Finite Group Databasebeta

← #26375701

Subgroups — #26375701

Order 12032.

2434 conjugacy classes of subgroups.

The subgroup lattice has 660 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#13093989C211yesmaximal, characteristic
2#20246342C214yesmaximal, complemented
3#20246341C212yesmaximal, complemented
4#20246302C212yesmaximal, complemented
5#20246266C212yesmaximal
6#20246265C211yesmaximal, characteristic
7#20246260C212yesmaximal
8C22⋊C4×Dic94C211yesmaximal, characteristic
9#20246220C211yesmaximal, characteristic
10#20245861C214yesmaximal, complemented
11#20245859C212yesmaximal, complemented
12#20245790C211yesmaximal, characteristic
13#20245790C212yesmaximal
14C4×#13093989C212yesmaximal, complemented
15C4×#13093980C211yesmaximal, characteristic
16C4×#13093980C212yesmaximal
17C47×#2345C211yesmaximal, characteristic
18C23×Dic94C2211yescharacteristic
19C2×C4×Dic94C2212yes—
20C2×#13093989C4116yescomplemented
21C2×#13093989C2212yes—
22C2×#13093989C2211yescharacteristic
23C2×#13093989C2211yescharacteristic
24C2×#13093989C2212yes—
25C2×#13093989C2211yescharacteristic
26C2×#13093989C2212yes—
27C2×#13093989C2211yescharacteristic
28C2×#13093980C2212yes—
29C2×#13093980C2211yescharacteristic
30C2×#13093980C2211yescharacteristic
31C22×Dic47⋊C4C2212yes—
32C22×C4×Dic47C2211yescharacteristic
33C22×C4×Dic47C2212yes—
34#15689029C2212yes—
35#15689028C2212yes—
36#15689027C2214yes—
37#15689027C2214yes—
38#15689022C2212yes—
39#15689020C2214yes—
40#15689018C2214yes—
41#15689010C2214yes—
42#15689009C2214yes—
43#15689007C2214yes—
44#15689005C2214yes—
45#15689004C2214yes—
46#15688992C2212yes—
47#15688992C2214yes—
48#15688987C2214yes—
49#15688987C2212yes—
50#15688983C2211yescharacteristic
51#15688983C2212yes—
52#15688981C2212yes—
53#15688981C2212yes—
54#15688981C2212yes—
55C4⋊C4×Dic47C2214yes—
56C4⋊C4×Dic47C2212yes—
57C4⋊C4×Dic47C2212yes—
58C4×Dic47⋊C4C2212yes—
59C4×Dic47⋊C4C2214yes—
60C4×Dic47⋊C4C2214yes—
61#15688974C2212yes—
62#15688974C2214yes—
63#15688974C2214yes—
64#15688974C2212yes—
65#15688974C2214yes—
66C22⋊C4×Dic47C2212yes—
67C22⋊C4×Dic47C2212yes—
68C22⋊C4×Dic47C2211yescharacteristic
69C22⋊C4×Dic47C2211yescharacteristic
70C4×C22⋊Dic47C2211yescharacteristic
71C4×C22⋊Dic47C2212yes—
72#15688968C2214yes—
73#15688968C2212yes—
74C42×Dic47C2212yes—
75C42×Dic47C2214yes—
76C22×C4⋊C4×C47C2211yescharacteristic
77C47×#389C2212yes—
78C47×#387C2214yes—
79C47×#380C2211yescharacteristic
80C47×#380C2212yes—
81C4×C4⋊C4×C47C2212yes—
82C4×C22⋊C4×C47C2211yescharacteristic
83C4×C22⋊C4×C47C2212yes—
84C22×Dic94—28no—
