Finite Group Databasebeta

← #26226878

Subgroups — #26226878

Order 12032.

762 conjugacy classes of subgroups.

The subgroup lattice has 378 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20246011C211yesmaximal, characteristic
2#20245997C211yesmaximal, characteristic
3#20244158C212yesmaximal, complemented
4#20244147C212yesmaximal, complemented
5#20243589C212yesmaximal
6#20243578C212yesmaximal
7#20243556C211yesmaximal, characteristic, complemented
8#20243548C211yesmaximal, characteristic, complemented
9#20243522C211yesmaximal, characteristic, complemented
10#20243519C211yesmaximal, characteristic, complemented
11C2×C47×#433C211yesmaximal, characteristic
12C2×#13093983C2211yescharacteristic
13C22×C4⋊Dic47C2211yescharacteristic
14C2×#13093958C2212yes—
15C2×#13093958C2212yes—
16C2×#13093958C2212yes—
17C2×#13093956C2212yes—
18#15689022—22no—
19#15689016—22no—
20#15689009—22no—
21#15688998—22no—
22#15688993C2211yescharacteristic
23#15688993C2211yescharacteristic
24#15688993C2211yescharacteristic
25#15688989C2211yescharacteristic
26#15688989C2211yescharacteristic
27#15688989C2211yescharacteristic
28#15688983C2212yes—
29C4×C4⋊Dic47C2212yes—
30#15688886C2212yes—
31#15688886C2212yes—
32#15688886C2212yes—
33#15688886C2212yes—
34C2×#117×C47C2211yescharacteristic
35C47×#433C418yescomplemented
36C2×C4⋊C8×C47C2212yes—
37C2×C22⋊C8×C47C2212yes—
38C2×#97×C47C2211yescharacteristic
39C2×C4×C8×C47C2211yescharacteristic
40C23×Dic47—21no—
41#13093983—42no—
42C2×C4⋊Dic47—21no—
43C2×C4⋊Dic47—22no—
44C2×C4⋊Dic47—22no—
45C2×C4⋊Dic47—22no—
46C2×C4⋊Dic47—42no—
47C2×C4⋊Dic47C2312yes—
48C2×C4⋊Dic47C2312yes—
49C2×C4⋊Dic47C2311yescharacteristic
50C2×C4⋊Dic47C2311yescharacteristic
51C2×C4⋊Dic47C2311yescharacteristic
52C2×C4⋊Dic47C2311yescharacteristic
53C2×C22⋊Dic47—21no—
54C2×C22⋊Dic47—22no—
55C2×C4×Dic47—21no—
56C2×C4×Dic47—22no—
57#13093958—42no—
58#13093958—42no—
59#13093958—42no—
60#13093956—42no—
61C47⋊#95—22no—
62C47⋊#95—22no—
63C47⋊#95—22no—
64C47⋊#95—22no—
65C47⋊#95—22no—
66C23×C4×C47C2311yescharacteristic
67C22×C8×C47C2312yes—
68C22×C8×C47C2312yes—
69#117×C47C2×C414yes—
70#117×C47Q814yes—
71C2×C4⋊C4×C47D412yes—
72C2×C22⋊C4×C47D412yes—
73C2×C42×C47C2311yescharacteristic
74C2×C42×C47C2311yescharacteristic
75C4⋊C8×C47C2×C418yescomplemented
76C22⋊C8×C47C2×C418yes—
77#97×C47C2×C414yescomplemented
78C4×C8×C47C2×C414yescomplemented
79C24×C47C2×D411yescharacteristic
80C22×C4×C47—22no—
81C22×C4×C47—22no—
82C22×C4×C47C22×C412yes—
83C22×C4×C47C2×D412yes—
84C22×C4×C47C2×D412yes—
85C22×C4×C47C2×D411yescharacteristic
86C22×C4×C47C2×Q812yes—
87C22×C4×C47C2411yescharacteristic
88C22×Dic47—21no—
89C22×Dic47—22no—
90C22×Dic47—22no—
91C22×Dic47—21no—
92C22×Dic47—21no—
93C22×Dic47—21no—
94C22×Dic47—42no—
95C22×Dic47—82no—
96C2×C8×C47—22no—
97C2×C8×C47—22no—
98C2×C8×C47—22no—
99C2×C8×C47—22no—
100C2×C8×C47C22×C414yes—
101C2×C8×C47C22×C414yes—
102C4⋊C4×C47C4⋊C418yes—
103C22⋊C4×C47C4⋊C418yes—
104C42×C47C22×C412yes—
105C42×C47C22×C412yes—
106C42×C47C2×Q812yes—
107C42×C47C2×Q812yes—
108C22⋊Dic47—42no—
109C22⋊Dic47—82no—
110C4⋊Dic47—22no—
111C4⋊Dic47—22no—
112C4⋊Dic47—22no—
113C4⋊Dic47—22no—
114C4⋊Dic47—22no—
