Finite Group Databasebeta

← C2×#20246815 #26384691

Subgroups — C2×#20246815 #26384691

Order 12032.

3526 conjugacy classes of subgroups.

The subgroup lattice has 652 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#13094004C211yesmaximal, characteristic, complemented
2C22×#13094004C211yesmaximal, characteristic, complemented
3C22×C4×D188C211yesmaximal, characteristic
4C22×#13093982C211yesmaximal, characteristic, complemented
5#20246815C2116yesmaximal, direct product factor
6C2×#15689050C212yesmaximal, complemented
7C2×#15689040C212yesmaximal
8C2×#15689029C211yesmaximal, characteristic, complemented
9C2×#15689022C212yesmaximal, complemented
10C2×#15689015C212yesmaximal, complemented
11C2×#15688980C211yesmaximal, characteristic, complemented
12C2×C47×#389C211yesmaximal, characteristic, complemented
13C23×Dic94C2211yescharacteristic
14C23×D188C2211yescharacteristic
15C23×C4×D47C2211yescharacteristic
16C23×C4×D47C2211yescharacteristic
17C2×#13094004—28no—
18C2×#13094004—28no—
19C2×#13094004C2218yescomplemented
20C2×#13094004C2218yescomplemented
21C2×C4×D188—28no—
22C2×C4×D188C2218yes—
23C2×#13093982—28no—
24C2×#13093982C2218yescomplemented
25C22×C4⋊Dic47C2211yescharacteristic
26C22×C4⋊Dic47C2211yescharacteristic
27C22×C4⋊Dic47C2212yes—
28C22×C4⋊Dic47C2212yes—
29C22×(C2×D47)⋊C4C2212yes—
30C22×(C2×D47)⋊C4C2211yescharacteristic
31C22×(C2×D47)⋊C4C2211yescharacteristic
32C22×(C2×D47)⋊C4C2212yes—
33C22×(C2×D47)⋊C4C2212yes—
34C22×(C2×D47)⋊C4C2212yes—
35C22×C4×Dic47C2212yes—
36#15689050C22116yescomplemented
37#15689040C22116yes—
38#15689029C2218yescomplemented
39#15689022C22116yescomplemented
40#15689015C22116yescomplemented
41#15688980C2218yescomplemented
42C2×C47⋊#95C2212yes—
43C2×C47⋊#95C2212yes—
44C2×C47⋊#95C2212yes—
45C22×C4⋊C4×C47C2212yes—
46C22×C4⋊C4×C47C2211yescharacteristic
47C22×C4⋊C4×C47C2211yescharacteristic
48C22×C42×C47C2211yescharacteristic
49C47×#389C2218yescomplemented
50C2×#95×C47C2212yes—
51C24×D47C2311yescharacteristic
52C24×D47C2311yescharacteristic
53C22×Dic94—24no—
54C22×Dic94—24no—
55C22×Dic94—24no—
56C22×Dic94C2314yes—
57C22×D188—24no—
58C22×D188—24no—
59C22×D188—24no—
60C22×D188C2314yes—
61C23×Dic47C2312yes—
62C23×Dic47C2312yes—
63C23×Dic47C2311yescharacteristic
64C23×Dic47C2311yescharacteristic
65C22×C4×D47—24no—
66C22×C4×D47—24no—
67C22×C4×D47—24no—
68C22×C4×D47—24no—
69C22×C4×D47—24no—
70C22×C4×D47D418yes—
71C22×C4×D47C2314yes—
72C22×C4×D47C2314yes—
73#13094004—232no—
74#13094004—232no—
75C4×D188—232no—
76#13093982—232no—
77C2×C4⋊Dic47—24no—
78C2×C4⋊Dic47—28no—
79C2×C4⋊Dic47—24no—
