Finite Group Databasebeta

← #26217253

Subgroups — #26217253

Order 12032.

1550 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×C4×#10625350C211yesmaximal, characteristic
2C2×#15689163C211yesmaximal, characteristic, complemented
3C2×#15689122C211yesmaximal, characteristic
4C2×#15689122C211yesmaximal, characteristic
5#20243655C212yesmaximal, complemented
6#20243648C212yesmaximal, complemented
7#20243641C212yesmaximal, complemented
8#20243634C212yesmaximal, complemented
9#20243107C211yesmaximal, characteristic, complemented
10#20243082C211yesmaximal, characteristic, complemented
11C2×C47×#389C211yesmaximal, characteristic, complemented
12C22×#10625350C2211yescharacteristic
13C22×#10625329C2211yescharacteristic
14C22×#10625329C2211yescharacteristic
15C4×#10625350—24no—
16C4×#10625350—24no—
17C2×C4×Dic94C2211yescharacteristic
18C2×C4×Dic94C2211yescharacteristic
19C2×C4×D188C2211yescharacteristic
20C2×C4×D188C2211yescharacteristic
21C2×C4×C22⋊D47—21no—
22C2×C4×C22⋊D47—21no—
23C2×#13093983C2211yescharacteristic
24C2×#13093981—21no—
25C2×C42×D47—21no—
26#15689163—24no—
27C2×#13093957C2212yes—
28C2×#13093955C2212yes—
29#15689122—24no—
30#15689122C418yescomplemented
31C2×#13093947C2212yes—
32C2×#13093947C2212yes—
33C2×#13093944C2212yes—
34C2×#13093944C2212yes—
35#15688885C2212yes—
36#15688885C2212yes—
37#15688878C2212yes—
38#15688878C2212yes—
39C22×C4⋊C4×C47—21no—
40C22×C4⋊C4×C47C2211yescharacteristic
41C22×C4⋊C4×C47C2211yescharacteristic
42C22×C42×C47C2211yescharacteristic
43C47×#389—21no—
44C47×#389—21no—
45C47×#389C2212yes—
46C47×#389C2212yes—
47C2×#95×C47—21no—
48C2×#10625350—22no—
49C2×#10625350—22no—
50C2×#10625350—22no—
51C2×#10625350C2×C414yes—
52C2×#10625350D414yes—
53C2×#10625350Q814yes—
54C22×Dic94C2311yescharacteristic
55C22×D188C2311yescharacteristic
56C22×C22⋊D47—21no—
57C22×C4×D47—21no—
58C22×C4×D47—21no—
59C22×C4×D47—21no—
60C2×#10625329—22no—
61C2×#10625329—22no—
62C2×#10625329—22no—
63C2×#10625329—22no—
64C2×#10625329—22no—
65C2×#10625329C2×C414yescomplemented
66C22×C47⋊C8C2312yes—
67C22×C47⋊C8C2312yes—
68C4×Dic94—22no—
69C4×Dic94—22no—
70C4×Dic94—22no—
71C4×Dic94—22no—
72C4×D188—22no—
73C4×D188—22no—
74C4×D188—22no—
75C4×D188—22no—
76C4×C22⋊D47—28no—
77C4×C22⋊D47—28no—
78#13093983—22no—
79#13093983D414yes—
80#13093981—28no—
81C2×C4⋊Dic47C2311yescharacteristic
82C2×C4⋊Dic47C2311yescharacteristic
83C2×Dic47⋊C4—21no—
84C2×Dic47⋊C4—21no—
85C2×C22⋊Dic47—21no—
86C2×(C2×D47)⋊C4—21no—
87C2×(C2×D47)⋊C4—21no—
88C2×C4×Dic47—21no—
89C2×C4×Dic47—21no—
90C2×C4×Dic47—21no—
91C42×D47—28no—
92#13093957—24no—
