Finite Group Databasebeta

← C2×#20253301 #27084045

Subgroups — C2×#20253301 #27084045

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×#15689634C211yesmaximal, characteristic
2#20253301C2116yesmaximal, direct product factor
3C2×#15689458C211yesmaximal, characteristic, complemented
4C2×#15689441C211yesmaximal, characteristic, complemented
5C2×#15689464C212yesmaximal, complemented
6C2×#15689447C212yesmaximal, complemented
7C2×#15689453C211yesmaximal, characteristic, complemented
8C2×#15689436C211yesmaximal, characteristic, complemented
9C2×#15689159C212yesmaximal, complemented
10C2×#15689117C212yesmaximal, complemented
11C22×C22⋊(C47⋊C8)C211yesmaximal, characteristic, complemented
12C22×#117×C47C211yesmaximal, characteristic, complemented
13C22×#10625350C2211yescharacteristic
14C23×C47⋊C8C2212yes—
15#15689634—24no—
16#15689634C2218yes—
17C2×#13094007—21no—
18C2×#13094003C2211yescharacteristic
19C2×#13094003C2211yescharacteristic
20C2×#13093998C2211yescharacteristic
21C2×#13093998C2211yescharacteristic
22C2×#13093993—21no—
23C2×C4×C22⋊D47—21no—
24C2×C4×C22⋊D47—21no—
25C2×#13093983C2211yescharacteristic
26#15689464C22116yescomplemented
27#15689458C2218yescomplemented
28#15689453C2218yescomplemented
29#15689447C22116yescomplemented
30#15689441C2218yescomplemented
31#15689436C2218yescomplemented
32#15689159C22116yescomplemented
33C2×#13093957C2212yescomplemented
34C2×#13093957C2212yescomplemented
35C2×#13093955C2212yescomplemented
36C2×#13093955C2212yescomplemented
37#15689117C22116yescomplemented
38C2×#13093947C2212yescomplemented
39C2×#13093947C2212yescomplemented
40C2×#13093944C2212yescomplemented
41C2×#13093944C2212yescomplemented
42C2×C22⋊(C47⋊C8)C2211yescharacteristic
43C2×C22⋊(C47⋊C8)C2211yescharacteristic
44C2×C22⋊(C47⋊C8)C2212yes—
45C2×C22⋊(C47⋊C8)C2218yescomplemented
46C24×C4×C47C2211yescharacteristic
47C2×#117×C47—21no—
48C2×#117×C47—21no—
49C2×#117×C47—24no—
50C2×#117×C47C2212yes—
51C2×#117×C47C2212yes—
52C2×#117×C47C2218yescomplemented
53C22×C4⋊C4×C47C2211yescharacteristic
54C22×C4⋊C4×C47C2211yescharacteristic
55C22×C22⋊C4×C47—21no—
56C22×C42×C47—21no—
57C2×#10625350—22no—
58C2×#10625350—28no—
59C2×#10625350C2314yes—
60C22×Dic94C2311yescharacteristic
61C22×D188C2311yescharacteristic
62C22×C22⋊D47—21no—
63C22×C22⋊D47—21no—
64C22×C22⋊D47—21no—
65C22×C4×D47—21no—
66C22×C47⋊C8—28no—
67C22×C47⋊C8C2312yes—
68C22×C47⋊C8C2312yes—
69C22×C47⋊C8C2318yes—
70#13094007—28no—
71#13094003—22no—
72#13094003—22no—
73#13094003C2314yes—
74#13094003C2314yes—
75#13093998—22no—
76#13093998—22no—
77#13093998C2314yes—
