Finite Group Databasebeta

← #32985

Subgroups — #32985

Order 256.

715 conjugacy classes of subgroups; only the 151 normal subgroups are shown.

normal subgroups only

This group has 715 conjugacy classes of subgroups — too many to draw. Only its 151 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×#117C211yesmaximal, characteristic
2C2×#485C212yesmaximal, complemented
3#2848C211yesmaximal, characteristic
4#1873C212yesmaximal, complemented
5#1872C211yesmaximal, characteristic, complemented
6#1872C211yesmaximal, characteristic, complemented
7#1854C212yesmaximal, complemented
8#1853C211yesmaximal, characteristic, complemented
9#1852C211yesmaximal, characteristic, complemented
10#1825C212yesmaximal, complemented
11#1779C211yesmaximal, characteristic
12C24×C4C2211yescharacteristic
13C2×C22⋊Q8C2211yescharacteristic
14C2×C22⋊Q8C2211yescharacteristic
15C2×C22⋊Q8C2211yescharacteristic
16C2×#117—21no—
17C2×#117—21no—
18C2×#117—28no—
19C2×#117C2212yes—
20C2×#117C2212yes—
21C22×C4⋊C4C2211yescharacteristic
22C22×C4⋊C4C2211yescharacteristic
23C22×C22⋊C4—21no—
24C22×C42—21no—
25#485—28no—
26C2×#107C2212yescomplemented
27C2×#106C2212yescomplemented
28C2×#103C2212yescomplemented
29C2×#103C2212yescomplemented
30C2×C22⋊C8C2212yes—
31C2×C22⋊C8C2211yescharacteristic
32C2×C22⋊C8C2211yescharacteristic
33C2×C22⋊C8C2212yes—
34#385—21no—
35#385—21no—
36#385—21no—
37#380C2211yescharacteristic
38#380C2211yescharacteristic
39#380C2211yescharacteristic
40#379—21no—
41#340C2212yescomplemented
42#340C2212yescomplemented
43#340C2212yescomplemented
44#340C2212yescomplemented
45C25D411yescharacteristic
46C22×Q8C2311yescharacteristic
47C23×C4—24no—
48C23×C4—24no—
49C23×C4—22no—
50C23×C4—22no—
51C23×C4—21no—
52C23×C4—21no—
53C23×C4D411yescharacteristic
54C23×C4D411yescharacteristic
55C23×C4D411yescharacteristic
56C23×C4C2311yescharacteristic
57C23×C4C2311yescharacteristic
58C23×C4C2311yescharacteristic
59C22×C8C2312yes—
60C22×C8C2312yes—
61C22⋊Q8—24no—
62C22⋊Q8—22no—
63C22⋊Q8—22no—
64C22⋊Q8—24no—
65#117—28no—
66#117—28no—
67#117—24no—
68#117—24no—
69#117—24no—
70#117D418yes—
71C2×C4⋊C4—21no—
72C2×C4⋊C4—21no—
73C2×C4⋊C4—24no—
74C2×C4⋊C4—24no—
75C2×C4⋊C4—21no—
76C2×C4⋊C4—21no—
77C2×C4⋊C4—21no—
78C2×C4⋊C4C2311yescharacteristic
79C2×C4⋊C4C2311yescharacteristic
80C2×C4⋊C4C2311yescharacteristic
81C2×C4⋊C4C2312yes—
82C2×C4⋊C4C2312yes—
83C2×C22⋊C4—28no—
84C2×C22⋊C4—22no—
85C2×C22⋊C4—22no—
86C2×C22⋊C4—21no—
87C2×C22⋊C4—21no—
88C2×C22⋊C4—21no—
89C2×C22⋊C4—21no—
90C2×C22⋊C4—21no—
91C2×C22⋊C4—21no—
92C2×C42—28no—
93C2×C42—22no—
94C2×C42—22no—
95#107—24no—
96#106—24no—
97#103—24no—
98#103—24no—
99C22⋊C8—24no—
100C22⋊C8—22no—
101C22⋊C8—22no—
102C22⋊C8—24no—
103#95—21no—
104#95—21no—
105#95—21no—
106C24—22no—
107C24—22no—
108C24—22no—
109C24—22no—
110C24—22no—
111C24—41no—
112C24—41no—
113C24C2×D411yescharacteristic
114C24C2×D411yescharacteristic
115C24C2×D411yescharacteristic
116C2×Q8—22no—
117C2×Q8—21no—
118C2×Q8—21no—
119C2×Q8—22no—
120C22×C4—216no—
121C22×C4—216no—
122C22×C4—22no—
123C22×C4—22no—
124C22×C4—22no—
125C22×C4—22no—
126C22×C4—22no—
127C22×C4—22no—
128C22×C4—22no—
129C22×C4—22no—
130C22×C4—22no—
131C22×C4—22no—
132C22×C4—22no—
133C22×C4—21no—
134C22×C4—21no—
135C22×C4—22no—
136C22×C4—22no—
137C22×C4—21no—
138C22×C4—21no—
139C22×C4—24no—
140C22×C4—24no—
141C22×C4—22no—
142C22×C4—22no—
143C22×C4—21no—
144C22×C4—21no—
145C22×C4—21no—
146C22×C4—21no—
