Finite Group Databasebeta

← C2×#20253209 #27083816

Subgroups — C2×#20253209 #27083816

Order 12032.

2142 conjugacy classes of subgroups.

The subgroup lattice has 600 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#13094015C211yesmaximal, characteristic, complemented
2C2×#15689591C211yesmaximal, characteristic
3#20253209C2116yesmaximal, direct product factor
4C2×#15689433C212yesmaximal, complemented
5C2×#15689429C212yesmaximal, complemented
6C2×#15689426C212yesmaximal, complemented
7C2×#15689422C212yesmaximal, complemented
8C2×#15689166C211yesmaximal, characteristic, complemented
9C2×#15689146C211yesmaximal, characteristic, complemented
10C2×#15689125C211yesmaximal, characteristic
11C2×#15689122C211yesmaximal, characteristic
12C2×C47×#428C211yesmaximal, characteristic, complemented
13C22×#10625350C2211yescharacteristic
14C22×#10625329C2211yescharacteristic
15C2×#13094015—21no—
16C2×#13094015—21no—
17C2×#13094015—24no—
18C2×#13094015C2212yes—
19C2×#13094015C2212yes—
20C2×#13094015C2218yescomplemented
21#15689591—24no—
22#15689591C2218yes—
23C2×#13093989C2211yescharacteristic
24C2×#13093989C2211yescharacteristic
25C2×#13093986C2211yescharacteristic
26C2×#13093986C2211yescharacteristic
27C2×(C22⋊D47)⋊C4—21no—
28C2×(C22⋊D47)⋊C4—21no—
29C2×#13093981—21no—
30C22×C4⋊Dic47—21no—
31C22×C4⋊Dic47C2211yescharacteristic
32C22×Dic47⋊C4—21no—
33C22×C4×Dic47C2211yescharacteristic
34C2×C42×D47—21no—
35#15689433C22116yescomplemented
36#15689429C22116yescomplemented
37#15689426C22116yescomplemented
38#15689422C22116yescomplemented
39#15689166C2218yescomplemented
40C2×#13093957C2212yes—
41C2×#13093955C2212yes—
42#15689146C2218yescomplemented
43C2×Dic47⋊C8C2212yes—
44C2×Dic47⋊C8C2212yes—
45#15689125C2218yes—
46#15689122C2218yes—
47C2×#13093949C2212yes—
48C2×#13093947C2212yes—
49C2×#13093946C2212yes—
50C2×#13093944C2212yes—
51C22×M16×C47C2211yescharacteristic
52C2×#117×C47C2211yescharacteristic
53C22×C4⋊C4×C47C2211yescharacteristic
54C47×#428C2218yescomplemented
55C2×#107×C47C2212yes—
56C2×#106×C47C2212yes—
57C2×#10625350—22no—
58C2×#10625350—24no—
59C2×#10625350Q818yes—
60C2×#10625350C2314yes—
61C22×Dic94C2311yescharacteristic
62C22×D188C2311yescharacteristic
63C22×C22⋊D47—21no—
64C23×Dic47—21no—
65C23×Dic47D411yescharacteristic
66C23×Dic47D411yescharacteristic
67C22×C4×D47—21no—
68C22×C4×D47—21no—
69C22×C4×D47—21no—
70C2×#10625329—24no—
71C2×#10625329C2314yes—
72C22×C47⋊C8C2312yes—
73#13094015—24no—
74#13094015—24no—
75#13094015—28no—
76#13094015—28no—
77#13094015C2318yes—
78#13094015C2318yes—
79#13093989—22no—
80#13093989—22no—
81#13093989C2314yes—
82#13093989C2314yes—
83#13093986—22no—
84#13093986—22no—
85#13093986C2314yes—
86#13093986C2314yes—
87(C22⋊D47)⋊C4—28no—
88(C22⋊D47)⋊C4—28no—
89#13093981—28no—
90C2×C4⋊Dic47—21no—
91C2×C4⋊Dic47—22no—
92C2×C4⋊Dic47—22no—
