Finite Group Databasebeta

← #26108911

Subgroups — #26108911

Order 12032.

1026 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 1026 conjugacy classes of subgroups — too many to draw. Only its 131 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
1C22×#13094023C211yesmaximal, characteristic
2#20243311C212yesmaximal
3#20241718C211yesmaximal, characteristic
4#20241717C212yesmaximal
5C47×#1463C211yesmaximal, characteristic
6C23×Dic94C411yescharacteristic
7C23×Dic94C2211yescharacteristic
8C2×#13094023—24no—
9C2×#13094023—24no—
10C2×#13094023—22no—
11C2×#13094023—22no—
12C22×C4⋊Dic47C411yescharacteristic
13C22×Dic47⋊C4C411yescharacteristic
14C22×Dic47⋊C4C2211yescharacteristic
15C22×C4×Dic47C411yescharacteristic
16C2×#13093942—24no—
17#15689087C2212yes—
18#15688849—22no—
19#15688848—24no—
20C22×C4⋊C4×C47C2211yescharacteristic
21C47×#407C2212yes—
22C47×#324—22no—
23C22×Dic94—22no—
24C22×Dic94—21no—
25C22×Dic94—22no—
26C22×Dic94—21no—
27C22×Dic94—22no—
28C22×Dic94—22no—
29C22×Dic94—44no—
30C22×Dic94—44no—
31C22×Dic94D414yes—
32C23×Dic47C2×C411yescharacteristic
33C23×Dic47C2×C411yescharacteristic
34C23×Dic47C2×C411yescharacteristic
35C23×Dic47C2×C411yescharacteristic
36C2×#10625329—22no—
37#13094023—216no—
38#13094023—216no—
39C2×C4⋊Dic47—22no—
40C2×C4⋊Dic47—22no—
41C2×C4⋊Dic47—22no—
42C2×Dic47⋊C4—22no—
43C2×Dic47⋊C4—22no—
44C2×Dic47⋊C4—21no—
45C2×Dic47⋊C4—21no—
46C2×Dic47⋊C4—22no—
47C2×Dic47⋊C4—22no—
48C2×Dic47⋊C4D414yes—
49C2×C4×Dic47—22no—
50C2×C4×Dic47—22no—
51C2×C4×Dic47—22no—
52#13093942—48no—
53C22⋊(C47⋊C8)—22no—
54C22⋊(C47⋊C8)—22no—
55C23×C4×C47C2×C411yescharacteristic
56C23×C4×C47C2×C411yescharacteristic
57C23×C4×C47C2311yescharacteristic
58C2×M16×C47—22no—
59C2×C4⋊C4×C47—22no—
60C2×C4⋊C4×C47—22no—
61C2×C4⋊C4×C47D412yes—
62C2×C4⋊C4×C47D412yes—
63C22⋊C8×C47—22no—
64C22⋊C8×C47—22no—
65C24×C47C22×C411yescharacteristic
66C22×C4×C47—21no—
67C22×C4×C47—21no—
68C22×C4×C47—24no—
69C22×C4×C47—21no—
70C22×C4×C47—21no—
71C22×C4×C47—42no—
72C22×C4×C47—42no—
73C22×C4×C47C22⋊C412yes—
74C22×C4×C47C22⋊C412yes—
75C22×C4×C47C2×D412yes—
76C22×C4×C47C2×D411yescharacteristic
77C22×C4×C47C2×D412yes—
78C22×C4×C47C2×D411yescharacteristic
79C22×Dic47—21no—
80C22×Dic47—21no—
81C22×Dic47—21no—
82C22×Dic47—21no—
83C22×Dic47—24no—
84C22×Dic47—24no—
85C22×Dic47—21no—
86C22×Dic47—21no—
87C22×Dic47—21no—
88C22×Dic47—21no—
89C22×Dic47—42no—
90C22×Dic47—42no—
91C22×Dic47C22⋊C412yes—
92C22×Dic47C22⋊C412yes—
93C22×Dic47C22⋊C412yes—
94C22×Dic47C22⋊C412yes—
95C2×Dic94—28no—
96C2×Dic94—24no—
97C2×Dic94—24no—
98C2×Dic94—28no—
99C2×Dic94—44no—
100C2×Dic94—44no—
101C2×Dic94—48no—
102C2×Dic94—44no—
103C2×Dic94—48no—
104C2×Dic94—48no—
105C2×Dic94—44no—
106C2×Dic94—44no—
107M16×C47—44no—
108C2×C8×C47—42no—
109C4⋊C4×C47—24no—
110C4⋊C4×C47—24no—
111C4⋊Dic47—28no—
112Dic47⋊C4—28no—
