Finite Group Databasebeta

← C2×#24915796 #35610301

Subgroups — C2×#24915796 #35610301

Order 21248.

4070 conjugacy classes of subgroups.

The subgroup lattice has 664 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14581777C212yesmaximal, complemented
2C22×#14581767C211yesmaximal, characteristic
3C22×C42×D83C211yesmaximal, characteristic, complemented
4#24915796C2116yesmaximal, direct product factor
5C2×#19127603C211yesmaximal, characteristic, complemented
6C2×#19127602C211yesmaximal, characteristic, complemented
7C2×#19127597C211yesmaximal, characteristic, complemented
8C2×#19127591C212yesmaximal, complemented
9C2×#19127590C212yesmaximal, complemented
10C2×#19127579C212yesmaximal, complemented
11C2×#19127558C211yesmaximal, characteristic, complemented
12C2×#389×C83C211yesmaximal, characteristic, complemented
13C23×D332C2212yes—
14C23×D332C2211yescharacteristic
15C23×C4×D83C2211yescharacteristic
16C23×C4×D83C2211yescharacteristic
17C23×C4×D83C2211yescharacteristic
18C2×#14581777—216no—
19C2×#14581777C22116yescomplemented
20C2×#14581767—28no—
21C2×#14581767C2218yes—
22C22×C4⋊Dic83C2211yescharacteristic
23C22×Dic83⋊C4C2212yes—
24C22×(C2×D83)⋊C4C2212yes—
25C22×(C2×D83)⋊C4C2212yes—
26C22×(C2×D83)⋊C4C2212yes—
27C22×(C2×D83)⋊C4C2212yes—
28C22×(C2×D83)⋊C4C2212yes—
29C22×(C2×D83)⋊C4C2212yes—
30C22×C4×Dic83C2211yescharacteristic
31C22×C4×Dic83C2211yescharacteristic
32C22×C4×Dic83C2211yescharacteristic
33C2×C42×D83—28no—
34C2×C42×D83C2218yescomplemented
35#19127603C2218yescomplemented
36#19127602C2218yescomplemented
37#19127597C2218yescomplemented
38#19127591C22116yescomplemented
39#19127590C22116yescomplemented
40#19127579C22116yescomplemented
41#19127558C2218yescomplemented
42C2×C83⋊#95C2212yes—
43C2×C83⋊#95C2212yes—
44C22×C4⋊C4×C83C2211yescharacteristic
45C22×C4⋊C4×C83C2212yes—
46C22×C4⋊C4×C83C2211yescharacteristic
47C22×C42×C83C2211yescharacteristic
48#389×C83C2218yescomplemented
49C2×#95×C83C2212yes—
50C24×D83C2312yes—
51C24×D83C2311yescharacteristic
52C22×D332—28no—
53C22×D332—28no—
54C22×D332—28no—
55C22×D332—24no—
56C22×D332—24no—
57C22×D332—24no—
58C22×D332C2318yes—
59C22×D332C2314yes—
60C23×Dic83C2312yes—
61C23×Dic83C2311yescharacteristic
62C23×Dic83C2311yescharacteristic
63C23×Dic83C2311yescharacteristic
64C22×C4×D83—24no—
65C22×C4×D83—24no—
66C22×C4×D83—24no—
67C22×C4×D83—24no—
68C22×C4×D83—24no—
69C22×C4×D83—24no—
70C22×C4×D83—24no—
71C22×C4×D83D418yes—
72C22×C4×D83D418yescomplemented
73C22×C4×D83C2314yes—
74C22×C4×D83C2314yes—
75C22×C4×D83C2314yes—
76#14581777—264no—
77#14581767—232no—
78C2×C4⋊Dic83—24no—
79C2×C4⋊Dic83C2314yes—
80C2×Dic83⋊C4—28no—
81C2×Dic83⋊C4C2318yes—
