Finite Group Databasebeta

← #26388708

Subgroups — #26388708

Order 12032.

1716 conjugacy classes of subgroups.

The subgroup lattice has 406 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20248132C214yesmaximal, complemented
2#20248063C214yesmaximal
3#20247471C214yesmaximal
4#20247372C214yesmaximal, complemented
5#20246693C211yesmaximal, characteristic, complemented
6#20246681C211yesmaximal, characteristic, complemented
7#20246656C212yesmaximal
8C4×#13094014C214yesmaximal, complemented
9C4×#13094017C214yesmaximal
10C42×Dic94C212yesmaximal, complemented
11C47×#2298C211yesmaximal, characteristic
12C2×#13094017C2214yes—
13C2×#13094014C2214yes—
14C2×C4×Dic94C2214yes—
15C2×C4×Dic94C2212yes—
16#15689065C2214yes—
17#15689062C2212yes—
18#15689059C2212yes—
19#15689057C2214yes—
20#15689056C2214yes—
21#15689055C2214yes—
22#15689046C2214yes—
23#15689044C2214yes—
24#15689043C2212yes—
25#15689035C2212yes—
26#15689020C2214yes—
27#15689018C2214yes—
28#15689017C2214yes—
29#15689016C2214yes—
30#15689011C2214yes—
31#15689010C2214yes—
32#15689009C2214yes—
33#15689008C2214yes—
34#15689008C2214yes—
35#15689005C2214yes—
36#15689005C2214yes—
37#15689004C2214yes—
38#15688990C2214yes—
39#15688987C2214yes—
40C4×Dic47⋊C4C2214yes—
41C4×Dic47⋊C4C2214yes—
42C4×Dic47⋊C4C2214yes—
43C4×Dic47⋊C4C2214yes—
44C4×Dic47⋊C4C2214yes—
45C4×C4⋊Dic47C2214yes—
46C4×C4⋊Dic47C2212yes—
47C4×C4⋊Dic47C2214yes—
48C4×C22⋊Dic47C2214yes—
49C42×Dic47C2214yes—
50C2×#117×C47C2211yescharacteristic
51C47×#388C2214yes—
52C47×#387C2214yes—
53C4×C4⋊C4×C47C2212yes—
54C4×C22⋊C4×C47C2212yes—
55C47×#376C2211yescharacteristic
56C43×C47C2211yescharacteristic
57C22×Dic94C2312yes—
58#13094017—216no—
59#13094014—216no—
60C4×Dic94—28no—
61C4×Dic94—216no—
62C2×C4⋊Dic47C2314yes—
63C2×C4⋊Dic47C2312yes—
64C2×C4⋊Dic47C2314yes—
65C2×C4⋊Dic47C2314yes—
66C2×Dic47⋊C4C2314yes—
67C2×Dic47⋊C4C2314yes—
68C2×Dic47⋊C4C2314yes—
69C2×Dic47⋊C4C2314yes—
70C2×Dic47⋊C4C2314yes—
71C2×Dic47⋊C4C2314yes—
72C2×Dic47⋊C4C2314yes—
73C2×C22⋊Dic47C2314yes—
74C2×C22⋊Dic47C2314yes—
75C2×C4×Dic47C2314yes—
76C2×C4×Dic47C2314yes—
77C2×C4×Dic47C2314yes—
78C2×C4×Dic47C2314yes—
79C2×C4×Dic47C2314yes—
80C2×C4×Dic47C2314yes—
81C47⋊#95C2314yes—
82C47⋊#95C2314yes—
83C47⋊#95C2314yes—
84C47⋊#95C2314yes—
85C47⋊#95C2312yes—
86C47⋊#95C2312yes—
87C47⋊#95C2314yes—
88C47⋊#95C2314yes—
89C47⋊#95C2314yes—
90C47⋊#95C2314yes—
91C47⋊#95C2314yes—
92C47⋊#95C2314yes—
93C23×C4×C47C2311yescharacteristic
94#117×C47—24no—
95C2×C4⋊C4×C47C2314yes—
96C2×C4⋊C4×C47C2312yes—
97C2×C22⋊C4×C47C2314yes—
98C2×C22⋊C4×C47C2312yes—
99C2×C42×C47C2312yes—
100C2×C42×C47C2312yes—
101C2×C42×C47C2314yes—
102C2×C42×C47C2311yescharacteristic
103C2×C42×C47C2311yescharacteristic
104#95×C47C2314yes—
105#95×C47C2314yes—
106#95×C47C2314yes—
107C24×C47C2411yescharacteristic
108C22×C4×C47—22no—
109C22×C4×C47—24no—
110C22×C4×C47C2412yes—
111C22×C4×C47C2414yes—
112C22×C4×C47C2411yescharacteristic
113C22×C4×C47C2414yes—
114C22×C4×C47C2412yes—
115C22×C4×C47C2411yescharacteristic
116C22×Dic47C2414yes—
117C22×Dic47C2414yes—
