Finite Group Databasebeta

← #23982

Subgroups — #23982

Order 256.

712 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 712 conjugacy classes of subgroups — too many to draw. Only its 133 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×#552C211yesmaximal, characteristic, complemented
2C2×#450C211yesmaximal, characteristic, complemented
3C2×#418C211yesmaximal, characteristic, complemented
4#2408C211yesmaximal, characteristic
5#2025C212yesmaximal, complemented
6#2000C212yesmaximal, complemented
7#1894C211yesmaximal, characteristic, complemented
8#1880C212yesmaximal, complemented
9#1874C212yesmaximal, complemented
10#1870C211yesmaximal, characteristic, complemented
11#1825C211yesmaximal, characteristic, complemented
12C23×D4C2211yescharacteristic
13C22×SD16C2212yes—
14C22×M16C2211yescharacteristic
15#552—24no—
16#552—24no—
17C2×#126—21no—
18C2×#125—21no—
19C2×#125C2211yescharacteristic, complemented
20C2×#123—21no—
21C2×C22⋊Q8C2211yescharacteristic
22C2×C22⋊Q8C2211yescharacteristic, complemented
23C2×#121C2212yes—
24C2×#120—21no—
25C2×C4×Q8C2211yescharacteristic, complemented
26C2×C4×D4C2211yescharacteristic
27C2×#117—21no—
28C2×#117C2211yescharacteristic
29C22×C4⋊C4C2211yescharacteristic, complemented
30#450—24no—
31#418—24no—
32C2×#103C2212yescomplemented
33C2×#102C2212yes—
34C2×#102C2212yes—
35C2×#102C2212yes—
36C2×C22⋊C8C2211yescharacteristic
37C2×C22⋊C8C2212yescomplemented
38#394—21no—
39#392—21no—
40#390C2212yes—
41#380C2211yescharacteristic
42#379—21no—
43#340C2212yescomplemented
44#340C2212yescomplemented
45#340C2212yescomplemented
46#336C2212yescomplemented
47C25—21no—
48C22×Q8C2311yescharacteristic
49C22×D4—24no—
50C22×D4—22no—
51C22×D4—22no—
52C22×D4—22no—
53C22×D4—21no—
54C22×D4—42no—
55C22×D4D411yescharacteristic
56C22×D4C2312yes—
57C23×C4—21no—
58C23×C4C2311yescharacteristic
59C23×C4C2311yescharacteristic
60C2×SD16—24no—
61C2×SD16—24no—
62C2×SD16—42no—
63C2×M16—22no—
64C2×M16—42no—
65C22×C8C2312yes—
66C22×C8C2312yes—
67#126—28no—
68#125—28no—
69#125—22no—
70#125—22no—
71#123—28no—
72C22⋊Q8—22no—
73C22⋊Q8—22no—
74C22⋊Q8—22no—
75C22⋊Q8D414yescomplemented
76#121—24no—
77#121—42no—
78#120—44no—
79C4×Q8—22no—
80C4×Q8—22no—
81C4×D4—22no—
82C4×D4—22no—
83#117—28no—
84#117—22no—
85#117D414yescomplemented
86C2×C4⋊C4—21no—
87C2×C4⋊C4—21no—
88C2×C4⋊C4—21no—
89C2×C4⋊C4—21no—
90C2×C4⋊C4—21no—
91C2×C4⋊C4—21no—
92C2×C4⋊C4—22no—
93C2×C4⋊C4—22no—
94C2×C4⋊C4D411yescharacteristic
95C2×C4⋊C4C2312yes—
96C2×C4⋊C4C2311yescharacteristic
97C2×C4⋊C4C2311yescharacteristic
98C2×C4⋊C4C2311yescharacteristic
99C2×C22⋊C4—21no—
100C2×C22⋊C4—21no—
101C2×C22⋊C4—21no—
102C2×C22⋊C4—21no—
103C2×C22⋊C4—22no—
104C2×C22⋊C4—22no—
105C2×C22⋊C4—22no—
106C2×C22⋊C4D411yescharacteristic
107C2×C22⋊C4D411yescharacteristic
108C2×C42—21no—
109C2×C42—21no—
110C2×C42C2311yescharacteristic
111#103—42no—
112#102—24no—
113#102—24no—
114#102—24no—
115C22⋊C8—22no—
116C22⋊C8—42no—
117#95—21no—
118C24—22no—
119C24—44no—
120C24—42no—
121C24—42no—
122C24—42no—
123C24—82no—
124C24C2×D411yescharacteristic
125C24C2×D411yescharacteristic
126C2×Q8—21no—
127C2×Q8—22no—
128C2×Q8—22no—
129C2×Q8C2×D412yes—
130C2×D4—28no—
131C2×D4—22no—
132C2×D4—22no—
133C2×D4—22no—
134C2×D4—22no—
135C2×D4—22no—
136C2×D4—22no—
137C2×D4—22no—
138C2×D4—48no—
139C2×D4—41no—
140C2×D4—44no—
141C2×D4—44no—
142C2×D4—44no—
143C2×D4—44no—
144C2×D4—42no—
145C2×D4—82no—
146C2×D4#4114yes—
147C22×C4—22no—
148C22×C4—22no—
149C22×C4—24no—
150C22×C4—21no—
