Finite Group Databasebeta

← #35404286

Subgroups — #35404286

Order 21248.

1974 conjugacy classes of subgroups.

The subgroup lattice has 680 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C42×D83C211yesmaximal, characteristic
2C2×#19127756C212yesmaximal, complemented
3#24907530C212yesmaximal, complemented
4#24907442C212yesmaximal, complemented
5#24907441C212yesmaximal, complemented
6#24907372C212yesmaximal
7#24907226C212yesmaximal, complemented
8#24907225C211yesmaximal, characteristic, complemented
9C83×#1749C211yesmaximal, characteristic, complemented
10C23×C4×D83C411yescharacteristic
11C23×C4×D83C411yescharacteristic
12C23×C4×D83C2211yescharacteristic
13C22×C4×Dic83C411yescharacteristic
14C22×C4×Dic83C411yescharacteristic
15C22×C4×Dic83C2211yescharacteristic
16C2×C42×D83—28no—
17C2×C42×D83C411yescharacteristic
18C2×C42×D83C411yescharacteristic
19C2×C42×D83C411yescharacteristic
20C2×C42×D83C411yescharacteristic
21C2×C42×D83C2211yescharacteristic
22C2×C42×D83C2211yescharacteristic
23C2×C42×D83C2211yescharacteristic
24C2×C42×D83C2211yescharacteristic
25#19127756—28no—
26C2×Dic83⋊C8C2214yescomplemented
27C2×C22⋊(C83⋊C8)C2212yes—
28C2×(C2×D83)⋊C8C2214yes—
29C2×(C2×D83)⋊C8C2214yes—
30C2×#14581703C2212yescomplemented
31#19127456C2212yescomplemented
32#19127455C2214yescomplemented
33C22×C42×C83C2211yescharacteristic
34C2×C22⋊C8×C83C2212yes—
35C2×#97×C83C2212yescomplemented
36#336×C83C2212yescomplemented
37C24×D83C2×C411yescharacteristic
38C23×Dic83C2×C411yescharacteristic
39C23×Dic83C2×C411yescharacteristic
40C23×Dic83C2×C411yescharacteristic
41C22×C4×D83—24no—
42C22×C4×D83—24no—
43C22×C4×D83—24no—
44C22×C4×D83—24no—
45C22×C4×D83—24no—
46C22×C4×D83—24no—
47C22×C4×D83—24no—
48C22×C4×D83—24no—
49C22×C4×D83—24no—
50C22×C4×D83C2×C411yescharacteristic
51C22×C4×D83C2×C411yescharacteristic
52C22×C4×D83C2×C411yescharacteristic
53C22×C4×D83C2×C411yescharacteristic
54C22×C4×D83C2×C411yescharacteristic
55C22×C4×D83C2×C411yescharacteristic
56C22×C4×D83C2×C411yescharacteristic
57C22×C4×D83C2×C411yescharacteristic
58C22×C4×D83C2×C411yescharacteristic
59C22×C4×D83C2×C411yescharacteristic
60C22×C4×D83C2311yescharacteristic
61C22×C4×D83C2311yescharacteristic
62C22×C83⋊C8C2314yes—
63C2×C4×Dic83—24no—
64C2×C4×Dic83—24no—
65C2×C4×Dic83—24no—
66C2×C4×Dic83C2×C411yescharacteristic
67C2×C4×Dic83C2×C411yescharacteristic
68C2×C4×Dic83C2×C411yescharacteristic
69C2×C4×Dic83C2×C411yescharacteristic
70C2×C4×Dic83C2×C411yescharacteristic
71C2×C4×Dic83C2×C411yescharacteristic
72C2×C4×Dic83C2×C411yescharacteristic
73C2×C4×Dic83C2×C411yescharacteristic
