Finite Group Databasebeta

← C2×#20252206 #27080351

Subgroups — C2×#20252206 #27080351

Order 12032.

2810 conjugacy classes of subgroups.

The subgroup lattice has 636 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#13094057C211yesmaximal, characteristic
2C2×#15689591C211yesmaximal, characteristic
3C22×#13093989C211yesmaximal, characteristic, complemented
4#20252206C2116yesmaximal, direct product factor
5C2×#15689229C212yesmaximal, complemented
6C2×#15689222C212yesmaximal, complemented
7C2×#15689217C212yesmaximal, complemented
8C2×#15689210C212yesmaximal, complemented
9C2×#15689122C211yesmaximal, characteristic
10C2×#15689131C211yesmaximal, characteristic, complemented
11C2×M16×Dic47C211yesmaximal, characteristic, complemented
12C2×C47×#428C211yesmaximal, characteristic, complemented
13C22×#10625350C2211yescharacteristic
14C23×Dic94C2211yescharacteristic
15C2×#13094057C4116yescomplemented
16C2×#13094057C2218yes—
17C22×Dic188C2212yes—
18C22×#10625325C2212yes—
19C22×#10625329C2211yescharacteristic
20#15689591—24no—
21#15689591C2218yes—
22C2×#13093989—21no—
23C2×#13093989—21no—
24C2×#13093989—24no—
25C2×#13093989C2212yes—
26C2×#13093989C2211yescharacteristic
27C2×#13093989C2212yes—
28C2×#13093989C2211yescharacteristic
29C2×#13093989C2218yescomplemented
30C2×#13093986C2211yescharacteristic
31C2×#13093986C2211yescharacteristic
32C2×(C22⋊D47)⋊C4—21no—
33C2×(C22⋊D47)⋊C4—21no—
34C2×#13093981—21no—
35C22×Dic47⋊C4—21no—
36C22×C4×Dic47—21no—
37C22×C4×Dic47C2211yescharacteristic
38C2×C42×D47—21no—
39#15689229C22116yescomplemented
40#15689222C22116yescomplemented
41#15689217C22116yescomplemented
42#15689210C22116yescomplemented
43#15689131C2218yescomplemented
44#15689122C2218yes—
45C2×#13093947C2212yes—
46C2×#13093947C2212yes—
47C2×#13093947C2212yes—
48C2×#13093944C2212yes—
49M16×Dic47C2218yescomplemented
50C2×#13093934C2212yes—
51C2×C8×Dic47C2212yes—
52C22×M16×C47C2211yescharacteristic
53C2×#117×C47C2211yescharacteristic
54C22×C4⋊C4×C47C2211yescharacteristic
55C47×#428C2218yescomplemented
56C2×#107×C47C2212yes—
57C2×#106×C47C2212yes—
58C2×#10625350—22no—
59C2×#10625350—24no—
60C2×#10625350C2×C418yes—
61C2×#10625350C2314yes—
62C22×Dic94—21no—
63C22×Dic94—22no—
64C22×Dic94—24no—
65C22×Dic94C2×C418yescomplemented
66C22×Dic94C2312yes—
67C22×Dic94C2311yescharacteristic
68C22×Dic94C2314yes—
69C22×D188C2311yescharacteristic
70C22×C22⋊D47—21no—
71C23×Dic47—21no—
72C23×Dic47D411yescharacteristic
73C23×Dic47D411yescharacteristic
74C22×C4×D47—21no—
75C22×C4×D47—21no—
76C22×C4×D47—21no—
77#13094057C2×C4164yescomplemented
