Finite Group Databasebeta

← C2×#20247743 #26385727

Subgroups — C2×#20247743 #26385727

Order 12032.

3158 conjugacy classes of subgroups.

The subgroup lattice has 632 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#13093999C211yesmaximal, characteristic
2#20247743C2116yesmaximal, direct product factor
3C2×#15689049C212yesmaximal, complemented
4C2×#15689040C212yesmaximal, complemented
5C2×#15689025C211yesmaximal, characteristic, complemented
6C2×#15689024C212yesmaximal, complemented
7C2×#15689024C211yesmaximal, characteristic, complemented
8C2×#15689013C212yesmaximal, complemented
9C2×#15688980C211yesmaximal, characteristic, complemented
10C2×C4⋊C4×Dic47C211yesmaximal, characteristic, complemented
11C2×C4×(C2×D47)⋊C4C211yesmaximal, characteristic, complemented
12C22×#125×C47C211yesmaximal, characteristic, complemented
13C23×D188C2211yescharacteristic
14C23×D188C2212yes—
15C23×C4×D47C2211yescharacteristic
16C2×#13093999—28no—
17C2×#13093999C2218yes—
18C22×C4⋊Dic47C2211yescharacteristic
19C22×(C2×D47)⋊C4C2212yes—
20C22×(C2×D47)⋊C4C2212yes—
21C22×(C2×D47)⋊C4C2211yescharacteristic
22C22×(C2×D47)⋊C4C2211yescharacteristic
23C22×(C2×D47)⋊C4C2212yes—
24C22×(C2×D47)⋊C4C2212yes—
25C22×(C2×D47)⋊C4C2212yes—
26C22×(C2×D47)⋊C4C2212yes—
27C22×C4×Dic47C2211yescharacteristic
28C22×C4×Dic47C2212yes—
29#15689049C22116yescomplemented
30#15689040C22116yescomplemented
31#15689025C2218yescomplemented
32#15689024C2218yescomplemented
33#15689024C22116yescomplemented
34#15689013C22116yescomplemented
35#15688980C2218yescomplemented
36C4⋊C4×Dic47C2218yescomplemented
37C4×(C2×D47)⋊C4C2218yescomplemented
38C2×C47⋊#95C2211yescharacteristic
39C2×C47⋊#95C2212yes—
40C2×C47⋊#95C2212yes—
41C2×C47⋊#95C2211yescharacteristic
42C2×#125×C47—28no—
43C2×#125×C47C2218yescomplemented
44C22×C4⋊C4×C47C2212yes—
45C22×C4⋊C4×C47C2212yes—
46C22×C4⋊C4×C47C2211yescharacteristic
47C22×C4⋊C4×C47C2211yescharacteristic
48C22×C42×C47C2211yescharacteristic
49C24×D47C2312yes—
50C24×D47C2311yescharacteristic
51C22×D188—28no—
52C22×D188—28no—
53C22×D188—24no—
54C22×D188—24no—
55C22×D188—24no—
56C22×D188—44no—
57C22×D188C2314yes—
58C22×D188C2318yes—
59C23×Dic47C2311yescharacteristic
60C23×Dic47C2311yescharacteristic
61C23×Dic47C2312yes—
62C23×Dic47C2311yescharacteristic
63C22×C4×D47—24no—
64C22×C4×D47—24no—
65C22×C4×D47—24no—
66C22×C4×D47C2314yes—
67#13093999—232no—
68C2×C4⋊Dic47—24no—
69C2×C4⋊Dic47C2314yes—
70C2×(C2×D47)⋊C4—24no—
71C2×(C2×D47)⋊C4—28no—
72C2×(C2×D47)⋊C4—24no—
