Finite Group Databasebeta

← #35330601

Subgroups — #35330601

Order 21248.

1470 conjugacy classes of subgroups.

The subgroup lattice has 588 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19127622C211yesmaximal, characteristic
2C2×#19127606C211yesmaximal, characteristic
3C2×#19127552C211yesmaximal, characteristic
4#24905733C211yesmaximal, characteristic
5#24905713C211yesmaximal, characteristic
6#24905697C212yesmaximal
7#24905695C212yesmaximal
8#24905694C212yesmaximal
9#24905692C212yesmaximal
10#95×Dic83C211yesmaximal, characteristic
11C83×#1444C211yesmaximal, characteristic
12C23×Dic166C2211yescharacteristic
13C22×C4⋊Dic83C2212yes—
14C22×Dic83⋊C4C2211yescharacteristic
15C22×Dic83⋊C4C2211yescharacteristic
16C22×C4×Dic83C2211yescharacteristic
17C22×C4×Dic83C2212yes—
18C22×C4×Dic83C2211yescharacteristic
19C22×C4×Dic83C2211yescharacteristic
20#19127622C418yescomplemented
21#19127606—24no—
22#19127552—24no—
23C2×C83⋊#95C2211yescharacteristic
24C2×C83⋊#95C2212yes—
25C2×C83⋊#95C2212yes—
26C2×C83⋊#95C2211yescharacteristic
27C2×C83⋊#95C2212yes—
28C2×C83⋊#95C2212yes—
29C2×C83⋊#95C2212yes—
30C2×C83⋊#95C2212yes—
31C2×C83⋊#95C2212yes—
32C2×C83⋊#95C2212yes—
33C22×C4⋊C4×C83C2212yes—
34C22×C4⋊C4×C83C2211yescharacteristic
35C2×#95×C83C2211yescharacteristic
36C2×#95×C83C2212yes—
37C2×#95×C83C2211yescharacteristic
38C22×Dic166—22no—
39C22×Dic166—22no—
40C22×Dic166—42no—
41C22×Dic166—42no—
42C22×Dic166C2×C414yes—
43C23×Dic83C2311yescharacteristic
44C23×Dic83C2312yes—
45C23×Dic83C2312yes—
46C23×Dic83C2311yescharacteristic
47C23×Dic83C2311yescharacteristic
48C23×Dic83C2311yescharacteristic
49C2×C4⋊Dic83—42no—
50C2×C4⋊Dic83—42no—
51C2×C4⋊Dic83C2×C418yes—
52C2×Dic83⋊C4—22no—
53C2×Dic83⋊C4—22no—
54C2×Dic83⋊C4—22no—
55C2×Dic83⋊C4—22no—
56C2×Dic83⋊C4C2×C414yes—
57C2×Dic83⋊C4Q814yescomplemented
58C2×C4×Dic83—22no—
59C2×C4×Dic83—22no—
60C2×C4×Dic83—22no—
61C2×C4×Dic83—24no—
62C2×C4×Dic83—22no—
63C2×C4×Dic83—24no—
64C2×C4×Dic83—42no—
65C2×C4×Dic83—42no—
66C2×C4×Dic83—42no—
67C2×C4×Dic83D414yes—
68C83⋊#95—22no—
69C83⋊#95—24no—
70C83⋊#95—24no—
71C83⋊#95—24no—
72C83⋊#95—22no—
73C83⋊#95—42no—
74C83⋊#95—42no—
75C83⋊#95—42no—
76C83⋊#95C2×C418yes—
77C83⋊#95Q818yescomplemented
78C23×C4×C83C2312yes—
79C23×C4×C83C2311yescharacteristic
80C23×C4×C83C2311yescharacteristic
81C23×C4×C83C2311yescharacteristic
82C23×C4×C83C2312yes—
83C2×C4⋊C4×C83—22no—
84C2×C4⋊C4×C83—24no—
85C2×C4⋊C4×C83—24no—
86C2×C4⋊C4×C83—42no—
87C2×C4⋊C4×C83—42no—
88#95×C83—22no—
89#95×C83—24no—
90#95×C83C2×C414yes—
91C2×Dic166—28no—
92C2×Dic166—48no—
93C2×Dic166—44no—
94C2×Dic166—44no—
