Finite Group Databasebeta

← C2×#20246965 #26384421

Subgroups — C2×#20246965 #26384421

Order 12032.

3894 conjugacy classes of subgroups.

The subgroup lattice has 660 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#13094012C211yesmaximal, characteristic, complemented
2C22×#13094003C211yesmaximal, characteristic, complemented
3C22×#13093998C211yesmaximal, characteristic
4C22×C4×D188C211yesmaximal, characteristic, complemented
5#20246965C2116yesmaximal, direct product factor
6C2×#15689052C212yesmaximal, complemented
7C2×#15689038C212yesmaximal, complemented
8C2×#15689029C211yesmaximal, characteristic, complemented
9C2×#15689021C211yesmaximal, characteristic, complemented
10C2×#15688999C212yesmaximal, complemented
11C2×#15688982C212yesmaximal, complemented
12C2×C4×C22⋊C4×C47C211yesmaximal, characteristic, complemented
13C23×Dic94C2211yescharacteristic
14C23×D188C2211yescharacteristic
15C23×C22⋊D47C2212yes—
16C23×C4×D47C2212yes—
17C2×#13094012—28no—
18C2×#13094012C2218yescomplemented
19C2×#13094003—28no—
20C2×#13094003C2218yescomplemented
21C2×#13093998—28no—
22C2×#13093998C2218yes—
23C2×C4×D188—28no—
24C2×C4×D188C2218yescomplemented
25C22×C4⋊Dic47C2211yescharacteristic
26C22×C4⋊Dic47C2211yescharacteristic
27C22×Dic47⋊C4C2212yes—
28C22×Dic47⋊C4C2212yes—
29C22×C22⋊Dic47C2212yes—
30C22×(C2×D47)⋊C4C2212yes—
31C22×(C2×D47)⋊C4C2212yes—
32C22×(C2×D47)⋊C4C2212yes—
33C22×(C2×D47)⋊C4C2212yes—
34#15689052C22116yescomplemented
35#15689038C22116yescomplemented
36#15689029C2218yescomplemented
37#15689021C2218yescomplemented
38#15688999C22116yescomplemented
39#15688982C22116yescomplemented
40C2×C47⋊#95C2212yes—
41C2×C47⋊#95C2212yes—
42C2×C47⋊#95C2212yes—
43C24×C4×C47C2211yescharacteristic
44C22×C22⋊C4×C47C2211yescharacteristic
45C22×C22⋊C4×C47C2211yescharacteristic
46C22×C42×C47C2211yescharacteristic
47C22×C42×C47C2211yescharacteristic
48C4×C22⋊C4×C47C2218yescomplemented
49C2×#95×C47C2211yescharacteristic
50C2×#95×C47C2211yescharacteristic
51C24×D47C2312yes—
52C22×Dic94—24no—
53C22×Dic94—24no—
54C22×Dic94—24no—
55C22×Dic94C2314yes—
56C22×D188—24no—
57C22×D188—24no—
58C22×D188D418yes—
59C22×D188C2314yes—
60C22×C22⋊D47—28no—
61C22×C22⋊D47—28no—
62C22×C22⋊D47—44no—
63C22×C22⋊D47C2318yes—
64C23×Dic47C2312yes—
65C23×Dic47C2312yes—
66C23×Dic47C2312yes—
67C22×C4×D47—28no—
68C22×C4×D47—28no—
69C22×C4×D47—44no—
70C22×C4×D47C2318yes—
71#13094012—232no—
72#13094003—232no—
73#13093998—232no—
74C4×D188—232no—
75C2×C4⋊Dic47—24no—
76C2×C4⋊Dic47D418yescomplemented
77C2×C4⋊Dic47C2314yes—
78C2×C4⋊Dic47C2314yes—
