Finite Group Databasebeta

← C2×#20248336 #26388054

Subgroups — C2×#20248336 #26388054

Order 12032.

4006 conjugacy classes of subgroups.

The subgroup lattice has 588 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#13094023C211yesmaximal, characteristic, complemented
2C22×#13094002C212yesmaximal, complemented
3C22×#13094001C212yesmaximal, complemented
4C22×(C22⋊D47)⋊C4C211yesmaximal, characteristic
5#20248336C2116yesmaximal, direct product factor
6C2×#15689060C211yesmaximal, characteristic, complemented
7C2×#15689059C211yesmaximal, characteristic, complemented
8C2×#15689058C211yesmaximal, characteristic, complemented
9C2×#15689023C212yesmaximal
10C2×#15688992C212yesmaximal, complemented
11C2×#15688974C211yesmaximal, characteristic, complemented
12C2×C47×#399C211yesmaximal, characteristic, complemented
13C23×Dic94C2212yes—
14C23×C22⋊D47C2211yescharacteristic
15C24×Dic47C2211yescharacteristic
16C23×C4×D47C2211yescharacteristic
17C2×#13094023—28no—
18C2×#13094023C2218yescomplemented
19C2×#13094002—216no—
20C2×#13094002C22116yescomplemented
21C2×#13094001—216no—
22C2×#13094001C22116yescomplemented
23C2×(C22⋊D47)⋊C4—28no—
24C2×(C22⋊D47)⋊C4C2218yes—
25C22×C4⋊Dic47C2212yes—
26C22×C4⋊Dic47C2211yescharacteristic
27C22×Dic47⋊C4C2211yescharacteristic
28C22×Dic47⋊C4C2212yes—
29C22×Dic47⋊C4C2212yes—
30C22×Dic47⋊C4C2212yes—
31C22×Dic47⋊C4C2212yes—
32C22×C22⋊Dic47C2212yes—
33C22×(C2×D47)⋊C4C2211yescharacteristic
34C22×(C2×D47)⋊C4C2212yes—
35C22×(C2×D47)⋊C4C2212yes—
36C22×C4×Dic47C2212yes—
37C22×C4×Dic47C2211yescharacteristic
38#15689060C2218yescomplemented
39#15689059C2218yescomplemented
40#15689058C2218yescomplemented
41#15689023C22116yes—
42#15688992C22116yescomplemented
43#15688974C2218yescomplemented
44C2×C47⋊#95C2211yescharacteristic
45C22×C4⋊C4×C47C2212yes—
46C22×C4⋊C4×C47C2211yescharacteristic
47C22×C22⋊C4×C47C2211yescharacteristic
48C22×C22⋊C4×C47C2212yes—
49C47×#399C2218yescomplemented
50C2×#95×C47C2211yescharacteristic
51C24×D47C2311yescharacteristic
52C22×Dic94—28no—
53C22×Dic94—28no—
54C22×Dic94—28no—
55C22×Dic94C2318yes—
56C22×C22⋊D47—24no—
57C22×C22⋊D47—24no—
58C22×C22⋊D47Q818yescomplemented
59C22×C22⋊D47C2314yes—
60C23×Dic47—24no—
61C23×Dic47—28no—
62C23×Dic47C2312yes—
63C23×Dic47C2311yescharacteristic
64C23×Dic47C2314yes—
65C23×Dic47C2312yes—
66C23×Dic47C2311yescharacteristic
