Finite Group Databasebeta

← C2×#20253334 #27084082

Subgroups — C2×#20253334 #27084082

Order 12032.

2326 conjugacy classes of subgroups.

The subgroup lattice has 648 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#13094000C211yesmaximal, characteristic
2C2×#117×D47C211yesmaximal, characteristic
3#20253334C2116yesmaximal, direct product factor
4C2×#15689457C212yesmaximal, complemented
5C2×#15689463C212yesmaximal, complemented
6C2×#15689462C212yesmaximal, complemented
7C2×#15689452C212yesmaximal, complemented
8C2×#15689166C211yesmaximal, characteristic, complemented
9C2×#15689165C211yesmaximal, characteristic, complemented
10C2×#15689132C211yesmaximal, characteristic, complemented
11C2×#15689086C211yesmaximal, characteristic
12C2×C47×#419C211yesmaximal, characteristic, complemented
13C23×C4×D47C2211yescharacteristic
14C22×#10625329C2211yescharacteristic
15C2×#13094000—21no—
16C2×#13094000—24no—
17C2×#13094000—21no—
18C2×#13094000C2212yes—
19C2×#13094000C2212yes—
20C2×#13094000C2218yes—
21C2×#13093983C2211yescharacteristic
22C2×#13093982C2211yescharacteristic
23C2×#13093982C2211yescharacteristic
24C2×#13093981—21no—
25C2×#13093980—21no—
26#117×D47—24no—
27#117×D47C2218yes—
28C22×C4⋊Dic47C2211yescharacteristic
29C22×Dic47⋊C4—21no—
30C2×C4⋊C4×D47C2211yescharacteristic
31C2×C4⋊C4×D47C2211yescharacteristic
32C22×(C2×D47)⋊C4—21no—
33C2×C22⋊C4×D47—21no—
34C2×C42×D47—21no—
35#15689463C22116yescomplemented
36#15689462C22116yescomplemented
37#15689457C22116yescomplemented
38#15689452C22116yescomplemented
39#15689166C2218yescomplemented
40#15689165C2218yescomplemented
41C2×#13093958C2212yescomplemented
42C2×#13093957C2212yescomplemented
43C2×#13093956C2212yescomplemented
44C2×#13093955C2212yescomplemented
45#15689132C2218yescomplemented
46C2×#13093948C2212yescomplemented
47C2×#13093948C2212yescomplemented
48#15689086C2218yes—
49C2×(C2×D47)⋊C8C2212yes—
50C2×(C2×D47)⋊C8C2212yes—
51C23×Q8×C47C2211yescharacteristic
52C22×M16×C47C2211yescharacteristic
53C2×#117×C47C2211yescharacteristic
54C47×#419C2218yescomplemented
55C2×#103×C47C2212yescomplemented
56C2×#103×C47C2212yescomplemented
57C24×D47D411yescharacteristic
58C23×Dic47—21no—
59C23×Dic47D411yescharacteristic
60C22×C4×D47—24no—
61C22×C4×D47—24no—
62C22×C4×D47—21no—
63C22×C4×D47—21no—
64C22×C4×D47D411yescharacteristic
65C22×C4×D47D414yes—
66C22×C4×D47D411yescharacteristic
67C22×C4×D47C2311yescharacteristic
68C22×C4×D47C2314yes—
69C22×C4×D47C2311yescharacteristic
70C2×#10625329—24no—
71C2×#10625329C2314yes—
72C22×C47⋊C8C2312yes—
73#13094000—28no—
74#13094000—24no—
75#13094000—28no—
76#13094000—24no—
77#13094000C2318yes—