82C2×(C2×D83)⋊C4—28no—
83C2×(C2×D83)⋊C4—28no—
84C2×(C2×D83)⋊C4—28no—
85C2×(C2×D83)⋊C4—28no—
86C2×(C2×D83)⋊C4—28no—
87C2×(C2×D83)⋊C4—28no—
88C2×(C2×D83)⋊C4C2318yes—
89C2×(C2×D83)⋊C4C2318yes—
90C2×(C2×D83)⋊C4C2318yes—
91C2×(C2×D83)⋊C4C2318yes—
92C2×(C2×D83)⋊C4C2318yes—
93C2×(C2×D83)⋊C4C2318yes—
94C2×C4×Dic83—24no—
95C2×C4×Dic83—24no—
96C2×C4×Dic83—24no—
97C2×C4×Dic83C2314yes—
98C2×C4×Dic83C2314yes—
99C2×C4×Dic83C2314yes—
100C42×D83—232no—
101C83⋊#95C2318yes—
102C83⋊#95C2318yes—
103C23×C4×C83C2312yes—
104C23×C4×C83C2312yes—
105C23×C4×C83C2311yescharacteristic
106C23×C4×C83C2311yescharacteristic
107C23×C4×C83C2311yescharacteristic
108C2×C4⋊C4×C83—24no—
109C2×C4⋊C4×C83—28no—
110C2×C4⋊C4×C83—24no—
111C2×C4⋊C4×C83C2314yes—
112C2×C4⋊C4×C83C2318yes—
113C2×C4⋊C4×C83C2314yes—
114C2×C42×C83—24no—
115C2×C42×C83C2314yes—
116#95×C83C2318yes—
117C23×D83—22no—
118C23×D83—22no—
119C23×D83—22no—
120C23×D83—22no—
121C23×D83—24no—
122C23×D83—24no—
123C23×D83—24no—
124C23×D83—24no—
125C23×D83—24no—
126C23×D83—24no—
127C23×D83—24no—
128C23×D83C2×D414yes—
129C23×D83C2×D414yes—
130C23×D83#4114yes—
131C23×D83C2414yes—
132C23×D83C2412yes—
133C2×D332—216no—
134C2×D332—216no—
135C2×D332—216no—
136C2×D332—28no—
137C2×D332—28no—
138C2×D332—28no—
139C2×D332—432no—
140C2×D332—416no—
141C22×Dic83—22no—
142C22×Dic83—22no—
143C22×Dic83—22no—
144C22×Dic83—22no—
145C22×Dic83—22no—
146C22×Dic83—22no—
147C22×Dic83—24no—
148C22×Dic83—24no—
149C22×Dic83—24no—
150C22×Dic83C2×D414yes—
151C22×Dic83C2×D414yes—
152C22×Dic83#4114yes—
153C22×Dic83C2414yes—
154C22×Dic83C2412yes—
155C22×Dic83C2412yes—
156C22×Dic83C2412yes—
157C2×C4×D83—232no—
158C2×C4×D83—232no—
159C2×C4×D83—28no—
160C2×C4×D83—28no—
161C2×C4×D83—28no—
162C2×C4×D83—232no—
163C2×C4×D83—28no—
164C2×C4×D83—28no—
165C2×C4×D83—28no—
166C2×C4×D83—28no—
167C2×C4×D83C2×D4116yes—
168C2×C4×D83C2×D4116yescomplemented
169C4⋊Dic83—28no—
170Dic83⋊C4—216no—
171(C2×D83)⋊C4—216no—
172(C2×D83)⋊C4—216no—
173(C2×D83)⋊C4—216no—
174(C2×D83)⋊C4—216no—
175(C2×D83)⋊C4—216no—
176(C2×D83)⋊C4—216no—
177C4×Dic83—28no—
178C4×Dic83—28no—
179C4×Dic83—28no—
180C24×C83C2411yescharacteristic
181C22×C4×C83—22no—
182C22×C4×C83—22no—
183C22×C4×C83—22no—
184C22×C4×C83—22no—
185C22×C4×C83—24no—
186C22×C4×C83—24no—
187C22×C4×C83—24no—
188C22×C4×C83—24no—
189C22×C4×C83—24no—
190C22×C4×C83—24no—
191C22×C4×C83C2×D414yes—
192C22×C4×C83C2×D414yes—
193C22×C4×C83#4114yes—
194C22×C4×C83#4114yes—
195C22×C4×C83#4114yes—
196C22×C4×C83C2414yes—
197C22×C4×C83C2414yes—
198C22×C4×C83C2412yes—
199C22×C4×C83C2412yes—
200C22×C4×C83C2412yes—
201C4⋊C4×C83—28no—
202C4⋊C4×C83—216no—
203C4⋊C4×C83—28no—
