Finite Group Databasebeta

← #32981

Subgroups — #32981

Order 256.

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

normal subgroups only

This group has 754 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, complemented
2C2×#482C212yesmaximal, complemented
3#2847C211yesmaximal, characteristic
4#1875C212yesmaximal, complemented
5#1874C211yesmaximal, characteristic, complemented
6#1874C211yesmaximal, characteristic, complemented
7#1854C212yesmaximal, complemented
8#1853C211yesmaximal, characteristic, complemented
9#1852C211yesmaximal, characteristic, complemented
10#1825C212yesmaximal, complemented
11#1779C211yesmaximal, characteristic, complemented
12C24×C4C2211yescharacteristic
13C2×#121C2211yescharacteristic
14C2×#121C2211yescharacteristic
15C2×#121C2211yescharacteristic
16C2×#120—21no—
17C2×#117—21no—
18C2×#117—21no—
19C2×#117—28no—
20C2×#117C2212yes—
21C2×#117C2212yes—
22C22×C4⋊C4C2211yescharacteristic
23C22×C4⋊C4C2211yescharacteristic
24C22×C22⋊C4—21no—
25C22×C42—21no—
26#482—28no—
27C2×#107C2212yescomplemented
28C2×#106C2212yescomplemented
29C2×#102C2212yescomplemented
30C2×#102C2212yescomplemented
31C2×C22⋊C8C2212yes—
32C2×C22⋊C8C2211yescharacteristic
33C2×C22⋊C8C2211yescharacteristic
34C2×C22⋊C8C2212yes—
35#386—21no—
36#386—21no—
37#386—21no—
38#380C2211yescharacteristic
39#380C2211yescharacteristic
40#380C2211yescharacteristic
41#340C2212yescomplemented
42#340C2212yescomplemented
43#340C2212yescomplemented
44#340C2212yescomplemented
45C25D411yescharacteristic
46C22×D4—21no—
47C22×D4—21no—
48C22×D4—21no—
49C22×D4C2311yescharacteristic
50C23×C4—24no—
51C23×C4—24no—
52C23×C4—22no—
53C23×C4—22no—
54C23×C4—21no—
55C23×C4—21no—
56C23×C4D411yescharacteristic
57C23×C4D411yescharacteristic
58C23×C4D411yescharacteristic
59C23×C4C2311yescharacteristic
60C23×C4C2311yescharacteristic
61C23×C4C2311yescharacteristic
62C22×C8C2312yes—
63C22×C8C2312yes—
64#121—24no—
65#121—22no—
66#121—22no—
67#121—24no—
68#120—44no—
69#117—28no—
70#117—28no—
71#117—24no—
72#117—24no—
73#117—24no—
74#117D418yescomplemented
75C2×C4⋊C4—21no—
76C2×C4⋊C4—21no—
77C2×C4⋊C4—24no—
78C2×C4⋊C4—24no—
79C2×C4⋊C4C2311yescharacteristic
80C2×C4⋊C4C2311yescharacteristic
81C2×C4⋊C4C2311yescharacteristic
82C2×C4⋊C4C2312yes—
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—28no—
94C2×C42—22no—
95C2×C42—22no—
96#107—24no—
97#106—24no—
98#102—24no—
99#102—24no—
100C22⋊C8—24no—
101C22⋊C8—22no—
102C22⋊C8—22no—
103C22⋊C8—24no—
104#95—21no—
105#95—21no—
106#95—21no—
107C24—22no—
108C24—22no—
109C24—22no—
110C24—22no—
111C24—22no—
112C24—21no—
113C24—41no—
114C24—41no—
115C24C2×D411yescharacteristic
116C24C2×D411yescharacteristic
117C24C2×D411yescharacteristic
118C2×D4—22no—
119C2×D4—21no—
120C2×D4—21no—
121C2×D4—22no—
122C2×D4—42no—
123C2×D4—42no—
124C2×D4—42no—
125C2×D4—42no—
126C2×D4—42no—
127C2×D4—42no—
128C2×D4—42no—
129C2×D4—42no—
130C2×D4—42no—
131C22×C4—216no—
132C22×C4—216no—
133C22×C4—22no—
134C22×C4—22no—
135C22×C4—22no—
136C22×C4—22no—
137C22×C4—22no—
138C22×C4—22no—
139C22×C4—22no—
140C22×C4—22no—
141C22×C4—22no—
142C22×C4—22no—
143C22×C4—22no—
144C22×C4—21no—
145C22×C4—21no—
146C22×C4—22no—
147C22×C4—22no—
148C22×C4—21no—
149C22×C4—21no—
150C22×C4—24no—
151C22×C4—24no—
152C22×C4—22no—
153C22×C4—22no—
154C22×C4—21no—
155C22×C4—21no—
156C22×C4—21no—
157C22×C4—42no—
158C22×C4—42no—
159C22×C4—42no—
