Finite Group Databasebeta

← #26244368

Subgroups — #26244368

Order 12032.

1110 conjugacy classes of subgroups.

The subgroup lattice has 668 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689753C211yesmaximal, characteristic
2C2×#15689164C211yesmaximal, characteristic
3C2×#15689158C211yesmaximal, characteristic
4#20246003C211yesmaximal, characteristic
5#20244728C211yesmaximal, characteristic, complemented
6#20244725C211yesmaximal, characteristic, complemented
7#20244723C211yesmaximal, characteristic, complemented
8#20244722C211yesmaximal, characteristic, complemented
9#20244720C211yesmaximal, characteristic, complemented
10#20244716C211yesmaximal, characteristic, complemented
11#20244714C211yesmaximal, characteristic, complemented
12#20243106C211yesmaximal, characteristic
13#340×D47C211yesmaximal, characteristic
14#20243083C211yesmaximal, characteristic
15C47×#2059C211yesmaximal, characteristic, complemented
16C22×#10625324C2211yescharacteristic
17C22×C8×D47C2211yescharacteristic
18#15689753—24no—
19#15689753—24no—
20C2×#13094024C2211yescharacteristic
21C2×#13094023—21no—
22C2×#13094023C2211yescharacteristic
23C2×#13094015C2211yescharacteristic
24C2×#13094014—21no—
25C2×#13094004—21no—
26C2×#13094002—21no—
27C2×#13093982C2211yescharacteristic
28C2×#13093982C2211yescharacteristic
29C2×#13093981C2211yescharacteristic
30C2×C4⋊C4×D47C2211yescharacteristic
31C2×C4⋊C4×D47C2211yescharacteristic
32#15689164—24no—
33#15689158—24no—
34C2×#13093957C2211yescharacteristic
35C2×#13093957C2211yescharacteristic
36C2×#13093957C2211yescharacteristic
37C2×#13093956C2211yescharacteristic
38C2×#13093956C2211yescharacteristic
39C2×#13093955C2211yescharacteristic
40#15689022C2211yescharacteristic
41#15689021C2211yescharacteristic
42#15689020—21no—
43#15689019C2211yescharacteristic
44#15689018—21no—
45#15688982—21no—
46#15688980—21no—
47#15688886C2211yescharacteristic
48#15688886C2211yescharacteristic
49#15688885C2211yescharacteristic
50#15688885C2211yescharacteristic
51#15688885C2211yescharacteristic
52#15688885C2211yescharacteristic
53#15688878C2211yescharacteristic
54#15688877C2211yescharacteristic
55#15688877C2211yescharacteristic
56C2×#125×C47C2211yescharacteristic
57C2×#107×C47C2211yescharacteristic
58C2×#106×C47C2211yescharacteristic
59C47×#389C2211yescharacteristic
60C47×#340C2211yescharacteristic
61C47×#340C2211yescharacteristic
62C47×#336C2211yescharacteristic
63C22×Dic94—21no—
64C22×Dic94—21no—
65C22×C4×D47—21no—
66C22×C4×D47—21no—
67C22×C4×D47C2311yescharacteristic
68C2×#10625324—22no—
69C2×#10625324—22no—
70C2×#10625324Q814yescomplemented
