Finite Group Databasebeta

← #26226868

Subgroups — #26226868

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
1C2×#15689184C211yesmaximal, characteristic
2#20245997C211yesmaximal, characteristic
3#20244154C212yesmaximal, complemented
4#20244149C212yesmaximal, complemented
5#20243587C212yesmaximal
6#20243581C212yesmaximal
7#20243557C211yesmaximal, characteristic, complemented
8#20243544C211yesmaximal, characteristic, complemented
9#20243523C211yesmaximal, characteristic, complemented
10#20243518C211yesmaximal, characteristic, complemented
11C47×#2317C211yesmaximal, characteristic
12#15689184C418yescomplemented
13C2×#13093957C2212yes—
14C2×#13093955C2212yes—
15C2×#13093955C2212yes—
16C2×#13093955C2212yes—
17C2×C4⋊(C47⋊C8)C2212yes—
18C2×C22⋊(C47⋊C8)C2212yes—
19C2×#13093935C2211yescharacteristic
20C2×C4×C47⋊C8C2211yescharacteristic
21#15689016—22no—
22#15689009—22no—
23#15688993C2211yescharacteristic
24#15688989C2211yescharacteristic
25#15688983C2212yes—
26C4×C4⋊Dic47C2212yes—
27#15688885C2212yes—
28#15688885C2212yes—
29#15688885C2212yes—
30#15688885C2212yes—
31C2×#117×C47C2211yescharacteristic
32C2×#117×C47C2211yescharacteristic
33C22×C4⋊C4×C47C2211yescharacteristic
34C47×#389—22no—
35C47×#385—22no—
36C47×#384C2211yescharacteristic
37C47×#384C2211yescharacteristic
38C47×#382C2211yescharacteristic
39C47×#382C2211yescharacteristic
40C22×C47⋊C8C2312yes—
41C22×C47⋊C8C2312yes—
42C2×C4⋊Dic47—22no—
43C2×C4⋊Dic47C2312yes—
44C2×C4⋊Dic47C2312yes—
45C2×C22⋊Dic47—22no—
46C2×C4×Dic47—22no—
47#13093957—42no—
48#13093955—42no—
49#13093955—42no—
50#13093955—42no—
51C4⋊(C47⋊C8)C2×C418yescomplemented
52C22⋊(C47⋊C8)C2×C418yes—
53#13093935C2×C414yescomplemented
54C4×C47⋊C8C2×C414yescomplemented
55C47⋊#95—22no—
56C47⋊#95—22no—
57C47⋊#95—22no—
58C23×C4×C47—21no—
59C23×C4×C47C2311yescharacteristic
60#117×C47—42no—
61#117×C47C2×C414yes—
62#117×C47Q814yes—
63C2×C4⋊C4×C47—21no—
64C2×C4⋊C4×C47—22no—
65C2×C4⋊C4×C47—22no—
66C2×C4⋊C4×C47—42no—
67C2×C4⋊C4×C47D412yes—
68C2×C4⋊C4×C47C2311yescharacteristic
69C2×C4⋊C4×C47C2311yescharacteristic
70C2×C4⋊C4×C47C2311yescharacteristic
71C2×C4⋊C4×C47C2311yescharacteristic
72C2×C22⋊C4×C47—21no—
73C2×C22⋊C4×C47D412yes—
74C2×C42×C47—21no—
75C2×C42×C47C2311yescharacteristic
76C2×C42×C47C2311yescharacteristic
77#95×C47—22no—
78#95×C47—22no—
79C24×C47C2×D411yescharacteristic
80C22×C4×C47—22no—
81C22×C4×C47—22no—
82C22×C4×C47—21no—
83C22×C4×C47—21no—
84C22×C4×C47—21no—
85C22×C4×C47—21no—
86C22×C4×C47—42no—
87C22×C4×C47—82no—
88C22×C4×C47C22×C412yes—
89C22×C4×C47C2×D411yescharacteristic
90C22×C4×C47C2×D412yes—
91C22×C4×C47C2×D412yes—
92C22×C4×C47C2×Q812yes—
93C22×C4×C47C2411yescharacteristic
94C22×Dic47—22no—
95C22×Dic47—22no—
96C4⋊C4×C47—22no—
97C4⋊C4×C47—22no—
98C4⋊C4×C47—22no—
99C4⋊C4×C47—22no—
100C4⋊C4×C47—42no—
101C4⋊C4×C47—82no—
102C4⋊C4×C47—82no—
103C4⋊C4×C47C4⋊C418yes—
104C22⋊C4×C47—42no—
105C22⋊C4×C47C4⋊C418yes—
106C42×C47—42no—
