Finite Group Databasebeta

← #26184679

Subgroups — #26184679

Order 12032.

972 conjugacy classes of subgroups.

The subgroup lattice has 560 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689678C211yesmaximal, characteristic
2#20252020C211yesmaximal, characteristic
3#431×D47C211yesmaximal, characteristic
4#20242085C214yesmaximal, complemented
5#20242074C211yesmaximal, characteristic, complemented
6#20242072C211yesmaximal, characteristic, complemented
7#20242060C211yesmaximal, characteristic, complemented
8#20242051C214yesmaximal, complemented
9C47×#1546C211yesmaximal, characteristic, complemented
10#15689678—21no—
11#15689678—21no—
12#15689678—21no—
13#15689678—21no—
14#15689678C412yes—
15#15689678C412yes—
16#15689678C2212yes—
17#15689678C2212yes—
18C2×#13094024C2211yescharacteristic
19C2×#13094015C412yes—
20C2×#13094004—21no—
21C2×#13094000—21no—
22C2×Q8×Dic47—21no—
23C2×#13093989—21no—
24C2×#13093982C412yes—
25C2×C42×D47C2211yescharacteristic
26#15689179—21no—
27#15689179C2212yes—
28#15689176C2211yescharacteristic
29#15689175—21no—
30#15689175C2212yes—
31#15689175C2212yes—
32#15689174—21no—
33#15689173—21no—
34#15689147C2211yescharacteristic
35C4⋊C8×D47C2212yes—
36#15689086—21no—
37C22⋊C8×D47—21no—
38M16×Dic47—21no—
39C4×C8×D47—21no—
40#15688870C2214yescomplemented
41#15688869C2214yescomplemented
42#15688868C2214yescomplemented
43C2×C4⋊Q8×C47C2211yescharacteristic
44C47×#431C2211yescharacteristic
45C47×#423C2211yescharacteristic
46C47×#332C2214yescomplemented
47C22×Dic94—21no—
48C22×C4×D47C2×C411yescharacteristic
49C22×C4×D47D411yescharacteristic
50C22×C4×D47D411yescharacteristic
51C2×#10625329—21no—
52C2×C8×D47—22no—
53#13094024—21no—
54#13094024—21no—
55#13094024C2×C412yes—
56#13094024C2312yes—
57#13094015—22no—
58#13094015—22no—
59#13094015C2×C414yes—
60#13094015C2×C414yes—
61#13094004—22no—
62#13094004—22no—
63#13094004—22no—
64#13094004—22no—
65#13094000—22no—
66#13094000—22no—
67#13094000—22no—
68#13094000—22no—
69Q8×Dic47—22no—
70Q8×Dic47—22no—
71Q8×Dic47—22no—
72Q8×Dic47—22no—
73#13093989—22no—
74#13093989—22no—
75#13093989—22no—
76#13093989—22no—
77#13093982—22no—
78#13093982—22no—
79#13093982C2×C414yes—
80#13093982C2×C414yes—
81C2×C4⋊Dic47—21no—
82C2×C4⋊Dic47—22no—
83C2×C4⋊Dic47C2×C412yes—
84C2×Dic47⋊C4—22no—
85C2×(C2×D47)⋊C4—22no—
86C2×C4×Dic47—22no—
87C2×C4×Dic47C2×C411yescharacteristic
88C2×C4×Dic47D411yescharacteristic
89C2×C4×Dic47D411yescharacteristic
90C42×D47C2×C411yescharacteristic
91C42×D47C2×C411yescharacteristic
92C42×D47D411yescharacteristic
93C42×D47D411yescharacteristic
94C42×D47D411yescharacteristic
95C42×D47D411yescharacteristic
