Finite Group Databasebeta

← #26227589

Subgroups — #26227589

Order 12032.

920 conjugacy classes of subgroups.

The subgroup lattice has 684 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689279C211yesmaximal, characteristic
2#20246070C211yesmaximal, characteristic
3#20244258C211yesmaximal, characteristic, complemented
4#20244248C211yesmaximal, characteristic, complemented
5#20244081C211yesmaximal, characteristic, complemented
6#20244067C211yesmaximal, characteristic, complemented
7#20244026C211yesmaximal, characteristic, complemented
8#20244019C211yesmaximal, characteristic, complemented
9#20243732C211yesmaximal, characteristic, complemented
10#20243732C211yesmaximal, characteristic, complemented
11#20243700C211yesmaximal, characteristic
12#20243587C211yesmaximal, characteristic
13#20243323C211yesmaximal, characteristic
14#20243154C211yesmaximal, characteristic
15C47×#2310C211yesmaximal, characteristic
16C22×#10625337C2211yescharacteristic
17C22×#10625337C2211yescharacteristic
18C2×#13094003C2211yescharacteristic
19C2×C4×Dic94C2211yescharacteristic
20#15689279C418yescomplemented
21C2×#13093957C2211yescharacteristic
22C2×#13093955C2211yescharacteristic
23C2×#13093948C2211yescharacteristic
24C2×#13093948C2211yescharacteristic
25C2×#13093948C2211yescharacteristic
26C2×#13093948C2211yescharacteristic
27C2×#13093948C2211yescharacteristic
28C2×#13093948C2211yescharacteristic
29C2×#13093947C2211yescharacteristic
30C2×#13093947C2211yescharacteristic
31C2×C22⋊(C47⋊C8)C2211yescharacteristic
32C2×C22⋊(C47⋊C8)C2211yescharacteristic
33C2×#13093935C2211yescharacteristic
34C2×C4×C47⋊C8C2211yescharacteristic
35#15689029C2211yescharacteristic
36#15689021C2211yescharacteristic
37#15689018—21no—
38#15689017—21no—
39#15689011—21no—
40#15689004—21no—
41#15688983C2211yescharacteristic
42C4×C4⋊Dic47C2211yescharacteristic
43#15688885C2211yescharacteristic
44#15688885C2211yescharacteristic
45#15688885C2211yescharacteristic
46#15688885C2211yescharacteristic
47#15688878C2211yescharacteristic
48#15688878C2211yescharacteristic
49C23×Q8×C47C2211yescharacteristic
50C2×#117×C47C2211yescharacteristic
51C2×#117×C47C2211yescharacteristic
52C47×#391—21no—
53C47×#391—21no—
54C47×#391C2211yescharacteristic
55C47×#391C2211yescharacteristic
56C47×#391C2211yescharacteristic
57C47×#391C2211yescharacteristic
58C47×#381—21no—
59C47×#381—21no—
60C22×Dic94C2311yescharacteristic
61C2×#10625337—41no—
62C2×#10625337—41no—
63C2×#10625337—41no—
64C2×#10625337—41no—
65C2×#10625337C2×C414yes—
66C2×#10625337C2×C414yes—
67C22×C47⋊C8C2311yescharacteristic
68C22×C47⋊C8C2311yescharacteristic
