Finite Group Databasebeta

← C2×#20249182 #26383308

Subgroups — C2×#20249182 #26383308

Order 12032.

3110 conjugacy classes of subgroups.

The subgroup lattice has 636 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#13093999C211yesmaximal, characteristic
2C22×#13093981C211yesmaximal, characteristic, complemented
3#20249182C2116yesmaximal, direct product factor
4C2×#15689058C212yesmaximal, complemented
5C2×#15689052C212yesmaximal, complemented
6C2×#15689041C212yesmaximal, complemented
7C2×#15689040C212yesmaximal, complemented
8C2×#15689025C211yesmaximal, characteristic, complemented
9C2×#15689024C211yesmaximal, characteristic, complemented
10C2×#15689019C211yesmaximal, characteristic, complemented
11C2×#15688980C211yesmaximal, characteristic, complemented
12C2×C47×#389C211yesmaximal, characteristic, complemented
13C23×D188C2211yescharacteristic
14C23×D188C2212yes—
15C23×C4×D47C2211yescharacteristic
16C2×#13093999—28no—
17C2×#13093999C2218yes—
18C2×#13093981—28no—
19C2×#13093981C2218yescomplemented
20C22×C4⋊Dic47C2211yescharacteristic
21C22×Dic47⋊C4C2211yescharacteristic
22C22×Dic47⋊C4C2211yescharacteristic
23C22×Dic47⋊C4C2212yes—
24C22×(C2×D47)⋊C4C2211yescharacteristic
25C22×(C2×D47)⋊C4C2211yescharacteristic
26C22×(C2×D47)⋊C4C2212yes—
27C22×(C2×D47)⋊C4C2212yes—
28C22×(C2×D47)⋊C4C2212yes—
29C22×(C2×D47)⋊C4C2212yes—
30C22×(C2×D47)⋊C4C2212yes—
31C22×(C2×D47)⋊C4C2212yes—
32C22×C4×Dic47C2211yescharacteristic
33#15689058C22116yescomplemented
34#15689052C22116yescomplemented
35#15689041C22116yescomplemented
36#15689040C22116yescomplemented
37#15689025C2218yescomplemented
38#15689024C2218yescomplemented
39#15689019C2218yescomplemented
40#15688980C2218yescomplemented
41C2×C47⋊#95C2212yes—
42C2×C47⋊#95C2212yes—
43C22×C4⋊C4×C47C2211yescharacteristic
44C22×C4⋊C4×C47C2211yescharacteristic
45C22×C4⋊C4×C47C2212yes—
46C22×C42×C47C2211yescharacteristic
47C47×#389C2218yescomplemented
48C2×#95×C47C2212yes—
49C24×D47C2312yes—
50C24×D47C2311yescharacteristic
51C22×D188—28no—
52C22×D188—24no—
53C22×D188—24no—
54C22×D188—24no—
55C22×D188—44no—
56C22×D188—44no—
57C22×D188C2314yes—
58C22×D188C2318yes—
59C23×Dic47C2311yescharacteristic
60C23×Dic47C2312yes—
61C23×Dic47C2311yescharacteristic
62C23×Dic47C2311yescharacteristic
63C22×C4×D47—24no—
64C22×C4×D47—24no—
65C22×C4×D47D418yes—
66C22×C4×D47C2314yes—
67#13093999—232no—
68#13093981—232no—
69C2×C4⋊Dic47—24no—
70C2×C4⋊Dic47C2314yes—
71C2×Dic47⋊C4—24no—
72C2×Dic47⋊C4—24no—
