Finite Group Databasebeta

← #20175

Subgroups — #20175

Order 256.

321 conjugacy classes of subgroups.

The subgroup lattice has 304 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3384C211yesmaximal, characteristic, complemented
2#3329C211yesmaximal, characteristic, complemented
3#3325C211yesmaximal, characteristic
4#3295C211yesmaximal, characteristic, complemented
5#1720C211yesmaximal, characteristic, complemented
6#1698C211yesmaximal, characteristic, complemented
7#1697C211yesmaximal, characteristic, complemented
8#1691C211yesmaximal, characteristic, complemented
9#1683C211yesmaximal, characteristic, complemented
10#1639C211yesmaximal, characteristic, complemented
11#1634C211yesmaximal, characteristic, complemented
12#1632C211yesmaximal, characteristic, complemented
13#1627C211yesmaximal, characteristic, complemented
14#1592C211yesmaximal, characteristic, complemented
15#1581C211yesmaximal, characteristic, complemented
16#555C2211yescharacteristic
17#554C2211yescharacteristic
18D4×Q8C2211yescharacteristic, complemented
19D42C2211yescharacteristic
20#501—21no—
21#498C2211yescharacteristic, complemented
22#494C2211yescharacteristic, complemented
23#487C2211yescharacteristic, complemented
24#486C2211yescharacteristic
25#486C2211yescharacteristic
26#486C2211yescharacteristic
27#485—21no—
28#482—21no—
29#481—21no—
30#478—21no—
31#477C2211yescharacteristic, complemented
32#476—21no—
33#476C2211yescharacteristic
34#476C2211yescharacteristic
35#476C2211yescharacteristic, complemented
36#474C2211yescharacteristic
37#469—21no—
38#468—21no—
39#465—21no—
40#461C2211yescharacteristic
41#460C2211yescharacteristic
42#459C2211yescharacteristic
43#450—21no—
44#448—21no—
45#447—21no—
46#446C2211yescharacteristic, complemented
47#442—21no—
48C4×SD16C2211yescharacteristic
49C4×D8C2211yescharacteristic
50#436C2211yescharacteristic
51#436C2211yescharacteristic, complemented
52C8×D4C2211yescharacteristic
53#419—21no—
54#418—21no—
55C2×#102—21no—
56#335C2211yescharacteristic, complemented
57#335C2211yescharacteristic, complemented
58#333C2211yescharacteristic, complemented
59#333C2211yescharacteristic, complemented
60#332C2211yescharacteristic, complemented
61#331C2211yescharacteristic
62Q8⋊C8C2211yescharacteristic, complemented
63D4⋊C8C2211yescharacteristic
64D4⋊C8C2211yescharacteristic
65D4⋊C8C2211yescharacteristic, complemented
66#322C2211yescharacteristic, complemented
67C22×Q8—21no—
68C22×D4—21no—
69C22×D4—21no—
70C2×Q16—41no—
71C2×SD16—21no—
72C2×SD16—21no—
73C2×SD16D411yescharacteristic
74C2×D8—21no—
75C2×D8D411yescharacteristic
76C2×M16—21no—
77C2×M16—21no—
78C22×C8—21no—
79C4⋊Q8—21no—
80C4⋊Q8C2311yescharacteristic
81C4⋊Q8C2311yescharacteristic
82C4⋊Q8C2311yescharacteristic
83#127C2311yescharacteristic
84#125—21no—
85#124—41no—
86C22⋊Q8—21no—
87C22⋊Q8—21no—
88C22⋊Q8—41no—
89C22⋊Q8—41no—
90#121—21no—
91#121—21no—
92#121—21no—
93#121—21no—
94#120—41no—
95C4×Q8C2311yescharacteristic
96C4×D4—21no—
97C4×D4—21no—
98C4×D4C2311yescharacteristic
99C4×D4C2311yescharacteristic
100C4×D4C2311yescharacteristic
101#117—21no—
102#117—21no—
103C2×C4⋊C4—21no—
104#107—21no—
105#107—21no—
106#107D411yescharacteristic
107#106—21no—
108#106—21no—
109#106—21no—
110#106—21no—
111#106—21no—
112#106D411yescharacteristic
113C4⋊C8—21no—
114C4⋊C8C2311yescharacteristic
115C4⋊C8C2311yescharacteristic
116C4⋊C8C2311yescharacteristic
117C4⋊C8C2311yescharacteristic
118#103—21no—
119#103—21no—
120#103—21no—
121#103—21no—
122#103D411yescharacteristic
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—41no—
