Finite Group Databasebeta

← #26138936

Subgroups — #26138936

Order 12032.

1342 conjugacy classes of subgroups.

The subgroup lattice has 388 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15688991C211yesmaximal, characteristic
2C2×#15688969C211yesmaximal, characteristic
3#20241651C212yesmaximal
4#20241647C212yesmaximal
5#20241621C214yesmaximal
6Dic47⋊#95C212yesmaximal
7#20241614C212yesmaximal
8C2×C4×C4⋊C4×C47C211yesmaximal, characteristic
9C22×Dic47⋊C4C2214yes—
10C22×C4×Dic47C2212yes—
11#15688991—24no—
12#15688969C418yescomplemented
13C2×C47⋊#95C2211yescharacteristic
14C2×C47⋊#95C2211yescharacteristic
15C2×C47⋊#95C2212yes—
16C2×C47⋊#95C2212yes—
17C2×C47⋊#95C2214yes—
18C2×C47⋊#95C2214yes—
19C2×C47⋊#95C2214yes—
20C2×C47⋊#95C2212yes—
21C2×C47⋊#95C2211yescharacteristic
22C2×C47⋊#95C2211yescharacteristic
23C22×C4⋊C4×C47C2212yes—
24C22×C42×C47C2212yes—
25C22×C42×C47C2211yescharacteristic
26C4×C4⋊C4×C47—24no—
27C2×#95×C47C2212yes—
28C23×Dic47C2312yes—
29C23×Dic47C2312yes—
30C23×Dic47C2312yes—
31C23×Dic47C2312yes—
32C2×Dic47⋊C4—28no—
33C2×Dic47⋊C4—28no—
34C2×Dic47⋊C4—44no—
35C2×C4×Dic47—28no—
36C2×C4×Dic47C2×C418yes—
37C47⋊#95—24no—
38C47⋊#95—24no—
39C47⋊#95—22no—
40C47⋊#95—22no—
41C47⋊#95—28no—
42C47⋊#95—28no—
43C47⋊#95—44no—
44C47⋊#95C2×C414yes—
45C47⋊#95C2×C418yes—
46C47⋊#95C2×C414yes—
47C23×C4×C47C2314yes—
48C23×C4×C47C2311yescharacteristic
49C23×C4×C47C2311yescharacteristic
50C23×C4×C47C2311yescharacteristic
51C2×C4⋊C4×C47—24no—
52C2×C4⋊C4×C47—28no—
53C2×C42×C47—24no—
54C2×C42×C47—42no—
55C2×C42×C47—42no—
56C2×C42×C47C2×C414yes—
57C2×C42×C47D414yes—
58C2×C42×C47Q814yes—
59#95×C47—24no—
60C24×C47C2411yescharacteristic
61C22×C4×C47—24no—
62C22×C4×C47—24no—
63C22×C4×C47—21no—
64C22×C4×C47—21no—
65C22×C4×C47—21no—
66C22×C4×C47—21no—
67C22×C4×C47—24no—
68C22×C4×C47—24no—
69C22×C4×C47—24no—
70C22×C4×C47—24no—
71C22×C4×C47—24no—
72C22×C4×C47—24no—
73C22×C4×C47C22×C412yes—
74C22×C4×C47C22×C412yes—
75C22×C4×C47C22×C412yes—
76C22×C4×C47C2×D412yes—
77C22×C4×C47C2×D412yes—
78C22×C4×C47C2×D412yes—
79C22×C4×C47C2×Q812yes—
80C22×C4×C47C2×Q812yes—
81C22×C4×C47C2×Q812yes—
82C22×C4×C47#4118yes—
83C22×Dic47—24no—
84C22×Dic47—22no—
85C22×Dic47—22no—
86C22×Dic47—24no—
87C22×Dic47—24no—
88C22×Dic47—22no—
89C22×Dic47—22no—
90C22×Dic47—22no—
91C22×Dic47—22no—
92C22×Dic47—24no—
93C22×Dic47—24no—
94C22×Dic47—24no—
95C22×Dic47—24no—
96C22×Dic47—22no—
97C22×Dic47—22no—
98C22×Dic47—22no—
99C22×Dic47—22no—
100C22×Dic47C22×C414yes—
101C22×Dic47C22×C414yes—
102C22×Dic47#4118yes—
103C4⋊C4×C47—216no—
104C42×C47—48no—
105C42×C47C4⋊C4116yescomplemented
106Dic47⋊C4—416no—
107C4×Dic47—216no—
108C23×C47C23×C411yescharacteristic
109C23×C47C22×D411yescharacteristic
110C23×C47C22×Q811yescharacteristic
111C23×C47C2×#4114yes—
112C23×C47C2×#4114yes—
113C23×C47H(5,𝔽2)11yescharacteristic
114C23×C47H(5,𝔽2)11yescharacteristic
115C23×C472−1+411yescharacteristic
116C23×C472−1+411yescharacteristic
117C2×C4×C47—24no—
118C2×C4×C47—24no—
119C2×C4×C47—22no—
120C2×C4×C47—22no—
121C2×C4×C47—24no—
122C2×C4×C47—24no—
