Finite Group Databasebeta

← #26183792

Subgroups — #26183792

Order 12032.

994 conjugacy classes of subgroups.

The subgroup lattice has 658 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689675C211yesmaximal, characteristic
2C2×#15689177C211yesmaximal, characteristic
3#20245783C211yesmaximal, characteristic
4#20241961C211yesmaximal, characteristic
5#20241960C211yesmaximal, characteristic
6#20241959C211yesmaximal, characteristic
7#20241937C214yesmaximal, complemented
8#20241898C214yesmaximal, complemented
9C47×#1520C211yesmaximal, characteristic
10#15689675—21no—
11#15689675—21no—
12#15689675—21no—
13#15689675—21no—
14#15689675C412yescomplemented
15#15689675C412yescomplemented
16#15689675C2212yes—
17#15689675C2212yes—
18C2×#13094023C411yescharacteristic, complemented
19C2×#13094022C411yescharacteristic, complemented
20C2×#13094014—21no—
21C2×#13094003C2211yescharacteristic
22C2×#13094001—21no—
23C2×Q8×Dic47C411yescharacteristic, complemented
24C2×#13093989—21no—
25C2×#13093989C411yescharacteristic, complemented
26C2×#13093980—21no—
27C22×C4×Dic47C2211yescharacteristic
28#15689177—21no—
29#15689177—21no—
30#15689177C2212yes—
31#15689177C2212yes—
32C2×Dic47⋊C8—21no—
33C2×C22⋊(C47⋊C8)C2211yescharacteristic
34C2×#13093934—21no—
35#15688990C2212yes—
36#15688988—21no—
37#15688984—21no—
38#15688983C2211yescharacteristic
39C4⋊C4×Dic47—21no—
40C22⋊C4×Dic47—21no—
41#15688968C2212yes—
42#15688861C2214yes—
43#15688860C2214yes—
44#15688859C2214yes—
45C2×C22⋊Q8×C47C2211yescharacteristic
46C2×C22⋊C8×C47C2211yescharacteristic
47C47×#380C2211yescharacteristic
48C47×#328C2214yes—
49C22×Dic94—21no—
50C22×Dic94C2×C411yescharacteristic
51C23×Dic47D411yescharacteristic
52C23×Dic47D411yescharacteristic
53C22×C47⋊C8—21no—
54#13094023—21no—
55#13094023—21no—
56#13094023C2×C412yes—
57#13094023C2×C412yes—
58#13094022—21no—
59#13094022—21no—
60#13094022C2×C412yes—
61#13094022C2×C412yes—
62#13094014—22no—
63#13094014—22no—
64#13094014—22no—
65#13094014—22no—
66#13094003—21no—
67#13094003—21no—
68#13094003C2×C412yes—
69#13094003C2312yes—
70#13094001—22no—
71#13094001—22no—
72#13094001—22no—
73#13094001—22no—
74Q8×Dic47—21no—
75Q8×Dic47—21no—
76Q8×Dic47C2×C412yes—
77Q8×Dic47C2×C412yes—
78#13093989—22no—
79#13093989—22no—
80#13093989—21no—
81#13093989—21no—
82#13093989—22no—
83#13093989—22no—
84#13093989C2×C412yes—
85#13093989C2×C412yes—
86#13093980—22no—
87#13093980—22no—
88#13093980—22no—
89#13093980—22no—
90C2×C4⋊Dic47—21no—
