Finite Group Databasebeta

← #35329734

Subgroups — #35329734

Order 21248.

1758 conjugacy classes of subgroups.

The subgroup lattice has 594 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×Q8×Dic83C211yesmaximal, characteristic
2C2×#19127620C211yesmaximal, characteristic
3C2×#19127588C212yesmaximal
4C2×#19127552C211yesmaximal, characteristic
5#24905719C211yesmaximal, characteristic
6#24905717C211yesmaximal, characteristic
7#24905698C212yesmaximal
8#24905697C212yesmaximal
9#24905694C212yesmaximal
10#24905688C211yesmaximal, characteristic
11C83×#1446C211yesmaximal, characteristic
12C2×Q8×Dic83—28no—
13C2×Q8×Dic83—24no—
14C22×C4⋊Dic83C2211yescharacteristic
15C22×C4⋊Dic83C2212yes—
16C22×Dic83⋊C4C2212yes—
17C22×Dic83⋊C4C2211yescharacteristic
18C22×Dic83⋊C4C2211yescharacteristic
19C22×C4×Dic83C2212yes—
20C22×C4×Dic83C2211yescharacteristic
21C22×C4×Dic83C2211yescharacteristic
22C22×C4×Dic83C2211yescharacteristic
23#19127620C418yescomplemented
24#19127588—28no—
25#19127552—24no—
26C2×C83⋊#95C2212yes—
27C2×C83⋊#95C2212yes—
28C2×C83⋊#95C2212yes—
29C2×C83⋊#95C2212yes—
30C2×C83⋊#95C2212yes—
31C2×C83⋊#95C2212yes—
32C2×C83⋊#95C2211yescharacteristic
33C2×C83⋊#95C2211yescharacteristic
34C2×C83⋊#95C2212yes—
35C23×Q8×C83C2211yescharacteristic
36C2×#95×C83C2212yes—
37C2×#95×C83C2211yescharacteristic
38C2×#95×C83C2212yes—
39C2×#95×C83C2211yescharacteristic
40C23×Dic83C2311yescharacteristic
41C23×Dic83C2312yes—
42C23×Dic83C2312yes—
43C23×Dic83C2311yescharacteristic
44C23×Dic83C2311yescharacteristic
45C23×Dic83C2311yescharacteristic
46Q8×Dic83—232no—
47C2×C4⋊Dic83—24no—
48C2×C4⋊Dic83—24no—
49C2×C4⋊Dic83—24no—
50C2×C4⋊Dic83—22no—
51C2×C4⋊Dic83D418yes—
52C2×Dic83⋊C4—22no—
53C2×Dic83⋊C4—22no—
54C2×Dic83⋊C4—22no—
55C2×Dic83⋊C4—24no—
56C2×Dic83⋊C4—22no—
57C2×Dic83⋊C4—24no—
58C2×Dic83⋊C4C2×C418yes—
59C2×Dic83⋊C4C2×C414yes—
60C2×Dic83⋊C4Q814yescomplemented
61C2×C4×Dic83—22no—
62C2×C4×Dic83—24no—
63C2×C4×Dic83—24no—
64C2×C4×Dic83—24no—
65C2×C4×Dic83—22no—
66C2×C4×Dic83—22no—
67C2×C4×Dic83—24no—
68C2×C4×Dic83—24no—
69C2×C4×Dic83—22no—
70C2×C4×Dic83D414yes—
71C83⋊#95—22no—
72C83⋊#95—24no—
73C83⋊#95—24no—
74C83⋊#95—24no—
75C83⋊#95—24no—
76C83⋊#95—24no—
77C83⋊#95—24no—
78C83⋊#95C2×C414yes—
79C83⋊#95C2×C418yes—
80C22×Q8×C83—24no—
81C22×Q8×C83—22no—
82C22×Q8×C83—22no—
83C22×Q8×C83—24no—
84C22×Q8×C83C2×C414yes—
85C23×C4×C83C2312yes—
86C23×C4×C83C2311yescharacteristic
87C23×C4×C83C2311yescharacteristic
88C23×C4×C83C2312yes—
89C23×C4×C83C2311yescharacteristic
90#95×C83—22no—
91#95×C83—24no—
92#95×C83—24no—
93#95×C83—22no—
94C22×Dic83—22no—
95C22×Dic83—21no—
96C22×Dic83—21no—
97C22×Dic83—21no—
98C22×Dic83—21no—