73C2×(C2×D47)⋊C4—28no—
74C2×(C2×D47)⋊C4—44no—
75C2×(C2×D47)⋊C4—44no—
76C2×(C2×D47)⋊C4—44no—
77C2×(C2×D47)⋊C4—44no—
78C2×(C2×D47)⋊C4C2314yes—
79C2×(C2×D47)⋊C4C2314yes—
80C2×(C2×D47)⋊C4C2318yes—
81C2×(C2×D47)⋊C4C2318yes—
82C2×(C2×D47)⋊C4C2318yes—
83C2×(C2×D47)⋊C4C2318yes—
84C2×(C2×D47)⋊C4C2318yes—
85C2×(C2×D47)⋊C4C2318yes—
86C2×C4×Dic47—24no—
87C2×C4×Dic47—44no—
88C2×C4×Dic47C2314yes—
89C2×C4×Dic47C2318yes—
90C47⋊#95C2314yes—
91C47⋊#95C2314yes—
92C47⋊#95C2318yes—
93C47⋊#95C2318yes—
94C23×C4×C47C2311yescharacteristic
95C23×C4×C47C2311yescharacteristic
96C23×C4×C47C2312yes—
97C23×C4×C47C2312yes—
98C23×C4×C47C2311yescharacteristic
99#125×C47—232no—
100C2×C4⋊C4×C47—28no—
101C2×C4⋊C4×C47—28no—
102C2×C4⋊C4×C47—24no—
103C2×C4⋊C4×C47D418yes—
104C2×C4⋊C4×C47C2314yes—
105C2×C4⋊C4×C47C2318yes—
106C2×C4⋊C4×C47C2318yes—
107C2×C4⋊C4×C47C2314yes—
108C2×C42×C47—24no—
109C2×C42×C47C2314yes—
110C24×C47C2411yescharacteristic
111C23×D47—22no—
112C23×D47—22no—
113C23×D47—22no—
114C23×D47—22no—
115C23×D47—22no—
116C23×D47—22no—
117C23×D47—24no—
118C23×D47—24no—
119C23×D47—24no—
120C23×D47—24no—
121C23×D47—24no—
122C23×D47—24no—
123C23×D47—24no—
124C23×D47#4114yes—
125C23×D47C2414yes—
126C23×D47C2412yes—
127C22×C4×C47—24no—
128C22×C4×C47—24no—
129C22×C4×C47—22no—
130C22×C4×C47—22no—
131C22×C4×C47—24no—
132C22×C4×C47—22no—
133C22×C4×C47—22no—
134C22×C4×C47—24no—
135C22×C4×C47—22no—
136C22×C4×C47—22no—
137C22×C4×C47—24no—
138C22×C4×C47C2×D418yes—
139C22×C4×C47C2×D414yes—
140C22×C4×C47#4114yes—
141C22×C4×C47#4114yes—
142C22×C4×C47C2412yes—
143C22×C4×C47C2412yes—
144C22×C4×C47C2414yes—
145C22×C4×C47C2414yes—
146C22×C4×C47C2412yes—
147C22×Dic47—22no—
148C22×Dic47—22no—
149C22×Dic47—22no—
150C22×Dic47—22no—
151C22×Dic47—22no—
152C22×Dic47—24no—
153C22×Dic47—22no—
154C22×Dic47—22no—
155C22×Dic47—24no—
156C22×Dic47—24no—
157C22×Dic47—22no—
158C22×Dic47#4114yes—
159C22×Dic47C2412yes—
160C22×Dic47C2412yes—
161C22×Dic47C2414yes—
162C22×Dic47C2412yes—
163C2×D188—216no—
164C2×D188—216no—
165C2×D188—28no—
166C2×D188—28no—
167C2×D188—28no—
168C2×D188—48no—
169C2×D188—432no—
170C2×D188—416no—
171C2×C4×D47—28no—
172C2×C4×D47—28no—
173C2×C4×D47—28no—
174C2×C4×D47—416no—
175C4⋊C4×C47—216no—
176C4⋊C4×C47—216no—
177C4⋊C4×C47—28no—
178C4⋊C4×C47C2×D4116yes—
179C42×C47—28no—
180(C2×D47)⋊C4—28no—
181(C2×D47)⋊C4—216no—
182(C2×D47)⋊C4—28no—
183(C2×D47)⋊C4—216no—
184(C2×D47)⋊C4—48no—
185(C2×D47)⋊C4—48no—
186(C2×D47)⋊C4—48no—
187(C2×D47)⋊C4—48no—
188C4⋊Dic47—28no—
