Finite Group Databasebeta

← C2×#24912457 #35576214

Subgroups — C2×#24912457 #35576214

Order 21248.

3526 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×#14581791C211yesmaximal, characteristic, complemented
2C22×#14581778C212yesmaximal
3C22×C4×C22⋊D83C211yesmaximal, characteristic
4#24912457C2116yesmaximal, direct product factor
5C2×#19127630C212yesmaximal, complemented
6C2×#19127629C212yesmaximal, complemented
7C2×#19127605C212yesmaximal, complemented
8C2×#19127561C211yesmaximal, characteristic, complemented
9C2×#19127558C211yesmaximal, characteristic, complemented
10C2×#19127558C211yesmaximal, characteristic, complemented
11C2×C4⋊C4×Dic83C211yesmaximal, characteristic, complemented
12C2×#380×C83C211yesmaximal, characteristic, complemented
13C23×Dic166C2212yes—
14C23×C22⋊D83C2211yescharacteristic
15C23×C4×D83C2211yescharacteristic
16C2×#14581791—28no—
17C2×#14581791C2218yescomplemented
18C2×#14581778—216no—
19C2×#14581778C22116yes—
20C2×C4×C22⋊D83—28no—
21C2×C4×C22⋊D83C2218yes—
22C22×C4⋊Dic83C2211yescharacteristic
23C22×C4⋊Dic83C2211yescharacteristic
24C22×Dic83⋊C4C2211yescharacteristic
25C22×Dic83⋊C4C2212yes—
26C22×C22⋊Dic83C2212yes—
27C22×C22⋊Dic83C2211yescharacteristic
28C22×(C2×D83)⋊C4C2212yes—
29C22×(C2×D83)⋊C4C2212yes—
30C22×(C2×D83)⋊C4C2211yescharacteristic
31C22×C4×Dic83C2212yes—
32C22×C4×Dic83C2211yescharacteristic
33#19127630C22116yescomplemented
34#19127629C22116yescomplemented
35#19127605C22116yescomplemented
36#19127561C2218yescomplemented
37#19127558C2218yescomplemented
38#19127558C2218yescomplemented
39C4⋊C4×Dic83C2218yescomplemented
40C2×C83⋊#95C2212yes—
41C2×C83⋊#95C2212yes—
42C2×C83⋊#95C2212yes—
43C2×C83⋊#95C2212yes—
44C24×C4×C83C2211yescharacteristic
45C22×C4⋊C4×C83C2211yescharacteristic
46C22×C4⋊C4×C83C2211yescharacteristic
47C22×C22⋊C4×C83C2212yes—
48#380×C83C2218yescomplemented
49C2×#95×C83C2212yes—
50C24×D83C2311yescharacteristic
51C22×Dic166—28no—
52C22×Dic166—28no—
53C22×Dic166—28no—
54C22×Dic166C2318yes—
55C22×C22⋊D83—24no—
56C22×C22⋊D83—24no—
57C22×C22⋊D83D418yes—
58C22×C22⋊D83C2314yes—
59C23×Dic83C2312yes—
60C23×Dic83C2312yes—
61C23×Dic83C2311yescharacteristic
62C23×Dic83C2311yescharacteristic
63C23×Dic83C2311yescharacteristic
64C22×C4×D83—24no—
65C22×C4×D83—24no—
66C22×C4×D83—24no—
67C22×C4×D83C2314yes—
68#14581791—232no—
69#14581778—264no—
70C4×C22⋊D83—232no—
71C2×C4⋊Dic83—24no—
72C2×C4⋊Dic83—24no—
73C2×C4⋊Dic83C2314yes—
74C2×C4⋊Dic83C2314yes—
75C2×Dic83⋊C4—24no—
76C2×Dic83⋊C4—28no—
77C2×Dic83⋊C4C2314yes—
78C2×Dic83⋊C4C2318yes—
