Finite Group Databasebeta

← #26225730

Subgroups — #26225730

Order 12032.

1130 conjugacy classes of subgroups; only the 235 normal subgroups are shown.

normal subgroups only

This group has 1130 conjugacy classes of subgroups — too many to draw. Only its 235 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
1C2×#15689749C211yesmaximal, characteristic
2#20245984C212yesmaximal
3#20244371C218yesmaximal, complemented
4#20243462C211yesmaximal, characteristic
5#20243460C212yesmaximal
6C47×#1819C211yesmaximal, characteristic
7#15689749—24no—
8#15689749—24no—
9C2×#13094003C414yescomplemented
10C2×#13094001C414yescomplemented
11C2×#13094001C2214yes—
12C22×C22⋊Dic47C2211yescharacteristic
13C22×C22⋊Dic47C2211yescharacteristic
14#15689017—24no—
15#15689011—24no—
16#15688981C2214yes—
17#15688979C2212yes—
18C22⋊C4×Dic47C2214yes—
19#15688890C2214yes—
20#15688889C2218yes—
21C22×C22⋊C4×C47C2211yescharacteristic
22C47×#379C2212yes—
23C47×#342C2214yes—
24C22×Dic94C2×C414yes—
25C23×Dic47C2×C412yes—
26C23×Dic47C2312yes—
27#13094003—28no—
28#13094003—28no—
29#13094001—28no—
30#13094001—28no—
31#13094001—28no—
32#13094001—28no—
33C2×C4⋊Dic47C2×C414yes—
34C2×Dic47⋊C4—24no—
35C2×Dic47⋊C4C2×C418yes—
36C2×C22⋊Dic47—22no—
37C2×C22⋊Dic47—22no—
38C2×C22⋊Dic47—22no—
39C2×C22⋊Dic47—22no—
40C2×C22⋊Dic47C2×C412yes—
41C2×C22⋊Dic47C2×C412yes—
42C2×C22⋊Dic47C2×C412yes—
43C2×C22⋊Dic47D414yes—
44C2×C22⋊Dic47D414yes—
45C2×C22⋊Dic47C2314yes—
46C2×C22⋊Dic47C2312yes—
47C2×C4×Dic47—24no—
48C47⋊#95—24no—
49C47⋊#95—24no—
50C25×C47C2311yescharacteristic
51C23×C4×C47C2×C412yes—
52C2×C22⋊C4×C47—22no—
53C2×C22⋊C4×C47—22no—
54C2×C22⋊C4×C47—22no—
55C2×C22⋊C4×C47—22no—
56C2×C22⋊C4×C47C2×C412yes—
57C2×C22⋊C4×C47C2314yes—
58C2×C22⋊C4×C47C2312yes—
59C24×C47—24no—
60C24×C47—82no—
61C24×C47C22×C411yescharacteristic
62C24×C47C22×C411yescharacteristic
63C24×C47C2×D412yes—
64C24×C47C2×D412yes—
65C24×C47C2411yescharacteristic
66C22×C4×C47—22no—
67C22×C4×C47—22no—
68C22×C4×C47—24no—
69C22×C4×C47—82no—
70C22×C4×C47C22×C414yes—
71C22×Dic47—22no—
72C22×Dic47—22no—
73C22×Dic47—24no—
74C22×Dic47—82no—
75C22×Dic47—82no—
76C22×Dic47C22×C414yes—
77C22×Dic47C22×C414yes—
78C22×Dic47C2×D414yes—
79C22×Dic47C2×D414yes—
80C2×Dic94—28no—
81C2×Dic94—84no—
82C22⋊C4×C47—22no—
83C22⋊C4×C47—22no—
84C22⋊C4×C47—24no—
85C22⋊C4×C47—28no—
86C22⋊C4×C47—82no—
87C22⋊C4×C47—82no—
88C22⋊Dic47—24no—
89C22⋊Dic47—24no—
90C22⋊Dic47—28no—
91C22⋊Dic47—82no—
92C22⋊Dic47—82no—
93C22⋊Dic47C22⋊C414yes—
94C22⋊Dic47C22⋊C414yes—
95C22⋊Dic47C22⋊C418yes—
96C4⋊Dic47—28no—
