Finite Group Databasebeta

← #19855

Subgroups — #19855

Order 256.

311 conjugacy classes of subgroups.

The subgroup lattice has 302 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3399C211yesmaximal, characteristic, complemented
2#3394C211yesmaximal, characteristic, complemented
3#3340C211yesmaximal, characteristic, complemented
4#3305C211yesmaximal, characteristic, complemented
5#1715C211yesmaximal, characteristic, complemented
6#1692C211yesmaximal, characteristic, complemented
7#1678C211yesmaximal, characteristic, complemented
8#1667C211yesmaximal, characteristic, complemented
9#1665C211yesmaximal, characteristic, complemented
10#1641C211yesmaximal, characteristic, complemented
11#1630C211yesmaximal, characteristic, complemented
12#1628C211yesmaximal, characteristic, complemented
13#1624C211yesmaximal, characteristic, complemented
14#1582C211yesmaximal, characteristic, complemented
15SD16⋊C8C211yesmaximal, characteristic, complemented
16#559C2211yescharacteristic
17#556C2211yescharacteristic
18#555C2211yescharacteristic
19#550C2211yescharacteristic
20#498C2211yescharacteristic
21#496—21no—
22#495—21no—
23#493C2211yescharacteristic
24#492C2211yescharacteristic
25#492C2211yescharacteristic
26#486C2211yescharacteristic
27#486C2211yescharacteristic
28#485—21no—
29#482—21no—
30#482—21no—
31#477C2211yescharacteristic
32#477C2211yescharacteristic
33#474C2211yescharacteristic
34#466—21no—
35#465—21no—
36#464—21no—
37#463—21no—
38#463—21no—
39#462C2211yescharacteristic
40#460C2211yescharacteristic
41#459—21no—
42#459C2211yescharacteristic
43#459C2211yescharacteristic
44#459C2211yescharacteristic
45#449—21no—
46#446C2211yescharacteristic
47C8×Q8C2211yescharacteristic
48#442—21no—
49C4×SD16—21no—
50C4×SD16C2211yescharacteristic
51C4×SD16C2211yescharacteristic
52C8×D4C2211yescharacteristic
53C8×D4C2211yescharacteristic
54#427—21no—
55#416—21no—
56#335C2211yescharacteristic
57#334C2211yescharacteristic
58#333C2211yescharacteristic
59#332C2211yescharacteristic
60#331C2211yescharacteristic
61#331C2211yescharacteristic
62Q8⋊C8C2211yescharacteristic
63Q8⋊C8C2211yescharacteristic
64D4⋊C8C2211yescharacteristic
65D4⋊C8C2211yescharacteristic
66#322C2211yescharacteristic
67C2×#41—21no—
68C2×Q16—21no—
69C2×SD16—21no—
70C2×SD16—21no—
71C2×SD16—21no—
72C2×SD16—41no—
73C2×SD16D411yescharacteristic
74C2×SD16D411yescharacteristic
75C2×D8—21no—
76C2×D8—21no—
77C2×D8—21no—
78C2×D8—21no—
79C2×D8—21no—
80C22×C8—21no—
81C22×C8—21no—
82C4⋊Q8C2311yescharacteristic
83C4⋊Q8C2311yescharacteristic
84#127—21no—
85#127C2311yescharacteristic
86#127C2311yescharacteristic
87#126—41no—
88#124—21no—
89#124—21no—
90#124—21no—
91#124—41no—
92#123—41no—
93C22⋊Q8—21no—
94#121—21no—
95#121—21no—
96#121—21no—
97#121—41no—
98C4×Q8—21no—
99C4×Q8C2311yescharacteristic
100C4×Q8C2311yescharacteristic
101C4×D4—21no—
102C4×D4—21no—
103C4×D4—21no—
104C4×D4C2311yescharacteristic
105C4×D4C2311yescharacteristic
106#117—21no—
107#117—21no—
108#117—21no—
109#107—21no—
110#107—21no—
111#107—21no—
112#106—21no—
113#106—21no—
114#106D411yescharacteristic
115#106D411yescharacteristic
116C4⋊C8—21no—
117C4⋊C8—21no—
118C4⋊C8C2311yescharacteristic
119C4⋊C8C2311yescharacteristic
120C4⋊C8C2311yescharacteristic
121C4⋊C8C2311yescharacteristic
122#103—21no—
123#103—21no—
124#103—21no—
125#103D411yescharacteristic
126#103D411yescharacteristic
127#102—21no—
128#102—21no—
129#102—21no—
130#102—21no—