71C22×C47⋊C8C2311yescharacteristic
72C22×C47⋊C8C2311yescharacteristic
73C2×C8×D47—22no—
74C2×C8×D47—22no—
75C2×C8×D47Q814yescomplemented
76#13094024—22no—
77#13094024—22no—
78#13094023—28no—
79#13094023—22no—
80#13094023—22no—
81#13094015—22no—
82#13094015—22no—
83#13094014—28no—
84#13094004—28no—
85#13094002—28no—
86#13093982—22no—
87#13093982—22no—
88#13093982—22no—
89#13093982D414yes—
90#13093981—22no—
91#13093981—22no—
92C2×C4⋊Dic47—21no—
93C2×C4⋊Dic47—21no—
94C2×C4⋊Dic47—21no—
95C2×C4⋊Dic47C2311yescharacteristic
96C2×C4⋊Dic47C2311yescharacteristic
97C2×C4⋊Dic47C2311yescharacteristic
98C2×C4⋊Dic47C2311yescharacteristic
99C2×Dic47⋊C4—21no—
100C2×Dic47⋊C4—21no—
101C2×Dic47⋊C4—21no—
102C2×Dic47⋊C4—21no—
103C2×Dic47⋊C4—21no—
104C2×Dic47⋊C4—21no—
105C2×Dic47⋊C4D411yescharacteristic
106C2×Dic47⋊C4D411yescharacteristic
107C4⋊C4×D47—22no—
108C4⋊C4×D47—22no—
109C4⋊C4×D47D414yes—
110C4⋊C4×D47Q814yescomplemented
111C2×(C2×D47)⋊C4—21no—
112C2×(C2×D47)⋊C4—21no—
113C2×(C2×D47)⋊C4D411yescharacteristic
114C2×(C2×D47)⋊C4D411yescharacteristic
115C2×C4×Dic47—21no—
116C2×C4×Dic47—21no—
117C2×C4×Dic47C2311yescharacteristic
118#13093957—22no—
119#13093957—22no—
120#13093957—22no—
121#13093956—22no—
122#13093956—22no—
123#13093955—22no—
124C47⋊#95—21no—
125C47⋊#95—21no—
126C47⋊#95—21no—
127C22×C8×C47C2311yescharacteristic
128C22×C8×C47C2311yescharacteristic
129#125×C47—22no—
130#125×C47—22no—
131C2×C4⋊C4×C47—21no—
132C2×C4⋊C4×C47—21no—
133C2×C4⋊C4×C47—21no—
134C2×C4⋊C4×C47C2311yescharacteristic
135C2×C4⋊C4×C47C2311yescharacteristic
136C2×C4⋊C4×C47C2311yescharacteristic
137C2×C4⋊C4×C47C2311yescharacteristic
138C2×C42×C47C2311yescharacteristic
139#107×C47—22no—
140#106×C47—22no—
141#95×C47—21no—
142C23×D47C2×D411yescharacteristic
143C22×C4×C47—21no—
144C22×C4×C47—21no—
145C22×C4×C47—21no—
146C22×C4×C47—21no—
147C22×C4×C47C2×D411yescharacteristic
148C22×C4×C47C2×D411yescharacteristic
149C22×C4×C47C2411yescharacteristic
150C22×Dic47—21no—
151C22×Dic47—21no—
152C22×Dic47—21no—
153C22×Dic47—21no—
154C22×Dic47C2×D411yescharacteristic
155C22×Dic47C2×D411yescharacteristic
156C22×Dic47C2×D411yescharacteristic
157C2×C4×D47—24no—
158C2×C4×D47—24no—
159C2×C4×D47—81no—
160C2×C4×D47—81no—
161C2×C4×D47—81no—
162C2×C4×D47—81no—
163C2×C4×D47C2×D412yes—
164C2×C4×D47C2×D412yes—
165C2×C4×D47C2×Q812yes—
166C2×C4×D47C2×Q812yes—
167C2×C4×D47C2×Q812yes—
168C2×C4×D47#4112yes—
169C2×Dic94—24no—
170C2×Dic94—24no—
171C2×Dic94—81no—
172C2×Dic94—81no—
173C2×Dic94—81no—
174C2×Dic94—81no—
175C2×C8×C47—21no—
176C2×C8×C47—21no—
177C2×C8×C47—21no—
178C2×C8×C47—21no—