123C2×C4×C47—24no—
124C2×C4×C47—24no—
125C2×C4×C47—22no—
126C2×C4×C47—22no—
127C2×C4×C47—22no—
128C2×C4×C47—22no—
129C2×C4×C47—28no—
130C2×C4×C47—28no—
131C2×C4×C47—28no—
132C2×C4×C47—44no—
133C2×C4×C47—44no—
134C2×C4×C47—44no—
135C2×C4×C47—44no—
136C2×C4×C47C2×C4⋊C414yes—
137C2×C4×C47C2×C4⋊C414yes—
138C2×C4×C47C2×C4⋊C414yes—
139C2×Dic47—24no—
140C2×Dic47—24no—
141C2×Dic47—28no—
142C2×Dic47—28no—
143C2×Dic47—24no—
144C2×Dic47—24no—
145C2×Dic47—24no—
146C2×Dic47—44no—
147C2×Dic47—44no—
148C2×Dic47—42no—
149C2×Dic47—42no—
150C2×Dic47—42no—
151C2×Dic47—42no—
152C2×Dic47—42no—
153C2×Dic47—42no—
154C2×Dic47—44no—
155C2×Dic47—44no—
156C2×Dic47—44no—
157C2×Dic47—44no—
158C2×Dic47#11718yes—
159#13952—471nomaximal, Sylow
160C22×C47C22×C4⋊C411yescharacteristic
161C22×C47C2×#11712yes—
162C22×C47C4×#4112yes—
163C22×C47#52011yescharacteristic
164C22×C47#52011yescharacteristic
165C22×C47#52512yes—
166C22×C47#53212yes—
167C22×C47#53312yes—
168C22×C47#53312yes—
169C22×C47#53511yescharacteristic
170C22×C47#53611yescharacteristic
171C22×C47#54411yescharacteristic
172C22×C47#54411yescharacteristic
173C22×C47#55114yes—
174C22×C47#55214yes—
175C22×C47#55314yes—
176C22×C47#55614yes—
177C4×C47—24no—
178C4×C47—24no—
179C4×C47—24no—
180C4×C47—48no—
181Dic47—28no—
182Dic47—44no—
183Dic47—82no—
184Dic47—82no—
185C2×#383—471no—
186C2×C4×C4⋊C4D4711yescharacteristic
187C2×#376—471no—
188#1444—472no—
189#1442—472no—
190#1436—474no—
191C4⋊#95—472no—
192#1434—472no—
193C2×C47#228511yescharacteristic
194C2×C47#229611yescharacteristic
195C2×C47#231711yescharacteristic
196C2×C47#234312yes—
197C2×C47#234712yes—
198C2×C47#260312yes—
199C2×C47#260412yes—
200C2×C47#260512yes—
201C2×C47#260612yes—
202C22×C4⋊C4—474no—
203C22×C4⋊C4C2×D4712yes—
204C22×C42—472no—
205C22×C42C2×D4712yes—
206C22×C42C2×D4711yescharacteristic
207#383—944no—
208C4×C4⋊C4—24no—
209#376—478no—
210C2×#95—472no—
211C2×#95—471no—
212C2×#95—471no—
213C2×#95—471no—
214C2×#95—474no—
215C2×#95—474no—
216C2×#95—471no—
217C2×#95—472no—
218C2×#95—472no—
219C2×#95—474no—
220C2×#95C2×D4712yes—
221C47#1395211yescharacteristic, complemented, Sylow
222C23×C4—472no—
223C23×C4—472no—
224C23×C4—472no—
225C23×C4—472no—
226C23×C4C22×D4714yes—
227C23×C4C22×D4711yescharacteristic
228C23×C4C22×D4711yescharacteristic
229C23×C4C22×D4711yescharacteristic
230C2×C4⋊C4—28no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4—948no—
233C2×C4⋊C4—948no—
234C2×C4⋊C4—1884no—
235C2×C42—24no—
236C2×C42—42no—
237C2×C42—42no—
238C2×C42—478no—
239C2×C42—948no—
240C2×C42Dic9414yes—
241C2×C42C4×D4714yes—
242C2×C42C22⋊D4714yes—
243#95—24no—
244#95—474no—
245#95—478no—
246#95—474no—
247#95—948no—
248#95—948no—
249#95—942no—
250#95—942no—
251#95—944no—
252#95—944no—
253#95—1884no—
254C24C23×D4711yescharacteristic
255C22×C4—24no—
256C22×C4—24no—
257C22×C4—21no—
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—21no—
261C22×C4—24no—
262C22×C4—24no—
263C22×C4—24no—
264C22×C4—24no—
265C22×C4—24no—
266C22×C4—24no—
267C22×C4—474no—
268C22×C4—478no—
269C22×C4—474no—
270C22×C4—944no—
