Finite Group Databasebeta

← #19956

Subgroups — #19956

Order 256.

294 conjugacy classes of subgroups.

The subgroup lattice has 286 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3360C211yesmaximal, characteristic, complemented
2#3357C211yesmaximal, characteristic, complemented
3#3353C211yesmaximal, characteristic, complemented
4#3303C211yesmaximal, characteristic
5#1710C211yesmaximal, characteristic, complemented
6#1695C211yesmaximal, characteristic, complemented
7#1693C211yesmaximal, characteristic, complemented
8#1684C211yesmaximal, characteristic, complemented
9#1683C211yesmaximal, characteristic, complemented
10#1646C211yesmaximal, characteristic, complemented
11#1643C211yesmaximal, characteristic, complemented
12#1638C211yesmaximal, characteristic, complemented
13#1635C211yesmaximal, characteristic, complemented
14#1589C211yesmaximal, characteristic, complemented
15C8×SD16C211yesmaximal, characteristic, complemented
16#556C2211yescharacteristic
17#555C2211yescharacteristic
18#554C2211yescharacteristic
19#547C2211yescharacteristic
20#500C2211yescharacteristic
21#500C2211yescharacteristic
22#488—21no—
23#487C2211yescharacteristic
24#487C2211yescharacteristic
25#487C2211yescharacteristic
26#487C2211yescharacteristic
27#485—21no—
28#485—21no—
29#484—21no—
30#482—21no—
31#480—21no—
32#479—21no—
33#477C2211yescharacteristic
34#476C2211yescharacteristic
35#476C2211yescharacteristic
36#476C2211yescharacteristic
37#475C2211yescharacteristic
38#474C2211yescharacteristic
39#468—21no—
40#468—21no—
41#465—21no—
42#463—21no—
43#461C2211yescharacteristic
44#461C2211yescharacteristic
45#449—21no—
46C8×Q8C2211yescharacteristic
47C4×SD16—21no—
48C4×SD16C2211yescharacteristic
49C4×D8C2211yescharacteristic
50#436C2211yescharacteristic
51#436C2211yescharacteristic
52C8×D4C2211yescharacteristic
53#428—21no—
54#428—21no—
55#416—21no—
56#335C2211yescharacteristic
57#334C2211yescharacteristic
58#333C2211yescharacteristic
59#333C2211yescharacteristic
60#332C2211yescharacteristic
61#332C2211yescharacteristic
62Q8⋊C8C2211yescharacteristic
63D4⋊C8C2211yescharacteristic
64D4⋊C8C2211yescharacteristic
65D4⋊C8C2211yescharacteristic
66C82C2211yescharacteristic
67C2×#41—21no—
68C2×SD16—21no—
69C2×SD16—21no—
70C2×SD16D411yescharacteristic
71C2×D8D411yescharacteristic
72C2×M16—21no—
73C2×M16—21no—
74C22×C8—21no—
75C4⋊Q8C2311yescharacteristic
76C4⋊Q8C2311yescharacteristic
77C4⋊Q8C2311yescharacteristic
78C4⋊Q8C2311yescharacteristic
79#126—41no—
80#125—21no—
81#124—21no—
82#124—41no—
83#123—41no—
84C22⋊Q8—21no—
85C22⋊Q8—21no—
86C22⋊Q8—21no—
87C22⋊Q8—41no—
88#121—21no—
89#121—21no—
90#121—21no—
91C4×Q8—21no—
92C4×Q8C2311yescharacteristic
93C4×D4—21no—
94C4×D4C2311yescharacteristic
95C4×D4C2311yescharacteristic
96C4×D4C2311yescharacteristic
97#117—21no—
98#117—21no—
99#117—21no—
100C2×C4⋊C4—21no—
101C2×C4⋊C4—21no—
102#107—21no—
103#107—21no—
104#107—21no—
105#107—21no—
106#107—21no—
107#107—21no—
108#107D411yescharacteristic
109#106—21no—
110#106—21no—
111#106—21no—
112#106—21no—
113#106D411yescharacteristic
114C4⋊C8—21no—
115C4⋊C8C2311yescharacteristic
116C4⋊C8C2311yescharacteristic
117C4⋊C8C2311yescharacteristic
118C4⋊C8C2311yescharacteristic
119#103—21no—
120#103—21no—
121#103—21no—
122#103—21no—
123#103—21no—
