Finite Group Databasebeta

← #26379755

Subgroups — #26379755

Order 12032.

2286 conjugacy classes of subgroups.

The subgroup lattice has 688 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689694C211yesmaximal, characteristic
2#20246567C212yesmaximal, complemented
3#20246531C212yesmaximal, complemented
4#20246521C212yesmaximal, complemented
5#20246476C212yesmaximal, complemented
6#20246415C212yesmaximal, complemented
7#20246393C212yesmaximal, complemented
8#20246283C212yesmaximal, complemented
9#20246220C212yesmaximal, complemented
10#20246197C214yesmaximal, complemented
11#20246175C214yesmaximal, complemented
12#20246066C212yesmaximal, complemented
13#20245916C212yesmaximal, complemented
14#20245879C211yesmaximal, characteristic
15C47×#2311C211yesmaximal, characteristic
16C23×Dic94C2211yescharacteristic
17#15689694C4116yescomplemented
18C2×#13094022C2212yes—
19C2×#13094011C2212yes—
20C2×#13094005C2212yes—
21C2×#13094005C2212yes—
22C2×#13094003C2212yes—
23C2×#13094003C2212yes—
24C2×#13093989C2212yes—
25C2×#13093989C2212yes—
26C2×#13093983C2211yescharacteristic
27C2×#13093983C2211yescharacteristic
28C22×C4×Dic47C2211yescharacteristic
29#15689030C2211yescharacteristic
30#15689030C2211yescharacteristic
31#15689028C2212yes—
32#15689028C2212yes—
33#15689027C2214yes—
34#15689027C2214yes—
35#15689026C2211yescharacteristic
36#15689026C2211yescharacteristic
37#15689020C2214yes—
38#15689019C2212yes—
39#15689019C2212yes—
40#15689018C2214yes—
41#15689017C2214yes—
42#15689017C2214yes—
43#15689011C2214yes—
44#15689011C2214yes—
45#15689011C2214yes—
46#15689011C2214yes—
47#15689010C2214yes—
48#15689007C2214yes—
49#15689005C2214yes—
50#15689004C2214yes—
51#15689003C2212yes—
52#15689003C2212yes—
53#15688996C2214yes—
54#15688995C2214yes—
55#15688990C2214yes—
56#15688981C2214yes—
57C4⋊C4×Dic47C2212yes—
58C4⋊C4×Dic47C2212yes—
59C4×Dic47⋊C4C2214yes—
60#15688974C2214yes—
61#15688974C2214yes—
62C22⋊C4×Dic47C2214yes—
63C22⋊C4×Dic47C2214yes—
64C4×C22⋊Dic47C2212yes—
65C4×C22⋊Dic47C2212yes—
66C4×C22⋊Dic47C2214yes—
67C22×#41×C47C2211yescharacteristic
68C2×#117×C47C2211yescharacteristic
69C22×C4⋊C4×C47C2211yescharacteristic
70C47×#391C2212yes—
71C47×#390C2212yes—
72C47×#386C2212yes—
73C47×#386C2212yes—
74C47×#380C2212yes—
75C47×#380C2212yes—
76C22×Dic94—42no—
77C22×Dic94—42no—
78C22×Dic94C2×C418yescomplemented
79C22×Dic94C2312yes—
80C22×Dic94C2312yes—
81C23×Dic47C2312yes—
82C23×Dic47C2312yes—
83#13094022C2×C4116yescomplemented
84#13094011C2×C4116yes—
85#13094005C2×C4116yescomplemented
86#13094005C2×C4116yescomplemented
87#13094003C2×C4116yescomplemented
