Finite Group Databasebeta

← C2×#20247663 #26386046

Subgroups — C2×#20247663 #26386046

Order 12032.

4166 conjugacy classes of subgroups.

The subgroup lattice has 612 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#13093982C211yesmaximal, characteristic, complemented
2C22×C22⋊C4×D47C211yesmaximal, characteristic
3#20247663C2116yesmaximal, direct product factor
4C2×#15689058C211yesmaximal, characteristic, complemented
5C2×#15689050C211yesmaximal, characteristic, complemented
6C2×#15689048C211yesmaximal, characteristic, complemented
7C2×#15689017C212yesmaximal, complemented
8C2×#15689013C212yesmaximal, complemented
9C2×#15688994C212yesmaximal, complemented
10C2×#15688985C211yesmaximal, characteristic, complemented
11C2×#15688978C212yesmaximal
12C2×C47×#399C211yesmaximal, characteristic, complemented
13C25×D47C2211yescharacteristic
14C23×C4×D47C2211yescharacteristic
15C23×C4×D47C2211yescharacteristic
16C2×#13093982—28no—
17C2×#13093982C2218yescomplemented
18C22×C4⋊Dic47C2211yescharacteristic, complemented
19C22×Dic47⋊C4C2212yescomplemented
20C22×C22⋊Dic47C2211yescharacteristic
21C22×C22⋊Dic47C2212yes—
22C22×(C2×D47)⋊C4C2212yes—
23C22×(C2×D47)⋊C4C2211yescharacteristic
24C22×(C2×D47)⋊C4C2211yescharacteristic
25C22×(C2×D47)⋊C4C2212yes—
26C22×(C2×D47)⋊C4C2212yes—
27C22×(C2×D47)⋊C4C2212yes—
28C2×C22⋊C4×D47—28no—
29C2×C22⋊C4×D47C2218yes—
30C22×C4×Dic47C2212yescomplemented
31#15689058C2218yescomplemented
32#15689050C2218yescomplemented
33#15689048C2218yescomplemented
34#15689017C22116yescomplemented
35#15689013C22116yescomplemented
36#15688994C22116yescomplemented
37#15688985C2218yescomplemented
38#15688978C22116yes—
39C2×C47⋊#95C2212yescomplemented
40C2×C47⋊#95C2211yescharacteristic, complemented
41C2×C47⋊#95C2212yescomplemented
42C2×C47⋊#95C2212yescomplemented
43C22×C4⋊C4×C47C2211yescharacteristic, complemented
44C22×C4⋊C4×C47C2212yescomplemented
45C22×C22⋊C4×C47C2211yescharacteristic
46C22×C22⋊C4×C47C2212yes—
47C47×#399C2218yescomplemented
48C2×#95×C47C2211yescharacteristic, complemented
49C24×D47—24no—
50C24×D47—24no—
51C24×D47—24no—
52C24×D47—28no—
53C24×D47—44no—
54C24×D47C2311yescharacteristic
55C24×D47C2314yes—
56C24×D47C2311yescharacteristic
57C23×Dic47C2312yes—
58C23×Dic47C2312yes—
59C23×Dic47C2311yescharacteristic
60C23×Dic47C2311yescharacteristic
61C22×C4×D47—28no—
62C22×C4×D47—24no—
63C22×C4×D47—44no—
64C22×C4×D47D418yes—
65C22×C4×D47C2314yes—
66C22×C4×D47C2314yes—
67#13093982—232no—
68C2×C4⋊Dic47—24no—
69C2×C4⋊Dic47C2314yescomplemented