85C22×Dic94C2×C418yes—
86C22×Dic94C2312yes—
87C22×Dic94C2311yescharacteristic
88C22×Dic94C2311yescharacteristic
89C23×Dic47C2312yes—
90C23×Dic47C2311yescharacteristic
91C23×Dic47C2311yescharacteristic
92C4×Dic94—28no—
93#13093989—28no—
94#13093989C2×C418yes—
95#13093989C2×C4116yes—
96#13093989C2×C418yes—
97#13093989C2×C4116yes—
98#13093989C2×C418yes—
99#13093989C2×C418yes—
100#13093980—24no—
101#13093980—28no—
102#13093980Q818yescomplemented
103C2×C4⋊Dic47C2312yes—
104C2×C4⋊Dic47C2314yes—
105C2×Dic47⋊C4C2×C4116yes—
106C2×Dic47⋊C4C2314yes—
107C2×Dic47⋊C4C2314yes—
108C2×Dic47⋊C4C2314yes—
109C2×Dic47⋊C4C2312yes—
110C2×Dic47⋊C4C2314yes—
111C2×Dic47⋊C4C2312yes—
112C2×Dic47⋊C4C2312yes—
113C2×Dic47⋊C4C2312yes—
114C2×Dic47⋊C4C2312yes—
115C2×Dic47⋊C4C2312yes—
116C2×C22⋊Dic47C2311yescharacteristic
117C2×C22⋊Dic47C2312yes—
118C2×C22⋊Dic47C2311yescharacteristic
119C2×C4×Dic47C2×C4116yes—
120C2×C4×Dic47C2×C418yes—
121C2×C4×Dic47C2312yes—
122C2×C4×Dic47C2314yes—
123C2×C4×Dic47C2312yes—
124C2×C4×Dic47C2314yes—
125C2×C4×Dic47C2314yes—
126C2×C4×Dic47C2314yes—
127C2×C4×Dic47C2312yes—
128C2×C4×Dic47C2312yes—
129C2×C4×Dic47C2312yes—
130C2×C4×Dic47C2311yescharacteristic
131C2×C4×Dic47C2311yescharacteristic
132C2×C4×Dic47C2312yes—
133C2×C4×Dic47C2311yescharacteristic
134C2×C4×Dic47C2312yes—
135C2×C4×Dic47C2312yes—
136C2×C4×Dic47C2311yescharacteristic
137C2×C4×Dic47C2312yes—
138C47⋊#95C2314yes—
139C47⋊#95C2312yes—
140C47⋊#95C2314yes—
141C47⋊#95C2312yes—
142C47⋊#95C2312yes—
143C47⋊#95C2314yes—
144C47⋊#95C2314yes—
145C47⋊#95C2314yes—
146C47⋊#95C2312yes—
147C47⋊#95C2314yes—
148C47⋊#95C2314yes—
149C23×C4×C47C2311yescharacteristic
150C23×C4×C47C2312yes—
151C2×C4⋊C4×C47C2×C418yes—
152C2×C4⋊C4×C47C2312yes—
153C2×C4⋊C4×C47C2314yes—
154C2×C4⋊C4×C47C2311yescharacteristic
155C2×C4⋊C4×C47C2312yes—
156C2×C4⋊C4×C47C2311yescharacteristic
157C2×C22⋊C4×C47C2312yes—
158C2×C22⋊C4×C47C2311yescharacteristic
159C2×C22⋊C4×C47C2311yescharacteristic
160C2×C42×C47C2312yes—
161C2×C42×C47C2314yes—
162#95×C47C2314yes—
163#95×C47C2312yes—
164#95×C47C2314yes—
165#95×C47C2312yes—
166C24×C47C2411yescharacteristic
167C22×C4×C47—24no—
168C22×C4×C47—44no—
169C22×C4×C47C22×C414yes—
170C22×C4×C47C22×C418yes—
171C22×C4×C47C2414yes—
172C22×C4×C47C2412yes—
173C22×C4×C47C2412yes—
174C22×C4×C47C2412yes—
175C22×C4×C47C2411yescharacteristic
176C22×C4×C47C2412yes—
177C22×C4×C47C2411yescharacteristic
178C22×Dic47—28no—
179C22×Dic47—42no—
180C22×Dic47C22×C418yes—
181C22×Dic47C22×C414yes—
182C22×Dic47C22×C414yes—
183C22×Dic47C2×Q814yes—
184C22×Dic47#4114yes—