88#13094003C2×C4116yescomplemented
89#13093989—44no—
90#13093989—44no—
91#13093983—42no—
92#13093983C2×C418yescomplemented
93C2×C4⋊Dic47C2311yescharacteristic
94C2×C4⋊Dic47C2311yescharacteristic
95C2×C4⋊Dic47C2312yes—
96C2×Dic47⋊C4C2314yes—
97C2×Dic47⋊C4C2314yes—
98C2×Dic47⋊C4C2314yes—
99C2×Dic47⋊C4C2314yes—
100C2×Dic47⋊C4C2314yes—
101C2×Dic47⋊C4C2314yes—
102C2×C22⋊Dic47C2312yes—
103C2×C22⋊Dic47C2312yes—
104C2×C22⋊Dic47C2312yes—
105C2×C22⋊Dic47C2312yes—
106C2×C22⋊Dic47C2312yes—
107C2×C22⋊Dic47C2314yes—
108C2×C22⋊Dic47C2314yes—
109C2×C22⋊Dic47C2312yes—
110C2×C4×Dic47—42no—
111C2×C4×Dic47C2312yes—
112C2×C4×Dic47C2312yes—
113C2×C4×Dic47C2314yes—
114C2×C4×Dic47C2314yes—
115C2×C4×Dic47C2314yes—
116C2×C4×Dic47C2311yescharacteristic
117C2×C4×Dic47C2314yes—
118C2×C4×Dic47C2311yescharacteristic
119C2×C4×Dic47C2312yes—
120C47⋊#95C2314yes—
121C47⋊#95C2312yes—
122C47⋊#95C2312yes—
123C47⋊#95C2312yes—
124C47⋊#95C2314yes—
125C47⋊#95C2312yes—
126C47⋊#95C2314yes—
127C47⋊#95C2314yes—
128C47⋊#95C2314yes—
129C47⋊#95C2314yes—
130C47⋊#95C2314yes—
131C47⋊#95C2314yes—
132C2×#41×C47—42no—
133C2×#41×C47—42no—
134C2×#41×C47C2×C418yes—
135C22×Q8×C47C2311yescharacteristic
136C22×D4×C47C2311yescharacteristic
137C22×D4×C47C2311yescharacteristic
138C22×D4×C47C2311yescharacteristic
139C23×C4×C47C2311yescharacteristic
140C23×C4×C47C2311yescharacteristic
141C23×C4×C47C2311yescharacteristic
142C23×C4×C47C2312yes—
143#117×C47—42no—
144C2×C4⋊C4×C47—42no—
145C2×C4⋊C4×C47C2312yes—
146C2×C4⋊C4×C47C2312yes—
147C2×C4⋊C4×C47C2312yes—
148C2×C22⋊C4×C47C2314yes—
149C2×C22⋊C4×C47C2314yes—
150C2×C22⋊C4×C47C2312yes—
151C2×C42×C47C2312yes—
152#95×C47C2314yes—
153#95×C47C2314yes—
154C24×C47C2411yescharacteristic
155C24×C47C2411yescharacteristic
156C24×C47C2411yescharacteristic
157#41×C47—416no—
158C2×Q8×C47—41no—
159C2×Q8×C47—41no—
160C2×Q8×C47C22×C414yes—
161C2×D4×C47—41no—
162C2×D4×C47—41no—
163C2×D4×C47—41no—
164C2×D4×C47—41no—
165C2×D4×C47—41no—
166C2×D4×C47—41no—
167C2×D4×C47C22×C414yes—
168C2×D4×C47C22×C414yes—
169C2×D4×C47C22×C414yes—
170C22×C4×C47—22no—
171C22×C4×C47—22no—
172C22×C4×C47—22no—
173C22×C4×C47—22no—
174C22×C4×C47—22no—
175C22×C4×C47—22no—
176C22×C4×C47—42no—
177C22×C4×C47—44no—
178C22×C4×C47C22×C414yes—
179C22×C4×C47C22×C414yes—
180C22×C4×C47C22×C414yes—
181C22×C4×C47C2414yes—
182C22×C4×C47C2414yes—
183C22×C4×C47C2411yescharacteristic
184C22×C4×C47C2411yescharacteristic
185C22×C4×C47C2411yescharacteristic
186C22×C4×C47C2411yescharacteristic
187C22×Dic47—24no—
188C22×Dic47—42no—
189C22×Dic47—42no—
190C22×Dic47—42no—
191C22×Dic47—42no—
192C22×Dic47C22×C418yes—
193C22×Dic47C2412yes—
