Finite Group Databasebeta

← #35326416

Subgroups — #35326416

Order 21248.

1862 conjugacy classes of subgroups.

The subgroup lattice has 678 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19127561C211yesmaximal, characteristic
2C2×C4×C22⋊Dic83C211yesmaximal, characteristic
3#24905725C212yesmaximal, complemented
4#24905720C212yesmaximal, complemented
5#24905697C212yesmaximal, complemented
6#24905688C212yesmaximal, complemented
7#24905686C211yesmaximal, characteristic
8#24905686C211yesmaximal, characteristic
9#24905686C211yesmaximal, characteristic
10#24905686C211yesmaximal, characteristic
11C22×#95×C83C211yesmaximal, characteristic
12C22×C4⋊Dic83C2212yes—
13C22×C22⋊Dic83C2211yescharacteristic
14C22×C22⋊Dic83C2211yescharacteristic
15C22×C22⋊Dic83C2211yescharacteristic
16C22×C22⋊Dic83C2211yescharacteristic
17C22×C4×Dic83C2212yes—
18#19127561—24no—
19C4×C22⋊Dic83—24no—
20C2×C83⋊#95C2212yes—
21C2×C83⋊#95C2212yes—
22C2×C83⋊#95C2212yes—
23C2×C83⋊#95C2212yes—
24C2×C83⋊#95C2212yes—
25C2×C83⋊#95C2212yes—
26C2×C83⋊#95C2212yes—
27C2×C83⋊#95C2212yes—
28C2×C83⋊#95C2212yes—
29C2×C83⋊#95C2212yes—
30C24×C4×C83C2211yescharacteristic
31C24×C4×C83C2211yescharacteristic
32C24×C4×C83C2211yescharacteristic
33C2×#95×C83C418yescomplemented
34C2×#95×C83C2211yescharacteristic
35C2×#95×C83C2211yescharacteristic
36C2×#95×C83C2211yescharacteristic
37C2×#95×C83C2211yescharacteristic
38C23×Dic83C2312yes—
39C23×Dic83C2312yes—
40C23×Dic83C2312yes—
41C23×Dic83C2312yes—
42C2×C4⋊Dic83—24no—
43C2×C4⋊Dic83—24no—
44C2×C4⋊Dic83—42no—
45C2×C22⋊Dic83—22no—
46C2×C22⋊Dic83—22no—
47C2×C22⋊Dic83—22no—
48C2×C22⋊Dic83—22no—
49C2×C22⋊Dic83—22no—
50C2×C22⋊Dic83—22no—
51C2×C22⋊Dic83—22no—
52C2×C22⋊Dic83—22no—
53C2×C22⋊Dic83—22no—
54C2×C22⋊Dic83—22no—
55C2×C22⋊Dic83D414yes—
56C2×C22⋊Dic83Q814yes—
57C2×C4×Dic83—24no—
58C2×C4×Dic83—24no—
59C2×C4×Dic83—42no—
60C83⋊#95—24no—
61C83⋊#95—24no—
62C83⋊#95—24no—
63C83⋊#95—24no—
64C83⋊#95—42no—
65C83⋊#95—42no—
66C83⋊#95—42no—
67C83⋊#95—42no—
68C83⋊#95—42no—
69C83⋊#95—42no—
70C25×C83C2311yescharacteristic
71C23×C4×C83—22no—
72C23×C4×C83—22no—
73C23×C4×C83—22no—
74C23×C4×C83—22no—
75C23×C4×C83—22no—
76C23×C4×C83—22no—
77C23×C4×C83—24no—
78C23×C4×C83—24no—
79C23×C4×C83—42no—
80C23×C4×C83—42no—
81C23×C4×C83—42no—
82C23×C4×C83—42no—
83C23×C4×C83C2×C414yes—
84C23×C4×C83C2×C414yes—
85C23×C4×C83C2×C414yes—
86C23×C4×C83C2311yescharacteristic
87C23×C4×C83C2311yescharacteristic
88C23×C4×C83C2311yescharacteristic
89C23×C4×C83C2311yescharacteristic
90C23×C4×C83C2311yescharacteristic
91C23×C4×C83C2311yescharacteristic
92#95×C83C2×C414yes—
93#95×C83C2×C414yes—
94#95×C83C2×C414yes—
95#95×C83C2×C414yes—
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—22no—
108C22×Dic83—22no—
109C22×Dic83—22no—
110C22×Dic83—22no—
111C22×Dic83—22no—
112C22×Dic83—22no—
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×Dic83—22no—
122C22×Dic83C2×D414yes—