69C22×C47⋊C8C2311yescharacteristic
70C22×C47⋊C8C2311yescharacteristic
71#13094003—22no—
72#13094003C2×C414yes—
73C4×Dic94—41no—
74C4×Dic94—41no—
75C2×C4⋊Dic47C2311yescharacteristic
76C2×C4⋊Dic47C2311yescharacteristic
77C2×C4⋊Dic47C2311yescharacteristic
78C2×Dic47⋊C4—21no—
79C2×Dic47⋊C4—21no—
80C2×Dic47⋊C4—21no—
81C2×C22⋊Dic47—21no—
82C2×C22⋊Dic47—21no—
83C2×C4×Dic47—21no—
84C2×C4×Dic47—21no—
85#13093957—41no—
86#13093955—41no—
87#13093948—41no—
88#13093948—41no—
89#13093948—41no—
90#13093948—41no—
91#13093948C2×C414yes—
92#13093948C2×C414yes—
93#13093947—41no—
94#13093947—41no—
95C22⋊(C47⋊C8)—22no—
96C22⋊(C47⋊C8)C2×C414yes—
97#13093935—41no—
98C4×C47⋊C8—41no—
99C47⋊#95—21no—
100C47⋊#95—21no—
101C47⋊#95—21no—
102C47⋊#95—21no—
103C47⋊#95—21no—
104C22×Q8×C47—21no—
105C22×Q8×C47—22no—
106C22×Q8×C47—41no—
107C22×Q8×C47—41no—
108C22×Q8×C47—41no—
109C22×Q8×C47—41no—
110C22×Q8×C47C2×C414yes—
111C22×Q8×C47C2311yescharacteristic
112C22×Q8×C47C2311yescharacteristic
113C23×C4×C47—21no—
114C23×C4×C47C2311yescharacteristic
115#117×C47—41no—
116#117×C47—41no—
117#117×C47—41no—
118#117×C47—41no—
119C2×C4⋊C4×C47D411yescharacteristic
120C2×C4⋊C4×C47D411yescharacteristic
121C2×C4⋊C4×C47C2311yescharacteristic
122C2×C4⋊C4×C47C2311yescharacteristic
123C2×C22⋊C4×C47—21no—
124C2×C22⋊C4×C47D411yescharacteristic
125C2×C22⋊C4×C47D411yescharacteristic
126C2×C42×C47—21no—
127C2×C42×C47C2311yescharacteristic
128C2×C42×C47C2311yescharacteristic
129#95×C47—21no—
130#95×C47—21no—
131#95×C47—21no—
132#95×C47—21no—
133#95×C47—21no—
134#95×C47—21no—
135#95×C47—21no—
136#95×C47—21no—
137C24×C47C2×D411yescharacteristic
138C2×Q8×C47—24no—
139C2×Q8×C47—21no—
140C2×Q8×C47—21no—
141C2×Q8×C47—21no—
142C2×Q8×C47—21no—
143C2×Q8×C47—21no—
144C2×Q8×C47—21no—
145C2×Q8×C47—21no—
146C2×Q8×C47—21no—
147C2×Q8×C47—21no—
148C2×Q8×C47—21no—
149C2×Q8×C47—44no—
150C2×Q8×C47—44no—
151C2×Q8×C47—42no—
152C2×Q8×C47—42no—
153C2×Q8×C47—44no—
154C2×Q8×C47—44no—
155C2×Q8×C47C22×C412yes—
156C2×Q8×C47C22×C412yes—
157C22×C4×C47—21no—
158C22×C4×C47—21no—
159C22×C4×C47—21no—
160C22×C4×C47—21no—
161C22×C4×C47—24no—
162C22×C4×C47—21no—
163C22×C4×C47—21no—
164C22×C4×C47—21no—
165C22×C4×C47—21no—
166C22×C4×C47—42no—
167C22×C4×C47—42no—
168C22×C4×C47C22×C412yes—
169C22×C4×C47C2×D411yescharacteristic
170C22×C4×C47C2×D411yescharacteristic
171C22×C4×C47C2×D411yescharacteristic
172C22×C4×C47C2×D411yescharacteristic
173C22×C4×C47C2×D411yescharacteristic
174C22×C4×C47#4112yes—
175C22×C4×C47C2411yescharacteristic
176C22×Dic47—21no—
177C22×Dic47—21no—
178C22×Dic47—21no—
179C22×Dic47—21no—
180C2×Dic94—21no—
181C2×Dic94—21no—
182C2×Dic94—21no—
