Finite Group Databasebeta

← #26218073

Subgroups — #26218073

Order 12032.

1994 conjugacy classes of subgroups.

The subgroup lattice has 652 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#13093983C211yesmaximal, characteristic, complemented
2C2×#15689097C212yesmaximal, complemented
3#20245771C211yesmaximal, characteristic, complemented
4#20243847C214yesmaximal, complemented
5#20243842C214yesmaximal, complemented
6#20243121C212yesmaximal, complemented
7C2×C47×#525C211yesmaximal, characteristic
8C2×#13093983—28no—
9C2×#13093983—21no—
10C2×#13093983—21no—
11C2×#13093983C2212yes—
12C2×#13093983C2212yes—
13C2×#13093983C2212yes—
14C22×C4⋊Dic47—21no—
15C22×C22⋊Dic47C2212yes—
16C22×C4×Dic47—21no—
17#15689097—28no—
18C2×#13093939C2218yes—
19#15688986C2212yes—
20#15688983—22no—
21#15688979C2212yes—
22C4×C22⋊Dic47—22no—
23#15688890C2218yes—
24C22×#41×C47C2211yescharacteristic
25C24×C4×C47C2211yescharacteristic
26C47×#525—22no—
27C47×#525—22no—
28C47×#525—21no—
29C47×#525—21no—
30C47×#525—21no—
31C47×#525—21no—
32C2×#123×C47C2212yes—
33C2×C22⋊Q8×C47C412yescomplemented
34C2×#121×C47C412yescomplemented
35C2×#120×C47C2212yes—
36C2×C4×D4×C47C414yescomplemented
37C2×#117×C47C2211yescharacteristic
38C23×Dic47—22no—
39#13093983—28no—
40#13093983—28no—
41#13093983—24no—
42#13093983—28no—
43#13093983—24no—
44#13093983—24no—
45#13093983D418yescomplemented
46C2×C4⋊Dic47—28no—
47C2×C4⋊Dic47—22no—
48C2×C4⋊Dic47—22no—
49C2×C4⋊Dic47—22no—
50C2×C22⋊Dic47—22no—
51C2×C22⋊Dic47—28no—
52C2×C22⋊Dic47—24no—
53C2×C22⋊Dic47C2314yes—
54C2×C22⋊Dic47C2314yes—
55C2×C4×Dic47—28no—
56C2×C4×Dic47—22no—
57C2×C4×Dic47—22no—
58C2×C4×Dic47—22no—
59#13093939—216no—
60C47⋊#95—24no—
61C25×C47C2311yescharacteristic
62C2×#41×C47—21no—
63C2×#41×C47—21no—
64C2×#41×C47—21no—
65C2×#41×C47—21no—
66C2×#41×C47—24no—
67C2×#41×C47—24no—
68C22×Q8×C47C2×C411yescharacteristic, complemented
69C22×D4×C47C2×C414yes—
70C22×D4×C47C2×C411yescharacteristic, complemented
71C22×D4×C47C2312yes—
72C23×C4×C47—24no—
73C23×C4×C47—24no—
74C23×C4×C47C2×C412yes—
75C23×C4×C47C2×C412yes—
76C23×C4×C47C2×C412yescomplemented
77C23×C4×C47D412yes—
78C23×C4×C47D412yes—
79C23×C4×C47D411yescharacteristic
80C23×C4×C47D411yescharacteristic
81C23×C4×C47D411yescharacteristic
82C23×C4×C47D411yescharacteristic
83C23×C4×C47C2311yescharacteristic
84C23×C4×C47C2311yescharacteristic
85#123×C47—24no—
86#123×C47—22no—
87#123×C47—22no—
88C22⋊Q8×C47—22no—
89C22⋊Q8×C47—22no—
90C22⋊Q8×C47—22no—
91C22⋊Q8×C47—22no—
92#121×C47—22no—
93#121×C47—22no—
94#121×C47—22no—
95#121×C47—22no—
96#120×C47—22no—
97#120×C47—24no—
98#120×C47—22no—
99C4×D4×C47—24no—
100C4×D4×C47—24no—
101C4×D4×C47—24no—
102C4×D4×C47—24no—
103#117×C47—21no—
104#117×C47—21no—
105#117×C47—21no—
106#117×C47—21no—
107C2×C4⋊C4×C47C2×C412yescomplemented
108C2×C4⋊C4×C47C2×C414yescomplemented
109C2×C22⋊C4×C47C2×C414yes—