96C42×D47C2311yescharacteristic
97C42×D47C2311yescharacteristic
98C4⋊(C47⋊C8)—21no—
99C4⋊(C47⋊C8)C2312yes—
100C4⋊(C47⋊C8)C2312yes—
101Dic47⋊C8—22no—
102C22⋊(C47⋊C8)—21no—
103(C2×D47)⋊C8—22no—
104(C2×D47)⋊C8—22no—
105(C2×D47)⋊C8—22no—
106#13093934—22no—
107C4×C47⋊C8—21no—
108C8×Dic47—22no—
109C8×Dic47—22no—
110C22×Q8×C47—21no—
111C2×M16×C47—21no—
112C4⋊Q8×C47—21no—
113C4⋊Q8×C47—21no—
114C4⋊Q8×C47C2×C412yes—
115C4⋊Q8×C47C2312yes—
116C2×C4⋊C4×C47—21no—
117C2×C4⋊C4×C47C2×C412yes—
118C2×C42×C47C2311yescharacteristic
119C4⋊C8×C47—21no—
120C4⋊C8×C47C2312yes—
121C4⋊C8×C47C2312yes—
122C22⋊C8×C47—21no—
123C4×C8×C47—21no—
124C23×D47C22⋊C411yescharacteristic
125C2×Q8×C47—22no—
126C2×Q8×C47—22no—
127C2×Q8×C47—42no—
128C2×Q8×C47—42no—
129C22×C4×C47—22no—
130C22×C4×C47C22×C411yescharacteristic
131C22×C4×C47C2×D411yescharacteristic
132C22×C4×C47C2×D411yescharacteristic
133C22×Dic47—22no—
134C22×Dic47C22⋊C411yescharacteristic
135C22×Dic47C22⋊C411yescharacteristic
136C22×Dic47C22⋊C411yescharacteristic
137C2×C4×D47—22no—
138C2×C4×D47—22no—
139C2×C4×D47—24no—
140C2×C4×D47—42no—
141C2×C4×D47C22⋊C411yescharacteristic
142C2×C4×D47C22⋊C411yescharacteristic
143C2×C4×D47C22⋊C411yescharacteristic
144C2×C4×D47C22⋊C411yescharacteristic
145C2×C4×D47C22⋊C411yescharacteristic
146C2×C4×D47C22⋊C411yescharacteristic
147C2×C4×D47C22×C411yescharacteristic
148C2×C4×D47C22×C411yescharacteristic
149C2×C4×D47C2×D411yescharacteristic
150C2×C4×D47C2×D411yescharacteristic
151C2×C4×D47C2×D411yescharacteristic
152C2×C4×D47C2×D411yescharacteristic
153C2×Dic94—22no—
154C2×Dic94—22no—
155C2×Dic94—44no—
156M16×C47—42no—
157C2×C8×C47—22no—
158C2×C8×C47—22no—
159C4⋊C4×C47—22no—
160C4⋊C4×C47—22no—
161C4⋊C4×C47—22no—
162C4⋊C4×C47C22×C414yes—
163C42×C47C22×C411yescharacteristic
164C42×C47C2×D411yescharacteristic
165C42×C47C2×D411yescharacteristic
166C42×C47C2411yescharacteristic
167(C2×D47)⋊C4—24no—
168(C2×D47)⋊C4—24no—
169C4⋊Dic47—22no—
170C4⋊Dic47—22no—
171C4⋊Dic47—22no—
172C4⋊Dic47—24no—
173C4⋊Dic47—24no—
174C4⋊Dic47C22×C414yes—
175Dic47⋊C4—24no—
176Dic47⋊C4—24no—
177C4×Dic47—24no—
178C4×Dic47—24no—
179C4×Dic47C22⋊C411yescharacteristic
180C4×Dic47C22⋊C411yescharacteristic
181C4×Dic47C22⋊C411yescharacteristic
182C4×Dic47C22⋊C411yescharacteristic
183C4×Dic47C22⋊C411yescharacteristic
184C4×Dic47C22⋊C411yescharacteristic
185C4×Dic47C22×C411yescharacteristic
186C4×Dic47C22×C411yescharacteristic
187C4×Dic47C2×D411yescharacteristic
188C4×Dic47C2×D411yescharacteristic
189C4×Dic47C2×D411yescharacteristic
190C4×Dic47C2×D411yescharacteristic
191#10625329—42no—
192C2×C47⋊C8—22no—
193C2×C47⋊C8—22no—
194C8×D47—44no—