123C22×Dic83C2×Q814yes—
124C4⋊Dic83—48no—
125C22⋊Dic83—28no—
126C22⋊Dic83—28no—
127C22⋊Dic83—44no—
128C22⋊Dic83—44no—
129C4×Dic83—48no—
130C24×C83—22no—
131C24×C83—22no—
132C24×C83—22no—
133C24×C83—22no—
134C24×C83C22×C412yes—
135C24×C83C2×D412yes—
136C24×C83C2×Q812yes—
137C24×C83#4112yes—
138C24×C83#4112yes—
139C24×C83#4112yes—
140C24×C83#4112yes—
141C24×C83C2411yescharacteristic
142C22×C4×C83—21no—
143C22×C4×C83—21no—
144C22×C4×C83—21no—
145C22×C4×C83—21no—
146C22×C4×C83—22no—
147C22×C4×C83—22no—
148C22×C4×C83—28no—
149C22×C4×C83—28no—
150C22×C4×C83—28no—
151C22×C4×C83—21no—
152C22×C4×C83—21no—
153C22×C4×C83—21no—
154C22×C4×C83—21no—
155C22×C4×C83—21no—
156C22×C4×C83—21no—
157C22×C4×C83—21no—
158C22×C4×C83—21no—
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×C83—22no—
167C22×C4×C83—28no—
168C22×C4×C83—28no—
169C22×C4×C83—44no—
170C22×C4×C83—44no—
171C22×C4×C83—44no—
172C22×C4×C83—44no—
173C22×C4×C83—48no—
174C22×C4×C83—48no—
175C22×C4×C83—48no—
176C22×C4×C83—48no—
177C22×C4×C83—48no—
178C22×C4×C83—48no—
179C22×C4×C83C22×C412yes—
180C22×C4×C83C22×C412yes—
181C22×C4×C83C22×C412yes—
182C22×C4×C83C22×C412yes—
183C22×C4×C83C22×C412yes—
184C22×C4×C83C22×C412yes—
185C22×C4×C83#4114yes—
186C22×C4×C83#4114yes—
187C23×C83—22no—
188C23×C83—22no—
189C23×C83—22no—
190C23×C83—22no—
191C23×C83—24no—
192C23×C83—24no—
193C23×C83—24no—
194C23×C83—24no—
195C23×C83—24no—
196C23×C83—24no—
197C23×C83—24no—
198C23×C83—24no—
199C23×C83—24no—
200C23×C83—24no—
201C23×C83—24no—
202C23×C83—24no—
203C23×C83#11714yes—
204C23×C83C4×D414yes—
205C23×C83C4×Q814yes—
206C23×C83C22⋊Q814yes—
207C23×C83#12314yes—
208C23×C83#12514yes—
209C23×C83#12614yes—
210C23×C83C23×C411yescharacteristic
211C23×C83C22×D411yescharacteristic
212C23×C83C22×Q811yescharacteristic
213C23×C83C2×#4111yescharacteristic
214C23×C83C2×#4111yescharacteristic
215C23×C83C2×#4111yescharacteristic
216C23×C83C2×#4112yes—
217C23×C83C2×#4112yes—
218C23×C83C2×#4111yescharacteristic
219C23×C83H(5,𝔽2)12yes—
220C23×C83H(5,𝔽2)12yes—
221C2×C4×C83—22no—
222C2×C4×C83—22no—
223C2×C4×C83—22no—
224C2×C4×C83—22no—
225C2×C4×C83—24no—
226C2×C4×C83—24no—
227C2×C4×C83—24no—
228C2×C4×C83—24no—
229C2×C4×C83—22no—
230C2×C4×C83—22no—
231C2×C4×C83—22no—
232C2×C4×C83—22no—
233C2×C4×C83—22no—
234C2×C4×C83—22no—
235C2×C4×C83—22no—
236C2×C4×C83—22no—
237C2×C4×C83—22no—
238C2×C4×C83—22no—
239C2×C4×C83—22no—
240C2×C4×C83—22no—
241C2×C4×C83—416no—
242C2×C4×C83—416no—
243C2×C4×C83—416no—
244C2×C4×C83—42no—
245C2×C4×C83—42no—
246C2×C4×C83—42no—
247C2×C4×C83—42no—
248C2×C4×C83—42no—
249C2×C4×C83—42no—
250C2×C4×C83—42no—
251C2×C4×C83—42no—
252C2×C4×C83#11714yes—
253C2×C4×C83#11714yes—
254C2×Dic83—24no—
255C2×Dic83—24no—
256C2×Dic83—24no—
257C2×Dic83—24no—
258C2×Dic83—24no—
259C2×Dic83—24no—
260C2×Dic83—42no—
261C2×Dic83—42no—
262C2×Dic83—42no—
263C2×Dic83—42no—
264C2×Dic83—42no—
265C2×Dic83—42no—
266C2×Dic83—42no—
267C2×Dic83—42no—
