Finite Group Databasebeta

← #26194810

Subgroups — #26194810

Order 12032.

966 conjugacy classes of subgroups.

The subgroup lattice has 476 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20254336C211yesmaximal, characteristic, complemented
2#20253396C211yesmaximal, characteristic
3#20253193C211yesmaximal, characteristic, complemented
4#20242595C214yesmaximal, complemented
5#20242522C214yesmaximal, complemented
6#20242069C211yesmaximal, characteristic, complemented
7#20241813C211yesmaximal, characteristic
8#20241783C211yesmaximal, characteristic, complemented
9C2×C47×#554C211yesmaximal, characteristic
10#15689680C2211yescharacteristic
11#15689502C2212yescomplemented
12#15689473C2212yescomplemented
13#15689453—21no—
14#15689430—22no—
15#15689418C2212yescomplemented
16#15689413C2212yes—
17#15689408C2212yescomplemented
18#15689330—22no—
19#15689328C2212yes—
20#15689279—22no—
21#15689274—21no—
22C4×#10625335—22no—
23#15689176C2211yescharacteristic
24#15689176C2211yescharacteristic
25#15689163—21no—
26#15689147C2211yescharacteristic
27#15689132—21no—
28#15688872C2214yescomplemented
29Q8⋊(C47⋊C8)C2214yescomplemented
30D4⋊(C47⋊C8)C2214yes—
31C47×#554—22no—
32C47×#554—22no—
33C47×#554—22no—
34C47×#554C2214yes—
35C2×C4⋊Q8×C47C2211yescharacteristic
36C2×#125×C47—21no—
37C2×C22⋊Q8×C47—21no—
38C2×C22⋊Q8×C47—22no—
39C2×C4×Q8×C47C2211yescharacteristic
40C2×C4×D4×C47—21no—
41C2×C4×D4×C47C2211yescharacteristic
42C22×C4⋊C4×C47—21no—
43C2×#10625337—24no—
44C2×#10625335—22no—
45C2×#10625329—21no—
46#13094024C2311yescharacteristic
47#13094024C2311yescharacteristic
48#13094016—21no—
49#13094003—21no—
50#13094003—21no—
51C4×Dic94—21no—
52C4×Dic94—21no—
53#13093983—21no—
54#13093957—22no—
55#13093955—22no—
56#13093955—22no—
57C4⋊(C47⋊C8)—21no—
58C4⋊(C47⋊C8)—21no—
59C4⋊(C47⋊C8)C2312yes—
60C4⋊(C47⋊C8)C2312yes—
61#13093948—24no—
62#13093948—22no—
63#13093948—22no—
64#13093947—22no—
65#13093945—22no—
66C22⋊(C47⋊C8)—21no—
67C22⋊(C47⋊C8)—21no—
68C4×C47⋊C8—21no—
69C4×C47⋊C8C2312yes—
70C22×Q8×C47D411yescharacteristic
71C22×Q8×C47D412yes—
72C22×D4×C47D411yescharacteristic
73C23×C4×C47—22no—
74C23×C4×C47—21no—
75C4⋊Q8×C47—21no—
76C4⋊Q8×C47—21no—
77C4⋊Q8×C47D412yes—
78C4⋊Q8×C47C2312yes—
79#125×C47—24no—
80#125×C47—24no—
81C22⋊Q8×C47—24no—
82C22⋊Q8×C47—28no—
83C22⋊Q8×C47—24no—
84C22⋊Q8×C47—28no—
85C4×Q8×C47—21no—
86C4×Q8×C47—21no—
87C4×Q8×C47D412yes—
88C4×Q8×C47C2312yes—
89C4×D4×C47—24no—
90C4×D4×C47—24no—
91C4×D4×C47—21no—
92C4×D4×C47—21no—
93C4×D4×C47D412yes—
94C4×D4×C47C2312yes—
95C2×C4⋊C4×C47—22no—
96C2×C4⋊C4×C47—24no—
97C2×C4⋊C4×C47—22no—
98C2×C4⋊C4×C47—21no—
99C2×C4⋊C4×C47—21no—
100C2×C4⋊C4×C47—21no—
101C2×C4⋊C4×C47—24no—
102C2×C4⋊C4×C47D411yescharacteristic
103C2×C4⋊C4×C47D411yescharacteristic