95C2×Dic166—48no—
96C22×Dic83—21no—
97C22×Dic83—21no—
98C22×Dic83—21no—
99C22×Dic83—21no—
100C22×Dic83—22no—
101C22×Dic83—21no—
102C22×Dic83—21no—
103C22×Dic83—22no—
104C22×Dic83—22no—
105C22×Dic83—22no—
106C22×Dic83—21no—
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×Dic83—22no—
120C22×Dic83—22no—
121C22×Dic83C22×C414yes—
122C22×Dic83C22×C412yes—
123C22×Dic83C22×C412yes—
124C22×Dic83C2×D412yes—
125C22×Dic83C2×D412yes—
126C22×Dic83C2×Q814yes—
127C22×Dic83C2×Q812yes—
128C22×Dic83C2×Q812yes—
129C22×Dic83C2×Q814yes—
130C22×Dic83#4112yes—
131C22×Dic83#4112yes—
132C4⋊Dic83—48no—
133Dic83⋊C4—28no—
134Dic83⋊C4—28no—
135C4×Dic83—28no—
136C4×Dic83—44no—
137C4×Dic83—48no—
138C4×Dic83—44no—
139C24×C83C2411yescharacteristic
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—21no—
147C22×C4×C83—21no—
148C22×C4×C83—22no—
149C22×C4×C83—22no—
150C22×C4×C83—21no—
151C22×C4×C83—21no—
152C22×C4×C83—22no—
153C22×C4×C83—22no—
154C22×C4×C83—22no—
155C22×C4×C83—22no—
156C22×C4×C83—22no—
157C22×C4×C83—22no—
158C22×C4×C83—22no—
159C22×C4×C83—22no—
160C22×C4×C83—22no—
161C22×C4×C83C22×C412yes—
162C22×C4×C83C22×C414yes—
163C22×C4×C83C2×D412yes—
164C22×C4×C83C2×Q812yes—
165C22×C4×C83C2×Q814yes—
166C22×C4×C83#4112yes—
167C22×C4×C83#4114yes—
168C22×C4×C83#4112yes—
169C4⋊C4×C83—48no—
170C4⋊C4×C83—44no—
171C23×C83C23×C411yescharacteristic
172C23×C83C22×D411yescharacteristic
173C23×C83C22×Q811yescharacteristic
174C23×C83C22×Q812yes—
175C23×C83C2×#4111yescharacteristic
176C23×C83C2×#4111yescharacteristic
177C23×C83C2×#4112yes—
178C23×C83C2×#4111yescharacteristic
179C23×C83C2×#4111yescharacteristic
180C23×C83H(5,𝔽2)12yes—
181C23×C83H(5,𝔽2)12yes—
182C2×C4×C83—24no—
183C2×C4×C83—24no—
184C2×C4×C83—24no—
185C2×C4×C83—22no—
186C2×C4×C83—22no—
187C2×C4×C83—24no—
188C2×C4×C83—24no—
189C2×C4×C83—24no—
190C2×C4×C83—24no—
191C2×C4×C83—24no—
192C2×C4×C83—22no—
193C2×C4×C83—22no—
194C2×C4×C83—24no—
195C2×C4×C83—24no—
196C2×C4×C83—41no—
197C2×C4×C83—41no—
198C2×C4×C83—41no—
199C2×C4×C83—41no—
200C2×C4×C83—42no—
201C2×C4×C83—42no—
202C2×C4×C83—42no—
203C2×C4×C83—42no—
204C2×C4×C83—42no—
205C2×C4×C83—42no—
206C2×C4×C83—42no—
207C2×C4×C83—42no—
208C2×C4×C83—42no—
209C2×C4×C83C4×D414yes—
210C2×C4×C83C4×Q818yes—
211C2×Dic83—22no—
212C2×Dic83—22no—
213C2×Dic83—22no—
214C2×Dic83—22no—
215C2×Dic83—22no—
216C2×Dic83—22no—
217C2×Dic83—22no—
218C2×Dic83—22no—
219C2×Dic83—22no—
220C2×Dic83—22no—
221C2×Dic83—24no—
222C2×Dic83—24no—
223C2×Dic83—24no—
224C2×Dic83—24no—
225C2×Dic83—22no—
226C2×Dic83—24no—
227C2×Dic83—24no—
228C2×Dic83—24no—
229C2×Dic83—24no—
