Finite Group Databasebeta

← #46783779

Subgroups — #46783779

Order 33664.

394 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 394 conjugacy classes of subgroups — too many to draw. Only its 75 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#22554503C211yesmaximal, characteristic, complemented
2#30682507C211yesmaximal, characteristic
3C2×#22554458C211yesmaximal, characteristic, complemented
4C2×#22554398C211yesmaximal, characteristic, complemented
5#30681946C211yesmaximal, characteristic, complemented
6C2×#22554386C211yesmaximal, characteristic, complemented
7#339×C263C211yesmaximal, characteristic, complemented
8C2×#16862376C2211yescharacteristic
9C22×Dic526C2211yescharacteristic
10C22×D1052C2211yescharacteristic
11#22554503—22no—
12#22554503—22no—
13C2×Dic1052—21no—
14C2×#16862365—21no—
15C2×#16862360C2211yescharacteristic
16#22554467—21no—
17#22554458—21no—
18#22554458—22no—
19#22554458—22no—
20#22554445—22no—
21#22554440—22no—
22C2×(C2×D263)⋊C4—22no—
23#22554398—22no—
24#22554393—22no—
25#22554386—22no—
26C2×M16×C263C2211yescharacteristic
27#117×C263C2211yescharacteristic
28C2×C42×C263C2211yescharacteristic
29C23×D263D412yes—
30#16862376—21no—
31#16862376—21no—
32#16862376—21no—
33#16862376—21no—
34C2×Dic526—21no—
35C2×Dic526—21no—
36C2×Dic526—21no—
37C2×Dic526—21no—
38C2×Dic526—21no—
39C2×Dic526D411yescharacteristic
40C2×Dic526D412yescomplemented
41C2×D1052—24no—
42C2×D1052—21no—
43C2×D1052—21no—
44C2×D1052—21no—
45C2×D1052—21no—
46C2×D1052—21no—
47C2×D1052—21no—
48C2×D1052D411yescharacteristic
49C2×C22⋊D263—21no—
50C22×Dic263—21no—
51C2×C4×D263—21no—
52Dic1052—24no—
53#16862365—24no—
54#16862360—21no—
55#16862360—21no—
56C2×C263⋊C8—21no—
57(C2×D263)⋊C4—28no—
58(C2×D263)⋊C4—24no—
59C22×C4×C263—21no—
60C22×C4×C263C2311yescharacteristic
61M16×C263—21no—
62M16×C263D412yescomplemented
63C2×C8×C263—21no—
64C4⋊C4×C263D412yes—
65C22⋊C4×C263—21no—
66C42×C263—21no—
67C42×C263—21no—
68C42×C263D412yescomplemented
69C22×D263—22no—
70C22×D263—22no—
71C22×D263—21no—
72C22×D263—22no—
73C22×D263—44no—
74Dic526—22no—
75Dic526—21no—
76Dic526—22no—
77Dic526—21no—
78Dic526—41no—
79Dic526—41no—
80D1052—21no—
81D1052—21no—
82D1052—41no—
83D1052—41no—
84D1052—41no—
85D1052—41no—
86D1052—84no—
87C22⋊D263—41no—
88C22⋊D263—41no—
89C2×Dic263—21no—
90C2×Dic263—22no—
91C2×Dic263—41no—
92C2×Dic263—41no—
93C4×D263—41no—
94C4×D263—41no—
95C263⋊C8—41no—
96C23×C263C2×D411yescharacteristic
97C2×C4×C263—22no—
98C2×C4×C263—22no—
99C2×C4×C263—42no—
100C2×C4×C263C2×D411yescharacteristic
101C2×C4×C263C2×D411yescharacteristic
102C2×C4×C263C2×D411yescharacteristic
103C2×C4×C263C2×D411yescharacteristic
104C2×C4×C263C2×D411yescharacteristic
105C2×C4×C263#4111yescharacteristic
106C8×C263—22no—
107C22×C263—22no—
108C22×C263#12011yescharacteristic
109C22×C263#12011yescharacteristic
110C22×C263#12111yescharacteristic
111C2×D263—44no—
112C2×D263—41no—
113C2×D263—42no—
114C2×D263—42no—
115C2×D263—44no—
116C2×D263—82no—
117C4×C263—42no—
118C4×C263—42no—
119C4×C263#12011yescharacteristic
120C4×C263#12111yescharacteristic
121C4×C263#12111yescharacteristic
122C4×C263#12711yescharacteristic
123Dic263—41no—
124Dic263—42no—
125C2×C263—21no—
126C2×C263—21no—
127C2×C263#39211yescharacteristic
128C2×C263#45512yes—
129D263—81no—
130D263—82no—
131C263#200411yescharacteristic, complemented, Sylow
132#2004—2631nomaximal, Sylow
