Finite Group Databasebeta

← #26126622

Subgroups — #26126622

Order 12032.

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

normal subgroups only

This group has 486 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
1C4×#13093939C211yesmaximal, characteristic, complemented
2#20241709C211yesmaximal, characteristic, complemented
3#20241412C212yesmaximal, complemented
4#20241385C212yesmaximal, complemented
5C47×#2635C211yesmaximal, characteristic
6#15689176—21no—
7C2×#13093939—21no—
8C4×C22⋊Dic47—21no—
9#15688890C2212yes—
10#15688847—21no—
11#15688847C2212yes—
12C2×#121×C47C411yescharacteristic, complemented
13C2×C4×D4×C47C2211yescharacteristic
14C47×#400C411yescharacteristic, complemented
15C47×#399C2211yescharacteristic
16C47×#394—21no—
17C47×#394C411yescharacteristic, complemented
18C47×#392—21no—
19C47×#392C2211yescharacteristic
20C47×#388—21no—
21C47×#386—21no—
22C47×#381C411yescharacteristic, complemented
23C2×C22⋊Dic47—22no—
24C2×C4×Dic47—41no—
25C4⋊(C47⋊C8)—41no—
26#13093939—24no—
27C22⋊(C47⋊C8)—22no—
28C4×C47⋊C8—41no—
29C47⋊#95—41no—
30C22×D4×C47—21no—
31C22×D4×C47—21no—
32C22×D4×C47C2×C411yescharacteristic
33C22×D4×C47D411yescharacteristic
34C23×C4×C47C2×C411yescharacteristic
35C23×C4×C47D411yescharacteristic
36#121×C47—41no—
37#121×C47—41no—
38C4×D4×C47—41no—
39C4×D4×C47—41no—
40C2×C4⋊C4×C47—21no—
41C2×C4⋊C4×C47C2×C411yescharacteristic
42C2×C4⋊C4×C47C2×C411yescharacteristic
43C2×C22⋊C4×C47—21no—
44C2×C22⋊C4×C47—21no—
45C2×C22⋊C4×C47—21no—
46C2×C22⋊C4×C47—21no—
47C2×C22⋊C4×C47—21no—
48C2×C22⋊C4×C47—21no—
49C2×C22⋊C4×C47D411yescharacteristic
50C2×C22⋊C4×C47C2311yescharacteristic
51C2×C42×C47D411yescharacteristic
52#95×C47—21no—
53#95×C47—21no—
54#95×C47—21no—
55#95×C47C2×C411yescharacteristic
56#95×C47C2×C411yescharacteristic
57C24×C47—21no—
58C24×C47C22⋊C411yescharacteristic
59C24×C47C2×D411yescharacteristic
60C2×D4×C47—22no—
61C2×D4×C47—22no—
62C2×D4×C47—41no—
63C2×D4×C47—41no—
64C2×D4×C47—42no—
65C2×D4×C47—42no—
66C2×D4×C47—81no—
67C2×D4×C47—81no—
68C2×D4×C47—81no—
69C2×D4×C47—81no—
70C2×D4×C47—81no—
71C22×C4×C47—21no—
72C22×C4×C47—21no—
73C22×C4×C47—21no—
74C22×C4×C47—21no—
75C22×C4×C47—21no—
76C22×C4×C47—21no—
77C22×C4×C47—21no—
78C22×C4×C47—81no—
79C22×C4×C47—81no—
80C22×C4×C47C22⋊C411yescharacteristic
81C22×C4×C47C22⋊C411yescharacteristic
82C22×C4×C47C22⋊C411yescharacteristic
83C22×C4×C47C22×C411yescharacteristic
84C22×C4×C47C2×D411yescharacteristic
85C22×Dic47—42no—
86C4⋊C4×C47—41no—
87C4⋊C4×C47—41no—
88C4⋊C4×C47—81no—
89C22⋊C4×C47—41no—
90C22⋊C4×C47—41no—
91C22⋊C4×C47—41no—
92C22⋊C4×C47—41no—
93C22⋊C4×C47—42no—
94C22⋊C4×C47—81no—
95C22⋊C4×C47—81no—
96C22⋊C4×C47—81no—
97C22⋊C4×C47—81no—
98C42×C47—21no—
99C42×C47—21no—
100C22⋊Dic47—24no—
101C22⋊Dic47—42no—
102C4×Dic47—82no—
103C2×C47⋊C8—42no—
104C23×C47—21no—
105C23×C47—21no—
106C23×C47—21no—
107C23×C47—21no—
108C23×C47—21no—
109C23×C47—41no—
110C23×C47—41no—
111C23×C47—41no—
