Finite Group Databasebeta

← #19899

Subgroups — #19899

Order 256.

333 conjugacy classes of subgroups.

The subgroup lattice has 314 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3330C211yesmaximal, characteristic, complemented
2#3325C211yesmaximal, characteristic, complemented
3#3303C211yesmaximal, characteristic, complemented
4#3294C211yesmaximal, characteristic, complemented
5#1711C211yesmaximal, characteristic, complemented
6#1699C211yesmaximal, characteristic, complemented
7#1697C211yesmaximal, characteristic, complemented
8#1683C211yesmaximal, characteristic, complemented
9#1681C211yesmaximal, characteristic, complemented
10#1646C211yesmaximal, characteristic, complemented
11#1639C211yesmaximal, characteristic, complemented
12#1635C211yesmaximal, characteristic, complemented
13#1625C211yesmaximal, characteristic, complemented
14#1589C211yesmaximal, characteristic, complemented
15C8×D8C211yesmaximal, characteristic, complemented
16#555C2211yescharacteristic
17#547C2211yescharacteristic
18#547C2211yescharacteristic, complemented
19D42C2211yescharacteristic
20#487C2211yescharacteristic, complemented
21#487C2211yescharacteristic, complemented
22#487C2211yescharacteristic, complemented
23#486C2211yescharacteristic
24#486C2211yescharacteristic
25#486C2211yescharacteristic
26#485—21no—
27#484—21no—
28#482—21no—
29#480—21no—
30#476C2211yescharacteristic
31#476C2211yescharacteristic, complemented
32#474C2211yescharacteristic
33#474C2211yescharacteristic
34#469—21no—
35#468—21no—
36#466—21no—
37#465—21no—
38#461C2211yescharacteristic
39#461C2211yescharacteristic, complemented
40#459C2211yescharacteristic
41#459C2211yescharacteristic
42#449—21no—
43#449—21no—
44#447—21no—
45#447—21no—
46C4×D8C2211yescharacteristic
47C4×D8C2211yescharacteristic
48#436C2211yescharacteristic
49#436C2211yescharacteristic, complemented
50C8×D4C2211yescharacteristic
51C8×D4C2211yescharacteristic, complemented
52#418—21no—
53#417—21no—
54#416—21no—
55C2×#102—21no—
56#335C2211yescharacteristic, complemented
57#335C2211yescharacteristic, complemented
58#333C2211yescharacteristic, complemented
59#332C2211yescharacteristic, complemented
60#332C2211yescharacteristic, complemented
61#331C2211yescharacteristic
62D4⋊C8C2211yescharacteristic
63D4⋊C8C2211yescharacteristic
64D4⋊C8C2211yescharacteristic, complemented
65D4⋊C8C2211yescharacteristic, complemented
66C82C2211yescharacteristic, complemented
67C2×#41—21no—
68C2×#41—21no—
69C22×D4—21no—
70C22×D4—21no—
71C2×SD16—21no—
72C2×SD16—21no—
73C2×D8—21no—
74C2×D8—21no—
75C2×D8D411yescharacteristic
76C2×D8D411yescharacteristic
77C2×M16—21no—
78C2×M16—21no—
79C22×C8—21no—
80C22×C8—21no—
81C4⋊Q8C2311yescharacteristic
82C4⋊Q8C2311yescharacteristic
83C4⋊Q8C2311yescharacteristic
84#127C2311yescharacteristic
85#124—41no—
86#123—41no—
87#123—41no—
88C22⋊Q8—21no—
89C22⋊Q8—21no—
90#121—21no—
91#121—21no—
92#121—21no—
93#121—21no—
94#121—21no—
95#121—21no—
96#120—41no—
97C4×D4C2311yescharacteristic
98C4×D4C2311yescharacteristic
99C4×D4C2311yescharacteristic
100C4×D4C2311yescharacteristic
101#117—21no—
102#117—21no—
103C2×C4⋊C4—21no—
104C2×C4⋊C4—21no—
105#107—21no—
106#107—21no—
107#107D411yescharacteristic
108#107D411yescharacteristic
109#106—21no—
110#106—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—41no—
122#102—21no—
123#102—21no—
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—41no—
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
144C24—41no—
145#41—42no—
146#41—42no—
147#41—81no—
148#41—81no—
149C2×Q8—21no—
150C2×Q8—21no—
151C2×Q8—21no—
