Finite Group Databasebeta

← #26376523

Subgroups — #26376523

Order 12032.

2314 conjugacy classes of subgroups.

The subgroup lattice has 396 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×#10625354C211yesmaximal, characteristic
2#20246363C213yesmaximal, complemented
3#20246359C213yesmaximal, complemented
4#20246309C211yesmaximal, characteristic, complemented
5#20246272C216yesmaximal, complemented
6#20246226C212yesmaximal
7#20246027C216yesmaximal, complemented
8#20245784C213yesmaximal, complemented
9#20245781C213yesmaximal, complemented
10#20245753C212yesmaximal, complemented
11C47×#2351C211yesmaximal, characteristic, complemented
12C22×#10625354C2211yescharacteristic
13C4×#10625354C4116yescomplemented
14C2×Q8×Dic47C2212yes—
15C2×Q8×Dic47C2211yescharacteristic
16C2×C4×D188C2213yes—
17C2×#13093986C2213yes—
18C2×#13093982C2213yes—
19C2×C42×D47C2213yes—
20#15689025C2216yes—
21#15689024C2216yes—
22#15689022C2216yes—
23#15689015C22112yes—
24#15689013C22112yes—
25#15688990C2213yes—
26#15688990C2211yescharacteristic
27#15688989C2213yes—
28#15688988C2216yes—
29#15688985C2216yes—
30#15688985C2213yes—
31#15688985C2213yes—
32#15688980C2216yes—
33#15688980C2216yes—
34C4⋊C4×Dic47C2216yes—
35C4⋊C4×Dic47C2216yes—
36C4×C4⋊Dic47C2216yes—
37C4×C4⋊Dic47C2213yes—
38C4×(C2×D47)⋊C4C2216yes—
39#15688970C2216yes—
40#15688969C2216yes—
41#15688968C2216yes—
42C2×C4×Q8×C47C2212yes—
43C2×C4×Q8×C47C2211yescharacteristic
44C47×#389C2216yes—
45C47×#382C2213yes—
46C4×C4⋊C4×C47C2213yes—
47C2×#10625354—24no—
48C2×#10625354—24no—
49C2×#10625354C2×C418yes—
50C22×D188C2313yes—
51C22×C4×D47C2313yes—
52C22×C4×D47C2313yes—
53C22×C4×D47C2313yes—
54Q8×Dic47—28no—
55Q8×Dic47C2×C418yes—
56C4×D188C2×C4124yes—
57#13093986C2×C4124yes—
58#13093982C2×C4124yes—
59C2×C4⋊Dic47C2316yes—
60C2×C4⋊Dic47C2316yes—
61C2×C4⋊Dic47C2313yes—
62C2×(C2×D47)⋊C4C23112yes—
63C2×(C2×D47)⋊C4C2316yes—
64C2×(C2×D47)⋊C4C2316yes—
65C2×(C2×D47)⋊C4C2316yes—
66C2×C4×Dic47C2316yes—
67C2×C4×Dic47C23112yes—
68C2×C4×Dic47C2313yes—
69C2×C4×Dic47C2313yes—
70C2×C4×Dic47C2311yescharacteristic
71C42×D47C2×C4124yes—
72C47⋊#95C2316yes—
73C47⋊#95C2316yes—
74C47⋊#95C2312yes—
75C47⋊#95C2316yes—
76C47⋊#95C2316yes—
77C47⋊#95C2316yes—
78C47⋊#95C2313yes—
79C47⋊#95C2312yes—
80C47⋊#95C2311yescharacteristic
81C22×Q8×C47C2311yescharacteristic
82C4×Q8×C47—28no—
83C4×Q8×C47C2×C418yes—
84C2×C4⋊C4×C47C2316yes—
85C2×C4⋊C4×C47C2316yes—
86C2×C4⋊C4×C47C2313yes—
87C2×C42×C47C2316yes—
88C2×C42×C47C2313yes—
89#95×C47C2316yes—
90#95×C47C2313yes—
91#95×C47C2311yescharacteristic
92C23×D47C2413yes—
93C2×Q8×C47C22×C414yes—
94C2×Q8×C47#4114yes—
95C2×Q8×C47#4114yes—
96C22×C4×C47C2412yes—
97C22×C4×C47C2416yes—
98C22×C4×C47C2411yescharacteristic
99C22×C4×C47C2413yes—
100C22×C4×C47C2413yes—
101C22×Dic47C2412yes—
102C22×Dic47C2416yes—
103C22×Dic47C2411yescharacteristic
104C22×Dic47C2411yescharacteristic
105C22×Dic47C2413yes—
106#10625354—232no—
107C2×D188—26no—
108C2×D188—26no—
109C2×D188C22×C4112yes—
110C2×C4×D47—26no—
111C2×C4×D47—26no—
112C2×C4×D47—46no—
113C2×C4×D47—46no—
114C2×C4×D47C22×C4112yes—
115C2×C4×D47C22×C4112yes—
116C2×C4×D47C22×C4112yes—
