Finite Group Databasebeta

← #26388704

Subgroups — #26388704

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#20248123C214yesmaximal, complemented
2#20248054C214yesmaximal
3#20247462C214yesmaximal
4#20247363C214yesmaximal, complemented
5#20246693C211yesmaximal, characteristic, complemented
6#20246658C212yesmaximal
7C4×#13094014C214yesmaximal, complemented
8C4×#13094019C214yesmaximal
9#20245883C211yesmaximal, characteristic
10C4×Q8×Dic47C212yesmaximal, complemented
11C47×#2399C211yesmaximal, characteristic, complemented
12C2×#13094019C2214yes—
13C2×#13094014C2214yes—
14C2×Q8×Dic47C2214yes—
15C2×#13093981C2211yescharacteristic
16#15689065C2214yes—
17#15689061C2212yes—
18#15689058C2212yes—
19#15689057C2214yes—
20#15689056C2214yes—
21#15689045C2214yes—
22#15689042C2212yes—
23#15689034C2212yes—
24#15689020C2214yes—
25#15689018C2214yes—
26#15689015C2214yes—
27#15689013C2214yes—
28#15689012C2214yes—
29#15689012C2214yes—
30#15689009C2214yes—
31#15689009C2214yes—
32#15689008C2214yes—
33#15689006C2214yes—
34#15689005C2214yes—
35#15689005C2214yes—
36#15689004C2214yes—
37#15689004C2214yes—
38#15688990C2214yes—
39#15688987C2214yes—
40C4⋊C4×Dic47C2214yes—
41C4⋊C4×Dic47C2214yes—
42C4⋊C4×Dic47C2214yes—
43C4×Dic47⋊C4C2212yes—
44C4×Dic47⋊C4C2214yes—
45C4×Dic47⋊C4C2214yes—
46C4×C4⋊Dic47C2212yes—
47C4×C4⋊Dic47C2214yes—
48C4×(C2×D47)⋊C4C2212yes—
49C4×(C2×D47)⋊C4C2214yes—
50C42×Dic47C2211yescharacteristic
51C42×Dic47C2214yes—
52C2×C4×Q8×C47C2212yes—
53C47×#398C2214yes—
54C47×#395C2214yes—
55C4×C4⋊C4×C47C2214yes—
56C47×#376C2211yescharacteristic
57C22×C4×D47C2311yescharacteristic
58#13094019—216no—
59#13094014—216no—
60Q8×Dic47—216no—
61#13093981—24no—
62C2×C4⋊Dic47C2314yes—
63C2×C4⋊Dic47C2314yes—
64C2×C4⋊Dic47C2314yes—
65C2×Dic47⋊C4C2314yes—
66C2×Dic47⋊C4C2314yes—
67C2×Dic47⋊C4C2314yes—
68C2×Dic47⋊C4C2314yes—
69C2×Dic47⋊C4C2314yes—
70C2×Dic47⋊C4C2312yes—
71C2×(C2×D47)⋊C4C2314yes—
72C2×(C2×D47)⋊C4C2314yes—
73C2×(C2×D47)⋊C4C2314yes—
74C2×(C2×D47)⋊C4C2312yes—
75C2×C4×Dic47C2312yes—
76C2×C4×Dic47C2314yes—
77C2×C4×Dic47C2314yes—
78C2×C4×Dic47C2314yes—
79C2×C4×Dic47C2314yes—
80C2×C4×Dic47C2314yes—
81C2×C4×Dic47C2314yes—
82C2×C4×Dic47C2311yescharacteristic
83C47⋊#95C2314yes—
84C47⋊#95C2314yes—
85C47⋊#95C2314yes—
86C47⋊#95C2314yes—
87C47⋊#95C2314yes—
88C47⋊#95C2312yes—
89C47⋊#95C2312yes—
90C47⋊#95C2314yes—
91C47⋊#95C2314yes—
92C47⋊#95C2314yes—
93C47⋊#95C2314yes—
94C47⋊#95C2314yes—
95C22×Q8×C47C2312yes—
96C4×Q8×C47—28no—
97C2×C4⋊C4×C47C2314yes—
98C2×C4⋊C4×C47C2312yes—
99C2×C4⋊C4×C47C2314yes—
100C2×C4⋊C4×C47C2314yes—
101C2×C42×C47C2314yes—
102C2×C42×C47C2312yes—
103C2×C42×C47C2311yescharacteristic
104#95×C47C2314yes—
105#95×C47C2314yes—
106#95×C47C2314yes—
107C23×D47C2411yescharacteristic
108C2×Q8×C47—24no—
109C2×Q8×C47—28no—
110C22×C4×C47C2414yes—
111C22×C4×C47C2414yes—
112C22×C4×C47C2412yes—
113C22×C4×C47C2414yes—
114C22×C4×C47C2411yescharacteristic
115C22×Dic47C2411yescharacteristic
116C22×Dic47C2414yes—