70C2×Dic47⋊C4—44no—
71C2×Dic47⋊C4C2318yescomplemented
72C2×C22⋊Dic47—28no—
73C2×C22⋊Dic47—24no—
74C2×C22⋊Dic47C2318yes—
75C2×C22⋊Dic47C2314yes—
76C2×(C2×D47)⋊C4—28no—
77C2×(C2×D47)⋊C4—28no—
78C2×(C2×D47)⋊C4—28no—
79C2×(C2×D47)⋊C4—24no—
80C2×(C2×D47)⋊C4—24no—
81C2×(C2×D47)⋊C4—44no—
82C2×(C2×D47)⋊C4C2318yes—
83C2×(C2×D47)⋊C4C2318yes—
84C2×(C2×D47)⋊C4C2318yes—
85C2×(C2×D47)⋊C4C2314yes—
86C2×(C2×D47)⋊C4C2314yes—
87C2×(C2×D47)⋊C4C2318yes—
88C22⋊C4×D47—232no—
89C2×C4×Dic47—28no—
90C2×C4×Dic47C2318yescomplemented
91C47⋊#95C2318yescomplemented
92C47⋊#95C2314yescomplemented
93C47⋊#95C2318yescomplemented
94C47⋊#95C2318yescomplemented
95C25×C47C2311yescharacteristic
96C23×C4×C47C2311yescharacteristic
97C23×C4×C47C2312yes—
98C23×C4×C47C2312yes—
99C23×C4×C47C2311yescharacteristic
100C2×C4⋊C4×C47—24no—
101C2×C4⋊C4×C47—44no—
102C2×C4⋊C4×C47C2314yescomplemented
103C2×C4⋊C4×C47C2318yescomplemented
104C2×C22⋊C4×C47—24no—
105C2×C22⋊C4×C47—44no—
106C2×C22⋊C4×C47C2314yes—
107C2×C22⋊C4×C47C2318yes—
108#95×C47C2314yescomplemented
109C24×C47—22no—
110C24×C47—22no—
111C24×C47—24no—
112C24×C47—24no—
113C24×C47#4114yes—
114C24×C47C2412yes—
115C24×C47C2411yescharacteristic
116C23×D47—22no—
117C23×D47—22no—
118C23×D47—24no—
119C23×D47—232no—
120C23×D47—22no—
121C23×D47—28no—
122C23×D47—22no—
123C23×D47—24no—
124C23×D47—24no—
125C23×D47—28no—
126C23×D47—28no—
127C23×D47—216no—
128C23×D47—48no—
129C23×D47—432no—
130C23×D47—432no—
131C23×D47—416no—
132C23×D47—416no—
133C23×D47C2×D414yes—
134C23×D47#4114yes—
135C23×D47#4118yes—
136C23×D47C2412yes—
137C23×D47C2412yes—
138C22×C4×C47—24no—
139C22×C4×C47—22no—
140C22×C4×C47—24no—
141C22×C4×C47—24no—
142C22×C4×C47—24no—
143C22×C4×C47—24no—
144C22×C4×C47—24no—
145C22×C4×C47—24no—
146C22×C4×C47—24no—
147C22×C4×C47C2×D414yes—
148C22×C4×C47C2412yes—
149C22×C4×C47C2414yes—
150C22×C4×C47C2414yes—
151C22×C4×C47C2412yes—
152C22×Dic47—24no—
153C22×Dic47—24no—
154C22×Dic47—22no—
155C22×Dic47—24no—
156C22×Dic47—24no—
157C22×Dic47—24no—
158C22×Dic47—24no—
159C22×Dic47—24no—
160C22×Dic47—24no—
161C22×Dic47C2×D414yes—
162C22×Dic47C2414yes—
163C22×Dic47C2414yes—
164C22×Dic47C2412yes—
165C22×Dic47C2412yes—
166C2×C4×D47—232no—
167C2×C4×D47—216no—
168C2×C4×D47—28no—
169C2×C4×D47—48no—
170C2×C4×D47—416no—
171C2×C4×D47C2×D4116yes—
172C4⋊C4×C47—28no—
173C4⋊C4×C47—48no—
174C22⋊C4×C47—28no—
175C22⋊C4×C47—48no—
176C22⋊Dic47—216no—
177C22⋊Dic47—28no—
178(C2×D47)⋊C4—216no—
179(C2×D47)⋊C4—216no—
180(C2×D47)⋊C4—216no—
181(C2×D47)⋊C4—28no—
182(C2×D47)⋊C4—28no—
