Finite Group Databasebeta

← #26183651

Subgroups — #26183651

Order 12032.

1058 conjugacy classes of subgroups.

The subgroup lattice has 664 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689675C211yesmaximal, characteristic
2C2×#15689173C211yesmaximal, characteristic
3#20245790C211yesmaximal, characteristic
4#20241961C211yesmaximal, characteristic
5#20241960C211yesmaximal, characteristic
6#20241959C211yesmaximal, characteristic
7#20241950C214yesmaximal, complemented
8#20241885C214yesmaximal, 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—
27C2×#13093980—21no—
28C22×C4×Dic47C2211yescharacteristic
29#15689173—21no—
30#15689173—21no—
31#15689173C2212yes—
32#15689173C2212yes—
33C2×Dic47⋊C8—21no—
34C2×C22⋊(C47⋊C8)C2211yescharacteristic
35C2×C8×Dic47—21no—
36#15688992—21no—
37#15688987C2212yes—
38#15688983C2211yescharacteristic
39#15688981—21no—
40C4⋊C4×Dic47—21no—
41#15688974C2212yes—
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—
90#13093980—44no—
91C2×C4⋊Dic47—21no—
92C2×C4⋊Dic47—21no—
93C2×C4⋊Dic47C2×C411yescharacteristic
94C2×Dic47⋊C4—21no—
95C2×Dic47⋊C4—21no—
96C2×Dic47⋊C4—21no—
97C2×Dic47⋊C4—21no—
98C2×Dic47⋊C4—21no—
99C2×Dic47⋊C4—22no—
100C2×Dic47⋊C4—22no—
101C2×C22⋊Dic47—21no—
102C2×C22⋊Dic47—21no—
103C2×C4×Dic47—21no—
104C2×C4×Dic47—21no—
105C2×C4×Dic47—22no—
106C2×C4×Dic47C2×C411yescharacteristic
107C2×C4×Dic47C2×C411yescharacteristic
108C2×C4×Dic47C2×C411yescharacteristic
109C2×C4×Dic47C2×C411yescharacteristic
110C2×C4×Dic47D411yescharacteristic
111C2×C4×Dic47D411yescharacteristic
112C2×C4×Dic47D411yescharacteristic
113C2×C4×Dic47D411yescharacteristic
114C2×C4×Dic47D411yescharacteristic
115C2×C4×Dic47D411yescharacteristic
116C2×C4×Dic47C2311yescharacteristic
117C2×C4×Dic47C2311yescharacteristic
118Dic47⋊C8—22no—
119Dic47⋊C8—22no—
120C22⋊(C47⋊C8)—21no—
121C22⋊(C47⋊C8)C2312yes—
122C8×Dic47—22no—
123C8×Dic47—22no—
124C47⋊#95—22no—
125C47⋊#95C2312yes—
126C22×Q8×C47C2×C411yescharacteristic
127C23×C4×C47C2311yescharacteristic
128C22×C8×C47—21no—
129C22⋊Q8×C47—21no—
130C22⋊Q8×C47—21no—
131C22⋊Q8×C47C2×C412yes—
132C22⋊Q8×C47C2312yes—
133C2×C4⋊C4×C47—21no—
134C2×C4⋊C4×C47—21no—
135C2×C4⋊C4×C47C2×C411yescharacteristic
136C2×C22⋊C4×C47—21no—
137C2×C22⋊C4×C47—21no—
138C22⋊C8×C47—21no—
139C22⋊C8×C47C2312yes—
140#95×C47C2312yes—
141C24×C47C2×D411yescharacteristic
142C2×Q8×C47—21no—
143C2×Q8×C47—42no—
144C2×Q8×C47C22×C412yes—
145C22×C4×C47—21no—
146C22×C4×C47—21no—
147C22×C4×C47—22no—
148C22×C4×C47—42no—
149C22×C4×C47C22×C411yescharacteristic