204C42×C83—28no—
205C23×C83C22×D412yes—
206C23×C83C22×D412yes—
207C23×C83C2×#4112yes—
208C23×C83C2×#4112yes—
209C23×C83C2×#4112yes—
210C23×C83C2×#4112yes—
211C23×C83C2×#4112yes—
212C23×C83C2511yescharacteristic
213C22×D83—28no—
214C22×D83—28no—
215C22×D83—28no—
216C22×D83—28no—
217C22×D83—28no—
218C22×D83—28no—
219C22×D83—22no—
220C22×D83—22no—
221C22×D83—22no—
222C22×D83—22no—
223C22×D83—24no—
224C22×D83—24no—
225C22×D83—24no—
226C22×D83—24no—
227C22×D83—24no—
228C22×D83—24no—
229C22×D83—24no—
230C22×D83—48no—
231C22×D83—48no—
232C22×D83—48no—
233C22×D83—48no—
234C22×D83—48no—
235C22×D83—48no—
236C22×D83—48no—
237C22×D83#121116yes—
238C22×D83C22×D414yes—
239C22×D83C22×D414yes—
240C22×D83C2×#4114yes—
241C2×C4×C83—28no—
242C2×C4×C83—28no—
243C2×C4×C83—22no—
244C2×C4×C83—22no—
245C2×C4×C83—22no—
246C2×C4×C83—22no—
247C2×C4×C83—24no—
248C2×C4×C83—24no—
249C2×C4×C83—24no—
250C2×C4×C83—24no—
251C2×C4×C83—24no—
252C2×C4×C83—24no—
253C2×C4×C83—48no—
254C2×C4×C83—48no—
255C2×C4×C83#123116yescomplemented
256C2×C4×C83C22×D414yes—
257C2×C4×C83C22×D414yes—
258C2×C4×C83C2×#4114yes—
259C2×C4×C83C2×#4114yes—
260C2×C4×C83C2×#4114yes—
261C2×Dic83—28no—
262C2×Dic83—28no—
263C2×Dic83—28no—
264C2×Dic83—22no—
265C2×Dic83—22no—
266C2×Dic83—22no—
267C2×Dic83—22no—
268C2×Dic83—22no—
269C2×Dic83—22no—
270C2×Dic83—24no—
271C2×Dic83—24no—
272C2×Dic83—24no—
273C2×Dic83—48no—
274C2×Dic83C22×D414yes—
275C2×Dic83C22×D414yes—
276C2×Dic83C2×#4114yes—
277D332—432no—
278D332—416no—
279C4×D83—232no—
280C4×D83—232no—
281C4×D83—232no—
282C22×C83C2×#12114yes—
283C22×C83C2×#12314yes—
284C22×C83#52514yes—
285C22×C83#52914yes—
286C22×C83#52914yes—
287C22×C83#53214yes—
288C22×C83#53214yes—
289C22×C83C23×D411yescharacteristic
290C22×C83C23×D411yescharacteristic
291C22×C83C22×#4111yescharacteristic
292C22×C83C22×#4111yescharacteristic
293C22×C83C22×#4111yescharacteristic
294C22×C83C22×#4111yescharacteristic
295C22×C83C22×#4111yescharacteristic
296C2×D83—232no—
297C2×D83—24no—
298C2×D83—24no—
299C2×D83—24no—
300C2×D83—24no—
301C2×D83—24no—
302C2×D83—24no—
303C2×D83—44no—
304C2×D83—44no—
305C2×D83—44no—
306C2×D83—44no—
307C2×D83—44no—
308C2×D83—44no—
309C2×D83—44no—
310C2×D83—816no—
311C2×D83C2×#12118yes—
312C4×C83—24no—
313C4×C83—24no—
314C4×C83—44no—
315C4×C83—44no—
316C4×C83C2×#12318yescomplemented
317Dic83—24no—
318Dic83—24no—
319Dic83—24no—
320Dic83—44no—
321C2×#2854—831nomaximal, Sylow
322C2×C83#285418yescomplemented
323C2×C83C22×#12111yescharacteristic
324C2×C83C22×#12311yescharacteristic
325C2×C83C2×#52511yescharacteristic
326C2×C83C2×#52911yescharacteristic
327C2×C83C2×#52911yescharacteristic
328C2×C83C2×#53211yescharacteristic