183(C2×D47)⋊C4—48no—
184C4⋊Dic47—28no—
185Dic47⋊C4—48no—
186C4×Dic47—216no—
187C23×C47—28no—
188C23×C47—216no—
189C23×C47—22no—
190C23×C47—22no—
191C23×C47—24no—
192C23×C47—24no—
193C23×C47—44no—
194C23×C47—44no—
195C23×C47—48no—
196C23×C47C22×D412yes—
197C23×C47C2×#4114yes—
198C23×C47C2×#4112yes—
199C23×C47C2×#4114yes—
200C23×C47C2×#4112yes—
201C23×C47H(5,𝔽2)14yes—
202C23×C47C2511yescharacteristic
203C22×D47—28no—
204C22×D47—28no—
205C22×D47—216no—
206C22×D47—216no—
207C22×D47—22no—
208C22×D47—22no—
209C22×D47—24no—
210C22×D47—232no—
211C22×D47—28no—
212C22×D47—216no—
213C22×D47—22no—
214C22×D47—24no—
215C22×D47—22no—
216C22×D47—24no—
217C22×D47—432no—
218C22×D47—432no—
219C22×D47—416no—
220C22×D47—4128no—
221C22×D47—44no—
222C22×D47—44no—
223C22×D47—48no—
224C22×D47—416no—
225C22×D47#123116yes—
226C22×D47C22×D414yes—
227C22×D47C2×#4114yes—
228C22×D47C2×#4118yes—
229C2×C4×C47—28no—
230C2×C4×C47—24no—
231C2×C4×C47—24no—
232C2×C4×C47—22no—
233C2×C4×C47—24no—
234C2×C4×C47—24no—
235C2×C4×C47—24no—
236C2×C4×C47—24no—
237C2×C4×C47—24no—
238C2×C4×C47—24no—
239C2×C4×C47—44no—
240C2×C4×C47—48no—
241C2×C4×C47—48no—
242C2×C4×C47C22×D414yes—
243C2×Dic47—28no—
244C2×Dic47—24no—
245C2×Dic47—24no—
246C2×Dic47—22no—
247C2×Dic47—24no—
248C2×Dic47—24no—
249C2×Dic47—24no—
250C2×Dic47—24no—
251C2×Dic47—24no—
252C2×Dic47—24no—
253C2×Dic47—48no—
254C2×Dic47—44no—
255C2×Dic47—48no—
256C2×Dic47C22×D414yes—
257C4×D47—232no—
258C4×D47—416no—
259C2×#2482—471nomaximal, Sylow
260C22×C47—24no—
261C22×C47—28no—
262C22×C47—42no—
263C22×C47—42no—
264C22×C47—44no—
265C22×C47—416no—
266C22×C47C2×#12314yes—
267C22×C47#52514yes—
268C22×C47#53514yes—
269C22×C47#55118yes—
270C22×C47#55118yes—
271C22×C47C23×D411yescharacteristic
272C22×C47C22×#4112yes—
273C22×C47C22×#4111yescharacteristic
274C22×C47C22×#4111yescharacteristic
275C22×C47C2×H(5,𝔽2)12yes—
276C2×D47—24no—
277C2×D47—24no—
278C2×D47—28no—
279C2×D47—28no—
280C2×D47—232no—
281C2×D47—24no—
282C2×D47—28no—
283C2×D47—464no—
284C2×D47—42no—
285C2×D47—42no—
286C2×D47—44no—
287C2×D47—416no—
288C2×D47C2×#12318yes—
289C4×C47—24no—
290C4×C47—42no—
291C4×C47—44no—
292C4×C47—44no—
293Dic47—24no—
294Dic47—44no—
295Dic47—42no—
296Dic47—44no—
297C22×#117—471no—
298C23×C22⋊C4—471no—
299#2482—4716no—
300C2×#399—471no—
301C2×#399D4711yescharacteristic, complemented
302C2×#397—471no—
303C2×#397—471no—
304C2×#388—472no—
305C2×#388—472no—
306C2×#385—472no—
307C2×#381—471no—
308C2×#379—472no—
309C2×C47—44no—
310C2×C47#248218yescomplemented