78#13094000C2318yes—
79#13093983—22no—
80#13093983C2314yes—
81#13093982—22no—
82#13093982—22no—
83#13093982C2314yes—
84#13093982C2314yes—
85#13093981—28no—
86#13093980—28no—
87C2×C4⋊Dic47—21no—
88C2×C4⋊Dic47—22no—
89C2×C4⋊Dic47C2311yescharacteristic
90C2×C4⋊Dic47C2311yescharacteristic
91C2×C4⋊Dic47C2312yes—
92C2×C4⋊Dic47C2314yes—
93C2×Dic47⋊C4—21no—
94C2×Dic47⋊C4—21no—
95C2×Dic47⋊C4—22no—
96C2×Dic47⋊C4—22no—
97C2×Dic47⋊C4—28no—
98C4⋊C4×D47—22no—
99C4⋊C4×D47—22no—
100C4⋊C4×D47C2314yes—
101C4⋊C4×D47C2314yes—
102C2×C22⋊Dic47—21no—
103C2×(C2×D47)⋊C4—21no—
104C2×(C2×D47)⋊C4—21no—
105C2×(C2×D47)⋊C4—22no—
106C2×(C2×D47)⋊C4—28no—
107C2×(C2×D47)⋊C4—22no—
108C22⋊C4×D47—28no—
109C2×C4×Dic47—21no—
110C2×C4×Dic47—21no—
111C2×C4×Dic47—21no—
112C42×D47—28no—
113#13093958C2318yescomplemented
114#13093957C2318yescomplemented
115#13093956C2318yescomplemented
116#13093955C2318yescomplemented
117#13093948C2318yescomplemented
118#13093948C2318yescomplemented
119(C2×D47)⋊C8C2318yes—
120(C2×D47)⋊C8C2318yes—
121C22×Q8×C47—21no—
122C22×Q8×C47—22no—
123C22×Q8×C47—24no—
124C22×Q8×C47—24no—
125C22×Q8×C47C2312yes—
126C22×Q8×C47C2314yes—
127C23×C4×C47—21no—
128C23×C4×C47C2311yescharacteristic
129C2×M16×C47—24no—
130C2×M16×C47C2314yes—
131C22×C8×C47C2312yes—
132#117×C47—22no—
133#117×C47C2314yes—
134C2×C4⋊C4×C47C2311yescharacteristic
135C2×C4⋊C4×C47C2311yescharacteristic
136C2×C22⋊C4×C47—21no—
137C2×C42×C47—21no—
138#103×C47C2318yescomplemented
139#103×C47C2318yescomplemented
140C24×C47C2×D411yescharacteristic
141C23×D47—24no—
142C23×D47—24no—
143C23×D47—42no—
144C23×D47C2×D411yescharacteristic
145C23×D47C2×D414yes—
146C23×D47C2×D411yescharacteristic
147C2×Q8×C47—24no—
148C2×Q8×C47—22no—
149C2×Q8×C47—24no—
150C2×Q8×C47—28no—
151C2×Q8×C47—24no—
152C2×Q8×C47—28no—
153C2×Q8×C47—48no—
154C2×Q8×C47—44no—
155C2×Q8×C47C2414yes—
156C22×C4×C47—21no—
157C22×C4×C47—21no—
158C22×C4×C47—24no—
159C22×C4×C47—22no—
160C22×C4×C47—44no—
161C22×C4×C47C2×D411yescharacteristic
162C22×C4×C47C2×D412yes—
163C22×C4×C47#4114yes—
164C22×C4×C47#4114yes—
165C22×C4×C47C2411yescharacteristic
166C22×C4×C47C2412yes—
167C22×Dic47—24no—
168C22×Dic47—21no—
169C22×Dic47—21no—
170C22×Dic47—24no—
171C22×Dic47—22no—
172C22×Dic47—44no—
173C22×Dic47C2×D411yescharacteristic
174C22×Dic47C2×D411yescharacteristic
175C22×Dic47C2×D414yes—
176C2×C4×D47—24no—
177C2×C4×D47—216no—
178C2×C4×D47—24no—
179C2×C4×D47—22no—
180C2×C4×D47—22no—
181C2×C4×D47—28no—
182C2×C4×D47—22no—
183C2×C4×D47—22no—
184C2×C4×D47—28no—
185C2×C4×D47—24no—
186C2×C4×D47—24no—
187C2×C4×D47—44no—
188C2×C4×D47—44no—
189C2×C4×D47C2×D414yes—
190C2×C4×D47C2×D412yes—
