Finite Group Databasebeta

← #26243661

Subgroups — #26243661

Order 12032.

1306 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689127C212yesmaximal
2#20245968C211yesmaximal, characteristic
3#20244541C218yesmaximal, complemented
4#20243127C211yesmaximal, characteristic, complemented
5#20243097C212yesmaximal, complemented
6C47×#3458C211yesmaximal, characteristic, complemented
7C23×D188C2211yescharacteristic
8C22×#10625329C2212yes—
9C2×#13093983C2211yescharacteristic
10#15689127—24no—
11#15689127C2218yes—
12C2×#13093944—24no—
13#15689025—22no—
14#15689024C2214yes—
15#15688978—22no—
16#15688892C2214yescomplemented
17#15688882C2218yescomplemented
18C47×#544—24no—
19C47×#544C2214yescomplemented
20C47×#543—22no—
21C47×#533—24no—
22C47×#533—22no—
23C2×#117×C47C2212yes—
24C2×#117×C47C2211yescharacteristic
25C24×D47—21no—
26C22×D188—22no—
27C22×D188—22no—
28C22×D188—24no—
29C22×D188—21no—
30C22×D188—22no—
31C22×D188Q814yescomplemented
32C22×D188C2312yes—
33C2×#10625329—22no—
34C2×#10625329—24no—
35C2×#10625329—24no—
36C2×#10625329C2314yes—
37C22×C47⋊C8—22no—
38#13093983—24no—
39C2×C4⋊Dic47—21no—
40C2×C22⋊Dic47—21no—
41C2×(C2×D47)⋊C4—28no—
42C2×C4×Dic47C2312yes—
43#13093944—216no—
44C23×C4×C47C2311yescharacteristic
45C4⋊Q8×C47—24no—
46C4⋊Q8×C47—28no—
47#125×C47—24no—
48#125×C47—28no—
49C22⋊Q8×C47—216no—
50C22⋊Q8×C47—28no—
51C4×Q8×C47—28no—
52#117×C47—22no—
53#117×C47—22no—
54#117×C47—22no—
55#117×C47—22no—
56#117×C47D414yes—
57#117×C47C2314yes—
58C2×C4⋊C4×C47D414yes—
59C2×C4⋊C4×C47C2312yes—
60C2×C22⋊C4×C47—22no—
61C2×C22⋊C4×C47D412yes—
62C2×C42×C47—22no—
63C2×C42×C47D412yes—
64C24×C47C2×D411yescharacteristic
65C23×D47—22no—
66C23×D47—24no—
67C23×D47—42no—
68C23×D47—42no—
69C23×D47—84no—
70C2×Q8×C47—28no—
71C22×C4×C47—24no—
72C22×C4×C47C2×D414yes—
73C22×C4×C47C2×D411yescharacteristic
74C22×C4×C47C2×D412yes—
75C22×C4×C47C2×D412yes—
76C22×C4×C47C2×Q812yes—
77C22×C4×C47#4114yes—
78C22×C4×C47#4112yes—
79C22×C4×C47C2411yescharacteristic
80C22×Dic47—22no—
81C2×D188—28no—
82C2×D188—28no—
83C2×D188—24no—
84C2×D188—24no—
85C2×D188—24no—
86C2×D188—22no—
87C2×D188—41no—
88C2×D188—41no—
89C2×D188—44no—
90C2×D188—44no—
91C2×D188—42no—
92C2×D188C2×Q814yes—
93C4⋊C4×C47—216no—
94C4⋊C4×C47—28no—
95C4⋊C4×C47—24no—
96C4⋊C4×C47—24no—
97C4⋊C4×C47C2×D418yes—
98C22⋊C4×C47—24no—
99C22⋊C4×C47—24no—
100C22⋊C4×C47—24no—
101C42×C47—24no—
102C42×C47—22no—
103C42×C47—22no—
104C42×C47—24no—
105C22⋊Dic47—42no—
106(C2×D47)⋊C4—88no—
107C4⋊Dic47—42no—
108C4×Dic47—24no—
109#10625329—24no—
110#10625329—24no—
111#10625329—44no—
112C2×C47⋊C8—24no—
113C2×C47⋊C8—44no—
114C23×C47—42no—
115C23×C47#12012yes—
116C23×C47C22⋊Q812yes—
117C23×C47#12412yes—
118C23×C47C22×D411yescharacteristic
119C22×D47—24no—
