Finite Group Databasebeta

← C2×#24917854 #36273691

Subgroups — C2×#24917854 #36273691

Order 21248.

2398 conjugacy classes of subgroups.

The subgroup lattice has 662 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19128247C211yesmaximal, characteristic
2C22×#14581791C211yesmaximal, characteristic, complemented
3#24917854C2116yesmaximal, direct product factor
4C2×#19128062C212yesmaximal, complemented
5C2×#19128059C212yesmaximal, complemented
6C2×#19128050C212yesmaximal, complemented
7C2×#19128046C212yesmaximal, complemented
8C2×#19127703C211yesmaximal, characteristic, complemented
9C2×#19127702C211yesmaximal, characteristic, complemented
10C2×#19127700C211yesmaximal, characteristic, complemented
11C2×M16×Dic83C211yesmaximal, characteristic, complemented
12C2×#417×C83C211yesmaximal, characteristic, complemented
13C22×#13050590C2211yescharacteristic
14C23×Dic166—21no—
15C22×#13050574C2211yescharacteristic
16#19128247—24no—
17#19128247C2218yes—
18C2×#14581791—24no—
19C2×#14581791—21no—
20C2×#14581791—21no—
21C2×#14581791C2212yes—
22C2×#14581791C2212yes—
23C2×#14581791C2218yescomplemented
24C2×#14581790C2211yescharacteristic
25C2×#14581788C2211yescharacteristic
26C2×#14581781C2211yescharacteristic
27C2×#14581779C2211yescharacteristic
28C2×#14581764—21no—
29C2×#14581764—21no—
30C2×C4×C22⋊D83—21no—
31C2×C4×C22⋊D83—21no—
32C22×C4⋊Dic83C2211yescharacteristic
33C22×Dic83⋊C4—21no—
34C22×C4×Dic83C2211yescharacteristic
35#19128062C22116yescomplemented
36#19128059C22116yescomplemented
37#19128050C22116yescomplemented
38#19128046C22116yescomplemented
39#19127703C2218yescomplemented
40#19127702C2218yescomplemented
41#19127700C2218yescomplemented
42C2×#14581717C2212yes—
43C2×#14581716C2212yes—
44C2×#14581715C2212yes—
45C2×#14581714C2212yes—
46C2×#14581713C2212yes—
47C2×#14581712C2212yes—
48M16×Dic83C2218yescomplemented
49C2×#14581702C2212yes—
50C2×C8×Dic83C2212yes—
51C22×#41×C83C2211yescharacteristic
52C22×M16×C83C2211yescharacteristic
53C22×C4⋊C4×C83C2211yescharacteristic
54#417×C83C2218yescomplemented
55C2×#103×C83C2212yes—
56C2×#102×C83C2212yes—
57C2×#13050590—22no—
58C2×#13050590—24no—
59C2×#13050590—24no—
60C2×#13050590C2314yes—
61C22×Dic166—22no—
62C22×Dic166—22no—
63C22×Dic166—28no—
64C22×Dic166—48no—
65C22×Dic166C2311yescharacteristic
66C22×D332C2311yescharacteristic
67C22×C22⋊D83—21no—
68C22×C22⋊D83—21no—
69C22×C22⋊D83—21no—
70C23×Dic83—21no—
71C23×Dic83D411yescharacteristic
72C23×Dic83D411yescharacteristic
73C22×C4×D83—21no—
74C2×#13050574—24no—
75C2×#13050574C2314yes—
76C22×C83⋊C8C2312yes—
77#14581791—24no—