117C22×Dic47C2412yes—
118C22×Dic47C2414yes—
119C22×Dic47C2414yes—
120C2×C4×D47—22no—
121C2×C4×D47—24no—
122C4⋊C4×C47—28no—
123C4⋊C4×C47—24no—
124C4⋊C4×C47—28no—
125C4⋊C4×C47—28no—
126C42×C47—28no—
127C42×C47—24no—
128C42×C47#4114yes—
129(C2×D47)⋊C4—24no—
130(C2×D47)⋊C4—28no—
131(C2×D47)⋊C4—28no—
132(C2×D47)⋊C4—28no—
133C4⋊Dic47—28no—
134C4⋊Dic47—28no—
135C4⋊Dic47#41116yes—
136Dic47⋊C4—24no—
137Dic47⋊C4—28no—
138Dic47⋊C4—28no—
139Dic47⋊C4—28no—
140Dic47⋊C4—28no—
141Dic47⋊C4—28no—
142C4×Dic47—22no—
143C4×Dic47—28no—
144C4×Dic47—24no—
145C4×Dic47—28no—
146C4×Dic47—28no—
147C4×Dic47—28no—
148C4×Dic47—28no—
149C4×Dic47—28no—
150C23×C47C2511yescharacteristic
151C22×D47—24no—
152C22×D47—21no—
153C22×D47—22no—
154Q8×C47—216no—
155C2×C4×C47—24no—
156C2×C4×C47—24no—
157C2×C4×C47—24no—
158C2×C4×C47—24no—
159C2×C4×C47—24no—
160C2×C4×C47—24no—
161C2×C4×C47—24no—
162C2×C4×C47—24no—
163C2×C4×C47—24no—
164C2×C4×C47C2×#4118yes—
165C2×C4×C47C2×#4114yes—
166C2×C4×C47C2×#4112yes—
167C2×C4×C472−1+414yes—
168C2×Dic47—24no—
169C2×Dic47—24no—
170C2×Dic47—24no—
171C2×Dic47—24no—
172C2×Dic47—24no—
173C2×Dic47—22no—
174C2×Dic47—24no—
175C2×Dic47—24no—
176C2×Dic47—21no—
177C2×Dic47—24no—
178C2×Dic47—22no—
179C2×Dic47C2×#4118yes—
180C2×Dic47C2×#4118yes—
181C4×D47—44no—
182#34973—471nomaximal, Sylow
183C22×C47C22×#4114yes—
184C22×C47C22×#4111yescharacteristic
185C22×C47C2×2−1+412yes—
186C2×D47—42no—
187C2×D47—41no—
188C2×D47—44no—
189C4×C47—28no—
190C4×C47—24no—
191C4×C47—44no—
192C4×C47—44no—
193C4×C47#54114yes—
194Dic47—28no—
195Dic47—28no—
196Dic47—42no—
197Dic47—41no—
198Dic47—44no—
199#2524—474no—
200#2519—474no—
201#2469—474no—
202#2462—474no—
203#2402—471no—
204#2399D4711yescharacteristic, complemented
205#2396—472no—
206C4×#125—474no—
207C4×#126—474no—
208#2298—471no—
209C42×Q8—472no—
210C2×C47#344412yes—
211C2×C47C2×#54111yescharacteristic
212C2×C47#348214yes—
213D47—81no—
214C2×#126—474no—
215C2×#125—474no—
216C2×C4×Q8—474no—
217C2×C4×Q8C2×D4712yes—
218C2×#117—471no—
219#400—472no—
220#400—474no—
221#399—472no—
222#398—474no—
223#398—474no—
224#398C2×D4714yes—
225#396—474no—
226#395—472no—
227#395C2×D4714yes—
228#393—472no—
229#389—474no—
230#389—474no—
231#388—474no—
232#388—474no—
233#388—474no—
234#388—474no—
235#387—474no—
236#387—474no—
237#387—474no—
238#387—474no—
239#387—474no—
240#387—474no—
241#387—474no—
242#387—474no—
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—474no—
250C4×C4⋊C4—474no—
251C4×C4⋊C4—472no—
252C4×C4⋊C4—472no—
253C4×C4⋊C4C2×D4714yes—
254C4×C22⋊C4—474no—
255C4×C22⋊C4—472no—
256#376C2×D4711yescharacteristic
257C43—474no—
258C43—471no—
259C47#3497311yescharacteristic, complemented, Sylow
260C22×Q8C22×D4712yes—
261C23×C4—471no—
262#126—9416no—
263#125—9416no—
264C4×Q8—28no—
265C4×Q8—9416no—
266#117—944no—
267C2×C4⋊C4—474no—
268C2×C4⋊C4—472no—
269C2×C4⋊C4—474no—
270C2×C4⋊C4—474no—
271C2×C4⋊C4—474no—
272C2×C4⋊C4—474no—
273C2×C4⋊C4—474no—
274C2×C4⋊C4—474no—