194C22×Dic47C2412yes—
195C22×Dic47C2412yes—
196C22×Dic47C2414yes—
197C22×Dic47C2414yes—
198C22×Dic47C2412yes—
199C2×Dic94—24no—
200C2×Dic94—24no—
201C2×Dic94—416no—
202C2×Dic94—42no—
203C2×Dic94—42no—
204C2×Dic94C22×C418yes—
205C2×Dic94C22×C418yes—
206C4⋊C4×C47—24no—
207C4⋊C4×C47—24no—
208C4⋊C4×C47—42no—
209C22⋊C4×C47—44no—
210C22⋊C4×C47—44no—
211C22⋊C4×C47—42no—
212C42×C47—42no—
213C22⋊Dic47—24no—
214C22⋊Dic47—24no—
215C22⋊Dic47—42no—
216C22⋊Dic47—42no—
217C22⋊Dic47—42no—
218C22⋊Dic47C22×C418yes—
219C22⋊Dic47C22×C4116yes—
220C22⋊Dic47C22×C4116yes—
221C4⋊Dic47—41no—
222C4⋊Dic47—41no—
223C4⋊Dic47C22×C418yes—
224Dic47⋊C4—28no—
225Dic47⋊C4—44no—
226Dic47⋊C4—44no—
227Dic47⋊C4—44no—
228Dic47⋊C4C22×C4116yes—
229Dic47⋊C4C22×C4116yes—
230C4×Dic47—24no—
231C4×Dic47—24no—
232C4×Dic47—41no—
233C4×Dic47—41no—
234C4×Dic47—44no—
235C4×Dic47—44no—
236C4×Dic47—44no—
237C4×Dic47—44no—
238C4×Dic47C22×C418yes—
239C23×C47—21no—
240C23×C47—21no—
241C23×C47—21no—
242C23×C47—21no—
243C23×C47—21no—
244C23×C47—21no—
245C23×C47—22no—
246C23×C47—22no—
247C23×C47—22no—
248C23×C47—22no—
249C23×C47—22no—
250C23×C47—22no—
251C23×C47C23×C412yes—
252C23×C47C23×C412yes—
253C23×C47C23×C412yes—
254C23×C47C2511yescharacteristic
255Q8×C47—44no—
256D4×C47—44no—
257D4×C47—44no—
258D4×C47—44no—
259C2×C4×C47—21no—
260C2×C4×C47—21no—
261C2×C4×C47—21no—
262C2×C4×C47—21no—
263C2×C4×C47—21no—
264C2×C4×C47—21no—
265C2×C4×C47—28no—
266C2×C4×C47—28no—
267C2×C4×C47—28no—
268C2×C4×C47—24no—
269C2×C4×C47—24no—
270C2×C4×C47—24no—
271C2×C4×C47—24no—
272C2×C4×C47—24no—
273C2×C4×C47—24no—
274C2×C4×C47—84no—
275C2×C4×C47C23×C412yes—
276C2×C4×C47C23×C412yes—
277C2×C4×C47C23×C412yes—
278C2×C4×C47C23×C412yes—
279C2×C4×C47H(5,𝔽2)12yes—
280C2×C4×C47H(5,𝔽2)12yes—
281C2×Dic47—22no—
282C2×Dic47—24no—
283C2×Dic47—24no—
284C2×Dic47—24no—
285C2×Dic47—24no—
286C2×Dic47—22no—
287C2×Dic47—22no—
288C2×Dic47—22no—
289C2×Dic47—22no—
290C2×Dic47—22no—
291C2×Dic47—22no—
292C2×Dic47—22no—
293C2×Dic47—22no—
294C2×Dic47—22no—
295C2×Dic47—22no—
296C2×Dic47—48no—
297C2×Dic47—48no—
298C2×Dic47C23×C418yes—
299C2×Dic47C23×C418yes—
300C2×Dic472−1+414yes—
301Dic94—216no—
302Dic94—48no—
303#34413—471nomaximal, Sylow
304C22×C47—22no—
305C22×C47—22no—
306C22×C47—22no—
307C22×C47—24no—
308C22×C47—24no—
309C22×C47—24no—
310C22×C47—42no—
311C22×C47—42no—
312C22×C47—42no—
313C22×C47—41no—
314C22×C47—41no—
315C22×C47—41no—
316C22×C47—41no—
317C22×C47—41no—
318C22×C47—41no—
319C22×C47C24×C411yescharacteristic
320C22×C47C2×H(5,𝔽2)11yescharacteristic
321C22×C47C2×H(5,𝔽2)11yescharacteristic
322C22×C47C2×2−1+412yes—
323C22×C47#58512yes—
