Finite Group Databasebeta

← #20287

Subgroups — #20287

Order 256.

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

normal subgroups only

This group has 317 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#3367C211yesmaximal, characteristic, complemented
2#3329C211yesmaximal, characteristic, complemented
3#3325C211yesmaximal, characteristic
4#3324C211yesmaximal, characteristic, complemented
5#1711C211yesmaximal, characteristic, complemented
6#1703C211yesmaximal, characteristic, complemented
7#1701C211yesmaximal, characteristic, complemented
8#1699C211yesmaximal, characteristic, complemented
9#1698C211yesmaximal, characteristic, complemented
10#1646C211yesmaximal, characteristic, complemented
11#1639C211yesmaximal, characteristic, complemented
12#1638C211yesmaximal, characteristic, complemented
13#1627C211yesmaximal, characteristic, complemented
14#1590C211yesmaximal, characteristic, complemented
15#1589C211yesmaximal, characteristic, complemented
16#556C2211yescharacteristic
17#555C2211yescharacteristic
18D4×Q8C2211yescharacteristic, complemented
19D42C2211yescharacteristic
20#489—21no—
21#487C2211yescharacteristic, complemented
22#487C2211yescharacteristic, complemented
23#487C2211yescharacteristic, complemented
24#486C2211yescharacteristic
25#486C2211yescharacteristic
26#486C2211yescharacteristic
27#485—21no—
28#482—21no—
29#482—21no—
30#477—21no—
31#477C2211yescharacteristic, complemented
32#476C2211yescharacteristic
33#474C2211yescharacteristic
34#474C2211yescharacteristic
35#469—21no—
36#468—21no—
37#468—21no—
38#463—21no—
39#461C2211yescharacteristic
40#461C2211yescharacteristic, complemented
41#460C2211yescharacteristic
42#459C2211yescharacteristic
43#450—21no—
44#448—21no—
45#447—21no—
46#446C2211yescharacteristic, complemented
47#440—21no—
48C4×SD16C2211yescharacteristic
49C4×D8C2211yescharacteristic
50#436C2211yescharacteristic
51#436C2211yescharacteristic
52#436C2211yescharacteristic, complemented
53#419—21no—
54#418—21no—
55#418—21no—
56#335C2211yescharacteristic, complemented
57#334C2211yescharacteristic, complemented
58#333C2211yescharacteristic, complemented
59#332C2211yescharacteristic, complemented
60#332C2211yescharacteristic, complemented
61#331C2211yescharacteristic
62Q8⋊C8C2211yescharacteristic, complemented
63D4⋊C8C2211yescharacteristic
64D4⋊C8C2211yescharacteristic
65D4⋊C8C2211yescharacteristic, complemented
66C82C2211yescharacteristic, complemented
67C22×Q8—21no—
68C22×D4—21no—
69C22×D4—21no—
70C2×SD16—21no—
71C2×SD16—21no—
72C2×SD16—21no—
73C2×SD16D411yescharacteristic
74C2×D8—21no—
75C2×D8D411yescharacteristic
76C2×M16—21no—
77C2×M16—21no—
78C2×M16—21no—
79C4⋊Q8—21no—
80C4⋊Q8C2311yescharacteristic
81C4⋊Q8C2311yescharacteristic
82C4⋊Q8C2311yescharacteristic
83#127C2311yescharacteristic
84#126—41no—
85#124—21no—
86#124—41no—
87C22⋊Q8—21no—
88C22⋊Q8—21no—
89C22⋊Q8—41no—
90#121—21no—
91#121—21no—
92#121—21no—
93#121—21no—
94#120—41no—
95C4×Q8—21no—
96C4×Q8C2311yescharacteristic
97C4×D4—21no—
98C4×D4C2311yescharacteristic
99C4×D4C2311yescharacteristic
100C4×D4C2311yescharacteristic
101#117—21no—
102#117—21no—
103#117—21no—
104#107—21no—