79C2×Dic47⋊C4—28no—
80C2×Dic47⋊C4—44no—
81C2×Dic47⋊C4C2318yes—
82C2×Dic47⋊C4C2318yes—
83C2×C22⋊Dic47—28no—
84C2×C22⋊Dic47C2318yes—
85C2×(C2×D47)⋊C4—28no—
86C2×(C2×D47)⋊C4—28no—
87C2×(C2×D47)⋊C4—28no—
88C2×(C2×D47)⋊C4—28no—
89C2×(C2×D47)⋊C4C2318yes—
90C2×(C2×D47)⋊C4C2318yes—
91C2×(C2×D47)⋊C4C2318yes—
92C2×(C2×D47)⋊C4C2318yes—
93C47⋊#95C2318yes—
94C47⋊#95C2318yes—
95C47⋊#95C2318yes—
96C25×C47C2311yescharacteristic
97C23×C4×C47—24no—
98C23×C4×C47—24no—
99C23×C4×C47—24no—
100C23×C4×C47C2311yescharacteristic
101C23×C4×C47C2311yescharacteristic
102C23×C4×C47C2311yescharacteristic
103C23×C4×C47C2311yescharacteristic
104C23×C4×C47C2314yes—
105C23×C4×C47C2311yescharacteristic
106C23×C4×C47C2311yescharacteristic
107C2×C22⋊C4×C47—24no—
108C2×C22⋊C4×C47—24no—
109C2×C22⋊C4×C47C2314yes—
110C2×C22⋊C4×C47C2314yes—
111C2×C42×C47—24no—
112C2×C42×C47—24no—
113C2×C42×C47C2314yes—
114C2×C42×C47C2314yes—
115#95×C47C2314yes—
116#95×C47C2314yes—
117C24×C47—22no—
118C24×C47—22no—
119C24×C47—22no—
120C24×C47—22no—
121C24×C47—22no—
122C24×C47—22no—
123C24×C47#4114yes—
124C24×C47C2412yes—
125C24×C47C2411yescharacteristic
126C23×D47—24no—
127C23×D47—24no—
128C23×D47—24no—
129C23×D47—24no—
130C23×D47—24no—
131C23×D47—24no—
132C23×D47C2×D418yes—
133C23×D47C2414yes—
134C22×C4×C47—22no—
135C22×C4×C47—22no—
136C22×C4×C47—22no—
137C22×C4×C47—22no—
138C22×C4×C47—22no—
139C22×C4×C47—22no—
140C22×C4×C47—22no—
141C22×C4×C47—22no—
142C22×C4×C47—22no—
143C22×C4×C47—22no—
144C22×C4×C47—22no—
145C22×C4×C47—22no—
146C22×C4×C47—22no—
147C22×C4×C47—22no—
148C22×C4×C47—28no—
149C22×C4×C47—28no—
150C22×C4×C47—28no—
151C22×C4×C47—416no—
152C22×C4×C47C2×D414yes—
153C22×C4×C47C2×D414yes—
154C22×C4×C47#4114yes—
155C22×C4×C47#4114yes—
156C22×C4×C47C2412yes—
157C22×C4×C47C2412yes—
158C22×C4×C47C2412yes—
159C22×C4×C47C2412yes—
160C22×C4×C47C2412yes—
161C22×C4×C47C2412yes—
162C2×C22⋊D47—216no—
163C2×C22⋊D47—216no—
164C2×C22⋊D47—48no—
165C2×C22⋊D47—432no—
166C22×Dic47—24no—
167C22×Dic47—24no—
168C22×Dic47—24no—
169C22×Dic47—24no—
170C22×Dic47—24no—
171C22×Dic47—24no—
172C22×Dic47—24no—
173C22×Dic47—24no—
174C22×Dic47C2×D418yes—
175C22×Dic47C2414yes—
176C22×Dic47C2414yes—
177C22×Dic47C2414yes—
178C2×D188—232no—
179C2×D188—28no—
180C2×D188—28no—
181C2×D188C2×D4116yes—
182C2×C4×D47—216no—
183C2×C4×D47—216no—
184C2×C4×D47—48no—
185C2×C4×D47—432no—
186C2×Dic94—28no—
187C2×Dic94—28no—
188C2×Dic94—28no—
189C2×Dic94—416no—
190C22⋊C4×C47—28no—
191C22⋊C4×C47—28no—
192C42×C47—28no—
193C42×C47—28no—
194C22⋊Dic47—216no—
195(C2×D47)⋊C4—216no—
196(C2×D47)⋊C4—216no—
