Finite Group Databasebeta

← #26223910

Subgroups — #26223910

Order 12032.

988 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×#15689401C211yesmaximal, characteristic
2C4×#13094023C211yesmaximal, characteristic, complemented
3C4×D4×Dic47C211yesmaximal, characteristic
4#20244338C211yesmaximal, characteristic, complemented
5#20244150C211yesmaximal, characteristic, complemented
6#20243983C211yesmaximal, characteristic, complemented
7#20243979C211yesmaximal, characteristic, complemented
8#20243651C211yesmaximal, characteristic
9#20243647C211yesmaximal, characteristic
10#20243386C211yesmaximal, characteristic
11#20243286C211yesmaximal, characteristic, complemented
12#20243285C211yesmaximal, characteristic, complemented
13#20243242C211yesmaximal, characteristic, complemented
14#20243213C211yesmaximal, characteristic
15C47×#1762C211yesmaximal, characteristic
16C2×#13094023C2211yescharacteristic
17C2×C4×Dic94—21no—
18C2×D4×Dic47C2211yescharacteristic
19C2×D4×Dic47C2211yescharacteristic
20C22×C4×Dic47—21no—
21#15689401C418yescomplemented
22C2×#13093958C2211yescharacteristic
23C2×#13093957C2211yescharacteristic
24C2×#13093956C2211yescharacteristic
25C2×#13093955C2211yescharacteristic
26C2×Dic47⋊C8C2211yescharacteristic
27C2×Dic47⋊C8C2211yescharacteristic
28C2×#13093945C2211yescharacteristic
29C2×#13093945C2211yescharacteristic
30C2×#13093935C2211yescharacteristic
31#15689027—21no—
32#15689022C2211yescharacteristic
33#15689020—21no—
34#15689019C2211yescharacteristic
35#15688992C2211yescharacteristic
36C4⋊C4×Dic47C2211yescharacteristic
37C4⋊C4×Dic47C2211yescharacteristic
38C4×Dic47⋊C4—21no—
39C4×C4⋊Dic47C2211yescharacteristic
40C4×C4⋊Dic47C2211yescharacteristic
41C22⋊C4×Dic47—21no—
42C22⋊C4×Dic47—21no—
43C4×C22⋊Dic47—21no—
44C42×Dic47C2211yescharacteristic
45#15688886C2211yescharacteristic
46#15688886C2211yescharacteristic
47#15688885C2211yescharacteristic
48#15688885C2211yescharacteristic
49#15688885C2211yescharacteristic
50#15688885C2211yescharacteristic
51#15688877C2211yescharacteristic
52#15688877C2211yescharacteristic
53C2×C4×D4×C47C2211yescharacteristic
54C2×#102×C47C2211yescharacteristic
55C2×#102×C47C2211yescharacteristic
56C2×#97×C47C2211yescharacteristic
57C4×C4⋊C4×C47C2211yescharacteristic
58C47×#340C2211yescharacteristic
59C47×#340C2211yescharacteristic
60C22×Dic94—21no—
61C23×Dic47—21no—
62C23×Dic47—21no—
63C22×C47⋊C8C2311yescharacteristic
64C22×C47⋊C8C2311yescharacteristic
65#13094023—22no—
66#13094023C2×C414yescomplemented
67C4×Dic94—44no—
68D4×Dic47—22no—
69D4×Dic47—22no—
70D4×Dic47—22no—
71D4×Dic47C2×C414yes—
72C2×C4⋊Dic47—21no—
73C2×C4⋊Dic47C2311yescharacteristic
74C2×C4⋊Dic47C2311yescharacteristic
75C2×C4⋊Dic47C2311yescharacteristic
