Finite Group Databasebeta

← #35330489

Subgroups — #35330489

Order 21248.

1670 conjugacy classes of subgroups.

The subgroup lattice has 654 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19127594C211yesmaximal, characteristic
2C2×#19127594C212yesmaximal
3C2×#19127587C212yesmaximal, complemented
4C2×C4⋊C4×Dic83C211yesmaximal, characteristic, complemented
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
12C22×C4⋊Dic83C2211yescharacteristic
13C22×C4⋊Dic83C2211yescharacteristic
14C22×C4⋊Dic83C2212yes—
15C22×C22⋊Dic83C2211yescharacteristic
16C22×C22⋊Dic83C2211yescharacteristic
17C22×C22⋊Dic83C2212yes—
18C22×C4×Dic83C2212yes—
19C22×C4×Dic83C2212yes—
20C22×C4×Dic83C2211yescharacteristic
21#19127594—24no—
22#19127594—28no—
23#19127587—28no—
24C4⋊C4×Dic83—24no—
25C2×C83⋊#95C2211yescharacteristic
26C2×C83⋊#95C2212yes—
27C2×C83⋊#95C2212yes—
28C2×C83⋊#95C2212yes—
29C2×C83⋊#95C2211yescharacteristic
30C2×C83⋊#95C2212yes—
31C2×C83⋊#95C2211yescharacteristic
32C2×C83⋊#95C2212yes—
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×Dic83C2311yescharacteristic
41C23×Dic83C2311yescharacteristic
42C23×Dic83C2312yes—
43C23×Dic83C2312yes—
44C23×Dic83C2311yescharacteristic
45C23×Dic83C2311yescharacteristic
46C2×C4⋊Dic83—24no—
47C2×C4⋊Dic83—24no—
48C2×C4⋊Dic83—22no—
49C2×C4⋊Dic83—24no—
50C2×C4⋊Dic83—22no—
51C2×C4⋊Dic83—24no—
52C2×C4⋊Dic83—42no—
53C2×C22⋊Dic83—24no—
54C2×C22⋊Dic83—22no—
55C2×C22⋊Dic83—22no—
56C2×C22⋊Dic83—24no—
57C2×C22⋊Dic83—24no—
58C2×C22⋊Dic83—24no—
59C2×C22⋊Dic83—42no—
60C2×C4×Dic83—24no—
61C2×C4×Dic83—24no—
62C2×C4×Dic83—24no—
63C2×C4×Dic83—24no—
64C2×C4×Dic83—24no—
65C2×C4×Dic83—24no—
66C2×C4×Dic83—42no—
67C2×C4×Dic83D414yes—
68C83⋊#95—24no—
69C83⋊#95—24no—
70C83⋊#95—24no—
71C83⋊#95—24no—
72C83⋊#95—22no—
73C83⋊#95—24no—
74C83⋊#95—22no—
75C83⋊#95—22no—
76C83⋊#95D418yes—
77C25×C83C2311yescharacteristic
78C23×C4×C83C2312yes—
79C23×C4×C83C2311yescharacteristic
80C23×C4×C83C2311yescharacteristic
81C23×C4×C83C2312yes—
82C2×C4⋊C4×C83—22no—
83C2×C4⋊C4×C83—24no—
84C2×C4⋊C4×C83—22no—
85C2×C4⋊C4×C83—24no—
86C2×C4⋊C4×C83C2×C414yes—
87C2×C4⋊C4×C83C2×C418yes—
88C2×C22⋊C4×C83—24no—
89C2×C22⋊C4×C83—22no—
90C2×C22⋊C4×C83—22no—
91C2×C22⋊C4×C83—24no—
92C2×C22⋊C4×C83C2×C418yes—
93C2×C22⋊C4×C83C2×C414yes—
94#95×C83C2×C414yes—
95C22×Dic83—22no—
96C22×Dic83—22no—
97C22×Dic83—21no—
98C22×Dic83—21no—
99C22×Dic83—21no—
100C22×Dic83—21no—
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—21no—
114C22×Dic83—22no—
115C22×Dic83—21no—
116C22×Dic83—22no—
117C22×Dic83—22no—
118C22×Dic83—22no—
119C22×Dic83—21no—
120C22×Dic83—21no—
121C22×Dic83—21no—
122C22×Dic83—21no—
123C22×Dic83C2×D414yes—
124C22×Dic83C2×D412yes—
125C22×Dic83C2×D414yes—
126C22×Dic83C2×D412yes—
127C22×Dic83#4114yes—
128C22×Dic83#4114yes—
129C4⋊Dic83—28no—
