Finite Group Databasebeta

← C2×#20249194 #26382625

Subgroups — C2×#20249194 #26382625

Order 12032.

4142 conjugacy classes of subgroups.

The subgroup lattice has 624 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#13094004C211yesmaximal, characteristic, complemented
2C22×C22⋊C4×D47C211yesmaximal, characteristic
3#20249194C2116yesmaximal, direct product factor
4C2×#15689058C211yesmaximal, characteristic, complemented
5C2×#15689046C212yesmaximal, complemented
6C2×#15689045C212yesmaximal, complemented
7C2×#15689036C211yesmaximal, characteristic, complemented
8C2×#15689035C211yesmaximal, characteristic, complemented
9C2×#15688994C212yesmaximal, complemented
10C2×#15688982C211yesmaximal, characteristic, complemented
11C2×#15688978C212yesmaximal
12C2×C47×#399C211yesmaximal, characteristic, complemented
13C25×D47C2211yescharacteristic
14C23×Dic94C2211yescharacteristic, complemented
15C23×C4×D47C2211yescharacteristic
16C23×C4×D47C2211yescharacteristic
17C2×#13094004—28no—
18C2×#13094004C2218yescomplemented
19C22×C4⋊Dic47C2212yescomplemented
20C22×Dic47⋊C4C2212yescomplemented
21C22×Dic47⋊C4C2212yescomplemented
22C22×C22⋊Dic47C2211yescharacteristic
23C22×C22⋊Dic47C2212yes—
24C22×(C2×D47)⋊C4C2212yes—
25C22×(C2×D47)⋊C4C2211yescharacteristic
26C22×(C2×D47)⋊C4C2211yescharacteristic
27C22×(C2×D47)⋊C4C2212yes—
28C22×(C2×D47)⋊C4C2212yes—
29C22×(C2×D47)⋊C4C2212yes—
30C2×C22⋊C4×D47—28no—
31C2×C22⋊C4×D47C2218yes—
32#15689058C2218yescomplemented
33#15689046C22116yescomplemented
34#15689045C22116yescomplemented
35#15689036C2218yescomplemented
36#15689035C2218yescomplemented
37#15688994C22116yescomplemented
38#15688982C2218yescomplemented
39#15688978C22116yes—
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—44no—
52C24×D47—44no—
53C24×D47Q818yescomplemented
54C24×D47C2311yescharacteristic
55C24×D47C2311yescharacteristic
56C24×D47C2314yes—
57C22×Dic94—24no—
58C22×Dic94—44no—
59C22×Dic94C2314yescomplemented
60C23×Dic47C2312yes—
61C23×Dic47C2312yes—
62C23×Dic47C2311yescharacteristic
63C23×Dic47C2311yescharacteristic
64C22×C4×D47—24no—
65C22×C4×D47—44no—
66C22×C4×D47—44no—
67C22×C4×D47D418yes—
68C22×C4×D47C2314yes—
69C22×C4×D47C2314yes—
70#13094004—232no—
71C2×C4⋊Dic47—28no—
72C2×C4⋊Dic47C2318yescomplemented
73C2×Dic47⋊C4—44no—
74C2×Dic47⋊C4—44no—
75C2×Dic47⋊C4C2318yescomplemented
76C2×Dic47⋊C4C2318yescomplemented
77C2×C22⋊Dic47—24no—
78C2×C22⋊Dic47—44no—
79C2×C22⋊Dic47C2314yes—
80C2×C22⋊Dic47C2318yes—
81C2×(C2×D47)⋊C4—28no—