151C22×C4—21no—
152C22×C4—21no—
153C22×C4—21no—
154C22×C4—21no—
155C22×C4—21no—
156C22×C4—22no—
157C22×C4—22no—
158C22×C4—22no—
159C22×C4—42no—
160C22×C4—42no—
161C22×C4C2×D411yescharacteristic
162C22×C4C2×D411yescharacteristic
163C22×C4C2×D411yescharacteristic
164C22×C4C2×D411yescharacteristic
165C22×C4C2×D412yes—
166C22×C4C2×D412yes—
167C22×C4C2411yescharacteristic
168SD16—48no—
169M16—44no—
170C2×C8—22no—
171C2×C8—22no—
172C2×C8—22no—
173C2×C8—22no—
174C2×C8—22no—
175C2×C8—22no—
176C4⋊C4—24no—
177C4⋊C4—24no—
178C4⋊C4—24no—
179C4⋊C4—24no—
180C4⋊C4—24no—
181C4⋊C4—24no—
182C4⋊C4—22no—
183C4⋊C4—21no—
184C4⋊C4—21no—
185C4⋊C4—21no—
186C4⋊C4—22no—
187C4⋊C4—22no—
188C4⋊C4C2×D412yes—
189C4⋊C4C2×D412yes—
190C4⋊C4C2×D412yes—
191C22⋊C4—24no—
192C22⋊C4—24no—
193C22⋊C4—24no—
194C22⋊C4—22no—
195C22⋊C4—22no—
196C22⋊C4—44no—
197C22⋊C4—44no—
198C22⋊C4—42no—
199C22⋊C4—82no—
200C42—24no—
201C42—24no—
202C42—21no—
203C42—21no—
204C23—21no—
205C23—22no—
206C23—22no—
207C23—21no—
208C23—21no—
209C23—21no—
210C23—22no—
211C23—22no—
212C23—24no—
213C23—48no—
214C23—42no—
215C23—42no—
216C23—42no—
217C23—42no—
218C23—42no—
219C23—42no—
220C23—84no—
221C23—84no—
222C23—88no—
223C23—88no—
224C23#12012yes—
225C23AGL(1,ℤ/8)12yes—
226C23C22×D411yescharacteristic
227Q8—24no—
228Q8—41no—
229Q8—41no—
230D4—24no—
231D4—24no—
232D4—48no—
233D4—44no—
234D4—42no—
235D4—42no—
236D4—88no—
237C2×C4—22no—
238C2×C4—22no—
239C2×C4—22no—
240C2×C4—24no—
241C2×C4—21no—
242C2×C4—22no—
243C2×C4—22no—
244C2×C4—22no—
245C2×C4—21no—
246C2×C4—21no—
247C2×C4—22no—
248C2×C4—22no—
249C2×C4—21no—
250C2×C4—21no—
251C2×C4—22no—
252C2×C4—48no—
253C2×C4—42no—
254C2×C4—41no—
255C2×C4—41no—
256C2×C4—41no—
257C2×C4—41no—
258C2×C4—42no—
259C2×C4—41no—
260C2×C4—41no—
261C2×C4—42no—
262C2×C4—42no—
263C2×C4—42no—
264C2×C4—42no—
265C2×C4#12012yes—
266C2×C4C22×D411yescharacteristic
267C2×C4C22×D411yescharacteristic
268C2×C4C2×#4112yes—
269C2×C4H(5,𝔽2)12yes—
270C8—42no—
271C8—42no—
272C22—22no—
273C22—22no—
274C22—22no—
275C22—22no—
276C22—24no—
277C22—41no—
278C22—41no—
279C22—42no—
280C22—42no—
281C22—42no—
282C22—44no—
283C22—44no—
284C22—44no—
285C22—816no—
286C22—82no—
287C22—82no—
288C22—82no—
289C22—82no—
290C22C2×#12011yescharacteristic
291C22#54612yes—
292C22C2×AGL(1,ℤ/8)11yescharacteristic
293C22#57511yescharacteristic
294C22#57712yes—
295C4—22no—
296C4—41no—
297C4—42no—
298C4—41no—
299C4—42no—
300C4—42no—
301C4—41no—
302C4—42no—
303C4—82no—
304C4#52511yescharacteristic
305C4#53511yescharacteristic
306C4#54612yes—
307C2—42no—
308C2—42no—
309C2—84no—
310C2#240411yescharacteristic
311C2#300611yescharacteristic
312C2#301011yescharacteristic
313C2#331012yes—
314C2#332312yes—
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radical#2408C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#552C2semidirect
2. C2×#450C2semidirect
3. C2×#418C2semidirect
5. #2025C2semidirect
6. #2000C2semidirect
7. #1894C2semidirect
8. #1880C2semidirect
9. #1874C2semidirect
10. #1870C2semidirect
11. #1825C2semidirect
19. C2×#125C22semidirect
22. C2×C22⋊Q8C22semidirect
25. C2×C4×Q8C22semidirect
29. C22×C4⋊C4C22semidirect
32. C2×#103C22semidirect
37. C2×C22⋊C8C22semidirect
43. #340C22semidirect
44. #340C22semidirect
45. #340C22semidirect
46. #336C22semidirect
75. C22⋊Q8D4semidirect
85. #117D4semidirect

Maximal subgroups

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

subgroupindex
C2×#5522
C2×#4502
C2×#4182
#24082
#20252
#20002
#18942
#18802
#18742
#18702
#18252