Finite Group Databasebeta

← #20290

Subgroups — #20290

Order 256.

313 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 313 conjugacy classes of subgroups — too many to draw. Only its 99 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
1#3366C211yesmaximal, characteristic, complemented
2#3332C211yesmaximal, characteristic, complemented
3#3331C211yesmaximal, characteristic, complemented
4#3330C211yesmaximal, characteristic, complemented
5#1711C211yesmaximal, characteristic, complemented
6#1703C211yesmaximal, characteristic, complemented
7#1701C211yesmaximal, characteristic, complemented
8#1700C211yesmaximal, characteristic, complemented
9#1697C211yesmaximal, characteristic, complemented
10#1643C211yesmaximal, characteristic, complemented
11#1641C211yesmaximal, characteristic, complemented
12#1635C211yesmaximal, characteristic, complemented
13#1625C211yesmaximal, characteristic, complemented
14#1590C211yesmaximal, characteristic, complemented
15#1589C211yesmaximal, characteristic, complemented
16#554C2211yescharacteristic
17#550C2211yescharacteristic
18#547C2211yescharacteristic
19#547C2211yescharacteristic
20#490—21no—
21#487C2211yescharacteristic
22#487C2211yescharacteristic
23#487C2211yescharacteristic
24#486C2211yescharacteristic
25#486C2211yescharacteristic
26#486C2211yescharacteristic
27#484—21no—
28#483—21no—
29#481—21no—
30#478—21no—
31#476C2211yescharacteristic
32#476C2211yescharacteristic
33#475C2211yescharacteristic
34#474C2211yescharacteristic
35#470—21no—
36#469—21no—
37#468—21no—
38#462C2211yescharacteristic
39#461C2211yescharacteristic
40#459—21no—
41#459C2211yescharacteristic
42#459C2211yescharacteristic
43#452—21no—
44#449—21no—
45#449—21no—
46#446C2211yescharacteristic
47#442—21no—
48C4×SD16C2211yescharacteristic
49C4×D8C2211yescharacteristic
50#436C2211yescharacteristic
51#436C2211yescharacteristic
52#436C2211yescharacteristic
53#417—21no—
54#417—21no—
55#417—21no—
56#335C2211yescharacteristic
57#334C2211yescharacteristic
58#333C2211yescharacteristic
59#332C2211yescharacteristic
60#332C2211yescharacteristic
61#331C2211yescharacteristic
62Q8⋊C8C2211yescharacteristic
63D4⋊C8C2211yescharacteristic
64D4⋊C8C2211yescharacteristic
65D4⋊C8C2211yescharacteristic
66C82C2211yescharacteristic
67C2×#41—21no—
68C2×#41—21no—
69C2×#41—21no—
70C2×Q16—21no—
71C2×SD16—21no—
72C2×SD16D411yescharacteristic
73C2×D8—21no—
74C2×D8—21no—
75C2×D8D411yescharacteristic
76C2×M16—21no—
77C2×M16—21no—
78C2×M16—21no—
79C4⋊Q8C2311yescharacteristic
80C4⋊Q8C2311yescharacteristic
81C4⋊Q8C2311yescharacteristic
82#127—21no—
83#127C2311yescharacteristic
84#125—21no—
85#123—41no—
86#123—41no—
87C22⋊Q8—21no—
88C22⋊Q8—21no—
89C22⋊Q8—21no—
90C22⋊Q8—41no—
91#121—21no—
92#121—21no—
93#121—21no—
94#121—41no—
95C4×Q8C2311yescharacteristic
96C4×D4—21no—
97C4×D4—21no—
98C4×D4C2311yescharacteristic
99C4×D4C2311yescharacteristic
100C4×D4C2311yescharacteristic
101C2×C4⋊C4—21no—
102C2×C4⋊C4—21no—
103C2×C4⋊C4—21no—
104#107—21no—
105#107D411yescharacteristic
106#106—21no—
107#106—21no—
108#106—21no—
109#106D411yescharacteristic
110C4⋊C8—21no—
111C4⋊C8C2311yescharacteristic
112C4⋊C8C2311yescharacteristic
113C4⋊C8C2311yescharacteristic
114C4⋊C8C2311yescharacteristic