183C2×Dic94—21no—
184C2×Dic94—21no—
185C2×Dic94C22×C412yes—
186C4⋊C4×C47—21no—
187C4⋊C4×C47—21no—
188C4⋊C4×C47—21no—
189C4⋊C4×C47—21no—
190C4⋊C4×C47—41no—
191C4⋊C4×C47—41no—
192C22⋊C4×C47—41no—
193C22⋊C4×C47—41no—
194C22⋊C4×C47—42no—
195C42×C47—21no—
196C42×C47—21no—
197C42×C47—21no—
198C42×C47—21no—
199C42×C47—42no—
200C22⋊Dic47—24no—
201C22⋊Dic47—81no—
202#10625337—44no—
203#10625337—44no—
204C4⋊Dic47—21no—
205C4⋊Dic47—21no—
206C4⋊Dic47—21no—
207C4⋊Dic47—21no—
208C4⋊Dic47—21no—
209C4⋊Dic47C22×C412yes—
210Dic47⋊C4—24no—
211Dic47⋊C4—42no—
212Dic47⋊C4—81no—
213C4×Dic47—42no—
214C4×Dic47—81no—
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⋊C8C22×C412yes—
226C2×C47⋊C8C22×C412yes—
227C23×C47—21no—
228C23×C47—21no—
229C23×C47—21no—
230C23×C47—21no—
231C23×C47—21no—
232C23×C47—21no—
233C23×C47C4×D412yes—
234C23×C47C22×D411yescharacteristic
235Q8×C47—22no—
236Q8×C47—22no—
237Q8×C47—22no—
238Q8×C47—22no—
239Q8×C47—48no—
240Q8×C47—41no—
241Q8×C47—41no—
242Q8×C47—41no—
243Q8×C47—41no—
244C2×C4×C47—21no—
245C2×C4×C47—21no—
246C2×C4×C47—22no—
247C2×C4×C47—22no—
248C2×C4×C47—22no—
249C2×C4×C47—22no—
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—21no—
257C2×C4×C47—21no—
258C2×C4×C47—21no—
259C2×C4×C47—21no—
260C2×C4×C47—21no—
261C2×C4×C47—21no—
262C2×C4×C47—21no—
263C2×C4×C47—21no—
264C2×C4×C47—48no—
265C2×C4×C47—41no—
266C2×C4×C47—41no—
267C2×C4×C47—41no—
268C2×C4×C47—41no—
269C2×C4×C47—41no—
270C2×C4×C47—41no—
271C2×C4×C47—41no—
272C2×C4×C47—41no—
273C2×C4×C47—41no—
274C2×C4×C47—41no—
275C2×C4×C47C4×D412yes—
276C2×C4×C47C23×C411yescharacteristic
277C2×C4×C47C2×#4111yescharacteristic
278C2×C4×C47H(5,𝔽2)11yescharacteristic
279C2×C4×C47H(5,𝔽2)11yescharacteristic
280C2×C4×C47H(5,𝔽2)11yescharacteristic
281C2×C4×C47H(5,𝔽2)11yescharacteristic
282C2×Dic47—22no—
283C2×Dic47—22no—
284C2×Dic47—41no—
285C2×Dic47—41no—
286C2×Dic47—41no—
287C2×Dic47—41no—
288C2×Dic47—41no—
289C2×Dic47—41no—
290C2×Dic47—41no—
291C2×Dic47—41no—
292C2×Dic47—41no—
293C2×Dic47—41no—
294Dic94—22no—
295Dic94—22no—
296Dic94—41no—
297Dic94—41no—
298C47⋊C8—22no—
299C47⋊C8—22no—
300C47⋊C8—41no—
301C47⋊C8—41no—
302#22054—471nomaximal, Sylow
303C22×C47—22no—
304C22×C47—22no—
305C22×C47—22no—
306C22×C47—41no—
307C22×C47—41no—
308C22×C47—41no—
309C22×C47—41no—
310C22×C47C2×C4×D411yescharacteristic
311C22×C47#53411yescharacteristic
312C22×C47#53511yescharacteristic
313C22×C47#57711yescharacteristic
314C22×C47#57811yescharacteristic
315C22×C47#57811yescharacteristic
316C22×C47#57811yescharacteristic
317C4×C47—22no—
318C4×C47—42no—
319C4×C47—42no—
320C4×C47—41no—
321C4×C47—41no—
