Finite Group Databasebeta

← C2×#20253304 #27084048

Subgroups — C2×#20253304 #27084048

Order 12032.

2378 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×#15689636C211yesmaximal, characteristic
2C22×#13093982C211yesmaximal, characteristic, complemented
3#20253304C2116yesmaximal, direct product factor
4C2×#15689441C211yesmaximal, characteristic, complemented
5C2×#15689463C212yesmaximal, complemented
6C2×#15689450C212yesmaximal, complemented
7C2×#15689451C211yesmaximal, characteristic, complemented
8C2×#15689437C211yesmaximal, characteristic, complemented
9C2×#15689160C212yesmaximal, complemented
10C2×#15689118C212yesmaximal, complemented
11C2×C22⋊C8×D47C211yesmaximal, characteristic, complemented
12C2×C47×#484C211yesmaximal, characteristic, complemented
13C22×#10625354C2211yescharacteristic
14C23×C4×D47C2211yescharacteristic
15C22×C8×D47C2212yes—
16#15689636—24no—
17#15689636C2218yes—
18C2×#13094009—21no—
19C2×#13094000C2211yescharacteristic
20C2×#13093999C2211yescharacteristic
21C2×#13093998C2211yescharacteristic
22C2×#13093994—21no—
23C2×#13093986—21no—
24C2×(C22⋊D47)⋊C4—21no—
25C2×#13093982—21no—
26C2×#13093982—21no—
27C2×#13093982—24no—
28C2×#13093982C2212yes—
29C2×#13093982C2212yes—
30C2×#13093982C2211yescharacteristic
31C2×#13093982C2218yescomplemented
32C22×C4⋊Dic47C2211yescharacteristic
33C22×(C2×D47)⋊C4—21no—
34C22×C4×Dic47—21no—
35#15689463C22116yescomplemented
36#15689451C2218yescomplemented
37#15689450C22116yescomplemented
38#15689441C2218yescomplemented
39#15689437C2218yescomplemented
40#15689160C22116yescomplemented
41C2×#13093958C2212yescomplemented
42C2×#13093955C2212yescomplemented
43C2×#13093955C2212yescomplemented
44#15689118C22116yescomplemented
45C2×#13093948C2212yescomplemented
46C2×#13093946C2212yescomplemented
47C2×#13093944C2212yescomplemented
48C2×C22⋊(C47⋊C8)C2211yescharacteristic
49C2×(C2×D47)⋊C8C2212yes—
50C22⋊C8×D47C2218yescomplemented
51C2×C22⋊Q8×C47C2211yescharacteristic
52C22×C4⋊C4×C47C2211yescharacteristic
53C47×#484C2218yescomplemented
54C2×#107×C47C2212yescomplemented
55C2×#103×C47C2212yescomplemented
56C2×C22⋊C8×C47C2211yescharacteristic
57C24×D47D411yescharacteristic
58C2×#10625354—22no—
59C2×#10625354—28no—
60C2×#10625354C2314yes—
61C22×D188—21no—
62C22×D188—21no—
63C22×D188C2311yescharacteristic
64C22×C22⋊D47—21no—
65C23×Dic47—21no—
66C23×Dic47D411yescharacteristic
67C22×C4×D47—28no—
68C22×C4×D47—21no—
69C22×C4×D47D411yescharacteristic
70C22×C4×D47D411yescharacteristic
71C22×C4×D47D414yes—
72C22×C4×D47C2311yescharacteristic
73C22×C4×D47C2314yes—
74C22×C4×D47C2311yescharacteristic