91C2×C4⋊Dic47—21no—
92C2×C4⋊Dic47C2×C411yescharacteristic
93C2×Dic47⋊C4—21no—
94C2×Dic47⋊C4—21no—
95C2×Dic47⋊C4—21no—
96C2×Dic47⋊C4—21no—
97C2×Dic47⋊C4—21no—
98C2×C22⋊Dic47—21no—
99C2×C22⋊Dic47—21no—
100C2×C4×Dic47—21no—
101C2×C4×Dic47—21no—
102C2×C4×Dic47—22no—
103C2×C4×Dic47C2×C411yescharacteristic
104C2×C4×Dic47C2×C411yescharacteristic
105C2×C4×Dic47C2×C411yescharacteristic
106C2×C4×Dic47C2×C411yescharacteristic
107C2×C4×Dic47D411yescharacteristic
108C2×C4×Dic47D411yescharacteristic
109C2×C4×Dic47D411yescharacteristic
110C2×C4×Dic47D411yescharacteristic
111C2×C4×Dic47D411yescharacteristic
112C2×C4×Dic47D411yescharacteristic
113C2×C4×Dic47C2311yescharacteristic
114C2×C4×Dic47C2311yescharacteristic
115Dic47⋊C8—22no—
116Dic47⋊C8—22no—
117C22⋊(C47⋊C8)—21no—
118C22⋊(C47⋊C8)C2312yes—
119#13093934—22no—
120#13093934—22no—
121C47⋊#95—22no—
122C47⋊#95—22no—
123C47⋊#95—22no—
124C47⋊#95C2312yes—
125C22×Q8×C47C2×C411yescharacteristic
126C23×C4×C47C2311yescharacteristic
127C22×C8×C47—21no—
128C22⋊Q8×C47—21no—
129C22⋊Q8×C47—21no—
130C22⋊Q8×C47C2×C412yes—
131C22⋊Q8×C47C2312yes—
132C2×C4⋊C4×C47—21no—
133C2×C4⋊C4×C47—21no—
134C2×C4⋊C4×C47C2×C411yescharacteristic
135C2×C22⋊C4×C47—21no—
136C2×C22⋊C4×C47—21no—
137C22⋊C8×C47—21no—
138C22⋊C8×C47C2312yes—
139#95×C47C2312yes—
140C24×C47C2×D411yescharacteristic
141C2×Q8×C47—21no—
142C2×Q8×C47—42no—
143C2×Q8×C47C22×C412yes—
144C22×C4×C47—21no—
145C22×C4×C47—21no—
146C22×C4×C47—22no—
147C22×C4×C47—42no—
148C22×C4×C47C22×C411yescharacteristic
149C22×C4×C47C22×C411yescharacteristic
150C22×C4×C47C2×D411yescharacteristic
151C22×C4×C47C2×D411yescharacteristic
152C22×C4×C47C2×D411yescharacteristic
153C22×C4×C47C2411yescharacteristic
154C22×Dic47—22no—
155C22×Dic47—22no—
156C22×Dic47—21no—
157C22×Dic47—21no—
158C22×Dic47—22no—
159C22×Dic47—42no—
160C22×Dic47C22⋊C411yescharacteristic
161C22×Dic47C22⋊C411yescharacteristic
162C22×Dic47C22⋊C411yescharacteristic
163C22×Dic47C22⋊C411yescharacteristic
164C22×Dic47C22⋊C411yescharacteristic
165C22×Dic47C22⋊C411yescharacteristic
166C22×Dic47C22⋊C411yescharacteristic
167C22×Dic47C22⋊C411yescharacteristic
168C22×Dic47C2×D411yescharacteristic
169C22×Dic47C2×D411yescharacteristic
170C22×Dic47C2×D411yescharacteristic
171C22×Dic47C2×D411yescharacteristic
172C2×Dic94—21no—
173C2×Dic94—22no—
174C2×Dic94—22no—
175C2×Dic94—44no—
176C2×Dic94—42no—
177C2×Dic94C22×C412yes—
178C2×C8×C47—22no—
179C2×C8×C47—42no—
180C4⋊C4×C47—22no—
181C4⋊C4×C47—21no—
182C4⋊C4×C47—22no—
