Finite Group Databasebeta

← #26182317

Subgroups — #26182317

Order 12032.

976 conjugacy classes of subgroups.

The subgroup lattice has 388 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689738C211yesmaximal, characteristic
2#20252429C211yesmaximal, characteristic, complemented
3#20241751C214yesmaximal, complemented
4#20241734C214yesmaximal, complemented
5#20241711C212yesmaximal, complemented
6#20241700C212yesmaximal, complemented
7C47×#2972C211yesmaximal, characteristic, complemented
8#15689738—24no—
9#15689738—24no—
10C2×#13094023C411yescharacteristic
11C2×#13094017C412yes—
12C2×#13094011C411yescharacteristic
13C2×#13094010C412yes—
14C2×#13094007C2212yes—
15C2×#13094004C411yescharacteristic
16C2×#13094003C2212yes—
17C2×#13093995C411yescharacteristic
18C2×C4×Dic94C2211yescharacteristic
19C2×#13093981C2211yescharacteristic
20#15689201—24no—
21#15689176C2212yes—
22#15689087C2214yes—
23#15688848C2214yes—
24#15688848C2218yes—
25#15688846C2214yes—
26C2×C4×D4×C47C2211yescharacteristic
27C8×D4×C47—24no—
28C47×#431C2212yes—
29C2×C22⋊C8×C47C2214yes—
30C22×Dic94C2×C411yescharacteristic
31C22×Dic94C2×C412yes—
32C22×C22⋊D47C2×C412yes—
33C22×C4×D47C2×C411yescharacteristic
34C2×#10625329—24no—
35#13094023—24no—
36#13094017—28no—
37#13094011—24no—
38#13094010—28no—
39#13094007—24no—
40#13094007—24no—
41#13094004—24no—
42#13094003—24no—
43#13094003—24no—
44#13093995—24no—
45C4×Dic94—22no—
46C4×Dic94—22no—
47#13093981—22no—
48#13093981—22no—
49C2×C4⋊Dic47C2×C411yescharacteristic
50C2×C4⋊Dic47C2×C412yes—
51C2×Dic47⋊C4C2×C411yescharacteristic
52C2×Dic47⋊C4C2×C412yes—
53C2×Dic47⋊C4C2311yescharacteristic
54C2×Dic47⋊C4C2312yes—
55C2×C22⋊Dic47C2×C412yes—
56C2×C22⋊Dic47C2×C412yes—
57C2×C22⋊Dic47C2×C412yes—
58C2×(C2×D47)⋊C4C2×C411yescharacteristic
59C2×(C2×D47)⋊C4C2×C412yes—
60C2×(C2×D47)⋊C4C2311yescharacteristic
61C2×C4×Dic47C2×C411yescharacteristic
62C2×C4×Dic47C2×C412yes—
63C4⋊(C47⋊C8)—22no—
64C22⋊(C47⋊C8)—24no—
65C22⋊(C47⋊C8)C2314yes—
66C4×C47⋊C8—22no—
67C22×D4×C47C2×C411yescharacteristic
68C23×C4×C47C2312yes—
69C22×C8×C47—24no—
70C4×D4×C47D414yes—
71C4×D4×C47D414yes—
72C2×C4⋊C4×C47C2×C411yescharacteristic
73C2×C22⋊C4×C47C2×C412yes—
74C2×C42×C47C2311yescharacteristic
75C4⋊C8×C47—22no—
76C22⋊C8×C47—24no—
77C22⋊C8×C47C2314yes—
78C4×C8×C47—22no—
79C24×C47C22×C412yes—
80C23×D47C22×C411yescharacteristic
81C2×D4×C47—24no—
82C2×D4×C47C22⋊C414yes—
83C22×C4×C47—82no—
84C22×C4×C47C22×C411yescharacteristic
85C22×C4×C47C22×C412yes—
86C22×C4×C47C22×C411yescharacteristic
87C22×C4×C47C2×D414yes—
88C22×C4×C47C2×D414yes—
89C22×C4×C47C2411yescharacteristic
90C2×C22⋊D47—24no—
91C2×C22⋊D47—82no—
92C22×Dic47C22×C412yes—
93C22×Dic47C22×C411yescharacteristic
94C22×Dic47C22×C411yescharacteristic
95C22×Dic47C22×C411yescharacteristic
96C22×Dic47C22×C412yes—
97C2×C4×D47—22no—
98C2×C4×D47—24no—
99C2×Dic94—22no—
100C2×Dic94—24no—
101C2×Dic94—24no—
102C2×Dic94—82no—
103C2×C8×C47—24no—
104C2×C8×C47—84no—
105C4⋊C4×C47C22⋊C414yes—
106C22⋊C4×C47C22⋊C418yes—
107C42×C47C2×D412yes—
108C42×C47C2×D412yes—
