Finite Group Databasebeta

← #26225995

Subgroups — #26225995

Order 12032.

1106 conjugacy classes of subgroups.

The subgroup lattice has 374 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689680C211yesmaximal, characteristic
2C2×#15689176C211yesmaximal, characteristic
3#20244348C212yesmaximal, complemented
4#20244189C212yesmaximal, complemented
5#20243700C214yesmaximal
6#20243400C214yesmaximal, complemented
7C47×#2285C211yesmaximal, characteristic
8#15689680—24no—
9#15689680C418yescomplemented
10C2×#13094024C2211yescharacteristic
11C2×#13094024C2211yescharacteristic
12C2×#13094016—21no—
13C2×#13094003—21no—
14C2×#13094003C2212yes—
15C2×C4×Dic94—21no—
16C2×C4×Dic94C2212yes—
17C2×#13093983—21no—
18#15689176—24no—
19C2×C4⋊(C47⋊C8)C2212yes—
20C2×#13093947C2218yes—
21C2×C22⋊(C47⋊C8)C2212yes—
22C2×C4×C47⋊C8C2212yes—
23#15688885C2218yes—
24C2×#117×C47C2212yes—
25C22×C42×C47C2211yescharacteristic
26C47×#384C2212yes—
27C47×#382C2212yes—
28C47×#380—21no—
29C47×#380—21no—
30C4×C4⋊C4×C47—21no—
31C4×C4⋊C4×C47—21no—
32C22×Dic94—21no—
33C22×Dic94C2312yes—
34C22×C47⋊C8C2314yes—
35#13094024—22no—
36#13094024—22no—
37#13094024C2×C414yes—
38#13094024C2×C414yes—
39#13094016—28no—
40#13094003—28no—
41#13094003—24no—
42#13094003C2×C418yes—
43C4×Dic94—28no—
44C4×Dic94—24no—
45C4×Dic94C2×C418yes—
46#13093983—28no—
47C2×C4⋊Dic47—21no—
48C2×C4⋊Dic47—21no—
49C2×C4⋊Dic47—21no—
50C2×C4⋊Dic47C2312yes—
51C2×Dic47⋊C4—22no—
52C2×Dic47⋊C4—22no—
53C2×C22⋊Dic47—22no—
54C2×C4×Dic47—22no—
55C4⋊(C47⋊C8)—24no—
56#13093947—48no—
57C22⋊(C47⋊C8)—24no—
58C4×C47⋊C8—24no—
59C23×C4×C47D411yescharacteristic
60C23×C4×C47D411yescharacteristic
61C23×C4×C47C2311yescharacteristic
62#117×C47—44no—
63C2×C4⋊C4×C47—22no—
64C2×C4⋊C4×C47—22no—
65C2×C4⋊C4×C47C2314yes—
66C2×C22⋊C4×C47—22no—
67C2×C42×C47—22no—
68C2×C42×C47C2×C414yes—
69C2×C42×C47D411yescharacteristic
70C2×C42×C47D411yescharacteristic
71C2×C42×C47D414yes—
72C2×C42×C47C2311yescharacteristic
73C2×C42×C47C2311yescharacteristic
74#95×C47—22no—
75#95×C47—22no—
76C24×C47C2×D411yescharacteristic
77C22×C4×C47—24no—
78C22×C4×C47—24no—
79C22×C4×C47—22no—
80C22×C4×C47—22no—
81C22×C4×C47—24no—
82C22×C4×C47C22⋊C414yes—
83C22×C4×C47C22⋊C414yes—
84C22×C4×C47C22×C412yes—
85C22×C4×C47C2×D411yescharacteristic
86C22×C4×C47C2×D411yescharacteristic
87C22×C4×C47C2×D411yescharacteristic
88C22×C4×C47C2×D412yes—
89C22×C4×C47C2×D411yescharacteristic
90C22×C4×C47C2×D411yescharacteristic
91C22×C4×C47C2411yescharacteristic
92C22×Dic47—22no—
93C22×Dic47—22no—
94C2×Dic94—24no—
95C2×Dic94—24no—
96C2×Dic94—24no—
97C2×Dic94—22no—
98C2×Dic94—44no—
99C2×Dic94C22×C414yes—
100C4⋊C4×C47—28no—
101C4⋊C4×C47—44no—
102C4⋊C4×C47—44no—
103C22⋊C4×C47—44no—
104C42×C47—44no—
105C42×C47C22⋊C414yes—
106C42×C47C22⋊C414yes—
107C42×C47C22×C412yes—
108C42×C47C22×C412yes—
109C42×C47C2×D412yes—
110C42×C47C2×D412yes—
111C22⋊Dic47—28no—
112C4⋊Dic47—24no—
113C4⋊Dic47—24no—
114C4⋊Dic47—24no—
115C4⋊Dic47—22no—
116C4⋊Dic47C22×C414yes—
117Dic47⋊C4—28no—
118Dic47⋊C4—28no—
