Finite Group Databasebeta

← #26223839

Subgroups — #26223839

Order 12032.

1000 conjugacy classes of subgroups.

The subgroup lattice has 624 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689402C211yesmaximal, characteristic
2#20245959C211yesmaximal, characteristic, complemented
3C4×#13093986C211yesmaximal, characteristic
4#20244338C211yesmaximal, characteristic, complemented
5#20244145C211yesmaximal, characteristic, complemented
6#20243959C211yesmaximal, characteristic, complemented
7#20243638C211yesmaximal, characteristic
8#20243634C211yesmaximal, characteristic
9#20243387C211yesmaximal, characteristic
10#107×Dic47C211yesmaximal, characteristic, complemented
11#20243280C211yesmaximal, characteristic, complemented
12#20243236C211yesmaximal, characteristic, complemented
13#20243202C211yesmaximal, characteristic
14#20243202C211yesmaximal, characteristic
15C47×#1918C211yesmaximal, characteristic, complemented
16C2×#13094023C2211yescharacteristic
17C2×#13093989—21no—
18C2×C4×D188C2211yescharacteristic
19C2×#13093986C2211yescharacteristic
20C2×#13093986C2211yescharacteristic
21C2×C42×D47—21no—
22#15689402C418yescomplemented
23C2×#13093958C2211yescharacteristic
24C2×#13093957C2211yescharacteristic
25C2×#13093957C2211yescharacteristic
26C2×Dic47⋊C8C2211yescharacteristic
27C2×Dic47⋊C8C2211yescharacteristic
28C2×#13093946C2211yescharacteristic
29C2×#13093946C2211yescharacteristic
30C2×#13093944C2211yescharacteristic
31C2×#13093944C2211yescharacteristic
32C2×C8×Dic47C2211yescharacteristic
33C2×C8×Dic47C2211yescharacteristic
34#15689027—21no—
35#15689021C2211yescharacteristic
36#15689018—21no—
37#15688992C2211yescharacteristic
38C4⋊C4×Dic47C2211yescharacteristic
39C4⋊C4×Dic47C2211yescharacteristic
40C4⋊C4×Dic47C2211yescharacteristic
41C4⋊C4×Dic47C2211yescharacteristic
42#15688974—21no—
43C4×(C2×D47)⋊C4—21no—
44#15688970—21no—
45#15688970—21no—
46C42×Dic47C2211yescharacteristic
47#15688886C2211yescharacteristic
48#15688885C2211yescharacteristic
49#15688885C2211yescharacteristic
50#15688885C2211yescharacteristic
51#15688885C2211yescharacteristic
52#15688878C2211yescharacteristic
53C2×#107×C47C2211yescharacteristic
54C47×#389C2211yescharacteristic
55C4×C4⋊C4×C47C2211yescharacteristic
56C47×#340C2211yescharacteristic
57C47×#340C2211yescharacteristic
58C47×#340C2211yescharacteristic
59C47×#336C2211yescharacteristic
60C22×Dic94—21no—
61C22×D188C2311yescharacteristic
62C22×C4×D47—21no—
63C22×C4×D47—21no—
64C22×C4×D47—21no—
65C22×C47⋊C8C2311yescharacteristic
66C22×C47⋊C8C2311yescharacteristic
67#13094023—22no—
68#13094023C2×C414yescomplemented
69#13093989—44no—
70C4×D188—22no—
71C4×D188—22no—
72#13093986—22no—
73#13093986—22no—
74#13093986C2×C414yes—
75#13093986Q814yescomplemented
76C2×C4⋊Dic47C2311yescharacteristic
77C2×C4⋊Dic47C2311yescharacteristic
78C2×C4⋊Dic47C2311yescharacteristic
79C2×Dic47⋊C4—21no—
80C2×Dic47⋊C4—21no—
81C2×Dic47⋊C4—21no—
82C2×Dic47⋊C4—21no—
83C2×(C2×D47)⋊C4—21no—
84C2×(C2×D47)⋊C4—21no—
85C2×(C2×D47)⋊C4—21no—