150C22×C4×C47C22×C411yescharacteristic
151C22×C4×C47C2×D411yescharacteristic
152C22×C4×C47C2×D411yescharacteristic
153C22×C4×C47C2×D411yescharacteristic
154C22×C4×C47C2411yescharacteristic
155C22×Dic47—21no—
156C22×Dic47—21no—
157C22×Dic47—22no—
158C22×Dic47—42no—
159C22×Dic47C22⋊C411yescharacteristic
160C22×Dic47C22⋊C411yescharacteristic
161C22×Dic47C22⋊C411yescharacteristic
162C22×Dic47C22⋊C411yescharacteristic
163C22×Dic47C22⋊C411yescharacteristic
164C22×Dic47C22⋊C411yescharacteristic
165C22×Dic47C22⋊C411yescharacteristic
166C22×Dic47C22⋊C411yescharacteristic
167C22×Dic47SD1614yes—
168C22×Dic47Q1614yes—
169C22×Dic47C2×D411yescharacteristic
170C22×Dic47C2×D411yescharacteristic
171C22×Dic47C2×D411yescharacteristic
172C22×Dic47C2×D411yescharacteristic
173C2×Dic94—21no—
174C2×Dic94—22no—
175C2×Dic94—22no—
176C2×Dic94—44no—
177C2×Dic94—42no—
178C2×Dic94C22×C412yes—
179C2×C8×C47—22no—
180C2×C8×C47—42no—
181C4⋊C4×C47—22no—
182C4⋊C4×C47—21no—
183C4⋊C4×C47—22no—
184C4⋊C4×C47—42no—
185C4⋊C4×C47C22×C412yes—
186C22⋊C4×C47—22no—
187C22⋊C4×C47—22no—
188C22⋊C4×C47—42no—
189C22⋊Dic47—22no—
190C22⋊Dic47—22no—
191C22⋊Dic47—42no—
192C4⋊Dic47—21no—
193C4⋊Dic47—22no—
194C4⋊Dic47—22no—
195C4⋊Dic47—42no—
196C4⋊Dic47C22×C412yes—
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—
206Dic47⋊C4—22no—
207Dic47⋊C4—44no—
208Dic47⋊C4—44no—
209C4×Dic47—22no—
210C4×Dic47—22no—
211C4×Dic47—22no—
212C4×Dic47—22no—
213C4×Dic47—44no—
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×Dic47C22×C411yescharacteristic
224C4×Dic47C22×C411yescharacteristic
225C4×Dic47C22×C411yescharacteristic
226C4×Dic47C2×D411yescharacteristic
227C4×Dic47C2×D411yescharacteristic
228C4×Dic47C2×D411yescharacteristic
229C4×Dic47C2×D411yescharacteristic
230C2×C47⋊C8—22no—
231C2×C47⋊C8—42no—
232C23×C47—41no—
233C23×C47—41no—
234C23×C47C2×C22⋊C411yescharacteristic
235C23×C47C2×C22⋊C411yescharacteristic
236C23×C47C2×SD1612yes—
237C23×C47C2×Q1612yes—
238C23×C47C22×D411yescharacteristic
239Q8×C47—42no—
240Q8×C47—42no—
241C2×C4×C47—22no—
242C2×C4×C47—22no—
243C2×C4×C47—22no—
244C2×C4×C47—22no—
245C2×C4×C47—22no—
246C2×C4×C47—41no—
247C2×C4×C47—41no—
248C2×C4×C47—41no—
249C2×C4×C47—42no—
250C2×C4×C47—42no—
251C2×C4×C47—42no—
252C2×C4×C47—42no—
253C2×C4×C47—42no—
254C2×C4×C47C2×C22⋊C411yescharacteristic
255C2×C4×C47C2×C22⋊C411yescharacteristic
256C2×C4×C47C23×C411yescharacteristic
257C2×C4×C47C22×D411yescharacteristic
258C2×Dic47—22no—
259C2×Dic47—22no—
260C2×Dic47—22no—
261C2×Dic47—22no—
262C2×Dic47—22no—
263C2×Dic47—41no—
264C2×Dic47—41no—
