Finite Group Databasebeta

← C2×#20247694 #26385330

Subgroups — C2×#20247694 #26385330

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×#13094009C212yesmaximal
2C22×C4×D188C211yesmaximal, characteristic
3C22×C4⋊C4×D47C211yesmaximal, characteristic, complemented
4#20247694C2116yesmaximal, direct product factor
5C2×#15689056C212yesmaximal, complemented
6C2×#15689052C212yesmaximal, complemented
7C2×#15689021C211yesmaximal, characteristic, complemented
8C2×#15688985C212yesmaximal, complemented
9C2×#15688980C211yesmaximal, characteristic, complemented
10C2×#15688980C211yesmaximal, characteristic, complemented
11C2×#15688980C211yesmaximal, characteristic, complemented
12C2×C47×#389C211yesmaximal, characteristic, complemented
13C23×D188C2211yescharacteristic
14C23×C4×D47C2212yes—
15C23×C4×D47C2211yescharacteristic
16C23×C4×D47C2211yescharacteristic
17C2×#13094009—216no—
18C2×#13094009C22116yes—
19C2×C4×D188—28no—
20C2×C4×D188C2218yes—
21C22×C4⋊Dic47C2211yescharacteristic
22C22×C4⋊Dic47C2211yescharacteristic
23C22×C4⋊Dic47C2211yescharacteristic
24C22×Dic47⋊C4C2212yes—
25C22×Dic47⋊C4C2212yes—
26C2×C4⋊C4×D47—28no—
27C2×C4⋊C4×D47C2218yescomplemented
28C22×(C2×D47)⋊C4C2211yescharacteristic
29C22×(C2×D47)⋊C4C2212yes—
30C22×(C2×D47)⋊C4C2211yescharacteristic
31C22×(C2×D47)⋊C4C2212yes—
32C22×(C2×D47)⋊C4C2212yes—
33#15689056C22116yescomplemented
34#15689052C22116yescomplemented
35#15689021C2218yescomplemented
36#15688985C22116yescomplemented
37#15688980C2218yescomplemented
38#15688980C2218yescomplemented
39#15688980C2218yescomplemented
40C2×C47⋊#95C2212yes—
41C2×C47⋊#95C2212yes—
42C2×C47⋊#95C2212yes—
43C2×C47⋊#95C2212yes—
44C22×C4⋊C4×C47C2211yescharacteristic
45C22×C4⋊C4×C47C2212yes—
46C22×C4⋊C4×C47C2211yescharacteristic
47C22×C42×C47C2211yescharacteristic
48C47×#389C2218yescomplemented
49C2×#95×C47C2212yes—
50C24×D47C2311yescharacteristic
51C24×D47C2311yescharacteristic
52C22×D188—24no—
53C22×D188—24no—
54C22×D188D418yes—
55C22×D188C2314yes—
56C23×Dic47C2311yescharacteristic
57C23×Dic47C2312yes—
58C23×Dic47C2312yes—
59C23×Dic47C2311yescharacteristic
60C22×C4×D47—24no—
61C22×C4×D47—24no—
62C22×C4×D47—28no—
63C22×C4×D47—28no—
64C22×C4×D47—28no—
65C22×C4×D47—24no—
66C22×C4×D47—24no—
67C22×C4×D47—24no—
68C22×C4×D47D418yes—
69C22×C4×D47C2314yes—
70C22×C4×D47C2318yes—
71C22×C4×D47C2314yes—
72#13094009—264no—
73C4×D188—232no—
74C2×C4⋊Dic47—24no—
75C2×C4⋊Dic47—24no—
76C2×C4⋊Dic47—24no—
77C2×C4⋊Dic47C2314yes—
78C2×C4⋊Dic47C2314yes—