191C2×C4×D47C2×D414yes—
192C2×C4×D47C2×D412yes—
193C2×C4×D47C2412yes—
194C2×C4×D47C2412yes—
195M16×C47—28no—
196C2×C8×C47—24no—
197C2×C8×C47C2414yes—
198C4⋊C4×C47—21no—
199C4⋊C4×C47—21no—
200C4⋊C4×C47C2412yes—
201C4⋊C4×C47C2412yes—
202C22⋊C4×C47—24no—
203C42×C47—24no—
204C22⋊Dic47—24no—
205(C2×D47)⋊C4—24no—
206(C2×D47)⋊C4—24no—
207(C2×D47)⋊C4—28no—
208(C2×D47)⋊C4—28no—
209C4⋊Dic47—21no—
210C4⋊Dic47—21no—
211C4⋊Dic47—24no—
212C4⋊Dic47—22no—
213C4⋊Dic47C2412yes—
214C4⋊Dic47C2412yes—
215C4⋊Dic47C2414yes—
216Dic47⋊C4—24no—
217Dic47⋊C4—24no—
218Dic47⋊C4—28no—
219Dic47⋊C4—28no—
220C4×Dic47—24no—
221C4×Dic47—24no—
222C4×Dic47—24no—
223#10625329—28no—
224C2×C47⋊C8—24no—
225C2×C47⋊C8C2414yes—
226C23×C47—22no—
227C23×C47—22no—
228C23×C47#12314yes—
229C23×C47C22×D412yes—
230C23×C47C22×D411yescharacteristic
231C22×D47—22no—
232C22×D47—22no—
233C22×D47—28no—
234C22×D47—22no—
235C22×D47—22no—
236C22×D47—22no—
237C22×D47—28no—
238C22×D47—44no—
239C22×D47—416no—
240C22×D47#13714yes—
241C22×D47C22×D412yes—
242C22×D47C22×D412yes—
243Q8×C47—24no—
244Q8×C47—24no—
245Q8×C47—48no—
246Q8×C47—44no—
247C2×C4×C47—22no—
248C2×C4×C47—22no—
249C2×C4×C47—24no—
250C2×C4×C47—48no—
251C2×C4×C47—44no—
252C2×C4×C47—42no—
253C2×C4×C47—42no—
254C2×C4×C47#12318yes—
255C2×C4×C47#12314yes—
256C2×C4×C47C22×D412yes—
257C2×C4×C47C22×D411yescharacteristic
258C2×C4×C47C2×#4112yes—
259C2×C4×C47C2×#4114yes—
260C2×C4×C47C2×#4112yes—
261C2×C4×C47C2×#4114yes—
262C2×C4×C47C2511yescharacteristic
263C2×Dic47—28no—
264C2×Dic47—22no—
265C2×Dic47—22no—
266C2×Dic47—22no—
267C2×Dic47—22no—
268C2×Dic47—24no—
269C2×Dic47—48no—
270C2×Dic47—42no—
271C2×Dic47—42no—
272C2×Dic47—44no—
273C2×Dic47C22×D412yes—
274C2×Dic47C22×D412yes—
275C4×D47—28no—
276C4×D47—24no—
277C4×D47—28no—
278C4×D47—24no—
279C4×D47—22no—
280C4×D47—22no—
281C4×D47—22no—
282C4×D47—22no—
283C4×D47—48no—
284C4×D47—48no—
285C8×C47—24no—
286C47⋊C8—24no—
287C2×#3095—471nomaximal, Sylow
288C22×C47—28no—
289C22×C47—22no—
290C22×C47—22no—
291C22×C47C2×#12314yes—
292C22×C47C2×#12312yes—
293C22×C47C2×#13712yes—
294C22×C47#57512yes—
295C22×C47C23×D411yescharacteristic
296C2×D47—28no—
297C2×D47—22no—
298C2×D47—22no—
299C2×D47—22no—
300C2×D47—22no—
301C2×D47—22no—
302C2×D47—416no—
303C2×D47—44no—
304C2×D47C2×#13714yes—
305C4×C47—44no—
306C4×C47—42no—
307C4×C47—42no—
308C4×C47C2×#12314yes—
309C4×C47C2×#12312yes—
310C4×C47C22×#4111yescharacteristic
311C4×C47C22×#4111yescharacteristic
312Dic47—22no—
313Dic47—22no—
314Dic47—42no—
315Dic47—42no—
316Dic47—44no—
317C22×C22⋊Q8—471no—
318C22×#117—471no—
319#3095—4716no—
320C2×#484—472no—
321C2×#485—472no—
322C2×#485—472no—
