Finite Group Databasebeta

← #26112302

Subgroups — #26112302

Order 12032.

524 conjugacy classes of subgroups.

The subgroup lattice has 330 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20251952C211yesmaximal, characteristic
2#20242060C212yesmaximal, complemented
3#20241702C211yesmaximal, characteristic
4#20241246C212yesmaximal, complemented
5C47×#2621C211yesmaximal, characteristic
6C22×#10625329—21no—
7#15689176C2212yes—
8#15689150—24no—
9#15689147—21no—
10#15689147C2212yes—
11#15688869—24no—
12#15688849—21no—
13#15688847—21no—
14C2×C4⋊Q8×C47C2211yescharacteristic
15C2×C22⋊Q8×C47—21no—
16C2×#117×C47C2211yescharacteristic
17C47×#398—21no—
18C47×#398C412yes—
19C47×#397—21no—
20C47×#397C412yes—
21C47×#391—21no—
22C47×#382C2211yescharacteristic
23C2×#10625329—22no—
24C2×#10625329—22no—
25C2×#10625329—42no—
26C2×#10625329—42no—
27C22×C47⋊C8—22no—
28C4⋊(C47⋊C8)—24no—
29C4⋊(C47⋊C8)—22no—
30C4⋊(C47⋊C8)—22no—
31C4⋊(C47⋊C8)D414yes—
32C22⋊(C47⋊C8)—22no—
33C4×C47⋊C8—22no—
34C22×Q8×C47—21no—
35C23×C4×C47D411yescharacteristic
36C4⋊Q8×C47—22no—
37C4⋊Q8×C47—22no—
38C22⋊Q8×C47—42no—
39C22⋊Q8×C47—42no—
40#117×C47—22no—
41#117×C47—22no—
42C2×C4⋊C4×C47—22no—
43C2×C4⋊C4×C47—21no—
44C2×C4⋊C4×C47—21no—
45C2×C4⋊C4×C47C2×C412yes—
46C2×C4⋊C4×C47C2×C411yescharacteristic
47C2×C4⋊C4×C47D411yescharacteristic
48C2×C22⋊C4×C47—22no—
49C2×C22⋊C4×C47C2×C411yescharacteristic
50C2×C22⋊C4×C47D411yescharacteristic
51C2×C42×C47D411yescharacteristic
52C2×C42×C47C2311yescharacteristic
53#95×C47—22no—
54#95×C47—22no—
55#95×C47—21no—
56#95×C47C2×C412yes—
57C24×C47C22⋊C411yescharacteristic
58C2×Q8×C47—41no—
59C2×Q8×C47—41no—
60C2×Q8×C47—41no—
61C2×Q8×C47—41no—
62C2×Q8×C47—41no—
63C2×Q8×C47—41no—
64C22×C4×C47—21no—
65C22×C4×C47—21no—
66C22×C4×C47—21no—
67C22×C4×C47—21no—
68C22×C4×C47—22no—
69C22×C4×C47—22no—
70C22×C4×C47C22⋊C411yescharacteristic
71C22×C4×C47C22⋊C411yescharacteristic
72C22×C4×C47C22⋊C411yescharacteristic
73C22×C4×C47D812yes—
74C22×C4×C47SD1612yes—
75C22×C4×C47C22×C411yescharacteristic
76C22×C4×C47C2×D411yescharacteristic
77C22×C4×C47C2×D411yescharacteristic
78C4⋊C4×C47—22no—
79C4⋊C4×C47—22no—
80C4⋊C4×C47—24no—
81C4⋊C4×C47—44no—
82C4⋊C4×C47—42no—
83C4⋊C4×C47—41no—
84C4⋊C4×C47—41no—
85C22⋊C4×C47—22no—
86C22⋊C4×C47—22no—
87C22⋊C4×C47—44no—
88C42×C47—21no—
89C42×C47—21no—
90C42×C47C2×D412yes—
91C42×C47#4112yes—
92#10625329—44no—
93#10625329—42no—
94#10625329—42no—
95C2×C47⋊C8—24no—
96C2×C47⋊C8—44no—
97C2×C47⋊C8—42no—
98C23×C47—21no—
99C23×C47—21no—
100C23×C47—41no—
101C23×C47—41no—
102C23×C47#10212yes—
103C23×C47C2×C22⋊C411yescharacteristic
104Q8×C47—81no—
105Q8×C47—81no—
106Q8×C47—81no—