230C2×Dic83—24no—
231C2×Dic83—24no—
232C2×Dic83—24no—
233C2×Dic83—42no—
234C2×Dic83—41no—
235C2×Dic83—42no—
236C2×Dic83—41no—
237C2×Dic83—42no—
238C2×Dic83—42no—
239C2×Dic83—42no—
240C2×Dic83—42no—
241C2×Dic83—42no—
242C2×Dic83—42no—
243C2×Dic83—42no—
244C2×Dic83C4×Q814yes—
245C2×Dic83C22⋊Q814yes—
246C2×Dic83C4⋊Q814yes—
247Dic166—416no—
248C22×C83C2×C4×D411yescharacteristic
249C22×C83C2×C4×Q811yescharacteristic
250C22×C83C2×C4×Q812yes—
251C22×C83C4×#4111yescharacteristic
252C22×C83#51812yes—
253C22×C83C2×C22⋊Q811yescharacteristic
254C22×C83#52511yescharacteristic
255C22×C83#52911yescharacteristic
256C22×C83C2×C4⋊Q811yescharacteristic
257C22×C83#53211yescharacteristic
258C22×C83#53312yes—
259C22×C83#53311yescharacteristic
260C22×C83#53411yescharacteristic
261C22×C83#53511yescharacteristic
262C22×C83#53812yes—
263C22×C83#55112yes—
264C22×C83#55112yes—
265C22×C83#55412yes—
266C22×C83#55412yes—
267C22×C83#55412yes—
268C22×C83#55412yes—
269C22×C83#55512yes—
270C22×C83Q8212yes—
271C4×C83—24no—
272C4×C83—28no—
273C4×C83—42no—
274C4×C83—42no—
275C4×C83—82no—
276Dic83—24no—
277Dic83—24no—
278Dic83—24no—
279Dic83—44no—
280Dic83—44no—
281Dic83—42no—
282#14241—831nomaximal, Sylow
283C2×C83C4×C22⋊Q811yescharacteristic
284C2×C83C4×C4⋊Q811yescharacteristic
285C2×C83#232111yescharacteristic
286C2×C83#236812yes—
287C2×C83#236912yes—
288C2×C83#256411yescharacteristic
289C2×C83#256711yescharacteristic
290C2×C83#257412yes—
291C2×C83#258311yescharacteristic
292C2×C83#258612yes—
293C2×C83#261011yescharacteristic
294C2×#395—831no—
295C2×#391—831no—
296C2×C4×C4⋊C4—831no—
297#1446—831no—
298#1444D8311yescharacteristic
299#1441—831no—
300#1436—832no—
301#1436—832no—
302#1436—832no—
303C4⋊#95—832no—
304C4×#95—831no—
305C83#1424111yescharacteristic, complemented, Sylow
306C23×Q8—831no—
307C22×C4⋊C4—832no—
308C22×C4⋊C4—831no—
309C22×C4⋊C4—831no—
310C22×C4⋊C4C2×D8312yes—
311C22×C4⋊C4C2×D8311yescharacteristic
312C22×C42—831no—
313C22×C42—831no—
314C22×C42—832no—
315C22×C42—831no—
316#395—838no—
317#391—1664no—
318C4×C4⋊C4—1664no—
319C2×#95—832no—
320C2×#95—832no—
321C2×#95—832no—
322C2×#95—832no—
323C2×#95—832no—
324C2×#95—831no—
325C2×#95—831no—
326C2×#95—832no—
327C2×#95—832no—
328C2×#95—832no—
329C2×#95C2×D8312yes—
330C2×#95C2×D8311yescharacteristic
331C2×#95C2×D8311yescharacteristic
332C22×Q8—3322no—
333C22×Q8—3322no—
334C22×Q8—834no—
335C22×Q8—1662no—
336C22×Q8—1662no—
337C23×C4—831no—
338C23×C4—831no—
339C23×C4—831no—
340C23×C4—831no—
341C23×C4—832no—
342C23×C4—832no—
343C23×C4C22×D8312yes—
344C23×C4C22×D8311yescharacteristic
345C23×C4C22×D8311yescharacteristic