78#14581791—24no—
79#14581791—28no—
80#14581791—28no—
81#14581791C2318yes—
82#14581791C2318yes—
83#14581790—22no—
84#14581790C2314yes—
85#14581788—22no—
86#14581788C2314yes—
87#14581781—22no—
88#14581781C2314yes—
89#14581779—22no—
90#14581779C2314yes—
91#14581764—28no—
92#14581764—28no—
93C4×C22⋊D83—28no—
94C4×C22⋊D83—28no—
95C2×C4⋊Dic83—21no—
96C2×C4⋊Dic83—22no—
97C2×C4⋊Dic83C2312yes—
98C2×C4⋊Dic83C2314yes—
99C2×Dic83⋊C4—22no—
100C2×Dic83⋊C4—22no—
101C2×Dic83⋊C4—28no—
102C2×Dic83⋊C4—21no—
103C2×Dic83⋊C4—21no—
104C2×C22⋊Dic83—21no—
105C2×C22⋊Dic83—21no—
106C2×(C2×D83)⋊C4—21no—
107C2×(C2×D83)⋊C4—21no—
108C2×C4×Dic83D411yescharacteristic
109C2×C4×Dic83D411yescharacteristic
110C2×C4×Dic83D414yes—
111C2×C4×Dic83C2311yescharacteristic
112C2×C4×Dic83C2311yescharacteristic
113C2×C4×Dic83C2314yes—
114#14581717C2318yes—
115#14581716C2318yes—
116#14581715C2318yes—
117#14581714C2318yes—
118#14581713C2318yes—
119#14581712C2318yes—
120#14581702C2318yes—
121C8×Dic83C2318yes—
122C2×#41×C83—22no—
123C2×#41×C83—24no—
124C2×#41×C83—24no—
125C2×#41×C83C2314yes—
126C22×Q8×C83C2311yescharacteristic
127C22×D4×C83—21no—
128C22×D4×C83C2311yescharacteristic
129C23×C4×C83—21no—
130C23×C4×C83—21no—
131C23×C4×C83C2311yescharacteristic
132C2×M16×C83—24no—
133C2×M16×C83C2314yes—
134C22×C8×C83C2312yes—
135C2×C4⋊C4×C83—21no—
136C2×C4⋊C4×C83—22no—
137C2×C4⋊C4×C83C2312yes—
138C2×C4⋊C4×C83C2314yes—
139#103×C83C2318yes—
140#102×C83C2318yes—
141C23×D83—21no—
142#13050590—28no—
143#13050590—28no—
144#13050590—48no—
145C2×Dic166—22no—
146C2×Dic166—22no—
147C2×Dic166—21no—
148C2×Dic166—28no—
149C2×Dic166—28no—
150C2×Dic166—48no—
151C2×Dic166—432no—
152C2×Dic166—48no—
153C2×Dic166C2412yes—
154C2×D332—22no—
155C2×D332—22no—
156C2×D332—21no—
157C2×D332C2412yes—
158C2×C22⋊D83—24no—
159C2×C22⋊D83—24no—
160C2×C22⋊D83—24no—
161C2×C22⋊D83—44no—
162C2×C22⋊D83—44no—
163C2×C22⋊D83—44no—
164C22×Dic83—24no—
165C22×Dic83—24no—
166C22×Dic83—21no—
167C22×Dic83—24no—
168C22×Dic83—22no—
169C22×Dic83—44no—
170C22×Dic83C2×D414yes—
171C22×Dic83C2×D414yes—
172C22×Dic83C2×D411yescharacteristic
173C22×Dic83C2×D411yescharacteristic
174C22×Dic83C2×D411yescharacteristic
175C22×Dic83C2×D411yescharacteristic
176C2×C4×D83—24no—
177C2×C4×D83—44no—
178#13050574—28no—
179C2×C83⋊C8—24no—
180C2×C83⋊C8C2414yes—
181C4⋊Dic83—24no—
182C4⋊Dic83—22no—
183C4⋊Dic83C2414yes—
184Dic83⋊C4—24no—
185Dic83⋊C4—24no—
186Dic83⋊C4—28no—
187Dic83⋊C4—28no—
188C22⋊Dic83—24no—
189C22⋊Dic83—24no—
190(C2×D83)⋊C4—24no—
191(C2×D83)⋊C4—24no—
192C4×Dic83C2×D412yes—
193C4×Dic83C2×D412yes—
