Finite Group Databasebeta

← #46783234

Subgroups — #46783234

Order 33664.

346 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 346 conjugacy classes of subgroups — too many to draw. Only its 89 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×#22554496C211yesmaximal, characteristic
2#30681956C211yesmaximal, characteristic, complemented
3#30681943C211yesmaximal, characteristic, complemented
4#30681911C211yesmaximal, characteristic, complemented
5#30681708C211yesmaximal, characteristic, complemented
6#30681694C211yesmaximal, characteristic, complemented
7#518×C263C211yesmaximal, characteristic, complemented
8C2×#16862376C2211yescharacteristic
9#22554496—22no—
10#22554496—22no—
11C2×#16862364C412yescomplemented
12C2×#16862361C412yescomplemented
13C2×#16862360C2211yescharacteristic
14C2×#16862360C2211yescharacteristic
15#22554429C2211yescharacteristic
16#22554398—22no—
17#22554391—22no—
18#22554384—22no—
19C22×D4×C263C2211yescharacteristic
20C4×D4×C263—22no—
21C4×D4×C263—22no—
22#117×C263C2211yescharacteristic
23#117×C263C2211yescharacteristic
24C2×C22⋊C4×C263—22no—
25#16862376—21no—
26#16862376—21no—
27#16862376—21no—
28#16862376—21no—
29C2×Dic526C2×C411yescharacteristic, complemented
30C2×D1052C2×C411yescharacteristic, complemented
31C2×C22⋊D263—21no—
32C2×C4×D263—21no—
33#16862364—24no—
34#16862361—24no—
35#16862360—21no—
36#16862360—21no—
37#16862360—21no—
38#16862360—21no—
39C2×C263⋊C8—21no—
40C2×C263⋊C8C2×C412yescomplemented
41C4⋊Dic263—21no—
42C22⋊Dic263—21no—
43C4×Dic263—21no—
44C24×C263—21no—
45C2×D4×C263—21no—
46C2×D4×C263—21no—
47C2×D4×C263—21no—
48C2×D4×C263C2×C412yes—
49C2×D4×C263D412yes—
50C2×D4×C263D412yes—
51C22×C4×C263—24no—
52C22×C4×C263C2311yescharacteristic
53C4⋊C4×C263—21no—
54C4⋊C4×C263—21no—
55C22⋊C4×C263—24no—
56C22⋊C4×C263—21no—
57C22⋊C4×C263D412yes—
58C42×C263—21no—
59C42×C263D412yes—
60C22×D263—21no—
61Dic526—21no—
62Dic526—21no—
63D1052—21no—
64D1052—21no—
65C22⋊D263—41no—
66C22⋊D263—41no—
67C2×Dic263—21no—
68C2×Dic263—41no—
69C4×D263—41no—
70C4×D263—41no—
71C263⋊C8—22no—
72C263⋊C8—41no—
73C23×C263—22no—
74C23×C263—22no—
75C23×C263—22no—
76C23×C263—82no—
77C23×C263C2×D411yescharacteristic
78D4×C263—22no—
79D4×C263—22no—
80D4×C263—22no—
81D4×C263C22⋊C414yes—
82C2×C4×C263—22no—
83C2×C4×C263—22no—
84C2×C4×C263—84no—
85C2×C4×C263C22×C411yescharacteristic
86C2×C4×C263C2×D411yescharacteristic
87C2×C4×C263C2×D411yescharacteristic
88C2×C4×C263C2×D411yescharacteristic
89C2×C4×C263#4111yescharacteristic
90C2×C4×C263#4111yescharacteristic
91C22×C263—24no—
92C22×C263—41no—
93C22×C263—44no—
94C22×C263—82no—
95C22×C263—82no—
96C22×C263C4×D411yescharacteristic
97C22×C263#12011yescharacteristic
98C22×C263#12111yescharacteristic
99C2×D263—41no—
100C2×D263—81no—
101C4×C263—42no—
102C4×C263—42no—
103C4×C263C2×C22⋊C411yescharacteristic
104C4×C263C4×D411yescharacteristic
105C4×C263#12111yescharacteristic
106C4×C263#12311yescharacteristic
107Dic263—41no—
108Dic263—81no—
109C2×C263—21no—
110C2×C263—21no—
111C2×C263—21no—
112C2×C263—44no—
113C2×C263#38611yescharacteristic
114D263—81no—
115C263#190411yescharacteristic, complemented, Sylow
