Finite Group Databasebeta

← #46783249

Subgroups — #46783249

Order 33664.

332 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 332 conjugacy classes of subgroups — too many to draw. Only its 79 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
1#30682738C211yesmaximal, characteristic
2#30682406C211yesmaximal, characteristic, complemented
3#30681954C211yesmaximal, characteristic, complemented
4C2×#22554396C211yesmaximal, characteristic, complemented
5#30681911C211yesmaximal, characteristic, complemented
6#30681699C211yesmaximal, characteristic, complemented
7#344×C263C211yesmaximal, characteristic, complemented
8C2×#16862376C2211yescharacteristic
9C2×#16862376C2211yescharacteristic
10#22554503—21no—
11#22554503C412yescomplemented
12#22554498—21no—
13#22554498C412yescomplemented
14#22554492—22no—
15#22554435—21no—
16#22554432—21no—
17(C22⋊D263)⋊C4—22no—
18#22554429C2211yescharacteristic
19#22554427—21no—
20C2×C4×Dic263C2211yescharacteristic
21C42×D263—21no—
22#22554396—22no—
23#22554396—21no—
24#22554396—21no—
25#22554384—22no—
26C2×M16×C263C2211yescharacteristic
27C2×M16×C263C2211yescharacteristic
28#117×C263C2211yescharacteristic
29#16862376—21no—
30#16862376—21no—
31#16862376—21no—
32#16862376—21no—
33#16862376—21no—
34#16862376—21no—
35#16862376C2×C412yes—
36#16862376D412yes—
37C2×Dic526C2×C411yescharacteristic, complemented
38C2×Dic526D411yescharacteristic
39C2×D1052C2×C411yescharacteristic, complemented
40C2×D1052D411yescharacteristic
41C2×C22⋊D263—21no—
42C2×C22⋊D263—21no—
43C22×Dic263—21no—
44C2×C4×D263—22no—
45C2×C4×D263—21no—
46C2×C4×D263—21no—
47Dic1052—22no—
48#16862365—22no—
49#16862365—22no—
50D2104—22no—
51C4⋊Dic263—21no—
52Dic263⋊C4—22no—
53C22⋊Dic263—21no—
54(C2×D263)⋊C4—22no—
55C4×Dic263—22no—
56C4×Dic263—21no—
57C4×Dic263—21no—
58C4×Dic263D411yescharacteristic
59C4×Dic263D411yescharacteristic
60C22×C4×C263C2311yescharacteristic
61M16×C263—21no—
62M16×C263—21no—
63M16×C263—21no—
64M16×C263C2×C412yescomplemented
65C2×C8×C263—21no—
66C2×C8×C263—21no—
67C4⋊C4×C263—21no—
68C22⋊C4×C263D412yes—
69C42×C263—21no—
70C22×D263—21no—
71C22×D263—21no—
72Dic526—21no—
73Dic526—21no—
74Dic526—21no—
75Dic526C22⋊C412yes—
76D1052—21no—
77D1052—21no—
78D1052—21no—
79D1052C22⋊C412yes—
80C22⋊D263—22no—
81C22⋊D263—22no—
82C22⋊D263—41no—
83C22⋊D263—41no—
84C2×Dic263—21no—
85C2×Dic263—21no—
86C2×Dic263—21no—
87C2×Dic263—21no—
88C2×Dic263—42no—
89C2×Dic263—41no—
90C4×D263—22no—
91C4×D263—22no—
92C4×D263—44no—
93C4×D263—41no—
94C4×D263—41no—
95C23×C263C2×D411yescharacteristic
96C2×C4×C263—22no—
97C2×C4×C263C22×C411yescharacteristic
98C2×C4×C263C2×D411yescharacteristic
99C2×C4×C263C2×D411yescharacteristic
100C2×C4×C263C2×D411yescharacteristic
101C2×C4×C263#4111yescharacteristic
102C2×C4×C263#4111yescharacteristic
103C8×C263—22no—
104C8×C263—41no—
105C22×C263—41no—
106C22×C263C4×D411yescharacteristic
107C22×C263#12011yescharacteristic
108C22×C263#12111yescharacteristic
109C2×D263—22no—
110C2×D263—42no—
111C2×D263—41no—
112C2×D263—81no—
113C4×C263—42no—
114C4×C263C2×C22⋊C411yescharacteristic
115C4×C263C4×D411yescharacteristic
116C4×C263#12111yescharacteristic
117C4×C263#12311yescharacteristic
118Dic263—22no—
119Dic263—22no—
120Dic263—41no—
121Dic263—41no—
122Dic263—81no—
123C2×C263—21no—
124C2×C263—21no—
125C2×C263—21no—
126C2×C263#38611yescharacteristic
127D263—42no—
128D263—81no—
129C263#190611yescharacteristic, complemented, Sylow