82C2×(C2×D47)⋊C4—28no—
83C2×(C2×D47)⋊C4—24no—
84C2×(C2×D47)⋊C4—24no—
85C2×(C2×D47)⋊C4—44no—
86C2×(C2×D47)⋊C4—44no—
87C2×(C2×D47)⋊C4C2318yes—
88C2×(C2×D47)⋊C4C2318yes—
89C2×(C2×D47)⋊C4C2314yes—
90C2×(C2×D47)⋊C4C2314yes—
91C2×(C2×D47)⋊C4C2318yes—
92C2×(C2×D47)⋊C4C2318yes—
93C22⋊C4×D47—232no—
94C47⋊#95C2314yescomplemented
95C47⋊#95C2318yescomplemented
96C47⋊#95C2318yescomplemented
97C25×C47C2311yescharacteristic
98C23×C4×C47C2311yescharacteristic
99C23×C4×C47C2312yes—
100C23×C4×C47C2312yes—
101C23×C4×C47C2311yescharacteristic
102C2×C4⋊C4×C47—24no—
103C2×C4⋊C4×C47—44no—
104C2×C4⋊C4×C47C2314yescomplemented
105C2×C4⋊C4×C47C2318yescomplemented
106C2×C22⋊C4×C47—24no—
107C2×C22⋊C4×C47—44no—
108C2×C22⋊C4×C47C2314yes—
109C2×C22⋊C4×C47C2318yes—
110#95×C47C2314yescomplemented
111C24×C47—22no—
112C24×C47—22no—
113C24×C47—24no—
114C24×C47—24no—
115C24×C47C2×Q814yes—
116C24×C47C2412yes—
117C24×C47C2411yescharacteristic
118C23×D47—24no—
119C23×D47—24no—
120C23×D47—22no—
121C23×D47—22no—
122C23×D47—24no—
123C23×D47—24no—
124C23×D47—232no—
125C23×D47—28no—
126C23×D47—28no—
127C23×D47—432no—
128C23×D47—48no—
129C23×D47—48no—
130C23×D47—416no—
131C23×D47—416no—
132C23×D47—432no—
133C23×D47C2×D414yes—
134C23×D47C2×Q814yes—
135C23×D47C2×Q814yes—
136C23×D47C2×Q8116yescomplemented
137C23×D47#4114yes—
138C23×D47C2412yes—
139C23×D47C2412yes—
140C22×C4×C47—24no—
141C22×C4×C47—22no—
142C22×C4×C47—24no—
143C22×C4×C47—24no—
144C22×C4×C47—24no—
145C22×C4×C47—24no—
146C22×C4×C47—24no—
147C22×C4×C47—24no—
148C22×C4×C47—24no—
149C22×C4×C47C2×D414yes—
150C22×C4×C47C2412yes—
151C22×C4×C47C2414yes—
152C22×C4×C47C2414yes—
153C22×C4×C47C2412yes—
154C22×Dic47—24no—
155C22×Dic47—22no—
156C22×Dic47—24no—
157C22×Dic47—24no—
158C22×Dic47—24no—
159C22×Dic47—24no—
160C22×Dic47—24no—
161C22×Dic47—24no—
162C22×Dic47—24no—
163C22×Dic47C2×D414yes—
164C22×Dic47C2414yes—
165C22×Dic47C2414yes—
166C22×Dic47C2412yes—
167C22×Dic47C2412yes—
168C2×C4×D47—28no—
169C2×C4×D47—416no—
170C2×C4×D47—48no—
171C2×C4×D47—48no—
172C2×C4×D47—416no—
173C2×C4×D47C2×D4116yes—
174C2×Dic94—28no—
175C2×Dic94—48no—
176C2×Dic94—416no—
177C4⋊C4×C47—28no—
178C4⋊C4×C47—48no—
179C22⋊C4×C47—28no—
180C22⋊C4×C47—48no—
181C22⋊Dic47—28no—
182C22⋊Dic47—48no—
183(C2×D47)⋊C4—216no—
184(C2×D47)⋊C4—216no—
185(C2×D47)⋊C4—28no—
186(C2×D47)⋊C4—28no—
187(C2×D47)⋊C4—48no—
188(C2×D47)⋊C4—48no—
189C4⋊Dic47—216no—
190Dic47⋊C4—48no—
191Dic47⋊C4—48no—
192C23×C47—28no—
193C23×C47—216no—
194C23×C47—22no—
195C23×C47—22no—
