Finite Group Databasebeta

← #19784

Subgroups — #19784

Order 256.

311 conjugacy classes of subgroups.

The subgroup lattice has 299 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3358C211yesmaximal, characteristic, complemented
2#3341C211yesmaximal, characteristic, complemented
3#3329C211yesmaximal, characteristic, complemented
4#3305C211yesmaximal, characteristic, complemented
5#1719C211yesmaximal, characteristic, complemented
6#1697C211yesmaximal, characteristic, complemented
7#1683C211yesmaximal, characteristic, complemented
8#1681C211yesmaximal, characteristic, complemented
9#1676C211yesmaximal, characteristic, complemented
10#1646C211yesmaximal, characteristic, complemented
11#1637C211yesmaximal, characteristic, complemented
12#1631C211yesmaximal, characteristic, complemented
13#1624C211yesmaximal, characteristic, complemented
14#1581C211yesmaximal, characteristic, complemented
15SD16⋊C8C211yesmaximal, characteristic, complemented
16#555C2211yescharacteristic
17#555C2211yescharacteristic
18#550C2211yescharacteristic
19D4×Q8C2211yescharacteristic
20#500C2211yescharacteristic
21#492C2211yescharacteristic
22#487C2211yescharacteristic
23#487C2211yescharacteristic
24#486—21no—
25#486C2211yescharacteristic
26#486C2211yescharacteristic
27#485—21no—
28#482—21no—
29#482—21no—
30#482—21no—
31#477C2211yescharacteristic
32#477C2211yescharacteristic
33#476C2211yescharacteristic
34#476C2211yescharacteristic
35#474—21no—
36#474C2211yescharacteristic
37#468—21no—
38#466—21no—
39#466—21no—
40#461C2211yescharacteristic
41#459—21no—
42#459C2211yescharacteristic
43#459C2211yescharacteristic
44#449—21no—
45#448—21no—
46C8×Q8C2211yescharacteristic
47C4×SD16C2211yescharacteristic
48C4×D8—21no—
49C4×D8C2211yescharacteristic
50#436C2211yescharacteristic
51C8×D4C2211yescharacteristic
52C8×D4C2211yescharacteristic
53#427—21no—
54#419—21no—
55#416—21no—
56#335C2211yescharacteristic
57#335C2211yescharacteristic
58#333C2211yescharacteristic
59#333C2211yescharacteristic
60#332C2211yescharacteristic
61#331C2211yescharacteristic
62Q8⋊C8C2211yescharacteristic
63D4⋊C8C2211yescharacteristic
64D4⋊C8C2211yescharacteristic
65D4⋊C8C2211yescharacteristic
66#322C2211yescharacteristic
67C2×#41—21no—
68C22×Q8—21no—
69C2×SD16—21no—
70C2×SD16—41no—
71C2×SD16D411yescharacteristic
72C2×D8—21no—
73C2×D8—21no—
74C2×D8D411yescharacteristic
75C2×M16—21no—
76C22×C8—21no—
77C22×C8—21no—
78C4⋊Q8—21no—
79C4⋊Q8C2311yescharacteristic
80C4⋊Q8C2311yescharacteristic
81C4⋊Q8C2311yescharacteristic
82#127—21no—
83#127C2311yescharacteristic
84#124—41no—
85#124—41no—
86C22⋊Q8—21no—
87C22⋊Q8—41no—
88#121—21no—
89#121—21no—
90#121—21no—
91#121—21no—
92#121—21no—
93#121—41no—
94C4×Q8C2311yescharacteristic
95C4×D4—21no—
96C4×D4—21no—
97C4×D4C2311yescharacteristic
98C4×D4C2311yescharacteristic
99C4×D4C2311yescharacteristic
100#117—21no—
101#117—21no—
102#117—21no—
103#117—21no—
104#107—21no—
105#107—21no—
106#107—21no—
107#107—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#103—41no—
124#103D411yescharacteristic