265C2×Dic47—41no—
266C2×Dic47—42no—
267C2×Dic47—42no—
268C2×Dic47—42no—
269C2×Dic47—42no—
270C2×Dic47—42no—
271C2×Dic47#10314yes—
272C2×Dic47#10314yes—
273C2×Dic47#10314yes—
274C2×Dic47#10314yes—
275C2×Dic47C2×C22⋊C411yescharacteristic
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×SD1612yes—
284C2×Dic47C2×SD1612yes—
285C2×Dic47C2×Q1612yes—
286C2×Dic47C2×Q1612yes—
287Dic94—44no—
288Dic94—42no—
289Dic94—44no—
290Dic94—42no—
291C8×C47—42no—
292C47⋊C8—42no—
293#17575—471nomaximal, Sylow
294C22×C47—21no—
295C22×C47—21no—
296C22×C47—21no—
297C22×C47—21no—
298C22×C47—21no—
299C22×C47—21no—
300C22×C47—42no—
301C22×C47—42no—
302C22×C47C2×#10312yes—
303C22×C47C2×#10312yes—
304C22×C47C22×C22⋊C411yescharacteristic
305C22×C47C22×SD1611yescharacteristic
306C22×C47C22×Q1611yescharacteristic
307C4×C47—21no—
308C4×C47—21no—
309C4×C47—21no—
310C4×C47—42no—
311C4×C47—42no—
312C4×C47—82no—
313C4×C47#41012yes—
314Dic47—21no—
315Dic47—21no—
316Dic47—21no—
317Dic47—42no—
318Dic47—42no—
319Dic47—82no—
320Dic47C2×#10312yes—
321Dic47C2×#10312yes—
322Dic47C2×#10312yes—
323Dic47C2×#10312yes—
324Dic47#42112yes—
325C2×#533—471no—
326C2×#431—471no—
327#2285—471no—
328#1520—471no—
329#1520—471no—
330#1520—471no—
331#1520D4711yescharacteristic
332#1518—474no—
333#1503—474no—
334C2×C47—21no—
335C2×C47—21no—
336C2×C47—22no—
337C2×C47—22no—
338C2×C47C2×#41011yescharacteristic
339C2×C47C22×#10311yescharacteristic
340C2×C47C2×#42111yescharacteristic
341#533—472no—
342#533—472no—
343#533—472no—
344#533—472no—
345#533—941no—
346#533—941no—
347#533—941no—
348#533—941no—
349C2×C4⋊Q8—471no—
350C2×C4⋊Q8—471no—
351C2×#125—941no—
352C2×C22⋊Q8—471no—
353C2×C22⋊Q8—941no—
354C2×C22⋊Q8C2×D4711yescharacteristic
355C2×C4×Q8—471no—
356C2×C4×Q8—471no—
357C2×C4×Q8—941no—
358C2×#117—941no—
359C2×#117—941no—
360C22×C42—471no—
361#431—472no—
362#431—472no—
363#431—941no—
364#431—941no—
365C2×C4⋊C8—941no—
366C2×C22⋊C8—471no—
367C2×C22⋊C8C2×D4711yescharacteristic
368C2×C4×C8—941no—
369#384—941no—
370#382—472no—
371#380—471no—
372#380—941no—
373#380C2×D4711yescharacteristic
374C4×C4⋊C4—472no—
375C4×C4⋊C4—941no—
376#328—474no—
377#328—474no—
378#328—474no—
379#328C2×D4714yes—
380C47#1757511yescharacteristic, complemented, Sylow
381C22×Q8—471no—
382C22×Q8—941no—
383C22×Q8C4×D4711yescharacteristic
384C23×C4—471no—
385C23×C4—471no—
386C23×C4C22×D4711yescharacteristic
387C22×C8—21no—
388C22×C8—941no—
389C4⋊Q8—472no—
390C4⋊Q8—472no—
391C4⋊Q8—472no—
392C4⋊Q8—472no—
393C4⋊Q8—941no—
394C4⋊Q8—941no—
395C4⋊Q8—941no—
396C4⋊Q8—941no—