185C22×Dic47C2412yes—
186C22×Dic47C2414yes—
187C22×Dic47C2412yes—
188C22×Dic47C2411yescharacteristic
189C22×Dic47C2411yescharacteristic
190C22×Dic47C2412yes—
191C22×Dic47C2411yescharacteristic
192C22×Dic47C2411yescharacteristic
193C22×Dic47C2412yes—
194C2×Dic94—24no—
195C2×Dic94—28no—
196C2×Dic94—232no—
197C2×Dic94C22×C414yes—
198C2×Dic94C22×C418yes—
199C2×Dic94C22×C414yes—
200C2×Dic94#4118yes—
201C4⋊C4×C47—24no—
202C4⋊C4×C47—44no—
203C4⋊C4×C47C22×C418yes—
204C4⋊C4×C47C22×C414yes—
205C4⋊C4×C47C22×C414yes—
206C22⋊C4×C47—24no—
207C22⋊C4×C47—22no—
208C22⋊C4×C47C2×Q814yes—
209C42×C47—24no—
210C42×C47—44no—
211C22⋊Dic47—22no—
212C22⋊Dic47—24no—
213C22⋊Dic47C2×Q814yes—
214C4⋊Dic47—24no—
215C4⋊Dic47—44no—
216Dic47⋊C4—28no—
217Dic47⋊C4—24no—
218Dic47⋊C4—28no—
219Dic47⋊C4—28no—
220Dic47⋊C4—44no—
221Dic47⋊C4C22×C418yes—
222Dic47⋊C4C22×C418yes—
223Dic47⋊C4C22×C418yes—
224Dic47⋊C4C22×C418yes—
225Dic47⋊C4C2×Q818yescomplemented
226C4×Dic47—24no—
227C4×Dic47—24no—
228C4×Dic47—28no—
229C4×Dic47—28no—
230C4×Dic47—28no—
231C4×Dic47—42no—
232C4×Dic47—44no—
233C4×Dic47—42no—
234C4×Dic47C22×C414yes—
235C4×Dic47C22×C414yes—
236C4×Dic47C22×C414yes—
237C4×Dic47C22×C414yes—
238C4×Dic47C22×C418yes—
239C4×Dic47C22×C418yes—
240C4×Dic47C22×C418yes—
241C4×Dic47C22×C418yes—
242C4×Dic47C2×Q818yescomplemented
243C23×C47—22no—
244C23×C47—22no—
245C23×C47C23×C412yes—
246C23×C47C22×Q812yes—
247C23×C47C2×#4112yes—
248C23×C47C2511yescharacteristic
249C2×C4×C47—22no—
250C2×C4×C47—22no—
251C2×C4×C47—22no—
252C2×C4×C47—24no—
253C2×C4×C47—24no—
254C2×C4×C47—28no—
255C2×C4×C47—24no—
256C2×C4×C47—22no—
257C2×C4×C47—24no—
258C2×C4×C47—22no—
259C2×C4×C47—22no—
260C2×C4×C47—24no—
261C2×C4×C47—48no—
262C2×C4×C47C23×C414yes—
263C2×C4×C47C23×C414yes—
264C2×C4×C47C23×C412yes—
265C2×C4×C47C23×C412yes—
266C2×C4×C47C22×Q814yes—
267C2×C4×C47C2×#4114yes—
268C2×Dic47—22no—
269C2×Dic47—24no—
270C2×Dic47—22no—
271C2×Dic47—22no—
272C2×Dic47—22no—
273C2×Dic47—216no—
274C2×Dic47—22no—
275C2×Dic47—24no—
276C2×Dic47—22no—
277C2×Dic47—24no—
278C2×Dic47—24no—
279C2×Dic47—22no—
280C2×Dic47—44no—
281C2×Dic47C4×Q8116yescomplemented
282C2×Dic47C23×C412yes—
283C2×Dic47C23×C412yes—
284C2×Dic47C23×C414yes—
285C2×Dic47C23×C414yes—
286C2×Dic47C23×C412yes—
287C2×Dic47C23×C412yes—
288C2×Dic47C22×Q814yes—
289C2×Dic47C22×Q812yes—
290C2×Dic47C22×Q812yes—
291C2×Dic47C2×#4118yes—
292C2×Dic47C2×#4112yes—
293C2×Dic47C2×#4112yes—
294Dic94—28no—
295Dic94—216no—
296Dic94#117116yescomplemented
297Q8×#117—471nomaximal, Sylow
298C22×C47—22no—
299C22×C47—22no—
300C22×C47—24no—
