Finite Group Databasebeta

← #35326232

Subgroups — #35326232

Order 21248.

1966 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×#19127636C211yesmaximal, characteristic
2C2×C4⋊C4×Dic83C211yesmaximal, characteristic, complemented
3C2×#19127552C212yesmaximal, complemented
4C2×#19127548C211yesmaximal, characteristic
5C2×#19127548C212yesmaximal
6#24905729C212yesmaximal, complemented
7#24905727C211yesmaximal, characteristic, complemented
8#24905709C211yesmaximal, characteristic
9#24905707C212yesmaximal
10#95×Dic83C211yesmaximal, characteristic, complemented
11C83×#1445C211yesmaximal, characteristic, complemented
12C23×C4×D83C2211yescharacteristic
13C22×C4⋊Dic83C2211yescharacteristic
14C22×Dic83⋊C4C2212yes—
15C22×Dic83⋊C4C2212yes—
16C22×(C2×D83)⋊C4C2211yescharacteristic
17C22×(C2×D83)⋊C4C2212yes—
18C22×(C2×D83)⋊C4C2212yes—
19C22×(C2×D83)⋊C4C2211yescharacteristic
20C22×C4×Dic83C2211yescharacteristic
21C22×C4×Dic83C2212yes—
22C22×C4×Dic83C2212yes—
23C22×C4×Dic83C2211yescharacteristic
24#19127636C418yescomplemented
25C4⋊C4×Dic83—24no—
26#19127552—28no—
27#19127548—28no—
28#19127548—24no—
29C2×C83⋊#95C2211yescharacteristic
30C2×C83⋊#95C2212yes—
31C2×C83⋊#95C2212yes—
32C2×C83⋊#95C2212yes—
33C2×C83⋊#95C2212yes—
34C2×C83⋊#95C2211yescharacteristic
35C22×C4⋊C4×C83C2211yescharacteristic
36C2×#95×C83C2212yes—
37C2×#95×C83C2211yescharacteristic
38C2×#95×C83C2211yescharacteristic
39C2×#95×C83C2212yes—
40C24×D83C2311yescharacteristic
41C23×Dic83C2311yescharacteristic
42C23×Dic83C2311yescharacteristic
43C23×Dic83C2312yes—
44C23×Dic83C2312yes—
45C23×Dic83C2311yescharacteristic
46C22×C4×D83—24no—
47C22×C4×D83—22no—
48C22×C4×D83—24no—
49C22×C4×D83—22no—
50C22×C4×D83Q814yes—
51C2×C4⋊Dic83—24no—
52C2×C4⋊Dic83—22no—
53C2×Dic83⋊C4—24no—
54C2×Dic83⋊C4—24no—
55C2×Dic83⋊C4—24no—
56C2×Dic83⋊C4—24no—
57C2×Dic83⋊C4—24no—
58C2×Dic83⋊C4C2×C418yescomplemented
59C2×(C2×D83)⋊C4—22no—
60C2×(C2×D83)⋊C4—24no—
61C2×(C2×D83)⋊C4—22no—
62C2×(C2×D83)⋊C4—24no—
63C2×(C2×D83)⋊C4—24no—
64C2×(C2×D83)⋊C4—24no—
65C2×(C2×D83)⋊C4—24no—
66C2×(C2×D83)⋊C4—22no—
67C2×(C2×D83)⋊C4—22no—
68C2×(C2×D83)⋊C4—22no—
69C2×(C2×D83)⋊C4C2×C418yes—
70C2×(C2×D83)⋊C4C2×C414yes—
71C2×C4×Dic83—24no—
72C2×C4×Dic83—24no—
73C2×C4×Dic83—24no—
74C2×C4×Dic83—24no—
75C2×C4×Dic83—24no—
76C2×C4×Dic83—24no—
77C2×C4×Dic83—22no—
78C2×C4×Dic83—24no—
79C2×C4×Dic83D418yes—
80C2×C4×Dic83D414yes—
81C83⋊#95—24no—
82C83⋊#95—22no—
83C83⋊#95—24no—
84C83⋊#95—24no—
85C83⋊#95—24no—
86C83⋊#95C2×C414yescomplemented
