Finite Group Databasebeta

← #26217377

Subgroups — #26217377

Order 12032.

1458 conjugacy classes of subgroups.

The subgroup lattice has 446 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689097C211yesmaximal, characteristic
2#20246008C212yesmaximal, complemented
3#20245980C211yesmaximal, characteristic
4#20243669C218yesmaximal, complemented
5#20243121C212yesmaximal, complemented
6C47×#3428C211yesmaximal, characteristic
7C2×#13093983C2211yescharacteristic
8C2×#13093983C2212yes—
9C22×C4⋊Dic47C2211yescharacteristic
10#15689097C418yescomplemented
11C2×#13093939C2214yes—
12#15689030—21no—
13#15689026—21no—
14#15689021—24no—
15#15689003C2214yes—
16#15688995C2218yes—
17#15688992—22no—
18#15688988—22no—
19#15688983—22no—
20#15688890C2218yes—
21C22×#41×C47C2211yescharacteristic
22C47×#520—22no—
23C47×#520—22no—
24C47×#519—21no—
25C47×#518—21no—
26C47×#518C2214yes—
27C4×#41×C47—24no—
28C2×#117×C47—22no—
29C2×#117×C47C2212yes—
30C23×Dic47C2312yes—
31#13093983—24no—
32#13093983—24no—
33#13093983C2×C418yescomplemented
34C2×C4⋊Dic47—22no—
35C2×C4⋊Dic47—21no—
36C2×C4⋊Dic47—21no—
37C2×C4⋊Dic47—22no—
38C2×C4⋊Dic47—22no—
39C2×C4⋊Dic47—21no—
40C2×Dic47⋊C4—28no—
41C2×C22⋊Dic47—24no—
42C2×C22⋊Dic47C2312yes—
43C2×C22⋊Dic47C2314yes—
44C2×C4×Dic47—22no—
45C2×C4×Dic47—21no—
46#13093939C2×C4116yescomplemented
47C47⋊#95—24no—
48C47⋊#95—28no—
49C2×#41×C47—22no—
50C2×#41×C47—22no—
51C2×#41×C47—24no—
52C2×#41×C47C2×C414yes—
53C2×#41×C47Q814yes—
54C22×Q8×C47D411yescharacteristic
55C22×D4×C47D411yescharacteristic
56C22×D4×C47C2312yes—
57C23×C4×C47D412yes—
58C23×C4×C47C2311yescharacteristic
59C4×Q8×C47—24no—
60C4×Q8×C47—24no—
61C4×D4×C47—24no—
62C4×D4×C47—24no—
63C4×D4×C47—28no—
64C4×D4×C47—28no—
65#117×C47—24no—
66#117×C47—24no—
67#117×C47—22no—
68#117×C47—24no—
69#117×C47—22no—
70#117×C47—22no—
71#117×C47—22no—
72C2×C4⋊C4×C47—22no—
73C2×C4⋊C4×C47—22no—
74C2×C4⋊C4×C47D412yes—
75C2×C4⋊C4×C47D412yes—
76C2×C22⋊C4×C47—24no—
77C2×C22⋊C4×C47C2314yes—
78C2×C42×C47—24no—
79C2×C42×C47D414yes—
80C24×C47C2×D412yes—
81C24×C47C2411yescharacteristic
82#41×C47—28no—
83#41×C47—28no—
84#41×C47—28no—
85#41×C47—28no—
86C2×Q8×C47—22no—
87C2×Q8×C47—22no—
88C2×Q8×C47C4⋊C414yescomplemented
89C2×D4×C47—24no—
90C2×D4×C47—24no—
91C2×D4×C47—22no—
92C2×D4×C47—22no—
93C2×D4×C47C4⋊C414yescomplemented
94C2×D4×C47C22×C414yes—
95C2×D4×C47C2×Q814yes—
96C22×C4×C47—22no—
97C22×C4×C47—22no—
98C22×C4×C47—24no—
99C22×C4×C47—24no—
100C22×C4×C47—24no—
101C22×C4×C47C4⋊C418yescomplemented
102C22×C4×C47C22×C412yes—
103C22×C4×C47C2×D412yes—
104C22×C4×C47C2×D414yes—
