—22no—
303C4—41no—
304C4—44no—
305C4—44no—
306C4—81no—
307C4—81no—
308C4—81no—
309C4—81no—
310C4#54612yes—
311C4#54612yes—
312C2—24no—
313C2—42no—
314C2—42no—
315C2#240511yescharacteristic
316C2#300211yescharacteristic
317C2#300411yescharacteristic
318C2C2×#48511yescharacteristic
319C2#309511yescharacteristic
320C2#319011yescharacteristic
321C2#319611yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC25D4
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
2. C2×#485C2semidirect
4. #2036C2semidirect
5. #2035C2semidirect
6. #1873C2semidirect
7. #1853C2semidirect
8. #1852C2semidirect
9. #1787C2semidirect
10. #1787C2semidirect
25. C2×#107C22semidirect
26. C2×#106C22semidirect
27. C2×#103C22semidirect
28. C2×#103C22semidirect
29. C2×#103C22semidirect
30. C2×#103C22semidirect
43. #340C22semidirect
44. #340C22semidirect
293. C22#485semidirect

Maximal subgroups

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

subgroupindex
C22×C22⋊Q82
C2×#4852
#22802
#20362
#20352
#18732
#18532
#18522
#17872
#17792