130#1906—2631nomaximal, Sylow
131#575—2631no—
132C4×#41—2631no—
133#421—2631no—
134C2×#104—2631no—
135#410—2631no—
136#344D26311yescharacteristic, complemented
137#339—2631no—
138C2×#41—2631no—
139C2×#41—2631no—
140#137—2632no—
141#137—5261no—
142AGL(1,ℤ/8)—2632no—
143AGL(1,ℤ/8)—5261no—
144#135—5262no—
145C2×M16C2×D26311yescharacteristic
146C2×M16C2×D26311yescharacteristic
147C4×Q8—5261no—
148C4×D4—5261no—
149C4×D4—5262no—
150#117—2631no—
151#117—5261no—
152#117C2×D26311yescharacteristic
153C2×C42—2631no—
154C2×C42—5261no—
155#104—5261no—
156#104—5261no—
157#104—5262no—
158#99—5262no—
159#41—2632no—
160#41—2632no—
161#41—5261no—
162#41—5261no—
163#41—5261no—
164#41—5261no—
165#41—5261no—
166#41—5261no—
167C2×Q8—2631no—
168C2×Q8—2631no—
169C2×D4—2631no—
170C2×D4—2631no—
171C2×D4—5261no—
172C2×D4—5261no—
173C22×C4—5262no—
174C22×C4—5261no—
175C22×C4—5261no—
176C22×C4—5261no—
177C22×C4C22×D26311yescharacteristic
178Q16—5262no—
179SD16—5262no—
180SD16—5262no—
181D8—5262no—
182M16—21no—
183M16—21no—
184M16—21no—
185M16C4×D26312yescomplemented
186C2×C8—21no—
187C2×C8—21no—
188C4⋊C4—21no—
189C4⋊C4—5262no—
190C4⋊C4—5261no—
191C22⋊C4—5262no—
192C22⋊C4—5261no—
193C22⋊C4D105212yes—
194C42—21no—
195C42—2631no—
196C42—2631no—
197C42—5262no—
198C42—5261no—
199C42—5261no—
200C23—5261no—
201C23—5261no—
202C23C2×D105211yescharacteristic
203Q8—2632no—
204Q8—5261no—
205Q8—5261no—
206Q8—5261no—
207D4—2632no—
208D4—5262no—
209D4—5262no—
210D4—5261no—
211D4—5261no—
212D4—5261no—
213D4—10521no—
214D4—10521no—
215C2×C4—22no—
216C2×C4—5261no—
217C2×C4—5261no—
218C2×C4—5262no—
219C2×C4—5262no—
220C2×C4—5261no—
221C2×C4—5261no—
222C2×C4—10524no—
223C2×C4—10522no—
224C2×C4—10521no—
225C2×C4—10521no—
226C2×C4—10521no—
227C2×C4C2×C4×D26311yescharacteristic
228C2×C4D4×D26311yescharacteristic
229C2×C4D4×D26311yescharacteristic
230C2×C4D4×D26311yescharacteristic
231C2×C4#1686237611yescharacteristic
232C2×C4#1686237711yescharacteristic
233C8—22no—
234C8—41no—
235C22—41no—
236C22—5262no—
237C22—10522no—
238C22—10521no—
239C22—21041no—
240C22C4×D105211yescharacteristic
241C22#2255444011yescharacteristic
242C22#2255444511yescharacteristic
243C4—42no—
244C4—5262no—
245C4—5262no—
246C4—10521no—
247C4—10521no—
248C4—21041no—
249C4C22⋊C4×D26311yescharacteristic
250C4#2255443211yescharacteristic
251C4#2255444311yescharacteristic
252C4#2255445511yescharacteristic
253C2—21no—
254C2—21no—
255C2—21no—
256C2—10522no—
257C2—21041no—
258C2#3068181611yescharacteristic
subgroupG/H
centerC4#22554443
commutatorC2×C4×C263C22×C4
FrattiniC22×C4C22×D263
Fitting#344×C263C2
socleC2×C263#386
prime radical#30682738C2
cosocleC22×C4×C263C23
hypercenter#344D263
hypocenterC263#1906
LCSG ▷ C2×C4×C263 ▷ C2×C263 ▷ C263
UCSC1 ◁ C4 ◁ C22×C4 ◁ #344

Complements

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

NH ≅ G/Ntype
2. #30682406C2semidirect
3. #30681954C2semidirect
4. C2×#22554396C2semidirect
5. #30681911C2semidirect
6. #30681699C2semidirect
7. #344×C263C2semidirect
11. #22554503C4semidirect
13. #22554498C4semidirect
37. C2×Dic526C2×C4semidirect
39. C2×D1052C2×C4semidirect
64. M16×C263C2×C4semidirect
129. C263#1906semidirect
136. #344D263semidirect
185. M16C4×D263semidirect

Maximal subgroups

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

subgroupindex
#306827382
#306824062
#306819542
C2×#225543962
#306819112
#306816992
#344×C2632
#1906263