329C2×C83C2×#53211yescharacteristic
330D83—28no—
331D83—84no—
332C22×#123—832no—
333C22×#121—831no—
334C23×C42—831no—
335#2854—8316no—
336C2×#390—831no—
337C2×#390—831no—
338C2×#389—831no—
339C2×#389D8311yescharacteristic, complemented
340C2×#388—832no—
341C2×#388—832no—
342C2×#386—832no—
343C2×#380—831no—
344C83C2×#285411yescharacteristic, complemented, Sylow
345C23×D4—832no—
346C23×D4—831no—
347C24×C4—831no—
348C24×C4—831no—
349C24×C4—831no—
350C2×#123—8316no—
351C2×#123—16616no—
352C2×#121—838no—
353C2×#121—1668no—
354C22×C4⋊C4—832no—
355C22×C4⋊C4—831no—
356C22×C4⋊C4C2×D8311yescharacteristic
357C22×C4⋊C4C2×D8312yes—
358C22×C4⋊C4C2×D8311yescharacteristic
359C22×C22⋊C4—832no—
360C22×C22⋊C4—832no—
361C22×C22⋊C4—832no—
362C22×C22⋊C4—832no—
363C22×C22⋊C4—832no—
364C22×C22⋊C4—832no—
365C22×C42—831no—
366C22×C42—831no—
367C22×C42—831no—
368C22×C42—838no—
369C22×C42—1668no—
370C22×C42C2×D8311yescharacteristic
371#390—838no—
372#390—838no—
373#389—838no—
374#389C2×D8318yescomplemented
375#388—8316no—
376#388—8316no—
377#386—8316no—
378#380—838no—
379C2×#95—832no—
380C2×#95—832no—
381C2×#95C2×D8312yes—
382C25—831no—
383C25—832no—
384C22×D4—838no—
385C22×D4—834no—
386C22×D4—1668no—
387C22×D4—1668no—
388C22×D4—1668no—
389C22×D4—1664no—
390C22×D4—1664no—
391C22×D4—1664no—
392C23×C4—831no—
393C23×C4—831no—
394C23×C4—831no—
395C23×C4—838no—
396C23×C4—838no—
397C23×C4—834no—
398C23×C4—834no—
399C23×C4—832no—
400C23×C4—834no—
401C23×C4—1664no—
402C23×C4—1664no—
403C23×C4—1664no—
404C23×C4—1664no—
405C23×C4—1664no—
406C23×C4—1664no—
407C23×C4—1664no—
408C23×C4C22×D8312yes—
409C23×C4C22×D8312yes—
410C23×C4C22×D8311yescharacteristic
411C23×C4C22×D8311yescharacteristic
412C23×C4C22×D8311yescharacteristic
413#123—16664no—
414#121—16632no—
415C2×C4⋊C4—24no—
416C2×C4⋊C4—28no—
417C2×C4⋊C4—24no—
418C2×C4⋊C4—838no—
419C2×C4⋊C4—834no—
420C2×C4⋊C4—1668no—
421C2×C4⋊C4—1664no—
422C2×C4⋊C4C22×D8314yes—
423C2×C4⋊C4C22×D8318yes—
424C2×C4⋊C4C22×D8314yes—
425C2×C22⋊C4—838no—
426C2×C22⋊C4—838no—
427C2×C22⋊C4—838no—
428C2×C22⋊C4—838no—
429C2×C22⋊C4—838no—
430C2×C22⋊C4—838no—
431C2×C22⋊C4—1668no—
432C2×C22⋊C4—1668no—
433C2×C22⋊C4—1668no—
434C2×C22⋊C4—1668no—
435C2×C22⋊C4—1668no—
436C2×C22⋊C4—1668no—
437C2×C42—24no—
438C2×C42—834no—
439C2×C42—834no—
440C2×C42—834no—
441C2×C42—16632no—
442C2×C42—1664no—
443C2×C42—1664no—
444C2×C42—1664no—
445C2×C42C22×D8314yes—
446#95—838no—
447#95—838no—
448#95C22×D8318yes—
449C24—834no—
450C24—834no—
451C24—834no—
452C24—832no—
453C24—834no—
454C24—1664no—
455C24—1664no—
456C24—1664no—
457C24—1664no—
458C24—1664no—
459C24—1664no—
460C24—1664no—
461C24—1662no—
