Finite Group Databasebeta

← #26422353

Subgroups — #26422353

Order 12032.

2782 conjugacy classes of subgroups.

The subgroup lattice has 354 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689680C211yesmaximal, characteristic, complemented
2C2×#15689673C211yesmaximal, characteristic
3#20251766C212yesmaximal, complemented
4#20251760C214yesmaximal, complemented
5#20251745C212yesmaximal, complemented
6#20251689C218yesmaximal, complemented
7C4×#13094012C218yesmaximal, complemented
8#20245878C214yesmaximal, complemented
9C2×C43×C47C211yesmaximal, characteristic, complemented
10C22×#10625350C2212yes—
11#15689680—28no—
12#15689673—28no—
13C2×#13094024C2212yes—
14C2×#13094024C2211yescharacteristic
15C2×#13094021C2211yescharacteristic
16C2×#13094016C2212yes—
17C2×#13094012C2218yes—
18C2×#13094012C2212yes—
19C2×#13094003C2214yes—
20C2×#13093998C2214yes—
21C2×C4×Dic94C2214yes—
22C2×C4×D188C2214yes—
23C2×#13093983C2212yes—
24#15689029C2218yes—
25#15689029C2218yes—
26#15689025C2218yes—
27#15689025C2218yes—
28#15689014C22132yes—
29#15689002C22116yes—
30#15688989C2218yes—
31C4×(C2×D47)⋊C4C22116yes—
32C22×C42×C47C2214yes—
33C22×C42×C47C2212yes—
34C22×C42×C47C2211yescharacteristic
35C43×C47C2212yes—
36C43×C47C2212yes—
37C43×C47C2214yes—
38C2×#10625350—28no—
39C2×#10625350—216no—
40C22×Dic94C2312yes—
41C22×Dic94C2314yes—
42C22×D188C2312yes—
43C22×D188C2314yes—
44C22×C22⋊D47C2314yes—
45C22×C4×D47C2314yes—
46#13094024—24no—
47#13094024—28no—
48#13094021—24no—
49#13094016—28no—
50#13094012—28no—
51#13094012—232no—
52#13094003—216no—
53#13093998—216no—
54C4×Dic94—216no—
55C4×D188—216no—
56#13093983—28no—
57C2×C4⋊Dic47C2314yes—
58C2×C4⋊Dic47C2314yes—
59C2×C4⋊Dic47C2314yes—
60C2×Dic47⋊C4C2318yes—
61C2×C22⋊Dic47C2314yes—
62C2×(C2×D47)⋊C4C23132yes—
63C2×(C2×D47)⋊C4C2318yes—
64C2×C4×Dic47C2314yes—
65C47⋊#95C23132yes—
66C23×C4×C47C2314yes—
67C23×C4×C47C2312yes—
68C23×C4×C47C2311yescharacteristic
69C2×C42×C47—28no—
70C2×C42×C47—216no—
71C2×C42×C47D418yes—
72C2×C42×C47C2314yes—
73C2×C42×C47C2318yes—
74C2×C42×C47C2314yes—
75C2×C42×C47C2314yes—
76C2×C42×C47C2312yes—
77C2×C42×C47C2311yescharacteristic
78C2×C42×C47C2311yescharacteristic
79C2×C42×C47C2314yes—
80C24×C47C2411yescharacteristic
81C23×D47C2414yes—
82C22×C4×C47—28no—
83C22×C4×C47—216no—
84C22×C4×C47—28no—
85C22×C4×C47C2×D418yes—
86C22×C4×C47C2×D414yes—
87C22×C4×C47#4118yes—
88C22×C4×C47C2412yes—
89C22×C4×C47C2418yes—
90C22×C4×C47C2411yescharacteristic
91C22×C4×C47C2412yes—
92C22×C4×C47C2411yescharacteristic
93C2×C22⋊D47—28no—
94C2×C22⋊D47—216no—
95C22×Dic47C2414yes—
96C22×Dic47C2418yes—
97#10625350—432no—
98C2×D188—28no—
99C2×D188—24no—
100C2×D188—216no—
101C2×D188—28no—
102C2×C4×D47—28no—
103C2×C4×D47—216no—
104C2×Dic94—28no—
105C2×Dic94—24no—
106C2×Dic94—216no—
107C2×Dic94—28no—
108C42×C47—216no—
109C42×C47—28no—
110C42×C47—28no—
111C42×C47—28no—