74C2×C4×Dic83C2×C411yescharacteristic
75C2×C4×Dic83C2×C411yescharacteristic
76C2×C4×Dic83C2311yescharacteristic
77C2×C4×Dic83C2311yescharacteristic
78C42×D83—24no—
79C42×D83—24no—
80C42×D83—24no—
81C42×D83—24no—
82C42×D83—24no—
83C42×D83—24no—
84C42×D83—24no—
85C42×D83D418yes—
86Dic83⋊C8—28no—
87C22⋊(C83⋊C8)—24no—
88(C2×D83)⋊C8—28no—
89(C2×D83)⋊C8—28no—
90#14581703—24no—
91C23×C4×C83C2×C411yescharacteristic
92C23×C4×C83C2×C411yescharacteristic
93C23×C4×C83C2311yescharacteristic
94C22×C8×C83C2314yes—
95C2×C42×C83—24no—
96C2×C42×C83C2×C411yescharacteristic
97C2×C42×C83C2×C411yescharacteristic
98C2×C42×C83C2311yescharacteristic
99C2×C42×C83C2311yescharacteristic
100C22⋊C8×C83—24no—
101#97×C83—24no—
102C23×D83—22no—
103C23×D83—22no—
104C23×D83—22no—
105C23×D83—22no—
106C23×D83—22no—
107C23×D83—22no—
108C23×D83M1614yes—
109C23×D83C22×C411yescharacteristic
110C23×D83C22×C411yescharacteristic
111C22×Dic83—22no—
112C22×Dic83—22no—
113C22×Dic83—22no—
114C22×Dic83—22no—
115C22×Dic83—22no—
116C22×Dic83—22no—
117C22×Dic83—22no—
118C22×Dic83—22no—
119C22×Dic83M1614yes—
120C22×Dic83C22×C411yescharacteristic
121C22×Dic83C22×C411yescharacteristic
122C22×Dic83C22×C411yescharacteristic
123C22×Dic83C22×C411yescharacteristic
124C22×Dic83C22×C411yescharacteristic
125C22×Dic83C22×C411yescharacteristic
126C2×C4×D83—232no—
127C2×C4×D83—232no—
128C2×C4×D83—232no—
129C2×C4×D83—22no—
130C2×C4×D83—22no—
131C2×C4×D83—22no—
132C2×C4×D83—22no—
133C2×C4×D83—22no—
134C2×C4×D83—22no—
135C2×C4×D83—22no—
136C2×C4×D83—22no—
137C2×C4×D83—22no—
138C2×C4×D83—22no—
139C2×C4×D83—22no—
140C2×C4×D83—22no—
141C2×C4×D83—22no—
142C2×C4×D83—22no—
143C2×C4×D83—22no—
144C2×C4×D83—22no—
145C2×C4×D83—22no—
146C2×C4×D83—22no—
147C2×C4×D83—22no—
148C2×C4×D83—22no—
149C2×C4×D83—22no—
150C2×C4×D83—22no—
151C2×C4×D83—22no—
152C2×C4×D83—22no—
153C2×C4×D83—22no—
154C2×C4×D83—22no—
155C2×C4×D83—22no—
156C2×C4×D83—22no—
157C2×C4×D83—22no—
158C2×C4×D83—22no—
159C2×C4×D83C22⋊C414yes—
160C2×C4×D83C22⋊C414yes—
161C2×C4×D83M1614yes—
162C2×C4×D83M1614yes—
163C2×C4×D83C2×D414yes—
164C2×C4×D83#4114yes—
165C2×C83⋊C8—24no—
166C2×C83⋊C8—24no—
167C2×C83⋊C8—24no—
168C4×Dic83—22no—
169C4×Dic83—22no—
170C4×Dic83—22no—
171C4×Dic83—22no—
172C4×Dic83—22no—
173C4×Dic83—22no—
174C4×Dic83—22no—
175C4×Dic83—22no—
176C4×Dic83—22no—
177C4×Dic83C22⋊C414yes—
178C4×Dic83C22⋊C414yes—
179C4×Dic83C2×D414yes—
180C24×C83C22×C411yescharacteristic
181C22×C4×C83—22no—
182C22×C4×C83—22no—
183C22×C4×C83—22no—
184C22×C4×C83—22no—
185C22×C4×C83—22no—
186C22×C4×C83—22no—
187C22×C4×C83—22no—
188C22×C4×C83—22no—
189C22×C4×C83C22×C411yescharacteristic