131#102—21no—
132#102—21no—
133#102—21no—
134#102—41no—
135#102D411yescharacteristic
136#102D411yescharacteristic
137C22⋊C8—21no—
138C22⋊C8—21no—
139#97—21no—
140C4×C8—21no—
141C4×C8C2311yescharacteristic
142C4×C8C2311yescharacteristic
143C4×C8C2311yescharacteristic
144#41—42no—
145#41—81no—
146C2×Q8—21no—
147C2×Q8—21no—
148C2×Q8C2×D411yescharacteristic
149C2×Q8C2×D411yescharacteristic
150C2×D4—21no—
151C2×D4—21no—
152C2×D4—21no—
153C2×D4—21no—
154C2×D4—21no—
155C2×D4—21no—
156C2×D4—41no—
157C2×D4—41no—
158C2×D4—41no—
159C2×D4C2×D411yescharacteristic
160C2×D4C2×D411yescharacteristic
161C22×C4—21no—
162C22×C4—21no—
163C22×C4—41no—
164C22×C4—41no—
165Q16—81no—
166SD16—22no—
167SD16—41no—
168SD16—42no—
169SD16—81no—
170SD16—81no—
171SD16—82no—
172D8—42no—
173D8—81no—
174D8—81no—
175D8—81no—
176D8—81no—
177C2×C8—21no—
178C2×C8—21no—
179C2×C8—21no—
180C2×C8—21no—
181C2×C8—21no—
182C2×C8—42no—
183C2×C8—81no—
184C2×C8C2×D411yescharacteristic
185C2×C8C2×D411yescharacteristic
186C2×C8C2×D411yescharacteristic
187C2×C8C2×D411yescharacteristic
188C4⋊C4—21no—
189C4⋊C4—21no—
190C4⋊C4—21no—
191C4⋊C4—21no—
192C4⋊C4—21no—
193C4⋊C4—21no—
194C4⋊C4—41no—
195C4⋊C4C2×D411yescharacteristic
196C4⋊C4C2×D411yescharacteristic
197C4⋊C4C2×D411yescharacteristic
198C4⋊C4C2×D411yescharacteristic
199C22⋊C4—21no—
200C22⋊C4—21no—
201C22⋊C4—41no—
202C22⋊C4—41no—
203C22⋊C4—41no—
204C22⋊C4—41no—
205C22⋊C4—41no—
206C22⋊C4—41no—
207C22⋊C4—41no—
208C22⋊C4—41no—
209C42—21no—
210C42—21no—
211C42—41no—
212C42—41no—
213C42C2411yescharacteristic
214C23—21no—
215C23—21no—
216C23—41no—
217C23—41no—
218Q8—21no—
219Q8—21no—
220Q8—21no—
221Q8—21no—
222Q8—41no—
223Q8—41no—
224Q8—81no—
225D4—21no—
226D4—21no—
227D4—21no—
228D4—21no—
229D4—41no—
230D4—41no—
231D4—41no—
232D4—41no—
233D4—41no—
234D4—41no—
235D4—41no—
236D4—41no—
237D4—81no—
238D4—81no—
239D4—82no—
240D4—161no—
241D4—161no—
242C2×C4—21no—
243C2×C4—21no—
244C2×C4—21no—
245C2×C4—21no—
246C2×C4—21no—
247C2×C4—21no—
248C2×C4—41no—
249C2×C4—41no—
250C2×C4—41no—
251C2×C4—41no—
252C2×C4—41no—
253C2×C4—41no—
254C2×C4—82no—
255C2×C4—161no—
256C2×C4C22×D411yescharacteristic
257C2×C4C22×D411yescharacteristic
258C2×C4H(5,𝔽2)11yescharacteristic
259C8—21no—
260C8—21no—
261C8—21no—
262C8—41no—
263C8—41no—
264C8—41no—
265C8—41no—
266C8—41no—
267C8#13512yes—
268C22—41no—
269C22—41no—
270C22—41no—
271C22—41no—
272C22—41no—
273C22—41no—
274C22—81no—
275C22—81no—
276C22—81no—
277C22—81no—
278C22—81no—
279C22—81no—
280C22D4211yescharacteristic
281C4—21no—
282C4—41no—
283C4—41no—
284C4—41no—
285C4—41no—
286C4—41no—
287C4—41no—
288C4—81no—
289C4—81no—
290C4C2×#13511yescharacteristic
291C4#57511yescharacteristic
292C4#57611yescharacteristic
293C4#57711yescharacteristic
294C2—81no—
295C2—81no—
296C2—161no—
297C2—161no—
298C2#326811yescharacteristic
299C2#328411yescharacteristic
300C2#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. #3399C2semidirect
2. #3394C2semidirect
3. #3340C2semidirect
4. #3305C2semidirect
5. #1715C2semidirect
6. #1692C2semidirect
7. #1678C2semidirect
8. #1667C2semidirect
9. #1665C2semidirect
10. #1641C2semidirect
11. #1630C2semidirect
12. #1628C2semidirect
13. #1624C2semidirect
14. #1582C2semidirect
15. SD16⋊C8C2semidirect

Maximal subgroups

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

subgroupindex
#33992
#33942
#33402
#33052
#17152
#16922
#16782
#16672
#16652
#16412
#16302
#16282
#16242
#15822
SD16⋊C82