301C22×C47—24no—
302C22×C47C2×C4×Q814yes—
303C22×C47C24×C411yescharacteristic
304C22×C47C23×Q811yescharacteristic
305C22×C47C22×#4112yes—
306C22×C47C22×#4111yescharacteristic
307C22×C47#58512yes—
308C4×C47—22no—
309C4×C47—24no—
310C4×C47—24no—
311C4×C47—24no—
312C4×C47—44no—
313C4×C47—42no—
314C4×C47C2×#11714yes—
315Dic47—24no—
316Dic47—22no—
317Dic47—22no—
318Dic47—24no—
319Dic47—42no—
320Dic47—44no—
321Dic47C2×#11718yes—
322Dic47C2×C4×Q814yes—
323Dic47C2×C4×Q814yes—
324C22×C4×Q8—471no—
325#2355—474no—
326#2355—472no—
327Q8×C4⋊C4—472no—
328#2345—472no—
329#2345—472no—
330#2345—471no—
331#2345D4711yescharacteristic
332Q8×C22⋊C4—471no—
333Q8×C22⋊C4—471no—
334#2295—472no—
335#2295—474no—
336#2285—471no—
337#2285—472no—
338C42×Q8—472no—
339C4×#117—471no—
340C4×#117—472no—
341C2×C47—24no—
342C2×C47C22×#11711yescharacteristic
343C2×C47C22×C4×Q811yescharacteristic
344C2×C47#342811yescharacteristic
345C2×C47Q8×#4112yes—
346C2×C47#348012yes—
347C23×Q8—471no—
348C2×C4×Q8—4716no—
349C2×C4×Q8—471no—
350C2×C4×Q8—472no—
351C2×C4×Q8—471no—
352C2×C4×Q8—471no—
353C2×C4×Q8—471no—
354C2×C4×Q8—472no—
355C2×C4×Q8—472no—
356C2×C4×Q8—472no—
357C2×#117—471no—
358C2×#117—472no—
359C2×#117—471no—
360C22×C4⋊C4—472no—
361C22×C4⋊C4C2×D4711yescharacteristic
362C22×C42—472no—
363C22×C42—471no—
364#391—472no—
365#391—472no—
366#391—474no—
367#391—474no—
368#389—474no—
369#389—472no—
370#389—474no—
371#389C2×D4712yes—
372#387—474no—
373#387—474no—
374#387—474no—
375#387—474no—
376#387—474no—
377#387C2×D4714yes—
378#384—474no—
379#384—472no—
380#382—472no—
381#382—474no—
382#380—471no—
383#380—472no—
384#380—472no—
385#380—472no—
386#380—472no—
387#380C2×D4711yescharacteristic
388#380C2×D4712yes—
389C4×C4⋊C4—472no—
390C4×C4⋊C4—472no—
391C4×C4⋊C4—474no—
392C4×C4⋊C4—474no—
393C4×C4⋊C4—472no—
394C4×C4⋊C4—474no—
395C4×C4⋊C4—474no—
396C4×C4⋊C4—474no—
397C4×C4⋊C4—472no—
398C4×C4⋊C4—474no—
399C4×C4⋊C4—472no—
400C4×C4⋊C4C2×D4712yes—
401C4×C22⋊C4—471no—
402C4×C22⋊C4—472no—
403C4×C22⋊C4—472no—
404C4×C22⋊C4—472no—
405C4×C22⋊C4—471no—
406C4×C22⋊C4—471no—
407C4×C22⋊C4C2×D4711yescharacteristic
408C4×C22⋊C4C2×D4712yes—
409#376—472no—
410#376—474no—
411C43—474no—
412C43—472no—
413C47Q8×#11711yescharacteristic, complemented, Sylow
414C22×Q8—478no—
415C22×Q8—471no—
416C22×Q8—472no—
417C22×Q8—471no—
418C22×Q8—948no—
419C23×C4—471no—
420C23×C4—472no—
421C23×C4—471no—
422C23×C4C22×D4711yescharacteristic
423C23×C4C22×D4712yes—
424C4×Q8—478no—
425C4×Q8—478no—
426C4×Q8—4716no—
427C4×Q8—478no—
428C4×Q8—4716no—
429C4×Q8—478no—
430C4×Q8—948no—