99C22×Dic83—21no—
100C22×Dic83—21no—
101C22×Dic83—22no—
102C22×Dic83—22no—
103C22×Dic83—22no—
104C22×Dic83—21no—
105C22×Dic83—22no—
106C22×Dic83—22no—
107C22×Dic83—22no—
108C22×Dic83—22no—
109C22×Dic83—22no—
110C22×Dic83—22no—
111C22×Dic83—22no—
112C22×Dic83—21no—
113C22×Dic83—22no—
114C22×Dic83—22no—
115C22×Dic83—22no—
116C22×Dic83—22no—
117C22×Dic83—22no—
118C22×Dic83—22no—
119C22×Dic83C22×C414yes—
120C22×Dic83C22×C412yes—
121C22×Dic83C22×C412yes—
122C22×Dic83C2×D414yes—
123C22×Dic83C2×D414yes—
124C22×Dic83C2×D412yes—
125C22×Dic83C2×D412yes—
126C22×Dic83C2×Q812yes—
127C22×Dic83C2×Q812yes—
128C22×Dic83#4112yes—
129C22×Dic83#4112yes—
130C4⋊Dic83—28no—
131C4⋊Dic83—216no—
132Dic83⋊C4—28no—
133Dic83⋊C4—216no—
134Dic83⋊C4—28no—
135C4×Dic83—28no—
136C4×Dic83—216no—
137C4×Dic83—28no—
138C4×Dic83—44no—
139C24×C83C2411yescharacteristic
140C2×Q8×C83—28no—
141C2×Q8×C83—216no—
142C2×Q8×C83—28no—
143C2×Q8×C83—28no—
144C2×Q8×C83—216no—
145C22×C4×C83—22no—
146C22×C4×C83—22no—
147C22×C4×C83—22no—
148C22×C4×C83—22no—
149C22×C4×C83—22no—
150C22×C4×C83—22no—
151C22×C4×C83—21no—
152C22×C4×C83—22no—
153C22×C4×C83—21no—
154C22×C4×C83—22no—
155C22×C4×C83—21no—
156C22×C4×C83—21no—
157C22×C4×C83—22no—
158C22×C4×C83—22no—
159C22×C4×C83—22no—
160C22×C4×C83—22no—
161C22×C4×C83—22no—
162C22×C4×C83—22no—
163C22×C4×C83—22no—
164C22×C4×C83—22no—
165C22×C4×C83—22no—
166C22×C4×C83C22×C414yes—
167C22×C4×C83C22×C412yes—
168C22×C4×C83C2×D414yes—
169C22×C4×C83C2×D412yes—
170C22×C4×C83C2×Q812yes—
171C22×C4×C83#4112yes—
172C22×C4×C83#4114yes—
173C22×C4×C83#4112yes—
174C23×C83C23×C411yescharacteristic
175C23×C83C22×D411yescharacteristic
176C23×C83C22×D412yes—
177C23×C83C22×Q811yescharacteristic
178C23×C83C2×#4111yescharacteristic
179C23×C83C2×#4111yescharacteristic
180C23×C83C2×#4111yescharacteristic
181C23×C83C2×#4112yes—
182C23×C83C2×#4111yescharacteristic
183C23×C832−1+412yes—
184C23×C832−1+412yes—
185Q8×C83—232no—
186C2×C4×C83—24no—
187C2×C4×C83—24no—
188C2×C4×C83—24no—
189C2×C4×C83—24no—
190C2×C4×C83—22no—
191C2×C4×C83—22no—
192C2×C4×C83—24no—
193C2×C4×C83—24no—
194C2×C4×C83—24no—
195C2×C4×C83—24no—
196C2×C4×C83—22no—
197C2×C4×C83—22no—
198C2×C4×C83—24no—
199C2×C4×C83—24no—
200C2×C4×C83—41no—
201C2×C4×C83—41no—
202C2×C4×C83—41no—
203C2×C4×C83—41no—
204C2×C4×C83—42no—
205C2×C4×C83—42no—
206C2×C4×C83—42no—
207C2×C4×C83—42no—
208C2×C4×C83—42no—
209C2×C4×C83—42no—
210C2×C4×C83—42no—
211C2×C4×C83—42no—
212C2×C4×C83—42no—
213C2×C4×C83C4×D414yes—
214C2×C4×C83C4×D418yes—
215C2×Dic83—22no—
216C2×Dic83—22no—
217C2×Dic83—22no—
218C2×Dic83—22no—
219C2×Dic83—22no—
220C2×Dic83—22no—
221C2×Dic83—22no—
222C2×Dic83—22no—