115C4⋊Dic47—22no—
116C4⋊Dic47—24no—
117C4⋊Dic47—42no—
118C4⋊Dic47—82no—
119C4⋊Dic47—82no—
120C4⋊Dic47—82no—
121C4×Dic47—42no—
122C4×Dic47—82no—
123C23×C47—21no—
124C23×C47—21no—
125C23×C47—22no—
126C23×C47—22no—
127C23×C47C2×C4⋊C412yes—
128C23×C47C22×D411yescharacteristic
129C2×C4×C47—21no—
130C2×C4×C47—21no—
131C2×C4×C47—22no—
132C2×C4×C47—22no—
133C2×C4×C47—22no—
134C2×C4×C47—22no—
135C2×C4×C47—24no—
136C2×C4×C47—24no—
137C2×C4×C47C2×C4⋊C414yes—
138C2×C4×C47C2×C4⋊C414yes—
139C2×C4×C47C2×C4⋊C412yes—
140C2×C4×C47C23×C411yescharacteristic
141C2×C4×C47C22×Q811yescharacteristic
142C2×C4×C47H(5,𝔽2)12yes—
143C2×C4×C472−1+412yes—
144C2×Dic47—41no—
145C2×Dic47—41no—
146C2×Dic47—41no—
147C2×Dic47—41no—
148C2×Dic47—42no—
149C2×Dic47—42no—
150C2×Dic47—42no—
151C2×Dic47—42no—
152C2×Dic47—42no—
153C2×Dic47—42no—
154C2×Dic47—42no—
155C2×Dic47—42no—
156C2×Dic47—42no—
157C2×Dic47—42no—
158C2×Dic47—84no—
159C8×C47—24no—
160C8×C47—24no—
161#21991—471nomaximal, Sylow
162C22×C47—22no—
163C22×C47—24no—
164C22×C47—41no—
165C22×C47—41no—
166C22×C47—42no—
167C22×C47C22×C4⋊C411yescharacteristic
168C22×C47#53511yescharacteristic
169C22×C47#53611yescharacteristic
170C22×C47#57612yes—
171C22×C47#57812yes—
172C4×C47—22no—
173C4×C47—24no—
174C4×C47—24no—
175C4×C47#52012yes—
176C4×C47#54412yes—
177Dic47—81no—
178Dic47—82no—
179Dic47—81no—
180Dic47—81no—
181Dic47—81no—
182Dic47—82no—
183C2×#433D4711yescharacteristic
184#2317—471no—
185#2314—471no—
186#1949—472no—
187#1947—472no—
188#1854—472no—
189#1852—472no—
190#1847—471no—
191#1845—471no—
192#1836—471no—
193#1835—471no—
194C2×C47—42no—
195C2×C47#231711yescharacteristic
196C2×C47#291312yes—
197C2×C47#322011yescharacteristic
198C2×C47#322311yescharacteristic
199C2×C47#324312yes—
200C2×#117—471no—
201C2×#117C2×D4711yescharacteristic
202C22×C4⋊C4—471no—
203#433Dic4718yescomplemented
204C2×#107—472no—
205C2×#107—472no—
206C2×#107—472no—
207C2×#106—472no—
208C2×C4⋊C8C2×D4712yes—
209C2×C22⋊C8C2×D4712yes—
210C2×#97C2×D4711yescharacteristic
211C2×C4×C8C2×D4711yescharacteristic
212#389—942no—
213#388—942no—
214#387—942no—
215#385—942no—
216#384—471no—
217#384—471no—
218#384—471no—
219#382—471no—
220#382—471no—
221#382—471no—
222#380—472no—
223C4×C4⋊C4—472no—
224#340—472no—
225#340—472no—
226#340—472no—
227#340—472no—
228C47#2199111yescharacteristic, complemented, Sylow
229C23×C4—941no—
230C23×C4C22×D4711yescharacteristic
231C22×C8C22×D4712yes—
232C22×C8C22×D4712yes—
233#117—1882no—
234#117Dic9414yes—
235#117C2×Dic4714yes—
236C2×C4⋊C4—471no—
237C2×C4⋊C4—471no—
238C2×C4⋊C4—471no—
239C2×C4⋊C4—471no—
240C2×C4⋊C4—472no—
241C2×C4⋊C4—472no—
242C2×C4⋊C4—942no—
243C2×C4⋊C4—942no—
244C2×C4⋊C4—942no—
245C2×C4⋊C4—941no—
246C2×C4⋊C4—1882no—
247C2×C4⋊C4D18812yes—
248C2×C22⋊C4—942no—
249C2×C22⋊C4—941no—
250C2×C22⋊C4D18812yes—
251C2×C42—942no—
252C2×C42—941no—
253C2×C42C22×D4711yescharacteristic
254C2×C42C22×D4711yescharacteristic
255#107—1882no—
256#107—1882no—
257#107—1882no—
258#106—1882no—
259C4⋊C8C2×Dic4718yescomplemented
260C22⋊C8C2×Dic4718yes—
261#97C2×Dic4714yescomplemented
262C4×C8C2×Dic4714yescomplemented
263#95—942no—
264#95—942no—
265#95—942no—
266#95—942no—
