Finite Group Databasebeta

← #35329657

Subgroups — #35329657

Order 21248.

1646 conjugacy classes of subgroups.

The subgroup lattice has 614 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19127562C211yesmaximal, characteristic
2C2×C22⋊C4×Dic83C212yesmaximal
3#24905708C211yesmaximal, characteristic
4#24905706C212yesmaximal
5#24905698C212yesmaximal, complemented
6#24905697C212yesmaximal, complemented
7#24905695C212yesmaximal, complemented
8#24905694C212yesmaximal, complemented
9C83×#1439C211yesmaximal, characteristic
10C24×Dic83C2211yescharacteristic
11C22×C4⋊Dic83C2211yescharacteristic
12C22×Dic83⋊C4C2212yes—
13C22×C22⋊Dic83C2212yes—
14C22×C22⋊Dic83C2211yescharacteristic
15C22×C4×Dic83C2212yes—
16C22×C4×Dic83C2212yes—
17#19127562C418yescomplemented
18C22⋊C4×Dic83—28no—
19C2×C83⋊#95C2212yes—
20C2×C83⋊#95C2212yes—
21C2×C83⋊#95C2212yes—
22C2×C83⋊#95C2212yes—
23C2×C83⋊#95C2212yes—
24C2×C83⋊#95C2212yes—
25C2×C83⋊#95C2211yescharacteristic
26C2×C83⋊#95C2212yes—
27C2×C83⋊#95C2212yes—
28C22×C4⋊C4×C83C2211yescharacteristic
29C22×C22⋊C4×C83C2212yes—
30C22×C22⋊C4×C83C2211yescharacteristic
31C2×#95×C83C2212yes—
32C2×#95×C83C2211yescharacteristic
33C23×Dic83—41no—
34C23×Dic83—41no—
35C23×Dic83—41no—
36C23×Dic83—41no—
37C23×Dic83C2×C414yes—
38C23×Dic83D414yes—
39C23×Dic83Q814yes—
40C23×Dic83C2312yes—
41C23×Dic83C2312yes—
42C23×Dic83C2312yes—
43C23×Dic83C2312yes—
44C2×C4⋊Dic83—24no—
45C2×C4⋊Dic83—41no—
46C2×Dic83⋊C4—24no—
47C2×Dic83⋊C4—24no—
48C2×Dic83⋊C4—42no—
49C2×C22⋊Dic83—24no—
50C2×C22⋊Dic83—24no—
51C2×C22⋊Dic83—24no—
52C2×C22⋊Dic83—41no—
53C2×C22⋊Dic83—41no—
54C2×C22⋊Dic83C2×C414yes—
55C2×C4×Dic83—24no—
56C2×C4×Dic83—24no—
57C2×C4×Dic83—42no—
58C2×C4×Dic83—42no—
59C2×C4×Dic83—42no—
60C2×C4×Dic83—42no—
61C83⋊#95—24no—
62C83⋊#95—24no—
63C83⋊#95—24no—
64C83⋊#95—24no—
65C83⋊#95—24no—
66C83⋊#95—41no—
67C83⋊#95—42no—
68C83⋊#95C2×C418yes—
69C83⋊#95C2×C418yes—
70C25×C83C2311yescharacteristic
71C23×C4×C83C2311yescharacteristic
72C23×C4×C83C2311yescharacteristic
73C23×C4×C83C2312yes—
74C23×C4×C83C2312yes—
75C2×C4⋊C4×C83—24no—
76C2×C4⋊C4×C83—41no—
77C2×C22⋊C4×C83—24no—
78C2×C22⋊C4×C83—24no—
79C2×C22⋊C4×C83—24no—
80C2×C22⋊C4×C83—41no—
81C2×C22⋊C4×C83—41no—
82C2×C22⋊C4×C83C2×C414yes—
83#95×C83—24no—
84#95×C83—41no—
85C22×Dic83—22no—
86C22×Dic83—22no—
87C22×Dic83—22no—
88C22×Dic83—22no—
89C22×Dic83—22no—
90C22×Dic83—22no—
91C22×Dic83—22no—
92C22×Dic83—22no—
93C22×Dic83—22no—
94C22×Dic83—22no—
95C22×Dic83—22no—
96C22×Dic83—22no—
97C22×Dic83—22no—
98C22×Dic83—22no—
99C22×Dic83—22no—
100C22×Dic83—22no—
101C22×Dic83—22no—
102C22×Dic83—22no—
103C22×Dic83—22no—
104C22×Dic83—22no—