67C23×Dic47C2311yescharacteristic
68C22×C4×D47—24no—
69C22×C4×D47—28no—
70C22×C4×D47C2314yes—
71#13094023—232no—
72#13094002—264no—
73#13094001—264no—
74(C22⋊D47)⋊C4—232no—
75C2×C4⋊Dic47—28no—
76C2×C4⋊Dic47—24no—
77C2×C4⋊Dic47C2318yes—
78C2×C4⋊Dic47C2314yes—
79C2×Dic47⋊C4—24no—
80C2×Dic47⋊C4—28no—
81C2×Dic47⋊C4—28no—
82C2×Dic47⋊C4—28no—
83C2×Dic47⋊C4D4116yescomplemented
84C2×Dic47⋊C4C2314yes—
85C2×Dic47⋊C4C2318yes—
86C2×Dic47⋊C4C2318yes—
87C2×Dic47⋊C4C2318yes—
88C2×Dic47⋊C4C2318yes—
89C2×C22⋊Dic47—28no—
90C2×C22⋊Dic47C2318yes—
91C2×(C2×D47)⋊C4—24no—
92C2×(C2×D47)⋊C4—28no—
93C2×(C2×D47)⋊C4—28no—
94C2×(C2×D47)⋊C4C2318yes—
95C2×(C2×D47)⋊C4C2318yes—
96C2×(C2×D47)⋊C4C2314yes—
97C2×C4×Dic47—28no—
98C2×C4×Dic47D418yes—
99C2×C4×Dic47C2318yes—
100C2×C4×Dic47C2314yes—
101C47⋊#95C2314yes—
102C25×C47C2311yescharacteristic
103C23×C4×C47C2312yes—
104C23×C4×C47C2311yescharacteristic
105C23×C4×C47C2311yescharacteristic
106C23×C4×C47C2312yes—
107C2×C4⋊C4×C47—28no—
108C2×C4⋊C4×C47—24no—
109C2×C4⋊C4×C47C2318yes—
110C2×C4⋊C4×C47C2314yes—
111C2×C22⋊C4×C47—24no—
112C2×C22⋊C4×C47—28no—
113C2×C22⋊C4×C47C2318yes—
114C2×C22⋊C4×C47C2314yes—
115#95×C47C2314yes—
116C24×C47—24no—
117C24×C47—22no—
118C24×C47—22no—
119C24×C47—24no—
120C24×C47C2×Q814yes—
121C24×C47C2412yes—
122C24×C47C2411yescharacteristic
123C23×D47—24no—
124C23×D47—22no—
125C23×D47—22no—
126C23×D47—24no—
127C23×D47C2×Q814yes—
128C23×D47C2412yes—
129C22×C4×C47—24no—
130C22×C4×C47—24no—
131C22×C4×C47—24no—
132C22×C4×C47—22no—
133C22×C4×C47—24no—
134C22×C4×C47—24no—
135C22×C4×C47—24no—
136C22×C4×C47—24no—
137C22×C4×C47C2×D414yes—
138C22×C4×C47C2×D418yes—
139C22×C4×C47C2414yes—
140C22×C4×C47C2412yes—
141C22×C4×C47C2412yes—
142C22×C4×C47C2414yes—
143C2×C22⋊D47—232no—
144C2×C22⋊D47—28no—
145C2×C22⋊D47—28no—
146C2×C22⋊D47C2×Q8116yescomplemented
147C22×Dic47—24no—
148C22×Dic47—24no—
149C22×Dic47—22no—
150C22×Dic47—24no—
151C22×Dic47—24no—
152C22×Dic47—24no—
153C22×Dic47—24no—
154C22×Dic47—28no—
155C22×Dic47—216no—
156C22×Dic47—416no—
157C22×Dic47C2×D418yes—
158C22×Dic47C2×D414yes—
159C22×Dic47C2×D414yes—
160C22×Dic47C2×D418yes—
161C22×Dic47C2×Q814yes—
162C22×Dic47#4114yes—
163C22×Dic47C2414yes—
164C22×Dic47C2412yes—
165C22×Dic47C2414yes—
166C22×Dic47C2412yes—
167C22×Dic47C2412yes—
168C2×C4×D47—28no—
169C2×C4×D47—216no—
170C2×C4×D47—416no—
171C2×Dic94—216no—
172C2×Dic94—216no—
