Finite Group Databasebeta

← #26376522

Subgroups — #26376522

Order 12032.

2314 conjugacy classes of subgroups.

The subgroup lattice has 392 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×#10625354C211yesmaximal, characteristic
2#20246357C216yesmaximal, complemented
3#20246306C212yesmaximal, complemented
4#20246269C216yesmaximal, complemented
5#20246226C212yesmaximal
6#20246032C213yesmaximal, complemented
7#20246026C213yesmaximal, complemented
8#20245784C213yesmaximal, complemented
9#20245780C213yesmaximal, complemented
10#20245753C211yesmaximal, characteristic, complemented
11C47×#2276C211yesmaximal, 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#15689021C2216yes—
23#15689019C2216yes—
24#15689014C22112yes—
25#15689012C22112yes—
26#15688990C2211yescharacteristic
27#15688990C2213yes—
28#15688988C2216yes—
29#15688987C2216yes—
30#15688985C2216yes—
31#15688985C2216yes—
32#15688982C2213yes—
33#15688982C2213yes—
34#15688980C2216yes—
35C4⋊C4×Dic47C2216yes—
36C4×Dic47⋊C4C2216yes—
37C4×C4⋊Dic47C2213yes—
38C4×(C2×D47)⋊C4C2216yes—
39C4×(C2×D47)⋊C4C2216yes—
40#15688969C2213yes—
41#15688969C2216yes—
42C2×C4×Q8×C47C2212yes—
43C2×C4×Q8×C47C2211yescharacteristic
44C4×C4⋊C4×C47C2213yes—
45C4×C4⋊C4×C47C2216yes—
46C47×#376C2213yes—
47C2×#10625354—28no—
48C2×#10625354C2×C418yes—
49C22×D188C2313yes—
50C22×C4×D47C2313yes—
51C22×C4×D47C2313yes—
52C22×C4×D47C2313yes—
53Q8×Dic47—28no—
54Q8×Dic47C2×C418yes—
55C4×D188C2×C4124yes—
56#13093986C2×C4124yes—
57#13093982C2×C4124yes—
58C2×C4⋊Dic47C2316yes—
59C2×C4⋊Dic47C2313yes—
60C2×Dic47⋊C4C2316yes—
61C2×Dic47⋊C4C2316yes—
62C2×(C2×D47)⋊C4C23112yes—
63C2×(C2×D47)⋊C4C2316yes—
64C2×(C2×D47)⋊C4C2316yes—
65C2×(C2×D47)⋊C4C2316yes—
66C2×C4×Dic47C2316yes—
67C2×C4×Dic47C2316yes—
68C2×C4×Dic47C2313yes—
69C2×C4×Dic47C2313yes—
70C2×C4×Dic47C2311yescharacteristic
71C42×D47C2×C4124yes—
72C47⋊#95C2311yescharacteristic
73C47⋊#95C2316yes—
74C47⋊#95C2313yes—
75C47⋊#95C2316yes—
76C47⋊#95C23112yes—
77C47⋊#95C2312yes—
78C47⋊#95C2316yes—
79C47⋊#95C2312yes—
80C22×Q8×C47C2311yescharacteristic
81C4×Q8×C47—28no—
82C4×Q8×C47C2×C418yes—
83C2×C4⋊C4×C47C2316yes—
84C2×C4⋊C4×C47C2313yes—
85C2×C42×C47C2316yes—
86C2×C42×C47C2316yes—
87C2×C42×C47C2313yes—
88#95×C47C2311yescharacteristic
89#95×C47C2313yes—
90#95×C47C2316yes—
91C23×D47C2413yes—
92C2×Q8×C47C22×C414yes—
93C2×Q8×C47#4118yes—
94C22×C4×C47C2416yes—
95C22×C4×C47C2412yes—
96C22×C4×C47C2413yes—
97C22×C4×C47C2411yescharacteristic
98C22×C4×C47C2413yes—
99C22×Dic47C2412yes—
100C22×Dic47C2416yes—
101C22×Dic47C2411yescharacteristic
102C22×Dic47C2411yescharacteristic
103C22×Dic47C2413yes—
104#10625354—232no—
105C2×D188—212no—
106C2×D188C22×C4112yes—
107C2×C4×D47—212no—
108C2×C4×D47—46no—
109C2×C4×D47—43no—
110C2×C4×D47—43no—
111C2×C4×D47C22×C4112yes—
112C2×C4×D47C22×C4112yes—
113C2×C4×D47C22×C4112yes—
114C4⋊C4×C47—212no—
115C4⋊C4×C47C22×C4112yes—
116C42×C47—212no—
117C42×C47—46no—
118C42×C47C22×C4112yes—
119(C2×D47)⋊C4—46no—
120(C2×D47)⋊C4—412no—
121(C2×D47)⋊C4—46no—
