Finite Group Databasebeta

← #26126615

Subgroups — #26126615

Order 12032.

470 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 470 conjugacy classes of subgroups — too many to draw. Only its 87 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#20246672C211yesmaximal, characteristic
2#20241706C211yesmaximal, characteristic
3#20241408C212yesmaximal, complemented
4#20241387C212yesmaximal, complemented
5C4×#99×C47C211yesmaximal, characteristic
6C2×C4×Dic94C411yescharacteristic, complemented
7C2×C22⋊(C47⋊C8)—21no—
8#15689061—21no—
9#15689048C411yescharacteristic, complemented
10#15689048C2211yescharacteristic
11#15689037—21no—
12#15689035C411yescharacteristic, complemented
13#15689035C2211yescharacteristic
14C4×C22⋊Dic47—21no—
15C4×C22⋊Dic47—21no—
16#15688969C411yescharacteristic, complemented
17#15688849—21no—
18#15688847C2212yes—
19C2×C4×D4×C47C2211yescharacteristic
20C2×#99×C47—21no—
21C4×C22⋊C4×C47—21no—
22C47×#342C2212yes—
23C22×Dic94C2×C411yescharacteristic
24C22×C47⋊C8—41no—
25C4×Dic94—41no—
26C4×Dic94—41no—
27C2×C4⋊Dic47C2×C411yescharacteristic
28C2×Dic47⋊C4—21no—
29C2×C22⋊Dic47—21no—
30C2×C22⋊Dic47—21no—
31C2×C22⋊Dic47—21no—
32C2×C22⋊Dic47—21no—
33C2×C4×Dic47—21no—
34C2×C4×Dic47—21no—
35C22⋊(C47⋊C8)—22no—
36C22⋊(C47⋊C8)—41no—
37C47⋊#95—21no—
38C47⋊#95—21no—
39C47⋊#95—21no—
40C47⋊#95—21no—
41C47⋊#95—21no—
42C47⋊#95C2×C411yescharacteristic
43C47⋊#95C2×C411yescharacteristic
44C22×D4×C47D411yescharacteristic
45C23×C4×C47D411yescharacteristic
46C23×C4×C47D411yescharacteristic
47C4×D4×C47—41no—
48C4×D4×C47—41no—
49C2×C4⋊C4×C47D411yescharacteristic
50C2×C22⋊C4×C47—22no—
51C2×C22⋊C4×C47C2×C411yescharacteristic
52C2×C22⋊C4×C47C2311yescharacteristic
53C2×C42×C47—41no—
54C2×C42×C47C2×C411yescharacteristic
55#99×C47—24no—
56#95×C47—41no—
57C24×C47C22⋊C411yescharacteristic
58C24×C47C2×D411yescharacteristic
59C2×D4×C47—22no—
60C2×D4×C47—22no—
61C2×D4×C47—41no—
62C22×C4×C47—21no—
63C22×C4×C47—21no—
64C22×C4×C47—21no—
65C22×C4×C47—21no—
66C22×C4×C47—42no—
67C22×C4×C47—42no—
68C22×C4×C47—42no—
69C22×C4×C47C22⋊C411yescharacteristic
70C22×C4×C47C22⋊C411yescharacteristic
71C22×C4×C47C22⋊C411yescharacteristic
72C22×C4×C47C22×C411yescharacteristic
73C22×C4×C47C2×D411yescharacteristic
74C22×Dic47—21no—
75C22×Dic47—21no—
76C22×Dic47—21no—
77C22×Dic47—21no—
78C2×Dic94—41no—
79C2×Dic94—42no—
80C4⋊C4×C47—41no—
81C22⋊C4×C47—24no—
82C22⋊C4×C47—41no—
83C22⋊C4×C47—41no—
84C22⋊C4×C47—42no—
85C42×C47—21no—
86C42×C47—21no—
87C42×C47—82no—
88C22⋊Dic47—42no—
89C22⋊Dic47—42no—
90C22⋊Dic47—42no—
91C22⋊Dic47—42no—
92C4⋊Dic47—41no—
93Dic47⋊C4—42no—
94C4×Dic47—42no—
95C4×Dic47—41no—
96C4×Dic47—41no—
97C2×C47⋊C8—42no—
98C2×C47⋊C8—82no—
99C23×C47—21no—
100C23×C47—21no—
101C23×C47—21no—
102C23×C47—21no—
103C23×C47—41no—