73C2×Dic47⋊C4—44no—
74C2×Dic47⋊C4C2314yes—
75C2×Dic47⋊C4C2314yes—
76C2×Dic47⋊C4C2318yes—
77C2×(C2×D47)⋊C4—24no—
78C2×(C2×D47)⋊C4—24no—
79C2×(C2×D47)⋊C4—28no—
80C2×(C2×D47)⋊C4—44no—
81C2×(C2×D47)⋊C4—44no—
82C2×(C2×D47)⋊C4—44no—
83C2×(C2×D47)⋊C4—44no—
84C2×(C2×D47)⋊C4—44no—
85C2×(C2×D47)⋊C4C2314yes—
86C2×(C2×D47)⋊C4C2314yes—
87C2×(C2×D47)⋊C4C2318yes—
88C2×(C2×D47)⋊C4C2318yes—
89C2×(C2×D47)⋊C4C2318yes—
90C2×(C2×D47)⋊C4C2318yes—
91C2×(C2×D47)⋊C4C2318yes—
92C2×(C2×D47)⋊C4C2318yes—
93C2×C4×Dic47—24no—
94C2×C4×Dic47C2314yes—
95C47⋊#95C2318yes—
96C47⋊#95C2318yes—
97C23×C4×C47C2312yes—
98C23×C4×C47C2312yes—
99C23×C4×C47C2311yescharacteristic
100C23×C4×C47C2311yescharacteristic
101C23×C4×C47C2311yescharacteristic
102C2×C4⋊C4×C47—24no—
103C2×C4⋊C4×C47—24no—
104C2×C4⋊C4×C47—44no—
105C2×C4⋊C4×C47C2314yes—
106C2×C4⋊C4×C47C2314yes—
107C2×C4⋊C4×C47C2318yes—
108C2×C42×C47—24no—
109C2×C42×C47C2314yes—
110#95×C47C2318yes—
111C24×C47C2411yescharacteristic
112C23×D47—22no—
113C23×D47—22no—
114C23×D47—22no—
115C23×D47—22no—
116C23×D47—22no—
117C23×D47—22no—
118C23×D47—24no—
119C23×D47—24no—
120C23×D47—24no—
121C23×D47—24no—
122C23×D47—24no—
123C23×D47—24no—
124C23×D47—24no—
125C23×D47C2×D414yes—
126C23×D47C2414yes—
127C23×D47C2412yes—
128C22×C4×C47—22no—
129C22×C4×C47—22no—
130C22×C4×C47—24no—
131C22×C4×C47—24no—
132C22×C4×C47—22no—
133C22×C4×C47—24no—
134C22×C4×C47—24no—
135C22×C4×C47—22no—
136C22×C4×C47—22no—
137C22×C4×C47—24no—
138C22×C4×C47—24no—
139C22×C4×C47—22no—
140C22×C4×C47C2×D414yes—
141C22×C4×C47#4114yes—
142C22×C4×C47#4114yes—
143C22×C4×C47C2412yes—
144C22×C4×C47C2412yes—
145C22×C4×C47C2414yes—
146C22×C4×C47C2414yes—
147C22×C4×C47C2412yes—
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—22no—
159C22×Dic47C2×D414yes—
160C22×Dic47C2412yes—
161C22×Dic47C2414yes—
162C22×Dic47C2412yes—
163C22×Dic47C2412yes—
164C2×D188—216no—
165C2×D188—28no—
166C2×D188—28no—
167C2×D188—28no—
168C2×D188—48no—
169C2×D188—48no—
170C2×D188—432no—
171C2×D188—416no—
172C2×C4×D47—232no—
173C2×C4×D47—28no—
174C2×C4×D47—28no—
175C2×C4×D47C2×D4116yes—
176C4⋊C4×C47—28no—
177C4⋊C4×C47—28no—
178C4⋊C4×C47—48no—
179C42×C47—28no—
180(C2×D47)⋊C4—28no—
181(C2×D47)⋊C4—28no—
182(C2×D47)⋊C4—216no—
183(C2×D47)⋊C4—48no—
184(C2×D47)⋊C4—48no—
185(C2×D47)⋊C4—48no—
186(C2×D47)⋊C4—48no—
187(C2×D47)⋊C4—48no—
188C4⋊Dic47—28no—
189Dic47⋊C4—28no—
190Dic47⋊C4—28no—
191Dic47⋊C4—48no—
192C4×Dic47—28no—
193C23×C47C22×D412yes—
194C23×C47C2×#4112yes—
195C23×C47C2×#4112yes—