93#13093955—24no—
94#13093947—24no—
95#13093947C2×C418yescomplemented
96#13093944—24no—
97#13093944C2×C418yescomplemented
98C23×C4×C47—21no—
99C23×C4×C47—21no—
100C23×C4×C47D411yescharacteristic
101C23×C4×C47D411yescharacteristic
102C23×C4×C47C2311yescharacteristic
103C2×C4⋊C4×C47—21no—
104C2×C4⋊C4×C47—22no—
105C2×C4⋊C4×C47—22no—
106C2×C4⋊C4×C47—22no—
107C2×C4⋊C4×C47—22no—
108C2×C4⋊C4×C47—22no—
109C2×C4⋊C4×C47—21no—
110C2×C4⋊C4×C47—44no—
111C2×C4⋊C4×C47C2×C414yescomplemented
112C2×C4⋊C4×C47C2312yes—
113C2×C4⋊C4×C47C2312yes—
114C2×C42×C47—22no—
115C2×C42×C47—22no—
116C2×C42×C47D411yescharacteristic
117C2×C42×C47D411yescharacteristic
118C2×C42×C47C2311yescharacteristic
119C2×C42×C47C2311yescharacteristic
120#95×C47—22no—
121#95×C47—22no—
122C24×C47C2×D411yescharacteristic
123C23×D47—21no—
124C22×C4×C47—24no—
125C22×C4×C47—22no—
126C22×C4×C47—22no—
127C22×C4×C47—22no—
128C22×C4×C47—22no—
129C22×C4×C47—22no—
130C22×C4×C47—22no—
131C22×C4×C47—41no—
132C22×C4×C47—41no—
133C22×C4×C47—42no—
134C22×C4×C47—42no—
135C22×C4×C47—42no—
136C22×C4×C47—42no—
137C22×C4×C47—42no—
138C22×C4×C47C22×C412yes—
139C22×C4×C47C2×D411yescharacteristic
140C22×C4×C47C2×D411yescharacteristic
141C22×C4×C47C2×D411yescharacteristic
142C22×C4×C47C2×D411yescharacteristic
143C22×C4×C47C2×D412yes—
144C22×C4×C47C2×D411yescharacteristic
145C22×C4×C47C2×D412yes—
146C22×C4×C47C2×Q812yes—
147C22×C4×C47#4112yes—
148C22×C4×C47#4112yes—
149C22×C4×C47C2411yescharacteristic
150C2×C22⋊D47—24no—
151C2×C22⋊D47—24no—
152C2×C22⋊D47—24no—
153C22×Dic47—21no—
154C22×Dic47—21no—
155C22×Dic47—21no—
156#10625350—28no—
157#10625350—28no—
158#10625350—28no—
159#10625350C4⋊C4116yescomplemented
160C2×D188—21no—
161C2×D188—21no—
162C2×D188—21no—
163C2×D188C22×C412yes—
164C2×D188C2×D412yes—
165C2×D188C2×Q812yes—
166C2×C4×D47—24no—
167C2×C4×D47—24no—
168C2×C4×D47—24no—
169C2×C4×D47—24no—
170C2×C4×D47—24no—
171C2×C4×D47—42no—
172C2×C4×D47—42no—
173C2×C4×D47—42no—
174C2×C4×D47—42no—
175C2×Dic94—21no—
176C2×Dic94—21no—
177C2×Dic94—21no—
178C2×Dic94C22×C412yes—
179C2×Dic94C2×D412yes—
180C2×Dic94C2×Q812yes—
181C4⋊C4×C47—24no—
182C4⋊C4×C47—22no—
183C4⋊C4×C47—22no—
184C4⋊C4×C47—22no—
185C4⋊C4×C47—44no—
186C4⋊C4×C47—42no—
187C4⋊C4×C47—44no—
188C4⋊C4×C47C22×C414yes—
189C42×C47—22no—
190C42×C47—22no—
191C42×C47—21no—
192C42×C47—21no—
193C42×C47—21no—
194C42×C47—21no—
195C22⋊Dic47—24no—
196(C2×D47)⋊C4—24no—
197(C2×D47)⋊C4—24no—
198C4⋊Dic47—21no—
199C4⋊Dic47—21no—
200C4⋊Dic47C2×D412yes—
201C4⋊Dic47C2×D412yes—
202Dic47⋊C4—24no—
203Dic47⋊C4—24no—
204C4×Dic47—24no—
205C4×Dic47—24no—
206C4×Dic47—24no—