120C22×D47—48no—
121C22×D47—42no—
122C22×D47—42no—
123C22×D47—48no—
124C22×D47—88no—
125C22×D47—88no—
126C22×D47—88no—
127Q8×C47—88no—
128C2×C4×C47—24no—
129C2×C4×C47—28no—
130C2×C4×C47—44no—
131C2×C4×C47—84no—
132C2×C4×C47#12011yescharacteristic
133C2×C4×C47#12011yescharacteristic
134C2×C4×C47C22⋊Q814yes—
135C2×C4×C47C22⋊Q814yes—
136C2×C4×C47C22⋊Q812yes—
137C2×C4×C47#12414yes—
138C2×C4×C47#12414yes—
139C2×C4×C47#12412yes—
140C2×C4×C47C22×D412yes—
141C2×C4×C47C22×Q811yescharacteristic
142C2×C4×C47C2×#4112yes—
143C2×C4×C47C2×#4111yescharacteristic
144C2×Dic47—44no—
145C2×Dic47—42no—
146D188—28no—
147D188—44no—
148D188—44no—
149D188—44no—
150C47⋊C8—44no—
151#23463—471nomaximal, Sylow
152C22×C47—22no—
153C22×C47—24no—
154C22×C47—42no—
155C22×C47—42no—
156C22×C47#39314yes—
157C22×C47C2×#12011yescharacteristic
158C22×C47C2×C22⋊Q811yescharacteristic
159C22×C47C2×#12411yescharacteristic
160C2×D47—48no—
161C2×D47—44no—
162C2×D47—84no—
163C2×D47—816no—
164C2×D47—84no—
165C4×C47—44no—
166C4×C47—48no—
167C4×C47#39312yes—
168C4×C47#39312yes—
169C4×C47C2×C22⋊Q812yes—
170C4×C47C2×#12412yes—
171Dic47—82no—
172#3458D4711yescharacteristic, complemented
173C2×#418—472no—
174#2309—471no—
175#2027—478no—
176#1742—471no—
177#1733—472no—
178C2×C47—21no—
179C2×C47—21no—
180C2×C47—24no—
181C2×C47C2×#39311yescharacteristic
182C2×C47#302112yes—
183D47—84no—
184C23×D4—471no—
185C22×M16—472no—
186#544—24no—
187#544C2×D4714yescomplemented
188#543—22no—
189#533—24no—
190#533—22no—
191C2×#117—471no—
192C2×#117C2×D4712yes—
193C2×#117C2×D4711yescharacteristic
194#418—478no—
195#418—944no—
196C2×#102—944no—
197#390—474no—
198#390—942no—
199#379—942no—
200#343—474no—
201#339—478no—
202C47#2346311yescharacteristic, complemented, Sylow
203C25—941no—
204C22×D4—474no—
205C22×D4—472no—
206C22×D4—944no—
207C22×D4—942no—
208C22×D4—942no—
209C22×D4—942no—
210C22×D4—941no—
211C23×C4C22×D4711yescharacteristic
212C2×M16—474no—
213C2×M16—942no—
214C2×M16—944no—
215C2×M16—944no—
216C22×C8—942no—
217C4⋊Q8—24no—
218C4⋊Q8—28no—
219#125—24no—
220#125—28no—
221C22⋊Q8—216no—
222C22⋊Q8—28no—
223C4×Q8—28no—
224#117—22no—
225#117—22no—
226#117—22no—
227#117—22no—
228#117—944no—
229#117C22⋊D4714yes—
230#117C22×D4714yes—
231C2×C4⋊C4—941no—
232C2×C4⋊C4C22⋊D4714yes—
233C2×C4⋊C4C22×D4712yes—
234C2×C22⋊C4—22no—
235C2×C22⋊C4—948no—
236C2×C22⋊C4—941no—
237C2×C22⋊C4C22⋊D4712yes—
238C2×C42—22no—
239C2×C42—472no—
240C2×C42C22⋊D4712yes—
241#102—9416no—
242C24—944no—
243C24—942no—
244C24—3764no—
245C24—1882no—
246C24—1882no—
247C24C2×C22⋊D4711yescharacteristic
248C2×Q8—28no—
249C2×D4—474no—
250C2×D4—948no—
251C2×D4—948no—
252C2×D4—944no—
253C2×D4—944no—
254C2×D4—944no—
255C2×D4—942no—
256C2×D4—1881no—
257C2×D4—1881no—