110C2×C22⋊C4×C47C2×C414yescomplemented
111C2×C22⋊C4×C47C2312yes—
112C2×C42×C47C2×C412yescomplemented
113C24×C47—22no—
114C24×C47—22no—
115C24×C47—44no—
116C24×C47C22×C412yes—
117C24×C47C22×C412yes—
118C24×C47C2×D411yescharacteristic
119C24×C47C2×D412yes—
120C24×C47C2×D411yescharacteristic
121C24×C47C2411yescharacteristic
122#41×C47—42no—
123#41×C47—42no—
124#41×C47—44no—
125#41×C47—44no—
126#41×C47—42no—
127#41×C47—42no—
128C2×Q8×C47—21no—
129C2×Q8×C47—21no—
130C2×Q8×C47—24no—
131C2×D4×C47—24no—
132C2×D4×C47—24no—
133C2×D4×C47—22no—
134C2×D4×C47—22no—
135C2×D4×C47—24no—
136C2×D4×C47—24no—
137C2×D4×C47—21no—
138C2×D4×C47—21no—
139C2×D4×C47—24no—
140C2×D4×C47—48no—
141C22×C4×C47—22no—
142C22×C4×C47—22no—
143C22×C4×C47—28no—
144C22×C4×C47—28no—
145C22×C4×C47—22no—
146C22×C4×C47—22no—
147C22×C4×C47—22no—
148C22×C4×C47—22no—
149C22×C4×C47—28no—
150C22×C4×C47—28no—
151C22×C4×C47—22no—
152C22×C4×C47—22no—
153C22×C4×C47—22no—
154C22×C4×C47—22no—
155C22×C4×C47—22no—
156C22×C4×C47—22no—
157C22×C4×C47—28no—
158C22×C4×C47C22⋊C412yes—
159C22×C4×C47C22⋊C412yes—
160C22×C4×C47C22⋊C412yes—
161C22×C4×C47C22⋊C412yes—
162C22×C4×C47C22⋊C412yes—
163C22×C4×C47C22⋊C412yes—
164C22×C4×C47C22⋊C412yes—
165C22×C4×C47C22⋊C412yes—
166C22×C4×C47C22×C412yes—
167C22×C4×C47C22×C412yes—
168C22×C4×C47C22×C414yes—
169C22×C4×C47C22×C411yescharacteristic
170C22×C4×C47C22×C411yescharacteristic
171C22×C4×C47C2×D412yes—
172C22×C4×C47C2×D411yescharacteristic
173C22×C4×C47C2×D411yescharacteristic
174C22×C4×C47C2×D411yescharacteristic
175C22×C4×C47C2×D411yescharacteristic
176C22×C4×C47C2×D411yescharacteristic
177C22×C4×C47C2×D411yescharacteristic
178C22×C4×C47C2×D414yes—
179C22×C4×C47#4114yes—
180C22×Dic47—28no—
181C22×Dic47—24no—
182C22×Dic47—24no—
183C22×Dic47—48no—
184C4⋊C4×C47—22no—
185C4⋊C4×C47—22no—
186C4⋊C4×C47—24no—
187C4⋊C4×C47—24no—
188C22⋊C4×C47—24no—
189C22⋊C4×C47—24no—
190C22⋊C4×C47—22no—
191C22⋊C4×C47—22no—
192C22⋊C4×C47—24no—
193C22⋊C4×C47—24no—
194C42×C47—22no—
195C42×C47—22no—
196C22⋊Dic47—28no—
197C22⋊Dic47—24no—
198C22⋊Dic47—24no—
199C22⋊Dic47—24no—
200C22⋊Dic47—48no—
201C22⋊Dic47C2×D418yes—
202C4⋊Dic47—28no—
203C4⋊Dic47—28no—
204C4⋊Dic47—44no—
205C4×Dic47—28no—
206C4×Dic47—28no—
207C4×Dic47—44no—
208C23×C47—21no—
209C23×C47—21no—
210C23×C47—24no—
211C23×C47—24no—
212C23×C47—21no—
213C23×C47—24no—
214C23×C47—21no—
215C23×C47—24no—
216C23×C47—24no—
217C23×C47—22no—
218C23×C47—24no—
219C23×C47—42no—
220C23×C47—42no—
221C23×C47—42no—
222C23×C47—42no—
223C23×C47—48no—
224C23×C47—48no—
225C23×C47—44no—
226C23×C47C2×C22⋊C412yes—
227C23×C47C2×C22⋊C412yes—
228C23×C47C23×C411yescharacteristic
229C23×C47C22×D412yes—
230C23×C47C22×D411yescharacteristic
231C23×C47C22×D411yescharacteristic
232C23×C47C2×#4112yes—
233Q8×C47—42no—
234Q8×C47—42no—
235D4×C47—48no—
236D4×C47—42no—
237D4×C47—44no—
238D4×C47—42no—