207#10625329—28no—
208#10625329—44no—
209C2×C47⋊C8—22no—
210C2×C47⋊C8—22no—
211C2×C47⋊C8—22no—
212C2×C47⋊C8—22no—
213C2×C47⋊C8—22no—
214C2×C47⋊C8C22×C414yes—
215C23×C47—21no—
216C23×C47—21no—
217C23×C47—21no—
218C23×C47—21no—
219C23×C47C4×D412yes—
220C23×C47#12012yes—
221C23×C47C22⋊Q812yes—
222C23×C47C22×D411yescharacteristic
223C22×D47—22no—
224C22×D47—22no—
225C22×D47—22no—
226C22×D47—41no—
227C22×D47—41no—
228C22×D47—41no—
229C22×D47—41no—
230C2×C4×C47—21no—
231C2×C4×C47—21no—
232C2×C4×C47—21no—
233C2×C4×C47—21no—
234C2×C4×C47—24no—
235C2×C4×C47—21no—
236C2×C4×C47—21no—
237C2×C4×C47—21no—
238C2×C4×C47—21no—
239C2×C4×C47—21no—
240C2×C4×C47—21no—
241C2×C4×C47—44no—
242C2×C4×C47—44no—
243C2×C4×C47—48no—
244C2×C4×C47—42no—
245C2×C4×C47—42no—
246C2×C4×C47—42no—
247C2×C4×C47—42no—
248C2×C4×C47—42no—
249C2×C4×C47—84no—
250C2×C4×C47C2×C4⋊C414yes—
251C2×C4×C47C4×D412yes—
252C2×C4×C47C4×D414yes—
253C2×C4×C47#12012yes—
254C2×C4×C47C22⋊Q812yes—
255C2×C4×C47C22⋊Q814yes—
256C2×C4×C47#12314yes—
257C2×C4×C47AGL(1,ℤ/8)12yes—
258C2×C4×C47#13712yes—
259C2×C4×C47C23×C411yescharacteristic
260C2×C4×C47C22×D411yescharacteristic
261C2×C4×C47C22×D411yescharacteristic
262C2×C4×C47C22×Q811yescharacteristic
263C2×C4×C47C2×#4111yescharacteristic
264C2×C4×C47C2×#4111yescharacteristic
265C22⋊D47—216no—
266C2×Dic47—22no—
267C2×Dic47—22no—
268C2×Dic47—22no—
269C2×Dic47—22no—
270C2×Dic47—22no—
271C2×Dic47—41no—
272C2×Dic47—41no—
273C2×Dic47—41no—
274C2×Dic47—41no—
275D188—22no—
276D188—22no—
277D188—22no—
278D188C2×C4⋊C414yescomplemented
279C4×D47—216no—
280C4×D47—48no—
281C4×D47—48no—
282Dic94—22no—
283Dic94—22no—
284Dic94—22no—
285Dic94C2×C4⋊C414yescomplemented
286C47⋊C8—24no—
287C47⋊C8—42no—
288#20875—471nomaximal, Sylow
289C22×C47—22no—
290C22×C47—22no—
291C22×C47—22no—
292C22×C47—22no—
293C22×C47—22no—
294C22×C47—22no—
295C22×C47#38514yes—
296C22×C47C2×C4×D411yescharacteristic
297C22×C47C2×#12011yescharacteristic
298C22×C47C2×C22⋊Q811yescharacteristic
299C22×C47C2×AGL(1,ℤ/8)11yescharacteristic
300C22×C47C2×#13711yescharacteristic
301C22×C47#57511yescharacteristic
302C22×C47#57511yescharacteristic
303C2×D47—24no—
304C2×D47—42no—
305C2×D47—42no—
306C2×D47—42no—
307C2×D47—42no—
308C2×D47—42no—
309C2×D47—42no—
310C4×C47—22no—
311C4×C47—22no—
312C4×C47—44no—
313C4×C47—41no—
314C4×C47—41no—
315C4×C47—82no—
316C4×C47#38514yes—
317C4×C47C22×C4⋊C411yescharacteristic
318C4×C47C2×C4×D411yescharacteristic
319C4×C47C2×C22⋊Q811yescharacteristic
320C4×C47C2×#12311yescharacteristic
321Dic47—24no—
322Dic47—42no—
323Dic47—42no—
324C2×C4×#41—471no—
