Finite Group Databasebeta

← #26376468

Subgroups — #26376468

Order 12032.

2250 conjugacy classes of subgroups.

The subgroup lattice has 394 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×Q8×D47C211yesmaximal, characteristic
2#20246355C213yesmaximal, complemented
3#20246351C213yesmaximal, complemented
4#20246309C211yesmaximal, characteristic, complemented
5#20246264C216yesmaximal
6Q8×(C2×D47)⋊C4C212yesmaximal
7#20246035C216yesmaximal, complemented
8#20245784C213yesmaximal
9#20245781C213yesmaximal
10#20245753C212yesmaximal, complemented
11C47×#2351C211yesmaximal, characteristic, complemented
12C22×Q8×D47C2211yescharacteristic
13C2×Q8×Dic47C2212yes—
14C2×Q8×Dic47C2211yescharacteristic
15C2×C4×Dic94C2213yes—
16C2×#13093989C2213yes—
17C4×Q8×D47C4116yescomplemented
18C2×C4⋊C4×D47C2213yes—
19C2×C42×D47C2213yes—
20#15689029C2216yes—
21#15689028C2216yes—
22#15689022C2216yes—
23#15689010C22112yes—
24#15689004C22112yes—
25#15688990C2216yes—
26#15688990C2213yes—
27#15688990C2213yes—
28#15688989C2213yes—
29#15688988C2216yes—
30#15688985C2213yes—
31#15688985C2211yescharacteristic
32#15688980C2216yes—
33#15688980C2216yes—
34C4⋊C4×Dic47C2216yes—
35C4⋊C4×Dic47C2216yes—
36C4×C4⋊Dic47C2213yes—
37C4×C4⋊Dic47C2216yes—
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—
47C22×Dic94C2313yes—
48C2×Q8×D47—24no—
49C2×Q8×D47—24no—
50C2×Q8×D47C2×C418yes—
51C22×C4×D47C2313yes—
52C22×C4×D47C2311yescharacteristic
53C22×C4×D47C2313yes—
54Q8×Dic47—28no—
55Q8×Dic47C2×C418yes—
56C4×Dic94C2×C4124yes—
57#13093989C2×C4124yes—
58C2×C4⋊Dic47C2316yes—
59C2×C4⋊Dic47C2316yes—
60C2×C4⋊Dic47C2313yes—
61C2×Dic47⋊C4C2316yes—
62C4⋊C4×D47C2×C4124yes—
63C2×(C2×D47)⋊C4C2316yes—
64C2×(C2×D47)⋊C4C2312yes—
65C2×C4×Dic47C2316yes—
66C2×C4×Dic47C23112yes—
67C2×C4×Dic47C2313yes—
68C2×C4×Dic47C2313yes—
69C2×C4×Dic47C2313yes—
70C42×D47C2×C4124yes—
71C47⋊#95C2313yes—
72C47⋊#95C23112yes—
73C47⋊#95C2312yes—
74C47⋊#95C2316yes—
75C47⋊#95C2316yes—
76C47⋊#95C2316yes—
77C47⋊#95C2316yes—
78C47⋊#95C2316yes—
79C47⋊#95C2316yes—
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×C47C2313yes—
90#95×C47C2316yes—
91#95×C47C2311yescharacteristic
92C23×D47C2411yescharacteristic
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×Dic47C2413yes—
105C22×Dic47C2413yes—
106Q8×D47—232no—
107C2×C4×D47—24no—
108C2×C4×D47—26no—
109C2×C4×D47—26no—
110C2×C4×D47—46no—
111C2×C4×D47C22×C414yes—
112C2×C4×D47C22×C4112yes—
113C2×C4×D47C22×C4112yes—
114C2×Dic94—26no—
115C2×Dic94—26no—
116C2×Dic94C22×C4112yes—
117C4⋊C4×C47—212no—
118C4⋊C4×C47—46no—
119C4⋊C4×C47C22×C4112yes—
120C42×C47—212no—
121C42×C47C22×C4112yes—
122(C2×D47)⋊C4—212no—
123(C2×D47)⋊C4—24no—
124C4⋊Dic47—212no—
125C4⋊Dic47—46no—
126C4⋊Dic47C22×C4112yes—
127Dic47⋊C4C22×C4124yes—
128C4×Dic47—212no—
129C4×Dic47—412no—
130C4×Dic47C22×C4112yes—
131C4×Dic47C22×C4112yes—
132C4×Dic47C22×C4112yes—
133C23×C47C2511yescharacteristic