173C2×Dic94—216no—
174C2×Dic94—432no—
175C4⋊C4×C47—216no—
176C4⋊C4×C47—28no—
177C22⋊C4×C47—28no—
178C22⋊C4×C47—216no—
179C22⋊Dic47—216no—
180(C2×D47)⋊C4—28no—
181(C2×D47)⋊C4—216no—
182(C2×D47)⋊C4—216no—
183C4⋊Dic47—216no—
184C4⋊Dic47—28no—
185Dic47⋊C4—28no—
186Dic47⋊C4—216no—
187Dic47⋊C4—216no—
188Dic47⋊C4—216no—
189Dic47⋊C4C2×D4132yescomplemented
190C4×Dic47—216no—
191C4×Dic47C2×D4116yes—
192C23×C47—28no—
193C23×C47—216no—
194C23×C47—24no—
195C23×C47—22no—
196C23×C47—22no—
197C23×C47—24no—
198C23×C47—44no—
199C23×C47—44no—
200C23×C47—48no—
201C23×C47C22×D412yes—
202C23×C47C22×D414yes—
203C23×C47C22×Q814yes—
204C23×C47C22×Q812yes—
205C23×C47C2×#4112yes—
206C23×C472−1+414yes—
207C23×C47C2511yescharacteristic
208C22×D47—28no—
209C22×D47—216no—
210C22×D47—24no—
211C22×D47—22no—
212C22×D47—22no—
213C22×D47—24no—
214C22×D47—44no—
215C22×D47—44no—
216C22×D47—48no—
217C22×D47C22×Q814yes—
218C2×C4×C47—28no—
219C2×C4×C47—216no—
220C2×C4×C47—24no—
221C2×C4×C47—24no—
222C2×C4×C47—24no—
223C2×C4×C47—24no—
224C2×C4×C47—22no—
225C2×C4×C47—24no—
226C2×C4×C47—24no—
227C2×C4×C47—24no—
228C2×C4×C47—48no—
229C2×C4×C47—44no—
230C2×C4×C47C22×D414yes—
231C2×C4×C47C22×D418yes—
232C22⋊D47—232no—
233C2×Dic47—216no—
234C2×Dic47—24no—
235C2×Dic47—24no—
236C2×Dic47—24no—
237C2×Dic47—24no—
238C2×Dic47—24no—
239C2×Dic47—22no—
240C2×Dic47—24no—
241C2×Dic47—48no—
242C2×Dic47—44no—
243C2×Dic47—416no—
244C2×Dic47C22⋊Q8116yescomplemented
245C2×Dic47#127116yes—
246C2×Dic47C22×D418yes—
247C2×Dic47C22×D414yes—
248C2×Dic47C22×D414yes—
249C2×Dic47C22×D418yes—
250C2×Dic47C22×Q814yes—
251C2×Dic47C2×#4114yes—
252C4×D47—416no—
253Dic94—432no—
254C2×#2542—471nomaximal, Sylow
255C22×C47—24no—
256C22×C47—28no—
257C22×C47—42no—
258C22×C47—42no—
259C22×C47—44no—
260C22×C47—416no—
261C22×C47C2×C22⋊Q814yes—
262C22×C47C2×#12714yes—
263C22×C47#53614yes—
264C22×C47#54718yes—
265C22×C47D4×Q818yes—
266C22×C47C23×D411yescharacteristic
267C22×C47C23×D412yes—
268C22×C47C23×Q811yescharacteristic
269C22×C47C22×#4111yescharacteristic
270C22×C47C2×2−1+412yes—
271C2×D47—24no—
272C2×D47—28no—
273C2×D47—42no—
274C2×D47—42no—
275C2×D47—44no—
276C2×D47—416no—
277C4×C47—24no—
278C4×C47—28no—
279C4×C47—44no—
280C4×C47—42no—
281Dic47—28no—
282Dic47—44no—
283Dic47—42no—
284Dic47C2×C22⋊Q818yescomplemented
285Dic47C2×#12718yes—
286C22×C4⋊Q8—471no—
287C22×C22⋊Q8—472no—