190C22×C4×C83C22×C411yescharacteristic
191C22×C4×C83C22×C411yescharacteristic
192C22×C4×C83C22×C411yescharacteristic
193C22×C4×C83C22×C411yescharacteristic
194C22×C4×C83#4114yes—
195C22×C4×C83C2411yescharacteristic
196C2×C8×C83—24no—
197C2×C8×C83—24no—
198C2×C8×C83—24no—
199C42×C83—22no—
200C42×C83—22no—
201C42×C83—22no—
202C42×C83C2×D414yes—
203C23×C83—21no—
204C23×C83—21no—
205C23×C83—21no—
206C23×C83—21no—
207C23×C83#11712yes—
208C23×C83C2×M1612yes—
209C23×C83C2×M1612yes—
210C23×C83C23×C411yescharacteristic
211C22×D83—28no—
212C22×D83—28no—
213C22×D83—28no—
214C22×D83—28no—
215C22×D83—21no—
216C22×D83—21no—
217C22×D83—21no—
218C22×D83—21no—
219C22×D83—21no—
220C22×D83—21no—
221C22×D83—21no—
222C22×D83—21no—
223C22×D83—28no—
224C22×D83—28no—
225C22×D83—28no—
226C22×D83C2×C22⋊C412yes—
227C22×D83#11712yes—
228C22×D83C2×M1612yes—
229C22×D83C2×M1612yes—
230C22×D83C2×M1612yes—
231C22×D83C2×M1612yes—
232C2×C4×C83—28no—
233C2×C4×C83—28no—
234C2×C4×C83—21no—
235C2×C4×C83—21no—
236C2×C4×C83—21no—
237C2×C4×C83—21no—
238C2×C4×C83—21no—
239C2×C4×C83—21no—
240C2×C4×C83—21no—
241C2×C4×C83—21no—
242C2×C4×C83—21no—
243C2×C4×C83—21no—
244C2×C4×C83—28no—
245C2×C4×C83C2×C22⋊C412yes—
246C2×C4×C83C2×C22⋊C412yes—
247C2×C4×C83#11712yes—
248C2×C4×C83C2×M1612yes—
249C2×C4×C83C2×M1612yes—
250C2×C4×C83C22×D412yes—
251C2×C4×C83C2×#4112yes—
252C2×C4×C83C2×#4112yes—
253C2×Dic83—28no—
254C2×Dic83—28no—
255C2×Dic83—21no—
256C2×Dic83—21no—
257C2×Dic83—21no—
258C2×Dic83—21no—
259C2×Dic83—21no—
260C2×Dic83—21no—
261C2×Dic83—21no—
262C2×Dic83—21no—
263C2×Dic83—21no—
264C2×Dic83—21no—
265C2×Dic83—28no—
266C2×Dic83C2×C22⋊C412yes—
267C2×Dic83C2×C22⋊C412yes—
268C2×Dic83C2×C22⋊C412yes—
269C2×Dic83#11712yes—
270C2×Dic83C2×M1612yes—
271C2×Dic83C2×M1612yes—
272C2×Dic83C2×M1612yes—
273C2×Dic83C2×M1612yes—
274C4×D83—28no—
275C4×D83—28no—
276C4×D83—28no—
277C4×D83—28no—
278C4×D83—28no—
279C4×D83—28no—
280C4×D83—28no—
281C4×D83—28no—
282C4×D83—28no—
283C4×D83—28no—
284C4×D83—28no—
285C4×D83—28no—
286C8×C83—44no—
287C83⋊C8—44no—
288C22×C83—22no—
289C22×C83—22no—
290C22×C83—22no—
291C22×C83—22no—
292C22×C83—22no—
293C22×C83—22no—
294C22×C83#43214yescomplemented
295C22×C83C22×C22⋊C411yescharacteristic
296C22×C83C2×#11711yescharacteristic
297C22×C83C2×#11711yescharacteristic
298C22×C83C22×M1611yescharacteristic
299C22×C83C22×M1611yescharacteristic
300C22×C83C22×M1611yescharacteristic
301C22×C83C22×M1611yescharacteristic
302C2×D83—232no—
303C2×D83—22no—
304C2×D83—22no—
305C2×D83—22no—
306C2×D83—22no—
307C2×D83—22no—
308C2×D83—22no—
309C2×D83—22no—
310C2×D83—22no—
311C2×D83—22no—
312C2×D83—22no—