79C2×C22⋊Dic83—24no—
80C2×C22⋊Dic83—28no—
81C2×C22⋊Dic83C2318yes—
82C2×C22⋊Dic83C2314yes—
83C2×(C2×D83)⋊C4—28no—
84C2×(C2×D83)⋊C4—28no—
85C2×(C2×D83)⋊C4—24no—
86C2×(C2×D83)⋊C4C2318yes—
87C2×(C2×D83)⋊C4C2318yes—
88C2×(C2×D83)⋊C4C2314yes—
89C2×C4×Dic83—28no—
90C2×C4×Dic83D418yes—
91C2×C4×Dic83C2318yes—
92C2×C4×Dic83C2314yes—
93C83⋊#95C2318yes—
94C83⋊#95C2318yes—
95C83⋊#95C2318yes—
96C83⋊#95C2318yes—
97C25×C83C2311yescharacteristic
98C23×C4×C83—24no—
99C23×C4×C83—24no—
100C23×C4×C83—24no—
101C23×C4×C83C2314yes—
102C23×C4×C83C2312yes—
103C23×C4×C83C2312yes—
104C23×C4×C83C2311yescharacteristic
105C23×C4×C83C2311yescharacteristic
106C2×C4⋊C4×C83—24no—
107C2×C4⋊C4×C83—24no—
108C2×C4⋊C4×C83C2314yes—
109C2×C4⋊C4×C83C2314yes—
110C2×C22⋊C4×C83—28no—
111C2×C22⋊C4×C83C2318yes—
112#95×C83C2318yes—
113C23×D83—22no—
114C23×D83—22no—
115C23×D83—22no—
116C23×D83—22no—
117C23×D83—22no—
118C23×D83—22no—
119C23×D83C2×D414yes—
120C23×D83C2412yes—
121C2×Dic166—216no—
122C2×Dic166—216no—
123C2×Dic166—216no—
124C2×Dic166—432no—
125C2×C22⋊D83—232no—
126C2×C22⋊D83—28no—
127C2×C22⋊D83—28no—
128C2×C22⋊D83C2×D4116yes—
129C22×Dic83—22no—
130C22×Dic83—22no—
131C22×Dic83—22no—
132C22×Dic83—24no—
133C22×Dic83—24no—
134C22×Dic83—24no—
135C22×Dic83—22no—
136C22×Dic83—22no—
137C22×Dic83—24no—
138C22×Dic83—24no—
139C22×Dic83—24no—
140C22×Dic83C2×D414yes—
141C22×Dic83C2×D414yes—
142C22×Dic83C2×D414yes—
143C22×Dic83#4114yes—
144C22×Dic83C2412yes—
145C22×Dic83C2414yes—
146C22×Dic83C2414yes—
147C22×Dic83C2412yes—
148C22×Dic83C2412yes—
149C2×C4×D83—28no—
150C2×C4×D83—28no—
151C2×C4×D83—28no—
152C2×C4×D83—416no—
153C4⋊Dic83—28no—
154C4⋊Dic83—28no—
155Dic83⋊C4—216no—
156Dic83⋊C4—28no—
157C22⋊Dic83—216no—
158C22⋊Dic83—28no—
159(C2×D83)⋊C4—216no—
160(C2×D83)⋊C4—216no—
161(C2×D83)⋊C4—28no—
162C4×Dic83—216no—
163C4×Dic83C2×D4116yes—
164C24×C83—22no—
165C24×C83—22no—
166C24×C83—22no—
167C24×C83—22no—
168C24×C83—22no—
169C24×C83—22no—
170C24×C83C2×D414yes—
171C24×C83C2412yes—
172C24×C83C2411yescharacteristic
173C22×C4×C83—24no—
174C22×C4×C83—24no—
175C22×C4×C83—24no—
176C22×C4×C83—22no—
177C22×C4×C83—24no—
178C22×C4×C83—24no—
179C22×C4×C83—22no—
180C22×C4×C83—22no—
181C22×C4×C83—24no—
182C22×C4×C83—28no—
183C22×C4×C83—28no—
184C22×C4×C83—28no—
185C22×C4×C83—416no—
186C22×C4×C83C2×D414yes—
187C22×C4×C83#4114yes—
188C22×C4×C83#4114yes—
189C22×C4×C83C2414yes—
190C22×C4×C83C2412yes—
191C22×C4×C83C2414yes—
192C22×C4×C83C2412yes—
193C4⋊C4×C83—28no—
194C4⋊C4×C83—28no—
195C22⋊C4×C83—216no—
196C23×C83—28no—
197C23×C83—28no—
198C23×C83—28no—
