Finite Group Databasebeta

← #26222581

Subgroups — #26222581

Order 12032.

1004 conjugacy classes of subgroups.

The subgroup lattice has 666 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689342C211yesmaximal, characteristic
2C2×#15689228C211yesmaximal, characteristic, complemented
3C2×#15689223C211yesmaximal, characteristic
4C42×Dic94C211yesmaximal, characteristic, complemented
5#20244321C211yesmaximal, characteristic
6#20244154C211yesmaximal, characteristic, complemented
7#20244026C211yesmaximal, characteristic, complemented
8#20244012C211yesmaximal, characteristic
9#20243746C211yesmaximal, characteristic
10#20243732C211yesmaximal, characteristic, complemented
11#20243400C211yesmaximal, characteristic, complemented
12#20243241C211yesmaximal, characteristic, complemented
13#20243218C211yesmaximal, characteristic, complemented
14#20243213C211yesmaximal, characteristic
15C4×#124×C47C211yesmaximal, characteristic
16C22×#10625337C2211yescharacteristic
17C22×#10625335C2211yescharacteristic
18C2×C4×Dic94C2211yescharacteristic
19C2×C4×Dic94C2211yescharacteristic
20C2×C4×Dic94C2211yescharacteristic
21#15689342C418yescomplemented
22#15689228—24no—
23#15689223—24no—
24C2×#13093957C2211yescharacteristic
25C2×#13093955C2211yescharacteristic
26C2×C4⋊(C47⋊C8)C2211yescharacteristic
27C2×C4⋊(C47⋊C8)C2211yescharacteristic
28C2×#13093948C2211yescharacteristic
29C2×#13093948C2211yescharacteristic
30C2×#13093948C2211yescharacteristic
31C2×#13093947C2211yescharacteristic
32C2×#13093947C2211yescharacteristic
33C2×#13093945C2211yescharacteristic
34C2×#13093945C2211yescharacteristic
35C2×#13093945C2211yescharacteristic
36C2×#13093935C2211yescharacteristic
37C2×#13093935C2211yescharacteristic
38C4×Dic47⋊C4—21no—
39C4×Dic47⋊C4—21no—
40C4×Dic47⋊C4—21no—
41C4×C4⋊Dic47C2211yescharacteristic
42C4×C4⋊Dic47C2211yescharacteristic
43C4×C4⋊Dic47C2211yescharacteristic
44C42×Dic47—21no—
45#15688885C2211yescharacteristic
46#15688885C2211yescharacteristic
47#15688885C2211yescharacteristic
48#15688885C2211yescharacteristic
49#15688878C2211yescharacteristic
50#15688878C2211yescharacteristic
51C2×#124×C47C2211yescharacteristic
52C2×C4×Q8×C47C2211yescharacteristic
53C2×C4×D4×C47C2211yescharacteristic
54C47×#391C2211yescharacteristic
55C47×#390C2211yescharacteristic
56C47×#388—21no—
57C47×#388—21no—
58C47×#382C2211yescharacteristic
59C4×C22⋊C4×C47—21no—
60C4×C22⋊C4×C47—21no—
61C43×C47C2211yescharacteristic
62C22×Dic94C2311yescharacteristic
63C2×#10625337—22no—
64C2×#10625337—22no—
65C2×#10625337C2×C414yescomplemented
66C2×#10625335—22no—
