Finite Group Databasebeta

← #26216296

Subgroups — #26216296

Order 12032.

1326 conjugacy classes of subgroups.

The subgroup lattice has 558 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×Q8×D47C211yesmaximal, characteristic
2C2×#15689175C211yesmaximal, characteristic
3#20245781C211yesmaximal, characteristic
4#20243473C212yesmaximal
5#20243404C211yesmaximal, characteristic, complemented
6#20243403C211yesmaximal, characteristic, complemented
7#20243400C211yesmaximal, characteristic, complemented
8#20243321C212yesmaximal
9#103×Dic47C212yesmaximal, complemented
10#20243207C212yesmaximal, complemented
11C47×#1805C211yesmaximal, characteristic, complemented
12C22×Q8×D47C2211yescharacteristic
13C2×Q8×Dic47C2211yescharacteristic
14C2×C4×Dic94—21no—
15C2×C4×Dic94C2211yescharacteristic
16C2×#13093989—21no—
17C2×#13093989C2211yescharacteristic
18C4×Q8×D47—24no—
19C4×Q8×D47C418yescomplemented
20C2×C4⋊C4×D47—21no—
21C2×C4⋊C4×D47C2211yescharacteristic
22C2×C42×D47—21no—
23C2×C42×D47C2211yescharacteristic
24#15689175—24no—
25C2×C4⋊(C47⋊C8)C2211yescharacteristic
26C2×#13093949C2212yes—
27C2×#13093948C2212yes—
28C2×#13093947C2212yes—
29C2×(C2×D47)⋊C8C2212yes—
30C2×C8×Dic47C2212yes—
31#15688990—21no—
32#15688989C2211yescharacteristic
33#15688985—21no—
34#15688980C2212yes—
35C4⋊C4×Dic47C2212yes—
36#15688970—21no—
37#15688968—21no—
38#15688886C2212yes—
39#15688885C2212yes—
40#15688885C2212yes—
41C2×C4×Q8×C47C2211yescharacteristic
42C2×C4⋊C8×C47C2211yescharacteristic
43C2×#103×C47C2212yes—
44C47×#382C2211yescharacteristic
45C47×#340C2212yes—
46C22×Dic94—21no—
47C22×Dic94C2311yescharacteristic
48C2×Q8×D47—22no—
49C2×Q8×D47—22no—
50C2×Q8×D47—22no—
51C2×Q8×D47—22no—
52C2×Q8×D47—22no—
53C2×Q8×D47C2×C414yes—
54C22×C4×D47—21no—
55C22×C4×D47—21no—
56C22×C4×D47D411yescharacteristic
57C22×C4×D47D411yescharacteristic
58C22×C4×D47C2311yescharacteristic
59C22×C47⋊C8C2312yes—
60Q8×Dic47—22no—
61Q8×Dic47C2×C414yes—
62C4×Dic94—22no—
63C4×Dic94—28no—
64C4×Dic94C2×C414yes—
65#13093989—28no—
66#13093989—22no—
67#13093989C2×C414yes—
68C2×C4⋊Dic47—21no—
69C2×C4⋊Dic47C2312yes—
70C2×C4⋊Dic47C2311yescharacteristic
71C2×Dic47⋊C4—21no—
72C2×Dic47⋊C4—21no—
73C2×Dic47⋊C4—21no—
74C4⋊C4×D47—28no—
75C4⋊C4×D47—22no—
76C4⋊C4×D47C2×C414yes—
77C2×(C2×D47)⋊C4—22no—
78C2×C4×Dic47—21no—
79C2×C4×Dic47—21no—
80C2×C4×Dic47—21no—
81C2×C4×Dic47—22no—
82C2×C4×Dic47D411yescharacteristic
83C2×C4×Dic47D411yescharacteristic
84C2×C4×Dic47C2311yescharacteristic
85C42×D47—28no—
86C42×D47C2×C414yes—
87C42×D47D414yes—