196C23×C47H(5,𝔽2)12yes—
197C23×C47H(5,𝔽2)12yes—
198C23×C47H(5,𝔽2)12yes—
199C23×C472−1+412yes—
200C23×C47C2511yescharacteristic
201C22×D47—28no—
202C22×D47—28no—
203C22×D47—28no—
204C22×D47—22no—
205C22×D47—22no—
206C22×D47—22no—
207C22×D47—22no—
208C22×D47—22no—
209C22×D47—22no—
210C22×D47—24no—
211C22×D47—24no—
212C22×D47—24no—
213C22×D47—24no—
214C22×D47—24no—
215C22×D47—24no—
216C22×D47—24no—
217C22×D47—44no—
218C22×D47—44no—
219C22×D47—44no—
220C22×D47—44no—
221C22×D47—48no—
222C22×D47—48no—
223C22×D47—48no—
224C22×D47—48no—
225C22×D47—48no—
226C22×D47—48no—
227C22×D47—48no—
228C22×D47C22×D414yes—
229C2×C4×C47—22no—
230C2×C4×C47—22no—
231C2×C4×C47—24no—
232C2×C4×C47—24no—
233C2×C4×C47—24no—
234C2×C4×C47—22no—
235C2×C4×C47—22no—
236C2×C4×C47—24no—
237C2×C4×C47—24no—
238C2×C4×C47—22no—
239C2×C4×C47—24no—
240C2×C4×C47—22no—
241C2×C4×C47—48no—
242C2×C4×C47—44no—
243C2×C4×C47—44no—
244C2×C4×C47—48no—
245C2×C4×C47#123116yes—
246C2×C4×C47C22×D414yes—
247C2×C4×C47C2×#4114yes—
248C2×C4×C47C2×#4114yes—
249C2×Dic47—28no—
250C2×Dic47—22no—
251C2×Dic47—22no—
252C2×Dic47—22no—
253C2×Dic47—22no—
254C2×Dic47—24no—
255C2×Dic47—22no—
256C2×Dic47—24no—
257C2×Dic47—22no—
258C2×Dic47—22no—
259C2×Dic47—24no—
260C2×Dic47—22no—
261C2×Dic47—44no—
262C2×Dic47—44no—
263C2×Dic47—48no—
264C2×Dic47C22×D414yes—
265D188—432no—
266D188—416no—
267C4×D47—232no—
268C2×#2624—471nomaximal, Sylow
269C22×C47C2×#12314yes—
270C22×C47#53514yes—
271C22×C47#53714yes—
272C22×C47#54014yes—
273C22×C47#54014yes—
274C22×C47#54214yes—
275C22×C47#54214yes—
276C22×C47C23×D411yescharacteristic
277C22×C47C22×#4111yescharacteristic
278C22×C47C22×#4111yescharacteristic
279C22×C47C2×H(5,𝔽2)11yescharacteristic
280C22×C47C2×H(5,𝔽2)11yescharacteristic
281C22×C47C2×H(5,𝔽2)11yescharacteristic
282C22×C47C2×2−1+411yescharacteristic
283C2×D47—24no—
284C2×D47—24no—
285C2×D47—24no—
286C2×D47—416no—
287C2×D47—42no—
288C2×D47—42no—
289C2×D47—42no—
290C2×D47—42no—
291C2×D47—44no—
292C2×D47—44no—
293C2×D47—44no—
294C2×D47—44no—
295C2×D47—44no—
296C2×D47—44no—
297C2×D47—44no—
298C2×D47—816no—
299C4×C47—44no—
300C4×C47—42no—
301C4×C47—42no—
302C4×C47—44no—
303C4×C47C2×#12318yes—
304Dic47—24no—
305Dic47—42no—
306Dic47—42no—
307Dic47—44no—
308C22×#121—471no—
309C22×#117—471no—
310#2624—4716no—
311C2×#399—472no—
312C2×#397—472no—
313C2×#394—472no—
314C2×#394—472no—
315C2×#390—471no—
316C2×#390—471no—
317C2×#389—471no—
318C2×#389D4711yescharacteristic, complemented
319C2×#380—471no—
