Finite Group Databasebeta

← #26108691

Subgroups — #26108691

Order 12032.

616 conjugacy classes of subgroups.

The subgroup lattice has 502 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20242098C211yesmaximal, characteristic, complemented
2#20242019C211yesmaximal, characteristic
3#20241965C211yesmaximal, characteristic
4#20241939C211yesmaximal, characteristic, complemented
5#20241939C211yesmaximal, characteristic, complemented
6#20241861C211yesmaximal, characteristic, complemented
7C47×#3555C211yesmaximal, characteristic
8#15689180C2211yescharacteristic
9#15689180C2211yescharacteristic
10#15689176C2211yescharacteristic
11#15688991C2211yescharacteristic
12#15688872—21no—
13#15688869—21no—
14#15688865—21no—
15#15688863—21no—
16#15688860—21no—
17#15688860—21no—
18#15688860—22no—
19#15688860—21no—
20#15688857—21no—
21Q8⋊(C47⋊C8)—22no—
22C47×#563—22no—
23Q82×C47C411yescharacteristic, complemented
24C47×#557—21no—
25C47×#557C411yescharacteristic, complemented
26C47×#556—21no—
27C47×#556—21no—
28C47×#556C411yescharacteristic, complemented
29C47×#554C411yescharacteristic, complemented
30C47×#552—21no—
31C47×#544—21no—
32C47×#542—21no—
33C47×#541—21no—
34C47×#541—21no—
35C47×#541—22no—
36C47×#533C2211yescharacteristic
37C47×#533C2211yescharacteristic
38C47×#529C2211yescharacteristic
39C4⋊(C47⋊C8)—21no—
40C4⋊(C47⋊C8)—21no—
41C4⋊(C47⋊C8)—21no—
42C22⋊(C47⋊C8)—21no—
43C22⋊(C47⋊C8)—21no—
44C22⋊(C47⋊C8)—21no—
45#13093935—21no—
46#13093935—21no—
47C4×C47⋊C8—21no—
48C47⋊#95—21no—
49C47⋊#95—21no—
50C47⋊#95—21no—
51C4⋊Q8×C47—22no—
52C4⋊Q8×C47—21no—
53C4⋊Q8×C47—22no—
54C4⋊Q8×C47—21no—
55C4⋊Q8×C47—21no—
56C4⋊Q8×C47C2×C411yescharacteristic
57C4⋊Q8×C47C2×C411yescharacteristic
58C4⋊Q8×C47D411yescharacteristic
59C4⋊Q8×C47D411yescharacteristic
60#126×C47—22no—
61#126×C47—22no—
62#126×C47—22no—
63#126×C47—22no—
64#126×C47—22no—
65#126×C47—22no—
66#126×C47—21no—
67#125×C47—22no—
68#125×C47—22no—
69#125×C47—22no—
70#125×C47—22no—
71#125×C47—21no—
72#125×C47—22no—
73#125×C47—21no—
74#125×C47—21no—
75#125×C47—21no—
76#125×C47D411yescharacteristic
77#124×C47—22no—
78#124×C47—21no—
79#124×C47D411yescharacteristic
80#123×C47—22no—
81#123×C47—21no—
82C22⋊Q8×C47—22no—
83C22⋊Q8×C47—22no—
84C22⋊Q8×C47—21no—
85C22⋊Q8×C47—21no—
86C22⋊Q8×C47—22no—
87C22⋊Q8×C47—21no—
88C22⋊Q8×C47—21no—
89C22⋊Q8×C47D411yescharacteristic
90C22⋊Q8×C47D411yescharacteristic
91C22⋊Q8×C47D411yescharacteristic
92C22⋊Q8×C47D412yes—
93#121×C47D411yescharacteristic
94C4×Q8×C47—22no—
95C4×Q8×C47—21no—
96C4×Q8×C47—21no—
97C4×Q8×C47—21no—
98C4×Q8×C47—21no—
99C4×Q8×C47—21no—
100C4×Q8×C47—21no—
101C4×Q8×C47—21no—
102C4×Q8×C47C2×C411yescharacteristic
103C4×Q8×C47C2×C411yescharacteristic
104C4×Q8×C47C2×C411yescharacteristic
105C4×Q8×C47D412yes—
106C4×D4×C47—21no—
107C4×D4×C47—21no—
108C4×D4×C47C2×C411yescharacteristic
109#117×C47—22no—
110#117×C47—21no—
111#117×C47—21no—
112#117×C47—21no—