78C2×Dic188C2×C4116yescomplemented
79C2×Dic188C2318yes—
80C2×#10625325C2×C4116yescomplemented
81C2×#10625325C2318yes—
82C2×#10625329—24no—
83C2×#10625329C2314yes—
84C22×C47⋊C8C2312yes—
85#13093989—24no—
86#13093989—24no—
87#13093989—28no—
88#13093989—28no—
89#13093989—22no—
90#13093989—22no—
91#13093989C2314yes—
92#13093989C2318yes—
93#13093989C2318yes—
94#13093989C2314yes—
95#13093986—22no—
96#13093986—22no—
97#13093986C2314yes—
98#13093986C2314yes—
99(C22⋊D47)⋊C4—28no—
100(C22⋊D47)⋊C4—28no—
101#13093981—28no—
102C2×Dic47⋊C4—22no—
103C2×Dic47⋊C4—22no—
104C2×Dic47⋊C4—28no—
105C2×Dic47⋊C4—21no—
106C2×Dic47⋊C4—21no—
107C2×(C2×D47)⋊C4—21no—
108C2×(C2×D47)⋊C4—21no—
109C2×C4×Dic47—21no—
110C2×C4×Dic47—21no—
111C2×C4×Dic47—22no—
112C2×C4×Dic47—22no—
113C2×C4×Dic47—28no—
114C2×C4×Dic47D411yescharacteristic
115C2×C4×Dic47D411yescharacteristic
116C2×C4×Dic47D414yes—
117C2×C4×Dic47C2311yescharacteristic
118C2×C4×Dic47C2311yescharacteristic
119C2×C4×Dic47C2314yes—
120C42×D47—28no—
121#13093947C2318yes—
122#13093947C2318yes—
123#13093947C2318yes—
124#13093944C2318yes—
125#13093934C2318yes—
126C8×Dic47C2318yes—
127C23×C4×C47—21no—
128C23×C4×C47C2311yescharacteristic
129C2×M16×C47C2×C418yescomplemented
130C2×M16×C47C2314yes—
131C22×C8×C47C2312yes—
132#117×C47—22no—
133#117×C47C2314yes—
134C2×C4⋊C4×C47—21no—
135C2×C4⋊C4×C47—22no—
136C2×C4⋊C4×C47C2312yes—
137C2×C4⋊C4×C47C2314yes—
138C2×C4⋊C4×C47C2311yescharacteristic
139C2×C4⋊C4×C47C2311yescharacteristic
140C2×C22⋊C4×C47—21no—
141C2×C42×C47—21no—
142#107×C47C2318yes—
143#106×C47C2318yes—
144C24×C47C2×D411yescharacteristic
145C23×D47—21no—
146C22×C4×C47—21no—
147C22×C4×C47—21no—
148C22×C4×C47—24no—
149C22×C4×C47—22no—
150C22×C4×C47—44no—
151C22×C4×C47C22×C414yes—
152C22×C4×C47C2×D411yescharacteristic
153C22×C4×C47C2×D412yes—
154C22×C4×C47#4114yes—
155C22×C4×C47C2411yescharacteristic
156C22×C4×C47C2412yes—
157C2×C22⋊D47—28no—
158C2×C22⋊D47—24no—
159C22×Dic47—28no—
160C22×Dic47—21no—
161C22×Dic47—22no—
162C22×Dic47—24no—
163C22×Dic47—24no—
164C22×Dic47—42no—
165C22×Dic47C2×D411yescharacteristic
166C22×Dic47C2×D414yes—
167C22×Dic47C2×D411yescharacteristic
168C22×Dic47C2×D411yescharacteristic
169C22×Dic47C2×D414yes—
170C22×Dic47C2×D411yescharacteristic
171#10625350—216no—
172#10625350—28no—
173#10625350C22×C4116yes—
174C2×D188—21no—
175C2×D188—22no—
176C2×D188C22×C414yes—
177C2×D188C2412yes—
178C2×C4×D47—28no—
179C2×C4×D47—24no—
180C2×C4×D47—24no—
181C2×C4×D47—24no—
182C2×C4×D47—44no—
183C2×C4×D47—44no—
184C2×Dic94—216no—
185C2×Dic94—28no—
186C2×Dic94—22no—
187C2×Dic94—24no—