320C2×C47#262418yescomplemented
321C2×C47C22×#12311yescharacteristic
322C2×C47C2×#53511yescharacteristic
323C2×C47C2×#53711yescharacteristic
324C2×C47C2×#54011yescharacteristic
325C2×C47C2×#54011yescharacteristic
326C2×C47C2×#54211yescharacteristic
327C2×C47C2×#54211yescharacteristic
328D47—44no—
329D47—84no—
330C23×D4—471no—
331C23×D4—472no—
332C24×C4—471no—
333C2×#121—478no—
334C2×#121—948no—
335C2×#117—478no—
336C2×#117—948no—
337C22×C4⋊C4—471no—
338C22×C4⋊C4—471no—
339C22×C4⋊C4—471no—
340C22×C4⋊C4—472no—
341C22×C4⋊C4C2×D4711yescharacteristic
342C22×C4⋊C4C2×D4711yescharacteristic
343C22×C4⋊C4C2×D4712yes—
344C22×C22⋊C4—471no—
345C22×C22⋊C4—471no—
346C22×C22⋊C4—472no—
347C22×C22⋊C4—472no—
348C22×C22⋊C4—472no—
349C22×C22⋊C4—472no—
350C22×C22⋊C4—472no—
351C22×C22⋊C4—472no—
352C22×C42—471no—
353C22×C42C2×D4711yescharacteristic
354#399—4716no—
355#397—4716no—
356#394—4716no—
357#394—4716no—
358#390—478no—
359#390—478no—
360#389—478no—
361#389C2×D4718yescomplemented
362#380—478no—
363C2×#95—472no—
364C2×#95—472no—
365C2×#95C2×D4712yes—
366C47C2×#262411yescharacteristic, complemented, Sylow
367C25—471no—
368C25—472no—
369C22×D4—474no—
370C22×D4—478no—
371C22×D4—948no—
372C22×D4—944no—
373C22×D4—944no—
374C22×D4—944no—
375C22×D4—1884no—
376C22×D4—1884no—
377C23×C4—471no—
378C23×C4—478no—
379C23×C4—474no—
380C23×C4—471no—
381C23×C4—472no—
382C23×C4—471no—
383C23×C4—944no—
384C23×C4—944no—
385C23×C4C22×D4711yescharacteristic
386C23×C4C22×D4711yescharacteristic
387C23×C4C22×D4712yes—
388C23×C4C22×D4712yes—
389C23×C4C22×D4711yescharacteristic
390#121—9432no—
391#117—9432no—
392C2×C4⋊C4—24no—
393C2×C4⋊C4—24no—
394C2×C4⋊C4—44no—
395C2×C4⋊C4—474no—
396C2×C4⋊C4—474no—
397C2×C4⋊C4—474no—
398C2×C4⋊C4—478no—
399C2×C4⋊C4—944no—
400C2×C4⋊C4—944no—
401C2×C4⋊C4—944no—
402C2×C4⋊C4—1884no—
403C2×C4⋊C4C22×D4714yes—
404C2×C4⋊C4C22×D4714yes—
405C2×C4⋊C4C22×D4718yes—
406C2×C22⋊C4—474no—
407C2×C22⋊C4—474no—
408C2×C22⋊C4—478no—
409C2×C22⋊C4—478no—
410C2×C22⋊C4—478no—
411C2×C22⋊C4—478no—
412C2×C22⋊C4—478no—
413C2×C22⋊C4—478no—
414C2×C22⋊C4—948no—
415C2×C22⋊C4—944no—
416C2×C22⋊C4—944no—
417C2×C22⋊C4—1884no—
418C2×C22⋊C4—1884no—
419C2×C22⋊C4—1884no—
420C2×C22⋊C4—1884no—
421C2×C22⋊C4—1884no—
422C2×C42—24no—
423C2×C42—474no—
424C2×C42—944no—
425C2×C42C22×D4714yes—
426#95—478no—
427#95—478no—
428#95C22×D4718yes—
429C24—474no—
430C24—472no—
431C24—474no—
432C24—942no—
433C24—942no—
434C24—942no—
435C24—942no—
436C24—942no—
437C24—942no—
438C24—944no—