313C2×D83—22no—
314C2×D83—22no—
315C2×D83#40714yes—
316C2×D83#43214yescomplemented
317C4×C83—22no—
318C4×C83—22no—
319C4×C83—22no—
320C4×C83—22no—
321C4×C83—22no—
322C4×C83#52514yes—
323Dic83—22no—
324Dic83—22no—
325Dic83—22no—
326Dic83—22no—
327Dic83—22no—
328Dic83#40714yes—
329#20715—831nomaximal, Sylow
330C2×C83—24no—
331C2×C83#228011yescharacteristic
332C2×C83C2×#40711yescharacteristic
333C2×C83C2×#40711yescharacteristic
334C2×C83C2×#43211yescharacteristic
335C2×C83C2×#43211yescharacteristic
336C2×C83#292011yescharacteristic
337C2×C83#292011yescharacteristic
338D83—24no—
339D83—24no—
340C23×C42—831no—
341C2×#432—832no—
342#1817—832no—
343#1797—832no—
344#1797—832no—
345#1779—832no—
346#1749—831no—
347#1749—832no—
348#1749D8311yescharacteristic, complemented
349C83#2071511yescharacteristic, complemented, Sylow
350C24×C4—831no—
351C24×C4—831no—
352C24×C4—831no—
353C22×C42—831no—
354C22×C42—831no—
355C22×C42—831no—
356C22×C42—831no—
357C22×C42—831no—
358C22×C42—831no—
359C22×C42—831no—
360C22×C42—831no—
361C22×C42—831no—
362C22×C42—831no—
363C22×C42—831no—
364C22×C42—1668no—
365C22×C42C2×D8311yescharacteristic
366#432—1668no—
367C2×C4⋊C8—834no—
368C2×C22⋊C8—832no—
369C2×C22⋊C8—834no—
370C2×C22⋊C8—834no—
371C2×C22⋊C8C2×D8312yes—
372C2×#97—832no—
373C2×#97C2×D8312yescomplemented
374#336—834no—
375#336—832no—
376#336C2×D8312yescomplemented
377C25—831no—
378C23×C4—831no—
379C23×C4—831no—
380C23×C4—831no—
381C23×C4—831no—
382C23×C4—831no—
383C23×C4—831no—
384C23×C4—831no—
385C23×C4—831no—
386C23×C4—831no—
387C23×C4—831no—
388C23×C4—831no—
389C23×C4—831no—
390C23×C4—831no—
391C23×C4—831no—
392C23×C4—831no—
393C23×C4—1664no—
394C23×C4—1664no—
395C23×C4—1664no—
396C23×C4—1664no—
397C23×C4—1664no—
398C23×C4—1664no—
399C23×C4—1664no—
400C23×C4—1664no—
401C23×C4—1664no—
402C23×C4C4×D8311yescharacteristic
403C23×C4C4×D8311yescharacteristic
404C23×C4C22×D8311yescharacteristic
405C22×C8—834no—
406C22×C8C22×D8314yes—
407C2×C42—24no—
408C2×C42—831no—
409C2×C42—831no—
410C2×C42—831no—
411C2×C42—838no—
412C2×C42—831no—
413C2×C42—831no—
414C2×C42—831no—
415C2×C42—831no—
416C2×C42—831no—
417C2×C42—831no—
418C2×C42—831no—
419C2×C42—831no—
420C2×C42—831no—
421C2×C42—1664no—
422C2×C42—1664no—
423C2×C42—1664no—
424C2×C42—1664no—
425C2×C42—1664no—
426C2×C42—1664no—
427C2×C42—1664no—
428C2×C42—1664no—
429C2×C42—1664no—
430C2×C42—1664no—
431C2×C42C4×D8311yescharacteristic
432C2×C42C4×D8311yescharacteristic
433C2×C42C22×D8311yescharacteristic
434C2×C42C22×D8311yescharacteristic
435C4⋊C8—1668no—
436C22⋊C8—24no—
437C22⋊C8—1668no—
438C22⋊C8—1668no—
439C22⋊C8—1664no—