188C2×Dic94—21no—
189C2×Dic94—24no—
190C2×Dic94—28no—
191C2×Dic94—22no—
192C2×Dic94C22×C418yes—
193C2×Dic94C22×C414yes—
194C2×Dic94C22×C4116yescomplemented
195C2×Dic94C2414yes—
196C2×Dic94C2412yes—
197M16×C47C22×C4116yescomplemented
198C2×C8×C47C22×C418yes—
199C2×C8×C47C2414yes—
200C4⋊C4×C47—24no—
201C4⋊C4×C47—22no—
202C4⋊C4×C47—21no—
203C4⋊C4×C47—21no—
204C4⋊C4×C47C2412yes—
205C4⋊C4×C47C2414yes—
206C4⋊C4×C47C2412yes—
207C22⋊C4×C47—24no—
208C42×C47—24no—
209(C2×D47)⋊C4—24no—
210(C2×D47)⋊C4—24no—
211Dic47⋊C4—28no—
212Dic47⋊C4—28no—
213Dic47⋊C4—24no—
214Dic47⋊C4—24no—
215C4×Dic47—24no—
216C4×Dic47—24no—
217C4×Dic47—28no—
218C4×Dic47—28no—
219C4×Dic47C2×D414yes—
220C4×Dic47C2×D414yes—
221C4×Dic47C2×D412yes—
222C4×Dic47C2×D412yes—
223C4×Dic47C2412yes—
224C4×Dic47C2412yes—
225#10625329—28no—
226C2×C47⋊C8—24no—
227C2×C47⋊C8C2414yes—
228Dic188C22×C4132yescomplemented
229#10625325C22×C4132yescomplemented
230C23×C47—22no—
231C23×C47—22no—
232C23×C47C4×D414yes—
233C23×C47C22×D412yes—
234C23×C47C22×D411yescharacteristic
235C22×D47—24no—
236C22×D47—22no—
237C22×D47—42no—
238C22×D47—42no—
239C2×C4×C47—24no—
240C2×C4×C47—22no—
241C2×C4×C47—22no—
242C2×C4×C47—48no—
243C2×C4×C47—42no—
244C2×C4×C47—42no—
245C2×C4×C47—44no—
246C2×C4×C47C4×D414yes—
247C2×C4×C47C4×D418yes—
248C2×C4×C47C23×C412yes—
249C2×C4×C47C23×C414yes—
250C2×C4×C47C22×D412yes—
251C2×C4×C47C22×D411yescharacteristic
252C2×C4×C47C2×#4112yes—
253C2×C4×C47C2×#4114yes—
254C2×C4×C47C2511yescharacteristic
255C22⋊D47—216no—
256C2×Dic47—28no—
257C2×Dic47—216no—
258C2×Dic47—24no—
259C2×Dic47—22no—
260C2×Dic47—22no—
261C2×Dic47—22no—
262C2×Dic47—24no—
263C2×Dic47—22no—
264C2×Dic47—28no—
265C2×Dic47—44no—
266C2×Dic47#13714yescomplemented
267C2×Dic47C22×D412yes—
268C2×Dic47C22×D412yes—
269C2×Dic47C22×D412yes—
270C2×Dic47C22×D412yes—
271D188—24no—
272D188—22no—
273D188C23×C414yes—
274C4×D47—216no—
275C4×D47—48no—
276C4×D47—48no—
277Dic94—28no—
278Dic94—216no—
279Dic94—24no—
280Dic94—24no—
281Dic94—22no—
282Dic94C23×C418yes—
283Dic94C23×C414yes—
284C8×C47C23×C418yes—
285C47⋊C8—24no—
286C2×C4×#137—471nomaximal, Sylow
287C22×C47—28no—
288C22×C47—22no—
289C22×C47—22no—
290C22×C47C2×C4×D414yes—
291C22×C47C2×C4×D412yes—
292C22×C47C2×#13712yes—
293C22×C47#57512yes—
294C22×C47C23×D411yescharacteristic
295C2×D47—24no—
296C2×D47—48no—
297C2×D47—42no—
298C2×D47—42no—
299C4×C47—42no—
300C4×C47—42no—
301C4×C47—44no—
302C4×C47C2×C4×D412yes—
303C4×C47C2×C4×D414yes—
304C4×C47C24×C411yescharacteristic
305C4×C47C22×#4111yescharacteristic
306Dic47—28no—
307Dic47—24no—
308Dic47—22no—
309Dic47—22no—