87C23×C4×C83C2312yes—
88C23×C4×C83C2312yes—
89C23×C4×C83C2311yescharacteristic
90C23×C4×C83C2311yescharacteristic
91C23×C4×C83C2311yescharacteristic
92C2×C4⋊C4×C83—22no—
93C2×C4⋊C4×C83—22no—
94C2×C4⋊C4×C83C2×C414yescomplemented
95#95×C83—24no—
96#95×C83—24no—
97#95×C83—22no—
98#95×C83—22no—
99C23×D83—21no—
100C23×D83—22no—
101C23×D83—22no—
102C23×D83—21no—
103C23×D83—22no—
104C23×D83—22no—
105C23×D83—21no—
106C23×D83—21no—
107C23×D83C22×C412yes—
108C23×D83C2×Q812yes—
109C23×D83#4112yes—
110C22×Dic83—22no—
111C22×Dic83—22no—
112C22×Dic83—22no—
113C22×Dic83—22no—
114C22×Dic83—21no—
115C22×Dic83—22no—
116C22×Dic83—22no—
117C22×Dic83—22no—
118C22×Dic83—22no—
119C22×Dic83—22no—
120C22×Dic83—22no—
121C22×Dic83—21no—
122C22×Dic83—22no—
123C22×Dic83—21no—
124C22×Dic83—21no—
125C22×Dic83—22no—
126C22×Dic83—22no—
127C22×Dic83—22no—
128C22×Dic83C22×C412yes—
129C22×Dic83C22×C414yes—
130C22×Dic83C2×D412yes—
131C22×Dic83C2×D414yes—
132C22×Dic83C2×D414yes—
133C22×Dic83C2×D412yes—
134C22×Dic83C2×Q812yes—
135C22×Dic83#4114yes—
136C22×Dic83#4114yes—
137C22×Dic83#4114yes—
138C22×Dic83#4112yes—
139C2×C4×D83—28no—
140C2×C4×D83—216no—
141C2×C4×D83—48no—
142C2×C4×D83—44no—
143C2×C4×D83—44no—
144C4⋊Dic83—28no—
145Dic83⋊C4—216no—
146Dic83⋊C4—216no—
147(C2×D83)⋊C4—28no—
148(C2×D83)⋊C4—216no—
149(C2×D83)⋊C4—28no—
150(C2×D83)⋊C4—216no—
151C4×Dic83—216no—
152C4×Dic83—28no—
153C4×Dic83—44no—
154C4×Dic83—48no—
155C24×C83C2411yescharacteristic
156C22×C4×C83—22no—
157C22×C4×C83—22no—
158C22×C4×C83—22no—
159C22×C4×C83—21no—
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×C83—21no—
167C22×C4×C83—22no—
168C22×C4×C83—22no—
169C22×C4×C83—22no—
170C22×C4×C83—22no—
171C22×C4×C83—22no—
172C22×C4×C83—22no—
173C22×C4×C83—22no—
174C22×C4×C83—22no—
175C22×C4×C83—21no—
176C22×C4×C83—22no—
177C22×C4×C83—22no—
178C22×C4×C83C22×C414yes—
179C22×C4×C83C22×C412yes—
180C22×C4×C83C2×D414yes—
181C22×C4×C83C2×D412yes—
182C22×C4×C83C2×Q812yes—
183C22×C4×C83#4112yes—
184C22×C4×C83#4112yes—
185C4⋊C4×C83—28no—
186C23×C83C23×C411yescharacteristic
187C23×C83C22×D412yes—
188C23×C83C22×D411yescharacteristic
189C23×C83C22×Q811yescharacteristic
190C23×C83C2×#4112yes—
191C23×C83C2×#4111yescharacteristic
192C23×C83C2×#4112yes—
193C23×C83C2×#4111yescharacteristic
194C23×C83C2×#4112yes—
195C23×C83C2×#4111yescharacteristic
196C23×C83C2×#4111yescharacteristic
197C22×D83—24no—
198C22×D83—22no—
199C22×D83—24no—
200C22×D83—22no—
201C22×D83—22no—
202C22×D83—24no—
203C22×D83—24no—