88C4⋊(C47⋊C8)—22no—
89#13093949—24no—
90#13093948—24no—
91#13093947—24no—
92(C2×D47)⋊C8—24no—
93C8×Dic47—24no—
94C47⋊#95—21no—
95C47⋊#95—21no—
96C47⋊#95—22no—
97C47⋊#95—22no—
98C22×Q8×C47C2311yescharacteristic
99C22×C8×C47C2312yes—
100C4×Q8×C47—22no—
101C4×Q8×C47C2×C414yes—
102C2×C4⋊C4×C47—21no—
103C2×C4⋊C4×C47C2312yes—
104C2×C4⋊C4×C47C2311yescharacteristic
105C2×C42×C47—21no—
106C2×C42×C47C2311yescharacteristic
107C4⋊C8×C47—22no—
108#103×C47—24no—
109#95×C47—21no—
110#95×C47—21no—
111C23×D47C2×D411yescharacteristic
112C2×Q8×C47—21no—
113C2×Q8×C47—21no—
114C2×Q8×C47—21no—
115C2×Q8×C47C22×C412yes—
116C2×Q8×C47#4112yes—
117C2×Q8×C47#4112yes—
118C22×C4×C47—21no—
119C22×C4×C47—21no—
120C22×C4×C47—22no—
121C22×C4×C47C2×D411yescharacteristic
122C22×C4×C47C2×D411yescharacteristic
123C22×C4×C47C2411yescharacteristic
124C22×Dic47—21no—
125C22×Dic47—21no—
126C22×Dic47—22no—
127C22×Dic47C2×D411yescharacteristic
128C22×Dic47C2×D411yescharacteristic
129C22×Dic47C2×D411yescharacteristic
130Q8×D47—28no—
131Q8×D47—28no—
132Q8×D47—28no—
133Q8×D47—28no—
134C2×C4×D47—24no—
135C2×C4×D47—24no—
136C2×C4×D47—24no—
137C2×C4×D47—24no—
138C2×C4×D47—21no—
139C2×C4×D47—21no—
140C2×C4×D47—21no—
141C2×C4×D47—21no—
142C2×C4×D47—24no—
143C2×C4×D47—24no—
144C2×C4×D47—44no—
145C2×C4×D47C22⋊C414yes—
146C2×C4×D47C22⋊C414yes—
147C2×C4×D47C22×C412yes—
148C2×C4×D47C2×D412yes—
149C2×Dic94—24no—
150C2×Dic94—24no—
151C2×Dic94—21no—
152C2×Dic94—21no—
153C2×Dic94—21no—
154C2×Dic94—24no—
155C2×Dic94C22×C412yes—
156C2×Dic94#4112yes—
157C2×Dic94#4112yes—
158C2×C8×C47—22no—
159C2×C8×C47—22no—
160C2×C8×C47—22no—
161C4⋊C4×C47—24no—
162C4⋊C4×C47—21no—
163C4⋊C4×C47—22no—
164C4⋊C4×C47—22no—
165C4⋊C4×C47C22×C412yes—
166C42×C47—24no—
167C42×C47C22×C412yes—
168C42×C47C2×D412yes—
169(C2×D47)⋊C4—44no—
170C4⋊Dic47—21no—
171C4⋊Dic47—24no—
172C4⋊Dic47—22no—
173C4⋊Dic47—22no—
174C4⋊Dic47C22×C412yes—
175Dic47⋊C4—24no—
176Dic47⋊C4—24no—
177Dic47⋊C4—24no—
178C4×Dic47—24no—
179C4×Dic47—24no—
180C4×Dic47—24no—
181C4×Dic47—44no—
182C4×Dic47C22⋊C414yes—
183C4×Dic47C22⋊C414yes—
184C4×Dic47C22×C412yes—
185C4×Dic47C2×D412yes—
186C2×C47⋊C8—22no—
187C2×C47⋊C8—22no—
188C2×C47⋊C8—22no—
189C23×C47C22×D411yescharacteristic
190C22×D47—21no—
191C22×D47—21no—
192C22×D47—22no—
193C22×D47C2×C22⋊C412yes—
194C22×D47#13514yes—
195Q8×C47—22no—
196Q8×C47—22no—
197Q8×C47—22no—
198Q8×C47#11714yes—
199C2×C4×C47—22no—
200C2×C4×C47—22no—
201C2×C4×C47—22no—
202C2×C4×C47—22no—
203C2×C4×C47—22no—
204C2×C4×C47—22no—
205C2×C4×C47—42no—
206C2×C4×C47—42no—
