Finite Group Databasebeta

← #26223474

Subgroups — #26223474

Order 12032.

964 conjugacy classes of subgroups.

The subgroup lattice has 638 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689455C211yesmaximal, characteristic
2C4×#13094004C211yesmaximal, characteristic
3C4×C4⋊C4×D47C211yesmaximal, characteristic
4#20244028C211yesmaximal, characteristic, complemented
5#20244027C211yesmaximal, characteristic, complemented
6#20243969C211yesmaximal, characteristic, complemented
7#20243968C211yesmaximal, characteristic, complemented
8#20243694C211yesmaximal, characteristic
9#20243574C211yesmaximal, characteristic
10#20243473C211yesmaximal, characteristic
11#20243286C211yesmaximal, characteristic, complemented
12#20243221C211yesmaximal, characteristic, complemented
13#20243219C211yesmaximal, characteristic, complemented
14#20243151C211yesmaximal, characteristic
15C47×#1776C211yesmaximal, characteristic, complemented
16C2×#13094004C2211yescharacteristic
17C2×C4×Dic94C2211yescharacteristic
18C2×C4⋊C4×D47C2211yescharacteristic
19C2×C4⋊C4×D47C2211yescharacteristic
20C2×C42×D47—21no—
21C2×C42×D47C2211yescharacteristic
22#15689455C418yescomplemented
23C2×#13093957C2211yescharacteristic
24C2×#13093955C2211yescharacteristic
25C2×#13093949C2211yescharacteristic
26C2×#13093949C2211yescharacteristic
27C2×#13093947C2211yescharacteristic
28C2×#13093947C2211yescharacteristic
29C2×(C2×D47)⋊C8C2211yescharacteristic
30C2×(C2×D47)⋊C8C2211yescharacteristic
31C2×#13093935C2211yescharacteristic
32#15689028C2211yescharacteristic
33#15689022C2211yescharacteristic
34#15689008—21no—
35#15688997—21no—
36#15688980C2211yescharacteristic
37C4⋊C4×Dic47C2211yescharacteristic
38C4⋊C4×Dic47C2211yescharacteristic
39C4×Dic47⋊C4—21no—
40C4×C4⋊Dic47—21no—
41C4×C4⋊Dic47C2211yescharacteristic
42#15688974—21no—
43C4×(C2×D47)⋊C4—21no—
44#95×D47—21no—
45#15688886C2211yescharacteristic
46#15688886C2211yescharacteristic
47#15688885C2211yescharacteristic
48#15688885C2211yescharacteristic
49#15688885C2211yescharacteristic
50#15688885C2211yescharacteristic
51#15688877C2211yescharacteristic
52#15688877C2211yescharacteristic
53C2×#107×C47C2211yescharacteristic
54C2×#106×C47C2211yescharacteristic
55C2×#97×C47C2211yescharacteristic
56C4×C4⋊C4×C47C2211yescharacteristic
57C4×C4⋊C4×C47C2211yescharacteristic
58C47×#340C2211yescharacteristic
59C47×#340C2211yescharacteristic
60C22×Dic94C2311yescharacteristic
61C22×C4×D47—21no—
62C22×C4×D47—21no—
63C22×C4×D47D411yescharacteristic
64C22×C4×D47D411yescharacteristic
65C22×C4×D47C2311yescharacteristic
66C22×C47⋊C8C2311yescharacteristic
67C22×C47⋊C8C2311yescharacteristic
68#13094004—22no—
69#13094004C2×C414yes—
70C4×Dic94—22no—
71C4×Dic94—22no—
72C2×C4⋊Dic47—21no—
73C2×C4⋊Dic47—21no—
74C2×C4⋊Dic47C2311yescharacteristic
75C2×C4⋊Dic47C2311yescharacteristic
76C2×C4⋊Dic47C2311yescharacteristic
77C2×Dic47⋊C4—21no—
78C2×Dic47⋊C4—21no—
79C2×Dic47⋊C4—21no—
80C4⋊C4×D47—22no—
81C4⋊C4×D47—22no—
82C4⋊C4×D47—22no—
83C4⋊C4×D47C2×C414yes—
