Finite Group Databasebeta

← #19853

Subgroups — #19853

Order 256.

294 conjugacy classes of subgroups.

The subgroup lattice has 283 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3398C211yesmaximal, characteristic, complemented
2#3381C211yesmaximal, characteristic, complemented
3#3335C211yesmaximal, characteristic
4#3302C211yesmaximal, characteristic
5#1716C211yesmaximal, characteristic, complemented
6#1692C211yesmaximal, characteristic, complemented
7#1680C211yesmaximal, characteristic, complemented
8#1671C211yesmaximal, characteristic, complemented
9#1669C211yesmaximal, characteristic, complemented
10#1654C211yesmaximal, characteristic
11#1653C211yesmaximal, characteristic, complemented
12#1644C211yesmaximal, characteristic, complemented
13#1638C211yesmaximal, characteristic, complemented
14#1582C211yesmaximal, characteristic, complemented
15SD16⋊C8C211yesmaximal, characteristic, complemented
16#557C2211yescharacteristic
17#556C2211yescharacteristic
18#554C2211yescharacteristic
19#547C2211yescharacteristic
20#499—21no—
21#498C2211yescharacteristic
22#498C2211yescharacteristic
23#498C2211yescharacteristic
24#497—21no—
25#494C2211yescharacteristic
26#487C2211yescharacteristic
27#487C2211yescharacteristic
28#485—21no—
29#483—21no—
30#481—21no—
31#479—21no—
32#478—21no—
33#477C2211yescharacteristic
34#476C2211yescharacteristic
35#475C2211yescharacteristic
36#474C2211yescharacteristic
37#474C2211yescharacteristic
38#467—21no—
39#465—21no—
40#464—21no—
41#462—21no—
42#462C2211yescharacteristic
43#462C2211yescharacteristic
44#461C2211yescharacteristic
45#452—21no—
46#446C2211yescharacteristic
47C8×Q8C2211yescharacteristic
48#441—21no—
49C4×SD16—21no—
50C4×SD16C2211yescharacteristic
51C4×SD16C2211yescharacteristic
52C8×D4C2211yescharacteristic
53C8×D4C2211yescharacteristic
54C2×#106—21no—
55#416—21no—
56#335C2211yescharacteristic
57#334C2211yescharacteristic
58#333C2211yescharacteristic
59#332C2211yescharacteristic
60#332C2211yescharacteristic
61#332C2211yescharacteristic
62Q8⋊C8C2211yescharacteristic
63Q8⋊C8C2211yescharacteristic
64D4⋊C8C2211yescharacteristic
65D4⋊C8C2211yescharacteristic
66#322C2211yescharacteristic
67C2×#41—21no—
68C2×Q16—21no—
69C2×Q16—21no—
70C2×Q16—21no—
71C2×Q16—21no—
72C2×SD16—21no—
73C2×SD16D411yescharacteristic
74C2×SD16D411yescharacteristic
75C22×C8—21no—
76C22×C8—21no—
77C4⋊Q8—21no—
78C4⋊Q8C2311yescharacteristic
79C4⋊Q8C2311yescharacteristic
80C4⋊Q8C2311yescharacteristic
81C4⋊Q8C2311yescharacteristic
82#126—41no—
83#125—21no—
84#125—21no—
85#125—41no—
86#124—21no—
87#123—41no—
88C22⋊Q8—21no—
89C22⋊Q8—21no—
90C22⋊Q8—21no—
91C22⋊Q8—41no—
92#121—21no—
93C4×Q8—21no—
94C4×Q8—21no—
95C4×Q8—21no—
96C4×Q8C2311yescharacteristic
97C4×Q8C2311yescharacteristic
98C4×D4—21no—
99C4×D4C2311yescharacteristic
100C4×D4C2311yescharacteristic
101#117—21no—
102C2×C4⋊C4—21no—
103C2×C4⋊C4—21no—
104#107—21no—
105#107—21no—
106#107—21no—
107#106—21no—
