Finite Group Databasebeta

← #26111998

Subgroups — #26111998

Order 12032.

738 conjugacy classes of subgroups.

The subgroup lattice has 338 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20251727C211yesmaximal, characteristic
2#20243456C211yesmaximal, characteristic
3C2×#15688859C212yesmaximal
4#20241267C212yesmaximal
5C2×C4×C4⋊C4×C47C211yesmaximal, characteristic
6C2×#13094003C2211yescharacteristic
7C2×C22⋊(C47⋊C8)C2212yes—
8#15689029C412yescomplemented
9#15689021C412yescomplemented
10#15689007—22no—
11#15688996—21no—
12#15688986—21no—
13#15688983C2211yescharacteristic
14#15688878—22no—
15#15688859—24no—
16C22×C4⋊C4×C47—21no—
17C22×C42×C47—21no—
18C22×C42×C47C2211yescharacteristic
19C4×C4⋊C4×C47—24no—
20C2×#95×C47C2212yes—
21C22×Dic94C2×C411yescharacteristic
22C22×C47⋊C8—22no—
23#13094003—22no—
24#13094003—22no—
25C2×C4⋊Dic47C2×C411yescharacteristic
26C2×C4⋊Dic47C2×C412yes—
27C2×Dic47⋊C4—22no—
28C2×Dic47⋊C4—21no—
29C2×C22⋊Dic47—21no—
30C2×C22⋊Dic47—21no—
31C22⋊(C47⋊C8)—22no—
32C22⋊(C47⋊C8)—22no—
33C47⋊#95—22no—
34C47⋊#95—22no—
35C47⋊#95—22no—
36C47⋊#95—21no—
37C23×C4×C47—22no—
38C23×C4×C47D411yescharacteristic
39C23×C4×C47D411yescharacteristic
40C23×C4×C47C2311yescharacteristic
41C2×C4⋊C4×C47—28no—
42C2×C4⋊C4×C47—24no—
43C2×C42×C47—22no—
44C2×C42×C47—22no—
45C2×C42×C47—24no—
46C2×C42×C47—42no—
47C2×C42×C47—42no—
48C2×C42×C47C2×C412yes—
49C2×C42×C47D412yes—
50#95×C47—22no—
51#95×C47D414yes—
52C24×C47C2×D411yescharacteristic
53C22×C4×C47—21no—
54C22×C4×C47—21no—
55C22×C4×C47—21no—
56C22×C4×C47—21no—
57C22×C4×C47—21no—
58C22×C4×C47—21no—
59C22×C4×C47—24no—
60C22×C4×C47—28no—
61C22×C4×C47—24no—
62C22×C4×C47—42no—
63C22×C4×C47—44no—
64C22×C4×C47—42no—
65C22×C4×C47—42no—
66C22×C4×C47C22⋊C412yes—
67C22×C4×C47C22⋊C412yes—
68C22×C4×C47SD1612yes—
69C22×C4×C47Q1612yes—
70C22×C4×C47C22×C411yescharacteristic
71C22×C4×C47C2×D411yescharacteristic
72C22×C4×C47C2×D412yes—
73C22×C4×C47#4112yes—
74C22×Dic47—22no—
75C22×Dic47—21no—
76C22×Dic47—21no—
77C2×Dic94—22no—
78C2×Dic94—42no—
79C4⋊C4×C47—216no—
80C42×C47—24no—
81C42×C47—24no—
82C42×C47—48no—
83C22⋊Dic47—41no—
84C22⋊Dic47—41no—
85C22⋊Dic47—41no—
86C22⋊Dic47—41no—
87C4⋊Dic47—22no—
88C4⋊Dic47—42no—
89Dic47⋊C4—44no—
90Dic47⋊C4—42no—
91C2×C47⋊C8—44no—
92C2×C47⋊C8—42no—
93C23×C47—22no—
94C23×C47—22no—
95C23×C47C2×C22⋊C411yescharacteristic
96C23×C47#12011yescharacteristic
97C23×C47#12311yescharacteristic
98C23×C47C2×SD1611yescharacteristic
99C23×C47C2×Q1611yescharacteristic
100C23×C47AGL(1,ℤ/8)11yescharacteristic
101C23×C47#13711yescharacteristic
102C2×C4×C47—24no—
103C2×C4×C47—24no—
104C2×C4×C47—22no—
105C2×C4×C47—22no—
106C2×C4×C47—22no—