204C22×D83—22no—
205C22×D83—22no—
206C22×D83—24no—
207C22×D83—22no—
208C22×D83—24no—
209C22×D83—42no—
210C22×D83—41no—
211C22×D83—41no—
212C22×D83—42no—
213C22×D83—42no—
214C22×D83—42no—
215C22×D83—42no—
216C22×D83—41no—
217C22×D83—42no—
218C22×D83—41no—
219C22×D83C4×Q814yes—
220C2×C4×C83—24no—
221C2×C4×C83—22no—
222C2×C4×C83—22no—
223C2×C4×C83—24no—
224C2×C4×C83—24no—
225C2×C4×C83—24no—
226C2×C4×C83—24no—
227C2×C4×C83—22no—
228C2×C4×C83—24no—
229C2×C4×C83—24no—
230C2×C4×C83—24no—
231C2×C4×C83—22no—
232C2×C4×C83—24no—
233C2×C4×C83—24no—
234C2×C4×C83—42no—
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×C83—42no—
243C2×C4×C83—41no—
244C2×C4×C83—41no—
245C2×C4×C83—42no—
246C2×C4×C83—42no—
247C2×C4×C83—42no—
248C2×C4×C83C4×D414yes—
249C2×Dic83—24no—
250C2×Dic83—24no—
251C2×Dic83—24no—
252C2×Dic83—24no—
253C2×Dic83—24no—
254C2×Dic83—24no—
255C2×Dic83—22no—
256C2×Dic83—24no—
257C2×Dic83—22no—
258C2×Dic83—22no—
259C2×Dic83—24no—
260C2×Dic83—24no—
261C2×Dic83—24no—
262C2×Dic83—22no—
263C2×Dic83—42no—
264C2×Dic83—42no—
265C2×Dic83—42no—
266C2×Dic83—42no—
267C2×Dic83—42no—
268C2×Dic83—41no—
269C2×Dic83—42no—
270C2×Dic83—42no—
271C2×Dic83—41no—
272C2×Dic83C4×D418yes—
273C2×Dic83C22⋊Q814yes—
274C2×Dic83C22⋊Q818yes—
275C2×Dic83#12318yes—
276C2×Dic83#12314yes—
277C2×Dic83#12714yes—
278C4×D83—416no—
279C22×C83C2×C4×D412yes—
280C22×C83C2×C4×D411yescharacteristic
281C22×C83C2×C4×Q811yescharacteristic
282C22×C83C4×#4112yes—
283C22×C83C4×#4111yescharacteristic
284C22×C83C2×C22⋊Q811yescharacteristic
285C22×C83C2×C22⋊Q812yes—
286C22×C83C2×#12312yes—
287C22×C83C2×#12311yescharacteristic
288C22×C83#52512yes—
289C22×C83#52511yescharacteristic
290C22×C83#52511yescharacteristic
291C22×C83#52512yes—
292C22×C83#52912yes—
293C22×C83#52911yescharacteristic
294C22×C83#52912yes—
295C22×C83#52912yes—
296C22×C83#52912yes—
297C22×C83C2×#12711yescharacteristic
298C22×C83#53212yes—
299C22×C83#53211yescharacteristic
300C22×C83#53311yescharacteristic
301C22×C83#53312yes—
302C2×D83—28no—
303C2×D83—24no—
304C2×D83—42no—
305C2×D83—42no—
306C2×D83—44no—
307C2×D83—42no—
308C2×D83—42no—
309C2×D83—44no—
310C2×D83—42no—
311C2×D83—42no—
312C2×D83—44no—
313C4×C83—24no—
314C4×C83—44no—
315C4×C83—42no—
316C4×C83—44no—
317C4×C83—42no—
318Dic83—28no—
319Dic83—42no—
320Dic83—42no—
321Dic83—44no—
322Dic83#39918yescomplemented
323C4×#399—831nomaximal, Sylow
324C2×C83C4×#12311yescharacteristic
325C2×C83C4×#12312yes—
326C2×C83C4×C22⋊Q812yes—
327C2×C83C4×C22⋊Q811yescharacteristic
