Finite Group Databasebeta

← C2×#24908471 #35431200

Subgroups — C2×#24908471 #35431200

Order 21248.

2182 conjugacy classes of subgroups.

The subgroup lattice has 632 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#13050580C211yesmaximal, characteristic, complemented
2C22×#14581771C211yesmaximal, characteristic
3C22×#14581716C211yesmaximal, characteristic, complemented
4C22×#14581715C211yesmaximal, characteristic, complemented
5C22×C22⋊(C83⋊C8)C211yesmaximal, characteristic
6#24908471C218yesmaximal, direct product factor
7C2×#19127463C211yesmaximal, characteristic, complemented
8C22×C22⋊Q8×C83C211yesmaximal, characteristic
9C23×Dic166C2211yescharacteristic
10C22×#13050580—24no—
11C22×#13050580—24no—
12C22×#13050580—24no—
13C22×#13050580C2214yescomplemented
14C23×C83⋊C8C2211yescharacteristic
15C23×C83⋊C8C2211yescharacteristic
16C2×#14581771—24no—
17C2×#14581771—22no—
18C2×#14581771—24no—
19C2×#14581771C2214yes—
20C22×C4⋊Dic83C2211yescharacteristic
21C22×Dic83⋊C4—21no—
22C22×C22⋊Dic83—21no—
23C2×#14581716—24no—
24C2×#14581716C2214yescomplemented
25C2×#14581715—24no—
26C2×#14581715C2214yescomplemented
27C2×C22⋊(C83⋊C8)—24no—
28C2×C22⋊(C83⋊C8)C2214yes—
29#19127463C2214yescomplemented
30C23×Q8×C83C2211yescharacteristic
31C24×C4×C83C2211yescharacteristic
32C2×C22⋊Q8×C83—24no—
33C2×C22⋊Q8×C83—22no—
34C2×C22⋊Q8×C83—24no—
35C2×C22⋊Q8×C83C2214yes—
36C22×C4⋊C4×C83—21no—
37C22×C4⋊C4×C83C2211yescharacteristic
38C22×C22⋊C4×C83—21no—
39C22×Dic166—22no—
40C22×Dic166—21no—
41C22×Dic166—22no—
42C22×Dic166—22no—
43C22×Dic166—22no—
44C22×Dic166—22no—
45C22×Dic166D414yes—
46C22×Dic166C2312yes—
47C23×Dic83—21no—
48C23×Dic83—21no—
49C2×#13050580—232no—
50C2×#13050580—28no—
51C2×#13050580—28no—
52C2×#13050580—28no—
53C22×C83⋊C8—22no—
54C22×C83⋊C8—22no—
55C22×C83⋊C8—22no—
56C22×C83⋊C8—22no—
57C22×C83⋊C8—22no—
58C22×C83⋊C8D414yes—
59C22×C83⋊C8C2312yes—
60C22×C83⋊C8C2312yes—
61#14581771—216no—
62#14581771—28no—
63#14581771—28no—
64C2×C4⋊Dic83—22no—
65C2×C4⋊Dic83—21no—
66C2×C4⋊Dic83D414yes—
67C2×C4⋊Dic83C2312yes—
68C2×Dic83⋊C4—28no—
69C2×Dic83⋊C4—24no—
70C2×C22⋊Dic83—28no—
71C2×C22⋊Dic83—24no—
72#14581716—28no—
73#14581715—28no—
74C22⋊(C83⋊C8)—28no—
75C25×C83D411yescharacteristic
76C22×Q8×C83—22no—
77C22×Q8×C83—21no—
78C22×Q8×C83—22no—
79C22×Q8×C83—22no—
80C22×Q8×C83—22no—
81C22×Q8×C83—22no—
82C22×Q8×C83D414yes—
83C22×Q8×C83C2312yes—
84C23×C4×C83—21no—
85C23×C4×C83—21no—
86C23×C4×C83—22no—
87C23×C4×C83—22no—