105C22×Dic83—22no—
106C22×Dic83—22no—
107C22×Dic83—44no—
108C22×Dic83—44no—
109C22×Dic83—44no—
110C22×Dic83—44no—
111C22×Dic83—44no—
112C22×Dic83—44no—
113C22×Dic83C4⋊C4116yescomplemented
114C22×Dic83C22×C414yes—
115C22×Dic83C22×C414yes—
116C22×Dic83C2×D414yes—
117C22×Dic83C2×Q814yes—
118C22×Dic83#4114yes—
119C22×Dic83#4114yes—
120C4⋊Dic83—44no—
121Dic83⋊C4—48no—
122C22⋊Dic83—216no—
123C22⋊Dic83—44no—
124C4×Dic83—48no—
125C4×Dic83—48no—
126C24×C83—22no—
127C24×C83—22no—
128C24×C83—22no—
129C24×C83—22no—
130C24×C83—21no—
131C24×C83—21no—
132C24×C83—21no—
133C24×C83—21no—
134C24×C83C22×C412yes—
135C24×C83C2×D412yes—
136C24×C83C2×Q812yes—
137C24×C83C2411yescharacteristic
138C22×C4×C83—22no—
139C22×C4×C83—22no—
140C22×C4×C83—22no—
141C22×C4×C83—22no—
142C22×C4×C83—22no—
143C22×C4×C83—22no—
144C22×C4×C83—22no—
145C22×C4×C83—22no—
146C22×C4×C83—22no—
147C22×C4×C83—22no—
148C22×C4×C83—22no—
149C22×C4×C83—22no—
150C22×C4×C83—21no—
151C22×C4×C83—22no—
152C22×C4×C83—21no—
153C22×C4×C83—21no—
154C22×C4×C83—21no—
155C22×C4×C83—22no—
156C22×C4×C83—22no—
157C22×C4×C83—22no—
158C22×C4×C83C22×C412yes—
159C22×C4×C83C22×C412yes—
160C22×C4×C83#4114yes—
161C22×C4×C83#4114yes—
162C4⋊C4×C83—44no—
163C22⋊C4×C83—216no—
164C22⋊C4×C83—44no—
165C23×C83—24no—
166C23×C83—24no—
167C23×C83—22no—
168C23×C83—22no—
169C23×C83—24no—
170C23×C83—24no—
171C23×C83—24no—
172C23×C83—24no—
173C23×C83—22no—
174C23×C83—22no—
175C23×C83—22no—
176C23×C83—22no—
177C23×C83—42no—
178C23×C83—41no—
179C23×C83—41no—
180C23×C83—42no—
181C23×C83—42no—
182C23×C83—42no—
183C23×C83—42no—
184C23×C83—42no—
185C23×C83—41no—
186C23×C83—41no—
187C23×C83C2×C4⋊C414yes—
188C23×C83C23×C411yescharacteristic
189C23×C83C22×D411yescharacteristic
190C23×C83C22×Q811yescharacteristic
191C23×C83C2×#4112yes—
192C23×C83C2×#4112yes—
193C23×C83C2×#4112yes—
194C23×C83C2×#4112yes—
195C23×C83H(5,𝔽2)11yescharacteristic
196C23×C83H(5,𝔽2)11yescharacteristic
197C23×C83H(5,𝔽2)11yescharacteristic
198C23×C83H(5,𝔽2)11yescharacteristic
199C2×C4×C83—24no—
200C2×C4×C83—24no—
201C2×C4×C83—22no—
202C2×C4×C83—22no—
203C2×C4×C83—24no—
204C2×C4×C83—24no—
205C2×C4×C83—24no—
206C2×C4×C83—24no—
207C2×C4×C83—24no—
208C2×C4×C83—24no—
209C2×C4×C83—42no—
210C2×C4×C83—42no—
211C2×C4×C83—42no—
212C2×C4×C83—42no—
213C2×C4×C83—41no—
214C2×C4×C83—41no—
215C2×C4×C83—42no—
216C2×C4×C83—42no—
217C2×C4×C83—42no—
218C2×C4×C83—42no—
219C2×C4×C83—42no—
220C2×C4×C83—41no—
221C2×C4×C83—42no—
222C2×C4×C83—41no—
223C2×Dic83—24no—
224C2×Dic83—24no—
225C2×Dic83—24no—
226C2×Dic83—24no—
227C2×Dic83—24no—
228C2×Dic83—24no—
229C2×Dic83—24no—
230C2×Dic83—24no—
231C2×Dic83—24no—
