Finite Group Databasebeta

← #35412249

Subgroups — #35412249

Order 21248.

730 conjugacy classes of subgroups.

The subgroup lattice has 638 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#24917251C211yesmaximal, characteristic
2#24916400C211yesmaximal, characteristic, complemented
3#24915891C211yesmaximal, characteristic
4#24908406C211yesmaximal, characteristic, complemented
5#24908236C211yesmaximal, characteristic, complemented
6#24908060C211yesmaximal, characteristic, complemented
7#24908060C211yesmaximal, characteristic, complemented
8#24907736C211yesmaximal, characteristic
9#24907736C211yesmaximal, characteristic
10#24907346C211yesmaximal, characteristic, complemented
11#24907341C211yesmaximal, characteristic, complemented
12#24907338C211yesmaximal, characteristic, complemented
13#24907308C211yesmaximal, characteristic
14#24907301C211yesmaximal, characteristic
15C83×#1806C211yesmaximal, characteristic
16C2×#14581799C2211yescharacteristic
17C2×#14581799C2211yescharacteristic
18#19128258C2211yescharacteristic
19#19128008C411yescharacteristic, complemented
20#19128008C411yescharacteristic, complemented
21#19128001C411yescharacteristic, complemented
22#19128001C411yescharacteristic, complemented
23#19127996C411yescharacteristic, complemented
24#19127991C411yescharacteristic, complemented
25#19127986C411yescharacteristic, complemented
26#19127981C411yescharacteristic, complemented
27#19127830—21no—
28#19127830—21no—
29#19127830—21no—
30#19127830—21no—
31C2×#14581727C2211yescharacteristic
32C2×#14581727C2211yescharacteristic
33#19127737C2211yescharacteristic
34#19127737C2211yescharacteristic
35#19127729C2211yescharacteristic
36#19127729C2211yescharacteristic
37C2×C4⋊(C83⋊C8)C2211yescharacteristic
38#19127697C2211yescharacteristic
39#19127697C2211yescharacteristic
40#19127659C2211yescharacteristic
41#19127659C2211yescharacteristic
42#19127657—21no—
43#19127657—21no—
44#19127657C2211yescharacteristic
45#19127657C2211yescharacteristic
46C4×#13050574—21no—
47C4×#13050574C2211yescharacteristic
48C2×#14581703—21no—
49C2×#14581703C2211yescharacteristic
50C2×#14581703C2211yescharacteristic
51C2×C4×C83⋊C8C2211yescharacteristic
52#19127465C2211yescharacteristic
53#19127465C2211yescharacteristic
54#19127461C2211yescharacteristic
55#19127461C2211yescharacteristic
56#19127461C2211yescharacteristic
57#19127461C2211yescharacteristic
58#19127456C2211yescharacteristic
59#19127456C2211yescharacteristic
60C4×#41×C83C2211yescharacteristic
61#423×C83C2211yescharacteristic
62C2×#104×C83C2211yescharacteristic
63C2×#104×C83C2211yescharacteristic
64C4×C4⋊C4×C83C2211yescharacteristic
65#339×C83C2211yescharacteristic
