Finite Group Databasebeta

← #26252488

Subgroups — #26252488

Order 12032.

1012 conjugacy classes of subgroups.

The subgroup lattice has 668 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689751C211yesmaximal, characteristic
2C2×#15689430C211yesmaximal, characteristic, complemented
3C2×#15689418C211yesmaximal, characteristic, complemented
4#20244726C211yesmaximal, characteristic, complemented
5#20244716C211yesmaximal, characteristic, complemented
6#20244539C211yesmaximal, characteristic, complemented
7#20244529C211yesmaximal, characteristic, complemented
8#20244345C211yesmaximal, characteristic, complemented
9#20244340C211yesmaximal, characteristic, complemented
10#20243701C211yesmaximal, characteristic
11#20243696C211yesmaximal, characteristic
12#20243585C211yesmaximal, characteristic
13#20243575C211yesmaximal, characteristic
14#20243372C211yesmaximal, characteristic
15C47×#2323C211yesmaximal, characteristic
16#15689751—24no—
17#15689751—24no—
18C2×#13094024C2211yescharacteristic
19C2×#13094024C2211yescharacteristic
20C2×#13094023—21no—
21C2×#13094016C2211yescharacteristic
22C2×#13094016C2211yescharacteristic
23C2×#13094015—21no—
24C2×#13094001—21no—
25C2×#13094001—21no—
26C2×#13093983C2211yescharacteristic
27C22×C4⋊Dic47C2211yescharacteristic
28#15689430—24no—
29#15689418—24no—
30C2×#13093957C2211yescharacteristic
31C2×#13093957C2211yescharacteristic
32C2×#13093957C2211yescharacteristic
33C2×#13093955C2211yescharacteristic
34C2×C4⋊(C47⋊C8)C2211yescharacteristic
35C2×C4⋊(C47⋊C8)C2211yescharacteristic
36C2×#13093948C2211yescharacteristic
37C2×#13093948C2211yescharacteristic
38C2×#13093948C2211yescharacteristic
39C2×#13093948C2211yescharacteristic
40C2×C22⋊(C47⋊C8)C2211yescharacteristic
41C2×C22⋊(C47⋊C8)C2211yescharacteristic
42#15688885C2211yescharacteristic
43#15688885C2211yescharacteristic
44#15688885C2211yescharacteristic
45#15688885C2211yescharacteristic
46#15688885C2211yescharacteristic
47#15688885C2211yescharacteristic
48#15688885C2211yescharacteristic
49#15688885C2211yescharacteristic
50#15688878C2211yescharacteristic
51#15688878C2211yescharacteristic
52C2×C22⋊Q8×C47C2211yescharacteristic
53C2×C4×Q8×C47C2211yescharacteristic
54C2×#117×C47C2211yescharacteristic
55C47×#391C2211yescharacteristic
56C47×#389—21no—
57C47×#389C2211yescharacteristic
58C47×#388—21no—
59C47×#388—21no—
60C47×#387—21no—
61C47×#380C2211yescharacteristic
62C4×C4⋊C4×C47C2211yescharacteristic
63C22×Dic94—21no—
64C22×Dic94—21no—
65C23×Dic47—21no—
66C22×C47⋊C8C2311yescharacteristic
67C22×C47⋊C8C2311yescharacteristic
68C22×C47⋊C8C2311yescharacteristic
