Finite Group Databasebeta

← C2×#20253338 #27084086

Subgroups — C2×#20253338 #27084086

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×#13094004C211yesmaximal, characteristic
2C2×#117×D47C211yesmaximal, characteristic
3#20253338C2116yesmaximal, direct product factor
4C2×#15689460C212yesmaximal, complemented
5C2×#15689467C212yesmaximal, complemented
6C2×#15689466C212yesmaximal, complemented
7C2×#15689455C212yesmaximal, complemented
8C2×#15689163C211yesmaximal, characteristic, complemented
9C2×#15689133C211yesmaximal, characteristic, complemented
10C2×#15689131C211yesmaximal, characteristic, complemented
11C2×#15689086C211yesmaximal, characteristic
12C2×C47×#428C211yesmaximal, characteristic, complemented
13C23×Dic94C2211yescharacteristic
14C23×C4×D47C2211yescharacteristic
15C22×#10625329C2211yescharacteristic
16C2×#13094004—21no—
17C2×#13094004—24no—
18C2×#13094004—21no—
19C2×#13094004C2212yes—
20C2×#13094004C2212yes—
21C2×#13094004C2218yes—
22C2×#13093983C2211yescharacteristic
23C2×#13093982C2211yescharacteristic
24C2×#13093982C2211yescharacteristic
25C2×#13093981—21no—
26C2×#13093980—21no—
27#117×D47—24no—
28#117×D47C2218yes—
29C22×C4⋊Dic47—21no—
30C2×C4⋊C4×D47C2211yescharacteristic
31C2×C4⋊C4×D47C2211yescharacteristic
32C22×(C2×D47)⋊C4—21no—
33C2×C22⋊C4×D47—21no—
34C2×C42×D47—21no—
35#15689467C22116yescomplemented
36#15689466C22116yescomplemented
37#15689460C22116yescomplemented
38#15689455C22116yescomplemented
39#15689163C2218yescomplemented
40C2×#13093957C2212yescomplemented
41C2×#13093955C2212yescomplemented
42#15689133C2218yescomplemented
43#15689131C2218yescomplemented
44C2×#13093949C2212yescomplemented
45C2×#13093949C2212yescomplemented
46C2×#13093947C2212yescomplemented
47C2×#13093947C2212yescomplemented
48#15689086C2218yes—
49C2×(C2×D47)⋊C8C2212yes—
50C2×(C2×D47)⋊C8C2212yes—
51C22×M16×C47C2211yescharacteristic
52C2×#117×C47C2211yescharacteristic
53C22×C4⋊C4×C47C2211yescharacteristic
54C47×#428C2218yescomplemented
55C2×#107×C47C2212yescomplemented
56C2×#106×C47C2212yescomplemented
57C24×D47D411yescharacteristic
58C22×Dic94—24no—
59C22×Dic94—24no—
60C22×Dic94—21no—
61C22×Dic94—22no—
62C22×Dic94C2312yes—
63C22×Dic94C2314yes—
64C23×Dic47—21no—
65C23×Dic47D411yescharacteristic
66C22×C4×D47—24no—
67C22×C4×D47—24no—
68C22×C4×D47—21no—
69C22×C4×D47—21no—
70C22×C4×D47D411yescharacteristic
71C22×C4×D47D414yes—
72C22×C4×D47D411yescharacteristic
73C22×C4×D47C2311yescharacteristic
74C22×C4×D47C2314yes—
75C22×C4×D47C2311yescharacteristic
76C2×#10625329—24no—
77C2×#10625329C2314yes—
78C22×C47⋊C8C2312yes—
79#13094004—28no—
80#13094004—24no—
81#13094004—28no—
82#13094004—24no—
83#13094004C2318yes—
84#13094004C2318yes—
85#13093983—22no—
86#13093983C2314yes—
87#13093982—22no—
88#13093982—22no—
89#13093982C2314yes—
90#13093982C2314yes—
91#13093981—28no—
92#13093980—28no—
93C2×C4⋊Dic47—22no—
94C2×C4⋊Dic47—22no—
95C2×C4⋊Dic47—28no—
96C2×C4⋊Dic47C2311yescharacteristic
97C2×C4⋊Dic47C2311yescharacteristic
98C2×Dic47⋊C4—21no—
