Finite Group Databasebeta

← C2×#20247692 #26385326

Subgroups — C2×#20247692 #26385326

Order 12032.

3678 conjugacy classes of subgroups.

The subgroup lattice has 644 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#13094006C212yesmaximal
2C22×C4×C22⋊D47C211yesmaximal, characteristic
3C22×C4⋊C4×D47C211yesmaximal, characteristic, complemented
4#20247692C2116yesmaximal, direct product factor
5C2×#15689057C212yesmaximal, complemented
6C2×#15689051C212yesmaximal, complemented
7C2×#15689022C211yesmaximal, characteristic, complemented
8C2×#15689019C211yesmaximal, characteristic, complemented
9C2×#15688985C212yesmaximal, complemented
10C2×#15688983C211yesmaximal, characteristic, complemented
11C2×#15688980C211yesmaximal, characteristic, complemented
12C2×C47×#380C211yesmaximal, characteristic, complemented
13C23×C22⋊D47C2211yescharacteristic
14C23×C4×D47C2212yes—
15C23×C4×D47C2211yescharacteristic
16C2×#13094006—216no—
17C2×#13094006C22116yes—
18C2×C4×C22⋊D47—28no—
19C2×C4×C22⋊D47C2218yes—
20C22×C4⋊Dic47C2211yescharacteristic
21C22×C4⋊Dic47C2211yescharacteristic
22C22×C4⋊Dic47C2212yes—
23C22×Dic47⋊C4C2212yes—
24C22×Dic47⋊C4C2211yescharacteristic
25C22×Dic47⋊C4C2212yes—
26C2×C4⋊C4×D47—28no—
27C2×C4⋊C4×D47C2218yescomplemented
28C22×C22⋊Dic47C2212yes—
29C22×C22⋊Dic47C2211yescharacteristic
30C22×(C2×D47)⋊C4C2211yescharacteristic
31C22×(C2×D47)⋊C4C2212yes—
32C22×C4×Dic47C2211yescharacteristic
33#15689057C22116yescomplemented
34#15689051C22116yescomplemented
35#15689022C2218yescomplemented
36#15689019C2218yescomplemented
37#15688985C22116yescomplemented
38#15688983C2218yescomplemented
39#15688980C2218yescomplemented
40C2×C47⋊#95C2212yes—
41C2×C47⋊#95C2212yes—
42C2×C47⋊#95C2212yes—
43C2×C47⋊#95C2212yes—
44C24×C4×C47C2211yescharacteristic
45C22×C4⋊C4×C47C2211yescharacteristic
46C22×C4⋊C4×C47C2211yescharacteristic
47C22×C22⋊C4×C47C2212yes—
48C47×#380C2218yescomplemented
49C2×#95×C47C2212yes—
50C24×D47C2311yescharacteristic
51C22×C22⋊D47—24no—
52C22×C22⋊D47—24no—
53C22×C22⋊D47D418yes—
54C22×C22⋊D47C2314yes—
55C23×Dic47C2311yescharacteristic
56C23×Dic47C2312yes—
57C23×Dic47C2312yes—
58C23×Dic47C2311yescharacteristic
59C23×Dic47C2311yescharacteristic
60C22×C4×D47—24no—
61C22×C4×D47—24no—
62C22×C4×D47—28no—
63C22×C4×D47—28no—
64C22×C4×D47—28no—
65C22×C4×D47D418yes—
66C22×C4×D47C2318yes—
67C22×C4×D47C2314yes—
68#13094006—264no—
69C4×C22⋊D47—232no—
70C2×C4⋊Dic47—24no—
71C2×C4⋊Dic47—24no—
72C2×C4⋊Dic47—28no—
73C2×C4⋊Dic47C2314yes—
74C2×C4⋊Dic47C2314yes—
75C2×C4⋊Dic47C2318yes—
