Finite Group Databasebeta

← #35341115

Subgroups — #35341115

Order 21248.

1934 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
1C22×#14581791C211yesmaximal, characteristic
2C2×#19127622C212yesmaximal
3C2×#19127621C212yesmaximal
4C2×#19127598C211yesmaximal, characteristic
5C2×#19127570C212yesmaximal
6#24905728C211yesmaximal, characteristic
7#24905727C211yesmaximal, characteristic
8#24905726C211yesmaximal, characteristic
9Dic83⋊#95C212yesmaximal
10#95×Dic83C211yesmaximal, characteristic
11C83×#1444C211yesmaximal, characteristic
12C23×Dic166C2212yes—
13C2×#14581791—28no—
14C2×#14581791—24no—
15C22×C4⋊Dic83C2211yescharacteristic
16C22×C4⋊Dic83C2211yescharacteristic
17C22×C4⋊Dic83C2212yes—
18C22×Dic83⋊C4C2211yescharacteristic
19C22×Dic83⋊C4C2211yescharacteristic
20C22×Dic83⋊C4C2212yes—
21C22×Dic83⋊C4C2212yes—
22C22×Dic83⋊C4C2212yes—
23C22×Dic83⋊C4C2211yescharacteristic
24C22×C4×Dic83C2211yescharacteristic
25C22×C4×Dic83C2212yes—
26#19127622—28no—
27#19127621—28no—
28#19127598—24no—
29#19127570—28no—
30C2×C83⋊#95C2212yes—
31C2×C83⋊#95C2211yescharacteristic
32C2×C83⋊#95C2212yes—
33C2×C83⋊#95C2212yes—
34C2×C83⋊#95C2212yes—
35C2×C83⋊#95C2211yescharacteristic
36C22×C4⋊C4×C83C2211yescharacteristic
37C22×C4⋊C4×C83C2212yes—
38C2×#95×C83C2212yes—
39C2×#95×C83C2211yescharacteristic
40C2×#95×C83C2211yescharacteristic
41C22×Dic166—24no—
42C22×Dic166—24no—
43C22×Dic166—24no—
44C22×Dic166—24no—
45C22×Dic166—24no—
46C22×Dic166—42no—
47C22×Dic166—42no—
48C23×Dic83C2311yescharacteristic
49C23×Dic83C2312yes—
50C23×Dic83C2311yescharacteristic
51C23×Dic83C2311yescharacteristic
52C23×Dic83C2312yes—
53C23×Dic83C2311yescharacteristic
54#14581791—232no—
55C2×C4⋊Dic83—24no—
56C2×C4⋊Dic83—24no—
57C2×C4⋊Dic83—22no—
58C2×C4⋊Dic83—24no—
59C2×C4⋊Dic83—22no—
60C2×C4⋊Dic83—24no—
61C2×C4⋊Dic83—42no—
62C2×Dic83⋊C4—22no—
63C2×Dic83⋊C4—24no—
64C2×Dic83⋊C4—24no—
65C2×Dic83⋊C4—22no—
66C2×Dic83⋊C4—24no—
67C2×Dic83⋊C4—24no—
68C2×Dic83⋊C4—22no—
69C2×Dic83⋊C4—22no—
70C2×Dic83⋊C4—24no—
71C2×Dic83⋊C4—24no—
72C2×Dic83⋊C4—24no—
73C2×Dic83⋊C4—24no—
74C2×Dic83⋊C4—42no—
75C2×Dic83⋊C4—42no—
76C2×Dic83⋊C4D418yes—
77C2×Dic83⋊C4Q814yescomplemented
78C2×C4×Dic83—24no—
79C2×C4×Dic83—24no—
80C2×C4×Dic83—24no—
81C2×C4×Dic83D414yes—
82C2×C4×Dic83Q818yescomplemented
83C83⋊#95—24no—
84C83⋊#95—24no—
85C83⋊#95—22no—
86C83⋊#95—24no—
87C83⋊#95D418yes—
88C83⋊#95Q814yescomplemented
89C23×C4×C83C2311yescharacteristic
90C23×C4×C83C2312yes—
91C23×C4×C83C2311yescharacteristic
92C23×C4×C83C2311yescharacteristic
93C23×C4×C83C2312yes—
94C2×C4⋊C4×C83—24no—
95C2×C4⋊C4×C83—22no—
96C2×C4⋊C4×C83—24no—
97C2×C4⋊C4×C83—24no—
98C2×C4⋊C4×C83—42no—
99#95×C83—22no—
100#95×C83—24no—
101#95×C83Q814yescomplemented
102C2×Dic166—216no—
103C2×Dic166—48no—