195C23×C47C2×C22⋊C411yescharacteristic
196C22×D47—22no—
197C22×D47—22no—
198C22×D47—41no—
199C22×D47C2×C22⋊C411yescharacteristic
200C22×D47C2×C22⋊C411yescharacteristic
201Q8×C47—42no—
202Q8×C47—42no—
203Q8×C47—44no—
204C2×C4×C47—22no—
205C2×C4×C47—22no—
206C2×C4×C47—24no—
207C2×C4×C47—44no—
208C2×C4×C47C2×C22⋊C411yescharacteristic
209C2×C4×C47C2×C22⋊C411yescharacteristic
210C2×C4×C47C2×C22⋊C411yescharacteristic
211C2×C4×C47#13512yes—
212C2×C4×C47#13512yes—
213C2×C4×C47C23×C411yescharacteristic
214C2×C4×C47C22×D411yescharacteristic
215C2×C4×C47C22×D411yescharacteristic
216C2×Dic47—22no—
217C2×Dic47—22no—
218C2×Dic47—24no—
219C2×Dic47—41no—
220C2×Dic47—44no—
221C2×Dic47C2×C22⋊C411yescharacteristic
222C2×Dic47C2×C22⋊C411yescharacteristic
223C2×Dic47C2×C22⋊C411yescharacteristic
224C2×Dic47C2×C22⋊C411yescharacteristic
225C2×Dic47C2×C22⋊C411yescharacteristic
226C2×Dic47C2×C22⋊C411yescharacteristic
227C4×D47—22no—
228C4×D47—21no—
229C4×D47—21no—
230C4×D47—21no—
231C4×D47—21no—
232C4×D47—22no—
233C4×D47—22no—
234C4×D47—22no—
235C4×D47—22no—
236C4×D47—22no—
237C4×D47—22no—
238C4×D47—41no—
239C4×D47—41no—
240C4×D47#10114yes—
241Dic94—44no—
242Dic94—44no—
243C8×C47—42no—
244C8×C47—42no—
245C47⋊C8—42no—
246C47⋊C8—42no—
247#17715—471nomaximal, Sylow
248C22×C47—21no—
249C22×C47—21no—
250C22×C47#41612yes—
251C22×C47C22×C22⋊C411yescharacteristic
252C2×D47—21no—
253C2×D47—21no—
254C2×D47—21no—
255C2×D47—21no—
256C2×D47—21no—
257C2×D47—44no—
258C2×D47C2×#10112yes—
259C4×C47—21no—
260C4×C47—21no—
261C4×C47—44no—
262C4×C47C2×#10112yes—
263C4×C47#41612yes—
264C4×C47C2×#13511yescharacteristic
265C4×C47C2×#13511yescharacteristic
266C4×C47#57512yes—
267Dic47—21no—
268Dic47—21no—
269Dic47—21no—
270Dic47—21no—
271Dic47—21no—
272Dic47—44no—
273Dic47C2×#10112yes—
274C2×#533—471no—
275#2920—471no—
276C2×#431—471no—
277#1552—474no—
278#1549—471no—
279#1549—471no—
280#1546—471no—
281#1546D4711yescharacteristic, complemented
282#1544—474no—
283C2×C47—22no—
284C2×C47C22×#10111yescharacteristic
285C2×C47C2×#41611yescharacteristic
286C2×C47#289611yescharacteristic
287D47—22no—
288D47—41no—
289#533—472no—
290#533—472no—
291#533—472no—
292#533—472no—
293#533—941no—
294#533—941no—
295#533—941no—
296#533—941no—
297C2×C4⋊Q8—471no—
298C2×C4⋊Q8C2×D4711yescharacteristic
299C2×#125—472no—
300C2×C22⋊Q8—941no—
301C2×C22⋊Q8—941no—
302C2×C4×Q8—941no—
303C2×C4×Q8—941no—
304C2×#117—472no—
305C22×C42—471no—
306#432—472no—
307#432—941no—
308#431—471no—
309#431—472no—
310#431—472no—
311#431—941no—
312#431—941no—
313#431—941no—
314#431C2×D4711yescharacteristic
315#423—471no—
316#423C2×D4711yescharacteristic
317C2×C4⋊C8—472no—