76C2×Dic47⋊C4—28no—
77C2×Dic47⋊C4—24no—
78C2×Dic47⋊C4—28no—
79C2×Dic47⋊C4C2318yes—
80C2×Dic47⋊C4C2318yes—
81C2×Dic47⋊C4C2314yes—
82C4⋊C4×D47—232no—
83C2×C22⋊Dic47—28no—
84C2×C22⋊Dic47—24no—
85C2×C22⋊Dic47C2318yes—
86C2×C22⋊Dic47C2314yes—
87C2×(C2×D47)⋊C4—24no—
88C2×(C2×D47)⋊C4—28no—
89C2×(C2×D47)⋊C4C2318yes—
90C2×(C2×D47)⋊C4C2314yes—
91C2×C4×Dic47—24no—
92C2×C4×Dic47C2314yes—
93C47⋊#95C2318yes—
94C47⋊#95C2318yes—
95C47⋊#95C2318yes—
96C47⋊#95C2318yes—
97C25×C47C2311yescharacteristic
98C23×C4×C47—24no—
99C23×C4×C47—24no—
100C23×C4×C47—24no—
101C23×C4×C47C2312yes—
102C23×C4×C47C2311yescharacteristic
103C23×C4×C47C2312yes—
104C23×C4×C47C2314yes—
105C23×C4×C47C2311yescharacteristic
106C2×C4⋊C4×C47—24no—
107C2×C4⋊C4×C47—24no—
108C2×C4⋊C4×C47C2314yes—
109C2×C4⋊C4×C47C2314yes—
110C2×C22⋊C4×C47—28no—
111C2×C22⋊C4×C47C2318yes—
112#95×C47C2318yes—
113C24×C47—22no—
114C24×C47—22no—
115C24×C47—22no—
116C24×C47—22no—
117C24×C47—22no—
118C24×C47—22no—
119C24×C47C2×D414yes—
120C24×C47C2412yes—
121C24×C47C2411yescharacteristic
122C23×D47—22no—
123C23×D47—22no—
124C23×D47—22no—
125C23×D47—22no—
126C23×D47C2×D414yes—
127C23×D47C2×D414yes—
128C23×D47#4114yes—
129C23×D47C2412yes—
130C22×C4×C47—24no—
131C22×C4×C47—24no—
132C22×C4×C47—24no—
133C22×C4×C47—22no—
134C22×C4×C47—24no—
135C22×C4×C47—22no—
136C22×C4×C47—22no—
137C22×C4×C47—24no—
138C22×C4×C47—24no—
139C22×C4×C47—28no—
140C22×C4×C47—28no—
141C22×C4×C47—28no—
142C22×C4×C47—416no—
143C22×C4×C47C2×D414yes—
144C22×C4×C47#4114yes—
145C22×C4×C47#4114yes—
146C22×C4×C47C2414yes—
147C22×C4×C47C2414yes—
148C22×C4×C47C2412yes—
149C22×C4×C47C2412yes—
150C2×C22⋊D47—232no—
151C2×C22⋊D47—28no—
152C2×C22⋊D47—28no—
153C2×C22⋊D47C2×D4116yes—
154C22×Dic47—22no—
155C22×Dic47—22no—
156C22×Dic47—22no—
157C22×Dic47—22no—
158C22×Dic47—22no—
159C22×Dic47—24no—
160C22×Dic47—24no—
161C22×Dic47—24no—
162C22×Dic47—24no—
163C22×Dic47—24no—
164C22×Dic47—22no—
165C22×Dic47—22no—
166C22×Dic47—24no—
167C22×Dic47C2×D414yes—
168C22×Dic47C2×D414yes—
169C22×Dic47C2412yes—
170C22×Dic47C2414yes—
171C22×Dic47C2414yes—
172C22×Dic47C2412yes—
173C22×Dic47C2412yes—
174C2×C4×D47—232no—
175C2×C4×D47—28no—
176C2×C4×D47—28no—
177C2×C4×D47—216no—
178C2×C4×D47—216no—
179C2×C4×D47—216no—
180C2×C4×D47—432no—
181C2×C4×D47C2×D4116yes—
182C4⋊C4×C47—28no—
183C4⋊C4×C47—28no—
184C22⋊C4×C47—216no—
185C22⋊Dic47—216no—
186C22⋊Dic47—28no—
187(C2×D47)⋊C4—28no—
188(C2×D47)⋊C4—216no—
189C4⋊Dic47—28no—
190C4⋊Dic47—28no—
191C4⋊Dic47—216no—
192Dic47⋊C4—216no—
193Dic47⋊C4—28no—
