Finite Group Databasebeta

← C2×#20251721 #26422175

Subgroups — C2×#20251721 #26422175

Order 12032.

4070 conjugacy classes of subgroups.

The subgroup lattice has 664 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#13094009C212yesmaximal, complemented
2C22×#13093999C211yesmaximal, characteristic
3C22×C42×D47C211yesmaximal, characteristic, complemented
4#20251721C2116yesmaximal, direct product factor
5C2×#15689025C211yesmaximal, characteristic, complemented
6C2×#15689024C211yesmaximal, characteristic, complemented
7C2×#15689019C211yesmaximal, characteristic, complemented
8C2×#15689013C212yesmaximal, complemented
9C2×#15689012C212yesmaximal, complemented
10C2×#15689001C212yesmaximal, complemented
11C2×#15688980C211yesmaximal, characteristic, complemented
12C2×C47×#389C211yesmaximal, characteristic, complemented
13C23×D188C2212yes—
14C23×D188C2211yescharacteristic
15C23×C4×D47C2211yescharacteristic
16C23×C4×D47C2211yescharacteristic
17C23×C4×D47C2211yescharacteristic
18C2×#13094009—216no—
19C2×#13094009C22116yescomplemented
20C2×#13093999—28no—
21C2×#13093999C2218yes—
22C22×C4⋊Dic47C2211yescharacteristic
23C22×Dic47⋊C4C2212yes—
24C22×(C2×D47)⋊C4C2212yes—
25C22×(C2×D47)⋊C4C2212yes—
26C22×(C2×D47)⋊C4C2212yes—
27C22×(C2×D47)⋊C4C2212yes—
28C22×(C2×D47)⋊C4C2212yes—
29C22×(C2×D47)⋊C4C2212yes—
30C22×C4×Dic47C2211yescharacteristic
31C22×C4×Dic47C2211yescharacteristic
32C22×C4×Dic47C2211yescharacteristic
33C2×C42×D47—28no—
34C2×C42×D47C2218yescomplemented
35#15689025C2218yescomplemented
36#15689024C2218yescomplemented
37#15689019C2218yescomplemented
38#15689013C22116yescomplemented
39#15689012C22116yescomplemented
40#15689001C22116yescomplemented
41#15688980C2218yescomplemented
42C2×C47⋊#95C2212yes—
43C2×C47⋊#95C2212yes—
44C22×C4⋊C4×C47C2212yes—
45C22×C4⋊C4×C47C2211yescharacteristic
46C22×C4⋊C4×C47C2211yescharacteristic
47C22×C42×C47C2211yescharacteristic
48C47×#389C2218yescomplemented
49C2×#95×C47C2212yes—
50C24×D47C2312yes—
51C24×D47C2311yescharacteristic
52C22×D188—28no—
53C22×D188—28no—
54C22×D188—28no—
55C22×D188—24no—
56C22×D188—24no—
57C22×D188—24no—
58C22×D188C2318yes—
59C22×D188C2314yes—
60C23×Dic47C2312yes—
61C23×Dic47C2311yescharacteristic
62C23×Dic47C2311yescharacteristic
63C23×Dic47C2311yescharacteristic
64C22×C4×D47—24no—
65C22×C4×D47—24no—
66C22×C4×D47—24no—
67C22×C4×D47—24no—
68C22×C4×D47—24no—
69C22×C4×D47—24no—
70C22×C4×D47—24no—
71C22×C4×D47D418yes—
72C22×C4×D47D418yescomplemented
73C22×C4×D47C2314yes—
74C22×C4×D47C2314yes—
75C22×C4×D47C2314yes—
76#13094009—264no—
77#13093999—232no—
78C2×C4⋊Dic47—24no—
79C2×C4⋊Dic47C2314yes—
80C2×Dic47⋊C4—28no—
