Finite Group Databasebeta

← #20242

Subgroups — #20242

Order 256.

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

normal subgroups only

This group has 311 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#3354C211yesmaximal, characteristic, complemented
2#3330C211yesmaximal, characteristic, complemented
3#3330C211yesmaximal, characteristic, complemented
4#3303C211yesmaximal, characteristic, complemented
5#1718C211yesmaximal, characteristic, complemented
6#1703C211yesmaximal, characteristic, complemented
7#1699C211yesmaximal, characteristic, complemented
8#1691C211yesmaximal, characteristic, complemented
9#1679C211yesmaximal, characteristic, complemented
10#1647C211yesmaximal, characteristic, complemented
11#1647C211yesmaximal, characteristic, complemented
12#1646C211yesmaximal, characteristic, complemented
13#1635C211yesmaximal, characteristic, complemented
14#1594C211yesmaximal, characteristic, complemented
15#1581C211yesmaximal, characteristic, complemented
16#555C2211yescharacteristic
17#547C2211yescharacteristic
18#547C2211yescharacteristic
19#547C2211yescharacteristic
20#500C2211yescharacteristic
21#498C2211yescharacteristic
22#487C2211yescharacteristic
23#487C2211yescharacteristic
24#487C2211yescharacteristic
25#487C2211yescharacteristic
26#485—21no—
27#484—21no—
28#484—21no—
29#482—21no—
30#481—21no—
31#476C2211yescharacteristic
32#476C2211yescharacteristic
33#474C2211yescharacteristic
34#474C2211yescharacteristic
35#474C2211yescharacteristic
36#469—21no—
37#468—21no—
38#468—21no—
39#465—21no—
40#461C2211yescharacteristic
41#461C2211yescharacteristic
42#461C2211yescharacteristic
43#449—21no—
44#449—21no—
45#449—21no—
46C4×D8C2211yescharacteristic
47C4×D8C2211yescharacteristic
48#436C2211yescharacteristic
49#436C2211yescharacteristic
50#436C2211yescharacteristic
51C8×D4C2211yescharacteristic
52#428—21no—
53#417—21no—
54#417—21no—
55#416—21no—
56#335C2211yescharacteristic
57#334C2211yescharacteristic
58#333C2211yescharacteristic
59#332C2211yescharacteristic
60#332C2211yescharacteristic
61#332C2211yescharacteristic
62D4⋊C8C2211yescharacteristic
63D4⋊C8C2211yescharacteristic
64D4⋊C8C2211yescharacteristic
65D4⋊C8C2211yescharacteristic
66#322C2211yescharacteristic
67C2×#41—21no—
68C2×#41—21no—
69C2×#41—21no—
70C2×SD16—21no—
71C2×SD16—21no—
72C2×SD16—21no—
73C2×D8D411yescharacteristic
74C2×D8D411yescharacteristic
75C2×M16—21no—
76C2×M16—21no—
77C2×M16—21no—
78C22×C8—21no—
79C4⋊Q8C2311yescharacteristic
80C4⋊Q8C2311yescharacteristic
81C4⋊Q8C2311yescharacteristic
82C4⋊Q8C2311yescharacteristic
83#124—41no—
84#123—41no—
85#123—41no—
86#123—41no—
87C22⋊Q8—21no—
88C22⋊Q8—21no—
89C22⋊Q8—21no—
90#121—21no—
91#121—21no—
92#121—21no—
93#121—21no—
94#121—21no—
95C4×D4C2311yescharacteristic
96C4×D4C2311yescharacteristic
97C4×D4C2311yescharacteristic
98C4×D4C2311yescharacteristic
99#117—21no—
100#117—21no—
101C2×C4⋊C4—21no—
102C2×C4⋊C4—21no—
103C2×C4⋊C4—21no—
104#107—21no—
105#107—21no—
106#107—21no—
107#107—41no—
108#107D411yescharacteristic
109#107D411yescharacteristic
110#106—21no—
111#106—21no—
112#106—21no—
113#106—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#103—21no—
124#103—21no—
125#103—21no—
126#103—21no—
127#103—41no—