462C24—1662no—
463C24—1662no—
464C24—1662no—
465C24C23×D8311yescharacteristic
466C2×D4—33232no—
467C2×D4—33216no—
468C2×D4—16616no—
469C2×D4—16616no—
470C2×D4—16616no—
471C2×D4—1668no—
472C2×D4—1668no—
473C2×D4—1668no—
474C22×C4—22no—
475C22×C4—22no—
476C22×C4—22no—
477C22×C4—22no—
478C22×C4—24no—
479C22×C4—24no—
480C22×C4—24no—
481C22×C4—24no—
482C22×C4—24no—
483C22×C4—24no—
484C22×C4—834no—
485C22×C4—834no—
486C22×C4—834no—
487C22×C4—832no—
488C22×C4—832no—
489C22×C4—832no—
490C22×C4—8316no—
491C22×C4—8316no—
492C22×C4—834no—
493C22×C4—1668no—
494C22×C4—1668no—
495C22×C4—1668no—
496C22×C4—16632no—
497C22×C4—1668no—
498C22×C4—16632no—
499C22×C4—1664no—
500C22×C4—1664no—
501C22×C4—1664no—
502C22×C4—1662no—
503C22×C4—1662no—
504C22×C4—1662no—
505C22×C4—1662no—
506C22×C4—1662no—
507C22×C4—1662no—
508C22×C4—16632no—
509C22×C4—1668no—
510C22×C4—1668no—
511C22×C4—1668no—
512C22×C4D4×D8314yes—
513C22×C4D4×D8314yes—
514C22×C4#1305059014yes—
515C22×C4#1305059114yes—
516C22×C4#1305059114yes—
517C22×C4C23×D8314yes—
518C22×C4C23×D8314yes—
519C22×C4C23×D8312yes—
520C22×C4C23×D8312yes—
521C22×C4C23×D8312yes—
522C4⋊C4—28no—
523C4⋊C4—216no—
524C4⋊C4—28no—
525C4⋊C4—16616no—
526C4⋊C4—1668no—
527C22⋊C4—16616no—
528C22⋊C4—16616no—
529C22⋊C4—16616no—
530C22⋊C4—16616no—
531C22⋊C4—16616no—
532C22⋊C4—16616no—
533C42—28no—
534C42—1668no—
535C42—1668no—
536C42—1668no—
537C23—3328no—
538C23—3328no—
539C23—3328no—
540C23—3328no—
541C23—3328no—
542C23—3328no—
543C23—3328no—
544C23—8316no—
545C23—834no—
546C23—834no—
547C23—834no—
548C23—1664no—
549C23—1664no—
550C23—1664no—
551C23—1664no—
552C23—1664no—
553C23—1664no—
554C23—1664no—
555C23—1668no—
556C23—1668no—
557C23—1668no—
558C23—1668no—
559C23—1668no—
560C23—1668no—
561C23—1662no—
562C23—1662no—
563C23—1662no—
564C23—1662no—
565C23C2×D4×D8312yes—
566C23C2×D4×D8312yes—
567C23#41×D8312yes—
568C23#41×D8312yes—
569C23C2×#1305059012yes—
570C23C2×#1305059112yes—
571C23C2×#1305059112yes—
572C23C24×D8311yescharacteristic
573D4—33232no—
574D4—33216no—
575C2×C4—28no—
576C2×C4—28no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—24no—
582C2×C4—24no—
583C2×C4—24no—
584C2×C4—24no—
585C2×C4—24no—
586C2×C4—24no—
587C2×C4—48no—
588C2×C4—48no—
589C2×C4—3328no—
590C2×C4—834no—
591C2×C4—834no—
592C2×C4—834no—
593C2×C4—16632no—
594C2×C4—16632no—
595C2×C4—1664no—
596C2×C4—1664no—
597C2×C4—1664no—
598C2×C4—1668no—
599C2×C4—1668no—
600C2×C4—1668no—
601C2×C4—1662no—
602C2×C4—1662no—
603C2×C4—1662no—
604C2×C4—1662no—
605C2×C4—1662no—
606C2×C4—1662no—
607C2×C4—16632no—
608C2×C4#14581777116yescomplemented
609C2×C4C2×D4×D8314yes—
610C2×C4C2×D4×D8314yes—
