Finite Group Databasebeta

← #35463203

Subgroups — #35463203

Order 21248.

826 conjugacy classes of subgroups.

The subgroup lattice has 622 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19128105C211yesmaximal, characteristic, complemented
2C2×#19127743C211yesmaximal, characteristic
3#24911166C211yesmaximal, characteristic
4#24911155C211yesmaximal, characteristic
5#24908949C211yesmaximal, characteristic, complemented
6#24908938C211yesmaximal, characteristic, complemented
7#24908795C211yesmaximal, characteristic, complemented
8#24908790C211yesmaximal, characteristic, complemented
9#24908199C211yesmaximal, characteristic, complemented
10#24908192C211yesmaximal, characteristic, complemented
11#24907660C211yesmaximal, characteristic
12#24907650C211yesmaximal, characteristic
13#24907630C211yesmaximal, characteristic, complemented
14#24907244C211yesmaximal, characteristic
15C83×#1809C211yesmaximal, characteristic
16C2×#14581791C2211yescharacteristic
17C2×#14581791C2211yescharacteristic
18C2×#14581769—21no—
19C2×#14581751C2211yescharacteristic
20C22×C4⋊Dic83C2211yescharacteristic
21#19128105—24no—
22#19127743—24no—
23C2×#14581726C2211yescharacteristic
24C2×#14581726C2211yescharacteristic
25C2×#14581725C2211yescharacteristic
26C2×#14581725C2211yescharacteristic
27C2×#14581725C2211yescharacteristic
28C2×#14581724C2211yescharacteristic
29C2×#14581724C2211yescharacteristic
30C2×#14581723C2211yescharacteristic
31C2×C22⋊(C83⋊C8)C2211yescharacteristic
32C2×#14581702C2211yescharacteristic
33C2×C8×Dic83C2211yescharacteristic
34#19127635—21no—
35#19127624—21no—
36#19127622—21no—
37#19127613—21no—
38#19127605—21no—
39#19127600C2211yescharacteristic
40#19127600C2211yescharacteristic
41#19127597C2211yescharacteristic
42#19127597C2211yescharacteristic
43#19127584—21no—
44#19127573—21no—
45#19127570C2211yescharacteristic
46#19127566C2211yescharacteristic
47C4×C22⋊Dic83C2211yescharacteristic
48#19127463C2211yescharacteristic
49#19127463C2211yescharacteristic
50#19127463C2211yescharacteristic
51#19127463C2211yescharacteristic
52#19127455C2211yescharacteristic
53#19127455C2211yescharacteristic
54C22×M16×C83C2211yescharacteristic
55C2×#107×C83C2211yescharacteristic
56C2×#106×C83C2211yescharacteristic
57#380×C83C2211yescharacteristic
58#380×C83C2211yescharacteristic
59#340×C83C2211yescharacteristic
60#340×C83C2211yescharacteristic
61C22×Dic166—21no—
62C22×Dic166—21no—
63C23×Dic83—21no—
64C22×C83⋊C8C2311yescharacteristic
65C22×C83⋊C8C2311yescharacteristic
66#14581791—22no—
67#14581791—22no—
68#14581791—41no—
69#14581791—41no—
70#14581769—44no—
71#14581751—22no—
72#14581751—41no—
73C2×C4⋊Dic83—22no—
74C2×C4⋊Dic83—21no—
75C2×C4⋊Dic83—21no—
76C2×C4⋊Dic83—21no—
77C2×C4⋊Dic83—41no—
78C2×C4⋊Dic83C2311yescharacteristic
79C2×C4⋊Dic83C2311yescharacteristic
80C2×C4⋊Dic83C2311yescharacteristic
81C2×C4⋊Dic83C2311yescharacteristic
82C2×Dic83⋊C4—21no—
83C2×Dic83⋊C4—21no—
84C2×Dic83⋊C4—21no—
85C2×Dic83⋊C4—21no—
86C2×Dic83⋊C4—21no—
87C2×Dic83⋊C4—21no—