232C2×Dic83—24no—
233C2×Dic83—24no—
234C2×Dic83—24no—
235C2×Dic83—24no—
236C2×Dic83—24no—
237C2×Dic83—24no—
238C2×Dic83—416no—
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×Dic83—42no—
249C2×Dic83—42no—
250C2×Dic83—42no—
251C2×Dic83C2×C4⋊C418yes—
252C22×C83—24no—
253C22×C83—28no—
254C22×C83—42no—
255C22×C83—44no—
256C22×C83—42no—
257C22×C83—42no—
258C22×C83—42no—
259C22×C83—44no—
260C22×C83—44no—
261C22×C83—42no—
262C22×C83—42no—
263C22×C83C22×C4⋊C411yescharacteristic
264C22×C83C4×#4112yes—
265C22×C83C4×#4112yes—
266C22×C83#51811yescharacteristic
267C22×C83#51811yescharacteristic
268C22×C83#52512yes—
269C22×C83#52512yes—
270C22×C83#53312yes—
271C22×C83#53312yes—
272C22×C83#53411yescharacteristic
273C22×C83#53411yescharacteristic
274C22×C83#54311yescharacteristic
275C22×C83#54311yescharacteristic
276C22×C83#55112yes—
277C22×C83#55111yescharacteristic
278C22×C83#55111yescharacteristic
279C22×C83#55312yes—
280C22×C83#55312yes—
281C22×C83#55312yes—
282C22×C83#55312yes—
283C22×C83#55912yes—
284C22×C83#55911yescharacteristic
285C22×C83#55911yescharacteristic
286C4×C83—44no—
287C4×C83—44no—
288C4×C83—42no—
289C4×C83—42no—
290Dic83—28no—
291Dic83—44no—
292Dic83—44no—
293Dic83—44no—
294#14165—831nomaximal, Sylow
295C2×C83—44no—
296C2×C83#229612yes—
297C2×C83#231511yescharacteristic
298C2×C83#237412yes—
299C2×C83#237411yescharacteristic
300C2×C83#237411yescharacteristic
301C2×C83#260011yescharacteristic
302C2×C83#260012yes—
303C2×C83#260011yescharacteristic
304C2×C83#260711yescharacteristic
305C2×C83#260711yescharacteristic
306C2×C83#260812yes—
307C2×#381—831no—
308C2×C4×C22⋊C4—832no—
309#1439—832no—
310#1439—831no—
311#1439D8311yescharacteristic
312#1436—832no—
313#1436—832no—
314#1436—832no—
315#1436—832no—
316C83#1416511yescharacteristic, complemented, Sylow
317C24×C4—831no—
318C22×C4⋊C4—832no—
319C22×C4⋊C4—831no—
320C22×C4⋊C4C2×D8311yescharacteristic
321C22×C22⋊C4—831no—
322C22×C22⋊C4—832no—
323C22×C22⋊C4C2×D8312yes—
324C22×C22⋊C4C2×D8311yescharacteristic
325C22×C42—832no—
326C22×C42—832no—
327#381—838no—
328C4×C22⋊C4—1668no—
329C2×#95—832no—
330C2×#95—831no—
331C2×#95—832no—
332C2×#95—832no—
333C2×#95—832no—
334C2×#95—832no—
335C2×#95—832no—
336C2×#95—832no—
337C2×#95—832no—
338C2×#95C2×D8312yes—
339C2×#95C2×D8311yescharacteristic
340C25C22×D8311yescharacteristic
341C23×C4—3321no—
342C23×C4—3321no—
343C23×C4—3321no—
344C23×C4—3321no—
345C23×C4—834no—
346C23×C4—834no—
347C23×C4—834no—
348C23×C4—832no—
349C23×C4—832no—
350C23×C4—832no—
351C23×C4—832no—
352C23×C4C22×D8311yescharacteristic
353C23×C4C22×D8311yescharacteristic
354C23×C4C22×D8312yes—
355C23×C4C22×D8312yes—