440#97—24no—
441#97—1664no—
442C24—831no—
443C24—831no—
444C24—834no—
445C24—1662no—
446C24—1662no—
447C24—1662no—
448C24—1662no—
449C24—1662no—
450C24—1662no—
451C24C2×C4×D8311yescharacteristic
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—831no—
461C22×C4—834no—
462C22×C4—831no—
463C22×C4—831no—
464C22×C4—831no—
465C22×C4—831no—
466C22×C4—831no—
467C22×C4—834no—
468C22×C4—834no—
469C22×C4—834no—
470C22×C4—834no—
471C22×C4—834no—
472C22×C4—834no—
473C22×C4—1662no—
474C22×C4—1662no—
475C22×C4—1662no—
476C22×C4—1662no—
477C22×C4—1662no—
478C22×C4—1662no—
479C22×C4—1662no—
480C22×C4—1662no—
481C22×C4—1662no—
482C22×C4—1662no—
483C22×C4—1662no—
484C22×C4—1662no—
485C22×C4—1662no—
486C22×C4—1662no—
487C22×C4—1662no—
488C22×C4—1662no—
489C22×C4—1662no—
490C22×C4—1662no—
491C22×C4—1662no—
492C22×C4—1662no—
493C22×C4—16632no—
494C22×C4—16632no—
495C22×C4—16632no—
496C22×C4—1662no—
497C22×C4—1662no—
498C22×C4—1662no—
499C22×C4—1662no—
500C22×C4—1662no—
501C22×C4—1662no—
502C22×C4—1662no—
503C22×C4—1662no—
504C22×C4—1662no—
505C22×C4—1662no—
506C22×C4—1662no—
507C22×C4—1662no—
508C22×C4—1662no—
509C22×C4—1662no—
510C22×C4—1662no—
511C22×C4—1662no—
512C22×C4—1662no—
513C22×C4—1662no—
514C22×C4C2×C4×D8311yescharacteristic
515C22×C4C2×C4×D8311yescharacteristic
516C22×C4C2×C4×D8311yescharacteristic
517C22×C4C2×C4×D8311yescharacteristic
518C22×C4C2×C4×D8311yescharacteristic
519C22×C4#1305059014yes—
520C22×C4C23×D8311yescharacteristic
521C2×C8—24no—
522C2×C8—24no—
523C2×C8—24no—
524C2×C8—1664no—
525C2×C8—1664no—
526C2×C8—1664no—
527C42—22no—
528C42—22no—
529C42—22no—
530C42—834no—
531C42—834no—
532C42—834no—
533C42—1662no—
534C42—1662no—
535C42—1662no—
536C42—1662no—
537C42—1662no—
538C42—1662no—
539C42—1662no—
540C42—1662no—
541C42—1662no—
542C42D4×D8314yes—
543C23—21no—
544C23—21no—
545C23—21no—
546C23—21no—
547C23—832no—
548C23—832no—
549C23—832no—
550C23—832no—
551C23—832no—
552C23—832no—
553C23—1668no—
554C23—1668no—
555C23—1668no—
556C23—1668no—
557C23—1668no—
558C23—1668no—
559C23—1661no—
560C23—1661no—
561C23—1661no—
562C23—1661no—
563C23—1661no—
564C23—1661no—
565C23—1661no—
566C23—1661no—
567C23—1668no—
568C23#1458174912yes—
569C23M16×D8312yes—
570C23M16×D8312yes—
571C23C22×C4×D8311yescharacteristic
572C2×C4—28no—
573C2×C4—28no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—28no—
585C2×C4—832no—
586C2×C4—832no—
587C2×C4—832no—
588C2×C4—832no—
589C2×C4—832no—
590C2×C4—832no—
591C2×C4—832no—
592C2×C4—832no—
593C2×C4—1668no—
594C2×C4—1668no—
595C2×C4—1668no—
596C2×C4—1668no—
597C2×C4—1668no—
598C2×C4—1668no—
599C2×C4—1668no—