107Q8×C47—81no—
108C2×C4×C47—21no—
109C2×C4×C47—21no—
110C2×C4×C47—22no—
111C2×C4×C47—22no—
112C2×C4×C47—22no—
113C2×C4×C47—22no—
114C2×C4×C47—21no—
115C2×C4×C47—21no—
116C2×C4×C47—41no—
117C2×C4×C47—41no—
118C2×C4×C47—41no—
119C2×C4×C47—41no—
120C2×C4×C47—42no—
121C2×C4×C47—42no—
122C2×C4×C47—42no—
123C2×C4×C47—42no—
124C2×C4×C47—42no—
125C2×C4×C47—42no—
126C2×C4×C47#10212yes—
127C2×C4×C47C4×D412yes—
128C2×C4×C47#12011yescharacteristic
129C2×C4×C47#12112yes—
130C2×C4×C47#12311yescharacteristic
131C2×C4×C47C2×D811yescharacteristic
132C2×C4×C47C2×SD1611yescharacteristic
133C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
134C2×C4×C47#13711yescharacteristic
135C47⋊C8—44no—
136C47⋊C8—82no—
137#9195—471nomaximal, Sylow
138C22×C47—21no—
139C22×C47—21no—
140C22×C47—22no—
141C22×C47—42no—
142C22×C47—41no—
143C22×C47—41no—
144C22×C47—81no—
145C22×C47#38611yescharacteristic
146C22×C47C2×#10211yescharacteristic
147C22×C47#41711yescharacteristic
148C4×C47—22no—
149C4×C47—22no—
150C4×C47—22no—
151C4×C47—41no—
152C4×C47—41no—
153C4×C47—82no—
154C4×C47—82no—
155C4×C47#44711yescharacteristic
156C4×C47#44811yescharacteristic
157C4×C47#48011yescharacteristic
158C4×C47#48111yescharacteristic
159#2905—471no—
160#2621D4711yescharacteristic
161#1546—472no—
162#1458—471no—
163#1286—472no—
164C2×C47—21no—
165C2×C47—21no—
166C2×C47—42no—
167C2×C47#187411yescharacteristic
168C2×C47#190911yescharacteristic
169C2×C47#191511yescharacteristic
170C22×M16—941no—
171C2×C4⋊Q8C2×D4711yescharacteristic
172C2×C22⋊Q8—21no—
173C2×#117C2×D4711yescharacteristic
174#431—472no—
175#424—944no—
176#423—472no—
177#423—941no—
178#398—21no—
179#398Dic4712yes—
180#397—21no—
181#397Dic4712yes—
182#391—21no—
183#382C2×D4711yescharacteristic
184#332—944no—
185#324—941no—
186#323—941no—
187C47#919511yescharacteristic, complemented, Sylow
188C22×Q8—21no—
189C23×C4C22⋊D4711yescharacteristic
190C2×M16—942no—
191C2×M16—942no—
192C2×M16—1882no—
193C2×M16—1882no—
194C22×C8—942no—
195C4⋊Q8—22no—
196C4⋊Q8—22no—
197C22⋊Q8—42no—
198C22⋊Q8—42no—
199#117—22no—
200#117—22no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4C2×Dic4712yes—
205C2×C4⋊C4C2×Dic4711yescharacteristic
206C2×C4⋊C4C22⋊D4711yescharacteristic
207C2×C22⋊C4—22no—
208C2×C22⋊C4C2×Dic4711yescharacteristic
209C2×C22⋊C4C22⋊D4711yescharacteristic
210C2×C42C22⋊D4711yescharacteristic
211C2×C42C22×D4711yescharacteristic
212C4⋊C8—474no—
213C4⋊C8—944no—
214C4⋊C8—942no—
215C4⋊C8—942no—
216C22⋊C8—942no—
217C4×C8—942no—
218#95—22no—
219#95—22no—
220#95—21no—
221#95C2×Dic4712yes—
222C24C22⋊Dic4711yescharacteristic
223C2×Q8—41no—
224C2×Q8—41no—
225C2×Q8—41no—