112C42×C47C2×D418yes—
113C42×C47C2×D414yes—
114C42×C47C2×D414yes—
115C42×C47#41116yescomplemented
116C22⋊Dic47—28no—
117(C2×D47)⋊C4—216no—
118(C2×D47)⋊C4—264no—
119C4⋊Dic47—28no—
120C4⋊Dic47—28no—
121C4⋊Dic47—28no—
122Dic47⋊C4—216no—
123C4×Dic47—28no—
124C23×C47—24no—
125C23×C47C22×D412yes—
126C23×C47C2×#4114yes—
127C23×C47C2511yescharacteristic
128C22×D47—24no—
129C22×D47—216no—
130C22×D47—28no—
131C2×C4×C47—232no—
132C2×C4×C47—24no—
133C2×C4×C47—28no—
134C2×C4×C47—28no—
135C2×C4×C47—216no—
136C2×C4×C47#124116yescomplemented
137C2×C4×C47C22×D414yes—
138C2×C4×C47C22×D412yes—
139C2×C4×C47C22×D412yes—
140C2×C4×C47C22×D414yes—
141C2×C4×C47C2×#41116yes—
142C2×C4×C47C2×#4114yes—
143C2×C4×C47C2×#4114yes—
144C2×C4×C47C2×#4118yes—
145C22⋊D47—416no—
146C2×Dic47—28no—
147C2×Dic47—24no—
148C2×Dic47—216no—
149C2×Dic47—28no—
150D188—48no—
151D188—416no—
152C4×D47—416no—
153Dic94—48no—
154Dic94—416no—
155#36639—471nomaximal, Sylow
156C22×C47—24no—
157C22×C47—28no—
158C22×C47C2×#12414yes—
159C22×C47C23×D411yescharacteristic
160C22×C47C22×#4112yes—
161C22×C47C22×#4114yes—
162C2×D47—44no—
163C2×D47—48no—
164C2×D47—416no—
165C4×C47—24no—
166C4×C47—216no—
167C4×C47C2×#12414yes—
168C4×C47C2×#12414yes—
169C4×C47#53218yes—
170Dic47—44no—
171Dic47—48no—
172C2×#533—471no—
173C2×#532—471no—
174#2862—472no—
175#2861—474no—
176#2858—472no—
177#2850—478no—
178C4×#124—478no—
179#2297—474no—
180C2×C43D4711yescharacteristic, complemented
181C2×C47—24no—
182C2×C47C22×#12411yescharacteristic
183C2×C47C2×#53212yes—
184C2×C47#344414yes—
185D47—84no—
186C22×#41—472no—
187#533—948no—
188#532—948no—
189C2×C4⋊Q8—472no—
190C2×C4⋊Q8—471no—
191C2×#127—471no—
192C2×#125—472no—
193C2×#124—478no—
194C2×#124—472no—
195C2×C22⋊Q8—474no—
196C2×#121—474no—
197C2×C4×Q8—474no—
198C2×C4×D4—474no—
199C2×#117—472no—
200C22×C42C2×D4714yes—
201C22×C42C2×D4712yes—
202C22×C42C2×D4711yescharacteristic
203#391—478no—
204#391—478no—
205#390—478no—
206#390—478no—
207#388—4732no—
208#386—4716no—
209#382—478no—
210C4×C22⋊C4—4716no—
211C43C2×D4712yes—
212C43C2×D4712yes—
213C43C2×D4714yes—
214C47#3663911yescharacteristic, complemented, Sylow
215C2×#41—948no—
216C2×#41—9416no—
217C22×Q8—474no—
218C22×Q8—472no—
219C22×D4—474no—
220C22×D4—474no—
221C22×D4—472no—
222C23×C4—474no—
223C23×C4C22×D4714yes—
224C23×C4C22×D4712yes—
225C23×C4C22×D4711yescharacteristic
226C4⋊Q8—944no—
227C4⋊Q8—948no—
228#127—944no—
229#125—948no—
230#124—948no—
231#124—9432no—
232C22⋊Q8—9416no—
233#121—9416no—
234C4×Q8—9416no—
235C4×D4—9416no—
236#117—948no—
237C2×C4⋊C4—474no—
238C2×C4⋊C4—474no—
239C2×C4⋊C4—478no—
240C2×C4⋊C4—474no—
241C2×C22⋊C4—474no—
242C2×C22⋊C4—478no—
243C2×C22⋊C4—4732no—
244C2×C42—28no—
245C2×C42—216no—
246C2×C42—474no—
247C2×C42D18818yes—
248C2×C42C22×D4718yes—
249C2×C42C22×D4714yes—
250C2×C42C22×D4714yes—
251C2×C42C22×D4714yes—
252C2×C42C22×D4711yescharacteristic