288C22×C22⋊Q8—472no—
289C22×C4×D4—471no—
290#2542—4716no—
291C2×#399—471no—
292C2×#399—471no—
293C2×#399—471no—
294C2×#399D4711yescharacteristic, complemented
295C2×#390—472no—
296C2×#384—472no—
297C2×C4×C4⋊C4—471no—
298C2×C47—44no—
299C2×C47#254218yescomplemented
300C2×C47C22×C22⋊Q811yescharacteristic
301C2×C47C22×#12711yescharacteristic
302C2×C47C2×#53611yescharacteristic
303C2×C47C2×#54712yes—
304C2×C47C2×D4×Q812yes—
305D47—44no—
306C23×Q8—472no—
307C23×D4—471no—
308C24×C4—471no—
309C24×C4—471no—
310C2×C4⋊Q8—478no—
311C2×C4⋊Q8—948no—
312C2×C22⋊Q8—4716no—
313C2×C22⋊Q8—4716no—
314C2×C22⋊Q8—9416no—
315C2×C22⋊Q8—9416no—
316C2×C4×D4—478no—
317C2×C4×D4—948no—
318C22×C4⋊C4—472no—
319C22×C4⋊C4—471no—
320C22×C4⋊C4—472no—
321C22×C4⋊C4—472no—
322C22×C4⋊C4—472no—
323C22×C4⋊C4—471no—
324C22×C4⋊C4—472no—
325C22×C4⋊C4C2×D4711yescharacteristic
326C22×C4⋊C4C2×D4712yes—
327C22×C22⋊C4—472no—
328C22×C22⋊C4—471no—
329C22×C22⋊C4—472no—
330C22×C22⋊C4—472no—
331C22×C22⋊C4C2×D4711yescharacteristic
332C22×C22⋊C4C2×D4712yes—
333C22×C42—471no—
334C22×C42—472no—
335#399—478no—
336#399—478no—
337#399—478no—
338#399C2×D4718yescomplemented
339#390—4716no—
340#384—4716no—
341C4×C4⋊C4—478no—
342C2×#95—471no—
343C2×#95C2×D4711yescharacteristic
344C47C2×#254211yescharacteristic, complemented, Sylow
345C25—471no—
346C25C22×D4711yescharacteristic
347C22×Q8—478no—
348C22×Q8—948no—
349C22×Q8—948no—
350C22×Q8—948no—
351C22×D4—478no—
352C22×D4—474no—
353C22×D4—944no—
354C22×D4—944no—
355C23×C4—471no—
356C23×C4—471no—
357C23×C4—472no—
358C23×C4—474no—
359C23×C4—472no—
360C23×C4—471no—
361C23×C4—474no—
362C23×C4—944no—
363C23×C4—948no—
364C23×C4—944no—
365C23×C4—948no—
366C23×C4C22×D4712yes—
367C23×C4C22×D4711yescharacteristic
368C23×C4C22×D4712yes—
369C23×C4C22×D4711yescharacteristic
370C4⋊Q8—9432no—
371C22⋊Q8—9464no—
372C22⋊Q8—9464no—
373C4×D4—9432no—
374C2×C4⋊C4—28no—
375C2×C4⋊C4—24no—
376C2×C4⋊C4—4716no—
377C2×C4⋊C4—478no—
378C2×C4⋊C4—474no—
379C2×C4⋊C4—478no—
380C2×C4⋊C4—478no—
381C2×C4⋊C4—478no—
382C2×C4⋊C4—478no—
383C2×C4⋊C4—474no—
384C2×C4⋊C4—944no—
385C2×C4⋊C4—948no—
386C2×C4⋊C4—948no—
387C2×C4⋊C4—948no—
388C2×C4⋊C4—948no—
389C2×C4⋊C4—944no—
390C2×C4⋊C4C22×D4718yes—
391C2×C4⋊C4C22×D4714yes—
392C2×C22⋊C4—24no—
393C2×C22⋊C4—28no—
394C2×C22⋊C4—478no—
395C2×C22⋊C4—474no—
396C2×C22⋊C4—478no—
397C2×C22⋊C4—478no—
398C2×C22⋊C4—948no—