318#407—941no—
319C2×C22⋊C8—941no—
320C4×M16—941no—
321C2×C4×C8—941no—
322#332—474no—
323#332—474no—
324#332—474no—
325#332C2×D4714yescomplemented
326C47#1771511yescharacteristic, complemented, Sylow
327C22×Q8—21no—
328C22×Q8—941no—
329C23×C4—471no—
330C23×C4—471no—
331C23×C4—471no—
332C2×M16—21no—
333C2×M16—941no—
334C22×C8—942no—
335C4⋊Q8—21no—
336C4⋊Q8—21no—
337C4⋊Q8—472no—
338C4⋊Q8—472no—
339C4⋊Q8—941no—
340C4⋊Q8—941no—
341C4⋊Q8C4×D4712yes—
342C4⋊Q8C22×D4712yes—
343#125—474no—
344#125—474no—
345#125—942no—
346#125—942no—
347C22⋊Q8—942no—
348C22⋊Q8—942no—
349C22⋊Q8—942no—
350C22⋊Q8—942no—
351C22⋊Q8—942no—
352C22⋊Q8—942no—
353C22⋊Q8—942no—
354C22⋊Q8—942no—
355C4×Q8—942no—
356C4×Q8—942no—
357C4×Q8—942no—
358C4×Q8—942no—
359C4×Q8—942no—
360C4×Q8—942no—
361C4×Q8—942no—
362C4×Q8—942no—
363#117—474no—
364#117—474no—
365#117—942no—
366#117—942no—
367C2×C4⋊C4—21no—
368C2×C4⋊C4—472no—
369C2×C4⋊C4—942no—
370C2×C4⋊C4—942no—
371C2×C4⋊C4—941no—
372C2×C4⋊C4C4×D4712yes—
373C2×C22⋊C4—942no—
374C2×C42—471no—
375C2×C42—471no—
376C2×C42—471no—
377C2×C42—471no—
378C2×C42—471no—
379C2×C42—471no—
380C2×C42—471no—
381C2×C42—471no—
382C2×C42—471no—
383C2×C42—471no—
384C2×C42—471no—
385C2×C42—942no—
386C2×C42C22×D4711yescharacteristic
387C4⋊C8—21no—
388C4⋊C8—472no—
389C4⋊C8—472no—
390C4⋊C8—942no—
391C4⋊C8—941no—
392C4⋊C8C22×D4712yes—
393C4⋊C8C22×D4712yes—
394C22⋊C8—21no—
395C22⋊C8—942no—
396C22⋊C8—942no—
397C22⋊C8—942no—
398C22⋊C8—941no—
399#97—942no—
400C4×C8—21no—
401C4×C8—942no—
402C4×C8—942no—
403C4×C8—941no—
404C24—471no—
405C2×Q8—22no—
406C2×Q8—22no—
407C2×Q8—42no—
408C2×Q8—42no—
409C2×Q8—942no—
410C2×Q8—942no—
411C2×Q8—1884no—
412C22×C4—22no—
413C22×C4—471no—
414C22×C4—471no—
415C22×C4—471no—
416C22×C4—471no—
417C22×C4—471no—
418C22×C4—471no—
419C22×C4—471no—
420C22×C4—471no—
421C22×C4—471no—
422C22×C4—471no—
423C22×C4—471no—
424C22×C4—471no—
425C22×C4—471no—
426C22×C4—471no—
427C22×C4—471no—
428C22×C4—944no—
429C22×C4—942no—
430C22×C4—942no—
431C22×C4—942no—
432C22×C4—1882no—
433C22×C4C2×C4×D4711yescharacteristic
434C22×C4D4×D4711yescharacteristic
435C22×C4D4×D4711yescharacteristic
436M16—42no—
437M16—1882no—
438C2×C8—22no—
439C2×C8—22no—
440C2×C8—942no—
441C2×C8—942no—
442C2×C8—1884no—
443C4⋊C4—22no—
444C4⋊C4—22no—
445C4⋊C4—22no—
446C4⋊C4—474no—
447C4⋊C4—944no—
448C4⋊C4—944no—
449C4⋊C4—944no—
450C4⋊C4—944no—
451C4⋊C4—942no—
452C4⋊C4—942no—
453C4⋊C4—942no—
454C4⋊C4C2×C4×D4714yes—
455C22⋊C4—944no—
456C22⋊C4—944no—
457C42—471no—
458C42—471no—
459C42—471no—
460C42—471no—
461C42—471no—
462C42—471no—
463C42—471no—
464C42—471no—
465C42—471no—
466C42—471no—
467C42—471no—
468C42—471no—