113#117×C47—21no—
114C2×C4⋊C4×C47—21no—
115C2×C42×C47C2311yescharacteristic
116C2×Q8×C47—22no—
117C2×Q8×C47—21no—
118C2×Q8×C47—21no—
119C2×Q8×C47C22⋊C411yescharacteristic
120C2×Q8×C47C22⋊C411yescharacteristic
121C2×Q8×C47C22⋊C411yescharacteristic
122C2×Q8×C47C22⋊C412yes—
123C2×D4×C47—21no—
124C2×D4×C47C22⋊C411yescharacteristic
125C22×C4×C47—22no—
126C22×C4×C47—21no—
127C22×C4×C47C2×D411yescharacteristic
128C22×C4×C47C2×D411yescharacteristic
129C22×C4×C47C2×D411yescharacteristic
130C22×Dic47—41no—
131C4⋊C4×C47—22no—
132C4⋊C4×C47—22no—
133C4⋊C4×C47—22no—
134C4⋊C4×C47—22no—
135C4⋊C4×C47—22no—
136C4⋊C4×C47—22no—
137C4⋊C4×C47—22no—
138C4⋊C4×C47—22no—
139C4⋊C4×C47—21no—
140C4⋊C4×C47—21no—
141C4⋊C4×C47—21no—
142C4⋊C4×C47—21no—
143C4⋊C4×C47—21no—
144C4⋊C4×C47—21no—
145C4⋊C4×C47—21no—
146C4⋊C4×C47—21no—
147C4⋊C4×C47—21no—
148C4⋊C4×C47—21no—
149C4⋊C4×C47—21no—
150C4⋊C4×C47—21no—
151C4⋊C4×C47—21no—
152C4⋊C4×C47—22no—
153C4⋊C4×C47—21no—
154C4⋊C4×C47—22no—
155C4⋊C4×C47C22⋊C411yescharacteristic
156C4⋊C4×C47C22⋊C411yescharacteristic
157C4⋊C4×C47C22⋊C411yescharacteristic
158C4⋊C4×C47C22⋊C411yescharacteristic
159C4⋊C4×C47C22⋊C412yes—
160C22⋊C4×C47—22no—
161C22⋊C4×C47—22no—
162C22⋊C4×C47—22no—
163C22⋊C4×C47—21no—
164C22⋊C4×C47—22no—
165C22⋊C4×C47—21no—
166C22⋊C4×C47—21no—
167C22⋊C4×C47—21no—
168C22⋊C4×C47—21no—
169C22⋊C4×C47—21no—
170C22⋊C4×C47—21no—
171C42×C47—22no—
172C42×C47—22no—
173C42×C47—22no—
174C42×C47—21no—
175C42×C47—21no—
176C42×C47—21no—
177C42×C47—22no—
178C42×C47C22×C411yescharacteristic
179C42×C47C2×D411yescharacteristic
180C42×C47C2×D411yescharacteristic
181C42×C47C2×D411yescharacteristic
182C2×C47⋊C8—41no—
183C2×C47⋊C8—41no—
184C2×C47⋊C8—41no—
185C23×C47—21no—
186C23×C47#12011yescharacteristic
187Q8×C47—42no—
188Q8×C47—41no—
189Q8×C47—41no—
190Q8×C47—41no—
191Q8×C47—42no—
192Q8×C47—81no—
193Q8×C47—82no—
194D4×C47—41no—
195D4×C47—81no—
196C2×C4×C47—21no—
197C2×C4×C47—21no—
198C2×C4×C47—21no—
199C2×C4×C47—22no—
200C2×C4×C47—21no—
201C2×C4×C47—21no—
202C2×C4×C47—21no—
203C2×C4×C47—22no—
204C2×C4×C47—21no—
205C2×C4×C47—41no—
206C2×C4×C47—41no—
207C2×C4×C47—41no—
208C2×C4×C47—42no—
209C2×C4×C47—82no—
210C2×C4×C47C2×C22⋊C411yescharacteristic
211C2×C4×C47C2×C22⋊C411yescharacteristic
212C2×C4×C47C2×C22⋊C411yescharacteristic
213C2×C4×C47#12011yescharacteristic
214C2×C4×C47#12011yescharacteristic
215C2×C4×C47#12011yescharacteristic
216C2×Dic47—81no—
217C2×Dic47—81no—
218C2×Dic47—81no—
219C47⋊C8—81no—
220C47⋊C8—81no—
221C47⋊C8—81no—
222#8501—471nomaximal, Sylow
223C22×C47—21no—
224C22×C47—21no—
225C22×C47—21no—
226C22×C47—41no—
227C22×C47—41no—
228C22×C47—41no—
229C22×C47#37911yescharacteristic
230C4×C47—21no—
231C4×C47—21no—
232C4×C47—21no—
233C4×C47—21no—
234C4×C47—21no—
235C4×C47—21no—
236C4×C47—41no—
237C4×C47—42no—
238C4×C47—41no—
239C4×C47—41no—
240C4×C47—41no—
241C4×C47—41no—
242C4×C47—41no—
243C4×C47—41no—
244C4×C47—42no—
245Dic47—161no—