104C2×Dic166—48no—
105C2×Dic166—48no—
106C2×Dic166—48no—
107C2×Dic166—48no—
108C2×Dic166—48no—
109C22×Dic83—22no—
110C22×Dic83—22no—
111C22×Dic83—22no—
112C22×Dic83—21no—
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—21no—
122C22×Dic83—21no—
123C22×Dic83—22no—
124C22×Dic83—22no—
125C22×Dic83—22no—
126C22×Dic83—21no—
127C22×Dic83—22no—
128C22×Dic83—22no—
129C22×Dic83—21no—
130C22×Dic83—21no—
131C22×Dic83C2×D414yes—
132C22×Dic83C2×D414yes—
133C22×Dic83C2×D412yes—
134C22×Dic83C2×D414yes—
135C22×Dic83C2×D414yes—
136C22×Dic83C2×D412yes—
137C22×Dic83C2×Q812yes—
138C22×Dic83C2×Q814yes—
139C22×Dic83C2×Q814yes—
140C22×Dic83C2×Q814yes—
141C22×Dic83C2×Q812yes—
142C22×Dic83#4112yes—
143C4⋊Dic83—28no—
144C4⋊Dic83—48no—
145C4⋊Dic83—44no—
146Dic83⋊C4—216no—
147Dic83⋊C4—28no—
148Dic83⋊C4—48no—
149Dic83⋊C4—48no—
150Dic83⋊C4—44no—
151Dic83⋊C4—44no—
152C4×Dic83—28no—
153C4×Dic83—216no—
154C24×C83C2411yescharacteristic
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×C83—22no—
162C22×C4×C83—22no—
163C22×C4×C83—22no—
164C22×C4×C83—22no—
165C22×C4×C83—22no—
166C22×C4×C83—21no—
167C22×C4×C83—21no—
168C22×C4×C83—22no—
169C22×C4×C83—22no—
170C22×C4×C83—22no—
171C22×C4×C83—21no—
172C22×C4×C83—22no—
173C22×C4×C83—22no—
174C22×C4×C83—21no—
175C22×C4×C83—21no—
176C22×C4×C83C2×D414yes—
177C22×C4×C83C2×D412yes—
178C22×C4×C83C2×D414yes—
179C22×C4×C83C2×Q812yes—
180C22×C4×C83C2×Q812yes—
181C22×C4×C83C2×Q814yes—
182C22×C4×C83C2×Q812yes—
183C22×C4×C83C2×Q814yes—
184C4⋊C4×C83—28no—
185C4⋊C4×C83—48no—
186C23×C83C22×D411yescharacteristic
187C23×C83C22×D412yes—
188C23×C83C22×D412yes—
189C23×C83C22×Q812yes—
190C23×C83C22×Q811yescharacteristic
191C23×C83C22×Q811yescharacteristic
192C23×C83C22×Q811yescharacteristic
193C23×C83C2×#4111yescharacteristic
194C23×C83H(5,𝔽2)11yescharacteristic
195C23×C83H(5,𝔽2)11yescharacteristic
196C23×C832−1+412yes—
197C2×C4×C83—22no—
198C2×C4×C83—24no—
199C2×C4×C83—22no—
200C2×C4×C83—24no—
201C2×C4×C83—24no—
202C2×C4×C83—24no—
203C2×C4×C83—24no—
204C2×C4×C83—24no—
205C2×C4×C83—24no—
206C2×C4×C83—24no—
207C2×C4×C83—24no—
208C2×C4×C83—22no—
209C2×C4×C83—24no—
210C2×C4×C83—24no—
211C2×C4×C83—24no—
212C2×C4×C83—41no—
213C2×C4×C83—42no—
214C2×C4×C83—42no—
215C2×C4×C83—41no—
216C2×C4×C83—42no—
217C2×C4×C83—42no—
218C2×C4×C83—42no—
219C2×C4×C83—41no—
220C2×C4×C83—41no—
221C2×C4×C83—41no—
222C2×C4×C83—42no—
223C2×C4×C83—42no—
224C2×C4×C83C4⋊Q814yes—
225C2×C4×C83C4⋊Q818yescomplemented
226C2×Dic83—22no—
227C2×Dic83—24no—
228C2×Dic83—22no—
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—22no—
238C2×Dic83—24no—
239C2×Dic83—24no—
240C2×Dic83—24no—
241C2×Dic83—41no—
242C2×Dic83—42no—
243C2×Dic83—42no—
244C2×Dic83—41no—
245C2×Dic83—42no—
246C2×Dic83—42no—
247C2×Dic83—42no—