324C4×C47—22no—
325C4×C47—22no—
326C4×C47—22no—
327C4×C47—44no—
328C4×C47—44no—
329C4×C47#51814yes—
330Dic47—28no—
331Dic47—28no—
332Dic47—24no—
333Dic47—42no—
334Dic47—42no—
335Dic47—42no—
336C2×#536—471no—
337#2381—472no—
338#2376—472no—
339#2374—472no—
340#2369—472no—
341#2362—472no—
342#2360—472no—
343#2347—472no—
344Q8×C22⋊C4—472no—
345#2337—474no—
346#2335—474no—
347#2323—472no—
348#2311D4711yescharacteristic
349C4×C22⋊Q8—472no—
350#2297—471no—
351C2×C47—42no—
352C2×C47—42no—
353C2×C47—42no—
354C2×C47C2×#51811yescharacteristic
355C2×C47C4×2−1+412yes—
356C2×C47#354112yes—
357C2×C47#355011yescharacteristic
358C2×C47#355111yescharacteristic
359C22×#41C2×D4711yescharacteristic
360C23×Q8—471no—
361#536—4716no—
362C2×C4⋊Q8—472no—
363C2×#124—472no—
364C2×#123—472no—
365C2×#123—472no—
366C2×C22⋊Q8—472no—
367C2×C22⋊Q8—472no—
368C2×C4×Q8—472no—
369C2×C4×Q8—472no—
370C2×#117—471no—
371C2×#117—471no—
372C2×#117C2×D4711yescharacteristic
373C22×C4⋊C4C2×D4711yescharacteristic
374C22×C42—471no—
375#391—471no—
376#391—471no—
377#391—472no—
378#391—472no—
379#391—474no—
380#391—474no—
381#391C2×D4712yes—
382#390—471no—
383#390—471no—
384#390C2×D4712yes—
385#389—472no—
386#389—474no—
387#389—472no—
388#389—474no—
389#388—474no—
390#388—474no—
391#388—474no—
392#388—474no—
393#388—474no—
394#388—474no—
395#387—474no—
396#387—474no—
397#387—474no—
398#387—474no—
399#386—472no—
400#386—472no—
401#386C2×D4712yes—
402#386C2×D4712yes—
403#385—474no—
404#385—474no—
405#383—474no—
406#380—474no—
407#380C2×D4712yes—
408#380C2×D4712yes—
409C4×C4⋊C4—472no—
410C4×C4⋊C4—472no—
411C4×C4⋊C4—474no—
412C4×C4⋊C4—474no—
413C4×C4⋊C4—474no—
414C4×C22⋊C4—474no—
415C4×C22⋊C4—472no—
416C4×C22⋊C4—472no—
417C4×C22⋊C4—474no—
418C4×C22⋊C4—474no—
419C47#3441311yescharacteristic, complemented, Sylow
420C2×#41—42no—
421C2×#41—42no—
422C2×#41C4×D4718yes—
423C22×Q8—478no—
424C22×Q8—472no—
425C22×Q8—472no—
426C22×Q8—1882no—
427C22×Q8—1882no—
428C22×Q8C22×D4711yescharacteristic
429C22×D4C22×D4711yescharacteristic
430C22×D4C22×D4711yescharacteristic
431C22×D4C22×D4711yescharacteristic
432C23×C4—472no—
433C23×C4—472no—
434C23×C4C22×D4711yescharacteristic
435C23×C4C22×D4711yescharacteristic
436C23×C4C22×D4711yescharacteristic
437C23×C4C22×D4712yes—
438C4⋊Q8—4716no—
439#124—4716no—
440#123—4716no—
441#123—4716no—
442C22⋊Q8—4716no—
443C22⋊Q8—4716no—
444C4×Q8—1884no—
445C4×Q8—1884no—
446#117—42no—
447#117—478no—
448#117—1882no—
449C2×C4⋊C4—42no—
450C2×C4⋊C4—472no—
451C2×C4⋊C4—474no—
452C2×C4⋊C4—474no—
453C2×C4⋊C4—474no—
454C2×C4⋊C4—471no—
455C2×C4⋊C4—471no—
456C2×C4⋊C4—474no—
457C2×C4⋊C4—474no—