104C23×C47—41no—
105C23×C47—41no—
106C23×C47—41no—
107C23×C47#10412yes—
108C23×C47C2×C22⋊C411yescharacteristic
109C23×C47#13512yes—
110D4×C47—44no—
111C2×C4×C47—22no—
112C2×C4×C47—21no—
113C2×C4×C47—21no—
114C2×C4×C47—21no—
115C2×C4×C47—21no—
116C2×C4×C47—21no—
117C2×C4×C47—21no—
118C2×C4×C47—42no—
119C2×C4×C47—42no—
120C2×C4×C47—41no—
121C2×C4×C47—41no—
122C2×C4×C47—41no—
123C2×C4×C47—44no—
124C2×C4×C47—81no—
125C2×C4×C47—81no—
126C2×C4×C47—82no—
127C2×C4×C47—82no—
128C2×C4×C47#9911yescharacteristic
129C2×C4×C47#9911yescharacteristic
130C2×Dic47—22no—
131C2×Dic47—41no—
132C2×Dic47—41no—
133C2×Dic47—41no—
134C2×Dic47—41no—
135C2×Dic47—41no—
136C2×Dic47—41no—
137C2×Dic47—41no—
138C2×Dic47—41no—
139C2×Dic47—41no—
140C2×Dic47—41no—
141C2×Dic47—41no—
142Dic94—42no—
143Dic94—42no—
144C47⋊C8—82no—
145#11728—471nomaximal, Sylow
146C22×C47—21no—
147C22×C47—21no—
148C22×C47—22no—
149C22×C47—22no—
150C22×C47—42no—
151C22×C47—42no—
152C22×C47—41no—
153C22×C47—41no—
154C22×C47—41no—
155C22×C47—41no—
156C22×C47—81no—
157C22×C47—81no—
158C22×C47C2×#9911yescharacteristic
159C22×C47#41611yescharacteristic
160C22×C47C2×#10411yescharacteristic
161C4×C47—21no—
162C4×C47—41no—
163C4×C47—41no—
164C4×C47—41no—
165C4×C47—41no—
166C4×C47—84no—
167C4×C47#35412yes—
168Dic47—42no—
169Dic47—81no—
170Dic47—81no—
171Dic47—81no—
172#2398—471no—
173C4×#99D4711yescharacteristic
174#1459—471no—
175#1351—472no—
176#1344—472no—
177C2×C47—21no—
178C2×C47—21no—
179C2×C47—42no—
180C2×C47—42no—
181C2×C47#150111yescharacteristic
182C2×C47#212011yescharacteristic
183C2×C47C2×#35411yescharacteristic
184C2×C4×Q8—471no—
185C2×C4×D4C2×D4711yescharacteristic
186C2×#99—21no—
187C2×C22⋊C8—941no—
188#400—941no—
189#397—471no—
190#397—471no—
191#394—941no—
192#393—471no—
193#393—471no—
194C4×C22⋊C4—21no—
195C4×C22⋊C4—941no—
196C4×C22⋊C4—941no—
197#376—471no—
198#342C2×D4712yes—
199#324—941no—
200#323—472no—
201C47#1172811yescharacteristic, complemented, Sylow
202C22×Q8—471no—
203C22×D4C22⋊D4711yescharacteristic
204C23×C4D18811yescharacteristic
205C23×C4C22⋊D4711yescharacteristic
206C22×C8—1881no—
207C4×Q8—1881no—
208C4×Q8—1881no—
209C4×D4—41no—
210C4×D4—41no—
211C2×C4⋊C4—471no—
212C2×C4⋊C4—941no—
213C2×C4⋊C4D18811yescharacteristic
214C2×C22⋊C4—22no—
215C2×C22⋊C4—941no—
216C2×C22⋊C4—941no—
217C2×C22⋊C4—941no—
218C2×C22⋊C4—941no—
219C2×C22⋊C4C4×D4711yescharacteristic
220C2×C22⋊C4C22×D4711yescharacteristic
221C2×C42—41no—
222C2×C42—941no—
223C2×C42—941no—
224C2×C42C4×D4711yescharacteristic
225#99—24no—
226C22⋊C8—942no—
227C22⋊C8—1881no—
228#95—41no—
229#95—471no—
230#95—471no—
231#95—941no—
232#95—941no—
233#95—941no—
234#95—941no—
235#95—941no—
236C24(C2×D47)⋊C411yescharacteristic
237C24C2×C22⋊D4711yescharacteristic
238C2×Q8—1881no—
