Finite Group Databasebeta

← #46783766

Subgroups — #46783766

Order 33664.

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

normal subgroups only

This group has 382 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
1#30682509C211yesmaximal, characteristic, complemented
2C2×#22554457C211yesmaximal, characteristic, complemented
3C2×#22554397C211yesmaximal, characteristic, complemented
4#30681943C211yesmaximal, characteristic, complemented
5C2×#22554388C211yesmaximal, characteristic, complemented
6#30681699C211yesmaximal, characteristic, complemented
7C2×AGL(1,ℤ/8)×C263C211yesmaximal, characteristic
8C22×Dic526C2211yescharacteristic
9C2×#16862360C2211yescharacteristic
10#22554468—21no—
11#22554457—22no—
12#22554457—22no—
13#22554457—21no—
14#22554451—22no—
15#22554449—22no—
16#22554429C2211yescharacteristic
17C2×C22⋊Dic263—22no—
18C2×C4×Dic263C2211yescharacteristic
19#22554397—22no—
20#22554391—22no—
21#22554388—22no—
22C2×#41×C263C2211yescharacteristic
23C22×D4×C263C2211yescharacteristic
24AGL(1,ℤ/8)×C263—22no—
25AGL(1,ℤ/8)×C263—22no—
26C2×SD16×C263—21no—
27C2×D8×C263—21no—
28C2×M16×C263C2211yescharacteristic
29C2×Dic526—21no—
30C2×Dic526—21no—
31C2×Dic526—21no—
32C2×Dic526—21no—
33C2×Dic526—21no—
34C2×Dic526—21no—
35C22×Dic263—21no—
36C22×Dic263D412yes—
37#16862360—21no—
38#16862360—21no—
39C2×C263⋊C8—21no—
40C4⋊Dic263D412yes—
41Dic263⋊C4—24no—
42C22⋊Dic263—28no—
43C22⋊Dic263—24no—
44C22⋊Dic263—21no—
45C4×Dic263—21no—
46C4×Dic263—21no—
47C4×Dic263D412yes—
48C24×C263—21no—
49#41×C263—21no—
50#41×C263—21no—
51#41×C263—21no—
52#41×C263—21no—
53C2×Q8×C263D411yescharacteristic
54C2×D4×C263—21no—
55C2×D4×C263—21no—
56C2×D4×C263—21no—
57C2×D4×C263—21no—
58C2×D4×C263—21no—
59C2×D4×C263—21no—
60C2×D4×C263D412yes—
61C2×D4×C263D411yescharacteristic
62C22×C4×C263—21no—
63C22×C4×C263C2311yescharacteristic
64SD16×C263—24no—
65D8×C263—24no—
66M16×C263—21no—
67M16×C263D412yes—
68C2×C8×C263—21no—
69Dic526—41no—
70Dic526—41no—
71Dic526—41no—
72Dic526—41no—
73C2×Dic263—22no—
74C2×Dic263—22no—
75C2×Dic263—22no—
76C2×Dic263—22no—
77C2×Dic263—22no—
78C2×Dic263—42no—
79C263⋊C8—41no—
80C23×C263—21no—
81C23×C263—22no—
82C23×C263—44no—
83C23×C263—41no—
84C23×C263—41no—
85C23×C263C2×D411yescharacteristic
86Q8×C263—21no—
87Q8×C263—21no—
88D4×C263—21no—
89D4×C263—22no—
90D4×C263—22no—
91D4×C263—21no—
92D4×C263—41no—
93D4×C263—41no—
94D4×C263—41no—
95D4×C263—41no—
96C2×C4×C263—21no—
97C2×C4×C263—41no—
98C2×C4×C263—41no—
99C2×C4×C263C2×D411yescharacteristic
100C2×C4×C263C2×D411yescharacteristic
101C2×C4×C263C2×D411yescharacteristic
102C2×C4×C263C2×D411yescharacteristic
103C2×C4×C263C2×D411yescharacteristic
104C2×C4×C263#4111yescharacteristic
105C8×C263—22no—
106C22×C263—22no—
107C22×C263—44no—
108C22×C263—41no—
109C22×C263—42no—
110C22×C263—42no—
111C22×C263—82no—
112C22×C263—82no—
113C22×C263#12011yescharacteristic
114C22×C263#12011yescharacteristic
115C22×C263#12111yescharacteristic
116C4×C263—41no—
117C4×C263#12011yescharacteristic
118C4×C263#12111yescharacteristic
119C4×C263#12111yescharacteristic
120C4×C263#12711yescharacteristic
121Dic263—42no—
122Dic263—42no—
123Dic263—42no—
124C2×C263—21no—
125C2×C263—21no—
126C2×C263—81no—
127C2×C263—82no—
128C2×C263#39211yescharacteristic