81C2×Dic47⋊C4C2318yes—
82C2×(C2×D47)⋊C4—28no—
83C2×(C2×D47)⋊C4—28no—
84C2×(C2×D47)⋊C4—28no—
85C2×(C2×D47)⋊C4—28no—
86C2×(C2×D47)⋊C4—28no—
87C2×(C2×D47)⋊C4—28no—
88C2×(C2×D47)⋊C4C2318yes—
89C2×(C2×D47)⋊C4C2318yes—
90C2×(C2×D47)⋊C4C2318yes—
91C2×(C2×D47)⋊C4C2318yes—
92C2×(C2×D47)⋊C4C2318yes—
93C2×(C2×D47)⋊C4C2318yes—
94C2×C4×Dic47—24no—
95C2×C4×Dic47—24no—
96C2×C4×Dic47—24no—
97C2×C4×Dic47C2314yes—
98C2×C4×Dic47C2314yes—
99C2×C4×Dic47C2314yes—
100C42×D47—232no—
101C47⋊#95C2318yes—
102C47⋊#95C2318yes—
103C23×C4×C47C2312yes—
104C23×C4×C47C2312yes—
105C23×C4×C47C2311yescharacteristic
106C23×C4×C47C2311yescharacteristic
107C23×C4×C47C2311yescharacteristic
108C2×C4⋊C4×C47—24no—
109C2×C4⋊C4×C47—28no—
110C2×C4⋊C4×C47—24no—
111C2×C4⋊C4×C47C2314yes—
112C2×C4⋊C4×C47C2318yes—
113C2×C4⋊C4×C47C2314yes—
114C2×C42×C47—24no—
115C2×C42×C47C2314yes—
116#95×C47C2318yes—
117C24×C47C2411yescharacteristic
118C23×D47—22no—
119C23×D47—22no—
120C23×D47—22no—
121C23×D47—22no—
122C23×D47—24no—
123C23×D47—24no—
124C23×D47—24no—
125C23×D47—24no—
126C23×D47—24no—
127C23×D47—24no—
128C23×D47—24no—
129C23×D47C2×D414yes—
130C23×D47C2×D414yes—
131C23×D47#4114yes—
132C23×D47C2414yes—
133C23×D47C2412yes—
134C22×C4×C47—22no—
135C22×C4×C47—22no—
136C22×C4×C47—22no—
137C22×C4×C47—22no—
138C22×C4×C47—24no—
139C22×C4×C47—24no—
140C22×C4×C47—24no—
141C22×C4×C47—24no—
142C22×C4×C47—24no—
143C22×C4×C47—24no—
144C22×C4×C47C2×D414yes—
145C22×C4×C47C2×D414yes—
146C22×C4×C47#4114yes—
147C22×C4×C47#4114yes—
148C22×C4×C47#4114yes—
149C22×C4×C47C2414yes—
150C22×C4×C47C2414yes—
151C22×C4×C47C2412yes—
152C22×C4×C47C2412yes—
153C22×C4×C47C2412yes—
154C22×Dic47—22no—
155C22×Dic47—22no—
156C22×Dic47—22no—
157C22×Dic47—22no—
158C22×Dic47—22no—
159C22×Dic47—22no—
160C22×Dic47—24no—
161C22×Dic47—24no—
162C22×Dic47—24no—
163C22×Dic47C2×D414yes—
164C22×Dic47C2×D414yes—
165C22×Dic47#4114yes—
166C22×Dic47C2414yes—
167C22×Dic47C2412yes—
168C22×Dic47C2412yes—
169C22×Dic47C2412yes—
170C2×D188—216no—
171C2×D188—216no—
172C2×D188—216no—
173C2×D188—28no—
174C2×D188—28no—
175C2×D188—28no—
176C2×D188—432no—
177C2×D188—416no—
178C2×C4×D47—232no—
179C2×C4×D47—232no—
180C2×C4×D47—28no—
181C2×C4×D47—28no—
182C2×C4×D47—28no—
183C2×C4×D47—232no—
184C2×C4×D47—28no—
185C2×C4×D47—28no—
186C2×C4×D47—28no—
187C2×C4×D47—28no—
188C2×C4×D47C2×D4116yes—
189C2×C4×D47C2×D4116yescomplemented
190C4⋊C4×C47—28no—
191C4⋊C4×C47—216no—
192C4⋊C4×C47—28no—
193C42×C47—28no—
194(C2×D47)⋊C4—216no—
195(C2×D47)⋊C4—216no—
196(C2×D47)⋊C4—216no—
197(C2×D47)⋊C4—216no—
198(C2×D47)⋊C4—216no—
199(C2×D47)⋊C4—216no—
200C4⋊Dic47—28no—
201Dic47⋊C4—216no—