113Dic47⋊C4—24no—
114Dic47⋊C4—24no—
115C4×Dic47—28no—
116#10625329—44no—
117C2×C47⋊C8—42no—
118C23×C47—22no—
119C23×C47—22no—
120C23×C47—22no—
121C23×C47C2×C22⋊C411yescharacteristic
122C23×C47C2×C22⋊C411yescharacteristic
123C23×C47C2×C22⋊C411yescharacteristic
124C2×C4×C47—24no—
125C2×C4×C47—24no—
126C2×C4×C47—22no—
127C2×C4×C47—24no—
128C2×C4×C47—22no—
129C2×C4×C47—22no—
130C2×C4×C47—41no—
131C2×C4×C47—42no—
132C2×C4×C47—41no—
133C2×C4×C47—42no—
134C2×C4×C47—42no—
135C2×C4×C47—42no—
136C2×C4×C47#10114yescomplemented
137C2×C4×C47#12012yes—
138C2×C4×C47#12012yes—
139C2×Dic47—24no—
140C2×Dic47—24no—
141C2×Dic47—24no—
142C2×Dic47—24no—
143C2×Dic47—22no—
144C2×Dic47—22no—
145C2×Dic47—22no—
146C2×Dic47—22no—
147C2×Dic47—22no—
148C2×Dic47—22no—
149C2×Dic47—24no—
150C2×Dic47—42no—
151C2×Dic47—42no—
152C2×Dic47—42no—
153C2×Dic47—42no—
154C2×Dic47—42no—
155C2×Dic47#9914yes—
156C2×Dic47#10114yescomplemented
157Dic94—48no—
158Dic94—48no—
159Dic94—416no—
160C8×C47—82no—
161C47⋊C8—82no—
162#8551—471nomaximal, Sylow
163C22×C47—28no—
164C22×C47—21no—
165C22×C47—21no—
166C22×C47—21no—
167C22×C47—21no—
168C22×C47—21no—
169C22×C47—21no—
170C22×C47#37911yescharacteristic
171C22×C47C2×#9912yes—
172C22×C47C2×#10112yes—
173C22×C47C2×#10112yes—
174C4×C47—22no—
175C4×C47—22no—
176C4×C47—24no—
177C4×C47—42no—
178Dic47—24no—
179Dic47—24no—
180Dic47—42no—
181Dic47—42no—
182C22×C4⋊Q8—471no—
183#1784—472no—
184#1463—472no—
185#1463—471no—
186#1463D4711yescharacteristic
187C2×C47—22no—
188C2×C47—22no—
189C2×C47—22no—
190C2×C47#178111yescharacteristic
191C2×C47#178411yescharacteristic
192C2×C47#178411yescharacteristic
193C23×Q8—471no—
194C23×Q8—471no—
195C2×C4⋊Q8—944no—
196C2×C4⋊Q8—944no—
197C2×C4⋊Q8—942no—
198C2×C4⋊Q8—942no—
199C22×C4⋊C4—471no—
200C22×C4⋊C4—471no—
201C22×C4⋊C4—471no—
202C22×C4⋊C4C2×D4711yescharacteristic
203C22×C42—471no—
204C2×#101—944no—
205#407—472no—
206#407C2×D4712yes—
207#324—22no—
208#324—944no—
209#324—942no—
210C47#855111yescharacteristic, complemented, Sylow
211C22×Q8—474no—
212C22×Q8—942no—
213C22×Q8—942no—
214C22×Q8—942no—
215C22×Q8—941no—
216C22×Q8—942no—
217C22×Q8—941no—
218C22×Q8—1884no—
219C22×Q8—1884no—
220C23×C4—471no—
221C23×C4—471no—
222C23×C4—471no—
223C23×C4—471no—
224C23×C4C4×D4711yescharacteristic
225C23×C4C4×D4711yescharacteristic
226C23×C4C22×D4711yescharacteristic
227C2×M16—22no—
228C2×M16—942no—
229C4⋊Q8—9416no—
230C4⋊Q8—9416no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—474no—
234C2×C4⋊C4—942no—
235C2×C4⋊C4—942no—
236C2×C4⋊C4—942no—
237C2×C4⋊C4—942no—
238C2×C4⋊C4—942no—
239C2×C4⋊C4—942no—
240C2×C4⋊C4—941no—
241C2×C4⋊C4—942no—
242C2×C4⋊C4—941no—
243C2×C4⋊C4D18812yes—
244C2×C4⋊C4C22⋊D4712yes—
245C2×C42—942no—
246C2×C42—942no—
247C2×C42—942no—
248#101—1888no—
249C22⋊C8—22no—
250C22⋊C8—22no—
251C22⋊C8—942no—