119C4×Dic47—28no—
120C2×C47⋊C8—24no—
121C2×C47⋊C8—24no—
122C2×C47⋊C8—24no—
123C23×C47—21no—
124C23×C47—21no—
125C23×C47—22no—
126C23×C47—22no—
127C23×C47C2×C22⋊C412yes—
128C23×C47C22×D411yescharacteristic
129C2×C4×C47—21no—
130C2×C4×C47—21no—
131C2×C4×C47—22no—
132C2×C4×C47—22no—
133C2×C4×C47—24no—
134C2×C4×C47—24no—
135C2×C4×C47—28no—
136C2×C4×C47—28no—
137C2×C4×C47—44no—
138C2×C4×C47—44no—
139C2×C4×C47—44no—
140C2×C4×C47C2×C22⋊C412yes—
141C2×C4×C47C2×C22⋊C412yes—
142C2×C4×C47C2×C22⋊C412yes—
143C2×C4×C47C2×C22⋊C412yes—
144C2×C4×C47C2×C22⋊C412yes—
145C2×C4×C47#13514yes—
146C2×C4×C47#13714yes—
147C2×C4×C47C23×C411yescharacteristic
148C2×C4×C47C22×D411yescharacteristic
149C2×C4×C47H(5,𝔽2)12yes—
150C2×C4×C472−1+412yes—
151C2×Dic47—24no—
152C2×Dic47—24no—
153C2×Dic47—42no—
154C2×Dic47—42no—
155C2×Dic47—44no—
156Dic94—28no—
157Dic94—44no—
158Dic94—48no—
159C47⋊C8—44no—
160#21864—471nomaximal, Sylow
161C22×C47—22no—
162C22×C47—22no—
163C22×C47—22no—
164C22×C47—42no—
165C22×C47—42no—
166C22×C47C22×C22⋊C411yescharacteristic
167C22×C47#53511yescharacteristic
168C22×C47#53611yescharacteristic
169C22×C47C2×#13512yes—
170C22×C47C2×#13712yes—
171C4×C47—22no—
172C4×C47—22no—
173C4×C47—22no—
174C4×C47—84no—
175C4×C47#41614yes—
176C4×C47#41914yes—
177C4×C47#52012yes—
178C4×C47#53712yes—
179Dic47—44no—
180Dic47—44no—
181C2×#533—471no—
182C2×#431—471no—
183#2285D4711yescharacteristic
184#1990—472no—
185#1956—472no—
186#1873—474no—
187#1805—474no—
188C2×C47—42no—
189C2×C47#231111yescharacteristic
190C2×C47C2×#41611yescharacteristic
191C2×C47C2×#41911yescharacteristic
192C2×C47#321012yes—
193C2×C47#323112yes—
194#533—478no—
195#533—944no—
196C2×C4⋊Q8—471no—
197C2×C4⋊Q8—471no—
198C2×#125—941no—
199C2×C22⋊Q8—472no—
200C2×C22⋊Q8—941no—
201C2×C4×Q8—472no—
202C2×C4×Q8—941no—
203C2×#117—941no—
204C2×#117C2×D4712yes—
205C22×C42C2×D4711yescharacteristic
206#431—944no—
207C2×C4⋊C8—472no—
208C2×#103—478no—
209C2×C22⋊C8—472no—
210C2×C4×C8—472no—
211#384C2×D4712yes—
212#382C2×D4712yes—
213#380—21no—
214#380—21no—
215C4×C4⋊C4—21no—
216C4×C4⋊C4—21no—
217#340—478no—
218C47#2186411yescharacteristic, complemented, Sylow
219C22×Q8—472no—
220C22×Q8—941no—
221C23×C4C22⋊D4711yescharacteristic
222C23×C4C22⋊D4711yescharacteristic
223C23×C4C22×D4711yescharacteristic
224C22×C8—474no—
225C4⋊Q8—474no—
226C4⋊Q8—474no—
227C4⋊Q8—942no—
228C4⋊Q8—942no—
229#125—948no—
230C22⋊Q8—478no—
231C22⋊Q8—948no—
232C22⋊Q8—944no—
233C4×Q8—478no—
234C4×Q8—948no—
235C4×Q8—944no—
236#117—44no—
237#117—948no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—472no—
241C2×C4⋊C4—941no—
242C2×C4⋊C4—941no—
243C2×C4⋊C4—942no—
244C2×C4⋊C4—942no—
245C2×C4⋊C4—941no—
246C2×C4⋊C4C22×D4714yes—
247C2×C22⋊C4—22no—
248C2×C22⋊C4—942no—
249C2×C42—22no—
250C2×C42—942no—
251C2×C42C4×D4714yes—
252C2×C42D18814yes—
253C2×C42C22⋊D4711yescharacteristic
254C2×C42C22⋊D4711yescharacteristic
255C2×C42C22×D4711yescharacteristic
256C2×C42C22×D4711yescharacteristic
257C4⋊C8—944no—
258#103—1888no—
259C22⋊C8—944no—