325C2×#428—471no—
326C2×#417—471no—
327C2×#417—471no—
328C2×#389D4711yescharacteristic, complemented
329#1865—472no—
330#1864—472no—
331#1863—472no—
332#1862—472no—
333#1736—471no—
334#1728—471no—
335C2×C47—24no—
336C2×C47C2×#38511yescharacteristic
337C2×C47C4×AGL(1,ℤ/8)11yescharacteristic
338C2×C47C4×#13711yescharacteristic
339C2×C47#300611yescharacteristic
340C2×C47#300711yescharacteristic
341C2×C47#307911yescharacteristic
342C2×C47#308011yescharacteristic
343D47—44no—
344C22×#41—471no—
345C22×M16—471no—
346C22×M16—471no—
347C4×#41—944no—
348C4×#41—944no—
349C2×C4×Q8—471no—
350C2×C4×Q8—471no—
351C2×C4×D4—471no—
352C2×C4×D4—471no—
353C2×C4×D4—941no—
354C2×C4×D4—941no—
355C2×#117—471no—
356C2×#117—941no—
357C22×C4⋊C4—21no—
358C22×C4⋊C4C2×D4711yescharacteristic
359C22×C4⋊C4C2×D4711yescharacteristic
360C22×C42—941no—
361C22×C42C2×D4711yescharacteristic
362#428—944no—
363C2×#107—472no—
364C2×#106—472no—
365#417—478no—
366#417—944no—
367C2×#103—472no—
368C2×#103—472no—
369C2×#102—472no—
370C2×#102—472no—
371#389—21no—
372#389—21no—
373#389C2×D4712yes—
374#389C2×D4712yes—
375C2×#95—21no—
376#340—472no—
377#340—472no—
378#336—472no—
379#336—472no—
380C47#2087511yescharacteristic, complemented, Sylow
381C2×#41—474no—
382C2×#41—474no—
383C2×#41—474no—
384C2×#41—942no—
385C2×#41—942no—
386C2×#41—942no—
387C22×Q8—471no—
388C22×D4—471no—
389C22×D4—941no—
390C23×C4—21no—
391C23×C4—21no—
392C23×C4—941no—
393C23×C4—941no—
394C23×C4—941no—
395C23×C4C22⋊D4711yescharacteristic
396C23×C4C22⋊D4711yescharacteristic
397C23×C4C22×D4711yescharacteristic
398C2×M16—474no—
399C2×M16—942no—
400C2×M16—942no—
401C2×M16—942no—
402C2×M16—942no—
403C2×M16—942no—
404C22×C8—472no—
405C22×C8—472no—
406C4×Q8—942no—
407C4×Q8—942no—
408C4×Q8—942no—
409C4×Q8—942no—
410C4×D4—942no—
411C4×D4—942no—
412C4×D4—942no—
413C4×D4—942no—
414C4×D4—948no—
415C4×D4—948no—
416#117—474no—
417#117—948no—
418#117—942no—
419C2×C4⋊C4—22no—
420C2×C4⋊C4—21no—
421C2×C4⋊C4—22no—
422C2×C4⋊C4—22no—
423C2×C4⋊C4—22no—
424C2×C4⋊C4—22no—
425C2×C4⋊C4—21no—
426C2×C4⋊C4—44no—
427C2×C4⋊C4—471no—
428C2×C4⋊C4—471no—
429C2×C4⋊C4—941no—
430C2×C4⋊C4—941no—
431C2×C4⋊C4C4×D4714yescomplemented
432C2×C4⋊C4C22×D4712yes—
433C2×C4⋊C4C22×D4712yes—
434C2×C22⋊C4—941no—
435C2×C22⋊C4—941no—
436C2×C22⋊C4—941no—
437C2×C42—22no—
438C2×C42—22no—
439C2×C42—948no—
440C2×C42—941no—
441C2×C42—941no—
442C2×C42—941no—
443C2×C42C22⋊D4711yescharacteristic
444C2×C42C22⋊D4711yescharacteristic
445C2×C42C22×D4711yescharacteristic
446C2×C42C22×D4711yescharacteristic
447#107—944no—
448#106—944no—
449#103—478no—
450#103—944no—