129C2×C263#45412yes—
130C263#200311yescharacteristic, complemented, Sylow
131#2003—2631nomaximal, Sylow
132C2×AGL(1,ℤ/8)D26311yescharacteristic
133#536—2631no—
134C2×#124—2631no—
135C2×#104—2631no—
136#418—2631no—
137C2×#101—2631no—
138#339—2631no—
139C2×#41C2×D26311yescharacteristic
140C22×Q8—2631no—
141C22×D4C2×D26311yescharacteristic
142AGL(1,ℤ/8)—22no—
143AGL(1,ℤ/8)—22no—
144C2×SD16—21no—
145C2×D8—21no—
146C2×M16—2631no—
147C2×M16C2×D26311yescharacteristic
148C4⋊Q8—5261no—
149#124—5262no—
150#124—5262no—
151#124—5261no—
152#123—5262no—
153C22⋊Q8—5262no—
154#117—2631no—
155C2×C22⋊C4—5262no—
156C2×C42—2631no—
157#104—5262no—
158#102—5262no—
159#101—5262no—
160C24—21no—
161#41—21no—
162#41—21no—
163#41—21no—
164#41—21no—
165C2×Q8—5261no—
166C2×Q8—5261no—
167C2×Q8—5261no—
168C2×Q8—5261no—
169C2×Q8—5261no—
170C2×Q8—5261no—
171C2×Q8C22⋊D26311yescharacteristic
172C2×D4—21no—
173C2×D4—21no—
174C2×D4—21no—
175C2×D4—21no—
176C2×D4—21no—
177C2×D4—21no—
178C2×D4C22⋊D26311yescharacteristic
179C2×D4C22⋊D26312yes—
180C22×C4—21no—
181C22×C4—2632no—
182C22×C4—5261no—
183C22×C4C22×D26311yescharacteristic
184SD16—24no—
185D8—24no—
186M16—21no—
187M16—5261no—
188M16—5261no—
189M16C22⋊D26312yes—
190C2×C8—21no—
191C2×C8—5261no—
192C4⋊C4—2632no—
193C4⋊C4—5264no—
194C22⋊C4—5268no—
195C22⋊C4—5264no—
196C22⋊C4—5261no—
197C42—2632no—
198C42—5261no—
199C42—5261no—
200C23—22no—
201C23—21no—
202C23—44no—
203C23—41no—
204C23—41no—
205C23D4×D26311yescharacteristic
206Q8—21no—
207Q8—21no—
208Q8—10521no—
209Q8—10521no—
210Q8—10521no—
211Q8—10521no—
212D4—21no—
213D4—22no—
214D4—22no—
215D4—21no—
216D4—41no—
217D4—41no—
218D4—41no—
219D4—41no—
220C2×C4—21no—
221C2×C4—41no—
222C2×C4—41no—
223C2×C4—5262no—
224C2×C4—5262no—
225C2×C4—5262no—
226C2×C4—5262no—
227C2×C4—5262no—
228C2×C4—10522no—
229C2×C4D4×D26311yescharacteristic
230C2×C4D4×D26311yescharacteristic
231C2×C4C2×C22⋊D26311yescharacteristic
232C2×C4C2×C22⋊D26311yescharacteristic
233C2×C4C2×C22⋊D26311yescharacteristic
234C2×C4#1686237511yescharacteristic
235C8—22no—
236C8—10521no—
237C22—22no—
238C22—44no—
239C22—41no—
240C22—42no—
241C22—42no—
242C22—82no—
243C22—82no—
244C22#2255443811yescharacteristic
245C22#2255443811yescharacteristic
246C22#2255444111yescharacteristic
247C4—41no—
248C4—10522no—
249C4—10522no—
250C4—10522no—
251C4#2255443911yescharacteristic
252C4#2255444211yescharacteristic
253C4#2255444211yescharacteristic
254C4#2255446611yescharacteristic
255C2—21no—
256C2—21no—
257C2—82no—
258C2—81no—
259C2#3068184611yescharacteristic
260C2#3068210912yes—
subgroupG/H
centerC22#22554438
commutator, cosocleC22×C4×C263C23
FrattiniC22×C4C22×D263
Fitting, prime radicalC2×AGL(1,ℤ/8)×C263C2
socleC22×C263#120
hypercenterC2×AGL(1,ℤ/8)D263
hypocenterC263#2003
LCSG ▷ C22×C4×C263 ▷ C2×C263 ▷ C263
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. #30682509C2semidirect
2. C2×#22554457C2semidirect
3. C2×#22554397C2semidirect
4. #30681943C2semidirect
5. C2×#22554388C2semidirect
6. #30681699C2semidirect
130. C263#2003semidirect

Maximal subgroups

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

subgroupindex
#306825092
C2×#225544572
C2×#225543972
#306819432
C2×#225543882
#306816992
C2×AGL(1,ℤ/8)×C2632
#2003263