311C2×C47C22×#12311yescharacteristic
312C2×C47C2×#52511yescharacteristic
313C2×C47C2×#53511yescharacteristic
314C2×C47C2×#55112yes—
315C2×C47C2×#55112yes—
316D47—28no—
317D47—44no—
318C26—471no—
319C24×C4—471no—
320C24×C4—471no—
321C2×#117—478no—
322C2×#117—948no—
323C22×C4⋊C4—471no—
324C22×C4⋊C4—472no—
325C22×C4⋊C4C2×D4711yescharacteristic, complemented
326C22×C4⋊C4C2×D4712yescomplemented
327C22×C22⋊C4—472no—
328C22×C22⋊C4—471no—
329C22×C22⋊C4—471no—
330C22×C22⋊C4—472no—
331C22×C22⋊C4—471no—
332C22×C22⋊C4—472no—
333C22×C22⋊C4—472no—
334C22×C22⋊C4—472no—
335C22×C22⋊C4—478no—
336C22×C22⋊C4—948no—
337C22×C22⋊C4C2×D4711yescharacteristic
338C22×C22⋊C4C2×D4712yes—
339C22×C42—472no—
340#399—478no—
341#399C2×D4718yescomplemented
342#397—478no—
343#397—478no—
344#388—4716no—
345#388—4716no—
346#385—4716no—
347#381—478no—
348#379—4716no—
349C2×#95—471no—
350C2×#95—472no—
351C2×#95—472no—
352C2×#95—472no—
353C2×#95C2×D4711yescharacteristic, complemented
354C47C2×#248211yescharacteristic, complemented, Sylow
355C25—471no—
356C25—471no—
357C25—474no—
358C25—944no—
359C25—944no—
360C25—948no—
361C25—944no—
362C25—1884no—
363C25C22×D4711yescharacteristic
364C23×C4—471no—
365C23×C4—478no—
366C23×C4—474no—
367C23×C4—472no—
368C23×C4—472no—
369C23×C4—471no—
370C23×C4—474no—
371C23×C4—948no—
372C23×C4—944no—
373C23×C4—1884no—
374C23×C4C22×D4712yes—
375C23×C4C22×D4712yes—
376C23×C4C22×D4711yescharacteristic
377C23×C4C22×D4711yescharacteristic
378#117—9432no—
379C2×C4⋊C4—24no—
380C2×C4⋊C4—44no—
381C2×C4⋊C4—474no—
382C2×C4⋊C4—478no—
383C2×C4⋊C4—944no—
384C2×C4⋊C4—1884no—
385C2×C4⋊C4C22×D4714yescomplemented
386C2×C4⋊C4C22×D4718yescomplemented
387C2×C22⋊C4—24no—
388C2×C22⋊C4—44no—
389C2×C22⋊C4—474no—
390C2×C22⋊C4—478no—
391C2×C22⋊C4—474no—
392C2×C22⋊C4—474no—
393C2×C22⋊C4—478no—
394C2×C22⋊C4—478no—
395C2×C22⋊C4—478no—
396C2×C22⋊C4—478no—
397C2×C22⋊C4—9432no—
398C2×C22⋊C4—944no—
399C2×C22⋊C4—948no—
400C2×C22⋊C4—948no—
401C2×C22⋊C4—948no—
402C2×C22⋊C4—944no—
403C2×C22⋊C4—948no—
404C2×C22⋊C4—944no—
405C2×C22⋊C4—1884no—
406C2×C22⋊C4C22×D4714yes—
407C2×C22⋊C4C22×D4718yes—
408C2×C42—478no—
409C2×C42—948no—
410#95—478no—
411#95—474no—
412#95—478no—
413#95—478no—
414#95C22×D4714yescomplemented
415C24—22no—
416C24—24no—
417C24—22no—
418C24—24no—
419C24—474no—
420C24—478no—
421C24—472no—
422C24—474no—
423C24—472no—
424C24—948no—
425C24—948no—
426C24—9416no—
427C24—9432no—
428C24—942no—
429C24—942no—
430C24—944no—
431C24—942no—
432C24—944no—
433C24—948no—
434C24—942no—
