Finite Group Databasebeta

← #26368887

Subgroups — #26368887

Order 12032.

806 conjugacy classes of subgroups.

The subgroup lattice has 334 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689531C212yesmaximal
2C2×#15689523C211yesmaximal, characteristic
3C2×#15689520C211yesmaximal, characteristic
4#20249512C212yesmaximal
5#20243557C212yesmaximal
6#20243555C212yesmaximal
7#20243549C211yesmaximal, characteristic
8#20243545C211yesmaximal, characteristic
9#20243543C211yesmaximal, characteristic
10#20243074C211yesmaximal, characteristic
11C47×#1846C211yesmaximal, characteristic
12C2×#13094023C2212yes—
13C2×#13094015C2212yes—
14#15689531—28no—
15#15689523—24no—
16#15689520—24no—
17C2×#13093957C2212yes—
18C2×#13093957C2212yes—
19C2×#13093956C2211yescharacteristic
20C2×#13093956C2211yescharacteristic
21C2×#13093956C2212yes—
22C2×#13093955C2212yes—
23C2×#13093955C2212yes—
24C2×#13093935C2211yescharacteristic
25C2×#13093934C2211yescharacteristic
26C2×#13093934C2211yescharacteristic
27C2×C8×Dic47C2211yescharacteristic
28C2×C8×Dic47C2211yescharacteristic
29#15689057—24no—
30#15689044—24no—
31#15688993C2212yes—
32#15688992C2212yes—
33#15688988C2212yes—
34#15688969C2211yescharacteristic
35C2×#107×C47C2212yes—
36C2×#106×C47C2211yescharacteristic
37C2×#106×C47C2211yescharacteristic
38C2×C4×C8×C47C2211yescharacteristic
39C47×#382C2212yes—
40C22×Dic94—22no—
41C22×C47⋊C8C2311yescharacteristic
42C22×C47⋊C8C2311yescharacteristic
43#13094023—24no—
44#13094023—24no—
45#13094015—24no—
46#13094015—24no—
47C2×C4⋊Dic47—22no—
48C2×C4⋊Dic47—24no—
49C2×C4⋊Dic47C2312yes—
50C2×C4⋊Dic47C2312yes—
51C2×Dic47⋊C4—22no—
52C2×Dic47⋊C4—22no—
53C2×Dic47⋊C4—24no—
54C2×C4×Dic47C2311yescharacteristic
55C2×C4×Dic47C2311yescharacteristic
56#13093957—24no—
57#13093957—24no—
58#13093956—22no—
59#13093956—22no—
60#13093956—24no—
61#13093955—24no—
62#13093955—24no—
63#13093935Q814yescomplemented
64#13093934D414yes—
65#13093934Q814yescomplemented
66C8×Dic47D414yes—
67C8×Dic47Q814yescomplemented
68C47⋊#95—24no—
69C47⋊#95D414yes—
70C22×C8×C47C2311yescharacteristic
71C22×C8×C47C2311yescharacteristic
72C2×C4⋊C4×C47C2312yes—
73C2×C4⋊C4×C47C2312yes—
74C2×C42×C47C2311yescharacteristic
75#107×C47—24no—
76#106×C47—22no—
77#106×C47—22no—
78C4×C8×C47Q814yescomplemented
79#95×C47—24no—
80C22×C4×C47—22no—
81C22×C4×C47—22no—
82C22×C4×C47C2×D412yes—
83C22×C4×C47C2411yescharacteristic
84C22×Dic47—22no—
85C22×Dic47—22no—
86C22×Dic47C2×D412yes—
87C22×Dic47C2×D412yes—
88C2×Dic94—44no—
89C2×Dic94—84no—
90C2×C8×C47C2×D412yes—
91C2×C8×C47C2×D412yes—
92C2×C8×C47C2×Q812yes—
93C2×C8×C47C2×Q812yes—
94C2×C8×C47C2×Q812yes—
95C2×C8×C47C2×Q812yes—
96C4⋊C4×C47—22no—
97C4⋊C4×C47—22no—
98C4⋊C4×C47—22no—
99C4⋊C4×C47—22no—
100C42×C47C2×Q812yes—
101C42×C47C2×Q812yes—
102C4⋊Dic47—22no—