323C2×#483—472no—
324C2×#428—471no—
325C2×#428—471no—
326C2×#419—471no—
327C2×#419D4711yescharacteristic, complemented
328C2×#407—471no—
329C2×C47—24no—
330C2×C47#309514yescomplemented
331C2×C47C22×#12311yescharacteristic
332C2×C47C22×#13711yescharacteristic
333C2×C47C2×#57511yescharacteristic
334D47—24no—
335D47—42no—
336C23×Q8C2×D4711yescharacteristic
337C24×C4—471no—
338C22×M16—471no—
339C22×M16C2×D4711yescharacteristic
340C2×C22⋊Q8—472no—
341C2×C22⋊Q8—472no—
342C2×C22⋊Q8—478no—
343C2×C22⋊Q8—944no—
344C2×C22⋊Q8—941no—
345C2×C22⋊Q8—941no—
346C2×#117—471no—
347C2×#117—478no—
348C2×#117—471no—
349C2×#117—471no—
350C2×#117—941no—
351C2×#117—941no—
352C2×#117—944no—
353C2×#117C2×D4711yescharacteristic
354C22×C4⋊C4—471no—
355C22×C4⋊C4—471no—
356C22×C4⋊C4—471no—
357C22×C4⋊C4—941no—
358C22×C22⋊C4—941no—
359C22×C22⋊C4—941no—
360C22×C42—941no—
361#485—4716no—
362#485—4716no—
363#484—4716no—
364#483—4716no—
365#428—478no—
366#428—478no—
367C2×#107—472no—
368C2×#107—472no—
369C2×#106—472no—
370C2×#106—472no—
371#419—478no—
372#419C2×D4718yescomplemented
373C2×#103—472no—
374C2×#103—472no—
375C2×#103C2×D4712yescomplemented
376C2×#103C2×D4712yescomplemented
377#407—478no—
378C2×C22⋊C8—472no—
379C2×C22⋊C8—472no—
380C47C2×#309511yescharacteristic, complemented, Sylow
381C25—471no—
382C22×Q8—22no—
383C22×Q8—21no—
384C22×Q8—24no—
385C22×Q8—24no—
386C22×Q8C22×D4712yes—
387C22×Q8C22×D4714yes—
388C23×C4—21no—
389C23×C4—471no—
390C23×C4—474no—
391C23×C4—471no—
392C23×C4—471no—
393C23×C4—471no—
394C23×C4—471no—
395C23×C4—474no—
396C23×C4—941no—
397C23×C4—941no—
398C23×C4—944no—
399C23×C4—944no—
400C23×C4—941no—
401C23×C4C22×D4711yescharacteristic
402C2×M16—24no—
403C2×M16—474no—
404C2×M16—944no—
405C2×M16C22×D4714yes—
406C22×C8—472no—
407C22×C8C22×D4712yes—
408C22⋊Q8—478no—
409C22⋊Q8—478no—
410C22⋊Q8—948no—
411C22⋊Q8—944no—
412C22⋊Q8—944no—
413C22⋊Q8—948no—
414#117—22no—
415#117—474no—
416#117—474no—
417#117—474no—
418#117—948no—
419#117—948no—
420#117—942no—
421#117—942no—
422#117—942no—
423#117C22×D4714yes—
424C2×C4⋊C4—471no—
425C2×C4⋊C4—471no—
426C2×C4⋊C4—472no—
427C2×C4⋊C4—474no—
428C2×C4⋊C4—474no—
429C2×C4⋊C4—474no—
430C2×C4⋊C4—942no—
431C2×C4⋊C4—942no—
432C2×C4⋊C4—941no—
433C2×C4⋊C4—941no—
434C2×C4⋊C4—948no—
435C2×C4⋊C4—942no—
436C2×C4⋊C4—942no—
437C2×C4⋊C4—942no—
438C2×C4⋊C4—941no—
439C2×C4⋊C4C22×D4711yescharacteristic
440C2×C4⋊C4C22×D4711yescharacteristic
441C2×C22⋊C4—21no—
442C2×C22⋊C4—948no—
443C2×C22⋊C4—941no—
444C2×C22⋊C4—941no—
445C2×C22⋊C4—948no—
446C2×C22⋊C4—942no—
447C2×C22⋊C4—942no—
448C2×C22⋊C4—941no—
449C2×C42—21no—
450C2×C42—948no—
451C2×C42—941no—
452C2×C42—941no—