80C2×C4⋊Dic47—28no—
81C2×C4⋊Dic47C2314yes—
82C2×C4⋊Dic47C2318yes—
83C2×C4⋊Dic47C2314yes—
84C2×C4⋊Dic47C2318yes—
85C2×(C2×D47)⋊C4—24no—
86C2×(C2×D47)⋊C4—28no—
87C2×(C2×D47)⋊C4—28no—
88C2×(C2×D47)⋊C4—28no—
89C2×(C2×D47)⋊C4—28no—
90C2×(C2×D47)⋊C4—24no—
91C2×(C2×D47)⋊C4C2318yes—
92C2×(C2×D47)⋊C4C2314yes—
93C2×(C2×D47)⋊C4C2318yes—
94C2×(C2×D47)⋊C4C2318yes—
95C2×(C2×D47)⋊C4C2318yes—
96C2×(C2×D47)⋊C4C2314yes—
97C2×C4×Dic47—28no—
98C2×C4×Dic47C2318yes—
99C47⋊#95C2318yes—
100C47⋊#95C2318yes—
101C47⋊#95C2318yes—
102C23×C4×C47C2311yescharacteristic
103C23×C4×C47C2312yes—
104C23×C4×C47C2312yes—
105C23×C4×C47C2311yescharacteristic
106C23×C4×C47C2311yescharacteristic
107C2×C4⋊C4×C47—24no—
108C2×C4⋊C4×C47—28no—
109C2×C4⋊C4×C47D418yescomplemented
110C2×C4⋊C4×C47C2314yes—
111C2×C4⋊C4×C47C2318yes—
112C2×C4⋊C4×C47C2314yes—
113C2×C42×C47—24no—
114C2×C42×C47C2314yes—
115#95×C47C2318yes—
116C24×C47C2411yescharacteristic
117C23×D47—22no—
118C23×D47—22no—
119C23×D47—22no—
120C23×D47—22no—
121C23×D47—22no—
122C23×D47—22no—
123C23×D47—22no—
124C23×D47—22no—
125C23×D47—22no—
126C23×D47—22no—
127C23×D47—22no—
128C23×D47—22no—
129C23×D47C2×D414yes—
130C23×D47#4114yes—
131C23×D47C2412yes—
132C23×D47C2412yes—
133C22×C4×C47—22no—
134C22×C4×C47—24no—
135C22×C4×C47—22no—
136C22×C4×C47—24no—
137C22×C4×C47—22no—
138C22×C4×C47—24no—
139C22×C4×C47—24no—
140C22×C4×C47—24no—
141C22×C4×C47—22no—
142C22×C4×C47—22no—
143C22×C4×C47C2×D418yes—
144C22×C4×C47C2×D414yes—
145C22×C4×C47C2×D414yes—
146C22×C4×C47#4114yes—
147C22×C4×C47#4114yes—
148C22×C4×C47C2412yes—
149C22×C4×C47C2414yes—
150C22×C4×C47C2414yes—
151C22×C4×C47C2412yes—
152C22×C4×C47C2412yes—
153C22×Dic47—22no—
154C22×Dic47—22no—
155C22×Dic47—24no—
156C22×Dic47—22no—
157C22×Dic47—24no—
158C22×Dic47—24no—
159C22×Dic47—22no—
160C22×Dic47—24no—
161C22×Dic47—22no—
162C22×Dic47—24no—
163C22×Dic47—24no—
164C22×Dic47C2×D414yes—
165C22×Dic47C2414yes—
166C22×Dic47C2414yes—
167C22×Dic47C2412yes—
168C22×Dic47C2412yes—
169C2×D188—232no—
170C2×D188—28no—
171C2×D188—28no—
172C2×D188—28no—
173C2×C4×D47—232no—
174C2×C4×D47—28no—
175C2×C4×D47—28no—
176C2×C4×D47—28no—
177C2×C4×D47—28no—
178C2×C4×D47—28no—
179C2×C4×D47—416no—
180C2×C4×D47C2×D4116yes—
181C2×Dic94—28no—
182C2×Dic94—28no—
183C2×Dic94—28no—
184C2×Dic94—416no—
185C4⋊C4×C47—28no—
186C4⋊C4×C47—216no—
187C4⋊C4×C47C2×D4116yescomplemented
188C42×C47—28no—
189(C2×D47)⋊C4—28no—
190(C2×D47)⋊C4—216no—
191(C2×D47)⋊C4—216no—
192(C2×D47)⋊C4—216no—
193(C2×D47)⋊C4—216no—
194(C2×D47)⋊C4—28no—
195C4⋊Dic47—28no—
196C4⋊Dic47—216no—
197C4⋊Dic47—28no—