179C2×C8×C47C2×Q812yes—
180C2×C8×C47C2×Q812yes—
181C4⋊C4×C47—24no—
182C4⋊C4×C47—24no—
183C4⋊C4×C47—21no—
184C4⋊C4×C47—21no—
185C4⋊C4×C47—21no—
186C4⋊C4×C47—21no—
187C4⋊C4×C47—21no—
188C4⋊C4×C47—81no—
189C4⋊C4×C47C2×D412yes—
190C4⋊C4×C47C2×D412yes—
191C4⋊C4×C47C2×Q812yes—
192C42×C47—21no—
193C42×C47—21no—
194(C2×D47)⋊C4—24no—
195(C2×D47)⋊C4—22no—
196(C2×D47)⋊C4—22no—
197(C2×D47)⋊C4—42no—
198C4⋊Dic47—24no—
199C4⋊Dic47—24no—
200C4⋊Dic47—21no—
201C4⋊Dic47—21no—
202C4⋊Dic47—21no—
203C4⋊Dic47—21no—
204C4⋊Dic47—21no—
205C4⋊Dic47—81no—
206C4⋊Dic47C2×D412yes—
207C4⋊Dic47C2×D412yes—
208C4⋊Dic47C2×Q812yes—
209Dic47⋊C4—24no—
210Dic47⋊C4—24no—
211Dic47⋊C4—24no—
212Dic47⋊C4—22no—
213Dic47⋊C4—22no—
214Dic47⋊C4—24no—
215Dic47⋊C4—81no—
216Dic47⋊C4—81no—
217C4×Dic47—21no—
218C4×Dic47—21no—
219C4×Dic47—24no—
220C4×Dic47—42no—
221C2×C47⋊C8—21no—
222C2×C47⋊C8—21no—
223C2×C47⋊C8—21no—
224C2×C47⋊C8—21no—
225C2×C47⋊C8C2×Q812yes—
226C2×C47⋊C8C2×Q812yes—
227#10625324—28no—
228C8×D47—28no—
229C23×C47C22×D411yescharacteristic
230C22×D47—21no—
231C22×D47—21no—
232C22×D47—21no—
233C22×D47—21no—
234C22×D47#12012yes—
235C22×D47C22⋊Q812yes—
236C22×D47C4⋊Q812yes—
237C2×C4×C47—22no—
238C2×C4×C47—22no—
239C2×C4×C47—21no—
240C2×C4×C47—21no—
241C2×C4×C47—22no—
242C2×C4×C47—21no—
243C2×C4×C47—21no—
244C2×C4×C47—21no—
245C2×C4×C47—21no—
246C2×C4×C47—41no—
247C2×C4×C47—41no—
248C2×C4×C47—41no—
249C2×C4×C47—41no—
250C2×C4×C47—41no—
251C2×C4×C47—41no—
252C2×C4×C47—41no—
253C2×C4×C47—41no—
254C2×C4×C47—41no—
255C2×C4×C47C22×D411yescharacteristic
256C2×C4×C47C22×D411yescharacteristic
257C2×C4×C47C22×Q811yescharacteristic
258C2×C4×C47C22×Q811yescharacteristic
259C2×C4×C47C22×Q811yescharacteristic
260C2×C4×C47C2×#4111yescharacteristic
261C2×Dic47—21no—
262C2×Dic47—21no—
263C2×Dic47—22no—
264C2×Dic47—22no—
265C2×Dic47—22no—
266C2×Dic47—21no—
267C2×Dic47—21no—
268C2×Dic47—21no—
269C2×Dic47—21no—
270C2×Dic47—41no—
271C2×Dic47—41no—
272C2×Dic47—41no—
273C2×Dic47—41no—
274C2×Dic47—41no—
275C2×Dic47—41no—
276C2×Dic47—41no—
277C2×Dic47—41no—
278C2×Dic47—41no—
279C2×Dic47#12012yes—
280C2×Dic47C22⋊Q812yes—
281C2×Dic47C4⋊Q812yes—
282C4×D47—84no—
283C4×D47—84no—
284C4×D47C22⋊Q814yes—
285C4×D47C22⋊Q814yes—
286C4×D47C4⋊Q814yes—
287C4×D47C4⋊Q814yes—
288Dic94—84no—
289Dic94—84no—
290C8×C47—22no—
291C8×C47—22no—
292C47⋊C8—22no—
293C47⋊C8—22no—
294#23557—471nomaximal, Sylow
295C22×C47C2×#12011yescharacteristic
296C22×C47C2×C22⋊Q811yescharacteristic
297C22×C47C2×C4⋊Q811yescharacteristic
298C22×C47#57611yescharacteristic
299C22×C47#57711yescharacteristic
300C22×C47#57711yescharacteristic