130C4⋊Dic83—48no—
131C4⋊Dic83—44no—
132C22⋊Dic83—48no—
133C22⋊Dic83—44no—
134C22⋊Dic83—44no—
135C4×Dic83—28no—
136C4×Dic83—48no—
137C4×Dic83—48no—
138C24×C83—21no—
139C24×C83—22no—
140C24×C83—21no—
141C24×C83—22no—
142C24×C83—22no—
143C24×C83—22no—
144C24×C83—21no—
145C24×C83—21no—
146C24×C83—21no—
147C24×C83—21no—
148C24×C83C22×C412yes—
149C24×C83C2411yescharacteristic
150C22×C4×C83—22no—
151C22×C4×C83—22no—
152C22×C4×C83—22no—
153C22×C4×C83—22no—
154C22×C4×C83—22no—
155C22×C4×C83—22no—
156C22×C4×C83—22no—
157C22×C4×C83—21no—
158C22×C4×C83—22no—
159C22×C4×C83—22no—
160C22×C4×C83—22no—
161C22×C4×C83—21no—
162C22×C4×C83—22no—
163C22×C4×C83—22no—
164C22×C4×C83—22no—
165C22×C4×C83—22no—
166C22×C4×C83C22×C414yes—
167C22×C4×C83C22×C412yes—
168C22×C4×C83C22×C412yes—
169C22×C4×C83C22×C414yes—
170C22×C4×C83C2×D414yes—
171C22×C4×C83C2×D412yes—
172C22×C4×C83#4112yes—
173C22×C4×C83#4114yes—
174C4⋊C4×C83—28no—
175C4⋊C4×C83—216no—
176C22⋊C4×C83—28no—
177C22⋊C4×C83—216no—
178C23×C83—22no—
179C23×C83—24no—
180C23×C83—24no—
181C23×C83—22no—
182C23×C83—22no—
183C23×C83—41no—
184C23×C83—41no—
185C23×C83—42no—
186C23×C83—41no—
187C23×C83—41no—
188C23×C83—42no—
189C23×C83—41no—
190C23×C83—41no—
191C23×C83—41no—
192C23×C83—41no—
193C23×C83—42no—
194C23×C83—42no—
195C23×C83—42no—
196C23×C83—42no—
197C23×C83—42no—
198C23×C83—42no—
199C23×C83—42no—
200C23×C83—42no—
201C23×C83C23×C411yescharacteristic
202C23×C83C22×D411yescharacteristic
203C23×C83C22×D412yes—
204C23×C83C2×#4112yes—
205C23×C83C2×#4112yes—
206C23×C83C2×#4111yescharacteristic
207C23×C83C2×#4112yes—
208C23×C83H(5,𝔽2)11yescharacteristic
209C23×C832−1+411yescharacteristic
210C23×C832−1+411yescharacteristic
211C23×C832−1+411yescharacteristic
212C2×C4×C83—24no—
213C2×C4×C83—24no—
214C2×C4×C83—24no—
215C2×C4×C83—22no—
216C2×C4×C83—22no—
217C2×C4×C83—24no—
218C2×C4×C83—24no—
219C2×C4×C83—24no—
220C2×C4×C83—24no—
221C2×C4×C83—24no—
222C2×C4×C83—22no—
223C2×C4×C83—24no—
224C2×C4×C83—24no—
225C2×C4×C83—24no—
226C2×C4×C83—24no—
227C2×C4×C83—42no—
228C2×C4×C83—41no—
229C2×C4×C83—41no—
230C2×C4×C83—42no—
231C2×C4×C83—42no—
232C2×C4×C83—42no—
233C2×C4×C83—42no—
234C2×C4×C83C4×D418yes—
235C2×C4×C83C4×D414yes—
236C2×Dic83—22no—
237C2×Dic83—22no—
238C2×Dic83—22no—
239C2×Dic83—22no—
240C2×Dic83—24no—
241C2×Dic83—24no—
242C2×Dic83—24no—
243C2×Dic83—24no—
244C2×Dic83—24no—
245C2×Dic83—24no—
246C2×Dic83—24no—
247C2×Dic83—24no—
248C2×Dic83—24no—
249C2×Dic83—24no—
250C2×Dic83—42no—
251C2×Dic83—42no—
252C2×Dic83—42no—
253C2×Dic83—42no—
254C2×Dic83—42no—
255C2×Dic83—42no—
256C2×Dic83—42no—
257C2×Dic83—42no—
258C2×Dic83—41no—
259C2×Dic83—41no—
260C2×Dic83—41no—
261C2×Dic83—41no—
262C2×Dic83—41no—
263C2×Dic83—41no—
264C2×Dic83—42no—
265C2×Dic83—42no—
266C2×Dic83—42no—
267C2×Dic83—42no—