356C2×C4⋊C4—24no—
357C2×C4⋊C4—41no—
358C2×C4⋊C4—3322no—
359C2×C4⋊C4—3321no—
360C2×C4⋊C4—1664no—
361C2×C4⋊C4—1664no—
362C2×C4⋊C4—1664no—
363C2×C22⋊C4—24no—
364C2×C22⋊C4—24no—
365C2×C22⋊C4—24no—
366C2×C22⋊C4—41no—
367C2×C22⋊C4—41no—
368C2×C22⋊C4—3321no—
369C2×C22⋊C4—3321no—
370C2×C22⋊C4—834no—
371C2×C22⋊C4—1664no—
372C2×C22⋊C4—1664no—
373C2×C22⋊C4—1664no—
374C2×C22⋊C4C4×D8314yes—
375C2×C42—3322no—
376C2×C42—3322no—
377C2×C42—3322no—
378C2×C42—3322no—
379C2×C42—1664no—
380C2×C42—1664no—
381#95—24no—
382#95—41no—
383#95—3321no—
384#95—3322no—
385#95—838no—
386#95—838no—
387#95—1664no—
388#95—1664no—
389#95—1664no—
390#95—1664no—
391#95—1664no—
392C24—22no—
393C24—22no—
394C24—22no—
395C24—22no—
396C24—21no—
397C24—21no—
398C24—21no—
399C24—21no—
400C24C2×C4×D8312yes—
401C24D4×D8312yes—
402C24Q8×D8312yes—
403C24C23×D8311yescharacteristic
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
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—21no—
418C22×C4—21no—
419C22×C4—22no—
420C22×C4—21no—
421C22×C4—21no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—3324no—
425C22×C4—3324no—
426C22×C4—3324no—
427C22×C4—3324no—
428C22×C4—3324no—
429C22×C4—3324no—
430C22×C4—8316no—
431C22×C4—834no—
432C22×C4—834no—
433C22×C4—834no—
434C22×C4—834no—
435C22×C4—834no—
436C22×C4—834no—
437C22×C4—1662no—
438C22×C4—1662no—
439C22×C4—1662no—
440C22×C4—1662no—
441C22×C4—1662no—
442C22×C4—1662no—
443C22×C4—1662no—
444C22×C4—1662no—
445C22×C4—1662no—
446C22×C4—1662no—
447C22×C4—1662no—
448C22×C4—1662no—
449C22×C4—1662no—
450C22×C4—1662no—
451C22×C4—1662no—
452C22×C4—1662no—
453C22×C4—1662no—
454C22×C4—1662no—
455C22×C4—1662no—
456C22×C4—1662no—
457C22×C4—1662no—
458C22×C4—1662no—
459C22×C4C2×C4×D8312yes—
460C22×C4C2×C4×D8312yes—
461C22×C4#1305058914yes—
462C22×C4#1305058914yes—
463C4⋊C4—44no—
464C4⋊C4—3328no—
465C4⋊C4—3324no—
466C22⋊C4—216no—
467C22⋊C4—44no—
468C22⋊C4—3324no—
469C22⋊C4—16616no—
470C42—3328no—
471C42—3328no—
472C23—22no—
473C23—22no—
474C23—24no—
475C23—24no—
476C23—22no—
477C23—22no—
478C23—24no—
479C23—24no—
480C23—24no—
481C23—24no—
482C23—22no—
483C23—22no—
484C23—42no—
485C23—41no—
486C23—41no—
487C23—42no—
488C23—42no—
489C23—42no—
490C23—42no—
491C23—42no—
492C23—41no—
493C23—41no—
494C23C4⋊C4×D8314yes—
495C23C22×C4×D8311yescharacteristic
496C23C2×D4×D8311yescharacteristic
497C23C2×Q8×D8311yescharacteristic
498C23#41×D8312yes—
499C23#41×D8312yes—
500C23C2×#1305058912yes—
501C23C2×#1305058912yes—
502C23#1458183711yescharacteristic
503C23#1458183711yescharacteristic
504C23#1458183811yescharacteristic
505C23#1458183811yescharacteristic
506C2×C4—24no—