116#1904—2631nomaximal, Sylow
117C2×AGL(1,ℤ/8)—2631no—
118#518D26311yescharacteristic, complemented
119#421—2631no—
120#418—2631no—
121#410—2631no—
122#344—2631no—
123H(3,ℤ/4)—2631no—
124C2×#41—2631no—
125C22×D4C2×D26311yescharacteristic
126AGL(1,ℤ/8)—5262no—
127AGL(1,ℤ/8)—5262no—
128C2×SD16—2632no—
129C2×D8—2632no—
130C2×M16—2631no—
131C2×M16—2631no—
132C4×D4—22no—
133C4×D4—22no—
134#117—2631no—
135#117C2×D26311yescharacteristic
136#117C2×D26311yescharacteristic
137C2×C22⋊C4—22no—
138#104—5262no—
139#102—5262no—
140#99—5262no—
141C24—21no—
142#41—5261no—
143#41—5261no—
144#41—5261no—
145#41—5261no—
146C2×Q8—2631no—
147C2×D4—21no—
148C2×D4—21no—
149C2×D4—21no—
150C2×D4—2631no—
151C2×D4—5261no—
152C2×D4C4×D26312yes—
153C2×D4C22⋊D26312yes—
154C2×D4D105212yes—
155C22×C4—24no—
156C22×C4—5261no—
157C22×C4C22×D26311yescharacteristic
158SD16—5264no—
159D8—5264no—
160M16—5261no—
161M16—5261no—
162M16—5261no—
163M16—5261no—
164C2×C8—2632no—
165C2×C8—5261no—
166C4⋊C4—21no—
167C4⋊C4—21no—
168C4⋊C4—5261no—
169C22⋊C4—24no—
170C22⋊C4—21no—
171C22⋊C4—5261no—
172C22⋊C4C22⋊D26312yes—
173C42—21no—
174C42—5261no—
175C42C22⋊D26312yes—
176C23—22no—
177C23—22no—
178C23—22no—
179C23—82no—
180C23—5261no—
181C23C2×C22⋊D26311yescharacteristic
182Q8—5261no—
183Q8—5261no—
184D4—22no—
185D4—22no—
186D4—22no—
187D4—5261no—
188D4—5261no—
189D4—10521no—
190D4—10521no—
191D4(C2×D263)⋊C414yes—
192C2×C4—22no—
193C2×C4—22no—
194C2×C4—84no—
195C2×C4—5261no—
196C2×C4—10521no—
197C2×C4—10521no—
198C2×C4—10521no—
199C2×C4C2×C4×D26311yescharacteristic
200C2×C4C2×C22⋊D26311yescharacteristic
201C2×C4C2×C22⋊D26311yescharacteristic
202C2×C4C2×D105211yescharacteristic
203C2×C4#1686237611yescharacteristic
204C2×C4#1686237611yescharacteristic
205C8—5262no—
206C8—10521no—
207C22—24no—
208C22—41no—
209C22—44no—
210C22—82no—
211C22—82no—
212C22—10521no—
213C22—21041no—
214C22C4×C22⋊D26311yescharacteristic
215C22#2255443911yescharacteristic
216C22#2255444411yescharacteristic
217C4—42no—
218C4—42no—
219C4—10521no—
220C4—21041no—
221C4C2×(C2×D263)⋊C411yescharacteristic
222C4C4×C22⋊D26311yescharacteristic
223C4#2255444411yescharacteristic
224C4#2255445311yescharacteristic
225C2—21no—
226C2—21no—
227C2—21no—
228C2—44no—
229C2—21041no—
230C2#3068181711yescharacteristic
subgroupG/H
centerC2#30681817
commutatorC2×C4×C263C22×C4
FrattiniC22×C4C22×D263
Fitting#518×C263C2
socleC2×C263#386
prime radicalC2×#22554496C2
cosocleC22×C4×C263C23
hypercenter#518D263
hypocenterC263#1904
LCSG ▷ C2×C4×C263 ▷ C2×C263 ▷ C263
UCSC1 ◁ C2 ◁ C22×C4 ◁ #518

Complements

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

NH ≅ G/Ntype
2. #30681956C2semidirect
3. #30681943C2semidirect
4. #30681911C2semidirect
5. #30681708C2semidirect
6. #30681694C2semidirect
7. #518×C263C2semidirect
11. C2×#16862364C4semidirect
12. C2×#16862361C4semidirect
29. C2×Dic526C2×C4semidirect
30. C2×D1052C2×C4semidirect
40. C2×C263⋊C8C2×C4semidirect
115. C263#1904semidirect
118. #518D263semidirect

Maximal subgroups

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

subgroupindex
C2×#225544962
#306819562
#306819432
#306819112
#306817082
#306816942
#518×C2632
#1904263