Finite Group Databasebeta

← #20243

Subgroups — #20243

Order 256.

337 conjugacy classes of subgroups.

The subgroup lattice has 313 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3354C211yesmaximal, characteristic, complemented
2#3341C211yesmaximal, characteristic, complemented
3#3325C211yesmaximal, characteristic, complemented
4#3272C211yesmaximal, characteristic, complemented
5#1714C211yesmaximal, characteristic, complemented
6#1697C211yesmaximal, characteristic, complemented
7#1695C211yesmaximal, characteristic, complemented
8#1691C211yesmaximal, characteristic, complemented
9#1665C211yesmaximal, characteristic, complemented
10#1646C211yesmaximal, characteristic, complemented
11#1635C211yesmaximal, characteristic, complemented
12#1629C211yesmaximal, characteristic, complemented
13#1624C211yesmaximal, characteristic, complemented
14#1594C211yesmaximal, characteristic, complemented
15#1581C211yesmaximal, characteristic, complemented
16C2×AGL(1,ℤ/8)—21no—
17#555C2211yescharacteristic
18#555C2211yescharacteristic, complemented
19#547C2211yescharacteristic
20D42C2211yescharacteristic
21#500C2211yescharacteristic, complemented
22#498C2211yescharacteristic, complemented
23#492C2211yescharacteristic
24#492C2211yescharacteristic
25#487C2211yescharacteristic, complemented
26#486C2211yescharacteristic
27#482—21no—
28#482—21no—
29#482—21no—
30#482—21no—
31#481—21no—
32#476C2211yescharacteristic
33#476C2211yescharacteristic
34#476C2211yescharacteristic
35#476C2211yescharacteristic, complemented
36#474C2211yescharacteristic, complemented
37#469—21no—
38#469—21no—
39#468—21no—
40#466—21no—
41#461C2211yescharacteristic
42#459C2211yescharacteristic
43#459C2211yescharacteristic
44#450—21no—
45#449—21no—
46#447—21no—
47C4×D8C2211yescharacteristic
48C4×D8C2211yescharacteristic
49#436C2211yescharacteristic
50#436C2211yescharacteristic
51#436C2211yescharacteristic, complemented
52C8×D4C2211yescharacteristic, complemented
53#428—21no—
54#427—21no—
55#418—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—
70AGL(1,ℤ/8)—44no—
71C2×SD16—21no—
72C2×SD16—21no—
73C2×SD16—21no—
74C2×SD16—41no—
75C2×D8—21no—
76C2×D8—21no—
77C2×D8D411yescharacteristic
78C2×D8D411yescharacteristic
79C2×M16—21no—
80C2×M16—21no—
81C2×M16—21no—
82C22×C8—21no—
83C4⋊Q8C2311yescharacteristic
84C4⋊Q8C2311yescharacteristic
85C4⋊Q8C2311yescharacteristic
86#127C2311yescharacteristic
87#124—41no—
88#124—41no—
89#123—41no—
90C22⋊Q8—21no—
91#121—21no—
92#121—21no—
93#121—21no—
94#121—21no—
95#121—21no—
96#121—21no—
97#121—21no—
98#120—41no—
99C4×D4C2311yescharacteristic
100C4×D4C2311yescharacteristic
101C4×D4C2311yescharacteristic
102C4×D4C2311yescharacteristic
103#117—21no—
104#117—21no—
105#117—21no—
106#117—21no—
107C2×C4⋊C4—21no—
108#107—21no—
109#107—21no—
110#107—21no—
111#107D411yescharacteristic
112#107D411yescharacteristic
113#106—21no—
114#106—21no—
115#106—21no—
116#106—21no—
117#106—21no—
118#106—21no—
119C4⋊C8C2311yescharacteristic
120C4⋊C8C2311yescharacteristic
121C4⋊C8C2311yescharacteristic
122C4⋊C8C2311yescharacteristic
123#103—21no—
124#103—41no—
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
137#102D411yescharacteristic