252C22⋊C8—942no—
253C24C2×C4×D4711yescharacteristic
254C2×Q8—948no—
255C2×Q8—948no—
256C2×Q8—944no—
257C2×Q8—944no—
258C2×Q8—1884no—
259C2×Q8—1884no—
260C2×Q8—1884no—
261C2×Q8—1888no—
262C2×Q8—1884no—
263C2×Q8—1888no—
264C2×Q8—1884no—
265C2×Q8—1888no—
266C22×C4—21no—
267C22×C4—21no—
268C22×C4—24no—
269C22×C4—21no—
270C22×C4—21no—
271C22×C4—42no—
272C22×C4—42no—
273C22×C4—472no—
274C22×C4—472no—
275C22×C4—472no—
276C22×C4—472no—
277C22×C4—944no—
278C22×C4—944no—
279C22×C4—941no—
280C22×C4—941no—
281C22×C4—941no—
282C22×C4—941no—
283C22×C4—941no—
284C22×C4—941no—
285C22×C4—941no—
286C22×C4—941no—
287C22×C4—1882no—
288C22×C4—1882no—
289C22×C4(C2×D47)⋊C412yes—
290C22×C4(C2×D47)⋊C412yes—
291C22×C4C2×D18811yescharacteristic
292C22×C4D4×D4712yes—
293C22×C4D4×D4712yes—
294C22×C4C2×C22⋊D4711yescharacteristic
295M16—44no—
296M16—1884no—
297C2×C8—42no—
298C2×C8—1882no—
299C4⋊C4—24no—
300C4⋊C4—24no—
301C4⋊C4—948no—
302C4⋊C4—948no—
303C4⋊C4—944no—
304C4⋊C4—944no—
305C42—948no—
306C23—22no—
307C23—22no—
308C23—22no—
309C23C22⋊C4×D4711yescharacteristic
310C23C22⋊C4×D4711yescharacteristic
311C23C2×(C2×D47)⋊C411yescharacteristic
312Q8—1888no—
313Q8—1888no—
314Q8—18816no—
315C2×C4—24no—
316C2×C4—24no—
317C2×C4—22no—
318C2×C4—24no—
319C2×C4—22no—
320C2×C4—22no—
321C2×C4—41no—
322C2×C4—42no—
323C2×C4—41no—
324C2×C4—42no—
325C2×C4—42no—
326C2×C4—42no—
327C2×C4—474no—
328C2×C4—474no—
329C2×C4—944no—
330C2×C4—944no—
331C2×C4—944no—
332C2×C4—944no—
333C2×C4—942no—
334C2×C4—942no—
335C2×C4—942no—
336C2×C4—942no—
337C2×C4—942no—
338C2×C4—942no—
339C2×C4—944no—
340C2×C4—1882no—
341C2×C4—1882no—
342C2×C4—1882no—
343C2×C4—1882no—
344C2×C4—1882no—
345C2×C4#1309394214yescomplemented
346C2×C4#1309399212yes—
347C2×C4#1309399412yes—
348C8—82no—
349C8—3762no—
350C22—28no—
351C22—21no—
352C22—21no—
353C22—21no—
354C22—21no—
355C22—21no—
356C22—21no—
357C22#1568897811yescharacteristic
358C22#99×D4712yes—
359C22#101×D4712yes—
360C22C2×#1309394212yes—
361C4—22no—
362C4—22no—
363C4—24no—
364C4—42no—
365C4—944no—
366C4—944no—
367C4—1882no—
368C4—1882no—
369C2—22no—
370C2—22no—
371C2—22no—
372C2#2024330111yescharacteristic
373C2#2024331011yescharacteristic
374C2#2024331111yescharacteristic
subgroupG/H
centerC22#15688978
commutator, prime radicalC24×C47C22×C4
FrattiniC23×C4C22×D47
FittingC47×#1463C2
socleC22×C47#379
cosocleC23×C4×C47C23
hypercenter#1463D47
hypocenterC47#8551
LCSG ▷ C24×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1463

Complements

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

NH ≅ G/Ntype
136. C2×C4×C47#101semidirect
156. C2×Dic47#101semidirect
210. C47#8551semidirect
345. C2×C4#13093942semidirect

Maximal subgroups

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

subgroupindex
C22×#130940232
#202433112
#202417182
#202417172
C47×#14632
#855147