268C2×Dic83#12314yes—
269C2×Dic83#12714yes—
270C22×C83—42no—
271C22×C83—42no—
272C22×C83—44no—
273C22×C83—44no—
274C22×C83—42no—
275C22×C83—81no—
276C22×C83—81no—
277C22×C83—81no—
278C22×C83—82no—
279C22×C83—82no—
280C22×C83—81no—
281C22×C83C2×C4×D411yescharacteristic
282C22×C83C2×C4×D412yes—
283C22×C83C4×#4112yes—
284C22×C83#51911yescharacteristic
285C22×C83#52011yescharacteristic
286C22×C83C2×#12311yescharacteristic
287C22×C83#52511yescharacteristic
288C22×C83#52912yes—
289C22×C83C2×#12711yescharacteristic
290C22×C83#53211yescharacteristic
291C22×C83#53212yes—
292C22×C83#53611yescharacteristic
293C22×C83#53711yescharacteristic
294C22×C83#54111yescharacteristic
295C22×C83#54211yescharacteristic
296C22×C83#54612yes—
297C22×C83#54712yes—
298C22×C83#54712yes—
299C22×C83#54812yes—
300C22×C83#55112yes—
301C22×C83#55212yes—
302C22×C83#55212yes—
303C22×C83#55612yes—
304C4×C83—28no—
305C4×C83—24no—
306C4×C83—44no—
307C4×C83—42no—
308Dic83—24no—
309Dic83—24no—
310Dic83—44no—
311Dic83—44no—
312Dic83—81no—
313Dic83—81no—
314#14231—831nomaximal, Sylow
315C2×C83—82no—
316C2×C83C4×#12311yescharacteristic
317C2×C83C4×#12711yescharacteristic
318C2×C83#232311yescharacteristic
319C2×C83#235912yes—
320C2×C83#236212yes—
321C2×C83#249811yescharacteristic
322C2×C83#250111yescharacteristic
323C2×C83#251912yes—
324C2×C83#252112yes—
325C2×C83#254011yescharacteristic
326C2×C83#260411yescharacteristic
327C2×#399D8311yescharacteristic
328C2×#388—832no—
329C2×#388—831no—
330C2×#387—832no—
331C2×C4×C4⋊C4—831no—
332#1445—831no—
333#1444—831no—
334#1439—831no—
335#1437—832no—
336C4⋊#95—832no—
337C4×#95—831no—
338C83#1423111yescharacteristic, complemented, Sylow
339C22×C4⋊C4—832no—
340C22×C4⋊C4—831no—
341C22×C4⋊C4—831no—
342C22×C4⋊C4C2×D8312yes—
343C22×C4⋊C4C2×D8311yescharacteristic
344C22×C22⋊C4—832no—
345C22×C22⋊C4—831no—
346C22×C22⋊C4—831no—
347C22×C22⋊C4C2×D8312yes—
348C22×C22⋊C4C2×D8311yescharacteristic
349C22×C42—832no—
350C22×C42—831no—
351C22×C42—832no—
352#399Dic8318yescomplemented
353#388—1668no—
354#388—1664no—
355#387—1668no—
356C4×C4⋊C4—1664no—
357C2×#95—832no—
358C2×#95—831no—
359C2×#95—831no—
360C2×#95—832no—
361C2×#95—832no—
362C2×#95—832no—
363C2×#95—832no—
364C2×#95—832no—
365C2×#95—831no—
366C2×#95C2×D8311yescharacteristic
367C25C22×D8311yescharacteristic
368C23×C4—831no—
369C23×C4—831no—
370C23×C4—832no—
371C23×C4—832no—
372C23×C4—831no—
373C23×C4—831no—
374C23×C4C22×D8311yescharacteristic
375C23×C4C22×D8312yes—
376C23×C4C22×D8311yescharacteristic
377C23×C4C22×D8312yes—
378C2×C4⋊C4—22no—
379C2×C4⋊C4—24no—
380C2×C4⋊C4—22no—
381C2×C4⋊C4—24no—
382C2×C4⋊C4—3322no—
383C2×C4⋊C4—1664no—
384C2×C4⋊C4—1662no—
385C2×C4⋊C4—1664no—
386C2×C4⋊C4—1664no—
387C2×C4⋊C4—1664no—
388C2×C4⋊C4—1662no—
389C2×C4⋊C4C2×Dic8318yes—
390C2×C4⋊C4C2×Dic8314yes—