66#339×C83C2211yescharacteristic
67#14581799—22no—
68#14581799—22no—
69#14581799—22no—
70#14581799Q814yescomplemented
71C2×#13050574—21no—
72C2×#13050574—21no—
73C2×#13050574C2311yescharacteristic
74C2×#13050574C2311yescharacteristic
75C22×C83⋊C8—21no—
76C22×C83⋊C8—21no—
77C22×C83⋊C8C2311yescharacteristic
78C22×C83⋊C8C2311yescharacteristic
79C22×C83⋊C8C2311yescharacteristic
80C22×C83⋊C8C2311yescharacteristic
81#14581792C2×C411yescharacteristic
82#14581792C2×C411yescharacteristic
83#14581784C2×C411yescharacteristic
84#14581784C2×C411yescharacteristic
85#14581771—21no—
86#14581771—21no—
87C4×Dic166—21no—
88C4×Dic166—21no—
89#14581751C2311yescharacteristic
90#14581751C2311yescharacteristic
91#14581727—22no—
92#14581727—22no—
93#14581725C2×C411yescharacteristic
94#14581725C2×C411yescharacteristic
95#14581725C2×C411yescharacteristic
96#14581725C2×C411yescharacteristic
97#14581723C2×C411yescharacteristic
98#14581723C2×C411yescharacteristic
99#14581723C2×C411yescharacteristic
100#14581723C2×C411yescharacteristic
101C4⋊(C83⋊C8)C2×C411yescharacteristic
102C4⋊(C83⋊C8)C2×C411yescharacteristic
103C4⋊(C83⋊C8)C2×C411yescharacteristic
104C4⋊(C83⋊C8)C2×C411yescharacteristic
105Dic83⋊C8—21no—
106Dic83⋊C8—21no—
107Dic83⋊C8—21no—
108Dic83⋊C8—21no—
109#14581716C2×C411yescharacteristic
110#14581716C2×C411yescharacteristic
111#14581716C2×C411yescharacteristic
112#14581716C2×C411yescharacteristic
113#14581713C2×C411yescharacteristic
114#14581713C2×C411yescharacteristic
115#14581713C2×C411yescharacteristic
116#14581713C2×C411yescharacteristic
117#14581703—21no—
118#14581703—21no—
119#14581703—22no—
120#14581703—21no—
121#14581703D411yescharacteristic
122#14581703D411yescharacteristic
123#14581703D411yescharacteristic
124#14581703D411yescharacteristic
125#14581702—21no—
126#14581702—21no—
127C4×C83⋊C8—21no—
128C4×C83⋊C8—21no—
129C4×C83⋊C8—22no—
130C4×C83⋊C8—21no—
131C8×Dic83—21no—
132C8×Dic83—21no—
133C2×#41×C83C2311yescharacteristic
134C2×M16×C83C2311yescharacteristic
135C2×M16×C83C2311yescharacteristic
136C4×Q8×C83C2×C411yescharacteristic
137C4×Q8×C83C2×C411yescharacteristic
138C4×D4×C83—21no—
139C4×D4×C83—21no—
140C4×D4×C83C2×C411yescharacteristic
141C4×D4×C83C2×C411yescharacteristic
142#117×C83—21no—
143#117×C83C2311yescharacteristic
144C2×C4⋊C4×C83—21no—
145C2×C42×C83—21no—
146C2×C42×C83C2311yescharacteristic
147C2×C42×C83C2311yescharacteristic
148C2×C42×C83C2311yescharacteristic
149C4⋊C8×C83—21no—
150C4⋊C8×C83—21no—
151#104×C83—22no—
152#104×C83—22no—
153#95×C83—21no—
154C2×Dic166—21no—
155C2×Dic166—21no—
156#13050574—24no—
157#13050574—21no—
158#13050574—21no—
159#13050574—24no—
160#13050574C2×Q812yes—