202C4×Dic47—28no—
203C4×Dic47—28no—
204C4×Dic47—28no—
205C23×C47C22×D412yes—
206C23×C47C22×D412yes—
207C23×C47C2×#4112yes—
208C23×C47C2×#4112yes—
209C23×C47C2×#4112yes—
210C23×C47C2×#4112yes—
211C23×C47C2×#4112yes—
212C23×C47C2511yescharacteristic
213C22×D47—28no—
214C22×D47—28no—
215C22×D47—28no—
216C22×D47—28no—
217C22×D47—28no—
218C22×D47—28no—
219C22×D47—22no—
220C22×D47—22no—
221C22×D47—22no—
222C22×D47—22no—
223C22×D47—24no—
224C22×D47—24no—
225C22×D47—24no—
226C22×D47—24no—
227C22×D47—24no—
228C22×D47—24no—
229C22×D47—24no—
230C22×D47—48no—
231C22×D47—48no—
232C22×D47—48no—
233C22×D47—48no—
234C22×D47—48no—
235C22×D47—48no—
236C22×D47—48no—
237C22×D47#121116yes—
238C22×D47C22×D414yes—
239C22×D47C22×D414yes—
240C22×D47C2×#4114yes—
241C2×C4×C47—28no—
242C2×C4×C47—28no—
243C2×C4×C47—22no—
244C2×C4×C47—22no—
245C2×C4×C47—22no—
246C2×C4×C47—22no—
247C2×C4×C47—24no—
248C2×C4×C47—24no—
249C2×C4×C47—24no—
250C2×C4×C47—24no—
251C2×C4×C47—24no—
252C2×C4×C47—24no—
253C2×C4×C47—48no—
254C2×C4×C47—48no—
255C2×C4×C47#123116yescomplemented
256C2×C4×C47C22×D414yes—
257C2×C4×C47C22×D414yes—
258C2×C4×C47C2×#4114yes—
259C2×C4×C47C2×#4114yes—
260C2×C4×C47C2×#4114yes—
261C2×Dic47—28no—
262C2×Dic47—28no—
263C2×Dic47—28no—
264C2×Dic47—22no—
265C2×Dic47—22no—
266C2×Dic47—22no—
267C2×Dic47—22no—
268C2×Dic47—22no—
269C2×Dic47—22no—
270C2×Dic47—24no—
271C2×Dic47—24no—
272C2×Dic47—24no—
273C2×Dic47—48no—
274C2×Dic47C22×D414yes—
275C2×Dic47C22×D414yes—
276C2×Dic47C2×#4114yes—
277D188—432no—
278D188—416no—
279C4×D47—232no—
280C4×D47—232no—
281C4×D47—232no—
282C2×#2854—471nomaximal, Sylow
283C22×C47C2×#12114yes—
284C22×C47C2×#12314yes—
285C22×C47#52514yes—
286C22×C47#52914yes—
287C22×C47#52914yes—
288C22×C47#53214yes—
289C22×C47#53214yes—
290C22×C47C23×D411yescharacteristic
291C22×C47C23×D411yescharacteristic
292C22×C47C22×#4111yescharacteristic
293C22×C47C22×#4111yescharacteristic
294C22×C47C22×#4111yescharacteristic
295C22×C47C22×#4111yescharacteristic
296C22×C47C22×#4111yescharacteristic
297C2×D47—232no—
298C2×D47—24no—
299C2×D47—24no—
300C2×D47—24no—
301C2×D47—24no—
302C2×D47—24no—
303C2×D47—24no—
304C2×D47—44no—
305C2×D47—44no—
306C2×D47—44no—
307C2×D47—44no—
308C2×D47—44no—
309C2×D47—44no—
310C2×D47—44no—
311C2×D47—816no—
312C2×D47C2×#12118yes—
313C4×C47—24no—
314C4×C47—24no—
315C4×C47—44no—
316C4×C47—44no—
317C4×C47C2×#12318yescomplemented
318Dic47—24no—
319Dic47—24no—
320Dic47—24no—
321Dic47—44no—
322C22×#123—472no—
323C22×#121—471no—
324C23×C42—471no—
325#2854—4716no—
326C2×#390—471no—
327C2×#390—471no—
328C2×#389—471no—
329C2×#389D4711yescharacteristic, complemented
