Finite Group Databasebeta

← #35330477

Subgroups — #35330477

Order 21248.

1822 conjugacy classes of subgroups.

The subgroup lattice has 672 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19127637C211yesmaximal, characteristic
2C2×#19127599C211yesmaximal, characteristic, complemented
3C2×#19127585C212yesmaximal, complemented
4C2×#19127574C212yesmaximal
5#24905732C211yesmaximal, characteristic, complemented
6#24905727C211yesmaximal, characteristic, complemented
7#24905708C211yesmaximal, characteristic
8#24905700C212yesmaximal
9Dic83⋊#95C212yesmaximal, complemented
10C4×C83⋊#95C211yesmaximal, characteristic, complemented
11C2×C4×C22⋊C4×C83C211yesmaximal, characteristic
12C22×C4⋊Dic83C2211yescharacteristic
13C22×C4⋊Dic83C2211yescharacteristic
14C22×Dic83⋊C4C2212yes—
15C22×Dic83⋊C4C2212yes—
16C22×Dic83⋊C4C2212yes—
17C22×C22⋊Dic83C2212yes—
18C22×C22⋊Dic83C2212yes—
19C22×C4×Dic83C2212yes—
20#19127637C418yescomplemented
21#19127599—24no—
22#19127585—28no—
23#19127574—28no—
24C2×C83⋊#95C2212yes—
25C2×C83⋊#95C2212yes—
26C2×C83⋊#95C2211yescharacteristic
27C2×C83⋊#95C2212yes—
28C2×C83⋊#95C2212yes—
29C2×C83⋊#95C2212yes—
30C2×C83⋊#95C2211yescharacteristic
31C2×C83⋊#95C2212yes—
32C24×C4×C83C2211yescharacteristic
33C22×C22⋊C4×C83C2211yescharacteristic
34C22×C22⋊C4×C83C2211yescharacteristic
35C22×C42×C83C2211yescharacteristic
36C22×C42×C83C2211yescharacteristic
37C4×C22⋊C4×C83—24no—
38C2×#95×C83C2211yescharacteristic
39C2×#95×C83C2211yescharacteristic
40C23×Dic83C2312yes—
41C23×Dic83C2312yes—
42C23×Dic83C2312yes—
43C23×Dic83C2312yes—
44C2×C4⋊Dic83—22no—
45C2×C4⋊Dic83—22no—
46C2×C4⋊Dic83—22no—
47C2×C4⋊Dic83—42no—
48C2×C4⋊Dic83C2×C414yescomplemented
49C2×Dic83⋊C4—24no—
50C2×Dic83⋊C4—24no—
51C2×Dic83⋊C4—24no—
52C2×Dic83⋊C4—24no—
53C2×Dic83⋊C4—24no—
54C2×Dic83⋊C4—24no—
55C2×Dic83⋊C4—24no—
56C2×Dic83⋊C4—42no—
57C2×Dic83⋊C4C2×C418yescomplemented
58C2×C22⋊Dic83—24no—
59C2×C22⋊Dic83—24no—
60C2×C22⋊Dic83—24no—
61C2×C22⋊Dic83—24no—
62C2×C22⋊Dic83—42no—
63C2×C22⋊Dic83C2×C418yes—
64C2×C4×Dic83—24no—
65C2×C4×Dic83—24no—
66C2×C4×Dic83—42no—
67C83⋊#95—24no—
68C83⋊#95—24no—
69C83⋊#95—24no—
70C83⋊#95—24no—
71C83⋊#95—24no—
72C83⋊#95—22no—
73C83⋊#95C2×C414yescomplemented
74C83⋊#95D418yes—
75C25×C83C2311yescharacteristic
76C23×C4×C83—22no—
77C23×C4×C83—22no—
78C23×C4×C83—22no—
79C23×C4×C83—24no—
80C23×C4×C83—42no—
81C23×C4×C83C2311yescharacteristic
82C23×C4×C83C2311yescharacteristic
83C23×C4×C83C2311yescharacteristic
84C23×C4×C83C2311yescharacteristic
85C23×C4×C83C2311yescharacteristic
86C23×C4×C83C2311yescharacteristic
87C2×C22⋊C4×C83—22no—
88C2×C22⋊C4×C83—22no—
89C2×C22⋊C4×C83—22no—
90C2×C22⋊C4×C83—22no—
91C2×C22⋊C4×C83C2×C414yes—
92C2×C22⋊C4×C83Q814yes—
93C2×C42×C83—24no—
94C2×C42×C83—22no—
95C2×C42×C83—42no—
96C2×C42×C83D414yes—
97#95×C83—22no—
98#95×C83—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×Dic83C22×C414yes—