246#3555D4711yescharacteristic
247#1554—471no—
248#1535—471no—
249#1521—471no—
250#1515—471no—
251#1515—471no—
252#1497—471no—
253C2×C47—41no—
254C2×C47—81no—
255C2×C47#179411yescharacteristic
256C2×C47#179611yescharacteristic
257C2×C47#179611yescharacteristic
258#563—22no—
259Q82Dic4711yescharacteristic, complemented
260#557—21no—
261#557Dic4711yescharacteristic, complemented
262#556—21no—
263#556—21no—
264#556Dic4711yescharacteristic, complemented
265#554Dic4711yescharacteristic, complemented
266#552—21no—
267#544—21no—
268#542—21no—
269#541—21no—
270#541—21no—
271#541—22no—
272#533C2×D4711yescharacteristic
273#533C2×D4711yescharacteristic
274#529C2×D4711yescharacteristic
275#432—471no—
276#432—471no—
277#431—471no—
278#383—471no—
279#333—941no—
280#332—941no—
281#330—941no—
282#329—941no—
283#328—941no—
284#328—941no—
285#328—942no—
286#328—941no—
287#327—941no—
288Q8⋊C8—942no—
289C47#850111yescharacteristic, complemented, Sylow
290C4⋊Q8—22no—
291C4⋊Q8—21no—
292C4⋊Q8—22no—
293C4⋊Q8—21no—
294C4⋊Q8—21no—
295C4⋊Q8C2×Dic4711yescharacteristic
296C4⋊Q8C2×Dic4711yescharacteristic
297C4⋊Q8C22⋊D4711yescharacteristic
298C4⋊Q8C22⋊D4711yescharacteristic
299#126—22no—
300#126—22no—
301#126—22no—
302#126—22no—
303#126—22no—
304#126—22no—
305#126—21no—
306#125—22no—
307#125—22no—
308#125—22no—
309#125—22no—
310#125—21no—
311#125—22no—
312#125—21no—
313#125—21no—
314#125—21no—
315#125C22⋊D4711yescharacteristic
316#124—22no—
317#124—21no—
318#124C22⋊D4711yescharacteristic
319#123—22no—
320#123—21no—
321C22⋊Q8—22no—
322C22⋊Q8—22no—
323C22⋊Q8—21no—
324C22⋊Q8—21no—
325C22⋊Q8—22no—
326C22⋊Q8—21no—
327C22⋊Q8—21no—
328C22⋊Q8C22⋊D4711yescharacteristic
329C22⋊Q8C22⋊D4711yescharacteristic
330C22⋊Q8C22⋊D4711yescharacteristic
331C22⋊Q8C22⋊D4712yes—
332#121C22⋊D4711yescharacteristic
333C4×Q8—22no—
334C4×Q8—21no—
335C4×Q8—21no—
336C4×Q8—21no—
337C4×Q8—21no—
338C4×Q8—21no—
339C4×Q8—21no—
340C4×Q8—21no—
341C4×Q8C2×Dic4711yescharacteristic
342C4×Q8C2×Dic4711yescharacteristic
343C4×Q8C2×Dic4711yescharacteristic
344C4×Q8C22⋊D4712yes—
345C4×D4—21no—
346C4×D4—21no—
347C4×D4C2×Dic4711yescharacteristic
348#117—22no—
349#117—21no—
350#117—21no—
351#117—21no—
352#117—21no—
353C2×C4⋊C4—21no—
354C2×C42C22×D4711yescharacteristic
355C4⋊C8—941no—
356C4⋊C8—941no—
357C4⋊C8—941no—
358C22⋊C8—941no—
359C22⋊C8—941no—
360C22⋊C8—941no—
361#97—941no—
362#97—941no—
363C4×C8—941no—
364#95—941no—
365#95—941no—
366#95—941no—
367C2×Q8—22no—
368C2×Q8—21no—
369C2×Q8—21no—
370C2×Q8C22⋊Dic4711yescharacteristic
371C2×Q8C22⋊Dic4711yescharacteristic
372C2×Q8C22⋊Dic4711yescharacteristic
373C2×Q8C22⋊Dic4712yes—
374C2×D4—21no—
375C2×D4C22⋊Dic4711yescharacteristic
376C22×C4—22no—
377C22×C4—21no—
378C22×C4—1881no—
379C22×C4C2×C22⋊D4711yescharacteristic
380C22×C4C2×C22⋊D4711yescharacteristic
381C22×C4C2×C22⋊D4711yescharacteristic
382C2×C8—1881no—
383C2×C8—1881no—
384C2×C8—1881no—
385C4⋊C4—22no—