223C2×Dic83—22no—
224C2×Dic83—22no—
225C2×Dic83—24no—
226C2×Dic83—24no—
227C2×Dic83—24no—
228C2×Dic83—24no—
229C2×Dic83—24no—
230C2×Dic83—22no—
231C2×Dic83—24no—
232C2×Dic83—24no—
233C2×Dic83—24no—
234C2×Dic83—24no—
235C2×Dic83—24no—
236C2×Dic83—24no—
237C2×Dic83—41no—
238C2×Dic83—41no—
239C2×Dic83—42no—
240C2×Dic83—42no—
241C2×Dic83—42no—
242C2×Dic83—42no—
243C2×Dic83—42no—
244C2×Dic83—42no—
245C2×Dic83—42no—
246C2×Dic83—42no—
247C2×Dic83—42no—
248C2×Dic83C4×Q814yescomplemented
249C2×Dic83#12014yes—
250C2×Dic83C22⋊Q814yes—
251C22×C83C2×C4×D411yescharacteristic
252C22×C83C2×C4×D412yes—
253C22×C83C2×C4×Q811yescharacteristic
254C22×C83C4×#4111yescharacteristic
255C22×C83#51912yes—
256C22×C83C2×#12011yescharacteristic
257C22×C83C2×C22⋊Q811yescharacteristic
258C22×C83#52511yescharacteristic
259C22×C83#52512yes—
260C22×C83#52511yescharacteristic
261C22×C83#52911yescharacteristic
262C22×C83#53311yescharacteristic
263C22×C83#53611yescharacteristic
264C22×C83#53611yescharacteristic
265C22×C83#54112yes—
266C22×C83#54712yes—
267C22×C83#54712yes—
268C22×C83#54812yes—
269C22×C83D4×Q812yes—
270C22×C83#55212yes—
271C22×C83#55612yes—
272C22×C83#55612yes—
273C22×C83#55712yes—
274C4×C83—28no—
275C4×C83—24no—
276C4×C83—42no—
277C4×C83—42no—
278C4×C83—82no—
279Dic83—24no—
280Dic83—24no—
281Dic83—24no—
282Dic83—44no—
283Dic83—44no—
284Dic83—42no—
285#14174—831nomaximal, Sylow
286C2×C83C4×#12011yescharacteristic
287C2×C83C4×C22⋊Q811yescharacteristic
288C2×C83#232711yescharacteristic
289C2×C83#236212yes—
290C2×C83#236512yes—
291C2×C83#240611yescharacteristic
292C2×C83#240911yescharacteristic
293C2×C83#246212yes—
294C2×C83#246912yes—
295C2×C83#258111yescharacteristic
296C2×C83#259911yescharacteristic
297C22×C4×Q8—831no—
298C2×#395—831no—
299C2×#387—832no—
300C2×C4×C4⋊C4—831no—
301#1446D8311yescharacteristic
302#1442—831no—
303#1441—831no—
304#1436—832no—
305#1436—832no—
306#1436—832no—
307#1434—831no—
308C83#1417411yescharacteristic, complemented, Sylow
309C23×Q8C2×D8311yescharacteristic
310C2×C4×Q8—1664no—
311C2×C4×Q8—1668no—
312C22×C4⋊C4—832no—
313C22×C4⋊C4—831no—
314C22×C4⋊C4—831no—
315C22×C4⋊C4—832no—
316C22×C4⋊C4—831no—
317C22×C42—831no—
318C22×C42—832no—
319C22×C42—831no—
320C22×C42—831no—
321#395—838no—
322#387—1668no—
323C4×C4⋊C4—1664no—
324C2×#95—831no—
325C2×#95—832no—
326C2×#95—832no—
327C2×#95—832no—
328C2×#95—832no—
329C2×#95—832no—
330C2×#95—831no—
331C2×#95—832no—
332C2×#95—832no—
333C2×#95C2×D8311yescharacteristic
334C2×#95C2×D8312yes—
335C2×#95C2×D8312yes—
336C2×#95C2×D8311yescharacteristic
337C22×Q8—22no—
338C22×Q8—24no—
339C22×Q8—24no—
340C22×Q8—22no—
341C22×Q8C4×D8314yes—
342C23×C4—831no—