86C2×C4×Dic47—21no—
87C2×C4×Dic47—21no—
88C2×C4×Dic47—21no—
89C2×C4×Dic47—21no—
90C2×C4×Dic47D411yescharacteristic
91C2×C4×Dic47D411yescharacteristic
92C2×C4×Dic47D411yescharacteristic
93C2×C4×Dic47D411yescharacteristic
94C2×C4×Dic47C2311yescharacteristic
95C2×C4×Dic47C2311yescharacteristic
96C42×D47—44no—
97#13093958—22no—
98#13093957—22no—
99#13093957C2×C414yescomplemented
100Dic47⋊C8—22no—
101Dic47⋊C8C2×C414yescomplemented
102#13093946—22no—
103#13093946C2×C414yes—
104#13093944—22no—
105#13093944C2×C414yes—
106C8×Dic47—22no—
107C8×Dic47—22no—
108C47⋊#95—21no—
109C47⋊#95—21no—
110C47⋊#95—21no—
111C47⋊#95—21no—
112C47⋊#95—21no—
113C22×C8×C47C2311yescharacteristic
114C22×C8×C47C2311yescharacteristic
115C2×C4⋊C4×C47—21no—
116C2×C4⋊C4×C47C2311yescharacteristic
117C2×C4⋊C4×C47C2311yescharacteristic
118C2×C4⋊C4×C47C2311yescharacteristic
119C2×C4⋊C4×C47C2311yescharacteristic
120C2×C42×C47—21no—
121C2×C42×C47C2311yescharacteristic
122#107×C47C2×C414yescomplemented
123#95×C47—21no—
124#95×C47—21no—
125C23×D47—21no—
126C22×C4×C47—21no—
127C22×C4×C47—21no—
128C22×C4×C47—21no—
129C22×C4×C47—21no—
130C22×C4×C47C2×D411yescharacteristic
131C22×C4×C47C2×D411yescharacteristic
132C22×C4×C47C2411yescharacteristic
133C22×Dic47—21no—
134C22×Dic47—21no—
135C22×Dic47—21no—
136C22×Dic47C2×D411yescharacteristic
137C22×Dic47C2×D411yescharacteristic
138C22×Dic47C2×D411yescharacteristic
139C22×Dic47C2×D411yescharacteristic
140C2×D188—21no—
141C2×D188—21no—
142C2×D188—21no—
143C2×D188C22×C412yes—
144C2×D188C2×Q812yes—
145C2×D188#4112yes—
146C2×C4×D47—24no—
147C2×C4×D47—24no—
148C2×C4×D47—24no—
149C2×C4×D47—42no—
150C2×C4×D47—42no—
151C2×C4×D47—42no—
152C2×C4×D47—42no—
153C2×C4×D47—42no—
154C2×C4×D47—42no—
155C2×Dic94—24no—
156C2×Dic94—42no—
157C2×Dic94—42no—
158C2×C8×C47—21no—
159C2×C8×C47—21no—
160C2×C8×C47—21no—
161C2×C8×C47—21no—
162C2×C8×C47—21no—
163C2×C8×C47C22×C412yes—
164C4⋊C4×C47—21no—
165C4⋊C4×C47—21no—
166C4⋊C4×C47—21no—
167C4⋊C4×C47—21no—
168C4⋊C4×C47—21no—
169C4⋊C4×C47—42no—
170C4⋊C4×C47C22×C412yes—
171C4⋊C4×C47C22×C412yes—
172C4⋊C4×C47C2×Q812yes—
173C42×C47—21no—
174C42×C47—21no—
175C42×C47—42no—
176(C2×D47)⋊C4—24no—
177(C2×D47)⋊C4—24no—
178(C2×D47)⋊C4—24no—
179C4⋊Dic47—21no—
180C4⋊Dic47—21no—
181C4⋊Dic47—21no—
182C4⋊Dic47—21no—
183C4⋊Dic47C22×C412yes—
184C4⋊Dic47#4112yes—
185Dic47⋊C4—24no—
186Dic47⋊C4—42no—
187Dic47⋊C4—42no—
188Dic47⋊C4—42no—
189C4×Dic47—22no—
190C4×Dic47—22no—
191C4×Dic47—41no—
192C4×Dic47—42no—
193C4×Dic47—42no—
194C4×Dic47—42no—
195C4×Dic47—42no—
196C4×Dic47D814yes—
197C4×Dic47C22×C412yes—
198C4×Dic47C2×Q812yes—
199C4×Dic47#4112yes—
200C4×Dic47#4112yes—
201C2×C47⋊C8—21no—
202C2×C47⋊C8—21no—
203C2×C47⋊C8—21no—
204C2×C47⋊C8—21no—
205C2×C47⋊C8—21no—
206C2×C47⋊C8C22×C412yes—
207C23×C47C22×D411yescharacteristic
208C22×D47—22no—
209C22×D47—22no—
210C22×D47—22no—
211C22×D47—41no—
212C22×D47—41no—
213C22×D47—41no—
214C22×D47—41no—
215C2×C4×C47—21no—
216C2×C4×C47—21no—
217C2×C4×C47—22no—
218C2×C4×C47—22no—