435C24—944no—
436C24—1888no—
437C24—18832no—
438C24—18816no—
439C24—18832no—
440C24—18816no—
441C24#1062535414yes—
442C24C23×D4712yes—
443C24C23×D4711yescharacteristic
444C22×C4—24no—
445C22×C4—24no—
446C22×C4—22no—
447C22×C4—24no—
448C22×C4—24no—
449C22×C4—24no—
450C22×C4—24no—
451C22×C4—24no—
452C22×C4—24no—
453C22×C4—474no—
454C22×C4—472no—
455C22×C4—4716no—
456C22×C4—474no—
457C22×C4—474no—
458C22×C4—472no—
459C22×C4—9432no—
460C22×C4—9416no—
461C22×C4—948no—
462C22×C4—944no—
463C22×C4—944no—
464C22×C4—942no—
465C22×C4—944no—
466C22×C4—944no—
467C22×C4—944no—
468C22×C4—944no—
469C22×C4—944no—
470C22×C4—944no—
471C22×C4—1888no—
472C22×C4—18816no—
473C22×C4D4×D4714yes—
474C22×C4C23×D4714yes—
475C22×C4C23×D4712yes—
476C22×C4C23×D4714yes—
477C22×C4C23×D4712yes—
478C4⋊C4—28no—
479C4⋊C4—48no—
480C4⋊C4—948no—
481C4⋊C4—1888no—
482C22⋊C4—28no—
483C22⋊C4—48no—
484C22⋊C4—948no—
485C22⋊C4—9416no—
486C22⋊C4—9416no—
487C22⋊C4—9416no—
488C22⋊C4—948no—
489C22⋊C4—9416no—
490C22⋊C4—948no—
491C22⋊C4—1888no—
492C42—9416no—
493C23—216no—
494C23—28no—
495C23—22no—
496C23—24no—
497C23—22no—
498C23—24no—
499C23—48no—
500C23—44no—
501C23—44no—
502C23—4716no—
503C23—474no—
504C23—478no—
505C23—474no—
506C23—9432no—
507C23—9416no—
508C23—9416no—
509C23—948no—
510C23—948no—
511C23—942no—
512C23—942no—
513C23—944no—
514C23—9416no—
515C23—948no—
516C23—942no—
517C23—944no—
518C23—942no—
519C23—944no—
520C23—18832no—
521C23—18816no—
522C23—18832no—
523C23—188128no—
524C23—18816no—
525C23—1888no—
526C23—1884no—
527C23—1884no—
528C23C2×D4×D4712yes—
529C23#41×D4714yes—
530C23#41×D4712yes—
531C23C2×#1062535414yes—
532C23C2×#1062535412yes—
533C23#1309406914yes—
534C23C24×D4711yescharacteristic
535C2×C4—28no—
536C2×C4—24no—
537C2×C4—24no—
538C2×C4—22no—
539C2×C4—24no—
540C2×C4—24no—
541C2×C4—24no—
542C2×C4—24no—
543C2×C4—24no—
544C2×C4—24no—
545C2×C4—48no—
546C2×C4—48no—
547C2×C4—44no—
548C2×C4—474no—
549C2×C4—9432no—
550C2×C4—948no—
551C2×C4—944no—
552C2×C4—944no—
553C2×C4—942no—
554C2×C4—944no—
555C2×C4—944no—
556C2×C4—944no—
557C2×C4—944no—
558C2×C4—944no—
559C2×C4—944no—
560C2×C4—18816no—
561C2×C4—1888no—
562C2×C4—1888no—
563C2×C4—1884no—
564C2×C4C2×D4×D4714yes—
565C22—28no—
566C22—24no—
567C22—42no—
568C22—416no—
569C22—44no—
570C22—42no—
571C22—478no—
572C22—9432no—
573C22—948no—
574C22—948no—
575C22—944no—
576C22—944no—
577C22—948no—
578C22—944no—
579C22—18864no—
580C22—1882no—
581C22—18816no—
582C22—1884no—
583C22—1882no—
584C22#123×D4714yes—
585C22#1568963614yes—