118C22×Dic47C2414yes—
119C22×Dic47C2414yes—
120C2×Dic94—24no—
121C2×Dic94—28no—
122C4⋊C4×C47—24no—
123C4⋊C4×C47—28no—
124C22⋊C4×C47—24no—
125C22⋊C4×C47—28no—
126C42×C47—22no—
127C42×C47—28no—
128C42×C47—24no—
129C42×C47—24no—
130C42×C47#4114yes—
131C22⋊Dic47—28no—
132C22⋊Dic47—28no—
133C4⋊Dic47—24no—
134C4⋊Dic47—28no—
135C4⋊Dic47—28no—
136C4⋊Dic47—28no—
137Dic47⋊C4—28no—
138Dic47⋊C4—28no—
139Dic47⋊C4—28no—
140Dic47⋊C4—28no—
141Dic47⋊C4—28no—
142Dic47⋊C4—28no—
143Dic47⋊C4#41116yes—
144C4×Dic47—28no—
145C4×Dic47—28no—
146C4×Dic47—28no—
147C4×Dic47—28no—
148C4×Dic47—28no—
149C4×Dic47—28no—
150C23×C47—24no—
151C23×C47—21no—
152C23×C47—22no—
153C23×C47C2511yescharacteristic
154C2×C4×C47—24no—
155C2×C4×C47—24no—
156C2×C4×C47—24no—
157C2×C4×C47—24no—
158C2×C4×C47—24no—
159C2×C4×C47—24no—
160C2×C4×C47—22no—
161C2×C4×C47—21no—
162C2×C4×C47—24no—
163C2×C4×C47—22no—
164C2×C4×C47—44no—
165C2×C4×C47C2×#4118yes—
166C2×C4×C47C2×#4114yes—
167C2×C4×C47C2×#4112yes—
168C2×C4×C472−1+414yes—
169C2×Dic47—24no—
170C2×Dic47—24no—
171C2×Dic47—24no—
172C2×Dic47—24no—
173C2×Dic47—24no—
174C2×Dic47—24no—
175C2×Dic47—24no—
176C2×Dic47—24no—
177C2×Dic47—24no—
178C2×Dic47—24no—
179C2×Dic47C2×#4118yes—
180C2×Dic47C2×#4118yes—
181Dic94—216no—
182#34973—471nomaximal, Sylow
183C22×C47—42no—
184C22×C47—41no—
185C22×C47—44no—
186C22×C47C22×#4114yes—
187C22×C47C22×#4111yescharacteristic
188C22×C47C2×2−1+412yes—
189C4×C47—24no—
190C4×C47—28no—
191C4×C47—42no—
192C4×C47—41no—
193C4×C47—44no—
194C4×C47#54114yes—
195Dic47—28no—
196Dic47—28no—
197Dic47—44no—
198Dic47—44no—
199#2524—474no—
200#2519—474no—
201#2469—474no—
202#2462—474no—
203#2402—471no—
204#2399—471no—
205#2396—472no—
206C4×#125—474no—
207C4×#126—474no—
208#2298D4711yescharacteristic
209C42×Q8—472no—
210C2×C47—81no—
211C2×C47#344412yes—
212C2×C47C2×#54111yescharacteristic
213C2×C47#348214yes—
214C2×#126—474no—
215C2×#125—474no—
216C2×C4×Q8—472no—
217C2×C4×Q8—474no—
218C2×#117C2×D4711yescharacteristic
219#400—472no—
220#400—474no—
221#399—472no—
222#398—474no—
223#398—474no—
224#398—474no—
225#396—474no—
226#395—472no—
227#395—474no—
228#393—472no—
229#389—474no—
230#389—474no—
231#388—474no—
232#388—474no—
233#388—474no—
234#388C2×D4714yes—
235#387—474no—
236#387—474no—
237#387—474no—
238#387—474no—
239#387—474no—
240#387—474no—
241#387—474no—
242#387C2×D4714yes—
243#383—474no—
244#382—474no—
245C4×C4⋊C4—474no—
246C4×C4⋊C4—474no—
247C4×C4⋊C4—474no—
248C4×C4⋊C4—474no—
249C4×C4⋊C4—472no—
250C4×C4⋊C4—474no—
251C4×C4⋊C4—474no—
252C4×C4⋊C4—474no—
253C4×C4⋊C4C2×D4712yes—
254C4×C22⋊C4—474no—
255C4×C22⋊C4C2×D4712yes—
256#376C2×D4711yescharacteristic
257C43—474no—
258C43C2×D4711yescharacteristic
259C47#3497311yescharacteristic, complemented, Sylow
260C22×Q8—472no—
261C23×C4C22×D4711yescharacteristic
262#126—9416no—
263#125—9416no—
264C4×Q8—9416no—
265C4×Q8—948no—
266#117—24no—
267C2×C4⋊C4—474no—
268C2×C4⋊C4—474no—
269C2×C4⋊C4—474no—
270C2×C4⋊C4—474no—
271C2×C4⋊C4—472no—
272C2×C4⋊C4—474no—
273C2×C4⋊C4—474no—
274C2×C4⋊C4—474no—
275C2×C4⋊C4—474no—