88C23×C4×C83—22no—
89C23×C4×C83—22no—
90C23×C4×C83—22no—
91C23×C4×C83—22no—
92C23×C4×C83D411yescharacteristic
93C23×C4×C83D412yes—
94C23×C4×C83C2312yes—
95C23×C4×C83C2311yescharacteristic
96C22⋊Q8×C83—216no—
97C22⋊Q8×C83—28no—
98C22⋊Q8×C83—28no—
99C2×C4⋊C4×C83—28no—
100C2×C4⋊C4×C83—24no—
101C2×C4⋊C4×C83—21no—
102C2×C4⋊C4×C83—22no—
103C2×C4⋊C4×C83D414yes—
104C2×C4⋊C4×C83C2312yes—
105C2×C22⋊C4×C83—28no—
106C2×C22⋊C4×C83—24no—
107C2×Dic166—28no—
108C2×Dic166—28no—
109C2×Dic166—22no—
110C2×Dic166—22no—
111C2×Dic166—22no—
112C2×Dic166—22no—
113C2×Dic166—24no—
114C2×Dic166—22no—
115C2×Dic166—42no—
116C2×Dic166—44no—
117C2×Dic166—42no—
118C2×Dic166—44no—
119C2×Dic166C2×D414yes—
120C22×Dic83—24no—
121C22×Dic83—24no—
122C22×Dic83—22no—
123C22×Dic83—22no—
124C22×Dic83—42no—
125C22×Dic83—42no—
126C22×Dic83—42no—
127C22×Dic83—42no—
128#13050580—232no—
129C2×C83⋊C8—28no—
130C2×C83⋊C8—22no—
131C2×C83⋊C8—22no—
132C2×C83⋊C8—22no—
133C2×C83⋊C8—22no—
134C2×C83⋊C8—22no—
135C2×C83⋊C8—44no—
136C2×C83⋊C8C2×D414yes—
137C4⋊Dic83—24no—
138C4⋊Dic83—22no—
139C4⋊Dic83C2×D414yes—
140Dic83⋊C4—216no—
141C22⋊Dic83—216no—
142C24×C83—22no—
143C24×C83—22no—
144C24×C83—22no—
145C24×C83—22no—
146C24×C83—22no—
147C24×C83—22no—
148C24×C83Q1614yescomplemented
149C24×C83C2×D412yes—
150C24×C83C2×D411yescharacteristic
151C2×Q8×C83—24no—
152C2×Q8×C83—22no—
153C2×Q8×C83—28no—
154C2×Q8×C83—28no—
155C2×Q8×C83—22no—
156C2×Q8×C83—22no—
157C2×Q8×C83—22no—
158C2×Q8×C83—22no—
159C2×Q8×C83—42no—
160C2×Q8×C83—44no—
161C2×Q8×C83—42no—
162C2×Q8×C83—44no—
163C2×Q8×C83C2×D414yes—
164C22×C4×C83—24no—
165C22×C4×C83—24no—
166C22×C4×C83—22no—
167C22×C4×C83—22no—
168C22×C4×C83—22no—
169C22×C4×C83—22no—
170C22×C4×C83—22no—
171C22×C4×C83—24no—
172C22×C4×C83—24no—
173C22×C4×C83—24no—
174C22×C4×C83—22no—
175C22×C4×C83—22no—
176C22×C4×C83—22no—
177C22×C4×C83—22no—
178C22×C4×C83—22no—
179C22×C4×C83—22no—
180C22×C4×C83—42no—
181C22×C4×C83—42no—
182C22×C4×C83—42no—
183C22×C4×C83—42no—
184C22×C4×C83—44no—
185C22×C4×C83—44no—
186C22×C4×C83—44no—
187C22×C4×C83—44no—
188C22×C4×C83C2×D412yes—
189C22×C4×C83C2×D412yes—
190C22×C4×C83C2×D412yes—
191C22×C4×C83C2×D412yes—
192C22×C4×C83C2×D411yescharacteristic
193C22×C4×C83C2×D412yes—
194C22×C4×C83C2×D412yes—
195C22×C4×C83#4112yes—
196C22×C4×C83C2411yescharacteristic
197C4⋊C4×C83—216no—
198C4⋊C4×C83—24no—
199C4⋊C4×C83—22no—
200C4⋊C4×C83C2×D414yes—
201C22⋊C4×C83—216no—
202C23×C83—22no—
203C23×C83—22no—
204C23×C83—22no—