79C2×C4⋊Dic47C2314yes—
80C2×Dic47⋊C4—28no—
81C2×Dic47⋊C4—28no—
82C2×Dic47⋊C4C2318yes—
83C2×Dic47⋊C4C2318yes—
84C4⋊C4×D47—232no—
85C2×(C2×D47)⋊C4—24no—
86C2×(C2×D47)⋊C4—28no—
87C2×(C2×D47)⋊C4—28no—
88C2×(C2×D47)⋊C4—28no—
89C2×(C2×D47)⋊C4—24no—
90C2×(C2×D47)⋊C4C2318yes—
91C2×(C2×D47)⋊C4C2318yes—
92C2×(C2×D47)⋊C4C2318yes—
93C2×(C2×D47)⋊C4C2314yes—
94C2×(C2×D47)⋊C4C2314yes—
95C47⋊#95C2318yes—
96C47⋊#95C2318yes—
97C47⋊#95C2318yes—
98C47⋊#95C2318yes—
99C23×C4×C47C2311yescharacteristic
100C23×C4×C47C2312yes—
101C23×C4×C47C2312yes—
102C23×C4×C47C2311yescharacteristic
103C23×C4×C47C2311yescharacteristic
104C2×C4⋊C4×C47—24no—
105C2×C4⋊C4×C47—28no—
106C2×C4⋊C4×C47—24no—
107C2×C4⋊C4×C47C2314yes—
108C2×C4⋊C4×C47C2314yes—
109C2×C4⋊C4×C47C2318yes—
110C2×C42×C47—24no—
111C2×C42×C47C2314yes—
112#95×C47C2318yes—
113C24×C47C2411yescharacteristic
114C23×D47—22no—
115C23×D47—22no—
116C23×D47—22no—
117C23×D47—22no—
118C23×D47—22no—
119C23×D47—22no—
120C23×D47—22no—
121C23×D47—22no—
122C23×D47—22no—
123C23×D47—22no—
124C23×D47C2×D414yes—
125C23×D47C2×D414yes—
126C23×D47C2×D414yes—
127C23×D47#4114yes—
128C23×D47C2412yes—
129C23×D47C2412yes—
130C22×C4×C47—22no—
131C22×C4×C47—22no—
132C22×C4×C47—22no—
133C22×C4×C47—24no—
134C22×C4×C47—24no—
135C22×C4×C47—24no—
136C22×C4×C47—24no—
137C22×C4×C47—24no—
138C22×C4×C47—22no—
139C22×C4×C47—22no—
140C22×C4×C47—24no—
141C22×C4×C47C2×D414yes—
142C22×C4×C47C2×D414yes—
143C22×C4×C47#4114yes—
144C22×C4×C47#4114yes—
145C22×C4×C47C2412yes—
146C22×C4×C47C2414yes—
147C22×C4×C47C2414yes—
148C22×C4×C47C2412yes—
149C22×C4×C47C2412yes—
150C22×Dic47—22no—
151C22×Dic47—22no—
152C22×Dic47—24no—
153C22×Dic47—24no—
154C22×Dic47—22no—
155C22×Dic47—24no—
156C22×Dic47—24no—
157C22×Dic47—24no—
158C22×Dic47—22no—
159C22×Dic47—22no—
160C22×Dic47—24no—
161C22×Dic47C2×D414yes—
162C22×Dic47C2412yes—
163C22×Dic47C2414yes—
164C22×Dic47C2414yes—
165C22×Dic47C2412yes—
166C2×D188—232no—
167C2×D188—28no—
168C2×D188—28no—
169C2×D188C2×D4116yes—
170C2×C4×D47—232no—
171C2×C4×D47—28no—
172C2×C4×D47—28no—
173C2×C4×D47—216no—
174C2×C4×D47—216no—
175C2×C4×D47—216no—
176C2×C4×D47—28no—
177C2×C4×D47—28no—
178C2×C4×D47—28no—
179C2×C4×D47—432no—
180C2×C4×D47—416no—
181C2×C4×D47C2×D4116yes—
182C4⋊C4×C47—28no—
183C4⋊C4×C47—216no—
184C4⋊C4×C47—28no—
185C42×C47—28no—
186(C2×D47)⋊C4—28no—
187(C2×D47)⋊C4—216no—
188(C2×D47)⋊C4—216no—
189(C2×D47)⋊C4—216no—
190(C2×D47)⋊C4—28no—
191C4⋊Dic47—28no—
192C4⋊Dic47—28no—
193C4⋊Dic47—28no—