207C2×C4×C47—42no—
208C2×C4×C47—42no—
209C2×C4×C47C2×C22⋊C412yes—
210C2×C4×C47C2×C22⋊C412yes—
211C2×C4×C47C23×C411yescharacteristic
212C2×C4×C47C22×D411yescharacteristic
213C2×C4×C47C2×#4111yescharacteristic
214C2×C4×C47C2×#4111yescharacteristic
215C2×C4×C47C2×#4111yescharacteristic
216C2×C4×C47C2×#4111yescharacteristic
217C2×Dic47—22no—
218C2×Dic47—21no—
219C2×Dic47—21no—
220C2×Dic47—22no—
221C2×Dic47—22no—
222C2×Dic47—22no—
223C2×Dic47—22no—
224C2×Dic47—42no—
225C2×Dic47—42no—
226C2×Dic47—42no—
227C2×Dic47—42no—
228C2×Dic47C2×C22⋊C412yes—
229C2×Dic47C2×C22⋊C412yes—
230C2×Dic47C2×C22⋊C412yes—
231C2×Dic47#13714yes—
232C4×D47—216no—
233C4×D47—28no—
234C4×D47—22no—
235C4×D47—22no—
236C4×D47—22no—
237C4×D47—22no—
238C4×D47—28no—
239C4×D47—48no—
240Dic94—216no—
241Dic94—22no—
242Dic94—22no—
243Dic94—22no—
244Dic94#11714yes—
245C8×C47—42no—
246C47⋊C8—42no—
247#20735—471nomaximal, Sylow
248C22×C47C22×C22⋊C411yescharacteristic
249C22×C47#52511yescharacteristic
250C22×C47#52511yescharacteristic
251C22×C47C2×#13512yes—
252C22×C47C2×#13712yes—
253C2×D47—24no—
254C2×D47—24no—
255C2×D47—22no—
256C2×D47—22no—
257C2×D47#41614yes—
258C4×C47—22no—
259C4×C47—22no—
260C4×C47—24no—
261C4×C47—42no—
262C4×C47—82no—
263C4×C47C2×#11711yescharacteristic
264C4×C47C2×#11711yescharacteristic
265C4×C47#52511yescharacteristic
266C4×C47#52511yescharacteristic
267Dic47—22no—
268Dic47—22no—
269Dic47—24no—
270Dic47—42no—
271Dic47—82no—
272Dic47#41914yes—
273C22×C4×Q8—471no—
274C2×#431—471no—
275#2283—471no—
276#1823—472no—
277#1805—471no—
278#1805—471no—
279#1805—471no—
280#1805D4711yescharacteristic, complemented
281#1787—472no—
282C4×#103—472no—
283C4×#103—472no—
284C2×C47#228011yescharacteristic
285C2×C47C2×#41611yescharacteristic
286C2×C47C2×#41911yescharacteristic
287C2×C47#310812yes—
288C2×C47#310812yes—
289D47—24no—
290C23×Q8—471no—
291C2×C4×Q8—471no—
292C2×C4×Q8—478no—
293C2×C4×Q8—471no—
294C2×C4×Q8—471no—
295C2×C4×Q8—941no—
296C2×C4×Q8—941no—
297C2×C4×Q8—944no—
298C2×C4×Q8C2×D4711yescharacteristic
299C22×C4⋊C4—471no—
300C22×C4⋊C4—941no—
301C22×C42—471no—
302C22×C42—941no—
303#431—944no—
304C2×C4⋊C8—471no—
305C2×C4⋊C8C2×D4711yescharacteristic
306C2×#103—472no—
307C2×#103—472no—
308C2×#103—472no—
309C2×#103C2×D4712yes—
310C2×C22⋊C8—472no—
311C2×C4×C8—472no—
312#383—941no—
313#382—471no—
314#382C2×D4711yescharacteristic
315#381—941no—
316#380—472no—
317C4×C4⋊C4—472no—
318C4×C22⋊C4—941no—
319#376—941no—
320#340—472no—
321#340—472no—
322#340—472no—
323#340C2×D4712yes—
324C47#2073511yescharacteristic, complemented, Sylow
