Finite Group Databasebeta

← #26422355

Subgroups — #26422355

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×#15689673C211yesmaximal, characteristic
2#20251765C212yesmaximal, complemented
3#20251761C214yesmaximal, complemented
4#20251746C212yesmaximal, complemented
5#20251691C218yesmaximal, complemented
6C4×#13094013C218yesmaximal, complemented
7#20245876C214yesmaximal, complemented
8C2×C42×Dic47C211yesmaximal, characteristic, complemented
9C2×C47×#533C211yesmaximal, characteristic, complemented
10C22×#10625350C2212yes—
11#15689673—28no—
12C2×#13094024C2211yescharacteristic
13C2×#13094021C2211yescharacteristic
14C2×#13094013C2218yes—
15C2×#13094012C2212yes—
16C2×#13093998C2214yes—
17C2×C4×D188C2214yes—
18C22×C4×Dic47C2214yes—
19C22×C4×Dic47C2212yes—
20#15689030C2218yes—
21#15689028C2218yes—
22#15689024C2218yes—
23#15689024C2218yes—
24#15689013C22132yes—
25#15689000C22116yes—
26#15688988C2218yes—
27C4×(C2×D47)⋊C4C22116yes—
28C42×Dic47C2212yes—
29C42×Dic47C2212yes—
30C42×Dic47C2214yes—
31C47×#533—28no—
32C2×C4⋊Q8×C47C2212yes—
33C2×#125×C47C2212yes—
34C2×C22⋊Q8×C47C2214yes—
35C2×C4×Q8×C47C2214yes—
36C2×#117×C47C2212yes—
37C22×C42×C47C2211yescharacteristic
38C2×#10625350—28no—
39C2×#10625350—216no—
40C22×Dic94C2312yes—
41C22×D188C2312yes—
42C22×D188C2314yes—
43C22×C22⋊D47C2314yes—
44C23×Dic47C2314yes—
45C22×C4×D47C2314yes—
46#13094024—24no—
47#13094021—24no—
48#13094013—232no—
49#13094012—28no—
50#13093998—216no—
51C4×D188—216no—
52C2×C4⋊Dic47C2314yes—
53C2×(C2×D47)⋊C4C23132yes—
54C2×(C2×D47)⋊C4C2318yes—
55C2×C4×Dic47—28no—
56C2×C4×Dic47—216no—
57C2×C4×Dic47C2314yes—
58C2×C4×Dic47C2318yes—
59C2×C4×Dic47C2314yes—
60C2×C4×Dic47C2314yes—
61C2×C4×Dic47C2314yes—
62C47⋊#95C23132yes—
63C22×Q8×C47C2314yes—
64C23×C4×C47C2312yes—
65C23×C4×C47C2311yescharacteristic
66C4⋊Q8×C47—28no—
67#125×C47—28no—
68C22⋊Q8×C47—216no—
69C4×Q8×C47—216no—
70#117×C47—28no—
71C2×C4⋊C4×C47C2314yes—
72C2×C4⋊C4×C47C2318yes—
73C2×C4⋊C4×C47C2314yes—
74C2×C22⋊C4×C47C2314yes—
75C2×C42×C47D418yes—
76C2×C42×C47C2314yes—
77C2×C42×C47C2312yes—
78C2×C42×C47C2311yescharacteristic
79C2×C42×C47C2311yescharacteristic
80C24×C47C2411yescharacteristic
81C23×D47C2414yes—
82C2×Q8×C47—28no—
83C2×Q8×C47—216no—
84C22×C4×C47—28no—
85C22×C4×C47C2×D418yes—
86C22×C4×C47C2×D414yes—
87C22×C4×C47#4118yes—
88C22×C4×C47C2418yes—
89C22×C4×C47C2412yes—
90C22×C4×C47C2411yescharacteristic
91C22×C4×C47C2412yes—
92C22×C4×C47C2411yescharacteristic
93C2×C22⋊D47—28no—
94C2×C22⋊D47—216no—
95C22×Dic47—28no—
96C22×Dic47—216no—
97C22×Dic47C2414yes—
98C22×Dic47C2418yes—
99#10625350—432no—
100C2×D188—28no—
101C2×D188—24no—
102C2×D188—216no—
103C2×D188—28no—
104C2×C4×D47—28no—
105C2×C4×D47—216no—
106C2×Dic94—24no—
107C2×Dic94—28no—