105C22×C4×C47C2×D414yes—
106C22×C4×C47C2×D411yescharacteristic
107C22×C4×C47C2×D411yescharacteristic
108C22×C4×C47C2×D412yes—
109C22×C4×C47C2×D412yes—
110C22×C4×C47C2×Q812yes—
111C22×C4×C47#4114yes—
112C22×Dic47—24no—
113C22×Dic47—22no—
114C22×Dic47—22no—
115C22×Dic47—24no—
116C22×Dic47—82no—
117C4⋊C4×C47—24no—
118C4⋊C4×C47—24no—
119C4⋊C4×C47—22no—
120C4⋊C4×C47—22no—
121C4⋊C4×C47—22no—
122C4⋊C4×C47—22no—
123C22⋊C4×C47—28no—
124C22⋊C4×C47—24no—
125C22⋊C4×C47—24no—
126C42×C47—28no—
127C42×C47—24no—
128C42×C47—24no—
129C22⋊Dic47—24no—
130C22⋊Dic47—24no—
131C22⋊Dic47—84no—
132C22⋊Dic47C22×C418yes—
133C4⋊Dic47—28no—
134C4⋊Dic47—41no—
135C4⋊Dic47—42no—
136C4⋊Dic47—41no—
137C4⋊Dic47—42no—
138Dic47⋊C4—88no—
139C4×Dic47—28no—
140C4×Dic47—42no—
141C23×C47—22no—
142C23×C47—22no—
143C23×C47—41no—
144C23×C47—41no—
145C23×C47—42no—
146C23×C47—42no—
147C23×C47C2×C4⋊C414yes—
148C23×C47C23×C411yescharacteristic
149C23×C47C22×D411yescharacteristic
150C23×C47C22×D412yes—
151C23×C47C22×Q811yescharacteristic
152C23×C47C2×#4112yes—
153Q8×C47—28no—
154D4×C47—28no—
155D4×C47—28no—
156D4×C47—28no—
157C2×C4×C47—22no—
158C2×C4×C47—22no—
159C2×C4×C47—28no—
160C2×C4×C47—28no—
161C2×C4×C47—24no—
162C2×C4×C47—24no—
163C2×C4×C47—44no—
164C2×C4×C47—44no—
165C2×C4×C47—84no—
166C2×C4×C47—84no—
167C2×C4×C47C2×C4⋊C414yes—
168C2×C4×C47C2×C4⋊C412yes—
169C2×C4×C47C2×C4⋊C412yes—
170C2×C4×C47C4×D414yes—
171C2×C4×C47C4×D414yes—
172C2×C4×C47#12011yescharacteristic
173C2×C4×C47#12012yes—
174C2×C4×C47#12011yescharacteristic
175C2×C4×C47C22⋊Q814yes—
176C2×C4×C47C22⋊Q814yes—
177C2×C4×C47#12312yes—
178C2×C4×C47#12312yes—
179C2×Dic47—28no—
180C2×Dic47—44no—
181C2×Dic47—44no—
182C2×Dic47—48no—
183C2×Dic47—42no—
184C2×Dic47—44no—
185C2×Dic47—42no—
186C2×Dic47—42no—
187C2×Dic47—42no—
188#20890—471nomaximal, Sylow
189C22×C47—22no—
190C22×C47—21no—
191C22×C47—22no—
192C22×C47—21no—
193C22×C47—24no—
194C22×C47—41no—
195C22×C47—41no—
196C22×C47—42no—
197C22×C47—44no—
198C22×C47—42no—
199C22×C47—42no—
200C22×C47—82no—
201C22×C47C22×C4⋊C411yescharacteristic
202C22×C47C2×C4×D412yes—
203C22×C47C2×#12011yescharacteristic
204C22×C47C2×C22⋊Q812yes—
205C22×C47C2×#12311yescharacteristic
206C4×C47—24no—
207C4×C47—44no—
208C4×C47—44no—
209C4×C47#38512yes—
210C4×C47#38512yes—
211C4×C47#38514yes—
212Dic47—48no—
213Dic47—82no—
214Dic47—82no—
215#3428D4711yescharacteristic
216C2×#410—471no—
217#2317—472no—
218#2311—471no—
219#1867—478no—
220#1740—472no—
221C2×C47—22no—
222C2×C47—21no—
223C2×C47—21no—
224C2×C47—22no—
225C2×C47—44no—
226C2×C47C2×#38511yescharacteristic
