Finite Group Databasebeta

← #35330478

Subgroups — #35330478

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×#19127600C211yesmaximal, characteristic, complemented
2C2×#19127587C212yesmaximal, complemented
3C2×#19127576C212yesmaximal
4C2×C22⋊C4×Dic83C211yesmaximal, characteristic
5#24905731C211yesmaximal, characteristic, complemented
6#24905728C211yesmaximal, characteristic, complemented
7#24905708C211yesmaximal, characteristic
8#24905700C212yesmaximal
9#24905692C212yesmaximal, complemented
10#95×Dic83C211yesmaximal, characteristic, complemented
11C2×#399×C83C211yesmaximal, characteristic
12C24×Dic83C2211yescharacteristic
13C22×C4⋊Dic83C2212yes—
14C22×C4⋊Dic83C2212yes—
15C22×C4⋊Dic83C2211yescharacteristic
16C22×C22⋊Dic83C2212yes—
17C22×C22⋊Dic83C2211yescharacteristic
18C22×C4×Dic83C2211yescharacteristic
19C22×C4×Dic83C2212yes—
20C22×C4×Dic83C2211yescharacteristic
21#19127600—24no—
22#19127587—28no—
23#19127576—28no—
24C22⋊C4×Dic83—24no—
25C2×C83⋊#95C2211yescharacteristic
26C2×C83⋊#95C2212yes—
27C2×C83⋊#95C2212yes—
28C2×C83⋊#95C2212yes—
29C2×C83⋊#95C2211yescharacteristic
30C2×C83⋊#95C2212yes—
31C2×C83⋊#95C2212yes—
32C2×C83⋊#95C2211yescharacteristic
33C2×C83⋊#95C2212yes—
34C22×C4⋊C4×C83C2211yescharacteristic
35C22×C4⋊C4×C83C2212yes—
36C22×C22⋊C4×C83C2212yes—
37C22×C22⋊C4×C83C2211yescharacteristic
38#399×C83C418yescomplemented
39C2×#95×C83C2211yescharacteristic
40C23×Dic83—22no—
41C23×Dic83—22no—
42C23×Dic83—24no—
43C23×Dic83—22no—
44C23×Dic83—42no—
45C23×Dic83C2312yes—
46C23×Dic83C2312yes—
47C23×Dic83C2311yescharacteristic
48C23×Dic83C2311yescharacteristic
49C23×Dic83C2311yescharacteristic
50C23×Dic83C2311yescharacteristic
51C2×C4⋊Dic83—24no—
52C2×C4⋊Dic83—24no—
53C2×C4⋊Dic83—24no—
54C2×C4⋊Dic83—24no—
55C2×C4⋊Dic83—24no—
56C2×C4⋊Dic83—22no—
57C2×C4⋊Dic83—42no—
58C2×C4⋊Dic83—42no—
59C2×C22⋊Dic83—22no—
60C2×C22⋊Dic83—22no—
61C2×C22⋊Dic83—24no—
62C2×C22⋊Dic83—24no—
63C2×C22⋊Dic83—42no—
64C2×C22⋊Dic83Q814yes—
65C2×C4×Dic83—24no—
66C2×C4×Dic83—22no—
67C2×C4×Dic83—24no—
68C2×C4×Dic83—24no—
69C2×C4×Dic83—42no—
70C2×C4×Dic83—42no—
71C2×C4×Dic83D414yes—
72C83⋊#95—24no—
73C83⋊#95—24no—
74C83⋊#95—24no—
75C83⋊#95—24no—
76C83⋊#95—22no—
77C83⋊#95—24no—
78C83⋊#95—22no—
79C83⋊#95—22no—
80C83⋊#95D418yes—
81C25×C83C2311yescharacteristic
82C23×C4×C83C2311yescharacteristic
83C23×C4×C83C2312yes—
84C23×C4×C83C2312yes—
85C23×C4×C83C2311yescharacteristic
86C2×C4⋊C4×C83—24no—
87C2×C4⋊C4×C83—22no—
88C2×C4⋊C4×C83—22no—
89C2×C4⋊C4×C83—24no—
90C2×C4⋊C4×C83C2×C414yescomplemented
91C2×C4⋊C4×C83C2×C418yescomplemented
92C2×C22⋊C4×C83—24no—
93C2×C22⋊C4×C83—22no—