391C2×C22⋊C4—24no—
392C2×C22⋊C4—22no—
393C2×C22⋊C4—22no—
394C2×C22⋊C4—24no—
395C2×C22⋊C4—3322no—
396C2×C22⋊C4—1664no—
397C2×C22⋊C4—1662no—
398C2×C22⋊C4—1662no—
399C2×C22⋊C4—1664no—
400C2×C22⋊C4—1664no—
401C2×C22⋊C4—1664no—
402C2×C22⋊C4C2×Dic8318yes—
403C2×C22⋊C4C2×Dic8314yes—
404C2×C42—3322no—
405C2×C42—834no—
406C2×C42—1664no—
407C2×C42—1664no—
408C2×C42—1664no—
409C2×C42—1664no—
410C2×C42—1664no—
411C2×C42—1664no—
412#95—838no—
413#95—1664no—
414#95—1664no—
415#95—1664no—
416#95—1664no—
417#95—1662no—
418#95—1664no—
419#95—1662no—
420#95—1662no—
421#95C2×Dic8314yes—
422C24—22no—
423C24—21no—
424C24—22no—
425C24—22no—
426C24—21no—
427C24—22no—
428C24—21no—
429C24—21no—
430C24—21no—
431C24—21no—
432C24C22×Dic8312yes—
433C24C23×D8311yescharacteristic
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—21no—
444C22×C4—22no—
445C22×C4—21no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—834no—
451C22×C4—832no—
452C22×C4—832no—
453C22×C4—834no—
454C22×C4—834no—
455C22×C4—834no—
456C22×C4—1661no—
457C22×C4—1662no—
458C22×C4—1662no—
459C22×C4—1661no—
460C22×C4—1662no—
461C22×C4—1662no—
462C22×C4—1661no—
463C22×C4—1661no—
464C22×C4—1661no—
465C22×C4—1661no—
466C22×C4—1662no—
467C22×C4—1662no—
468C22×C4—1662no—
469C22×C4—1662no—
470C22×C4—1662no—
471C22×C4—1662no—
472C22×C4—1662no—
473C22×C4—1662no—
474C22×C4—1662no—
475C22×C4—1662no—
476C22×C4—1662no—
477C22×C4—1662no—
478C22×C4—1662no—
479C22×C4—1662no—
480C22×C4—1661no—
481C22×C4—1661no—
482C22×C4—1661no—
483C22×C4—1661no—
484C22×C4C22×Dic8312yes—
485C22×C4C22×Dic8314yes—
486C22×C4C22×Dic8312yes—
487C22×C4C22×Dic8314yes—
488C22×C4D4×D8314yes—
489C22×C4D4×D8312yes—
490C22×C4#1305058912yes—
491C22×C4#1305058914yes—
492C4⋊C4—28no—
493C4⋊C4—216no—
494C4⋊C4—3328no—
495C4⋊C4—3324no—
496C4⋊C4—1668no—
497C22⋊C4—28no—
498C22⋊C4—216no—
499C22⋊C4—3324no—
500C22⋊C4—3324no—
501C22⋊C4—3328no—
502C42—3328no—
503C42—3328no—
504C42—1668no—
505C23—22no—
506C23—24no—
507C23—24no—
508C23—22no—
509C23—22no—
510C23—41no—
511C23—42no—
512C23—42no—
513C23—42no—
514C23—42no—
515C23—42no—
516C23—42no—
517C23—42no—
518C23—42no—
519C23—41no—
520C23—41no—
521C23—42no—
522C23—41no—
523C23—42no—
524C23—41no—
525C23—41no—
526C23—41no—
527C23—41no—
528C23C23×Dic8311yescharacteristic
529C23C2×D4×D8312yes—
530C23C2×D4×D8311yescharacteristic
531C23#41×D8312yes—
532C23#41×D8312yes—
533C23C2×#1305058911yescharacteristic
534C23C2×#1305058912yes—
535C23#1458183811yescharacteristic
536C23#1458183911yescharacteristic
537C23#1458184011yescharacteristic
538C23#1458184011yescharacteristic
539C2×C4—24no—
540C2×C4—24no—
541C2×C4—24no—
542C2×C4—24no—
543C2×C4—24no—
544C2×C4—24no—
545C2×C4—24no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—24no—
549C2×C4—22no—
550C2×C4—24no—
551C2×C4—24no—
