Finite Group Databasebeta

← #26107177

Subgroups — #26107177

Order 12032.

720 conjugacy classes of subgroups.

The subgroup lattice has 418 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20243150C211yesmaximal, characteristic
2C2×#15688860C212yesmaximal
3C4×C47⋊#95C211yesmaximal, characteristic
4#20241266C212yesmaximal
5C4×C22⋊Q8×C47C211yesmaximal, characteristic
6C22×C4×Dic47—21no—
7C2×C22⋊(C47⋊C8)C2212yes—
8C2×#13093935—21no—
9C2×C47⋊#95—21no—
10C2×C47⋊#95C2212yes—
11#15688878—21no—
12#15688860C418yescomplemented
13C2×C22⋊Q8×C47C2211yescharacteristic
14C2×C4×Q8×C47C411yescharacteristic, complemented
15C22×C42×C47C2211yescharacteristic
16C47×#391C411yescharacteristic, complemented
17C47×#389C411yescharacteristic, complemented
18C47×#387—21no—
19C47×#385—21no—
20C47×#380C2211yescharacteristic
21C4×C4⋊C4×C47—21no—
22C4×C4⋊C4×C47C411yescharacteristic, complemented
23C4×C22⋊C4×C47—21no—
24C23×Dic47—22no—
25C22×C47⋊C8—22no—
26C2×C4×Dic47—24no—
27C2×C4×Dic47—42no—
28C2×C4×Dic47—42no—
29C22⋊(C47⋊C8)—22no—
30C22⋊(C47⋊C8)C2×C414yes—
31#13093935—42no—
32C47⋊#95—24no—
33C47⋊#95—22no—
34C47⋊#95C2×C414yes—
35C22×Q8×C47C2×C411yescharacteristic
36C23×C4×C47D411yescharacteristic
37C23×C4×C47D411yescharacteristic
38C23×C4×C47C2311yescharacteristic
39C22⋊Q8×C47—22no—
40C22⋊Q8×C47C2×C414yes—
41C4×Q8×C47—42no—
42C2×C4⋊C4×C47—21no—
43C2×C4⋊C4×C47—21no—
44C2×C4⋊C4×C47—21no—
45C2×C4⋊C4×C47C2×C411yescharacteristic
46C2×C4⋊C4×C47C2×C411yescharacteristic
47C2×C4⋊C4×C47C2×C411yescharacteristic
48C2×C22⋊C4×C47—21no—
49C2×C22⋊C4×C47—21no—
50C2×C42×C47—21no—
51C2×C42×C47—21no—
52C2×C42×C47—42no—
53C2×C42×C47C2×C411yescharacteristic
54C2×C42×C47C2×C411yescharacteristic
55C2×C42×C47D411yescharacteristic
56C2×C42×C47D411yescharacteristic
57#95×C47—21no—
58#95×C47—21no—
59#95×C47—21no—
60#95×C47—21no—
61#95×C47—21no—
62C24×C47C2×D411yescharacteristic
63C2×Q8×C47—21no—
64C2×Q8×C47—41no—
65C2×Q8×C47—41no—
66C2×Q8×C47C4212yescomplemented
67C22×C4×C47—21no—
68C22×C4×C47—21no—
69C22×C4×C47—22no—
70C22×C4×C47—22no—
71C22×C4×C47—21no—
72C22×C4×C47—21no—
73C22×C4×C47—22no—
74C22×C4×C47—22no—
75C22×C4×C47—21no—
76C22×C4×C47—21no—
77C22×C4×C47—41no—
78C22×C4×C47—41no—
79C22×C4×C47—41no—
80C22×C4×C47C22⋊C411yescharacteristic
81C22×C4×C47C22⋊C411yescharacteristic
82C22×C4×C47C22⋊C411yescharacteristic
83C22×C4×C47C22⋊C411yescharacteristic
84C22×C4×C47C22×C411yescharacteristic
85C22×C4×C47C22×C412yes—
86C22×C4×C47C2×D411yescharacteristic
87C22×C4×C47#4112yes—
88C22×Dic47—24no—
89C22×Dic47—24no—
90C22×Dic47—24no—
91C22×Dic47—42no—