194Dic47⋊C4—216no—
195Dic47⋊C4—216no—
196C23×C47C22×D412yes—
197C23×C47C22×D412yes—
198C23×C47C2×#4112yes—
199C23×C47C2×#4112yes—
200C23×C47C2×#4112yes—
201C23×C472−1+412yes—
202C23×C472−1+412yes—
203C23×C47C2511yescharacteristic
204C22×D47—28no—
205C22×D47—28no—
206C22×D47—28no—
207C22×D47—28no—
208C22×D47—28no—
209C22×D47—28no—
210C22×D47—22no—
211C22×D47—22no—
212C22×D47—22no—
213C22×D47—22no—
214C22×D47—28no—
215C22×D47—28no—
216C22×D47—28no—
217C22×D47—22no—
218C22×D47—22no—
219C22×D47—22no—
220C22×D47—22no—
221C22×D47—22no—
222C22×D47—22no—
223C22×D47—44no—
224C22×D47—44no—
225C22×D47—44no—
226C22×D47—44no—
227C22×D47#121116yes—
228C22×D47C22×D414yes—
229C22×D47C22×D414yes—
230C22×D47C22×D414yes—
231C22×D47C2×#4114yes—
232C2×C4×C47—28no—
233C2×C4×C47—22no—
234C2×C4×C47—22no—
235C2×C4×C47—24no—
236C2×C4×C47—22no—
237C2×C4×C47—24no—
238C2×C4×C47—24no—
239C2×C4×C47—24no—
240C2×C4×C47—24no—
241C2×C4×C47—22no—
242C2×C4×C47—24no—
243C2×C4×C47—22no—
244C2×C4×C47—48no—
245C2×C4×C47—44no—
246C2×C4×C47—48no—
247C2×C4×C47#123116yescomplemented
248C2×C4×C47C22×D414yes—
249C2×C4×C47C22×D414yes—
250C2×C4×C47C2×#4114yes—
251C2×C4×C47C2×#4114yes—
252C2×Dic47—28no—
253C2×Dic47—22no—
254C2×Dic47—22no—
255C2×Dic47—24no—
256C2×Dic47—24no—
257C2×Dic47—24no—
258C2×Dic47—22no—
259C2×Dic47—24no—
260C2×Dic47—24no—
261C2×Dic47—22no—
262C2×Dic47—22no—
263C2×Dic47—24no—
264C2×Dic47—48no—
265C2×Dic47—44no—
266C2×Dic47—48no—
267C2×Dic47C22×D414yes—
268D188—232no—
269C4×D47—232no—
270C4×D47—432no—
271C4×D47—416no—
272C2×#2486—471nomaximal, Sylow
273C22×C47C2×#12114yes—
274C22×C47C2×#12314yes—
275C22×C47#53614yes—
276C22×C47#54714yes—
277C22×C47#54714yes—
278C22×C47#55214yes—
279C22×C47#55214yes—
280C22×C47C23×D411yescharacteristic
281C22×C47C23×D411yescharacteristic
282C22×C47C22×#4111yescharacteristic
283C22×C47C22×#4111yescharacteristic
284C22×C47C22×#4111yescharacteristic
285C22×C47C2×2−1+411yescharacteristic
286C22×C47C2×2−1+411yescharacteristic
287C2×D47—24no—
288C2×D47—24no—
289C2×D47—24no—
290C2×D47—24no—
291C2×D47—24no—
292C2×D47—24no—
293C2×D47—232no—
294C2×D47—24no—
295C2×D47—24no—
296C2×D47—24no—
297C2×D47—42no—
298C2×D47—42no—
299C2×D47—42no—
300C2×D47—42no—
301C2×D47—416no—
302C2×D47C2×#12118yes—
303C4×C47—24no—
304C4×C47—44no—
305C4×C47—42no—
306C4×C47—44no—
307C4×C47C2×#12318yescomplemented
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×#389D4711yescharacteristic, complemented
320C2×#381—472no—
321C2×#380—471no—