600C2×C4—1668no—
601C2×C4—1668no—
602C2×C4—1668no—
603C2×C4—1668no—
604C2×C4—1668no—
605C2×C4—1668no—
606C2×C4—1661no—
607C2×C4—1661no—
608C2×C4—1661no—
609C2×C4—1661no—
610C2×C4—1661no—
611C2×C4—1661no—
612C2×C4—1661no—
613C2×C4—1661no—
614C2×C4—1661no—
615C2×C4—1661no—
616C2×C4—1668no—
617C2×C4—1668no—
618C2×C4C22⋊C4×D8312yes—
619C2×C4C22⋊C4×D8312yes—
620C2×C4#1458174912yes—
621C2×C4M16×D8312yes—
622C2×C4M16×D8312yes—
623C2×C4C2×D4×D8312yes—
624C2×C4#41×D8312yes—
625C2×C4C2×#1305059012yes—
626C8—44no—
627C8—3324no—
628C22—22no—
629C22—22no—
630C22—22no—
631C22—22no—
632C22—22no—
633C22—22no—
634C22—834no—
635C22—834no—
636C22—16632no—
637C22—1662no—
638C22—1662no—
639C22—1662no—
640C22—1662no—
641C22—1662no—
642C22—1662no—
643C22—1662no—
644C22—1662no—
645C22—1662no—
646C22—1662no—
647C22—1662no—
648C22—1662no—
649C22#1912775614yescomplemented
650C22C2×C22⋊C4×D8311yescharacteristic
651C22#117×D8311yescharacteristic
652C22C2×#1458174911yescharacteristic
653C22C2×M16×D8311yescharacteristic
654C22C2×M16×D8311yescharacteristic
655C22C2×M16×D8311yescharacteristic
656C22C2×M16×D8311yescharacteristic
657C4—22no—
658C4—22no—
659C4—22no—
660C4—22no—
661C4—22no—
662C4—834no—
663C4—1662no—
664C4—1662no—
665C4—1662no—
666C4—1662no—
667C4—1662no—
668C4#1912821014yes—
669C2—24no—
670C2—1664no—
671C2—1664no—
672C2#2490984411yescharacteristic
673C2#407×D8311yescharacteristic
674C2#407×D8311yescharacteristic
675C2#432×D8311yescharacteristic
676C2C2×#1912775611yescharacteristic
677C2#2491609311yescharacteristic
678C2#2491609411yescharacteristic
subgroupG/H
center, FrattiniC22×C4C23×D83
commutator, socleC23×C83C23×C4
FittingC83×#1749C2
prime radicalC24×D83C2×C4
cosocleC22×C4×C83C24
hypercenter#1749D83
hypocenterC83#20715
LCSG ▷ C23×C83 ▷ C83
UCSC1 ◁ C22×C4 ◁ #1749

Complements

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

NH ≅ G/Ntype
2. C2×#19127756C2semidirect
3. #24907530C2semidirect
4. #24907442C2semidirect
5. #24907441C2semidirect
7. #24907226C2semidirect
8. #24907225C2semidirect
9. C83×#1749C2semidirect
26. C2×Dic83⋊C8C22semidirect
30. C2×#14581703C22semidirect
31. #19127456C22semidirect
32. #19127455C22semidirect
35. C2×#97×C83C22semidirect
36. #336×C83C22semidirect
294. C22×C83#432semidirect
316. C2×D83#432semidirect
348. #1749D83semidirect
349. C83#20715semidirect
373. C2×#97C2×D83semidirect
376. #336C2×D83semidirect
649. C22#19127756semidirect

Maximal subgroups

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

subgroupindex
C22×C42×D832
C2×#191277562
#249075302
#249074422
#249074412
#249073722
#249072262
#249072252
C83×#17492
#2071583