253C2×C42C22×D4711yescharacteristic
254C2×C42C22×D4712yes—
255C2×C42C22×D4714yes—
256#95—4732no—
257C24—474no—
258C24C23×D4711yescharacteristic
259#41—18832no—
260C2×Q8—948no—
261C2×Q8—944no—
262C2×Q8—9416no—
263C2×Q8—948no—
264C2×D4—948no—
265C2×D4—9416no—
266C2×D4—948no—
267C2×D4—944no—
268C2×D4—9416no—
269C2×D4—948no—
270C22×C4—28no—
271C22×C4—216no—
272C22×C4—28no—
273C22×C4—474no—
274C22×C4—478no—
275C22×C4—948no—
276C22×C4—9416no—
277C22×C4C2×D18818yes—
278C22×C4C2×D18814yes—
279C22×C4#1062535018yes—
280C22×C4C23×D4712yes—
281C22×C4C23×D4718yes—
282C22×C4C23×D4711yescharacteristic
283C22×C4C23×D4712yes—
284C22×C4C23×D4711yescharacteristic
285C4⋊C4—9416no—
286C4⋊C4—948no—
287C4⋊C4—948no—
288C4⋊C4—948no—
289C22⋊C4—948no—
290C22⋊C4—9416no—
291C22⋊C4—9464no—
292C42—216no—
293C42—28no—
294C42—28no—
295C42—28no—
296C42—948no—
297C42C2×D18818yes—
298C42C2×D18814yes—
299C42C2×D18814yes—
300C42#10625350116yescomplemented
301C23—24no—
302C23—944no—
303C23—9416no—
304C23—948no—
305C23C22×D18812yes—
306C23C2×#1062535014yes—
307C23C24×D4711yescharacteristic
308Q8—1888no—
309Q8—18816no—
310D4—18816no—
311D4—1888no—
312D4—18816no—
313C2×C4—232no—
314C2×C4—24no—
315C2×C4—28no—
316C2×C4—28no—
317C2×C4—216no—
318C2×C4—948no—
319C2×C4—944no—
320C2×C4—9416no—
321C2×C4—948no—
322C2×C4—18816no—
323C2×C4#13094012116yescomplemented
324C2×C4C22×D18814yes—
325C2×C4C22×D18812yes—
326C2×C4C22×D18812yes—
327C2×C4C22×D18814yes—
328C2×C4C2×#10625350116yes—
329C2×C4C2×#1062535014yes—
330C2×C4C2×#1062535014yes—
331C2×C4C2×#1062535018yes—
332C22—24no—
333C22—28no—
334C22—1884no—
335C22—1888no—
336C22—18816no—
337C22C2×#1309401214yes—
338C22C23×D18811yescharacteristic
339C22C22×#1062535012yes—
340C22C22×#1062535014yes—
341C4—24no—
342C4—216no—
343C4—1884no—
344C4—1888no—
345C4C2×#1309401214yes—
346C4C2×#1309401214yes—
347C4#1568967318yes—
348C2—24no—
349C2—3764no—
350C2C22×#1309401211yescharacteristic
351C2C2×#1568967312yes—
352C2#2025668414yes—
subgroupG/H
centerC22×C4C23×D47
commutator, socle, cosocleC23×C47C25
FrattiniC23C24×D47
FittingC2×C43×C47C2
prime radicalC2×#15689673C2
hypercenterC2×C43D47
hypocenterC47#36639
LCSG ▷ C23×C47 ▷ C47
UCSC1 ◁ C22×C4 ◁ C2×C43

Complements

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

NH ≅ G/Ntype
1. C2×#15689680C2semidirect
3. #20251766C2semidirect
4. #20251760C2semidirect
5. #20251745C2semidirect
6. #20251689C2semidirect
7. C4×#13094012C2semidirect
8. #20245878C2semidirect
9. C2×C43×C47C2semidirect
115. C42×C47#41semidirect
136. C2×C4×C47#124semidirect
180. C2×C43D47semidirect
214. C47#36639semidirect
300. C42#10625350semidirect
323. C2×C4#13094012semidirect

Maximal subgroups

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

subgroupindex
C2×#156896802
C2×#156896732
#202517662
#202517602
#202517452
#202516892
C4×#130940122
#202458782
C2×C43×C472
#3663947