197(C2×D47)⋊C4—216no—
198(C2×D47)⋊C4—216no—
199C4⋊Dic47—28no—
200C4⋊Dic47C2×D4116yescomplemented
201Dic47⋊C4—216no—
202Dic47⋊C4—48no—
203C23×C47—28no—
204C23×C47—28no—
205C23×C47—28no—
206C23×C47—22no—
207C23×C47—22no—
208C23×C47—22no—
209C23×C47—22no—
210C23×C47—22no—
211C23×C47—22no—
212C23×C47—44no—
213C23×C47—44no—
214C23×C47—44no—
215C23×C47—44no—
216C23×C47C22×D412yes—
217C23×C47C22×D412yes—
218C23×C47C2×#4114yes—
219C23×C47C2×#4112yes—
220C23×C47C2×#4112yes—
221C23×C47C2×#4112yes—
222C23×C47H(5,𝔽2)12yes—
223C23×C472−1+412yes—
224C23×C47C2511yescharacteristic
225C22×D47—216no—
226C22×D47—216no—
227C22×D47—216no—
228C22×D47—24no—
229C22×D47—24no—
230C22×D47—24no—
231C22×D47—24no—
232C22×D47—24no—
233C22×D47—24no—
234C22×D47—48no—
235C22×D47—48no—
236C22×D47—48no—
237C22×D47—48no—
238C22×D47C22×D418yes—
239C2×C4×C47—28no—
240C2×C4×C47—22no—
241C2×C4×C47—22no—
242C2×C4×C47—22no—
243C2×C4×C47—22no—
244C2×C4×C47—22no—
245C2×C4×C47—22no—
246C2×C4×C47—22no—
247C2×C4×C47—22no—
248C2×C4×C47—22no—
249C2×C4×C47—22no—
250C2×C4×C47—22no—
251C2×C4×C47—22no—
252C2×C4×C47—22no—
253C2×C4×C47—22no—
254C2×C4×C47—44no—
255C2×C4×C47—44no—
256C2×C4×C47—44no—
257C2×C4×C47—44no—
258C2×C4×C47—416no—
259C2×C4×C47#121116yes—
260C2×C4×C47C22×D414yes—
261C2×C4×C47C22×D414yes—
262C2×C4×C47C2×#4114yes—
263C2×C4×C47C2×#4114yes—
264C22⋊D47—432no—
265C2×Dic47—216no—
266C2×Dic47—24no—
267C2×Dic47—24no—
268C2×Dic47—24no—
269C2×Dic47—24no—
270C2×Dic47—24no—
271C2×Dic47—24no—
272C2×Dic47—24no—
273C2×Dic47—24no—
274C2×Dic47—48no—
275C2×Dic47—48no—
276C2×Dic47C22×D418yes—
277D188—232no—
278C4×D47—432no—
279Dic94—416no—
280C2×#2427—471nomaximal, Sylow
281C22×C47—24no—
282C22×C47—24no—
283C22×C47—24no—
284C22×C47—42no—
285C22×C47—416no—
286C22×C47—42no—
287C22×C47—42no—
288C22×C47—42no—
289C22×C47C2×#12114yes—
290C22×C47#52914yes—
291C22×C47#54214yes—
292C22×C47#54614yes—
293C22×C47#54614yes—
294C22×C47#54714yes—
295C22×C47#54814yes—
296C22×C47C23×D411yescharacteristic
297C22×C47C23×D411yescharacteristic
298C22×C47C22×#4111yescharacteristic
299C22×C47C22×#4111yescharacteristic
300C22×C47C22×#4111yescharacteristic
301C22×C47C2×H(5,𝔽2)11yescharacteristic
302C22×C47C2×2−1+411yescharacteristic
303C2×D47—28no—
304C2×D47—28no—
305C2×D47—28no—
306C2×D47—44no—
307C2×D47—432no—
308C2×D47—44no—
309C2×D47—44no—
310C2×D47—44no—
311C4×C47—24no—
312C4×C47—42no—
313C4×C47—42no—
314C4×C47—42no—
315C4×C47—42no—
316C4×C47C2×#12118yes—
317Dic47—28no—
318Dic47—44no—
319Dic47—44no—
320C22×#124—471no—
321C22×C22⋊Q8—471no—
322C22×#121—471no—