205C23×C83—22no—
206C23×C83—22no—
207C23×C83—22no—
208C23×C83—28no—
209C23×C83—28no—
210C23×C83—28no—
211C23×C83—44no—
212C23×C83—44no—
213C23×C83—44no—
214C23×C83—44no—
215C23×C83#12012yes—
216C23×C83#12012yes—
217C23×C83#12112yes—
218C23×C83C2×Q1612yes—
219C23×C83C2×Q1614yescomplemented
220C23×C83#13512yes—
221C23×C83#13712yes—
222C23×C83#13712yes—
223C23×C83C22×D411yescharacteristic
224Q8×C83—24no—
225Q8×C83—24no—
226Q8×C83—42no—
227Q8×C83—42no—
228Q8×C83—48no—
229C2×C4×C83—24no—
230C2×C4×C83—24no—
231C2×C4×C83—22no—
232C2×C4×C83—22no—
233C2×C4×C83—22no—
234C2×C4×C83—42no—
235C2×C4×C83—42no—
236C2×C4×C83—42no—
237C2×C4×C83—42no—
238C2×C4×C83—44no—
239C2×C4×C83—48no—
240C2×C4×C83—48no—
241C2×C4×C83—42no—
242C2×C4×C83—42no—
243C2×C4×C83—42no—
244C2×C4×C83—42no—
245C2×C4×C83—48no—
246C2×C4×C83#12014yes—
247C2×C4×C83#12012yes—
248C2×C4×C83#12012yes—
249C2×C4×C83#12114yes—
250C2×C4×C83#12114yes—
251C2×C4×C83#12112yes—
252C2×C4×C83#12714yes—
253C2×C4×C83C22×D411yescharacteristic
254C2×C4×C83C22×D411yescharacteristic
255C2×C4×C83C22×D411yescharacteristic
256C2×C4×C83C22×D411yescharacteristic
257C2×C4×C83C22×D411yescharacteristic
258C2×C4×C83C2×#4111yescharacteristic
259C2×Dic83—24no—
260C2×Dic83—24no—
261C2×Dic83—42no—
262C2×Dic83—42no—
263C2×Dic83—42no—
264C2×Dic83—42no—
265C2×Dic83—48no—
266C2×Dic83—48no—
267Dic166—24no—
268Dic166—24no—
269Dic166—42no—
270Dic166—42no—
271Dic166—48no—
272C83⋊C8—24no—
273C83⋊C8—42no—
274C22×C83—24no—
275C22×C83—24no—
276C22×C83—24no—
277C22×C83—42no—
278C22×C83—42no—
279C22×C83—42no—
280C22×C83—416no—
281C22×C83—42no—
282C22×C83#39214yes—
283C22×C83#45114yes—
284C22×C83#45114yes—
285C22×C83#45214yes—
286C22×C83#45214yes—
287C22×C83#46714yes—
288C22×C83#47014yes—
289C22×C83C2×#12011yescharacteristic
290C22×C83C2×#12011yescharacteristic
291C22×C83C2×#12111yescharacteristic
292C22×C83C22×Q1611yescharacteristic
293C22×C83C2×#13511yescharacteristic
294C22×C83C2×#13711yescharacteristic
295C22×C83C2×#13711yescharacteristic
296C4×C83—42no—
297C4×C83—44no—
298C4×C83—44no—
299C4×C83#39214yes—
300C4×C83C2×#12011yescharacteristic
301C4×C83C2×#12111yescharacteristic
302C4×C83C2×#12111yescharacteristic
303C4×C83C2×#12711yescharacteristic
304Dic83—44no—
305Dic83—44no—
306C2×#2001—831nomaximal, Sylow
307C2×C83—44no—
308C2×C83#200118yescomplemented
309C2×C83C2×#39211yescharacteristic
310C2×C83C2×#45111yescharacteristic
311C2×C83C2×#45111yescharacteristic
312C2×C83C2×#45211yescharacteristic