239D4×C47—48no—
240D4×C47—42no—
241D4×C47—42no—
242C2×C4×C47—24no—
243C2×C4×C47—21no—
244C2×C4×C47—21no—
245C2×C4×C47—24no—
246C2×C4×C47—21no—
247C2×C4×C47—21no—
248C2×C4×C47—24no—
249C2×C4×C47—24no—
250C2×C4×C47—24no—
251C2×C4×C47—24no—
252C2×C4×C47—24no—
253C2×C4×C47—24no—
254C2×C4×C47—24no—
255C2×C4×C47—24no—
256C2×C4×C47—24no—
257C2×C4×C47—22no—
258C2×C4×C47—24no—
259C2×C4×C47—44no—
260C2×C4×C47—44no—
261C2×C4×C47—44no—
262C2×C4×C47C2×C22⋊C412yes—
263C2×C4×C47C2×C22⋊C412yes—
264C2×C4×C47C2×C22⋊C411yescharacteristic
265C2×C4×C47C2×C22⋊C411yescharacteristic
266C2×C4×C47C2×C22⋊C411yescharacteristic
267C2×C4×C47C2×C22⋊C411yescharacteristic
268C2×C4×C47C4×D414yes—
269C2×C4×C47C4×D414yes—
270C2×C4×C47#12012yes—
271C2×C4×C47#12012yes—
272C2×C4×C47#12114yes—
273C2×C4×C47#12114yes—
274C2×C4×C47#12312yes—
275C2×C4×C47#12312yes—
276C2×Dic47—28no—
277C2×Dic47—416no—
278C2×Dic47—48no—
279C2×Dic47—48no—
280C2×Dic47—44no—
281#20983—471nomaximal, Sylow
282C22×C47—24no—
283C22×C47—24no—
284C22×C47—22no—
285C22×C47—22no—
286C22×C47—22no—
287C22×C47—22no—
288C22×C47—22no—
289C22×C47—22no—
290C22×C47—22no—
291C22×C47—22no—
292C22×C47—42no—
293C22×C47—42no—
294C22×C47—48no—
295C22×C47—48no—
296C22×C47—42no—
297C22×C47—42no—
298C22×C47—44no—
299C22×C47—42no—
300C22×C47—84no—
301C22×C47#41014yescomplemented
302C22×C47C22×C22⋊C411yescharacteristic
303C22×C47C2×C4×D412yes—
304C22×C47C2×#12011yescharacteristic
305C22×C47C2×#12112yes—
306C22×C47C2×#12311yescharacteristic
307C4×C47—22no—
308C4×C47—22no—
309C4×C47—44no—
310C4×C47—42no—
311C4×C47#38612yes—
312C4×C47#38612yes—
313C4×C47#38612yes—
314C4×C47#38612yes—
315Dic47—48no—
316Dic47—84no—
317C2×#525D4711yescharacteristic
318C22×#117—471no—
319C2×#410—472no—
320#2280—471no—
321#1899—474no—
322#1898—474no—
323#1740—472no—
324C2×C47—21no—
325C2×C47—22no—
326C2×C47—21no—
327C2×C47—24no—
328C2×C47—22no—
329C2×C47—42no—
330C2×C47—82no—
331C2×C47C2×#38611yescharacteristic
332C2×C47C2×#41012yes—
333C22×#41C2×D4711yescharacteristic
334C24×C4C2×D4711yescharacteristic
335#525—21no—
336#525—21no—
337#525—21no—
338#525—22no—
339#525—22no—
340#525—21no—
341C2×#123C2×D4712yes—
342C2×C22⋊Q8Dic4712yescomplemented
343C2×#121Dic4712yescomplemented
344C2×#120C2×D4712yes—
345C2×C4×D4Dic4714yescomplemented
346C2×#117—472no—
347C2×#117—472no—
348C2×#117—472no—
349C2×#117—948no—
350C2×#117—941no—
351C2×#117—941no—
352C2×#117C2×D4711yescharacteristic
353C22×C4⋊C4—941no—
354C22×C22⋊C4—472no—
355C22×C42—941no—
356#410—948no—
357C2×#99—478no—
358#381—472no—
359#380—942no—
360#379—472no—
361C4×C22⋊C4—942no—
362#342—478no—
363C47#2098311yescharacteristic, complemented, Sylow
364C25C22×D4711yescharacteristic
365C2×#41—24no—
366C2×#41—24no—
367C2×#41—21no—
368C2×#41—21no—
369C2×#41—21no—
370C2×#41—21no—
371C22×Q8C2×Dic4711yescharacteristic, complemented
372C22×D4C2×Dic4714yes—