322C2×#380—471no—
323C2×#380—471no—
324C2×C47#248618yescomplemented
325C2×C47C22×#12111yescharacteristic
326C2×C47C22×#12311yescharacteristic
327C2×C47C2×#53611yescharacteristic
328C2×C47C2×#54711yescharacteristic
329C2×C47C2×#54711yescharacteristic
330C2×C47C2×#55211yescharacteristic
331C2×C47C2×#55211yescharacteristic
332D47—28no—
333D47—44no—
334C23×D4—471no—
335C24×C4—471no—
336C24×C4—472no—
337C24×C4—471no—
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—471no—
346C22×C4⋊C4—471no—
347C22×C4⋊C4—478no—
348C22×C4⋊C4—948no—
349C22×C4⋊C4C2×D4712yes—
350C22×C4⋊C4C2×D4711yescharacteristic
351C22×C4⋊C4C2×D4711yescharacteristic
352C22×C22⋊C4—471no—
353C22×C22⋊C4—472no—
354C22×C22⋊C4—472no—
355C22×C22⋊C4—471no—
356C22×C22⋊C4—472no—
357C22×C42C2×D4711yescharacteristic
358#398—4716no—
359#397—4716no—
360#389—478no—
361#389C2×D4718yescomplemented
362#381—4716no—
363#380—478no—
364#380—478no—
365#380—478no—
366C2×#95—472no—
367C2×#95—472no—
368C2×#95—472no—
369C2×#95—472no—
370C2×#95C2×D4712yes—
371C47C2×#248611yescharacteristic, complemented, Sylow
372C25—471no—
373C25—471no—
374C22×D4—478no—
375C22×D4—474no—
376C22×D4—944no—
377C22×D4—944no—
378C23×C4—471no—
379C23×C4—478no—
380C23×C4—474no—
381C23×C4—472no—
382C23×C4—472no—
383C23×C4—471no—
384C23×C4—478no—
385C23×C4—474no—
386C23×C4—948no—
387C23×C4—948no—
388C23×C4—948no—
389C23×C4—944no—
390C23×C4—944no—
391C23×C4—944no—
392C23×C4—944no—
393C23×C4—944no—
394C23×C4C22×D4712yes—
395C23×C4C22×D4711yescharacteristic
396C23×C4C22×D4712yes—
397C23×C4C22×D4711yescharacteristic
398C23×C4C22×D4711yescharacteristic
399#123—9464no—
400C4×D4—9432no—
401C2×C4⋊C4—24no—
402C2×C4⋊C4—24no—
403C2×C4⋊C4—28no—
404C2×C4⋊C4—478no—
405C2×C4⋊C4—474no—
406C2×C4⋊C4—474no—
407C2×C4⋊C4—478no—
408C2×C4⋊C4—474no—
409C2×C4⋊C4—9432no—
410C2×C4⋊C4—948no—
411C2×C4⋊C4—944no—
412C2×C4⋊C4—944no—
413C2×C4⋊C4—948no—
414C2×C4⋊C4—944no—
415C2×C4⋊C4C22×D4714yes—
416C2×C4⋊C4C22×D4718yes—
417C2×C4⋊C4C22×D4714yes—
418C2×C22⋊C4—474no—
419C2×C22⋊C4—474no—
420C2×C22⋊C4—478no—
421C2×C22⋊C4—478no—
422C2×C22⋊C4—478no—
423C2×C22⋊C4—944no—
424C2×C22⋊C4—944no—
425C2×C22⋊C4—948no—
426C2×C22⋊C4—948no—
427C2×C22⋊C4—948no—
428C2×C42—24no—
429C2×C42C22×D4714yes—
430#95—478no—
431#95—478no—
432#95—478no—
433#95—478no—
434#95C22×D4718yes—
435C24—474no—
436C24—474no—
437C24—474no—
438C24—472no—
439C24—474no—
440C24—472no—
441C24—942no—
442C24—942no—
443C24—942no—
444C24—942no—
445C24—942no—
446C24—942no—
447C24—942no—
448C24—942no—