330C2×#388—472no—
331C2×#388—472no—
332C2×#386—472no—
333C2×#380—471no—
334C2×C47#285418yescomplemented
335C2×C47C22×#12111yescharacteristic
336C2×C47C22×#12311yescharacteristic
337C2×C47C2×#52511yescharacteristic
338C2×C47C2×#52911yescharacteristic
339C2×C47C2×#52911yescharacteristic
340C2×C47C2×#53211yescharacteristic
341C2×C47C2×#53211yescharacteristic
342D47—28no—
343D47—84no—
344C23×D4—472no—
345C23×D4—471no—
346C24×C4—471no—
347C24×C4—471no—
348C24×C4—471no—
349C2×#123—4716no—
350C2×#123—9416no—
351C2×#121—478no—
352C2×#121—948no—
353C22×C4⋊C4—472no—
354C22×C4⋊C4—471no—
355C22×C4⋊C4C2×D4711yescharacteristic
356C22×C4⋊C4C2×D4712yes—
357C22×C4⋊C4C2×D4711yescharacteristic
358C22×C22⋊C4—472no—
359C22×C22⋊C4—472no—
360C22×C22⋊C4—472no—
361C22×C22⋊C4—472no—
362C22×C22⋊C4—472no—
363C22×C22⋊C4—472no—
364C22×C42—471no—
365C22×C42—471no—
366C22×C42—471no—
367C22×C42—478no—
368C22×C42—948no—
369C22×C42C2×D4711yescharacteristic
370#390—478no—
371#390—478no—
372#389—478no—
373#389C2×D4718yescomplemented
374#388—4716no—
375#388—4716no—
376#386—4716no—
377#380—478no—
378C2×#95—472no—
379C2×#95—472no—
380C2×#95C2×D4712yes—
381C47C2×#285411yescharacteristic, complemented, Sylow
382C25—471no—
383C25—472no—
384C22×D4—478no—
385C22×D4—474no—
386C22×D4—948no—
387C22×D4—948no—
388C22×D4—948no—
389C22×D4—944no—
390C22×D4—944no—
391C22×D4—944no—
392C23×C4—471no—
393C23×C4—471no—
394C23×C4—471no—
395C23×C4—478no—
396C23×C4—478no—
397C23×C4—474no—
398C23×C4—474no—
399C23×C4—472no—
400C23×C4—474no—
401C23×C4—944no—
402C23×C4—944no—
403C23×C4—944no—
404C23×C4—944no—
405C23×C4—944no—
406C23×C4—944no—
407C23×C4—944no—
408C23×C4C22×D4712yes—
409C23×C4C22×D4712yes—
410C23×C4C22×D4711yescharacteristic
411C23×C4C22×D4711yescharacteristic
412C23×C4C22×D4711yescharacteristic
413#123—9464no—
414#121—9432no—
415C2×C4⋊C4—24no—
416C2×C4⋊C4—24no—
417C2×C4⋊C4—28no—
418C2×C4⋊C4—478no—
419C2×C4⋊C4—474no—
420C2×C4⋊C4—948no—
421C2×C4⋊C4—944no—
422C2×C4⋊C4C22×D4714yes—
423C2×C4⋊C4C22×D4718yes—
424C2×C4⋊C4C22×D4714yes—
425C2×C22⋊C4—478no—
426C2×C22⋊C4—478no—
427C2×C22⋊C4—478no—
428C2×C22⋊C4—478no—
429C2×C22⋊C4—478no—
430C2×C22⋊C4—478no—
431C2×C22⋊C4—948no—
432C2×C22⋊C4—948no—
433C2×C22⋊C4—948no—
434C2×C22⋊C4—948no—
435C2×C22⋊C4—948no—
436C2×C22⋊C4—948no—
437C2×C42—24no—
438C2×C42—474no—
439C2×C42—474no—
440C2×C42—474no—
441C2×C42—9432no—
442C2×C42—944no—
443C2×C42—944no—
444C2×C42—944no—
445C2×C42C22×D4714yes—
446#95—478no—
447#95—478no—
448#95C22×D4718yes—
449C24—474no—
450C24—474no—
451C24—474no—
452C24—472no—
453C24—474no—
454C24—942no—
455C24—942no—
456C24—942no—
457C24—942no—
458C24—944no—