194Dic47⋊C4—216no—
195C4×Dic47—28no—
196C23×C47—28no—
197C23×C47—28no—
198C23×C47—28no—
199C23×C47—22no—
200C23×C47—22no—
201C23×C47—22no—
202C23×C47—22no—
203C23×C47—22no—
204C23×C47—22no—
205C23×C47—44no—
206C23×C47—44no—
207C23×C47—44no—
208C23×C47—44no—
209C23×C47C22×D414yes—
210C23×C47C22×D412yes—
211C23×C47C22×D412yes—
212C23×C47C2×#4112yes—
213C23×C47C2×#4112yes—
214C23×C47C2×#4112yes—
215C23×C472−1+412yes—
216C23×C472−1+412yes—
217C23×C47C2511yescharacteristic
218C22×D47—28no—
219C22×D47—28no—
220C22×D47—28no—
221C22×D47—28no—
222C22×D47—28no—
223C22×D47—28no—
224C22×D47—22no—
225C22×D47—22no—
226C22×D47—22no—
227C22×D47—22no—
228C22×D47#121116yes—
229C22×D47C22×D414yes—
230C22×D47C22×D414yes—
231C22×D47C2×#4114yes—
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—22no—
239C2×C4×C47—22no—
240C2×C4×C47—24no—
241C2×C4×C47—48no—
242C2×C4×C47—44no—
243C2×C4×C47—48no—
244C2×C4×C47—416no—
245C2×C4×C47#123116yescomplemented
246C2×C4×C47C22×D414yes—
247C2×C4×C47C2×#4114yes—
248C2×C4×C47C2×#4114yes—
249C22⋊D47—232no—
250C2×Dic47—28no—
251C2×Dic47—28no—
252C2×Dic47—22no—
253C2×Dic47—22no—
254C2×Dic47—22no—
255C2×Dic47—22no—
256C2×Dic47—22no—
257C2×Dic47—24no—
258C2×Dic47—24no—
259C2×Dic47—24no—
260C2×Dic47—24no—
261C2×Dic47—24no—
262C2×Dic47—22no—
263C2×Dic47—22no—
264C2×Dic47—24no—
265C2×Dic47—48no—
266C2×Dic47—44no—
267C2×Dic47—48no—
268C2×Dic47C22×D414yes—
269C2×Dic47C22×D414yes—
270C4×D47—232no—
271C4×D47—432no—
272C2×#2486—471nomaximal, Sylow
273C22×C47—24no—
274C22×C47—24no—
275C22×C47—24no—
276C22×C47—42no—
277C22×C47—42no—
278C22×C47—42no—
279C22×C47—42no—
280C22×C47—416no—
281C22×C47C2×#12114yes—
282C22×C47C2×#12314yes—
283C22×C47#53614yes—
284C22×C47#54714yes—
285C22×C47#54714yes—
286C22×C47#55214yes—
287C22×C47#55214yes—
288C22×C47C23×D411yescharacteristic
289C22×C47C23×D411yescharacteristic
290C22×C47C22×#4111yescharacteristic
291C22×C47C22×#4111yescharacteristic
292C22×C47C22×#4111yescharacteristic
293C22×C47C2×2−1+411yescharacteristic
294C22×C47C2×2−1+411yescharacteristic
295C2×D47—24no—
296C2×D47—24no—
297C2×D47—24no—
298C2×D47—24no—
299C2×D47—24no—
300C2×D47—24no—
301C2×D47—232no—
302C2×D47C2×#12118yes—
303C4×C47—44no—
304C4×C47—42no—
305C4×C47—44no—
306C4×C47C2×#12318yescomplemented
307Dic47—24no—
308Dic47—24no—
309Dic47—44no—
310Dic47—42no—
311Dic47—44no—
312C22×#123—472no—
313C22×C4×D4—471no—
314C23×C4⋊C4—471no—
315#2486—4716no—
316C2×#398—472no—
317C2×#397—472no—
318C2×#389—471no—
319C2×#389—471no—
320C2×#381—472no—
321C2×#380—471no—