104C2×C22⋊C4×C47—22no—
105C2×C22⋊C4×C47—21no—
106C2×C42×C47—21no—
107C2×C42×C47C2311yescharacteristic
108C24×C47—21no—
109C2×Q8×C47—21no—
110C2×Q8×C47—22no—
111C2×Q8×C47—22no—
112C2×Q8×C47—22no—
113C2×Q8×C47—82no—
114C2×Q8×C47C2×D412yes—
115C2×Q8×C47C2×D414yes—
116C2×D4×C47—24no—
117C2×D4×C47—21no—
118C2×D4×C47C2×D412yes—
119C22×C4×C47—21no—
120C22×C4×C47—21no—
121C22×C4×C47—21no—
122C22×C4×C47—22no—
123C22×C4×C47—28no—
124C22×C4×C47—22no—
125C22×C4×C47—22no—
126C22×C4×C47—42no—
127C22×C4×C47—42no—
128C22×C4×C47—84no—
129C22×C4×C47C2×D411yescharacteristic
130C22×C4×C47C2×D411yescharacteristic
131C22×C4×C47C2×D411yescharacteristic
132C2×Dic94—21no—
133C2×Dic94—21no—
134C4⋊C4×C47—28no—
135C4⋊C4×C47—28no—
136C4⋊C4×C47—21no—
137C4⋊C4×C47—22no—
138C4⋊C4×C47—22no—
139C4⋊C4×C47—22no—
140C4⋊C4×C47—21no—
141C4⋊C4×C47—22no—
142C4⋊C4×C47—22no—
143C4⋊C4×C47—22no—
144C4⋊C4×C47C2×D412yes—
145C4⋊C4×C47C2×D412yes—
146C22⋊C4×C47—28no—
147C22⋊C4×C47—22no—
148C22⋊C4×C47—22no—
149C42×C47—22no—
150C42×C47—22no—
151C42×C47C2×D411yescharacteristic
152C42×C47C2×D411yescharacteristic
153C42×C47C2×D411yescharacteristic
154C42×C47C2411yescharacteristic
155C22⋊Dic47—41no—
156#10625337—84no—
157#10625335—44no—
158C4⋊Dic47—21no—
159C4⋊Dic47—21no—
160C4⋊Dic47—21no—
161C4⋊Dic47—21no—
162Dic47⋊C4—41no—
163Dic47⋊C4—41no—
164C4×Dic47—41no—
165#10625329—42no—
166C2×C47⋊C8—22no—
167C2×C47⋊C8—22no—
168C2×C47⋊C8—22no—
169C23×C47—24no—
170C23×C47—22no—
171C23×C47—42no—
172C23×C47—42no—
173C23×C47#12011yescharacteristic
174Q8×C47—22no—
175Q8×C47—22no—
176Q8×C47—44no—
177Q8×C47—42no—
178Q8×C47—82no—
179D4×C47—24no—
180D4×C47—22no—
181D4×C47#13714yes—
182C2×C4×C47—22no—
183C2×C4×C47—22no—
184C2×C4×C47—22no—
185C2×C4×C47—22no—
186C2×C4×C47—24no—
187C2×C4×C47—41no—
188C2×C4×C47—42no—
189C2×C4×C47—44no—
190C2×C4×C47—42no—
191C2×C4×C47—42no—
192C2×C4×C47—42no—
193C2×C4×C47—42no—
194C2×C4×C47—44no—
195C2×C4×C47—88no—
196C2×C4×C47#12011yescharacteristic
197C2×C4×C47#12011yescharacteristic
198C2×C4×C47#12011yescharacteristic
199C2×C4×C47#13512yes—
200C2×C4×C47#13712yes—
201C2×C4×C47C22×D411yescharacteristic
202C2×C4×C47C22×D411yescharacteristic
203C2×C4×C47C22×D411yescharacteristic
204C2×Dic47—41no—
205C2×Dic47—41no—
206Dic94—42no—
207Dic94—42no—
208C47⋊C8—42no—
209C47⋊C8—42no—
210C47⋊C8—42no—
211#18753—471nomaximal, Sylow
212C22×C47—21no—
213C22×C47—21no—
214C22×C47—24no—
215C22×C47—48no—
216C22×C47—42no—
217C22×C47—42no—
218C22×C47#45212yes—
219C22×C47C2×#12011yescharacteristic
220C4×C47—21no—
221C4×C47—21no—
222C4×C47—21no—
223C4×C47—42no—
224C4×C47—42no—
225C4×C47—42no—
226C4×C47—44no—
227C4×C47#45212yes—
228C4×C47C2×#13511yescharacteristic
229C4×C47C2×#13711yescharacteristic
230C4×C47C2×#13712yes—
231Dic47—81no—