67C2×#10625335—22no—
68C2×#10625335C2×C414yes—
69C22×C47⋊C8C2311yescharacteristic
70C22×C47⋊C8C2311yescharacteristic
71C22×C47⋊C8C2311yescharacteristic
72C22×C47⋊C8C2311yescharacteristic
73C4×Dic94—22no—
74C4×Dic94—22no—
75C4×Dic94—22no—
76C4×Dic94—22no—
77C4×Dic94C2×C414yescomplemented
78C4×Dic94D414yes—
79C2×C4⋊Dic47C2311yescharacteristic
80C2×C4⋊Dic47C2311yescharacteristic
81C2×C4⋊Dic47C2311yescharacteristic
82C2×Dic47⋊C4—21no—
83C2×Dic47⋊C4—21no—
84C2×Dic47⋊C4—21no—
85C2×C4×Dic47—21no—
86C2×C4×Dic47—21no—
87C2×C4×Dic47—21no—
88C2×C4×Dic47—21no—
89C2×C4×Dic47—21no—
90C2×C4×Dic47—21no—
91#13093957—22no—
92#13093955—22no—
93C4⋊(C47⋊C8)—22no—
94C4⋊(C47⋊C8)C2×C414yescomplemented
95#13093948—22no—
96#13093948—22no—
97#13093948C2×C414yescomplemented
98#13093947—22no—
99#13093947—22no—
100#13093945—22no—
101#13093945—22no—
102#13093945C2×C414yes—
103#13093935—22no—
104#13093935—22no—
105C47⋊#95—21no—
106C47⋊#95—21no—
107C47⋊#95—21no—
108C22×Q8×C47C2311yescharacteristic
109C22×D4×C47C2311yescharacteristic
110C23×C4×C47—21no—
111#124×C47—22no—
112#124×C47C2×C414yes—
113C4×Q8×C47—22no—
114C4×Q8×C47—22no—
115C4×D4×C47—22no—
116C4×D4×C47—22no—
117C2×C4⋊C4×C47—21no—
118C2×C4⋊C4×C47C2311yescharacteristic
119C2×C4⋊C4×C47C2311yescharacteristic
120C2×C22⋊C4×C47—21no—
121C2×C22⋊C4×C47—21no—
122C2×C22⋊C4×C47—21no—
123C2×C22⋊C4×C47—21no—
124C2×C22⋊C4×C47—21no—
125C2×C42×C47—21no—
126C2×C42×C47D411yescharacteristic
127C2×C42×C47D411yescharacteristic
128C2×C42×C47D411yescharacteristic
129C2×C42×C47D411yescharacteristic
130C2×C42×C47C2311yescharacteristic
131C2×C42×C47C2311yescharacteristic
132C2×C42×C47C2311yescharacteristic
133#95×C47—21no—
134#95×C47—21no—
135#95×C47—21no—
136#95×C47—21no—
137C24×C47—21no—
138C2×Q8×C47—21no—
139C2×Q8×C47—21no—
140C2×Q8×C47—21no—
141C2×Q8×C47—21no—
142C2×Q8×C47—21no—
143C2×Q8×C47C22×C412yes—
144C2×D4×C47—21no—
145C2×D4×C47—21no—
146C2×D4×C47—21no—
147C2×D4×C47—21no—
148C2×D4×C47—21no—
149C2×D4×C47C22×C412yes—
150C22×C4×C47—21no—
151C22×C4×C47—21no—
152C22×C4×C47—21no—
153C22×C4×C47—42no—
154C22×C4×C47—42no—
155C22×C4×C47—42no—
156C22×C4×C47C2×D411yescharacteristic
157C22×C4×C47C2×D411yescharacteristic
158C22×C4×C47C2×D411yescharacteristic
159C22×C4×C47C2×D411yescharacteristic
160C22×C4×C47C2×D411yescharacteristic
161C22×C4×C47C2×D411yescharacteristic
162C22×C4×C47C2411yescharacteristic
163C22×Dic47—21no—
164C22×Dic47—21no—
165C22×Dic47—21no—
166C22×Dic47—21no—
167C2×Dic94—21no—
168C2×Dic94—21no—
169C2×Dic94—21no—
170C2×Dic94C22×C412yes—
171C2×Dic94C2×D412yes—
172C2×Dic94#4112yes—