92C22×Dic47—42no—
93C22×Dic47—42no—
94C22×Dic47—42no—
95C4⋊C4×C47—21no—
96C4⋊C4×C47—24no—
97C4⋊C4×C47—41no—
98C4⋊C4×C47—41no—
99C4⋊C4×C47—42no—
100C4⋊C4×C47—42no—
101C4⋊C4×C47C4212yescomplemented
102C22⋊C4×C47—24no—
103C22⋊C4×C47—41no—
104C22⋊C4×C47—41no—
105C42×C47—22no—
106C42×C47—22no—
107C42×C47—41no—
108C42×C47—41no—
109C42×C47—42no—
110C42×C47—42no—
111C4×Dic47—48no—
112C2×C47⋊C8—24no—
113C2×C47⋊C8—42no—
114C2×C47⋊C8—42no—
115C23×C47—21no—
116C23×C47—21no—
117C23×C47—21no—
118C23×C47—21no—
119C23×C47C2×C22⋊C411yescharacteristic
120C23×C47C4×D412yes—
121C23×C47AGL(1,ℤ/8)11yescharacteristic
122C23×C47#13711yescharacteristic
123C23×C47#13712yes—
124Q8×C47—42no—
125Q8×C47—42no—
126C2×C4×C47—24no—
127C2×C4×C47—24no—
128C2×C4×C47—21no—
129C2×C4×C47—21no—
130C2×C4×C47—21no—
131C2×C4×C47—21no—
132C2×C4×C47—21no—
133C2×C4×C47—21no—
134C2×C4×C47—21no—
135C2×C4×C47—21no—
136C2×C4×C47—21no—
137C2×C4×C47—21no—
138C2×C4×C47—22no—
139C2×C4×C47—22no—
140C2×C4×C47—21no—
141C2×C4×C47—21no—
142C2×C4×C47—21no—
143C2×C4×C47—42no—
144C2×C4×C47—42no—
145C2×C4×C47—44no—
146C2×C4×C47—41no—
147C2×C4×C47—41no—
148C2×C4×C47—41no—
149C2×C4×C47—41no—
150C2×C4×C47—41no—
151C2×C4×C47—41no—
152C2×C4×C47—41no—
153C2×C4×C47—41no—
154C2×C4×C47—41no—
155C2×C4×C47—41no—
156C2×C4×C47#9912yes—
157C2×C4×C47#10412yes—
158C2×C4×C47#10412yes—
159C2×C4×C47C2×C4211yescharacteristic
160C2×C4×C47#11711yescharacteristic
161C2×C4×C47C4×D412yes—
162C2×Dic47—28no—
163C2×Dic47—44no—
164C2×Dic47—44no—
165C2×Dic47—44no—
166C2×Dic47—44no—
167C2×Dic47—44no—
168C2×Dic47—44no—
169C47⋊C8—44no—
170#8219—471nomaximal, Sylow
171C22×C47—22no—
172C22×C47—22no—
173C22×C47—21no—
174C22×C47—21no—
175C22×C47—22no—
176C22×C47—22no—
177C22×C47—21no—
178C22×C47—21no—
179C22×C47C4×C22⋊C411yescharacteristic
180C22×C47C2×#9911yescharacteristic
181C22×C47#41011yescharacteristic
182C22×C47#41711yescharacteristic
183C22×C47#41911yescharacteristic
184C22×C47C2×#10411yescharacteristic
185C22×C47C2×#10411yescharacteristic
186C22×C47#44012yes—
187C22×C47#44112yes—
188C4×C47—22no—
189C4×C47—21no—
190C4×C47—21no—
191C4×C47—22no—
192C4×C47—22no—
193C4×C47—22no—
194C4×C47—42no—
195C4×C47—42no—
196C4×C47—41no—
197C4×C47—81no—
198C4×C47—81no—
199Dic47—48no—
200C4×C22⋊Q8D4711yescharacteristic
201#1749—471no—
202C2×#328—472no—
203C4×#95—471no—
204#1294—472no—
205C2×C47—22no—
206C2×C47—22no—
207C2×C47#150411yescharacteristic
208C2×C47#150611yescharacteristic
209C2×C47#150811yescharacteristic
210C2×C47#150911yescharacteristic
211C2×C47C4×#9911yescharacteristic
212C2×C47C4×#10411yescharacteristic