69C22×C47⋊C8C2311yescharacteristic
70#13094024—22no—
71#13094024—22no—
72#13094024—22no—
73#13094024D414yes—
74#13094023—28no—
75#13094016—22no—
76#13094016—22no—
77#13094016—22no—
78#13094016D414yes—
79#13094015—28no—
80#13094001—28no—
81#13094001—28no—
82#13093983—22no—
83#13093983—22no—
84C2×C4⋊Dic47—21no—
85C2×C4⋊Dic47—21no—
86C2×C4⋊Dic47—21no—
87C2×C4⋊Dic47—22no—
88C2×C4⋊Dic47—22no—
89C2×C4⋊Dic47C2311yescharacteristic
90C2×C4⋊Dic47C2311yescharacteristic
91C2×C4⋊Dic47C2311yescharacteristic
92C2×C4⋊Dic47C2311yescharacteristic
93C2×Dic47⋊C4—21no—
94C2×Dic47⋊C4—21no—
95C2×Dic47⋊C4—21no—
96C2×Dic47⋊C4—21no—
97C2×C22⋊Dic47—21no—
98C2×C4×Dic47—21no—
99#13093957—22no—
100#13093957—22no—
101#13093957—22no—
102#13093955—22no—
103C4⋊(C47⋊C8)—22no—
104C4⋊(C47⋊C8)—22no—
105#13093948—22no—
106#13093948—22no—
107#13093948—22no—
108#13093948—22no—
109C22⋊(C47⋊C8)—22no—
110C22⋊(C47⋊C8)—22no—
111C22×Q8×C47C2311yescharacteristic
112C23×C4×C47C2311yescharacteristic
113C22⋊Q8×C47—22no—
114C22⋊Q8×C47Q814yescomplemented
115C4×Q8×C47—22no—
116C4×Q8×C47—22no—
117#117×C47—22no—
118#117×C47—22no—
119C2×C4⋊C4×C47—21no—
120C2×C4⋊C4×C47—21no—
121C2×C4⋊C4×C47—21no—
122C2×C4⋊C4×C47D411yescharacteristic
123C2×C4⋊C4×C47D411yescharacteristic
124C2×C4⋊C4×C47C2311yescharacteristic
125C2×C4⋊C4×C47C2311yescharacteristic
126C2×C4⋊C4×C47C2311yescharacteristic
127C2×C22⋊C4×C47—21no—
128C2×C22⋊C4×C47—21no—
129C2×C22⋊C4×C47D411yescharacteristic
130C2×C22⋊C4×C47D411yescharacteristic
131C2×C42×C47—21no—
132C2×C42×C47—21no—
133C2×C42×C47C2311yescharacteristic
134C2×C42×C47C2311yescharacteristic
135#95×C47—21no—
136#95×C47—21no—
137#95×C47—21no—
138#95×C47—21no—
139#95×C47—21no—
140C24×C47C2×D411yescharacteristic
141C2×Q8×C47—21no—
142C2×Q8×C47—21no—
143C2×Q8×C47—21no—
144C2×Q8×C47—21no—
145C2×Q8×C47—21no—
146C2×Q8×C47C2×Q812yes—
147C22×C4×C47—21no—
148C22×C4×C47—21no—
149C22×C4×C47—21no—
150C22×C4×C47—21no—
151C22×C4×C47—21no—
152C22×C4×C47—21no—
153C22×C4×C47—21no—
154C22×C4×C47—21no—
155C22×C4×C47C2×D411yescharacteristic
156C22×C4×C47C2×D411yescharacteristic
157C22×C4×C47C2×D411yescharacteristic
158C22×C4×C47C2×D411yescharacteristic
159C22×C4×C47C2×D411yescharacteristic
160C22×C4×C47C2×Q812yes—
161C22×C4×C47#4112yes—
162C22×C4×C47C2411yescharacteristic
163C22×Dic47—21no—
164C22×Dic47—21no—
165C22×Dic47—21no—
166C22×Dic47—21no—
167C22×Dic47—24no—
168C22×Dic47—81no—
169C22×Dic47—81no—
170C2×Dic94—24no—
171C2×Dic94—24no—
172C2×Dic94—81no—
173C2×Dic94—81no—
174C2×Dic94—81no—
175C2×Dic94—81no—
176C4⋊C4×C47—22no—
177C4⋊C4×C47—22no—
178C4⋊C4×C47—24no—
179C4⋊C4×C47—21no—
