Finite Group Databasebeta

← #26127166

Subgroups — #26127166

Order 12032.

506 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 506 conjugacy classes of subgroups — too many to draw. Only its 91 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20243462C211yesmaximal, characteristic
2C2×#15688847C211yesmaximal, characteristic
3#20241409C212yesmaximal, complemented
4#20241388C212yesmaximal, complemented
5C47×#2561C211yesmaximal, characteristic
6C2×C22⋊(C47⋊C8)—21no—
7#15688979—21no—
8#15688890—21no—
9#15688890C2212yes—
10#15688847—21no—
11#15688847C2212yes—
12C2×C22⋊Q8×C47C411yescharacteristic, complemented
13C22×C22⋊C4×C47C2211yescharacteristic
14C47×#397C411yescharacteristic, complemented
15C47×#397C2211yescharacteristic
16C47×#393C411yescharacteristic, complemented
17C47×#393C2211yescharacteristic
18C47×#388—21no—
19C47×#388—21no—
20C47×#381—21no—
21C47×#379—21no—
22C4×C22⋊C4×C47C411yescharacteristic, complemented
23C22×C47⋊C8—41no—
24C2×C22⋊Dic47—22no—
25C2×C22⋊Dic47—41no—
26C2×C22⋊Dic47—41no—
27C22⋊(C47⋊C8)—22no—
28C22⋊(C47⋊C8)—41no—
29C25×C47D411yescharacteristic
30C22×Q8×C47C2×C411yescharacteristic
31C23×C4×C47C2×C411yescharacteristic
32C23×C4×C47D411yescharacteristic
33C22⋊Q8×C47—41no—
34C22⋊Q8×C47—41no—
35C2×C4⋊C4×C47—21no—
36C2×C4⋊C4×C47C2×C411yescharacteristic
37C2×C22⋊C4×C47—21no—
38C2×C22⋊C4×C47—21no—
39C2×C22⋊C4×C47—21no—
40C2×C22⋊C4×C47—21no—
41C2×C22⋊C4×C47—21no—
42C2×C22⋊C4×C47—21no—
43C2×C22⋊C4×C47—41no—
44C2×C22⋊C4×C47—41no—
45C2×C22⋊C4×C47C2×C411yescharacteristic
46C2×C22⋊C4×C47D411yescharacteristic
47C2×C22⋊C4×C47D411yescharacteristic
48C2×C22⋊C4×C47C2311yescharacteristic
49C2×C42×C47—21no—
50#95×C47—21no—
51#95×C47—21no—
52#95×C47—21no—
53#95×C47—21no—
54#95×C47C2×C411yescharacteristic
55#95×C47C2×C411yescharacteristic
56C24×C47—21no—
57C24×C47—21no—
58C24×C47—22no—
59C24×C47—41no—
60C24×C47—42no—
61C24×C47—81no—
62C24×C47C22⋊C411yescharacteristic
63C24×C47C22⋊C411yescharacteristic
64C24×C47C2×D411yescharacteristic
65C2×Q8×C47—41no—
66C2×Q8×C47—81no—
67C22×C4×C47—21no—
68C22×C4×C47—21no—
69C22×C4×C47—21no—
70C22×C4×C47—21no—
71C22×C4×C47—21no—
72C22×C4×C47—21no—
73C22×C4×C47—21no—
74C22×C4×C47—21no—
75C22×C4×C47—42no—
76C22×C4×C47—81no—
77C22×C4×C47C22⋊C411yescharacteristic
78C22×C4×C47C22⋊C411yescharacteristic
79C22×C4×C47C22×C411yescharacteristic
80C22×C4×C47C2×D411yescharacteristic
81C22×Dic47—42no—
82C4⋊C4×C47—42no—
83C4⋊C4×C47—41no—
84C22⋊C4×C47—41no—
85C22⋊C4×C47—41no—
86C22⋊C4×C47—41no—
87C22⋊C4×C47—41no—
88C22⋊C4×C47—42no—
89C22⋊C4×C47—42no—
90C22⋊C4×C47—41no—
91C22⋊C4×C47—41no—
92C22⋊C4×C47—42no—
93C22⋊C4×C47—81no—
94C22⋊C4×C47—81no—
95C42×C47—81no—
96C22⋊Dic47—42no—
97C22⋊Dic47—42no—
98C22⋊Dic47—82no—
99C22⋊Dic47—82no—
100C2×C47⋊C8—42no—
101C2×C47⋊C8—82no—
102C23×C47—22no—
103C23×C47—21no—
104C23×C47—21no—
105C23×C47—21no—
106C23×C47—21no—
107C23×C47—21no—
108C23×C47—21no—
109C23×C47—21no—
110C23×C47—41no—
111C23×C47—42no—
112C23×C47—42no—
113C23×C47—41no—
114C23×C47—41no—
115C23×C47—41no—
116C23×C47—41no—
117C23×C47—42no—
118C23×C47—42no—
119C23×C47—81no—
120C23×C47—81no—
121C23×C47—84no—
