Finite Group Databasebeta

← #19783

Subgroups — #19783

Order 256.

328 conjugacy classes of subgroups.

The subgroup lattice has 314 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3391C211yesmaximal, characteristic, complemented
2#3332C211yesmaximal, characteristic, complemented
3#3305C211yesmaximal, characteristic, complemented
4#3303C211yesmaximal, characteristic, complemented
5#1717C211yesmaximal, characteristic, complemented
6#1697C211yesmaximal, characteristic, complemented
7#1684C211yesmaximal, characteristic, complemented
8#1681C211yesmaximal, characteristic, complemented
9#1667C211yesmaximal, characteristic, complemented
10#1642C211yesmaximal, characteristic, complemented
11#1631C211yesmaximal, characteristic, complemented
12#1625C211yesmaximal, characteristic, complemented
13#1624C211yesmaximal, characteristic, complemented
14#1581C211yesmaximal, characteristic, complemented
15SD16⋊C8C211yesmaximal, characteristic, complemented
16#555C2211yescharacteristic
17#550C2211yescharacteristic
18#550C2211yescharacteristic
19#547C2211yescharacteristic
20#493—21no—
21#493C2211yescharacteristic
22#492C2211yescharacteristic
23#487C2211yescharacteristic
24#486C2211yescharacteristic
25#486C2211yescharacteristic
26#486C2211yescharacteristic
27#485—21no—
28#482—21no—
29#481—21no—
30#477C2211yescharacteristic
31#476C2211yescharacteristic
32#475C2211yescharacteristic
33#469—21no—
34#466—21no—
35#465—21no—
36#461C2211yescharacteristic
37#459—21no—
38#459—21no—
39#459C2211yescharacteristic
40#459C2211yescharacteristic
41#459C2211yescharacteristic
42#459C2211yescharacteristic
43#449—21no—
44#449—21no—
45#449—21no—
46C8×Q8C2211yescharacteristic
47C4×SD16C2211yescharacteristic
48C4×D8—21no—
49C4×D8C2211yescharacteristic
50#436C2211yescharacteristic
51C8×D4C2211yescharacteristic
52C8×D4C2211yescharacteristic
53#417—21no—
54#416—21no—
55#416—21no—
56#335C2211yescharacteristic
57#335C2211yescharacteristic
58#333C2211yescharacteristic
59#332C2211yescharacteristic
60#331C2211yescharacteristic
61#331C2211yescharacteristic
62Q8⋊C8C2211yescharacteristic
63D4⋊C8C2211yescharacteristic
64D4⋊C8C2211yescharacteristic
65D4⋊C8C2211yescharacteristic
66#322C2211yescharacteristic
67C2×#41—21no—
68C2×#41—21no—
69C2×#41—21no—
70C2×SD16—21no—
71C2×SD16—41no—
72C2×SD16D411yescharacteristic
73C2×D8—21no—
74C2×D8—21no—
75C2×D8—21no—
76C2×D8—21no—
77C2×D8—21no—
78C2×D8—21no—
79C2×D8D411yescharacteristic
80C2×M16—21no—
81C22×C8—21no—
82C22×C8—21no—
83C4⋊Q8C2311yescharacteristic
84C4⋊Q8C2311yescharacteristic
85#127—21no—
86#127—21no—
87#127C2311yescharacteristic
88#127C2311yescharacteristic
89#124—41no—
90#123—41no—
91C22⋊Q8—21no—
92#121—21no—
93#121—21no—
94#121—21no—
95#121—21no—
96#121—21no—
97#121—41no—
98#121—41no—
99C4×Q8C2311yescharacteristic
100C4×D4—21no—
101C4×D4—21no—
102C4×D4C2311yescharacteristic
103C4×D4C2311yescharacteristic
104C4×D4C2311yescharacteristic
105#117—21no—
106#117—21no—
107C2×C4⋊C4—21no—
108#107—21no—
109#107D411yescharacteristic
110#106—21no—
111#106—21no—
112#106D411yescharacteristic
113C4⋊C8—21no—
114C4⋊C8C2311yescharacteristic
115C4⋊C8C2311yescharacteristic
116C4⋊C8C2311yescharacteristic
117C4⋊C8C2311yescharacteristic
118#103—21no—
119#103—21no—
120#103—21no—
121#103—21no—
122#103—21no—
123#103D411yescharacteristic
124#102—21no—
125#102—21no—
126#102—21no—
127#102—21no—
128#102—21no—
129#102—21no—
130#102—21no—
131#102—21no—
132#102—21no—
133#102—41no—
134#102D411yescharacteristic
135#102D411yescharacteristic
136#102D411yescharacteristic
137C22⋊C8—21no—