180C4⋊C4×C47—21no—
181C4⋊C4×C47—21no—
182C4⋊C4×C47—21no—
183C4⋊C4×C47—21no—
184C4⋊C4×C47—42no—
185C4⋊C4×C47—81no—
186C4⋊C4×C47C2×Q812yes—
187C22⋊C4×C47—24no—
188C22⋊C4×C47—22no—
189C22⋊C4×C47—22no—
190C22⋊C4×C47—81no—
191C42×C47—21no—
192C42×C47—21no—
193C42×C47—42no—
194C42×C47—81no—
195C42×C47C2×D412yes—
196C42×C47C2×D412yes—
197C22⋊Dic47—24no—
198C4⋊Dic47—24no—
199C4⋊Dic47—24no—
200C4⋊Dic47—24no—
201C4⋊Dic47—21no—
202C4⋊Dic47—21no—
203C4⋊Dic47—21no—
204C4⋊Dic47—21no—
205C4⋊Dic47C2×D412yes—
206C4⋊Dic47C2×D412yes—
207C4⋊Dic47C2×D412yes—
208C4⋊Dic47C2×D412yes—
209Dic47⋊C4—24no—
210Dic47⋊C4—24no—
211Dic47⋊C4—24no—
212Dic47⋊C4—24no—
213C4×Dic47—24no—
214C2×C47⋊C8—21no—
215C2×C47⋊C8—21no—
216C2×C47⋊C8—21no—
217C2×C47⋊C8—21no—
218C2×C47⋊C8—21no—
219C2×C47⋊C8—21no—
220C2×C47⋊C8—21no—
221C2×C47⋊C8—21no—
222C2×C47⋊C8—21no—
223C2×C47⋊C8—21no—
224C2×C47⋊C8—21no—
225C2×C47⋊C8—21no—
226C23×C47—21no—
227C23×C47—21no—
228C23×C47—21no—
229C23×C47—21no—
230C23×C47—21no—
231C23×C47—21no—
232C23×C47C22⋊Q812yes—
233C23×C47C22×D411yescharacteristic
234Q8×C47—22no—
235Q8×C47—22no—
236Q8×C47—41no—
237Q8×C47—41no—
238C2×C4×C47—21no—
239C2×C4×C47—21no—
240C2×C4×C47—21no—
241C2×C4×C47—21no—
242C2×C4×C47—21no—
243C2×C4×C47—21no—
244C2×C4×C47—22no—
245C2×C4×C47—22no—
246C2×C4×C47—22no—
247C2×C4×C47—22no—
248C2×C4×C47—21no—
249C2×C4×C47—21no—
250C2×C4×C47—22no—
251C2×C4×C47—22no—
252C2×C4×C47—21no—
253C2×C4×C47—21no—
254C2×C4×C47—21no—
255C2×C4×C47—21no—
256C2×C4×C47—41no—
257C2×C4×C47—41no—
258C2×C4×C47—41no—
259C2×C4×C47—41no—
260C2×C4×C47—41no—
261C2×C4×C47—41no—
262C2×C4×C47—41no—
263C2×C4×C47—41no—
264C2×C4×C47—41no—
265C2×C4×C47—41no—
266C2×C4×C47#12012yes—
267C2×C4×C47#12012yes—
268C2×C4×C47C22⋊Q812yes—
269C2×C4×C47C22×D411yescharacteristic
270C2×C4×C47C22×D411yescharacteristic
271C2×C4×C47C22×Q811yescharacteristic
272C2×C4×C47C2×#4111yescharacteristic
273C2×C4×C472−1+411yescharacteristic
274C2×C4×C472−1+411yescharacteristic
275C2×Dic47—22no—
276C2×Dic47—22no—
277C2×Dic47—22no—
278C2×Dic47—22no—
279C2×Dic47—41no—
280C2×Dic47—41no—
281C2×Dic47—41no—
282C2×Dic47—41no—
283C2×Dic47—41no—
284C2×Dic47—41no—
285C2×Dic47—41no—
286C2×Dic47—41no—
287C2×Dic47—84no—
288Dic94—84no—
289Dic94—84no—
290C47⋊C8—41no—
291C47⋊C8—41no—
292C47⋊C8—41no—
293C47⋊C8—41no—
294#24311—471nomaximal, Sylow
295C22×C47—22no—
296C22×C47—22no—
297C22×C47—22no—
298C22×C47—41no—
299C22×C47—41no—
300C22×C47—41no—
301C22×C47—41no—
302C22×C47C2×#12011yescharacteristic
303C22×C47C2×C22⋊Q811yescharacteristic
304C22×C47#53611yescharacteristic
305C22×C47#57611yescharacteristic