94C2×C22⋊C4×C83—22no—
95C2×C22⋊C4×C83—24no—
96C2×C22⋊C4×C83C2×C418yes—
97C2×C22⋊C4×C83C2×C414yes—
98#95×C83C2×C414yescomplemented
99C22×Dic83—21no—
100C22×Dic83—21no—
101C22×Dic83—22no—
102C22×Dic83—22no—
103C22×Dic83—22no—
104C22×Dic83—28no—
105C22×Dic83—21no—
106C22×Dic83—22no—
107C22×Dic83—22no—
108C22×Dic83—21no—
109C22×Dic83—21no—
110C22×Dic83—22no—
111C22×Dic83—22no—
112C22×Dic83—22no—
113C22×Dic83—21no—
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×Dic83—22no—
123C22×Dic83—22no—
124C22×Dic83—22no—
125C22×Dic83—22no—
126C22×Dic83—48no—
127C22×Dic83—48no—
128C22×Dic83—44no—
129C22×Dic83—44no—
130C22×Dic83C2×D414yes—
131C22×Dic83C2×D412yes—
132C22×Dic83C2×D414yes—
133C22×Dic83C2×D412yes—
134C22×Dic83C2×Q812yes—
135C22×Dic83C2×Q812yes—
136C22×Dic83#4112yes—
137C22×Dic83#4112yes—
138C4⋊Dic83—48no—
139C4⋊Dic83—44no—
140C4⋊Dic83—48no—
141C22⋊Dic83—28no—
142C22⋊Dic83—48no—
143C4×Dic83—44no—
144C4×Dic83—48no—
145C4×Dic83—44no—
146C24×C83—21no—
147C24×C83—22no—
148C24×C83—21no—
149C24×C83—21no—
150C24×C83—22no—
151C24×C83—21no—
152C24×C83—22no—
153C24×C83—22no—
154C24×C83C22×C412yes—
155C24×C83C2×Q812yes—
156C24×C83#4112yes—
157C24×C83C2411yescharacteristic
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—21no—
165C22×C4×C83—22no—
166C22×C4×C83—22no—
167C22×C4×C83—22no—
168C22×C4×C83—22no—
169C22×C4×C83—21no—
170C22×C4×C83—22no—
171C22×C4×C83—22no—
172C22×C4×C83—22no—
173C22×C4×C83—22no—
174C22×C4×C83C22×C412yes—
175C22×C4×C83C22×C414yes—
176C22×C4×C83C22×C414yes—
177C22×C4×C83C22×C412yes—
178C22×C4×C83C2×D414yes—
179C22×C4×C83C2×D412yes—
180C22×C4×C83#4112yes—
181C22×C4×C83#4114yes—
182C4⋊C4×C83—216no—
183C4⋊C4×C83—28no—
184C22⋊C4×C83—216no—
185C22⋊C4×C83—28no—
186C23×C83—24no—
187C23×C83—22no—
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×D411yescharacteristic
211C23×C83C22×D412yes—
212C23×C83C22×Q811yescharacteristic
213C23×C83C2×#4111yescharacteristic
214C23×C83C2×#4111yescharacteristic
215C23×C83C2×#4111yescharacteristic
216C23×C83C2×#4112yes—
217C23×C83C2×#4111yescharacteristic
218C23×C83H(5,𝔽2)12yes—
219C23×C832−1+412yes—
220C2×C4×C83—24no—
221C2×C4×C83—24no—
222C2×C4×C83—24no—
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—24no—
230C2×C4×C83—22no—
231C2×C4×C83—24no—
232C2×C4×C83—24no—
233C2×C4×C83—24no—
234C2×C4×C83—24no—
235C2×C4×C83—41no—
236C2×C4×C83—42no—
237C2×C4×C83—41no—
238C2×C4×C83—42no—
239C2×C4×C83—42no—
240C2×C4×C83—42no—
241C2×C4×C83—42no—
242C2×C4×C83C4×D414yes—
243C2×C4×C83C4×D418yes—
244C2×Dic83—22no—
245C2×Dic83—22no—
246C2×Dic83—22no—
247C2×Dic83—22no—
248C2×Dic83—24no—
249C2×Dic83—24no—