76C2×C4⋊Dic47C2311yescharacteristic
77C2×Dic47⋊C4—21no—
78C2×Dic47⋊C4—21no—
79C2×Dic47⋊C4—21no—
80C2×Dic47⋊C4—21no—
81C2×C22⋊Dic47—21no—
82C2×C22⋊Dic47—21no—
83C2×C4×Dic47—21no—
84C2×C4×Dic47—21no—
85C2×C4×Dic47—21no—
86C2×C4×Dic47—21no—
87C2×C4×Dic47—44no—
88C2×C4×Dic47D411yescharacteristic
89C2×C4×Dic47D411yescharacteristic
90C2×C4×Dic47D411yescharacteristic
91C2×C4×Dic47D411yescharacteristic
92C2×C4×Dic47C2311yescharacteristic
93C2×C4×Dic47C2311yescharacteristic
94#13093958C2×C414yescomplemented
95#13093957C2×C414yescomplemented
96#13093956—22no—
97#13093955—22no—
98Dic47⋊C8—22no—
99Dic47⋊C8C2×C414yescomplemented
100#13093945—22no—
101#13093945C2×C414yes—
102#13093935—22no—
103C47⋊#95—21no—
104C47⋊#95—21no—
105C47⋊#95—21no—
106C47⋊#95—21no—
107C47⋊#95—21no—
108C47⋊#95—21no—
109C22×D4×C47C2311yescharacteristic
110C23×C4×C47—21no—
111C22×C8×C47C2311yescharacteristic
112C22×C8×C47C2311yescharacteristic
113C4×D4×C47—22no—
114C4×D4×C47Q814yescomplemented
115C2×C4⋊C4×C47C2311yescharacteristic
116C2×C4⋊C4×C47C2311yescharacteristic
117C2×C4⋊C4×C47C2311yescharacteristic
118C2×C22⋊C4×C47—21no—
119C2×C42×C47—21no—
120C2×C42×C47C2311yescharacteristic
121#102×C47—22no—
122#102×C47C2×C414yes—
123#97×C47—22no—
124#95×C47—21no—
125C24×C47—21no—
126C2×D4×C47—21no—
127C2×D4×C47—21no—
128C2×D4×C47—21no—
129C2×D4×C47C22×C412yes—
130C2×D4×C47C2×Q812yes—
131C2×D4×C47#4112yes—
132C22×C4×C47—21no—
133C22×C4×C47—21no—
134C22×C4×C47—24no—
135C22×C4×C47—21no—
136C22×C4×C47—42no—
137C22×C4×C47—42no—
138C22×C4×C47C2×D411yescharacteristic
139C22×C4×C47C2×D411yescharacteristic
140C22×C4×C47C2411yescharacteristic
141C22×Dic47—24no—
142C22×Dic47—21no—
143C22×Dic47—21no—
144C22×Dic47—21no—
145C22×Dic47—24no—
146C22×Dic47—21no—
147C22×Dic47—42no—
148C22×Dic47—42no—
149C22×Dic47—42no—
150C22×Dic47—42no—
151C22×Dic47C2×D411yescharacteristic
152C22×Dic47C2×D411yescharacteristic
153C22×Dic47C2×D411yescharacteristic
154C22×Dic47C2×D411yescharacteristic
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×C47C22×C412yes—
169C4⋊C4×C47C2×Q812yes—
170C22⋊C4×C47—24no—
171C42×C47—42no—
172C42×C47C2×Q812yes—
173C42×C47#4112yes—
174C22⋊Dic47—24no—
175C22⋊Dic47—24no—
176C4⋊Dic47—21no—
177C4⋊Dic47—21no—
178C4⋊Dic47—21no—
179C4⋊Dic47—21no—
180C4⋊Dic47—21no—
181C4⋊Dic47—42no—
182C4⋊Dic47C22×C412yes—
183C4⋊Dic47C22×C412yes—
184C4⋊Dic47#4112yes—
185Dic47⋊C4—24no—
186Dic47⋊C4—42no—
187Dic47⋊C4—42no—
188Dic47⋊C4—42no—
189C4×Dic47—21no—
190C4×Dic47—21no—
191C4×Dic47—22no—
192C4×Dic47—22no—
193C4×Dic47—22no—
194C4×Dic47—41no—
195C4×Dic47—42no—
196C4×Dic47—42no—
197C4×Dic47—42no—
198C4×Dic47—42no—
199C4×Dic47C22×C412yes—
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×C47—22no—