107C42×C47C22×C412yes—
108C42×C47C22×C412yes—
109C42×C47C2×Q812yes—
110C42×C47C2×Q812yes—
111C22⋊Dic47—82no—
112C4⋊Dic47—22no—
113C4⋊Dic47—22no—
114C4⋊Dic47—24no—
115C4⋊Dic47—82no—
116C4×Dic47—82no—
117C2×C47⋊C8—22no—
118C2×C47⋊C8—22no—
119C2×C47⋊C8—22no—
120C2×C47⋊C8—22no—
121C2×C47⋊C8C22×C414yes—
122C2×C47⋊C8C22×C414yes—
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×C47—41no—
138C2×C4×C47—41no—
139C2×C4×C47—41no—
140C2×C4×C47—41no—
141C2×C4×C47—42no—
142C2×C4×C47—42no—
143C2×C4×C47—42no—
144C2×C4×C47—42no—
145C2×C4×C47—84no—
146C2×C4×C47C2×C4⋊C414yes—
147C2×C4×C47C2×C4⋊C414yes—
148C2×C4×C47C2×C4⋊C412yes—
149C2×C4×C47C23×C411yescharacteristic
150C2×C4×C47C22×Q811yescharacteristic
151C2×C4×C47H(5,𝔽2)12yes—
152C2×C4×C472−1+412yes—
153C2×Dic47—42no—
154C2×Dic47—42no—
155C2×Dic47—42no—
156C2×Dic47—42no—
157C2×Dic47—42no—
158C2×Dic47—42no—
159C47⋊C8—24no—
160C47⋊C8—24no—
161#21990—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#57712yes—
171C22×C47#57712yes—
172C4×C47—22no—
173C4×C47—24no—
174C4×C47—24no—
175C4×C47—81no—
176C4×C47—81no—
177C4×C47—81no—
178C4×C47—81no—
179C4×C47#52012yes—
180C4×C47#54412yes—
181Dic47—82no—
182Dic47—82no—
183C2×#433—471no—
184#2317D4711yescharacteristic
185#2314—471no—
186#1949—472no—
187#1948—472no—
188#1854—472no—
189#1853—472no—
190#1847—471no—
191#1844—471no—
192#1836—471no—
193#1835—471no—
194C2×C47—42no—
195C2×C47#231711yescharacteristic
196C2×C47#291212yes—
197C2×C47#322111yescharacteristic
198C2×C47#322211yescharacteristic
199C2×C47#324412yes—
200C2×#117C2×D4711yescharacteristic
201C2×#117C2×D4711yescharacteristic
202C22×C4⋊C4C2×D4711yescharacteristic
203#433—478no—
204C2×#107—472no—
205C2×#106—472no—
206C2×#106—472no—
207C2×#106—472no—
208C2×C4⋊C8—472no—
209C2×C22⋊C8—472no—
210C2×#97—471no—
211C2×C4×C8—471no—
212#389—22no—
213#388—942no—
214#387—942no—
215#385—22no—
216#384—471no—
217#384C2×D4711yescharacteristic
218#384C2×D4711yescharacteristic
219#382—471no—
220#382C2×D4711yescharacteristic
221#382C2×D4711yescharacteristic
222#380—472no—
223C4×C4⋊C4—472no—
224#340—472no—
225#340—472no—
226#340—472no—
227#340—472no—
228C47#2199011yescharacteristic, complemented, Sylow
229C23×C4—21no—
230C23×C4C22×D4711yescharacteristic
231C22×C8—472no—
232C22×C8—472no—
233#117—42no—
234#117Dic9414yes—
235#117C4×D4714yes—
236C2×C4⋊C4—21no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—42no—
240C2×C4⋊C4—472no—
241C2×C4⋊C4—472no—
242C2×C4⋊C4—942no—
243C2×C4⋊C4C22⋊D4712yes—
244C2×C4⋊C4C22×D4711yescharacteristic
245C2×C4⋊C4C22×D4711yescharacteristic
246C2×C4⋊C4C22×D4711yescharacteristic
247C2×C4⋊C4C22×D4711yescharacteristic
248C2×C22⋊C4—21no—
249C2×C22⋊C4—942no—
250C2×C22⋊C4C22⋊D4712yes—
251C2×C42—21no—
252C2×C42—942no—
253C2×C42C22×D4711yescharacteristic
254C2×C42C22×D4711yescharacteristic
255#107—1882no—
256#106—1882no—
257#106—1882no—
258#106—1882no—
259C4⋊C8—478no—
260C22⋊C8—478no—
261#97—474no—