138C22⋊C8—21no—
139C22⋊C8—21no—
140C22⋊C8—21no—
141C22⋊C8—21no—
142C4×C8C2311yescharacteristic
143C4×C8C2311yescharacteristic
144C4×C8C2311yescharacteristic
145C24—41no—
146#41—42no—
147#41—81no—
148C2×Q8—21no—
149C2×Q8—21no—
150C2×Q8—21no—
151C2×D4—21no—
152C2×D4—21no—
153C2×D4—21no—
154C2×D4—41no—
155C2×D4—41no—
156C2×D4—42no—
157C2×D4—42no—
158C2×D4—41no—
159C2×D4—41no—
160C2×D4—41no—
161C2×D4—81no—
162C2×D4—81no—
163C2×D4C2×D411yescharacteristic
164C2×D4C2×D411yescharacteristic
165C2×D4C2×D411yescharacteristic
166C2×D4C2×D411yescharacteristic
167C22×C4—21no—
168C22×C4—21no—
169C22×C4—21no—
170C22×C4—21no—
171C22×C4—41no—
172SD16—42no—
173SD16—42no—
174SD16—81no—
175SD16—82no—
176D8—22no—
177D8—41no—
178D8—42no—
179D8—81no—
180M16—42no—
181M16—42no—
182M16—81no—
183C2×C8—21no—
184C2×C8—21no—
185C2×C8—21no—
186C2×C8—21no—
187C2×C8—21no—
188C2×C8—42no—
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⋊C4C2×D411yescharacteristic
201C4⋊C4C2×D411yescharacteristic
202C4⋊C4C2×D411yescharacteristic
203C4⋊C4C2×D411yescharacteristic
204C22⋊C4—21no—
205C22⋊C4—21no—
206C22⋊C4—21no—
207C22⋊C4—21no—
208C22⋊C4—41no—
209C22⋊C4—41no—
210C22⋊C4—41no—
211C22⋊C4—41no—
212C22⋊C4—41no—
213C22⋊C4—41no—
214C42—41no—
215C42—41no—
216C42C2411yescharacteristic
217C23—21no—
218C23—21no—
219C23—21no—
220C23—21no—
221C23—41no—
222C23—82no—
223C23—82no—
224C23—161no—
225Q8—41no—
226Q8—41no—
227Q8—81no—
228Q8—81no—
229D4—21no—
230D4—21no—
231D4—21no—
232D4—21no—
233D4—21no—
234D4—21no—
235D4—21no—
236D4—21no—
237D4—41no—
238D4—41no—
239D4—41no—
240D4—41no—
241D4—81no—
242D4—82no—
243D4—82no—
244D4—82no—
245D4—161no—
246D4—161no—
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×C4C22×D411yescharacteristic
264C2×C4C22×D411yescharacteristic
265C2×C4H(5,𝔽2)11yescharacteristic
266C8—21no—
267C8—21no—
268C8—21no—
269C8—41no—
270C8—41no—
271C8—41no—
272C8—41no—
273C8—41no—
274C8AGL(1,ℤ/8)12yes—
275C22—41no—
276C22—41no—
277C22—41no—
278C22—41no—
279C22—41no—
280C22—41no—
281C22—41no—
282C22—41no—
283C22—41no—
284C22—41no—
285C22—41no—
286C22—41no—
287C22—81no—
288C22—81no—
289C22—81no—
290C22—164no—
291C22D4211yescharacteristic
292C4—21no—
293C4—41no—
294C4—41no—
295C4—41no—
296C4—41no—
297C4—81no—
298C4—81no—
299C4—81no—
300C4C2×AGL(1,ℤ/8)11yescharacteristic
301C4#57511yescharacteristic
302C4#57711yescharacteristic
303C4#57711yescharacteristic
304C2—81no—
305C2—81no—
306C2—81no—
307C2—81no—
308C2—161no—
309C2#326711yescharacteristic
310C2#327211yescharacteristic
311C2#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. #3341C2semidirect
3. #3325C2semidirect
4. #3272C2semidirect
5. #1714C2semidirect
6. #1697C2semidirect
7. #1695C2semidirect
8. #1691C2semidirect
9. #1665C2semidirect
10. #1646C2semidirect
11. #1635C2semidirect
12. #1629C2semidirect
13. #1624C2semidirect
14. #1594C2semidirect
15. #1581C2semidirect
18. #555C22semidirect
21. #500C22semidirect
22. #498C22semidirect
25. #487C22semidirect
35. #476C22semidirect
36. #474C22semidirect
51. #436C22semidirect
52. 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
#33412
#33252
#32722
#17142
#16972
#16952
#16912
#16652
#16462
#16352
#16292
#16242
#15942
#15812