451#102—478no—
452#102—944no—
453#95—22no—
454#95—22no—
455C24—941no—
456C24C2×C22⋊D4711yescharacteristic
457#41—4716no—
458#41—948no—
459#41—948no—
460#41—948no—
461C2×Q8—472no—
462C2×Q8—472no—
463C2×Q8—472no—
464C2×Q8—941no—
465C2×Q8—941no—
466C2×Q8—941no—
467C2×D4—472no—
468C2×D4—472no—
469C2×D4—472no—
470C2×D4—944no—
471C2×D4—944no—
472C2×D4—944no—
473C2×D4—941no—
474C2×D4—941no—
475C2×D4—941no—
476C22×C4—24no—
477C22×C4—22no—
478C22×C4—22no—
479C22×C4—22no—
480C22×C4—22no—
481C22×C4—22no—
482C22×C4—22no—
483C22×C4—41no—
484C22×C4—41no—
485C22×C4—42no—
486C22×C4—42no—
487C22×C4—42no—
488C22×C4—42no—
489C22×C4—42no—
490C22×C4—944no—
491C22×C4—944no—
492C22×C4—941no—
493C22×C4—944no—
494C22×C4—944no—
495C22×C4—944no—
496C22×C4—941no—
497C22×C4—941no—
498C22×C4—1882no—
499C22×C4—1882no—
500C22×C4—1882no—
501C22×C4—1882no—
502C22×C4C2×C4×D4712yes—
503C22×C4#1062535012yes—
504C22×C4D4×D4712yes—
505C22×C4D4×D4712yes—
506C22×C4Q8×D4712yes—
507C22×C4#1062535412yes—
508C22×C4C2×C22⋊D4711yescharacteristic
509C22×C4C2×C22⋊D4711yescharacteristic
510C22×C4C2×C22⋊D4711yescharacteristic
511C22×C4C2×C22⋊D4711yescharacteristic
512C22×C4C2×C22⋊D4711yescharacteristic
513C22×C4C23×D4711yescharacteristic
514M16—948no—
515M16—1884no—
516C2×C8—474no—
517C2×C8—942no—
518C2×C8—942no—
519C2×C8—942no—
520C2×C8—942no—
521C2×C8—942no—
522C4⋊C4—24no—
523C4⋊C4—22no—
524C4⋊C4—22no—
525C4⋊C4—22no—
526C4⋊C4—44no—
527C4⋊C4—42no—
528C4⋊C4—44no—
529C4⋊C4—472no—
530C4⋊C4—472no—
531C4⋊C4—944no—
532C4⋊C4—944no—
533C4⋊C4—941no—
534C4⋊C4—941no—
535C4⋊C4C2×C4×D4714yes—
536C22⋊C4—944no—
537C22⋊C4—944no—
538C22⋊C4—944no—
539C42—22no—
540C42—22no—
541C42—21no—
542C42—21no—
543C42—21no—
544C42—21no—
545C42—944no—
546C42—944no—
547C42—944no—
548C23—21no—
549C23—21no—
550C23—21no—
551C23—21no—
552C23—942no—
553C23—942no—
554C23—942no—
555C23—1881no—
556C23—1881no—
557C23—1881no—
558C23—1881no—
559C23C4×C22⋊D4712yes—
560C23#1309399212yes—
561C23#1309400012yes—
562C23C22×C22⋊D4711yescharacteristic
563Q8—474no—
564Q8—942no—
565Q8—942no—
566Q8—942no—
567D4—474no—
568D4—9416no—
569D4—942no—
570D4—942no—
571D4—942no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—24no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—44no—
584C2×C4—44no—
585C2×C4—48no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—84no—
592C2×C4—942no—
593C2×C4—942no—
594C2×C4—942no—
595C2×C4—9416no—
596C2×C4—942no—
597C2×C4—942no—
598C2×C4—1888no—
599C2×C4—1888no—
600C2×C4—1881no—
601C2×C4—1881no—
602C2×C4—1881no—
603C2×C4—1881no—
604C2×C4C4⋊C4×D4714yes—
605C2×C4C4×C22⋊D4712yes—