310Dic47—22no—
311Dic47C2×#13714yescomplemented
312C22×#137—471no—
313C2×C4×#41—471no—
314C22×C4×Q8—471no—
315C4×#137—4716no—
316C2×#441—472no—
317C2×#440—472no—
318C2×C4×Q16—472no—
319C2×C4×SD16—472no—
320C2×#428D4711yescharacteristic, complemented
321C2×#417—471no—
322C2×#419—471no—
323C2×C4×M16—471no—
324C2×C47—24no—
325C2×C47C4×#13714yescomplemented
326C2×C47C22×C4×D411yescharacteristic
327C2×C47C22×#13711yescharacteristic
328C2×C47C2×#57511yescharacteristic
329D47—44no—
330C22×#41—471no—
331C23×Q8—471no—
332C2×#137—4716no—
333C2×#137—478no—
334C22×Q16—472no—
335C22×SD16—472no—
336C22×M16—471no—
337C22×M16C2×D4711yescharacteristic
338C4×#41—478no—
339C4×#41—944no—
340C2×C4×Q8—472no—
341C2×C4×Q8—472no—
342C2×C4×Q8—478no—
343C2×C4×Q8—471no—
344C2×C4×Q8—471no—
345C2×C4×Q8—941no—
346C2×C4×Q8—944no—
347C2×C4×Q8—941no—
348C2×C4×D4—471no—
349C2×C4×D4—471no—
350C2×C4×D4—941no—
351C2×C4×D4—941no—
352C2×#117—941no—
353C2×#117C2×D4711yescharacteristic
354C22×C4⋊C4—941no—
355C22×C4⋊C4C2×D4711yescharacteristic
356C22×C42—471no—
357C22×C42—941no—
358C22×C42—941no—
359#441—4716no—
360#440—4716no—
361C4×Q16—4716no—
362C4×SD16—4716no—
363#428C2×D4718yescomplemented
364C2×#107C2×D4712yes—
365C2×#106C2×D4712yes—
366#419—478no—
367#417—478no—
368C2×#103—472no—
369C2×#103—472no—
370C2×#103—472no—
371C2×#102—472no—
372C4×M16—478no—
373C2×#97—472no—
374C2×C4×C8—472no—
375C47C2×C4×#13711yescharacteristic, complemented, Sylow
376C2×#41—478no—
377C2×#41—474no—
378C2×#41—942no—
379C2×#41—944no—
380C22×Q8—478no—
381C22×Q8—472no—
382C22×Q8—471no—
383C22×Q8—474no—
384C22×Q8—942no—
385C22×Q8—941no—
386C22×Q8—944no—
387C22×D4—471no—
388C22×D4—941no—
389C23×C4—21no—
390C23×C4—471no—
391C23×C4—471no—
392C23×C4—941no—
393C23×C4—941no—
394C23×C4—941no—
395C23×C4—941no—
396C23×C4C22×D4711yescharacteristic
397#137—4764no—
398C2×Q16—4716no—
399C2×Q16—478no—
400C2×SD16—4716no—
401C2×SD16—478no—
402C2×M16—474no—
403C2×M16—944no—
404C2×M16C4×D4718yescomplemented
405C2×M16C22×D4714yes—
406C22×C8—472no—
407C22×C8C22×D4712yes—
408C4×Q8—478no—
409C4×Q8—478no—
410C4×Q8—474no—
411C4×Q8—474no—
412C4×Q8—948no—
413C4×Q8—948no—
414C4×Q8—944no—
415C4×Q8—944no—
416C4×Q8—942no—
417C4×Q8—942no—
418C4×D4—474no—
419C4×D4—474no—
420C4×D4—942no—
421C4×D4—942no—
422C4×D4—948no—
423C4×D4—948no—
424#117—22no—
425#117—948no—
426#117C22×D4714yes—
427C2×C4⋊C4—22no—
428C2×C4⋊C4—21no—
429C2×C4⋊C4—948no—
430C2×C4⋊C4—942no—
431C2×C4⋊C4—942no—
432C2×C4⋊C4—941no—
433C2×C4⋊C4—941no—
434C2×C4⋊C4C22×D4712yes—
435C2×C4⋊C4C22×D4714yes—
436C2×C4⋊C4C22×D4711yescharacteristic