328C2×C83C4×#12711yescharacteristic
329C2×C83C2×#39911yescharacteristic
330C2×C83#239512yes—
331C2×C83#239511yescharacteristic
332C2×C83#239611yescharacteristic
333C2×C83#239612yes—
334C2×C83#239711yescharacteristic
335D83—44no—
336C2×#399—831no—
337C2×C4×C4⋊C4—832no—
338C2×C4×C4⋊C4—831no—
339C2×C4×C22⋊C4—832no—
340C2×C4×C22⋊C4—831no—
341#1445—832no—
342#1445D8311yescharacteristic, complemented
343#1444—831no—
344#1439—832no—
345#1439—831no—
346C4×#95—831no—
347C83C4×#39911yescharacteristic, complemented, Sylow
348C24×C4—831no—
349C22×C4⋊C4—832no—
350C22×C4⋊C4—832no—
351C22×C4⋊C4—831no—
352C22×C4⋊C4C2×D8311yescharacteristic
353C22×C22⋊C4—832no—
354C22×C22⋊C4—831no—
355C22×C22⋊C4—831no—
356C22×C22⋊C4—832no—
357C22×C42—832no—
358C22×C42—831no—
359C22×C42—832no—
360C22×C42—831no—
361#399—838no—
362C4×C4⋊C4—1668no—
363C4×C4⋊C4—1664no—
364C4×C22⋊C4—1668no—
365C4×C22⋊C4—1664no—
366C2×#95—831no—
367C2×#95—832no—
368C2×#95—832no—
369C2×#95—832no—
370C2×#95—832no—
371C2×#95—831no—
372C2×#95C2×D8311yescharacteristic
373C2×#95C2×D8312yes—
374C2×#95C2×D8311yescharacteristic
375C2×#95C2×D8312yes—
376C25—831no—
377C23×C4—831no—
378C23×C4—832no—
379C23×C4—831no—
380C23×C4—832no—
381C23×C4—831no—
382C23×C4—834no—
383C23×C4—1664no—
384C23×C4—1662no—
385C23×C4—1664no—
386C23×C4—1662no—
387C23×C4C22×D8312yes—
388C23×C4C22×D8312yes—
389C23×C4C22×D8311yescharacteristic
390C23×C4C22×D8311yescharacteristic
391C23×C4C22×D8311yescharacteristic
392C2×C4⋊C4—22no—
393C2×C4⋊C4—22no—
394C2×C4⋊C4—838no—
395C2×C4⋊C4—1664no—
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⋊C4C4×D8314yescomplemented
403C2×C22⋊C4—838no—
404C2×C22⋊C4—834no—
405C2×C22⋊C4—1664no—
406C2×C22⋊C4—1662no—
407C2×C22⋊C4—1664no—
408C2×C22⋊C4—1664no—
409C2×C22⋊C4—1664no—
410C2×C22⋊C4—1662no—
411C2×C22⋊C4—1664no—
412C2×C22⋊C4—1662no—
413C2×C22⋊C4—1662no—
414C2×C22⋊C4—1662no—
415C2×C42—838no—
416C2×C42—834no—
417C2×C42—1664no—
418C2×C42—1662no—
419C2×C42—1664no—
420C2×C42—1664no—
421C2×C42—1664no—
422C2×C42—1664no—
423C2×C42—1664no—
424C2×C42—1664no—
425#95—24no—
426#95—22no—
427#95—24no—
428#95—22no—
429#95—834no—
430#95—1664no—
431#95—1662no—
432#95—1664no—
433#95—1664no—
434#95—1664no—
435C24—832no—
436C24—832no—
437C24—832no—
438C24—1661no—
439C24—1662no—
440C24—1662no—
441C24—1662no—
442C24—1662no—
443C24—1661no—
444C24—1661no—
445C24—1661no—
446C24C23×D8311yescharacteristic
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—21no—
451C22×C4—22no—