313C2×C83C2×#45211yescharacteristic
314C2×C83C2×#46711yescharacteristic
315C2×C83C2×#47011yescharacteristic
316C23×Q16—831no—
317C22×C22⋊Q8—831no—
318C22×C22⋊Q8D8311yescharacteristic
319C22×#103—831no—
320C22×#103—831no—
321C22×C22⋊C8—831no—
322#2001—838no—
323C2×#340—831no—
324C83C2×#200111yescharacteristic, complemented, Sylow
325C23×Q8—831no—
326C23×Q8C2×D8311yescharacteristic
327C24×C4C2×D8311yescharacteristic
328C22×Q16—834no—
329C22×Q16—1664no—
330C22×Q16—1664no—
331C22×Q16—1664no—
332C23×C8—831no—
333C23×C8—831no—
334C2×C22⋊Q8—24no—
335C2×C22⋊Q8—22no—
336C2×C22⋊Q8—24no—
337C2×C22⋊Q8—834no—
338C2×C22⋊Q8—1664no—
339C2×C22⋊Q8—1664no—
340C2×C22⋊Q8—1662no—
341C2×C22⋊Q8C2×D8314yes—
342C22×C4⋊C4—21no—
343C22×C4⋊C4—831no—
344C22×C4⋊C4—1661no—
345C22×C4⋊C4C2×D8311yescharacteristic
346C22×C22⋊C4—21no—
347C22×C22⋊C4—1661no—
348C2×#103—834no—
349C2×#103—834no—
350C2×#103—1664no—
351C2×#103—1664no—
352C2×C22⋊C8—834no—
353C2×C22⋊C8—1664no—
354#340—834no—
355C25C22⋊D8311yescharacteristic
356C22×Q8—22no—
357C22×Q8—21no—
358C22×Q8—22no—
359C22×Q8—22no—
360C22×Q8—22no—
361C22×Q8—22no—
362C22×Q8—834no—
363C22×Q8—832no—
364C22×Q8—1662no—
365C22×Q8—1662no—
366C22×Q8—1662no—
367C22×Q8—1662no—
368C22×Q8—1662no—
369C22×Q8—1661no—
370C22×Q8C22⋊D8314yes—
371C22×Q8C22×D8312yes—
372C23×C4—21no—
373C23×C4—21no—
374C23×C4—22no—
375C23×C4—22no—
376C23×C4—22no—
377C23×C4—22no—
378C23×C4—22no—
379C23×C4—22no—
380C23×C4—1661no—
381C23×C4—1661no—
382C23×C4C22⋊D8312yes—
383C23×C4C22⋊D8311yescharacteristic
384C23×C4C22×D8312yes—
385C23×C4C22×D8311yescharacteristic
386C2×Q16—1668no—
387C2×Q16—1668no—
388C2×Q16—16632no—
389C2×Q16—1668no—
390C22×C8—834no—
391C22×C8—832no—
392C22×C8—832no—
393C22×C8—1662no—
394C22×C8—1662no—
395C22×C8—1662no—
396C22×C8—1662no—
397C22×C8—1662no—
398C22⋊Q8—216no—
399C22⋊Q8—28no—
400C22⋊Q8—28no—
401C22⋊Q8—1668no—
402C22⋊Q8—1668no—
403C22⋊Q8—16616no—
404C2×C4⋊C4—28no—
405C2×C4⋊C4—24no—
406C2×C4⋊C4—21no—
407C2×C4⋊C4—22no—
408C2×C4⋊C4—834no—
409C2×C4⋊C4—832no—
410C2×C4⋊C4—1668no—
411C2×C4⋊C4—1662no—
412C2×C4⋊C4—1661no—
413C2×C4⋊C4—1664no—
414C2×C4⋊C4C22⋊D8314yes—
415C2×C4⋊C4C22×D8312yes—
416C2×C22⋊C4—28no—
417C2×C22⋊C4—24no—
418C2×C22⋊C4—1668no—
419C2×C22⋊C4—1664no—
420#103—1668no—
421#103—1668no—
422C22⋊C8—1668no—
423C24—22no—
424C24—22no—
425C24—22no—
426C24—22no—
427C24—22no—
428C24—22no—
429C24#1305058014yescomplemented
430C24C2×C22⋊D8312yes—
431C24C2×C22⋊D8311yescharacteristic