78#13093998C2314yes—
79#13093993—28no—
80C4×C22⋊D47—28no—
81C4×C22⋊D47—28no—
82#13093983—22no—
83#13093983C2314yes—
84C2×C4⋊Dic47C2311yescharacteristic
85C2×C4⋊Dic47C2311yescharacteristic
86C2×Dic47⋊C4—21no—
87C2×Dic47⋊C4—21no—
88C2×C22⋊Dic47—21no—
89C2×C22⋊Dic47—21no—
90C2×C22⋊Dic47—21no—
91C2×(C2×D47)⋊C4—21no—
92C2×(C2×D47)⋊C4—21no—
93C2×C4×Dic47—21no—
94#13093957C2318yescomplemented
95#13093957C2318yescomplemented
96#13093955C2318yescomplemented
97#13093955C2318yescomplemented
98#13093947C2318yescomplemented
99#13093947C2318yescomplemented
100#13093944C2318yescomplemented
101#13093944C2318yescomplemented
102C22⋊(C47⋊C8)C2314yes—
103C22⋊(C47⋊C8)C2314yes—
104C22⋊(C47⋊C8)C2318yes—
105C25×C47D411yescharacteristic
106C23×C4×C47—28no—
107C23×C4×C47—21no—
108C23×C4×C47—21no—
109C23×C4×C47D411yescharacteristic
110C23×C4×C47D411yescharacteristic
111C23×C4×C47D411yescharacteristic
112C23×C4×C47D414yes—
113C23×C4×C47C2311yescharacteristic
114C23×C4×C47C2311yescharacteristic
115C23×C4×C47C2314yes—
116C23×C4×C47C2311yescharacteristic
117#117×C47—24no—
118#117×C47—24no—
119#117×C47—28no—
120#117×C47—28no—
121#117×C47C2318yes—
122#117×C47C2318yes—
123C2×C4⋊C4×C47—21no—
124C2×C4⋊C4×C47—21no—
125C2×C4⋊C4×C47—22no—
126C2×C4⋊C4×C47—22no—
127C2×C4⋊C4×C47C2312yes—
128C2×C4⋊C4×C47C2312yes—
129C2×C4⋊C4×C47C2314yes—
130C2×C4⋊C4×C47C2314yes—
131C2×C22⋊C4×C47—22no—
132C2×C22⋊C4×C47—22no—
133C2×C22⋊C4×C47—28no—
134C2×C42×C47—22no—
135C2×C42×C47—22no—
136C2×C42×C47—28no—
137C24×C47—24no—
138C24×C47—24no—
139C24×C47—42no—
140C24×C47C2×D411yescharacteristic
141C24×C47C2×D414yes—
142C24×C47C2×D411yescharacteristic
143C24×C47C2×D411yescharacteristic
144C23×D47—21no—
145C22×C4×C47—24no—
146C22×C4×C47—24no—
147C22×C4×C47—24no—
148C22×C4×C47—216no—
149C22×C4×C47—24no—
150C22×C4×C47—24no—
151C22×C4×C47—216no—
152C22×C4×C47—24no—
153C22×C4×C47—24no—
154C22×C4×C47—22no—
155C22×C4×C47—22no—
156C22×C4×C47—44no—
157C22×C4×C47—44no—
158C22×C4×C47C2×D411yescharacteristic
159C22×C4×C47C2×D414yes—
160C22×C4×C47C2×D411yescharacteristic
161C22×C4×C47C2×D412yes—
162C22×C4×C47C2×D414yes—
163C22×C4×C47C2×D412yes—
164C22×C4×C47C2×D414yes—
165C22×C4×C47C2×D412yes—
166C22×C4×C47C2×D411yescharacteristic
167C22×C4×C47#4118yes—
168C22×C4×C47C2412yes—
169C22×C4×C47C2412yes—
170C22×C4×C47C2411yescharacteristic
171C22×C4×C47C2412yes—
172C2×C22⋊D47—24no—
173C2×C22⋊D47—24no—
174C2×C22⋊D47—24no—
175C2×C22⋊D47—44no—
176C2×C22⋊D47—44no—
177C2×C22⋊D47—44no—
178C22×Dic47—21no—
179C22×Dic47—21no—
180C22×Dic47—21no—
181#10625350—216no—
182#10625350—48no—
183C2×D188—24no—
184C2×D188—21no—