196C23×C47—24no—
197C23×C47—24no—
198C23×C47—44no—
199C23×C47—44no—
200C23×C47—48no—
201C23×C47C22×D412yes—
202C23×C47C22×Q814yes—
203C23×C47C22×Q812yes—
204C23×C47C2×#4112yes—
205C23×C47H(5,𝔽2)14yes—
206C23×C47H(5,𝔽2)14yes—
207C23×C47C2511yescharacteristic
208C22×D47—28no—
209C22×D47—28no—
210C22×D47—216no—
211C22×D47—216no—
212C22×D47—24no—
213C22×D47—24no—
214C22×D47—28no—
215C22×D47—216no—
216C22×D47—22no—
217C22×D47—22no—
218C22×D47—24no—
219C22×D47—24no—
220C22×D47—232no—
221C22×D47—432no—
222C22×D47—4128no—
223C22×D47—416no—
224C22×D47—416no—
225C22×D47—432no—
226C22×D47—44no—
227C22×D47—44no—
228C22×D47—48no—
229C22×D47C22⋊Q8116yes—
230C22×D47C22×D414yes—
231C22×D47C22×Q814yes—
232C22×D47C22×Q814yes—
233C22×D47C2×#4114yes—
234C2×C4×C47—28no—
235C2×C4×C47—24no—
236C2×C4×C47—24no—
237C2×C4×C47—22no—
238C2×C4×C47—24no—
239C2×C4×C47—24no—
240C2×C4×C47—24no—
241C2×C4×C47—24no—
242C2×C4×C47—24no—
243C2×C4×C47—24no—
244C2×C4×C47—44no—
245C2×C4×C47—48no—
246C2×C4×C47—48no—
247C2×C4×C47C22×D414yes—
248C2×Dic47—28no—
249C2×Dic47—24no—
250C2×Dic47—24no—
251C2×Dic47—22no—
252C2×Dic47—24no—
253C2×Dic47—24no—
254C2×Dic47—24no—
255C2×Dic47—24no—
256C2×Dic47—24no—
257C2×Dic47—24no—
258C2×Dic47—48no—
259C2×Dic47—44no—
260C2×Dic47—48no—
261C2×Dic47C22×D414yes—
262C4×D47—416no—
263C4×D47—416no—
264Dic94—416no—
265C2×#2626—471nomaximal, Sylow
266C22×C47—24no—
267C22×C47—28no—
268C22×C47—416no—
269C22×C47—42no—
270C22×C47—42no—
271C22×C47—44no—
272C22×C47C2×C22⋊Q814yes—
273C22×C47#53414yes—
274C22×C47#53514yes—
275C22×C47#53818yes—
276C22×C47#54318yes—
277C22×C47C23×D411yescharacteristic
278C22×C47C23×Q811yescharacteristic
279C22×C47C22×#4111yescharacteristic
280C22×C47C2×H(5,𝔽2)12yes—
281C22×C47C2×H(5,𝔽2)12yes—
282C2×D47—232no—
283C2×D47—24no—
284C2×D47—24no—
285C2×D47—28no—
286C2×D47—28no—
287C2×D47—24no—
288C2×D47—28no—
289C2×D47—464no—
290C2×D47—416no—
291C2×D47—42no—
292C2×D47—42no—
293C2×D47—44no—
294C2×D47C2×C22⋊Q818yes—
295C4×C47—24no—
296C4×C47—42no—
297C4×C47—44no—
298C4×C47—44no—
299Dic47—24no—
300Dic47—44no—
301Dic47—42no—
302Dic47—44no—
303C22×C22⋊Q8—471no—
304C23×C22⋊C4—471no—
305#2626—4716no—
306C2×#399—471no—
307C2×#399D4711yescharacteristic, complemented
308C2×#396—472no—
309C2×#396—472no—
310C2×#393—471no—
311C2×#393—471no—
312C2×#385—472no—
313C2×#380—471no—
314C2×#379—472no—
315C2×C47—44no—
316C2×C47#262618yescomplemented
317C2×C47C22×C22⋊Q811yescharacteristic
318C2×C47C2×#53411yescharacteristic
319C2×C47C2×#53511yescharacteristic