306C22×C47#57711yescharacteristic
307C22×C47#57811yescharacteristic
308C22×C47#57811yescharacteristic
309C4×C47—22no—
310C4×C47—22no—
311C4×C47—22no—
312C4×C47—42no—
313C4×C47—42no—
314C4×C47—41no—
315C4×C47—41no—
316C4×C47—81no—
317C4×C47—81no—
318C4×C47#54811yescharacteristic
319C4×C47#54811yescharacteristic
320C4×C47D4×Q811yescharacteristic
321C4×C47D4×Q811yescharacteristic
322Dic47—42no—
323Dic47—42no—
324Dic47—42no—
325Dic47—42no—
326C2×#544—471no—
327C2×#479—471no—
328C2×#477—471no—
329#2323D4711yescharacteristic
330#2059—471no—
331#2057—471no—
332#2026—471no—
333#2024—471no—
334#1989—471no—
335#1988—471no—
336#1873—471no—
337#1872—471no—
338#1854—471no—
339#1852—471no—
340#1798—471no—
341C2×C47—42no—
342C2×C47#240911yescharacteristic
343C2×C47#301111yescharacteristic
344C2×C47#301211yescharacteristic
345C2×C47#308211yescharacteristic
346C2×C47#308311yescharacteristic
347C2×C47#324211yescharacteristic
348C2×C47#324311yescharacteristic
349#544—944no—
350#544—944no—
351C2×C4⋊Q8—471no—
352C2×C4⋊Q8—471no—
353C2×C4⋊Q8—941no—
354C2×#125—471no—
355C2×#125—471no—
356C2×#125—941no—
357C2×C22⋊Q8—941no—
358C2×C22⋊Q8—941no—
359C2×C22⋊Q8C2×D4711yescharacteristic
360C2×C4×Q8C2×D4711yescharacteristic
361C2×#117—471no—
362C2×#117C2×D4711yescharacteristic
363C22×C4⋊C4—471no—
364#479—944no—
365#477—944no—
366C2×#107—471no—
367C2×#107—471no—
368C2×#107—471no—
369C2×#106—471no—
370C2×C4⋊C8—471no—
371C2×C4⋊C8—471no—
372C2×#103—471no—
373C2×#103—471no—
374C2×#103—471no—
375C2×#103—471no—
376C2×C22⋊C8—471no—
377C2×C22⋊C8—471no—
378#391C2×D4711yescharacteristic
379#389—21no—
380#389C2×D4711yescharacteristic
381#388—21no—
382#388—21no—
383#387—21no—
384#380C2×D4711yescharacteristic
385C4×C4⋊C4C2×D4711yescharacteristic
386#340—471no—
387#340—471no—
388#340—471no—
389#340—471no—
390#340—471no—
391#340—471no—
392#340—471no—
393#340—471no—
394#336—471no—
395#336—471no—
396C47#2431111yescharacteristic, complemented, Sylow
397C22×Q8—941no—
398C22×Q8—941no—
399C22×Q8C22×D4711yescharacteristic
400C23×C4—941no—
401C23×C4C22×D4711yescharacteristic
402C22×C8—471no—
403C22×C8—471no—
404C22×C8—471no—
405C22×C8—471no—
406C4⋊Q8—474no—
407C4⋊Q8—942no—
408C4⋊Q8—942no—
409C4⋊Q8—948no—
410C4⋊Q8—942no—
411#125—474no—
412#125—942no—
413#125—942no—
414#125—948no—
415#125—942no—
416C22⋊Q8—22no—
417C22⋊Q8—948no—
418C22⋊Q8—948no—
419C22⋊Q8Dic9414yescomplemented
420C4×Q8—22no—
421C4×Q8—22no—
422#117—22no—
423#117—22no—
424#117—942no—
425#117—942no—
426C2×C4⋊C4—21no—
427C2×C4⋊C4—21no—
428C2×C4⋊C4—21no—
429C2×C4⋊C4—471no—
430C2×C4⋊C4—471no—
431C2×C4⋊C4—471no—
432C2×C4⋊C4—471no—
