Finite Group Databasebeta

← #20239

Subgroups — #20239

Order 256.

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

normal subgroups only

This group has 324 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#3354C211yesmaximal, characteristic, complemented
2#3325C211yesmaximal, characteristic, complemented
3#3325C211yesmaximal, characteristic, complemented
4#3303C211yesmaximal, characteristic, complemented
5#1720C211yesmaximal, characteristic, complemented
6#1703C211yesmaximal, characteristic, complemented
7#1697C211yesmaximal, characteristic, complemented
8#1691C211yesmaximal, characteristic, complemented
9#1677C211yesmaximal, characteristic, complemented
10#1639C211yesmaximal, characteristic, complemented
11#1634C211yesmaximal, characteristic, complemented
12#1634C211yesmaximal, characteristic, complemented
13#1625C211yesmaximal, characteristic, complemented
14#1594C211yesmaximal, characteristic, complemented
15#1581C211yesmaximal, characteristic, complemented
16#555C2211yescharacteristic
17#555C2211yescharacteristic
18#547C2211yescharacteristic, complemented
19D42C2211yescharacteristic
20#498C2211yescharacteristic, complemented
21#494C2211yescharacteristic, complemented
22#487C2211yescharacteristic, complemented
23#486C2211yescharacteristic
24#486C2211yescharacteristic
25#486C2211yescharacteristic
26#485—21no—
27#482—21no—
28#482—21no—
29#482—21no—
30#481—21no—
31#476C2211yescharacteristic
32#476C2211yescharacteristic
33#476C2211yescharacteristic, complemented
34#476C2211yescharacteristic, complemented
35#474C2211yescharacteristic
36#469—21no—
37#469—21no—
38#469—21no—
39#465—21no—
40#461C2211yescharacteristic
41#459C2211yescharacteristic
42#459C2211yescharacteristic
43#449—21no—
44#447—21no—
45#447—21no—
46C4×D8C2211yescharacteristic
47C4×D8C2211yescharacteristic
48#436C2211yescharacteristic
49#436C2211yescharacteristic
50#436C2211yescharacteristic, complemented
51C8×D4C2211yescharacteristic, complemented
52#428—21no—
53#418—21no—
54#418—21no—
55#416—21no—
56#335C2211yescharacteristic, complemented
57#334C2211yescharacteristic, complemented
58#333C2211yescharacteristic, complemented
59#333C2211yescharacteristic, complemented
60#332C2211yescharacteristic, complemented
61#331C2211yescharacteristic
62D4⋊C8C2211yescharacteristic
63D4⋊C8C2211yescharacteristic
64D4⋊C8C2211yescharacteristic, complemented
65D4⋊C8C2211yescharacteristic, complemented
66#322C2211yescharacteristic, complemented
67C2×#41—21no—
68C22×D4—21no—
69C22×D4—21no—
70C2×Q16—41no—
71C2×SD16—21no—
72C2×D8—21no—
73C2×D8—21no—
74C2×D8D411yescharacteristic
75C2×D8D411yescharacteristic
76C2×M16—21no—
77C2×M16—21no—
78C2×M16—21no—
79C22×C8—21no—
80C4⋊Q8C2311yescharacteristic
81C4⋊Q8C2311yescharacteristic
82C4⋊Q8C2311yescharacteristic
83#127C2311yescharacteristic
84#124—41no—
85#124—41no—
86#123—41no—
87C22⋊Q8—21no—
88#121—21no—
89#121—21no—
90#121—21no—
91#121—21no—
92#121—21no—
93#121—21no—
94#121—21no—
95#120—41no—
96C4×D4C2311yescharacteristic
97C4×D4C2311yescharacteristic
98C4×D4C2311yescharacteristic
99C4×D4C2311yescharacteristic
100#117—21no—
101#117—21no—
102#117—21no—
103#117—21no—
104C2×C4⋊C4—21no—
105#107—21no—
106#107D411yescharacteristic
107#107D411yescharacteristic
108#106—21no—
109#106—21no—
110#106—21no—
111#106—21no—
112#106—21no—
113#106—21no—
114C4⋊C8C2311yescharacteristic
115C4⋊C8C2311yescharacteristic
116C4⋊C8C2311yescharacteristic
117C4⋊C8C2311yescharacteristic
118#103—21no—