449C24—942no—
450C24—942no—
451C24C23×D4711yescharacteristic
452C2×D4—4716no—
453C2×D4—948no—
454C2×D4—9432no—
455C2×D4—948no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—24no—
460C22×C4—24no—
461C22×C4—24no—
462C22×C4—24no—
463C22×C4—24no—
464C22×C4—24no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—474no—
468C22×C4—472no—
469C22×C4—4716no—
470C22×C4—472no—
471C22×C4—474no—
472C22×C4—474no—
473C22×C4—9416no—
474C22×C4—9416no—
475C22×C4—9416no—
476C22×C4—942no—
477C22×C4—942no—
478C22×C4—9432no—
479C22×C4—948no—
480C22×C4—948no—
481C22×C4—944no—
482C22×C4—944no—
483C22×C4—942no—
484C22×C4—944no—
485C22×C4—944no—
486C22×C4—942no—
487C22×C4—944no—
488C22×C4—942no—
489C22×C4—944no—
490C22×C4—948no—
491C22×C4—948no—
492C22×C4—948no—
493C22×C4—18832no—
494C22×C4—18816no—
495C22×C4#1062535014yes—
496C22×C4D4×D4714yes—
497C22×C4D4×D4714yes—
498C22×C4#1062535414yes—
499C22×C4C23×D4712yes—
500C22×C4C23×D4714yes—
501C22×C4C23×D4714yes—
502C22×C4C23×D4712yes—
503C22×C4C23×D4712yes—
504C4⋊C4—28no—
505C4⋊C4—216no—
506C4⋊C4—28no—
507C4⋊C4—9416no—
508C4⋊C4—948no—
509C4⋊C4—9416no—
510C4⋊C4—948no—
511C4⋊C4—948no—
512C22⋊C4—948no—
513C22⋊C4—948no—
514C22⋊C4—9416no—
515C22⋊C4—9416no—
516C22⋊C4—9416no—
517C42—28no—
518C23—4716no—
519C23—474no—
520C23—474no—
521C23—474no—
522C23—474no—
523C23—948no—
524C23—948no—
525C23—948no—
526C23—948no—
527C23—948no—
528C23—942no—
529C23—942no—
530C23—948no—
531C23—942no—
532C23—942no—
533C23—948no—
534C23—948no—
535C23—948no—
536C23—942no—
537C23—942no—
538C23—942no—
539C23—942no—
540C23—942no—
541C23—942no—
542C23—1884no—
543C23—1884no—
544C23—1884no—
545C23—1884no—
546C23C2×D4×D4712yes—
547C23C2×D4×D4712yes—
548C23#41×D4712yes—
549C23C2×#1062535012yes—
550C23C2×#1062535412yes—
551C23#1309407112yes—
552C23#1309407212yes—
553C23C24×D4711yescharacteristic
554D4—9432no—
555C2×C4—28no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—24no—
560C2×C4—24no—
561C2×C4—24no—
562C2×C4—24no—
563C2×C4—24no—
564C2×C4—24no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—48no—
568C2×C4—48no—
569C2×C4—44no—
570C2×C4—474no—
571C2×C4—948no—
572C2×C4—942no—
573C2×C4—942no—
574C2×C4—9432no—
575C2×C4—944no—
576C2×C4—944no—
577C2×C4—942no—
578C2×C4—944no—
579C2×C4—944no—
580C2×C4—942no—
581C2×C4—944no—
582C2×C4—942no—
583C2×C4—944no—
584C2×C4—18832no—
585C2×C4—1888no—
586C2×C4—1888no—
587C2×C4—1884no—
588C2×C4—18816no—
589C2×C4#13094009116yescomplemented
590C2×C4C2×D4×D4714yes—