75C22×C47⋊C8C2312yes—
76C2×C8×D47—28no—
77C2×C8×D47C2318yes—
78#13094009—28no—
79#13094000—22no—
80#13094000C2314yes—
81#13093999—22no—
82#13093999C2314yes—
83#13093998—22no—
84#13093998C2314yes—
85#13093994—28no—
86#13093986—28no—
87(C22⋊D47)⋊C4—28no—
88#13093982—24no—
89#13093982—24no—
90#13093982—28no—
91#13093982—28no—
92#13093982—22no—
93#13093982C2318yes—
94#13093982C2318yes—
95#13093982C2314yes—
96C2×C4⋊Dic47—21no—
97C2×C4⋊Dic47—22no—
98C2×C4⋊Dic47C2312yes—
99C2×C4⋊Dic47C2314yes—
100C2×C4⋊Dic47C2311yescharacteristic
101C2×Dic47⋊C4—21no—
102C2×(C2×D47)⋊C4—22no—
103C2×(C2×D47)⋊C4—22no—
104C2×(C2×D47)⋊C4—28no—
105C2×(C2×D47)⋊C4—21no—
106C2×(C2×D47)⋊C4—21no—
107C2×(C2×D47)⋊C4—21no—
108C2×(C2×D47)⋊C4—21no—
109C2×C4×Dic47—22no—
110C2×C4×Dic47—22no—
111C2×C4×Dic47—28no—
112C2×C4×Dic47—21no—
113#13093958C2318yescomplemented
114#13093955C2318yescomplemented
115#13093955C2318yescomplemented
116#13093948C2318yescomplemented
117#13093946C2318yescomplemented
118#13093944C2318yescomplemented
119C22⋊(C47⋊C8)C2314yes—
120(C2×D47)⋊C8C2318yes—
121C22×Q8×C47C2311yescharacteristic
122C23×C4×C47—21no—
123C23×C4×C47C2311yescharacteristic
124C22×C8×C47C2312yes—
125C22⋊Q8×C47—22no—
126C22⋊Q8×C47C2314yes—
127C2×C4⋊C4×C47—21no—
128C2×C4⋊C4×C47—22no—
129C2×C4⋊C4×C47—21no—
130C2×C4⋊C4×C47C2314yes—
131C2×C4⋊C4×C47C2312yes—
132C2×C4⋊C4×C47C2311yescharacteristic
133C2×C22⋊C4×C47—21no—
134#107×C47C2318yescomplemented
135#103×C47C2318yescomplemented
136C22⋊C8×C47C2314yes—
137C24×C47C2×D411yescharacteristic
138C23×D47—24no—
139C23×D47—24no—
140C23×D47—21no—
141C23×D47—42no—
142C23×D47C2×D411yescharacteristic
143C23×D47C2×D414yes—
144C23×D47C2×D411yescharacteristic
145C2×Q8×C47—21no—
146C2×Q8×C47—24no—
147C2×Q8×C47C2412yes—
148C22×C4×C47—24no—
149C22×C4×C47—21no—
150C22×C4×C47—21no—
151C22×C4×C47—24no—
152C22×C4×C47—22no—
153C22×C4×C47—44no—
154C22×C4×C47C2×D411yescharacteristic
155C22×C4×C47C2×D412yes—
156C22×C4×C47C2412yes—
157C22×C4×C47C2411yescharacteristic
158C2×C22⋊D47—24no—
159C2×C22⋊D47—44no—
160C22×Dic47—24no—
161C22×Dic47—24no—
162C22×Dic47—22no—
163C22×Dic47—21no—
164C22×Dic47—44no—
165C22×Dic47C2×D411yescharacteristic
166C22×Dic47C2×D414yes—
167C22×Dic47C2×D411yescharacteristic
168#10625354—216no—
169#10625354—48no—
170C2×D188—24no—
171C2×D188—24no—
172C2×D188—24no—
173C2×D188—21no—
174C2×D188—44no—
175C2×D188—44no—
176C2×D188C2412yes—
177C2×C4×D47—24no—
178C2×C4×D47—24no—
179C2×C4×D47—24no—
180C2×C4×D47—216no—
181C2×C4×D47—216no—
182C2×C4×D47—24no—
183C2×C4×D47—44no—
184C2×C4×D47C2×D414yes—
185C2×C4×D47C2×D412yes—