322C2×#380—471no—
323C2×#380D4711yescharacteristic, complemented
324C2×C47—44no—
325C2×C47#248618yescomplemented
326C2×C47C22×#12111yescharacteristic
327C2×C47C22×#12311yescharacteristic
328C2×C47C2×#53611yescharacteristic
329C2×C47C2×#54711yescharacteristic
330C2×C47C2×#54711yescharacteristic
331C2×C47C2×#55211yescharacteristic
332C2×C47C2×#55211yescharacteristic
333D47—28no—
334C23×D4—471no—
335C24×C4—471no—
336C24×C4—472no—
337C24×C4C2×D4711yescharacteristic
338C2×#123—4716no—
339C2×#123—9416no—
340C2×C4×D4—478no—
341C2×C4×D4—948no—
342C22×C4⋊C4—472no—
343C22×C4⋊C4—471no—
344C22×C4⋊C4—472no—
345C22×C4⋊C4—472no—
346C22×C4⋊C4—471no—
347C22×C4⋊C4—471no—
348C22×C4⋊C4—478no—
349C22×C4⋊C4—948no—
350C22×C4⋊C4C2×D4711yescharacteristic
351C22×C4⋊C4C2×D4711yescharacteristic
352C22×C22⋊C4—471no—
353C22×C22⋊C4—472no—
354C22×C22⋊C4—471no—
355C22×C22⋊C4—472no—
356C22×C22⋊C4C2×D4712yes—
357C22×C42—471no—
358#398—4716no—
359#397—4716no—
360#389—478no—
361#389—478no—
362#381—4716no—
363#380—478no—
364#380—478no—
365#380C2×D4718yescomplemented
366C2×#95—472no—
367C2×#95—472no—
368C2×#95—472no—
369C2×#95—472no—
370C2×#95C2×D4712yes—
371C47C2×#248611yescharacteristic, complemented, Sylow
372C25—471no—
373C25C22×D4711yescharacteristic
374C22×D4—478no—
375C22×D4—474no—
376C22×D4—944no—
377C22×D4—944no—
378C23×C4—24no—
379C23×C4—24no—
380C23×C4—24no—
381C23×C4—471no—
382C23×C4—471no—
383C23×C4—478no—
384C23×C4—474no—
385C23×C4—472no—
386C23×C4—471no—
387C23×C4—472no—
388C23×C4—478no—
389C23×C4—948no—
390C23×C4—948no—
391C23×C4—948no—
392C23×C4—944no—
393C23×C4—944no—
394C23×C4C22×D4714yes—
395C23×C4C22×D4712yes—
396C23×C4C22×D4711yescharacteristic
397C23×C4C22×D4712yes—
398C23×C4C22×D4711yescharacteristic
399#123—9464no—
400C4×D4—9432no—
401C2×C4⋊C4—24no—
402C2×C4⋊C4—24no—
403C2×C4⋊C4—478no—
404C2×C4⋊C4—474no—
405C2×C4⋊C4—474no—
406C2×C4⋊C4—478no—
407C2×C4⋊C4—478no—
408C2×C4⋊C4—474no—
409C2×C4⋊C4—9432no—
410C2×C4⋊C4—948no—
411C2×C4⋊C4—944no—
412C2×C4⋊C4—948no—
413C2×C4⋊C4—948no—
414C2×C4⋊C4—944no—
415C2×C4⋊C4—944no—
416C2×C4⋊C4C22×D4714yes—
417C2×C4⋊C4C22×D4714yes—
418C2×C22⋊C4—28no—
419C2×C22⋊C4—474no—
420C2×C22⋊C4—478no—
421C2×C22⋊C4—474no—
422C2×C22⋊C4—478no—
423C2×C22⋊C4—944no—
424C2×C22⋊C4—948no—
425C2×C22⋊C4—944no—
426C2×C22⋊C4—948no—
427C2×C22⋊C4C22×D4718yes—
428C2×C42—474no—
429C2×C42—944no—
430#95—478no—
431#95—478no—
432#95—478no—
433#95—478no—
434#95C22×D4718yes—
435C24—22no—
436C24—22no—
437C24—22no—
438C24—22no—
439C24—22no—
440C24—22no—
441C24—474no—
442C24—474no—