123C22×Dic83C22×C414yes—
124C22×Dic83C2×D414yes—
125C22×Dic83C2×D414yes—
126C22×Dic83#4114yes—
127C4⋊Dic83—28no—
128C4⋊Dic83—44no—
129Dic83⋊C4—216no—
130Dic83⋊C4—48no—
131Dic83⋊C4—48no—
132C22⋊Dic83—216no—
133C22⋊Dic83—48no—
134C4×Dic83—48no—
135C24×C83—21no—
136C24×C83—22no—
137C24×C83—21no—
138C24×C83—21no—
139C24×C83—21no—
140C24×C83—22no—
141C24×C83—22no—
142C24×C83—22no—
143C24×C83C22×C412yes—
144C24×C83C2×Q812yes—
145C24×C83#4112yes—
146C24×C83C2411yescharacteristic
147C22×C4×C83—21no—
148C22×C4×C83—21no—
149C22×C4×C83—22no—
150C22×C4×C83—22no—
151C22×C4×C83—22no—
152C22×C4×C83—22no—
153C22×C4×C83—22no—
154C22×C4×C83—28no—
155C22×C4×C83—21no—
156C22×C4×C83—21no—
157C22×C4×C83—21no—
158C22×C4×C83—22no—
159C22×C4×C83—22no—
160C22×C4×C83—22no—
161C22×C4×C83—21no—
162C22×C4×C83—21no—
163C22×C4×C83—21no—
164C22×C4×C83—22no—
165C22×C4×C83—22no—
166C22×C4×C83—22no—
167C22×C4×C83—48no—
168C22×C4×C83—48no—
169C22×C4×C83—44no—
170C22×C4×C83—44no—
171C22×C4×C83C22×C412yes—
172C22×C4×C83C22×C412yes—
173C22×C4×C83C2×D414yes—
174C22×C4×C83C2×D412yes—
175C22×C4×C83C2×D412yes—
176C22×C4×C83C2×D412yes—
177C22×C4×C83C2×Q812yes—
178C22×C4×C83C2×Q812yes—
179C22×C4×C83#4112yes—
180C22×C4×C83#4112yes—
181C22×C4×C83#4112yes—
182C22⋊C4×C83—28no—
183C22⋊C4×C83—28no—
184C42×C83—44no—
185C42×C83—44no—
186C23×C83—22no—
187C23×C83—24no—
188C23×C83—22no—
189C23×C83—24no—
190C23×C83—22no—
191C23×C83—22no—
192C23×C83—22no—
193C23×C83—22no—
194C23×C83—24no—
195C23×C83—24no—
196C23×C83—24no—
197C23×C83—24no—
198C23×C83—41no—
199C23×C83—41no—
200C23×C83—42no—
201C23×C83—41no—
202C23×C83—41no—
203C23×C83—42no—
204C23×C83—42no—
205C23×C83—42no—
206C23×C83—42no—
207C23×C83—42no—
208C23×C83C4×Q814yes—
209C23×C83C23×C411yescharacteristic
210C23×C83C22×D412yes—
211C23×C83C22×D411yescharacteristic
212C23×C83C22×Q811yescharacteristic
213C23×C83C2×#4112yes—
214C23×C83C2×#4111yescharacteristic
215C23×C83C2×#4111yescharacteristic
216C23×C83C2×#4111yescharacteristic
217C23×C83C2×#4111yescharacteristic
218C23×C83H(5,𝔽2)12yes—
219C23×C832−1+412yes—
220C2×C4×C83—24no—
221C2×C4×C83—22no—
222C2×C4×C83—22no—
223C2×C4×C83—22no—
224C2×C4×C83—22no—
225C2×C4×C83—22no—
226C2×C4×C83—22no—
227C2×C4×C83—24no—
228C2×C4×C83—24no—
229C2×C4×C83—24no—
230C2×C4×C83—24no—
231C2×C4×C83—24no—
232C2×C4×C83—22no—
233C2×C4×C83—24no—
234C2×C4×C83—22no—
235C2×C4×C83—22no—
236C2×C4×C83—24no—
237C2×C4×C83—24no—
238C2×C4×C83—416no—
239C2×C4×C83—41no—
240C2×C4×C83—41no—
241C2×C4×C83—41no—
242C2×C4×C83—41no—
243C2×C4×C83—42no—
244C2×C4×C83—41no—
245C2×C4×C83—41no—
246C2×C4×C83—42no—
247C2×C4×C83—42no—
248C2×C4×C83C4×D414yes—
249C2×C4×C83C22⋊Q814yes—
250C2×C4×C83#12714yes—
251C2×Dic83—24no—
252C2×Dic83—24no—
253C2×Dic83—24no—
254C2×Dic83—24no—
255C2×Dic83—24no—
256C2×Dic83—24no—
257C2×Dic83—24no—
258C2×Dic83—24no—
259C2×Dic83—24no—
