Finite Group Databasebeta

← #46782714

Subgroups — #46782714

Order 33664.

588 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 588 conjugacy classes of subgroups — too many to draw. Only its 97 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×#22554515C211yesmaximal, characteristic
2#30681956C211yesmaximal, characteristic, complemented
3#30681954C211yesmaximal, characteristic, complemented
4#30681943C211yesmaximal, characteristic, complemented
5#30681920C211yesmaximal, characteristic, complemented
6#30681911C211yesmaximal, characteristic, complemented
7#418×C263C211yesmaximal, characteristic, complemented
8#22554515—22no—
9#22554515—22no—
10#22554515—22no—
11#22554515—22no—
12C2×#16862376C2211yescharacteristic
13C2×#16862376C2211yescharacteristic
14C2×#16862375—21no—
15C2×#16862375C412yescomplemented
16C22×C22⋊D263—22no—
17C2×D4×D263—21no—
18C2×D4×D263C412yescomplemented
19C2×#16862360C2211yescharacteristic, complemented
20#22554429C2211yescharacteristic, complemented
21#22554398—22no—
22#22554396—22no—
23#22554391—22no—
24#22554388—21no—
25#22554386—21no—
26#22554384—22no—
27C22×D4×C263C2211yescharacteristic
28C2×M16×C263C2211yescharacteristic, complemented
29#117×C263C2211yescharacteristic, complemented
30#102×C263—22no—
31C23×D263—22no—
32#16862376—21no—
33#16862376—21no—
34#16862376—21no—
35#16862376—21no—
36#16862376—21no—
37#16862376—21no—
38#16862376D412yes—
39#16862376D412yes—
40#16862375—24no—
41#16862375—24no—
42#16862375—24no—
43#16862375—24no—
44C2×Dic526C2×C411yescharacteristic
45C2×Dic526D411yescharacteristic
46C2×D1052C2×C411yescharacteristic
47C2×D1052D411yescharacteristic
48C2×C22⋊D263—28no—
49C2×C22⋊D263—28no—
50C2×C22⋊D263—22no—
51C2×C22⋊D263—22no—
52C2×C22⋊D263—21no—
53C2×C22⋊D263D412yes—
54D4×D263—24no—
55D4×D263—24no—
56D4×D263—24no—
57D4×D263—24no—
58C22×Dic263—22no—
59C2×C4×D263—21no—
60C2×C4×D263C2×C412yes—
61#16862360—21no—
62#16862360—21no—
63C2×C263⋊C8—21no—
64C4⋊Dic263—21no—
65C22⋊Dic263—21no—
66C4×Dic263—21no—
67C24×C263—21no—
68C2×D4×C263—21no—
69C2×D4×C263—21no—
70C2×D4×C263—21no—
71C2×D4×C263C2×C412yes—
72C2×D4×C263D412yes—
73C2×D4×C263D412yes—
74C22×C4×C263C2311yescharacteristic
75M16×C263—21no—
76M16×C263—21no—
77C2×C8×C263—21no—
78C4⋊C4×C263—21no—
79C22⋊C4×C263—21no—
80C42×C263—21no—
81C22×D263—21no—
82C22×D263—28no—
83C22×D263—84no—
84C22×D263C22⋊C412yes—
85Dic526—21no—
86Dic526—21no—
87Dic526—21no—
88Dic526C22⋊C412yes—
89D1052—21no—
90D1052—21no—
91D1052—21no—
92D1052C22⋊C412yes—
93C22⋊D263—28no—
94C22⋊D263—28no—
95C22⋊D263—22no—
96C22⋊D263—22no—
97C22⋊D263—22no—
98C22⋊D263—22no—
99C2×Dic263—28no—
100C2×Dic263—21no—
101C2×Dic263—41no—
102C2×Dic263C22⋊C412yes—
103C4×D263—22no—
104C4×D263—22no—
105C4×D263—22no—
106C4×D263—22no—
107C263⋊C8—41no—
108C23×C263—22no—
109C23×C263—22no—
110C23×C263—22no—
111C23×C263—82no—
112C23×C263C2×D411yescharacteristic
113D4×C263—22no—
114D4×C263—22no—
115D4×C263—22no—
116D4×C263C22⋊C414yes—
117C2×C4×C263—41no—
118C2×C4×C263C22×C411yescharacteristic
119C2×C4×C263C2×D411yescharacteristic
120C2×C4×C263C2×D411yescharacteristic
121C2×C4×C263C2×D411yescharacteristic
122C2×C4×C263C2×D411yescharacteristic
123C2×C4×C263C2×D411yescharacteristic
124C8×C263—41no—
125C22×C263—24no—
126C22×C263—41no—
127C22×C263—44no—
128C22×C263—82no—
129C22×C263—82no—
130C22×C263C2×C22⋊C411yescharacteristic
131C22×C263#12011yescharacteristic
132C22×C263#12011yescharacteristic
133C2×D263—22no—
134C2×D263—22no—
135C2×D263—42no—
136C2×D263—42no—
137C2×D263—88no—
138C4×C263—81no—
139C4×C263C2×C22⋊C411yescharacteristic
140C4×C263C2×C22⋊C411yescharacteristic
141C4×C263#12011yescharacteristic
142C4×C263#12011yescharacteristic
143Dic263—22no—
144Dic263—22no—
145Dic263—81no—
146C2×C263—21no—
147C2×C263—21no—
148C2×C263—21no—
149C2×C263—44no—
150C2×C263#37911yescharacteristic
151D263—42no—
152D263—42no—