208C23×C47—22no—
209C23×C47—22no—
210C23×C47—41no—
211C23×C47—41no—
212C23×C47—41no—
213C23×C47—41no—
214C23×C47C22×D411yescharacteristic
215D4×C47—22no—
216D4×C47—22no—
217D4×C47—22no—
218D4×C47C4×Q814yescomplemented
219C2×C4×C47—21no—
220C2×C4×C47—21no—
221C2×C4×C47—21no—
222C2×C4×C47—21no—
223C2×C4×C47—22no—
224C2×C4×C47—22no—
225C2×C4×C47—48no—
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×C47C22⋊Q812yes—
234C2×C4×C47C22⋊Q812yes—
235C2×C4×C47C23×C411yescharacteristic
236C2×C4×C47C22×Q811yescharacteristic
237C2×C4×C47C2×#4111yescharacteristic
238C2×C4×C47C2×#4111yescharacteristic
239C2×C4×C47C2×#4111yescharacteristic
240C2×C4×C47C2×#4111yescharacteristic
241C2×Dic47—21no—
242C2×Dic47—21no—
243C2×Dic47—21no—
244C2×Dic47—21no—
245C2×Dic47—22no—
246C2×Dic47—22no—
247C2×Dic47—21no—
248C2×Dic47—22no—
249C2×Dic47—21no—
250C2×Dic47—21no—
251C2×Dic47—48no—
252C2×Dic47—41no—
253C2×Dic47—41no—
254C2×Dic47—41no—
255C2×Dic47—41no—
256C2×Dic47—41no—
257C2×Dic47—41no—
258C2×Dic47—48no—
259C2×Dic47—41no—
260C2×Dic47—41no—
261C2×Dic47—41no—
262C2×Dic47C4×D412yes—
263C2×Dic47C4×D412yes—
264C2×Dic47AGL(1,ℤ/8)12yes—
265C2×Dic47AGL(1,ℤ/8)12yes—
266C2×Dic47#13712yes—
267Dic94—48no—
268C8×C47—22no—
269C8×C47—41no—
270C47⋊C8—22no—
271C47⋊C8—41no—
272#21613—471nomaximal, Sylow
273C22×C47—24no—
274C22×C47—42no—
275C22×C47—42no—
276C22×C47—42no—
277C22×C47—42no—
278C22×C47—42no—
279C22×C47—42no—
280C22×C47C2×C4×D411yescharacteristic
281C22×C47C2×C22⋊Q811yescharacteristic
282C22×C47#52511yescharacteristic
283C22×C47C2×AGL(1,ℤ/8)11yescharacteristic
284C22×C47C2×AGL(1,ℤ/8)11yescharacteristic
285C22×C47C2×#13711yescharacteristic
286C22×C47#57511yescharacteristic
287C4×C47—22no—
288C4×C47—22no—
289C4×C47—42no—
290C4×C47—42no—
291C4×C47—81no—
292C4×C47C2×C4×Q811yescharacteristic
293C4×C47C4×#4111yescharacteristic
294C4×C47#52911yescharacteristic
295C4×C47#53311yescharacteristic
296Dic47—22no—
297Dic47—22no—
298Dic47—42no—
299Dic47—41no—
300Dic47—42no—
301Dic47—42no—
302Dic47—81no—
303Dic47#44214yes—
304C2×#474—471no—
305C4×C4⋊Q8—471no—
306C42×D4—471no—
307#1988—471no—
308#1948—471no—
309#1921—471no—
310#1921—471no—
311#1864—471no—
312#1864—471no—
313#1802—471no—
314#1776—471no—
315#1776—471no—
316#1767—471no—
317#1762—471no—
318#1762D4711yescharacteristic
319C2×C47—44no—
320C2×C47C4×C22⋊Q811yescharacteristic
321C2×C47C2×#44211yescharacteristic
322C2×C47C4×#13711yescharacteristic
323C2×C47#307711yescharacteristic
324C2×C47#307911yescharacteristic
325C2×C47#311011yescharacteristic
326C2×C47#311511yescharacteristic
327C2×C4⋊Q8—471no—
328C2×C4×Q8—941no—
329C2×C4×D4—471no—
330C2×C4×D4—471no—
331C2×C4×D4C2×D4711yescharacteristic
332C22×C42—941no—
333#474—478no—
334C2×#107—471no—