227C2×C47#302711yescharacteristic
228C2×C47#302811yescharacteristic
229C22×#41C2×D4711yescharacteristic
230#520—22no—
231#520—22no—
232#519—21no—
233#518—21no—
234#518C2×D4714yes—
235C4×#41—24no—
236C2×#117—22no—
237C2×#117—472no—
238C2×#117—471no—
239C2×#117C2×D4712yes—
240C22×C4⋊C4—471no—
241#410—478no—
242C2×#99—474no—
243#391—941no—
244#390—941no—
245#389—944no—
246#386—474no—
247#385—478no—
248#384—942no—
249#382—942no—
250#380—942no—
251#342—478no—
252C47#2089011yescharacteristic, complemented, Sylow
253C2×#41—24no—
254C2×#41—22no—
255C2×#41—22no—
256C2×#41Dic9414yes—
257C2×#41C4×D4714yes—
258C22×Q8C22⋊D4711yescharacteristic
259C22×D4C22⋊D4711yescharacteristic
260C22×D4C22×D4712yes—
261C23×C4—472no—
262C23×C4C22⋊D4712yes—
263C23×C4C22×D4711yescharacteristic
264C4×Q8—24no—
265C4×Q8—24no—
266C4×D4—24no—
267C4×D4—24no—
268C4×D4—28no—
269C4×D4—28no—
270#117—24no—
271#117—24no—
272#117—22no—
273#117—24no—
274#117—22no—
275#117—22no—
276#117—22no—
277#117—478no—
278#117—944no—
279#117—944no—
280C2×C4⋊C4—22no—
281C2×C4⋊C4—22no—
282C2×C4⋊C4—942no—
283C2×C4⋊C4—948no—
284C2×C4⋊C4—941no—
285C2×C4⋊C4—942no—
286C2×C4⋊C4—942no—
287C2×C4⋊C4—941no—
288C2×C4⋊C4—941no—
289C2×C4⋊C4C22⋊D4712yes—
290C2×C4⋊C4C22⋊D4712yes—
291C2×C22⋊C4—24no—
292C2×C22⋊C4—474no—
293C2×C22⋊C4—472no—
294C2×C22⋊C4—944no—
295C2×C22⋊C4C22×D4714yes—
296C2×C42—24no—
297C2×C42—942no—
298C2×C42—941no—
299C2×C42C22⋊D4714yes—
300#99—4716no—
301#95—944no—
302#95—948no—
303C24C2×C22⋊D4712yes—
304C24C23×D4711yescharacteristic
305#41—28no—
306#41—28no—
307#41—28no—
308#41—28no—
309C2×Q8—22no—
310C2×Q8—22no—
311C2×Q8Dic47⋊C414yescomplemented
312C2×D4—24no—
313C2×D4—24no—
314C2×D4—22no—
315C2×D4—22no—
316C2×D4Dic47⋊C414yescomplemented
317C2×D4C2×Dic9414yes—
318C2×D4C2×C4×D4714yes—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—24no—
322C22×C4—24no—
323C22×C4—24no—
324C22×C4—944no—
325C22×C4—942no—
326C22×C4—942no—
327C22×C4—944no—
328C22×C4—3762no—
329C22×C4Dic47⋊C418yescomplemented
330C22×C4C2×Dic9412yes—
331C22×C4C2×C4×D4712yes—
332C22×C4#1062535014yes—
333C22×C4C2×C22⋊D4712yes—
334C22×C4C2×C22⋊D4714yes—
335C22×C4C2×C22⋊D4714yes—
336C22×C4C2×C22⋊D4711yescharacteristic
337C22×C4C2×C22⋊D4711yescharacteristic
338C22×C4C2×C22⋊D4712yes—
339C22×C4C2×C22⋊D4712yes—
340C4⋊C4—24no—
341C4⋊C4—24no—
342C4⋊C4—22no—
343C4⋊C4—22no—
344C4⋊C4—22no—
345C4⋊C4—22no—
346C4⋊C4—948no—
347C4⋊C4—3768no—
348C4⋊C4—1881no—
349C4⋊C4—1882no—
350C4⋊C4—1881no—
351C4⋊C4—1882no—
352C22⋊C4—28no—
353C22⋊C4—24no—
354C22⋊C4—24no—
355C22⋊C4—478no—
356C22⋊C4—944no—
357C22⋊C4—944no—