108C4⋊C4×C47—28no—
109C4⋊C4×C47—216no—
110C4⋊C4×C47—28no—
111C22⋊C4×C47—28no—
112C42×C47—28no—
113C42×C47C2×D418yes—
114C42×C47C2×D414yes—
115C42×C47C2×D414yes—
116(C2×D47)⋊C4—216no—
117(C2×D47)⋊C4—264no—
118C4⋊Dic47—28no—
119C4×Dic47—28no—
120C4×Dic47—28no—
121C4×Dic47—28no—
122C4×Dic47—216no—
123C4×Dic47#41116yescomplemented
124C23×C47—24no—
125C23×C47C22×D412yes—
126C23×C47C2×#4114yes—
127C23×C47C2511yescharacteristic
128C22×D47—24no—
129C22×D47—216no—
130C22×D47—28no—
131Q8×C47—416no—
132C2×C4×C47—24no—
133C2×C4×C47—216no—
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×#4118yes—
142C2×C4×C47C2×#4114yes—
143C2×C4×C47C2×#4114yes—
144C22⋊D47—416no—
145C2×Dic47—232no—
146C2×Dic47—28no—
147C2×Dic47—28no—
148C2×Dic47—24no—
149C2×Dic47—28no—
150C2×Dic47C2×#41116yes—
151D188—48no—
152D188—416no—
153C4×D47—416no—
154Dic94—48no—
155#36639—471nomaximal, Sylow
156C22×C47—24no—
157C22×C47—28no—
158C22×C47C2×#12414yes—
159C22×C47C23×D411yescharacteristic
160C22×C47C22×#4114yes—
161C22×C47C22×#4112yes—
162C2×D47—44no—
163C2×D47—48no—
164C2×D47—416no—
165C4×C47—24no—
166C4×C47—48no—
167C4×C47C2×#12414yes—
168C4×C47C2×#12414yes—
169C4×C47#53218yes—
170Dic47—216no—
171Dic47—44no—
172C2×#533D4711yescharacteristic, complemented
173C2×#532—471no—
174#2862—472no—
175#2861—474no—
176#2858—472no—
177#2850—478no—
178C4×#124—478no—
179#2297—474no—
180C2×C43—471no—
181C2×C47—24no—
182C2×C47C22×#12411yescharacteristic
183C2×C47C2×#53212yes—
184C2×C47#344414yes—
185D47—84no—
186C22×#41—472no—
187#533—28no—
188#532—948no—
189C2×C4⋊Q8—471no—
190C2×C4⋊Q8C2×D4712yes—
191C2×#127—471no—
192C2×#125C2×D4712yes—
193C2×#124—478no—
194C2×#124—472no—
195C2×C22⋊Q8C2×D4714yes—
196C2×#121—474no—
197C2×C4×Q8C2×D4714yes—
198C2×C4×D4—474no—
199C2×#117C2×D4712yes—
200C22×C42—474no—
201C22×C42—472no—
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—
211C43—472no—
212C43—472no—
213C43—474no—
214C47#3663911yescharacteristic, complemented, Sylow
215C2×#41—9416no—
216C2×#41—948no—
217C22×Q8—472no—
218C22×Q8C22×D4714yes—
219C22×D4—474no—
220C22×D4—474no—
221C22×D4—472no—
222C23×C4—474no—
223C23×C4—474no—
224C23×C4C22×D4712yes—
225C23×C4C22×D4711yescharacteristic
226C4⋊Q8—28no—
227C4⋊Q8—944no—
228#127—944no—
229#125—28no—
230#124—9432no—
231#124—948no—
232C22⋊Q8—216no—
233#121—9416no—
234C4×Q8—216no—
235C4×D4—9416no—
236#117—28no—
237C2×C4⋊C4—474no—
238C2×C4⋊C4C22×D4714yes—
239C2×C4⋊C4C22×D4718yes—
240C2×C4⋊C4C22×D4714yes—
241C2×C22⋊C4—478no—
242C2×C22⋊C4—4732no—
243C2×C22⋊C4C22×D4714yes—
244C2×C42—474no—
245C2×C42—478no—
246C2×C42—474no—
247C2×C42—474no—
248C2×C42—474no—
249C2×C42—9416no—
250C2×C42—948no—
251C2×C42C22⋊D4718yes—
252C2×C42C22×D4714yes—
253C2×C42C22×D4711yescharacteristic