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×Dic47C2311yescharacteristic
93C42×D47—22no—
94C42×D47—22no—
95C42×D47—44no—
96#13093957—22no—
97#13093955C2×C414yescomplemented
98#13093949—22no—
99#13093949C2×C414yescomplemented
100#13093947—22no—
101#13093947C2×C414yescomplemented
102(C2×D47)⋊C8—22no—
103(C2×D47)⋊C8C2×C414yes—
104#13093935—22no—
105C47⋊#95—21no—
106C47⋊#95—21no—
107C47⋊#95—21no—
108C47⋊#95—21no—
109C47⋊#95—21no—
110C47⋊#95—21no—
111C47⋊#95—21no—
112C22×C8×C47C2311yescharacteristic
113C22×C8×C47C2311yescharacteristic
114C2×C4⋊C4×C47C2311yescharacteristic
115C2×C4⋊C4×C47C2311yescharacteristic
116C2×C4⋊C4×C47C2311yescharacteristic
117C2×C4⋊C4×C47C2311yescharacteristic
118C2×C42×C47—21no—
119C2×C42×C47—21no—
120C2×C42×C47C2311yescharacteristic
121#107×C47—22no—
122#106×C47C2×C414yescomplemented
123#97×C47—22no—
124#95×C47—21no—
125#95×C47—21no—
126C23×D47C2×D411yescharacteristic
127C22×C4×C47—21no—
128C22×C4×C47—21no—
129C22×C4×C47—21no—
130C22×C4×C47—21no—
131C22×C4×C47C2×D411yescharacteristic
132C22×C4×C47C2×D411yescharacteristic
133C22×C4×C47C2411yescharacteristic
134C22×Dic47—21no—
135C22×Dic47—21no—
136C22×Dic47—21no—
137C22×Dic47—21no—
138C22×Dic47C2×D411yescharacteristic
139C22×Dic47C2×D411yescharacteristic
140C22×Dic47C2×D411yescharacteristic
141C2×C4×D47—22no—
142C2×C4×D47—22no—
143C2×C4×D47—21no—
144C2×C4×D47—21no—
145C2×C4×D47—21no—
146C2×C4×D47—21no—
147C2×C4×D47—22no—
148C2×C4×D47—24no—
149C2×C4×D47—24no—
150C2×C4×D47—22no—
151C2×C4×D47—22no—
152C2×C4×D47—41no—
153C2×C4×D47—42no—
154C2×C4×D47—42no—
155C2×C4×D47—42no—
156C2×C4×D47—42no—
157C2×C4×D47C22×C412yes—
158C2×C4×D47#4112yes—
159C2×Dic94—21no—
160C2×Dic94—21no—
161C2×Dic94—21no—
162C2×Dic94—21no—
163C2×Dic94—21no—
164C2×Dic94C22×C412yes—
165C2×C8×C47—21no—
166C2×C8×C47—21no—
167C2×C8×C47—21no—
168C2×C8×C47—21no—
169C2×C8×C47—21no—
170C2×C8×C47C22×C412yes—
171C4⋊C4×C47—21no—
172C4⋊C4×C47—21no—
173C4⋊C4×C47—21no—
174C4⋊C4×C47—21no—
175C4⋊C4×C47—21no—
176C4⋊C4×C47C22×C412yes—
177C4⋊C4×C47C22×C412yes—
178C4⋊C4×C47#4112yes—
179C42×C47—42no—
180C42×C47—42no—
181C42×C47#4112yes—
182C42×C47#4112yes—
183(C2×D47)⋊C4—24no—
184(C2×D47)⋊C4—42no—
185C4⋊Dic47—21no—
186C4⋊Dic47—21no—
187C4⋊Dic47—24no—
188C4⋊Dic47—21no—
189C4⋊Dic47—21no—
190C4⋊Dic47—42no—
191C4⋊Dic47C22×C412yes—
192C4⋊Dic47#4112yes—
193Dic47⋊C4—24no—
194Dic47⋊C4—24no—
195Dic47⋊C4—42no—
196C4×Dic47—22no—
197C4×Dic47—22no—
198C4×Dic47—21no—
199C4×Dic47—21no—
200C4×Dic47—42no—
201C4×Dic47—42no—
202C4×Dic47—42no—
203C4×Dic47—42no—
204C2×C47⋊C8—21no—
205C2×C47⋊C8—21no—
206C2×C47⋊C8—21no—
207C2×C47⋊C8—21no—
208C2×C47⋊C8—21no—
209C2×C47⋊C8C22×C412yes—
210C23×C47C22×D411yescharacteristic
211C22×D47—21no—
212C22×D47—21no—
213C22×D47—21no—
214C22×D47—21no—
215C22×D47C4×D412yes—
216C22×D47AGL(1,ℤ/8)12yes—
217C22×D47#13712yes—
218C2×C4×C47—21no—
219C2×C4×C47—21no—
220C2×C4×C47—21no—
221C2×C4×C47—21no—
222C2×C4×C47—22no—
223C2×C4×C47—22no—
224C2×C4×C47—22no—