507C2×C4—24no—
508C2×C4—24no—
509C2×C4—24no—
510C2×C4—24no—
511C2×C4—24no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—24no—
515C2×C4—24no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—3322no—
531C2×C4—3322no—
532C2×C4—3322no—
533C2×C4—3322no—
534C2×C4—3322no—
535C2×C4—3322no—
536C2×C4—3322no—
537C2×C4—3322no—
538C2×C4—3322no—
539C2×C4—3322no—
540C2×C4—3322no—
541C2×C4—3322no—
542C2×C4—33216no—
543C2×C4—838no—
544C2×C4—1664no—
545C2×C4—1664no—
546C2×C4—1664no—
547C2×C4—1664no—
548C2×C4—1664no—
549C2×C4—1664no—
550C2×C4—1664no—
551C2×C4—1664no—
552C2×C4—1664no—
553C2×C4—1664no—
554C2×C4—1664no—
555C2×C4—1664no—
556C2×C4—1664no—
557C2×C4—1664no—
558C2×C4—1664no—
559C22—28no—
560C22—24no—
561C22—44no—
562C22—42no—
563C22—42no—
564C22—42no—
565C22—44no—
566C22—42no—
567C22—42no—
568C22—42no—
569C22—44no—
570C22C2×C4⋊C4×D8311yescharacteristic
571C22C4×#1305058912yes—
572C22C4×#1305058912yes—
573C22#1912817411yescharacteristic
574C22#1912817511yescharacteristic
575C22#1912820712yes—
576C22#1912820912yes—
577C22#1912825312yes—
578C22#1912825312yes—
579C22#1912825911yescharacteristic
580C22#1912825911yescharacteristic
581C22#1912832611yescharacteristic
582C22#1912832611yescharacteristic
583C22#1912837412yes—
584C22#1912838111yescharacteristic
585C22#1912838211yescharacteristic
586C22#1912839412yes—
587C22#1912839812yes—
588C22#1912840212yes—
589C22#1912840312yes—
590C22#1912843312yes—
591C22#1912843511yescharacteristic
592C22#1912843711yescharacteristic
593C4—42no—
594C4—42no—
595C4—44no—
596C4—44no—
597C4—3324no—
598C4—3324no—
599C4—3324no—
600C4—1668no—
601C2—44no—
602C2#2490994212yes—
603C2#2491007611yescharacteristic
604C2#2491059312yes—
605C2#2491059611yescharacteristic
606C2#2491059811yescharacteristic
607C2#2491304411yescharacteristic
608C2#2491304612yes—
609C2#2491305111yescharacteristic
610C2#2491312311yescharacteristic
611C2#2491312511yescharacteristic
612C2#2491312912yes—
subgroupG/H
center, FrattiniC24C23×D83
commutatorC23×C83C23×C4
FittingC83×#1439C2
socle, cosocleC24×C83C24
prime radicalC25×C83C23
hypercenter#1439D83
hypocenterC83#14165
LCSG ▷ C23×C83 ▷ C83
UCSC1 ◁ C24 ◁ #1439

Complements

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

NH ≅ G/Ntype
5. #24905698C2semidirect
6. #24905697C2semidirect
7. #24905695C2semidirect
8. #24905694C2semidirect
17. #19127562C4semidirect
113. C22×Dic83C4⋊C4semidirect
316. C83#14165semidirect

Maximal subgroups

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

subgroupindex
C2×#191275622
C2×C22⋊C4×Dic832
#249057082
#249057062
#249056982
#249056972
#249056952
#249056942
C83×#14392
#1416583