232Dic47—81no—
233C2×#554D4711yescharacteristic
234#3209—471no—
235#3101—471no—
236#3078—471no—
237#1643—474no—
238#1632—474no—
239#1548—471no—
240#1486—471no—
241#1479—471no—
242C2×C47—22no—
243C2×C47—44no—
244C2×C47C2×#45211yescharacteristic
245C2×C47#300411yescharacteristic
246C2×C47#301911yescharacteristic
247#554—22no—
248#554—22no—
249#554—22no—
250#554C2×D4714yes—
251#533—471no—
252C2×C4⋊Q8C2×D4711yescharacteristic
253C2×#125—21no—
254C2×C22⋊Q8—21no—
255C2×C22⋊Q8—22no—
256C2×C4×Q8C2×D4711yescharacteristic
257C2×C4×D4—21no—
258C2×C4×D4C2×D4711yescharacteristic
259C22×C4⋊C4—21no—
260#494—472no—
261#487—472no—
262#483—941no—
263#479—942no—
264#477—472no—
265#476—472no—
266#475—472no—
267#462—942no—
268#461—472no—
269#451—942no—
270#450—941no—
271C4×SD16—942no—
272#431—471no—
273#431—471no—
274#428—941no—
275#423—471no—
276#419—941no—
277#333—474no—
278Q8⋊C8—474no—
279D4⋊C8—474no—
280C47#1875311yescharacteristic, complemented, Sylow
281C22×Q8C22⋊D4711yescharacteristic
282C22×Q8C22⋊D4712yes—
283C22×D4C22⋊D4711yescharacteristic
284C23×C4—22no—
285C23×C4—21no—
286C2×Q16—944no—
287C2×SD16—942no—
288C2×M16—941no—
289C4⋊Q8—21no—
290C4⋊Q8—21no—
291C4⋊Q8—471no—
292C4⋊Q8—471no—
293C4⋊Q8C22⋊D4712yes—
294C4⋊Q8C22×D4712yes—
295#125—24no—
296#125—24no—
297#125—941no—
298C22⋊Q8—24no—
299C22⋊Q8—28no—
300C22⋊Q8—24no—
301C22⋊Q8—28no—
302C22⋊Q8—941no—
303C22⋊Q8—941no—
304C4×Q8—21no—
305C4×Q8—21no—
306C4×Q8—941no—
307C4×Q8—941no—
308C4×Q8C22⋊D4712yes—
309C4×Q8C22×D4712yes—
310C4×D4—24no—
311C4×D4—24no—
312C4×D4—21no—
313C4×D4—21no—
314C4×D4C22⋊D4712yes—
315C4×D4C22×D4712yes—
316#117—941no—
317C2×C4⋊C4—22no—
318C2×C4⋊C4—24no—
319C2×C4⋊C4—22no—
320C2×C4⋊C4—21no—
321C2×C4⋊C4—21no—
322C2×C4⋊C4—21no—
323C2×C4⋊C4—24no—
324C2×C4⋊C4C22⋊D4711yescharacteristic
325C2×C4⋊C4C22⋊D4711yescharacteristic
326C2×C22⋊C4—22no—
327C2×C22⋊C4—21no—
328C2×C42—21no—
329C2×C42C22×D4711yescharacteristic
330#107—942no—
331#106—942no—
332#106—942no—
333C4⋊C8—472no—
334C4⋊C8—472no—
335C4⋊C8—941no—
336C4⋊C8—941no—
337#103—942no—
338#103—942no—
339#103—944no—
340#103—942no—
341#102—942no—
342C22⋊C8—941no—
343C22⋊C8—941no—
344C4×C8—472no—
345C4×C8—941no—
346C24—21no—
347C2×Q8—21no—
348C2×Q8—22no—
349C2×Q8—22no—
350C2×Q8—22no—
351C2×Q8—82no—
352C2×Q8—941no—
353C2×Q8—941no—
354C2×Q8C2×C22⋊D4712yes—
355C2×Q8C2×C22⋊D4714yes—
356C2×D4—24no—
357C2×D4—21no—
358C2×D4C2×C22⋊D4712yes—
359C22×C4—21no—
360C22×C4—21no—
361C22×C4—21no—
362C22×C4—22no—
363C22×C4—28no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—42no—
367C22×C4—42no—
368C22×C4—84no—
369C22×C4C2×C22⋊D4711yescharacteristic
370C22×C4C2×C22⋊D4711yescharacteristic
371C22×C4C2×C22⋊D4711yescharacteristic
372Q16—3764no—
373SD16—1884no—
374M16—1882no—
375C2×C8—942no—
376C2×C8—942no—