552C2×C4—24no—
553C2×C4—24no—
554C2×C4—42no—
555C2×C4—41no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—41no—
561C2×C4—3322no—
562C2×C4—3322no—
563C2×C4—3322no—
564C2×C4—3322no—
565C2×C4—3322no—
566C2×C4—3322no—
567C2×C4—3322no—
568C2×C4—3322no—
569C2×C4—3322no—
570C2×C4—3322no—
571C2×C4—3322no—
572C2×C4—3322no—
573C2×C4—3321no—
574C2×C4—3321no—
575C2×C4—3321no—
576C2×C4—3321no—
577C2×C4—3321no—
578C2×C4—3321no—
579C2×C4—834no—
580C2×C4—834no—
581C2×C4—1664no—
582C2×C4—1664no—
583C2×C4—1664no—
584C2×C4—1664no—
585C2×C4—1664no—
586C2×C4—1664no—
587C2×C4—1664no—
588C2×C4—1664no—
589C2×C4—1664no—
590C2×C4—1664no—
591C2×C4—1662no—
592C2×C4—1662no—
593C2×C4—1662no—
594C2×C4—1662no—
595C2×C4D4×Dic8318yes—
596C2×C4D4×Dic8314yes—
597C22—42no—
598C22—44no—
599C22—44no—
600C22—42no—
601C22—42no—
602C22—82no—
603C22—81no—
604C22—81no—
605C22—82no—
606C22—81no—
607C22—81no—
608C22C2×D4×Dic8311yescharacteristic
609C22C2×D4×Dic8312yes—
610C22#41×Dic8312yes—
611C22#1912817911yescharacteristic
612C22#1912818411yescharacteristic
613C22#123×D8311yescharacteristic
614C22#1912820911yescharacteristic
615C22#1912822812yes—
616C22#127×D8311yescharacteristic
617C22#1912824912yes—
618C22#1912824911yescharacteristic
619C22#1912827411yescharacteristic
620C22#1912828111yescharacteristic
621C22#1912831111yescharacteristic
622C22#1912832411yescharacteristic
623C22#1912834312yes—
624C22#1912834912yes—
625C22#1912835312yes—
626C22#1912836112yes—
627C22#1912837412yes—
628C22#1912838512yes—
629C22#1912838812yes—
630C22#1912842212yes—
631C4—28no—
632C4—24no—
633C4—44no—
634C4—42no—
635C4—3324no—
636C4—3324no—
637C4—6641no—
638C4—6641no—
639C4—1664no—
640C4—1664no—
641C2—82no—
642C2#123×Dic8311yescharacteristic
643C2#127×Dic8311yescharacteristic
644C2#2491014711yescharacteristic
645C2#2491045912yes—
646C2#2491049412yes—
647C2#2491190011yescharacteristic
648C2#2491192911yescharacteristic
649C2#2491213112yes—
650C2#2491216112yes—
651C2#2491239811yescharacteristic
652C2#2491310211yescharacteristic
subgroupG/H
center, FrattiniC24C23×D83
commutatorC23×C83C23×C4
FittingC2×#399×C83C2
socle, cosocleC24×C83C24
prime radicalC25×C83C23
hypercenterC2×#399D83
hypocenterC83#14231
LCSG ▷ C23×C83 ▷ C83
UCSC1 ◁ C24 ◁ C2×#399

Complements

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

NH ≅ G/Ntype
3. C2×#19127587C2semidirect
4. C2×C4⋊C4×Dic83C2semidirect
5. #24905731C2semidirect
6. #24905728C2semidirect
9. #24905692C2semidirect
10. #95×Dic83C2semidirect
38. #399×C83C4semidirect
338. C83#14231semidirect
352. #399Dic83semidirect

Maximal subgroups

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

subgroupindex
C2×#191275942
C2×#191275872
C2×C4⋊C4×Dic832
#249057312
#249057282
#249057082
#249057002
#249056922
#95×Dic832
C2×#399×C832
#1423183