107C2×C4×C47—22no—
108C2×C4×C47—22no—
109C2×C4×C47—22no—
110C2×C4×C47—22no—
111C2×C4×C47—22no—
112C2×C4×C47—22no—
113C2×C4×C47—22no—
114C2×C4×C47—28no—
115C2×C4×C47—42no—
116C2×C4×C47—42no—
117C2×C4×C47—44no—
118C2×C4×C47—48no—
119C2×C4×C47—48no—
120C2×C4×C47—44no—
121C2×C4×C47#10314yes—
122C2×C4×C47#10414yes—
123C2×C4×C47C4×D412yes—
124C2×C4×C47#12112yes—
125C2×Dic47—41no—
126C2×Dic47—41no—
127C2×Dic47—42no—
128C2×Dic47—42no—
129C2×Dic47—42no—
130C2×Dic47—42no—
131C2×Dic47—42no—
132Dic94—82no—
133C47⋊C8—82no—
134#9144—471nomaximal, Sylow
135C22×C47—21no—
136C22×C47—21no—
137C22×C47—21no—
138C22×C47—21no—
139C22×C47—22no—
140C22×C47—22no—
141C22×C47—22no—
142C22×C47—22no—
143C22×C47#38611yescharacteristic
144C22×C47#41012yes—
145C22×C47C2×#10311yescharacteristic
146C22×C47#41711yescharacteristic
147C22×C47C2×#10412yes—
148C22×C47#45011yescharacteristic
149C22×C47#45111yescharacteristic
150C22×C47#48311yescharacteristic
151C22×C47#48411yescharacteristic
152C4×C47—24no—
153C4×C47—24no—
154C4×C47—22no—
155C4×C47—22no—
156C4×C47—48no—
157Dic47—81no—
158Dic47—81no—
159Dic47—82no—
160#2855—471no—
161C2×C4×C4⋊C4D4711yescharacteristic
162#1817—471no—
163C2×#328—472no—
164#1294—472no—
165C2×C47—21no—
166C2×C47—21no—
167C2×C47—21no—
168C2×C47—21no—
169C2×C47#150312yes—
170C2×C47#150912yes—
171C2×C47#187211yescharacteristic
172C2×C47#189611yescharacteristic
173C2×C47#189911yescharacteristic
174C2×C22⋊Q8—471no—
175C22×C4⋊C4—21no—
176C22×C42—21no—
177C22×C42C2×D4711yescharacteristic
178C2×C22⋊C8—472no—
179#391—472no—
180#389—472no—
181#387—942no—
182#385—941no—
183#381—941no—
184#380—471no—
185C4×C4⋊C4—24no—
186C2×#95C2×D4712yes—
187#336—942no—
188#328—944no—
189C47#914411yescharacteristic, complemented, Sylow
190C22×Q8—471no—
191C23×C4—22no—
192C23×C4C22⋊D4711yescharacteristic
193C23×C4C22⋊D4711yescharacteristic
194C23×C4C22×D4711yescharacteristic
195C22×C8—942no—
196C22⋊Q8—942no—
197C22⋊Q8—942no—
198C2×C4⋊C4—28no—
199C2×C4⋊C4—24no—
200C2×C4⋊C4—471no—
201C2×C4⋊C4—472no—
202C2×C4⋊C4—942no—
203C2×C4⋊C4—941no—
204C2×C22⋊C4—941no—
205C2×C22⋊C4—941no—
206C2×C42—22no—
207C2×C42—22no—
208C2×C42—24no—
209C2×C42—42no—
210C2×C42—42no—
211C2×C42C4×D4712yes—
212C2×C42D18812yes—
213C22⋊C8—942no—
214C22⋊C8—942no—
215#95—22no—
216#95—942no—
217#95—942no—
218#95—942no—
219#95—941no—
220#95C22⋊D4714yes—
221C24C2×C22⋊D4711yescharacteristic
222C2×Q8—942no—
223C2×Q8—1882no—
224C22×C4—21no—
225C22×C4—21no—
226C22×C4—21no—
227C22×C4—21no—
228C22×C4—21no—
229C22×C4—21no—
230C22×C4—24no—
231C22×C4—28no—
232C22×C4—24no—
233C22×C4—42no—
234C22×C4—44no—
235C22×C4—42no—
236C22×C4—42no—
237C22×C4—942no—
