Finite Group Databasebeta

← #20301

Subgroups — #20301

Order 256.

342 conjugacy classes of subgroups.

The subgroup lattice has 317 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3354C211yesmaximal, characteristic, complemented
2#3330C211yesmaximal, characteristic, complemented
3#3325C211yesmaximal, characteristic, complemented
4#3272C211yesmaximal, characteristic, complemented
5#1708C211yesmaximal, characteristic, complemented
6#1699C211yesmaximal, characteristic, complemented
7#1697C211yesmaximal, characteristic, complemented
8#1691C211yesmaximal, characteristic, complemented
9#1685C211yesmaximal, characteristic, complemented
10#1646C211yesmaximal, characteristic, complemented
11#1639C211yesmaximal, characteristic, complemented
12#1635C211yesmaximal, characteristic, complemented
13#1625C211yesmaximal, characteristic, complemented
14#1589C211yesmaximal, characteristic, complemented
15#1589C211yesmaximal, characteristic, complemented
16C2×AGL(1,ℤ/8)—21no—
17#555C2211yescharacteristic
18#547C2211yescharacteristic
19#547C2211yescharacteristic, complemented
20D42C2211yescharacteristic
21#498C2211yescharacteristic, complemented
22#492C2211yescharacteristic
23#487C2211yescharacteristic, complemented
24#487C2211yescharacteristic, complemented
25#486C2211yescharacteristic
26#486C2211yescharacteristic
27#484—21no—
28#482—21no—
29#482—21no—
30#481—21no—
31#476C2211yescharacteristic
32#476C2211yescharacteristic, complemented
33#474C2211yescharacteristic
34#474C2211yescharacteristic, complemented
35#469—21no—
36#469—21no—
37#468—21no—
38#468—21no—
39#461C2211yescharacteristic
40#461C2211yescharacteristic
41#459C2211yescharacteristic
42#459C2211yescharacteristic
43#450—21no—
44#449—21no—
45#449—21no—
46#447—21no—
47C4×D8C2211yescharacteristic
48C4×D8C2211yescharacteristic
49#436C2211yescharacteristic
50#436C2211yescharacteristic
51#436C2211yescharacteristic, complemented
52#436C2211yescharacteristic, complemented
53#428—21no—
54#418—21no—
55#417—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—
71AGL(1,ℤ/8)—44no—
72C2×SD16—21no—
73C2×SD16—21no—
74C2×SD16—21no—
75C2×SD16—21no—
76C2×D8—21no—
77C2×D8—21no—
78C2×D8D411yescharacteristic
79C2×D8D411yescharacteristic
80C2×M16—21no—
81C2×M16—21no—
82C2×M16—21no—
83C2×M16—21no—
84C4⋊Q8C2311yescharacteristic
85C4⋊Q8C2311yescharacteristic
86C4⋊Q8C2311yescharacteristic
87#127C2311yescharacteristic
88#124—41no—
89#123—41no—
90#123—41no—
91C22⋊Q8—21no—
92C22⋊Q8—21no—
93#121—21no—
94#121—21no—
95#121—21no—
96#121—21no—
97#121—21no—
98#121—21no—
99#120—41no—
100C4×D4C2311yescharacteristic
101C4×D4C2311yescharacteristic
102C4×D4C2311yescharacteristic
103C4×D4C2311yescharacteristic
104#117—21no—
105#117—21no—
106C2×C4⋊C4—21no—
107C2×C4⋊C4—21no—
108#107—21no—
109#107—21no—
110#107D411yescharacteristic
111#107D411yescharacteristic
112#106—21no—
113#106—21no—
114#106—21no—
115#106—21no—
116C4⋊C8C2311yescharacteristic
117C4⋊C8C2311yescharacteristic
118C4⋊C8C2311yescharacteristic
119C4⋊C8C2311yescharacteristic
120#103—21no—
121#103—21no—
122#103—21no—
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—41no—
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—42no—
148#41—81no—
149#41—81no—
150C2×Q8—21no—
151C2×Q8—21no—
152C2×Q8—21no—
153C2×D4—21no—
154C2×D4—21no—