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×C47—41no—
233C2×C4×C47—41no—
234C2×C4×C47#12312yes—
235C2×C4×C47#12312yes—
236C2×C4×C47C23×C411yescharacteristic
237C2×C4×C47C2×#4111yescharacteristic
238C2×C4×C47C2×#4111yescharacteristic
239C2×C4×C47C2×#4111yescharacteristic
240C2×C4×C47C2×#4111yescharacteristic
241C2×C4×C47C2×#4111yescharacteristic
242C2×Dic47—21no—
243C2×Dic47—21no—
244C2×Dic47—21no—
245C2×Dic47—21no—
246C2×Dic47—22no—
247C2×Dic47—22no—
248C2×Dic47—21no—
249C2×Dic47—22no—
250C2×Dic47—21no—
251C2×Dic47—21no—
252C2×Dic47—41no—
253C2×Dic47—41no—
254C2×Dic47—41no—
255C2×Dic47—41no—
256C2×Dic47—41no—
257C2×Dic47—41no—
258C2×Dic47—41no—
259C2×Dic47—41no—
260C2×Dic47—41no—
261C2×Dic47C4×D412yes—
262C2×Dic47#13712yes—
263C4×D47—28no—
264C4×D47—22no—
265C4×D47—22no—
266C4×D47—22no—
267C4×D47—22no—
268C4×D47—44no—
269C4×D47—48no—
270C4×D47—48no—
271Dic94—22no—
272Dic94—22no—
273Dic94—41no—
274Dic94—41no—
275C8×C47—22no—
276C8×C47—41no—
277C47⋊C8—22no—
278C47⋊C8—41no—
279#21579—471nomaximal, Sylow
280C22×C47C2×C4×D411yescharacteristic
281C22×C47C2×#12311yescharacteristic
282C22×C47#52511yescharacteristic
283C22×C47C2×AGL(1,ℤ/8)11yescharacteristic
284C22×C47C2×#13711yescharacteristic
285C22×C47C2×#13711yescharacteristic
286C22×C47#57511yescharacteristic
287C2×D47—22no—
288C2×D47—22no—
289C2×D47—22no—
290C2×D47—22no—
291C2×D47—22no—
292C2×D47—22no—
293C2×D47#44014yes—
294C4×C47—22no—
295C4×C47—22no—
296C4×C47—42no—
297C4×C47—42no—
298C4×C47—42no—
299C4×C47—81no—
300C4×C47C4×#4111yescharacteristic
301C4×C47C4×#4111yescharacteristic
302C4×C47#52911yescharacteristic
303C4×C47#52911yescharacteristic
304Dic47—22no—
305Dic47—22no—
306Dic47—41no—
307Dic47—42no—
308Dic47—42no—
309Dic47—42no—
310Dic47—81no—
311C2×#483—471no—
312C4×C22⋊Q8—471no—
313C2×C4×C4⋊C4—471no—
314#1928—471no—
315#1928—471no—
316#1919—471no—
317#1919—471no—
318#1872—471no—
319#1852—471no—
320#1823—471no—
321#1776—471no—
322#1776D4711yescharacteristic, complemented
323#1763—471no—
324#1763—471no—
325#1749—471no—
326C2×C47C4×#12311yescharacteristic
327C2×C47C2×#44011yescharacteristic
328C2×C47C4×#13711yescharacteristic
329C2×C47#309211yescharacteristic
330C2×C47#309511yescharacteristic
331C2×C47#311311yescharacteristic
332C2×C47#311411yescharacteristic
333D47—24no—
334C2×C22⋊Q8—471no—
335C2×C4×Q8—471no—
336C22×C4⋊C4—471no—
337C22×C4⋊C4—471no—
338C22×C42—471no—
339C22×C42—941no—
340#483—478no—
341C2×#107—471no—
342C2×#107C2×D4711yescharacteristic
343C2×#106—471no—
344C2×#106C2×D4711yescharacteristic
345C2×#103—471no—
346C2×#103—471no—
347C2×#103—471no—
348C2×#103—471no—
349C2×C22⋊C8—471no—
350C2×C22⋊C8—471no—
351C2×#97—471no—
352C2×#97C2×D4711yescharacteristic
353#391—471no—
354#389—471no—
355#387—941no—
356#385—941no—
357#380—471no—
358C4×C4⋊C4—471no—
359C4×C4⋊C4—471no—
360C4×C4⋊C4—471no—
361C4×C4⋊C4—941no—
362C4×C4⋊C4—941no—
363C4×C4⋊C4—941no—
364C4×C4⋊C4C2×D4711yescharacteristic
365C4×C4⋊C4C2×D4711yescharacteristic
366C4×C22⋊C4—941no—
367C2×#95—941no—