452C22×C4—21no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—21no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—21no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—3328no—
470C22×C4—3324no—
471C22×C4—3324no—
472C22×C4—834no—
473C22×C4—832no—
474C22×C4—832no—
475C22×C4—834no—
476C22×C4—832no—
477C22×C4—834no—
478C22×C4—834no—
479C22×C4—834no—
480C22×C4—832no—
481C22×C4—832no—
482C22×C4—834no—
483C22×C4—1668no—
484C22×C4—16616no—
485C22×C4—1662no—
486C22×C4—1661no—
487C22×C4—1662no—
488C22×C4—1662no—
489C22×C4—1662no—
490C22×C4—1662no—
491C22×C4—1662no—
492C22×C4—1662no—
493C22×C4—1661no—
494C22×C4—1662no—
495C22×C4—1662no—
496C22×C4—1661no—
497C22×C4—1661no—
498C22×C4—1662no—
499C22×C4—1662no—
500C22×C4—1662no—
501C22×C4—1662no—
502C22×C4—1662no—
503C22×C4C2×C4×D8314yes—
504C22×C4C2×C4×D8312yes—
505C22×C4D4×D8314yes—
506C22×C4D4×D8312yes—
507C22×C4Q8×D8312yes—
508C22×C4#1305058912yes—
509C22×C4#1305059112yes—
510C4⋊C4—28no—
511C4⋊C4—16616no—
512C4⋊C4—16616no—
513C4⋊C4—1668no—
514C22⋊C4—16616no—
515C22⋊C4—16616no—
516C22⋊C4—1668no—
517C22⋊C4—1668no—
518C42—3324no—
519C42—3328no—
520C42—1668no—
521C42—16616no—
522C23—3322no—
523C23—3321no—
524C23—3321no—
525C23—3322no—
526C23—3322no—
527C23—3322no—
528C23—3322no—
529C23—3321no—
530C23—3322no—
531C23—3321no—
532C23—834no—
533C23—1664no—
534C23—1662no—
535C23—1664no—
536C23—1662no—
537C23—1662no—
538C23—1664no—
539C23—1664no—
540C23—1662no—
541C23—1662no—
542C23—1664no—
543C23—1662no—
544C23—1664no—
545C23C22×C4×D8311yescharacteristic
546C23C2×D4×D8312yes—
547C23C2×D4×D8311yescharacteristic
548C23C2×Q8×D8311yescharacteristic
549C23#41×D8312yes—
550C23#41×D8311yescharacteristic
551C23#41×D8312yes—
552C23#41×D8312yes—
553C23#41×D8311yescharacteristic
554C23C2×#1305058911yescharacteristic
555C23C2×#1305059111yescharacteristic
556C2×C4—24no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—24no—
560C2×C4—24no—
561C2×C4—24no—
562C2×C4—24no—
563C2×C4—22no—
564C2×C4—24no—
565C2×C4—24no—
566C2×C4—24no—
567C2×C4—22no—
568C2×C4—24no—
569C2×C4—24no—
570C2×C4—42no—
571C2×C4—41no—
572C2×C4—42no—
573C2×C4—41no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—41no—
579C2×C4—42no—
580C2×C4—41no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—33216no—
585C2×C4—3322no—
586C2×C4—3322no—
587C2×C4—3322no—
588C2×C4—3322no—
589C2×C4—3322no—
590C2×C4—3322no—
591C2×C4—3321no—
592C2×C4—3322no—
593C2×C4—3321no—
594C2×C4—834no—
595C2×C4—838no—
596C2×C4—838no—
597C2×C4—834no—
598C2×C4—834no—