325C22×Q8—474no—
326C22×Q8—471no—
327C22×Q8—941no—
328C22×Q8—942no—
329C22×Q8—942no—
330C22×Q8—942no—
331C22×Q8—942no—
332C22×Q8—942no—
333C22×Q8C22×D4711yescharacteristic
334C23×C4—471no—
335C23×C4—471no—
336C23×C4—471no—
337C23×C4—941no—
338C23×C4—941no—
339C22×C8—472no—
340C22×C8C22×D4712yes—
341C4×Q8—22no—
342C4×Q8—474no—
343C4×Q8—474no—
344C4×Q8—474no—
345C4×Q8—948no—
346C4×Q8—948no—
347C4×Q8—942no—
348C4×Q8—942no—
349C4×Q8—942no—
350C4×Q8C4×D4714yes—
351C2×C4⋊C4—21no—
352C2×C4⋊C4—471no—
353C2×C4⋊C4—474no—
354C2×C4⋊C4—472no—
355C2×C4⋊C4—942no—
356C2×C4⋊C4—948no—
357C2×C4⋊C4—941no—
358C2×C4⋊C4—941no—
359C2×C4⋊C4—941no—
360C2×C4⋊C4—941no—
361C2×C4⋊C4C22×D4712yes—
362C2×C4⋊C4C22×D4711yescharacteristic
363C2×C22⋊C4—942no—
364C2×C42—21no—
365C2×C42—471no—
366C2×C42—471no—
367C2×C42—471no—
368C2×C42—474no—
369C2×C42—474no—
370C2×C42—948no—
371C2×C42—941no—
372C2×C42—941no—
373C2×C42—941no—
374C2×C42—942no—
375C2×C42C22×D4711yescharacteristic
376C4⋊C8—22no—
377C4⋊C8—942no—
378#103—24no—
379#103—944no—
380#103—944no—
381#103—944no—
382C22⋊C8—944no—
383C4×C8—944no—
384#95—21no—
385#95—21no—
386#95—942no—
387#95—942no—
388#95—941no—
389#95—941no—
390C24—471no—
391C2×Q8—21no—
392C2×Q8—21no—
393C2×Q8—21no—
394C2×Q8—472no—
395C2×Q8—472no—
396C2×Q8—472no—
397C2×Q8—944no—
398C2×Q8—944no—
399C2×Q8—944no—
400C2×Q8—948no—
401C2×Q8—948no—
402C2×Q8—948no—
403C2×Q8—948no—
404C2×Q8—941no—
405C2×Q8—941no—
406C2×Q8—941no—
407C2×Q8C2×C4×D4712yes—
408C2×Q8#1062535212yes—
409C2×Q8#1062535412yes—
410C22×C4—21no—
411C22×C4—21no—
412C22×C4—22no—
413C22×C4—471no—
414C22×C4—471no—
415C22×C4—471no—
416C22×C4—474no—
417C22×C4—474no—
418C22×C4—472no—
419C22×C4—472no—
420C22×C4—944no—
421C22×C4—944no—
422C22×C4—944no—
423C22×C4—944no—
424C22×C4—944no—
425C22×C4—944no—
426C22×C4—941no—
427C22×C4—941no—
428C22×C4—941no—
429C22×C4—941no—
430C22×C4—941no—
431C22×C4—941no—
432C22×C4—942no—
433C22×C4—1884no—
434C22×C4D4×D4711yescharacteristic
435C22×C4D4×D4711yescharacteristic
436C22×C4C23×D4711yescharacteristic
437C2×C8—22no—
438C2×C8—22no—
439C2×C8—22no—
440C2×C8—942no—
441C2×C8—942no—
442C2×C8—942no—
443C4⋊C4—24no—
444C4⋊C4—21no—
445C4⋊C4—22no—
446C4⋊C4—22no—
447C4⋊C4—472no—
448C4⋊C4—944no—
449C4⋊C4—944no—
450C4⋊C4—944no—
451C4⋊C4—944no—
452C4⋊C4—941no—
453C4⋊C4—942no—
454C4⋊C4—942no—
455C4⋊C4C2×C4×D4712yes—
456C22⋊C4—1884no—
457C42—24no—
458C42—474no—
459C42—474no—
460C42—472no—
461C42—472no—
462C42—944no—
463C42—944no—
464C42—944no—
465C42—1884no—
466C42C2×C4×D4712yes—