453C2×C42—941no—
454#107—478no—
455#107—478no—
456#106—478no—
457#106—478no—
458#103—478no—
459#103—478no—
460#103C22×D4718yescomplemented
461#103C22×D4718yescomplemented
462C22⋊C8—478no—
463C22⋊C8—478no—
464C24—471no—
465C24—471no—
466C24—474no—
467C24—944no—
468C24—944no—
469C24—1882no—
470C24D4×D4711yescharacteristic
471C2×Q8—24no—
472C2×Q8—22no—
473C2×Q8—24no—
474C2×Q8—28no—
475C2×Q8—24no—
476C2×Q8—28no—
477C2×Q8—48no—
478C2×Q8—44no—
479C2×Q8C23×D4714yes—
480C22×C4—24no—
481C22×C4—21no—
482C22×C4—21no—
483C22×C4—22no—
484C22×C4—44no—
485C22×C4—474no—
486C22×C4—471no—
487C22×C4—471no—
488C22×C4—474no—
489C22×C4—472no—
490C22×C4—472no—
491C22×C4—474no—
492C22×C4—472no—
493C22×C4—472no—
494C22×C4—944no—
495C22×C4—944no—
496C22×C4—944no—
497C22×C4—944no—
498C22×C4—9416no—
499C22×C4—944no—
500C22×C4—942no—
501C22×C4—942no—
502C22×C4—948no—
503C22×C4—942no—
504C22×C4—942no—
505C22×C4—948no—
506C22×C4—944no—
507C22×C4—941no—
508C22×C4—941no—
509C22×C4—942no—
510C22×C4—1884no—
511C22×C4—1884no—
512C22×C4—1884no—
513C22×C4#1062535014yes—
514C22×C4D4×D4712yes—
515C22×C4D4×D4711yescharacteristic
516C22×C4#1062535214yes—
517C22×C4C23×D4712yes—
518C22×C4C23×D4711yescharacteristic
519M16—28no—
520M16—948no—
521C2×C8—24no—
522C2×C8—474no—
523C2×C8—944no—
524C2×C8C23×D4714yes—
525C4⋊C4—21no—
526C4⋊C4—21no—
527C4⋊C4—472no—
528C4⋊C4—472no—
529C4⋊C4—474no—
530C4⋊C4—944no—
531C4⋊C4—944no—
532C4⋊C4—948no—
533C4⋊C4—948no—
534C4⋊C4—941no—
535C4⋊C4—941no—
536C4⋊C4—944no—
537C4⋊C4—942no—
538C4⋊C4C23×D4712yes—
539C4⋊C4C23×D4712yes—
540C22⋊C4—24no—
541C22⋊C4—944no—
542C22⋊C4—944no—
543C22⋊C4—948no—
544C22⋊C4—948no—
545C22⋊C4—944no—
546C42—24no—
547C42—944no—
548C42—944no—
549C42—944no—
550C23—22no—
551C23—22no—
552C23—474no—
553C23—472no—
554C23—472no—
555C23—948no—
556C23—942no—
557C23—942no—
558C23—948no—
559C23—942no—
560C23—942no—
561C23—942no—
562C23—1884no—
563C23—18816no—
564C23#1309400614yes—
565C23C2×D4×D4712yes—
566C23C2×D4×D4711yescharacteristic
567Q8—24no—
568Q8—24no—
569Q8—48no—
570Q8—44no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—24no—
574C2×C4—48no—
575C2×C4—44no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—472no—
579C2×C4—472no—
580C2×C4—948no—
581C2×C4—942no—
582C2×C4—942no—
583C2×C4—948no—
584C2×C4—944no—
585C2×C4—944no—
586C2×C4—948no—
587C2×C4—942no—
588C2×C4—942no—
589C2×C4—942no—
590C2×C4—942no—
591C2×C4—942no—
592C2×C4—942no—
593C2×C4—944no—
594C2×C4—1888no—
595C2×C4—1888no—
596C2×C4—1888no—
597C2×C4—1882no—
598C2×C4—1882no—
599C2×C4—1884no—
600C2×C4#1309400618yes—
601C2×C4#1309400614yes—
602C2×C4C2×D4×D4712yes—
603C2×C4C2×D4×D4711yescharacteristic