109C22⋊Dic47—24no—
110C22⋊Dic47—24no—
111C22⋊Dic47—24no—
112(C2×D47)⋊C4—22no—
113(C2×D47)⋊C4—24no—
114(C2×D47)⋊C4—21no—
115(C2×D47)⋊C4—21no—
116C4⋊Dic47—22no—
117C4⋊Dic47—24no—
118Dic47⋊C4—22no—
119Dic47⋊C4—24no—
120Dic47⋊C4—21no—
121Dic47⋊C4—21no—
122Dic47⋊C4—22no—
123Dic47⋊C4—22no—
124C4×Dic47—22no—
125C4×Dic47—24no—
126#10625329—84no—
127C2×C47⋊C8—24no—
128C23×C47—24no—
129C23×C47—42no—
130C23×C47—42no—
131C23×C47C2×C22⋊C414yes—
132C23×C47C23×C411yescharacteristic
133C22×D47—21no—
134C22×D47—22no—
135C22×D47—41no—
136C22×D47—41no—
137D4×C47—28no—
138C2×C4×C47—44no—
139C2×C4×C47—44no—
140C2×C4×C47—42no—
141C2×C4×C47—41no—
142C2×C4×C47—41no—
143C2×C4×C47C2×C22⋊C412yes—
144C2×C4×C47C2×C22⋊C414yes—
145C2×C4×C47C2×C22⋊C412yes—
146C2×C4×C47#13114yes—
147C2×C4×C47C22×D411yescharacteristic
148C2×C4×C47C22×D411yescharacteristic
149C22⋊D47—84no—
150C2×Dic47—21no—
151C2×Dic47—22no—
152C2×Dic47—22no—
153C2×Dic47—21no—
154C2×Dic47—21no—
155C2×Dic47—24no—
156C2×Dic47—22no—
157C2×Dic47—41no—
158C2×Dic47—42no—
159C2×Dic47—41no—
160C4×D47—44no—
161Dic94—44no—
162Dic94—84no—
163C8×C47—44no—
164C47⋊C8—44no—
165#17388—471nomaximal, Sylow
166C22×C47—21no—
167C22×C47—21no—
168C22×C47—24no—
169C22×C47—24no—
170C22×C47—24no—
171C22×C47—84no—
172C22×C47C22×C22⋊C411yescharacteristic
173C22×C47C2×#13112yes—
174C2×D47—41no—
175C2×D47—41no—
176C2×D47—41no—
177C2×D47—82no—
178C4×C47—21no—
179C4×C47—21no—
180C4×C47—24no—
181C4×C47—22no—
182C4×C47#40814yes—
183Dic47—42no—
184Dic47—41no—
185Dic47—41no—
186Dic47—42no—
187Dic47—41no—
188C2×#542—471no—
189#2975—471no—
190#2972D4711yescharacteristic, complemented
191#1472—474no—
192#1468—474no—
193#1461—472no—
194#1458—472no—
195C2×C47—22no—
196C2×C47—44no—
197C2×C47C2×#40811yescharacteristic
198C2×C47#288311yescharacteristic
199C2×C47#288911yescharacteristic
200D47—81no—
201#542—944no—
202#542—944no—
203C2×C4⋊Q8—471no—
204C2×#126—472no—
205C2×#124—471no—
206C2×#124—472no—
207C2×#123—472no—
208C2×C22⋊Q8—472no—
209C2×C22⋊Q8—471no—
210C2×#121—471no—
211C2×C4×Q8—471no—
212C2×C4×D4C2×D4711yescharacteristic
213C2×#117—471no—
214#436—944no—
215C8×D4—24no—
216#431—472no—
217#431C2×D4712yes—
218#407—474no—
219C2×C22⋊C8C2×D4714yes—
220#324—474no—
221#324—478no—
222#323—474no—
223C47#1738811yescharacteristic, complemented, Sylow
224C22×Q8—471no—
225C22×Q8—472no—
226C22×D4—472no—
227C22×D4C4×D4711yescharacteristic
228C23×C4—471no—
229C23×C4C22×D4712yes—
230C2×M16—944no—
231C22×C8—24no—
232C4⋊Q8—944no—
233#126—948no—
234#124—944no—
235#124—948no—
236#123—944no—
237#123—944no—
238C22⋊Q8—944no—
239C22⋊Q8—944no—
240C22⋊Q8—944no—
241#121—944no—
242C4×Q8—942no—
243C4×Q8—942no—
244C4×D4D18814yes—
245C4×D4C22⋊D4714yes—
246#117—942no—
247#117—942no—
248C2×C4⋊C4—471no—
249C2×C4⋊C4—471no—
250C2×C4⋊C4—472no—
251C2×C4⋊C4—472no—
252C2×C4⋊C4—471no—
253C2×C4⋊C4—472no—
254C2×C4⋊C4C4×D4711yescharacteristic
255C2×C22⋊C4—472no—
256C2×C22⋊C4—472no—
257C2×C22⋊C4—472no—
258C2×C22⋊C4—471no—
259C2×C22⋊C4—472no—
260C2×C22⋊C4—471no—
261C2×C22⋊C4C4×D4712yes—
262C2×C42—471no—
263C2×C42—472no—