250C2×Dic83—24no—
251C2×Dic83—24no—
252C2×Dic83—22no—
253C2×Dic83—22no—
254C2×Dic83—24no—
255C2×Dic83—24no—
256C2×Dic83—24no—
257C2×Dic83—24no—
258C2×Dic83—24no—
259C2×Dic83—416no—
260C2×Dic83—42no—
261C2×Dic83—41no—
262C2×Dic83—41no—
263C2×Dic83—41no—
264C2×Dic83—41no—
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×Dic83C22⋊Q814yes—
278C2×Dic83#12714yes—
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×D411yescharacteristic
291C22×C83C2×C4×D412yes—
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—28no—
315C4×C83—42no—
316C4×C83—44no—
317Dic83—24no—
318Dic83—24no—
319Dic83—42no—
320Dic83—42no—
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×#399D8311yescharacteristic
337C2×#389—831no—
338C2×#387—832no—
339C2×#385—832no—
340C2×C4×C22⋊C4—831no—
341#1445—831no—
342#1444—831no—
343#1439—831no—
344#1437—832no—
345C4⋊#95—832no—
346C4×#95—831no—
347C83#1423011yescharacteristic, complemented, Sylow
348C24×C4—831no—
349C22×C4⋊C4—831no—
350C22×C4⋊C4—832no—
351C22×C4⋊C4—832no—
352C22×C4⋊C4C2×D8312yes—
353C22×C4⋊C4C2×D8311yescharacteristic
354C22×C22⋊C4—832no—
355C22×C22⋊C4—831no—
356C22×C22⋊C4C2×D8312yes—
357C22×C22⋊C4C2×D8311yescharacteristic
358C22×C42—831no—
359C22×C42—832no—
360C22×C42—831no—
361#399Dic8318yescomplemented
362#389—1664no—
363#387—1668no—
364#385—1668no—
365C4×C22⋊C4—1664no—
366C2×#95—832no—
367C2×#95—831no—
368C2×#95—831no—
369C2×#95—832no—
370C2×#95—832no—
371C2×#95—832no—
372C2×#95—832no—
373C2×#95—832no—
374C2×#95—831no—
375C2×#95C2×D8311yescharacteristic
376C25C22×D8311yescharacteristic
377C23×C4—3322no—
378C23×C4—831no—
379C23×C4—831no—
380C23×C4—832no—
381C23×C4—831no—
382C23×C4—831no—
383C23×C4—832no—
384C23×C4—1664no—
385C23×C4—1662no—
386C23×C4—1662no—
387C23×C4—1662no—
388C23×C4C22×D8312yes—
389C23×C4C22×D8311yescharacteristic
390C23×C4C22×D8312yes—
391C23×C4C22×D8311yescharacteristic
392C2×C4⋊C4—24no—
393C2×C4⋊C4—22no—
394C2×C4⋊C4—22no—
395C2×C4⋊C4—24no—
396C2×C4⋊C4—3322no—
397C2×C4⋊C4—3322no—
398C2×C4⋊C4—1664no—
399C2×C4⋊C4—1664no—
400C2×C4⋊C4—1664no—
401C2×C4⋊C4—1664no—
402C2×C4⋊C4—1664no—
403C2×C4⋊C4—1662no—
404C2×C4⋊C4C2×Dic8318yescomplemented
405C2×C4⋊C4C2×Dic8314yescomplemented
406C2×C22⋊C4—22no—
407C2×C22⋊C4—24no—
408C2×C22⋊C4—22no—
409C2×C22⋊C4—24no—
410C2×C22⋊C4—3322no—
411C2×C22⋊C4—834no—
412C2×C22⋊C4—1664no—
413C2×C22⋊C4—1664no—
414C2×C22⋊C4—1662no—
415C2×C22⋊C4—1662no—
416C2×C22⋊C4C2×Dic8318yes—
417C2×C22⋊C4C2×Dic8314yes—
418C2×C42—3322no—
419C2×C42—3322no—
420C2×C42—834no—
421C2×C42—1662no—
422C2×C42—1664no—
423C2×C42—1664no—
424C2×C42—1664no—
425#95—838no—
426#95—1662no—
427#95—1664no—