267#95—942no—
268C24C2×D18811yescharacteristic
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—942no—
272C22×C4—941no—
273C22×C4—942no—
274C22×C4—941no—
275C22×C4—941no—
276C22×C4—941no—
277C22×C4—3762no—
278C22×C4—1882no—
279C22×C4C2×Dic9412yes—
280C22×C4C2×D18812yes—
281C22×C4C2×D18811yescharacteristic
282C22×C4C2×D18812yes—
283C22×C4C22×Dic4712yes—
284C22×C4C23×D4711yescharacteristic
285C2×C8—22no—
286C2×C8—22no—
287C2×C8—22no—
288C2×C8—22no—
289C2×C8C22×Dic4714yes—
290C2×C8C22×Dic4714yes—
291C4⋊C4—942no—
292C4⋊C4—942no—
293C4⋊C4—942no—
294C4⋊C4—942no—
295C4⋊C4—942no—
296C4⋊C4—942no—
297C4⋊C4—944no—
298C4⋊C4—3762no—
299C4⋊C4—3762no—
300C4⋊C4—3762no—
301C4⋊C4—1882no—
302C4⋊C4C4⋊Dic4718yes—
303C22⋊C4—3762no—
304C22⋊C4—1882no—
305C22⋊C4C4⋊Dic4718yes—
306C42—3762no—
307C42—1882no—
308C42C2×Dic9412yes—
309C42C2×Dic9412yes—
310C42C22×Dic4712yes—
311C42C22×Dic4712yes—
312C23—22no—
313C23—22no—
314C23—21no—
315C23—21no—
316C23C2×C4⋊Dic4712yes—
317C23C22×D18811yescharacteristic
318C2×C4—22no—
319C2×C4—22no—
320C2×C4—22no—
321C2×C4—22no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—24no—
325C2×C4—24no—
326C2×C4—3764no—
327C2×C4—1882no—
328C2×C4—1882no—
329C2×C4—1882no—
330C2×C4—1882no—
331C2×C4—1882no—
332C2×C4—1882no—
333C2×C4—1882no—
334C2×C4—1882no—
335C2×C4—1881no—
336C2×C4—1881no—
337C2×C4—1881no—
338C2×C4—1881no—
339C2×C4—1882no—
340C2×C4—1882no—
341C2×C4C2×C4⋊Dic4714yes—
342C2×C4C2×C4⋊Dic4714yes—
343C2×C4C2×C4⋊Dic4712yes—
344C2×C4C23×Dic4711yescharacteristic
345C2×C4C22×Dic9411yescharacteristic
346C2×C4#1309407012yes—
347C2×C4#1309407112yes—
348C8—24no—
349C8—24no—
350C22—24no—
351C22—22no—
352C22—41no—
353C22—42no—
354C22—41no—
355C22C22×C4⋊Dic4711yescharacteristic
356C22#1568969211yescharacteristic
357C22#1568969911yescharacteristic
358C22#1568992812yes—
359C22#1568994012yes—
360C4—24no—
361C4—24no—
362C4—22no—
363C4—3761no—
364C4—3762no—
365C4—3761no—
366C4—3761no—
367C4—3762no—
368C4—3761no—
369C4#1568960612yes—
370C4#1568975112yes—
371C2—42no—
372C2#2024601111yescharacteristic
373C2#2025198812yes—
374C2#2025444011yescharacteristic
375C2#2025447311yescharacteristic
376C2#2025467312yes—
subgroupG/H
centerC23C22×D188
commutatorC2×C4×C47C23×C4
FrattiniC22×C4C23×D47
FittingC2×C47×#433C2
socleC23×C47C22×D4
prime radicalC24×C47C2×D4
cosocleC22×C4×C47C24
hypercenterC2×#433D47
hypocenterC47#21991
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C2×#117 ◁ C2×#433

Complements

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

NH ≅ G/Ntype
3. #20244158C2semidirect
4. #20244147C2semidirect
7. #20243556C2semidirect
8. #20243548C2semidirect
9. #20243522C2semidirect
10. #20243519C2semidirect
35. C47×#433C4semidirect
75. C4⋊C8×C47C2×C4semidirect
77. #97×C47C2×C4semidirect
78. C4×C8×C47C2×C4semidirect
203. #433Dic47semidirect
228. C47#21991semidirect
259. C4⋊C8C2×Dic47semidirect
261. #97C2×Dic47semidirect
262. C4×C8C2×Dic47semidirect

Maximal subgroups

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

subgroupindex
#202460112
#202459972
#202441582
#202441472
#202435892
#202435782
#202435562
#202435482
#202435222
#202435192
C2×C47×#4332
#2199147