88C2×C22⋊Dic83—21no—
89C2×C22⋊Dic83D411yescharacteristic
90C2×C22⋊Dic83D411yescharacteristic
91C2×C4×Dic83—21no—
92C2×C4×Dic83C2311yescharacteristic
93C2×C4×Dic83C2311yescharacteristic
94#14581726—22no—
95#14581726D414yes—
96#14581725—22no—
97#14581725—22no—
98#14581725—41no—
99#14581724—22no—
100#14581724—22no—
101#14581723—41no—
102C22⋊(C83⋊C8)Q814yescomplemented
103#14581702—22no—
104C8×Dic83—22no—
105C83⋊#95—21no—
106C83⋊#95—21no—
107C83⋊#95—21no—
108C83⋊#95—21no—
109C83⋊#95—21no—
110C83⋊#95—21no—
111C83⋊#95D411yescharacteristic
112C83⋊#95D411yescharacteristic
113C23×C4×C83C2311yescharacteristic
114C2×M16×C83—22no—
115C2×M16×C83—22no—
116C2×M16×C83Q814yescomplemented
117C22×C8×C83C2311yescharacteristic
118C22×C8×C83C2311yescharacteristic
119C2×C4⋊C4×C83C2311yescharacteristic
120C2×C4⋊C4×C83C2311yescharacteristic
121C2×C4⋊C4×C83C2311yescharacteristic
122C2×C4⋊C4×C83C2311yescharacteristic
123C2×C22⋊C4×C83—21no—
124C2×C22⋊C4×C83—21no—
125#107×C83—22no—
126#106×C83—22no—
127#95×C83—21no—
128#95×C83—21no—
129C2×Dic166—42no—
130C2×Dic166—42no—
131C2×Dic166—42no—
132C2×Dic166—42no—
133C2×Dic166—81no—
134C2×Dic166—81no—
135C22×Dic83—21no—
136C22×Dic83—21no—
137C22×Dic83—21no—
138C22×Dic83—21no—
139C22×Dic83—42no—
140C22×Dic83—42no—
141C22×Dic83—81no—
142C22×Dic83C2×D411yescharacteristic
143C22×Dic83C2×D411yescharacteristic
144C22×Dic83C2×D411yescharacteristic
145C22×Dic83C2×D411yescharacteristic
146C2×C83⋊C8—21no—
147C2×C83⋊C8—21no—
148C2×C83⋊C8—21no—
149C2×C83⋊C8—21no—
150C2×C83⋊C8C2×Q812yes—
151C2×C83⋊C8C2×Q812yes—
152C4⋊Dic83—21no—
153C4⋊Dic83—21no—
154C4⋊Dic83—21no—
155C4⋊Dic83—21no—
156C4⋊Dic83—21no—
157C4⋊Dic83—21no—
158C4⋊Dic83—42no—
159C4⋊Dic83—42no—
160C4⋊Dic83—42no—
161C4⋊Dic83C2×D412yes—
162C4⋊Dic83C2×D412yes—
163Dic83⋊C4—42no—
164Dic83⋊C4—42no—
165Dic83⋊C4—42no—
166Dic83⋊C4—42no—
167Dic83⋊C4—81no—
168Dic83⋊C4—81no—
169C22⋊Dic83—22no—
170C22⋊Dic83—22no—
171C22⋊Dic83—42no—
172C4×Dic83—21no—
173C4×Dic83—21no—
174C4×Dic83—21no—
175C4×Dic83—21no—
176C4×Dic83—42no—
177C24×C83C2×D411yescharacteristic
178C22×C4×C83—21no—
179C22×C4×C83—21no—
180C22×C4×C83—21no—
181C22×C4×C83—21no—
182C22×C4×C83—21no—
183C22×C4×C83—21no—
184C22×C4×C83—21no—
185C22×C4×C83—21no—
186C22×C4×C83C2×D411yescharacteristic
187C22×C4×C83C2×Q812yes—
188C22×C4×C83C2×Q812yes—
189C22×C4×C83C2411yescharacteristic
190M16×C83—28no—
191C2×C8×C83—21no—
192C2×C8×C83—21no—
193C2×C8×C83C2×D412yes—
194C2×C8×C83C2×Q812yes—
195C2×C8×C83C2×Q812yes—
196C2×C8×C83#4112yes—
197C4⋊C4×C83—21no—
198C4⋊C4×C83—21no—
199C4⋊C4×C83—21no—
200C4⋊C4×C83—21no—
201C4⋊C4×C83—21no—
202C4⋊C4×C83—21no—
203C4⋊C4×C83—21no—
204C4⋊C4×C83—21no—
205C22⋊C4×C83—42no—
206C22⋊C4×C83—81no—
207C23×C83—21no—
208C23×C83—21no—
209C23×C83—21no—
210C23×C83—21no—
211C23×C83—21no—
212C23×C83—21no—
213C23×C83C4⋊Q812yes—
214C23×C83C22×D411yescharacteristic
215C2×C4×C83—21no—
216C2×C4×C83—21no—
217C2×C4×C83—22no—
218C2×C4×C83—22no—
219C2×C4×C83—22no—
220C2×C4×C83—22no—
221C2×C4×C83—41no—
222C2×C4×C83—41no—
223C2×C4×C83—41no—