301C22×C47#57811yescharacteristic
302C2×D47—22no—
303C2×D47—22no—
304C2×D47—22no—
305C2×D47—22no—
306C2×D47—22no—
307C2×D47—22no—
308C2×D47#39514yes—
309C4×C47—42no—
310C4×C47—41no—
311C4×C47—42no—
312C4×C47—41no—
313C4×C47—42no—
314C4×C47—81no—
315C4×C47C2×C22⋊Q811yescharacteristic
316C4×C47C2×C22⋊Q811yescharacteristic
317C4×C47C2×C4⋊Q811yescharacteristic
318C4×C47C2×C4⋊Q811yescharacteristic
319Dic47—42no—
320Dic47—41no—
321Dic47—42no—
322Dic47—42no—
323Dic47—41no—
324Dic47—81no—
325Dic47#39514yes—
326C2×#544—471no—
327C2×#428—471no—
328C2×#427—471no—
329#2316—471no—
330#2059—471no—
331#2059—471no—
332#2059—471no—
333#2059D4711yescharacteristic, complemented
334#2058—471no—
335#2058—471no—
336#2057—471no—
337#2057—471no—
338#1736—471no—
339C2×#340—471no—
340#1729—471no—
341C2×C47C2×#39511yescharacteristic
342C2×C47#301211yescharacteristic
343C2×C47#301311yescharacteristic
344C2×C47#308111yescharacteristic
345C2×C47#308211yescharacteristic
346C2×C47#316311yescharacteristic
347C2×C47#316411yescharacteristic
348D47—24no—
349C22×M16—471no—
350C23×C8—471no—
351#544—944no—
352#544—944no—
353C2×C4⋊Q8—471no—
354C2×C4⋊Q8—471no—
355C2×C4⋊Q8—941no—
356C2×#125—471no—
357C2×#125—941no—
358C2×#125C2×D4711yescharacteristic
359C2×C22⋊Q8—941no—
360C2×C22⋊Q8—941no—
361C2×#117—471no—
362C2×#117—471no—
363C2×#117—471no—
364C22×C4⋊C4—471no—
365C22×C4⋊C4—471no—
366#428—944no—
367#427—944no—
368C2×#107—471no—
369C2×#107—471no—
370C2×#107—471no—
371C2×#107C2×D4711yescharacteristic
372C2×#106—471no—
373C2×#106—471no—
374C2×#106—471no—
375C2×#106C2×D4711yescharacteristic
376#389—471no—
377#389—471no—
378#389—471no—
379#389—941no—
380#389—941no—
381#389C2×D4711yescharacteristic
382#380—941no—
383#380—941no—
384#340—471no—
385#340—471no—
386#340—471no—
387#340—471no—
388#340—471no—
389#340—471no—
390#340C2×D4711yescharacteristic
391#340C2×D4711yescharacteristic
392#336—471no—
393#336—471no—
394#336—471no—
395#336C2×D4711yescharacteristic
396C47#2355711yescharacteristic, complemented, Sylow
397C22×Q8—941no—
398C22×Q8—941no—
399C23×C4—471no—
400C23×C4—941no—
401C23×C4—941no—
402C2×M16—474no—
403C2×M16—942no—
404C2×M16—942no—
405C22×C8—471no—
406C22×C8—471no—
407C22×C8—474no—
408C22×C8—942no—
409C22×C8—942no—
410C22×C8C22×D4711yescharacteristic
411C22×C8C22×D4711yescharacteristic
412C4⋊Q8—942no—
413C4⋊Q8—942no—
414C4⋊Q8—942no—
415C4⋊Q8—948no—
416C4⋊Q8—942no—
417#125—22no—
418#125—22no—
419#125—942no—
420#125—948no—
421#125—942no—
422C22⋊Q8—948no—
423C22⋊Q8—948no—
424#117—474no—
425#117—942no—
426#117—942no—
427#117—942no—
428#117—942no—
429#117—942no—