428#95—1664no—
429#95—1664no—
430#95—1664no—
431#95—1662no—
432#95—1664no—
433#95—1662no—
434#95C2×Dic8314yescomplemented
435C24—22no—
436C24—21no—
437C24—21no—
438C24—21no—
439C24—22no—
440C24—21no—
441C24—22no—
442C24—22no—
443C24C22×Dic8312yes—
444C24Q8×D8312yes—
445C24#1305059112yes—
446C24C23×D8311yescharacteristic
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—21no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—21no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—3324no—
464C22×C4—3328no—
465C22×C4—3324no—
466C22×C4—3328no—
467C22×C4—834no—
468C22×C4—832no—
469C22×C4—832no—
470C22×C4—832no—
471C22×C4—832no—
472C22×C4—834no—
473C22×C4—832no—
474C22×C4—832no—
475C22×C4—1662no—
476C22×C4—1662no—
477C22×C4—1662no—
478C22×C4—1662no—
479C22×C4—1662no—
480C22×C4—1662no—
481C22×C4—1662no—
482C22×C4—1668no—
483C22×C4—1662no—
484C22×C4—1661no—
485C22×C4—1661no—
486C22×C4—1662no—
487C22×C4—1662no—
488C22×C4—1661no—
489C22×C4—1662no—
490C22×C4—1662no—
491C22×C4—1662no—
492C22×C4—1661no—
493C22×C4—1662no—
494C22×C4—1662no—
495C22×C4—1662no—
496C22×C4—1662no—
497C22×C4—1661no—
498C22×C4—1662no—
499C22×C4—1661no—
500C22×C4—1662no—
501C22×C4—1662no—
502C22×C4C22×Dic8314yes—
503C22×C4C22×Dic8312yes—
504C22×C4C22×Dic8314yes—
505C22×C4C22×Dic8312yes—
506C22×C4D4×D8314yes—
507C22×C4D4×D8312yes—
508C22×C4#1305058912yes—
509C22×C4#1305058914yes—
510C4⋊C4—216no—
511C4⋊C4—28no—
512C4⋊C4—3324no—
513C4⋊C4—3328no—
514C4⋊C4—3328no—
515C22⋊C4—28no—
516C22⋊C4—216no—
517C22⋊C4—3328no—
518C22⋊C4—1668no—
519C42—3328no—
520C42—3324no—
521C42—3324no—
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—
544C23Q8×Dic8314yes—
545C23C23×Dic8311yescharacteristic
546C23C2×D4×D8312yes—
547C23C2×D4×D8311yescharacteristic
548C23C2×Q8×D8311yescharacteristic
549C23#41×D8311yescharacteristic
550C23#41×D8311yescharacteristic
551C23C2×#1305058912yes—
552C23C2×#1305058911yescharacteristic
553C23C2×#1305059111yescharacteristic
554C23#1458183812yes—
555C23#1458183912yes—
556C2×C4—24no—
557C2×C4—24no—
558C2×C4—24no—
559C2×C4—24no—
560C2×C4—24no—
561C2×C4—24no—
562C2×C4—24no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—24no—
566C2×C4—22no—
567C2×C4—24no—
568C2×C4—24no—
569C2×C4—24no—
570C2×C4—24no—
571C2×C4—41no—
572C2×C4—42no—
573C2×C4—41no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—3322no—
579C2×C4—3322no—
580C2×C4—3322no—
581C2×C4—3322no—
582C2×C4—3322no—
583C2×C4—3321no—
584C2×C4—3321no—
585C2×C4—3321no—
586C2×C4—3321no—
587C2×C4—3322no—
588C2×C4—33216no—
589C2×C4—3322no—
590C2×C4—3322no—
591C2×C4—3322no—
592C2×C4—3322no—
593C2×C4—3322no—