467C42D4×D4712yes—
468C23—472no—
469C23—474no—
470C23—941no—
471C23—941no—
472C23—942no—
473C23C2×D4×D4711yescharacteristic
474Q8—22no—
475Q8—22no—
476Q8—22no—
477Q8—474no—
478Q8—9416no—
479Q8—942no—
480Q8—942no—
481Q8—942no—
482Q8#1309398214yes—
483C2×C4—22no—
484C2×C4—22no—
485C2×C4—22no—
486C2×C4—22no—
487C2×C4—22no—
488C2×C4—22no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—472no—
494C2×C4—474no—
495C2×C4—472no—
496C2×C4—472no—
497C2×C4—948no—
498C2×C4—948no—
499C2×C4—9416no—
500C2×C4—942no—
501C2×C4—941no—
502C2×C4—942no—
503C2×C4—941no—
504C2×C4—942no—
505C2×C4—942no—
506C2×C4—942no—
507C2×C4—942no—
508C2×C4—942no—
509C2×C4—942no—
510C2×C4—942no—
511C2×C4—1888no—
512C2×C4—1882no—
513C2×C4—1882no—
514C2×C4—1882no—
515C2×C4—1882no—
516C2×C4C22⋊C4×D4712yes—
517C2×C4C22⋊C4×D4712yes—
518C2×C4C22×C4×D4711yescharacteristic
519C2×C4C2×D4×D4711yescharacteristic
520C2×C4#41×D4711yescharacteristic
521C2×C4#41×D4711yescharacteristic
522C2×C4C2×#1062535211yescharacteristic
523C2×C4C2×#1062535411yescharacteristic
524C8—42no—
525C8—1882no—
526C22—474no—
527C22—942no—
528C22—944no—
529C22—944no—
530C22—942no—
531C22C2×C22⋊C4×D4711yescharacteristic
532C22#1568963311yescharacteristic
533C22#1568963611yescharacteristic
534C22#135×D4712yes—
535C22#137×D4712yes—
536C4—24no—
537C4—22no—
538C4—22no—
539C4—42no—
540C4—82no—
541C4—474no—
542C4—942no—
543C4—944no—
544C4—942no—
545C4—3762no—
546C4—1882no—
547C4#117×D4711yescharacteristic
548C4C2×#1309398211yescharacteristic
549C4#1568963311yescharacteristic
550C4#1568963611yescharacteristic
551C2—944no—
552C2#2024577011yescharacteristic
553C2#416×D4711yescharacteristic
554C2#419×D4711yescharacteristic
555C2#2025346712yes—
556C2#2025347312yes—
subgroupG/H
centerC23C2×D4×D47
commutatorC2×C4×C47C23×C4
FrattiniC22×C4C23×D47
FittingC47×#1805C2
socleC23×C47C22×D4
prime radicalC23×D47C2×D4
cosocleC22×C4×C47C24
hypercenter#1805D47
hypocenterC47#20735
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1805

Complements

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

NH ≅ G/Ntype
5. #20243404C2semidirect
6. #20243403C2semidirect
7. #20243400C2semidirect
9. #103×Dic47C2semidirect
10. #20243207C2semidirect
11. C47×#1805C2semidirect
19. C4×Q8×D47C4semidirect
280. #1805D47semidirect
324. C47#20735semidirect

Maximal subgroups

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

subgroupindex
C2×C4×Q8×D472
C2×#156891752
#202457812
#202434732
#202434042
#202434032
#202434002
#202433212
#103×Dic472
#202432072
C47×#18052
#2073547