399C2×C22⋊C4—948no—
400C2×C22⋊C4—944no—
401C2×C22⋊C4—948no—
402C2×C22⋊C4C22×D4714yes—
403C2×C22⋊C4C22×D4718yes—
404C2×C42—478no—
405C2×C42—474no—
406C2×C42—478no—
407C2×C42—948no—
408#95—474no—
409#95C22×D4714yes—
410C24—24no—
411C24—22no—
412C24—22no—
413C24—24no—
414C24—474no—
415C24—472no—
416C24—944no—
417C24—942no—
418C24—942no—
419C24—944no—
420C24Q8×D4714yes—
421C24C23×D4712yes—
422C24C23×D4711yescharacteristic
423C2×Q8—9416no—
424C2×Q8—9416no—
425C2×Q8—9416no—
426C2×Q8—18832no—
427C2×D4—4716no—
428C2×D4—948no—
429C2×D4—9432no—
430C2×D4—948no—
431C22×C4—24no—
432C22×C4—24no—
433C22×C4—24no—
434C22×C4—22no—
435C22×C4—24no—
436C22×C4—24no—
437C22×C4—24no—
438C22×C4—24no—
439C22×C4—474no—
440C22×C4—474no—
441C22×C4—478no—
442C22×C4—474no—
443C22×C4—474no—
444C22×C4—472no—
445C22×C4—472no—
446C22×C4—478no—
447C22×C4—474no—
448C22×C4—474no—
449C22×C4—472no—
450C22×C4—948no—
451C22×C4—9416no—
452C22×C4—948no—
453C22×C4—9416no—
454C22×C4—944no—
455C22×C4—944no—
456C22×C4—942no—
457C22×C4—944no—
458C22×C4—944no—
459C22×C4—944no—
460C22×C4—944no—
461C22×C4—18816no—
462C22×C4—18816no—
463C22×C4D4×D4714yes—
464C22×C4D4×D4718yes—
465C22×C4C23×D4714yes—
466C22×C4C23×D4712yes—
467C22×C4C23×D4712yes—
468C22×C4C23×D4714yes—
469C4⋊C4—216no—
470C4⋊C4—28no—
471C4⋊C4—4732no—
472C4⋊C4—948no—
473C4⋊C4—9416no—
474C4⋊C4—9416no—
475C4⋊C4—9416no—
476C4⋊C4—9416no—
477C4⋊C4—948no—
478C22⋊C4—28no—
479C22⋊C4—216no—
480C22⋊C4—9416no—
481C22⋊C4—9416no—
482C22⋊C4—948no—
483C22⋊C4—9416no—
484C42—4716no—
485C42—9416no—
486C23—216no—
487C23—28no—
488C23—24no—
489C23—22no—
490C23—22no—
491C23—24no—
492C23—48no—
493C23—44no—
494C23—44no—
495C23—474no—
496C23—9416no—
497C23—948no—
498C23—944no—
499C23—942no—
500C23—942no—
501C23—944no—
502C23—1888no—
503C23—1884no—
504C23—1884no—
505C23C2×D4×D4714yes—
506C23C2×D4×D4712yes—
507C23C2×Q8×D4714yes—
508C23C2×Q8×D4712yes—
509C23#41×D4712yes—
510C23#1309407114yes—
511C23C24×D4711yescharacteristic
512Q8—18832no—
513D4—9432no—
514C2×C4—216no—
515C2×C4—24no—
516C2×C4—28no—
517C2×C4—24no—
518C2×C4—24no—
519C2×C4—22no—
520C2×C4—24no—
521C2×C4—24no—
522C2×C4—24no—
523C2×C4—24no—
524C2×C4—44no—
525C2×C4—48no—
526C2×C4—4716no—
527C2×C4—4716no—
528C2×C4—474no—
529C2×C4—474no—
530C2×C4—478no—
531C2×C4—474no—
532C2×C4—474no—
533C2×C4—478no—
534C2×C4—9416no—
535C2×C4—944no—
536C2×C4—944no—
537C2×C4—942no—