198C4⋊Dic47—216no—
199C4×Dic47—216no—
200C23×C47C22×D412yes—
201C23×C47C22×D412yes—
202C23×C47C2×#4112yes—
203C23×C47C2×#4112yes—
204C23×C47C2×#4112yes—
205C23×C472−1+412yes—
206C23×C472−1+412yes—
207C23×C47C2511yescharacteristic
208C22×D47—28no—
209C22×D47—28no—
210C22×D47—28no—
211C22×D47—28no—
212C22×D47—28no—
213C22×D47—28no—
214C22×D47—22no—
215C22×D47—22no—
216C22×D47—22no—
217C22×D47—22no—
218C22×D47—22no—
219C22×D47—22no—
220C22×D47—22no—
221C22×D47—22no—
222C22×D47—22no—
223C22×D47—22no—
224C22×D47—22no—
225C22×D47—22no—
226C22×D47—44no—
227C22×D47—44no—
228C22×D47—44no—
229C22×D47—44no—
230C22×D47—44no—
231C22×D47—44no—
232C22×D47—44no—
233C22×D47—44no—
234C22×D47C22×D414yes—
235C22×D47C2×#4114yes—
236C2×C4×C47—28no—
237C2×C4×C47—216no—
238C2×C4×C47—22no—
239C2×C4×C47—24no—
240C2×C4×C47—22no—
241C2×C4×C47—24no—
242C2×C4×C47—24no—
243C2×C4×C47—22no—
244C2×C4×C47—24no—
245C2×C4×C47—24no—
246C2×C4×C47—22no—
247C2×C4×C47—22no—
248C2×C4×C47—48no—
249C2×C4×C47—44no—
250C2×C4×C47#121116yes—
251C2×C4×C47C22×D414yes—
252C2×C4×C47C22×D418yes—
253C2×C4×C47C22×D414yes—
254C2×C4×C47C2×#4114yes—
255C2×C4×C47C2×#4114yes—
256C2×Dic47—28no—
257C2×Dic47—22no—
258C2×Dic47—22no—
259C2×Dic47—24no—
260C2×Dic47—22no—
261C2×Dic47—24no—
262C2×Dic47—24no—
263C2×Dic47—22no—
264C2×Dic47—24no—
265C2×Dic47—24no—
266C2×Dic47—22no—
267C2×Dic47—24no—
268C2×Dic47—48no—
269C2×Dic47—44no—
270C2×Dic47—48no—
271C2×Dic47C22×D414yes—
272D188—232no—
273C4×D47—232no—
274C4×D47—416no—
275Dic94—416no—
276C2×#2415—471nomaximal, Sylow
277C22×C47C2×#12114yes—
278C22×C47#52514yes—
279C22×C47#53614yes—
280C22×C47#54714yes—
281C22×C47#54814yes—
282C22×C47#55214yes—
283C22×C47#55614yes—
284C22×C47C23×D411yescharacteristic
285C22×C47C23×D411yescharacteristic
286C22×C47C22×#4111yescharacteristic
287C22×C47C22×#4111yescharacteristic
288C22×C47C22×#4111yescharacteristic
289C22×C47C2×2−1+411yescharacteristic
290C22×C47C2×2−1+411yescharacteristic
291C2×D47—24no—
292C2×D47—24no—
293C2×D47—24no—
294C2×D47—24no—
295C2×D47—24no—
296C2×D47—24no—
297C2×D47—42no—
298C2×D47—416no—
299C2×D47—416no—
300C2×D47—42no—
301C2×D47—42no—
302C2×D47—42no—
303C2×D47—42no—
304C2×D47—42no—
305C2×D47—42no—
306C2×D47—42no—
307C4×C47—24no—
308C4×C47—28no—
309C4×C47—44no—
310C4×C47—42no—
311C4×C47C2×#12118yes—
312Dic47—24no—
313Dic47—44no—
314Dic47—42no—
315Dic47—44no—
316C22×C22⋊Q8—471no—
317C22×C22⋊Q8—471no—
318C22×C4×D4—471no—
319C22×#117—471no—
320#2415—4716no—
321C2×#397—472no—
322C2×#394—472no—
323C2×#391—471no—
324C2×#389—472no—
325C2×#389D4711yescharacteristic, complemented