276C2×C4⋊C4—474no—
277C2×C4⋊C4—474no—
278C2×C4⋊C4C22×D4712yes—
279C2×C4⋊C4C22×D4714yes—
280C2×C22⋊C4—474no—
281C2×C22⋊C4—474no—
282C2×C22⋊C4C22×D4712yes—
283C2×C22⋊C4C22×D4714yes—
284C2×C42—474no—
285C2×C42—474no—
286C2×C42—474no—
287C2×C42—474no—
288C2×C42—474no—
289C2×C42—474no—
290C2×C42C22×D4714yes—
291C2×C42C22×D4711yescharacteristic
292C2×C42C22×D4712yes—
293C2×C42C22×D4712yes—
294C2×C42C22×D4711yescharacteristic
295#95—474no—
296#95—474no—
297#95—474no—
298#95—474no—
299#95—472no—
300#95—472no—
301#95—474no—
302#95—474no—
303#95—474no—
304#95—474no—
305#95—474no—
306#95—474no—
307#95C22×D4714yes—
308#95C22×D4714yes—
309#95C22×D4714yes—
310C24C23×D4711yescharacteristic
311C2×Q8—948no—
312C2×Q8—944no—
313C22×C4—22no—
314C22×C4—24no—
315C22×C4—474no—
316C22×C4—474no—
317C22×C4—474no—
318C22×C4—474no—
319C22×C4C23×D4711yescharacteristic
320C22×C4C23×D4714yes—
321C22×C4C23×D4712yes—
322C22×C4C23×D4714yes—
323C22×C4C23×D4712yes—
324C22×C4C23×D4711yescharacteristic
325C4⋊C4—28no—
326C4⋊C4—24no—
327C4⋊C4—4716no—
328C4⋊C4—948no—
329C4⋊C4—948no—
330C4⋊C4—948no—
331C4⋊C4—948no—
332C4⋊C4—948no—
333C4⋊C4—948no—
334C4⋊C4—948no—
335C4⋊C4—948no—
336C4⋊C4—948no—
337C4⋊C4—944no—
338C22⋊C4—28no—
339C22⋊C4—24no—
340C22⋊C4—948no—
341C22⋊C4—948no—
342C42—28no—
343C42—22no—
344C42—24no—
345C42—24no—
346C42—948no—
347C42—948no—
348C42—948no—
349C42—948no—
350C42—948no—
351C42—948no—
352C42#1062535014yes—
353C23—24no—
354C23—21no—
355C23—22no—
356C23C24×D4711yescharacteristic
357Q8—9416no—
358C2×C4—24no—
359C2×C4—24no—
360C2×C4—24no—
361C2×C4—24no—
362C2×C4—24no—
363C2×C4—24no—
364C2×C4—22no—
365C2×C4—21no—
366C2×C4—24no—
367C2×C4—22no—
368C2×C4—44no—
369C2×C4—478no—
370C2×C4—478no—
371C2×C4—944no—
372C2×C4—944no—
373C2×C4—944no—
374C2×C4—944no—
375C2×C4—944no—
376C2×C4—944no—
377C2×C4—944no—
378C2×C4—944no—
379C2×C4—944no—
380C2×C4—944no—
381C2×C4#41×D4718yes—
382C2×C4C2×#1062535014yes—
383C2×C4C2×#1062535012yes—
384C2×C4#1309407114yes—
385C22—42no—
386C22—44no—
387C22—41no—
388C22C2×#41×D4714yes—
389C22C22×#1062535011yescharacteristic
390C22C2×#1309407112yes—
391C4—24no—
392C4—28no—
393C4—42no—
394C4—44no—
395C4—41no—
396C4—948no—
397C4—948no—
398C4—1884no—
399C4—1884no—
400C4#1568973514yes—
401C2—81no—
402C2#2025668112yes—
403C2C2×#1568973511yescharacteristic
404C2#2025700114yes—
subgroupG/H
centerC22×C4C23×D47
commutator, socle, cosocleC23×C47C25
FrattiniC23C24×D47
FittingC47×#2298C2
prime radicalC24×C47C24
hypercenter#2298D47
hypocenterC47#34973
LCSG ▷ C23×C47 ▷ C47
UCSC1 ◁ C22×C4 ◁ #2298

Complements

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

NH ≅ G/Ntype
1. #20248132C2semidirect
4. #20247372C2semidirect
5. #20246693C2semidirect
6. #20246681C2semidirect
8. C4×#13094014C2semidirect
10. C42×Dic94C2semidirect
259. C47#34973semidirect

Maximal subgroups

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

subgroupindex
#202481322
#202480632
#202474712
#202473722
#202466932
#202466812
#202466562
C4×#130940142
C4×#130940172
C42×Dic942
C47×#22982
#3497347