185C2×D188C2412yes—
186C2×C4×D47—24no—
187C2×C4×D47—44no—
188C2×Dic94—24no—
189C2×Dic94—21no—
190C2×Dic94C2412yes—
191C4⋊C4×C47—24no—
192C4⋊C4×C47—24no—
193C4⋊C4×C47—22no—
194C4⋊C4×C47—22no—
195C4⋊C4×C47C2414yes—
196C4⋊C4×C47C2414yes—
197C22⋊C4×C47—28no—
198C22⋊C4×C47—28no—
199C42×C47—28no—
200C42×C47—28no—
201C22⋊Dic47—24no—
202C22⋊Dic47—24no—
203C22⋊Dic47—24no—
204(C2×D47)⋊C4—24no—
205(C2×D47)⋊C4—24no—
206C4⋊Dic47—21no—
207C4⋊Dic47—21no—
208C4⋊Dic47C2412yes—
209C4⋊Dic47C2412yes—
210Dic47⋊C4—24no—
211Dic47⋊C4—24no—
212C4×Dic47—24no—
213C2×C47⋊C8—24no—
214C2×C47⋊C8—24no—
215C2×C47⋊C8—216no—
216C2×C47⋊C8C2414yes—
217C2×C47⋊C8C2414yes—
218C23×C47—22no—
219C23×C47—22no—
220C23×C47—22no—
221C23×C47—22no—
222C23×C47—28no—
223C23×C47—22no—
224C23×C47—28no—
225C23×C47—22no—
226C23×C47—22no—
227C23×C47—44no—
228C23×C47—416no—
229C23×C47#12314yes—
230C23×C47#13514yes—
231C23×C47C22×D412yes—
232C23×C47C22×D412yes—
233C23×C47C22×D412yes—
234C23×C47C22×D411yescharacteristic
235C22×D47—22no—
236C22×D47—42no—
237C22×D47—42no—
238C22×D47—42no—
239C2×C4×C47—28no—
240C2×C4×C47—22no—
241C2×C4×C47—22no—
242C2×C4×C47—24no—
243C2×C4×C47—24no—
244C2×C4×C47—28no—
245C2×C4×C47—28no—
246C2×C4×C47—24no—
247C2×C4×C47—24no—
248C2×C4×C47—24no—
249C2×C4×C47—24no—
250C2×C4×C47—48no—
251C2×C4×C47—48no—
252C2×C4×C47—44no—
253C2×C4×C47—44no—
254C2×C4×C47#12314yes—
255C2×C4×C47#12318yes—
256C2×C4×C47C22×D412yes—
257C2×C4×C47C22×D412yes—
258C2×C4×C47C22×D412yes—
259C2×C4×C47C22×D411yescharacteristic
260C2×C4×C47C2×#4114yes—
261C2×C4×C47C2×#4118yes—
262C2×C4×C47C2511yescharacteristic
263C22⋊D47—48no—
264C22⋊D47—48no—
265C22⋊D47—48no—
266C2×Dic47—22no—
267C2×Dic47—22no—
268C2×Dic47—22no—
269C2×Dic47—42no—
270C2×Dic47—42no—
271C2×Dic47—42no—
272D188—24no—
273D188—42no—
274C4×D47—48no—
275Dic94—24no—
276Dic94—42no—
277C47⋊C8—24no—
278C47⋊C8—24no—
279C2×#3091—471nomaximal, Sylow
280C22×C47—28no—
281C22×C47—28no—
282C22×C47—22no—
283C22×C47—22no—
284C22×C47—22no—
285C22×C47—22no—
286C22×C47—22no—
287C22×C47—22no—
288C22×C47—22no—
289C22×C47—416no—
290C22×C47—44no—
291C22×C47C2×#12314yes—
292C22×C47C2×#12312yes—
293C22×C47C2×#13514yes—
294C22×C47C2×#13512yes—
295C22×C47#57512yes—
296C22×C47C23×D411yescharacteristic
297C2×D47—42no—
298C2×D47—42no—
299C2×D47—42no—
300C2×D47—84no—
301C4×C47—22no—
302C4×C47—22no—
303C4×C47—44no—
304C4×C47—44no—
305C4×C47C2×#12312yes—
306C4×C47C2×#12314yes—
307C4×C47C22×#4112yes—
308Dic47—42no—
309Dic47—42no—
310Dic47—42no—
311C2×#525—471no—
312C22×#117D4711yescharacteristic, complemented
313#3091—4716no—
314C2×#484—471no—