186C2×C4×D47C2×D414yes—
187C2×C4×D47C2×D412yes—
188C2×C4×D47#4118yes—
189C2×C4×D47C2412yes—
190C2×C4×D47C2412yes—
191C2×C8×C47—24no—
192C2×C8×C47C2414yes—
193C4⋊C4×C47—24no—
194C4⋊C4×C47—22no—
195C4⋊C4×C47—24no—
196C4⋊C4×C47—21no—
197C4⋊C4×C47C2414yes—
198C4⋊C4×C47C2412yes—
199C22⋊C4×C47—24no—
200(C2×D47)⋊C4—28no—
201(C2×D47)⋊C4—28no—
202(C2×D47)⋊C4—24no—
203(C2×D47)⋊C4—24no—
204(C2×D47)⋊C4—24no—
205(C2×D47)⋊C4—24no—
206C4⋊Dic47—24no—
207C4⋊Dic47—22no—
208C4⋊Dic47—21no—
209C4⋊Dic47C2414yes—
210C4⋊Dic47C2412yes—
211Dic47⋊C4—24no—
212C4×Dic47—28no—
213C4×Dic47—28no—
214C4×Dic47—24no—
215C2×C47⋊C8—24no—
216C2×C47⋊C8C2414yes—
217C8×D47—216no—
218C23×C47—22no—
219C23×C47—22no—
220C23×C47#13514yes—
221C23×C47C22×D412yes—
222C23×C47C22×D411yescharacteristic
223C22×D47—22no—
224C22×D47—22no—
225C22×D47—22no—
226C22×D47—28no—
227C22×D47—22no—
228C22×D47—22no—
229C22×D47—28no—
230C22×D47—22no—
231C22×D47—44no—
232C22×D47—416no—
233C22×D47—42no—
234C22×D47—42no—
235C22×D47—42no—
236C22×D47#12314yes—
237C22×D47C22×D412yes—
238C22×D47C22×D412yes—
239Q8×C47—24no—
240Q8×C47—42no—
241C2×C4×C47—22no—
242C2×C4×C47—24no—
243C2×C4×C47—24no—
244C2×C4×C47—24no—
245C2×C4×C47—22no—
246C2×C4×C47—22no—
247C2×C4×C47—48no—
248C2×C4×C47—42no—
249C2×C4×C47—42no—
250C2×C4×C47—44no—
251C2×C4×C47C22×D412yes—
252C2×C4×C47C22×D411yescharacteristic
253C2×C4×C47C2×#4114yes—
254C2×C4×C47C2511yescharacteristic
255C22⋊D47—48no—
256C2×Dic47—22no—
257C2×Dic47—28no—
258C2×Dic47—24no—
259C2×Dic47—22no—
260C2×Dic47—48no—
261C2×Dic47—44no—
262C2×Dic47—42no—
263C2×Dic47#12314yes—
264C2×Dic47C22×D412yes—
265C2×Dic47C22×D412yes—
266D188—24no—
267D188—48no—
268D188—48no—
269D188—42no—
270C4×D47—28no—
271C4×D47—24no—
272C4×D47—24no—
273C4×D47—28no—
274C4×D47—48no—
275C4×D47#12318yes—
276C4×D47C2×#4118yes—
277C8×C47—24no—
278C47⋊C8—24no—
279C2×#3091—471nomaximal, Sylow
280C22×C47—28no—
281C22×C47—22no—
282C22×C47—22no—
283C22×C47C2×#12312yes—
284C22×C47C2×#13514yes—
285C22×C47C2×#13512yes—
286C22×C47#57512yes—
287C22×C47C23×D411yescharacteristic
288C2×D47—28no—
289C2×D47—22no—
290C2×D47—22no—
291C2×D47—22no—
292C2×D47—22no—
293C2×D47—22no—
294C2×D47—416no—
295C2×D47—44no—
296C2×D47—42no—
297C2×D47—42no—
298C2×D47—42no—
299C2×D47—84no—
300C2×D47C2×#12314yes—
301C4×C47—22no—
302C4×C47—42no—
303C4×C47—42no—
304C4×C47—44no—
305C4×C47C2×#12312yes—
306C4×C47C22×#4112yes—
307Dic47—22no—
308Dic47—44no—
309Dic47—42no—
310Dic47C2×#12314yes—
311C2×#525—471no—
312C22×#117—471no—
313#3091—4716no—
314C2×#484D4711yescharacteristic, complemented
315C2×#481—471no—
316C2×#485—472no—