117C4⋊C4×C47—212no—
118C4⋊C4×C47—46no—
119C4⋊C4×C47C22×C4112yes—
120C42×C47—212no—
121C42×C47C22×C4112yes—
122(C2×D47)⋊C4—46no—
123(C2×D47)⋊C4—412no—
124(C2×D47)⋊C4—46no—
125(C2×D47)⋊C4C22×C4124yes—
126C4⋊Dic47—212no—
127C4⋊Dic47—46no—
128C4⋊Dic47C22×C4112yes—
129C4×Dic47—212no—
130C4×Dic47—412no—
131C4×Dic47C22×C414yes—
132C4×Dic47C22×C4112yes—
133C4×Dic47C22×C4112yes—
134C23×C47C2511yescharacteristic
135C22×D47—23no—
136C22×D47—23no—
137C22×D47—26no—
138C22×D47—26no—
139C22×D47C23×C416yes—
140Q8×C47#117116yes—
141C2×C4×C47—26no—
142C2×C4×C47—22no—
143C2×C4×C47—22no—
144C2×C4×C47—26no—
145C2×C4×C47—26no—
146C2×C4×C47—22no—
147C2×C4×C47—26no—
148C2×C4×C47—22no—
149C2×C4×C47C23×C416yes—
150C2×C4×C47C23×C412yes—
151C2×C4×C47C23×C416yes—
152C2×C4×C47C2×#4116yes—
153C2×C4×C47C2×#4116yes—
154C2×Dic47—21no—
155C2×Dic47—21no—
156C2×Dic47—26no—
157C2×Dic47—22no—
158C2×Dic47—22no—
159C2×Dic47—26no—
160C2×Dic47—26no—
161C2×Dic47—22no—
162C2×Dic47—26no—
163C2×Dic47—22no—
164C2×Dic47C23×C412yes—
165C2×Dic47C23×C412yes—
166C2×Dic47C23×C416yes—
167D188—224no—
168C4×D47—224no—
169C4×D47—412no—
170C4×D47—412no—
171#34197—471nomaximal, Sylow
172C22×C47C24×C411yescharacteristic
173C22×C47C22×#4111yescharacteristic
174C22×C47C22×#4111yescharacteristic
175C22×C47#58512yes—
176C22×C47#58512yes—
177C2×D47—26no—
178C2×D47—26no—
179C2×D47—26no—
180C2×D47—46no—
181C2×D47—46no—
182C4×C47—22no—
183C4×C47—26no—
184C4×C47—42no—
185C4×C47—46no—
186C4×C47C2×#117112yes—
187Dic47—22no—
188Dic47—22no—
189Dic47—26no—
190Dic47—42no—
191Dic47—46no—
192C2×C4×#41—471no—
193#2357—473no—
194#2357—473no—
195#2351—471no—
196#2351D4711yescharacteristic, complemented
197#2346—476no—
198#2341—472no—
199#2319—476no—
200#2283—473no—
201#2283—473no—
202#2276—472no—
203C2×C47C22×#11711yescharacteristic
204C2×C47#342811yescharacteristic
205C2×C47#342811yescharacteristic
206C2×C47#352412yes—
207C2×C47#352412yes—
208D47—46no—
209C22×#41—471no—
210C4×#41—4716no—
211C2×C4×Q8—471no—
212C2×C4×Q8—472no—
213C2×C4×Q8C2×D4712yes—
214C2×C4×Q8C2×D4711yescharacteristic
215C2×C4×D4—473no—
216C2×C4×D4—473no—
217C2×#117—473no—
218C22×C42—473no—
219#390—476no—
220#390—476no—
221#389—476no—
222#389C2×D4716yes—
223#388—4712no—
224#388—4712no—
225#383—471no—
226#383—473no—
227#382—476no—
228#382—473no—
229#382C2×D4713yes—
230#381—476no—
231#381—473no—
232#381—473no—
233#380—476no—
234#380—476no—
235C4×C4⋊C4—476no—
236C4×C4⋊C4—476no—
237C4×C4⋊C4—476no—
238C4×C4⋊C4—473no—
239C4×C4⋊C4C2×D4713yes—
240C4×C22⋊C4—476no—
241C4×C22⋊C4—476no—
242#376—476no—
243#376—476no—
244C47#3419711yescharacteristic, complemented, Sylow
245C2×#41—478no—
246C2×#41—944no—
247C2×#41—944no—
248C22×Q8C22×D4711yescharacteristic
249C22×D4—473no—
250C23×C4—473no—
251C23×C4—473no—
252C23×C4—473no—
253C4×Q8—28no—
254C4×Q8—478no—
255C4×Q8—948no—
256C4×Q8C4×D4718yes—
257C4×D4—4724no—
258C4×D4—4724no—
259#117—4724no—
260C2×C4⋊C4—473no—
261C2×C4⋊C4—476no—
262C2×C4⋊C4—476no—
263C2×C4⋊C4C22×D4716yes—
264C2×C4⋊C4C22×D4716yes—
265C2×C4⋊C4C22×D4713yes—
266C2×C22⋊C4—476no—
267C2×C22⋊C4—476no—
268C2×C22⋊C4—4712no—
269C2×C22⋊C4—476no—
270C2×C42—473no—
271C2×C42—473no—
