Finite Group Databasebeta

← C2×#20253179 #27083782

Subgroups — C2×#20253179 #27083782

Order 12032.

2162 conjugacy classes of subgroups.

The subgroup lattice has 596 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689678C211yesmaximal, characteristic, complemented
2C2×C4×#10625354C211yesmaximal, characteristic
3#20253179C2116yesmaximal, direct product factor
4C2×#15689432C212yesmaximal, complemented
5C2×#15689408C211yesmaximal, characteristic, complemented
6C2×#15689426C212yesmaximal, complemented
7C2×#15689410C211yesmaximal, characteristic, complemented
8C2×#15689404C211yesmaximal, characteristic, complemented
9C2×#15689160C212yesmaximal, complemented
10C2×C4⋊C8×D47C211yesmaximal, characteristic, complemented
11C2×#15689118C212yesmaximal
12C2×C47×#477C211yesmaximal, characteristic, complemented
13C22×#10625354C2211yescharacteristic
14C22×C8×D47C2212yes—
15#15689678—24no—
16#15689678C2218yescomplemented
17C2×#13094024C2211yescharacteristic
18C2×#13094015C2212yes—
19C2×#13094004—21no—
20C2×#13094000—21no—
21C4×#10625354—24no—
22C4×#10625354C2218yes—
23C2×Q8×Dic47—21no—
24C2×Q8×Dic47C2211yescharacteristic
25C2×#13093989—21no—
26C2×C4×D188—21no—
27C2×C4×D188C2211yescharacteristic
28C2×#13093986—21no—
29C2×#13093986C2211yescharacteristic
30C2×#13093982—21no—
31C2×#13093982C2211yescharacteristic
32C2×#13093982C2212yes—
33C2×C42×D47—21no—
34C2×C42×D47C2211yescharacteristic
35#15689432C22116yescomplemented
36#15689426C22116yescomplemented
37#15689410C2218yescomplemented
38#15689408C2218yescomplemented
39#15689404C2218yescomplemented
40#15689160C22116yescomplemented
41C2×#13093958C2212yes—
42C2×#13093955C2212yes—
43C2×#13093955C2212yes—
44C2×C4⋊(C47⋊C8)C2211yescharacteristic
45C2×Dic47⋊C8C2212yes—
46C4⋊C8×D47C2218yescomplemented
47#15689118C22116yes—
48C2×#13093948C2212yes—
49C2×#13093946C2212yes—
50C2×#13093944C2212yes—
51C2×C4⋊Q8×C47C2211yescharacteristic
52C2×C4×Q8×C47C2211yescharacteristic
53C47×#477C2218yescomplemented
54C2×#107×C47C2212yes—
55C2×C4⋊C8×C47C2211yescharacteristic
56C2×#103×C47C2212yes—
57C2×#10625354—22no—
58C2×#10625354—24no—
59C2×#10625354Q818yescomplemented
60C2×#10625354C2314yes—
61C22×Dic94—21no—
62C22×D188—21no—
63C22×D188C2311yescharacteristic
64C22×C4×D47—21no—
65C22×C4×D47—21no—
66C22×C4×D47—21no—
67C22×C4×D47D411yescharacteristic
68C22×C4×D47D411yescharacteristic
69C22×C4×D47C2311yescharacteristic
70C22×C47⋊C8C2312yes—
71C2×C8×D47—28no—
72C2×C8×D47C2318yes—
73#13094024—22no—
74#13094024C2314yes—
75#13094015—24no—
76#13094015C2318yes—
77#13094004—28no—
78#13094000—28no—
79Q8×Dic47—28no—
80Q8×Dic47—22no—
81Q8×Dic47C2314yes—
82#13093989—28no—
83C4×D188—28no—
84C4×D188—22no—
85C4×D188C2314yes—
86#13093986—28no—
87#13093986—22no—
88#13093986C2314yes—
89#13093982—22no—
90#13093982—24no—
91#13093982—28no—
92#13093982C2314yes—