183C4⋊C4×C47—42no—
184C4⋊C4×C47C22×C412yes—
185C22⋊C4×C47—22no—
186C22⋊C4×C47—22no—
187C22⋊C4×C47—42no—
188C22⋊Dic47—22no—
189C22⋊Dic47—22no—
190C22⋊Dic47—42no—
191C4⋊Dic47—21no—
192C4⋊Dic47—22no—
193C4⋊Dic47—22no—
194C4⋊Dic47—42no—
195C4⋊Dic47C22×C412yes—
196Dic47⋊C4—22no—
197Dic47⋊C4—22no—
198Dic47⋊C4—22no—
199Dic47⋊C4—22no—
200Dic47⋊C4—22no—
201Dic47⋊C4—22no—
202Dic47⋊C4—22no—
203Dic47⋊C4—22no—
204Dic47⋊C4—22no—
205Dic47⋊C4—22no—
206C4×Dic47—22no—
207C4×Dic47—22no—
208C4×Dic47—22no—
209C4×Dic47—22no—
210C4×Dic47—44no—
211C4×Dic47C22⋊C411yescharacteristic
212C4×Dic47C22⋊C411yescharacteristic
213C4×Dic47C22⋊C411yescharacteristic
214C4×Dic47C22⋊C411yescharacteristic
215C4×Dic47C22⋊C411yescharacteristic
216C4×Dic47C22⋊C411yescharacteristic
217C4×Dic47C22⋊C411yescharacteristic
218C4×Dic47C22⋊C411yescharacteristic
219C4×Dic47C22×C411yescharacteristic
220C4×Dic47C22×C411yescharacteristic
221C4×Dic47C22×C411yescharacteristic
222C4×Dic47C22×C411yescharacteristic
223C4×Dic47C2×D411yescharacteristic
224C4×Dic47C2×D411yescharacteristic
225C4×Dic47C2×D411yescharacteristic
226C4×Dic47C2×D411yescharacteristic
227C2×C47⋊C8—22no—
228C2×C47⋊C8—42no—
229C23×C47—41no—
230C23×C47—41no—
231C23×C47C2×C22⋊C411yescharacteristic
232C23×C47C2×C22⋊C411yescharacteristic
233C23×C47#13712yes—
234C23×C47#13712yes—
235C23×C47C22×D411yescharacteristic
236Q8×C47—42no—
237Q8×C47—42no—
238C2×C4×C47—22no—
239C2×C4×C47—22no—
240C2×C4×C47—22no—
241C2×C4×C47—22no—
242C2×C4×C47—22no—
243C2×C4×C47—41no—
244C2×C4×C47—41no—
245C2×C4×C47—41no—
246C2×C4×C47—42no—
247C2×C4×C47—42no—
248C2×C4×C47—42no—
249C2×C4×C47—42no—
250C2×C4×C47—42no—
251C2×C4×C47C2×C22⋊C411yescharacteristic
252C2×C4×C47C2×C22⋊C411yescharacteristic
253C2×C4×C47C23×C411yescharacteristic
254C2×C4×C47C22×D411yescharacteristic
255C2×Dic47—21no—
256C2×Dic47—21no—
257C2×Dic47—21no—
258C2×Dic47—21no—
259C2×Dic47—22no—
260C2×Dic47—22no—
261C2×Dic47—22no—
262C2×Dic47—22no—
263C2×Dic47—22no—
264C2×Dic47—22no—
265C2×Dic47—22no—
266C2×Dic47—22no—
267C2×Dic47—22no—
268C2×Dic47—41no—
269C2×Dic47—41no—
270C2×Dic47—41no—
271C2×Dic47—42no—
272C2×Dic47—42no—
273C2×Dic47—42no—
274C2×Dic47—42no—
275C2×Dic47—42no—
276C2×Dic47C2×C22⋊C411yescharacteristic
277C2×Dic47C2×C22⋊C411yescharacteristic
278C2×Dic47C2×C22⋊C411yescharacteristic
279C2×Dic47C2×C22⋊C411yescharacteristic
280C2×Dic47C2×C22⋊C411yescharacteristic
281C2×Dic47C2×C22⋊C411yescharacteristic
282C2×Dic47C2×C22⋊C411yescharacteristic
283C2×Dic47C2×C22⋊C411yescharacteristic