153C263#179511yescharacteristic, complemented, Sylow
154#1795—2631nomaximal, Sylow
155C2×H(5,𝔽2)—2631no—
156#421—2631no—
157#421—2631no—
158#418—2631no—
159#418D26311yescharacteristic, complemented
160#413—2631no—
161#410—2631no—
162H(5,𝔽2)—5262no—
163H(5,𝔽2)—5262no—
164H(5,𝔽2)—5262no—
165H(5,𝔽2)—5262no—
166C2×#41—2631no—
167C2×#41—2632no—
168C2×#41—2631no—
169C2×#41—5261no—
170C22×D4—2632no—
171C22×D4—5261no—
172C22×D4—5262no—
173C22×D4C2×D26311yescharacteristic
174C2×M16—2631no—
175C2×M16C2×D26311yescharacteristic, complemented
176#117—2631no—
177#117C2×D26311yescharacteristic, complemented
178#104—5262no—
179#104—5262no—
180#102—22no—
181#102—5262no—
182#101—5261no—
183#100—5261no—
184#99—5262no—
185C24—21no—
186C24—5262no—
187#41—2632no—
188#41—2632no—
189#41—5264no—
190#41—5264no—
191#41—5264no—
192#41—5264no—
193#41—5261no—
194#41—5261no—
195#41—5261no—
196#41—5261no—
197#41—5261no—
198#41—5261no—
199C2×Q8—2631no—
200C2×Q8—2631no—
201C2×D4—21no—
202C2×D4—21no—
203C2×D4—21no—
204C2×D4—2632no—
205C2×D4—2631no—
206C2×D4—2631no—
207C2×D4—5262no—
208C2×D4—5264no—
209C2×D4—5264no—
210C2×D4—5264no—
211C2×D4—5262no—
212C2×D4—5268no—
213C2×D4—5268no—
214C2×D4—5261no—
215C2×D4—5264no—
216C2×D4C4×D26312yes—
217C2×D4C22⋊D26312yes—
218C2×D4D105212yes—
219C22×C4—2632no—
220C22×C4—5262no—
221C22×C4—5261no—
222C22×C4C22×D26311yescharacteristic
223M16—21no—
224M16—21no—
225M16—5261no—
226M16—5261no—
227C2×C8—21no—
228C2×C8—5261no—
229C4⋊C4—21no—
230C4⋊C4—5261no—
231C22⋊C4—21no—
232C22⋊C4—5261no—
233C42—21no—
234C42—5261no—
235C23—22no—
236C23—22no—
237C23—22no—
238C23—2632no—
239C23—82no—
240C23—5268no—
241C23—5261no—
242C23—21044no—
243C23D4×D26311yescharacteristic
244Q8—2632no—
245Q8—5261no—
246Q8—5261no—
247Q8—5261no—
248D4—22no—
249D4—22no—
250D4—22no—
251D4—2632no—
252D4—5268no—
253D4—5268no—
254D4—5262no—
255D4—5262no—
256D4—5262no—
257D4—5262no—
258D4—5261no—
259D4—5261no—
260D4—5261no—
261D4(C2×D263)⋊C414yes—
262C2×C4—41no—
263C2×C4—2632no—
264C2×C4—5268no—
265C2×C4—5261no—
266C2×C4—5262no—
267C2×C4—5262no—
268C2×C4—5262no—
269C2×C4—5262no—
270C2×C4—10521no—
271C2×C4C2×C4×D26311yescharacteristic
272C2×C4D4×D26311yescharacteristic
273C2×C4D4×D26311yescharacteristic
274C2×C4D4×D26311yescharacteristic
275C2×C4C2×C22⋊D26311yescharacteristic
276C2×C4C2×D105211yescharacteristic
277C8—41no—
278C8—10521no—
279C22—24no—
280C22—41no—
281C22—44no—
282C22—82no—
283C22—82no—
284C22—5262no—
285C22—5262no—
286C22—10522no—
287C22—10522no—
288C22—21048no—
289C22C22⋊C4×D26311yescharacteristic
290C22#2255443811yescharacteristic
291C22#2255444011yescharacteristic
292C4—81no—
293C4—5262no—
294C4—5262no—
295C4—21041no—
296C4C22⋊C4×D26311yescharacteristic
297C4C2×(C2×D263)⋊C411yescharacteristic
298C4#2255443811yescharacteristic
299C4#2255444011yescharacteristic
300C2—21no—
301C2—21no—
302C2—21no—
303C2—44no—
304C2—10522no—
305C2—10522no—
306C2#3068179311yescharacteristic
subgroupG/H
centerC2#30681793
commutatorC2×C4×C263C22×C4
FrattiniC22×C4C22×D263
Fitting#418×C263C2
socleC2×C263#379
prime radicalC2×#22554515C2
cosocleC22×C4×C263C23
hypercenter#418D263
hypocenterC263#1795
LCSG ▷ C2×C4×C263 ▷ C2×C263 ▷ C263
UCSC1 ◁ C2 ◁ C22×C4 ◁ #418

Complements

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

NH ≅ G/Ntype
2. #30681956C2semidirect
3. #30681954C2semidirect
4. #30681943C2semidirect
5. #30681920C2semidirect
6. #30681911C2semidirect
7. #418×C263C2semidirect
15. C2×#16862375C4semidirect
18. C2×D4×D263C4semidirect
19. C2×#16862360C22semidirect
20. #22554429C22semidirect
28. C2×M16×C263C22semidirect
29. #117×C263C22semidirect
153. C263#1795semidirect
159. #418D263semidirect
175. C2×M16C2×D263semidirect
177. #117C2×D263semidirect

Maximal subgroups

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

subgroupindex
C2×#225545152
#306819562
#306819542
#306819432
#306819202
#306819112
#418×C2632
#1795263