317C2×#482—472no—
318C2×#483—471no—
319C2×#480—471no—
320C2×#427—472no—
321C2×#416—472no—
322C22×C22⋊C8—471no—
323C2×C47—24no—
324C2×C47#309114yescomplemented
325C2×C47C22×#12311yescharacteristic
326C2×C47C22×#13511yescharacteristic
327C2×C47C2×#57511yescharacteristic
328D47—24no—
329D47—42no—
330D47—82no—
331C22×#41—471no—
332C24×C4—471no—
333C23×C8—472no—
334#525—478no—
335#525—944no—
336C2×#123—941no—
337C2×C22⋊Q8—471no—
338C2×C22⋊Q8C2×D4711yescharacteristic
339C2×#121—471no—
340C2×#121—471no—
341C2×#120—941no—
342C2×C4×D4—941no—
343C2×C4×D4—941no—
344C2×#117—472no—
345C2×#117—478no—
346C2×#117—472no—
347C2×#117—471no—
348C2×#117—941no—
349C2×#117—941no—
350C2×#117—944no—
351C22×C4⋊C4—471no—
352C22×C4⋊C4C2×D4711yescharacteristic
353C22×C22⋊C4—941no—
354C22×C42—941no—
355#485—4716no—
356#484C2×D4718yescomplemented
357#483—478no—
358#482—4716no—
359#481—478no—
360#480—478no—
361#427—4716no—
362C2×#107—472no—
363C2×#107C2×D4712yescomplemented
364C2×#106—472no—
365C2×#106—472no—
366#416—4716no—
367C2×#103—472no—
368C2×#103C2×D4712yescomplemented
369C2×#102—472no—
370C2×#102—472no—
371C2×C22⋊C8—471no—
372C2×C22⋊C8—472no—
373C2×C22⋊C8—478no—
374C2×C22⋊C8C2×D4711yescharacteristic
375C47C2×#309111yescharacteristic, complemented, Sylow
376C25—471no—
377C2×#41—474no—
378C2×#41—942no—
379C2×#41—948no—
380C22×Q8C22×D4711yescharacteristic
381C22×D4—471no—
382C22×D4—941no—
383C22×D4—941no—
384C22×D4—941no—
385C23×C4—21no—
386C23×C4—471no—
387C23×C4—471no—
388C23×C4—471no—
389C23×C4—471no—
390C23×C4—474no—
391C23×C4—471no—
392C23×C4—474no—
393C23×C4—948no—
394C23×C4—941no—
395C23×C4—941no—
396C23×C4C22×D4711yescharacteristic
397C22×C8—472no—
398C22×C8—478no—
399C22×C8—948no—
400C22×C8C22×D4712yes—
401#123—948no—
402C22⋊Q8—22no—
403C22⋊Q8—474no—
404C22⋊Q8—942no—
405C22⋊Q8C22×D4714yes—
406#121—474no—
407#121—474no—
408#121—942no—
409#121—942no—
410#120—948no—
411C4×D4—948no—
412C4×D4—948no—
413#117—478no—
414#117—478no—
415#117—474no—
416#117—944no—
417#117—948no—
418#117—944no—
419#117—948no—
420#117—942no—
421C2×C4⋊C4—22no—
422C2×C4⋊C4—21no—
423C2×C4⋊C4—21no—
424C2×C4⋊C4—472no—
425C2×C4⋊C4—474no—
426C2×C4⋊C4—471no—
427C2×C4⋊C4—941no—
428C2×C4⋊C4—942no—
429C2×C4⋊C4—941no—
430C2×C4⋊C4C22×D4712yes—
431C2×C4⋊C4C22×D4714yes—
432C2×C4⋊C4C22×D4711yescharacteristic
433C2×C22⋊C4—21no—
434C2×C22⋊C4—941no—
435C2×C22⋊C4—948no—
436C2×C22⋊C4—942no—
437C2×C22⋊C4—942no—
438C2×C22⋊C4—941no—
439C2×C22⋊C4—941no—
440C2×C22⋊C4—941no—
441C2×C42—948no—
442C2×C42—942no—
443C2×C42—942no—
444C2×C42—941no—
445#107—478no—
446#107C22×D4718yescomplemented