323C22×C4×D4—471no—
324#2427—4716no—
325C2×#397—472no—
326C2×#394—472no—
327C2×#391—471no—
328C2×#389—471no—
329C2×#386—472no—
330C2×#380—472no—
331C2×C4×C22⋊C4D4711yescharacteristic, complemented
332C2×C47—44no—
333C2×C47#242718yescomplemented
334C2×C47C22×#12111yescharacteristic
335C2×C47C2×#52911yescharacteristic
336C2×C47C2×#54211yescharacteristic
337C2×C47C2×#54611yescharacteristic
338C2×C47C2×#54611yescharacteristic
339C2×C47C2×#54711yescharacteristic
340C2×C47C2×#54811yescharacteristic
341D47—48no—
342C23×Q8—471no—
343C23×D4—472no—
344C23×D4—471no—
345C24×C4—472no—
346C24×C4C2×D4711yescharacteristic
347C2×#124—478no—
348C2×#124—948no—
349C2×C22⋊Q8—478no—
350C2×C22⋊Q8—948no—
351C2×#121—478no—
352C2×#121—948no—
353C2×C4×D4—478no—
354C2×C4×D4—948no—
355C22×C4⋊C4—471no—
356C22×C4⋊C4—472no—
357C22×C4⋊C4—471no—
358C22×C4⋊C4—472no—
359C22×C22⋊C4—472no—
360C22×C22⋊C4—472no—
361C22×C22⋊C4—472no—
362C22×C22⋊C4—472no—
363C22×C22⋊C4—472no—
364C22×C22⋊C4C2×D4711yescharacteristic
365C22×C22⋊C4C2×D4711yescharacteristic
366C22×C42C2×D4711yescharacteristic
367C22×C42C2×D4711yescharacteristic
368#397—4716no—
369#394—4716no—
370#391—478no—
371#389—478no—
372#386—4716no—
373#380—4716no—
374C4×C22⋊C4C2×D4718yescomplemented
375C2×#95—472no—
376C2×#95—472no—
377C2×#95—472no—
378C2×#95C2×D4711yescharacteristic
379C2×#95C2×D4711yescharacteristic
380C47C2×#242711yescharacteristic, complemented, Sylow
381C25—472no—
382C25C22×D4711yescharacteristic
383C22×Q8—474no—
384C22×Q8—944no—
385C22×Q8—944no—
386C22×Q8—944no—
387C22×D4—478no—
388C22×D4—478no—
389C22×D4—474no—
390C22×D4—948no—
391C22×D4—948no—
392C22×D4—944no—
393C22×D4—944no—
394C22×D4—1884no—
395C23×C4—24no—
396C23×C4—24no—
397C23×C4—24no—
398C23×C4—472no—
399C23×C4—472no—
400C23×C4—472no—
401C23×C4—478no—
402C23×C4—948no—
403C23×C4—948no—
404C23×C4—1884no—
405C23×C4C22×D4714yes—
406C23×C4C22×D4711yescharacteristic
407C23×C4C22×D4711yescharacteristic
408C23×C4C22×D4711yescharacteristic
409C23×C4C22×D4711yescharacteristic
410C23×C4C22×D4711yescharacteristic
411C23×C4C22×D4711yescharacteristic
412#124—9432no—
413C22⋊Q8—9432no—
414#121—9432no—
415C4×D4—9432no—
416C2×C4⋊C4—478no—
417C2×C4⋊C4—474no—
418C2×C4⋊C4—474no—
419C2×C4⋊C4—478no—
420C2×C4⋊C4—478no—
421C2×C4⋊C4—948no—
422C2×C4⋊C4—944no—
423C2×C4⋊C4—1884no—
424C2×C22⋊C4—24no—
425C2×C22⋊C4—24no—
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⋊C4C22×D4714yes—
437C2×C22⋊C4C22×D4714yes—
438C2×C42—24no—
439C2×C42—24no—
440C2×C42C22×D4714yes—
441C2×C42C22×D4714yes—
442#95—478no—
443#95—478no—
444#95—478no—