93#13093982C2318yes—
94C2×C4⋊Dic47—21no—
95C2×C4⋊Dic47—21no—
96C2×C4⋊Dic47—22no—
97C2×C4⋊Dic47C2311yescharacteristic
98C2×C4⋊Dic47C2312yes—
99C2×Dic47⋊C4—22no—
100C2×(C2×D47)⋊C4—21no—
101C2×(C2×D47)⋊C4—22no—
102C2×(C2×D47)⋊C4—21no—
103C2×(C2×D47)⋊C4—21no—
104C2×C4×Dic47—21no—
105C2×C4×Dic47—21no—
106C2×C4×Dic47—22no—
107C2×C4×Dic47D411yescharacteristic
108C2×C4×Dic47D411yescharacteristic
109C2×C4×Dic47C2311yescharacteristic
110C42×D47—28no—
111C42×D47D414yes—
112C42×D47C2314yes—
113#13093958C2318yes—
114#13093955C2318yes—
115#13093955C2318yes—
116C4⋊(C47⋊C8)C2314yes—
117Dic47⋊C8C2318yes—
118#13093948C2318yes—
119#13093946C2318yes—
120#13093944C2318yes—
121C22×Q8×C47—21no—
122C22×Q8×C47C2311yescharacteristic
123C22×C8×C47C2312yes—
124C4⋊Q8×C47—22no—
125C4⋊Q8×C47C2314yes—
126C4×Q8×C47—22no—
127C4×Q8×C47C2314yes—
128C2×C4⋊C4×C47—21no—
129C2×C4⋊C4×C47—21no—
130C2×C4⋊C4×C47C2311yescharacteristic
131C2×C4⋊C4×C47C2312yes—
132C2×C42×C47—21no—
133C2×C42×C47C2311yescharacteristic
134#107×C47C2318yes—
135C4⋊C8×C47C2314yes—
136#103×C47C2318yes—
137C23×D47—21no—
138C23×D47C2×D411yescharacteristic
139C2×Q8×C47—21no—
140C2×Q8×C47—24no—
141C2×Q8×C47—22no—
142C2×Q8×C47—44no—
143C2×Q8×C47C2×Q814yes—
144C2×Q8×C47C2412yes—
145C22×C4×C47—21no—
146C22×C4×C47—21no—
147C22×C4×C47—22no—
148C22×C4×C47C2×D411yescharacteristic
149C22×C4×C47C2×D411yescharacteristic
150C22×C4×C47C2411yescharacteristic
151C22×Dic47—21no—
152C22×Dic47—22no—
153C22×Dic47C2×D411yescharacteristic
154C22×Dic47C2×D411yescharacteristic
155C22×Dic47C2×D411yescharacteristic
156#10625354—216no—
157#10625354—28no—
158#10625354C2×Q8116yescomplemented
159C2×D188—28no—
160C2×D188—24no—
161C2×D188—21no—
162C2×D188—22no—
163C2×D188C2×Q814yes—
164C2×D188C2412yes—
165C2×C4×D47—24no—
166C2×C4×D47—28no—
167C2×C4×D47—24no—
168C2×C4×D47—24no—
169C2×C4×D47—24no—
170C2×C4×D47—42no—
171C2×C4×D47—44no—
172C2×C4×D47—44no—
173C2×C4×D47C2×D414yes—
174C2×C4×D47C2×D414yes—
175C2×C4×D47C2×D412yes—
176C2×C4×D47C2×Q814yes—
177C2×C4×D47#4114yes—
178C2×C4×D47C2412yes—
179C2×Dic94—24no—
180C2×Dic94—44no—
181C2×C8×C47—24no—
182C2×C8×C47C2414yes—
183C4⋊C4×C47—24no—
184C4⋊C4×C47—24no—
185C4⋊C4×C47—21no—
186C4⋊C4×C47—22no—
187C4⋊C4×C47C2414yes—
188C4⋊C4×C47C2412yes—
189C42×C47—24no—
190C42×C47C2×D412yes—
191C42×C47C2412yes—
192(C2×D47)⋊C4—24no—
193(C2×D47)⋊C4—28no—
194(C2×D47)⋊C4—24no—
195(C2×D47)⋊C4—24no—
196C4⋊Dic47—24no—
197C4⋊Dic47—21no—
198C4⋊Dic47—22no—
199C4⋊Dic47—24no—
200C4⋊Dic47—28no—
201C4⋊Dic47C2414yes—
202C4⋊Dic47C2412yes—
203Dic47⋊C4—28no—
204C4×Dic47—24no—
205C4×Dic47—24no—
206C4×Dic47—28no—
207C4×Dic47C2×D414yes—
208C4×Dic47C2×D412yes—
209C4×Dic47C2×D414yes—
210C4×Dic47C2412yes—
211C2×C47⋊C8—24no—
212C2×C47⋊C8C2414yes—
213C8×D47—216no—
214C23×C47C22×D411yescharacteristic
215C22×D47—22no—
216C22×D47—22no—
217C22×D47—24no—