368#340—471no—
369#340—471no—
370#340—471no—
371#340—471no—
372#340—471no—
373#340—471no—
374#340C2×D4711yescharacteristic
375#340C2×D4711yescharacteristic
376#336—471no—
377#336—471no—
378C47#2157911yescharacteristic, complemented, Sylow
379C22×Q8—471no—
380C23×C4—471no—
381C23×C4—471no—
382C23×C4—471no—
383C23×C4—941no—
384C23×C4—941no—
385C22×C8—471no—
386C22×C8—471no—
387C22×C8C22×D4711yescharacteristic
388C22×C8C22×D4711yescharacteristic
389C22⋊Q8—474no—
390C22⋊Q8—942no—
391C4×Q8—942no—
392C4×Q8—942no—
393C2×C4⋊C4—471no—
394C2×C4⋊C4—471no—
395C2×C4⋊C4—474no—
396C2×C4⋊C4—471no—
397C2×C4⋊C4—942no—
398C2×C4⋊C4—942no—
399C2×C4⋊C4—941no—
400C2×C4⋊C4—941no—
401C2×C4⋊C4—941no—
402C2×C4⋊C4—942no—
403C2×C4⋊C4—941no—
404C2×C4⋊C4—941no—
405C2×C4⋊C4C22×D4711yescharacteristic
406C2×C4⋊C4C22×D4711yescharacteristic
407C2×C4⋊C4C22×D4711yescharacteristic
408C2×C4⋊C4C22×D4711yescharacteristic
409C2×C22⋊C4—941no—
410C2×C22⋊C4—941no—
411C2×C42—21no—
412C2×C42—21no—
413C2×C42—471no—
414C2×C42—471no—
415C2×C42—471no—
416C2×C42—942no—
417C2×C42—942no—
418C2×C42—941no—
419C2×C42—941no—
420C2×C42—941no—
421C2×C42—941no—
422C2×C42—1884no—
423C2×C42C22×D4711yescharacteristic
424#107—22no—
425#107—942no—
426#106—474no—
427#106C4×D4714yescomplemented
428#103—474no—
429#103—474no—
430#103—942no—
431#103—942no—
432C22⋊C8—474no—
433C22⋊C8—942no—
434#97—22no—
435#97—942no—
436#95—21no—
437#95—21no—
438#95—941no—
439#95—941no—
440#95—941no—
441#95—941no—
442#95—941no—
443#95—941no—
444#95—941no—
445C24—471no—
446C2×Q8—472no—
447C2×Q8—941no—
448C2×Q8—941no—
449C2×Q8—941no—
450C2×Q8—941no—
451C2×Q8—941no—
452C22×C4—21no—
453C22×C4—21no—
454C22×C4—21no—
455C22×C4—21no—
456C22×C4—471no—
457C22×C4—471no—
458C22×C4—472no—
459C22×C4—472no—
460C22×C4—471no—
461C22×C4—942no—
462C22×C4—942no—
463C22×C4—944no—
464C22×C4—942no—
465C22×C4—944no—
466C22×C4—942no—
467C22×C4—941no—
468C22×C4—941no—
469C22×C4—941no—
470C22×C4—941no—
471C22×C4—941no—
472C22×C4—941no—
473C22×C4—941no—
474C22×C4—942no—
475C22×C4—941no—
476C22×C4—1881no—
477C22×C4—1882no—
478C22×C4—1882no—
479C22×C4—1882no—
480C22×C4—1882no—
481C22×C4D4×D4711yescharacteristic
482C22×C4D4×D4711yescharacteristic
483C22×C4C23×D4711yescharacteristic
484C2×C8—21no—
485C2×C8—21no—
486C2×C8—21no—
487C2×C8—21no—
488C2×C8—21no—
489C2×C8—472no—
490C2×C8—941no—
491C2×C8—941no—
492C2×C8—941no—
493C2×C8—941no—
494C2×C8—941no—
495C2×C8C2×C4×D4712yes—
496C4⋊C4—21no—
497C4⋊C4—21no—
498C4⋊C4—21no—
499C4⋊C4—21no—
500C4⋊C4—21no—
501C4⋊C4—472no—
502C4⋊C4—472no—
503C4⋊C4—944no—
504C4⋊C4—944no—
505C4⋊C4—944no—
506C4⋊C4—941no—
507C4⋊C4—941no—
508C4⋊C4—941no—
509C4⋊C4—941no—
510C4⋊C4—1882no—
511C4⋊C4—1882no—
512C4⋊C4C2×C4×D4712yes—
513C4⋊C4C2×C4×D4712yes—
514C4⋊C4#1062535012yes—
515C22⋊C4—944no—
516C22⋊C4—1882no—
517C42—42no—
518C42—42no—
519C42—942no—
520C42—942no—
521C42—941no—
522C42—941no—
523C42—1882no—
524C42—1882no—
525C42—1882no—
526C42—1882no—
527C42#1062535012yes—
528C42#1062535412yes—
529C23—472no—