224C2×C4×C83—41no—
225C2×C4×C83—41no—
226C2×C4×C83—41no—
227C2×C4×C83—41no—
228C2×C4×C83—41no—
229C2×C4×C83—41no—
230C2×C4×C83—41no—
231C2×C4×C83—41no—
232C2×C4×C83—41no—
233C2×C4×C83C4⋊Q812yes—
234C2×C4×C83C22×D411yescharacteristic
235C2×C4×C83C22×Q811yescharacteristic
236C2×C4×C83C22×Q811yescharacteristic
237C2×C4×C83C2×#4111yescharacteristic
238C2×C4×C83H(5,𝔽2)11yescharacteristic
239C2×C4×C832−1+411yescharacteristic
240C2×Dic83—21no—
241C2×Dic83—21no—
242C2×Dic83—21no—
243C2×Dic83—21no—
244C2×Dic83—22no—
245C2×Dic83—22no—
246C2×Dic83—21no—
247C2×Dic83—21no—
248C2×Dic83—21no—
249C2×Dic83—21no—
250C2×Dic83—21no—
251C2×Dic83—21no—
252C2×Dic83—41no—
253C2×Dic83—41no—
254C2×Dic83—41no—
255C2×Dic83—41no—
256C2×Dic83—41no—
257C2×Dic83—41no—
258C2×Dic83—41no—
259C2×Dic83—41no—
260C2×Dic83—41no—
261C2×Dic83—41no—
262C2×Dic83—84no—
263C2×Dic83AGL(1,ℤ/8)12yes—
264C2×Dic83#13712yes—
265Dic166—84no—
266Dic166—84no—
267C8×C83—22no—
268C8×C83C22⋊Q814yescomplemented
269C83⋊C8—22no—
270C83⋊C8—22no—
271C22×C83—22no—
272C22×C83—22no—
273C22×C83—22no—
274C22×C83—41no—
275C22×C83—41no—
276C22×C83—41no—
277C22×C83—41no—
278C22×C83C2×C4⋊Q811yescharacteristic
279C22×C83D4211yescharacteristic
280C22×C83#54711yescharacteristic
281C22×C83C2×AGL(1,ℤ/8)11yescharacteristic
282C22×C83C2×#13711yescharacteristic
283C22×C83#57611yescharacteristic
284C22×C83#57811yescharacteristic
285C4×C83—22no—
286C4×C83—81no—
287C4×C83—81no—
288C4×C83—81no—
289C4×C83—81no—
290C4×C83C2×C22⋊Q811yescharacteristic
291C4×C83#54411yescharacteristic
292C4×C83D4×Q811yescharacteristic
293C4×C83#55411yescharacteristic
294Dic83—22no—
295Dic83—22no—
296Dic83—41no—
297Dic83—41no—
298Dic83—42no—
299Dic83—42no—
300Dic83—81no—
301Dic83—81no—
302#26074—831nomaximal, Sylow
303C2×C83—42no—
304C2×C83#244011yescharacteristic
305C2×C83#316411yescharacteristic
306C2×C83#316511yescharacteristic
307C2×C83#326411yescharacteristic
308C2×C83#327011yescharacteristic
309C2×C83#335311yescharacteristic
310C2×C83#335711yescharacteristic
311C2×#500—831no—
312C2×#428—831no—
313#2440—831no—
314#2439—831no—
315#2085—831no—
316#2083—831no—
317#2058—831no—
318#2057—831no—
319#1943—831no—
320#1941—831no—
321#1854—831no—
322#1852—831no—
323#1847—831no—
324#1809D8311yescharacteristic
325#1753—831no—
326C83#2607411yescharacteristic, complemented, Sylow
327C22×M16C2×D8311yescharacteristic
328C2×C4⋊Q8—831no—
329C2×C4⋊Q8—831no—
330C2×C22⋊Q8—1661no—
331C2×#117—831no—
332C22×C4⋊C4—831no—
333#500—1664no—
334#428—1664no—
335C2×#107—831no—
336C2×#107—831no—
337C2×#107—831no—
338C2×#107—831no—
339C2×#107—831no—
340C2×#107C2×D8311yescharacteristic
341C2×#106—831no—
342C2×#106—831no—
343C2×#106—831no—
344C2×#106C2×D8311yescharacteristic
345C2×C22⋊C8—831no—
346C2×#97—831no—
347C2×C4×C8—831no—
348#398—1661no—
349#396—1661no—
350#395—1661no—
351#393—1661no—
352#391—1661no—
353#389—831no—
354#389—831no—
355#389—831no—
356#389—831no—
357#387—1661no—
358#385—1661no—
359#384—831no—
360#382—831no—
361#380C2×D8311yescharacteristic
