Finite Group Databasebeta

← #20286

Subgroups — #20286

Order 256.

342 conjugacy classes of subgroups.

The subgroup lattice has 316 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3366C211yesmaximal, characteristic, complemented
2#3357C211yesmaximal, characteristic, complemented
3#3325C211yesmaximal, characteristic, complemented
4#3274C211yesmaximal, characteristic, complemented
5#1706C211yesmaximal, characteristic, complemented
6#1698C211yesmaximal, characteristic, complemented
7#1697C211yesmaximal, characteristic, complemented
8#1695C211yesmaximal, characteristic, complemented
9#1687C211yesmaximal, characteristic, complemented
10#1631C211yesmaximal, characteristic, complemented
11#1629C211yesmaximal, characteristic, complemented
12#1624C211yesmaximal, characteristic, complemented
13#1622C211yesmaximal, characteristic, complemented
14#1590C211yesmaximal, characteristic, complemented
15#1589C211yesmaximal, characteristic, complemented
16C2×AGL(1,ℤ/8)—21no—
17#555C2211yescharacteristic
18#554C2211yescharacteristic, complemented
19#550C2211yescharacteristic
20D42C2211yescharacteristic
21#500C2211yescharacteristic, complemented
22#493C2211yescharacteristic
23#492C2211yescharacteristic
24#492C2211yescharacteristic, complemented
25#490—21no—
26#486C2211yescharacteristic
27#486C2211yescharacteristic, complemented
28#485—21no—
29#482—21no—
30#480—21no—
31#478—21no—
32#477C2211yescharacteristic, complemented
33#476C2211yescharacteristic
34#476C2211yescharacteristic
35#476C2211yescharacteristic
36#469—21no—
37#469—21no—
38#468—21no—
39#460C2211yescharacteristic
40#459—21no—
41#459C2211yescharacteristic
42#459C2211yescharacteristic
43#459C2211yescharacteristic, complemented
44#449—21no—
45#447—21no—
46#447—21no—
47#446C2211yescharacteristic, complemented
48#442—21no—
49C4×SD16C2211yescharacteristic
50C4×D8C2211yescharacteristic
51#436C2211yescharacteristic
52#436C2211yescharacteristic
53#436C2211yescharacteristic, complemented
54#428—21no—
55#418—21no—
56#335C2211yescharacteristic, complemented
57#334C2211yescharacteristic, complemented
58#333C2211yescharacteristic, complemented
59#333C2211yescharacteristic, complemented
60#331C2211yescharacteristic
61#331C2211yescharacteristic, complemented
62Q8⋊C8C2211yescharacteristic, complemented
63D4⋊C8C2211yescharacteristic
64D4⋊C8C2211yescharacteristic
65D4⋊C8C2211yescharacteristic, complemented
66C82C2211yescharacteristic, complemented
67C2×#41—21no—
68C22×D4—21no—
69C22×D4—21no—
70AGL(1,ℤ/8)—44no—
71C2×SD16—21no—
72C2×SD16—41no—
73C2×SD16—41no—
74C2×SD16D411yescharacteristic
75C2×D8—21no—
76C2×D8—21no—
77C2×D8—21no—
78C2×D8—21no—
79C2×D8—21no—
80C2×D8D411yescharacteristic
81C2×M16—21no—
82C2×M16—21no—
83C2×M16—21no—
84C4⋊Q8C2311yescharacteristic
85C4⋊Q8C2311yescharacteristic
86#127—21no—
87#127C2311yescharacteristic
88#127C2311yescharacteristic
89#125—21no—
90#124—41no—
91C22⋊Q8—21no—
92C22⋊Q8—41no—
93#121—21no—
94#121—21no—
95#121—21no—
96#121—21no—
97#121—21no—
98#121—41no—
99#120—41no—
100C4×Q8C2311yescharacteristic
101C4×D4—21no—
102C4×D4—21no—
103C4×D4C2311yescharacteristic
104C4×D4C2311yescharacteristic
105C4×D4C2311yescharacteristic
106#117—21no—
107#117—21no—
108C2×C4⋊C4—21no—
109#107—21no—
110#107—21no—
111#107—21no—
112#107D411yescharacteristic
113#106—21no—
114#106—21no—
115#106—21no—
116#106D411yescharacteristic
117C4⋊C8—21no—
118C4⋊C8C2311yescharacteristic
119C4⋊C8C2311yescharacteristic
120C4⋊C8C2311yescharacteristic
121C4⋊C8C2311yescharacteristic
122#103—21no—