97Dic47⋊C4—216no—
98Dic47⋊C4—84no—
99C4×Dic47—84no—
100C23×C47—22no—
101C23×C47—22no—
102C23×C47—24no—
103C23×C47—24no—
104C23×C47—42no—
105C23×C47—42no—
106C23×C47—42no—
107C23×C47—42no—
108C23×C47—42no—
109C23×C47—42no—
110C23×C47—42no—
111C23×C47—88no—
112C23×C47C2×C22⋊C412yes—
113C23×C47C2×C22⋊C411yescharacteristic
114C23×C47C2×C22⋊C411yescharacteristic
115C23×C47C23×C411yescharacteristic
116C23×C47C22×D412yes—
117C23×C472−1+414yes—
118C2×C4×C47—24no—
119C2×C4×C47—44no—
120C2×C4×C47—44no—
121C2×C4×C47—44no—
122C2×C4×C47—44no—
123C2×C4×C47—82no—
124C2×C4×C47—82no—
125C2×Dic47—24no—
126C2×Dic47—44no—
127C2×Dic47—44no—
128C2×Dic47—44no—
129C2×Dic47—44no—
130C2×Dic47—44no—
131C2×Dic47—44no—
132C2×Dic47—44no—
133C2×Dic47—82no—
134C2×Dic47—82no—
135C2×Dic47C2×C22⋊C418yes—
136Dic94—88no—
137#21827—471nomaximal, Sylow
138C22×C47—22no—
139C22×C47—22no—
140C22×C47—22no—
141C22×C47—22no—
142C22×C47—21no—
143C22×C47—21no—
144C22×C47—22no—
145C22×C47—22no—
146C22×C47—22no—
147C22×C47—22no—
148C22×C47—44no—
149C22×C47—44no—
150C22×C47—44no—
151C22×C47—84no—
152C22×C47—84no—
153C22×C47C22×C22⋊C411yescharacteristic
154C22×C47#51912yes—
155C22×C47#53614yes—
156C4×C47—44no—
157C4×C47—84no—
158Dic47—28no—
159Dic47—44no—
160Dic47—84no—
161C2×#543—471no—
162#2312—472no—
163#1996—478no—
164#1819—472no—
165#1819—471no—
166#1819D4711yescharacteristic
167C2×C47—24no—
168C2×C47—22no—
169C2×C47—41no—
170C2×C47—41no—
171C2×C47—42no—
172C2×C47#231011yescharacteristic
173C2×C47#288312yes—
174#543—944no—
175#543—944no—
176C2×C22⋊Q8—474no—
177C2×C22⋊Q8—474no—
178C2×C22⋊Q8—474no—
179C22×C22⋊C4—471no—
180C22×C22⋊C4—471no—
181C22×C22⋊C4C2×D4711yescharacteristic
182#388—944no—
183#388—944no—
184#380—474no—
185#379—472no—
186#379C2×D4712yes—
187C4×C22⋊C4—474no—
188#342—478no—
189#342—474no—
190#342C2×D4714yes—
191C47#2182711yescharacteristic, complemented, Sylow
192C25C22×D4711yescharacteristic
193C22×Q8—474no—
194C23×C4—472no—
195C23×C4—472no—
196C23×C4C4×D4712yes—
197C22⋊Q8—948no—
198C22⋊Q8—948no—
199C22⋊Q8—948no—
200C22⋊Q8—948no—
201C22⋊Q8—948no—
202C22⋊Q8—948no—
203C2×C4⋊C4—478no—
204C2×C4⋊C4—474no—
205C2×C4⋊C4—944no—
206C2×C22⋊C4—22no—
207C2×C22⋊C4—22no—
208C2×C22⋊C4—22no—
209C2×C22⋊C4—22no—
210C2×C22⋊C4—472no—
211C2×C22⋊C4—472no—
212C2×C22⋊C4—474no—
213C2×C22⋊C4—474no—
214C2×C22⋊C4—472no—
215C2×C22⋊C4—472no—
216C2×C22⋊C4—474no—
217C2×C22⋊C4—942no—
218C2×C22⋊C4—942no—
219C2×C22⋊C4—942no—
220C2×C22⋊C4—942no—
221C2×C22⋊C4C4×D4712yes—
222C2×C22⋊C4C22×D4714yes—
223C2×C22⋊C4C22×D4712yes—