173C4⋊C4×C47—21no—
174C4⋊C4×C47—21no—
175C4⋊C4×C47—21no—
176C4⋊C4×C47—21no—
177C4⋊C4×C47—42no—
178C22⋊C4×C47—24no—
179C22⋊C4×C47—24no—
180C22⋊C4×C47—42no—
181C22⋊C4×C47—42no—
182C22⋊C4×C47—42no—
183C42×C47—22no—
184C42×C47—21no—
185C42×C47—21no—
186C42×C47—41no—
187C42×C47—41no—
188C42×C47—41no—
189C42×C47—42no—
190C42×C47C22×C412yes—
191C42×C47C2×D412yes—
192C42×C47#4112yes—
193C42×C47#4112yes—
194#10625337—28no—
195#10625335—28no—
196C4⋊Dic47—21no—
197C4⋊Dic47—21no—
198C4⋊Dic47—21no—
199C4⋊Dic47C22×C412yes—
200C4⋊Dic47C2×D412yes—
201C4⋊Dic47#4112yes—
202Dic47⋊C4—24no—
203Dic47⋊C4—24no—
204Dic47⋊C4—24no—
205C4×Dic47—24no—
206C4×Dic47—24no—
207C4×Dic47—24no—
208C4×Dic47—42no—
209C4×Dic47—42no—
210C4×Dic47—42no—
211C2×C47⋊C8—21no—
212C2×C47⋊C8—21no—
213C2×C47⋊C8—21no—
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⋊C8C22×C412yes—
222C2×C47⋊C8C22×C412yes—
223C23×C47—22no—
224C23×C47—41no—
225C23×C47—41no—
226C23×C47—41no—
227C23×C47—41no—
228C23×C47—41no—
229C23×C47—41no—
230C23×C47C22×D411yescharacteristic
231Q8×C47—22no—
232Q8×C47—22no—
233Q8×C47—41no—
234Q8×C47—41no—
235D4×C47—22no—
236D4×C47—22no—
237D4×C47—41no—
238D4×C47—41no—
239C2×C4×C47—21no—
240C2×C4×C47—21no—
241C2×C4×C47—21no—
242C2×C4×C47—21no—
243C2×C4×C47—21no—
244C2×C4×C47—21no—
245C2×C4×C47—21no—
246C2×C4×C47—21no—
247C2×C4×C47—21no—
248C2×C4×C47—21no—
249C2×C4×C47—22no—
250C2×C4×C47—21no—
251C2×C4×C47—21no—
252C2×C4×C47—21no—
253C2×C4×C47—41no—
254C2×C4×C47—41no—
255C2×C4×C47—41no—
256C2×C4×C47—41no—
257C2×C4×C47—41no—
258C2×C4×C47—41no—
259C2×C4×C47—41no—
260C2×C4×C47—41no—
261C2×C4×C47—84no—
262C2×C4×C47C4×D412yes—
263C2×C4×C47C4×D412yes—
264C2×C4×C47#12112yes—
265C2×C4×C47#12112yes—
266C2×C4×C47#13712yes—
267C2×C4×C47C23×C411yescharacteristic
268C2×C4×C47C22×D411yescharacteristic
269C2×C4×C47C2×#4111yescharacteristic
270C2×C4×C47C2×#4111yescharacteristic
271C2×C4×C47C2×#4111yescharacteristic
272C2×C4×C47C2×#4111yescharacteristic
273C2×Dic47—22no—
274C2×Dic47—22no—
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—
285Dic94—22no—
286Dic94—22no—
287Dic94—22no—
288Dic94C4×D414yes—
289C47⋊C8—22no—
290C47⋊C8—22no—
291C47⋊C8—41no—
292C47⋊C8—41no—
293#21508—471nomaximal, Sylow
294C22×C47—42no—
295C22×C47—42no—
296C22×C47—42no—
297C22×C47—81no—
298C22×C47—81no—
299C22×C47—81no—
300C22×C47—81no—
301C22×C47C2×C4×D411yescharacteristic
302C22×C47C2×#12111yescharacteristic
303C22×C47#52511yescharacteristic
304C22×C47C2×#13711yescharacteristic
305C22×C47#57511yescharacteristic
306C22×C47#57511yescharacteristic
307C22×C47#57511yescharacteristic