538C2×C4—944no—
539C2×C4—944no—
540C2×C4—944no—
541C2×C4—944no—
542C2×C4—18816no—
543C2×C4—18816no—
544C2×C4—1884no—
545C2×C4—1888no—
546C2×C4C2×D4×D4714yes—
547C2×C4C2×D4×D4718yes—
548C22—28no—
549C22—24no—
550C22—416no—
551C22—44no—
552C22—42no—
553C22—42no—
554C22—948no—
555C22—944no—
556C22—18816no—
557C22—1884no—
558C22—1882no—
559C22—1882no—
560C22C22⋊Q8×D4714yes—
561C22#127×D4714yes—
562C22#1568969514yes—
563C22#1568977118yes—
564C22#1568978618yes—
565C22C22×D4×D4712yes—
566C22C22×D4×D4711yescharacteristic
567C22C22×Q8×D4711yescharacteristic
568C22C2×#41×D4711yescharacteristic
569C22C2×#1309407112yes—
570C4—28no—
571C4—24no—
572C4—42no—
573C4—44no—
574C4—478no—
575C4—478no—
576C4—948no—
577C4—1882no—
578C4—1884no—
579C2—44no—
580C2—1884no—
581C2#2024833618yesdirect product factor
582C2C2×C22⋊Q8×D4711yescharacteristic
583C2C2×#127×D4711yescharacteristic
584C2C2×#1568969511yescharacteristic
585C2C2×#1568977112yes—
586C2C2×#1568978612yes—
subgroupG/H
centerC24C23×D47
commutator, cosocleC23×C47C25
FrattiniC23C24×D47
FittingC2×C47×#399C2
socleC24×C47C24
prime radicalC23×C22⋊D47C22
hypercenterC2×#399D47
hypocenterC47C2×#2542
LCSG ▷ C23×C47 ▷ C47
UCSC1 ◁ C24 ◁ C2×#399

Complements

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

NH ≅ G/Ntype
1. C22×#13094023C2semidirect
2. C22×#13094002C2semidirect
3. C22×#13094001C2semidirect
5. #20248336C2direct (N × H)
6. C2×#15689060C2semidirect
7. C2×#15689059C2semidirect
8. C2×#15689058C2semidirect
10. C2×#15688992C2semidirect
11. C2×#15688974C2semidirect
12. C2×C47×#399C2semidirect
18. C2×#13094023C22semidirect
20. C2×#13094002C22semidirect
22. C2×#13094001C22semidirect
38. #15689060C22semidirect
39. #15689059C22semidirect
40. #15689058C22semidirect
42. #15688992C22semidirect
43. #15688974C22semidirect
49. C47×#399C22semidirect
58. C22×C22⋊D47Q8semidirect
83. C2×Dic47⋊C4D4semidirect
146. C2×C22⋊D47C2×Q8semidirect
189. Dic47⋊C4C2×D4semidirect
244. C2×Dic47C22⋊Q8semidirect
284. Dic47C2×C22⋊Q8semidirect
294. C2×#399D47semidirect
299. C2×C47#2542semidirect
338. #399C2×D47semidirect
344. C47C2×#2542semidirect
581. C2#20248336direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#130940232
C22×#130940022
C22×#130940012
C22×(C22⋊D47)⋊C42
#202483362
C2×#156890602
C2×#156890592
C2×#156890582
C2×#156890232
C2×#156889922
C2×#156889742
C2×C47×#3992
C2×#254247