105#107—21no—
106#107—21no—
107#107D411yescharacteristic
108#106—21no—
109#106D411yescharacteristic
110C4⋊C8—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—21no—
122#103—41no—
123#103D411yescharacteristic
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—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
144C24—41no—
145C2×Q8—21no—
146C2×Q8—21no—
147C2×Q8—21no—
148C2×Q8—41no—
149C2×Q8—42no—
150C2×Q8—81no—
151C2×Q8C2×D411yescharacteristic
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—81no—
161C2×D4—81no—
162C2×D4C2×D411yescharacteristic
163C2×D4C2×D411yescharacteristic
164C2×D4C2×D411yescharacteristic
165C22×C4—21no—
166C22×C4—21no—
167C22×C4—21no—
168C22×C4—41no—
169SD16—41no—
170SD16—42no—
171SD16—81no—
172SD16—81no—
173D8—41no—
174D8—81no—
175M16—81no—
176M16—81no—
177M16—81no—
178C2×C8—21no—
179C2×C8—21no—
180C2×C8—21no—
181C2×C8—21no—
182C2×C8—21no—
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—
217C23—82no—
218C23—82no—
219C23—161no—
220Q8—21no—
221Q8—21no—
222Q8—41no—
223Q8—41no—
224Q8—41no—
225Q8—41no—
226Q8—81no—
227Q8—82no—
228D4—21no—
229D4—21no—
230D4—21no—
231D4—21no—
232D4—21no—
233D4—21no—
234D4—41no—
235D4—41no—
236D4—41no—
237D4—41no—
238D4—81no—
239D4—82no—
240D4—82no—
241D4—161no—
242C2×C4—21no—
243C2×C4—21no—
244C2×C4—21no—
245C2×C4—21no—
246C2×C4—21no—
247C2×C4—41no—
248C2×C4—41no—
249C2×C4—41no—
250C2×C4—41no—
251C2×C4—41no—
252C2×C4—41no—
253C2×C4—41no—
254C2×C4—41no—
255C2×C4—41no—
256C2×C4—82no—
257C2×C4C22×D411yescharacteristic
258C2×C4C22×D411yescharacteristic
259C2×C4H(5,𝔽2)11yescharacteristic
260C8—21no—
261C8—21no—
262C8—21no—
263C8—21no—
264C8—41no—
265C8—41no—
266C8—41no—
267C8—41no—
268C8—41no—
269C22—41no—
270C22—41no—
271C22—41no—
272C22—41no—
273C22—41no—
274C22—41no—
275C22—41no—
276C22—41no—
277C22—41no—
278C22—81no—
279C22—81no—
280C22—81no—
281C22—164no—
282C22D4211yescharacteristic
283C4—21no—
284C4—41no—
285C4—41no—
286C4—41no—
287C4—41no—
288C4—41no—
289C4—81no—
290C4—81no—
291C4—81no—
292C4#57511yescharacteristic
293C4#57511yescharacteristic
294C4#57611yescharacteristic
295C4#57711yescharacteristic
296C2—81no—
297C2—81no—
298C2—81no—
299C2—161no—
300C2#327311yescharacteristic
301C2#327511yescharacteristic
302C2#329011yescharacteristic
subgroupG/H
center, socleC22D42
commutator, Frattini, cosocleC42C24
prime radical#3325C2
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. #3367C2semidirect
2. #3329C2semidirect
4. #3324C2semidirect
5. #1711C2semidirect
6. #1703C2semidirect
7. #1701C2semidirect
8. #1699C2semidirect
9. #1698C2semidirect
10. #1646C2semidirect
11. #1639C2semidirect
12. #1638C2semidirect
13. #1627C2semidirect
14. #1590C2semidirect
15. #1589C2semidirect
18. D4×Q8C22semidirect
21. #487C22semidirect
22. #487C22semidirect
23. #487C22semidirect
31. #477C22semidirect
40. #461C22semidirect
46. #446C22semidirect
52. #436C22semidirect
56. #335C22semidirect
57. #334C22semidirect
58. #333C22semidirect
59. #332C22semidirect
60. #332C22semidirect
62. Q8⋊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
#33672
#33292
#33252
#33242
#17112
#17032
#17012
#16992
#16982
#16462
#16392
#16382
#16272
#15902
#15892