397#125—942no—
398#125—942no—
399#125—942no—
400#125—942no—
401C22⋊Q8—21no—
402C22⋊Q8—21no—
403C22⋊Q8—472no—
404C22⋊Q8—472no—
405C22⋊Q8—942no—
406C22⋊Q8—942no—
407C22⋊Q8—942no—
408C22⋊Q8—942no—
409C22⋊Q8—941no—
410C22⋊Q8—941no—
411C22⋊Q8C4×D4712yes—
412C22⋊Q8C22×D4712yes—
413C4×Q8—472no—
414C4×Q8—472no—
415C4×Q8—472no—
416C4×Q8—472no—
417C4×Q8—941no—
418C4×Q8—941no—
419C4×Q8—942no—
420C4×Q8—942no—
421C4×Q8—942no—
422C4×Q8—942no—
423C4×Q8—941no—
424C4×Q8—941no—
425#117—942no—
426#117—942no—
427#117—942no—
428#117—942no—
429#117—1884no—
430C2×C4⋊C4—21no—
431C2×C4⋊C4—21no—
432C2×C4⋊C4—471no—
433C2×C4⋊C4—941no—
434C2×C4⋊C4—941no—
435C2×C4⋊C4—941no—
436C2×C4⋊C4—941no—
437C2×C4⋊C4—941no—
438C2×C4⋊C4—941no—
439C2×C4⋊C4—942no—
440C2×C4⋊C4—942no—
441C2×C4⋊C4—941no—
442C2×C4⋊C4C4×D4711yescharacteristic
443C2×C22⋊C4—21no—
444C2×C22⋊C4—21no—
445C2×C22⋊C4—941no—
446C2×C22⋊C4—941no—
447C2×C42—471no—
448C2×C42—471no—
449C2×C42—471no—
450C2×C42—471no—
451C2×C42—471no—
452C2×C42—471no—
453C2×C42—471no—
454C2×C42—471no—
455C2×C42—471no—
456C2×C42—471no—
457C2×C42—471no—
458C2×C42—471no—
459C2×C42—941no—
460C2×C42—941no—
461C2×C42—942no—
462C4⋊C8—942no—
463C4⋊C8—942no—
464C22⋊C8—21no—
465C22⋊C8—472no—
466C22⋊C8—941no—
467C22⋊C8C22×D4712yes—
468C4×C8—942no—
469C4×C8—942no—
470#95—472no—
471#95—942no—
472#95C22×D4712yes—
473C24D4×D4711yescharacteristic
474C2×Q8—21no—
475C2×Q8—42no—
476C2×Q8—472no—
477C2×Q8—942no—
478C2×Q8—942no—
479C2×Q8—941no—
480C2×Q8—1882no—
481C2×Q8—1884no—
482C2×Q8C2×C4×D4712yes—
483C22×C4—21no—
484C22×C4—21no—
485C22×C4—22no—
486C22×C4—42no—
487C22×C4—471no—
488C22×C4—471no—
489C22×C4—471no—
490C22×C4—471no—
491C22×C4—474no—
492C22×C4—474no—
493C22×C4—471no—
494C22×C4—471no—
495C22×C4—471no—
496C22×C4—471no—
497C22×C4—471no—
498C22×C4—471no—
499C22×C4—471no—
500C22×C4—471no—
501C22×C4—941no—
502C22×C4—941no—
503C22×C4—942no—
504C22×C4—1882no—
505C22×C4C2×C4×D4711yescharacteristic
506C22×C4C2×C4×D4711yescharacteristic
507C22×C4D4×D4711yescharacteristic
508C22×C4D4×D4711yescharacteristic
509C22×C4D4×D4711yescharacteristic
510C22×C4C23×D4711yescharacteristic
511C2×C8—22no—
512C2×C8—42no—
513C2×C8—942no—
514C2×C8—1882no—
515C4⋊C4—22no—
516C4⋊C4—21no—
517C4⋊C4—22no—
518C4⋊C4—42no—
519C4⋊C4—472no—
520C4⋊C4—942no—
521C4⋊C4—942no—
522C4⋊C4—942no—
523C4⋊C4—942no—
524C4⋊C4—942no—
525C4⋊C4—942no—
526C4⋊C4—942no—
527C4⋊C4—942no—
528C4⋊C4—942no—
529C4⋊C4—942no—
530C4⋊C4—942no—
531C4⋊C4—941no—
532C4⋊C4—942no—
533C4⋊C4—1882no—
534C4⋊C4—1884no—
535C4⋊C4—1884no—
536C4⋊C4C2×C4×D4712yes—