308C4×C47—22no—
309C4×C47—22no—
310C4×C47—22no—
311C4×C47—22no—
312C4×C47—22no—
313C4×C47—41no—
314C4×C47—42no—
315C4×C47—81no—
316C4×C47—81no—
317C4×C47C2×C4×D411yescharacteristic
318C4×C47C4×#4111yescharacteristic
319C4×C47#52911yescharacteristic
320C4×C47#53211yescharacteristic
321Dic47—24no—
322Dic47—42no—
323Dic47—42no—
324Dic47—42no—
325C2×#464—471no—
326C2×#441—471no—
327C2×#440—471no—
328C4×#124D4711yescharacteristic
329C42×Q8—471no—
330#1983—471no—
331#1949—471no—
332#1928—471no—
333#1926—471no—
334#1880—471no—
335#1878—471no—
336#1805—471no—
337#1767—471no—
338#1763—471no—
339#1762—471no—
340C2×C47—82no—
341C2×C47C4×#12111yescharacteristic
342C2×C47#294311yescharacteristic
343C2×C47C4×#13711yescharacteristic
344C2×C47#303711yescharacteristic
345C2×C47#304011yescharacteristic
346C2×C47#311111yescharacteristic
347C2×C47#311711yescharacteristic
348C22×Q16—471no—
349C22×SD16—471no—
350C2×#124C2×D4711yescharacteristic
351C2×C4×Q8—471no—
352C2×C4×Q8—471no—
353C2×C4×Q8—471no—
354C2×C4×Q8C2×D4711yescharacteristic
355C2×C4×D4C2×D4711yescharacteristic
356#464—478no—
357#441—944no—
358#440—944no—
359C2×#107—471no—
360C2×#106—471no—
361C2×C4⋊C8—471no—
362C2×C4⋊C8—471no—
363C2×#103—471no—
364C2×#103—471no—
365C2×#103—471no—
366C2×#103—471no—
367C2×#103—471no—
368C2×#102—471no—
369C2×#102—471no—
370C2×#102—471no—
371C2×#97—471no—
372C2×#97—471no—
373#391C2×D4711yescharacteristic
374#390C2×D4711yescharacteristic
375#388—21no—
376#388—21no—
377#382C2×D4711yescharacteristic
378C4×C4⋊C4—471no—
379C4×C4⋊C4—471no—
380C4×C4⋊C4—471no—
381C4×C4⋊C4—941no—
382C4×C4⋊C4—941no—
383C4×C4⋊C4—941no—
384C4×C22⋊C4—21no—
385C4×C22⋊C4—21no—
386C43—941no—
387C43C2×D4711yescharacteristic
388#340—471no—
389#340—471no—
390#340—471no—
391#340—471no—
392#336—471no—
393#336—471no—
394C47#2150811yescharacteristic, complemented, Sylow
395C22×Q8—471no—
396C22×Q8C22×D4711yescharacteristic
397C22×D4C22×D4711yescharacteristic
398C23×C4—21no—
399C2×Q16—474no—
400C2×Q16—942no—
401C2×Q16—942no—
402C2×SD16—474no—
403C2×SD16—942no—
404C2×SD16—942no—
405C22×C8—471no—
406C22×C8—471no—
407C22×C8—471no—
408C22×C8—471no—
409#124—22no—
410#124C4×D4714yes—
411C4×Q8—22no—
412C4×Q8—22no—
413C4×Q8—474no—
414C4×Q8—474no—
415C4×Q8—942no—
416C4×Q8—942no—
417C4×Q8—942no—
418C4×Q8—942no—
419C4×D4—22no—
420C4×D4—22no—
421C2×C4⋊C4—21no—
422C2×C4⋊C4—471no—
423C2×C4⋊C4—471no—
424C2×C4⋊C4—471no—
425C2×C4⋊C4—941no—
426C2×C4⋊C4—941no—
427C2×C4⋊C4—941no—
428C2×C4⋊C4C22×D4711yescharacteristic
429C2×C4⋊C4C22×D4711yescharacteristic
430C2×C22⋊C4—21no—
431C2×C22⋊C4—21no—