219C2×C4×C47—22no—
220C2×C4×C47—21no—
221C2×C4×C47—21no—
222C2×C4×C47—21no—
223C2×C4×C47—41no—
224C2×C4×C47—41no—
225C2×C4×C47—41no—
226C2×C4×C47—41no—
227C2×C4×C47—41no—
228C2×C4×C47—41no—
229C2×C4×C47—41no—
230C2×C4×C47—41no—
231C2×C4×C47—41no—
232C2×C4×C47C2×D812yes—
233C2×C4×C47C23×C411yescharacteristic
234C2×C4×C47C22×Q811yescharacteristic
235C2×C4×C47C2×#4111yescharacteristic
236C2×C4×C47C2×#4111yescharacteristic
237C2×C4×C47C2×#4111yescharacteristic
238C2×C4×C47C2×#4111yescharacteristic
239C2×Dic47—21no—
240C2×Dic47—21no—
241C2×Dic47—21no—
242C2×Dic47—21no—
243C2×Dic47—22no—
244C2×Dic47—22no—
245C2×Dic47—41no—
246C2×Dic47—41no—
247C2×Dic47—41no—
248C2×Dic47—41no—
249C2×Dic47—41no—
250C2×Dic47—41no—
251C2×Dic47—41no—
252C2×Dic47C4×D412yes—
253C2×Dic47C4×D412yes—
254C2×Dic47C22⋊Q812yes—
255C2×Dic47C22⋊Q812yes—
256C2×Dic47C2×D812yes—
257C2×Dic47C2×D812yes—
258C2×Dic47#13512yes—
259C2×Dic47#13712yes—
260D188—22no—
261D188—22no—
262D188—22no—
263D188C4×Q814yescomplemented
264C4×D47—48no—
265C4×D47—48no—
266C4×D47—48no—
267Dic94—48no—
268C8×C47—22no—
269C8×C47—41no—
270C47⋊C8—22no—
271C47⋊C8—41no—
272C4×#474—471nomaximal, Sylow
273C22×C47C2×C4×D411yescharacteristic
274C22×C47C2×C22⋊Q811yescharacteristic
275C22×C47#52511yescharacteristic
276C22×C47C22×D811yescharacteristic
277C22×C47C2×#13511yescharacteristic
278C22×C47C2×#13711yescharacteristic
279C22×C47#57511yescharacteristic
280C2×D47—24no—
281C2×D47—42no—
282C2×D47—42no—
283C2×D47—42no—
284C2×D47—42no—
285C2×D47—42no—
286C2×D47—42no—
287C4×C47—22no—
288C4×C47—41no—
289C4×C47—42no—
290C4×C47—42no—
291C4×C47—42no—
292C4×C47—81no—
293C4×C47C2×C4×Q811yescharacteristic
294C4×C47C4×#4111yescharacteristic
295C4×C47#52911yescharacteristic
296C4×C47#53311yescharacteristic
297Dic47—22no—
298Dic47—22no—
299Dic47—42no—
300Dic47—42no—
301Dic47—81no—
302Dic47C4×D814yes—
303Dic47#47414yescomplemented
304C2×#474—471no—
305C4×C4⋊Q8—471no—
306C42×D4—471no—
307#1988—471no—
308#1947—471no—
309#1918—471no—
310#1918D4711yescharacteristic, complemented
311#1862—471no—
312#1862—471no—
313#1802—471no—
314C4×#107—471no—
315C4×#107—471no—
316C4×C4⋊C8—471no—
317C4×#102—471no—
318C4×#102—471no—
319C2×C47C4×C22⋊Q811yescharacteristic
320C2×C47C2×C4×D811yescharacteristic
321C2×C47C4×#13711yescharacteristic
322C2×C47C2×#47411yescharacteristic
323C2×C47#307611yescharacteristic
324C2×C47#310011yescharacteristic
325C2×C47#310711yescharacteristic
326D47—44no—
327C2×C4⋊Q8—471no—
328C2×C4×Q8—941no—
329C2×C4×D4—471no—
330C2×C4×D4—471no—
331C2×C4×D4—471no—
332C22×C42—941no—
333#474—478no—
334C2×#107—471no—
335C2×#107—471no—
336C2×#107—471no—
337C2×#107C2×D4711yescharacteristic
338C2×C4⋊C8—471no—
339C2×C4⋊C8—471no—
340C2×#102—471no—
341C2×#102—471no—
342C2×#102—471no—
343C2×#102—471no—
344C2×C4×C8—471no—
345C2×C4×C8—471no—
346#391—941no—
347#389—471no—
348#389—941no—
349#389C2×D4711yescharacteristic
350#384—471no—
351C4×C4⋊C4—471no—
352C4×C4⋊C4—471no—
353C4×C4⋊C4—471no—
354C4×C4⋊C4—471no—
355C4×C4⋊C4—941no—