134C22×D47—21no—
135C22×D47—21no—
136C22×D47—22no—
137C22×D47C23×C412yes—
138C22×D472−1+414yes—
139Q8×C47#117116yes—
140C2×C4×C47—22no—
141C2×C4×C47—26no—
142C2×C4×C47—22no—
143C2×C4×C47—26no—
144C2×C4×C47—22no—
145C2×C4×C47—26no—
146C2×C4×C47—26no—
147C2×C4×C47—22no—
148C2×C4×C47C23×C412yes—
149C2×C4×C47C23×C416yes—
150C2×C4×C47C23×C416yes—
151C2×C4×C47C2×#4116yes—
152C2×C4×C47C2×#4116yes—
153C2×Dic47—23no—
154C2×Dic47—23no—
155C2×Dic47—22no—
156C2×Dic47—26no—
157C2×Dic47—22no—
158C2×Dic47—26no—
159C2×Dic47—22no—
160C2×Dic47—26no—
161C2×Dic47—26no—
162C2×Dic47—22no—
163C2×Dic47C23×C412yes—
164C2×Dic47C23×C416yes—
165C2×Dic47C23×C416yes—
166C4×D47—28no—
167C4×D47—224no—
168C4×D47—412no—
169Dic94—224no—
170#34192—471nomaximal, Sylow
171C22×C47C24×C411yescharacteristic
172C22×C47C22×#4111yescharacteristic
173C22×C47C22×#4111yescharacteristic
174C22×C47C2×2−1+412yes—
175C22×C47#58512yes—
176C2×D47—22no—
177C2×D47—22no—
178C2×D47—24no—
179C2×D47—24no—
180C2×D47#51914yes—
181C4×C47—22no—
182C4×C47—26no—
183C4×C47—42no—
184C4×C47—46no—
185C4×C47C2×#117112yes—
186Dic47—26no—
187Dic47—22no—
188Dic47—26no—
189Dic47—42no—
190Dic47—46no—
191C22×C4×Q8—471no—
192#2356—473no—
193#2356—473no—
194#2351—471no—
195#2351D4711yescharacteristic, complemented
196#2345—476no—
197Q8×C22⋊C4—472no—
198#2320—476no—
199#2283—473no—
200#2283—473no—
201#2276—472no—
202C2×C47C22×#11711yescharacteristic
203C2×C47C2×#51911yescharacteristic
204C2×C47#342811yescharacteristic
205C2×C47#351612yes—
206C2×C47#351612yes—
207D47—24no—
208C23×Q8—471no—
209C2×C4×Q8—471no—
210C2×C4×Q8—4716no—
211C2×C4×Q8—472no—
212C2×C4×Q8—473no—
213C2×C4×Q8—473no—
214C2×C4×Q8C2×D4712yes—
215C2×C4×Q8C2×D4711yescharacteristic
216C22×C4⋊C4—473no—
217C22×C42—473no—
218#391—476no—
219#391—476no—
220#389—476no—
221#389C2×D4716yes—
222#387—4712no—
223#387—4712no—
224#383—476no—
225#383—473no—
226#383—473no—
227#382—476no—
228#382—473no—
229#382C2×D4713yes—
230#381—471no—
231#381—473no—
232#380—476no—
233#380—476no—
234C4×C4⋊C4—476no—
235C4×C4⋊C4—476no—
236C4×C4⋊C4—476no—
237C4×C4⋊C4—473no—
238C4×C4⋊C4C2×D4713yes—
239C4×C22⋊C4—476no—
240C4×C22⋊C4—476no—
241#376—476no—
242#376—476no—
243C47#3419211yescharacteristic, complemented, Sylow
244C22×Q8—478no—
245C22×Q8—473no—
246C22×Q8—944no—
247C22×Q8—944no—
248C22×Q8C22×D4711yescharacteristic
249C23×C4—473no—
250C23×C4—471no—
251C23×C4—473no—
252C4×Q8—28no—
253C4×Q8—478no—
254C4×Q8—4724no—
255C4×Q8—4724no—
256C4×Q8—948no—
257C4×Q8C4×D4718yes—
258C2×C4⋊C4—476no—
259C2×C4⋊C4—473no—
260C2×C4⋊C4—4724no—
261C2×C4⋊C4—476no—
262C2×C4⋊C4—476no—
263C2×C4⋊C4C22×D4716yes—
264C2×C4⋊C4C22×D4716yes—
265C2×C4⋊C4C22×D4713yes—
266C2×C22⋊C4—472no—
267C2×C22⋊C4—476no—
268C2×C42—473no—
269C2×C42—473no—
270C2×C42—473no—
271C2×C42—4724no—
272C2×C42—476no—
273C2×C42—4712no—
274C2×C42C22×D4716yes—
275C2×C42C22×D4713yes—
276#95—472no—
277#95—473no—