469C42—944no—
470C42—944no—
471C42C2×C4×D4711yescharacteristic
472C42D4×D4711yescharacteristic
473C42D4×D4711yescharacteristic
474C42C23×D4711yescharacteristic
475C23—471no—
476C23—471no—
477C23—942no—
478C23—942no—
479C23—1881no—
480C23C22⋊C4×D4711yescharacteristic
481Q8—44no—
482Q8—42no—
483Q8—42no—
484Q8—1884no—
485Q8—1884no—
486C2×C4—22no—
487C2×C4—22no—
488C2×C4—24no—
489C2×C4—44no—
490C2×C4—471no—
491C2×C4—474no—
492C2×C4—471no—
493C2×C4—471no—
494C2×C4—471no—
495C2×C4—471no—
496C2×C4—471no—
497C2×C4—942no—
498C2×C4—942no—
499C2×C4—942no—
500C2×C4—942no—
501C2×C4—942no—
502C2×C4—942no—
503C2×C4—942no—
504C2×C4—941no—
505C2×C4—941no—
506C2×C4—941no—
507C2×C4—941no—
508C2×C4—942no—
509C2×C4—942no—
510C2×C4—944no—
511C2×C4—1881no—
512C2×C4—1881no—
513C2×C4—1881no—
514C2×C4—1884no—
515C2×C4C22⋊C4×D4711yescharacteristic
516C2×C4C22⋊C4×D4711yescharacteristic
517C2×C4C22⋊C4×D4711yescharacteristic
518C2×C4#1309404312yes—
519C2×C4#1309404512yes—
520C2×C4C22×C4×D4711yescharacteristic
521C2×C4C2×D4×D4711yescharacteristic
522C2×C4C2×D4×D4711yescharacteristic
523C8—42no—
524C8—42no—
525C8—1882no—
526C8—1882no—
527C22—21no—
528C22—21no—
529C22—472no—
530C22—941no—
531C22—941no—
532C22—941no—
533C22—941no—
534C22—941no—
535C22—1884no—
536C22#1568911612yes—
537C22C2×C22⋊C4×D4711yescharacteristic
538C4—21no—
539C4—21no—
540C4—44no—
541C4—472no—
542C4—941no—
543C4—941no—
544C4—941no—
545C4—941no—
546C4—941no—
547C4—1884no—
548C4#101×D4712yes—
549C4#1568911612yes—
550C4C2×#1309404311yescharacteristic
551C4C2×#1309404511yescharacteristic
552C4#1568992212yes—
553C2—22no—
554C2—942no—
555C2—1881no—
556C2C2×#101×D4711yescharacteristic
557C2C2×#1568911611yescharacteristic
558C2#2025190611yescharacteristic
subgroupG/H
centerC22C2×C22⋊C4×D47
commutatorC2×C4×C47C23×C4
FrattiniC42C23×D47
FittingC47×#1546C2
socleC22×C47C22×C22⋊C4
prime radicalC23×D47C22⋊C4
cosocleC42×C47C24
hypercenter#1546D47
hypocenterC47#17715
LCSG ▷ C2×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1546

Complements

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

NH ≅ G/Ntype
4. #20242085C2semidirect
5. #20242074C2semidirect
6. #20242072C2semidirect
7. #20242060C2semidirect
8. #20242051C2semidirect
9. C47×#1546C2semidirect
40. #15688870C22semidirect
41. #15688869C22semidirect
42. #15688868C22semidirect
46. C47×#332C22semidirect
281. #1546D47semidirect
325. #332C2×D47semidirect
326. C47#17715semidirect

Maximal subgroups

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

subgroupindex
C2×#156896782
#202520202
#431×D472
#202420852
#202420742
#202420722
#202420602
#202420512
C47×#15462
#1771547