432C2×C22⋊C4—21no—
433C2×C22⋊C4—21no—
434C2×C22⋊C4—21no—
435C2×C42—21no—
436C2×C42—941no—
437C2×C42—941no—
438C2×C42—941no—
439C2×C42—941no—
440C2×C42—941no—
441C2×C42—941no—
442C2×C42C22⋊D4711yescharacteristic
443C2×C42C22⋊D4711yescharacteristic
444C2×C42C22⋊D4711yescharacteristic
445C2×C42C22⋊D4711yescharacteristic
446C2×C42C22×D4711yescharacteristic
447C2×C42C22×D4711yescharacteristic
448C2×C42C22×D4711yescharacteristic
449#107—942no—
450#106—942no—
451C4⋊C8—474no—
452C4⋊C8—942no—
453#103—474no—
454#103—942no—
455#103—942no—
456#103—942no—
457#103—942no—
458#102—474no—
459#102—942no—
460#102—942no—
461#97—942no—
462#97—942no—
463#95—21no—
464#95—21no—
465#95—21no—
466#95—21no—
467#95—941no—
468#95—941no—
469#95—941no—
470C24—21no—
471C2×Q8—21no—
472C2×Q8—21no—
473C2×Q8—21no—
474C2×Q8—21no—
475C2×Q8—21no—
476C2×Q8—472no—
477C2×Q8—472no—
478C2×Q8—472no—
479C2×Q8—941no—
480C2×Q8—941no—
481C2×Q8—941no—
482C2×Q8C2×C4×D4712yes—
483C2×D4—21no—
484C2×D4—21no—
485C2×D4—21no—
486C2×D4—21no—
487C2×D4—21no—
488C2×D4C2×C4×D4712yes—
489C22×C4—21no—
490C22×C4—21no—
491C22×C4—21no—
492C22×C4—42no—
493C22×C4—42no—
494C22×C4—42no—
495C22×C4—941no—
496C22×C4—941no—
497C22×C4—941no—
498C22×C4—941no—
499C22×C4C2×C22⋊D4711yescharacteristic
500C22×C4C2×C22⋊D4711yescharacteristic
501C22×C4C2×C22⋊D4711yescharacteristic
502C22×C4C2×C22⋊D4711yescharacteristic
503C22×C4C2×C22⋊D4711yescharacteristic
504C22×C4C2×C22⋊D4711yescharacteristic
505C22×C4C23×D4711yescharacteristic
506Q16—948no—
507SD16—948no—
508C2×C8—472no—
509C2×C8—472no—
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—
520C4⋊C4—21no—
521C4⋊C4—21no—
522C4⋊C4—21no—
523C4⋊C4—21no—
524C4⋊C4—42no—
525C4⋊C4—472no—
526C4⋊C4—472no—
527C4⋊C4—472no—
528C4⋊C4—944no—
529C4⋊C4—944no—
530C4⋊C4—944no—
531C4⋊C4—941no—
532C4⋊C4—941no—
533C4⋊C4—941no—
534C22⋊C4—24no—
535C22⋊C4—24no—
536C22⋊C4—42no—
537C22⋊C4—42no—
538C22⋊C4—42no—
539C42—22no—
540C42—21no—
541C42—21no—
542C42—41no—
543C42—41no—
544C42—41no—
545C42—42no—
546C42—944no—
547C42—944no—
548C42—944no—
549C42—1882no—
550C42—1882no—
551C42—1882no—
552C42C2×C4×D4712yes—
553C42#1062535012yes—
554C42D4×D4712yes—
555C42#1062535212yes—
556C23—22no—
557C23—41no—
558C23—41no—
559C23—41no—
560C23—41no—
561C23—41no—
562C23—41no—
563C23C22×C22⋊D4711yescharacteristic
564Q8—22no—
565Q8—22no—
566Q8—41no—
567Q8—41no—
568Q8—474no—
569Q8—942no—
570Q8—942no—
571Q8—942no—
572D4—22no—
573D4—22no—
574D4—41no—
575D4—41no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—22no—
587C2×C4—21no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—41no—