226C2×Q8—41no—
227C2×Q8—41no—
228C2×Q8—41no—
229C22×C4—21no—
230C22×C4—21no—
231C22×C4—21no—
232C22×C4—21no—
233C22×C4—22no—
234C22×C4—22no—
235C22×C4#1062533412yes—
236C22×C4#1062533512yes—
237C22×C4C22⋊Dic4711yescharacteristic
238C22×C4C22⋊Dic4711yescharacteristic
239C22×C4C22⋊Dic4711yescharacteristic
240C22×C4C22×Dic4711yescharacteristic
241C22×C4C2×C22⋊D4711yescharacteristic
242C22×C4C2×C22⋊D4711yescharacteristic
243M16—1884no—
244M16—1882no—
245M16—1882no—
246C2×C8—944no—
247C2×C8—1882no—
248C2×C8—1884no—
249C4⋊C4—22no—
250C4⋊C4—22no—
251C4⋊C4—24no—
252C4⋊C4—44no—
253C4⋊C4—42no—
254C4⋊C4—41no—
255C4⋊C4—41no—
256C22⋊C4—22no—
257C22⋊C4—22no—
258C22⋊C4—44no—
259C42—21no—
260C42—21no—
261C42D4×D4712yes—
262C42#1062535212yes—
263C23—21no—
264C23—21no—
265C23—41no—
266C23—41no—
267C23#1309394512yes—
268C23C2×C22⋊Dic4711yescharacteristic
269Q8—81no—
270Q8—81no—
271Q8—81no—
272Q8—81no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—22no—
276C2×C4—22no—
277C2×C4—22no—
278C2×C4—22no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—41no—
285C2×C4—42no—
286C2×C4—42no—
287C2×C4—42no—
288C2×C4—42no—
289C2×C4—42no—
290C2×C4—42no—
291C2×C4#1309394512yes—
292C2×C4D4×Dic4712yes—
293C2×C4#1309399211yescharacteristic
294C2×C4#1309399612yes—
295C2×C4#1309400511yescharacteristic
296C2×C4C2×#1062533411yescharacteristic
297C2×C4C2×#1062533511yescharacteristic
298C2×C4#1309404911yescharacteristic
299C2×C4#1309405411yescharacteristic
300C8—3762no—
301C8—1884no—
302C22—21no—
303C22—21no—
304C22—22no—
305C22—42no—
306C22—41no—
307C22—41no—
308C22—81no—
309C22#1568900311yescharacteristic
310C22C2×#1309394511yescharacteristic
311C22#1568912411yescharacteristic
312C4—22no—
313C4—22no—
314C4—22no—
315C4—41no—
316C4—41no—
317C4—82no—
318C4—82no—
319C4#1568925611yescharacteristic
320C4#1568926111yescharacteristic
321C4#1568943411yescharacteristic
322C4#1568943911yescharacteristic
323C2—21no—
324C2—21no—
325C2—42no—
326C2#2024370611yescharacteristic
327C2#2024391111yescharacteristic
328C2#2024394111yescharacteristic
subgroupG/H
centerC22#15689003
commutatorC22×C4×C47C22×C4
FrattiniC2×C42C22×D47
FittingC47×#2621C2
socleC22×C47#386
prime radicalC24×C47C22⋊C4
cosocleC2×C42×C47C23
hypercenter#2621D47
hypocenterC47#9195
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2621

Complements

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

NH ≅ G/Ntype
2. #20242060C2semidirect
4. #20241246C2semidirect
187. C47#9195semidirect

Maximal subgroups

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

subgroupindex
#202519522
#202420602
#202417022
#202412462
C47×#26212
#919547