335C2×#107—471no—
336C2×#106—471no—
337C2×#106—471no—
338C2×C4⋊C8—471no—
339C2×C4⋊C8—471no—
340C2×#102—471no—
341C2×#102—471no—
342C2×#102C2×D4711yescharacteristic
343C2×#102C2×D4711yescharacteristic
344C2×#97—471no—
345C2×#97C2×D4711yescharacteristic
346#391—941no—
347#389—471no—
348#389—471no—
349#389—941no—
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#340—471no—
367#340C2×D4711yescharacteristic
368#340C2×D4711yescharacteristic
369#336—471no—
370#336—471no—
371C47#2161311yescharacteristic, complemented, Sylow
372C22×Q8—941no—
373C22×D4C22×D4711yescharacteristic
374C23×C4—21no—
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—22no—
385C4×D4—474no—
386C4×D4—942no—
387C4×D4—942no—
388C4×D4—942no—
389C4×D4Dic9414yescomplemented
390C2×C4⋊C4—471no—
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⋊C4—941no—
399C2×C4⋊C4C22×D4711yescharacteristic
400C2×C4⋊C4C22×D4711yescharacteristic
401C2×C4⋊C4C22×D4711yescharacteristic
402C2×C22⋊C4—21no—
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—474no—
420#106—942no—
421#106—942no—
422C4⋊C8—474no—
423C4⋊C8—942no—
424#102—22no—
425#102—474no—
426#102—942no—
427#102C4×D4714yes—
428#97—22no—
429#97—942no—
430#95—21no—
431#95—941no—
432#95—941no—
433#95—941no—
434#95—941no—
435#95—941no—
436#95—941no—
437C24—21no—
438C2×Q8—944no—
439C2×Q8—1882no—
440C2×Q8—1882no—
441C2×D4—21no—
442C2×D4—21no—
443C2×D4—21no—
444C2×D4C2×Dic9412yes—
445C2×D4C2×C4×D4712yes—
446C2×D4#1062535012yes—
447C22×C4—21no—
448C22×C4—21no—
449C22×C4—24no—
450C22×C4—21no—
451C22×C4—42no—
452C22×C4—42no—
453C22×C4—471no—
454C22×C4—471no—
455C22×C4—471no—
456C22×C4—471no—
457C22×C4—944no—
458C22×C4—944no—
459C22×C4—941no—
460C22×C4—941no—
461C22×C4—941no—
462C22×C4—941no—
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—472no—
487C4⋊C4—472no—
488C4⋊C4—472no—
489C4⋊C4—944no—
490C4⋊C4—941no—
491C4⋊C4—941no—
492C4⋊C4—941no—
493C4⋊C4—941no—
494C4⋊C4—941no—
495C4⋊C4—1882no—
496C4⋊C4—1882no—
497C4⋊C4—1882no—
498C4⋊C4—1882no—
499C4⋊C4C2×Dic9412yes—
500C4⋊C4C2×C4×D4712yes—
501C22⋊C4—24no—
502C22⋊C4—944no—
503C22⋊C4—944no—
504C42—42no—
505C42—472no—
506C42—472no—
507C42—942no—
508C42—942no—
509C42—942no—
510C42—941no—
511C42—941no—
512C42—1881no—
513C42—1882no—
514C42—1882no—
515C42—1882no—
516C42—1882no—
517C42C2×Dic9412yes—
518C42#1062535212yes—
519C23—22no—
520C23—22no—
521C23—22no—
522C23—41no—
523C23—41no—
524C23—41no—
525C23—41no—
526C23C2×D4×D4711yescharacteristic
527Q8—1888no—
528D4—22no—
529D4—22no—
530D4—22no—
531D4C4×Dic9414yescomplemented
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—48no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—472no—
547C2×C4—472no—
548C2×C4—472no—
549C2×C4—472no—