189C4×Dic47—28no—
190C4×Dic47—48no—
191C23×C47C22×D412yes—
192C23×C47C2×#4112yes—
193C23×C47C2×#4112yes—
194C23×C47C2×#4112yes—
195C23×C47C2×#4112yes—
196C23×C47H(5,𝔽2)12yes—
197C23×C47H(5,𝔽2)12yes—
198C23×C47C2511yescharacteristic
199C22×D47—28no—
200C22×D47—28no—
201C22×D47—28no—
202C22×D47—22no—
203C22×D47—22no—
204C22×D47—22no—
205C22×D47—22no—
206C22×D47—22no—
207C22×D47—22no—
208C22×D47—24no—
209C22×D47—24no—
210C22×D47—24no—
211C22×D47—24no—
212C22×D47—24no—
213C22×D47—24no—
214C22×D47—24no—
215C22×D47—44no—
216C22×D47—44no—
217C22×D47—44no—
218C22×D47—44no—
219C22×D47—48no—
220C22×D47—48no—
221C22×D47—48no—
222C22×D47—48no—
223C22×D47—48no—
224C22×D47—48no—
225C22×D47—48no—
226C22×D47C2×#4114yes—
227C2×C4×C47—216no—
228C2×C4×C47—24no—
229C2×C4×C47—24no—
230C2×C4×C47—22no—
231C2×C4×C47—22no—
232C2×C4×C47—24no—
233C2×C4×C47—22no—
234C2×C4×C47—22no—
235C2×C4×C47—24no—
236C2×C4×C47—22no—
237C2×C4×C47—22no—
238C2×C4×C47—24no—
239C2×C4×C47—44no—
240C2×C4×C47—44no—
241C2×C4×C47—48no—
242C2×C4×C47#124116yes—
243C2×C4×C47C22×D418yes—
244C2×C4×C47C22×D414yes—
245C2×C4×C47C2×#4114yes—
246C2×C4×C47C2×#4114yes—
247C2×Dic47—28no—
248C2×Dic47—22no—
249C2×Dic47—22no—
250C2×Dic47—22no—
251C2×Dic47—22no—
252C2×Dic47—24no—
253C2×Dic47—22no—
254C2×Dic47—22no—
255C2×Dic47—22no—
256C2×Dic47—24no—
257C2×Dic47—24no—
258C2×Dic47—22no—
259C2×Dic47—44no—
260C2×Dic47—48no—
261C2×Dic47—44no—
262C2×Dic47C2×#4114yes—
263D188—432no—
264D188—416no—
265C4×D47—416no—
266C2×#2490—471nomaximal, Sylow
267C22×C47C2×#12414yes—
268C22×C47#52914yes—
269C22×C47#54014yes—
270C22×C47#54614yes—
271C22×C47#55014yes—
272C22×C47#55314yes—
273C22×C47#55314yes—
274C22×C47C23×D411yescharacteristic
275C22×C47C22×#4111yescharacteristic
276C22×C47C22×#4111yescharacteristic
277C22×C47C22×#4111yescharacteristic
278C22×C47C22×#4111yescharacteristic
279C22×C47C2×H(5,𝔽2)11yescharacteristic
280C22×C47C2×H(5,𝔽2)11yescharacteristic
281C2×D47—24no—
282C2×D47—24no—
283C2×D47—24no—
284C2×D47—42no—
285C2×D47—42no—
286C2×D47—42no—
287C2×D47—42no—
288C2×D47—416no—
289C2×D47—44no—
290C2×D47—44no—
291C2×D47—44no—
292C2×D47—44no—
293C2×D47—44no—
294C2×D47—44no—
295C2×D47—44no—
296C2×D47—816no—
297C4×C47—28no—
298C4×C47—42no—
299C4×C47—42no—
300C4×C47—44no—
301C4×C47C2×#12418yes—
302Dic47—24no—
303Dic47—42no—
304Dic47—44no—
305Dic47—42no—
306C22×#125D4711yescharacteristic, complemented
307C22×#121—471no—
308#2490—4716no—
309C2×#397—472no—
310C2×#394—472no—
311C2×#390—471no—
312C2×#390—472no—