530C23—472no—
531C23—472no—
532C23—941no—
533C23—941no—
534C23—941no—
535C23—941no—
536C23C2×D4×D4711yescharacteristic
537Q8—942no—
538Q8—942no—
539Q8—1881no—
540Q8—1881no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—472no—
558C2×C4—472no—
559C2×C4—948no—
560C2×C4—941no—
561C2×C4—941no—
562C2×C4—941no—
563C2×C4—941no—
564C2×C4—942no—
565C2×C4—942no—
566C2×C4—942no—
567C2×C4—942no—
568C2×C4—942no—
569C2×C4—942no—
570C2×C4—941no—
571C2×C4—942no—
572C2×C4—941no—
573C2×C4—941no—
574C2×C4—1884no—
575C2×C4—1888no—
576C2×C4—1888no—
577C2×C4—1881no—
578C2×C4—1881no—
579C2×C4—1881no—
580C2×C4—1881no—
581C2×C4—1881no—
582C2×C4—1881no—
583C2×C4—1881no—
584C2×C4—1881no—
585C2×C4—1881no—
586C2×C4#1309400912yes—
587C2×C4#1309400912yes—
588C2×C4C22×C4×D4711yescharacteristic
589C2×C4#41×D4711yescharacteristic
590C2×C4#41×D4711yescharacteristic
591C2×C4C2×#1062535011yescharacteristic
592C2×C4C2×#1062535011yescharacteristic
593C2×C4C2×#1062535411yescharacteristic
594C8—22no—
595C8—41no—
596C8—942no—
597C8—1881no—
598C22—474no—
599C22—942no—
600C22—942no—
601C22—942no—
602C22—942no—
603C22—942no—
604C22—942no—
605C22C4×D4×D4711yescharacteristic
606C22C2×#1309400911yescharacteristic
607C22#1568963211yescharacteristic
608C22AGL(1,ℤ/8)×D4711yescharacteristic
609C22#137×D4711yescharacteristic
610C22#137×D4711yescharacteristic
611C22#1568991911yescharacteristic
612C4—22no—
613C4—22no—
614C4—42no—
615C4—42no—
616C4—42no—
617C4—81no—
618C4—942no—
619C4—942no—
620C4—3761no—
621C4—1882no—
622C4—1882no—
623C4—1881no—
624C4—1882no—
625C4C4×#1062535011yescharacteristic
626C4C4×#1062535411yescharacteristic
627C4#1568965111yescharacteristic
628C4#1568966011yescharacteristic
629C2—944no—
630C2C4×#1309400911yescharacteristic
631C2#440×D4711yescharacteristic
632C2#2025221011yescharacteristic
633C2#2025331411yescharacteristic
634C2#2025333811yescharacteristic
635C2#2025351611yescharacteristic
636C2#2025352411yescharacteristic
subgroupG/H
centerC23C2×D4×D47
commutatorC2×C4×C47C23×C4
FrattiniC22×C4C23×D47
FittingC47×#1776C2
socleC23×C47C22×D4
prime radicalC23×D47C2×D4
cosocleC22×C4×C47C24
hypercenter#1776D47
hypocenterC47#21579
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1776

Complements

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

NH ≅ G/Ntype
4. #20244028C2semidirect
5. #20244027C2semidirect
6. #20243969C2semidirect
7. #20243968C2semidirect
11. #20243286C2semidirect
12. #20243221C2semidirect
13. #20243219C2semidirect
15. C47×#1776C2semidirect
22. #15689455C4semidirect
97. #13093955C2×C4semidirect
99. #13093949C2×C4semidirect
101. #13093947C2×C4semidirect
122. #106×C47C2×C4semidirect
322. #1776D47semidirect
378. C47#21579semidirect
427. #106C4×D47semidirect

Maximal subgroups

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

subgroupindex
C2×#156894552
C4×#130940042
C4×C4⋊C4×D472
#202440282
#202440272
#202439692
#202439682
#202436942
#202435742
#202434732
#202432862
#202432212
#202432192
#202431512
C47×#17762
#2157947