377C2×C8—942no—
378C4⋊C4—28no—
379C4⋊C4—28no—
380C4⋊C4—21no—
381C4⋊C4—22no—
382C4⋊C4—22no—
383C4⋊C4—22no—
384C4⋊C4—21no—
385C4⋊C4—22no—
386C4⋊C4—22no—
387C4⋊C4—22no—
388C4⋊C4—941no—
389C4⋊C4—941no—
390C4⋊C4—941no—
391C4⋊C4—941no—
392C4⋊C4—1881no—
393C4⋊C4—1881no—
394C4⋊C4C2×C22⋊D4712yes—
395C4⋊C4C2×C22⋊D4712yes—
396C22⋊C4—28no—
397C22⋊C4—22no—
398C22⋊C4—22no—
399C22⋊C4—1881no—
400C42—22no—
401C42—22no—
402C42—1881no—
403C42C2×C22⋊D4711yescharacteristic
404C42C2×C22⋊D4711yescharacteristic
405C42C2×C22⋊D4711yescharacteristic
406C42C23×D4711yescharacteristic
407C23—24no—
408C23—22no—
409C23—42no—
410C23—42no—
411C23#1309399311yescharacteristic
412Q8—22no—
413Q8—22no—
414Q8—44no—
415Q8—42no—
416Q8—82no—
417Q8—1882no—
418Q8—1882no—
419D4—24no—
420D4—22no—
421D4#1309405514yes—
422C2×C4—22no—
423C2×C4—22no—
424C2×C4—22no—
425C2×C4—22no—
426C2×C4—24no—
427C2×C4—41no—
428C2×C4—42no—
429C2×C4—44no—
430C2×C4—42no—
431C2×C4—42no—
432C2×C4—44no—
433C2×C4—42no—
434C2×C4—42no—
435C2×C4—88no—
436C2×C4—1881no—
437C2×C4—1881no—
438C2×C4#1309399311yescharacteristic
439C2×C4#1309399311yescharacteristic
440C2×C4#1309399311yescharacteristic
441C2×C4#1309404412yes—
442C2×C4#1309405412yes—
443C2×C4C22×C22⋊D4711yescharacteristic
444C2×C4C22×C22⋊D4711yescharacteristic
445C2×C4C22×C22⋊D4711yescharacteristic
446C8—1882no—
447C8—1882no—
448C8—1882no—
449C22—21no—
450C22—21no—
451C22—24no—
452C22—48no—
453C22—42no—
454C22—42no—
455C22#1568928512yes—
456C22C2×#1309399311yescharacteristic
457C4—21no—
458C4—21no—
459C4—21no—
460C4—42no—
461C4—44no—
462C4—42no—
463C4—42no—
464C4—3761no—
465C4—3761no—
466C4#1568928512yes—
467C4C2×#1309404411yescharacteristic
468C4C2×#1309405411yescharacteristic
469C4C2×#1309405512yes—
470C2—22no—
471C2—44no—
472C2C2×#1568928511yescharacteristic
473C2#2025262211yescharacteristic
474C2#2025274511yescharacteristic
subgroupG/H
centerC22C2×#13093993
commutator, cosocleC42×C47C24
FrattiniC42C23×D47
FittingC2×C47×#554C2
socleC22×C47C2×#120
prime radicalC22×D4×C47D4
hypercenterC2×#554D47
hypocenterC47#18753
LCSG ▷ C42×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#554

Complements

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

NH ≅ G/Ntype
1. #20254336C2semidirect
3. #20253193C2semidirect
4. #20242595C2semidirect
5. #20242522C2semidirect
6. #20242069C2semidirect
8. #20241783C2semidirect
11. #15689502C22semidirect
12. #15689473C22semidirect
15. #15689418C22semidirect
17. #15689408C22semidirect
28. #15688872C22semidirect
29. Q8⋊(C47⋊C8)C22semidirect
280. C47#18753semidirect

Maximal subgroups

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

subgroupindex
#202543362
#202533962
#202531932
#202425952
#202425222
#202420692
#202418132
#202417832
C2×C47×#5542
#1875347