128#102—21no—
129#102—21no—
130#102—21no—
131#102—21no—
132#102—21no—
133#102D411yescharacteristic
134#102D411yescharacteristic
135#102D411yescharacteristic
136#102D411yescharacteristic
137C22⋊C8—21no—
138C22⋊C8—21no—
139C22⋊C8—21no—
140C22⋊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×Q8—21no—
153C2×Q8—21no—
154C2×D4—41no—
155C2×D4—41no—
156C2×D4—41no—
157C2×D4—41no—
158C2×D4C2×D411yescharacteristic
159C2×D4C2×D411yescharacteristic
160C2×D4C2×D411yescharacteristic
161C2×D4C2×D411yescharacteristic
162C22×C4—21no—
163C22×C4—21no—
164C22×C4—21no—
165C22×C4—21no—
166C22×C4—41no—
167C22×C4—41no—
168C22×C4—41no—
169SD16—81no—
170SD16—81no—
171SD16—81no—
172D8—41no—
173D8—41no—
174M16—42no—
175M16—81no—
176M16—81no—
177C2×C8—21no—
178C2×C8—21no—
179C2×C8—21no—
180C2×C8—21no—
181C2×C8—21no—
182C2×C8—81no—
183C2×C8C2×D411yescharacteristic
184C2×C8C2×D411yescharacteristic
185C2×C8C2×D411yescharacteristic
186C2×C8C2×D411yescharacteristic
187C4⋊C4—21no—
188C4⋊C4—21no—
189C4⋊C4—21no—
190C4⋊C4—21no—
191C4⋊C4—21no—
192C4⋊C4—21no—
193C4⋊C4—21no—
194C4⋊C4—21no—
195C4⋊C4—41no—
196C4⋊C4—41no—
197C4⋊C4—41no—
198C4⋊C4C2×D411yescharacteristic
199C4⋊C4C2×D411yescharacteristic
200C4⋊C4C2×D411yescharacteristic
201C4⋊C4C2×D411yescharacteristic
202C22⋊C4—21no—
203C22⋊C4—21no—
204C22⋊C4—21no—
205C22⋊C4—21no—
206C22⋊C4—41no—
207C22⋊C4—41no—
208C22⋊C4—41no—
209C22⋊C4—41no—
210C22⋊C4—41no—
211C42—41no—
212C42C2411yescharacteristic
213C23—21no—
214C23—21no—
215C23—21no—
216C23—21no—
217Q8—41no—
218Q8—41no—
219Q8—41no—
220Q8—41no—
221Q8—41no—
222Q8—41no—
223Q8—81no—
224D4—21no—
225D4—21no—
226D4—21no—
227D4—21no—
228D4—21no—
229D4—21no—
230D4—21no—
231D4—21no—
232D4—82no—
233D4—82no—
234D4—82no—
235D4—161no—
236C2×C4—21no—
237C2×C4—21no—
238C2×C4—21no—
239C2×C4—21no—
240C2×C4—41no—
241C2×C4—41no—
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—82no—
253C2×C4—82no—
254C2×C4—82no—
255C2×C4C22×D411yescharacteristic
256C2×C4C22×D411yescharacteristic
257C2×C4H(5,𝔽2)11yescharacteristic
258C8—21no—
259C8—21no—
260C8—21no—
261C8—21no—
262C8—41no—
263C8—41no—
264C8—41no—
265C8—41no—
266C8—41no—
267C22—41no—
268C22—41no—
269C22—41no—
270C22—41no—
271C22—41no—
272C22—41no—
273C22—41no—
274C22—41no—
275C22—41no—
276C22—41no—
277C22—41no—
278C22—41no—
279C22D4211yescharacteristic
280C4—21no—
281C4—41no—
282C4—41no—
283C4—41no—
284C4—41no—
285C4—81no—
286C4—81no—
287C4—81no—
288C4—81no—
289C4#57511yescharacteristic
290C4#57511yescharacteristic
291C4#57611yescharacteristic
292C4#57711yescharacteristic
293C2—81no—
294C2—81no—
295C2—81no—
296C2—81no—
297C2#326711yescharacteristic
298C2#327311yescharacteristic
299C2#328911yescharacteristic
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. #3354C2semidirect
2. #3330C2semidirect
3. #3330C2semidirect
4. #3303C2semidirect
5. #1718C2semidirect
6. #1703C2semidirect
7. #1699C2semidirect
8. #1691C2semidirect
9. #1679C2semidirect
10. #1647C2semidirect
11. #1647C2semidirect
12. #1646C2semidirect
13. #1635C2semidirect
14. #1594C2semidirect
15. #1581C2semidirect

Maximal subgroups

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

subgroupindex
#33542
#33302
#33032
#17182
#17032
#16992
#16912
#16792
#16472
#16462
#16352
#15942
#15812