445#95C22×D4714yes—
446#95C22×D4714yes—
447C24—22no—
448C24—22no—
449C24—22no—
450C24—22no—
451C24—22no—
452C24—22no—
453C24—478no—
454C24—474no—
455C24—944no—
456C24—944no—
457C24—944no—
458C24—944no—
459C24—944no—
460C24—944no—
461C24#1062535014yes—
462C24C23×D4712yes—
463C24C23×D4711yescharacteristic
464C2×Q8—948no—
465C2×Q8—948no—
466C2×Q8—948no—
467C2×Q8—18816no—
468C2×D4—4716no—
469C2×D4—9416no—
470C2×D4—9416no—
471C2×D4—9432no—
472C2×D4—948no—
473C2×D4—948no—
474C2×D4—1888no—
475C2×D4—18832no—
476C22×C4—22no—
477C22×C4—22no—
478C22×C4—22no—
479C22×C4—22no—
480C22×C4—22no—
481C22×C4—22no—
482C22×C4—22no—
483C22×C4—22no—
484C22×C4—22no—
485C22×C4—22no—
486C22×C4—22no—
487C22×C4—22no—
488C22×C4—22no—
489C22×C4—22no—
490C22×C4—28no—
491C22×C4—28no—
492C22×C4—28no—
493C22×C4—416no—
494C22×C4—478no—
495C22×C4—474no—
496C22×C4—474no—
497C22×C4—474no—
498C22×C4—9416no—
499C22×C4—944no—
500C22×C4—944no—
501C22×C4—944no—
502C22×C4—944no—
503C22×C4—944no—
504C22×C4—944no—
505C22×C4—944no—
506C22×C4—944no—
507C22×C4—9416no—
508C22×C4—1888no—
509C22×C4—18832no—
510C22×C4#1062535014yes—
511C22×C4#1062535014yes—
512C22×C4D4×D4714yes—
513C22×C4D4×D4714yes—
514C22×C4C23×D4712yes—
515C22×C4C23×D4712yes—
516C22×C4C23×D4712yes—
517C22×C4C23×D4712yes—
518C22×C4C23×D4712yes—
519C22×C4C23×D4712yes—
520C4⋊C4—4716no—
521C4⋊C4—9416no—
522C4⋊C4—948no—
523C4⋊C4—1888no—
524C22⋊C4—28no—
525C22⋊C4—28no—
526C22⋊C4—9416no—
527C22⋊C4—9416no—
528C22⋊C4—9416no—
529C22⋊C4—9416no—
530C22⋊C4—9416no—
531C42—28no—
532C42—28no—
533C23—28no—
534C23—28no—
535C23—28no—
536C23—22no—
537C23—22no—
538C23—22no—
539C23—22no—
540C23—22no—
541C23—22no—
542C23—44no—
543C23—44no—
544C23—44no—
545C23—44no—
546C23—478no—
547C23—9416no—
548C23—9416no—
549C23—9416no—
550C23—944no—
551C23—944no—
552C23—944no—
553C23—944no—
554C23—944no—
555C23—944no—
556C23—1888no—
557C23—1888no—
558C23—1888no—
559C23—1888no—
560C23C2×D4×D4712yes—
561C23C2×D4×D4712yes—
562C23C2×#1062535014yes—
563C23C2×#1062535012yes—
564C23C2×#1062535012yes—
565C23C2×#1062535012yes—
566C23#1309407012yes—
567C23#1309407112yes—
568C23C24×D4711yescharacteristic
569Q8—18816no—
570D4—9432no—
571D4—18832no—
572C2×C4—28no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—44no—
588C2×C4—44no—
589C2×C4—44no—
590C2×C4—44no—
591C2×C4—416no—
592C2×C4—478no—
593C2×C4—9416no—
594C2×C4—944no—
595C2×C4—944no—
596C2×C4—944no—
597C2×C4—944no—
598C2×C4—944no—
599C2×C4—944no—
600C2×C4—944no—
601C2×C4—944no—
602C2×C4—18832no—
603C2×C4—1888no—
604C2×C4—1888no—
605C2×C4#13093997116yes—
606C2×C4C2×D4×D4714yes—