431C4×Q8—948no—
432#117—478no—
433#117—948no—
434#117—944no—
435C2×C4⋊C4—4716no—
436C2×C4⋊C4—472no—
437C2×C4⋊C4—472no—
438C2×C4⋊C4—472no—
439C2×C4⋊C4—472no—
440C2×C4⋊C4—472no—
441C2×C4⋊C4—472no—
442C2×C4⋊C4—474no—
443C2×C4⋊C4—474no—
444C2×C4⋊C4—474no—
445C2×C4⋊C4—472no—
446C2×C4⋊C4—474no—
447C2×C4⋊C4—474no—
448C2×C4⋊C4C4×D4718yes—
449C2×C4⋊C4C22×D4712yes—
450C2×C4⋊C4C22×D4714yes—
451C2×C4⋊C4C22×D4711yescharacteristic
452C2×C4⋊C4C22×D4712yes—
453C2×C4⋊C4C22×D4711yescharacteristic
454C2×C22⋊C4—471no—
455C2×C22⋊C4—471no—
456C2×C22⋊C4—472no—
457C2×C22⋊C4C22×D4712yes—
458C2×C22⋊C4C22×D4711yescharacteristic
459C2×C22⋊C4C22×D4711yescharacteristic
460C2×C42—4716no—
461C2×C42—478no—
462C2×C42—472no—
463C2×C42—472no—
464C2×C42—471no—
465C2×C42—472no—
466C2×C42—471no—
467C2×C42—472no—
468C2×C42—471no—
469C2×C42—472no—
470C2×C42—471no—
471C2×C42—474no—
472C2×C42—472no—
473C2×C42—474no—
474C2×C42—474no—
475C2×C42—472no—
476C2×C42—472no—
477C2×C42—474no—
478C2×C42—472no—
479C2×C42C22×D4712yes—
480C2×C42C22×D4714yes—
481#95—472no—
482#95—474no—
483#95—472no—
484#95—474no—
485#95—474no—
486#95—472no—
487#95—474no—
488#95—474no—
489#95—474no—
490#95—474no—
491#95—472no—
492#95C22×D4714yes—
493#95C22×D4712yes—
494#95C22×D4714yes—
495#95C22×D4712yes—
496C24C23×D4711yescharacteristic
497C2×Q8—474no—
498C2×Q8—478no—
499C2×Q8—474no—
500C2×Q8—478no—
501C2×Q8—948no—
502C2×Q8—9432no—
503C2×Q8—944no—
504C22×C4—24no—
505C22×C4—44no—
506C22×C4—474no—
507C22×C4—474no—
508C22×C4—474no—
509C22×C4—471no—
510C22×C4—471no—
511C22×C4—472no—
512C22×C4—478no—
513C22×C4—472no—
514C22×C4—472no—
515C22×C4—471no—
516C22×C4—471no—
517C22×C4—474no—
518C22×C4—472no—
519C22×C4—474no—
520C22×C4—948no—
521C22×C4—1882no—
522C22×C4C2×C4×D4714yes—
523C22×C4C2×C4×D4718yes—
524C22×C4C23×D4712yes—
525C22×C4C23×D4714yes—
526C22×C4C23×D4711yescharacteristic
527C22×C4C23×D4712yes—
528C22×C4C23×D4712yes—
529C22×C4C23×D4712yes—
530C22×C4C23×D4711yescharacteristic
531C4⋊C4—24no—
532C4⋊C4—44no—
533C4⋊C4—478no—
534C4⋊C4—478no—
535C4⋊C4—478no—
536C4⋊C4—478no—
537C4⋊C4—478no—
538C4⋊C4—944no—
539C4⋊C4—948no—
540C4⋊C4—948no—
541C4⋊C4—948no—
542C4⋊C4—944no—
543C4⋊C4—1884no—
544C4⋊C4—1884no—
545C4⋊C4C2×C4×D4718yes—
546C4⋊C4C2×C4×D4714yes—
547C4⋊C4C2×C4×D4714yes—
548C22⋊C4—24no—
549C22⋊C4—22no—
550C22⋊C4—474no—
551C22⋊C4—942no—
552C22⋊C4—944no—
553C22⋊C4Q8×D4714yes—
554C42—24no—
555C42—44no—
556C42—478no—
557C42—478no—
558C42—474no—
559C42—474no—
560C42—474no—
561C42—474no—
562C42—478no—
563C42—478no—
564C42—478no—
565C42—948no—
566C42—944no—
567C42—948no—