112C23×C47—41no—
113C23×C47—41no—
114C23×C47—41no—
115C23×C47—41no—
116C23×C47—41no—
117C23×C47—41no—
118C23×C47—41no—
119C23×C47—41no—
120C23×C47C2×C22⋊C411yescharacteristic
121C23×C47#13512yes—
122D4×C47—44no—
123D4×C47—82no—
124D4×C47—84no—
125D4×C47—84no—
126C2×C4×C47—21no—
127C2×C4×C47—21no—
128C2×C4×C47—21no—
129C2×C4×C47—21no—
130C2×C4×C47—21no—
131C2×C4×C47—41no—
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—81no—
146C2×C4×C47—81no—
147C2×C4×C47—81no—
148C2×C4×C47#9911yescharacteristic
149C2×C4×C47#9911yescharacteristic
150C2×C4×C47#10412yes—
151C2×Dic47—44no—
152C2×Dic47—82no—
153C2×Dic47—82no—
154C47⋊C8—82no—
155#11729—471nomaximal, Sylow
156C22×C47—21no—
157C22×C47—21no—
158C22×C47—22no—
159C22×C47—42no—
160C22×C47—41no—
161C22×C47—41no—
162C22×C47—41no—
163C22×C47—41no—
164C22×C47—41no—
165C22×C47—81no—
166C22×C47—81no—
167C22×C47—81no—
168C22×C47—81no—
169C22×C47—81no—
170C22×C47—81no—
171C22×C47—81no—
172C22×C47—81no—
173C22×C47—81no—
174C22×C47—81no—
175C22×C47C2×#9911yescharacteristic
176C22×C47#41611yescharacteristic
177C22×C47C2×#10411yescharacteristic
178C4×C47—21no—
179C4×C47—21no—
180C4×C47—22no—
181C4×C47—41no—
182C4×C47—41no—
183C4×C47—41no—
184C4×C47—81no—
185C4×C47—81no—
186C4×C47—81no—
187Dic47—84no—
188#2635D4711yescharacteristic
189C4×#99—471no—
190#1460—471no—
191#1352—472no—
192#1343—472no—
193C2×C47—21no—
194C2×C47—21no—
195C2×C47—42no—
196C2×C47—81no—
197C2×C47—161no—
198C2×C47#150111yescharacteristic
199C2×C47#212111yescharacteristic
200C2×C47#212811yescharacteristic
201C2×#121Dic4711yescharacteristic, complemented
202C2×C4×D4C2×D4711yescharacteristic
203#431—941no—
204C2×#99—941no—
205#400Dic4711yescharacteristic, complemented
206#399C2×D4711yescharacteristic
207#394—21no—
208#394Dic4711yescharacteristic, complemented
209#392—21no—
210#392C2×D4711yescharacteristic
211#388—21no—
212#386—21no—
213#381Dic4711yescharacteristic, complemented
214C4×C22⋊C4—941no—
215#342—472no—
216#323—472no—
217#323—941no—
218C47#1172911yescharacteristic, complemented, Sylow
219C22×D4—21no—
220C22×D4—21no—
221C22×D4C2×Dic4711yescharacteristic
222C22×D4C22⋊D4711yescharacteristic
223C23×C4C2×Dic4711yescharacteristic
224C23×C4C22⋊D4711yescharacteristic
225#121—41no—
226#121—41no—
227C4×D4—41no—
228C4×D4—41no—
229C2×C4⋊C4—21no—
230C2×C4⋊C4C2×Dic4711yescharacteristic
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—942no—
239C2×C22⋊C4C22⋊D4711yescharacteristic
240C2×C22⋊C4C22×D4711yescharacteristic
241C2×C42—1881no—
242C2×C42C22⋊D4711yescharacteristic
243C4⋊C8—1881no—
244#99—944no—
245C22⋊C8—942no—
246C4×C8—1881no—
247#95—21no—
248#95—21no—
249#95—21no—
250#95—1881no—
251#95C2×Dic4711yescharacteristic
252#95C2×Dic4711yescharacteristic
253C24—21no—
254C24C22⋊Dic4711yescharacteristic
255C24C2×C22⋊D4711yescharacteristic
256C2×D4—22no—
257C2×D4—22no—
258C2×D4—41no—
259C2×D4—41no—
260C2×D4—42no—
261C2×D4—42no—
262C2×D4—81no—