239C2×Q8—1882no—
240C2×D4—22no—
241C2×D4—22no—
242C2×D4—41no—
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—42no—
248C22×C4—42no—
249C22×C4—42no—
250C22×C4—941no—
251C22×C4—941no—
252C22×C4—941no—
253C22×C4—941no—
254C22×C4(C2×D47)⋊C411yescharacteristic
255C22×C4(C2×D47)⋊C411yescharacteristic
256C22×C4(C2×D47)⋊C411yescharacteristic
257C22×C4C2×C4×D4711yescharacteristic
258C22×C4C2×D18811yescharacteristic
259C2×C8—3762no—
260C2×C8—1882no—
261C4⋊C4—41no—
262C4⋊C4—1881no—
263C4⋊C4—1882no—
264C22⋊C4—24no—
265C22⋊C4—41no—
266C22⋊C4—41no—
267C22⋊C4—42no—
268C22⋊C4—1882no—
269C22⋊C4—1882no—
270C22⋊C4—1882no—
271C22⋊C4—1882no—
272C42—21no—
273C42—21no—
274C42—82no—
275C42—1882no—
276C42—1881no—
277C42—1881no—
278C23—21no—
279C23—21no—
280C23—21no—
281C23—21no—
282C23—41no—
283C23—41no—
284C23—41no—
285C23—41no—
286C23#1309395212yes—
287C23C2×(C2×D47)⋊C411yescharacteristic
288C23#1309404412yes—
289Q8—1882no—
290Q8—1882no—
291D4—44no—
292C2×C4—22no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—42no—
300C2×C4—42no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—44no—
305C2×C4—81no—
306C2×C4—81no—
307C2×C4—82no—
308C2×C4—82no—
309C2×C4—942no—
310C2×C4—1881no—
311C2×C4—1881no—
312C2×C4—1881no—
313C2×C4—1881no—
314C2×C4—1881no—
315C2×C4—1881no—
316C2×C4—1881no—
317C2×C4—1881no—
318C2×C4—1881no—
319C2×C4—1881no—
320C2×C4—1881no—
321C2×C4#1309393811yescharacteristic
322C2×C4#1309393811yescharacteristic
323C8—3762no—
324C22—21no—
325C22—21no—
326C22—22no—
327C22—22no—
328C22—42no—
329C22—42no—
330C22—41no—
331C22—41no—
332C22—41no—
333C22—41no—
334C22—81no—
335C22—81no—
336C22C2×#1309393811yescharacteristic
337C22#1568911711yescharacteristic
338C22C2×#1309395211yescharacteristic
339C4—21no—
340C4—41no—
341C4—41no—
342C4—41no—
343C4—41no—
344C4—84no—
345C4—3761no—
346C4—3761no—
347C4—3761no—
348C4—1882no—
349C4#1568891712yes—
350C2—21no—
351C2—21no—
352C2—42no—
353C2—42no—
354C2#2024187711yescharacteristic
355C2#2024502711yescharacteristic
356C2C2×#1568891711yescharacteristic
subgroupG/H
centerC2×C4#13093938
commutatorC22×C4×C47C22×C4
FrattiniC2×C22⋊C4C22×D47
FittingC4×#99×C47C2
socleC22×C47C2×#99
prime radicalC22×D4×C47D4
cosocleC2×C22⋊C4×C47C23
hypercenterC4×#99D47
hypocenterC47#11728
LCSG ▷ C22×C4×C47 ▷ C23×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ C2×C4×D4 ◁ C4×#99

Complements

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

NH ≅ G/Ntype
3. #20241408C2semidirect
4. #20241387C2semidirect
6. C2×C4×Dic94C4semidirect
9. #15689048C4semidirect
12. #15689035C4semidirect
16. #15688969C4semidirect
201. C47#11728semidirect

Maximal subgroups

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

subgroupindex
#202466722
#202417062
#202414082
#202413872
C4×#99×C472
#1172847