433C2×C4⋊C4—942no—
434C2×C4⋊C4—942no—
435C2×C4⋊C4—941no—
436C2×C4⋊C4—941no—
437C2×C4⋊C4—941no—
438C2×C4⋊C4—941no—
439C2×C4⋊C4—941no—
440C2×C4⋊C4—941no—
441C2×C4⋊C4—941no—
442C2×C4⋊C4C22⋊D4711yescharacteristic
443C2×C4⋊C4C22⋊D4711yescharacteristic
444C2×C4⋊C4C22×D4711yescharacteristic
445C2×C4⋊C4C22×D4711yescharacteristic
446C2×C4⋊C4C22×D4711yescharacteristic
447C2×C22⋊C4—21no—
448C2×C22⋊C4—21no—
449C2×C22⋊C4—941no—
450C2×C22⋊C4C22⋊D4711yescharacteristic
451C2×C22⋊C4C22⋊D4711yescharacteristic
452C2×C42—21no—
453C2×C42—21no—
454C2×C42—941no—
455C2×C42C22×D4711yescharacteristic
456C2×C42C22×D4711yescharacteristic
457#107—942no—
458#107—942no—
459#107—942no—
460#106—942no—
461C4⋊C8—942no—
462C4⋊C8—942no—
463#103—942no—
464#103—942no—
465#103—942no—
466#103—942no—
467C22⋊C8—942no—
468C22⋊C8—942no—
469#95—21no—
470#95—21no—
471#95—21no—
472#95—21no—
473#95—21no—
474C24C2×C22⋊D4711yescharacteristic
475C2×Q8—21no—
476C2×Q8—21no—
477C2×Q8—21no—
478C2×Q8—21no—
479C2×Q8—21no—
480C2×Q8—944no—
481C2×Q8—944no—
482C2×Q8—3761no—
483C2×Q8—3761no—
484C2×Q8—3761no—
485C2×Q8—3761no—
486C2×Q8C2×Dic9412yes—
487C22×C4—21no—
488C22×C4—21no—
489C22×C4—21no—
490C22×C4—21no—
491C22×C4—21no—
492C22×C4—21no—
493C22×C4—21no—
494C22×C4—21no—
495C22×C4—944no—
496C22×C4—941no—
497C22×C4—941no—
498C22×C4—941no—
499C22×C4—941no—
500C22×C4—3761no—
501C22×C4—3761no—
502C22×C4C2×Dic9412yes—
503C22×C4#1062535012yes—
504C22×C4C2×C22⋊D4711yescharacteristic
505C22×C4C2×C22⋊D4711yescharacteristic
506C22×C4C2×C22⋊D4711yescharacteristic
507C22×C4C2×C22⋊D4711yescharacteristic
508C22×C4C2×C22⋊D4711yescharacteristic
509C22×C4C23×D4711yescharacteristic
510C2×C8—941no—
511C2×C8—941no—
512C2×C8—941no—
513C2×C8—941no—
514C2×C8—941no—
515C2×C8—941no—
516C2×C8—941no—
517C2×C8—941no—
518C2×C8—941no—
519C2×C8—941no—
520C2×C8—941no—
521C2×C8—941no—
522C4⋊C4—22no—
523C4⋊C4—22no—
524C4⋊C4—24no—
525C4⋊C4—21no—
526C4⋊C4—21no—
527C4⋊C4—21no—
528C4⋊C4—21no—
529C4⋊C4—21no—
530C4⋊C4—42no—
531C4⋊C4—81no—
532C4⋊C4—472no—
533C4⋊C4—472no—
534C4⋊C4—472no—
535C4⋊C4—472no—
536C4⋊C4—944no—
537C4⋊C4—944no—
538C4⋊C4—944no—
539C4⋊C4—944no—
540C4⋊C4—944no—
541C4⋊C4—944no—
542C4⋊C4—944no—
543C4⋊C4—941no—
544C4⋊C4—941no—
545C4⋊C4—941no—
546C4⋊C4—941no—
547C4⋊C4C2×Dic9412yes—
548C22⋊C4—24no—
549C22⋊C4—22no—
550C22⋊C4—22no—
551C22⋊C4—81no—
552C22⋊C4—944no—
553C42—21no—
554C42—21no—
555C42—42no—
556C42—81no—
557C42—944no—
558C42D4×D4712yes—
559C42D4×D4712yes—
560C23—21no—
561C23—21no—
562C23—21no—
563C23—21no—
564C23—21no—
565C23—21no—
566C23#1309400312yes—
567C23C22×C22⋊D4711yescharacteristic
568Q8—22no—
569Q8—22no—