268C2×Dic83—42no—
269C2×Dic83—42no—
270C2×Dic83—42no—
271C2×Dic83—42no—
272C2×Dic83—42no—
273C2×Dic83—42no—
274C2×Dic83—42no—
275C2×Dic83—42no—
276C2×Dic83—42no—
277C2×Dic83—42no—
278C2×Dic83—42no—
279C2×Dic83—42no—
280C2×Dic83—42no—
281C2×Dic83—42no—
282C22×C83—24no—
283C22×C83—24no—
284C22×C83—24no—
285C22×C83—24no—
286C22×C83—24no—
287C22×C83—28no—
288C22×C83—28no—
289C22×C83—28no—
290C22×C83—28no—
291C22×C83—24no—
292C22×C83#38718yes—
293C22×C83C2×#11711yescharacteristic
294C22×C83C2×#11711yescharacteristic
295C22×C83C2×#11711yescharacteristic
296C22×C83C2×C4×D411yescharacteristic
297C22×C83C2×C4×Q811yescharacteristic
298C22×C83#51811yescharacteristic
299C22×C83#51811yescharacteristic
300C22×C83C2×C22⋊Q811yescharacteristic
301C22×C83C2×#12311yescharacteristic
302C22×C83C2×#12511yescharacteristic
303C22×C83C2×#12611yescharacteristic
304C22×C83#52912yes—
305C22×C83#52912yes—
306C22×C83#53812yes—
307C22×C83#53812yes—
308C22×C83#54612yes—
309C22×C83#54612yes—
310C22×C83#55112yes—
311C22×C83#55112yes—
312C22×C83#55312yes—
313C22×C83#55312yes—
314C22×C83#55412yes—
315C22×C83#55412yes—
316C4×C83—24no—
317C4×C83—24no—
318C4×C83—42no—
319C4×C83—42no—
320C4×C83—42no—
321C4×C83—42no—
322Dic83—44no—
323Dic83—44no—
324Dic83—82no—
325Dic83—82no—
326#13883—831nomaximal, Sylow
327C2×C83—28no—
328C2×C83#228311yescharacteristic
329C2×C83C2×#38711yescharacteristic
330C2×C83#231211yescharacteristic
331C2×C83#234211yescharacteristic
332C2×C83#234211yescharacteristic
333C2×C83#234511yescharacteristic
334C2×C83#234511yescharacteristic
335C2×C83#244412yes—
336C2×C83#245012yes—
337C2×C83#250512yes—
338C2×C83#251012yes—
339C2×#380—831no—
340C2×C4×C22⋊C4—831no—
341C22×#95D8311yescharacteristic
342#1443—832no—
343#1442—832no—
344#1436—832no—
345#1434—832no—
346C22⋊#95—831no—
347C22⋊#95—831no—
348C22⋊#95—831no—
349C22⋊#95—831no—
350C83#1388311yescharacteristic, complemented, Sylow
351C24×C4C2×D8311yescharacteristic
352C24×C4C2×D8311yescharacteristic
353C24×C4C2×D8311yescharacteristic
354C22×C4⋊C4—832no—
355C22×C22⋊C4—831no—
356C22×C22⋊C4—831no—
357C22×C22⋊C4—831no—
358C22×C22⋊C4—831no—
359C22×C42—832no—
360#380—1664no—
361C4×C22⋊C4—1664no—
362C2×#95—832no—
363C2×#95—832no—
364C2×#95—832no—
365C2×#95—832no—
366C2×#95—832no—
367C2×#95—832no—
368C2×#95—832no—
369C2×#95—832no—
370C2×#95—832no—
371C2×#95—832no—
372C2×#95Dic8318yescomplemented
373C2×#95C2×D8311yescharacteristic
374C2×#95C2×D8311yescharacteristic
375C2×#95C2×D8311yescharacteristic
376C2×#95C2×D8311yescharacteristic
377C25C22×D8311yescharacteristic
378C23×C4—22no—
379C23×C4—22no—
380C23×C4—22no—
381C23×C4—22no—
382C23×C4—22no—
383C23×C4—22no—
384C23×C4—24no—
385C23×C4—24no—
386C23×C4—42no—
387C23×C4—42no—
388C23×C4—42no—
389C23×C4—42no—
390C23×C4—832no—
391C23×C4—832no—
392C23×C4—832no—
393C23×C4—832no—
394C23×C4C2×Dic8314yes—
395C23×C4C2×Dic8314yes—
396C23×C4C2×Dic8314yes—
397C23×C4C22×D8311yescharacteristic