271C22×C4—942no—
272C22×C4—942no—
273C22×C4—944no—
274C22×C4—944no—
275C22×C4—942no—
276C22×C4—942no—
277C22×C4—942no—
278C22×C4—942no—
279C22×C4—944no—
280C22×C4—944no—
281C22×C4—944no—
282C22×C4—944no—
283C22×C4—942no—
284C22×C4—942no—
285C22×C4—942no—
286C22×C4—942no—
287C22×C4C2×Dic9412yes—
288C22×C4C2×Dic9412yes—
289C22×C4C2×Dic9412yes—
290C22×C4C2×C4×D4712yes—
291C22×C4C2×C4×D4712yes—
292C22×C4C2×C4×D4712yes—
293C22×C4#1062535018yes—
294C22×C4C2×C22⋊D4712yes—
295C22×C4C2×C22⋊D4712yes—
296C22×C4C2×C22⋊D4712yes—
297C4⋊C4—216no—
298C4⋊C4—18816no—
299C42—48no—
300C42—9416no—
301C42Dic47⋊C4116yescomplemented
302C23C22×C4×D4711yescharacteristic
303C23C22×C22⋊D4711yescharacteristic
304C23C22×Dic9411yescharacteristic
305C23#41×D4714yes—
306C23C2×#1062535014yes—
307C23#1309406911yescharacteristic
308C23#1309406911yescharacteristic
309C23#1309407211yescharacteristic
310C23#1309407211yescharacteristic
311C2×C4—22no—
312C2×C4—22no—
313C2×C4—24no—
314C2×C4—24no—
315C2×C4—22no—
316C2×C4—22no—
317C2×C4—24no—
318C2×C4—24no—
319C2×C4—24no—
320C2×C4—24no—
321C2×C4—22no—
322C2×C4—22no—
323C2×C4—28no—
324C2×C4—28no—
325C2×C4—28no—
326C2×C4—44no—
327C2×C4—44no—
328C2×C4—44no—
329C2×C4—44no—
330C2×C4—478no—
331C2×C4—948no—
332C2×C4—948no—
333C2×C4—944no—
334C2×C4—944no—
335C2×C4—944no—
336C2×C4—944no—
337C2×C4—944no—
338C2×C4—1884no—
339C2×C4—1884no—
340C2×C4—1884no—
341C2×C4—1884no—
342C2×C4—1882no—
343C2×C4—1882no—
344C2×C4—1882no—
345C2×C4—1882no—
346C2×C4—1882no—
347C2×C4—1882no—
348C2×C4—1884no—
349C2×C4—1884no—
350C2×C4C2×Dic47⋊C414yes—
351C2×C4C2×Dic47⋊C414yes—
352C2×C4C2×Dic47⋊C414yes—
353C22C22×Dic47⋊C411yescharacteristic
354C22#117×D4712yes—
355C22C4×#1062535012yes—
356C22#1568960411yescharacteristic
357C22#1568960411yescharacteristic
358C22#1568963412yes—
359C22#1568966912yes—
360C22#1568967612yes—
361C22#1568968012yes—
362C22#1568969011yescharacteristic
363C22#1568969711yescharacteristic
364C22#1568975411yescharacteristic
365C22#1568975411yescharacteristic
366C22#1568980214yes—
367C22#1568981414yes—
368C22#1568982214yes—
369C22#1568984514yes—
370C4—24no—
371C4—24no—
372C4—24no—
373C4—48no—
374C4—948no—
375C4—3762no—
376C4—3762no—
377C4—1884no—
378C2#2024579111yescharacteristic
379C2#2024587111yescharacteristic
380C2#2024601211yescharacteristic
381C2#2024625112yes—
382C2#2024628212yes—
383C2#2024900912yes—
384C2#2024902012yes—
385C2#2024903112yes—
386C2#2024904012yes—
subgroupG/H
center, FrattiniC24C23×D47
commutatorC23×C47C23×C4
FittingC2×C4×C4⋊C4×C47C2
socle, prime radical, cosocleC24×C47C24
hypercenterC2×C4×C4⋊C4D47
hypocenterC47#13952
LCSG ▷ C23×C47 ▷ C47
UCSC1 ◁ C24 ◁ C2×C4×C4⋊C4

Complements

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

NH ≅ G/Ntype
12. #15688969C4semidirect
105. C42×C47C4⋊C4semidirect
221. C47#13952semidirect
301. C42Dic47⋊C4semidirect

Maximal subgroups

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

subgroupindex
C2×#156889912
C2×#156889692
#202416512
#202416472
#202416212
Dic47⋊#952
#202416142
C2×C4×C4⋊C4×C472
#1395247