224C2×C42—944no—
225#95—944no—
226#95—944no—
227C24—24no—
228C24—82no—
229C24C2×C4×D4711yescharacteristic
230C24C2×C4×D4711yescharacteristic
231C24D4×D4712yes—
232C24D4×D4712yes—
233C24C23×D4711yescharacteristic
234C2×Q8—948no—
235C2×Q8—3764no—
236C22×C4—22no—
237C22×C4—22no—
238C22×C4—24no—
239C22×C4—82no—
240C22×C4—474no—
241C22×C4—474no—
242C22×C4—474no—
243C22×C4—474no—
244C22×C4—942no—
245C22×C4—942no—
246C22×C4—944no—
247C22×C4—3762no—
248C22×C4—3762no—
249C22×C4C2×C4×D4714yes—
250C4⋊C4—9416no—
251C4⋊C4—948no—
252C4⋊C4—3764no—
253C22⋊C4—22no—
254C22⋊C4—22no—
255C22⋊C4—24no—
256C22⋊C4—28no—
257C22⋊C4—82no—
258C22⋊C4—82no—
259C22⋊C4—474no—
260C22⋊C4—474no—
261C22⋊C4—478no—
262C22⋊C4—944no—
263C22⋊C4—944no—
264C22⋊C4—948no—
265C22⋊C4—3762no—
266C22⋊C4—3762no—
267C42—3764no—
268C23—22no—
269C23—22no—
270C23—24no—
271C23—24no—
272C23—42no—
273C23—42no—
274C23—42no—
275C23—42no—
276C23—42no—
277C23—42no—
278C23—42no—
279C23—88no—
280C23C22⋊C4×D4711yescharacteristic
281C23C22⋊C4×D4711yescharacteristic
282C23C22⋊C4×D4712yes—
283C23C22×C4×D4711yescharacteristic
284C23C2×D4×D4712yes—
285C23#1309407114yes—
286Q8—3768no—
287C2×C4—24no—
288C2×C4—44no—
289C2×C4—44no—
290C2×C4—44no—
291C2×C4—44no—
292C2×C4—82no—
293C2×C4—82no—
294C2×C4—478no—
295C2×C4—944no—
296C2×C4—3762no—
297C2×C4—3762no—
298C2×C4—1884no—
299C2×C4—1884no—
300C2×C4—1884no—
301C2×C4—1884no—
302C2×C4—1884no—
303C2×C4—1884no—
304C2×C4—1884no—
305C22—22no—
306C22—22no—
307C22—22no—
308C22—22no—
309C22—22no—
310C22—22no—
311C22—22no—
312C22—21no—
313C22—21no—
314C22—22no—
315C22—44no—
316C22—44no—
317C22—44no—
318C22—84no—
319C22—84no—
320C22C2×C22⋊C4×D4711yescharacteristic
321C22#1568960012yes—
322C22#1568969514yes—
323C4—44no—
324C4—84no—
325C4—948no—
326C4—3764no—
327C4—1884no—
328C2—24no—
329C2—22no—
330C2—42no—
331C2—41no—
332C2—41no—
333C2#2024597311yescharacteristic
334C2#2025184612yes—
subgroupG/H
centerC22C2×C22⋊C4×D47
commutatorC23×C47C23×C4
FrattiniC24C23×D47
FittingC47×#1819C2
socleC22×C47C22×C22⋊C4
prime radicalC25×C47C23
cosocleC24×C47C24
hypercenter#1819D47
hypocenterC47#21827
LCSG ▷ C23×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C25 ◁ #1819

Complements

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

NH ≅ G/Ntype
3. #20244371C2semidirect
9. C2×#13094003C4semidirect
10. C2×#13094001C4semidirect
191. C47#21827semidirect

Maximal subgroups

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

subgroupindex
C2×#156897492
#202459842
#202443712
#202434622
#202434602
C47×#18192
#2182747