591C2×C4C2×D4×D4714yes—
592C2×C4C2×#1062535014yes—
593C2×C4C2×#1062535414yes—
594C22—478no—
595C22—944no—
596C22—944no—
597C22—944no—
598C22—944no—
599C22—944no—
600C22—9432no—
601C22—944no—
602C22—944no—
603C22—944no—
604C22—944no—
605C22—1882no—
606C22—1882no—
607C22—1882no—
608C22—18816no—
609C22—1882no—
610C22#121×D4714yes—
611C22C2×#1309400914yes—
612C22#1568969614yes—
613C22#1568977614yes—
614C22#1568977814yes—
615C22#1568981114yes—
616C22#1568981414yes—
617C22C22×D4×D4711yescharacteristic
618C22C22×D4×D4711yescharacteristic
619C22C2×#41×D4711yescharacteristic
620C22C22×#1062535011yescharacteristic
621C22C22×#1062535411yescharacteristic
622C22C2×#1309407111yescharacteristic
623C22C2×#1309407211yescharacteristic
624C4—24no—
625C4—44no—
626C4—44no—
627C4—42no—
628C4—944no—
629C4—1884no—
630C4—1884no—
631C4—1882no—
632C4C2×#1309400918yescomplemented
633C2—948no—
634C2—1884no—
635C2#2024769418yesdirect product factor
636C2C2×#121×D4711yescharacteristic
637C2C22×#1309400911yescharacteristic
638C2C2×#1568969611yescharacteristic
639C2C2×#1568977611yescharacteristic
640C2C2×#1568977811yescharacteristic
641C2C2×#1568981111yescharacteristic
642C2C2×#1568981411yescharacteristic
subgroupG/H
centerC24C23×D47
commutator, cosocleC23×C47C25
FrattiniC23C24×D47
FittingC2×C47×#389C2
socleC24×C47C24
prime radicalC23×D188C22
hypercenterC2×#389D47
hypocenterC47C2×#2486
LCSG ▷ C23×C47 ▷ C47
UCSC1 ◁ C24 ◁ C2×#389

Complements

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

NH ≅ G/Ntype
3. C22×C4⋊C4×D47C2semidirect
4. #20247694C2direct (N × H)
5. C2×#15689056C2semidirect
6. C2×#15689052C2semidirect
7. C2×#15689021C2semidirect
8. C2×#15688985C2semidirect
9. C2×#15688980C2semidirect
10. C2×#15688980C2semidirect
11. C2×#15688980C2semidirect
12. C2×C47×#389C2semidirect
27. C2×C4⋊C4×D47C22semidirect
33. #15689056C22semidirect
34. #15689052C22semidirect
35. #15689021C22semidirect
36. #15688985C22semidirect
37. #15688980C22semidirect
38. #15688980C22semidirect
39. #15688980C22semidirect
48. C47×#389C22semidirect
247. C2×C4×C47#123semidirect
307. C4×C47C2×#123semidirect
319. C2×#389D47semidirect
324. C2×C47#2486semidirect
361. #389C2×D47semidirect
371. C47C2×#2486semidirect
589. C2×C4#13094009semidirect
632. C4C2×#13094009semidirect
635. C2#20247694direct (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×C4×D1882
C22×C4⋊C4×D472
#202476942
C2×#156890562
C2×#156890522
C2×#156890212
C2×#156889852
C2×#156889802
C2×C47×#3892
C2×#248647