147C22×C4—42no—
148C22×C4—42no—
149C22×C4—42no—
150C22×C4—42no—
151C22×C4C2×D411yescharacteristic
152C22×C4C2×D414yes—
153C22×C4C2×D411yescharacteristic
154C22×C4C2×D411yescharacteristic
155C22×C4#4114yes—
156C22×C4#4112yes—
157C22×C4#4112yes—
158C22×C4C2411yescharacteristic
159C2×C8—22no—
160C2×C8—22no—
161C2×C8—22no—
162C2×C8—22no—
163C2×C8—22no—
164C2×C8—22no—
165C4⋊C4—24no—
166C4⋊C4—24no—
167C4⋊C4—22no—
168C4⋊C4—22no—
169C4⋊C4—22no—
170C4⋊C4—22no—
171C4⋊C4—21no—
172C4⋊C4—21no—
173C4⋊C4—42no—
174C4⋊C4—42no—
175C4⋊C4—42no—
176C4⋊C4C2×D414yes—
177C4⋊C4C2×D414yes—
178C22⋊C4—28no—
179C22⋊C4—28no—
180C22⋊C4—42no—
181C22⋊C4—42no—
182C22⋊C4—42no—
183C22⋊C4—42no—
184C22⋊C4—42no—
185C22⋊C4—42no—
186C42—28no—
187C42—28no—
188C23—21no—
189C23—21no—
190C23—21no—
191C23—21no—
192C23—28no—
193C23—21no—
194C23—21no—
195C23—21no—
196C23—21no—
197C23—21no—
198C23—21no—
199C23—21no—
200C23—21no—
201C23—21no—
202C23—21no—
203C23—21no—
204C23—21no—
205C23—44no—
206C23—44no—
207C23—44no—
208C23—44no—
209C23—44no—
210C23—44no—
211C23#12012yes—
212C23#12312yes—
213C23#12312yes—
214C23#13512yes—
215C23#13712yes—
216C23C22×D411yescharacteristic
217Q8—42no—
218Q8—42no—
219C2×C4—28no—
220C2×C4—28no—
221C2×C4—28no—
222C2×C4—21no—
223C2×C4—21no—
224C2×C4—21no—
225C2×C4—21no—
226C2×C4—21no—
227C2×C4—21no—
228C2×C4—24no—
229C2×C4—24no—
230C2×C4—24no—
231C2×C4—24no—
232C2×C4—24no—
233C2×C4—24no—
234C2×C4—24no—
235C2×C4—24no—
236C2×C4—48no—
237C2×C4—48no—
238C2×C4—42no—
239C2×C4—42no—
240C2×C4—42no—
241C2×C4—42no—
242C2×C4—41no—
243C2×C4—41no—
244C2×C4—41no—
245C2×C4—41no—
246C2×C4—41no—
247C2×C4—41no—
248C2×C4—41no—
249C2×C4—41no—
250C2×C4—41no—
251C2×C4—41no—
252C2×C4—41no—
253C2×C4—41no—
254C2×C4#12012yes—
255C2×C4#12312yes—
256C2×C4#12312yes—
257C2×C4C22×D412yes—
258C2×C4C2×#4111yescharacteristic
259C2×C4C2×#4111yescharacteristic
260C2×C4C2×#4112yes—
261C8—42no—
262C8—42no—
263C22—22no—
264C22—22no—
265C22—22no—
266C22—22no—
267C22—22no—
268C22—22no—
269C22—22no—
270C22—22no—
271C22—22no—
272C22—22no—
273C22—22no—
274C22—22no—
275C22—416no—
276C22—41no—
277C22—41no—
278C22—41no—
279C22—41no—
280C22—41no—
281C22—41no—
282C22—41no—
283C22—41no—
284C22#48514yescomplemented
285C22C2×#12011yescharacteristic
286C22C2×#12311yescharacteristic
287C22C2×#12311yescharacteristic
288C22C2×#13511yescharacteristic
289C22C2×#13711yescharacteristic
290C22#57511yescharacteristic
291C22#57511yescharacteristic
292C4—22no—
293C4—22no—
294C4—22no—
295C4—44no—
296C4—44no—
297C4—81no—
298C4—81no—
299C4—81no—
300C4—81no—
301C4#52512yes—
302C4#52512yes—
303C2—24no—
304C2—42no—
305C2—42no—
306C2#284911yescharacteristic
307C2#300211yescharacteristic
308C2#300711yescharacteristic
309C2C2×#48511yescharacteristic
310C2#309111yescharacteristic
311C2#309311yescharacteristic
312C2#309511yescharacteristic
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. #1873C2semidirect
5. #1872C2semidirect
6. #1872C2semidirect
7. #1854C2semidirect
8. #1853C2semidirect
9. #1852C2semidirect
10. #1825C2semidirect
26. C2×#107C22semidirect
27. C2×#106C22semidirect
28. C2×#103C22semidirect
29. C2×#103C22semidirect
41. #340C22semidirect
42. #340C22semidirect
43. #340C22semidirect
44. #340C22semidirect
284. C22#485semidirect

Maximal subgroups

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

subgroupindex
C22×#1172
C2×#4852
#28482
#18732
#18722
#18542
#18532
#18522
#18252
#17792