437C2×C4⋊C4C22×D4711yescharacteristic
438C2×C22⋊C4—21no—
439C2×C22⋊C4—941no—
440C2×C22⋊C4—941no—
441C2×C42—21no—
442C2×C42—471no—
443C2×C42—474no—
444C2×C42—471no—
445C2×C42—471no—
446C2×C42—471no—
447C2×C42—474no—
448C2×C42—948no—
449C2×C42—941no—
450C2×C42—941no—
451C2×C42—948no—
452C2×C42—942no—
453C2×C42—942no—
454#107C22×D4718yes—
455#106C22×D4718yes—
456#103—478no—
457#103—478no—
458#103—478no—
459#102—478no—
460#97—478no—
461C4×C8—478no—
462C24—941no—
463C24D4×D4711yescharacteristic
464#41—4716no—
465#41—9416no—
466#41—948no—
467C2×Q8—478no—
468C2×Q8—474no—
469C2×Q8—4716no—
470C2×Q8—474no—
471C2×Q8—472no—
472C2×Q8—948no—
473C2×Q8—9416no—
474C2×Q8—944no—
475C2×Q8—942no—
476C2×Q8—941no—
477C2×Q8—944no—
478C2×Q8—948no—
479C2×Q8—942no—
480C2×D4—474no—
481C2×D4—472no—
482C2×D4—948no—
483C2×D4—944no—
484C2×D4—941no—
485C2×D4—942no—
486C22×C4—24no—
487C22×C4—21no—
488C22×C4—21no—
489C22×C4—22no—
490C22×C4—44no—
491C22×C4—471no—
492C22×C4—474no—
493C22×C4—471no—
494C22×C4—471no—
495C22×C4—471no—
496C22×C4—474no—
497C22×C4—944no—
498C22×C4—944no—
499C22×C4—944no—
500C22×C4—948no—
501C22×C4—944no—
502C22×C4—941no—
503C22×C4—942no—
504C22×C4—948no—
505C22×C4—944no—
506C22×C4—1884no—
507C22×C4—1884no—
508C22×C4—1882no—
509C22×C4C2×C4×D4714yes—
510C22×C4D4×D4712yes—
511C22×C4D4×D4711yescharacteristic
512C22×C4#1062535414yes—
513C22×C4C23×D4712yes—
514C22×C4C23×D4711yescharacteristic
515Q16—4732no—
516SD16—4732no—
517M16—948no—
518M16C2×C4×D47116yescomplemented
519C2×C8—474no—
520C2×C8—944no—
521C2×C8C2×C4×D4718yes—
522C2×C8C23×D4714yes—
523C4⋊C4—24no—
524C4⋊C4—22no—
525C4⋊C4—21no—
526C4⋊C4—21no—
527C4⋊C4—948no—
528C4⋊C4—948no—
529C4⋊C4—944no—
530C4⋊C4—944no—
531C4⋊C4C23×D4714yes—
532C4⋊C4C23×D4712yes—
533C4⋊C4C23×D4712yes—
534C22⋊C4—24no—
535C22⋊C4—944no—
536C22⋊C4—944no—
537C42—24no—
538C42—474no—
539C42—474no—
540C42—472no—
541C42—472no—
542C42—472no—
543C42—472no—
544C42—944no—
545C42—944no—
546C42—948no—
547C42—948no—
548C23—22no—
549C23—22no—
550C23—944no—
551C23—942no—
552C23—1882no—
553C23—1882no—
554C23#1309398614yes—
555C23C2×D4×D4712yes—
556C23C2×D4×D4711yescharacteristic
557Q8—478no—
558Q8—474no—
559Q8—9416no—
560Q8—948no—
561Q8—944no—
562Q8—944no—
563Q8—942no—
564D4—474no—
565D4—9416no—
566D4—944no—
567D4—942no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—24no—
571C2×C4—48no—
572C2×C4—42no—
573C2×C4—44no—
574C2×C4—42no—
575C2×C4—474no—
576C2×C4—472no—
577C2×C4—472no—
578C2×C4—472no—
579C2×C4—472no—
580C2×C4—948no—
581C2×C4—944no—
582C2×C4—9416no—
583C2×C4—948no—
584C2×C4—942no—
585C2×C4—942no—