284Dic94—44no—
285Dic94—42no—
286Dic94—44no—
287Dic94—42no—
288C8×C47—42no—
289C47⋊C8—42no—
290#17596—471nomaximal, Sylow
291C22×C47—21no—
292C22×C47—21no—
293C22×C47—21no—
294C22×C47—21no—
295C22×C47—21no—
296C22×C47—21no—
297C22×C47—42no—
298C22×C47—42no—
299C22×C47#41912yes—
300C22×C47#41912yes—
301C22×C47C22×C22⋊C411yescharacteristic
302C22×C47C2×#13711yescharacteristic
303C22×C47C2×#13711yescharacteristic
304C4×C47—21no—
305C4×C47—21no—
306C4×C47—21no—
307C4×C47—42no—
308C4×C47—42no—
309C4×C47—82no—
310C4×C47#41012yes—
311Dic47—21no—
312Dic47—21no—
313Dic47—21no—
314Dic47—21no—
315Dic47—21no—
316Dic47—21no—
317Dic47—21no—
318Dic47—42no—
319Dic47—42no—
320Dic47—82no—
321Dic47#42112yes—
322C2×#533—471no—
323C2×#432—471no—
324#2283—471no—
325#1520—471no—
326#1520—471no—
327#1520—471no—
328#1520D4711yescharacteristic
329#1515—474no—
330#1506—474no—
331C2×C47—21no—
332C2×C47—21no—
333C2×C47—22no—
334C2×C47—22no—
335C2×C47C2×#41011yescharacteristic
336C2×C47C2×#41911yescharacteristic
337C2×C47C2×#42111yescharacteristic
338#533—472no—
339#533—472no—
340#533—472no—
341#533—472no—
342#533—941no—
343#533—941no—
344#533—941no—
345#533—941no—
346C2×C4⋊Q8—471no—
347C2×C4⋊Q8—471no—
348C2×#125—941no—
349C2×C22⋊Q8—471no—
350C2×C22⋊Q8—941no—
351C2×C22⋊Q8C2×D4711yescharacteristic
352C2×C4×Q8—471no—
353C2×C4×Q8—471no—
354C2×C4×Q8—941no—
355C2×#117—941no—
356C22×C42—471no—
357#432—472no—
358#432—472no—
359#432—941no—
360#432—941no—
361C2×C4⋊C8—941no—
362C2×C22⋊C8—471no—
363C2×C22⋊C8C2×D4711yescharacteristic
364C2×#97—941no—
365#383—472no—
366#382—941no—
367#381—941no—
368#380—471no—
369#380C2×D4711yescharacteristic
370C4×C4⋊C4—941no—
371C4×C22⋊C4—941no—
372#376—472no—
373#328—474no—
374#328—474no—
375#328—474no—
376#328C2×D4714yes—
377C47#1759611yescharacteristic, complemented, Sylow
378C22×Q8—471no—
379C22×Q8—941no—
380C22×Q8C4×D4711yescharacteristic
381C23×C4—471no—
382C23×C4—471no—
383C23×C4C22×D4711yescharacteristic
384C22×C8—21no—
385C22×C8—941no—
386C4⋊Q8—472no—
387C4⋊Q8—472no—
388C4⋊Q8—472no—
389C4⋊Q8—472no—
390C4⋊Q8—941no—
391C4⋊Q8—941no—
392C4⋊Q8—941no—
393C4⋊Q8—941no—
394#125—942no—
395#125—942no—
396#125—942no—
397#125—942no—
398C22⋊Q8—21no—
399C22⋊Q8—21no—
400C22⋊Q8—472no—
401C22⋊Q8—472no—
402C22⋊Q8—942no—
403C22⋊Q8—942no—
404C22⋊Q8—942no—
405C22⋊Q8—942no—
406C22⋊Q8—941no—
407C22⋊Q8—941no—
408C22⋊Q8C4×D4712yes—
409C22⋊Q8C22×D4712yes—
410C4×Q8—472no—
411C4×Q8—472no—
412C4×Q8—472no—
413C4×Q8—472no—