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—24no—
325C2×C47—42no—
326C2×C47#309114yescomplemented
327C2×C47C22×#12311yescharacteristic
328C2×C47C22×#13511yescharacteristic
329C2×C47C2×#57511yescharacteristic
330D47—82no—
331C22×#41—471no—
332C24×C4C2×D4711yescharacteristic
333C23×C8—472no—
334#525—478no—
335#525—944no—
336C2×#123—941no—
337C2×C22⋊Q8—471no—
338C2×C22⋊Q8—471no—
339C2×#121—471no—
340C2×#121—471no—
341C2×#120—941no—
342C2×C4×D4—941no—
343C2×C4×D4—941no—
344C2×#117—21no—
345C2×#117—24no—
346C2×#117—21no—
347C2×#117—471no—
348C2×#117C2×D4712yes—
349C2×#117C2×D4712yes—
350C2×#117C2×D4718yescomplemented
351C22×C4⋊C4C2×D4711yescharacteristic
352C22×C4⋊C4C2×D4711yescharacteristic
353C22×C22⋊C4—21no—
354C22×C42—21no—
355#485—4716no—
356#484—478no—
357#483—478no—
358#482—4716no—
359#481—478no—
360#480—478no—
361#427—4716no—
362C2×#107—472no—
363C2×#107—472no—
364C2×#106—472no—
365C2×#106—472no—
366#416—4716no—
367C2×#103—472no—
368C2×#103—472no—
369C2×#102—472no—
370C2×#102—472no—
371C2×C22⋊C8—471no—
372C2×C22⋊C8—471no—
373C2×C22⋊C8—472no—
374C2×C22⋊C8—478no—
375C47C2×#309111yescharacteristic, complemented, Sylow
376C25C22⋊D4711yescharacteristic
377C2×#41—474no—
378C2×#41—942no—
379C2×#41—948no—
380C22×Q8—471no—
381C22×D4—471no—
382C22×D4—941no—
383C22×D4—941no—
384C22×D4—941no—
385C23×C4—28no—
386C23×C4—21no—
387C23×C4—21no—
388C23×C4—941no—
389C23×C4C22⋊D4711yescharacteristic
390C23×C4C22⋊D4714yes—
391C23×C4C22⋊D4711yescharacteristic
392C23×C4C22⋊D4711yescharacteristic
393C23×C4C22×D4711yescharacteristic
394C23×C4C22×D4711yescharacteristic
395C23×C4C22×D4714yes—
396C23×C4C22×D4711yescharacteristic
397C22×C8—472no—
398C22×C8—472no—
399C22×C8—478no—
400C22×C8—948no—
401#123—948no—
402C22⋊Q8—474no—
403C22⋊Q8—474no—
404C22⋊Q8—942no—
405C22⋊Q8—942no—
406#121—474no—
407#121—474no—
408#121—942no—
409#121—942no—
410#120—948no—
411C4×D4—948no—
412C4×D4—948no—
413#117—24no—
414#117—24no—
415#117—28no—
416#117—28no—
417#117—474no—
418#117—942no—
419#117C22×D4718yes—
420#117C22×D4718yes—
421C2×C4⋊C4—21no—
422C2×C4⋊C4—21no—
423C2×C4⋊C4—22no—
424C2×C4⋊C4—22no—
425C2×C4⋊C4—471no—
426C2×C4⋊C4—471no—
427C2×C4⋊C4—941no—
428C2×C4⋊C4—941no—
429C2×C4⋊C4C22×D4712yes—
430C2×C4⋊C4C22×D4712yes—
431C2×C4⋊C4C22×D4714yes—
432C2×C4⋊C4C22×D4714yes—
433C2×C22⋊C4—22no—
434C2×C22⋊C4—22no—
435C2×C22⋊C4—28no—
436C2×C22⋊C4—941no—
437C2×C22⋊C4—941no—
438C2×C22⋊C4—941no—
439C2×C22⋊C4—941no—
440C2×C22⋊C4—941no—
441C2×C42—22no—
442C2×C42—28no—
443C2×C42—22no—
444C2×C42—941no—
445#107—478no—
446#107—478no—
447#106—478no—
448#106—478no—