320C2×C47C2×#53812yes—
321C2×C47C2×#54312yes—
322D47—28no—
323D47—44no—
324C26—471no—
325C23×Q8—471no—
326C24×C4—471no—
327C24×C4—471no—
328C2×C22⋊Q8—478no—
329C2×C22⋊Q8—948no—
330C22×C4⋊C4—472no—
331C22×C4⋊C4—472no—
332C22×C4⋊C4—472no—
333C22×C4⋊C4C2×D4711yescharacteristic, complemented
334C22×C4⋊C4C2×D4712yescomplemented
335C22×C22⋊C4—471no—
336C22×C22⋊C4—471no—
337C22×C22⋊C4—471no—
338C22×C22⋊C4—472no—
339C22×C22⋊C4—472no—
340C22×C22⋊C4—472no—
341C22×C22⋊C4—472no—
342C22×C22⋊C4—472no—
343C22×C22⋊C4—478no—
344C22×C22⋊C4—948no—
345C22×C22⋊C4C2×D4711yescharacteristic
346C22×C22⋊C4C2×D4712yes—
347#399—478no—
348#399C2×D4718yescomplemented
349#396—4716no—
350#396—4716no—
351#393—478no—
352#393—478no—
353#385—4716no—
354#380—478no—
355#379—4716no—
356C2×#95—472no—
357C2×#95—472no—
358C2×#95—471no—
359C2×#95C2×D4711yescharacteristic, complemented
360C47C2×#262611yescharacteristic, complemented, Sylow
361C25—471no—
362C25—471no—
363C25—478no—
364C25—474no—
365C25—944no—
366C25—944no—
367C25—1884no—
368C25—1884no—
369C25C22×D4711yescharacteristic
370C22×Q8—474no—
371C22×Q8—944no—
372C22×Q8—1884no—
373C23×C4—471no—
374C23×C4—478no—
375C23×C4—474no—
376C23×C4—472no—
377C23×C4—472no—
378C23×C4—471no—
379C23×C4—474no—
380C23×C4—944no—
381C23×C4—1884no—
382C23×C4—1884no—
383C23×C4C22×D4712yes—
384C23×C4C22×D4712yes—
385C23×C4C22×D4711yescharacteristic
386C23×C4C22×D4711yescharacteristic
387C22⋊Q8—9432no—
388C2×C4⋊C4—24no—
389C2×C4⋊C4—44no—
390C2×C4⋊C4—478no—
391C2×C4⋊C4—478no—
392C2×C4⋊C4—478no—
393C2×C4⋊C4—948no—
394C2×C4⋊C4—1884no—
395C2×C4⋊C4—1884no—
396C2×C4⋊C4C22×D4714yescomplemented
397C2×C4⋊C4C22×D4718yescomplemented
398C2×C22⋊C4—24no—
399C2×C22⋊C4—44no—
400C2×C22⋊C4—474no—
401C2×C22⋊C4—474no—
402C2×C22⋊C4—474no—
403C2×C22⋊C4—478no—
404C2×C22⋊C4—478no—
405C2×C22⋊C4—478no—
406C2×C22⋊C4—478no—
407C2×C22⋊C4—478no—
408C2×C22⋊C4—9432no—
409C2×C22⋊C4—944no—
410C2×C22⋊C4—948no—
411C2×C22⋊C4—948no—
412C2×C22⋊C4—944no—
413C2×C22⋊C4—944no—
414C2×C22⋊C4—1884no—
415C2×C22⋊C4—1884no—
416C2×C22⋊C4—1884no—
417C2×C22⋊C4C22×D4714yes—
418C2×C22⋊C4C22×D4718yes—
419#95—474no—
420#95—478no—
421#95—478no—
422#95C22×D4714yescomplemented
423C24—22no—
424C24—22no—
425C24—24no—
426C24—24no—
427C24—474no—
428C24—474no—
429C24—474no—
430C24—472no—
431C24—474no—
432C24—472no—
433C24—4716no—
434C24—948no—
435C24—9432no—
436C24—948no—
437C24—944no—
438C24—944no—
439C24—942no—
440C24—942no—
441C24—944no—
442C24—944no—
443C24—18832no—