373C22×D4C2×Dic4711yescharacteristic, complemented
374C22×D4C22×D4712yes—
375C23×C4—24no—
376C23×C4—24no—
377C23×C4—942no—
378C23×C4C2×Dic4712yes—
379C23×C4C2×Dic4712yes—
380C23×C4C2×Dic4712yescomplemented
381C23×C4C22⋊D4711yescharacteristic
382C23×C4C22⋊D4711yescharacteristic
383C23×C4C22⋊D4711yescharacteristic
384C23×C4C22⋊D4712yes—
385C23×C4C22⋊D4712yes—
386C23×C4C22⋊D4711yescharacteristic
387C23×C4C22×D4711yescharacteristic
388C23×C4C22×D4711yescharacteristic
389#123—24no—
390#123—22no—
391#123—22no—
392C22⋊Q8—22no—
393C22⋊Q8—22no—
394C22⋊Q8—22no—
395C22⋊Q8—22no—
396#121—22no—
397#121—22no—
398#121—22no—
399#121—22no—
400#120—22no—
401#120—24no—
402#120—22no—
403C4×D4—24no—
404C4×D4—24no—
405C4×D4—24no—
406C4×D4—24no—
407#117—21no—
408#117—21no—
409#117—21no—
410#117—21no—
411#117—478no—
412#117—948no—
413#117—948no—
414#117—948no—
415#117—944no—
416#117—944no—
417#117—944no—
418C2×C4⋊C4—948no—
419C2×C4⋊C4—942no—
420C2×C4⋊C4—942no—
421C2×C4⋊C4—942no—
422C2×C4⋊C4C2×Dic4712yescomplemented
423C2×C4⋊C4C2×Dic4714yescomplemented
424C2×C22⋊C4—474no—
425C2×C22⋊C4—474no—
426C2×C22⋊C4—948no—
427C2×C22⋊C4—942no—
428C2×C22⋊C4—944no—
429C2×C22⋊C4C2×Dic4714yes—
430C2×C22⋊C4C2×Dic4714yescomplemented
431C2×C22⋊C4C22×D4712yes—
432C2×C42—948no—
433C2×C42—942no—
434C2×C42—942no—
435C2×C42—942no—
436C2×C42C2×Dic4712yescomplemented
437#99—9416no—
438#95—944no—
439C24—22no—
440C24—22no—
441C24—44no—
442C24C22×Dic4712yes—
443C24C22×Dic4712yes—
444C24C2×C22⋊D4711yescharacteristic
445C24C2×C22⋊D4712yes—
446C24C2×C22⋊D4711yescharacteristic
447C24C23×D4711yescharacteristic
448#41—42no—
449#41—42no—
450#41—44no—
451#41—44no—
452#41—42no—
453#41—42no—
454C2×Q8—21no—
455C2×Q8—21no—
456C2×Q8—24no—
457C2×D4—24no—
458C2×D4—24no—
459C2×D4—24no—
460C2×D4—22no—
461C2×D4—22no—
462C2×D4—24no—
463C2×D4—21no—
464C2×D4—21no—
465C2×D4—24no—
466C2×D4—48no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—28no—
470C22×C4—28no—
471C22×C4—22no—
472C22×C4—22no—
473C22×C4—22no—
474C22×C4—22no—
475C22×C4—28no—
476C22×C4—28no—
477C22×C4—22no—
478C22×C4—22no—
479C22×C4—22no—
480C22×C4—22no—
481C22×C4—22no—
482C22×C4—22no—
483C22×C4—28no—
484C22×C4—948no—
485C22×C4—944no—
486C22×C4—944no—
487C22×C4—1888no—
488C22×C4C22⋊Dic4712yes—
489C22×C4C22⋊Dic4712yes—
490C22×C4C22⋊Dic4712yes—
491C22×C4C22⋊Dic4712yes—
492C22×C4C22⋊Dic4712yes—
493C22×C4C22⋊Dic4712yes—
494C22×C4C22⋊Dic4712yes—
495C22×C4C22⋊Dic4712yes—
496C22×C4D4×D4714yes—
497C22×C4#1062535214yes—
498C22×C4C22×Dic4712yes—
499C22×C4C22×Dic4712yes—
500C22×C4C22×Dic4714yes—
501C22×C4C22×Dic4711yescharacteristic
502C22×C4C22×Dic4711yescharacteristic
503C22×C4C2×C22⋊D4712yes—
504C22×C4C2×C22⋊D4711yescharacteristic
505C22×C4C2×C22⋊D4711yescharacteristic
506C22×C4C2×C22⋊D4711yescharacteristic
507C22×C4C2×C22⋊D4711yescharacteristic
508C22×C4C2×C22⋊D4711yescharacteristic
