Finite Group Databasebeta

← #19911

Subgroups — #19911

Order 256.

333 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#3366C211yesmaximal, characteristic, complemented
2#3325C211yesmaximal, characteristic, complemented
3#3305C211yesmaximal, characteristic, complemented
4#3295C211yesmaximal, characteristic, complemented
5#1709C211yesmaximal, characteristic, complemented
6#1698C211yesmaximal, characteristic, complemented
7#1697C211yesmaximal, characteristic, complemented
8#1677C211yesmaximal, characteristic, complemented
9#1677C211yesmaximal, characteristic, complemented
10#1634C211yesmaximal, characteristic, complemented
11#1632C211yesmaximal, characteristic, complemented
12#1621C211yesmaximal, characteristic, complemented
13#1619C211yesmaximal, characteristic, complemented
14#1590C211yesmaximal, characteristic, complemented
15C8×D8C211yesmaximal, characteristic, complemented
16#555C2211yescharacteristic
17#554C2211yescharacteristic
18#550C2211yescharacteristic, complemented
19D42C2211yescharacteristic
20#494C2211yescharacteristic, complemented
21#493C2211yescharacteristic
22#490—21no—
23#486C2211yescharacteristic
24#486C2211yescharacteristic
25#486C2211yescharacteristic, complemented
26#486C2211yescharacteristic, complemented
27#482—21no—
28#482—21no—
29#481—21no—
30#478—21no—
31#477C2211yescharacteristic, complemented
32#476C2211yescharacteristic
33#476C2211yescharacteristic
34#476C2211yescharacteristic, complemented
35#469—21no—
36#466—21no—
37#465—21no—
38#460C2211yescharacteristic
39#459—21no—
40#459C2211yescharacteristic
41#459C2211yescharacteristic
42#459C2211yescharacteristic
43#450—21no—
44#449—21no—
45#447—21no—
46#446C2211yescharacteristic, complemented
47#442—21no—
48C4×SD16C2211yescharacteristic
49C4×D8C2211yescharacteristic
50#436C2211yescharacteristic
51C8×D4C2211yescharacteristic
52C8×D4C2211yescharacteristic, complemented
53#418—21no—
54#416—21no—
55C2×#102—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—
70C2×Q16—41no—
71C2×SD16—21no—
72C2×SD16D411yescharacteristic
73C2×D8—21no—
74C2×D8—21no—
75C2×D8—21no—
76C2×D8—41no—
77C2×D8D411yescharacteristic
78C2×M16—21no—
79C22×C8—21no—
80C22×C8—21no—
81C4⋊Q8C2311yescharacteristic
82C4⋊Q8C2311yescharacteristic
83#127—21no—
84#127C2311yescharacteristic
85#127C2311yescharacteristic
86#125—21no—
87#124—41no—
88C22⋊Q8—21no—
89C22⋊Q8—41no—
90#121—21no—
91#121—21no—
92#121—21no—
93#121—21no—
94#121—21no—
95#121—41no—
96#120—41no—
97C4×Q8C2311yescharacteristic
98C4×D4—21no—
99C4×D4—21no—
100C4×D4C2311yescharacteristic
101C4×D4C2311yescharacteristic
102C4×D4C2311yescharacteristic
103#117—21no—
104#117—21no—
105C2×C4⋊C4—21no—
106#107—21no—
107#107D411yescharacteristic
108#106—21no—
109#106—21no—
110#106—21no—
111#106D411yescharacteristic
112C4⋊C8—21no—
113C4⋊C8C2311yescharacteristic
114C4⋊C8C2311yescharacteristic
115C4⋊C8C2311yescharacteristic
116C4⋊C8C2311yescharacteristic
117#103—21no—
118#103D411yescharacteristic
119#102—21no—
120#102—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—21no—
133#102—21no—
