Finite Group Databasebeta

← #26220364

Subgroups — #26220364

Order 12032.

1694 conjugacy classes of subgroups.

The subgroup lattice has 368 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#13094012C211yesmaximal, characteristic
2C2×#15689133C212yesmaximal, complemented
3C2×#15689129C212yesmaximal
4C4×#13093983C211yesmaximal, characteristic, complemented
5#20244318C218yesmaximal, complemented
6C2×C47×#423C211yesmaximal, characteristic, complemented
7C23×Dic94C2211yescharacteristic
8C23×D188C2211yescharacteristic
9C2×#13094012—21no—
10C2×#13094012—21no—
11C2×#13094012—24no—
12C2×#13094012C418yescomplemented
13C2×#13094012C2214yes—
14C2×#13093983C2212yes—
15C22×(C2×D47)⋊C4—22no—
16#15689133—28no—
17#15689129—28no—
18C2×#13093949C2218yescomplemented
19C2×#13093946C2218yes—
20C4×C4⋊Dic47C2214yescomplemented
21C4×C22⋊Dic47—22no—
22#15688969—21no—
23C42×Dic47—21no—
24C22×M16×C47C2212yes—
25C22×C42×C47C2211yescharacteristic
26C47×#423—24no—
27C2×C4⋊C8×C47C2214yescomplemented
28C24×D47—21no—
29C22×Dic94—24no—
30C22×Dic94—24no—
31C22×Dic94—21no—
32C22×Dic94—22no—
33C22×Dic94C2×C414yes—
34C22×Dic94C2312yes—
35C22×D188—24no—
36C22×D188—24no—
37C22×D188—21no—
38C22×D188—22no—
39C22×D188C2×C414yes—
40C22×D188C2312yes—
41C23×Dic47—21no—
42#13094012—28no—
43#13094012—28no—
44#13094012—28no—
45#13094012C2×C4116yes—
46#13093983—24no—
47#13093983D418yes—
48C2×C4⋊Dic47C2314yes—
49C2×C22⋊Dic47—22no—
50C2×(C2×D47)⋊C4—216no—
51C2×(C2×D47)⋊C4—24no—
52C2×(C2×D47)⋊C4—24no—
53C2×C4×Dic47—22no—
54C2×C4×Dic47—24no—
55#13093949—216no—
56#13093946—216no—
57C47⋊#95—24no—
58C23×C4×C47D412yes—
59C23×C4×C47C2311yescharacteristic
60C2×M16×C47—24no—
61C2×M16×C47—24no—
62C2×M16×C47—24no—
63C22×C8×C47C2314yes—
64C2×C42×C47C2×C414yes—
65C2×C42×C47D411yescharacteristic
66C2×C42×C47D411yescharacteristic
67C2×C42×C47D414yes—
68C2×C42×C47C2312yes—
69C4⋊C8×C47—28no—
70C24×C47C2×D411yescharacteristic
71C23×D47—24no—
72C23×D47—22no—
73C23×D47—48no—
74C23×D47—44no—
75C22×C4×C47—44no—
76C22×C4×C47C22⋊C418yes—
77C22×C4×C47C22×C412yes—
78C22×C4×C47C2×D412yes—
79C22×C4×C47C2×D412yes—
80C22×C4×C47C2×D411yescharacteristic
81C22×C4×C47C2×D412yes—
82C22×C4×C47C2×D414yes—
83C22×C4×C47#4114yes—
84C22×C4×C47C2411yescharacteristic
85C22×Dic47—24no—
86C22×Dic47—24no—
87C22×Dic47—22no—
88C22×Dic47—44no—
89C2×D188—216no—
90C2×D188—24no—
91C2×D188—24no—
92C2×D188—24no—
93C2×D188—22no—
94C2×D188—44no—
95C2×D188—44no—
96C2×D188—44no—
97C2×D188C22×C414yes—
98C2×Dic94—216no—
99C2×Dic94—24no—
100C2×Dic94—24no—
101C2×Dic94—24no—
102C2×Dic94—22no—
103C2×Dic94—44no—
104C2×Dic94—44no—
105C2×Dic94—44no—
106C2×Dic94C22×C414yes—
107M16×C47—48no—
108C2×C8×C47—24no—
109C2×C8×C47—24no—
110C2×C8×C47—24no—
111C42×C47C22⋊C414yes—
112C42×C47C22⋊C414yes—
113C42×C47C22×C414yes—
114C42×C47C2×D414yes—
115C22⋊Dic47—28no—
116(C2×D47)⋊C4—216no—
117(C2×D47)⋊C4—216no—
118C4⋊Dic47—24no—
119C4⋊Dic47C2×D418yes—