248C2×Dic83—41no—
249C2×Dic83—41no—
250C2×Dic83—41no—
251C2×Dic83—42no—
252C2×Dic83—42no—
253C2×Dic83#12014yes—
254C2×Dic83C22⋊Q818yes—
255C2×Dic83C22⋊Q814yes—
256C2×Dic83#12714yes—
257C2×Dic83C4⋊Q814yes—
258C2×Dic83C4⋊Q818yes—
259C2×Dic83C4⋊Q818yescomplemented
260Dic166—432no—
261C22×C83C2×#12011yescharacteristic
262C22×C83C2×C22⋊Q811yescharacteristic
263C22×C83C2×C22⋊Q812yes—
264C22×C83C2×#12711yescharacteristic
265C22×C83C2×C4⋊Q811yescharacteristic
266C22×C83C2×C4⋊Q812yes—
267C22×C83C2×C4⋊Q812yes—
268C22×C83C2×C4⋊Q811yescharacteristic
269C22×C83C2×C4⋊Q812yes—
270C22×C83#53511yescharacteristic
271C22×C83#53611yescharacteristic
272C22×C83#54412yes—
273C22×C83#54412yes—
274C22×C83D4212yes—
275C22×C83D4212yes—
276C22×C83#54712yes—
277C22×C83D4×Q812yes—
278C22×C83D4×Q812yes—
279C22×C83D4×Q812yes—
280C22×C83#55411yescharacteristic
281C22×C83#55411yescharacteristic
282C22×C83Q8211yescharacteristic
283C22×C83Q8211yescharacteristic
284C4×C83—24no—
285C4×C83—28no—
286C4×C83—44no—
287C4×C83—42no—
288C4×C83—81no—
289Dic83—24no—
290Dic83—28no—
291Dic83—44no—
292Dic83—42no—
293Dic83—81no—
294Dic83#39518yescomplemented
295#15143—831nomaximal, Sylow
296C2×C83C2×#39511yescharacteristic
297C2×C83#243311yescharacteristic
298C2×C83#243411yescharacteristic
299C2×C83#244012yes—
300C2×C83#244012yes—
301C2×C83#254211yescharacteristic
302C2×C83#255112yes—
303C2×C83#255112yes—
304C2×C83#256711yescharacteristic
305C2×C83#266011yescharacteristic
306C2×C83#286811yescharacteristic
307C22×C4⋊Q8—831no—
308C2×#395—832no—
309C2×#395—832no—
310C2×#389—831no—
311C2×#384—832no—
312#1444—831no—
313#1444—831no—
314#1444—831no—
315#1444D8311yescharacteristic
316C4⋊#95—832no—
317C4×#95—831no—
318C83#1514311yescharacteristic, complemented, Sylow
319C23×Q8—832no—
320C2×C4⋊Q8—1664no—
321C2×C4⋊Q8—1668no—
322C22×C4⋊C4—832no—
323C22×C4⋊C4—832no—
324C22×C4⋊C4—832no—
325C22×C4⋊C4—832no—
326C22×C4⋊C4—831no—
327C22×C4⋊C4—831no—
328C22×C4⋊C4—831no—
329C22×C4⋊C4—831no—
330C22×C4⋊C4—831no—
331C22×C4⋊C4C2×D8311yescharacteristic
332C22×C4⋊C4C2×D8312yes—
333C22×C42—831no—
334C22×C42—832no—
335#395—1668no—
336#395—1668no—
337#389—1664no—
338#384—1668no—
339C2×#95—832no—
340C2×#95—831no—
341C2×#95—832no—
342C2×#95—832no—
343C2×#95—832no—
344C2×#95—831no—
345C2×#95C2×D8312yes—
346C2×#95C2×D8311yescharacteristic
347C2×#95C2×D8311yescharacteristic
348C22×Q8—3322no—
349C22×Q8—3322no—
350C22×Q8—1664no—
351C22×Q8—1664no—
352C22×Q8—1664no—
353C22×Q8—1664no—
354C22×Q8—1664no—
355C23×C4—831no—
356C23×C4—831no—
357C23×C4—832no—
358C23×C4—832no—
359C23×C4—831no—
360C23×C4—831no—
361C23×C4C22×D8311yescharacteristic
362C23×C4C22×D8312yes—
363C23×C4C22×D8311yescharacteristic
364C23×C4C22×D8311yescharacteristic
365C23×C4C22×D8312yes—
366C4⋊Q8—16632no—