607C2×C4C2×D4×D4714yes—
608C2×C4C2×#1062535014yes—
609C2×C4C2×#1062535014yes—
610C22—24no—
611C22—24no—
612C22—24no—
613C22—42no—
614C22—42no—
615C22—42no—
616C22—416no—
617C22—42no—
618C22—948no—
619C22—948no—
620C22—948no—
621C22—1884no—
622C22—1884no—
623C22—1884no—
624C22—18832no—
625C22—1884no—
626C22C2×#1309399714yes—
627C22#1568965714yes—
628C22#1568974414yes—
629C22#1568976914yes—
630C22#1568976914yes—
631C22#1568977814yes—
632C22#1568978214yes—
633C22C22×D4×D4711yescharacteristic
634C22C22×D4×D4711yescharacteristic
635C22C22×#1062535011yescharacteristic
636C22C22×#1062535011yescharacteristic
637C22C22×#1062535011yescharacteristic
638C22C2×#1309407011yescharacteristic
639C22C2×#1309407111yescharacteristic
640C4—24no—
641C4—42no—
642C4—42no—
643C4—42no—
644C4—42no—
645C4—948no—
646C4—1884no—
647C4—1884no—
648C4C2×#1309399718yes—
649C2—44no—
650C2—1888no—
651C2#2024696518yesdirect product factor
652C2C22×#1309399711yescharacteristic
653C2C2×#1568965711yescharacteristic
654C2C2×#1568974411yescharacteristic
655C2C2×#1568976911yescharacteristic
656C2C2×#1568976911yescharacteristic
657C2C2×#1568977811yescharacteristic
658C2C2×#1568978211yescharacteristic
subgroupG/H
centerC24C23×D47
commutator, cosocleC23×C47C25
FrattiniC23C24×D47
FittingC2×C4×C22⋊C4×C47C2
socleC24×C47C24
prime radicalC22×#13093998C2
hypercenterC2×C4×C22⋊C4D47
hypocenterC47C2×#2427
LCSG ▷ C23×C47 ▷ C47
UCSC1 ◁ C24 ◁ C2×C4×C22⋊C4

Complements

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

NH ≅ G/Ntype
1. C22×#13094012C2semidirect
2. C22×#13094003C2semidirect
4. C22×C4×D188C2semidirect
5. #20246965C2direct (N × H)
6. C2×#15689052C2semidirect
7. C2×#15689038C2semidirect
8. C2×#15689029C2semidirect
9. C2×#15689021C2semidirect
10. C2×#15688999C2semidirect
11. C2×#15688982C2semidirect
12. C2×C4×C22⋊C4×C47C2semidirect
18. C2×#13094012C22semidirect
20. C2×#13094003C22semidirect
24. C2×C4×D188C22semidirect
34. #15689052C22semidirect
35. #15689038C22semidirect
36. #15689029C22semidirect
37. #15689021C22semidirect
38. #15688999C22semidirect
39. #15688982C22semidirect
48. C4×C22⋊C4×C47C22semidirect
76. C2×C4⋊Dic47D4semidirect
200. C4⋊Dic47C2×D4semidirect
331. C2×C4×C22⋊C4D47semidirect
333. C2×C47#2427semidirect
374. C4×C22⋊C4C2×D47semidirect
380. C47C2×#2427semidirect
651. C2#20246965direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#130940122
C22×#130940032
C22×#130939982
C22×C4×D1882
#202469652
C2×#156890522
C2×#156890382
C2×#156890292
C2×#156890212
C2×#156889992
C2×#156889822
C2×C4×C22⋊C4×C472
C2×#242747