313C2×#390—471no—
314C2×#388—472no—
315C2×#380—471no—
316C2×C4×C4⋊C4—471no—
317C2×C4×C22⋊C4—471no—
318C2×C47#249018yescomplemented
319C2×C47C22×#12411yescharacteristic
320C2×C47C2×#52911yescharacteristic
321C2×C47C2×#54011yescharacteristic
322C2×C47C2×#54611yescharacteristic
323C2×C47C2×#55011yescharacteristic
324C2×C47C2×#55311yescharacteristic
325C2×C47C2×#55311yescharacteristic
326D47—44no—
327D47—84no—
328C23×D4—471no—
329C23×D4—472no—
330C24×C4—471no—
331C2×#125—28no—
332C2×#125C2×D4718yescomplemented
333C2×#121—478no—
334C2×#121—948no—
335C22×C4⋊C4—471no—
336C22×C4⋊C4C2×D4712yes—
337C22×C4⋊C4C2×D4712yes—
338C22×C4⋊C4C2×D4711yescharacteristic
339C22×C4⋊C4C2×D4711yescharacteristic
340C22×C22⋊C4—471no—
341C22×C22⋊C4—472no—
342C22×C22⋊C4—471no—
343C22×C22⋊C4—472no—
344C22×C22⋊C4—472no—
345C22×C22⋊C4—472no—
346C22×C22⋊C4—472no—
347C22×C22⋊C4—472no—
348C22×C42—471no—
349C22×C42—472no—
350C22×C42C2×D4711yescharacteristic
351#397—4716no—
352#394—4716no—
353#390—478no—
354#390—4716no—
355#390—478no—
356#388—4716no—
357#380—478no—
358C4×C4⋊C4—478no—
359C4×C22⋊C4—478no—
360C2×#95—472no—
361C2×#95—471no—
362C2×#95—472no—
363C2×#95—471no—
364C47C2×#249011yescharacteristic, complemented, Sylow
365C25—471no—
366C25—472no—
367C22×D4—474no—
368C22×D4—478no—
369C22×D4—948no—
370C22×D4—944no—
371C22×D4—944no—
372C22×D4—948no—
373C22×D4—944no—
374C22×D4—1884no—
375C23×C4—471no—
376C23×C4—472no—
377C23×C4—471no—
378C23×C4—471no—
379C23×C4—474no—
380C23×C4—944no—
381C23×C4—944no—
382C23×C4—944no—
383C23×C4C22×D4712yes—
384C23×C4C22×D4711yescharacteristic
385C23×C4C22×D4711yescharacteristic
386C23×C4C22×D4712yes—
387C23×C4C22×D4711yescharacteristic
388#125—232no—
389#121—9432no—
390C2×C4⋊C4—28no—
391C2×C4⋊C4—28no—
392C2×C4⋊C4—24no—
393C2×C4⋊C4—474no—
394C2×C4⋊C4—944no—
395C2×C4⋊C4C22⋊D4718yes—
396C2×C4⋊C4C22×D4714yes—
397C2×C4⋊C4C22×D4718yes—
398C2×C4⋊C4C22×D4718yes—
399C2×C4⋊C4C22×D4714yes—
400C2×C22⋊C4—478no—
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—944no—
409C2×C22⋊C4—948no—
410C2×C22⋊C4—944no—
411C2×C22⋊C4—948no—
412C2×C22⋊C4—1884no—
413C2×C22⋊C4—1884no—
414C2×C22⋊C4—1884no—
415C2×C22⋊C4—1884no—
416C2×C42—24no—
417C2×C42—474no—
418C2×C42—478no—
419C2×C42—944no—
420C2×C42—1884no—
421C2×C42C22×D4714yes—
422#95—474no—
423#95—474no—
424#95—478no—
425#95—478no—
426C24—474no—
427C24—472no—
428C24—474no—
429C24—942no—
430C24—942no—
431C24—942no—
432C24—942no—
433C24—942no—
434C24—942no—
435C24—944no—
436C24—944no—
437C24—944no—