99C2×Dic47⋊C4—21no—
100C4⋊C4×D47—22no—
101C4⋊C4×D47—22no—
102C4⋊C4×D47C2314yes—
103C4⋊C4×D47C2314yes—
104C2×C22⋊Dic47—21no—
105C2×(C2×D47)⋊C4—21no—
106C2×(C2×D47)⋊C4—21no—
107C2×(C2×D47)⋊C4—22no—
108C2×(C2×D47)⋊C4—28no—
109C2×(C2×D47)⋊C4—22no—
110C22⋊C4×D47—28no—
111C2×C4×Dic47—21no—
112C2×C4×Dic47—21no—
113C2×C4×Dic47—21no—
114C42×D47—28no—
115#13093957C2318yescomplemented
116#13093955C2318yescomplemented
117#13093949C2318yescomplemented
118#13093949C2318yescomplemented
119#13093947C2318yescomplemented
120#13093947C2318yescomplemented
121(C2×D47)⋊C8C2318yes—
122(C2×D47)⋊C8C2318yes—
123C23×C4×C47—21no—
124C23×C4×C47C2311yescharacteristic
125C2×M16×C47—24no—
126C2×M16×C47C2314yes—
127C22×C8×C47C2312yes—
128#117×C47—22no—
129#117×C47C2314yes—
130C2×C4⋊C4×C47—21no—
131C2×C4⋊C4×C47—22no—
132C2×C4⋊C4×C47C2312yes—
133C2×C4⋊C4×C47C2314yes—
134C2×C4⋊C4×C47C2311yescharacteristic
135C2×C4⋊C4×C47C2311yescharacteristic
136C2×C22⋊C4×C47—21no—
137C2×C42×C47—21no—
138#107×C47C2318yescomplemented
139#106×C47C2318yescomplemented
140C24×C47C2×D411yescharacteristic
141C23×D47—24no—
142C23×D47—24no—
143C23×D47—42no—
144C23×D47C2×D411yescharacteristic
145C23×D47C2×D414yes—
146C23×D47C2×D411yescharacteristic
147C22×C4×C47—21no—
148C22×C4×C47—21no—
149C22×C4×C47—24no—
150C22×C4×C47—22no—
151C22×C4×C47—44no—
152C22×C4×C47C2×D411yescharacteristic
153C22×C4×C47C2×D412yes—
154C22×C4×C47#4114yes—
155C22×C4×C47#4114yes—
156C22×C4×C47C2411yescharacteristic
157C22×C4×C47C2412yes—
158C22×Dic47—24no—
159C22×Dic47—21no—
160C22×Dic47—21no—
161C22×Dic47—24no—
162C22×Dic47—22no—
163C22×Dic47—44no—
164C22×Dic47C2×D411yescharacteristic
165C22×Dic47C2×D411yescharacteristic
166C22×Dic47C2×D414yes—
167C2×C4×D47—24no—
168C2×C4×D47—216no—
169C2×C4×D47—24no—
170C2×C4×D47—22no—
171C2×C4×D47—22no—
172C2×C4×D47—28no—
173C2×C4×D47—22no—
174C2×C4×D47—22no—
175C2×C4×D47—28no—
176C2×C4×D47—24no—
177C2×C4×D47—24no—
178C2×C4×D47—44no—
179C2×C4×D47—44no—
180C2×C4×D47C2×D414yes—
181C2×C4×D47C2×D412yes—
182C2×C4×D47C2×D414yes—
183C2×C4×D47C2×D412yes—
184C2×C4×D47C2412yes—
185C2×C4×D47C2412yes—
186C2×Dic94—24no—
187C2×Dic94—28no—
188C2×Dic94—24no—
189C2×Dic94—28no—
190C2×Dic94—24no—
191C2×Dic94—22no—
192C2×Dic94—48no—
193C2×Dic94—44no—
194C2×Dic94C2414yes—
195M16×C47—28no—
196C2×C8×C47—24no—
197C2×C8×C47C2414yes—
198C4⋊C4×C47—24no—
199C4⋊C4×C47—22no—
200C4⋊C4×C47—21no—
201C4⋊C4×C47—21no—
202C4⋊C4×C47C2412yes—
203C4⋊C4×C47C2414yes—
204C4⋊C4×C47C2412yes—
205C22⋊C4×C47—24no—
206C42×C47—24no—
207C22⋊Dic47—24no—
208(C2×D47)⋊C4—24no—
209(C2×D47)⋊C4—24no—
210(C2×D47)⋊C4—28no—
211(C2×D47)⋊C4—28no—
212C4⋊Dic47—21no—
213C4⋊Dic47—21no—
214C4⋊Dic47—28no—
215C4⋊Dic47—28no—
216C4⋊Dic47C2412yes—
217C4⋊Dic47C2412yes—
218Dic47⋊C4—24no—
219Dic47⋊C4—24no—
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—
243C2×C4×C47—22no—
244C2×C4×C47—22no—
245C2×C4×C47—24no—
246C2×C4×C47—48no—
247C2×C4×C47—42no—
248C2×C4×C47—42no—
249C2×C4×C47—44no—
250C2×C4×C47#12318yes—
251C2×C4×C47#12314yes—
252C2×C4×C47C22×D412yes—