509C22×C4C2×C22⋊D4711yescharacteristic
510C4⋊C4—22no—
511C4⋊C4—22no—
512C4⋊C4—24no—
513C4⋊C4—24no—
514C4⋊C4—948no—
515C4⋊C4—948no—
516C4⋊C4—1884no—
517C22⋊C4—24no—
518C22⋊C4—24no—
519C22⋊C4—22no—
520C22⋊C4—22no—
521C22⋊C4—24no—
522C22⋊C4—24no—
523C22⋊C4—478no—
524C22⋊C4—948no—
525C22⋊C4—944no—
526C22⋊C4—944no—
527C22⋊C4—944no—
528C22⋊C4—1888no—
529C42—22no—
530C42—22no—
531C42—948no—
532C42—948no—
533C42—1884no—
534C23—21no—
535C23—24no—
536C23—21no—
537C23—24no—
538C23—21no—
539C23—24no—
540C23—24no—
541C23—24no—
542C23—21no—
543C23—22no—
544C23—24no—
545C23—42no—
546C23—42no—
547C23—42no—
548C23—42no—
549C23—48no—
550C23—48no—
551C23—44no—
552C23C2×C22⋊Dic4712yes—
553C23C2×C22⋊Dic4712yes—
554C23C23×Dic4711yescharacteristic
555C23C2×D4×D4712yes—
556C23C22×C22⋊D4711yescharacteristic
557C23C22×C22⋊D4711yescharacteristic
558C23C2×#1062535212yes—
559Q8—42no—
560Q8—42no—
561D4—48no—
562D4—42no—
563D4—48no—
564D4—44no—
565D4—42no—
566D4—42no—
567D4—42no—
568C2×C4—24no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—24no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—24no—
575C2×C4—24no—
576C2×C4—24no—
577C2×C4—24no—
578C2×C4—24no—
579C2×C4—24no—
580C2×C4—24no—
581C2×C4—24no—
582C2×C4—24no—
583C2×C4—22no—
584C2×C4—24no—
585C2×C4—44no—
586C2×C4—44no—
587C2×C4—44no—
588C2×C4—948no—
589C2×C4—1888no—
590C2×C4—1884no—
591C2×C4—18816no—
592C2×C4—1888no—
593C2×C4C2×C22⋊Dic4712yes—
594C2×C4C2×C22⋊Dic4712yes—
595C2×C4C2×C22⋊Dic4711yescharacteristic
596C2×C4C2×C22⋊Dic4711yescharacteristic
597C2×C4C2×C22⋊Dic4711yescharacteristic
598C2×C4C2×C22⋊Dic4711yescharacteristic
599C2×C4D4×Dic4714yes—
600C2×C4D4×Dic4714yes—
601C2×C4#1309399212yes—
602C2×C4#1309399212yes—
603C2×C4#1309399614yes—
604C2×C4#1309399614yes—
605C2×C4#1309400512yes—
606C2×C4#1309400512yes—
607C22—24no—
608C22—24no—
609C22—22no—
610C22—22no—
611C22—22no—
612C22—22no—
613C22—22no—
614C22—22no—
615C22—22no—
616C22—22no—
617C22—42no—
618C22—42no—
619C22—48no—
620C22—48no—
621C22—42no—
622C22—42no—
623C22—42no—
624C22—44no—
625C22—84no—
626C22#1568909714yescomplemented
627C22C22×C22⋊Dic4711yescharacteristic
628C22C2×D4×Dic4712yes—
629C22C2×#1309399211yescharacteristic
630C22C2×#1309399612yes—
631C22C2×#1309400511yescharacteristic
632C4—22no—
633C4—22no—
634C4—42no—
635C4—44no—
636C4—3764no—
637C4—1888no—
638C4#1568900312yes—
639C4#1568900312yes—
640C4#1568900312yes—
641C4#1568900312yes—
642C2—22no—
643C2—21no—
644C2—24no—
645C2—21no—
646C2—22no—
647C2—42no—
648C2—82no—
649C2C2×#1568900311yescharacteristic
650C2C2×#1568909712yes—
subgroupG/H
centerC2×C4C2×C22⋊Dic47
commutatorC23×C47C23×C4
FrattiniC24C23×D47
Fitting, prime radicalC2×C47×#525C2
socleC22×C47C22×C22⋊C4
cosocleC24×C47C24
hypercenterC2×#525D47
hypocenterC47#20983
LCSG ▷ C23×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C2×C4 ◁ C24×C4 ◁ C2×#525