272C2×C42—471no—
273C2×C42—4724no—
274C2×C42—476no—
275C2×C42—4712no—
276C2×C42C22×D4716yes—
277C2×C42C22×D4713yes—
278#95—472no—
279#95—476no—
280#95—476no—
281#95—476no—
282#95—476no—
283#95—476no—
284#95—472no—
285#95—473no—
286#95—471no—
287#95C22×D4716yes—
288#95C22×D4713yes—
289#95C22×D4711yescharacteristic
290C24—473no—
291#41—9432no—
292C2×Q8C2×C4×D4714yes—
293C2×Q8#1062535214yes—
294C2×Q8#1062535414yes—
295C2×D4—4712no—
296C2×D4—946no—
297C2×D4—946no—
298C22×C4—471no—
299C22×C4—471no—
300C22×C4—473no—
301C22×C4—4712no—
302C22×C4—4712no—
303C22×C4—4712no—
304C22×C4—472no—
305C22×C4—476no—
306C22×C4—946no—
307C22×C4—946no—
308C22×C4—1886no—
309C22×C4—1886no—
310C22×C4C23×D4712yes—
311C22×C4C23×D4716yes—
312C22×C4C23×D4711yescharacteristic
313C22×C4C23×D4713yes—
314C22×C4C23×D4713yes—
315C4⋊C4—212no—
316C4⋊C4—46no—
317C4⋊C4—4712no—
318C4⋊C4—9412no—
319C4⋊C4—1886no—
320C4⋊C4C2×C4×D47112yes—
321C22⋊C4—4724no—
322C22⋊C4—18812no—
323C22⋊C4—1886no—
324C22⋊C4—1886no—
325C42—212no—
326C42—474no—
327C42—4712no—
328C42—4712no—
329C42—9412no—
330C42—18812no—
331C42C2×C4×D47112yes—
332C23—476no—
333C23—943no—
334C23—943no—
335C23—946no—
336C23—946no—
337C23C24×D4711yescharacteristic
338Q8#13093982116yes—
339D4—9424no—
340C2×C4—26no—
341C2×C4—22no—
342C2×C4—22no—
343C2×C4—26no—
344C2×C4—26no—
345C2×C4—22no—
346C2×C4—26no—
347C2×C4—22no—
348C2×C4—472no—
349C2×C4—472no—
350C2×C4—476no—
351C2×C4—941no—
352C2×C4—941no—
353C2×C4—942no—
354C2×C4—946no—
355C2×C4—9424no—
356C2×C4—942no—
357C2×C4—946no—
358C2×C4—946no—
359C2×C4—942no—
360C2×C4—946no—
361C2×C4—942no—
362C2×C4—18812no—
363C2×C4—18812no—
364C2×C4C22×C4×D4712yes—
365C2×C4C22×C4×D4716yes—
366C2×C4C22×C4×D4716yes—
367C2×C4C2×#1062535216yes—
368C2×C4C2×#1062535416yes—
369C22—946no—
370C22—946no—
371C22—946no—
372C22—1886no—
373C22—1886no—
374C22C23×C4×D4711yescharacteristic
375C22C22×#1062535211yescharacteristic
376C22C22×#1062535411yescharacteristic
377C22#1568995912yes—
378C22#1568996012yes—
379C4—26no—
380C4—22no—
381C4—42no—
382C4—46no—
383C4—942no—
384C4—942no—
385C4—946no—
386C4—1882no—
387C4—1886no—
388C4C2×#13093982112yes—
389C2—1886no—
390C2C22×#1309398211yescharacteristic
391C2#2025658711yescharacteristic
392C2#2025659011yescharacteristic
393C2#2025749012yes—
394C2#2025750012yes—
subgroupG/H
center, FrattiniC23C24×D47
commutatorC22×C47C24×C4
FittingC47×#2351C2
socle, cosocleC23×C47C25
prime radicalC22×#10625354C22
hypercenter#2351D47
hypocenterC47#34197
LCSG ▷ C22×C47 ▷ C47
UCSC1 ◁ C23 ◁ #2351

Complements

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

NH ≅ G/Ntype
2. #20246363C2semidirect
3. #20246359C2semidirect
4. #20246309C2semidirect
5. #20246272C2semidirect
7. #20246027C2semidirect
8. #20245784C2semidirect
9. #20245781C2semidirect
10. #20245753C2semidirect
11. C47×#2351C2semidirect
13. C4×#10625354C4semidirect
196. #2351D47semidirect
244. C47#34197semidirect

Maximal subgroups

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

subgroupindex
C2×C4×#106253542
#202463632
#202463592
#202463092
#202462722
#202462262
#202460272
#202457842
#202457812
#202457532
C47×#23512
#3419747