253C2×C4×C47C22×D411yescharacteristic
254C2×C4×C47C2×#4112yes—
255C2×C4×C47C2×#4114yes—
256C2×C4×C47C2×#4112yes—
257C2×C4×C47C2×#4114yes—
258C2×C4×C47C2511yescharacteristic
259C2×Dic47—28no—
260C2×Dic47—22no—
261C2×Dic47—22no—
262C2×Dic47—22no—
263C2×Dic47—22no—
264C2×Dic47—24no—
265C2×Dic47—48no—
266C2×Dic47—44no—
267C2×Dic47—42no—
268C2×Dic47—42no—
269C2×Dic47C22×D412yes—
270C2×Dic47C22×D412yes—
271C4×D47—28no—
272C4×D47—24no—
273C4×D47—28no—
274C4×D47—24no—
275C4×D47—22no—
276C4×D47—22no—
277C4×D47—22no—
278C4×D47—22no—
279C4×D47—48no—
280C4×D47—48no—
281Dic94—24no—
282Dic94—24no—
283Dic94—48no—
284Dic94—44no—
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—42no—
306C4×C47—42no—
307C4×C47—44no—
308C4×C47C2×#12314yes—
309C4×C47C2×#12312yes—
310C4×C47C22×#4111yescharacteristic
311C4×C47C22×#4111yescharacteristic
312Dic47—22no—
313Dic47—22no—
314Dic47—44no—
315Dic47—42no—
316Dic47—42no—
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×#428D4711yescharacteristic, complemented
326C2×#419—471no—
327C2×#419—471no—
328C2×#407—471no—
329C2×C47—24no—
330C2×C47#309514yescomplemented
331C2×C47C22×#12311yescharacteristic
332C2×C47C22×#13711yescharacteristic
333C2×C47C2×#57511yescharacteristic
334D47—24no—
335D47—42no—
336C23×Q8—471no—
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—941no—
357C22×C4⋊C4C2×D4711yescharacteristic
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#428C2×D4718yescomplemented
367C2×#107—472no—
368C2×#107C2×D4712yescomplemented
369C2×#106—472no—
370C2×#106C2×D4712yescomplemented
371#419—478no—
372#419—478no—
373C2×#103—472no—
374C2×#103—472no—
375C2×#103—472no—
376C2×#103—472no—
377#407—478no—
378C2×C22⋊C8—472no—
379C2×C22⋊C8—472no—
380C47C2×#309511yescharacteristic, complemented, Sylow
381C25—471no—
382C22×Q8—472no—
383C22×Q8—474no—
384C22×Q8—942no—
385C22×Q8—941no—
386C22×Q8—944no—
387C22×Q8—944no—
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—22no—
425C2×C4⋊C4—21no—
426C2×C4⋊C4—471no—
427C2×C4⋊C4—471no—
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⋊C4C22×D4711yescharacteristic
438C2×C4⋊C4C22×D4711yescharacteristic
439C2×C4⋊C4C22×D4712yes—
440C2×C4⋊C4C22×D4714yes—
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#107C22×D4718yescomplemented
456#106—478no—
457#106C22×D4718yescomplemented
458#103—478no—
459#103—478no—
460#103—478no—
461#103—478no—
462C22⋊C8—478no—
463C22⋊C8—478no—
464C24—471no—
465C24—471no—
466C24—474no—
467C24—944no—
468C24—944no—
469C24—1882no—
470C24D4×D4711yescharacteristic
471C2×Q8—474no—
472C2×Q8—944no—
473C2×Q8—942no—
474C2×Q8—944no—
475C2×Q8—948no—
476C2×Q8—944no—
477C2×Q8—948no—
478C2×Q8—1888no—
479C2×Q8—1884no—
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#1062535414yes—
517C22×C4C23×D4712yes—
518C22×C4C23×D4711yescharacteristic
519M16—28no—
520M16—948no—
521C2×C8—24no—
522C2×C8—474no—
523C2×C8—944no—
524C2×C8C23×D4714yes—
525C4⋊C4—24no—
526C4⋊C4—22no—
527C4⋊C4—21no—
528C4⋊C4—21no—
529C4⋊C4—472no—
530C4⋊C4—472no—
531C4⋊C4—944no—