322C4×C47—41no—
323C4×C47—41no—
324C4×C47—81no—
325C4×C47—81no—
326C4×C47#51811yescharacteristic
327C4×C47#51811yescharacteristic
328C4×C47#53811yescharacteristic
329C4×C47#53811yescharacteristic
330Dic47—42no—
331Dic47—42no—
332Dic47—81no—
333Dic47—81no—
334C2×#451—471no—
335#2323—471no—
336#2310D4711yescharacteristic
337#1971—471no—
338#1969—471no—
339#1935—471no—
340#1933—471no—
341#1928—471no—
342#1927—471no—
343#1878—471no—
344#1878—471no—
345#1873—471no—
346#1854—471no—
347#1787—471no—
348#1750—471no—
349C2×C47—42no—
350C2×C47#232111yescharacteristic
351C2×C47#294711yescharacteristic
352C2×C47#295011yescharacteristic
353C2×C47#320111yescharacteristic
354C2×C47#320211yescharacteristic
355C2×C47#322311yescharacteristic
356C2×C47#322511yescharacteristic
357C23×Q8C2×D4711yescharacteristic
358C22×Q16—471no—
359C22×Q16—471no—
360C2×C22⋊Q8—471no—
361C2×C4×Q8—471no—
362C2×#117C2×D4711yescharacteristic
363C2×#117C2×D4711yescharacteristic
364#451—478no—
365C2×#107—471no—
366C2×#106—471no—
367C2×#103—471no—
368C2×#103—471no—
369C2×#103—471no—
370C2×#103—471no—
371C2×#103—471no—
372C2×#103—471no—
373C2×#103—471no—
374C2×#103—471no—
375C2×C22⋊C8—471no—
376C2×C22⋊C8—471no—
377C2×#97—471no—
378C2×C4×C8—471no—
379#391—21no—
380#391—21no—
381#391—471no—
382#391C2×D4711yescharacteristic
383#391C2×D4711yescharacteristic
384#391C2×D4711yescharacteristic
385#391C2×D4711yescharacteristic
386#389—471no—
387#389—941no—
388#388—941no—
389#388—941no—
390#387—941no—
391#381—21no—
392#381—21no—
393#380—471no—
394C4×C4⋊C4—471no—
395#340—471no—
396#340—471no—
397#340—471no—
398#340—471no—
399#336—471no—
400#336—471no—
401C47#2205411yescharacteristic, complemented, Sylow
402C22×Q8—22no—
403C22×Q8—21no—
404C22×Q8—41no—
405C22×Q8—41no—
406C22×Q8—41no—
407C22×Q8—41no—
408C22×Q8—471no—
409C22×Q8C4×D4714yes—
410C22×Q8C22×D4711yescharacteristic
411C22×Q8C22×D4711yescharacteristic
412C23×C4—21no—
413C23×C4C22×D4711yescharacteristic
414C2×Q16—474no—
415C2×Q16—474no—
416C2×Q16—1881no—
417C2×Q16—1881no—
418C2×Q16—1881no—
419C2×Q16—1881no—
420C22×C8—471no—
421C22×C8—471no—
422C22×C8—471no—
423C22×C8—471no—
424C22⋊Q8—474no—
425C22⋊Q8—942no—
426C4×Q8—1881no—
427C4×Q8—1881no—
428#117—41no—
429#117—41no—
430#117—41no—
431#117—41no—
432C2×C4⋊C4—471no—
433C2×C4⋊C4—471no—
434C2×C4⋊C4—471no—
435C2×C4⋊C4—941no—
436C2×C4⋊C4—941no—
437C2×C4⋊C4—941no—
438C2×C4⋊C4C22⋊D4711yescharacteristic
439C2×C4⋊C4C22⋊D4711yescharacteristic
440C2×C4⋊C4C22×D4711yescharacteristic
441C2×C4⋊C4C22×D4711yescharacteristic
442C2×C22⋊C4—21no—
443C2×C22⋊C4—941no—
444C2×C22⋊C4—941no—
445C2×C22⋊C4C22⋊D4711yescharacteristic