447#106—478no—
448#106—478no—
449#103—478no—
450#103C22×D4718yescomplemented
451#102—478no—
452#102—478no—
453C22⋊C8—474no—
454C22⋊C8—478no—
455C22⋊C8C22×D4714yes—
456C24—471no—
457C24—471no—
458C24—474no—
459C24—944no—
460C24—944no—
461C24—941no—
462C24—1882no—
463C24D4×D4711yescharacteristic
464#41—9416no—
465#41—1888no—
466C2×Q8—21no—
467C2×Q8—24no—
468C2×Q8C23×D4712yes—
469C2×D4—472no—
470C2×D4—944no—
471C2×D4—944no—
472C2×D4—944no—
473C2×D4—941no—
474C2×D4—944no—
475C2×D4—1884no—
476C2×D4—1884no—
477C2×D4—1884no—
478C22×C4—24no—
479C22×C4—24no—
480C22×C4—22no—
481C22×C4—21no—
482C22×C4—21no—
483C22×C4—44no—
484C22×C4—471no—
485C22×C4—472no—
486C22×C4—478no—
487C22×C4—472no—
488C22×C4—474no—
489C22×C4—471no—
490C22×C4—474no—
491C22×C4—472no—
492C22×C4—474no—
493C22×C4—472no—
494C22×C4—944no—
495C22×C4—9416no—
496C22×C4—9416no—
497C22×C4—944no—
498C22×C4—944no—
499C22×C4—944no—
500C22×C4—944no—
501C22×C4—944no—
502C22×C4—942no—
503C22×C4—941no—
504C22×C4—1884no—
505C22×C4—1884no—
506C22×C4D4×D4712yes—
507C22×C4D4×D4711yescharacteristic
508C22×C4C23×D4712yes—
509C22×C4C23×D4711yescharacteristic
510C2×C8—24no—
511C2×C8—474no—
512C2×C8—9416no—
513C2×C8—944no—
514C2×C8C23×D4714yes—
515C4⋊C4—24no—
516C4⋊C4—22no—
517C4⋊C4—24no—
518C4⋊C4—21no—
519C4⋊C4—474no—
520C4⋊C4—472no—
521C4⋊C4—944no—
522C4⋊C4—944no—
523C4⋊C4—942no—
524C4⋊C4—941no—
525C4⋊C4C23×D4714yes—
526C4⋊C4C23×D4712yes—
527C22⋊C4—24no—
528C22⋊C4—944no—
529C22⋊C4—948no—
530C22⋊C4—948no—
531C22⋊C4—944no—
532C22⋊C4—944no—
533C22⋊C4—944no—
534C42—948no—
535C42—948no—
536C42—944no—
537C23—22no—
538C23—22no—
539C23—474no—
540C23—472no—
541C23—472no—
542C23—948no—
543C23—942no—
544C23—942no—
545C23—942no—
546C23—948no—
547C23—942no—
548C23—942no—
549C23—942no—
550C23—1884no—
551C23—18816no—
552C23—1882no—
553C23—1882no—
554C23—1882no—
555C23#1309404714yes—
556C23C2×D4×D4712yes—
557C23C2×D4×D4711yescharacteristic
558Q8—24no—
559Q8—42no—
560D4—944no—
561D4—1888no—
562D4—1888no—
563D4—1888no—
564D4—1882no—
565C2×C4—22no—
566C2×C4—24no—
567C2×C4—24no—
568C2×C4—24no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—48no—
572C2×C4—42no—
573C2×C4—44no—
574C2×C4—42no—
575C2×C4—474no—
576C2×C4—472no—
577C2×C4—478no—
578C2×C4—478no—
579C2×C4—472no—
580C2×C4—948no—
581C2×C4—948no—
582C2×C4—948no—
583C2×C4—942no—
584C2×C4—944no—
585C2×C4—944no—
586C2×C4—944no—
587C2×C4—942no—
588C2×C4—1888no—
589C2×C4—1888no—
590C2×C4—1884no—
591C2×C4—1882no—
592C2×C4C2×D4×D4712yes—
593C2×C4C2×D4×D4711yescharacteristic
594C2×C4#41×D4714yes—
595C2×C4C24×D4711yescharacteristic
596C8—24no—