537C22⋊C4—22no—
538C22⋊C4—22no—
539C22⋊C4—42no—
540C22⋊C4—942no—
541C22⋊C4—942no—
542C22⋊C4—1882no—
543C42—471no—
544C42—471no—
545C42—471no—
546C42—471no—
547C42—471no—
548C42—471no—
549C42—471no—
550C42—471no—
551C42—471no—
552C42—471no—
553C42—471no—
554C42—471no—
555C42—471no—
556C42—471no—
557C42—471no—
558C42—471no—
559C42—942no—
560C42—942no—
561C42—942no—
562C42—942no—
563C42—1884no—
564C23—41no—
565C23—41no—
566C23C22⋊C4×D4711yescharacteristic
567C23C22⋊C4×D4711yescharacteristic
568C23SD16×D4712yes—
569C23Q16×D4712yes—
570C23C2×D4×D4711yescharacteristic
571Q8—42no—
572Q8—42no—
573Q8—1882no—
574Q8—1884no—
575Q8—1884no—
576Q8—1882no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—471no—
591C2×C4—471no—
592C2×C4—471no—
593C2×C4—471no—
594C2×C4—471no—
595C2×C4—474no—
596C2×C4—474no—
597C2×C4—474no—
598C2×C4—474no—
599C2×C4—472no—
600C2×C4—472no—
601C2×C4—472no—
602C2×C4—472no—
603C2×C4—471no—
604C2×C4—471no—
605C2×C4—471no—
606C2×C4—942no—
607C2×C4—942no—
608C2×C4—942no—
609C2×C4—942no—
610C2×C4—942no—
611C2×C4—1881no—
612C2×C4—1881no—
613C2×C4—1881no—
614C2×C4—1882no—
615C2×C4—1882no—
616C2×C4—1882no—
617C2×C4—1882no—
618C2×C4—1882no—
619C2×C4C22⋊C4×D4711yescharacteristic
620C2×C4C22⋊C4×D4711yescharacteristic
621C2×C4C22×C4×D4711yescharacteristic
622C2×C4C2×D4×D4711yescharacteristic
623C8—42no—
624C8—1882no—
625C22—21no—
626C22—21no—
627C22—21no—
628C22—21no—
629C22—21no—
630C22—21no—
631C22—42no—
632C22—42no—
633C22#103×D4712yes—
634C22#103×D4712yes—
635C22C2×C22⋊C4×D4711yescharacteristic
636C22C2×SD16×D4711yescharacteristic
637C22C2×Q16×D4711yescharacteristic
638C4—21no—
639C4—21no—
640C4—21no—
641C4—42no—
642C4—42no—
643C4—82no—
644C4—472no—
645C4—472no—
646C4—472no—
647C4—472no—
648C4—472no—
649C4—941no—
650C4—941no—
651C4—941no—
652C4—3762no—
653C4—1882no—
654C4—1882no—
655C4#1568909512yes—
656C2—21no—
657C2—22no—
658C2—22no—
659C2—21no—
660C2C2×#1568909511yescharacteristic
661C2C2×#103×D4711yescharacteristic
662C2#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#17575
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. #20241950C2semidirect
8. #20241885C2semidirect
14. #15689675C4semidirect
15. #15689675C4semidirect
18. C2×#13094023C4semidirect
19. C2×#13094022C4semidirect
23. C2×Q8×Dic47C4semidirect
25. C2×#13093989C4semidirect
380. C47#17575semidirect

Maximal subgroups

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

subgroupindex
C2×#156896752
C2×#156891732
#202457902
#202419612
#202419602
#202419592
#202419502
#202418852
C47×#15202
#1757547