254C2×C42C22×D4711yescharacteristic
255C2×C42C22×D4712yes—
256#95—4732no—
257C24—474no—
258C24C23×D4711yescharacteristic
259#41—18832no—
260C2×Q8—28no—
261C2×Q8—216no—
262C2×Q8—944no—
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—478no—
272C22×C4—474no—
273C22×C4—9416no—
274C22×C4—948no—
275C22×C4—948no—
276C22×C4—9416no—
277C22×C4#1062535418yes—
278C22×C4C2×C22⋊D4718yes—
279C22×C4C2×C22⋊D4714yes—
280C22×C4C23×D4718yes—
281C22×C4C23×D4712yes—
282C22×C4C23×D4712yes—
283C22×C4C23×D4711yescharacteristic
284C22×C4C23×D4711yescharacteristic
285C4⋊C4—28no—
286C4⋊C4—216no—
287C4⋊C4—28no—
288C4⋊C4—948no—
289C22⋊C4—28no—
290C22⋊C4—9464no—
291C22⋊C4—9416no—
292C42—28no—
293C42—4716no—
294C42—9416no—
295C42—948no—
296C42—948no—
297C42—948no—
298C42C2×C22⋊D4718yes—
299C42C2×C22⋊D4714yes—
300C42C2×C22⋊D4714yes—
301C23—24no—
302C23—944no—
303C23—9416no—
304C23—948no—
305C23C22×C22⋊D4712yes—
306C23C2×#1062535414yes—
307C23C24×D4711yescharacteristic
308Q8—416no—
309Q8—1888no—
310D4—18816no—
311D4—1888no—
312D4—18816no—
313C2×C4—24no—
314C2×C4—216no—
315C2×C4—28no—
316C2×C4—216no—
317C2×C4—4716no—
318C2×C4—9432no—
319C2×C4—948no—
320C2×C4—948no—
321C2×C4—944no—
322C2×C4—948no—
323C2×C4—18816no—
324C2×C4#13094013116yescomplemented
325C2×C4C22×C22⋊D4714yes—
326C2×C4C22×C22⋊D4712yes—
327C2×C4C22×C22⋊D4712yes—
328C2×C4C22×C22⋊D4714yes—
329C2×C4#41×D4718yes—
330C2×C4C2×#1062535414yes—
331C2×C4C2×#1062535414yes—
332C22—24no—
333C22—28no—
334C22—1884no—
335C22—1888no—
336C22—18816no—
337C22C2×#1309401314yes—
338C22C23×C22⋊D4711yescharacteristic
339C22C2×#41×D4714yes—
340C22C22×#1062535412yes—
341C4—24no—
342C4—48no—
343C4—9416no—
344C4—1884no—
345C4C2×#1309401314yes—
346C4C2×#1309401314yes—
347C4#1568966918yes—
348C2—24no—
349C2—3764no—
350C2C22×#1309401311yescharacteristic
351C2C2×#1568966912yes—
352C2#2025668514yes—
subgroupG/H
centerC22×C4C23×D47
commutator, socle, cosocleC23×C47C25
FrattiniC23C24×D47
FittingC2×C47×#533C2
prime radicalC2×#15689673C2
hypercenterC2×#533D47
hypocenterC47#36639
LCSG ▷ C23×C47 ▷ C47
UCSC1 ◁ C22×C4 ◁ C2×#533

Complements

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

NH ≅ G/Ntype
2. #20251765C2semidirect
3. #20251761C2semidirect
4. #20251746C2semidirect
5. #20251691C2semidirect
6. C4×#13094013C2semidirect
7. #20245876C2semidirect
8. C2×C42×Dic47C2semidirect
9. C2×C47×#533C2semidirect
123. C4×Dic47#41semidirect
136. C2×C4×C47#124semidirect
172. C2×#533D47semidirect
214. C47#36639semidirect
324. C2×C4#13094013semidirect

Maximal subgroups

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

subgroupindex
C2×#156896732
#202517652
#202517612
#202517462
#202516912
C4×#130940132
#202458762
C2×C42×Dic472
C2×C47×#5332
#3663947