598C2×C4—84no—
599C2×C4—942no—
600C2×C4—942no—
601C2×C4—942no—
602C2×C4—942no—
603C2×C4—942no—
604C2×C4—942no—
605C2×C4—1881no—
606C2×C4—1881no—
607C2×C4—1881no—
608C2×C4—1881no—
609C2×C4—1881no—
610C2×C4—1881no—
611C2×C4C4×C22⋊D4712yes—
612C2×C4C4×C22⋊D4712yes—
613C2×C4#1309399512yes—
614C2×C4#1309399512yes—
615C2×C4#1309405412yes—
616C2×C4C22×C4×D4711yescharacteristic
617C2×C4C2×D4×D4711yescharacteristic
618C2×C4#41×D4711yescharacteristic
619C2×C4#41×D4711yescharacteristic
620C2×C4C2×#1062535211yescharacteristic
621C2×C4C2×#1062535011yescharacteristic
622C8—942no—
623C8—942no—
624C8—1881no—
625C8—1881no—
626C22—42no—
627C22—42no—
628C22—42no—
629C22—81no—
630C22—81no—
631C22—81no—
632C22—81no—
633C22C2×C4×C22⋊D4711yescharacteristic
634C22C2×#1309399511yescharacteristic
635C22#1568963011yescharacteristic
636C22C2×#1309405411yescharacteristic
637C22#1568992111yescharacteristic
638C22#1568992111yescharacteristic
639C22#1568992111yescharacteristic
640C4—22no—
641C4—22no—
642C4—22no—
643C4—22no—
644C4—22no—
645C4—41no—
646C4—42no—
647C4—81no—
648C4—81no—
649C4—944no—
650C4—1882no—
651C4—1882no—
652C4—1882no—
653C4C4×D4×D4711yescharacteristic
654C4C4×#1062535211yescharacteristic
655C4#1568965511yescharacteristic
656C4#1568967011yescharacteristic
657C2—82no—
658C2C4×#1309399511yescharacteristic
659C2#2025218611yescharacteristic
660C2C4×#1309405411yescharacteristic
661C2#2025285611yescharacteristic
662C2#2025287811yescharacteristic
663C2#2025349111yescharacteristic
664C2#2025355511yescharacteristic
subgroupG/H
centerC23C22×C22⋊D47
commutatorC2×C4×C47C23×C4
FrattiniC22×C4C23×D47
FittingC4×#124×C47C2
socleC23×C47C22×D4
prime radicalC22×D4×C47C23
cosocleC22×C4×C47C24
hypercenterC4×#124D47
hypocenterC47#21508
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C43 ◁ C4×#124

Complements

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

NH ≅ G/Ntype
2. C2×#15689228C2semidirect
4. C42×Dic94C2semidirect
6. #20244154C2semidirect
7. #20244026C2semidirect
10. #20243732C2semidirect
11. #20243400C2semidirect
12. #20243241C2semidirect
13. #20243218C2semidirect
21. #15689342C4semidirect
65. C2×#10625337C2×C4semidirect
77. C4×Dic94C2×C4semidirect
94. C4⋊(C47⋊C8)C2×C4semidirect
97. #13093948C2×C4semidirect
394. C47#21508semidirect

Maximal subgroups

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

subgroupindex
C2×#156893422
C2×#156892282
C2×#156892232
C42×Dic942
#202443212
#202441542
#202440262
#202440122
#202437462
#202437322
#202434002
#202432412
#202432182
#202432132
C4×#124×C472
#2150847