432C2×Q8—24no—
433C2×Q8—22no—
434C2×Q8—28no—
435C2×Q8—28no—
436C2×Q8—22no—
437C2×Q8—22no—
438C2×Q8—22no—
439C2×Q8—22no—
440C2×Q8—42no—
441C2×Q8—44no—
442C2×Q8—42no—
443C2×Q8—44no—
444C2×Q8—3322no—
445C2×Q8—3324no—
446C2×Q8—3322no—
447C2×Q8—3324no—
448C2×Q8—834no—
449C2×Q8—1668no—
450C2×Q8—1668no—
451C2×Q8—1662no—
452C2×Q8—1662no—
453C2×Q8—1662no—
454C2×Q8—1662no—
455C2×Q8—1664no—
456C2×Q8—1662no—
457C2×Q8C2×C22⋊D8314yes—
458C22×C4—22no—
459C22×C4—24no—
460C22×C4—24no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—24no—
466C22×C4—24no—
467C22×C4—24no—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—22no—
473C22×C4—22no—
474C22×C4—42no—
475C22×C4—42no—
476C22×C4—42no—
477C22×C4—42no—
478C22×C4—44no—
479C22×C4—44no—
480C22×C4—44no—
481C22×C4—44no—
482C22×C4—3322no—
483C22×C4—3322no—
484C22×C4—3322no—
485C22×C4—3322no—
486C22×C4—1664no—
487C22×C4—1664no—
488C22×C4—1662no—
489C22×C4—1662no—
490C22×C4D4×D8312yes—
491C22×C4D4×D8312yes—
492C22×C4D4×D8312yes—
493C22×C4C2×C22⋊D8312yes—
494C22×C4C2×C22⋊D8311yescharacteristic
495C22×C4C2×C22⋊D8312yes—
496C22×C4C2×C22⋊D8312yes—
497C22×C4#1305058912yes—
498C22×C4C23×D8311yescharacteristic
499Q16—16632no—
500C2×C8—3324no—
501C2×C8—834no—
502C2×C8—1662no—
503C2×C8—1662no—
504C2×C8—1662no—
505C2×C8—1668no—
506C2×C8—1662no—
507C2×C8—1662no—
508C4⋊C4—216no—
509C4⋊C4—24no—
510C4⋊C4—22no—
511C4⋊C4—834no—
512C4⋊C4—16616no—
513C4⋊C4—1664no—
514C4⋊C4—1662no—
515C4⋊C4C2×C22⋊D8314yes—
516C22⋊C4—216no—
517C22⋊C4—16616no—
518C23—28no—
519C23—28no—
520C23—22no—
521C23—22no—
522C23—22no—
523C23—28no—
524C23—22no—
525C23—22no—
526C23—22no—
527C23—44no—
528C23—44no—
529C23—44no—
530C23—44no—
531C23#1458176012yes—
532C23#1458176112yes—
533C23#1458176412yes—
534C23C2×#1305058012yes—
535C23C2×#1305058014yescomplemented
536C23#1458181212yes—
537C23#1458182212yes—
538C23#1458182312yes—
539C23C22×C22⋊D8311yescharacteristic
540Q8—24no—
541Q8—24no—
542Q8—48no—
543Q8—42no—
544Q8—42no—
545Q8—3328no—
546Q8—3322no—
547Q8—3322no—
548Q8—1664no—
549Q8—1664no—
550C2×C4—22no—
551C2×C4—24no—
552C2×C4—24no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—48no—
556C2×C4—44no—
557C2×C4—48no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—48no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—3328no—
568C2×C4—3328no—
569C2×C4—3322no—
570C2×C4—3322no—
571C2×C4—3322no—
572C2×C4—3322no—
573C2×C4—1664no—
574C2×C4—1664no—
575C2×C4#1458176012yes—
576C2×C4#1458176014yes—
577C2×C4#1458176112yes—