122C23×C47—84no—
123C23×C47#9911yescharacteristic
124C23×C47#9911yescharacteristic
125C23×C47C2×C22⋊C411yescharacteristic
126C23×C47#13512yes—
127Q8×C47—82no—
128C2×C4×C47—21no—
129C2×C4×C47—21no—
130C2×C4×C47—21no—
131C2×C4×C47—21no—
132C2×C4×C47—42no—
133C2×C4×C47—41no—
134C2×C4×C47—41no—
135C2×C4×C47—41no—
136C2×C4×C47—41no—
137C2×C4×C47—41no—
138C2×C4×C47—41no—
139C2×C4×C47—41no—
140C2×C4×C47—41no—
141C2×C4×C47—41no—
142C2×C4×C47—41no—
143C2×C4×C47—41no—
144C2×C4×C47—41no—
145C2×C4×C47—41no—
146C2×C4×C47—41no—
147C2×C4×C47—41no—
148C2×C4×C47—41no—
149C2×C4×C47—81no—
150C2×C4×C47—81no—
151C2×C4×C47—81no—
152C2×Dic47—82no—
153C2×Dic47—82no—
154C2×Dic47—82no—
155C47⋊C8—82no—
156#11845—471nomaximal, Sylow
157C22×C47—21no—
158C22×C47—21no—
159C22×C47—21no—
160C22×C47—21no—
161C22×C47—21no—
162C22×C47—21no—
163C22×C47—22no—
164C22×C47—21no—
165C22×C47—42no—
166C22×C47—42no—
167C22×C47—42no—
168C22×C47—42no—
169C22×C47—41no—
170C22×C47—41no—
171C22×C47—41no—
172C22×C47—41no—
173C22×C47—41no—
174C22×C47—81no—
175C22×C47—84no—
176C22×C47—81no—
177C22×C47—84no—
178C22×C47#35412yes—
179C22×C47C2×#9911yescharacteristic
180C22×C47#41611yescharacteristic
181C22×C47#42111yescharacteristic
182C4×C47—41no—
183C4×C47—41no—
184C4×C47—41no—
185C4×C47—41no—
186C4×C47—81no—
187C4×C47—81no—
188C4×C47—81no—
189C4×C47—81no—
190Dic47—162no—
191#2561D4711yescharacteristic
192#1819—471no—
193C2×#323—471no—
194#1351—472no—
195#1344—472no—
196C2×C47—21no—
197C2×C47—21no—
198C2×C47—22no—
199C2×C47—42no—
200C2×C47—41no—
201C2×C47—81no—
202C2×C47#151611yescharacteristic
203C2×C47#212111yescharacteristic
204C2×C47C2×#35411yescharacteristic
205C2×C22⋊Q8Dic4711yescharacteristic, complemented
206C22×C22⋊C4C2×D4711yescharacteristic
207C2×C22⋊C8—941no—
208#397Dic4711yescharacteristic, complemented
209#397C2×D4711yescharacteristic
210#393Dic4711yescharacteristic, complemented
211#393C2×D4711yescharacteristic
212#388—21no—
213#388—21no—
214#381—21no—
215#379—21no—
216#379—941no—
217C4×C22⋊C4Dic4711yescharacteristic, complemented
218#342—472no—
219#342—941no—
220#323—472no—
221#323—941no—
222C47#1184511yescharacteristic, complemented, Sylow
223C25C22⋊D4711yescharacteristic
224C22×Q8C2×Dic4711yescharacteristic
225C23×C4C2×Dic4711yescharacteristic
226C23×C4C22⋊D4711yescharacteristic
227C22×C8—1881no—
228C22⋊Q8—41no—
229C22⋊Q8—41no—
230C2×C4⋊C4—21no—
231C2×C4⋊C4C2×Dic4711yescharacteristic
232C2×C22⋊C4—21no—
233C2×C22⋊C4—21no—
234C2×C22⋊C4—21no—
235C2×C22⋊C4—21no—
236C2×C22⋊C4—21no—
237C2×C22⋊C4—21no—
238C2×C22⋊C4—41no—
239C2×C22⋊C4—41no—
240C2×C22⋊C4—942no—
241C2×C22⋊C4—1881no—
242C2×C22⋊C4—1881no—
243C2×C22⋊C4C2×Dic4711yescharacteristic
244C2×C22⋊C4C22⋊D4711yescharacteristic
245C2×C22⋊C4C22⋊D4711yescharacteristic
246C2×C22⋊C4C22×D4711yescharacteristic
247C2×C42—21no—
248C22⋊C8—942no—
249C22⋊C8—1881no—
250#95—21no—
251#95—21no—
252#95—21no—
253#95—21no—
254#95C2×Dic4711yescharacteristic
255#95C2×Dic4711yescharacteristic
256C24—21no—
257C24—21no—
258C24—22no—
259C24—41no—
260C24—42no—
261C24—81no—
262C24C22⋊Dic4711yescharacteristic
263C24C22⋊Dic4711yescharacteristic