430C2×C4⋊C4—21no—
431C2×C4⋊C4—21no—
432C2×C4⋊C4—21no—
433C2×C4⋊C4—471no—
434C2×C4⋊C4—471no—
435C2×C4⋊C4—471no—
436C2×C4⋊C4—474no—
437C2×C4⋊C4—474no—
438C2×C4⋊C4—471no—
439C2×C4⋊C4—471no—
440C2×C4⋊C4—471no—
441C2×C4⋊C4—942no—
442C2×C4⋊C4—942no—
443C2×C4⋊C4—941no—
444C2×C4⋊C4—941no—
445C2×C4⋊C4—941no—
446C2×C4⋊C4—941no—
447C2×C4⋊C4—941no—
448C2×C4⋊C4—941no—
449C2×C4⋊C4—941no—
450C2×C4⋊C4—941no—
451C2×C4⋊C4—941no—
452C2×C4⋊C4C22×D4711yescharacteristic
453C2×C4⋊C4C22×D4711yescharacteristic
454C2×C4⋊C4C22×D4711yescharacteristic
455C2×C4⋊C4C22×D4711yescharacteristic
456C2×C22⋊C4—471no—
457C2×C22⋊C4—471no—
458C2×C22⋊C4—941no—
459C2×C22⋊C4—941no—
460C2×C42—471no—
461C2×C42—941no—
462C2×C42—941no—
463C2×C42C22×D4711yescharacteristic
464#107—22no—
465#107—942no—
466#107—942no—
467#107—942no—
468#106—22no—
469#106—942no—
470#106—942no—
471#106—942no—
472#95—21no—
473#95—941no—
474#95—941no—
475#95—941no—
476C24—471no—
477C2×Q8—944no—
478C2×Q8—944no—
479C2×Q8—3761no—
480C2×Q8—3761no—
481C2×Q8—3761no—
482C2×Q8—3761no—
483C22×C4—21no—
484C22×C4—21no—
485C22×C4—21no—
486C22×C4—21no—
487C22×C4—471no—
488C22×C4—472no—
489C22×C4—472no—
490C22×C4—472no—
491C22×C4—472no—
492C22×C4—472no—
493C22×C4—472no—
494C22×C4—471no—
495C22×C4—471no—
496C22×C4—944no—
497C22×C4—944no—
498C22×C4—941no—
499C22×C4—941no—
500C22×C4—941no—
501C22×C4—941no—
502C22×C4—3761no—
503C22×C4—3761no—
504C22×C4—3761no—
505C22×C4—3761no—
506C22×C4D4×D4711yescharacteristic
507C22×C4D4×D4711yescharacteristic
508C22×C4C23×D4711yescharacteristic
509M16—948no—
510C2×C8—21no—
511C2×C8—21no—
512C2×C8—21no—
513C2×C8—21no—
514C2×C8—472no—
515C2×C8—472no—
516C2×C8—948no—
517C2×C8—941no—
518C2×C8—941no—
519C2×C8—941no—
520C2×C8—941no—
521C2×C8Q8×D4712yes—
522C2×C8Q8×D4712yes—
523C4⋊C4—24no—
524C4⋊C4—24no—
525C4⋊C4—21no—
526C4⋊C4—21no—
527C4⋊C4—21no—
528C4⋊C4—21no—
529C4⋊C4—21no—
530C4⋊C4—81no—
531C4⋊C4—472no—
532C4⋊C4—472no—
533C4⋊C4—472no—
534C4⋊C4—944no—
535C4⋊C4—944no—
536C4⋊C4—942no—
537C4⋊C4—942no—
538C4⋊C4—944no—
539C4⋊C4—944no—
540C4⋊C4—944no—
541C4⋊C4—944no—
542C4⋊C4—941no—
543C4⋊C4—941no—
544C4⋊C4—941no—
545C4⋊C4—941no—
546C4⋊C4—941no—
547C4⋊C4—3761no—
548C4⋊C4—3761no—
549C4⋊C4—3761no—
550C4⋊C4D4×D4712yes—
551C4⋊C4D4×D4712yes—
552C4⋊C4Q8×D4712yes—
553C22⋊C4—942no—
554C22⋊C4—942no—
555C22⋊C4—944no—
556C22⋊C4—1882no—
557C42—21no—
558C42—21no—
559C42—944no—
560C42—941no—
561C42—941no—
562C42—1882no—
563C23—472no—
564C23—472no—
565C23—472no—
566C23—941no—
567C23—941no—
568C23—941no—
569C23—941no—
570C23C2×D4×D4711yescharacteristic
571Q8—3764no—
572Q8—3764no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—472no—