103C4⋊Dic47—22no—
104C4⋊Dic47—22no—
105C4⋊Dic47—22no—
106C4⋊Dic47—44no—
107C4⋊Dic47—84no—
108Dic47⋊C4—44no—
109Dic47⋊C4—44no—
110Dic47⋊C4—84no—
111C4×Dic47C2×D412yes—
112C4×Dic47C2×D412yes—
113C4×Dic47C2×Q812yes—
114C4×Dic47C2×Q812yes—
115C2×C47⋊C8C2×D412yes—
116C2×C47⋊C8C2×D412yes—
117C2×C47⋊C8C2×Q812yes—
118C2×C47⋊C8C2×Q812yes—
119C2×C47⋊C8C2×Q812yes—
120C2×C47⋊C8C2×Q812yes—
121C23×C47C22×D411yescharacteristic
122C2×C4×C47—24no—
123C2×C4×C47—42no—
124C2×C4×C47—42no—
125C2×C4×C47—44no—
126C2×C4×C47—44no—
127C2×C4×C47C4⋊Q814yes—
128C2×C4×C47C22×D411yescharacteristic
129C2×C4×C47C22×D411yescharacteristic
130C2×C4×C47C22×Q811yescharacteristic
131C2×C4×C47C22×Q811yescharacteristic
132C2×C4×C47C22×Q811yescharacteristic
133C2×C4×C47C22×Q811yescharacteristic
134C2×Dic47—24no—
135C2×Dic47—24no—
136C2×Dic47—42no—
137C2×Dic47—42no—
138C2×Dic47—44no—
139C2×Dic47—44no—
140C2×Dic47#12714yes—
141C2×Dic47C4⋊Q814yes—
142Dic94—88no—
143C8×C47C4⋊Q814yes—
144C8×C47C4⋊Q814yes—
145C47⋊C8C4⋊Q814yes—
146C47⋊C8C4⋊Q814yes—
147#33318—471nomaximal, Sylow
148C22×C47C2×#12711yescharacteristic
149C22×C47C2×C4⋊Q811yescharacteristic
150C22×C47C2×C4⋊Q811yescharacteristic
151C22×C47#57714yes—
152C4×C47—24no—
153C4×C47—82no—
154C4×C47—82no—
155C4×C47C2×C4⋊Q811yescharacteristic
156C4×C47C2×C4⋊Q811yescharacteristic
157C4×C47C2×C4⋊Q811yescharacteristic
158C4×C47C2×C4⋊Q811yescharacteristic
159Dic47—24no—
160Dic47—24no—
161Dic47—82no—
162Dic47—82no—
163C2×#501—472no—
164C2×#499—471no—
165C2×#498—471no—
166#2660—472no—
167#1847—472no—
168#1847—472no—
169#1846—471no—
170#1846D4711yescharacteristic
171#1844—471no—
172#1844—471no—
173#1726—471no—
174C2×C47#286811yescharacteristic
175C2×C47#315412yes—
176C2×C47#316312yes—
177C2×C47#316312yes—
178C2×C4⋊Q8—472no—
179C2×#125—472no—
180#501—948no—
181#499—944no—
182#498—944no—
183C2×#107—472no—
184C2×#107—472no—
185C2×#107C2×D4712yes—
186C2×#106—471no—
187C2×#106—471no—
188C2×#106—472no—
189C2×#106—472no—
190C2×#106—472no—
191C2×#106C2×D4711yescharacteristic
192C2×#106C2×D4711yescharacteristic
193C2×#97—471no—
194C2×#97—471no—
195C2×#97—471no—
196C2×C4×C8—471no—
197C2×C4×C8—471no—
198C2×C4×C8C2×D4711yescharacteristic
199#398—944no—
200#395—944no—
201#384—472no—
202#384—472no—
203#382—472no—
204#382C2×D4712yes—
205#376—471no—
206C47#3331811yescharacteristic, complemented, Sylow
207C22×Q8—942no—
208C22×C8—471no—
209C22×C8—471no—
210C22×C8C22×D4711yescharacteristic
211C22×C8C22×D4711yescharacteristic
212C4⋊Q8—944no—
213C4⋊Q8—944no—
214#125—944no—
215#125—944no—
216C2×C4⋊C4—472no—
217C2×C4⋊C4—472no—
218C2×C4⋊C4—944no—
219C2×C4⋊C4—944no—
220C2×C4⋊C4—942no—
221C2×C4⋊C4—942no—
222C2×C4⋊C4—942no—
223C2×C4⋊C4C22×D4712yes—
224C2×C4⋊C4C22×D4712yes—
225C2×C42—471no—
226C2×C42—471no—
227C2×C42C22×D4711yescharacteristic