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—21no—
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—81no—
146C2×Q8—21no—
147C2×Q8—21no—
148C2×Q8—21no—
149C2×Q8—41no—
150C2×Q8—42no—
151C2×Q8—81no—
152C2×Q8C2×D411yescharacteristic
153C2×D4—21no—
154C2×D4—21no—
155C2×D4—21no—
156C2×D4—41no—
157C2×D4—41no—
158C2×D4—41no—
159C2×D4—41no—
160C2×D4C2×D411yescharacteristic
161C2×D4C2×D411yescharacteristic
162C2×D4C2×D411yescharacteristic
163C22×C4—21no—
164C22×C4—21no—
165C22×C4—21no—
166C22×C4—41no—
167C22×C4—41no—
168SD16—22no—
169SD16—81no—
170SD16—82no—
171D8—41no—
172D8—42no—
173D8—81no—
174M16—81no—
175C2×C8—21no—
176C2×C8—21no—
177C2×C8—21no—
178C2×C8—21no—
179C2×C8—21no—
180C2×C8—42no—
181C2×C8—81no—
182C2×C8C2×D411yescharacteristic
183C2×C8C2×D411yescharacteristic
184C2×C8C2×D411yescharacteristic
185C2×C8C2×D411yescharacteristic
186C4⋊C4—21no—
187C4⋊C4—21no—
188C4⋊C4—21no—
189C4⋊C4—21no—
190C4⋊C4—21no—
191C4⋊C4—21no—
192C4⋊C4—21no—
193C4⋊C4—41no—
194C4⋊C4C2×D411yescharacteristic
195C4⋊C4C2×D411yescharacteristic
196C4⋊C4C2×D411yescharacteristic
197C4⋊C4C2×D411yescharacteristic
198C22⋊C4—21no—
199C22⋊C4—21no—
200C22⋊C4—21no—
201C22⋊C4—41no—
202C22⋊C4—41no—
203C22⋊C4—41no—
204C22⋊C4—41no—
205C22⋊C4—41no—
206C22⋊C4—41no—
207C42—21no—
208C42—41no—
209C42—41no—
210C42C2411yescharacteristic
211C23—21no—
212C23—21no—
213C23—21no—
214C23—41no—
215Q8—21no—
216Q8—41no—
217Q8—41no—
218Q8—82no—
219Q8—81no—
220Q8—81no—
221Q8#13512yes—
222D4—21no—
223D4—21no—
224D4—21no—
225D4—21no—
226D4—21no—
227D4—21no—
228D4—41no—
229D4—41no—
230D4—41no—
231D4—41no—
232D4—82no—
233D4—81no—
234D4—161no—
235D4—161no—
236D4—161no—
237C2×C4—21no—
238C2×C4—21no—
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—21no—
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—82no—
252C2×C4—82no—
253C2×C4C22×D411yescharacteristic
254C2×C4C22×D411yescharacteristic
255C2×C4H(5,𝔽2)11yescharacteristic
256C8—21no—
257C8—21no—
258C8—21no—
259C8—21no—
260C8—41no—
261C8—41no—
262C8—41no—
263C8—41no—
264C8—41no—
265C22—41no—
266C22—41no—
267C22—41no—
268C22—41no—
269C22—41no—
270C22—41no—
271C22—41no—
272C22—41no—
273C22—41no—
274C22—81no—
275C22—81no—
276C22—81no—
277C22D4211yescharacteristic
278C4—22no—
279C4—21no—
280C4—41no—
281C4—41no—
282C4—41no—
283C4—41no—
284C4—81no—
285C4—81no—
286C4—81no—
287C4C2×#13511yescharacteristic
288C4#57511yescharacteristic
289C4#57611yescharacteristic
290C4#57811yescharacteristic
291C2—81no—
292C2—81no—
293C2—81no—
294C2—161no—
295C2#326911yescharacteristic
296C2#328011yescharacteristic
297C2#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. #3358C2semidirect
2. #3341C2semidirect
3. #3329C2semidirect
4. #3305C2semidirect
5. #1719C2semidirect
6. #1697C2semidirect
7. #1683C2semidirect
8. #1681C2semidirect
9. #1676C2semidirect
10. #1646C2semidirect
11. #1637C2semidirect
12. #1631C2semidirect
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
#33582
#33412
#33292
#33052
#17192
#16972
#16832
#16812
#16762
#16462
#16372
#16312
#16242
#15812
SD16⋊C82