194C4×Dic83C2×D414yes—
195C4×Dic83C2×D414yes—
196C4×Dic83C2412yes—
197C4×Dic83C2412yes—
198C24×C83—21no—
199C24×C83C2×D411yescharacteristic
200#41×C83—28no—
201#41×C83—28no—
202#41×C83—48no—
203C2×Q8×C83—21no—
204C2×Q8×C83—22no—
205C2×Q8×C83—22no—
206C2×Q8×C83C2412yes—
207C2×D4×C83—24no—
208C2×D4×C83—21no—
209C2×D4×C83—22no—
210C2×D4×C83—22no—
211C2×D4×C83—44no—
212C2×D4×C83C2412yes—
213C22×C4×C83—24no—
214C22×C4×C83—22no—
215C22×C4×C83—21no—
216C22×C4×C83—24no—
217C22×C4×C83—44no—
218C22×C4×C83—44no—
219C22×C4×C83C2×D411yescharacteristic
220C22×C4×C83C2×D412yes—
221C22×C4×C83#4114yes—
222C22×C4×C83#4114yes—
223C22×C4×C83C2411yescharacteristic
224C22×C4×C83C2412yes—
225M16×C83—28no—
226C2×C8×C83—24no—
227C2×C8×C83C2414yes—
228C4⋊C4×C83—24no—
229C4⋊C4×C83—22no—
230C4⋊C4×C83C2414yes—
231C23×C83—22no—
232C23×C83—22no—
233C23×C83—22no—
234C23×C83—42no—
235C23×C83—42no—
236C23×C83—42no—
237C23×C83#12414yes—
238C23×C83C22×D412yes—
239C23×C83C22×D411yescharacteristic
240C22×D83—22no—
241C22×D83—42no—
242C22×D83—42no—
243C22×D83—42no—
244Q8×C83—22no—
245Q8×C83—22no—
246Q8×C83—42no—
247D4×C83—22no—
248D4×C83—22no—
249D4×C83—48no—
250D4×C83—42no—
251C2×C4×C83—24no—
252C2×C4×C83—22no—
253C2×C4×C83—48no—
254C2×C4×C83—42no—
255C2×C4×C83—44no—
256C2×C4×C83—48no—
257C2×C4×C83#12414yes—
258C2×C4×C83#12418yes—
259C2×C4×C83C22×D412yes—
260C2×C4×C83C22×D411yescharacteristic
261C2×C4×C83C2×#4112yes—
262C2×C4×C83C2×#4114yes—
263C2×C4×C83C2×#4112yes—
264C2×C4×C83C2×#4114yes—
265C2×C4×C83C2511yescharacteristic
266C22⋊D83—48no—
267C22⋊D83—48no—
268C22⋊D83—48no—
269C2×Dic83—22no—
270C2×Dic83—22no—
271C2×Dic83—28no—
272C2×Dic83—28no—
273C2×Dic83—22no—
274C2×Dic83—24no—
275C2×Dic83—48no—
276C2×Dic83—42no—
277C2×Dic83—44no—
278C2×Dic83AGL(1,ℤ/8)14yes—
279C2×Dic83#13714yes—
280C2×Dic83C22×D412yes—
281C2×Dic83C22×D412yes—
282C2×Dic83C22×D412yes—
283C2×Dic83C22×D412yes—
284D332—22no—
285D332—22no—
286D332—42no—
287C4×D83—48no—
288Dic166—22no—
289Dic166—22no—
290Dic166—416no—
291Dic166—42no—
292Dic166—416no—
293C8×C83—24no—
294C83⋊C8—24no—
295C22×C83—28no—
296C22×C83—22no—
297C22×C83—22no—
298C22×C83—42no—
299C22×C83—42no—
300C22×C83—42no—
301C22×C83—84no—
302C22×C83C2×#12414yes—
303C22×C83C2×#12412yes—
304C22×C83C2×AGL(1,ℤ/8)12yes—
305C22×C83C2×#13712yes—
306C22×C83C23×D411yescharacteristic
307C2×D83—42no—
308C2×D83—42no—
309C2×D83—42no—
310C2×D83—84no—
311C4×C83—42no—
312C4×C83—44no—
313C4×C83C2×#12412yes—
314C4×C83C2×#12414yes—
315C4×C83C22×#4111yescharacteristic
316C4×C83C22×#4111yescharacteristic
317Dic83—22no—
318Dic83—22no—
319Dic83—42no—