213C2×C47#176311yescharacteristic
214C2×C22⋊Q8C2×D4711yescharacteristic
215C2×C4×Q8Dic4711yescharacteristic, complemented
216C22×C42—941no—
217C22×C42C2×D4711yescharacteristic
218C2×C22⋊C8—472no—
219C2×#97—941no—
220#391Dic4711yescharacteristic, complemented
221#389Dic4711yescharacteristic, complemented
222#387—21no—
223#385—21no—
224#380C2×D4711yescharacteristic
225C4×C4⋊C4—21no—
226C4×C4⋊C4Dic4711yescharacteristic, complemented
227C4×C22⋊C4—21no—
228C2×#95—472no—
229C2×#95—941no—
230#336—941no—
231#328—478no—
232C47#821911yescharacteristic, complemented, Sylow
233C22×Q8C2×Dic4711yescharacteristic
234C23×C4—942no—
235C23×C4C22⋊D4711yescharacteristic
236C23×C4C22⋊D4711yescharacteristic
237C23×C4C22×D4711yescharacteristic
238C22×C8—942no—
239C22⋊Q8—22no—
240C22⋊Q8C4×D4714yes—
241C4×Q8—42no—
242C2×C4⋊C4—21no—
243C2×C4⋊C4—21no—
244C2×C4⋊C4—21no—
245C2×C4⋊C4C2×Dic4711yescharacteristic
246C2×C4⋊C4C2×Dic4711yescharacteristic
247C2×C4⋊C4C2×Dic4711yescharacteristic
248C2×C22⋊C4—21no—
249C2×C22⋊C4—21no—
250C2×C42—21no—
251C2×C42—21no—
252C2×C42—42no—
253C2×C42—944no—
254C2×C42—1882no—
255C2×C42—1882no—
256C2×C42C2×Dic4711yescharacteristic
257C2×C42C2×Dic4711yescharacteristic
258C2×C42C22⋊D4711yescharacteristic
259C2×C42C22⋊D4711yescharacteristic
260C22⋊C8—474no—
261C22⋊C8—942no—
262#97—1882no—
263#95—21no—
264#95—21no—
265#95—21no—
266#95—21no—
267#95—21no—
268#95—474no—
269#95—944no—
270#95—942no—
271C24C2×C22⋊D4711yescharacteristic
272C2×Q8—21no—
273C2×Q8—41no—
274C2×Q8—41no—
275C2×Q8C4×Dic4712yescomplemented
276C22×C4—21no—
277C22×C4—21no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—21no—
281C22×C4—21no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—21no—
285C22×C4—21no—
286C22×C4—41no—
287C22×C4—41no—
288C22×C4—41no—
289C22×C4—944no—
290C22×C4—944no—
291C22×C4—944no—
292C22×C4—1882no—
293C22×C4—1882no—
294C22×C4—1882no—
295C22×C4—1882no—
296C22×C4C22⋊Dic4711yescharacteristic
297C22×C4C22⋊Dic4711yescharacteristic
298C22×C4C22⋊Dic4711yescharacteristic
299C22×C4C22⋊Dic4711yescharacteristic
300C22×C4C2×C4×D4712yes—
301C22×C4#1062535012yes—
302C22×C4C22×Dic4711yescharacteristic
303C22×C4C2×C22⋊D4711yescharacteristic
304C2×C8—944no—
305C2×C8—1882no—
306C2×C8—1882no—
307C4⋊C4—21no—
308C4⋊C4—24no—
309C4⋊C4—41no—
310C4⋊C4—41no—
311C4⋊C4—42no—
312C4⋊C4—42no—
313C4⋊C4C4×Dic4712yescomplemented
314C22⋊C4—24no—
315C22⋊C4—41no—
316C22⋊C4—41no—
317C42—22no—
318C42—22no—
319C42—41no—
320C42—41no—
321C42—42no—
322C42—42no—
323C42—1888no—
324C23—21no—
325C23—21no—
326C23—21no—
327C23—21no—
328C23C2×C22⋊Dic4711yescharacteristic