108#106—21no—
109#106—21no—
110#106—21no—
111#106—21no—
112#106—21no—
113#106—41no—
114#106D411yescharacteristic
115#106D411yescharacteristic
116C4⋊C8—21no—
117C4⋊C8—21no—
118C4⋊C8C2311yescharacteristic
119C4⋊C8C2311yescharacteristic
120C4⋊C8C2311yescharacteristic
121C4⋊C8C2311yescharacteristic
122#103—21no—
123#103—21no—
124#103—21no—
125#103—21no—
126#103—21no—
127#103—21no—
128#103—21no—
129#103—41no—
130#103D411yescharacteristic
131#103D411yescharacteristic
132#102—21no—
133#102—21no—
134#102—21no—
135#102D411yescharacteristic
136#102D411yescharacteristic
137C22⋊C8—21no—
138C22⋊C8—21no—
139#97—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×Q8C2×D411yescharacteristic
152C2×D4—41no—
153C2×D4C2×D411yescharacteristic
154C2×D4C2×D411yescharacteristic
155C22×C4—21no—
156C22×C4—21no—
157C22×C4—41no—
158C22×C4—41no—
159Q16—42no—
160Q16—81no—
161Q16—81no—
162Q16—81no—
163SD16—22no—
164SD16—22no—
165SD16—81no—
166C2×C8—21no—
167C2×C8—21no—
168C2×C8—21no—
169C2×C8—21no—
170C2×C8—21no—
171C2×C8—42no—
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—21no—
187C4⋊C4—41no—
188C4⋊C4—41no—
189C4⋊C4—41no—
190C4⋊C4—41no—
191C4⋊C4—41no—
192C4⋊C4—41no—
193C4⋊C4—41no—
194C4⋊C4C2×D411yescharacteristic
195C4⋊C4C2×D411yescharacteristic
196C4⋊C4C2×D411yescharacteristic
197C4⋊C4C2×D411yescharacteristic
198C22⋊C4—21no—
199C22⋊C4—21no—
200C22⋊C4—41no—
201C22⋊C4—41no—
202C22⋊C4—41no—
203C22⋊C4—41no—
204C42—21no—
205C42—21no—
206C42—41no—
207C42—41no—
208C42C2411yescharacteristic
209C23—21no—
210C23—21no—
211Q8—21no—
212Q8—21no—
213Q8—21no—
214Q8—21no—
215Q8—41no—
216Q8—41no—
217Q8—41no—
218Q8—41no—
219Q8—41no—
220Q8—41no—
221Q8—81no—
222D4—21no—
223D4—21no—
224D4—21no—
225D4—82no—
226D4#13712yes—
227C2×C4—21no—
228C2×C4—21no—
229C2×C4—21no—
230C2×C4—21no—
231C2×C4—21no—
232C2×C4—21no—
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—22no—
256C22—41no—
257C22—41no—
258C22—41no—
259C22—41no—
260C22—41no—
261C22D4211yescharacteristic
262C4—21no—
263C4—41no—
264C4—41no—
265C4—41no—
266C4—41no—
267C4—41no—
268C4—41no—
269C4—81no—
270C4—81no—
271C4—81no—
272C4—81no—
273C4C2×#13511yescharacteristic
274C4C2×#13711yescharacteristic
275C4#57711yescharacteristic
276C4#57711yescharacteristic
277C2—42no—
278C2—81no—
279C2#326611yescharacteristic
280C2#328411yescharacteristic
281C2#328911yescharacteristic
subgroupG/H
center, socleC22D42
commutator, Frattini, cosocleC42C24
prime radical#547C22
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. #3398C2semidirect
2. #3381C2semidirect
5. #1716C2semidirect
6. #1692C2semidirect
7. #1680C2semidirect
8. #1671C2semidirect
9. #1669C2semidirect
11. #1653C2semidirect
12. #1644C2semidirect
13. #1638C2semidirect
14. #1582C2semidirect
15. SD16⋊C8C2semidirect

Maximal subgroups

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

subgroupindex
#33982
#33812
#33352
#33022
#17162
#16922
#16802
#16712
#16692
#16542
#16532
#16442
#16382
#15822
SD16⋊C82