458C2×C4⋊C4—474no—
459C2×C4⋊C4C22×D4712yes—
460C2×C4⋊C4C22×D4712yes—
461C2×C4⋊C4C22×D4712yes—
462C2×C22⋊C4—472no—
463C2×C22⋊C4—474no—
464C2×C22⋊C4—474no—
465C2×C22⋊C4—472no—
466C2×C22⋊C4—472no—
467C2×C22⋊C4—472no—
468C2×C22⋊C4—472no—
469C2×C22⋊C4—472no—
470C2×C22⋊C4C22×D4712yes—
471C2×C22⋊C4C22×D4714yes—
472C2×C22⋊C4C22×D4714yes—
473C2×C42—472no—
474C2×C42—472no—
475C2×C42—472no—
476C2×C42—474no—
477C2×C42—474no—
478C2×C42—474no—
479C2×C42—471no—
480C2×C42—474no—
481C2×C42—471no—
482C2×C42—1882no—
483C2×C42C22×D4712yes—
484#95—474no—
485#95—472no—
486#95—472no—
487#95—474no—
488#95—472no—
489#95—474no—
490#95—474no—
491#95—472no—
492#95—474no—
493#95—474no—
494#95—474no—
495#95—474no—
496#95C22×D4714yes—
497#95C22×D4714yes—
498C24C23×D4711yescharacteristic
499C24C23×D4711yescharacteristic
500C24C23×D4711yescharacteristic
501#41—416no—
502C2×Q8—41no—
503C2×Q8—41no—
504C2×Q8—478no—
505C2×Q8—478no—
506C2×Q8—944no—
507C2×Q8—944no—
508C2×Q8—1882no—
509C2×Q8—1882no—
510C2×Q8—18816no—
511C2×Q8C2×C4×D4714yes—
512C2×D4—41no—
513C2×D4—41no—
514C2×D4—41no—
515C2×D4—41no—
516C2×D4—41no—
517C2×D4—41no—
518C2×D4C2×C4×D4714yes—
519C2×D4C2×C4×D4714yes—
520C2×D4C2×C4×D4714yes—
521C22×C4—22no—
522C22×C4—22no—
523C22×C4—22no—
524C22×C4—22no—
525C22×C4—22no—
526C22×C4—22no—
527C22×C4—42no—
528C22×C4—44no—
529C22×C4—472no—
530C22×C4—474no—
531C22×C4—474no—
532C22×C4—478no—
533C22×C4—472no—
534C22×C4—472no—
535C22×C4—472no—
536C22×C4—944no—
537C22×C4—1882no—
538C22×C4—1882no—
539C22×C4—1882no—
540C22×C4—1882no—
541C22×C4C2×C4×D4714yes—
542C22×C4C2×C4×D4714yes—
543C22×C4C2×C4×D4714yes—
544C22×C4C23×D4714yes—
545C22×C4C23×D4714yes—
546C22×C4C23×D4711yescharacteristic
547C22×C4C23×D4711yescharacteristic
548C22×C4C23×D4711yescharacteristic
549C22×C4C23×D4711yescharacteristic
550C4⋊C4—24no—
551C4⋊C4—24no—
552C4⋊C4—42no—
553C4⋊C4—4716no—
554C4⋊C4—4716no—
555C4⋊C4—478no—
556C4⋊C4—948no—
557C4⋊C4—1884no—
558C4⋊C4—1884no—
559C4⋊C4—1884no—
560C4⋊C4—1881no—
561C4⋊C4—1881no—
562C22⋊C4—44no—
563C22⋊C4—44no—
564C22⋊C4—42no—
565C22⋊C4—478no—
566C22⋊C4—4716no—
567C22⋊C4—4716no—
568C22⋊C4—944no—
569C22⋊C4—944no—
570C22⋊C4—1882no—
571C22⋊C4—1882no—
572C22⋊C4—1882no—
573C42—42no—
574C42—478no—
575C42—944no—
576C42—944no—
577C42—1884no—
578C42—1884no—
579C42—1881no—
580C42—1881no—
581C42—1884no—
582C42—1884no—
583C23—21no—
584C23—21no—
585C23—21no—
586C23—21no—
587C23—21no—
588C23—21no—
589C23—22no—
590C23—22no—
591C23—22no—
592C23—22no—
593C23—22no—
594C23—22no—
595C23C22×C4×D4712yes—
596C23C22×C4×D4712yes—
597C23C22×C4×D4712yes—
598C23C24×D4711yescharacteristic
599Q8—44no—