346C23×C4C22×D8311yescharacteristic
347C23×C4C22×D8312yes—
348C2×C4⋊C4—22no—
349C2×C4⋊C4—24no—
350C2×C4⋊C4—24no—
351C2×C4⋊C4—42no—
352C2×C4⋊C4—42no—
353C2×C4⋊C4—3322no—
354C2×C4⋊C4—3322no—
355C2×C4⋊C4—838no—
356C2×C4⋊C4—834no—
357C2×C4⋊C4—834no—
358C2×C4⋊C4—1662no—
359C2×C4⋊C4—1662no—
360C2×C4⋊C4—1662no—
361C2×C4⋊C4—1662no—
362C2×C42—3322no—
363C2×C42—3322no—
364C2×C42—3322no—
365C2×C42—834no—
366C2×C42—1662no—
367C2×C42—1664no—
368C2×C42—1664no—
369C2×C42—1662no—
370C2×C42—1662no—
371C2×C42—1662no—
372#95—22no—
373#95—24no—
374#95—3322no—
375#95—3322no—
376#95—3322no—
377#95—838no—
378#95—838no—
379#95—1662no—
380#95—1664no—
381#95—1664no—
382#95—1664no—
383#95—1662no—
384#95C4×D8314yes—
385C24C23×D8311yescharacteristic
386C2×Q8—3324no—
387C2×Q8—3324no—
388C2×Q8—3328no—
389C2×Q8—3328no—
390C2×Q8—1668no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—21no—
398C22×C4—21no—
399C22×C4—21no—
400C22×C4—21no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
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—832no—
413C22×C4—832no—
414C22×C4—832no—
415C22×C4—834no—
416C22×C4—832no—
417C22×C4—832no—
418C22×C4—832no—
419C22×C4—832no—
420C22×C4—834no—
421C22×C4—832no—
422C22×C4—834no—
423C22×C4—1662no—
424C22×C4—1661no—
425C22×C4—1662no—
426C22×C4—1662no—
427C22×C4—1662no—
428C22×C4—1662no—
429C22×C4—1661no—
430C22×C4—1662no—
431C22×C4—1662no—
432C22×C4—1662no—
433C22×C4—1662no—
434C22×C4—1662no—
435C22×C4—1662no—
436C22×C4—1662no—
437C22×C4—1662no—
438C22×C4—1662no—
439C22×C4—1661no—
440C22×C4—1661no—
441C22×C4—1661no—
442C22×C4—1662no—
443C22×C4—1661no—
444C22×C4—1661no—
445C22×C4—1661no—
446C22×C4—1662no—
447C22×C4—1662no—
448C22×C4C2×C4×D8312yes—
449C22×C4C2×C4×D8314yes—
450C22×C4D4×D8312yes—
451C22×C4Q8×D8312yes—
452C22×C4Q8×D8314yes—
453C22×C4#1305058912yes—
454C22×C4#1305058914yes—
455C22×C4#1305059112yes—
456C4⋊C4—48no—
457C4⋊C4—44no—
458C4⋊C4—3328no—
459C4⋊C4—1668no—
460C4⋊C4—1668no—
461C42—3328no—
462C42—3324no—
463C42—3324no—
464C42—1668no—
465C23C22×C4×D8311yescharacteristic
466C23C2×D4×D8311yescharacteristic
467C23C2×Q8×D8312yes—
468C23C2×Q8×D8311yescharacteristic
469C23#41×D8311yescharacteristic
470C23#41×D8311yescharacteristic
471C23C2×#1305058912yes—
472C23C2×#1305058911yescharacteristic
473C23C2×#1305059111yescharacteristic
474C23#1458183712yes—
475C23#1458183812yes—
476Q8—33216no—
477C2×C4—24no—
478C2×C4—24no—
479C2×C4—24no—
480C2×C4—24no—
481C2×C4—24no—
482C2×C4—24no—
483C2×C4—24no—
484C2×C4—22no—
485C2×C4—22no—
486C2×C4—24no—
487C2×C4—24no—
488C2×C4—24no—
489C2×C4—22no—