597C8—944no—
598C22—22no—
599C22—22no—
600C22—28no—
601C22—474no—
602C22—942no—
603C22—942no—
604C22—948no—
605C22—942no—
606C22—942no—
607C22—942no—
608C22—3764no—
609C22—18816no—
610C22—1882no—
611C22—1882no—
612C22—1882no—
613C22—1884no—
614C22#123×D4712yes—
615C22C2×#1309404714yes—
616C22C2×#1309404712yes—
617C22#1568991912yes—
618C22C22×D4×D4711yescharacteristic
619C4—22no—
620C4—44no—
621C4—42no—
622C4—42no—
623C4—474no—
624C4—942no—
625C4—1884no—
626C4—1882no—
627C4#123×D4712yes—
628C4C2×#41×D4712yes—
629C2—24no—
630C2—944no—
631C2—3762no—
632C2—1882no—
633C2#2025330414yesdirect product factor
634C2C2×#123×D4711yescharacteristic
635C2C22×#1309404711yescharacteristic
636C2C2×#1568991911yescharacteristic
subgroupG/H
centerC23C2×D4×D47
commutator, cosocleC2×C4×C47C25
FrattiniC2×C4C24×D47
FittingC2×C47×#484C2
socleC23×C47C22×D4
prime radicalC2×#15689636C2
hypercenterC2×#484D47
hypocenterC47C2×#3091
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#484

Complements

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

NH ≅ G/Ntype
2. C22×#13093982C2semidirect
3. #20253304C2direct (N × H)
4. C2×#15689441C2semidirect
5. C2×#15689463C2semidirect
6. C2×#15689450C2semidirect
7. C2×#15689451C2semidirect
8. C2×#15689437C2semidirect
9. C2×#15689160C2semidirect
10. C2×#15689118C2semidirect
11. C2×C22⋊C8×D47C2semidirect
12. C2×C47×#484C2semidirect
31. C2×#13093982C22semidirect
35. #15689463C22semidirect
36. #15689451C22semidirect
37. #15689450C22semidirect
38. #15689441C22semidirect
39. #15689437C22semidirect
40. #15689160C22semidirect
41. C2×#13093958C22semidirect
42. C2×#13093955C22semidirect
43. C2×#13093955C22semidirect
44. #15689118C22semidirect
45. C2×#13093948C22semidirect
46. C2×#13093946C22semidirect
47. C2×#13093944C22semidirect
50. C22⋊C8×D47C22semidirect
53. C47×#484C22semidirect
54. C2×#107×C47C22semidirect
55. C2×#103×C47C22semidirect
113. #13093958C23semidirect
114. #13093955C23semidirect
115. #13093955C23semidirect
116. #13093948C23semidirect
117. #13093946C23semidirect
118. #13093944C23semidirect
134. #107×C47C23semidirect
135. #103×C47C23semidirect
314. C2×#484D47semidirect
324. C2×C47#3091semidirect
356. #484C2×D47semidirect
363. C2×#107C2×D47semidirect
368. C2×#103C2×D47semidirect
375. C47C2×#3091semidirect
446. #107C22×D47semidirect
450. #103C22×D47semidirect
633. C2#20253304direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#156896362
C22×#130939822
#202533042
C2×#156894412
C2×#156894632
C2×#156894502
C2×#156894512
C2×#156894372
C2×#156891602
C2×#156891182
C2×C22⋊C8×D472
C2×C47×#4842
C2×#309147