Finite Group Databasebeta

← #19946

Subgroups — #19946

Order 256.

311 conjugacy classes of subgroups.

The subgroup lattice has 301 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3397C211yesmaximal, characteristic, complemented
2#3354C211yesmaximal, characteristic, complemented
3#3305C211yesmaximal, characteristic, complemented
4#3302C211yesmaximal, characteristic, complemented
5#1708C211yesmaximal, characteristic, complemented
6#1692C211yesmaximal, characteristic, complemented
7#1685C211yesmaximal, characteristic, complemented
8#1679C211yesmaximal, characteristic, complemented
9#1677C211yesmaximal, characteristic, complemented
10#1646C211yesmaximal, characteristic, complemented
11#1641C211yesmaximal, characteristic, complemented
12#1638C211yesmaximal, characteristic, complemented
13#1625C211yesmaximal, characteristic, complemented
14#1590C211yesmaximal, characteristic, complemented
15C8×D8C211yesmaximal, characteristic, complemented
16#556C2211yescharacteristic
17#555C2211yescharacteristic
18#550C2211yescharacteristic
19#547C2211yescharacteristic
20#498C2211yescharacteristic
21#496—21no—
22#492C2211yescharacteristic
23#487C2211yescharacteristic
24#487C2211yescharacteristic
25#486C2211yescharacteristic
26#486C2211yescharacteristic
27#485—21no—
28#482—21no—
29#482—21no—
30#481—21no—
31#477C2211yescharacteristic
32#476C2211yescharacteristic
33#474C2211yescharacteristic
34#474C2211yescharacteristic
35#469—21no—
36#467—21no—
37#466—21no—
38#463—21no—
39#462C2211yescharacteristic
40#461C2211yescharacteristic
41#460—21no—
42#459C2211yescharacteristic
43#459C2211yescharacteristic
44#452—21no—
45#449—21no—
46#446C2211yescharacteristic
47#440—21no—
48C4×SD16C2211yescharacteristic
49C4×D8C2211yescharacteristic
50#436C2211yescharacteristic
51C8×D4C2211yescharacteristic
52C8×D4C2211yescharacteristic
53#428—21no—
54#416—21no—
55#416—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×Q16—21no—
70C2×SD16—21no—
71C2×SD16—21no—
72C2×SD16—21no—
73C2×SD16D411yescharacteristic
74C2×D8—21no—
75C2×D8—21no—
76C2×D8D411yescharacteristic
77C2×M16—21no—
78C22×C8—21no—
79C22×C8—21no—
80C4⋊Q8C2311yescharacteristic
81C4⋊Q8C2311yescharacteristic
82C4⋊Q8C2311yescharacteristic
83#127—21no—
84#127C2311yescharacteristic
85#126—41no—
86#124—21no—
87#124—41no—
88#123—41no—
89C22⋊Q8—21no—
90C22⋊Q8—21no—
91#121—21no—
92#121—21no—
93#121—21no—
94#121—21no—
95#121—41no—
96C4×Q8—21no—
97C4×Q8C2311yescharacteristic
98C4×D4—21no—
99C4×D4C2311yescharacteristic
100C4×D4C2311yescharacteristic
101C4×D4C2311yescharacteristic
102#117—21no—
103#117—21no—
104#117—21no—
105C2×C4⋊C4—21no—
106#107—21no—
107#107—21no—
108#107—21no—
109#107D411yescharacteristic
110#106—21no—
111#106—21no—
112#106—21no—
113#106D411yescharacteristic
114C4⋊C8—21no—
115C4⋊C8C2311yescharacteristic
116C4⋊C8C2311yescharacteristic
117C4⋊C8C2311yescharacteristic
118C4⋊C8C2311yescharacteristic
119#103—21no—
120#103—21no—
121#103—21no—
122#103—21no—
123#103—21no—
124#103—41no—
125#103D411yescharacteristic
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—81no—
147#41—81no—
148C2×Q8—21no—
149C2×Q8—21no—
150C2×Q8—21no—
151C2×Q8C2×D411yescharacteristic
152C2×D4—21no—
153C2×D4—21no—
154C2×D4—21no—
155C2×D4—41no—
156C2×D4—41no—
157C2×D4—41no—
158C2×D4—41no—
159C2×D4C2×D411yescharacteristic
160C2×D4C2×D411yescharacteristic
161C2×D4C2×D411yescharacteristic
162C22×C4—21no—
163C22×C4—21no—
164C22×C4—21no—
165C22×C4—41no—
166C22×C4—41no—
167Q16—81no—
168SD16—41no—
169SD16—42no—
170SD16—81no—
171SD16—81no—
172D8—22no—
173D8—81no—
174D8—81no—
175M16—42no—
176C2×C8—21no—
177C2×C8—21no—
178C2×C8—21no—
179C2×C8—21no—
180C2×C8—21no—
181C2×C8—81no—
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—41no—
195C4⋊C4—41no—
196C4⋊C4C2×D411yescharacteristic
197C4⋊C4C2×D411yescharacteristic
198C4⋊C4C2×D411yescharacteristic
199C4⋊C4C2×D411yescharacteristic
200C22⋊C4—21no—
201C22⋊C4—21no—
202C22⋊C4—21no—
203C22⋊C4—41no—
204C22⋊C4—41no—
205C22⋊C4—41no—
206C22⋊C4—41no—
207C22⋊C4—41no—
208C22⋊C4—41no—
209C42—21no—
210C42—41no—
211C42—41no—
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—161no—
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×C4C22×D411yescharacteristic
256C2×C4C22×D411yescharacteristic
257C2×C4H(5,𝔽2)11yescharacteristic
258C8—21no—
259C8—21no—
260C8—21no—
261C8—41no—
262C8—41no—
263C8—41no—
264C8—41no—
265C8—41no—
266C8#13512yes—
267C22—41no—
268C22—41no—
269C22—41no—
270C22—41no—
271C22—41no—
272C22—41no—
273C22—41no—
274C22—41no—
275C22—41no—
276C22—81no—
277C22—81no—
278C22—81no—
279C22D4211yescharacteristic
280C4—21no—
281C4—41no—
282C4—41no—
283C4—41no—
284C4—41no—
285C4—41no—
286C4—81no—
287C4—81no—
288C4—81no—
289C4C2×#13511yescharacteristic
290C4#57511yescharacteristic
291C4#57611yescharacteristic
292C4#57711yescharacteristic
293C2—81no—
294C2—81no—
295C2—81no—
296C2—161no—
297C2#327311yescharacteristic
298C2#328411yescharacteristic
299C2#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. #3397C2semidirect
2. #3354C2semidirect
3. #3305C2semidirect
4. #3302C2semidirect
5. #1708C2semidirect
6. #1692C2semidirect
7. #1685C2semidirect
8. #1679C2semidirect
9. #1677C2semidirect
10. #1646C2semidirect
11. #1641C2semidirect
12. #1638C2semidirect
13. #1625C2semidirect
14. #1590C2semidirect
15. C8×D8C2semidirect

Maximal subgroups

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

subgroupindex
#33972
#33542
#33052
#33022
#17082
#16922
#16852
#16792
#16772
#16462
#16412
#16382
#16252
#15902
C8×D82