459C24—944no—
460C24—944no—
461C24—944no—
462C24—944no—
463C24—944no—
464C24—944no—
465C24C23×D4711yescharacteristic
466C2×D4—9416no—
467C2×D4—9416no—
468C2×D4—9416no—
469C2×D4—948no—
470C2×D4—948no—
471C2×D4—948no—
472C2×D4—18832no—
473C2×D4—18816no—
474C22×C4—22no—
475C22×C4—22no—
476C22×C4—22no—
477C22×C4—22no—
478C22×C4—24no—
479C22×C4—24no—
480C22×C4—24no—
481C22×C4—24no—
482C22×C4—24no—
483C22×C4—24no—
484C22×C4—474no—
485C22×C4—474no—
486C22×C4—474no—
487C22×C4—472no—
488C22×C4—472no—
489C22×C4—472no—
490C22×C4—4716no—
491C22×C4—4716no—
492C22×C4—474no—
493C22×C4—948no—
494C22×C4—948no—
495C22×C4—948no—
496C22×C4—9432no—
497C22×C4—948no—
498C22×C4—9432no—
499C22×C4—942no—
500C22×C4—942no—
501C22×C4—942no—
502C22×C4—942no—
503C22×C4—942no—
504C22×C4—942no—
505C22×C4—9432no—
506C22×C4—948no—
507C22×C4—948no—
508C22×C4—948no—
509C22×C4—944no—
510C22×C4—944no—
511C22×C4—944no—
512C22×C4#1062535014yes—
513C22×C4D4×D4714yes—
514C22×C4D4×D4714yes—
515C22×C4#1062535414yes—
516C22×C4#1062535414yes—
517C22×C4C23×D4714yes—
518C22×C4C23×D4714yes—
519C22×C4C23×D4712yes—
520C22×C4C23×D4712yes—
521C22×C4C23×D4712yes—
522C4⋊C4—28no—
523C4⋊C4—216no—
524C4⋊C4—28no—
525C4⋊C4—9416no—
526C4⋊C4—948no—
527C22⋊C4—9416no—
528C22⋊C4—9416no—
529C22⋊C4—9416no—
530C22⋊C4—9416no—
531C22⋊C4—9416no—
532C22⋊C4—9416no—
533C42—28no—
534C42—948no—
535C42—948no—
536C42—948no—
537C23—4716no—
538C23—474no—
539C23—474no—
540C23—474no—
541C23—948no—
542C23—948no—
543C23—948no—
544C23—948no—
545C23—948no—
546C23—948no—
547C23—942no—
548C23—942no—
549C23—942no—
550C23—942no—
551C23—944no—
552C23—944no—
553C23—944no—
554C23—944no—
555C23—944no—
556C23—944no—
557C23—944no—
558C23—1888no—
559C23—1888no—
560C23—1888no—
561C23—1888no—
562C23—1888no—
563C23—1888no—
564C23—1888no—
565C23C2×D4×D4712yes—
566C23C2×D4×D4712yes—
567C23#41×D4712yes—
568C23#41×D4712yes—
569C23C2×#1062535012yes—
570C23C2×#1062535412yes—
571C23C2×#1062535412yes—
572C23C24×D4711yescharacteristic
573D4—18832no—
574D4—18816no—
575C2×C4—28no—
576C2×C4—28no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—24no—
582C2×C4—24no—
583C2×C4—24no—
584C2×C4—24no—
585C2×C4—24no—
586C2×C4—24no—
587C2×C4—48no—
588C2×C4—48no—
589C2×C4—474no—
590C2×C4—474no—
591C2×C4—474no—
592C2×C4—9432no—
593C2×C4—9432no—
594C2×C4—948no—
595C2×C4—948no—
596C2×C4—948no—
597C2×C4—942no—
598C2×C4—942no—
599C2×C4—942no—
600C2×C4—942no—
601C2×C4—942no—
602C2×C4—942no—
603C2×C4—9432no—
604C2×C4—944no—
605C2×C4—944no—
606C2×C4—944no—
607C2×C4—1888no—
608C2×C4#13094009116yescomplemented
609C2×C4C2×D4×D4714yes—
610C2×C4C2×D4×D4714yes—
611C2×C4C2×#1062535014yes—