592C2×C4—472no—
593C2×C4—472no—
594C2×C4—474no—
595C2×C4—474no—
596C2×C4—474no—
597C2×C4—474no—
598C2×C4—942no—
599C2×C4—941no—
600C2×C4—941no—
601C2×C4—942no—
602C2×C4—942no—
603C2×C4—941no—
604C2×C4—941no—
605C2×C4—941no—
606C2×C4—941no—
607C2×C4—3764no—
608C2×C4—3764no—
609C2×C4—1881no—
610C2×C4—1881no—
611C2×C4—1881no—
612C2×C4—1881no—
613C2×C4—1881no—
614C2×C4—1881no—
615C2×C4—1881no—
616C2×C4—1881no—
617C2×C4—1881no—
618C2×C4C2×D4×D4711yescharacteristic
619C2×C4C2×D4×D4711yescharacteristic
620C2×C4C2×Q8×D4711yescharacteristic
621C2×C4C2×Q8×D4711yescharacteristic
622C2×C4C2×Q8×D4711yescharacteristic
623C2×C4#41×D4711yescharacteristic
624C8—22no—
625C8—22no—
626C8—942no—
627C8—942no—
628C22—474no—
629C22—942no—
630C22—942no—
631C22—942no—
632C22—942no—
633C22—942no—
634C22—942no—
635C22#120×D4711yescharacteristic
636C22C22⋊Q8×D4711yescharacteristic
637C22C4⋊Q8×D4711yescharacteristic
638C22#1568992711yescharacteristic
639C22#1568993011yescharacteristic
640C22#1568993011yescharacteristic
641C22#1568993911yescharacteristic
642C4—41no—
643C4—42no—
644C4—42no—
645C4—41no—
646C4—42no—
647C4—81no—
648C4—474no—
649C4—3761no—
650C4—1881no—
651C4—1882no—
652C4—1882no—
653C4—1882no—
654C4—1881no—
655C4C22⋊Q8×D4711yescharacteristic
656C4C22⋊Q8×D4711yescharacteristic
657C4C4⋊Q8×D4711yescharacteristic
658C4C4⋊Q8×D4711yescharacteristic
659C2—944no—
660C2#395×D4711yescharacteristic
661C2#2025269011yescharacteristic
662C2#2025270711yescharacteristic
663C2#2025321811yescharacteristic
664C2#2025322711yescharacteristic
665C2#2025391711yescharacteristic
666C2#2025392311yescharacteristic
subgroupG/H
centerC23C2×D4×D47
commutator, cosocleC22×C4×C47C24
FrattiniC22×C4C23×D47
FittingC47×#2059C2
socleC23×C47C22×D4
prime radicalC23×D47C2×D4
hypercenter#2059D47
hypocenterC47#23557
LCSG ▷ C22×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2059

Complements

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

NH ≅ G/Ntype
5. #20244728C2semidirect
6. #20244725C2semidirect
7. #20244723C2semidirect
8. #20244722C2semidirect
9. #20244720C2semidirect
10. #20244716C2semidirect
11. #20244714C2semidirect
15. C47×#2059C2semidirect
70. C2×#10625324Q8semidirect
75. C2×C8×D47Q8semidirect
110. C4⋊C4×D47Q8semidirect
333. #2059D47semidirect
396. C47#23557semidirect

Maximal subgroups

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

subgroupindex
C2×#156897532
C2×#156891642
C2×#156891582
#202460032
#202447282
#202447252
#202447232
#202447222
#202447202
#202447162
#202447142
#202431062
#340×D472
#202430832
C47×#20592
#2355747