199C23×C83—22no—
200C23×C83—22no—
201C23×C83—22no—
202C23×C83—22no—
203C23×C83—22no—
204C23×C83—22no—
205C23×C83—44no—
206C23×C83—44no—
207C23×C83—44no—
208C23×C83—44no—
209C23×C83C22×D414yes—
210C23×C83C22×D412yes—
211C23×C83C22×D412yes—
212C23×C83C2×#4112yes—
213C23×C83C2×#4112yes—
214C23×C83C2×#4112yes—
215C23×C832−1+412yes—
216C23×C832−1+412yes—
217C23×C83C2511yescharacteristic
218C22×D83—28no—
219C22×D83—28no—
220C22×D83—28no—
221C22×D83—22no—
222C22×D83—22no—
223C22×D83—22no—
224C22×D83—22no—
225C22×D83—22no—
226C22×D83—22no—
227C22×D83—44no—
228C22×D83—44no—
229C22×D83—44no—
230C22×D83—44no—
231C22×D83C22×D414yes—
232C2×C4×C83—24no—
233C2×C4×C83—24no—
234C2×C4×C83—24no—
235C2×C4×C83—22no—
236C2×C4×C83—24no—
237C2×C4×C83—22no—
238C2×C4×C83—22no—
239C2×C4×C83—24no—
240C2×C4×C83—24no—
241C2×C4×C83—48no—
242C2×C4×C83—44no—
243C2×C4×C83—48no—
244C2×C4×C83—416no—
245C2×C4×C83#124116yescomplemented
246C2×C4×C83C22×D414yes—
247C2×C4×C83C2×#4114yes—
248C2×C4×C83C2×#4114yes—
249C22⋊D83—232no—
250C2×Dic83—28no—
251C2×Dic83—22no—
252C2×Dic83—22no—
253C2×Dic83—24no—
254C2×Dic83—24no—
255C2×Dic83—22no—
256C2×Dic83—24no—
257C2×Dic83—24no—
258C2×Dic83—22no—
259C2×Dic83—22no—
260C2×Dic83—24no—
261C2×Dic83—24no—
262C2×Dic83—48no—
263C2×Dic83—44no—
264C2×Dic83—48no—
265C2×Dic83#121116yes—
266C2×Dic83C22×D414yes—
267C2×Dic83C22×D414yes—
268C2×Dic83C22×D414yes—
269C2×Dic83C2×#4114yes—
270C4×D83—416no—
271Dic166—432no—
272C22×C83—24no—
273C22×C83—24no—
274C22×C83—24no—
275C22×C83—42no—
276C22×C83—42no—
277C22×C83—42no—
278C22×C83—42no—
279C22×C83—416no—
280C22×C83C2×#12114yes—
281C22×C83C2×#12414yes—
282C22×C83#53614yes—
283C22×C83#54714yes—
284C22×C83#54714yes—
285C22×C83#55614yes—
286C22×C83#55614yes—
287C22×C83C23×D411yescharacteristic
288C22×C83C23×D411yescharacteristic
289C22×C83C22×#4111yescharacteristic
290C22×C83C22×#4111yescharacteristic
291C22×C83C22×#4111yescharacteristic
292C22×C83C2×2−1+411yescharacteristic
293C22×C83C2×2−1+411yescharacteristic
294C2×D83—24no—
295C2×D83—24no—
296C2×D83—24no—
297C2×D83—42no—
298C2×D83—42no—
299C2×D83—42no—
300C2×D83—42no—
301C2×D83—416no—
302C4×C83—44no—
303C4×C83—42no—
304C4×C83—44no—
305C4×C83C2×#12418yescomplemented
306Dic83—24no—
307Dic83—44no—
308Dic83—42no—
309Dic83—44no—
310Dic83C2×#12118yes—
311C2×#2547—831nomaximal, Sylow
312C2×C83—44no—
313C2×C83#254718yescomplemented
314C2×C83C22×#12111yescharacteristic
315C2×C83C22×#12411yescharacteristic
316C2×C83C2×#53611yescharacteristic
317C2×C83C2×#54711yescharacteristic
318C2×C83C2×#54711yescharacteristic