320Dic83—44no—
321Dic83C2×AGL(1,ℤ/8)14yes—
322Dic83C2×#13714yes—
323C2×#3141—831nomaximal, Sylow
324C2×C83—24no—
325C2×C83—82no—
326C2×C83#314114yescomplemented
327C2×C83C22×#12411yescharacteristic
328C2×C83C22×AGL(1,ℤ/8)11yescharacteristic
329C2×C83C22×#13711yescharacteristic
330D83—82no—
331C2×#532—831no—
332C22×C4⋊Q8—831no—
333#3141—8316no—
334C2×#489—832no—
335C2×#489—832no—
336C2×#487—832no—
337C2×#486—832no—
338C2×#417—831no—
339C2×#417—831no—
340C2×#417—831no—
341C2×#417D8311yescharacteristic, complemented
342C2×C4×M16—831no—
343C83C2×#314111yescharacteristic, complemented, Sylow
344C22×#41—831no—
345C22×#41C2×D8311yescharacteristic
346C23×Q8—1661no—
347C22×M16—831no—
348C22×M16C2×D8311yescharacteristic
349#532—838no—
350#532—1664no—
351C2×C4⋊Q8—832no—
352C2×C4⋊Q8—832no—
353C2×C4⋊Q8—838no—
354C2×C4⋊Q8—831no—
355C2×C4⋊Q8—1661no—
356C2×C4⋊Q8—1661no—
357C2×C4⋊Q8—1664no—
358C2×#127—831no—
359C2×#124—831no—
360C2×#124—831no—
361C2×#121—1661no—
362C2×#121—1661no—
363C2×C4×D4—1661no—
364C2×C4×D4—1661no—
365C22×C4⋊C4—831no—
366C22×C4⋊C4—1661no—
367C22×C4⋊C4C2×D8311yescharacteristic
368C22×C42—831no—
369#489—8316no—
370#489—8316no—
371#487—8316no—
372#486—8316no—
373#417—838no—
374#417—838no—
375#417—838no—
376#417C2×D8318yescomplemented
377C2×#103—832no—
378C2×#103—832no—
379C2×#103—832no—
380C2×#103C2×D8312yes—
381C2×#102—832no—
382C2×#102—832no—
383C2×#102—832no—
384C2×#102C2×D8312yes—
385C4×M16—838no—
386C2×#97—832no—
387C2×C4×C8—832no—
388C2×#41—22no—
389C2×#41—24no—
390C2×#41—24no—
391C2×#41—834no—
392C2×#41—1664no—
393C2×#41—1664no—
394C2×#41—1662no—
395C2×#41C22×D8314yes—
396C22×Q8—3328no—
397C22×Q8—831no—
398C22×Q8—1668no—
399C22×Q8—1662no—
400C22×Q8—1662no—
401C22×Q8C22×D8311yescharacteristic
402C22×D4—21no—
403C22×D4—831no—
404C22×D4—1661no—
405C22×D4—1661no—
406C22×D4—1661no—
407C22×D4C22×D8311yescharacteristic
408C23×C4—21no—
409C23×C4—21no—
410C23×C4—831no—
411C23×C4—831no—
412C23×C4—1661no—
413C23×C4—1661no—
414C23×C4C22×D8311yescharacteristic
415C2×M16—24no—
416C2×M16—834no—
417C2×M16—1664no—
418C2×M16C22×D8314yes—
419C22×C8—832no—
420C22×C8C22×D8312yes—
421C4⋊Q8—838no—
422C4⋊Q8—838no—
423C4⋊Q8—834no—
424C4⋊Q8—1668no—
425C4⋊Q8—1668no—
426C4⋊Q8—1664no—
427C4⋊Q8—1664no—
428C4⋊Q8—1662no—
429#127—834no—
430#127—1662no—
431#124—834no—
432#124—834no—
433#124—1662no—
434#124—1662no—
435#121—1668no—
436#121—1668no—
437C4×D4—1668no—
438C4×D4—1668no—
439C2×C4⋊C4—22no—
440C2×C4⋊C4—21no—
441C2×C4⋊C4—832no—
442C2×C4⋊C4—834no—
443C2×C4⋊C4—1668no—
444C2×C4⋊C4—1662no—
445C2×C4⋊C4—1662no—
446C2×C4⋊C4—1661no—