260C2×Dic83—24no—
261C2×Dic83—24no—
262C2×Dic83—24no—
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×Dic83C4×D418yes—
279C22×C83—28no—
280C22×C83—24no—
281C22×C83—42no—
282C22×C83—42no—
283C22×C83—42no—
284C22×C83—42no—
285C22×C83—44no—
286C22×C83—42no—
287C22×C83—42no—
288C22×C83—44no—
289C22×C83—44no—
290C22×C83C2×C4×D412yes—
291C22×C83C2×C4×D411yescharacteristic
292C22×C83C2×C4×Q811yescharacteristic
293C22×C83C4×#4111yescharacteristic
294C22×C83#52012yes—
295C22×C83C2×C22⋊Q811yescharacteristic
296C22×C83#52511yescharacteristic
297C22×C83#52911yescharacteristic
298C22×C83C2×#12711yescharacteristic
299C22×C83#53211yescharacteristic
300C22×C83#53212yes—
301C22×C83#53311yescharacteristic
302C22×C83#53411yescharacteristic
303C22×C83#53611yescharacteristic
304C22×C83#53912yes—
305C22×C83#54612yes—
306C22×C83#54612yes—
307C22×C83#54712yes—
308C22×C83D4×Q812yes—
309C22×C83#55112yes—
310C22×C83#55212yes—
311C22×C83#55212yes—
312C22×C83#55412yes—
313C4×C83—24no—
314C4×C83—24no—
315C4×C83—24no—
316C4×C83—42no—
317C4×C83—42no—
318C4×C83—42no—
319Dic83—28no—
320Dic83—44no—
321Dic83—44no—
322Dic83—82no—
323#14230—831nomaximal, Sylow
324C2×C83—44no—
325C2×C83C4×C22⋊Q811yescharacteristic
326C2×C83C4×#12711yescharacteristic
327C2×C83#232211yescharacteristic
328C2×C83#235912yes—
329C2×C83#236012yes—
330C2×C83#253911yescharacteristic
331C2×C83#254211yescharacteristic
332C2×C83#255312yes—
333C2×C83#255612yes—
334C2×C83#256411yescharacteristic
335C2×C83#261111yescharacteristic
336C2×#399—831no—
337C2×#389—831no—
338C2×#387—832no—
339C2×#385—832no—
340C2×C4×C22⋊C4D8311yescharacteristic
341#1445—831no—
342#1444—831no—
343#1439—831no—
344#1437—832no—
345C4⋊#95—832no—
346C4×#95—831no—
347C83#1423011yescharacteristic, complemented, Sylow
348C24×C4C2×D8311yescharacteristic
349C22×C4⋊C4—831no—
350C22×C4⋊C4—832no—
351C22×C4⋊C4—832no—
352C22×C4⋊C4—831no—
353C22×C4⋊C4—832no—
354C22×C22⋊C4—832no—
355C22×C22⋊C4—832no—
356C22×C22⋊C4C2×D8311yescharacteristic
357C22×C22⋊C4C2×D8311yescharacteristic
358C22×C42—832no—
359C22×C42C2×D8311yescharacteristic
360C22×C42C2×D8311yescharacteristic
361#399—838no—
362#389—1664no—
363#387—1668no—
364#385—1668no—
365C4×C22⋊C4—24no—
366C2×#95—831no—
367C2×#95—832no—
368C2×#95—831no—
369C2×#95—832no—
370C2×#95—832no—
371C2×#95—832no—
372C2×#95—832no—
373C2×#95—832no—
374C2×#95C2×D8311yescharacteristic
375C2×#95C2×D8311yescharacteristic
376C25C22×D8311yescharacteristic
377C23×C4—22no—
378C23×C4—22no—
379C23×C4—22no—
380C23×C4—24no—
381C23×C4—42no—
382C23×C4—832no—
383C23×C4—832no—
384C23×C4—832no—
385C23×C4—832no—
386C23×C4C22×D8311yescharacteristic
387C23×C4C22×D8311yescharacteristic
388C23×C4C22×D8311yescharacteristic
389C23×C4C22×D8311yescharacteristic
390C23×C4C22×D8311yescharacteristic
391C23×C4C22×D8311yescharacteristic
392C2×C4⋊C4—3322no—
393C2×C4⋊C4—3322no—
394C2×C4⋊C4—838no—