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—21no—
134#102—21no—
135#102D411yescharacteristic
136#102D411yescharacteristic
137#102D411yescharacteristic
138C22⋊C8—21no—
139C22⋊C8—21no—
140C22⋊C8—21no—
141#97—21no—
142C4×C8C2311yescharacteristic
143C4×C8C2311yescharacteristic
144C4×C8C2311yescharacteristic
145C24—41no—
146#41—42no—
147#41—81no—
148C2×Q8—21no—
149C2×Q8—21no—
150C2×Q8C2×D411yescharacteristic
151C2×D4—21no—
152C2×D4—21no—
153C2×D4—21no—
154C2×D4—21no—
155C2×D4—21no—
156C2×D4—21no—
157C2×D4—41no—
158C2×D4—41no—
159C2×D4—41no—
160C2×D4—41no—
161C2×D4—42no—
162C2×D4—42no—
163C2×D4—41no—
164C2×D4—81no—
165C2×D4—81no—
166C2×D4C2×D411yescharacteristic
167C2×D4C2×D411yescharacteristic
168C2×D4C2×D411yescharacteristic
169C22×C4—21no—
170C22×C4—21no—
171C22×C4—21no—
172C22×C4—41no—
173C22×C4—41no—
174SD16—22no—
175SD16—42no—
176SD16—82no—
177SD16—82no—
178D8—41no—
179D8—42no—
180D8—42no—
181D8—42no—
182D8—81no—
183D8—81no—
184M16—42no—
185M16—42no—
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—41no—
202C4⋊C4—41no—
203C4⋊C4C2×D411yescharacteristic
204C4⋊C4C2×D411yescharacteristic
205C4⋊C4C2×D411yescharacteristic
206C4⋊C4C2×D411yescharacteristic
207C22⋊C4—21no—
208C22⋊C4—21no—
209C22⋊C4—21no—
210C22⋊C4—41no—
211C22⋊C4—41no—
212C22⋊C4—41no—
213C22⋊C4—41no—
214C22⋊C4—41no—
215C42—21no—
216C42—41no—
217C42C2411yescharacteristic
218C23—21no—
219C23—21no—
220C23—21no—
221C23—41no—
222C23—41no—
223C23—82no—
224C23—82no—
225C23—161no—
226Q8—21no—
227Q8—21no—
228Q8—81no—
229Q8—81no—
230D4—21no—
231D4—21no—
232D4—21no—
233D4—21no—
234D4—21no—
235D4—21no—
236D4—41no—
237D4—41no—
238D4—41no—
239D4—41no—
240D4—41no—
241D4—41no—
242D4—41no—
243D4—41no—
244D4—81no—
245D4—81no—
246D4—82no—
247D4—82no—
248D4—82no—
249D4—161no—
250D4—161no—
251C2×C4—21no—
252C2×C4—21no—
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—21no—
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—82no—
265C2×C4—161no—
266C2×C4C22×D411yescharacteristic
267C2×C4C22×D411yescharacteristic
268C2×C4H(5,𝔽2)11yescharacteristic
269C8—22no—
270C8—21no—
271C8—21no—
272C8—21no—
273C8—41no—
274C8—41no—
275C8—41no—
276C8—41no—
277C8AGL(1,ℤ/8)12yes—
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—81no—
291C22—81no—
292C22—81no—
293C22—164no—
294C22D4211yescharacteristic
295C4—21no—
296C4—41no—
297C4—41no—
298C4—41no—
299C4—41no—
300C4—41no—
301C4—81no—
302C4—81no—
303C4C2×AGL(1,ℤ/8)11yescharacteristic
304C4#57511yescharacteristic
305C4#57711yescharacteristic
306C4#57711yescharacteristic
307C2—81no—
308C2—81no—
309C2—81no—
310C2—161no—
311C2—161no—
312C2#327211yescharacteristic
313C2#327511yescharacteristic
314C2#329211yescharacteristic
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. #3366C2semidirect
2. #3357C2semidirect
3. #3325C2semidirect
4. #3274C2semidirect
5. #1706C2semidirect
6. #1698C2semidirect
7. #1697C2semidirect
8. #1695C2semidirect
9. #1687C2semidirect
10. #1631C2semidirect
11. #1629C2semidirect
12. #1624C2semidirect
13. #1622C2semidirect
14. #1590C2semidirect
15. #1589C2semidirect
18. #554C22semidirect
21. #500C22semidirect
24. #492C22semidirect
27. #486C22semidirect
32. #477C22semidirect
43. #459C22semidirect
47. #446C22semidirect
53. #436C22semidirect
56. #335C22semidirect
57. #334C22semidirect
58. #333C22semidirect
59. #333C22semidirect
61. #331C22semidirect
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
#33662
#33572
#33252
#32742
#17062
#16982
#16972
#16952
#16872
#16312
#16292
#16242
#16222
#15902
#15892