93C2×C4⋊Dic47—22no—
94C2×C4⋊Dic47—28no—
95C2×C4⋊Dic47C2312yes—
96C2×C4⋊Dic47C2314yes—
97C2×Dic47⋊C4—22no—
98C2×Dic47⋊C4—22no—
99C2×Dic47⋊C4—28no—
100C2×Dic47⋊C4—21no—
101C2×Dic47⋊C4—21no—
102C2×(C2×D47)⋊C4—21no—
103C2×(C2×D47)⋊C4—21no—
104C2×C4×Dic47—21no—
105C2×C4×Dic47—21no—
106C2×C4×Dic47D411yescharacteristic
107C2×C4×Dic47D411yescharacteristic
108C2×C4×Dic47D414yes—
109C2×C4×Dic47C2311yescharacteristic
110C2×C4×Dic47C2311yescharacteristic
111C2×C4×Dic47C2314yes—
112C42×D47—28no—
113#13093957C2318yes—
114#13093955C2318yes—
115Dic47⋊C8C2318yes—
116Dic47⋊C8C2318yes—
117#13093949C2318yes—
118#13093947C2318yes—
119#13093946C2318yes—
120#13093944C2318yes—
121C23×C4×C47—21no—
122C23×C4×C47C2311yescharacteristic
123C2×M16×C47—24no—
124C2×M16×C47C2314yes—
125C22×C8×C47C2312yes—
126#117×C47—22no—
127#117×C47C2314yes—
128C2×C4⋊C4×C47—21no—
129C2×C4⋊C4×C47—22no—
130C2×C4⋊C4×C47C2312yes—
131C2×C4⋊C4×C47C2314yes—
132C2×C4⋊C4×C47C2311yescharacteristic
133C2×C4⋊C4×C47C2311yescharacteristic
134C2×C22⋊C4×C47—21no—
135C2×C42×C47—21no—
136#107×C47C2318yes—
137#106×C47C2318yes—
138C24×C47C2×D411yescharacteristic
139C23×D47—21no—
140C22×C4×C47—21no—
141C22×C4×C47—21no—
142C22×C4×C47—24no—
143C22×C4×C47—22no—
144C22×C4×C47—44no—
145C22×C4×C47C2×D411yescharacteristic
146C22×C4×C47C2×D412yes—
147C22×C4×C47C2×Q814yes—
148C22×C4×C47#4114yes—
149C22×C4×C47C2411yescharacteristic
150C22×C4×C47C2412yes—
151C2×C22⋊D47—28no—
152C2×C22⋊D47—24no—
153C22×Dic47—21no—
154C22×Dic47—24no—
155C22×Dic47—22no—
156C22×Dic47—42no—
157C22×Dic47—42no—
158C22×Dic47—44no—
159C22×Dic47C2×D411yescharacteristic
160C22×Dic47C2×D411yescharacteristic
161C22×Dic47C2×D411yescharacteristic
162C22×Dic47C2×D411yescharacteristic
163C22×Dic47C2×D414yes—
164C22×Dic47C2×D414yes—
165#10625350—216no—
166#10625350—28no—
167#10625350C2×Q8116yes—
168C2×D188—21no—
169C2×D188—22no—
170C2×D188C2×Q814yes—
171C2×D188C2412yes—
172C2×C4×D47—28no—
173C2×C4×D47—24no—
174C2×C4×D47—24no—
175C2×C4×D47—24no—
176C2×C4×D47—44no—
177C2×C4×D47—44no—
178C2×Dic94—21no—
179C2×Dic94—22no—
180C2×Dic94C2×Q814yes—
181C2×Dic94C2412yes—
182M16×C47—28no—
183C2×C8×C47—24no—
184C2×C8×C47C2414yes—
185C4⋊C4×C47—24no—
186C4⋊C4×C47—22no—
187C4⋊C4×C47—21no—
188C4⋊C4×C47—21no—
189C4⋊C4×C47C2412yes—
190C4⋊C4×C47C2414yes—
191C4⋊C4×C47C2412yes—
192C22⋊C4×C47—24no—
193C42×C47—24no—
194(C2×D47)⋊C4—24no—
195(C2×D47)⋊C4—24no—
196C4⋊Dic47—24no—
197C4⋊Dic47—22no—
198C4⋊Dic47—28no—
199C4⋊Dic47—28no—
200C4⋊Dic47C2414yes—
201Dic47⋊C4—28no—
202Dic47⋊C4—28no—
203Dic47⋊C4—24no—
204Dic47⋊C4—24no—
205C4×Dic47—24no—
206C4×Dic47—24no—
207C4×Dic47C2×D414yes—
208C4×Dic47C2×D414yes—
209C4×Dic47C2×D412yes—
210C4×Dic47C2×D412yes—
211C4×Dic47C2412yes—
212C4×Dic47C2412yes—
213#10625329—28no—
214C2×C47⋊C8—24no—
215C2×C47⋊C8C2414yes—
216C23×C47—22no—
217C23×C47—22no—
218C23×C47C22⋊Q814yes—