414C4×Q8—941no—
415C4×Q8—941no—
416C4×Q8—942no—
417C4×Q8—942no—
418C4×Q8—942no—
419C4×Q8—942no—
420C4×Q8—941no—
421C4×Q8—941no—
422#117—942no—
423#117—942no—
424#117—942no—
425#117—942no—
426C2×C4⋊C4—21no—
427C2×C4⋊C4—21no—
428C2×C4⋊C4—471no—
429C2×C4⋊C4—941no—
430C2×C4⋊C4—941no—
431C2×C4⋊C4—941no—
432C2×C4⋊C4—941no—
433C2×C4⋊C4—941no—
434C2×C4⋊C4—941no—
435C2×C4⋊C4—941no—
436C2×C4⋊C4C4×D4711yescharacteristic
437C2×C22⋊C4—21no—
438C2×C22⋊C4—21no—
439C2×C22⋊C4—941no—
440C2×C22⋊C4—941no—
441C2×C42—471no—
442C2×C42—471no—
443C2×C42—471no—
444C2×C42—471no—
445C2×C42—471no—
446C2×C42—471no—
447C2×C42—471no—
448C2×C42—471no—
449C2×C42—471no—
450C2×C42—471no—
451C2×C42—471no—
452C2×C42—471no—
453C2×C42—941no—
454C2×C42—941no—
455C2×C42—942no—
456C4⋊C8—942no—
457C4⋊C8—942no—
458C22⋊C8—21no—
459C22⋊C8—472no—
460C22⋊C8—941no—
461C22⋊C8C22×D4712yes—
462#97—942no—
463#97—942no—
464#95—472no—
465#95—942no—
466#95—942no—
467#95—942no—
468#95C22×D4712yes—
469C24D4×D4711yescharacteristic
470C2×Q8—21no—
471C2×Q8—42no—
472C2×Q8—472no—
473C2×Q8—942no—
474C2×Q8—942no—
475C2×Q8—941no—
476C2×Q8—1882no—
477C2×Q8—1884no—
478C2×Q8C2×C4×D4712yes—
479C22×C4—21no—
480C22×C4—21no—
481C22×C4—22no—
482C22×C4—42no—
483C22×C4—471no—
484C22×C4—471no—
485C22×C4—471no—
486C22×C4—471no—
487C22×C4—471no—
488C22×C4—471no—
489C22×C4—471no—
490C22×C4—471no—
491C22×C4—471no—
492C22×C4—471no—
493C22×C4—471no—
494C22×C4—471no—
495C22×C4—942no—
496C22×C4—942no—
497C22×C4—941no—
498C22×C4—941no—
499C22×C4—942no—
500C22×C4—1882no—
501C22×C4C2×C4×D4711yescharacteristic
502C22×C4C2×C4×D4711yescharacteristic
503C22×C4D4×D4711yescharacteristic
504C22×C4D4×D4711yescharacteristic
505C22×C4D4×D4711yescharacteristic
506C22×C4C23×D4711yescharacteristic
507C2×C8—22no—
508C2×C8—42no—
509C2×C8—942no—
510C2×C8—1882no—
511C4⋊C4—22no—
512C4⋊C4—21no—
513C4⋊C4—22no—
514C4⋊C4—42no—
515C4⋊C4—472no—
516C4⋊C4—942no—
517C4⋊C4—942no—
518C4⋊C4—942no—
519C4⋊C4—942no—
520C4⋊C4—942no—
521C4⋊C4—942no—
522C4⋊C4—942no—
523C4⋊C4—942no—
524C4⋊C4—942no—
525C4⋊C4—942no—
526C4⋊C4—942no—
527C4⋊C4—941no—
528C4⋊C4—942no—
529C4⋊C4—1882no—
530C4⋊C4C2×C4×D4712yes—
531C22⋊C4—22no—
532C22⋊C4—22no—
533C22⋊C4—42no—
534C22⋊C4—942no—
535C22⋊C4—942no—
536C22⋊C4—1882no—
537C42—471no—
538C42—471no—
539C42—471no—
540C42—471no—
541C42—471no—
542C42—471no—
543C42—471no—
544C42—471no—
545C42—471no—
546C42—471no—
547C42—471no—
548C42—471no—
549C42—471no—
550C42—471no—
551C42—471no—
552C42—471no—
553C42—942no—
554C42—942no—