398C23×C4C22×D8311yescharacteristic
399C23×C4C22×D8311yescharacteristic
400C23×C4C22×D8311yescharacteristic
401C23×C4C22×D8311yescharacteristic
402C23×C4C22×D8311yescharacteristic
403C2×C4⋊C4—3322no—
404C2×C4⋊C4—1664no—
405C2×C4⋊C4—1664no—
406C2×C22⋊C4—834no—
407C2×C22⋊C4—834no—
408C2×C22⋊C4—1662no—
409C2×C22⋊C4—1662no—
410C2×C22⋊C4—1662no—
411C2×C22⋊C4—1662no—
412C2×C22⋊C4—1662no—
413C2×C22⋊C4—1662no—
414C2×C22⋊C4—1662no—
415C2×C22⋊C4—1662no—
416C2×C22⋊C4—1662no—
417C2×C22⋊C4—1662no—
418C2×C42—3322no—
419C2×C42—1664no—
420C2×C42—1664no—
421#95—3322no—
422#95—3322no—
423#95—3322no—
424#95—3322no—
425#95—3322no—
426#95—3322no—
427#95—1664no—
428#95—1664no—
429#95—1664no—
430#95—1664no—
431#95C2×Dic8314yes—
432#95C2×Dic8314yes—
433#95C2×Dic8314yes—
434#95C2×Dic8314yes—
435C24—22no—
436C24—22no—
437C24—22no—
438C24—22no—
439C24C22×Dic8312yes—
440C24D4×D8312yes—
441C24Q8×D8312yes—
442C24#1305058912yes—
443C24#1305059012yes—
444C24#1305059012yes—
445C24#1305059112yes—
446C24C23×D8311yescharacteristic
447C22×C4—21no—
448C22×C4—21no—
449C22×C4—21no—
450C22×C4—21no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—21no—
454C22×C4—21no—
455C22×C4—21no—
456C22×C4—21no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—21no—
460C22×C4—21no—
461C22×C4—21no—
462C22×C4—21no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—28no—
470C22×C4—28no—
471C22×C4—28no—
472C22×C4—28no—
473C22×C4—28no—
474C22×C4—44no—
475C22×C4—44no—
476C22×C4—44no—
477C22×C4—44no—
478C22×C4—48no—
479C22×C4—48no—
480C22×C4—48no—
481C22×C4—48no—
482C22×C4—48no—
483C22×C4—48no—
484C22×C4—834no—
485C22×C4—834no—
486C22×C4—1662no—
487C22×C4—1662no—
488C22×C4—1662no—
489C22×C4—1662no—
490C22×C4—1662no—
491C22×C4—1662no—
492C22×C4—1662no—
493C22×C4—1662no—
494C22×C4—1662no—
495C22×C4—1662no—
496C22×C4—1662no—
497C22×C4—1662no—
498C22×C4—1662no—
499C22×C4—1662no—
500C22×C4—1662no—
501C22×C4—1662no—
502C22×C4—1662no—
503C22×C4—1662no—
504C22×C4—1662no—
505C22×C4—1662no—
506C22×C4—1662no—
507C22×C4—1662no—
508C22×C4—1662no—
509C22×C4—1662no—
510C22×C4—1662no—
511C22×C4—1662no—
512C22×C4C22×Dic8312yes—
513C22×C4C22×Dic8312yes—
514C22×C4C22×Dic8312yes—
515C22×C4C22×Dic8312yes—
516C22×C4C22×Dic8312yes—
517C22×C4C22×Dic8312yes—
518C22×C4#1305059014yes—
519C22×C4#1305059014yes—
520C4⋊C4—3328no—
521C22⋊C4—3324no—
522C22⋊C4—3324no—
523C22⋊C4—1668no—
524C22⋊C4—1668no—
525C42—3328no—
526C23—24no—
527C23—24no—
528C23—24no—
529C23—24no—
530C23—24no—
531C23—24no—
532C23—24no—
533C23—24no—
534C23—22no—
535C23—22no—
536C23—22no—
537C23—22no—
538C23—24no—
539C23—24no—
540C23—24no—
541C23—24no—
542C23#1458175114yes—
543C23D4×Dic8314yes—
544C23Q8×Dic8314yes—
545C23#1458177014yes—
546C23#1458177714yes—
547C23#1458178214yes—
548C23#1458178714yes—
549C23C23×Dic8311yescharacteristic
550C23C2×D4×D8311yescharacteristic
551C23C2×Q8×D8311yescharacteristic
552C23C2×#1305058911yescharacteristic