447C2×C4⋊C4—1661no—
448C2×C4⋊C4—1662no—
449C2×C4⋊C4—1661no—
450C2×C4⋊C4C22×D8312yes—
451C2×C4⋊C4C22×D8314yes—
452C2×C22⋊C4—1661no—
453C2×C22⋊C4—1661no—
454C2×C22⋊C4—1661no—
455C2×C22⋊C4—1661no—
456C2×C42—831no—
457C2×C42—831no—
458C2×C42—834no—
459C2×C42—831no—
460C2×C42—831no—
461C2×C42—834no—
462#103—838no—
463#103—838no—
464#103—838no—
465#103C22×D8318yes—
466#102—838no—
467#102—838no—
468#102—838no—
469#102C22×D8318yes—
470#97—838no—
471C4×C8—838no—
472C24—21no—
473C24—1661no—
474C24D4×D8311yescharacteristic
475#41—28no—
476#41—28no—
477#41—48no—
478#41—3328no—
479#41—1668no—
480#41—1668no—
481C2×Q8—21no—
482C2×Q8—22no—
483C2×Q8—22no—
484C2×Q8—3328no—
485C2×Q8—33232no—
486C2×Q8—3328no—
487C2×Q8—832no—
488C2×Q8—1668no—
489C2×Q8—1668no—
490C2×Q8—1662no—
491C2×Q8—1662no—
492C2×Q8—1661no—
493C2×Q8C23×D8312yes—
494C2×D4—24no—
495C2×D4—21no—
496C2×D4—22no—
497C2×D4—22no—
498C2×D4—44no—
499C2×D4—3324no—
500C2×D4—3324no—
501C2×D4—3324no—
502C2×D4—832no—
503C2×D4—1664no—
504C2×D4—1664no—
505C2×D4—1664no—
506C2×D4—1661no—
507C2×D4—1662no—
508C2×D4—1662no—
509C2×D4C23×D8312yes—
510C22×C4—24no—
511C22×C4—21no—
512C22×C4—22no—
513C22×C4—24no—
514C22×C4—44no—
515C22×C4—44no—
516C22×C4—3324no—
517C22×C4—3324no—
518C22×C4—831no—
519C22×C4—831no—
520C22×C4—834no—
521C22×C4—834no—
522C22×C4—831no—
523C22×C4—831no—
524C22×C4—1664no—
525C22×C4—1661no—
526C22×C4—1662no—
527C22×C4—1664no—
528C22×C4—1664no—
529C22×C4—1664no—
530C22×C4D4×D8312yes—
531C22×C4D4×D8311yescharacteristic
532C22×C4#1305058914yes—
533C22×C4#1305059014yes—
534C22×C4C23×D8312yes—
535C22×C4C23×D8311yescharacteristic
536M16—28no—
537M16—1668no—
538C2×C8—24no—
539C2×C8—834no—
540C2×C8—1664no—
541C2×C8C23×D8314yes—
542C4⋊C4—24no—
543C4⋊C4—22no—
544C4⋊C4—834no—
545C4⋊C4—1668no—
546C4⋊C4—1668no—
547C4⋊C4—1664no—
548C4⋊C4—1664no—
549C4⋊C4—1664no—
550C4⋊C4—1662no—
551C4⋊C4C23×D8314yes—
552C22⋊C4—1664no—
553C22⋊C4—1664no—
554C22⋊C4—1664no—
555C22⋊C4—1664no—
556C42—834no—
557C42—834no—
558C42—832no—
559C42—832no—
560C42—832no—
561C42—832no—
562C23—22no—
563C23—22no—
564C23—22no—
565C23—42no—
566C23—42no—
567C23—42no—
568C23—3322no—
569C23—3322no—
570C23—3322no—
571C23—1662no—
572C23#1458177814yes—
573C23C2×D4×D8312yes—
574C23C2×D4×D8311yescharacteristic
575Q8—22no—
576Q8—22no—
577Q8—42no—
578Q8—33216no—
579Q8—33216no—
580Q8—3322no—
581Q8—1662no—
582Q8—1662no—
583D4—22no—
584D4—22no—
585D4—48no—
586D4—42no—
587D4—3328no—
588D4—3328no—
589D4—3328no—
590D4—3322no—
591D4—1662no—
592D4—1662no—
593C2×C4—22no—
594C2×C4—24no—
595C2×C4—48no—
596C2×C4—42no—