Complements

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

NH ≅ G/Ntype
1. C22×#13093983C2semidirect
2. C2×#15689097C2semidirect
3. #20245771C2semidirect
4. #20243847C2semidirect
5. #20243842C2semidirect
6. #20243121C2semidirect
33. C2×C22⋊Q8×C47C4semidirect
34. C2×#121×C47C4semidirect
36. C2×C4×D4×C47C4semidirect
45. #13093983D4semidirect
68. C22×Q8×C47C2×C4semidirect
70. C22×D4×C47C2×C4semidirect
76. C23×C4×C47C2×C4semidirect
107. C2×C4⋊C4×C47C2×C4semidirect
108. C2×C4⋊C4×C47C2×C4semidirect
110. C2×C22⋊C4×C47C2×C4semidirect
112. C2×C42×C47C2×C4semidirect
301. C22×C47#410semidirect
342. C2×C22⋊Q8Dic47semidirect
343. C2×#121Dic47semidirect
345. C2×C4×D4Dic47semidirect
363. C47#20983semidirect
371. C22×Q8C2×Dic47semidirect
373. C22×D4C2×Dic47semidirect
380. C23×C4C2×Dic47semidirect
422. C2×C4⋊C4C2×Dic47semidirect
423. C2×C4⋊C4C2×Dic47semidirect
430. C2×C22⋊C4C2×Dic47semidirect
436. C2×C42C2×Dic47semidirect
626. C22#15689097semidirect

Maximal subgroups

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

subgroupindex
C22×#130939832
C2×#156890972
#202457712
#202438472
#202438422
#202431212
C2×C47×#5252
#2098347