264C24C2×C22⋊D4711yescharacteristic
265C2×Q8—41no—
266C2×Q8—81no—
267C22×C4—21no—
268C22×C4—21no—
269C22×C4—21no—
270C22×C4—21no—
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—21no—
274C22×C4—21no—
275C22×C4—42no—
276C22×C4—81no—
277C22×C4—1882no—
278C22×C4C22⋊Dic4711yescharacteristic
279C22×C4C22⋊Dic4711yescharacteristic
280C22×C4C22×Dic4711yescharacteristic
281C22×C4C2×C22⋊D4711yescharacteristic
282C2×C8—3762no—
283C2×C8—1882no—
284C4⋊C4—41no—
285C4⋊C4—42no—
286C22⋊C4—41no—
287C22⋊C4—41no—
288C22⋊C4—41no—
289C22⋊C4—41no—
290C22⋊C4—42no—
291C22⋊C4—42no—
292C22⋊C4—41no—
293C22⋊C4—41no—
294C22⋊C4—42no—
295C22⋊C4—81no—
296C22⋊C4—81no—
297C22⋊C4—3762no—
298C22⋊C4—3762no—
299C22⋊C4—1882no—
300C22⋊C4—1882no—
301C42—81no—
302C23—21no—
303C23—21no—
304C23—21no—
305C23—22no—
306C23—21no—
307C23—21no—
308C23—21no—
309C23—21no—
310C23—41no—
311C23—42no—
312C23—42no—
313C23—41no—
314C23—41no—
315C23—41no—
316C23—41no—
317C23—42no—
318C23—42no—
319C23—81no—
320C23—81no—
321C23—84no—
322C23—84no—
323C23#1309393911yescharacteristic
324C23#1309393911yescharacteristic
325C23C2×C22⋊Dic4711yescharacteristic
326C23#1309404412yes—
327Q8—82no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—42no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—41no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4—41no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4—41no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—81no—
350C2×C4—81no—
351C2×C4—81no—
352C2×C4—3762no—
353C2×C4—3762no—
354C2×C4—3762no—
355C8—3762no—
356C22—21no—
357C22—21no—
358C22—21no—
359C22—21no—
360C22—22no—
361C22—21no—
362C22—21no—
363C22—21no—
364C22—42no—
365C22—42no—
366C22—42no—
367C22—42no—
368C22—41no—
369C22—41no—
370C22—41no—
371C22—41no—
372C22—41no—
373C22—81no—
374C22—84no—
375C22—81no—
376C22—84no—
377C22#1568891812yes—
378C22C2×#1309393911yescharacteristic
379C22#1568911911yescharacteristic
380C22#1568914011yescharacteristic
381C4—41no—
382C4—41no—
383C4—41no—
384C4—41no—
385C4—81no—
386C4—81no—
387C4—81no—
388C4—81no—
389C4—7522no—
390C2—21no—
391C2—21no—
392C2—22no—
393C2—42no—
394C2—41no—
395C2—81no—
396C2#2024194411yescharacteristic
397C2#2024503411yescharacteristic
398C2C2×#1568891811yescharacteristic
subgroupG/H
centerC22C2×#13093939
commutatorC22×C4×C47C22×C4
FrattiniC2×C22⋊C4C22×D47
FittingC47×#2561C2
socleC22×C47C2×#99
prime radicalC25×C47D4
cosocleC2×C22⋊C4×C47C23
hypercenter#2561D47
hypocenterC47#11845
LCSG ▷ C22×C4×C47 ▷ C23×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C24 ◁ C22×C22⋊C4 ◁ #2561

Complements

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

NH ≅ G/Ntype
3. #20241409C2semidirect
4. #20241388C2semidirect
12. C2×C22⋊Q8×C47C4semidirect
14. C47×#397C4semidirect
16. C47×#393C4semidirect
22. C4×C22⋊C4×C47C4semidirect
205. C2×C22⋊Q8Dic47semidirect
208. #397Dic47semidirect
210. #393Dic47semidirect
217. C4×C22⋊C4Dic47semidirect
222. C47#11845semidirect

Maximal subgroups

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

subgroupindex
#202434622
C2×#156888472
#202414092
#202413882
C47×#25612
#1184547