138C22⋊C8—21no—
139C22⋊C8—21no—
140C4×C8—21no—
141C4×C8C2311yescharacteristic
142C4×C8C2311yescharacteristic
143C4×C8C2311yescharacteristic
144#41—42no—
145#41—42no—
146#41—42no—
147#41—81no—
148#41—81no—
149#41—81no—
150C2×Q8—21no—
151C2×Q8—21no—
152C2×Q8C2×D411yescharacteristic
153C2×D4—21no—
154C2×D4—21no—
155C2×D4—21no—
156C2×D4—21no—
157C2×D4—21no—
158C2×D4—21no—
159C2×D4—41no—
160C2×D4—41no—
161C2×D4—41no—
162C2×D4—41no—
163C2×D4—41no—
164C2×D4—41no—
165C2×D4C2×D411yescharacteristic
166C2×D4C2×D411yescharacteristic
167C2×D4C2×D411yescharacteristic
168C22×C4—21no—
169C22×C4—21no—
170C22×C4—21no—
171C22×C4—41no—
172C22×C4—41no—
173C22×C4—41no—
174SD16—22no—
175SD16—81no—
176SD16—82no—
177D8—41no—
178D8—42no—
179D8—81no—
180D8—81no—
181D8—81no—
182D8—81no—
183D8—81no—
184M16—81no—
185C2×C8—21no—
186C2×C8—21no—
187C2×C8—21no—
188C2×C8—21no—
189C2×C8—21no—
190C2×C8—81no—
191C2×C8—81no—
192C2×C8C2×D411yescharacteristic
193C2×C8C2×D411yescharacteristic
194C2×C8C2×D411yescharacteristic
195C2×C8C2×D411yescharacteristic
196C4⋊C4—21no—
197C4⋊C4—21no—
198C4⋊C4—21no—
199C4⋊C4—21no—
200C4⋊C4—21no—
201C4⋊C4—41no—
202C4⋊C4C2×D411yescharacteristic
203C4⋊C4C2×D411yescharacteristic
204C4⋊C4C2×D411yescharacteristic
205C4⋊C4C2×D411yescharacteristic
206C22⋊C4—21no—
207C22⋊C4—21no—
208C22⋊C4—21no—
209C22⋊C4—41no—
210C22⋊C4—41no—
211C22⋊C4—41no—
212C22⋊C4—41no—
213C22⋊C4—41no—
214C42—21no—
215C42—41no—
216C42C2411yescharacteristic
217C23—21no—
218C23—21no—
219C23—21no—
220C23—41no—
221C23—41no—
222Q8—21no—
223Q8—41no—
224Q8—41no—
225Q8—81no—
226Q8#13512yes—
227D4—21no—
228D4—21no—
229D4—21no—
230D4—21no—
231D4—21no—
232D4—21no—
233D4—41no—
234D4—41no—
235D4—41no—
236D4—41no—
237D4—41no—
238D4—41no—
239D4—41no—
240D4—41no—
241D4—82no—
242D4—82no—
243D4—81no—
244D4—81no—
245D4—82no—
246D4—161no—
247D4—161no—
248D4—161no—
249C2×C4—21no—
250C2×C4—21no—
251C2×C4—21no—
252C2×C4—21no—
253C2×C4—21no—
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—82no—
263C2×C4—82no—
264C2×C4—82no—
265C2×C4C22×D411yescharacteristic
266C2×C4C22×D411yescharacteristic
267C2×C4H(5,𝔽2)11yescharacteristic
268C8—21no—
269C8—21no—
270C8—21no—
271C8—21no—
272C8—41no—
273C8—41no—
274C8—41no—
275C8—41no—
276C8—41no—
277C22—41no—
278C22—41no—
279C22—41no—
280C22—41no—
281C22—41no—
282C22—41no—
283C22—41no—
284C22—41no—
285C22—41no—
286C22—81no—
287C22—81no—
288C22—81no—
289C22—81no—
290C22—81no—
291C22—81no—
292C22D4211yescharacteristic
293C4—22no—
294C4—21no—
295C4—41no—
296C4—41no—
297C4—41no—
298C4—41no—
299C4—81no—
300C4—81no—
301C4C2×#13511yescharacteristic
302C4#57511yescharacteristic
303C4#57611yescharacteristic
304C4#57611yescharacteristic
305C2—81no—
306C2—81no—
307C2—81no—
308C2—161no—
309C2—161no—
310C2#326811yescharacteristic
311C2#328011yescharacteristic
312C2#328811yescharacteristic
subgroupG/H
center, socleC22D42
commutator, Frattini, cosocleC42C24
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C42 ◁ G
LECSG ▷ C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #3391C2semidirect
2. #3332C2semidirect
3. #3305C2semidirect
4. #3303C2semidirect
5. #1717C2semidirect
6. #1697C2semidirect
7. #1684C2semidirect
8. #1681C2semidirect
9. #1667C2semidirect
10. #1642C2semidirect
11. #1631C2semidirect
12. #1625C2semidirect
13. #1624C2semidirect
14. #1581C2semidirect
15. SD16⋊C8C2semidirect

Maximal subgroups

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

subgroupindex
#33912
#33322
#33052
#33032
#17172
#16972
#16842
#16812
#16672
#16422
#16312
#16252
#16242
#15812
SD16⋊C82