443C24—474no—
444C24—472no—
445C24—942no—
446C24—942no—
447C24—942no—
448C24—942no—
449C24D4×D4714yes—
450C24C23×D4712yes—
451C24C23×D4711yescharacteristic
452C2×D4—4716no—
453C2×D4—948no—
454C2×D4—9432no—
455C2×D4—948no—
456C22×C4—24no—
457C22×C4—24no—
458C22×C4—22no—
459C22×C4—24no—
460C22×C4—24no—
461C22×C4—24no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—24no—
465C22×C4—28no—
466C22×C4—28no—
467C22×C4—28no—
468C22×C4—416no—
469C22×C4—474no—
470C22×C4—474no—
471C22×C4—472no—
472C22×C4—472no—
473C22×C4—4716no—
474C22×C4—472no—
475C22×C4—474no—
476C22×C4—474no—
477C22×C4—9416no—
478C22×C4—9416no—
479C22×C4—9416no—
480C22×C4—942no—
481C22×C4—942no—
482C22×C4—942no—
483C22×C4—942no—
484C22×C4—9432no—
485C22×C4—948no—
486C22×C4—948no—
487C22×C4—942no—
488C22×C4—944no—
489C22×C4—944no—
490C22×C4—944no—
491C22×C4—944no—
492C22×C4—944no—
493C22×C4—942no—
494C22×C4—944no—
495C22×C4—942no—
496C22×C4—18832no—
497C22×C4#1062535014yes—
498C22×C4D4×D4714yes—
499C22×C4#1062535214yes—
500C22×C4C23×D4712yes—
501C22×C4C23×D4714yes—
502C22×C4C23×D4714yes—
503C22×C4C23×D4712yes—
504C4⋊C4—28no—
505C4⋊C4—28no—
506C4⋊C4—9416no—
507C4⋊C4—948no—
508C4⋊C4—9416no—
509C4⋊C4—9416no—
510C4⋊C4—948no—
511C4⋊C4—948no—
512C22⋊C4—216no—
513C22⋊C4—948no—
514C22⋊C4—9416no—
515C22⋊C4—948no—
516C22⋊C4—9416no—
517C42—948no—
518C23—28no—
519C23—28no—
520C23—28no—
521C23—22no—
522C23—22no—
523C23—22no—
524C23—22no—
525C23—22no—
526C23—22no—
527C23—44no—
528C23—44no—
529C23—44no—
530C23—44no—
531C23—4716no—
532C23—474no—
533C23—474no—
534C23—474no—
535C23—948no—
536C23—948no—
537C23—948no—
538C23—948no—
539C23—948no—
540C23—942no—
541C23—942no—
542C23—948no—
543C23—942no—
544C23—942no—
545C23C2×D4×D4714yes—
546C23C2×D4×D4712yes—
547C23C2×D4×D4712yes—
548C23#41×D4712yes—
549C23C2×#1062535212yes—
550C23C2×#1062535012yes—
551C23#1309407112yes—
552C23#1309407212yes—
553C23C24×D4711yescharacteristic
554D4—9432no—
555C2×C4—24no—
556C2×C4—24no—
557C2×C4—22no—
558C2×C4—24no—
559C2×C4—24no—
560C2×C4—24no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—24no—
564C2×C4—48no—
565C2×C4—48no—
566C2×C4—44no—
567C2×C4—416no—
568C2×C4—474no—
569C2×C4—474no—
570C2×C4—948no—
571C2×C4—948no—
572C2×C4—942no—
573C2×C4—942no—
574C2×C4—942no—
575C2×C4—942no—
576C2×C4—9432no—
577C2×C4—942no—
578C2×C4—944no—
579C2×C4—944no—
580C2×C4—944no—
581C2×C4—944no—
582C2×C4—944no—
583C2×C4—942no—
584C2×C4—944no—
585C2×C4—942no—
586C2×C4—18832no—
587C2×C4—1888no—
588C2×C4—1888no—
589C2×C4—1884no—
590C2×C4#13094006116yescomplemented