119#103—21no—
120#103—21no—
121#102—21no—
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—81no—
147C2×Q8—21no—
148C2×Q8—21no—
149C2×Q8—21no—
150C2×D4—21no—
151C2×D4—21no—
152C2×D4—21no—
153C2×D4—41no—
154C2×D4—41no—
155C2×D4—42no—
156C2×D4—42no—
157C2×D4—41no—
158C2×D4—41no—
159C2×D4—41no—
160C2×D4—81no—
161C2×D4—81no—
162C2×D4C2×D411yescharacteristic
163C2×D4C2×D411yescharacteristic
164C2×D4C2×D411yescharacteristic
165C2×D4C2×D411yescharacteristic
166C22×C4—21no—
167C22×C4—21no—
168C22×C4—21no—
169C22×C4—21no—
170C22×C4—41no—
171Q16—82no—
172SD16—81no—
173D8—41no—
174D8—41no—
175D8—81no—
176D8—81no—
177M16—42no—
178M16—81no—
179M16—81no—
180C2×C8—21no—
181C2×C8—21no—
182C2×C8—21no—
183C2×C8—21no—
184C2×C8—21no—
185C2×C8—81no—
186C2×C8C2×D411yescharacteristic
187C2×C8C2×D411yescharacteristic
188C2×C8C2×D411yescharacteristic
189C2×C8C2×D411yescharacteristic
190C4⋊C4—21no—
191C4⋊C4—21no—
192C4⋊C4—21no—
193C4⋊C4—21no—
194C4⋊C4—21no—
195C4⋊C4—21no—
196C4⋊C4—41no—
197C4⋊C4C2×D411yescharacteristic
198C4⋊C4C2×D411yescharacteristic
199C4⋊C4C2×D411yescharacteristic
200C4⋊C4C2×D411yescharacteristic
201C22⋊C4—21no—
202C22⋊C4—21no—
203C22⋊C4—21no—
204C22⋊C4—21no—
205C22⋊C4—41no—
206C22⋊C4—41no—
207C22⋊C4—41no—
208C22⋊C4—41no—
209C22⋊C4—41no—
210C22⋊C4—41no—
211C42—41no—
212C42—41no—
213C42C2411yescharacteristic
214C23—21no—
215C23—21no—
216C23—21no—
217C23—21no—
218C23—41no—
219C23—82no—
220C23—82no—
221C23—161no—
222Q8—41no—
223Q8—41no—
224Q8—81no—
225Q8—81no—
226D4—21no—
227D4—21no—
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—82no—
242D4—161no—
243D4—161no—
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—41no—
255C2×C4—41no—
256C2×C4—41no—
257C2×C4—41no—
258C2×C4—41no—
259C2×C4—82no—
260C2×C4C22×D411yescharacteristic
261C2×C4C22×D411yescharacteristic
262C2×C4H(5,𝔽2)11yescharacteristic
263C8—21no—
264C8—21no—
265C8—21no—
266C8—21no—
267C8—41no—
268C8—41no—
269C8—41no—
270C8—41no—
271C8—41no—
272C22—41no—
273C22—41no—
274C22—41no—
275C22—41no—
276C22—41no—
277C22—41no—
278C22—41no—
279C22—41no—
280C22—41no—
281C22—41no—
282C22—41no—
283C22—41no—
284C22—81no—
285C22—81no—
286C22—81no—
287C22—164no—
288C22D4211yescharacteristic
289C4—21no—
290C4—41no—
291C4—41no—
292C4—41no—
293C4—41no—
294C4—81no—
295C4—81no—
296C4—81no—
297C4#57511yescharacteristic
298C4#57511yescharacteristic
299C4#57611yescharacteristic
300C4#57711yescharacteristic
301C2—81no—
302C2—81no—
303C2—81no—
304C2—81no—
305C2—161no—
306C2#326711yescharacteristic
307C2#327311yescharacteristic
308C2#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. #3354C2semidirect
2. #3325C2semidirect
3. #3325C2semidirect
4. #3303C2semidirect
5. #1720C2semidirect
6. #1703C2semidirect
7. #1697C2semidirect
8. #1691C2semidirect
9. #1677C2semidirect
10. #1639C2semidirect
11. #1634C2semidirect
12. #1634C2semidirect
13. #1625C2semidirect
14. #1594C2semidirect
15. #1581C2semidirect
18. #547C22semidirect
20. #498C22semidirect
21. #494C22semidirect
22. #487C22semidirect
33. #476C22semidirect
34. #476C22semidirect
50. #436C22semidirect
51. C8×D4C22semidirect
56. #335C22semidirect
57. #334C22semidirect
58. #333C22semidirect
59. #333C22semidirect
60. #332C22semidirect
64. D4⋊C8C22semidirect
65. D4⋊C8C22semidirect
66. #322C22semidirect

Maximal subgroups

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

subgroupindex
#33542
#33252
#33032
#17202
#17032
#16972
#16912
#16772
#16392
#16342
#16252
#15942
#15812