219C23×C47C22×D412yes—
220C23×C47C22×D411yescharacteristic
221C22×D47—24no—
222C22×D47—22no—
223C22×D47—42no—
224C22×D47—42no—
225C2×C4×C47—22no—
226C2×C4×C47—22no—
227C2×C4×C47—24no—
228C2×C4×C47—48no—
229C2×C4×C47—42no—
230C2×C4×C47—42no—
231C2×C4×C47—44no—
232C2×C4×C47C22⋊Q814yes—
233C2×C4×C47C22⋊Q818yes—
234C2×C4×C47C22×D412yes—
235C2×C4×C47C22×D411yescharacteristic
236C2×C4×C47C22×Q812yes—
237C2×C4×C47C22×Q814yes—
238C2×C4×C47C2×#4112yes—
239C2×C4×C47C2×#4114yes—
240C2×C4×C47C2511yescharacteristic
241C22⋊D47—216no—
242C2×Dic47—24no—
243C2×Dic47—22no—
244C2×Dic47—22no—
245C2×Dic47—22no—
246C2×Dic47—22no—
247C2×Dic47—22no—
248C2×Dic47—24no—
249C2×Dic47—44no—
250C2×Dic47—44no—
251C2×Dic47—48no—
252C2×Dic47—44no—
253C2×Dic47C22×D412yes—
254C2×Dic47C22×D412yes—
255C2×Dic47C22×D412yes—
256C2×Dic47C22×D412yes—
257D188—24no—
258D188—22no—
259D188C22×Q814yes—
260C4×D47—216no—
261C4×D47—48no—
262C4×D47—48no—
263Dic94—24no—
264Dic94—22no—
265Dic94C22×Q814yes—
266C8×C47—24no—
267C47⋊C8—24no—
268C2×#3080—471nomaximal, Sylow
269C22×C47—28no—
270C22×C47—22no—
271C22×C47—22no—
272C22×C47C2×C22⋊Q814yes—
273C22×C47C2×C22⋊Q812yes—
274C22×C47#57512yes—
275C22×C47#57512yes—
276C22×C47C23×D411yescharacteristic
277C2×D47—24no—
278C2×D47—48no—
279C2×D47—42no—
280C2×D47—42no—
281C4×C47—42no—
282C4×C47—42no—
283C4×C47—44no—
284C4×C47C2×C22⋊Q812yes—
285C4×C47C2×C22⋊Q814yes—
286C4×C47C23×Q811yescharacteristic
287C4×C47C22×#4111yescharacteristic
288Dic47—24no—
289Dic47—22no—
290Dic47—22no—
291Dic47—22no—
292Dic47—22no—
293Dic47—44no—
294C22×#125—471no—
295C2×C4×#41—471no—
296#3080—4716no—
297C2×#479—472no—
298C2×#479—472no—
299C2×#478—472no—
300C2×#478—472no—
301C2×#428—471no—
302C2×#428D4711yescharacteristic, complemented
303C2×#423—471no—
304C2×#417—471no—
305C2×#417—471no—
306C2×C47—24no—
307C2×C47#308014yescomplemented
308C2×C47C22×C22⋊Q811yescharacteristic
309C2×C47C2×#57511yescharacteristic
310C2×C47C2×#57511yescharacteristic
311D47—44no—
312C22×#41—471no—
313C22×M16—471no—
314C22×M16C2×D4711yescharacteristic
315C2×#125—472no—
316C2×#125—472no—
317C2×#125—478no—
318C2×#125—941no—
319C2×#125—944no—
320C2×#125—941no—
321C4×#41—478no—
322C4×#41—944no—
323C2×C4×Q8—471no—
324C2×C4×Q8—471no—
325C2×C4×D4—471no—
326C2×C4×D4—471no—
327C2×C4×D4—941no—
328C2×C4×D4—941no—
329C2×#117—941no—
330C2×#117C2×D4711yescharacteristic
331C22×C4⋊C4—471no—
332C22×C4⋊C4—941no—
333C22×C4⋊C4—941no—
334C22×C4⋊C4C2×D4711yescharacteristic
335C22×C42—471no—
336C22×C42—941no—
337#479—4716no—
338#479—4716no—
339#478—4716no—
340#478—4716no—
341#428—478no—
342#428C2×D4718yescomplemented
343C2×#107—472no—
344C2×#107C2×D4712yes—
345C2×#106—472no—
346C2×#106C2×D4712yes—
347#423—478no—
348C2×C4⋊C8—472no—
349C2×C4⋊C8—472no—
350#417—478no—
351#417—478no—
352C2×#103—472no—
353C2×#103—472no—
354C2×#102—472no—
355C2×#102—472no—