343C23×C4—831no—
344C23×C4—831no—
345C23×C4—831no—
346C23×C4—832no—
347C23×C4—832no—
348C23×C4C22×D8312yes—
349C23×C4C22×D8311yescharacteristic
350C23×C4C22×D8311yescharacteristic
351C23×C4C22×D8311yescharacteristic
352C23×C4C22×D8312yes—
353C4×Q8—16632no—
354C2×C4⋊C4—834no—
355C2×C4⋊C4—838no—
356C2×C4⋊C4—838no—
357C2×C4⋊C4—834no—
358C2×C4⋊C4—1664no—
359C2×C4⋊C4—1664no—
360C2×C4⋊C4—1664no—
361C2×C4⋊C4—1662no—
362C2×C4⋊C4—1662no—
363C2×C4⋊C4—1662no—
364C2×C4⋊C4—1662no—
365C2×C4⋊C4—1664no—
366C2×C4⋊C4—1664no—
367C2×C4⋊C4—1662no—
368C2×C42—834no—
369C2×C42—1662no—
370C2×C42—1664no—
371C2×C42—1664no—
372C2×C42—1664no—
373C2×C42—1664no—
374C2×C42—1662no—
375C2×C42—1662no—
376C2×C42—1664no—
377C2×C42—1662no—
378#95—22no—
379#95—24no—
380#95—24no—
381#95—22no—
382#95—838no—
383#95—834no—
384#95—1662no—
385#95—1664no—
386#95—1664no—
387#95—1664no—
388#95—1664no—
389#95—1664no—
390#95—1664no—
391C24C23×D8311yescharacteristic
392C2×Q8—28no—
393C2×Q8—216no—
394C2×Q8—28no—
395C2×Q8—28no—
396C2×Q8—216no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—21no—
405C22×C4—21no—
406C22×C4—21no—
407C22×C4—21no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—832no—
419C22×C4—832no—
420C22×C4—834no—
421C22×C4—832no—
422C22×C4—832no—
423C22×C4—832no—
424C22×C4—832no—
425C22×C4—832no—
426C22×C4—834no—
427C22×C4—832no—
428C22×C4—834no—
429C22×C4—1662no—
430C22×C4—1662no—
431C22×C4—1661no—
432C22×C4—1662no—
433C22×C4—1662no—
434C22×C4—1662no—
435C22×C4—1662no—
436C22×C4—1662no—
437C22×C4—1662no—
438C22×C4—1661no—
439C22×C4—1662no—
440C22×C4—1662no—
441C22×C4—1662no—
442C22×C4—1662no—
443C22×C4—1662no—
444C22×C4—1662no—
445C22×C4—1662no—
446C22×C4—1661no—
447C22×C4—1661no—
448C22×C4—1661no—
449C22×C4—1661no—
450C22×C4—1661no—
451C22×C4—1661no—
452C22×C4—1662no—
453C22×C4—1662no—
454C22×C4C2×C4×D8314yes—
455C22×C4C2×C4×D8312yes—
456C22×C4D4×D8312yes—
457C22×C4D4×D8314yes—
458C22×C4Q8×D8312yes—
459C22×C4#1305058914yes—
460C22×C4#1305058912yes—
461C22×C4#1305059112yes—
462C4⋊C4—16616no—
463C4⋊C4—1668no—
464C4⋊C4—1668no—
465C4⋊C4—16616no—
466C4⋊C4—1668no—
467C42—3324no—
468C42—1668no—
469C42—16616no—
470C42—1668no—
471C23C22×C4×D8311yescharacteristic
472C23C2×D4×D8312yes—
473C23C2×D4×D8311yescharacteristic
474C23C2×Q8×D8311yescharacteristic
475C23#41×D8311yescharacteristic
476C23#41×D8311yescharacteristic
477C23C2×#1305058912yes—
478C23C2×#1305058911yescharacteristic
479C23C2×#1305059111yescharacteristic
480C23#1458183912yes—
481C23#1458184012yes—
482Q8—232no—
483C2×C4—24no—
484C2×C4—24no—
485C2×C4—24no—
486C2×C4—22no—
487C2×C4—24no—
488C2×C4—24no—
489C2×C4—22no—
490C2×C4—24no—
491C2×C4—24no—