161#13050574#4112yes—
162C2×C83⋊C8—24no—
163C2×C83⋊C8—21no—
164C2×C83⋊C8—21no—
165C2×C83⋊C8—21no—
166C2×C83⋊C8—21no—
167C2×C83⋊C8—21no—
168C2×C83⋊C8—21no—
169C2×C83⋊C8—21no—
170C2×C83⋊C8—21no—
171C2×C83⋊C8—24no—
172C2×C83⋊C8C22×C411yescharacteristic
173C2×C83⋊C8C22×C411yescharacteristic
174C2×C83⋊C8C22×C411yescharacteristic
175C2×C83⋊C8C22×C411yescharacteristic
176C2×C83⋊C8C2×D411yescharacteristic
177C2×C83⋊C8C2×D411yescharacteristic
178C2×C83⋊C8C2×D411yescharacteristic
179C2×C83⋊C8C2×D411yescharacteristic
180C2×C83⋊C8C2×Q812yes—
181C2×C83⋊C8#4112yes—
182C4⋊Dic83—21no—
183C4⋊Dic83—21no—
184C4⋊Dic83C22×C411yescharacteristic
185C4⋊Dic83C22×C411yescharacteristic
186C4⋊Dic83C22×C411yescharacteristic
187C4⋊Dic83C22×C411yescharacteristic
188Dic83⋊C4—21no—
189Dic83⋊C4—21no—
190Dic83⋊C4—21no—
191Dic83⋊C4—21no—
192C22⋊Dic83—21no—
193C22⋊Dic83—21no—
194C4×Dic83—21no—
195C4×Dic83—21no—
196#41×C83—21no—
197#41×C83—21no—
198#41×C83C2×Q812yes—
199#41×C83#4112yes—
200C2×Q8×C83C22×C411yescharacteristic
201C2×D4×C83—21no—
202C2×D4×C83C22×C411yescharacteristic
203C22×C4×C83—21no—
204C22×C4×C83—21no—
205C22×C4×C83—21no—
206C22×C4×C83—21no—
207C22×C4×C83—21no—
208C22×C4×C83#4111yescharacteristic
209C22×C4×C83#4111yescharacteristic
210C22×C4×C83C2411yescharacteristic
211M16×C83—21no—
212M16×C83—21no—
213M16×C83—21no—
214M16×C83—21no—
215C2×C8×C83—21no—
216C2×C8×C83—21no—
217C4⋊C4×C83—21no—
218C4⋊C4×C83—21no—
219C4⋊C4×C83—42no—
220C4⋊C4×C83C22×C411yescharacteristic
221C4⋊C4×C83C22×C411yescharacteristic
222C22⋊C4×C83—21no—
223C22⋊C4×C83—21no—
224C22⋊C4×C83—21no—
225C42×C83—21no—
226C42×C83—21no—
227C42×C83—21no—
228C42×C83—21no—
229C42×C83—21no—
230C42×C83—21no—
231C42×C83—21no—
232C42×C83C22×C411yescharacteristic
233C42×C83C22×C411yescharacteristic
234C42×C83C2×D411yescharacteristic
235C42×C83C2×D411yescharacteristic
236C23×C83—21no—
237C23×C83C2×#4111yescharacteristic
238Q8×C83—21no—
239Q8×C83C4×Q812yes—
240D4×C83—22no—
241D4×C83—22no—
242D4×C83—21no—
243D4×C83C4×Q812yes—
244C2×C4×C83—21no—
245C2×C4×C83—22no—
246C2×C4×C83—22no—
247C2×C4×C83—22no—
248C2×C4×C83—21no—
249C2×C4×C83—21no—
250C2×C4×C83—41no—
251C2×C4×C83—41no—
252C2×C4×C83—41no—
253C2×C4×C83—41no—
254C2×C4×C83—41no—
255C2×C4×C83—41no—
256C2×C4×C83—41no—
257C2×C4×C83—81no—
258C2×C4×C83—81no—
259C2×C4×C83C4×D411yescharacteristic
260C2×C4×C83C4×D411yescharacteristic
261C2×C4×C83C4×D411yescharacteristic
262C2×C4×C83C4×D411yescharacteristic
263C2×C4×C83C23×C411yescharacteristic
264C2×C4×C83C22×D411yescharacteristic
265C2×C4×C83C22×Q811yescharacteristic
266C2×C4×C83C2×#4111yescharacteristic
267C2×C4×C83C2×#4111yescharacteristic
268C2×C4×C83C2×#4111yescharacteristic