120C4×Dic47—28no—
121C4×Dic47—48no—
122C23×C47—24no—
123C23×C47C2×C22⋊C412yes—
124C23×C47#12114yes—
125C23×C47C22×D411yescharacteristic
126C22×D47—24no—
127C22×D47—416no—
128C22×D47—44no—
129C22×D47—48no—
130C22×D47—48no—
131C22×D47—816no—
132C2×C4×C47—24no—
133C2×C4×C47—24no—
134C2×C4×C47—48no—
135C2×C4×C47C2×C22⋊C414yes—
136C2×C4×C47C2×C22⋊C414yes—
137C2×C4×C47C2×C22⋊C412yes—
138C2×C4×C47C4×D418yes—
139C2×C4×C47#12114yes—
140C2×C4×C47#12414yes—
141C2×C4×C47#12714yes—
142C2×C4×C47C23×C411yescharacteristic
143C2×C4×C47C22×D412yes—
144C2×C4×C47C22×D411yescharacteristic
145C2×C4×C47C2×#4112yes—
146C2×Dic47—24no—
147C2×Dic47—28no—
148C2×Dic47—44no—
149C2×Dic47—48no—
150C2×Dic47—48no—
151D188—28no—
152D188—42no—
153D188—42no—
154D188—48no—
155Dic94—28no—
156Dic94—42no—
157Dic94—42no—
158Dic94—48no—
159C8×C47—44no—
160#21261—471nomaximal, Sylow
161C22×C47—24no—
162C22×C47—28no—
163C22×C47#39014yes—
164C22×C47C22×C22⋊C411yescharacteristic
165C22×C47C2×#12112yes—
166C22×C47#57514yes—
167C2×D47—48no—
168C2×D47—44no—
169C2×D47—816no—
170C2×D47—88no—
171C4×C47—24no—
172C4×C47—24no—
173C4×C47#39014yes—
174C4×C47C2×C4×D412yes—
175C4×C47C2×#12411yescharacteristic
176C4×C47C2×#12711yescharacteristic
177Dic47—48no—
178Dic47—44no—
179C22×#124—471no—
180C2×#423D4711yescharacteristic, complemented
181C2×#419—472no—
182C2×#418—472no—
183C4×#117—471no—
184#1983—478no—
185C2×C47—24no—
186C2×C47C2×#39011yescharacteristic
187C2×C47#289612yes—
188C2×C47#304014yes—
189D47—84no—
190C23×Q8—471no—
191C23×D4—471no—
192C22×M16C2×D4712yes—
193C2×#124—478no—
194C2×#124—474no—
195C2×#124—941no—
196C2×#124—941no—
197C2×#124—944no—
198C2×#117—472no—
199C22×C22⋊C4—942no—
200C22×C42C2×D4711yescharacteristic
201#423—24no—
202C2×C4⋊C8C2×D4714yescomplemented
203#419—948no—
204#418—948no—
205C2×#103—478no—
206C2×#102—478no—
207C4×C4⋊C4—474no—
208C4×C22⋊C4—942no—
209#376—941no—
210C43—941no—
211C47#2126111yescharacteristic, complemented, Sylow
212C25—941no—
213C22×Q8—474no—
214C22×Q8—472no—
215C22×Q8—941no—
216C22×Q8—942no—
217C22×Q8—944no—
218C22×Q8—944no—
219C22×D4—474no—
220C22×D4—472no—
221C22×D4—941no—
222C22×D4—944no—
223C22×D4—944no—
224C22×D4—942no—
225C23×C4—941no—
226C23×C4D18812yes—
227C23×C4C22×D4711yescharacteristic
228C2×M16—24no—
229C2×M16—24no—
230C2×M16—24no—
231C22×C8C22×D4714yes—
232#124—4716no—
233#124—948no—
234#124—948no—
235#124—948no—
236#117—478no—
237#117—944no—
238C2×C4⋊C4—474no—
239C2×C22⋊C4—942no—
240C2×C22⋊C4—9416no—
241C2×C22⋊C4—944no—
242C2×C22⋊C4—944no—
243C2×C42—942no—
244C2×C42—944no—
245C2×C42C4×D4714yes—
246C2×C42D18811yescharacteristic
247C2×C42D18811yescharacteristic
248C2×C42C22⋊D4714yes—
249C2×C42C22×D4712yes—
250C4⋊C8—28no—
251#103—9416no—
252#102—9416no—
253#95—944no—
254C24—944no—
255C24—942no—
256C24—1888no—
257C24—1884no—
258C24C2×D18811yescharacteristic
259C2×Q8—474no—
260C2×Q8—944no—
261C2×Q8—942no—
262C2×Q8—9416no—
263C2×Q8—944no—