326C2×#388—472no—
327C2×#380—471no—
328C2×C47#241518yescomplemented
329C2×C47C22×#12111yescharacteristic
330C2×C47C2×#52511yescharacteristic
331C2×C47C2×#53611yescharacteristic
332C2×C47C2×#54711yescharacteristic
333C2×C47C2×#54811yescharacteristic
334C2×C47C2×#55211yescharacteristic
335C2×C47C2×#55611yescharacteristic
336D47—44no—
337D47—44no—
338C23×Q8—471no—
339C23×D4—471no—
340C24×C4—471no—
341C24×C4—471no—
342C2×C22⋊Q8—478no—
343C2×C22⋊Q8—478no—
344C2×C22⋊Q8—948no—
345C2×C22⋊Q8—948no—
346C2×C4×D4—478no—
347C2×C4×D4—948no—
348C2×#117—478no—
349C2×#117—948no—
350C22×C4⋊C4—472no—
351C22×C4⋊C4—471no—
352C22×C4⋊C4—471no—
353C22×C4⋊C4—472no—
354C22×C4⋊C4C2×D4712yes—
355C22×C4⋊C4C2×D4711yescharacteristic
356C22×C4⋊C4C2×D4711yescharacteristic
357C22×C22⋊C4—472no—
358C22×C22⋊C4—472no—
359C22×C22⋊C4—472no—
360C22×C22⋊C4—472no—
361C22×C22⋊C4—471no—
362C22×C22⋊C4—471no—
363C22×C42—472no—
364C22×C42C2×D4711yescharacteristic
365#397—4716no—
366#394—4716no—
367#391—478no—
368#389—4716no—
369#389C2×D4718yescomplemented
370#388—4716no—
371#380—478no—
372C2×#95—472no—
373C2×#95—472no—
374C2×#95—472no—
375C2×#95C2×D4712yes—
376C47C2×#241511yescharacteristic, complemented, Sylow
377C25—471no—
378C25—471no—
379C22×Q8—474no—
380C22×Q8—944no—
381C22×Q8—944no—
382C22×Q8—944no—
383C22×D4—474no—
384C22×D4—944no—
385C22×D4—944no—
386C22×D4—944no—
387C23×C4—471no—
388C23×C4—478no—
389C23×C4—474no—
390C23×C4—472no—
391C23×C4—472no—
392C23×C4—471no—
393C23×C4—474no—
394C23×C4—944no—
395C23×C4—944no—
396C23×C4—944no—
397C23×C4—944no—
398C23×C4—944no—
399C23×C4C22×D4712yes—
400C23×C4C22×D4711yescharacteristic
401C23×C4C22×D4712yes—
402C23×C4C22×D4711yescharacteristic
403C23×C4C22×D4711yescharacteristic
404C22⋊Q8—9432no—
405C22⋊Q8—9432no—
406C4×D4—9432no—
407#117—9432no—
408C2×C4⋊C4—24no—
409C2×C4⋊C4—28no—
410C2×C4⋊C4—474no—
411C2×C4⋊C4—474no—
412C2×C4⋊C4—478no—
413C2×C4⋊C4—478no—
414C2×C4⋊C4—944no—
415C2×C4⋊C4—948no—
416C2×C4⋊C4—944no—
417C2×C4⋊C4—948no—
418C2×C4⋊C4D18818yescomplemented
419C2×C4⋊C4C22×D4714yes—
420C2×C4⋊C4C22×D4718yes—
421C2×C4⋊C4C22×D4714yes—
422C2×C22⋊C4—478no—
423C2×C22⋊C4—474no—
424C2×C22⋊C4—474no—
425C2×C22⋊C4—478no—
426C2×C22⋊C4—478no—
427C2×C22⋊C4—478no—
428C2×C22⋊C4—944no—
429C2×C22⋊C4—944no—
430C2×C22⋊C4—948no—
431C2×C22⋊C4—948no—
432C2×C22⋊C4—948no—
433C2×C22⋊C4—948no—
434C2×C42—24no—
435C2×C42—478no—
436C2×C42—948no—
437C2×C42C22×D4714yes—
438#95—478no—
439#95—478no—
440#95—478no—
441#95C22×D4718yes—
442C24—474no—
443C24—474no—
444C24—472no—
445C24—472no—
446C24—942no—
447C24—942no—