152C2×D4—21no—
153C2×D4—21no—
154C2×D4—21no—
155C2×D4—41no—
156C2×D4—41no—
157C2×D4—42no—
158C2×D4—42no—
159C2×D4—41no—
160C2×D4—41no—
161C2×D4—41no—
162C2×D4—81no—
163C2×D4—81no—
164C2×D4C2×D411yescharacteristic
165C2×D4C2×D411yescharacteristic
166C2×D4C2×D411yescharacteristic
167C2×D4C2×D411yescharacteristic
168C22×C4—21no—
169C22×C4—21no—
170C22×C4—21no—
171C22×C4—21no—
172C22×C4—41no—
173C22×C4—41no—
174SD16—81no—
175SD16—81no—
176D8—22no—
177D8—41no—
178D8—81no—
179D8—81no—
180M16—81no—
181M16—81no—
182C2×C8—21no—
183C2×C8—21no—
184C2×C8—21no—
185C2×C8—21no—
186C2×C8—21no—
187C2×C8—81no—
188C2×C8—81no—
189C2×C8C2×D411yescharacteristic
190C2×C8C2×D411yescharacteristic
191C2×C8C2×D411yescharacteristic
192C2×C8C2×D411yescharacteristic
193C4⋊C4—21no—
194C4⋊C4—21no—
195C4⋊C4—21no—
196C4⋊C4—21no—
197C4⋊C4—21no—
198C4⋊C4—21no—
199C4⋊C4—41no—
200C4⋊C4—41no—
201C4⋊C4C2×D411yescharacteristic
202C4⋊C4C2×D411yescharacteristic
203C4⋊C4C2×D411yescharacteristic
204C4⋊C4C2×D411yescharacteristic
205C22⋊C4—21no—
206C22⋊C4—21no—
207C22⋊C4—21no—
208C22⋊C4—21no—
209C22⋊C4—41no—
210C22⋊C4—41no—
211C22⋊C4—41no—
212C22⋊C4—41no—
213C22⋊C4—41no—
214C42—41no—
215C42C2411yescharacteristic
216C23—21no—
217C23—21no—
218C23—21no—
219C23—21no—
220C23—41no—
221C23—82no—
222C23—82no—
223C23—161no—
224Q8—41no—
225Q8—41no—
226Q8—41no—
227Q8—41no—
228Q8—81no—
229D4—21no—
230D4—21no—
231D4—21no—
232D4—21no—
233D4—21no—
234D4—21no—
235D4—21no—
236D4—41no—
237D4—41no—
238D4—41no—
239D4—41no—
240D4—81no—
241D4—82no—
242D4—82no—
243D4—82no—
244D4—82no—
245D4—161no—
246D4#13512yes—
247C2×C4—21no—
248C2×C4—21no—
249C2×C4—21no—
250C2×C4—21no—
251C2×C4—41no—
252C2×C4—41no—
253C2×C4—41no—
254C2×C4—41no—
255C2×C4—41no—
256C2×C4—41no—
257C2×C4—41no—
258C2×C4—41no—
259C2×C4—41no—
260C2×C4—41no—
261C2×C4—41no—
262C2×C4—82no—
263C2×C4—82no—
264C2×C4C22×D411yescharacteristic
265C2×C4C22×D411yescharacteristic
266C2×C4H(5,𝔽2)11yescharacteristic
267C8—21no—
268C8—21no—
269C8—21no—
270C8—21no—
271C8—41no—
272C8—41no—
273C8—41no—
274C8—41no—
275C8—41no—
276C22—22no—
277C22—41no—
278C22—41no—
279C22—41no—
280C22—41no—
281C22—41no—
282C22—41no—
283C22—41no—
284C22—41no—
285C22—41no—
286C22—41no—
287C22—41no—
288C22—81no—
289C22—81no—
290C22—81no—
291C22—164no—
292C22D4211yescharacteristic
293C4—21no—
294C4—41no—
295C4—41no—
296C4—41no—
297C4—41no—
298C4—81no—
299C4—81no—
300C4—81no—
301C4C2×#13511yescharacteristic
302C4#57511yescharacteristic
303C4#57611yescharacteristic
304C4#57611yescharacteristic
305C2—42no—
306C2—81no—
307C2—81no—
308C2—81no—
309C2—161no—
310C2#327311yescharacteristic
311C2#328011yescharacteristic
312C2#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. #3330C2semidirect
2. #3325C2semidirect
3. #3303C2semidirect
4. #3294C2semidirect
5. #1711C2semidirect
6. #1699C2semidirect
7. #1697C2semidirect
8. #1683C2semidirect
9. #1681C2semidirect
10. #1646C2semidirect
11. #1639C2semidirect
12. #1635C2semidirect
13. #1625C2semidirect
14. #1589C2semidirect
15. C8×D8C2semidirect
18. #547C22semidirect
20. #487C22semidirect
21. #487C22semidirect
22. #487C22semidirect
31. #476C22semidirect
39. #461C22semidirect
49. #436C22semidirect
51. C8×D4C22semidirect
56. #335C22semidirect
57. #335C22semidirect
58. #333C22semidirect
59. #332C22semidirect
60. #332C22semidirect
64. D4⋊C8C22semidirect
65. D4⋊C8C22semidirect
66. C82C22semidirect

Maximal subgroups

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

subgroupindex
#33302
#33252
#33032
#32942
#17112
#16992
#16972
#16832
#16812
#16462
#16392
#16352
#16252
#15892
C8×D82