578C2×C4#1458176314yes—
579C2×C4#1458176412yes—
580C2×C4#1458176414yes—
581C2×C4#1458178814yes—
582C2×C4C2×D4×D8311yescharacteristic
583C2×C4C2×D4×D8311yescharacteristic
584C2×C4C2×D4×D8311yescharacteristic
585C2×C4C22×C22⋊D8311yescharacteristic
586C2×C4C22×C22⋊D8311yescharacteristic
587C2×C4C2×#1305058911yescharacteristic
588C8—3322no—
589C8—1664no—
590C22—24no—
591C22—24no—
592C22—24no—
593C22—42no—
594C22—42no—
595C22—416no—
596C22—42no—
597C22—42no—
598C22#1912760914yes—
599C22#1912785614yes—
600C22#1912785714yes—
601C22#1912786114yes—
602C22#1912786314yes—
603C22#1912794014yes—
604C22#1912795714yes—
605C22C2×#1458176011yescharacteristic
606C22C2×#1458176111yescharacteristic
607C22C2×#1458176411yescharacteristic
608C22C22×#1305058011yescharacteristic
609C22C2×#1458181211yescharacteristic
610C22C2×#1458182211yescharacteristic
611C22C2×#1458182311yescharacteristic
612C4—42no—
613C4—44no—
614C4—44no—
615C4—3324no—
616C4—3324no—
617C4#1912760914yes—
618C4C2×#1458176011yescharacteristic
619C4C2×#1458176311yescharacteristic
620C4C2×#1458176411yescharacteristic
621C4C2×#1458178811yescharacteristic
622C2—44no—
623C2#2490847118yesdirect product factor
624C2C2×#1912760911yescharacteristic
625C2C2×#1912785611yescharacteristic
626C2C2×#1912785711yescharacteristic
627C2C2×#1912786111yescharacteristic
628C2C2×#1912786311yescharacteristic
629C2C2×#1912794011yescharacteristic
630C2C2×#1912795711yescharacteristic
subgroupG/H
centerC24C2×C22⋊D83
commutator, cosocleC22×C4×C83C24
FrattiniC22×C4C23×D83
FittingC22×C22⋊Q8×C83C2
socleC24×C83C2×D4
prime radicalC25×C83D4
hypercenterC22×C22⋊Q8D83
hypocenterC83C2×#2001
LCSG ▷ C22×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C24 ◁ C24×C4 ◁ C22×C22⋊Q8

Complements

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

NH ≅ G/Ntype
1. C23×#13050580C2semidirect
3. C22×#14581716C2semidirect
4. C22×#14581715C2semidirect
6. #24908471C2direct (N × H)
7. C2×#19127463C2semidirect
13. C22×#13050580C22semidirect
24. C2×#14581716C22semidirect
26. C2×#14581715C22semidirect
29. #19127463C22semidirect
148. C24×C83Q16semidirect
219. C23×C83C2×Q16semidirect
308. C2×C83#2001semidirect
324. C83C2×#2001semidirect
429. C24#13050580semidirect
535. C23C2×#13050580semidirect
623. C2#24908471direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#130505802
C22×#145817712
C22×#145817162
C22×#145817152
C22×C22⋊(C83⋊C8)2
#249084712
C2×#191274632
C22×C22⋊Q8×C832
C2×#200183