356C4×C4⋊C4C2×D4711yescharacteristic
357C4×C22⋊C4—941no—
358C4×C22⋊C4—941no—
359C4×C22⋊C4—941no—
360C43—471no—
361#340—471no—
362#340—471no—
363#340—471no—
364#340—471no—
365#340—471no—
366#340C2×D4711yescharacteristic
367#340C2×D4711yescharacteristic
368#340C2×D4711yescharacteristic
369#336—471no—
370#336C2×D4711yescharacteristic
371C47C4×#47411yescharacteristic, complemented, Sylow
372C22×Q8—941no—
373C22×D4—471no—
374C23×C4—941no—
375C23×C4—941no—
376C23×C4—941no—
377C22×C8—471no—
378C22×C8—471no—
379C22×C8C22×D4711yescharacteristic
380C22×C8C22×D4711yescharacteristic
381C4⋊Q8—474no—
382C4⋊Q8—942no—
383C4×Q8—1884no—
384C4×D4—474no—
385C4×D4—474no—
386C4×D4—942no—
387C4×D4—942no—
388C4×D4—942no—
389C4×D4—942no—
390C2×C4⋊C4—21no—
391C2×C4⋊C4—471no—
392C2×C4⋊C4—471no—
393C2×C4⋊C4—471no—
394C2×C4⋊C4—941no—
395C2×C4⋊C4—941no—
396C2×C4⋊C4—941no—
397C2×C4⋊C4—941no—
398C2×C4⋊C4C22×D4711yescharacteristic
399C2×C4⋊C4C22×D4711yescharacteristic
400C2×C4⋊C4C22×D4711yescharacteristic
401C2×C4⋊C4C22×D4711yescharacteristic
402C2×C22⋊C4—941no—
403C2×C22⋊C4—941no—
404C2×C22⋊C4—941no—
405C2×C42—21no—
406C2×C42—471no—
407C2×C42—471no—
408C2×C42—471no—
409C2×C42—471no—
410C2×C42—471no—
411C2×C42—471no—
412C2×C42—941no—
413C2×C42—941no—
414C2×C42—941no—
415C2×C42—941no—
416C2×C42—1884no—
417C2×C42C22×D4711yescharacteristic
418#107—474no—
419#107—942no—
420#107—942no—
421#107C4×D4714yescomplemented
422C4⋊C8—474no—
423C4⋊C8—942no—
424#102—474no—
425#102—474no—
426#102—942no—
427#102—942no—
428C4×C8—942no—
429C4×C8—942no—
430#95—21no—
431#95—21no—
432#95—941no—
433#95—941no—
434#95—941no—
435#95—941no—
436#95—941no—
437C24—941no—
438C2×Q8—944no—
439C2×Q8—1882no—
440C2×Q8—1882no—
441C2×D4—472no—
442C2×D4—472no—
443C2×D4—472no—
444C2×D4—941no—
445C2×D4—941no—
446C2×D4—941no—
447C22×C4—21no—
448C22×C4—21no—
449C22×C4—21no—
450C22×C4—21no—
451C22×C4—471no—
452C22×C4—471no—
453C22×C4—471no—
454C22×C4—471no—
455C22×C4—944no—
456C22×C4—944no—
457C22×C4—944no—
458C22×C4—941no—
459C22×C4—941no—
460C22×C4—941no—
461C22×C4—1882no—
462C22×C4—1882no—
463C22×C4—1882no—
464C22×C4—1882no—
465C22×C4—1882no—
466C22×C4—1882no—
467C22×C4D4×D4711yescharacteristic
468C22×C4D4×D4711yescharacteristic
469C22×C4C23×D4711yescharacteristic
470C2×C8—21no—
471C2×C8—21no—
472C2×C8—21no—
473C2×C8—21no—
474C2×C8—21no—
475C2×C8—472no—
476C2×C8—941no—
477C2×C8—941no—
478C2×C8—941no—
479C2×C8—941no—
480C2×C8—941no—
481C2×C8C2×C4×D4712yes—
482C4⋊C4—21no—
483C4⋊C4—21no—
484C4⋊C4—21no—
485C4⋊C4—21no—
486C4⋊C4—21no—
487C4⋊C4—42no—
488C4⋊C4—472no—
489C4⋊C4—472no—
490C4⋊C4—944no—
491C4⋊C4—941no—
492C4⋊C4—941no—
493C4⋊C4—941no—
494C4⋊C4—941no—
495C4⋊C4—1882no—
496C4⋊C4—1882no—
497C4⋊C4—1882no—
498C4⋊C4C2×C4×D4712yes—
499C4⋊C4C2×C4×D4712yes—
500C4⋊C4Q8×D4712yes—
501C22⋊C4—944no—
502C22⋊C4—944no—
503C22⋊C4—944no—
504C42—21no—
505C42—21no—
506C42—42no—
507C42—474no—
508C42—472no—
509C42—472no—
510C42—472no—
511C42—472no—
512C42—942no—
513C42—942no—
514C42—1881no—
515C42—1882no—