122(C2×D47)⋊C4C22×C4124yes—
123C4⋊Dic47—212no—
124C4⋊Dic47C22×C4112yes—
125Dic47⋊C4—46no—
126Dic47⋊C4—46no—
127C4×Dic47—212no—
128C4×Dic47—46no—
129C4×Dic47C22×C414yes—
130C4×Dic47C22×C4112yes—
131C4×Dic47C22×C4112yes—
132C23×C47C2511yescharacteristic
133C22×D47—26no—
134C22×D47—23no—
135C22×D47—23no—
136C22×D47—26no—
137C22×D47C23×C416yes—
138Q8×C47#117116yes—
139C2×C4×C47—23no—
140C2×C4×C47—23no—
141C2×C4×C47—22no—
142C2×C4×C47—22no—
143C2×C4×C47—26no—
144C2×C4×C47—22no—
145C2×C4×C47—26no—
146C2×C4×C47—22no—
147C2×C4×C47—26no—
148C2×C4×C47C23×C416yes—
149C2×C4×C47C23×C412yes—
150C2×C4×C47C23×C416yes—
151C2×C4×C47C2×#41112yes—
152C2×Dic47—22no—
153C2×Dic47—26no—
154C2×Dic47—21no—
155C2×Dic47—21no—
156C2×Dic47—22no—
157C2×Dic47—26no—
158C2×Dic47—26no—
159C2×Dic47—22no—
160C2×Dic47—26no—
161C2×Dic47—22no—
162C2×Dic47C23×C412yes—
163C2×Dic47C23×C412yes—
164C2×Dic47C23×C416yes—
165D188—224no—
166C4×D47—224no—
167C4×D47—412no—
168C4×D47—412no—
169#34197—471nomaximal, Sylow
170C22×C47C24×C411yescharacteristic
171C22×C47C22×#4112yes—
172C22×C47#58511yescharacteristic
173C22×C47#58511yescharacteristic
174C22×C47#58512yes—
175C2×D47—26no—
176C2×D47—26no—
177C2×D47—26no—
178C2×D47—46no—
179C2×D47—46no—
180C4×C47—22no—
181C4×C47—26no—
182C4×C47—46no—
183C4×C47—42no—
184C4×C47C2×#117112yes—
185Dic47—22no—
186Dic47—22no—
187Dic47—26no—
188Dic47—42no—
189Dic47—46no—
190C2×C4×#41—471no—
191#2357—476no—
192#2351—472no—
193#2346—476no—
194#2341—472no—
195#2319—473no—
196#2319—473no—
197#2283—473no—
198#2283—473no—
199#2276—471no—
200#2276D4711yescharacteristic, complemented
201C2×C47C22×#11711yescharacteristic
202C2×C47#342811yescharacteristic
203C2×C47#342811yescharacteristic
204C2×C47#352412yes—
205C2×C47#352412yes—
206D47—46no—
207C22×#41—471no—
208C4×#41—4716no—
209C2×C4×Q8—471no—
210C2×C4×Q8—472no—
211C2×C4×Q8C2×D4712yes—
212C2×C4×Q8C2×D4711yescharacteristic
213C2×C4×D4—473no—
214C2×C4×D4—473no—
215C2×#117—473no—
216C22×C42—473no—
217#390—476no—
218#390—476no—
219#389—476no—
220#389—476no—
221#388—4712no—
222#388—4712no—
223#383—471no—
224#383—473no—
225#382—476no—
226#382—476no—
227#381—476no—
228#381—476no—
229#380—476no—
230#380—473no—
231#380—473no—
232C4×C4⋊C4—476no—
233C4×C4⋊C4—473no—
234C4×C4⋊C4—476no—
235C4×C4⋊C4C2×D4713yes—
236C4×C4⋊C4C2×D4716yes—
237C4×C22⋊C4—476no—
238C4×C22⋊C4—476no—
239#376—473no—
240#376—476no—
241#376C2×D4713yes—
242C47#3419711yescharacteristic, complemented, Sylow
243C2×#41—478no—
244C2×#41—948no—
245C22×Q8C22×D4711yescharacteristic
246C22×D4—473no—
247C23×C4—473no—
248C23×C4—473no—
249C23×C4—473no—
250C4×Q8—28no—
251C4×Q8—478no—
252C4×Q8—948no—
253C4×Q8C4×D4718yes—
254C4×D4—4724no—
255C4×D4—4724no—
256#117—4724no—
257C2×C4⋊C4—473no—
258C2×C4⋊C4—476no—
259C2×C4⋊C4—476no—
260C2×C4⋊C4—476no—
261C2×C4⋊C4C22×D4716yes—
262C2×C4⋊C4C22×D4713yes—
263C2×C22⋊C4—476no—
264C2×C22⋊C4—4712no—
265C2×C22⋊C4—476no—
266C2×C22⋊C4—476no—
267C2×C42—473no—
268C2×C42—473no—
269C2×C42—471no—
270C2×C42—4724no—