275C2×C4⋊C4—474no—
276C2×C4⋊C4C22×D4714yes—
277C2×C4⋊C4C22×D4712yes—
278C2×C4⋊C4C22×D4714yes—
279C2×C4⋊C4C22×D4714yes—
280C2×C22⋊C4—472no—
281C2×C22⋊C4—474no—
282C2×C22⋊C4—474no—
283C2×C22⋊C4—474no—
284C2×C42—474no—
285C2×C42—474no—
286C2×C42—474no—
287C2×C42—474no—
288C2×C42—474no—
289C2×C42—471no—
290C2×C42—472no—
291C2×C42—474no—
292C2×C42C22×D4714yes—
293C2×C42C22×D4712yes—
294C2×C42C22×D4711yescharacteristic
295#95—474no—
296#95—474no—
297#95—474no—
298#95—474no—
299#95—474no—
300#95—472no—
301#95—472no—
302#95—474no—
303#95—474no—
304#95—474no—
305#95—474no—
306#95—474no—
307#95C22×D4714yes—
308#95C22×D4714yes—
309#95C22×D4714yes—
310C24—471no—
311C2×Q8—28no—
312C2×Q8—24no—
313C22×C4—474no—
314C22×C4—474no—
315C22×C4—471no—
316C22×C4—474no—
317C22×C4—472no—
318C22×C4—942no—
319C22×C4—944no—
320C22×C4C23×D4714yes—
321C22×C4C23×D4714yes—
322C22×C4C23×D4714yes—
323C22×C4C23×D4712yes—
324C22×C4C23×D4711yescharacteristic
325C4⋊C4—28no—
326C4⋊C4—28no—
327C4⋊C4—28no—
328C4⋊C4—24no—
329C4⋊C4—4716no—
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—948no—
339C22⋊C4—948no—
340C22⋊C4—948no—
341C22⋊C4—944no—
342C42—28no—
343C42—24no—
344C42—948no—
345C42—948no—
346C42—948no—
347C42—948no—
348C42—948no—
349C42—942no—
350C42—944no—
351C42—948no—
352C42#1062535214yes—
353C23—944no—
354C23—941no—
355C23—942no—
356C23C24×D4711yescharacteristic
357Q8—216no—
358C2×C4—24no—
359C2×C4—24no—
360C2×C4—24no—
361C2×C4—24no—
362C2×C4—24no—
363C2×C4—24no—
364C2×C4—24no—
365C2×C4—24no—
366C2×C4—24no—
367C2×C4—478no—
368C2×C4—478no—
369C2×C4—944no—
370C2×C4—944no—
371C2×C4—944no—
372C2×C4—944no—
373C2×C4—944no—
374C2×C4—944no—
375C2×C4—944no—
376C2×C4—942no—
377C2×C4—941no—
378C2×C4—944no—
379C2×C4—942no—
380C2×C4—1884no—
381C2×C4#41×D4718yes—
382C2×C4C2×#1062535214yes—
383C2×C4C2×#1062535212yes—
384C2×C4#1309407214yes—
385C22—1882no—
386C22—1884no—
387C22—1881no—
388C22C2×#41×D4714yes—
389C22C22×#1062535211yescharacteristic
390C22C2×#1309407212yes—
391C4—28no—
392C4—24no—
393C4—44no—
394C4—44no—
395C4—948no—
396C4—948no—
397C4—1882no—
398C4—1884no—
399C4—1881no—
400C4#1568973414yes—
401C2—3761no—
402C2#2025668012yes—
403C2C2×#1568973411yescharacteristic
404C2#2025700614yes—
subgroupG/H
centerC22×C4C23×D47
commutator, socle, cosocleC23×C47C25
FrattiniC23C24×D47
FittingC47×#2399C2
prime radicalC23×D47C24
hypercenter#2399D47
hypocenterC47#34973
LCSG ▷ C23×C47 ▷ C47
UCSC1 ◁ C22×C4 ◁ #2399

Complements

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

NH ≅ G/Ntype
1. #20248123C2semidirect
4. #20247363C2semidirect
5. #20246693C2semidirect
7. C4×#13094014C2semidirect
10. C4×Q8×Dic47C2semidirect
11. C47×#2399C2semidirect
204. #2399D47semidirect
259. C47#34973semidirect

Maximal subgroups

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

subgroupindex
#202481232
#202480542
#202474622
#202473632
#202466932
#202466582
C4×#130940142
C4×#130940192
#202458832
C4×Q8×Dic472
C47×#23992
#3497347