218C22×D47—22no—
219C22×D47—42no—
220C22×D47—42no—
221C22×D47C22⋊Q814yes—
222C22×D47C22×D412yes—
223Q8×C47—24no—
224Q8×C47—22no—
225Q8×C47—48no—
226Q8×C47C22×Q814yes—
227C2×C4×C47—24no—
228C2×C4×C47—22no—
229C2×C4×C47—22no—
230C2×C4×C47—24no—
231C2×C4×C47—22no—
232C2×C4×C47—42no—
233C2×C4×C47—44no—
234C2×C4×C47#13514yes—
235C2×C4×C47C22×D412yes—
236C2×C4×C47C22×D411yescharacteristic
237C2×C4×C47C22×D412yes—
238C2×C4×C47C22×Q812yes—
239C2×C4×C47C2×#4112yes—
240C2×C4×C47C2511yescharacteristic
241C2×Dic47—22no—
242C2×Dic47—22no—
243C2×Dic47—22no—
244C2×Dic47—24no—
245C2×Dic47—44no—
246C2×Dic47—42no—
247C2×Dic47C22⋊Q814yes—
248C2×Dic47C22×D412yes—
249C2×Dic47C22×D412yes—
250C2×Dic47C22×D412yes—
251D188—216no—
252D188—24no—
253D188—22no—
254D188C22×Q814yes—
255C4×D47—28no—
256C4×D47—216no—
257C4×D47—44no—
258C4×D47—48no—
259C4×D47—48no—
260C4×D47C22⋊Q818yes—
261C4×D47C22×Q814yes—
262C4×D47C2×#4114yes—
263Dic94—48no—
264C8×C47—24no—
265C47⋊C8—24no—
266C2×#3076—471nomaximal, Sylow
267C22×C47C2×C22⋊Q812yes—
268C22×C47C2×#13512yes—
269C22×C47#57512yes—
270C22×C47C23×D411yescharacteristic
271C2×D47—28no—
272C2×D47—22no—
273C2×D47—22no—
274C2×D47—24no—
275C2×D47—48no—
276C2×D47—42no—
277C2×D47—42no—
278C2×D47C2×C22⋊Q814yes—
279C4×C47—24no—
280C4×C47—22no—
281C4×C47—42no—
282C4×C47—44no—
283C4×C47C2×C22⋊Q812yes—
284C4×C47C2×#13514yes—
285C4×C47C23×Q811yescharacteristic
286C4×C47C22×#4111yescharacteristic
287Dic47—22no—
288Dic47—22no—
289Dic47—44no—
290Dic47—42no—
291Dic47C2×C22⋊Q814yes—
292C2×#533—471no—
293C2×C4×#41—471no—
294#3076—4716no—
295C2×#479—472no—
296C2×#477D4711yescharacteristic, complemented
297C2×#475—471no—
298C2×#478—472no—
299C2×#476—471no—
300C2×#474—471no—
301C2×#427—472no—
302C22×C4⋊C8—471no—
303C2×#416—472no—
304C2×C47#307614yescomplemented
305C2×C47C22×C22⋊Q811yescharacteristic
306C2×C47C22×#13511yescharacteristic
307C2×C47C2×#57511yescharacteristic
308D47—24no—
309D47—44no—
310C22×#41—471no—
311C23×C8—472no—
312#533—478no—
313#533—944no—
314C2×C4⋊Q8—471no—
315C2×C4⋊Q8C2×D4711yescharacteristic
316C2×#125—472no—
317C2×C22⋊Q8—941no—
318C2×C22⋊Q8—941no—
319C4×#41—478no—
320C4×#41—944no—
321C2×C4×Q8—471no—
322C2×C4×Q8—941no—
323C2×C4×Q8—941no—
324C2×C4×Q8C2×D4711yescharacteristic
325C2×C4×D4—471no—
326C2×C4×D4—471no—
327C2×C4×D4—941no—
328C2×C4×D4—941no—
329C2×#117—472no—
330C2×#117—471no—
331C2×#117—941no—
332C22×C42—471no—
333C22×C42—941no—
334#479—4716no—
335#478—4716no—
336#477C2×D4718yescomplemented
337#476—478no—
338#475—478no—
339#474—478no—
340#427—4716no—
341C2×#107—472no—
342C2×#107C2×D4712yes—
343C2×#106—472no—
344C2×#106—472no—
345C2×C4⋊C8—472no—
346C2×C4⋊C8—471no—
347C2×C4⋊C8—478no—
348C2×C4⋊C8C2×D4711yescharacteristic
349#416—4716no—
350C2×#103—472no—
351C2×#103C2×D4712yes—
352C2×#102—472no—
353C2×#102—472no—