606C2×C4#1309398614yes—
607C2×C4#1309399212yes—
608C2×C4#1309400012yes—
609C2×C4#1309400214yes—
610C2×C4#1309400914yes—
611C2×C4#1309405012yes—
612C2×C4#1309405512yes—
613C2×C4C22×C4×D4711yescharacteristic
614C2×C4C2×D4×D4711yescharacteristic
615C2×C4C2×D4×D4711yescharacteristic
616C2×C4C2×Q8×D4711yescharacteristic
617C2×C4C2×#1062535011yescharacteristic
618C2×C4C2×#1062535411yescharacteristic
619C8—944no—
620C8—1882no—
621C22—22no—
622C22—22no—
623C22—22no—
624C22—22no—
625C22—22no—
626C22—22no—
627C22—944no—
628C22—1882no—
629C22—1882no—
630C22—1882no—
631C22—1882no—
632C22—1882no—
633C22—1882no—
634C22#1568899414yes—
635C22C2×C4×C22⋊D4711yescharacteristic
636C22C2×#1309399211yescharacteristic
637C22C2×#1309400011yescharacteristic
638C22C2×#1309405011yescharacteristic
639C22C2×#1309405511yescharacteristic
640C22#1568992111yescharacteristic
641C22#1568992111yescharacteristic
642C4—22no—
643C4—22no—
644C4—44no—
645C4—41no—
646C4—41no—
647C4—82no—
648C4—944no—
649C4—1882no—
650C4—1882no—
651C4#1568899414yes—
652C4C2×C4⋊C4×D4711yescharacteristic
653C4C2×#1309398611yescharacteristic
654C4C2×#1309400211yescharacteristic
655C4C2×#1309400911yescharacteristic
656C2—24no—
657C2—1884no—
658C2C2×#1568899411yescharacteristic
659C2C4×#1309405011yescharacteristic
660C2C4×#1309405511yescharacteristic
661C2#2025263811yescharacteristic
662C2#2025264611yescharacteristic
663C2#2025319711yescharacteristic
664C2#2025320611yescharacteristic
subgroupG/H
centerC23C22×C22⋊D47
commutatorC2×C4×C47C23×C4
FrattiniC22×C4C23×D47
FittingC2×C47×#389C2
socleC23×C47C22×D4
prime radicalC22×#10625350C22
cosocleC22×C4×C47C24
hypercenterC2×#389D47
hypocenterC47#20875
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C22×C42 ◁ C2×#389

Complements

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

NH ≅ G/Ntype
2. C2×#15689163C2semidirect
5. #20243655C2semidirect
6. #20243648C2semidirect
7. #20243641C2semidirect
8. #20243634C2semidirect
9. #20243107C2semidirect
10. #20243082C2semidirect
11. C2×C47×#389C2semidirect
30. #15689122C4semidirect
65. C2×#10625329C2×C4semidirect
95. #13093947C2×C4semidirect
97. #13093944C2×C4semidirect
111. C2×C4⋊C4×C47C2×C4semidirect
159. #10625350C4⋊C4semidirect
278. D188C2×C4⋊C4semidirect
285. Dic94C2×C4⋊C4semidirect
328. C2×#389D47semidirect
380. C47#20875semidirect
431. C2×C4⋊C4C4×D47semidirect

Maximal subgroups

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

subgroupindex
C2×C4×#106253502
C2×#156891632
C2×#156891222
#202436552
#202436482
#202436412
#202436342
#202431072
#202430822
C2×C47×#3892
#2087547