228#107—24no—
229#107—944no—
230#107—944no—
231#106—22no—
232#106—22no—
233#106—944no—
234#106—944no—
235#106—942no—
236#106—942no—
237#106—944no—
238#97—474no—
239#97—474no—
240#97—474no—
241C4×C8—474no—
242C4×C8—474no—
243C4×C8Dic9414yescomplemented
244#95—24no—
245#95—474no—
246#95—944no—
247C2×Q8—3764no—
248C2×Q8—1884no—
249C22×C4—22no—
250C22×C4—22no—
251C22×C4—472no—
252C22×C4—472no—
253C22×C4—942no—
254C22×C4—942no—
255C22×C4D4×D4712yes—
256C22×C4C23×D4711yescharacteristic
257C2×C8—472no—
258C2×C8—472no—
259C2×C8—472no—
260C2×C8—472no—
261C2×C8—472no—
262C2×C8—472no—
263C2×C8C2×Dic9412yes—
264C2×C8C2×Dic9412yes—
265C2×C8D4×D4712yes—
266C2×C8D4×D4712yes—
267C2×C8Q8×D4712yes—
268C2×C8Q8×D4712yes—
269C4⋊C4—22no—
270C4⋊C4—22no—
271C4⋊C4—22no—
272C4⋊C4—22no—
273C4⋊C4—942no—
274C4⋊C4—942no—
275C4⋊C4—942no—
276C4⋊C4—942no—
277C4⋊C4—3764no—
278C4⋊C4—3764no—
279C4⋊C4—1884no—
280C4⋊C4—1884no—
281C4⋊C4—1884no—
282C42—472no—
283C42—472no—
284C42—472no—
285C42—472no—
286C42C2×Dic9412yes—
287C42Q8×D4712yes—
288C23C2×D4×D4711yescharacteristic
289Q8—3768no—
290C2×C4—24no—
291C2×C4—42no—
292C2×C4—42no—
293C2×C4—44no—
294C2×C4—44no—
295C2×C4—474no—
296C2×C4—474no—
297C2×C4—944no—
298C2×C4—944no—
299C2×C4—1882no—
300C2×C4—1882no—
301C2×C4—1884no—
302C2×C4—1884no—
303C2×C4#1309402314yes—
304C2×C4C2×D4×D4711yescharacteristic
305C2×C4C2×D4×D4711yescharacteristic
306C2×C4C2×Q8×D4711yescharacteristic
307C2×C4C2×Q8×D4711yescharacteristic
308C2×C4C2×Q8×D4711yescharacteristic
309C2×C4C22×Dic9411yescharacteristic
310C8—474no—
311C8—474no—
312C8#1309402314yes—
313C8#1309402314yes—
314C22#127×D4711yescharacteristic
315C22C4⋊Q8×D4711yescharacteristic
316C22C2×#1309402311yescharacteristic
317C22#1568993014yes—
318C4—24no—
319C4—82no—
320C4—82no—
321C4—944no—
322C4—944no—
323C4—3762no—
324C4—3762no—
325C4C4⋊Q8×D4711yescharacteristic
326C4C4⋊Q8×D4711yescharacteristic
327C4C2×#1309402311yescharacteristic
328C4C2×#1309402311yescharacteristic
329C2#2025178511yescharacteristic
330C2#2025385912yes—
331C2#2025391712yes—
332C2#2025392012yes—
subgroupG/H
centerC23C2×D4×D47
commutator, cosocleC22×C4×C47C24
FrattiniC22×C4C23×D47
FittingC47×#1846C2
socle, prime radicalC23×C47C22×D4
hypercenter#1846D47
hypocenterC47#33318
LCSG ▷ C22×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1846

Complements

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

NH ≅ G/Ntype
63. #13093935Q8semidirect
65. #13093934Q8semidirect
67. C8×Dic47Q8semidirect
78. C4×C8×C47Q8semidirect
206. C47#33318semidirect
243. C4×C8Dic94semidirect

Maximal subgroups

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

subgroupindex
C2×#156895312
C2×#156895232
C2×#156895202
#202495122
#202435572
#202435552
#202435492
#202435452
#202435432
#202430742
C47×#18462
#3331847