264C2×Q8—944no—
265C2×Q8—1884no—
266C2×Q8—1884no—
267C2×Q8—1884no—
268C2×D4—474no—
269C2×D4—944no—
270C2×D4—9416no—
271C2×D4—944no—
272C2×D4—942no—
273C2×D4—944no—
274C2×D4—1884no—
275C2×D4—1884no—
276C2×D4—1884no—
277C22×C4—44no—
278C22×C4—944no—
279C22×C4—944no—
280C22×C4—942no—
281C22×C4—1884no—
282C22×C4(C2×D47)⋊C418yes—
283C22×C4C2×C4×D4712yes—
284C22×C4C2×D18812yes—
285C22×C4C2×D18812yes—
286C22×C4C2×D18811yescharacteristic
287C22×C4D4×D4714yes—
288C22×C4#1062535414yes—
289C22×C4C2×C22⋊D4712yes—
290C22×C4C23×D4711yescharacteristic
291M16—48no—
292C2×C8—24no—
293C2×C8—24no—
294C2×C8—24no—
295C4⋊C4—478no—
296C4⋊C4—944no—
297C22⋊C4—948no—
298C22⋊C4—9416no—
299C22⋊C4—9416no—
300C42—948no—
301C42—1888no—
302C42(C2×D47)⋊C414yes—
303C42(C2×D47)⋊C414yes—
304C42C2×C4×D4714yes—
305C42C2×C22⋊D4714yes—
306C23—24no—
307C23—944no—
308C23—37616no—
309C23—18816no—
310C23—1888no—
311C23—1884no—
312C23—1888no—
313C23C2×(C2×D47)⋊C412yes—
314C23#1309399914yes—
315C23C22×D18811yescharacteristic
316Q8—948no—
317Q8—1888no—
318Q8—1882no—
319Q8—1882no—
320D4—948no—
321D4—1888no—
322D4—1882no—
323D4—1882no—
324C2×C4—24no—
325C2×C4—24no—
326C2×C4—48no—
327C2×C4—944no—
328C2×C4—948no—
329C2×C4—1888no—
330C2×C4—1884no—
331C2×C4—1888no—
332C2×C4C2×(C2×D47)⋊C414yes—
333C2×C4C2×(C2×D47)⋊C414yes—
334C2×C4C2×(C2×D47)⋊C412yes—
335C2×C4#1309398618yes—
336C2×C4#1309399914yes—
337C2×C4#1309401314yes—
338C2×C4#1309402014yes—
339C2×C4C22×C4×D4711yescharacteristic
340C2×C4C2×D4×D4712yes—
341C2×C4C22×C22⋊D4711yescharacteristic
342C2×C4C2×#1062535412yes—
343C8—44no—
344C22—24no—
345C22—28no—
346C22—37616no—
347C22—3768no—
348C22—1888no—
349C22—1884no—
350C22#1568902414yes—
351C22C22×(C2×D47)⋊C411yescharacteristic
352C22C2×#1309399912yes—
353C22#1568992314yes—
354C4—24no—
355C4—24no—
356C4—1884no—
357C4—1888no—
358C4#1568902414yes—
359C4C2×#1309398612yes—
360C4C2×#1309401311yescharacteristic
361C4C2×#1309402011yescharacteristic
362C2—24no—
363C2—3764no—
364C2C2×#1568902411yescharacteristic
365C2#2025190812yes—
366C2#2025288314yes—
subgroupG/H
centerC23C22×D188
commutatorC2×C4×C47C23×C4
FrattiniC22×C4C23×D47
FittingC2×C47×#423C2
socleC23×C47C22×D4
prime radicalC23×D188C22
cosocleC22×C4×C47C24
hypercenterC2×#423D47
hypocenterC47#21261
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C22×C42 ◁ C2×#423

Complements

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

NH ≅ G/Ntype
2. C2×#15689133C2semidirect
4. C4×#13093983C2semidirect
5. #20244318C2semidirect
6. C2×C47×#423C2semidirect
12. C2×#13094012C4semidirect
18. C2×#13093949C22semidirect
20. C4×C4⋊Dic47C22semidirect
27. C2×C4⋊C8×C47C22semidirect
180. C2×#423D47semidirect
202. C2×C4⋊C8C2×D47semidirect
211. C47#21261semidirect

Maximal subgroups

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

subgroupindex
C22×#130940122
C2×#156891332
C2×#156891292
C4×#130939832
#202443182
C2×C47×#4232
#2126147