516C42—1882no—
517C42—1882no—
518C42—1882no—
519C23—942no—
520C23—942no—
521C23—942no—
522C23—1881no—
523C23—1881no—
524C23—1881no—
525C23—1881no—
526C23C2×D4×D4711yescharacteristic
527Q8—1888no—
528D4—474no—
529D4—942no—
530D4—942no—
531D4—942no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—472no—
550C2×C4—472no—
551C2×C4—472no—
552C2×C4—472no—
553C2×C4—472no—
554C2×C4—472no—
555C2×C4—472no—
556C2×C4—472no—
557C2×C4—941no—
558C2×C4—941no—
559C2×C4—941no—
560C2×C4—941no—
561C2×C4—942no—
562C2×C4—942no—
563C2×C4—1888no—
564C2×C4—1888no—
565C2×C4—1888no—
566C2×C4—1881no—
567C2×C4—1881no—
568C2×C4—1881no—
569C2×C4—1881no—
570C2×C4—1881no—
571C2×C4—1881no—
572C2×C4—1881no—
573C2×C4D8×D4712yes—
574C2×C4C22×C4×D4711yescharacteristic
575C2×C4C2×Q8×D4711yescharacteristic
576C2×C4#41×D4711yescharacteristic
577C2×C4#41×D4711yescharacteristic
578C2×C4#41×D4711yescharacteristic
579C2×C4#41×D4711yescharacteristic
580C8—22no—
581C8—41no—
582C8—942no—
583C8—1881no—
584C22—944no—
585C22—1882no—
586C22—1882no—
587C22—1882no—
588C22—1882no—
589C22—1882no—
590C22—1882no—
591C22C4×D4×D4711yescharacteristic
592C22C22⋊Q8×D4711yescharacteristic
593C22#1568963111yescharacteristic
594C22C2×D8×D4711yescharacteristic
595C22#135×D4711yescharacteristic
596C22#137×D4711yescharacteristic
597C22#1568992211yescharacteristic
598C4—22no—
599C4—41no—
600C4—42no—
601C4—42no—
602C4—42no—
603C4—81no—
604C4—474no—
605C4—474no—
606C4—942no—
607C4—942no—
608C4—3761no—
609C4—1882no—
610C4—1882no—
611C4C4×Q8×D4711yescharacteristic
612C4#1568959111yescharacteristic
613C4#1568965611yescharacteristic
614C4#1568967911yescharacteristic
615C2—1884no—
616C2#2024592211yescharacteristic
617C2C4×D8×D4711yescharacteristic
618C2#2025220411yescharacteristic
619C2#474×D4711yescharacteristic
620C2#2025317511yescharacteristic
621C2#2025338511yescharacteristic
622C2#2025345411yescharacteristic
subgroupG/H
centerC23C2×D4×D47
commutatorC2×C4×C47C23×C4
FrattiniC22×C4C23×D47
FittingC47×#1918C2
socleC23×C47C22×D4
prime radicalC22×D188C23
cosocleC22×C4×C47C24
hypercenter#1918D47
hypocenterC47C4×#474
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1918

Complements

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

NH ≅ G/Ntype
2. #20245959C2semidirect
4. #20244338C2semidirect
5. #20244145C2semidirect
6. #20243959C2semidirect
10. #107×Dic47C2semidirect
11. #20243280C2semidirect
12. #20243236C2semidirect
15. C47×#1918C2semidirect
22. #15689402C4semidirect
68. #13094023C2×C4semidirect
75. #13093986Q8semidirect
99. #13093957C2×C4semidirect
101. Dic47⋊C8C2×C4semidirect
122. #107×C47C2×C4semidirect
263. D188C4×Q8semidirect
303. Dic47#474semidirect
310. #1918D47semidirect
371. C47C4×#474semidirect
421. #107C4×D47semidirect

Maximal subgroups

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

subgroupindex
C2×#156894022
#202459592
C4×#130939862
#202443382
#202441452
#202439592
#202436382
#202436342
#202433872
#107×Dic472
#202432802
#202432362
#202432022
C47×#19182
C4×#47447