449#103—478no—
450#103—478no—
451#102—478no—
452#102—478no—
453C22⋊C8—474no—
454C22⋊C8—474no—
455C22⋊C8—478no—
456C24—24no—
457C24—24no—
458C24—42no—
459C24—941no—
460C24C2×C22⋊D4711yescharacteristic
461C24C2×C22⋊D4714yes—
462C24C2×C22⋊D4711yescharacteristic
463C24C2×C22⋊D4711yescharacteristic
464#41—9416no—
465#41—1888no—
466C2×Q8—472no—
467C2×Q8—941no—
468C2×Q8—944no—
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—24no—
481C22×C4—216no—
482C22×C4—24no—
483C22×C4—24no—
484C22×C4—216no—
485C22×C4—24no—
486C22×C4—24no—
487C22×C4—22no—
488C22×C4—22no—
489C22×C4—44no—
490C22×C4—44no—
491C22×C4—941no—
492C22×C4—941no—
493C22×C4—941no—
494C22×C4—944no—
495C22×C4—1884no—
496C22×C4#1062535018yes—
497C22×C4C2×C22⋊D4711yescharacteristic
498C22×C4C2×C22⋊D4714yes—
499C22×C4C2×C22⋊D4711yescharacteristic
500C22×C4C2×C22⋊D4712yes—
501C22×C4C2×C22⋊D4714yes—
502C22×C4C2×C22⋊D4712yes—
503C22×C4C2×C22⋊D4714yes—
504C22×C4C2×C22⋊D4712yes—
505C22×C4C2×C22⋊D4711yescharacteristic
506C22×C4C23×D4712yes—
507C22×C4C23×D4712yes—
508C22×C4C23×D4711yescharacteristic
509C22×C4C23×D4712yes—
510C2×C8—474no—
511C2×C8—474no—
512C2×C8—944no—
513C2×C8—944no—
514C2×C8—9416no—
515C4⋊C4—24no—
516C4⋊C4—24no—
517C4⋊C4—22no—
518C4⋊C4—22no—
519C4⋊C4—472no—
520C4⋊C4—472no—
521C4⋊C4—944no—
522C4⋊C4—944no—
523C4⋊C4—941no—
524C4⋊C4—941no—
525C4⋊C4C23×D4714yes—
526C4⋊C4C23×D4714yes—
527C22⋊C4—28no—
528C22⋊C4—28no—
529C22⋊C4—944no—
530C22⋊C4—944no—
531C22⋊C4—944no—
532C22⋊C4—944no—
533C22⋊C4—944no—
534C42—28no—
535C42—28no—
536C42—944no—
537C23—22no—
538C23—22no—
539C23—22no—
540C23—22no—
541C23—28no—
542C23—22no—
543C23—28no—
544C23—22no—
545C23—22no—
546C23—44no—
547C23—416no—
548C23—942no—
549C23—1882no—
550C23—1882no—
551C23—1882no—
552C23#1309400714yes—
553C23#1309404414yes—
554C23C22×C22⋊D4712yes—
555C23C22×C22⋊D4712yes—
556C23C22×C22⋊D4712yes—
557C23C22×C22⋊D4711yescharacteristic
558Q8—944no—
559Q8—1882no—
560D4—944no—
561D4—1888no—
562D4—1888no—
563D4—1888no—
564D4—1882no—
565C2×C4—28no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—28no—
569C2×C4—28no—
570C2×C4—24no—
571C2×C4—24no—
572C2×C4—24no—
573C2×C4—24no—
574C2×C4—24no—
575C2×C4—24no—
576C2×C4—48no—
577C2×C4—48no—
578C2×C4—44no—
579C2×C4—44no—
580C2×C4—942no—
581C2×C4—942no—
582C2×C4—942no—
583C2×C4—1882no—
584C2×C4—1882no—
585C2×C4—1882no—
586C2×C4—1888no—
587C2×C4#1309400714yes—
588C2×C4#1309400718yes—
589C2×C4C22×C22⋊D4712yes—
590C2×C4C22×C22⋊D4712yes—
591C2×C4C22×C22⋊D4712yes—
592C2×C4C22×C22⋊D4711yescharacteristic
593C2×C4C2×#1062535014yes—
594C2×C4C2×#1062535018yes—