490C2×C4—22no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—41no—
494C2×C4—41no—
495C2×C4—41no—
496C2×C4—41no—
497C2×C4—42no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—3322no—
505C2×C4—3322no—
506C2×C4—3322no—
507C2×C4—3321no—
508C2×C4—3322no—
509C2×C4—3322no—
510C2×C4—3321no—
511C2×C4—3322no—
512C2×C4—3322no—
513C2×C4—3322no—
514C2×C4—3322no—
515C2×C4—834no—
516C2×C4—834no—
517C2×C4—834no—
518C2×C4—1664no—
519C2×C4—1664no—
520C2×C4—1662no—
521C2×C4—1664no—
522C2×C4—1664no—
523C2×C4—1664no—
524C2×C4—1664no—
525C2×C4—1664no—
526C2×C4—1664no—
527C2×C4—1664no—
528C2×C4—1664no—
529C2×C4—1664no—
530C2×C4—1662no—
531C2×C4—1662no—
532C2×C4—1662no—
533C2×C4—1662no—
534C2×C4—1662no—
535C2×C4—1662no—
536C2×C4—1662no—
537C2×C4—1662no—
538C2×C4—1662no—
539C2×C4—1662no—
540C2×C4#1458175414yes—
541C2×C4#1458175718yes—
542C22C2×#1458175411yescharacteristic
543C22C4×Q8×D8311yescharacteristic
544C22C2×#1458175712yes—
545C22C4×#1305058911yescharacteristic
546C22#1912817312yes—
547C22C22⋊Q8×D8311yescharacteristic
548C22#1912821111yescharacteristic
549C22#1912823611yescharacteristic
550C22C4⋊Q8×D8311yescharacteristic
551C22#1912824911yescharacteristic
552C22#1912825612yes—
553C22#1912825611yescharacteristic
554C22#1912826211yescharacteristic
555C22#1912826611yescharacteristic
556C22#1912829112yes—
557C22#1912837412yes—
558C22#1912838212yes—
559C22#1912840512yes—
560C22#1912840712yes—
561C22#1912840912yes—
562C22#1912840912yes—
563C22#1912841512yes—
564C22#1912843112yes—
565C4—24no—
566C4—28no—
567C4—42no—
568C4—42no—
569C4—82no—
570C4—3324no—
571C4—3322no—
572C4—3324no—
573C4—1664no—
574C4—1664no—
575C4—1664no—
576C2#2490999311yescharacteristic
577C2#2491003011yescharacteristic
578C2#2491012011yescharacteristic
579C2#2491054412yes—
580C2#2491054912yes—
581C2#2491263811yescharacteristic
582C2#2491266711yescharacteristic
583C2#2491274012yes—
584C2#2491283711yescharacteristic
585C2#2491287512yes—
586C2#2491314311yescharacteristic
subgroupG/H
center, FrattiniC24C23×D83
commutatorC23×C83C23×C4
FittingC83×#1444C2
socle, prime radical, cosocleC24×C83C24
hypercenter#1444D83
hypocenterC83#14241
LCSG ▷ C23×C83 ▷ C83
UCSC1 ◁ C24 ◁ #1444

Complements

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

NH ≅ G/Ntype
20. #19127622C4semidirect
57. C2×Dic83⋊C4Q8semidirect
77. C83⋊#95Q8semidirect
305. C83#14241semidirect

Maximal subgroups

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

subgroupindex
C2×#191276222
C2×#191276062
C2×#191275522
#249057332
#249057132
#249056972
#249056952
#249056942
#249056922
#95×Dic832
C83×#14442
#1424183