260C4×C8—944no—
261#95—22no—
262#95—22no—
263C24C2×C22⋊D4711yescharacteristic
264C2×Q8—474no—
265C2×Q8—944no—
266C2×Q8—942no—
267C2×Q8—944no—
268C2×Q8—944no—
269C2×Q8—1884no—
270C22×C4—24no—
271C22×C4—24no—
272C22×C4—22no—
273C22×C4—22no—
274C22×C4—24no—
275C22×C4—942no—
276C22×C4—942no—
277C22×C4(C2×D47)⋊C414yes—
278C22×C4(C2×D47)⋊C414yes—
279C22×C4C2×C4×D4712yes—
280C22×C4C2×D18812yes—
281C22×C4C2×C22⋊D4711yescharacteristic
282C22×C4C2×C22⋊D4711yescharacteristic
283C22×C4C2×C22⋊D4711yescharacteristic
284C22×C4C2×C22⋊D4711yescharacteristic
285C22×C4C2×C22⋊D4711yescharacteristic
286C22×C4C23×D4711yescharacteristic
287C2×C8—944no—
288C2×C8—944no—
289C2×C8—944no—
290C4⋊C4—28no—
291C4⋊C4—44no—
292C4⋊C4—44no—
293C4⋊C4—474no—
294C4⋊C4—948no—
295C4⋊C4—948no—
296C4⋊C4—944no—
297C4⋊C4—944no—
298C4⋊C4—944no—
299C4⋊C4—942no—
300C22⋊C4—44no—
301C22⋊C4—948no—
302C42—44no—
303C42—948no—
304C42(C2×D47)⋊C414yes—
305C42(C2×D47)⋊C414yes—
306C42C2×C4×D4712yes—
307C42C2×C4×D4712yes—
308C42C2×D18812yes—
309C42C2×D18812yes—
310C23—22no—
311C23—22no—
312C23—21no—
313C23—21no—
314C23C2×(C2×D47)⋊C412yes—
315C23C22×C22⋊D4711yescharacteristic
316Q8—948no—
317Q8—1888no—
318Q8—1884no—
319C2×C4—22no—
320C2×C4—22no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—24no—
324C2×C4—28no—
325C2×C4—28no—
326C2×C4—24no—
327C2×C4—44no—
328C2×C4—44no—
329C2×C4—44no—
330C2×C4—944no—
331C2×C4—944no—
332C2×C4—1884no—
333C2×C4—1882no—
334C2×C4—1882no—
335C2×C4C2×(C2×D47)⋊C412yes—
336C2×C4C2×(C2×D47)⋊C412yes—
337C2×C4C2×(C2×D47)⋊C412yes—
338C2×C4C2×(C2×D47)⋊C412yes—
339C2×C4C2×(C2×D47)⋊C412yes—
340C2×C4#1309404414yes—
341C2×C4#1309405414yes—
342C2×C4C22×C4×D4711yescharacteristic
343C2×C4C22×D18811yescharacteristic
344C2×C4#1309406912yes—
345C2×C4#1309407212yes—
346C8—1884no—
347C22—22no—
348C22—22no—
349C22—22no—
350C22—42no—
351C22—42no—
352C22C22×(C2×D47)⋊C411yescharacteristic
353C22#1568969011yescharacteristic
354C22#1568969711yescharacteristic
355C22C2×#1309404412yes—
356C22C2×#1309405412yes—
357C4—22no—
358C4—22no—
359C4—22no—
360C4—84no—
361C4—1884no—
362C4—1884no—
363C4#1568911714yes—
364C4#1568913114yes—
365C4#1568960412yes—
366C4#1568970412yes—
367C2—42no—
368C2#2024598111yescharacteristic
369C2C2×#1568911711yescharacteristic
370C2C2×#1568913111yescharacteristic
371C2#2025434412yes—
372C2#2025454712yes—
subgroupG/H
centerC23C22×C22⋊D47
commutatorC2×C4×C47C23×C4
FrattiniC22×C4C23×D47
FittingC47×#2285C2
socleC23×C47C22×D4
prime radicalC24×C47C2×D4
cosocleC22×C4×C47C24
hypercenter#2285D47
hypocenterC47#21864
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C22×C42 ◁ #2285

Complements

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

NH ≅ G/Ntype
3. #20244348C2semidirect
4. #20244189C2semidirect
6. #20243400C2semidirect
9. #15689680C4semidirect
218. C47#21864semidirect

Maximal subgroups

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

subgroupindex
C2×#156896802
C2×#156891762
#202443482
#202441892
#202437002
#202434002
C47×#22852
#2186447