319C2×C83C2×#55611yescharacteristic
320C2×C83C2×#55611yescharacteristic
321D83—44no—
322C22×C4⋊Q8—831no—
323C22×#124—832no—
324C22×C4×D4—831no—
325#2547—8316no—
326C2×#397—832no—
327C2×#397—832no—
328C2×#391—832no—
329C2×#380—831no—
330C2×#380—831no—
331C2×#380—831no—
332C2×#380D8311yescharacteristic, complemented
333C2×C4×C4⋊C4—831no—
334C83C2×#254711yescharacteristic, complemented, Sylow
335C23×Q8—832no—
336C23×D4—831no—
337C24×C4—831no—
338C24×C4C2×D8311yescharacteristic
339C2×C4⋊Q8—838no—
340C2×C4⋊Q8—1668no—
341C2×#124—8316no—
342C2×#124—16616no—
343C2×C4×D4—838no—
344C2×C4×D4—1668no—
345C22×C4⋊C4—832no—
346C22×C4⋊C4—831no—
347C22×C4⋊C4—831no—
348C22×C4⋊C4—831no—
349C22×C4⋊C4C2×D8311yescharacteristic
350C22×C4⋊C4C2×D8311yescharacteristic
351C22×C22⋊C4—831no—
352C22×C22⋊C4—832no—
353C22×C22⋊C4—831no—
354C22×C22⋊C4—832no—
355C22×C22⋊C4—832no—
356C22×C22⋊C4C2×D8312yes—
357C22×C42—831no—
358C22×C42—832no—
359#397—8316no—
360#397—8316no—
361#391—8316no—
362#380—838no—
363#380—838no—
364#380—838no—
365#380C2×D8318yescomplemented
366C4×C4⋊C4—838no—
367C2×#95—832no—
368C2×#95—832no—
369C2×#95—832no—
370C2×#95—832no—
371C2×#95C2×D8312yes—
372C25—831no—
373C25C22×D8311yescharacteristic
374C22×Q8—838no—
375C22×Q8—1668no—
376C22×Q8—1668no—
377C22×Q8—1668no—
378C22×D4—838no—
379C22×D4—834no—
380C22×D4—1664no—
381C22×D4—1664no—
382C23×C4—24no—
383C23×C4—24no—
384C23×C4—24no—
385C23×C4—831no—
386C23×C4—831no—
387C23×C4—832no—
388C23×C4—832no—
389C23×C4—831no—
390C23×C4—834no—
391C23×C4—1664no—
392C23×C4—1664no—
393C23×C4—1664no—
394C23×C4C22×D8314yes—
395C23×C4C22×D8312yes—
396C23×C4C22×D8312yes—
397C23×C4C22×D8311yescharacteristic
398C23×C4C22×D8311yescharacteristic
399C4⋊Q8—16632no—
400#124—16664no—
401C4×D4—16632no—
402C2×C4⋊C4—24no—
403C2×C4⋊C4—24no—
404C2×C4⋊C4—834no—
405C2×C4⋊C4—834no—
406C2×C4⋊C4—838no—
407C2×C4⋊C4—834no—
408C2×C4⋊C4—1664no—
409C2×C4⋊C4—1668no—
410C2×C4⋊C4—1664no—
411C2×C4⋊C4—1664no—
412C2×C4⋊C4C22×D8314yes—
413C2×C4⋊C4C22×D8314yes—
414C2×C22⋊C4—28no—
415C2×C22⋊C4—834no—
416C2×C22⋊C4—838no—
417C2×C22⋊C4—834no—
418C2×C22⋊C4—838no—
419C2×C22⋊C4—838no—
420C2×C22⋊C4—1664no—
421C2×C22⋊C4—1668no—
422C2×C22⋊C4—1668no—
423C2×C22⋊C4—1664no—
424C2×C22⋊C4—1668no—
425C2×C22⋊C4C22×D8318yes—
426C2×C42—838no—
427C2×C42—834no—
428C2×C42—838no—
429C2×C42—1668no—
430#95—838no—
431#95—838no—
432#95—838no—
433#95—838no—
434#95C22×D8318yes—
435C24—22no—
436C24—22no—
437C24—22no—
438C24—22no—
439C24—22no—
440C24—22no—
441C24—834no—
442C24—832no—
443C24—1662no—
444C24—1662no—