438C24—944no—
439C24—944no—
440C24—944no—
441C24—944no—
442C24C23×D4711yescharacteristic
443C2×D4—9416no—
444C2×D4—9416no—
445C2×D4—948no—
446C2×D4—948no—
447C2×D4—948no—
448C2×D4—1888no—
449C2×D4—18832no—
450C2×D4—18816no—
451C22×C4—24no—
452C22×C4—24no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—24no—
456C22×C4—22no—
457C22×C4—24no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—24no—
461C22×C4—22no—
462C22×C4—474no—
463C22×C4—472no—
464C22×C4—472no—
465C22×C4—472no—
466C22×C4—474no—
467C22×C4—942no—
468C22×C4—942no—
469C22×C4—942no—
470C22×C4—942no—
471C22×C4—942no—
472C22×C4—944no—
473C22×C4—942no—
474C22×C4—942no—
475C22×C4—944no—
476C22×C4—944no—
477C22×C4—942no—
478C22×C4—948no—
479C22×C4—948no—
480C22×C4—948no—
481C22×C4—18816no—
482C22×C4#1062535414yes—
483C22×C4#1062535414yes—
484C22×C4C2×C22⋊D4718yes—
485C22×C4C2×C22⋊D4714yes—
486C22×C4C23×D4712yes—
487C22×C4C23×D4712yes—
488C22×C4C23×D4714yes—
489C22×C4C23×D4714yes—
490C22×C4C23×D4712yes—
491C4⋊C4—216no—
492C4⋊C4—216no—
493C4⋊C4—28no—
494C4⋊C4—948no—
495C4⋊C4C2×C22⋊D47116yes—
496C22⋊C4—948no—
497C22⋊C4—9416no—
498C22⋊C4—948no—
499C22⋊C4—9416no—
500C22⋊C4—1888no—
501C22⋊C4—1888no—
502C22⋊C4—1888no—
503C22⋊C4—1888no—
504C42—28no—
505C42—948no—
506C42—1888no—
507C23—474no—
508C23—948no—
509C23—948no—
510C23—948no—
511C23—942no—
512C23—942no—
513C23—942no—
514C23—942no—
515C23—942no—
516C23—942no—
517C23—944no—
518C23—944no—
519C23—944no—
520C23—944no—
521C23—944no—
522C23—944no—
523C23—944no—
524C23—1888no—
525C23—1888no—
526C23—1888no—
527C23—1888no—
528C23—1888no—
529C23—1888no—
530C23—1888no—
531C23—1884no—
532C23—1884no—
533C23—1884no—
534C23—1884no—
535C23C22×C22⋊D4712yes—
536C23#41×D4712yes—
537C23#41×D4712yes—
538C23C2×#1062535412yes—
539C23C2×#1062535412yes—
540C23#1309407012yes—
541C23#1309407012yes—
542C23C24×D4711yescharacteristic
543D4—18832no—
544D4—18816no—
545C2×C4—216no—
546C2×C4—24no—
547C2×C4—24no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—24no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—24no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—24no—
557C2×C4—44no—
558C2×C4—44no—
559C2×C4—48no—
560C2×C4—474no—
561C2×C4—948no—
562C2×C4—942no—
563C2×C4—942no—
564C2×C4—942no—
565C2×C4—942no—
566C2×C4—942no—
567C2×C4—944no—
568C2×C4—942no—
569C2×C4—942no—
570C2×C4—944no—
571C2×C4—942no—
572C2×C4—944no—
573C2×C4—1888no—
574C2×C4—1884no—
575C2×C4—1884no—
576C2×C4—18816no—
577C2×C4#13094013116yes—
578C2×C4C22×C22⋊D4718yes—