358C22⋊C4—3764no—
359C42—28no—
360C42—24no—
361C42—24no—
362C42—948no—
363C42—1882no—
364C23—22no—
365C23—22no—
366C23—41no—
367C23—41no—
368C23—42no—
369C23—42no—
370C23C2×Dic47⋊C414yes—
371C23C22×C4×D4711yescharacteristic
372C23C22×C22⋊D4711yescharacteristic
373C23C22×C22⋊D4712yes—
374C23C22×Dic9411yescharacteristic
375C23C2×#1062535012yes—
376Q8—28no—
377D4—28no—
378D4—28no—
379D4—28no—
380C2×C4—22no—
381C2×C4—22no—
382C2×C4—28no—
383C2×C4—28no—
384C2×C4—24no—
385C2×C4—24no—
386C2×C4—44no—
387C2×C4—44no—
388C2×C4—84no—
389C2×C4—84no—
390C2×C4—948no—
391C2×C4—1882no—
392C2×C4—1884no—
393C2×C4—1888no—
394C2×C4—1882no—
395C2×C4—1882no—
396C2×C4—1882no—
397C2×C4—1884no—
398C2×C4—1884no—
399C2×C4C2×Dic47⋊C414yes—
400C2×C4C2×Dic47⋊C412yes—
401C2×C4C2×Dic47⋊C412yes—
402C2×C4C4×C22⋊D4714yes—
403C2×C4C4×C22⋊D4714yes—
404C2×C4#1309399311yescharacteristic
405C2×C4#1309399312yes—
406C2×C4#1309399311yescharacteristic
407C2×C4#1309400314yes—
408C2×C4#1309400314yes—
409C2×C4#1309400712yes—
410C2×C4#1309400712yes—
411C22—22no—
412C22—21no—
413C22—22no—
414C22—21no—
415C22—24no—
416C22—41no—
417C22—41no—
418C22—42no—
419C22—44no—
420C22—42no—
421C22—42no—
422C22—82no—
423C22C22×Dic47⋊C411yescharacteristic
424C22C2×C4×C22⋊D4712yes—
425C22C2×#1309399311yescharacteristic
426C22C2×#1309400312yes—
427C22C2×#1309400711yescharacteristic
428C4—24no—
429C4—44no—
430C4—44no—
431C4—3762no—
432C4—3762no—
433C4—1888no—
434C4#1568899612yes—
435C4#1568899612yes—
436C4#1568899614yes—
437C2—21no—
438C2—22no—
439C2—21no—
440C2—22no—
441C2—44no—
442C2C2×#1568899611yescharacteristic
443C2#2025279011yescharacteristic
444C2#2025279411yescharacteristic
subgroupG/H
centerC22C2×#13093993
commutatorC23×C47C23×C4
FrattiniC24C23×D47
FittingC47×#3428C2
socleC22×C47C2×#120
prime radicalC22×#41×C47C22
cosocleC24×C47C24
hypercenter#3428D47
hypocenterC47#20890
LCSG ▷ C23×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C23×C4 ◁ #3428

Complements

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

NH ≅ G/Ntype
2. #20246008C2semidirect
4. #20243669C2semidirect
5. #20243121C2semidirect
10. #15689097C4semidirect
33. #13093983C2×C4semidirect
46. #13093939C2×C4semidirect
88. C2×Q8×C47C4⋊C4semidirect
93. C2×D4×C47C4⋊C4semidirect
101. C22×C4×C47C4⋊C4semidirect
252. C47#20890semidirect
311. C2×Q8Dic47⋊C4semidirect
316. C2×D4Dic47⋊C4semidirect
329. C22×C4Dic47⋊C4semidirect

Maximal subgroups

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

subgroupindex
C2×#156890972
#202460082
#202459802
#202436692
#202431212
C47×#34282
#2089047