258C2×D4—1882no—
259C2×D4—1884no—
260C2×D4—1884no—
261C22×C4—24no—
262C22×C4—942no—
263C22×C4Q8×D4712yes—
264C22×C4#1062535414yes—
265C22×C4#1062535412yes—
266C22×C4C2×C22⋊D4714yes—
267C22×C4C2×C22⋊D4711yescharacteristic
268C22×C4C2×C22⋊D4712yes—
269C22×C4C2×C22⋊D4712yes—
270C22×C4C23×D4711yescharacteristic
271M16—944no—
272M16—944no—
273M16—1884no—
274C2×C8—944no—
275C2×C8—1884no—
276C4⋊C4—216no—
277C4⋊C4—28no—
278C4⋊C4—24no—
279C4⋊C4—24no—
280C4⋊C4—1882no—
281C4⋊C4C2×C22⋊D4718yes—
282C22⋊C4—24no—
283C22⋊C4—24no—
284C22⋊C4—24no—
285C22⋊C4—3768no—
286C22⋊C4—1882no—
287C42—24no—
288C42—22no—
289C42—22no—
290C42—24no—
291C42—944no—
292C23—42no—
293C23—944no—
294C23—3768no—
295C23—3768no—
296C23—3768no—
297C23—1888no—
298C23—1888no—
299C23—1882no—
300C23—1882no—
301C23#1309399312yes—
302C23#1309400012yes—
303C23#1309401312yes—
304C23C22×C22⋊D4711yescharacteristic
305Q8—88no—
306D4—948no—
307D4—1884no—
308D4—1884no—
309D4—1884no—
310C2×C4—24no—
311C2×C4—28no—
312C2×C4—44no—
313C2×C4—84no—
314C2×C4—1884no—
315C2×C4—1882no—
316C2×C4#1309399311yescharacteristic
317C2×C4#1309399311yescharacteristic
318C2×C4#1309400012yes—
319C2×C4#1309400014yes—
320C2×C4#1309400014yes—
321C2×C4#1309401312yes—
322C2×C4#1309401314yes—
323C2×C4#1309401314yes—
324C2×C4C22×C22⋊D4712yes—
325C2×C4C2×Q8×D4711yescharacteristic
326C2×C4C2×#1062535412yes—
327C2×C4C2×#1062535411yescharacteristic
328C8—1884no—
329C22—22no—
330C22—24no—
331C22—42no—
332C22—42no—
333C22—37616no—
334C22—3764no—
335C22—3764no—
336C22—1888no—
337C22—1884no—
338C22#1568903414yes—
339C22C2×#1309399311yescharacteristic
340C22C2×#1309400011yescharacteristic
341C22C2×#1309401311yescharacteristic
342C4—44no—
343C4—48no—
344C4—3762no—
345C4#1568903412yes—
346C4#1568903412yes—
347C4C2×#1309400012yes—
348C4C2×#1309401312yes—
349C2—21no—
350C2—24no—
351C2—21no—
352C2—3764no—
353C2C2×#1568903411yescharacteristic
354C2#2025276012yes—
subgroupG/H
centerC22C2×#13093993
commutator, cosocleC22×C4×C47C24
FrattiniC22×C4C23×D47
FittingC47×#3458C2
socleC22×C47C2×#120
prime radicalC23×D188C22
hypercenter#3458D47
hypocenterC47#23463
LCSG ▷ C22×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C23×C4 ◁ #3458

Complements

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

NH ≅ G/Ntype
3. #20244541C2semidirect
4. #20243127C2semidirect
5. #20243097C2semidirect
6. C47×#3458C2semidirect
16. #15688892C22semidirect
17. #15688882C22semidirect
19. C47×#544C22semidirect
31. C22×D188Q8semidirect
172. #3458D47semidirect
187. #544C2×D47semidirect
202. C47#23463semidirect

Maximal subgroups

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

subgroupindex
C2×#156891272
#202459682
#202445412
#202431272
#202430972
C47×#34582
#2346347