532C4⋊C4—944no—
533C4⋊C4—948no—
534C4⋊C4—948no—
535C4⋊C4—941no—
536C4⋊C4—941no—
537C4⋊C4C23×D4712yes—
538C4⋊C4C23×D4712yes—
539C4⋊C4C23×D4714yes—
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#1309400914yes—
565C23C2×D4×D4712yes—
566C23C2×D4×D4711yescharacteristic
567Q8—944no—
568Q8—944no—
569Q8—1888no—
570Q8—1884no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—24no—
574C2×C4—48no—
575C2×C4—42no—
576C2×C4—44no—
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—1884no—
598C2×C4—1882no—
599C2×C4—1882no—
600C2×C4#1309400918yes—
601C2×C4#1309400914yes—
602C2×C4C2×D4×D4712yes—
603C2×C4C2×D4×D4711yescharacteristic
604C2×C4C2×#1062535012yes—
605C2×C4C2×#1062535014yes—
606C2×C4C2×#1062535412yes—
607C2×C4C2×#1062535414yes—
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×#1309400914yes—
624C22C2×#1309400912yes—
625C22#137×D4712yes—
626C22#1568991912yes—
627C22C22×D4×D4711yescharacteristic
628C4—42no—
629C4—44no—
630C4—42no—
631C4—942no—
632C4—942no—
633C4—1882no—
634C4—1882no—
635C4—1884no—
636C4C2×#1309400914yes—
637C4C2×#1309400912yes—
638C4C22×#1062535011yescharacteristic
639C4C22×#1062535411yescharacteristic
640C2—24no—
641C2—944no—
642C2—1882no—
643C2#2025333814yesdirect product factor
644C2C22×#1309400911yescharacteristic
645C2C2×#137×D4711yescharacteristic
646C2C2×#1568991911yescharacteristic
subgroupG/H
centerC23C2×D4×D47
commutator, cosocleC2×C4×C47C25
FrattiniC2×C4C24×D47
FittingC2×C47×#428C2
socleC23×C47C22×D4
prime radicalC24×D47D4
hypercenterC2×#428D47
hypocenterC47C2×#3095
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
3. #20253338C2direct (N × H)
4. C2×#15689460C2semidirect
5. C2×#15689467C2semidirect
6. C2×#15689466C2semidirect
7. C2×#15689455C2semidirect
8. C2×#15689163C2semidirect
9. C2×#15689133C2semidirect
10. C2×#15689131C2semidirect
12. C2×C47×#428C2semidirect
35. #15689467C22semidirect
36. #15689466C22semidirect
37. #15689460C22semidirect
38. #15689455C22semidirect
39. #15689163C22semidirect
40. C2×#13093957C22semidirect
41. C2×#13093955C22semidirect
42. #15689133C22semidirect
43. #15689131C22semidirect
44. C2×#13093949C22semidirect
45. C2×#13093949C22semidirect
46. C2×#13093947C22semidirect
47. C2×#13093947C22semidirect
54. C47×#428C22semidirect
55. C2×#107×C47C22semidirect
56. C2×#106×C47C22semidirect
115. #13093957C23semidirect
116. #13093955C23semidirect
117. #13093949C23semidirect
118. #13093949C23semidirect
119. #13093947C23semidirect
120. #13093947C23semidirect
138. #107×C47C23semidirect
139. #106×C47C23semidirect
325. C2×#428D47semidirect
330. C2×C47#3095semidirect
366. #428C2×D47semidirect
368. C2×#107C2×D47semidirect
370. C2×#106C2×D47semidirect
380. C47C2×#3095semidirect
455. #107C22×D47semidirect
457. #106C22×D47semidirect
643. C2#20253338direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#130940042
C2×#117×D472
#202533382
C2×#156894602
C2×#156894672
C2×#156894662
C2×#156894552
C2×#156891632
C2×#156891332
C2×#156891312
C2×#156890862
C2×C47×#4282
C2×#309547