492C2×C4—24no—
493C2×C4—24no—
494C2×C4—24no—
495C2×C4—22no—
496C2×C4—22no—
497C2×C4—42no—
498C2×C4—42no—
499C2×C4—41no—
500C2×C4—41no—
501C2×C4—41no—
502C2×C4—41no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—3322no—
511C2×C4—3322no—
512C2×C4—3322no—
513C2×C4—3322no—
514C2×C4—3322no—
515C2×C4—3322no—
516C2×C4—3321no—
517C2×C4—3322no—
518C2×C4—3321no—
519C2×C4—3322no—
520C2×C4—3322no—
521C2×C4—834no—
522C2×C4—834no—
523C2×C4—834no—
524C2×C4—1664no—
525C2×C4—1664no—
526C2×C4—1662no—
527C2×C4—1664no—
528C2×C4—1664no—
529C2×C4—1664no—
530C2×C4—1664no—
531C2×C4—1664no—
532C2×C4—1664no—
533C2×C4—1664no—
534C2×C4—1664no—
535C2×C4—1664no—
536C2×C4—1662no—
537C2×C4—1662no—
538C2×C4—1662no—
539C2×C4—1662no—
540C2×C4—1662no—
541C2×C4—1662no—
542C2×C4—1662no—
543C2×C4—1662no—
544C2×C4—1662no—
545C2×C4—1662no—
546C2×C4(C22⋊D83)⋊C414yes—
547C2×C4(C22⋊D83)⋊C418yes—
548C22C2×(C22⋊D83)⋊C411yescharacteristic
549C22C2×(C22⋊D83)⋊C412yes—
550C22C4×Q8×D8311yescharacteristic
551C22C4×#1305059111yescharacteristic
552C22#1912817712yes—
553C22#120×D8311yescharacteristic
554C22C22⋊Q8×D8311yescharacteristic
555C22#1912820712yes—
556C22#1912820711yescharacteristic
557C22#1912821411yescharacteristic
558C22#1912823011yescharacteristic
559C22#1912825611yescharacteristic
560C22#1912827111yescharacteristic
561C22#1912827611yescharacteristic
562C22#1912831112yes—
563C22#1912835312yes—
564C22#1912835412yes—
565C22#1912835912yes—
566C22#1912836412yes—
567C22#1912839312yes—
568C22#1912842012yes—
569C22#1912842212yes—
570C22#1912842612yes—
571C4—28no—
572C4—24no—
573C4—42no—
574C4—42no—
575C4—82no—
576C4—3322no—
577C4—3324no—
578C4—3324no—
579C4—1664no—
580C4—1664no—
581C4—1664no—
582C2#2490996211yescharacteristic
583C2#2491000111yescharacteristic
584C2#2491018111yescharacteristic
585C2#2491049512yes—
586C2#2491052012yes—
587C2#2491079911yescharacteristic
588C2#2491082111yescharacteristic
589C2#2491144012yes—
590C2#2491153912yes—
591C2#2491282411yescharacteristic
592C2#2491303311yescharacteristic
subgroupG/H
center, FrattiniC24C23×D83
commutatorC23×C83C23×C4
FittingC83×#1446C2
socle, prime radical, cosocleC24×C83C24
hypercenter#1446D83
hypocenterC83#14174
LCSG ▷ C23×C83 ▷ C83
UCSC1 ◁ C24 ◁ #1446

Complements

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

NH ≅ G/Ntype
23. #19127620C4semidirect
60. C2×Dic83⋊C4Q8semidirect
248. C2×Dic83C4×Q8semidirect
308. C83#14174semidirect

Maximal subgroups

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

subgroupindex
C22×Q8×Dic832
C2×#191276202
C2×#191275882
C2×#191275522
#249057192
#249057172
#249056982
#249056972
#249056942
#249056882
C83×#14462
#1417483