597C2×C4—44no—
598C2×C4—48no—
599C2×C4—3328no—
600C2×C4—3322no—
601C2×C4—3324no—
602C2×C4—3328no—
603C2×C4—834no—
604C2×C4—834no—
605C2×C4—832no—
606C2×C4—832no—
607C2×C4—832no—
608C2×C4—832no—
609C2×C4—1662no—
610C2×C4—1664no—
611C2×C4—1668no—
612C2×C4—1668no—
613C2×C4—1662no—
614C2×C4—1662no—
615C2×C4#1458177814yes—
616C2×C4#1458177818yes—
617C2×C4C2×D4×D8312yes—
618C2×C4C2×D4×D8311yescharacteristic
619C2×C4C2×#1305058912yes—
620C2×C4C2×#1305058914yes—
621C2×C4C2×#1305059012yes—
622C2×C4C2×#1305059014yes—
623C2×C4C24×D8311yescharacteristic
624C8—24no—
625C8—1664no—
626C22—22no—
627C22—22no—
628C22—28no—
629C22—42no—
630C22—42no—
631C22—42no—
632C22—84no—
633C22—3322no—
634C22—3322no—
635C22—3322no—
636C22—6644no—
637C22C2×#1458177814yes—
638C22C2×#1458177812yes—
639C22AGL(1,ℤ/8)×D8312yes—
640C22#137×D8312yes—
641C22C22×D4×D8311yescharacteristic
642C4—44no—
643C4—42no—
644C4—3324no—
645C4—3322no—
646C4—834no—
647C4—834no—
648C4—1662no—
649C4—1662no—
650C4C2×#1458177814yes—
651C4C2×#1458177812yes—
652C4C22×#1305058911yescharacteristic
653C4C22×#1305059011yescharacteristic
654C2—24no—
655C2—82no—
656C2—6642no—
657C2#2491785414yesdirect product factor
658C2C22×#1458177811yescharacteristic
659C2C2×AGL(1,ℤ/8)×D8311yescharacteristic
660C2C2×#137×D8311yescharacteristic
subgroupG/H
centerC23C2×D4×D83
commutator, cosocleC2×C4×C83C25
FrattiniC2×C4C24×D83
FittingC2×#417×C83C2
socleC23×C83C22×D4
prime radicalC2×#19128247C2
hypercenterC2×#417D83
hypocenterC83C2×#3141
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#417

Complements

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

NH ≅ G/Ntype
2. C22×#14581791C2semidirect
3. #24917854C2direct (N × H)
4. C2×#19128062C2semidirect
5. C2×#19128059C2semidirect
6. C2×#19128050C2semidirect
7. C2×#19128046C2semidirect
8. C2×#19127703C2semidirect
9. C2×#19127702C2semidirect
10. C2×#19127700C2semidirect
11. C2×M16×Dic83C2semidirect
12. C2×#417×C83C2semidirect
23. C2×#14581791C22semidirect
35. #19128062C22semidirect
36. #19128059C22semidirect
37. #19128050C22semidirect
38. #19128046C22semidirect
39. #19127703C22semidirect
40. #19127702C22semidirect
41. #19127700C22semidirect
48. M16×Dic83C22semidirect
54. #417×C83C22semidirect
326. C2×C83#3141semidirect
341. C2×#417D83semidirect
343. C83C2×#3141semidirect
376. #417C2×D83semidirect
657. C2#24917854direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#191282472
C22×#145817912
#249178542
C2×#191280622
C2×#191280592
C2×#191280502
C2×#191280462
C2×#191277032
C2×#191277022
C2×#191277002
C2×M16×Dic832
C2×#417×C832
C2×#314183