264C2×C42C22×D4711yescharacteristic
265C4⋊C8—22no—
266C4⋊C8—942no—
267C22⋊C8—24no—
268C22⋊C8—474no—
269C22⋊C8—944no—
270C22⋊C8C22×D4714yes—
271C4×C8—22no—
272C4×C8—942no—
273C24—471no—
274C24C2×C4×D4712yes—
275C2×Q8—944no—
276C2×Q8—942no—
277C2×Q8—944no—
278C2×Q8—3762no—
279C2×D4—24no—
280C2×D4—944no—
281C2×D4—3762no—
282C2×D4(C2×D47)⋊C414yes—
283C22×C4—82no—
284C22×C4—472no—
285C22×C4—471no—
286C22×C4—471no—
287C22×C4—471no—
288C22×C4—472no—
289C22×C4—942no—
290C22×C4—944no—
291C22×C4C2×C4×D4711yescharacteristic
292C22×C4C2×C4×D4712yes—
293C22×C4C2×C4×D4711yescharacteristic
294C22×C4C2×D18814yes—
295C22×C4C2×C22⋊D4714yes—
296C22×C4C23×D4711yescharacteristic
297M16—3764no—
298C2×C8—24no—
299C2×C8—84no—
300C2×C8—944no—
301C4⋊C4—942no—
302C4⋊C4—944no—
303C4⋊C4—942no—
304C4⋊C4—944no—
305C4⋊C4—941no—
306C4⋊C4—942no—
307C4⋊C4—941no—
308C4⋊C4—942no—
309C4⋊C4(C2×D47)⋊C414yes—
310C22⋊C4—944no—
311C22⋊C4—944no—
312C22⋊C4—944no—
313C22⋊C4—942no—
314C22⋊C4—944no—
315C22⋊C4—941no—
316C22⋊C4—941no—
317C22⋊C4(C2×D47)⋊C418yes—
318C42—942no—
319C42—944no—
320C42C2×D18812yes—
321C42C2×C22⋊D4712yes—
322C23—24no—
323C23—42no—
324C23—42no—
325C23—942no—
326C23—941no—
327C23—1881no—
328C23—1881no—
329C23C2×(C2×D47)⋊C414yes—
330C23C22×C4×D4711yescharacteristic
331Q8—3764no—
332Q8—1884no—
333D4—28no—
334D4—3764no—
335C2×C4—44no—
336C2×C4—44no—
337C2×C4—41no—
338C2×C4—42no—
339C2×C4—41no—
340C2×C4—941no—
341C2×C4—942no—
342C2×C4—941no—
343C2×C4—942no—
344C2×C4—941no—
345C2×C4—942no—
346C2×C4—944no—
347C2×C4—1884no—
348C2×C4—1881no—
349C2×C4—1882no—
350C2×C4—1881no—
351C2×C4C2×(C2×D47)⋊C412yes—
352C2×C4C2×(C2×D47)⋊C414yes—
353C2×C4C2×(C2×D47)⋊C412yes—
354C2×C4#1309403014yes—
355C2×C4C22×C22⋊D4711yescharacteristic
356C2×C4C22×D18811yescharacteristic
357C8—44no—
358C8—1884no—
359C22—21no—
360C22—21no—
361C22—24no—
362C22—24no—
363C22—24no—
364C22—84no—
365C22—3762no—
366C22—1881no—
367C22—1881no—
368C22—1881no—
369C22C22×(C2×D47)⋊C411yescharacteristic
370C22C2×#1309403012yes—
371C4—21no—
372C4—21no—
373C4—24no—
374C4—22no—
375C4—1881no—
376C4—1882no—
377C4—1882no—
378C4—1881no—
379C4—1881no—
380C4#1568908914yes—
381C2—22no—
382C2—44no—
383C2—3761no—
384C2C2×#1568908911yescharacteristic
385C2#2025184811yescharacteristic
386C2#2025187011yescharacteristic
subgroupG/H
centerC22C22×(C2×D47)⋊C4
commutatorC23×C47C23×C4
FrattiniC22×C4C23×D47
FittingC47×#2972C2
socleC22×C47C22×C22⋊C4
prime radicalC2×#13093995C4
cosocleC22×C4×C47C24
hypercenter#2972D47
hypocenterC47#17388
LCSG ▷ C23×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C2×C4×D4 ◁ #2972

Complements

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

NH ≅ G/Ntype
2. #20252429C2semidirect
3. #20241751C2semidirect
4. #20241734C2semidirect
5. #20241711C2semidirect
6. #20241700C2semidirect
7. C47×#2972C2semidirect
190. #2972D47semidirect
223. C47#17388semidirect

Maximal subgroups

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

subgroupindex
C2×#156897382
#202524292
#202417512
#202417342
#202417112
#202417002
C47×#29722
#1738847