439C24—944no—
440C24—944no—
441C24—944no—
442C24—944no—
443C24—944no—
444C24—944no—
445C24C23×D4711yescharacteristic
446C2×D4—9416no—
447C2×D4—948no—
448C2×D4—948no—
449C2×D4—948no—
450C2×D4—1888no—
451C2×D4—1888no—
452C2×D4—18832no—
453C2×D4—18816no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—24no—
457C22×C4—24no—
458C22×C4—24no—
459C22×C4—22no—
460C22×C4—24no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—24no—
464C22×C4—24no—
465C22×C4—22no—
466C22×C4—474no—
467C22×C4—472no—
468C22×C4—4716no—
469C22×C4—472no—
470C22×C4—474no—
471C22×C4—472no—
472C22×C4—942no—
473C22×C4—942no—
474C22×C4—9432no—
475C22×C4—948no—
476C22×C4—948no—
477C22×C4—942no—
478C22×C4—942no—
479C22×C4—944no—
480C22×C4—942no—
481C22×C4—944no—
482C22×C4—942no—
483C22×C4—944no—
484C22×C4—942no—
485C22×C4—942no—
486C22×C4#1062535014yes—
487C22×C4D4×D4714yes—
488C22×C4#1062535414yes—
489C22×C4C23×D4714yes—
490C22×C4C23×D4714yes—
491C22×C4C23×D4712yes—
492C22×C4C23×D4712yes—
493C22×C4C23×D4712yes—
494C4⋊C4—28no—
495C4⋊C4—28no—
496C4⋊C4—48no—
497C4⋊C4—948no—
498C4⋊C4—948no—
499C4⋊C4—948no—
500C4⋊C4—1888no—
501C22⋊C4—9416no—
502C22⋊C4—948no—
503C22⋊C4—948no—
504C22⋊C4—1888no—
505C22⋊C4—1888no—
506C22⋊C4—1888no—
507C22⋊C4—1888no—
508C22⋊C4—1888no—
509C42—28no—
510C42—948no—
511C23—474no—
512C23—948no—
513C23—948no—
514C23—942no—
515C23—942no—
516C23—942no—
517C23—942no—
518C23—942no—
519C23—942no—
520C23—948no—
521C23—944no—
522C23—944no—
523C23—944no—
524C23—944no—
525C23—944no—
526C23—944no—
527C23—944no—
528C23—1888no—
529C23—1888no—
530C23—1888no—
531C23—1888no—
532C23—1888no—
533C23—1888no—
534C23—1888no—
535C23—1884no—
536C23—1884no—
537C23—1884no—
538C23—1884no—
539C23C2×D4×D4712yes—
540C23C2×#1062535012yes—
541C23C2×#1062535412yes—
542C23#1309406912yes—
543C23#1309407012yes—
544C23#1309407012yes—
545C23#1309407212yes—
546C23C24×D4711yescharacteristic
547D4—18832no—
548D4—18816no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—24no—
552C2×C4—24no—
553C2×C4—24no—
554C2×C4—22no—
555C2×C4—24no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—24no—
559C2×C4—24no—
560C2×C4—22no—
561C2×C4—44no—
562C2×C4—48no—
563C2×C4—44no—
564C2×C4—48no—
565C2×C4—474no—
566C2×C4—948no—
567C2×C4—942no—
568C2×C4—942no—
569C2×C4—9432no—
570C2×C4—942no—
571C2×C4—942no—
572C2×C4—944no—
573C2×C4—942no—
574C2×C4—942no—
575C2×C4—944no—
576C2×C4—944no—
577C2×C4—942no—
578C2×C4—942no—
579C2×C4—1888no—
580C2×C4—1884no—
581C2×C4—1884no—
582C2×C4#13094009116yes—