367C2×C4⋊C4—24no—
368C2×C4⋊C4—22no—
369C2×C4⋊C4—24no—
370C2×C4⋊C4—24no—
371C2×C4⋊C4—42no—
372C2×C4⋊C4—3322no—
373C2×C4⋊C4—3322no—
374C2×C4⋊C4—3322no—
375C2×C4⋊C4—838no—
376C2×C4⋊C4—834no—
377C2×C4⋊C4—1664no—
378C2×C4⋊C4—1664no—
379C2×C4⋊C4—1664no—
380C2×C4⋊C4—1662no—
381C2×C4⋊C4—1662no—
382C2×C4⋊C4—1662no—
383C2×C4⋊C4—1664no—
384C2×C4⋊C4—1664no—
385C2×C4⋊C4—1664no—
386C2×C4⋊C4—1664no—
387C2×C4⋊C4—1664no—
388C2×C4⋊C4—1664no—
389C2×C4⋊C4—1664no—
390C2×C4⋊C4—1664no—
391C2×C4⋊C4—1664no—
392C2×C4⋊C4—1662no—
393C2×C4⋊C4—1662no—
394C2×C4⋊C4—1662no—
395C2×C42—834no—
396C2×C42—838no—
397C2×C42—1664no—
398C2×C42—1664no—
399C2×C42—1664no—
400#95—22no—
401#95—24no—
402#95—838no—
403#95—834no—
404#95—1664no—
405#95—1664no—
406#95—1662no—
407#95—1664no—
408#95Dic16614yescomplemented
409C24C23×D8311yescharacteristic
410C2×Q8—3328no—
411C2×Q8—3328no—
412C2×Q8—3328no—
413C2×Q8—3328no—
414C2×Q8—3328no—
415C2×Q8—3328no—
416C2×Q8—16616no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—21no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—21no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—21no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—21no—
437C22×C4—21no—
438C22×C4—832no—
439C22×C4—834no—
440C22×C4—834no—
441C22×C4—832no—
442C22×C4—832no—
443C22×C4—834no—
444C22×C4—834no—
445C22×C4—832no—
446C22×C4—834no—
447C22×C4—834no—
448C22×C4—834no—
449C22×C4—832no—
450C22×C4—1661no—
451C22×C4—1662no—
452C22×C4—1662no—
453C22×C4—1661no—
454C22×C4—1661no—
455C22×C4—1662no—
456C22×C4—1661no—
457C22×C4—1662no—
458C22×C4—1662no—
459C22×C4—1662no—
460C22×C4—1662no—
461C22×C4—1661no—
462C22×C4—1662no—
463C22×C4—1662no—
464C22×C4—1662no—
465C22×C4—1662no—
466C22×C4—1662no—
467C22×C4—1662no—
468C22×C4—1661no—
469C22×C4—1662no—
470C22×C4—1661no—
471C22×C4—1662no—
472C22×C4D4×D8314yes—
473C22×C4D4×D8312yes—
474C22×C4D4×D8314yes—
475C22×C4Q8×D8312yes—
476C22×C4Q8×D8312yes—
477C22×C4Q8×D8314yes—
478C22×C4C2×Dic16612yes—
479C22×C4C2×Dic16614yes—
480C4⋊C4—28no—
481C4⋊C4—48no—
482C4⋊C4—3328no—
483C4⋊C4—3328no—
484C4⋊C4—3324no—
485C4⋊C4—3324no—
486C4⋊C4—3324no—
487C4⋊C4—3328no—
488C4⋊C4—16616no—
489C4⋊C4—1668no—
490C4⋊C4—1668no—
491C42—1668no—
492C42—16616no—
493C23C2×D4×D8312yes—
494C23C2×D4×D8312yes—
495C23C2×D4×D8311yescharacteristic
496C23C2×Q8×D8312yes—
497C23C2×Q8×D8311yescharacteristic
498C23C2×Q8×D8311yescharacteristic
499C23C22×Dic16611yescharacteristic
500C23#41×D8311yescharacteristic
501C23#1458183711yescharacteristic
502C23#1458183811yescharacteristic
503C23#1458183912yes—
504Q8—33232no—
505C2×C4—24no—
506C2×C4—24no—
507C2×C4—24no—
508C2×C4—24no—
509C2×C4—24no—
510C2×C4—24no—
511C2×C4—24no—
512C2×C4—22no—
513C2×C4—24no—
514C2×C4—22no—
515C2×C4—24no—
516C2×C4—24no—
517C2×C4—24no—