395C2×C4⋊C4—834no—
396C2×C4⋊C4—1664no—
397C2×C4⋊C4—1664no—
398C2×C4⋊C4—1664no—
399C2×C4⋊C4—1664no—
400C2×C4⋊C4—1662no—
401C2×C4⋊C4—1664no—
402C2×C4⋊C4—1664no—
403C2×C4⋊C4—1664no—
404C2×C4⋊C4—1662no—
405C2×C4⋊C4—1662no—
406C2×C22⋊C4—22no—
407C2×C22⋊C4—22no—
408C2×C22⋊C4—22no—
409C2×C22⋊C4—22no—
410C2×C22⋊C4—3322no—
411C2×C22⋊C4—838no—
412C2×C22⋊C4—1664no—
413C2×C22⋊C4—1664no—
414C2×C22⋊C4—1664no—
415C2×C22⋊C4—1664no—
416C2×C22⋊C4Dic16614yes—
417C2×C22⋊C4C4×D8314yes—
418C2×C42—24no—
419C2×C42—22no—
420C2×C42—42no—
421C2×C42—3322no—
422C2×C42—1664no—
423C2×C42—1664no—
424C2×C42C22⋊D8314yes—
425#95—22no—
426#95—22no—
427#95—834no—
428#95—838no—
429#95—1664no—
430#95—1664no—
431#95—1664no—
432#95—1664no—
433#95—1664no—
434#95—1662no—
435C24—22no—
436C24—21no—
437C24—22no—
438C24—21no—
439C24—21no—
440C24—21no—
441C24—22no—
442C24—22no—
443C24C2×C4×D8312yes—
444C24C2×Dic16612yes—
445C24#1305059012yes—
446C24C23×D8311yescharacteristic
447C22×C4—22no—
448C22×C4—21no—
449C22×C4—21no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—28no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—21no—
459C22×C4—21no—
460C22×C4—21no—
461C22×C4—22no—
462C22×C4—21no—
463C22×C4—21no—
464C22×C4—21no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—48no—
468C22×C4—48no—
469C22×C4—44no—
470C22×C4—44no—
471C22×C4—834no—
472C22×C4—834no—
473C22×C4—834no—
474C22×C4—834no—
475C22×C4—834no—
476C22×C4—1662no—
477C22×C4—1662no—
478C22×C4—1662no—
479C22×C4—1662no—
480C22×C4—1662no—
481C22×C4—1662no—
482C22×C4—1662no—
483C22×C4—1662no—
484C22×C4—1662no—
485C22×C4—1662no—
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×C4C2×C4×D8312yes—
500C22×C4C2×C4×D8312yes—
501C22×C4D4×D8314yes—
502C22×C4C2×C22⋊D8312yes—
503C22×C4C2×C22⋊D8312yes—
504C22×C4C2×C22⋊D8312yes—
505C22×C4C2×Dic16612yes—
506C22×C4C2×Dic16612yes—
507C22×C4#1305059012yes—
508C22×C4#1305059012yes—
509C22×C4#1305059012yes—
510C4⋊C4—3324no—
511C4⋊C4—3328no—
512C4⋊C4—3328no—
513C4⋊C4—16616no—
514C4⋊C4—1668no—
515C22⋊C4—28no—
516C22⋊C4—28no—
517C22⋊C4—3328no—
518C22⋊C4—16616no—
519C42—44no—
520C42—44no—
521C42—3328no—
522C23—24no—
523C23—24no—
524C23—24no—
525C23—24no—
526C23—24no—
527C23—22no—
528C23—22no—
529C23—24no—
530C23—22no—
531C23—22no—
532C23—22no—
533C23—22no—
534C23—42no—
535C23—42no—
536C23—42no—
537C23—42no—
538C23—41no—
539C23—41no—
540C23—42no—
541C23—41no—
542C23—41no—
543C23—42no—
544C23C4×Dic16614yes—
545C23C22×C4×D8311yescharacteristic
546C23C2×D4×D8312yes—
547C23C22×C22⋊D8311yescharacteristic
548C23C22×Dic16611yescharacteristic
549C23#41×D8312yes—
550C23C2×#1305059011yescharacteristic
551C23C2×#1305059011yescharacteristic
552C23C2×#1305059011yescharacteristic
553C23C2×#1305059011yescharacteristic
554C23#1458183712yes—
555C23#1458183912yes—
556C2×C4—24no—
557C2×C4—24no—
558C2×C4—24no—
559C2×C4—24no—