550C2×C4—472no—
551C2×C4—941no—
552C2×C4—941no—
553C2×C4—941no—
554C2×C4—941no—
555C2×C4—942no—
556C2×C4—942no—
557C2×C4—941no—
558C2×C4—942no—
559C2×C4—941no—
560C2×C4—941no—
561C2×C4—1888no—
562C2×C4—1888no—
563C2×C4—1881no—
564C2×C4—1881no—
565C2×C4—1881no—
566C2×C4—1881no—
567C2×C4—1881no—
568C2×C4—1881no—
569C2×C4—1881no—
570C2×C4—1881no—
571C2×C4—1881no—
572C2×C4#1309400112yes—
573C2×C4#1309400112yes—
574C2×C4C22×C4×D4711yescharacteristic
575C2×C4C22×Dic9411yescharacteristic
576C2×C4#41×D4711yescharacteristic
577C2×C4#41×D4711yescharacteristic
578C2×C4C2×#1062535211yescharacteristic
579C2×C4C2×#1062535011yescharacteristic
580C8—22no—
581C8—41no—
582C8—942no—
583C8—1881no—
584C22—24no—
585C22—42no—
586C22—42no—
587C22—42no—
588C22—42no—
589C22—42no—
590C22—42no—
591C22C4×D4×D4711yescharacteristic
592C22C2×#1309400111yescharacteristic
593C22#1568963211yescharacteristic
594C22AGL(1,ℤ/8)×D4711yescharacteristic
595C22AGL(1,ℤ/8)×D4711yescharacteristic
596C22#137×D4711yescharacteristic
597C22#1568991911yescharacteristic
598C4—22no—
599C4—22no—
600C4—42no—
601C4—42no—
602C4—81no—
603C4—474no—
604C4—942no—
605C4—942no—
606C4—3761no—
607C4—1882no—
608C4—1882no—
609C4—1882no—
610C4—1881no—
611C4C2×C4×Dic9411yescharacteristic
612C4C4×#1062535211yescharacteristic
613C4#1568965311yescharacteristic
614C4#1568967611yescharacteristic
615C2—44no—
616C2C4×#1309400111yescharacteristic
617C2#442×D4711yescharacteristic
618C2#2025221011yescharacteristic
619C2#2025318211yescharacteristic
620C2#2025319911yescharacteristic
621C2#2025348511yescharacteristic
622C2#2025353511yescharacteristic
subgroupG/H
centerC23C2×D4×D47
commutatorC2×C4×C47C23×C4
FrattiniC22×C4C23×D47
FittingC47×#1762C2
socleC23×C47C22×D4
prime radicalC22×D4×C47C23
cosocleC22×C4×C47C24
hypercenter#1762D47
hypocenterC47#21613
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1762

Complements

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

NH ≅ G/Ntype
2. C4×#13094023C2semidirect
4. #20244338C2semidirect
5. #20244150C2semidirect
6. #20243983C2semidirect
7. #20243979C2semidirect
11. #20243286C2semidirect
12. #20243285C2semidirect
13. #20243242C2semidirect
21. #15689401C4semidirect
66. #13094023C2×C4semidirect
94. #13093958C2×C4semidirect
95. #13093957C2×C4semidirect
99. Dic47⋊C8C2×C4semidirect
114. C4×D4×C47Q8semidirect
218. D4×C47C4×Q8semidirect
371. C47#21613semidirect
389. C4×D4Dic94semidirect
531. D4C4×Dic94semidirect

Maximal subgroups

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

subgroupindex
C2×#156894012
C4×#130940232
C4×D4×Dic472
#202443382
#202441502
#202439832
#202439792
#202436512
#202436472
#202433862
#202432862
#202432852
#202432422
#202432132
C47×#17622
#2161347