329C23C4×C22⋊D4712yes—
330C23#1309404911yescharacteristic
331C23#1309405411yescharacteristic
332C23#1309405512yes—
333Q8—42no—
334Q8—42no—
335C2×C4—24no—
336C2×C4—24no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—22no—
348C2×C4—22no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—42no—
353C2×C4—42no—
354C2×C4—44no—
355C2×C4—41no—
356C2×C4—41no—
357C2×C4—41no—
358C2×C4—41no—
359C2×C4—41no—
360C2×C4—41no—
361C2×C4—41no—
362C2×C4—41no—
363C2×C4—41no—
364C2×C4—41no—
365C2×C4—948no—
366C2×C4—1884no—
367C2×C4—1884no—
368C2×C4—1884no—
369C2×C4—1884no—
370C2×C4—1884no—
371C2×C4—1884no—
372C2×C4#1309393912yes—
373C2×C4#1309395112yes—
374C2×C4#1309395112yes—
375C2×C4C2×C4×Dic4711yescharacteristic
376C2×C4#1309398311yescharacteristic
377C2×C4C4×C22⋊D4712yes—
378C8—1884no—
379C22—21no—
380C22—21no—
381C22—22no—
382C22—22no—
383C22—22no—
384C22—21no—
385C22—21no—
386C22—22no—
387C22C4×C22⋊Dic4711yescharacteristic
388C22C2×#1309393911yescharacteristic
389C22#1568909711yescharacteristic
390C22#1568912411yescharacteristic
391C22#1568913211yescharacteristic
392C22C2×#1309395111yescharacteristic
393C22C2×#1309395111yescharacteristic
394C22#1568922112yes—
395C22#1568922812yes—
396C4—22no—
397C4—22no—
398C4—22no—
399C4—21no—
400C4—21no—
401C4—22no—
402C4—42no—
403C4—41no—
404C4—42no—
405C4—81no—
406C4—81no—
407C4—1888no—
408C2—22no—
409C2—22no—
410C2#2024189211yescharacteristic
411C2#2024190011yescharacteristic
412C2#2024190811yescharacteristic
413C2#2024191311yescharacteristic
414C2C4×#1309393911yescharacteristic
415C2C4×#1309395111yescharacteristic
416C2#2024321811yescharacteristic
subgroupG/H
centerC23C2×C22⋊Dic47
commutatorC2×C4×C47C2×C42
FrattiniC23×C4C22×D47
FittingC4×C22⋊Q8×C47C2
socleC23×C47C2×C22⋊C4
prime radicalC24×C47C2×D4
cosocleC23×C4×C47C23
hypercenterC4×C22⋊Q8D47
hypocenterC47#8219
LCSG ▷ C2×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C23 ◁ C22×C42 ◁ C4×C22⋊Q8

Complements

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

NH ≅ G/Ntype
12. #15688860C4semidirect
14. C2×C4×Q8×C47C4semidirect
16. C47×#391C4semidirect
17. C47×#389C4semidirect
22. C4×C4⋊C4×C47C4semidirect
66. C2×Q8×C47C42semidirect
101. C4⋊C4×C47C42semidirect
215. C2×C4×Q8Dic47semidirect
220. #391Dic47semidirect
221. #389Dic47semidirect
226. C4×C4⋊C4Dic47semidirect
232. C47#8219semidirect
275. C2×Q8C4×Dic47semidirect
313. C4⋊C4C4×Dic47semidirect

Maximal subgroups

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

subgroupindex
#202431502
C2×#156888602
C4×C47⋊#952
#202412662
C4×C22⋊Q8×C472
#821947