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—
145#41—42no—
146#41—81no—
147C2×Q8—21no—
148C2×Q8—21no—
149C2×Q8C2×D411yescharacteristic
150C2×D4—21no—
151C2×D4—21no—
152C2×D4—21no—
153C2×D4—21no—
154C2×D4—21no—
155C2×D4—21no—
156C2×D4—41no—
157C2×D4—41no—
158C2×D4—41no—
159C2×D4—41no—
160C2×D4—42no—
161C2×D4—42no—
162C2×D4—41no—
163C2×D4—81no—
164C2×D4—81no—
165C2×D4C2×D411yescharacteristic
166C2×D4C2×D411yescharacteristic
167C2×D4C2×D411yescharacteristic
168C22×C4—21no—
169C22×C4—21no—
170C22×C4—21no—
171C22×C4—41no—
172C22×C4—41no—
173Q16—82no—
174SD16—41no—
175SD16—81no—
176D8—22no—
177D8—42no—
178D8—82no—
179D8—81no—
180D8—81no—
181M16—81no—
182C2×C8—21no—
183C2×C8—21no—
184C2×C8—21no—
185C2×C8—21no—
186C2×C8—21no—
187C2×C8—81no—
188C2×C8—81no—
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—41no—
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—41no—
208C22⋊C4—41no—
209C22⋊C4—41no—
210C22⋊C4—41no—
211C22⋊C4—41no—
212C42—21no—
213C42—41no—
214C42C2411yescharacteristic
215C23—21no—
216C23—21no—
217C23—21no—
218C23—41no—
219C23—41no—
220C23—82no—
221C23—82no—
222C23—161no—
223Q8—21no—
224Q8—21no—
225Q8—81no—
226Q8—81no—
227D4—21no—
228D4—21no—
229D4—21no—
230D4—21no—
231D4—21no—
232D4—41no—
233D4—41no—
234D4—41no—
235D4—41no—
236D4—41no—
237D4—41no—
238D4—41no—
239D4—41no—
240D4—81no—
241D4—81no—
242D4—82no—
243D4—82no—
244D4—82no—
245D4—161no—
246D4—161no—
247D4#13512yes—
248C2×C4—21no—
249C2×C4—21no—
250C2×C4—21no—
251C2×C4—21no—
252C2×C4—21no—
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—82no—
262C2×C4—161no—
263C2×C4C22×D411yescharacteristic
264C2×C4C22×D411yescharacteristic
265C2×C4H(5,𝔽2)11yescharacteristic
266C8—22no—
267C8—21no—
268C8—21no—
269C8—21no—
270C8—21no—
271C8—41no—
272C8—41no—
273C8—41no—
274C8—41no—
275C22—22no—
276C22—41no—
277C22—41no—
278C22—41no—
279C22—41no—
280C22—41no—
281C22—41no—
282C22—41no—
283C22—41no—
284C22—81no—
285C22—81no—
286C22—81no—
287C22—81no—
288C22—81no—
289C22—81no—
290C22—164no—
291C22D4211yescharacteristic
292C4—21no—
293C4—41no—
294C4—41no—
295C4—41no—
296C4—41no—
297C4—41no—
298C4—81no—
299C4—81no—
300C4C2×#13511yescharacteristic
301C4#57511yescharacteristic
302C4#57611yescharacteristic
303C4#57711yescharacteristic
304C2—42no—
305C2—81no—
306C2—81no—
307C2—161no—
308C2—161no—
309C2#327311yescharacteristic
310C2#328211yescharacteristic
311C2#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. #3325C2semidirect
3. #3305C2semidirect
4. #3295C2semidirect
5. #1709C2semidirect
6. #1698C2semidirect
7. #1697C2semidirect
8. #1677C2semidirect
9. #1677C2semidirect
10. #1634C2semidirect
11. #1632C2semidirect
12. #1621C2semidirect
13. #1619C2semidirect
14. #1590C2semidirect
15. C8×D8C2semidirect
18. #550C22semidirect
20. #494C22semidirect
25. #486C22semidirect
26. #486C22semidirect
31. #477C22semidirect
34. #476C22semidirect
46. #446C22semidirect
52. C8×D4C22semidirect
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
#33252
#33052
#32952
#17092
#16982
#16972
#16772
#16342
#16322
#16212
#16192
#15902
C8×D82