356C47C2×#308011yescharacteristic, complemented, Sylow
357C2×#41—478no—
358C2×#41—474no—
359C2×#41—944no—
360C2×#41—942no—
361C22×Q8—471no—
362C22×D4—471no—
363C22×D4—941no—
364C23×C4—21no—
365C23×C4—471no—
366C23×C4—471no—
367C23×C4—941no—
368C23×C4—941no—
369C23×C4—941no—
370C23×C4—941no—
371C23×C4C22×D4711yescharacteristic
372C2×M16—24no—
373C2×M16—474no—
374C2×M16—944no—
375C2×M16C22×D4714yes—
376C22×C8—472no—
377C22×C8C22×D4712yes—
378#125—478no—
379#125—478no—
380#125—948no—
381#125—948no—
382#125—944no—
383#125—944no—
384C4×Q8—474no—
385C4×Q8—474no—
386C4×Q8—942no—
387C4×Q8—942no—
388C4×D4—474no—
389C4×D4—474no—
390C4×D4—942no—
391C4×D4—942no—
392C4×D4—948no—
393C4×D4—948no—
394#117—22no—
395#117—948no—
396#117C22×D4714yes—
397C2×C4⋊C4—22no—
398C2×C4⋊C4—21no—
399C2×C4⋊C4—472no—
400C2×C4⋊C4—474no—
401C2×C4⋊C4—948no—
402C2×C4⋊C4—942no—
403C2×C4⋊C4—942no—
404C2×C4⋊C4—948no—
405C2×C4⋊C4—942no—
406C2×C4⋊C4—942no—
407C2×C4⋊C4—941no—
408C2×C4⋊C4—941no—
409C2×C4⋊C4—942no—
410C2×C4⋊C4—941no—
411C2×C4⋊C4C22×D4712yes—
412C2×C4⋊C4C22×D4714yes—
413C2×C4⋊C4C22×D4711yescharacteristic
414C2×C4⋊C4C22×D4711yescharacteristic
415C2×C22⋊C4—21no—
416C2×C22⋊C4—941no—
417C2×C22⋊C4—941no—
418C2×C42—21no—
419C2×C42—471no—
420C2×C42—474no—
421C2×C42—471no—
422C2×C42—471no—
423C2×C42—471no—
424C2×C42—474no—
425C2×C42—948no—
426C2×C42—941no—
427C2×C42—941no—
428#107—478no—
429#107C22×D4718yes—
430#106—478no—
431#106C22×D4718yes—
432C4⋊C8—478no—
433C4⋊C8—478no—
434#103—478no—
435#103—478no—
436#102—478no—
437#102—478no—
438C24—941no—
439C24D4×D4711yescharacteristic
440#41—4716no—
441#41—9416no—
442#41—948no—
443C2×Q8—474no—
444C2×Q8—472no—
445C2×Q8—941no—
446C2×Q8—942no—
447C2×D4—474no—
448C2×D4—472no—
449C2×D4—948no—
450C2×D4—944no—
451C2×D4—941no—
452C2×D4—942no—
453C22×C4—24no—
454C22×C4—21no—
455C22×C4—21no—
456C22×C4—22no—
457C22×C4—44no—
458C22×C4—471no—
459C22×C4—471no—
460C22×C4—471no—
461C22×C4—471no—
462C22×C4—474no—
463C22×C4—474no—
464C22×C4—944no—
465C22×C4—944no—
466C22×C4—941no—
467C22×C4—948no—
468C22×C4—944no—
469C22×C4—944no—
470C22×C4—942no—
471C22×C4—1884no—
472C22×C4—1884no—
473C22×C4—1882no—
474C22×C4—1882no—
475C22×C4—1884no—
476C22×C4#1062535014yes—
477C22×C4D4×D4712yes—
478C22×C4D4×D4711yescharacteristic
479C22×C4Q8×D4714yes—
480C22×C4C23×D4712yes—
481C22×C4C23×D4711yescharacteristic
482M16—28no—
483M16—948no—
484C2×C8—24no—
485C2×C8—474no—
486C2×C8—944no—
487C2×C8C23×D4714yes—
488C4⋊C4—24no—
489C4⋊C4—22no—
490C4⋊C4—21no—
491C4⋊C4—21no—
492C4⋊C4—474no—
493C4⋊C4—948no—
494C4⋊C4—948no—
495C4⋊C4—944no—
496C4⋊C4—944no—
497C4⋊C4—948no—
498C4⋊C4—948no—
499C4⋊C4—944no—
500C4⋊C4—942no—
501C4⋊C4C23×D4714yes—
502C4⋊C4C23×D4712yes—
503C4⋊C4C23×D4712yes—
504C22⋊C4—24no—
505C22⋊C4—944no—
506C22⋊C4—944no—
507C42—24no—
508C42—474no—
509C42—474no—
510C42—472no—
511C42—472no—