555C42—942no—
556C42—942no—
557C42—1884no—
558C23—41no—
559C23—41no—
560C23C22⋊C4×D4711yescharacteristic
561C23C22⋊C4×D4711yescharacteristic
562C23#1309405312yes—
563C23#1309405612yes—
564C23C2×D4×D4711yescharacteristic
565Q8—42no—
566Q8—42no—
567Q8—1882no—
568Q8—1884no—
569Q8—1884no—
570Q8—1882no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—471no—
585C2×C4—471no—
586C2×C4—471no—
587C2×C4—471no—
588C2×C4—471no—
589C2×C4—471no—
590C2×C4—471no—
591C2×C4—471no—
592C2×C4—942no—
593C2×C4—942no—
594C2×C4—942no—
595C2×C4—942no—
596C2×C4—942no—
597C2×C4—942no—
598C2×C4—942no—
599C2×C4—941no—
600C2×C4—941no—
601C2×C4—941no—
602C2×C4—941no—
603C2×C4—942no—
604C2×C4—942no—
605C2×C4—1881no—
606C2×C4—1881no—
607C2×C4—1881no—
608C2×C4—1882no—
609C2×C4—1882no—
610C2×C4—1882no—
611C2×C4—1882no—
612C2×C4—1882no—
613C2×C4C22⋊C4×D4711yescharacteristic
614C2×C4C22⋊C4×D4711yescharacteristic
615C2×C4C22×C4×D4711yescharacteristic
616C2×C4C2×D4×D4711yescharacteristic
617C8—42no—
618C8—1882no—
619C22—21no—
620C22—21no—
621C22—21no—
622C22—21no—
623C22—21no—
624C22—21no—
625C22—42no—
626C22—42no—
627C22#1568913012yes—
628C22#1568913012yes—
629C22C2×C22⋊C4×D4711yescharacteristic
630C22C2×#1309405311yescharacteristic
631C22C2×#1309405611yescharacteristic
632C4—21no—
633C4—21no—
634C4—21no—
635C4—42no—
636C4—42no—
637C4—82no—
638C4—472no—
639C4—941no—
640C4—941no—
641C4—941no—
642C4—941no—
643C4—941no—
644C4—941no—
645C4—941no—
646C4—3762no—
647C4—1882no—
648C4—1882no—
649C4#1568909512yes—
650C2—21no—
651C2—21no—
652C2—22no—
653C2—22no—
654C2C2×#1568909511yescharacteristic
655C2C2×#1568913011yescharacteristic
656C2#421×D4711yescharacteristic
subgroupG/H
centerC22C2×C22⋊C4×D47
commutatorC2×C4×C47C23×C4
FrattiniC22×C4C23×D47
FittingC47×#1520C2
socleC22×C47C22×C22⋊C4
prime radicalC24×C47C2×D4
cosocleC22×C4×C47C24
hypercenter#1520D47
hypocenterC47#17596
LCSG ▷ C2×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1520

Complements

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

NH ≅ G/Ntype
7. #20241937C2semidirect
8. #20241898C2semidirect
14. #15689675C4semidirect
15. #15689675C4semidirect
18. C2×#13094023C4semidirect
19. C2×#13094022C4semidirect
23. C2×Q8×Dic47C4semidirect
25. C2×#13093989C4semidirect
377. C47#17596semidirect

Maximal subgroups

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

subgroupindex
C2×#156896752
C2×#156891772
#202457832
#202419612
#202419602
#202419592
#202419372
#202418982
C47×#15202
#1759647