386C4⋊C4—22no—
387C4⋊C4—22no—
388C4⋊C4—22no—
389C4⋊C4—22no—
390C4⋊C4—22no—
391C4⋊C4—22no—
392C4⋊C4—22no—
393C4⋊C4—21no—
394C4⋊C4—21no—
395C4⋊C4—21no—
396C4⋊C4—21no—
397C4⋊C4—21no—
398C4⋊C4—21no—
399C4⋊C4—21no—
400C4⋊C4—21no—
401C4⋊C4—21no—
402C4⋊C4—21no—
403C4⋊C4—21no—
404C4⋊C4—21no—
405C4⋊C4—21no—
406C4⋊C4—22no—
407C4⋊C4—21no—
408C4⋊C4—22no—
409C4⋊C4C22⋊Dic4711yescharacteristic
410C4⋊C4C22⋊Dic4711yescharacteristic
411C4⋊C4C22⋊Dic4711yescharacteristic
412C4⋊C4C22⋊Dic4711yescharacteristic
413C4⋊C4C22⋊Dic4712yes—
414C22⋊C4—22no—
415C22⋊C4—22no—
416C22⋊C4—22no—
417C22⋊C4—21no—
418C22⋊C4—22no—
419C22⋊C4—21no—
420C22⋊C4—21no—
421C22⋊C4—21no—
422C22⋊C4—21no—
423C22⋊C4—21no—
424C22⋊C4—21no—
425C42—22no—
426C42—22no—
427C42—22no—
428C42—21no—
429C42—21no—
430C42—21no—
431C42—22no—
432C42C22×Dic4711yescharacteristic
433C42C2×C22⋊D4711yescharacteristic
434C42C2×C22⋊D4711yescharacteristic
435C42C2×C22⋊D4711yescharacteristic
436C23—21no—
437C23#1309399311yescharacteristic
438Q8—42no—
439Q8—41no—
440Q8—41no—
441Q8—41no—
442Q8—42no—
443Q8—81no—
444Q8—82no—
445D4—41no—
446D4—81no—
447C2×C4—21no—
448C2×C4—21no—
449C2×C4—21no—
450C2×C4—22no—
451C2×C4—21no—
452C2×C4—21no—
453C2×C4—21no—
454C2×C4—22no—
455C2×C4—21no—
456C2×C4—41no—
457C2×C4—41no—
458C2×C4—41no—
459C2×C4—42no—
460C2×C4—82no—
461C2×C4—3761no—
462C2×C4—3761no—
463C2×C4—3761no—
464C2×C4C2×C22⋊Dic4711yescharacteristic
465C2×C4C2×C22⋊Dic4711yescharacteristic
466C2×C4C2×C22⋊Dic4711yescharacteristic
467C2×C4#1309399311yescharacteristic
468C2×C4#1309399311yescharacteristic
469C2×C4#1309399311yescharacteristic
470C8—3761no—
471C8—3761no—
472C8—3761no—
473C22—21no—
474C22—21no—
475C22—21no—
476C22—41no—
477C22—41no—
478C22—41no—
479C22#1568897911yescharacteristic
480C4—21no—
481C4—21no—
482C4—21no—
483C4—21no—
484C4—21no—
485C4—21no—
486C4—41no—
487C4—42no—
488C4—41no—
489C4—41no—
490C4—41no—
491C4—42no—
492C4—41no—
493C4—41no—
494C4—41no—
495C4—7521no—
496C2—41no—
497C2—81no—
498C2#2024335211yescharacteristic
499C2#2024336311yescharacteristic
500C2#2024336311yescharacteristic
subgroupG/H
centerC22#15688979
commutatorC42×C47C22×C4
FrattiniC2×C42C22×D47
FittingC47×#3555C2
socleC22×C47#379
prime radical#121×C47D4
cosocleC2×C42×C47C23
hypercenter#3555D47
hypocenterC47#8501
LCSG ▷ C42×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3555

Complements

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

NH ≅ G/Ntype
1. #20242098C2semidirect
4. #20241939C2semidirect
5. #20241939C2semidirect
6. #20241861C2semidirect
23. Q82×C47C4semidirect
25. C47×#557C4semidirect
28. C47×#556C4semidirect
29. C47×#554C4semidirect
259. Q82Dic47semidirect
261. #557Dic47semidirect
264. #556Dic47semidirect
265. #554Dic47semidirect
289. C47#8501semidirect

Maximal subgroups

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

subgroupindex
#202420982
#202420192
#202419652
#202419392
#202418612
C47×#35552
#850147