611C2×C4C2×#1305059014yes—
612C2×C4C2×#1305059114yes—
613C2×C4C2×#1305059114yes—
614C22—3324no—
615C22—3324no—
616C22—3324no—
617C22—3324no—
618C22—3324no—
619C22—3324no—
620C22—3324no—
621C22—838no—
622C22—66416no—
623C22—16632no—
624C22—1664no—
625C22—1664no—
626C22—1664no—
627C22—1664no—
628C22—1664no—
629C22—1664no—
630C22#121×D8314yes—
631C22C2×#1458177714yes—
632C22#1912821414yes—
633C22#1912823614yes—
634C22#1912823814yes—
635C22#1912824814yes—
636C22#1912825214yes—
637C22C22×D4×D8311yescharacteristic
638C22C22×D4×D8311yescharacteristic
639C22C2×#41×D8311yescharacteristic
640C22C2×#41×D8311yescharacteristic
641C22C22×#1305059011yescharacteristic
642C22C22×#1305059111yescharacteristic
643C22C22×#1305059111yescharacteristic
644C4—24no—
645C4—24no—
646C4—44no—
647C4—44no—
648C4—3324no—
649C4—1664no—
650C4—1664no—
651C4—1664no—
652C4C2×#1458177718yescomplemented
653C2—6644no—
654C2—1668no—
655C2#2491579618yesdirect product factor
656C2C2×#121×D8311yescharacteristic
657C2C22×#1458177711yescharacteristic
658C2C2×#1912821411yescharacteristic
659C2C2×#1912823611yescharacteristic
660C2C2×#1912823811yescharacteristic
661C2C2×#1912824811yescharacteristic
662C2C2×#1912825211yescharacteristic
subgroupG/H
centerC24C23×D83
commutator, cosocleC23×C83C25
FrattiniC23C24×D83
FittingC2×#389×C83C2
socleC24×C83C24
prime radicalC22×#14581767C2
hypercenterC2×#389D83
hypocenterC83C2×#2854
LCSG ▷ C23×C83 ▷ C83
UCSC1 ◁ C24 ◁ C2×#389

Complements

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

NH ≅ G/Ntype
1. C22×#14581777C2semidirect
3. C22×C42×D83C2semidirect
4. #24915796C2direct (N × H)
5. C2×#19127603C2semidirect
6. C2×#19127602C2semidirect
7. C2×#19127597C2semidirect
8. C2×#19127591C2semidirect
9. C2×#19127590C2semidirect
10. C2×#19127579C2semidirect
11. C2×#19127558C2semidirect
12. C2×#389×C83C2semidirect
19. C2×#14581777C22semidirect
34. C2×C42×D83C22semidirect
35. #19127603C22semidirect
36. #19127602C22semidirect
37. #19127597C22semidirect
38. #19127591C22semidirect
39. #19127590C22semidirect
40. #19127579C22semidirect
41. #19127558C22semidirect
48. #389×C83C22semidirect
72. C22×C4×D83D4semidirect
168. C2×C4×D83C2×D4semidirect
255. C2×C4×C83#123semidirect
316. C4×C83C2×#123semidirect
322. C2×C83#2854semidirect
339. C2×#389D83semidirect
344. C83C2×#2854semidirect
374. #389C2×D83semidirect
608. C2×C4#14581777semidirect
652. C4C2×#14581777semidirect
655. C2#24915796direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#145817772
C22×#145817672
C22×C42×D832
#249157962
C2×#191276032
C2×#191276022
C2×#191275972
C2×#191275912
C2×#191275902
C2×#191275792
C2×#191275582
C2×#389×C832
C2×#285483