612C2×C4C2×#1062535414yes—
613C2×C4C2×#1062535414yes—
614C22—478no—
615C22—9432no—
616C22—944no—
617C22—944no—
618C22—944no—
619C22—944no—
620C22—944no—
621C22—944no—
622C22—37616no—
623C22—1884no—
624C22—1884no—
625C22—1884no—
626C22—1884no—
627C22—1884no—
628C22—1884no—
629C22—1884no—
630C22#121×D4714yes—
631C22C2×#1309400914yes—
632C22#1568963614yes—
633C22#1568965814yes—
634C22#1568966014yes—
635C22#1568967014yes—
636C22#1568967414yes—
637C22C22×D4×D4711yescharacteristic
638C22C22×D4×D4711yescharacteristic
639C22C2×#41×D4711yescharacteristic
640C22C2×#41×D4711yescharacteristic
641C22C22×#1062535011yescharacteristic
642C22C22×#1062535411yescharacteristic
643C22C22×#1062535411yescharacteristic
644C4—24no—
645C4—24no—
646C4—44no—
647C4—44no—
648C4—944no—
649C4—944no—
650C4—944no—
651C4—1884no—
652C4C2×#1309400918yescomplemented
653C2—948no—
654C2—3764no—
655C2#2025172118yesdirect product factor
656C2C2×#121×D4711yescharacteristic
657C2C22×#1309400911yescharacteristic
658C2C2×#1568963611yescharacteristic
659C2C2×#1568965811yescharacteristic
660C2C2×#1568966011yescharacteristic
661C2C2×#1568967011yescharacteristic
662C2C2×#1568967411yescharacteristic
subgroupG/H
centerC24C23×D47
commutator, cosocleC23×C47C25
FrattiniC23C24×D47
FittingC2×C47×#389C2
socleC24×C47C24
prime radicalC22×#13093999C2
hypercenterC2×#389D47
hypocenterC47C2×#2854
LCSG ▷ C23×C47 ▷ C47
UCSC1 ◁ C24 ◁ C2×#389

Complements

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

NH ≅ G/Ntype
1. C22×#13094009C2semidirect
3. C22×C42×D47C2semidirect
4. #20251721C2direct (N × H)
5. C2×#15689025C2semidirect
6. C2×#15689024C2semidirect
7. C2×#15689019C2semidirect
8. C2×#15689013C2semidirect
9. C2×#15689012C2semidirect
10. C2×#15689001C2semidirect
11. C2×#15688980C2semidirect
12. C2×C47×#389C2semidirect
19. C2×#13094009C22semidirect
34. C2×C42×D47C22semidirect
35. #15689025C22semidirect
36. #15689024C22semidirect
37. #15689019C22semidirect
38. #15689013C22semidirect
39. #15689012C22semidirect
40. #15689001C22semidirect
41. #15688980C22semidirect
48. C47×#389C22semidirect
72. C22×C4×D47D4semidirect
189. C2×C4×D47C2×D4semidirect
255. C2×C4×C47#123semidirect
317. C4×C47C2×#123semidirect
329. C2×#389D47semidirect
334. C2×C47#2854semidirect
373. #389C2×D47semidirect
381. C47C2×#2854semidirect
608. C2×C4#13094009semidirect
652. C4C2×#13094009semidirect
655. C2#20251721direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#130940092
C22×#130939992
C22×C42×D472
#202517212
C2×#156890252
C2×#156890242
C2×#156890192
C2×#156890132
C2×#156890122
C2×#156890012
C2×#156889802
C2×C47×#3892
C2×#285447