133C2×#137—2631no—
134#535—2631no—
135C2×#124—2631no—
136C2×#104—2631no—
137#419—2631no—
138C2×#100—2631no—
139#339D26311yescharacteristic, complemented
140C2×#41—2631no—
141C22×Q8—2631no—
142C22×D4—2631no—
143#137—5262no—
144#137—5262no—
145C2×Q16—5261no—
146C2×SD16—5261no—
147C2×M16—2631no—
148C2×M16C2×D26311yescharacteristic
149#127—5261no—
150#124—5262no—
151#124—5261no—
152#124—5262no—
153#121—5262no—
154#120—5262no—
155#117C2×D26311yescharacteristic
156C2×C22⋊C4—5262no—
157C2×C42C2×D26311yescharacteristic
158#104—5262no—
159#103—5262no—
160#100—5262no—
161C24—2632no—
162#41—5261no—
163#41—5261no—
164#41—5261no—
165#41—5261no—
166C2×Q8—2631no—
167C2×Q8—2632no—
168C2×Q8—5261no—
169C2×Q8—5261no—
170C2×Q8—5261no—
171C2×Q8—5261no—
172C2×Q8—5261no—
173C2×D4—2631no—
174C2×D4—5264no—
175C2×D4—5261no—
176C2×D4—5261no—
177C2×D4—5261no—
178C2×D4—5261no—
179C2×D4—5261no—
180C2×D4—5261no—
181C2×D4—5261no—
182C22×C4—21no—
183C22×C4—5261no—
184C22×C4—5261no—
185C22×C4C22×D26311yescharacteristic
186Q16—5264no—
187SD16—5264no—
188M16—21no—
189M16—5261no—
190M16—5261no—
191M16D105212yescomplemented
192C2×C8—21no—
193C2×C8—5261no—
194C4⋊C4D105212yes—
195C22⋊C4—21no—
196C22⋊C4—5268no—
197C22⋊C4—5264no—
198C42—21no—
199C42—21no—
200C42D105212yescomplemented
201C23—5262no—
202C23—5262no—
203C23—5261no—
204C23—5262no—
205C23—10524no—
206C23D4×D26311yescharacteristic
207Q8—5262no—
208Q8—5261no—
209Q8—5262no—
210Q8—5261no—
211Q8—10521no—
212Q8—10521no—
213D4—5261no—
214D4—5261no—
215D4—10521no—
216D4—10521no—
217D4—10521no—
218D4—10521no—
219D4—10521no—
220D4—10521no—
221D4—21044no—
222C2×C4—22no—
223C2×C4—22no—
224C2×C4—42no—
225C2×C4—5262no—
226C2×C4—5261no—
227C2×C4—10521no—
228C2×C4—10521no—
229C2×C4—10521no—
230C2×C4—10521no—
231C2×C4D4×D26311yescharacteristic
232C2×C4D4×D26311yescharacteristic
233C2×C4C2×D105211yescharacteristic
234C2×C4C2×D105211yescharacteristic
235C2×C4C2×D105211yescharacteristic
236C2×C4#1686237711yescharacteristic
237C8—22no—
238C8—10521no—
239C22—22no—
240C22—10524no—
241C22—10522no—
242C22—10524no—
243C22—10521no—
244C22—10522no—
245C22—21042no—
246C22#2255444011yescharacteristic
247C22#2255444011yescharacteristic
248C22#2255444511yescharacteristic
249C4—42no—
250C4—42no—
251C4—10521no—
252C4—10522no—
253C4#2255444011yescharacteristic
254C4#2255444511yescharacteristic
255C4#2255444511yescharacteristic
256C4#2255446711yescharacteristic
257C2—21no—
258C2—21no—
259C2—21042no—
260C2—21041no—
261C2#3068184811yescharacteristic
262C2#3068211712yes—
subgroupG/H
centerC22#22554440
commutator, cosocleC22×C4×C263C23
FrattiniC22×C4C22×D263
Fitting#339×C263C2
socleC22×C263#120
prime radical#30682507C2
hypercenter#339D263
hypocenterC263#2004
LCSG ▷ C22×C4×C263 ▷ C2×C263 ▷ C263
UCSC1 ◁ C22 ◁ C22×C4 ◁ #339

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. C2×#22554503C2semidirect
3. C2×#22554458C2semidirect
4. C2×#22554398C2semidirect
5. #30681946C2semidirect
6. C2×#22554386C2semidirect
7. #339×C263C2semidirect
40. C2×Dic526D4semidirect
62. M16×C263D4semidirect
68. C42×C263D4semidirect
131. C263#2004semidirect
139. #339D263semidirect
191. M16D1052semidirect
200. C42D1052semidirect

Maximal subgroups

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

subgroupindex
C2×#225545032
#306825072
C2×#225544582
C2×#225543982
#306819462
C2×#225543862
#339×C2632
#2004263