512C42—472no—
513C42—472no—
514C42—944no—
515C42—944no—
516C23—22no—
517C23—22no—
518C23—944no—
519C23—942no—
520C23—1882no—
521C23—1882no—
522C23#1309400214yes—
523C23C2×D4×D4712yes—
524C23C2×D4×D4711yescharacteristic
525Q8—474no—
526Q8—944no—
527Q8—942no—
528D4—474no—
529D4—9416no—
530D4—944no—
531D4—942no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—24no—
535C2×C4—48no—
536C2×C4—44no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—472no—
540C2×C4—472no—
541C2×C4—472no—
542C2×C4—472no—
543C2×C4—944no—
544C2×C4—942no—
545C2×C4—942no—
546C2×C4—942no—
547C2×C4—942no—
548C2×C4—942no—
549C2×C4—9416no—
550C2×C4—944no—
551C2×C4—1888no—
552C2×C4—1888no—
553C2×C4—1884no—
554C2×C4—1884no—
555C2×C4—1888no—
556C2×C4—1884no—
557C2×C4#1309400214yes—
558C2×C4#1309400218yes—
559C2×C4C2×D4×D4712yes—
560C2×C4C2×D4×D4711yescharacteristic
561C2×C4C2×Q8×D4712yes—
562C2×C4C2×Q8×D4714yes—
563C2×C4C2×#1062535012yes—
564C2×C4C2×#1062535014yes—
565C2×C4C24×D4711yescharacteristic
566C8—24no—
567C8—944no—
568C22—28no—
569C22—22no—
570C22—22no—
571C22—944no—
572C22—1888no—
573C22—1882no—
574C22—1882no—
575C22C2×#1309400214yes—
576C22C2×#1309400212yes—
577C22#1568991912yes—
578C22#1568992212yes—
579C22C22×D4×D4711yescharacteristic
580C4—44no—
581C4—42no—
582C4—42no—
583C4—942no—
584C4—942no—
585C4—942no—
586C4—942no—
587C4—944no—
588C4—1884no—
589C4C2×#1309400214yes—
590C4C2×#1309400212yes—
591C4C22×Q8×D4711yescharacteristic
592C4C22×#1062535011yescharacteristic
593C2—24no—
594C2—1884no—
595C2#2025320914yesdirect product factor
596C2C22×#1309400211yescharacteristic
597C2C2×#1568991911yescharacteristic
598C2C2×#1568992211yescharacteristic
subgroupG/H
centerC23C2×D4×D47
commutator, cosocleC2×C4×C47C25
FrattiniC2×C4C24×D47
FittingC2×C47×#428C2
socleC23×C47C22×D4
prime radicalC22×#10625350C22
hypercenterC2×#428D47
hypocenterC47C2×#3080
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#428

Complements

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

NH ≅ G/Ntype
1. C22×#13094015C2semidirect
3. #20253209C2direct (N × H)
4. C2×#15689433C2semidirect
5. C2×#15689429C2semidirect
6. C2×#15689426C2semidirect
7. C2×#15689422C2semidirect
8. C2×#15689166C2semidirect
9. C2×#15689146C2semidirect
12. C2×C47×#428C2semidirect
20. C2×#13094015C22semidirect
35. #15689433C22semidirect
36. #15689429C22semidirect
37. #15689426C22semidirect
38. #15689422C22semidirect
39. #15689166C22semidirect
42. #15689146C22semidirect
54. C47×#428C22semidirect
302. C2×#428D47semidirect
307. C2×C47#3080semidirect
342. #428C2×D47semidirect
356. C47C2×#3080semidirect
595. C2#20253209direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#130940152
C2×#156895912
#202532092
C2×#156894332
C2×#156894292
C2×#156894262
C2×#156894222
C2×#156891662
C2×#156891462
C2×#156891252
C2×#156891222
C2×C47×#4282
C2×#308047