446C2×C22⋊C4C22⋊D4711yescharacteristic
447C2×C42—21no—
448C2×C42—941no—
449C2×C42—941no—
450C2×C42C22×D4711yescharacteristic
451C2×C42C22×D4711yescharacteristic
452#107—1881no—
453#106—1881no—
454#103—474no—
455#103—474no—
456#103—1881no—
457#103—1881no—
458#103—1881no—
459#103—1881no—
460#103—1881no—
461#103—1881no—
462C22⋊C8—474no—
463C22⋊C8—942no—
464#97—1881no—
465C4×C8—1881no—
466#95—21no—
467#95—21no—
468#95—21no—
469#95—21no—
470#95—21no—
471#95—21no—
472#95—21no—
473#95—21no—
474#95—941no—
475#95—941no—
476#95—941no—
477#95—941no—
478#95—941no—
479C24C2×C22⋊D4711yescharacteristic
480C2×Q8—24no—
481C2×Q8—21no—
482C2×Q8—21no—
483C2×Q8—21no—
484C2×Q8—21no—
485C2×Q8—21no—
486C2×Q8—21no—
487C2×Q8—21no—
488C2×Q8—21no—
489C2×Q8—21no—
490C2×Q8—21no—
491C2×Q8—44no—
492C2×Q8—44no—
493C2×Q8—42no—
494C2×Q8—42no—
495C2×Q8—44no—
496C2×Q8—44no—
497C2×Q8—472no—
498C2×Q8—941no—
499C2×Q8—941no—
500C2×Q8—941no—
501C2×Q8—941no—
502C2×Q8—941no—
503C2×Q8C2×C4×D4712yes—
504C2×Q8C2×C4×D4712yes—
505C22×C4—21no—
506C22×C4—21no—
507C22×C4—21no—
508C22×C4—21no—
509C22×C4—24no—
510C22×C4—21no—
511C22×C4—21no—
512C22×C4—21no—
513C22×C4—21no—
514C22×C4—42no—
515C22×C4—42no—
516C22×C4—941no—
517C22×C4—941no—
518C22×C4—941no—
519C22×C4—941no—
520C22×C4C2×C4×D4712yes—
521C22×C4#1062535012yes—
522C22×C4C2×C22⋊D4711yescharacteristic
523C22×C4C2×C22⋊D4711yescharacteristic
524C22×C4C2×C22⋊D4711yescharacteristic
525C22×C4C2×C22⋊D4711yescharacteristic
526C22×C4C2×C22⋊D4711yescharacteristic
527C22×C4C23×D4711yescharacteristic
528Q16—1884no—
529Q16—1884no—
530C2×C8—472no—
531C2×C8—472no—
532C2×C8—941no—
533C2×C8—941no—
534C2×C8—941no—
535C2×C8—941no—
536C2×C8—941no—
537C2×C8—941no—
538C2×C8—941no—
539C2×C8—941no—
540C2×C8—941no—
541C2×C8—941no—
542C4⋊C4—21no—
543C4⋊C4—21no—
544C4⋊C4—21no—
545C4⋊C4—21no—
546C4⋊C4—41no—
547C4⋊C4—41no—
548C4⋊C4—472no—
549C4⋊C4—944no—
550C4⋊C4—941no—
551C4⋊C4—941no—
552C4⋊C4—941no—
553C4⋊C4—941no—
554C4⋊C4—941no—
555C4⋊C4—3761no—
556C4⋊C4—1882no—
557C22⋊C4—41no—
558C22⋊C4—41no—
559C22⋊C4—42no—
560C22⋊C4—944no—
561C22⋊C4—3761no—
562C42—21no—
563C42—21no—
564C42—21no—
565C42—21no—
566C42—42no—
567C42—3761no—
568C42—1882no—
569C23—21no—
570C23—21no—
571C23—21no—
572C23—21no—
573C23—21no—
574C23—21no—
575C23C4×C22⋊D4712yes—
576C23C22×C22⋊D4711yescharacteristic
577Q8—22no—
578Q8—22no—
579Q8—22no—
580Q8—22no—
581Q8—48no—
582Q8—41no—
583Q8—41no—
584Q8—41no—
585Q8—41no—
586Q8—942no—
587Q8—942no—
588Q8—1881no—
589Q8—1881no—
590C2×C4—21no—
591C2×C4—21no—
592C2×C4—22no—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—22no—
596C2×C4—22no—
597C2×C4—22no—
598C2×C4—21no—