560C2×C4—24no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—24no—
568C2×C4—24no—
569C2×C4—24no—
570C2×C4—24no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—416no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—42no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—3322no—
585C2×C4—3322no—
586C2×C4—3322no—
587C2×C4—3322no—
588C2×C4—3322no—
589C2×C4—3322no—
590C2×C4—3322no—
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—838no—
600C2×C4—1664no—
601C2×C4—1664no—
602C2×C4—1664no—
603C2×C4—1664no—
604C2×C4—1664no—
605C2×C4—1664no—
606C2×C4—1664no—
607C2×C4—1664no—
608C2×C4—1664no—
609C2×C4—1664no—
610C2×C4—1664no—
611C2×C4—1664no—
612C2×C4C4×C22⋊D8314yes—
613C2×C4#1458177114yes—
614C2×C4#1458178814yes—
615C22—28no—
616C22—24no—
617C22—42no—
618C22—42no—
619C22—42no—
620C22—42no—
621C22—44no—
622C22—44no—
623C22—44no—
624C22—42no—
625C22—42no—
626C22C4×D4×D8312yes—
627C22C2×C4×C22⋊D8311yescharacteristic
628C22C2×C4×Dic16611yescharacteristic
629C22C4×#1305059011yescharacteristic
630C22#1912818012yes—
631C22C2×#1458177111yescharacteristic
632C22#1912821211yescharacteristic
633C22#1912823511yescharacteristic
634C22C2×#1458178811yescharacteristic
635C22#1912824711yescharacteristic
636C22#1912824812yes—
637C22#1912825811yescharacteristic
638C22#1912826011yescharacteristic
639C22#1912827211yescharacteristic
640C22#1912829312yes—
641C22#1912834112yes—
642C22#1912834112yes—
643C22#1912835612yes—
644C22D4×Dic16612yes—
645C22#1912838012yes—
646C22#1912838412yes—
647C22#1912839012yes—
648C22#1912841012yes—
649C4—24no—
650C4—24no—
651C4—24no—
652C4—42no—
653C4—42no—
654C4—42no—
655C4—3324no—
656C4—3324no—
657C4—6642no—
658C4—1668no—
659C2—44no—
660C2C4×#1458177111yescharacteristic
661C2C4×#1458178811yescharacteristic
662C2#2491012811yescharacteristic
663C2#2491045612yes—
664C2#2491046912yes—
665C2#2491239411yescharacteristic
666C2#2491241611yescharacteristic
667C2#2491250812yes—
668C2#2491254712yes—
669C2#2491263911yescharacteristic
670C2#2491316111yescharacteristic
subgroupG/H
center, FrattiniC24C23×D83
commutatorC23×C83C23×C4
FittingC2×C4×C22⋊C4×C83C2
socle, cosocleC24×C83C24
prime radicalC25×C83C23
hypercenterC2×C4×C22⋊C4D83
hypocenterC83#14230
LCSG ▷ C23×C83 ▷ C83
UCSC1 ◁ C24 ◁ C2×C4×C22⋊C4

Complements

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

NH ≅ G/Ntype
2. C2×#19127599C2semidirect
3. C2×#19127585C2semidirect
5. #24905732C2semidirect
6. #24905727C2semidirect
9. Dic83⋊#95C2semidirect
10. C4×C83⋊#95C2semidirect
20. #19127637C4semidirect
48. C2×C4⋊Dic83C2×C4semidirect
57. C2×Dic83⋊C4C2×C4semidirect
73. C83⋊#95C2×C4semidirect
347. C83#14230semidirect

Maximal subgroups

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

subgroupindex
C2×#191276372
C2×#191275992
C2×#191275852
C2×#191275742
#249057322
#249057272
#249057082
#249057002
Dic83⋊#952
C4×C83⋊#952
C2×C4×C22⋊C4×C832
#1423083