155C2×D4—21no—
156C2×D4—41no—
157C2×D4—41no—
158C2×D4—42no—
159C2×D4—42no—
160C2×D4—41no—
161C2×D4—41no—
162C2×D4—41no—
163C2×D4—81no—
164C2×D4—81no—
165C2×D4C2×D411yescharacteristic
166C2×D4C2×D411yescharacteristic
167C2×D4C2×D411yescharacteristic
168C2×D4C2×D411yescharacteristic
169C22×C4—21no—
170C22×C4—21no—
171C22×C4—21no—
172C22×C4—21no—
173C22×C4—41no—
174C22×C4—41no—
175SD16—42no—
176SD16—42no—
177SD16—81no—
178SD16—81no—
179D8—22no—
180D8—41no—
181D8—42no—
182D8—81no—
183M16—42no—
184M16—42no—
185M16—81no—
186M16—81no—
187C2×C8—21no—
188C2×C8—21no—
189C2×C8—21no—
190C2×C8—21no—
191C2×C8—21no—
192C2×C8C2×D411yescharacteristic
193C2×C8C2×D411yescharacteristic
194C2×C8C2×D411yescharacteristic
195C2×C8C2×D411yescharacteristic
196C4⋊C4—21no—
197C4⋊C4—21no—
198C4⋊C4—21no—
199C4⋊C4—21no—
200C4⋊C4—21no—
201C4⋊C4—21no—
202C4⋊C4—41no—
203C4⋊C4—41no—
204C4⋊C4C2×D411yescharacteristic
205C4⋊C4C2×D411yescharacteristic
206C4⋊C4C2×D411yescharacteristic
207C4⋊C4C2×D411yescharacteristic
208C22⋊C4—21no—
209C22⋊C4—21no—
210C22⋊C4—21no—
211C22⋊C4—21no—
212C22⋊C4—41no—
213C22⋊C4—41no—
214C22⋊C4—41no—
215C22⋊C4—41no—
216C22⋊C4—41no—
217C42—41no—
218C42C2411yescharacteristic
219C23—21no—
220C23—21no—
221C23—21no—
222C23—21no—
223C23—41no—
224C23—82no—
225C23—82no—
226C23—161no—
227Q8—41no—
228Q8—41no—
229Q8—41no—
230Q8—41no—
231Q8—81no—
232D4—21no—
233D4—21no—
234D4—21no—
235D4—21no—
236D4—21no—
237D4—21no—
238D4—21no—
239D4—21no—
240D4—41no—
241D4—41no—
242D4—41no—
243D4—41no—
244D4—81no—
245D4—82no—
246D4—82no—
247D4—82no—
248D4—82no—
249D4—161no—
250C2×C4—21no—
251C2×C4—21no—
252C2×C4—21no—
253C2×C4—21no—
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—41no—
263C2×C4—41no—
264C2×C4—41no—
265C2×C4—82no—
266C2×C4—82no—
267C2×C4C22×D411yescharacteristic
268C2×C4C22×D411yescharacteristic
269C2×C4H(5,𝔽2)11yescharacteristic
270C8—21no—
271C8—21no—
272C8—21no—
273C8—41no—
274C8—41no—
275C8—41no—
276C8—41no—
277C8—41no—
278C8AGL(1,ℤ/8)12yes—
279C22—41no—
280C22—41no—
281C22—41no—
282C22—41no—
283C22—41no—
284C22—41no—
285C22—41no—
286C22—41no—
287C22—41no—
288C22—41no—
289C22—41no—
290C22—41no—
291C22—81no—
292C22—81no—
293C22—81no—
294C22—164no—
295C22D4211yescharacteristic
296C4—21no—
297C4—41no—
298C4—41no—
299C4—41no—
300C4—41no—
301C4—81no—
302C4—81no—
303C4—81no—
304C4C2×AGL(1,ℤ/8)11yescharacteristic
305C4#57511yescharacteristic
306C4#57611yescharacteristic
307C4#57711yescharacteristic
308C2—81no—
309C2—81no—
310C2—81no—
311C2—81no—
312C2—161no—
313C2#327211yescharacteristic
314C2#327311yescharacteristic
315C2#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. #3354C2semidirect
2. #3330C2semidirect
3. #3325C2semidirect
4. #3272C2semidirect
5. #1708C2semidirect
6. #1699C2semidirect
7. #1697C2semidirect
8. #1691C2semidirect
9. #1685C2semidirect
10. #1646C2semidirect
11. #1639C2semidirect
12. #1635C2semidirect
13. #1625C2semidirect
14. #1589C2semidirect
15. #1589C2semidirect
19. #547C22semidirect
21. #498C22semidirect
23. #487C22semidirect
24. #487C22semidirect
32. #476C22semidirect
34. #474C22semidirect
51. #436C22semidirect
52. #436C22semidirect
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
#33542
#33302
#33252
#32722
#17082
#16992
#16972
#16912
#16852
#16462
#16392
#16352
#16252
#15892