269C2×Dic83—21no—
270C2×Dic83—21no—
271C2×Dic83—21no—
272C2×Dic83—21no—
273Dic166—81no—
274Dic166—81no—
275C8×C83—41no—
276C8×C83—41no—
277C83⋊C8—21no—
278C83⋊C8—21no—
279C83⋊C8—21no—
280C83⋊C8—21no—
281C83⋊C8—22no—
282C83⋊C8—21no—
283C83⋊C8—22no—
284C83⋊C8—21no—
285C83⋊C8C22⋊Q812yes—
286C83⋊C8C22⋊Q812yes—
287C22×C83—21no—
288C22×C83—21no—
289C22×C83—22no—
290C22×C83—41no—
291C22×C83—41no—
292C22×C83C2×C4×D411yescharacteristic
293C22×C83#52911yescharacteristic
294C22×C83#53311yescharacteristic
295C4×C83—21no—
296C4×C83—21no—
297C4×C83—21no—
298C4×C83—21no—
299C4×C83—22no—
300C4×C83—42no—
301C4×C83—41no—
302C4×C83—41no—
303C4×C83—81no—
304C4×C83C2×C4×Q811yescharacteristic
305C4×C83C4×#4111yescharacteristic
306C4×C83C2×C22⋊Q811yescharacteristic
307C4×C83#52511yescharacteristic
308Dic83—41no—
309Dic83—41no—
310Dic83—41no—
311Dic83—41no—
312#21629—831nomaximal, Sylow
313C2×C83—21no—
314C2×C83—21no—
315C2×C83—42no—
316C2×C83C4×C22⋊Q811yescharacteristic
317C2×C83#295611yescharacteristic
318C2×C83#295711yescharacteristic
319#3076—831no—
320#2961—831no—
321#2875—831no—
322#1986—831no—
323#1950—831no—
324#1922—831no—
325#1922—831no—
326#1866—831no—
327#1866—831no—
328#1806D8311yescharacteristic
329#1773—831no—
330#1772—831no—
331#1771—831no—
332#1765—831no—
333#1764—831no—
334C83#2162911yescharacteristic, complemented, Sylow
335C2×#131—831no—
336C2×#131—831no—
337#533—831no—
338C4×#41C2×D8311yescharacteristic
339#479—831no—
340#479—831no—
341#478—831no—
342#478—831no—
343#477—831no—
344#476—831no—
345#475—831no—
346#474—831no—
347#446—1661no—
348#446—1661no—
349#446—1661no—
350#446—1661no—
351C2×#108—831no—
352C2×#108—831no—
353#427—831no—
354#427—831no—
355#424—831no—
356#424—831no—
357#423C2×D8311yescharacteristic
358C2×C4⋊C8—831no—
359C2×#104C2×D8311yescharacteristic
360C2×#104C2×D8311yescharacteristic
361#416—831no—
362#416—831no—
363#405—831no—
364#405—831no—
365#405—831no—
366#405—831no—
367#405—1661no—
368#405—1661no—
369C4×M16—831no—
370C4×M16—1661no—
371C2×#97—831no—
372C2×#97—831no—
373C2×#97—1661no—
374C2×C4×C8—831no—
375C4×C4⋊C4C2×D8311yescharacteristic
376#341—831no—
377#341—831no—
378#339—831no—
379#339—831no—
380#339—831no—
381#339—831no—
382#339C2×D8311yescharacteristic
383#339C2×D8311yescharacteristic
384#336—831no—
385#336—831no—
386C2×#41C22×D8311yescharacteristic
387#131—834no—
388#131—1662no—
389#131—1662no—
390#131—1662no—
391C2×M16—831no—
392C2×M16—831no—
393C2×M16—1661no—
394C2×M16—1661no—
395C2×M16C22×D8311yescharacteristic
396C2×M16C22×D8311yescharacteristic
397C22×C8—831no—
398C22×C8—831no—
399C22×C8—831no—
400C22×C8—831no—
401C22×C8—1661no—
402C22×C8—1661no—
403C4⋊Q8—831no—