518C2×C4—24no—
519C2×C4—22no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—41no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—41no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—3322no—
533C2×C4—3322no—
534C2×C4—3322no—
535C2×C4—3321no—
536C2×C4—3322no—
537C2×C4—3322no—
538C2×C4—3321no—
539C2×C4—3322no—
540C2×C4—3322no—
541C2×C4—3321no—
542C2×C4—3321no—
543C2×C4—3321no—
544C2×C4—834no—
545C2×C4—838no—
546C2×C4—834no—
547C2×C4—838no—
548C2×C4—834no—
549C2×C4—834no—
550C2×C4—838no—
551C2×C4—1664no—
552C2×C4—1664no—
553C2×C4—1664no—
554C2×C4—1664no—
555C2×C4—1662no—
556C2×C4—1664no—
557C2×C4—1664no—
558C2×C4—1664no—
559C2×C4—1664no—
560C2×C4—1664no—
561C2×C4—1664no—
562C2×C4—1664no—
563C2×C4—1662no—
564C2×C4—1664no—
565C2×C4—1662no—
566C2×C4#1458179114yes—
567C2×C4#1458179118yescomplemented
568C22#120×D8311yescharacteristic
569C22C22⋊Q8×D8311yescharacteristic
570C22C22⋊Q8×D8312yes—
571C22#127×D8311yescharacteristic
572C22C4⋊Q8×D8312yes—
573C22C4⋊Q8×D8312yes—
574C22C4⋊Q8×D8311yescharacteristic
575C22C2×#1458179111yescharacteristic
576C22C2×#1458179112yes—
577C22#1912826711yescharacteristic
578C22#1912827311yescharacteristic
579C22#1912833012yes—
580C22#1912833112yes—
581C22#1912833312yes—
582C22#1912833512yes—
583C22#1912834912yes—
584C22#1912836412yes—
585C22#1912836412yes—
586C22D4×Dic16612yes—
587C22#1912840711yescharacteristic
588C22#1912841011yescharacteristic
589C22#1912843111yescharacteristic
590C22Q8×Dic16611yescharacteristic
591C4—24no—
592C4—28no—
593C4—42no—
594C4—44no—
595C4—81no—
596C4—3322no—
597C4—3324no—
598C4—838no—
599C4—6641no—
600C4—1668no—
601C4—1664no—
602C2#395×D8311yescharacteristic
603C2#2491109511yescharacteristic
604C2#2491110511yescharacteristic
605C2#2491116012yes—
606C2#2491116412yes—
607C2#2491241111yescharacteristic
608C2#2491248812yes—
609C2#2491249112yes—
610C2#2491267011yescharacteristic
611C2#2491358611yescharacteristic
612C2#2491586011yescharacteristic
subgroupG/H
center, FrattiniC24C23×D83
commutator, socle, prime radical, cosocleC24×C83C24
FittingC83×#1444C2
hypercenter#1444D83
hypocenterC83#15143
LCSG ▷ C24×C83 ▷ C83
UCSC1 ◁ C24 ◁ #1444

Complements

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

NH ≅ G/Ntype
77. C2×Dic83⋊C4Q8semidirect
82. C2×C4×Dic83Q8semidirect
88. C83⋊#95Q8semidirect
101. #95×C83Q8semidirect
225. C2×C4×C83C4⋊Q8semidirect
259. C2×Dic83C4⋊Q8semidirect
294. Dic83#395semidirect
318. C83#15143semidirect
408. #95Dic166semidirect
567. C2×C4#14581791semidirect

Maximal subgroups

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

subgroupindex
C22×#145817912
C2×#191276222
C2×#191276212
C2×#191275982
C2×#191275702
#249057282
#249057272
#249057262
Dic83⋊#952
#95×Dic832
C83×#14442
#1514383