600Q8—9416no—
601Q8—1888no—
602D4—44no—
603D4—44no—
604D4—44no—
605C2×C4—21no—
606C2×C4—21no—
607C2×C4—21no—
608C2×C4—21no—
609C2×C4—21no—
610C2×C4—21no—
611C2×C4—28no—
612C2×C4—28no—
613C2×C4—28no—
614C2×C4—24no—
615C2×C4—24no—
616C2×C4—24no—
617C2×C4—24no—
618C2×C4—24no—
619C2×C4—24no—
620C2×C4—84no—
621C2×C4—478no—
622C2×C4—478no—
623C2×C4—474no—
624C2×C4—942no—
625C2×C4—944no—
626C2×C4—944no—
627C2×C4—944no—
628C2×C4—944no—
629C2×C4—942no—
630C2×C4—942no—
631C2×C4—942no—
632C2×C4—942no—
633C2×C4—942no—
634C2×C4—942no—
635C2×C4—942no—
636C2×C4—942no—
637C2×C4—942no—
638C2×C4—942no—
639C2×C4—1888no—
640C2×C4—1888no—
641C2×C4C22×C4×D4712yes—
642C2×C4C22×C4×D4712yes—
643C2×C4C22×C4×D4712yes—
644C2×C4C22×C4×D4712yes—
645C2×C4#1309406912yes—
646C2×C4#1309406912yes—
647C22—22no—
648C22—22no—
649C22—22no—
650C22—24no—
651C22—24no—
652C22—24no—
653C22—41no—
654C22—41no—
655C22—41no—
656C22—41no—
657C22—41no—
658C22—41no—
659C22—42no—
660C22—42no—
661C22—42no—
662C22C23×C4×D4711yescharacteristic
663C22C2×#1309406911yescharacteristic
664C22C2×#1309406911yescharacteristic
665C222−1+4×D4712yes—
666C22#1568995812yes—
667C4—22no—
668C4—22no—
669C4—22no—
670C4—44no—
671C4—44no—
672C4—948no—
673C4—948no—
674C4—944no—
675C4—1882no—
676C4—1882no—
677C4—1882no—
678C4#1568959614yes—
679C2—42no—
680C2—42no—
681C2—42no—
682C2C2×#1568959611yescharacteristic
683C2#2025659912yes—
684C2#2025766912yes—
685C2#2025779711yescharacteristic
686C2#2025781211yescharacteristic
subgroupG/H
center, FrattiniC23C24×D47
commutatorC22×C47C24×C4
FittingC47×#2311C2
socle, cosocleC23×C47C25
prime radicalC22×#41×C47C22
hypercenter#2311D47
hypocenterC47#34413
LCSG ▷ C22×C47 ▷ C47
UCSC1 ◁ C23 ◁ #2311

Complements

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

NH ≅ G/Ntype
2. #20246567C2semidirect
3. #20246531C2semidirect
4. #20246521C2semidirect
5. #20246476C2semidirect
6. #20246415C2semidirect
7. #20246393C2semidirect
8. #20246283C2semidirect
9. #20246220C2semidirect
10. #20246197C2semidirect
11. #20246175C2semidirect
12. #20246066C2semidirect
13. #20245916C2semidirect
17. #15689694C4semidirect
78. C22×Dic94C2×C4semidirect
83. #13094022C2×C4semidirect
85. #13094005C2×C4semidirect
86. #13094005C2×C4semidirect
87. #13094003C2×C4semidirect
88. #13094003C2×C4semidirect
92. #13093983C2×C4semidirect
419. C47#34413semidirect

Maximal subgroups

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

subgroupindex
C2×#156896942
#202465672
#202465312
#202465212
#202464762
#202464152
#202463932
#202462832
#202462202
#202461972
#202461752
#202460662
#202459162
#202458792
C47×#23112
#3441347