586C22#1568968614yes—
587C22#1568980018yes—
588C22#1568980118yes—
589C22C22×D4×D4711yescharacteristic
590C22C2×#41×D4712yes—
591C22C2×#41×D4711yescharacteristic
592C22C22×#1062535411yescharacteristic
593C22C2×#1309406912yes—
594C4—24no—
595C4—44no—
596C4—42no—
597C4—44no—
598C4—944no—
599C4—1884no—
600C4—1884no—
601C4—1882no—
602C2—44no—
603C2—948no—
604C2—1884no—
605C2#2024766318yesdirect product factor
606C2C2×#123×D4711yescharacteristic
607C2C2×#1568963611yescharacteristic
608C2C2×#1568968611yescharacteristic
609C2C2×#1568980012yes—
610C2C2×#1568980112yes—
subgroupG/H
centerC24C23×D47
commutator, cosocleC23×C47C25
FrattiniC23C24×D47
FittingC2×C47×#399C2
socleC24×C47C24
prime radicalC25×D47C22
hypercenterC2×#399D47
hypocenterC47C2×#2482
LCSG ▷ C23×C47 ▷ C47
UCSC1 ◁ C24 ◁ C2×#399

Complements

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

NH ≅ G/Ntype
1. C22×#13093982C2semidirect
3. #20247663C2direct (N × H)
4. C2×#15689058C2semidirect
5. C2×#15689050C2semidirect
6. C2×#15689048C2semidirect
7. C2×#15689017C2semidirect
8. C2×#15689013C2semidirect
9. C2×#15688994C2semidirect
10. C2×#15688985C2semidirect
12. C2×C47×#399C2semidirect
17. C2×#13093982C22semidirect
18. C22×C4⋊Dic47C22semidirect
19. C22×Dic47⋊C4C22semidirect
30. C22×C4×Dic47C22semidirect
31. #15689058C22semidirect
32. #15689050C22semidirect
33. #15689048C22semidirect
34. #15689017C22semidirect
35. #15689013C22semidirect
36. #15688994C22semidirect
37. #15688985C22semidirect
39. C2×C47⋊#95C22semidirect
40. C2×C47⋊#95C22semidirect
41. C2×C47⋊#95C22semidirect
42. C2×C47⋊#95C22semidirect
43. C22×C4⋊C4×C47C22semidirect
44. C22×C4⋊C4×C47C22semidirect
47. C47×#399C22semidirect
48. C2×#95×C47C22semidirect
69. C2×C4⋊Dic47C23semidirect
71. C2×Dic47⋊C4C23semidirect
90. C2×C4×Dic47C23semidirect
91. C47⋊#95C23semidirect
92. C47⋊#95C23semidirect
93. C47⋊#95C23semidirect
94. C47⋊#95C23semidirect
102. C2×C4⋊C4×C47C23semidirect
103. C2×C4⋊C4×C47C23semidirect
108. #95×C47C23semidirect
301. C2×#399D47semidirect
310. C2×C47#2482semidirect
325. C22×C4⋊C4C2×D47semidirect
326. C22×C4⋊C4C2×D47semidirect
341. #399C2×D47semidirect
353. C2×#95C2×D47semidirect
354. C47C2×#2482semidirect
385. C2×C4⋊C4C22×D47semidirect
386. C2×C4⋊C4C22×D47semidirect
414. #95C22×D47semidirect
605. C2#20247663direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#130939822
C22×C22⋊C4×D472
#202476632
C2×#156890582
C2×#156890502
C2×#156890482
C2×#156890172
C2×#156890132
C2×#156889942
C2×#156889852
C2×#156889782
C2×C47×#3992
C2×#248247