115#103—21no—
116#103—21no—
117#103—21no—
118#103—21no—
119#103—21no—
120#103—21no—
121#103—21no—
122#103—41no—
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—
140#97—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×Q8—21no—
153C2×Q8C2×D411yescharacteristic
154C2×D4—21no—
155C2×D4—21no—
156C2×D4—21no—
157C2×D4—41no—
158C2×D4—41no—
159C2×D4—41no—
160C2×D4—41no—
161C2×D4C2×D411yescharacteristic
162C2×D4C2×D411yescharacteristic
163C2×D4C2×D411yescharacteristic
164C22×C4—21no—
165C22×C4—21no—
166C22×C4—21no—
167C22×C4—41no—
168C22×C4—41no—
169C22×C4—41no—
170C22×C4—41no—
171Q16—81no—
172SD16—41no—
173SD16—81no—
174D8—41no—
175D8—42no—
176D8—81no—
177M16—81no—
178M16—81no—
179M16—81no—
180C2×C8—21no—
181C2×C8—21no—
182C2×C8—21no—
183C2×C8—21no—
184C2×C8—21no—
185C2×C8C2×D411yescharacteristic
186C2×C8C2×D411yescharacteristic
187C2×C8C2×D411yescharacteristic
188C2×C8C2×D411yescharacteristic
189C4⋊C4—21no—
190C4⋊C4—21no—
191C4⋊C4—21no—
192C4⋊C4—21no—
193C4⋊C4—21no—
194C4⋊C4—21no—
195C4⋊C4—21no—
196C4⋊C4—41no—
197C4⋊C4—41no—
198C4⋊C4—41no—
199C4⋊C4—41no—
200C4⋊C4C2×D411yescharacteristic
201C4⋊C4C2×D411yescharacteristic
202C4⋊C4C2×D411yescharacteristic
203C4⋊C4C2×D411yescharacteristic
204C22⋊C4—21no—
205C22⋊C4—21no—
206C22⋊C4—21no—
207C22⋊C4—41no—
208C22⋊C4—41no—
209C22⋊C4—41no—
210C22⋊C4—41no—
211C42—21no—
212C42C2411yescharacteristic
213C23—21no—
214C23—21no—
215C23—21no—
216C23—41no—
217Q8—21no—
218Q8—21no—
219Q8—41no—
220Q8—41no—
221Q8—41no—
222Q8—41no—
223Q8—81no—
224D4—21no—
225D4—21no—
226D4—21no—
227D4—21no—
228D4—21no—
229D4—21no—
230D4—41no—
231D4—41no—
232D4—41no—
233D4—41no—
234D4—81no—
235D4—82no—
236D4—82no—
237D4—82no—
238D4—161no—
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—21no—
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—82no—
254C2×C4—82no—
255C2×C4—82no—
256C2×C4—161no—
257C2×C4C22×D411yescharacteristic
258C2×C4C22×D411yescharacteristic
259C2×C4H(5,𝔽2)11yescharacteristic
260C8—21no—
261C8—21no—
262C8—21no—
263C8—21no—
264C8—41no—
265C8—41no—
266C8—41no—
267C8—41no—
268C8—41no—
269C22—41no—
270C22—41no—
271C22—41no—
272C22—41no—
273C22—41no—
274C22—41no—
275C22—41no—
276C22—41no—
277C22—41no—
278C22—81no—
279C22—81no—
280C22—81no—
281C22D4211yescharacteristic
282C4—21no—
283C4—41no—
284C4—41no—
285C4—41no—
286C4—41no—
287C4—41no—
288C4—81no—
289C4—81no—
290C4—81no—
291C4#57511yescharacteristic
292C4#57511yescharacteristic
293C4#57611yescharacteristic
294C4#57711yescharacteristic
295C2—81no—
296C2—81no—
297C2—81no—
298C2—161no—
299C2#327311yescharacteristic
300C2#327511yescharacteristic
301C2#329011yescharacteristic
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. #3366C2semidirect
2. #3332C2semidirect
3. #3331C2semidirect
4. #3330C2semidirect
5. #1711C2semidirect
6. #1703C2semidirect
7. #1701C2semidirect
8. #1700C2semidirect
9. #1697C2semidirect
10. #1643C2semidirect
11. #1641C2semidirect
12. #1635C2semidirect
13. #1625C2semidirect
14. #1590C2semidirect
15. #1589C2semidirect

Maximal subgroups

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

subgroupindex
#33662
#33322
#33312
#33302
#17112
#17032
#17012
#17002
#16972
#16432
#16412
#16352
#16252
#15902
#15892