599C2×C4—838no—
600C2×C4—1664no—
601C2×C4—1662no—
602C2×C4—1662no—
603C2×C4—1664no—
604C2×C4—1664no—
605C2×C4—1662no—
606C2×C4—1664no—
607C2×C4—1662no—
608C2×C4—1664no—
609C2×C4—1664no—
610C2×C4—1664no—
611C2×C4—1664no—
612C2×C4—1664no—
613C2×C4—1664no—
614C2×C4(C22⋊D83)⋊C414yes—
615C22—3322no—
616C22—3324no—
617C22—3322no—
618C22—3322no—
619C22—3324no—
620C22—3322no—
621C22—3322no—
622C22—3324no—
623C22—3322no—
624C22—1668no—
625C22—1664no—
626C22C4×D4×D8312yes—
627C22C2×(C22⋊D83)⋊C411yescharacteristic
628C22C4×Q8×D8311yescharacteristic
629C22#1912816912yes—
630C22C4×#1305059111yescharacteristic
631C22C22⋊Q8×D8311yescharacteristic
632C22C22⋊Q8×D8312yes—
633C22#123×D8312yes—
634C22#123×D8311yescharacteristic
635C22#1912820912yes—
636C22#1912820911yescharacteristic
637C22#1912821411yescharacteristic
638C22#1912821412yes—
639C22#1912822812yes—
640C22#1912823011yescharacteristic
641C22#1912823412yes—
642C22#1912823412yes—
643C22#1912823612yes—
644C22#127×D8311yescharacteristic
645C22#1912824612yes—
646C22#1912825011yescharacteristic
647C22#1912825611yescharacteristic
648C22#1912825712yes—
649C4—24no—
650C4—42no—
651C4—44no—
652C4—42no—
653C4—44no—
654C4—3322no—
655C4—3324no—
656C4—3322no—
657C4—838no—
658C4—1668no—
659C2—3324no—
660C2#2490998011yescharacteristic
661C2#2490998412yes—
662C2#2490999712yes—
663C2#2491000111yescharacteristic
664C2#2491002711yescharacteristic
665C2#399×D8311yescharacteristic
666C2#2491072112yes—
667C2#2491072911yescharacteristic
668C2#2491073611yescharacteristic
669C2#2491074012yes—
670C2#2491074311yescharacteristic
subgroupG/H
center, FrattiniC24C23×D83
commutatorC23×C83C23×C4
FittingC83×#1445C2
socle, cosocleC24×C83C24
prime radicalC24×D83C23
hypercenter#1445D83
hypocenterC83C4×#399
LCSG ▷ C23×C83 ▷ C83
UCSC1 ◁ C24 ◁ #1445

Complements

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

NH ≅ G/Ntype
2. C2×C4⋊C4×Dic83C2semidirect
3. C2×#19127552C2semidirect
6. #24905729C2semidirect
7. #24905727C2semidirect
10. #95×Dic83C2semidirect
11. C83×#1445C2semidirect
24. #19127636C4semidirect
58. C2×Dic83⋊C4C2×C4semidirect
86. C83⋊#95C2×C4semidirect
94. C2×C4⋊C4×C83C2×C4semidirect
322. Dic83#399semidirect
342. #1445D83semidirect
347. C83C4×#399semidirect
402. C2×C4⋊C4C4×D83semidirect

Maximal subgroups

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

subgroupindex
C2×#191276362
C2×C4⋊C4×Dic832
C2×#191275522
C2×#191275482
#249057292
#249057272
#249057092
#249057072
#95×Dic832
C83×#14452
C4×#39983