238C22×C4—941no—
239C22×C4—941no—
240C22×C4(C2×D47)⋊C412yes—
241C22×C4(C2×D47)⋊C412yes—
242C22×C4#1062533512yes—
243C22×C4#1062533712yes—
244C22×C4C2×C4×D4711yescharacteristic
245C22×C4C2×D18811yescharacteristic
246C22×C4#1062535012yes—
247C22×C4C2×C22⋊D4712yes—
248C2×C8—1884no—
249C2×C8—1882no—
250C4⋊C4—216no—
251C4⋊C4—942no—
252C4⋊C4—1882no—
253C4⋊C4—1884no—
254C4⋊C4—1882no—
255C22⋊C4—1881no—
256C22⋊C4—1881no—
257C22⋊C4—1881no—
258C22⋊C4—1881no—
259C42—24no—
260C42—24no—
261C42—48no—
262C23—22no—
263C23—22no—
264C23C2×(C2×D47)⋊C411yescharacteristic
265C23#1309399311yescharacteristic
266C23#1309400711yescharacteristic
267C23C2×#1062533511yescharacteristic
268C23C2×#1062533711yescharacteristic
269C23#1309405011yescharacteristic
270C23#1309405511yescharacteristic
271Q8—3762no—
272C2×C4—24no—
273C2×C4—24no—
274C2×C4—22no—
275C2×C4—22no—
276C2×C4—22no—
277C2×C4—22no—
278C2×C4—22no—
279C2×C4—22no—
280C2×C4—22no—
281C2×C4—22no—
282C2×C4—22no—
283C2×C4—22no—
284C2×C4—28no—
285C2×C4—42no—
286C2×C4—42no—
287C2×C4—44no—
288C2×C4—48no—
289C2×C4—48no—
290C2×C4—44no—
291C2×C4—1881no—
292C2×C4—1881no—
293C2×C4—1882no—
294C2×C4—1882no—
295C2×C4—1882no—
296C2×C4—1882no—
297C2×C4—1882no—
298C2×C4#1309394714yes—
299C2×C4#1309395214yes—
300C2×C4C4×C22⋊D4712yes—
301C2×C4#1309399812yes—
302C8—3762no—
303C22—21no—
304C22—21no—
305C22—21no—
306C22—21no—
307C22—22no—
308C22—22no—
309C22—22no—
310C22—22no—
311C22#1568900211yescharacteristic
312C22#1568909612yes—
313C22C2×#1309394711yescharacteristic
314C22#1568912211yescharacteristic
315C22C2×#1309395212yes—
316C22#1568927411yescharacteristic
317C22#1568927911yescharacteristic
318C22#1568945311yescharacteristic
319C22#1568945811yescharacteristic
320C4—24no—
321C4—22no—
322C4—22no—
323C4—24no—
324C4—48no—
325C4—3761no—
326C4—3761no—
327C4—3762no—
328C2—21no—
329C2—21no—
330C2—21no—
331C2—21no—
332C2#2024188612yes—
333C2#2024191112yes—
334C2#2024369511yescharacteristic
335C2#2024383111yescharacteristic
336C2#2024384611yescharacteristic
subgroupG/H
centerC23C2×(C2×D47)⋊C4
commutatorC22×C4×C47C22×C4
FrattiniC23×C4C22×D47
FittingC2×C4×C4⋊C4×C47C2
socleC23×C47C2×C22⋊C4
prime radicalC24×C47C2×D4
cosocleC23×C4×C47C23
hypercenterC2×C4×C4⋊C4D47
hypocenterC47#9144
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C23 ◁ C22×C42 ◁ C2×C4×C4⋊C4

Complements

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

NH ≅ G/Ntype
8. #15689029C4semidirect
9. #15689021C4semidirect
189. C47#9144semidirect

Maximal subgroups

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

subgroupindex
#202517272
#202434562
C2×#156888592
#202412672
C2×C4×C4⋊C4×C472
#914447