262C4×C8—474no—
263#95—22no—
264#95—22no—
265#95—942no—
266#95—942no—
267#95—942no—
268C24C2×C22⋊D4711yescharacteristic
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—21no—
274C22×C4—21no—
275C22×C4—42no—
276C22×C4—82no—
277C22×C4—942no—
278C22×C4—942no—
279C22×C4C2×Dic9412yes—
280C22×C4C2×C4×D4712yes—
281C22×C4C2×C22⋊D4712yes—
282C22×C4C2×C22⋊D4712yes—
283C22×C4C2×C22⋊D4711yescharacteristic
284C22×C4C23×D4711yescharacteristic
285C2×C8—474no—
286C2×C8—474no—
287C2×C8—942no—
288C2×C8—942no—
289C2×C8—942no—
290C2×C8—942no—
291C4⋊C4—22no—
292C4⋊C4—22no—
293C4⋊C4—22no—
294C4⋊C4—22no—
295C4⋊C4—42no—
296C4⋊C4—82no—
297C4⋊C4—82no—
298C4⋊C4—942no—
299C4⋊C4—942no—
300C4⋊C4—944no—
301C4⋊C4—3762no—
302C4⋊C4Dic47⋊C418yes—
303C22⋊C4—42no—
304C22⋊C4—3762no—
305C22⋊C4Dic47⋊C418yes—
306C42—42no—
307C42—3762no—
308C42C2×Dic9412yes—
309C42C2×Dic9412yes—
310C42C2×C4×D4712yes—
311C42C2×C4×D4712yes—
312C23—22no—
313C23—22no—
314C23—21no—
315C23—21no—
316C23C2×Dic47⋊C412yes—
317C23C22×C22⋊D4711yescharacteristic
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—42no—
327C2×C4—42no—
328C2×C4—42no—
329C2×C4—42no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—84no—
335C2×C4—1882no—
336C2×C4—1882no—
337C2×C4—1882no—
338C2×C4—1882no—
339C2×C4—1882no—
340C2×C4—1882no—
341C2×C4C2×Dic47⋊C414yes—
342C2×C4C2×Dic47⋊C414yes—
343C2×C4C2×Dic47⋊C412yes—
344C2×C4C22×C4×D4711yescharacteristic
345C2×C4C22×Dic9411yescharacteristic
346C2×C4#1309406912yes—
347C2×C4#1309407212yes—
348C8—944no—
349C8—944no—
350C22—24no—
351C22—22no—
352C22—42no—
353C22—41no—
354C22—41no—
355C22C22×Dic47⋊C411yescharacteristic
356C22#1568969011yescharacteristic
357C22#1568969711yescharacteristic
358C22#1568993112yes—
359C22#1568993412yes—
360C4—24no—
361C4—24no—
362C4—22no—
363C4—81no—
364C4—81no—
365C4—81no—
366C4—81no—
367C4—3762no—
368C4—3762no—
369C4#1568960412yes—
370C4#1568975412yes—
371C2—42no—
372C2#2024601211yescharacteristic
373C2#2025198112yes—
374C2#2025444911yescharacteristic
375C2#2025446011yescharacteristic
376C2#2025468612yes—
subgroupG/H
centerC23C22×C22⋊D47
commutatorC2×C4×C47C23×C4
FrattiniC22×C4C23×D47
FittingC47×#2317C2
socleC23×C47C22×D4
prime radicalC24×C47C2×D4
cosocleC22×C4×C47C24
hypercenter#2317D47
hypocenterC47#21990
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C2×#117 ◁ #2317

Complements

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

NH ≅ G/Ntype
3. #20244154C2semidirect
4. #20244149C2semidirect
7. #20243557C2semidirect
8. #20243544C2semidirect
9. #20243523C2semidirect
10. #20243518C2semidirect
12. #15689184C4semidirect
51. C4⋊(C47⋊C8)C2×C4semidirect
53. #13093935C2×C4semidirect
54. C4×C47⋊C8C2×C4semidirect
228. C47#21990semidirect

Maximal subgroups

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

subgroupindex
C2×#156891842
#202459972
#202441542
#202441492
#202435872
#202435812
#202435572
#202435442
#202435232
#202435182
C47×#23172
#2199047