444C24—1888no—
445C24—1888no—
446C24—18816no—
447C24—18816no—
448C24—18832no—
449C24Q8×D4714yes—
450C24C23×D4712yes—
451C24C23×D4711yescharacteristic
452C2×Q8—948no—
453C2×Q8—1888no—
454C2×Q8—18816no—
455C22×C4—24no—
456C22×C4—22no—
457C22×C4—24no—
458C22×C4—24no—
459C22×C4—24no—
460C22×C4—24no—
461C22×C4—24no—
462C22×C4—24no—
463C22×C4—24no—
464C22×C4—474no—
465C22×C4—472no—
466C22×C4—4716no—
467C22×C4—474no—
468C22×C4—474no—
469C22×C4—472no—
470C22×C4—948no—
471C22×C4—944no—
472C22×C4—942no—
473C22×C4—944no—
474C22×C4—944no—
475C22×C4—944no—
476C22×C4—944no—
477C22×C4—944no—
478C22×C4—944no—
479C22×C4—944no—
480C22×C4—18816no—
481C22×C4—1888no—
482C22×C4—1888no—
483C22×C4—18816no—
484C22×C4D4×D4714yes—
485C22×C4C23×D4712yes—
486C22×C4C23×D4714yes—
487C22×C4C23×D4714yes—
488C22×C4C23×D4712yes—
489C4⋊C4—28no—
490C4⋊C4—48no—
491C4⋊C4—9416no—
492C4⋊C4—1888no—
493C4⋊C4—1888no—
494C22⋊C4—28no—
495C22⋊C4—48no—
496C22⋊C4—948no—
497C22⋊C4—9416no—
498C22⋊C4—9416no—
499C22⋊C4—948no—
500C22⋊C4—948no—
501C22⋊C4—1888no—
502C22⋊C4—1888no—
503C22⋊C4—1888no—
504C23—216no—
505C23—28no—
506C23—22no—
507C23—22no—
508C23—24no—
509C23—24no—
510C23—44no—
511C23—44no—
512C23—48no—
513C23—4716no—
514C23—474no—
515C23—474no—
516C23—474no—
517C23—474no—
518C23—9432no—
519C23—9416no—
520C23—948no—
521C23—948no—
522C23—944no—
523C23—944no—
524C23—9416no—
525C23—9416no—
526C23—948no—
527C23—942no—
528C23—942no—
529C23—944no—
530C23—944no—
531C23—18832no—
532C23—188128no—
533C23—18816no—
534C23—18816no—
535C23—18832no—
536C23—1884no—
537C23—1884no—
538C23—1888no—
539C23C2×D4×D4712yes—
540C23C2×Q8×D4714yes—
541C23C2×Q8×D4712yes—
542C23#41×D4712yes—
543C23#1309406914yes—
544C23#1309407014yes—
545C23C24×D4711yescharacteristic
546Q8—18816no—
547C2×C4—28no—
548C2×C4—24no—
549C2×C4—22no—
550C2×C4—24no—
551C2×C4—24no—
552C2×C4—24no—
553C2×C4—24no—
554C2×C4—24no—
555C2×C4—24no—
556C2×C4—24no—
557C2×C4—48no—
558C2×C4—44no—
559C2×C4—48no—
560C2×C4—474no—
561C2×C4—948no—
562C2×C4—944no—
563C2×C4—942no—
564C2×C4—944no—
565C2×C4—944no—
566C2×C4—944no—
567C2×C4—944no—
568C2×C4—944no—
569C2×C4—944no—
570C2×C4—944no—
571C2×C4—18816no—
572C2×C4—18816no—
573C2×C4—1888no—
574C2×C4—1884no—
575C2×C4—1888no—
576C2×C4C2×D4×D4714yes—
577C22—28no—
578C22—24no—
579C22—416no—
580C22—42no—
581C22—42no—
582C22—44no—
583C22—478no—
584C22—948no—
585C22—9432no—
586C22—944no—
587C22—944no—
588C22—948no—
589C22—948no—
590C22—944no—
591C22—18864no—
592C22—18816no—
593C22—1882no—
594C22—1882no—