568C42—948no—
569C42—944no—
570C42—1882no—
571C42—1884no—
572C42—1882no—
573C23—22no—
574C23—22no—
575C23C22×C4×D4712yes—
576C23C2×Q8×D4712yes—
577C23#41×D4712yes—
578C23C24×D4711yescharacteristic
579Q8—4716no—
580Q8—9416no—
581Q8—948no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—24no—
586C2×C4—24no—
587C2×C4—28no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—24no—
591C2×C4—24no—
592C2×C4—24no—
593C2×C4—22no—
594C2×C4—48no—
595C2×C4—4716no—
596C2×C4—472no—
597C2×C4—472no—
598C2×C4—474no—
599C2×C4—472no—
600C2×C4—472no—
601C2×C4—472no—
602C2×C4—478no—
603C2×C4—472no—
604C2×C4—472no—
605C2×C4—472no—
606C2×C4—474no—
607C2×C4—474no—
608C2×C4—9416no—
609C2×C4—942no—
610C2×C4—944no—
611C2×C4—942no—
612C2×C4—942no—
613C2×C4—942no—
614C2×C4—942no—
615C2×C4—944no—
616C2×C4—942no—
617C2×C4—944no—
618C2×C4—944no—
619C2×C4—942no—
620C2×C4—1884no—
621C2×C4C22×C4×D4714yes—
622C2×C4C22×C4×D4714yes—
623C2×C4C22×C4×D4712yes—
624C2×C4C22×C4×D4712yes—
625C2×C4C2×Q8×D4714yes—
626C2×C4#41×D4714yes—
627C22—22no—
628C22—22no—
629C22—24no—
630C22—24no—
631C22C4×Q8×D4714yes—
632C22C23×C4×D4711yescharacteristic
633C22C22×Q8×D4711yescharacteristic
634C22C2×#41×D4712yes—
635C22C2×#41×D4711yescharacteristic
636C22#1568995812yes—
637C4—22no—
638C4—24no—
639C4—24no—
640C4—24no—
641C4—42no—
642C4—44no—
643C4—478no—
644C4—474no—
645C4—474no—
646C4—944no—
647C4—942no—
648C4—942no—
649C4—944no—
650C4—1882no—
651C4—1884no—
652C4#117×D4714yes—
653C2—24no—
654C2C2×#117×D4711yescharacteristic
655C2C2×C4×Q8×D4711yescharacteristic
656C2#2025658611yescharacteristic
657C2#2025696712yes—
658C2#2025698212yes—
subgroupG/H
center, FrattiniC23C24×D47
commutatorC22×C47C24×C4
FittingC47×#2345C2
socle, cosocleC23×C47C25
prime radicalC24×C47C24
hypercenter#2345D47
hypocenterC47Q8×#117
LCSG ▷ C22×C47 ▷ C47
UCSC1 ◁ C23 ◁ #2345

Complements

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

NH ≅ G/Ntype
2. #20246342C2semidirect
3. #20246341C2semidirect
4. #20246302C2semidirect
10. #20245861C2semidirect
11. #20245859C2semidirect
14. C4×#13093989C2semidirect
20. C2×#13093989C4semidirect
102. #13093980Q8semidirect
225. Dic47⋊C4C2×Q8semidirect
242. C4×Dic47C2×Q8semidirect
281. C2×Dic47C4×Q8semidirect
296. Dic94#117semidirect
413. C47Q8×#117semidirect

Maximal subgroups

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

subgroupindex
C22×#130939892
#202463422
#202463412
#202463022
#202462662
#202462652
#202462602
C22⋊C4×Dic942
#202462202
#202458612
#202458592
#202457902
C4×#130939892
C4×#130939802
C47×#23452
Q8×#11747