445C24—1662no—
446C24—1662no—
447C24—1662no—
448C24—1662no—
449C24D4×D8314yes—
450C24C23×D8312yes—
451C24C23×D8311yescharacteristic
452C2×Q8—33232no—
453C2×Q8—16616no—
454C2×Q8—16616no—
455C2×Q8—16616no—
456C2×D4—8316no—
457C2×D4—1668no—
458C2×D4—16632no—
459C2×D4—1668no—
460C22×C4—24no—
461C22×C4—24no—
462C22×C4—22no—
463C22×C4—24no—
464C22×C4—24no—
465C22×C4—24no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—24no—
469C22×C4—28no—
470C22×C4—28no—
471C22×C4—28no—
472C22×C4—416no—
473C22×C4—33216no—
474C22×C4—834no—
475C22×C4—834no—
476C22×C4—834no—
477C22×C4—832no—
478C22×C4—834no—
479C22×C4—832no—
480C22×C4—832no—
481C22×C4—834no—
482C22×C4—834no—
483C22×C4—1664no—
484C22×C4—1664no—
485C22×C4—1662no—
486C22×C4—1664no—
487C22×C4—1664no—
488C22×C4—1662no—
489C22×C4—1664no—
490C22×C4—1662no—
491C22×C4—1664no—
492C22×C4—1668no—
493C22×C4—1668no—
494C22×C4—1668no—
495C22×C4—1662no—
496C22×C4—1662no—
497C22×C4D4×D8314yes—
498C22×C4#1305058914yes—
499C22×C4#1305059014yes—
500C22×C4C23×D8312yes—
501C22×C4C23×D8314yes—
502C22×C4C23×D8314yes—
503C22×C4C23×D8312yes—
504C4⋊C4—28no—
505C4⋊C4—28no—
506C4⋊C4—1668no—
507C4⋊C4—16616no—
508C4⋊C4—1668no—
509C4⋊C4—1668no—
510C22⋊C4—216no—
511C22⋊C4—1668no—
512C22⋊C4—16616no—
513C22⋊C4—16616no—
514C22⋊C4—1668no—
515C22⋊C4—16616no—
516C42—8316no—
517C42—16616no—
518C23—28no—
519C23—28no—
520C23—28no—
521C23—22no—
522C23—22no—
523C23—22no—
524C23—22no—
525C23—22no—
526C23—22no—
527C23—44no—
528C23—44no—
529C23—44no—
530C23—44no—
531C23—3324no—
532C23—3324no—
533C23—3324no—
534C23—3324no—
535C23—834no—
536C23—1668no—
537C23—1668no—
538C23—1668no—
539C23—1662no—
540C23—1662no—
541C23—1662no—
542C23—1662no—
543C23—1662no—
544C23—1662no—
545C23C2×D4×D8314yes—
546C23C2×D4×D8312yes—
547C23C2×D4×D8312yes—
548C23#41×D8312yes—
549C23C2×#1305058912yes—
550C23C2×#1305059012yes—
551C23#1458183912yes—
552C23#1458184012yes—
553C23C24×D8311yescharacteristic
554Q8—33232no—
555D4—16632no—
556C2×C4—24no—
557C2×C4—24no—
558C2×C4—22no—
559C2×C4—24no—
560C2×C4—24no—
561C2×C4—24no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—24no—
565C2×C4—48no—
566C2×C4—48no—
567C2×C4—44no—
568C2×C4—416no—
569C2×C4—3328no—
570C2×C4—3328no—
571C2×C4—3324no—
572C2×C4—33216no—
573C2×C4—8316no—
574C2×C4—834no—
575C2×C4—834no—
576C2×C4—834no—
577C2×C4—834no—
578C2×C4—1664no—
579C2×C4—1664no—
580C2×C4—1662no—
581C2×C4—1664no—
582C2×C4—1664no—
583C2×C4—1662no—
584C2×C4—1664no—
585C2×C4—1662no—
586C2×C4—1664no—
587C2×C4—1668no—
588C2×C4—1662no—
589C2×C4—1662no—