354C47C2×#307611yescharacteristic, complemented, Sylow
355C2×#41—478no—
356C2×#41—474no—
357C2×#41—942no—
358C2×#41—944no—
359C22×Q8—21no—
360C22×Q8—941no—
361C22×Q8C22×D4711yescharacteristic
362C22×D4—471no—
363C22×D4—941no—
364C23×C4—471no—
365C23×C4—471no—
366C23×C4—471no—
367C23×C4—941no—
368C23×C4—941no—
369C23×C4—941no—
370C22×C8—472no—
371C22×C8—478no—
372C22×C8—948no—
373C22×C8C22×D4712yes—
374C4⋊Q8—22no—
375C4⋊Q8—474no—
376C4⋊Q8—942no—
377C4⋊Q8C22×D4714yes—
378#125—478no—
379#125—944no—
380C22⋊Q8—948no—
381C22⋊Q8—948no—
382C4×Q8—22no—
383C4×Q8—474no—
384C4×Q8—948no—
385C4×Q8—942no—
386C4×Q8—948no—
387C4×Q8C22×D4714yes—
388C4×D4—474no—
389C4×D4—474no—
390C4×D4—948no—
391C4×D4—942no—
392C4×D4—948no—
393C4×D4—942no—
394#117—474no—
395#117—478no—
396#117—948no—
397#117—942no—
398#117—944no—
399C2×C4⋊C4—21no—
400C2×C4⋊C4—21no—
401C2×C4⋊C4—471no—
402C2×C4⋊C4—472no—
403C2×C4⋊C4—941no—
404C2×C4⋊C4—942no—
405C2×C4⋊C4—942no—
406C2×C4⋊C4—941no—
407C2×C4⋊C4C22×D4711yescharacteristic
408C2×C4⋊C4C22×D4712yes—
409C2×C22⋊C4—941no—
410C2×C22⋊C4—941no—
411C2×C22⋊C4—942no—
412C2×C22⋊C4—941no—
413C2×C42—21no—
414C2×C42—471no—
415C2×C42—471no—
416C2×C42—471no—
417C2×C42—474no—
418C2×C42—474no—
419C2×C42—948no—
420C2×C42—941no—
421C2×C42—942no—
422C2×C42—941no—
423C2×C42C22×D4711yescharacteristic
424#107—478no—
425#107C22×D4718yes—
426#106—478no—
427#106—478no—
428C4⋊C8—478no—
429C4⋊C8—474no—
430C4⋊C8C22×D4714yes—
431#103—478no—
432#103C22×D4718yes—
433#102—478no—
434#102—478no—
435C24—471no—
436C24—941no—
437#41—4716no—
438#41—9416no—
439#41—948no—
440C2×Q8—21no—
441C2×Q8—24no—
442C2×Q8—22no—
443C2×Q8—44no—
444C2×Q8—944no—
445C2×Q8—1884no—
446C2×Q8Q8×D4714yes—
447C2×Q8C23×D4712yes—
448C2×D4—474no—
449C2×D4—472no—
450C2×D4—944no—
451C2×D4—948no—
452C2×D4—941no—
453C2×D4—942no—
454C22×C4—21no—
455C22×C4—21no—
456C22×C4—22no—
457C22×C4—471no—
458C22×C4—472no—
459C22×C4—474no—
460C22×C4—474no—
461C22×C4—472no—
462C22×C4—471no—
463C22×C4—471no—
464C22×C4—474no—
465C22×C4—474no—
466C22×C4—944no—
467C22×C4—948no—
468C22×C4—944no—
469C22×C4—944no—
470C22×C4—941no—
471C22×C4—942no—
472C22×C4—944no—
473C22×C4—1882no—
474C22×C4—1884no—
475C22×C4—1884no—
476C22×C4D4×D4711yescharacteristic
477C22×C4D4×D4711yescharacteristic
478C22×C4C23×D4711yescharacteristic
479C2×C8—24no—
480C2×C8—474no—
481C2×C8—9416no—
482C2×C8—944no—
483C2×C8C23×D4714yes—
484C4⋊C4—24no—
485C4⋊C4—24no—
486C4⋊C4—21no—
487C4⋊C4—22no—
488C4⋊C4—474no—
489C4⋊C4—472no—
490C4⋊C4—948no—
491C4⋊C4—944no—
492C4⋊C4—948no—
493C4⋊C4—941no—
494C4⋊C4—942no—
495C4⋊C4—944no—
496C4⋊C4C23×D4714yes—
497C4⋊C4C23×D4712yes—
498C22⋊C4—944no—
499C22⋊C4—944no—
500C22⋊C4—948no—
501C22⋊C4—944no—
502C42—24no—
503C42—474no—
504C42—474no—
505C42—472no—
506C42—472no—
507C42—944no—
508C42—944no—
509C42—948no—
510C42D4×D4712yes—
511C42C23×D4712yes—