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—
145C2×Q8—21no—
146C2×Q8—21no—
147C2×Q8—21no—
148C2×Q8—41no—
149C2×Q8—42no—
150C2×Q8—81no—
151C2×Q8C2×D411yescharacteristic
152C2×D4—21no—
153C2×D4—21no—
154C2×D4—21no—
155C2×D4—41no—
156C2×D4—41no—
157C2×D4—42no—
158C2×D4—42no—
159C2×D4—41no—
160C2×D4—81no—
161C2×D4—81no—
162C2×D4C2×D411yescharacteristic
163C2×D4C2×D411yescharacteristic
164C2×D4C2×D411yescharacteristic
165C22×C4—21no—
166C22×C4—21no—
167C22×C4—21no—
168C22×C4—41no—
169C22×C4—41no—
170Q16—82no—
171SD16—41no—
172SD16—81no—
173SD16—81no—
174D8—22no—
175D8—81no—
176M16—81no—
177M16—81no—
178C2×C8—21no—
179C2×C8—21no—
180C2×C8—21no—
181C2×C8—21no—
182C2×C8—21no—
183C2×C8—81no—
184C2×C8C2×D411yescharacteristic
185C2×C8C2×D411yescharacteristic
186C2×C8C2×D411yescharacteristic
187C2×C8C2×D411yescharacteristic
188C4⋊C4—21no—
189C4⋊C4—21no—
190C4⋊C4—21no—
191C4⋊C4—21no—
192C4⋊C4—21no—
193C4⋊C4—21no—
194C4⋊C4—21no—
195C4⋊C4—41no—
196C4⋊C4—41no—
197C4⋊C4—41no—
198C4⋊C4C2×D411yescharacteristic
199C4⋊C4C2×D411yescharacteristic
200C4⋊C4C2×D411yescharacteristic
201C4⋊C4C2×D411yescharacteristic
202C22⋊C4—21no—
203C22⋊C4—21no—
204C22⋊C4—21no—
205C22⋊C4—41no—
206C22⋊C4—41no—
207C22⋊C4—41no—
208C22⋊C4—41no—
209C22⋊C4—41no—
210C42—21no—
211C42—41no—
212C42C2411yescharacteristic
213C23—21no—
214C23—21no—
215C23—21no—
216C23—41no—
217C23—82no—
218C23—82no—
219C23—161no—
220Q8—21no—
221Q8—21no—
222Q8—41no—
223Q8—41no—
224Q8—81no—
225Q8—81no—
226Q8—82no—
227D4—21no—
228D4—21no—
229D4—21no—
230D4—21no—
231D4—21no—
232D4—41no—
233D4—41no—
234D4—41no—
235D4—41no—
236D4—81no—
237D4—82no—
238D4—82no—
239D4—161no—
240D4#13712yes—
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—21no—
244C2×C4—21no—
245C2×C4—21no—
246C2×C4—41no—
247C2×C4—41no—
248C2×C4—41no—
249C2×C4—41no—
250C2×C4—41no—
251C2×C4—41no—
252C2×C4—41no—
253C2×C4—41no—
254C2×C4—41no—
255C2×C4—82no—
256C2×C4—161no—
257C2×C4C22×D411yescharacteristic
258C2×C4C22×D411yescharacteristic
259C2×C4H(5,𝔽2)11yescharacteristic
260C8—21no—
261C8—21no—
262C8—21no—
263C8—21no—
264C8—41no—
265C8—41no—
266C8—41no—
267C8—41no—
268C8—41no—
269C22—22no—
270C22—41no—
271C22—41no—
272C22—41no—
273C22—41no—
274C22—41no—
275C22—41no—
276C22—41no—
277C22—41no—
278C22—81no—
279C22—81no—
280C22—81no—
281C22—164no—
282C22D4211yescharacteristic
283C4—21no—
284C4—41no—
285C4—41no—
286C4—41no—
287C4—41no—
288C4—41no—
289C4—81no—
290C4—81no—
291C4—81no—
292C4C2×#13711yescharacteristic
293C4#57511yescharacteristic
294C4#57611yescharacteristic
295C4#57711yescharacteristic
296C2—42no—
297C2—81no—
298C2—81no—
299C2—161no—
300C2#326611yescharacteristic
301C2#327311yescharacteristic
302C2#328811yescharacteristic
subgroupG/H
center, socleC22D42
commutator, Frattini, cosocleC42C24
prime radical#3325C2
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. #3384C2semidirect
2. #3329C2semidirect
4. #3295C2semidirect
5. #1720C2semidirect
6. #1698C2semidirect
7. #1697C2semidirect
8. #1691C2semidirect
9. #1683C2semidirect
10. #1639C2semidirect
11. #1634C2semidirect
12. #1632C2semidirect
13. #1627C2semidirect
14. #1592C2semidirect
15. #1581C2semidirect
18. D4×Q8C22semidirect
21. #498C22semidirect
22. #494C22semidirect
23. #487C22semidirect
31. #477C22semidirect
35. #476C22semidirect
46. #446C22semidirect
51. #436C22semidirect
56. #335C22semidirect
57. #335C22semidirect
58. #333C22semidirect
59. #333C22semidirect
60. #332C22semidirect
62. Q8⋊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
#33842
#33292
#33252
#32952
#17202
#16982
#16972
#16912
#16832
#16392
#16342
#16322
#16272
#15922
#15812