590C2×C4#14581778116yescomplemented
591C2×C4C2×D4×D8314yes—
592C2×C4C2×#1305058914yes—
593C2×C4C2×#1305059014yes—
594C22—24no—
595C22—24no—
596C22—24no—
597C22—416no—
598C22—42no—
599C22—42no—
600C22—42no—
601C22—42no—
602C22—33216no—
603C22—3322no—
604C22—3322no—
605C22—3322no—
606C22—3322no—
607C22—1664no—
608C22—1664no—
609C22—1664no—
610C22#121×D8314yes—
611C22C2×#1458177814yes—
612C22#1912827414yes—
613C22#1912835114yes—
614C22#1912835214yes—
615C22#1912841914yes—
616C22#1912842214yes—
617C22C22×D4×D8311yescharacteristic
618C22C22×D4×D8311yescharacteristic
619C22C2×#41×D8311yescharacteristic
620C22C22×#1305058911yescharacteristic
621C22C22×#1305059011yescharacteristic
622C22C2×#1458183911yescharacteristic
623C22C2×#1458184011yescharacteristic
624C4—44no—
625C4—44no—
626C4—42no—
627C4—3324no—
628C4—3324no—
629C4—3322no—
630C4—838no—
631C4—1664no—
632C4C2×#1458177818yescomplemented
633C2—44no—
634C2—3324no—
635C2#2491245718yesdirect product factor
636C2C2×#121×D8311yescharacteristic
637C2C22×#1458177811yescharacteristic
638C2C2×#1912827411yescharacteristic
639C2C2×#1912835111yescharacteristic
640C2C2×#1912835211yescharacteristic
641C2C2×#1912841911yescharacteristic
642C2C2×#1912842211yescharacteristic
subgroupG/H
centerC24C23×D83
commutator, cosocleC23×C83C25
FrattiniC23C24×D83
FittingC2×#380×C83C2
socleC24×C83C24
prime radicalC23×C22⋊D83C22
hypercenterC2×#380D83
hypocenterC83C2×#2547
LCSG ▷ C23×C83 ▷ C83
UCSC1 ◁ C24 ◁ C2×#380

Complements

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

NH ≅ G/Ntype
1. C22×#14581791C2semidirect
4. #24912457C2direct (N × H)
5. C2×#19127630C2semidirect
6. C2×#19127629C2semidirect
7. C2×#19127605C2semidirect
8. C2×#19127561C2semidirect
9. C2×#19127558C2semidirect
10. C2×#19127558C2semidirect
11. C2×C4⋊C4×Dic83C2semidirect
12. C2×#380×C83C2semidirect
17. C2×#14581791C22semidirect
33. #19127630C22semidirect
34. #19127629C22semidirect
35. #19127605C22semidirect
36. #19127561C22semidirect
37. #19127558C22semidirect
38. #19127558C22semidirect
39. C4⋊C4×Dic83C22semidirect
48. #380×C83C22semidirect
245. C2×C4×C83#124semidirect
305. C4×C83C2×#124semidirect
313. C2×C83#2547semidirect
332. C2×#380D83semidirect
334. C83C2×#2547semidirect
365. #380C2×D83semidirect
590. C2×C4#14581778semidirect
632. C4C2×#14581778semidirect
635. C2#24912457direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#145817912
C22×#145817782
C22×C4×C22⋊D832
#249124572
C2×#191276302
C2×#191276292
C2×#191276052
C2×#191275612
C2×#191275582
C2×C4⋊C4×Dic832
C2×#380×C832
C2×#254783