160C22×C4—42no—
161C22×C4C2×D411yescharacteristic
162C22×C4C2×D414yes—
163C22×C4C2×D411yescharacteristic
164C22×C4C2×D411yescharacteristic
165C22×C4#4114yes—
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—24no—
177C4⋊C4—22no—
178C4⋊C4—22no—
179C4⋊C4—22no—
180C4⋊C4—21no—
181C4⋊C4—21no—
182C4⋊C4—22no—
183C4⋊C4C2×D414yes—
184C4⋊C4C2×D414yes—
185C22⋊C4—28no—
186C22⋊C4—28no—
187C22⋊C4—42no—
188C22⋊C4—42no—
189C22⋊C4—42no—
190C22⋊C4—42no—
191C22⋊C4—42no—
192C22⋊C4—42no—
193C42—28no—
194C42—28no—
195C23—21no—
196C23—21no—
197C23—21no—
198C23—21no—
199C23—28no—
200C23—21no—
201C23—21no—
202C23—21no—
203C23—21no—
204C23—21no—
205C23—21no—
206C23—21no—
207C23—21no—
208C23—21no—
209C23—21no—
210C23—21no—
211C23—21no—
212C23—44no—
213C23—44no—
214C23—44no—
215C23—44no—
216C23—44no—
217C23—44no—
218C23—41no—
219C23—41no—
220C23—41no—
221C23—41no—
222C23—41no—
223C23—41no—
224C23—41no—
225C23#12012yes—
226C23#12312yes—
227C23#12312yes—
228C23#13512yes—
229C23AGL(1,ℤ/8)12yes—
230C23C22×D411yescharacteristic
231D4—42no—
232D4—42no—
233D4—84no—
234D4—84no—
235D4—84no—
236C2×C4—28no—
237C2×C4—28no—
238C2×C4—28no—
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—21no—
244C2×C4—21no—
245C2×C4—24no—
246C2×C4—24no—
247C2×C4—24no—
248C2×C4—24no—
249C2×C4—24no—
250C2×C4—24no—
251C2×C4—24no—
252C2×C4—24no—
253C2×C4—48no—
254C2×C4—48no—
255C2×C4—42no—
256C2×C4—42no—
257C2×C4—42no—
258C2×C4—42no—
259C2×C4—41no—
260C2×C4—41no—
261C2×C4—41no—
262C2×C4—41no—
263C2×C4—41no—
264C2×C4—41no—
265C2×C4—41no—
266C2×C4—41no—
267C2×C4—41no—
268C2×C4#12012yes—
269C2×C4#12312yes—
270C2×C4#12312yes—
271C2×C4C22×D412yes—
272C2×C4C2×#4111yescharacteristic
273C2×C4C2×#4111yescharacteristic
274C2×C4C2×#4112yes—
275C8—42no—
276C8—42no—
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—416no—
290C22—41no—
291C22—41no—
292C22—41no—
293C22—41no—
294C22—41no—
295C22—41no—
296C22—41no—
297C22—41no—
298C22—81no—
299C22—81no—
300C22—81no—
301C22—81no—
302C22—81no—
303C22—81no—
304C22—81no—
305C22#48214yescomplemented
306C22C2×#12011yescharacteristic
307C22C2×#12311yescharacteristic
308C22C2×#12311yescharacteristic
309C22C2×#13511yescharacteristic
310C22C2×AGL(1,ℤ/8)11yescharacteristic
311C22#57511yescharacteristic
312C22#57511yescharacteristic
313C4—22no—
314C4—22no—
315C4—22no—
316C4—44no—
317C4—44no—
318C4—81no—
319C4—81no—
320C4—81no—
321C4#52512yes—
322C4#52512yes—
323C2—24no—
324C2—42no—
325C2—42no—
326C2—161no—
327C2#284911yescharacteristic
328C2#300211yescharacteristic
329C2#300611yescharacteristic
330C2C2×#48211yescharacteristic
331C2#309111yescharacteristic
332C2#309311yescharacteristic
333C2#309411yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radical#2847C2
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×#117C2semidirect
2. C2×#482C2semidirect
4. #1875C2semidirect
5. #1874C2semidirect
6. #1874C2semidirect
7. #1854C2semidirect
8. #1853C2semidirect
9. #1852C2semidirect
10. #1825C2semidirect
11. #1779C2semidirect
27. C2×#107C22semidirect
28. C2×#106C22semidirect
29. C2×#102C22semidirect
30. C2×#102C22semidirect
41. #340C22semidirect
42. #340C22semidirect
43. #340C22semidirect
44. #340C22semidirect
74. #117D4semidirect
305. C22#482semidirect

Maximal subgroups

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

subgroupindex
C22×#1172
C2×#4822
#28472
#18752
#18742
#18542
#18532
#18522
#18252
#17792