448C24—942no—
449C24—942no—
450C24—942no—
451C24—942no—
452C24—942no—
453C24—942no—
454C24—942no—
455C24—942no—
456C24—942no—
457C24—942no—
458C24C23×D4711yescharacteristic
459C2×Q8—948no—
460C2×Q8—948no—
461C2×Q8—948no—
462C2×Q8—18816no—
463C2×D4—948no—
464C2×D4—948no—
465C2×D4—9432no—
466C2×D4—948no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—24no—
470C22×C4—24no—
471C22×C4—22no—
472C22×C4—24no—
473C22×C4—24no—
474C22×C4—24no—
475C22×C4—22no—
476C22×C4—22no—
477C22×C4—474no—
478C22×C4—472no—
479C22×C4—4716no—
480C22×C4—474no—
481C22×C4—474no—
482C22×C4—472no—
483C22×C4—942no—
484C22×C4—942no—
485C22×C4—9432no—
486C22×C4—948no—
487C22×C4—948no—
488C22×C4—942no—
489C22×C4—944no—
490C22×C4—944no—
491C22×C4—944no—
492C22×C4—942no—
493C22×C4—944no—
494C22×C4—942no—
495C22×C4—944no—
496C22×C4—944no—
497C22×C4—948no—
498C22×C4—948no—
499C22×C4—948no—
500C22×C4—18816no—
501C22×C4C2×D18818yes—
502C22×C4C2×D18814yes—
503C22×C4D4×D4714yes—
504C22×C4#1062535214yes—
505C22×C4#1062535414yes—
506C22×C4C23×D4712yes—
507C22×C4C23×D4714yes—
508C22×C4C23×D4714yes—
509C22×C4C23×D4712yes—
510C22×C4C23×D4712yes—
511C4⋊C4—28no—
512C4⋊C4—216no—
513C4⋊C4—948no—
514C4⋊C4—9416no—
515C4⋊C4—948no—
516C4⋊C4—9416no—
517C4⋊C4C2×D188116yescomplemented
518C22⋊C4—948no—
519C22⋊C4—9416no—
520C22⋊C4—948no—
521C22⋊C4—9416no—
522C22⋊C4—9416no—
523C22⋊C4—9416no—
524C42—28no—
525C42—9416no—
526C23—474no—
527C23—474no—
528C23—948no—
529C23—948no—
530C23—948no—
531C23—948no—
532C23—948no—
533C23—942no—
534C23—942no—
535C23—942no—
536C23—942no—
537C23—942no—
538C23—942no—
539C23—942no—
540C23—942no—
541C23—948no—
542C23—942no—
543C23—942no—
544C23—942no—
545C23—942no—
546C23—1884no—
547C23—1884no—
548C23—1884no—
549C23—1884no—
550C23—1884no—
551C23—1884no—
552C23—1884no—
553C23—1884no—
554C23C2×D4×D4712yes—
555C23C22×D18812yes—
556C23#41×D4712yes—
557C23C2×#1062535212yes—
558C23C2×#1062535412yes—
559C23#1309407112yes—
560C23#1309407212yes—
561C23C24×D4711yescharacteristic
562Q8—18816no—
563D4—9432no—
564C2×C4—216no—
565C2×C4—28no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—24no—
569C2×C4—24no—
570C2×C4—22no—
571C2×C4—24no—
572C2×C4—24no—
573C2×C4—24no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—48no—
577C2×C4—44no—
578C2×C4—474no—
579C2×C4—948no—
580C2×C4—942no—
581C2×C4—942no—
582C2×C4—9432no—
583C2×C4—942no—
584C2×C4—944no—
585C2×C4—944no—
586C2×C4—944no—
587C2×C4—942no—
588C2×C4—944no—
589C2×C4—942no—
590C2×C4—944no—
591C2×C4—944no—
592C2×C4—1888no—
593C2×C4—1884no—
594C2×C4—1888no—
595C2×C4—18816no—