583C2×C4C2×D4×D4714yes—
584C2×C4C2×#1062535014yes—
585C2×C4C2×#1062535414yes—
586C22—944no—
587C22—944no—
588C22—944no—
589C22—37616no—
590C22—1884no—
591C22—1884no—
592C22—1884no—
593C22—1884no—
594C22—1884no—
595C22—1884no—
596C22—1884no—
597C22—1882no—
598C22—18816no—
599C22—1882no—
600C22—1882no—
601C22—1882no—
602C22C2×#1309400914yes—
603C22#1568968914yes—
604C22#1568970314yes—
605C22#1568972414yes—
606C22#1568972914yes—
607C22#1568974514yes—
608C22#1568974714yes—
609C22C22×D4×D4711yescharacteristic
610C22C22×#1062535011yescharacteristic
611C22C22×#1062535411yescharacteristic
612C22C2×#1309406911yescharacteristic
613C22C2×#1309407011yescharacteristic
614C22C2×#1309407011yescharacteristic
615C22C2×#1309407211yescharacteristic
616C4—44no—
617C4—42no—
618C4—42no—
619C4—44no—
620C4—944no—
621C4—1882no—
622C4—1882no—
623C4—1884no—
624C4C2×#1309400918yes—
625C2—3764no—
626C2—1884no—
627C2#2024918218yesdirect product factor
628C2C22×#1309400911yescharacteristic
629C2C2×#1568968911yescharacteristic
630C2C2×#1568970311yescharacteristic
631C2C2×#1568972411yescharacteristic
632C2C2×#1568972911yescharacteristic
633C2C2×#1568974511yescharacteristic
634C2C2×#1568974711yescharacteristic
subgroupG/H
centerC24C23×D47
commutator, cosocleC23×C47C25
FrattiniC23C24×D47
FittingC2×C47×#389C2
socleC24×C47C24
prime radicalC22×#13093999C2
hypercenterC2×#389D47
hypocenterC47C2×#2624
LCSG ▷ C23×C47 ▷ C47
UCSC1 ◁ C24 ◁ C2×#389

Complements

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

NH ≅ G/Ntype
2. C22×#13093981C2semidirect
3. #20249182C2direct (N × H)
4. C2×#15689058C2semidirect
5. C2×#15689052C2semidirect
6. C2×#15689041C2semidirect
7. C2×#15689040C2semidirect
8. C2×#15689025C2semidirect
9. C2×#15689024C2semidirect
10. C2×#15689019C2semidirect
11. C2×#15688980C2semidirect
12. C2×C47×#389C2semidirect
19. C2×#13093981C22semidirect
33. #15689058C22semidirect
34. #15689052C22semidirect
35. #15689041C22semidirect
36. #15689040C22semidirect
37. #15689025C22semidirect
38. #15689024C22semidirect
39. #15689019C22semidirect
40. #15688980C22semidirect
47. C47×#389C22semidirect
318. C2×#389D47semidirect
320. C2×C47#2624semidirect
361. #389C2×D47semidirect
366. C47C2×#2624semidirect
627. C2#20249182direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#130939992
C22×#130939812
#202491822
C2×#156890582
C2×#156890522
C2×#156890412
C2×#156890402
C2×#156890252
C2×#156890242
C2×#156890192
C2×#156889802
C2×C47×#3892
C2×#262447