594C2×C4—3322no—
595C2×C4—3322no—
596C2×C4—834no—
597C2×C4—834no—
598C2×C4—1664no—
599C2×C4—1664no—
600C2×C4—1664no—
601C2×C4—1664no—
602C2×C4—1662no—
603C2×C4—1662no—
604C2×C4—1664no—
605C2×C4—1664no—
606C2×C4—1664no—
607C2×C4—1664no—
608C2×C4—1664no—
609C2×C4—1662no—
610C2×C4—1662no—
611C2×C4—1662no—
612C2×C4—1662no—
613C2×C4D4×Dic8314yes—
614C2×C4D4×Dic8318yes—
615C22—28no—
616C22—24no—
617C22—42no—
618C22—42no—
619C22—42no—
620C22—42no—
621C22—44no—
622C22—44no—
623C22—44no—
624C22—42no—
625C22—42no—
626C22C2×D4×Dic8311yescharacteristic
627C22C2×D4×Dic8312yes—
628C22C2×Q8×Dic8311yescharacteristic
629C22#41×Dic8311yescharacteristic
630C22#1912818412yes—
631C22C22⋊Q8×D8311yescharacteristic
632C22#1912821411yescharacteristic
633C22#1912823711yescharacteristic
634C22#127×D8311yescharacteristic
635C22#1912824911yescharacteristic
636C22#1912824912yes—
637C22#1912825311yescharacteristic
638C22#1912826211yescharacteristic
639C22#1912827311yescharacteristic
640C22#1912829512yes—
641C22#1912834212yes—
642C22#1912834312yes—
643C22#1912834912yes—
644C22#1912836412yes—
645C22#1912837412yes—
646C22#1912838812yes—
647C22#1912839312yes—
648C22#1912840912yes—
649C4—24no—
650C4—28no—
651C4—42no—
652C4—44no—
653C4—3322no—
654C4—3322no—
655C4—3324no—
656C4—6642no—
657C4—1664no—
658C4—1664no—
659C2—44no—
660C2C22⋊Q8×Dic8311yescharacteristic
661C2#127×Dic8311yescharacteristic
662C2#2491013511yescharacteristic
663C2#2491045912yes—
664C2#2491047212yes—
665C2#2491239011yescharacteristic
666C2#2491241111yescharacteristic
667C2#2491250912yes—
668C2#2491254812yes—
669C2#2491263811yescharacteristic
670C2#2491315411yescharacteristic
subgroupG/H
center, FrattiniC24C23×D83
commutatorC23×C83C23×C4
FittingC2×#399×C83C2
socle, cosocleC24×C83C24
prime radicalC25×C83C23
hypercenterC2×#399D83
hypocenterC83#14230
LCSG ▷ C23×C83 ▷ C83
UCSC1 ◁ C24 ◁ C2×#399

Complements

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

NH ≅ G/Ntype
1. C2×#19127600C2semidirect
2. C2×#19127587C2semidirect
5. #24905731C2semidirect
6. #24905728C2semidirect
9. #24905692C2semidirect
10. #95×Dic83C2semidirect
38. #399×C83C4semidirect
90. C2×C4⋊C4×C83C2×C4semidirect
91. C2×C4⋊C4×C83C2×C4semidirect
98. #95×C83C2×C4semidirect
347. C83#14230semidirect
361. #399Dic83semidirect
404. C2×C4⋊C4C2×Dic83semidirect
405. C2×C4⋊C4C2×Dic83semidirect
434. #95C2×Dic83semidirect

Maximal subgroups

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

subgroupindex
C2×#191276002
C2×#191275872
C2×#191275762
C2×C22⋊C4×Dic832
#249057312
#249057282
#249057082
#249057002
#249056922
#95×Dic832
C2×#399×C832
#1423083