599C2×C4—21no—
600C2×C4—21no—
601C2×C4—21no—
602C2×C4—21no—
603C2×C4—21no—
604C2×C4—21no—
605C2×C4—21no—
606C2×C4—21no—
607C2×C4—21no—
608C2×C4—21no—
609C2×C4—21no—
610C2×C4—48no—
611C2×C4—41no—
612C2×C4—41no—
613C2×C4—41no—
614C2×C4—41no—
615C2×C4—41no—
616C2×C4—41no—
617C2×C4—41no—
618C2×C4—41no—
619C2×C4—41no—
620C2×C4—41no—
621C2×C4—942no—
622C2×C4—942no—
623C2×C4—1881no—
624C2×C4—1881no—
625C2×C4—1881no—
626C2×C4—1881no—
627C2×C4—1881no—
628C2×C4—1881no—
629C2×C4—1881no—
630C2×C4—1881no—
631C2×C4—1881no—
632C2×C4—1881no—
633C2×C4C4×C22⋊D4712yes—
634C2×C4C22×C4×D4711yescharacteristic
635C2×C4C2×#1062535011yescharacteristic
636C2×C4#1309406911yescharacteristic
637C2×C4#1309406911yescharacteristic
638C2×C4#1309406911yescharacteristic
639C2×C4#1309406911yescharacteristic
640C8—942no—
641C8—942no—
642C8—1881no—
643C8—1881no—
644C22—22no—
645C22—22no—
646C22—22no—
647C22—41no—
648C22—41no—
649C22—41no—
650C22—41no—
651C22C2×C4×C22⋊D4711yescharacteristic
652C22#1568968211yescharacteristic
653C22#1568969011yescharacteristic
654C22#1568993111yescharacteristic
655C22#1568993811yescharacteristic
656C22#1568993811yescharacteristic
657C22#1568993811yescharacteristic
658C4—22no—
659C4—42no—
660C4—42no—
661C4—41no—
662C4—41no—
663C4—41no—
664C4—41no—
665C4—81no—
666C4—81no—
667C4—3761no—
668C4—3761no—
669C4—1882no—
670C4—1882no—
671C4#1568959611yescharacteristic
672C4#1568959611yescharacteristic
673C4#1568971111yescharacteristic
674C4#1568971111yescharacteristic
675C2—42no—
676C2#2024604611yescharacteristic
677C2#2025223111yescharacteristic
678C2#2025225411yescharacteristic
679C2#2025427011yescharacteristic
680C2#2025427911yescharacteristic
681C2#2025447111yescharacteristic
682C2#2025449711yescharacteristic
subgroupG/H
centerC23C22×C22⋊D47
commutatorC2×C4×C47C23×C4
FrattiniC22×C4C23×D47
FittingC47×#2310C2
socleC23×C47C22×D4
prime radicalC24×C47C2×D4
cosocleC22×C4×C47C24
hypercenter#2310D47
hypocenterC47#22054
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C2×#117 ◁ #2310

Complements

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

NH ≅ G/Ntype
3. #20244258C2semidirect
4. #20244248C2semidirect
5. #20244081C2semidirect
6. #20244067C2semidirect
7. #20244026C2semidirect
8. #20244019C2semidirect
9. #20243732C2semidirect
10. #20243732C2semidirect
20. #15689279C4semidirect
401. C47#22054semidirect

Maximal subgroups

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

subgroupindex
C2×#156892792
#202460702
#202442582
#202442482
#202440812
#202440672
#202440262
#202440192
#202437322
#202437002
#202435872
#202433232
#202431542
C47×#23102
#2205447