124#103—21no—
125#103—21no—
126#103—21no—
127#103—41no—
128#103—41no—
129#103D411yescharacteristic
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—
140C4×C8—21no—
141C4×C8C2311yescharacteristic
142C4×C8C2311yescharacteristic
143C4×C8C2311yescharacteristic
144#41—42no—
145#41—81no—
146C2×Q8—21no—
147C2×Q8—21no—
148C2×Q8—21no—
149C2×Q8—21no—
150C2×Q8C2×D411yescharacteristic
151C2×D4—41no—
152C2×D4—41no—
153C2×D4C2×D411yescharacteristic
154C2×D4C2×D411yescharacteristic
155C2×D4C2×D411yescharacteristic
156C22×C4—21no—
157C22×C4—21no—
158C22×C4—21no—
159C22×C4—41no—
160C22×C4—41no—
161SD16—22no—
162SD16—42no—
163SD16—81no—
164D8—41no—
165M16—42no—
166M16—42no—
167C2×C8—21no—
168C2×C8—21no—
169C2×C8—21no—
170C2×C8—21no—
171C2×C8—21no—
172C2×C8—81no—
173C2×C8C2×D411yescharacteristic
174C2×C8C2×D411yescharacteristic
175C2×C8C2×D411yescharacteristic
176C2×C8C2×D411yescharacteristic
177C4⋊C4—21no—
178C4⋊C4—21no—
179C4⋊C4—21no—
180C4⋊C4—21no—
181C4⋊C4—21no—
182C4⋊C4—21no—
183C4⋊C4—21no—
184C4⋊C4—21no—
185C4⋊C4—21no—
186C4⋊C4—41no—
187C4⋊C4—41no—
188C4⋊C4—41no—
189C4⋊C4—41no—
190C4⋊C4C2×D411yescharacteristic
191C4⋊C4C2×D411yescharacteristic
192C4⋊C4C2×D411yescharacteristic
193C4⋊C4C2×D411yescharacteristic
194C22⋊C4—21no—
195C22⋊C4—21no—
196C22⋊C4—21no—
197C22⋊C4—41no—
198C22⋊C4—41no—
199C22⋊C4—41no—
200C22⋊C4—41no—
201C22⋊C4—41no—
202C22⋊C4—41no—
203C42—21no—
204C42—41no—
205C42—41no—
206C42C2411yescharacteristic
207C23—21no—
208C23—21no—
209C23—21no—
210Q8—21no—
211Q8—21no—
212Q8—41no—
213Q8—41no—
214Q8—41no—
215Q8—41no—
216Q8—81no—
217Q8—81no—
218D4—21no—
219D4—21no—
220D4—21no—
221D4—21no—
222D4—21no—
223D4—21no—
224D4—82no—
225D4—161no—
226C2×C4—21no—
227C2×C4—21no—
228C2×C4—21no—
229C2×C4—21no—
230C2×C4—21no—
231C2×C4—41no—
232C2×C4—41no—
233C2×C4—41no—
234C2×C4—41no—
235C2×C4—41no—
236C2×C4—41no—
237C2×C4—41no—
238C2×C4—41no—
239C2×C4—41no—
240C2×C4—41no—
241C2×C4—82no—
242C2×C4—161no—
243C2×C4C22×D411yescharacteristic
244C2×C4C22×D411yescharacteristic
245C2×C4H(5,𝔽2)11yescharacteristic
246C8—21no—
247C8—21no—
248C8—21no—
249C8—41no—
250C8—41no—
251C8—41no—
252C8—41no—
253C8—41no—
254C8#13512yes—
255C22—41no—
256C22—41no—
257C22—41no—
258C22—41no—
259C22—41no—
260C22—41no—
261C22—41no—
262C22—41no—
263C22—41no—
264C22D4211yescharacteristic
265C4—21no—
266C4—41no—
267C4—41no—
268C4—41no—
269C4—41no—
270C4—41no—
271C4—81no—
272C4—81no—
273C4—81no—
274C4—81no—
275C4C2×#13511yescharacteristic
276C4#57511yescharacteristic
277C4#57711yescharacteristic
278C4#57811yescharacteristic
279C2—81no—
280C2—81no—
281C2—81no—
282C2#327511yescharacteristic
283C2#328311yescharacteristic
284C2#329011yescharacteristic
subgroupG/H
center, socleC22D42
commutator, Frattini, cosocleC42C24
prime radical#3303C2
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. #3360C2semidirect
2. #3357C2semidirect
3. #3353C2semidirect
5. #1710C2semidirect
6. #1695C2semidirect
7. #1693C2semidirect
8. #1684C2semidirect
9. #1683C2semidirect
10. #1646C2semidirect
11. #1643C2semidirect
12. #1638C2semidirect
13. #1635C2semidirect
14. #1589C2semidirect
15. C8×SD16C2semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#33602
#33572
#33532
#33032
#17102
#16952
#16932
#16842
#16832
#16462
#16432
#16382
#16352
#15892
C8×SD162