595C2×C4C24×D4711yescharacteristic
596C8—944no—
597C8—944no—
598C22—22no—
599C22—22no—
600C22—22no—
601C22—22no—
602C22—28no—
603C22—28no—
604C22—22no—
605C22—22no—
606C22—22no—
607C22—416no—
608C22—44no—
609C22—3764no—
610C22—1882no—
611C22—1882no—
612C22—1882no—
613C22C2×#1309400714yes—
614C22C2×#1309400712yes—
615C22C2×#1309404414yes—
616C22C2×#1309404412yes—
617C22#1568992112yes—
618C22C23×C22⋊D4711yescharacteristic
619C4—22no—
620C4—22no—
621C4—44no—
622C4—44no—
623C4—1882no—
624C4—1882no—
625C4—1882no—
626C4C2×#1309400714yes—
627C4C2×#1309400712yes—
628C4C22×#1062535012yes—
629C2—24no—
630C2—24no—
631C2—42no—
632C2—3762no—
633C2#2025330114yesdirect product factor
634C2C22×#1309400711yescharacteristic
635C2C22×#1309404411yescharacteristic
636C2C2×#1568992111yescharacteristic
subgroupG/H
centerC22×C4C2×C22⋊D47
commutator, cosocleC2×C4×C47C25
FrattiniC2×C4C24×D47
FittingC22×#117×C47C2
socleC23×C47C22×D4
prime radicalC2×#15689634C2
hypercenterC22×#117D47
hypocenterC47C2×#3091
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22×C4 ◁ C24×C4 ◁ C22×#117

Complements

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

NH ≅ G/Ntype
2. #20253301C2direct (N × H)
3. C2×#15689458C2semidirect
4. C2×#15689441C2semidirect
5. C2×#15689464C2semidirect
6. C2×#15689447C2semidirect
7. C2×#15689453C2semidirect
8. C2×#15689436C2semidirect
9. C2×#15689159C2semidirect
10. C2×#15689117C2semidirect
11. C22×C22⋊(C47⋊C8)C2semidirect
12. C22×#117×C47C2semidirect
26. #15689464C22semidirect
27. #15689458C22semidirect
28. #15689453C22semidirect
29. #15689447C22semidirect
30. #15689441C22semidirect
31. #15689436C22semidirect
32. #15689159C22semidirect
33. C2×#13093957C22semidirect
34. C2×#13093957C22semidirect
35. C2×#13093955C22semidirect
36. C2×#13093955C22semidirect
37. #15689117C22semidirect
38. C2×#13093947C22semidirect
39. C2×#13093947C22semidirect
40. C2×#13093944C22semidirect
41. C2×#13093944C22semidirect
45. C2×C22⋊(C47⋊C8)C22semidirect
52. C2×#117×C47C22semidirect
94. #13093957C23semidirect
95. #13093957C23semidirect
96. #13093955C23semidirect
97. #13093955C23semidirect
98. #13093947C23semidirect
99. #13093947C23semidirect
100. #13093944C23semidirect
101. #13093944C23semidirect
312. C22×#117D47semidirect
326. C2×C47#3091semidirect
350. C2×#117C2×D47semidirect
375. C47C2×#3091semidirect
633. C2#20253301direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#156896342
#202533012
C2×#156894582
C2×#156894412
C2×#156894642
C2×#156894472
C2×#156894532
C2×#156894362
C2×#156891592
C2×#156891172
C22×C22⋊(C47⋊C8)2
C22×#117×C472
C2×#309147