271C2×C42—476no—
272C2×C42—476no—
273C2×C42C22×D4716yes—
274C2×C42C22×D4716yes—
275C2×C42C22×D4713yes—
276#95—471no—
277#95—476no—
278#95—473no—
279#95—476no—
280#95—4712no—
281#95—472no—
282#95—476no—
283#95—472no—
284#95C22×D4711yescharacteristic
285#95C22×D4713yes—
286#95C22×D4716yes—
287C24—473no—
288#41—9432no—
289C2×Q8C2×C4×D4714yes—
290C2×Q8#1062535018yes—
291C2×D4—4712no—
292C2×D4—9412no—
293C22×C4—471no—
294C22×C4—471no—
295C22×C4—473no—
296C22×C4—4712no—
297C22×C4—4712no—
298C22×C4—4712no—
299C22×C4—472no—
300C22×C4—476no—
301C22×C4—9412no—
302C22×C4—1883no—
303C22×C4—1883no—
304C22×C4—1886no—
305C22×C4C23×D4716yes—
306C22×C4C23×D4712yes—
307C22×C4C23×D4711yescharacteristic
308C22×C4C23×D4713yes—
309C22×C4C23×D4713yes—
310C4⋊C4—212no—
311C4⋊C4—4712no—
312C4⋊C4—9412no—
313C4⋊C4—1886no—
314C4⋊C4—1886no—
315C4⋊C4C2×C4×D47112yes—
316C22⋊C4—4724no—
317C22⋊C4—1886no—
318C22⋊C4—1886no—
319C22⋊C4—18812no—
320C42—212no—
321C42—46no—
322C42—474no—
323C42—4712no—
324C42—4712no—
325C42—9412no—
326C42—1886no—
327C42C2×C4×D47112yes—
328C23—476no—
329C23—946no—
330C23—943no—
331C23—943no—
332C23—946no—
333C23C24×D4711yescharacteristic
334Q8#13093981116yes—
335D4—9424no—
336C2×C4—23no—
337C2×C4—23no—
338C2×C4—22no—
339C2×C4—22no—
340C2×C4—26no—
341C2×C4—26no—
342C2×C4—22no—
343C2×C4—26no—
344C2×C4—22no—
345C2×C4—472no—
346C2×C4—472no—
347C2×C4—476no—
348C2×C4—942no—
349C2×C4—941no—
350C2×C4—946no—
351C2×C4—941no—
352C2×C4—9424no—
353C2×C4—942no—
354C2×C4—946no—
355C2×C4—942no—
356C2×C4—946no—
357C2×C4—946no—
358C2×C4—942no—
359C2×C4—18812no—
360C2×C4—18812no—
361C2×C4C22×C4×D4712yes—
362C2×C4C22×C4×D4716yes—
363C2×C4C22×C4×D4716yes—
364C2×C4C2×#10625350112yes—
365C22—946no—
366C22—946no—
367C22—946no—
368C22—1886no—
369C22—1886no—
370C22C23×C4×D4711yescharacteristic
371C22C22×#1062535012yes—
372C22#1568995811yescharacteristic
373C22#1568996011yescharacteristic
374C22#1568996012yes—
375C4—26no—
376C4—22no—
377C4—46no—
378C4—42no—
379C4—942no—
380C4—942no—
381C4—946no—
382C4—1882no—
383C4—1886no—
384C4C2×#13093981112yes—
385C2—1886no—
386C2C22×#1309398111yescharacteristic
387C2#2025658811yescharacteristic
388C2#2025659011yescharacteristic
389C2#2025748912yes—
390C2#2025749912yes—
subgroupG/H
center, FrattiniC23C24×D47
commutatorC22×C47C24×C4
FittingC47×#2276C2
socle, cosocleC23×C47C25
prime radicalC22×#10625354C22
hypercenter#2276D47
hypocenterC47#34197
LCSG ▷ C22×C47 ▷ C47
UCSC1 ◁ C23 ◁ #2276

Complements

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

NH ≅ G/Ntype
2. #20246357C2semidirect
3. #20246306C2semidirect
4. #20246269C2semidirect
6. #20246032C2semidirect
7. #20246026C2semidirect
8. #20245784C2semidirect
9. #20245780C2semidirect
10. #20245753C2semidirect
11. C47×#2276C2semidirect
13. C4×#10625354C4semidirect
200. #2276D47semidirect
242. 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
#202463572
#202463062
#202462692
#202462262
#202460322
#202460262
#202457842
#202457802
#202457532
C47×#22762
#3419747