553C23C2×#1305059012yes—
554C23C2×#1305059012yes—
555C23C2×#1305059011yescharacteristic
556C23C2×#1305059011yescharacteristic
557C23C2×#1305059111yescharacteristic
558C23#1458183712yes—
559C23#1458183712yes—
560C2×C4—24no—
561C2×C4—24no—
562C2×C4—24no—
563C2×C4—24no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—416no—
581C2×C4—416no—
582C2×C4—416no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—3322no—
592C2×C4—3322no—
593C2×C4—3322no—
594C2×C4—3322no—
595C2×C4—3322no—
596C2×C4—3322no—
597C2×C4—3322no—
598C2×C4—3322no—
599C2×C4—3322no—
600C2×C4—3322no—
601C2×C4—3322no—
602C2×C4—3322no—
603C2×C4—3322no—
604C2×C4—3322no—
605C2×C4—3322no—
606C2×C4—3322no—
607C2×C4—3322no—
608C2×C4—3322no—
609C2×C4—3322no—
610C2×C4—3322no—
611C2×C4—3322no—
612C2×C4—3322no—
613C2×C4—1664no—
614C2×C4—1664no—
615C2×C4—1664no—
616C2×C4—1664no—
617C2×C4—1664no—
618C2×C4—1664no—
619C2×C4#1458175114yes—
620C2×C4#1458175114yes—
621C22—24no—
622C22—24no—
623C22—24no—
624C22—28no—
625C22—28no—
626C22—28no—
627C22—28no—
628C22—24no—
629C22—24no—
630C22—24no—
631C22#1912758718yes—
632C22C2×#1458175111yescharacteristic
633C22C2×#1458175111yescharacteristic
634C22C2×#1458175111yescharacteristic
635C22C2×D4×Dic8311yescharacteristic
636C22C2×Q8×Dic8311yescharacteristic
637C22#1912817611yescharacteristic
638C22#1912817611yescharacteristic
639C22C2×#1458177011yescharacteristic
640C22C2×#1458177711yescharacteristic
641C22C2×#1458178211yescharacteristic
642C22C2×#1458178711yescharacteristic
643C22#1912823512yes—
644C22#1912823512yes—
645C22#1912828912yes—
646C22#1912828912yes—
647C22#1912834112yes—
648C22#1912834112yes—
649C22#1912837712yes—
650C22#1912837712yes—
651C22#1912839912yes—
652C22#1912839912yes—
653C22#1912840612yes—
654C22#1912840612yes—
655C4—24no—
656C4—24no—
657C4—42no—
658C4—42no—
659C4—42no—
660C4—42no—
661C4—3324no—
662C4—3324no—
663C4—6642no—
664C4—6642no—
665C2—28no—
666C2#2490986111yescharacteristic
667C2C2×#1912758711yescharacteristic
668C2#2491006111yescharacteristic
669C2#2491031211yescharacteristic
670C2#2491031211yescharacteristic
671C2#2491034011yescharacteristic
672C2#2491034011yescharacteristic
673C2#2491121312yes—
674C2#2491128712yes—
675C2#2491196412yes—
676C2#2491202312yes—
subgroupG/H
center, FrattiniC24C23×D83
commutatorC23×C83C23×C4
FittingC22×#95×C83C2
socle, cosocleC24×C83C24
prime radicalC25×C83C23
hypercenterC22×#95D83
hypocenterC83#13883
LCSG ▷ C23×C83 ▷ C83
UCSC1 ◁ C24 ◁ C22×#95

Complements

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

NH ≅ G/Ntype
3. #24905725C2semidirect
4. #24905720C2semidirect
5. #24905697C2semidirect
6. #24905688C2semidirect
33. C2×#95×C83C4semidirect
350. C83#13883semidirect
372. C2×#95Dic83semidirect

Maximal subgroups

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

subgroupindex
C2×#191275612
C2×C4×C22⋊Dic832
#249057252
#249057202
#249056972
#249056882
#249056862
C22×#95×C832
#1388383