404C4⋊Q8—831no—
405#125—831no—
406#125—831no—
407C22⋊Q8—1661no—
408C22⋊Q8—1661no—
409C4×Q8—1661no—
410C4×Q8—1661no—
411C4×Q8C4×D8311yescharacteristic
412C4×Q8C4×D8311yescharacteristic
413C4×D4—21no—
414C4×D4—21no—
415C4×D4C4×D8311yescharacteristic
416C4×D4C4×D8311yescharacteristic
417#117—21no—
418#117—831no—
419#117—831no—
420#117C22×D8311yescharacteristic
421C2×C4⋊C4—21no—
422C2×C42—21no—
423C2×C42C22×D8311yescharacteristic
424C2×C42C22×D8311yescharacteristic
425C2×C42C22×D8311yescharacteristic
426#108—1662no—
427#108—1662no—
428#107—831no—
429#107—831no—
430#107—831no—
431#107—831no—
432#106—831no—
433#106—831no—
434#106—831no—
435#106—831no—
436C4⋊C8—21no—
437C4⋊C8—21no—
438C4⋊C8—831no—
439C4⋊C8—831no—
440C4⋊C8—831no—
441C4⋊C8—831no—
442C4⋊C8—1661no—
443C4⋊C8—1661no—
444C4⋊C8—1661no—
445C4⋊C8—1661no—
446#104—22no—
447#104—22no—
448#103—831no—
449#103—831no—
450#103—831no—
451#103—831no—
452#102—831no—
453#102—831no—
454#102—831no—
455#102—831no—
456#97—831no—
457#97—831no—
458#97—831no—
459#97—831no—
460#97—1662no—
461#97—1661no—
462#97—1661no—
463#97—1661no—
464#97—1661no—
465#97—1661no—
466C4×C8—1662no—
467C4×C8—1661no—
468C4×C8—1661no—
469C4×C8—1661no—
470C4×C8—1661no—
471C4×C8—1661no—
472#95—21no—
473#41—21no—
474#41—21no—
475#41Q8×D8312yes—
476#41#1305058912yes—
477C2×Q8—1661no—
478C2×Q8—1661no—
479C2×Q8C2×C4×D8311yescharacteristic
480C2×D4—21no—
481C2×D4C2×C4×D8311yescharacteristic
482C22×C4—21no—
483C22×C4—21no—
484C22×C4—21no—
485C22×C4—21no—
486C22×C4—21no—
487C22×C4#1305058911yescharacteristic
488C22×C4#1305058911yescharacteristic
489C22×C4C23×D8311yescharacteristic
490M16—21no—
491M16—21no—
492M16—21no—
493M16—21no—
494M16—832no—
495M16—832no—
496M16—1664no—
497M16—1664no—
498M16—1661no—
499M16—1661no—
500C2×C8—21no—
501C2×C8—21no—
502C2×C8—831no—
503C2×C8—831no—
504C2×C8—832no—
505C2×C8—832no—
506C2×C8—831no—
507C2×C8—831no—
508C2×C8—831no—
509C2×C8—831no—
510C2×C8—831no—
511C2×C8—831no—
512C2×C8—1664no—
513C2×C8—1664no—
514C2×C8—1661no—
515C2×C8—1661no—
516C2×C8—1661no—
517C2×C8—1661no—
518C2×C8—1661no—
519C2×C8—1661no—
520C2×C8—1661no—
521C2×C8—1661no—
522C4⋊C4—21no—
523C4⋊C4—21no—
524C4⋊C4—42no—
525C4⋊C4—831no—
526C4⋊C4—831no—
527C4⋊C4—831no—
528C4⋊C4—831no—
529C4⋊C4—1661no—
530C4⋊C4—1661no—
531C4⋊C4—1661no—
532C4⋊C4—1661no—
533C4⋊C4—1661no—
534C4⋊C4—1661no—
535C4⋊C4C2×C4×D8311yescharacteristic
536C4⋊C4C2×C4×D8311yescharacteristic
537C22⋊C4—21no—
538C22⋊C4—21no—
539C22⋊C4—21no—
540C22⋊C4—1661no—
541C22⋊C4—1661no—
542C42—21no—
543C42—21no—
544C42—21no—
545C42—21no—
546C42—21no—
547C42—21no—
548C42—21no—
549C42—1661no—
550C42—1661no—
551C42C2×C4×D8311yescharacteristic