263C2×D4—81no—
264C2×D4—81no—
265C2×D4—81no—
266C2×D4—81no—
267C22×C4—21no—
268C22×C4—21no—
269C22×C4—21no—
270C22×C4—21no—
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—21no—
274C22×C4—81no—
275C22×C4—81no—
276C22×C4—1882no—
277C22×C4C22⋊Dic4711yescharacteristic
278C22×C4C22⋊Dic4711yescharacteristic
279C22×C4C22⋊Dic4711yescharacteristic
280C22×C4C22×Dic4711yescharacteristic
281C22×C4C2×C22⋊D4711yescharacteristic
282C2×C8—1882no—
283C4⋊C4—41no—
284C4⋊C4—41no—
285C4⋊C4—81no—
286C22⋊C4—41no—
287C22⋊C4—41no—
288C22⋊C4—41no—
289C22⋊C4—41no—
290C22⋊C4—42no—
291C22⋊C4—81no—
292C22⋊C4—81no—
293C22⋊C4—81no—
294C22⋊C4—81no—
295C22⋊C4—944no—
296C22⋊C4—1882no—
297C42—21no—
298C42—21no—
299C42—3762no—
300C23—21no—
301C23—21no—
302C23—21no—
303C23—21no—
304C23—21no—
305C23—41no—
306C23—41no—
307C23—41no—
308C23—41no—
309C23—41no—
310C23—41no—
311C23—41no—
312C23—41no—
313C23—41no—
314C23—41no—
315C23—41no—
316C23C2×C22⋊Dic4711yescharacteristic
317C23#1309404412yes—
318D4—44no—
319D4—82no—
320D4—84no—
321D4—84no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—41no—
328C2×C4—42no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—41no—
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—81no—
342C2×C4—81no—
343C2×C4—81no—
344C2×C4—3762no—
345C2×C4—3762no—
346C2×C4—1884no—
347C2×C4#1309393911yescharacteristic
348C2×C4#1309393911yescharacteristic
349C2×C4#1309395112yes—
350C8—3762no—
351C22—21no—
352C22—21no—
353C22—22no—
354C22—42no—
355C22—41no—
356C22—41no—
357C22—41no—
358C22—41no—
359C22—41no—
360C22—81no—
361C22—81no—
362C22—81no—
363C22—81no—
364C22—81no—
365C22—81no—
366C22—81no—
367C22—81no—
368C22—81no—
369C22—81no—
370C22C2×#1309393911yescharacteristic
371C22#1568911911yescharacteristic
372C22C2×#1309395111yescharacteristic
373C4—21no—
374C4—21no—
375C4—22no—
376C4—41no—
377C4—41no—
378C4—41no—
379C4—81no—
380C4—81no—
381C4—81no—
382C4—3764no—
383C2—21no—
384C2—21no—
385C2—42no—
386C2—81no—
387C2—161no—
388C2#2024187911yescharacteristic
389C2#2024503411yescharacteristic
390C2#2024506011yescharacteristic
subgroupG/H
centerC22C2×#13093939
commutatorC22×C4×C47C22×C4
FrattiniC2×C22⋊C4C22×D47
Fitting, prime radicalC47×#2635C2
socleC22×C47C2×#99
cosocleC2×C22⋊C4×C47C23
hypercenter#2635D47
hypocenterC47#11729
LCSG ▷ C22×C4×C47 ▷ C23×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #2635

Complements

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

NH ≅ G/Ntype
1. C4×#13093939C2semidirect
2. #20241709C2semidirect
3. #20241412C2semidirect
4. #20241385C2semidirect
12. C2×#121×C47C4semidirect
14. C47×#400C4semidirect
17. C47×#394C4semidirect
22. C47×#381C4semidirect
201. C2×#121Dic47semidirect
205. #400Dic47semidirect
208. #394Dic47semidirect
213. #381Dic47semidirect
218. C47#11729semidirect

Maximal subgroups

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

subgroupindex
C4×#130939392
#202417092
#202414122
#202413852
C47×#26352
#1172947