586C2×C4—942no—
587C2×C4—942no—
588C2×C4—944no—
589C2×C4—9416no—
590C2×C4—1888no—
591C2×C4—1888no—
592C2×C4—1884no—
593C2×C4#1309398614yes—
594C2×C4#1309398618yes—
595C2×C4C22×C4×D4712yes—
596C2×C4C22×C4×D4714yes—
597C2×C4C2×D4×D4712yes—
598C2×C4C2×D4×D4711yescharacteristic
599C2×C4C2×#1062535412yes—
600C2×C4C2×#1062535414yes—
601C2×C4C24×D4711yescharacteristic
602C8—944no—
603C8C22×C4×D4718yes—
604C22—28no—
605C22—22no—
606C22—22no—
607C22—944no—
608C22—1888no—
609C22—1882no—
610C22—1882no—
611C22C2×#1309398614yes—
612C22C2×#1309398612yes—
613C22#137×D4712yes—
614C22#1568991912yes—
615C22C22×D4×D4711yescharacteristic
616C4—42no—
617C4—44no—
618C4—42no—
619C4—474no—
620C4—944no—
621C4—948no—
622C4—942no—
623C4—942no—
624C4—942no—
625C4C2×#1309398614yes—
626C4C2×#1309398612yes—
627C4C23×C4×D4711yescharacteristic
628C4C22×#1062535411yescharacteristic
629C2—24no—
630C2—1884no—
631C2#2025220614yesdirect product factor
632C2C22×#1309398611yescharacteristic
633C2C2×#137×D4711yescharacteristic
634C2C2×#1568991911yescharacteristic
subgroupG/H
centerC23C2×D4×D47
commutatorC4×C47C24×C4
FrattiniC2×C4C24×D47
FittingC2×C47×#428C2
socleC23×C47C22×D4
prime radicalC22×#10625350C22
cosocleC2×C4×C47C25
hypercenterC2×#428D47
hypocenterC47C2×C4×#137
LCSG ▷ 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
3. C22×#13093989C2semidirect
4. #20252206C2direct (N × H)
5. C2×#15689229C2semidirect
6. C2×#15689222C2semidirect
7. C2×#15689217C2semidirect
8. C2×#15689210C2semidirect
10. C2×#15689131C2semidirect
11. C2×M16×Dic47C2semidirect
12. C2×C47×#428C2semidirect
15. C2×#13094057C4semidirect
29. C2×#13093989C22semidirect
39. #15689229C22semidirect
40. #15689222C22semidirect
41. #15689217C22semidirect
42. #15689210C22semidirect
43. #15689131C22semidirect
49. M16×Dic47C22semidirect
55. C47×#428C22semidirect
65. C22×Dic94C2×C4semidirect
77. #13094057C2×C4semidirect
78. C2×Dic188C2×C4semidirect
80. C2×#10625325C2×C4semidirect
129. C2×M16×C47C2×C4semidirect
194. C2×Dic94C22×C4semidirect
197. M16×C47C22×C4semidirect
228. Dic188C22×C4semidirect
229. #10625325C22×C4semidirect
266. C2×Dic47#137semidirect
311. Dic47C2×#137semidirect
320. C2×#428D47semidirect
325. C2×C47C4×#137semidirect
363. #428C2×D47semidirect
375. C47C2×C4×#137semidirect
404. C2×M16C4×D47semidirect
518. M16C2×C4×D47semidirect
631. C2#20252206direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#130940572
C2×#156895912
C22×#130939892
#202522062
C2×#156892292
C2×#156892222
C2×#156892172
C2×#156892102
C2×#156891222
C2×#156891312
C2×M16×Dic472
C2×C47×#4282
C2×C4×#13747