552C42C2×C4×D8311yescharacteristic
553C42D4×D8311yescharacteristic
554C42D4×D8311yescharacteristic
555C23—21no—
556C23C2×#1305058911yescharacteristic
557Q8—21no—
558Q8—6641no—
559Q8—6641no—
560Q8#1458175712yes—
561D4—22no—
562D4—22no—
563D4—21no—
564D4#1458175712yes—
565C2×C4—21no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—81no—
579C2×C4—81no—
580C2×C4—1661no—
581C2×C4—1661no—
582C2×C4—1661no—
583C2×C4—1661no—
584C2×C4(C22⋊D83)⋊C411yescharacteristic
585C2×C4(C22⋊D83)⋊C411yescharacteristic
586C2×C4(C22⋊D83)⋊C411yescharacteristic
587C2×C4(C22⋊D83)⋊C411yescharacteristic
588C2×C4C22×C4×D8311yescharacteristic
589C2×C4C2×D4×D8311yescharacteristic
590C2×C4C2×Q8×D8311yescharacteristic
591C2×C4#41×D8311yescharacteristic
592C2×C4#41×D8311yescharacteristic
593C2×C4C2×#1305058911yescharacteristic
594C8—41no—
595C8—41no—
596C8—832no—
597C8—832no—
598C8—1661no—
599C8—1661no—
600C8—1661no—
601C8—1661no—
602C8—1662no—
603C8—1661no—
604C8—1661no—
605C8—1662no—
606C22—21no—
607C22—21no—
608C22—22no—
609C22—41no—
610C22—41no—
611C22C2×(C22⋊D83)⋊C411yescharacteristic
612C22#1912823211yescharacteristic
613C22#1912825311yescharacteristic
614C4—21no—
615C4—21no—
616C4—21no—
617C4—21no—
618C4—22no—
619C4—42no—
620C4—41no—
621C4—41no—
622C4—81no—
623C4—3321no—
624C4—3321no—
625C4—3321no—
626C4—3321no—
627C4C2×#1458175711yescharacteristic
628C4#1912816911yescharacteristic
629C4C22⋊Q8×D8311yescharacteristic
630C4#1912821111yescharacteristic
631C2—21no—
632C2—21no—
633C2—42no—
634C2#2490999111yescharacteristic
635C2#2491636911yescharacteristic
636C2#2491637611yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D83
commutatorC2×C4×C83C23×C4
FrattiniC22×C4C23×D83
FittingC83×#1806C2
socleC22×C83C2×C4×D4
prime radicalC2×#41×C83C23
cosocleC22×C4×C83C24
hypercenter#1806D83
hypocenterC83#21629
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1806

Complements

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

NH ≅ G/Ntype
2. #24916400C2semidirect
4. #24908406C2semidirect
5. #24908236C2semidirect
6. #24908060C2semidirect
7. #24908060C2semidirect
10. #24907346C2semidirect
11. #24907341C2semidirect
12. #24907338C2semidirect
19. #19128008C4semidirect
20. #19128008C4semidirect
21. #19128001C4semidirect
22. #19128001C4semidirect
23. #19127996C4semidirect
24. #19127991C4semidirect
25. #19127986C4semidirect
26. #19127981C4semidirect
70. #14581799Q8semidirect
334. C83#21629semidirect

Maximal subgroups

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

subgroupindex
#249172512
#249164002
#249158912
#249084062
#249082362
#249080602
#249077362
#249073462
#249073412
#249073382
#249073082
#249073012
C83×#18062
#2162983