278#95—476no—
279#95—476no—
280#95—476no—
281#95—4712no—
282#95—476no—
283#95—476no—
284#95—476no—
285#95—471no—
286#95C22×D4713yes—
287#95C22×D4716yes—
288#95C22×D4711yescharacteristic
289C24—471no—
290C2×Q8—4712no—
291C2×Q8—9432no—
292C2×Q8—946no—
293C2×Q8—946no—
294C2×Q8C2×C4×D4714yes—
295C2×Q8#1062535214yes—
296C2×Q8#1062535414yes—
297C22×C4—471no—
298C22×C4—473no—
299C22×C4—473no—
300C22×C4—474no—
301C22×C4—4712no—
302C22×C4—4712no—
303C22×C4—472no—
304C22×C4—476no—
305C22×C4—944no—
306C22×C4—946no—
307C22×C4—946no—
308C22×C4—1886no—
309C22×C4C23×D4712yes—
310C22×C4C23×D4716yes—
311C22×C4C23×D4711yescharacteristic
312C22×C4C23×D4713yes—
313C22×C4C23×D4713yes—
314C4⋊C4—212no—
315C4⋊C4—46no—
316C4⋊C4—4724no—
317C4⋊C4—4712no—
318C4⋊C4—9412no—
319C4⋊C4—1886no—
320C4⋊C4C2×C4×D47112yes—
321C22⋊C4—944no—
322C22⋊C4—9412no—
323C42—212no—
324C42—4712no—
325C42—4712no—
326C42—4712no—
327C42—9412no—
328C42—18812no—
329C42C2×C4×D47112yes—
330C23—472no—
331C23—474no—
332C23—941no—
333C23—941no—
334C23—942no—
335C23C24×D4711yescharacteristic
336Q8—9424no—
337Q8#13093982116yes—
338C2×C4—22no—
339C2×C4—26no—
340C2×C4—22no—
341C2×C4—26no—
342C2×C4—22no—
343C2×C4—26no—
344C2×C4—26no—
345C2×C4—22no—
346C2×C4—472no—
347C2×C4—476no—
348C2×C4—476no—
349C2×C4—948no—
350C2×C4—943no—
351C2×C4—943no—
352C2×C4—942no—
353C2×C4—946no—
354C2×C4—9424no—
355C2×C4—942no—
356C2×C4—946no—
357C2×C4—942no—
358C2×C4—946no—
359C2×C4—946no—
360C2×C4—942no—
361C2×C4—18812no—
362C2×C4C22×C4×D4712yes—
363C2×C4C22×C4×D4716yes—
364C2×C4C22×C4×D4716yes—
365C2×C4C2×#1062535216yes—
366C2×C4C2×#1062535416yes—
367C22—474no—
368C22—942no—
369C22—942no—
370C22—944no—
371C22—944no—
372C22C23×C4×D4711yescharacteristic
373C22C22×#1062535211yescharacteristic
374C22C22×#1062535411yescharacteristic
375C222−1+4×D4712yes—
376C22#1568995912yes—
377C4—22no—
378C4—26no—
379C4—42no—
380C4—46no—
381C4—946no—
382C4—942no—
383C4—946no—
384C4—1882no—
385C4—1886no—
386C4C2×#13093982112yes—
387C2—944no—
388C2C22×#1309398211yescharacteristic
389C2#519×D4711yescharacteristic
390C2#2025658711yescharacteristic
391C2#2025737412yes—
392C2#2025738412yes—
subgroupG/H
center, FrattiniC23C24×D47
commutatorC22×C47C24×C4
FittingC47×#2351C2
socle, cosocleC23×C47C25
prime radicalC23×D47C24
hypercenter#2351D47
hypocenterC47#34192
LCSG ▷ C22×C47 ▷ C47
UCSC1 ◁ C23 ◁ #2351

Complements

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

NH ≅ G/Ntype
2. #20246355C2semidirect
3. #20246351C2semidirect
4. #20246309C2semidirect
7. #20246035C2semidirect
10. #20245753C2semidirect
11. C47×#2351C2semidirect
17. C4×Q8×D47C4semidirect
195. #2351D47semidirect
243. C47#34192semidirect

Maximal subgroups

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

subgroupindex
C2×C4×Q8×D472
#202463552
#202463512
#202463092
#202462642
Q8×(C2×D47)⋊C42
#202460352
#202457842
#202457812
#202457532
C47×#23512
#3419247