604C2×C4C2×#1062535212yes—
605C2×C4C2×#1062535214yes—
606C2×C4C2×#1062535012yes—
607C2×C4C2×#1062535014yes—
608C2×C4C24×D4711yescharacteristic
609C8—24no—
610C8—944no—
611C22—22no—
612C22—22no—
613C22—28no—
614C22—474no—
615C22—942no—
616C22—942no—
617C22—948no—
618C22—942no—
619C22—942no—
620C22—942no—
621C22—18816no—
622C22—1884no—
623C22C2×#1309400614yes—
624C22C2×#1309400612yes—
625C22#137×D4712yes—
626C22#1568992212yes—
627C22C22×D4×D4711yescharacteristic
628C4—42no—
629C4—42no—
630C4—44no—
631C4—942no—
632C4—942no—
633C4—1882no—
634C4—1882no—
635C4—1884no—
636C4C2×#1309400614yes—
637C4C2×#1309400612yes—
638C4C22×#1062535211yescharacteristic
639C4C22×#1062535011yescharacteristic
640C2—24no—
641C2—944no—
642C2—1882no—
643C2#2025333414yesdirect product factor
644C2C22×#1309400611yescharacteristic
645C2C2×#137×D4711yescharacteristic
646C2C2×#1568992211yescharacteristic
subgroupG/H
centerC23C2×D4×D47
commutator, cosocleC2×C4×C47C25
FrattiniC2×C4C24×D47
FittingC2×C47×#419C2
socleC23×C47C22×D4
prime radicalC24×D47D4
hypercenterC2×#419D47
hypocenterC47C2×#3095
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#419

Complements

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

NH ≅ G/Ntype
3. #20253334C2direct (N × H)
4. C2×#15689457C2semidirect
5. C2×#15689463C2semidirect
6. C2×#15689462C2semidirect
7. C2×#15689452C2semidirect
8. C2×#15689166C2semidirect
9. C2×#15689165C2semidirect
10. C2×#15689132C2semidirect
12. C2×C47×#419C2semidirect
35. #15689463C22semidirect
36. #15689462C22semidirect
37. #15689457C22semidirect
38. #15689452C22semidirect
39. #15689166C22semidirect
40. #15689165C22semidirect
41. C2×#13093958C22semidirect
42. C2×#13093957C22semidirect
43. C2×#13093956C22semidirect
44. C2×#13093955C22semidirect
45. #15689132C22semidirect
46. C2×#13093948C22semidirect
47. C2×#13093948C22semidirect
54. C47×#419C22semidirect
55. C2×#103×C47C22semidirect
56. C2×#103×C47C22semidirect
113. #13093958C23semidirect
114. #13093957C23semidirect
115. #13093956C23semidirect
116. #13093955C23semidirect
117. #13093948C23semidirect
118. #13093948C23semidirect
138. #103×C47C23semidirect
139. #103×C47C23semidirect
327. C2×#419D47semidirect
330. C2×C47#3095semidirect
372. #419C2×D47semidirect
375. C2×#103C2×D47semidirect
376. C2×#103C2×D47semidirect
380. C47C2×#3095semidirect
460. #103C22×D47semidirect
461. #103C22×D47semidirect
643. C2#20253334direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#130940002
C2×#117×D472
#202533342
C2×#156894572
C2×#156894632
C2×#156894622
C2×#156894522
C2×#156891662
C2×#156891652
C2×#156891322
C2×#156890862
C2×C47×#4192
C2×#309547