579C2×C4C22×C22⋊D4714yes—
580C2×C4C2×#1062535414yes—
581C2×C4C2×#1062535414yes—
582C22—944no—
583C22—944no—
584C22—944no—
585C22—37616no—
586C22—1884no—
587C22—1884no—
588C22—1884no—
589C22—1884no—
590C22—1884no—
591C22—1884no—
592C22—1884no—
593C22—1882no—
594C22—1882no—
595C22—1882no—
596C22—18816no—
597C22—1882no—
598C22C2×#1309401314yes—
599C22#1568965814yes—
600C22#1568972914yes—
601C22#1568976114yes—
602C22#1568979214yes—
603C22#1568982614yes—
604C22#1568982614yes—
605C22C23×C22⋊D4711yescharacteristic
606C22C2×#41×D4711yescharacteristic
607C22C2×#41×D4711yescharacteristic
608C22C22×#1062535411yescharacteristic
609C22C22×#1062535411yescharacteristic
610C22C2×#1309407011yescharacteristic
611C22C2×#1309407011yescharacteristic
612C4—28no—
613C4—42no—
614C4—42no—
615C4—44no—
616C4—944no—
617C4—1882no—
618C4—1882no—
619C4—1884no—
620C4C2×#1309401318yes—
621C2—3764no—
622C2—1884no—
623C2#2024774318yesdirect product factor
624C2C22×#1309401311yescharacteristic
625C2C2×#1568965811yescharacteristic
626C2C2×#1568972911yescharacteristic
627C2C2×#1568976111yescharacteristic
628C2C2×#1568979211yescharacteristic
629C2C2×#1568982611yescharacteristic
630C2C2×#1568982611yescharacteristic
subgroupG/H
centerC24C23×D47
commutator, cosocleC23×C47C25
FrattiniC23C24×D47
FittingC22×#125×C47C2
socleC24×C47C24
prime radicalC22×#13093999C2
hypercenterC22×#125D47
hypocenterC47C2×#2490
LCSG ▷ C23×C47 ▷ C47
UCSC1 ◁ C24 ◁ C22×#125

Complements

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

NH ≅ G/Ntype
2. #20247743C2direct (N × H)
3. C2×#15689049C2semidirect
4. C2×#15689040C2semidirect
5. C2×#15689025C2semidirect
6. C2×#15689024C2semidirect
7. C2×#15689024C2semidirect
8. C2×#15689013C2semidirect
9. C2×#15688980C2semidirect
10. C2×C4⋊C4×Dic47C2semidirect
11. C2×C4×(C2×D47)⋊C4C2semidirect
12. C22×#125×C47C2semidirect
29. #15689049C22semidirect
30. #15689040C22semidirect
31. #15689025C22semidirect
32. #15689024C22semidirect
33. #15689024C22semidirect
34. #15689013C22semidirect
35. #15688980C22semidirect
36. C4⋊C4×Dic47C22semidirect
37. C4×(C2×D47)⋊C4C22semidirect
43. C2×#125×C47C22semidirect
306. C22×#125D47semidirect
318. C2×C47#2490semidirect
332. C2×#125C2×D47semidirect
364. C47C2×#2490semidirect
623. C2#20247743direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#130939992
#202477432
C2×#156890492
C2×#156890402
C2×#156890252
C2×#156890242
C2×#156890132
C2×#156889802
C2×C4⋊C4×Dic472
C2×C4×(C2×D47)⋊C42
C22×#125×C472
C2×#249047