512C23—474no—
513C23—472no—
514C23—942no—
515C23—942no—
516C23—944no—
517C23—942no—
518C23—1882no—
519C23—1882no—
520C23C2×D4×D4711yescharacteristic
521Q8—24no—
522Q8—22no—
523Q8—48no—
524Q8—1888no—
525Q8C2×Q8×D4714yes—
526D4—474no—
527D4—9416no—
528D4—944no—
529D4—942no—
530C2×C4—24no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—24no—
535C2×C4—42no—
536C2×C4—44no—
537C2×C4—474no—
538C2×C4—472no—
539C2×C4—478no—
540C2×C4—474no—
541C2×C4—474no—
542C2×C4—472no—
543C2×C4—472no—
544C2×C4—9416no—
545C2×C4—942no—
546C2×C4—942no—
547C2×C4—948no—
548C2×C4—942no—
549C2×C4—944no—
550C2×C4—1884no—
551C2×C4—1888no—
552C2×C4—1882no—
553C2×C4—1884no—
554C2×C4—1888no—
555C2×C4#1309404714yes—
556C2×C4C2×D4×D4712yes—
557C2×C4C2×D4×D4711yescharacteristic
558C2×C4C2×D4×D4712yes—
559C2×C4C2×Q8×D4712yes—
560C2×C4#41×D4712yes—
561C2×C4C24×D4711yescharacteristic
562C8—24no—
563C8—944no—
564C22—474no—
565C22—942no—
566C22—942no—
567C22—948no—
568C22—944no—
569C22—1882no—
570C22—1882no—
571C22—1888no—
572C22C22⋊Q8×D4712yes—
573C22C2×#1309404712yes—
574C22#1568991912yes—
575C22C22×D4×D4711yescharacteristic
576C4—22no—
577C4—24no—
578C4—42no—
579C4—44no—
580C4—474no—
581C4—942no—
582C4—942no—
583C4—1882no—
584C4—1884no—
585C4C22⋊Q8×D4712yes—
586C4C2×#1309404714yes—
587C4C22×Q8×D4711yescharacteristic
588C4C2×#41×D4711yescharacteristic
589C2—944no—
590C2—1884no—
591C2#2025317914yesdirect product factor
592C2C2×C22⋊Q8×D4711yescharacteristic
593C2C22×#1309404711yescharacteristic
594C2C2×#1568991911yescharacteristic
subgroupG/H
centerC23C2×D4×D47
commutator, cosocleC2×C4×C47C25
FrattiniC2×C4C24×D47
FittingC2×C47×#477C2
socleC23×C47C22×D4
prime radicalC22×#10625354C22
hypercenterC2×#477D47
hypocenterC47C2×#3076
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#477

Complements

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

NH ≅ G/Ntype
1. C2×#15689678C2semidirect
3. #20253179C2direct (N × H)
4. C2×#15689432C2semidirect
5. C2×#15689408C2semidirect
6. C2×#15689426C2semidirect
7. C2×#15689410C2semidirect
8. C2×#15689404C2semidirect
9. C2×#15689160C2semidirect
10. C2×C4⋊C8×D47C2semidirect
12. C2×C47×#477C2semidirect
16. #15689678C22semidirect
35. #15689432C22semidirect
36. #15689426C22semidirect
37. #15689410C22semidirect
38. #15689408C22semidirect
39. #15689404C22semidirect
40. #15689160C22semidirect
46. C4⋊C8×D47C22semidirect
53. C47×#477C22semidirect
59. C2×#10625354Q8semidirect
158. #10625354C2×Q8semidirect
296. C2×#477D47semidirect
304. C2×C47#3076semidirect
336. #477C2×D47semidirect
354. C47C2×#3076semidirect
591. C2#20253179direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#156896782
C2×C4×#106253542
#202531792
C2×#156894322
C2×#156894082
C2×#156894262
C2×#156894102
C2×#156894042
C2×#156891602
C2×C4⋊C8×D472
C2×#156891182
C2×C47×#4772
C2×#307647