596C2×C4#13093999116yes—
597C2×C4C2×D4×D4714yes—
598C2×C4C22×D18818yes—
599C2×C4C22×D18814yes—
600C2×C4C2×#1062535214yes—
601C2×C4C2×#1062535414yes—
602C22—944no—
603C22—944no—
604C22—944no—
605C22—944no—
606C22—944no—
607C22—944no—
608C22—18816no—
609C22—18816no—
610C22—1882no—
611C22—1882no—
612C22—1882no—
613C22—1882no—
614C22—1882no—
615C22—1882no—
616C22—1882no—
617C22—1882no—
618C22C2×#1309399914yes—
619C22#1568963314yes—
620C22#1568969614yes—
621C22#1568977714yes—
622C22#1568978514yes—
623C22#1568981114yes—
624C22#1568984714yes—
625C22C22×D4×D4711yescharacteristic
626C22C23×D18811yescharacteristic
627C22C2×#41×D4711yescharacteristic
628C22C22×#1062535211yescharacteristic
629C22C22×#1062535411yescharacteristic
630C22C2×#1309407111yescharacteristic
631C22C2×#1309407211yescharacteristic
632C4—28no—
633C4—24no—
634C4—44no—
635C4—42no—
636C4—944no—
637C4—1884no—
638C4—1882no—
639C4—1884no—
640C4C2×#1309399918yes—
641C2—1884no—
642C2—1884no—
643C2#2024681518yesdirect product factor
644C2C22×#1309399911yescharacteristic
645C2C2×#1568963311yescharacteristic
646C2C2×#1568969611yescharacteristic
647C2C2×#1568977711yescharacteristic
648C2C2×#1568978511yescharacteristic
649C2C2×#1568981111yescharacteristic
650C2C2×#1568984711yescharacteristic
subgroupG/H
centerC24C23×D47
commutator, cosocleC23×C47C25
FrattiniC23C24×D47
FittingC2×C47×#389C2
socleC24×C47C24
prime radicalC23×D188C22
hypercenterC2×#389D47
hypocenterC47C2×#2415
LCSG ▷ C23×C47 ▷ C47
UCSC1 ◁ C24 ◁ C2×#389

Complements

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

NH ≅ G/Ntype
1. C22×#13094004C2semidirect
2. C22×#13094004C2semidirect
4. C22×#13093982C2semidirect
5. #20246815C2direct (N × H)
6. C2×#15689050C2semidirect
8. C2×#15689029C2semidirect
9. C2×#15689022C2semidirect
10. C2×#15689015C2semidirect
11. C2×#15688980C2semidirect
12. C2×C47×#389C2semidirect
19. C2×#13094004C22semidirect
20. C2×#13094004C22semidirect
24. C2×#13093982C22semidirect
36. #15689050C22semidirect
38. #15689029C22semidirect
39. #15689022C22semidirect
40. #15689015C22semidirect
41. #15688980C22semidirect
49. C47×#389C22semidirect
109. C2×C4⋊C4×C47D4semidirect
187. C4⋊C4×C47C2×D4semidirect
325. C2×#389D47semidirect
328. C2×C47#2415semidirect
369. #389C2×D47semidirect
376. C47C2×#2415semidirect
418. C2×C4⋊C4D188semidirect
517. C4⋊C4C2×D188semidirect
643. C2#20246815direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#130940042
C22×C4×D1882
C22×#130939822
#202468152
C2×#156890502
C2×#156890402
C2×#156890292
C2×#156890222
C2×#156890152
C2×#156889802
C2×C47×#3892
C2×#241547