591C2×C4C2×D4×D4714yes—
592C2×C4C2×#1062535214yes—
593C2×C4C2×#1062535014yes—
594C22—24no—
595C22—24no—
596C22—24no—
597C22—42no—
598C22—42no—
599C22—42no—
600C22—416no—
601C22—42no—
602C22—478no—
603C22—944no—
604C22—944no—
605C22—944no—
606C22—944no—
607C22—944no—
608C22—9432no—
609C22—944no—
610C22#121×D4714yes—
611C22C2×#1309400614yes—
612C22#1568969614yes—
613C22#1568977314yes—
614C22#1568977414yes—
615C22#1568980714yes—
616C22#1568981214yes—
617C22C22×D4×D4711yescharacteristic
618C22C22×D4×D4711yescharacteristic
619C22C2×#41×D4711yescharacteristic
620C22C22×#1062535211yescharacteristic
621C22C22×#1062535011yescharacteristic
622C22C2×#1309407111yescharacteristic
623C22C2×#1309407211yescharacteristic
624C4—44no—
625C4—44no—
626C4—42no—
627C4—944no—
628C4—944no—
629C4—1884no—
630C4—1884no—
631C4—1882no—
632C4C2×#1309400618yescomplemented
633C2—44no—
634C2—948no—
635C2#2024769218yesdirect product factor
636C2C2×#121×D4711yescharacteristic
637C2C22×#1309400611yescharacteristic
638C2C2×#1568969611yescharacteristic
639C2C2×#1568977311yescharacteristic
640C2C2×#1568977411yescharacteristic
641C2C2×#1568980711yescharacteristic
642C2C2×#1568981211yescharacteristic
subgroupG/H
centerC24C23×D47
commutator, cosocleC23×C47C25
FrattiniC23C24×D47
FittingC2×C47×#380C2
socleC24×C47C24
prime radicalC23×C22⋊D47C22
hypercenterC2×#380D47
hypocenterC47C2×#2486
LCSG ▷ C23×C47 ▷ C47
UCSC1 ◁ C24 ◁ C2×#380

Complements

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

NH ≅ G/Ntype
3. C22×C4⋊C4×D47C2semidirect
4. #20247692C2direct (N × H)
5. C2×#15689057C2semidirect
6. C2×#15689051C2semidirect
7. C2×#15689022C2semidirect
8. C2×#15689019C2semidirect
9. C2×#15688985C2semidirect
10. C2×#15688983C2semidirect
11. C2×#15688980C2semidirect
12. C2×C47×#380C2semidirect
27. C2×C4⋊C4×D47C22semidirect
33. #15689057C22semidirect
34. #15689051C22semidirect
35. #15689022C22semidirect
36. #15689019C22semidirect
37. #15688985C22semidirect
38. #15688983C22semidirect
39. #15688980C22semidirect
48. C47×#380C22semidirect
245. C2×C4×C47#123semidirect
306. C4×C47C2×#123semidirect
323. C2×#380D47semidirect
325. C2×C47#2486semidirect
365. #380C2×D47semidirect
371. C47C2×#2486semidirect
590. C2×C4#13094006semidirect
632. C4C2×#13094006semidirect
635. C2#20247692direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#130940062
C22×C4×C22⋊D472
C22×C4⋊C4×D472
#202476922
C2×#156890572
C2×#156890512
C2×#156890222
C2×#156890192
C2×#156889852
C2×#156889832
C2×#156889802
C2×C47×#3802
C2×#248647