595C22—1884no—
596C22C22⋊Q8×D4714yes—
597C22#1568968414yes—
598C22#1568968614yes—
599C22#1568970918yes—
600C22#1568974818yes—
601C22C22×D4×D4711yescharacteristic
602C22C22×Q8×D4711yescharacteristic
603C22C2×#41×D4711yescharacteristic
604C22C2×#1309406912yes—
605C22C2×#1309407012yes—
606C4—24no—
607C4—42no—
608C4—44no—
609C4—44no—
610C4—944no—
611C4—1884no—
612C4—1882no—
613C4—1884no—
614C2—44no—
615C2—948no—
616C2—1884no—
617C2#2024919418yesdirect product factor
618C2C2×C22⋊Q8×D4711yescharacteristic
619C2C2×#1568968411yescharacteristic
620C2C2×#1568968611yescharacteristic
621C2C2×#1568970912yes—
622C2C2×#1568974812yes—
subgroupG/H
centerC24C23×D47
commutator, cosocleC23×C47C25
FrattiniC23C24×D47
FittingC2×C47×#399C2
socleC24×C47C24
prime radicalC25×D47C22
hypercenterC2×#399D47
hypocenterC47C2×#2626
LCSG ▷ C23×C47 ▷ C47
UCSC1 ◁ C24 ◁ C2×#399

Complements

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

NH ≅ G/Ntype
1. C22×#13094004C2semidirect
3. #20249194C2direct (N × H)
4. C2×#15689058C2semidirect
5. C2×#15689046C2semidirect
6. C2×#15689045C2semidirect
7. C2×#15689036C2semidirect
8. C2×#15689035C2semidirect
9. C2×#15688994C2semidirect
10. C2×#15688982C2semidirect
12. C2×C47×#399C2semidirect
14. C23×Dic94C22semidirect
18. C2×#13094004C22semidirect
19. C22×C4⋊Dic47C22semidirect
20. C22×Dic47⋊C4C22semidirect
21. C22×Dic47⋊C4C22semidirect
32. #15689058C22semidirect
33. #15689046C22semidirect
34. #15689045C22semidirect
35. #15689036C22semidirect
36. #15689035C22semidirect
37. #15688994C22semidirect
38. #15688982C22semidirect
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
53. C24×D47Q8semidirect
59. C22×Dic94C23semidirect
72. C2×C4⋊Dic47C23semidirect
75. C2×Dic47⋊C4C23semidirect
76. C2×Dic47⋊C4C23semidirect
94. C47⋊#95C23semidirect
95. C47⋊#95C23semidirect
96. C47⋊#95C23semidirect
104. C2×C4⋊C4×C47C23semidirect
105. C2×C4⋊C4×C47C23semidirect
110. #95×C47C23semidirect
136. C23×D47C2×Q8semidirect
307. C2×#399D47semidirect
316. C2×C47#2626semidirect
333. C22×C4⋊C4C2×D47semidirect
334. C22×C4⋊C4C2×D47semidirect
348. #399C2×D47semidirect
359. C2×#95C2×D47semidirect
360. C47C2×#2626semidirect
396. C2×C4⋊C4C22×D47semidirect
397. C2×C4⋊C4C22×D47semidirect
422. #95C22×D47semidirect
617. C2#20249194direct (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×C22⋊C4×D472
#202491942
C2×#156890582
C2×#156890462
C2×#156890452
C2×#156890362
C2×#156890352
C2×#156889942
C2×#156889822
C2×#156889782
C2×C47×#3992
C2×#262647