570Q8—41no—
571Q8—41no—
572Q8—3764no—
573Q8—3764no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—21no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—41no—
600C2×C4—41no—
601C2×C4—41no—
602C2×C4—942no—
603C2×C4—942no—
604C2×C4—942no—
605C2×C4—942no—
606C2×C4—3764no—
607C2×C4—1881no—
608C2×C4—1881no—
609C2×C4—1881no—
610C2×C4—1881no—
611C2×C4—1881no—
612C2×C4—1881no—
613C2×C4—1881no—
614C2×C4—1881no—
615C2×C4#1309399212yes—
616C2×C4#1309399212yes—
617C2×C4#1309400312yes—
618C2×C4C2×D4×D4711yescharacteristic
619C2×C4C2×D4×D4711yescharacteristic
620C2×C4C22×Dic9411yescharacteristic
621C2×C4C2×#1062535011yescharacteristic
622C2×C4#1309407111yescharacteristic
623C2×C4#1309407111yescharacteristic
624C8—1881no—
625C8—1881no—
626C8—1881no—
627C8—1881no—
628C22—22no—
629C22—22no—
630C22—22no—
631C22—41no—
632C22—41no—
633C22—41no—
634C22—41no—
635C22C2×#1309399211yescharacteristic
636C22C2×#1309400311yescharacteristic
637C22#1568969411yescharacteristic
638C22#1568992611yescharacteristic
639C22#1568993111yescharacteristic
640C22#1568993811yescharacteristic
641C22#1568993811yescharacteristic
642C4—22no—
643C4—22no—
644C4—22no—
645C4—41no—
646C4—41no—
647C4—42no—
648C4—42no—
649C4—81no—
650C4—81no—
651C4—1882no—
652C4—1882no—
653C4—1882no—
654C4—1882no—
655C4#1568978211yescharacteristic
656C4#1568978211yescharacteristic
657C4D4×Dic9411yescharacteristic
658C4D4×Dic9411yescharacteristic
659C2—42no—
660C2#2024675311yescharacteristic
661C2#2025268111yescharacteristic
662C2#2025268911yescharacteristic
663C2#2025323011yescharacteristic
664C2#2025323911yescharacteristic
665C2#2025465011yescharacteristic
666C2#2025466511yescharacteristic
subgroupG/H
centerC23C22×C22⋊D47
commutator, cosocleC22×C4×C47C24
FrattiniC22×C4C23×D47
FittingC47×#2323C2
socleC23×C47C22×D4
prime radicalC24×C47C2×D4
hypercenter#2323D47
hypocenterC47#24311
LCSG ▷ C22×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C2×#117 ◁ #2323

Complements

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

NH ≅ G/Ntype
2. C2×#15689430C2semidirect
3. C2×#15689418C2semidirect
4. #20244726C2semidirect
5. #20244716C2semidirect
6. #20244539C2semidirect
7. #20244529C2semidirect
8. #20244345C2semidirect
9. #20244340C2semidirect
114. C22⋊Q8×C47Q8semidirect
396. C47#24311semidirect
419. C22⋊Q8Dic94semidirect

Maximal subgroups

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

subgroupindex
C2×#156897512
C2×#156894302
C2×#156894182
#202447262
#202447162
#202445392
#202445292
#202443452
#202443402
#202437012
#202436962
#202435852
#202435752
#202433722
C47×#23232
#2431147