362#380C2×D8311yescharacteristic
363C4×C22⋊C4—831no—
364#340—831no—
365#340—831no—
366#340—831no—
367#340—831no—
368#340C2×D8311yescharacteristic
369#340C2×D8311yescharacteristic
370#336—831no—
371#336—831no—
372C22×Q8—1661no—
373C22×Q8—1661no—
374C23×C4—1661no—
375C23×C4C22×D8311yescharacteristic
376C2×M16—22no—
377C2×M16—22no—
378C2×M16Dic16614yescomplemented
379C22×C8—831no—
380C22×C8—831no—
381C22×C8C22×D8311yescharacteristic
382C22×C8C22×D8311yescharacteristic
383C4⋊Q8—3321no—
384C4⋊Q8—3321no—
385C4⋊Q8—1662no—
386C4⋊Q8—1662no—
387C22⋊Q8—3324no—
388#117—3321no—
389#117—1662no—
390C2×C4⋊C4—3321no—
391C2×C4⋊C4—831no—
392C2×C4⋊C4—831no—
393C2×C4⋊C4—831no—
394C2×C4⋊C4—831no—
395C2×C4⋊C4—1661no—
396C2×C4⋊C4—1661no—
397C2×C4⋊C4—1661no—
398C2×C4⋊C4—1661no—
399C2×C4⋊C4—1661no—
400C2×C4⋊C4—1661no—
401C2×C4⋊C4—1661no—
402C2×C4⋊C4—1662no—
403C2×C4⋊C4—1661no—
404C2×C4⋊C4—1661no—
405C2×C4⋊C4C22×D8311yescharacteristic
406C2×C4⋊C4C22×D8311yescharacteristic
407C2×C4⋊C4C22×D8311yescharacteristic
408C2×C4⋊C4C22×D8311yescharacteristic
409C2×C22⋊C4—21no—
410C2×C22⋊C4—21no—
411C2×C22⋊C4—831no—
412C2×C22⋊C4—831no—
413C2×C22⋊C4—1661no—
414C2×C42—831no—
415C2×C42—831no—
416C2×C42—1661no—
417#107—22no—
418#107—3321no—
419#107—834no—
420#107—1662no—
421#107—1662no—
422#107—1662no—
423#106—22no—
424#106—3321no—
425#106—1662no—
426#106—1662no—
427C22⋊C8—834no—
428#97—1662no—
429C4×C8—1662no—
430#95—21no—
431#95—21no—
432#95—831no—
433#95—831no—
434#95—1661no—
435#95—1661no—
436#95—1661no—
437#95—1661no—
438#95—1661no—
439#95—1661no—
440C24D4×D8311yescharacteristic
441C2×Q8—3322no—
442C2×Q8—3322no—
443C2×Q8—3322no—
444C2×Q8—3322no—
445C2×Q8—6641no—
446C2×Q8—6641no—
447C22×C4—21no—
448C22×C4—21no—
449C22×C4—21no—
450C22×C4—21no—
451C22×C4—21no—
452C22×C4—21no—
453C22×C4—21no—
454C22×C4—21no—
455C22×C4—3322no—
456C22×C4—3322no—
457C22×C4—831no—
458C22×C4—831no—
459C22×C4—831no—
460C22×C4—831no—
461C22×C4—6641no—
462C22×C4—1661no—
463C22×C4—1661no—
464C22×C4—1661no—
465C22×C4—1661no—
466C22×C4D4×D8311yescharacteristic
467C22×C4Q8×D8312yes—
468C22×C4C2×Dic16612yes—
469C22×C4C23×D8311yescharacteristic
470M16—28no—
471C2×C8—21no—
472C2×C8—21no—
473C2×C8—832no—
474C2×C8—832no—
475C2×C8—1661no—
476C2×C8—1661no—
477C2×C8—1661no—
478C2×C8—1661no—
479C2×C8D4×D8312yes—
480C2×C8C2×Dic16612yes—
481C2×C8C2×Dic16612yes—
482C2×C8#1305058912yes—
483C4⋊C4—21no—
484C4⋊C4—21no—
485C4⋊C4—21no—
486C4⋊C4—21no—
487C4⋊C4—21no—
488C4⋊C4—21no—
489C4⋊C4—21no—
490C4⋊C4—21no—
491C4⋊C4—3322no—
492C4⋊C4—3322no—
493C4⋊C4—3322no—
494C4⋊C4—3322no—
495C4⋊C4—3322no—
496C4⋊C4—3322no—
497C4⋊C4—3322no—
498C4⋊C4—832no—
499C4⋊C4—832no—
500C4⋊C4—6641no—
501C4⋊C4—6641no—
502C4⋊C4—1661no—
503C4⋊C4—1661no—
504C4⋊C4—1661no—
505C4⋊C4—1661no—
506C4⋊C4—1661no—
507C4⋊C4—1661no—
508C22⋊C4—42no—
509C22⋊C4—81no—
510C22⋊C4—3322no—
511C22⋊C4—1662no—
512C22⋊C4—1662no—
513C42—3322no—
514C42—1661no—
515C42—1661no—
516C42—1661no—
517C42—1661no—
518C23—21no—
519C23—21no—
520C23—21no—
521C23—21no—
522C23—21no—
523C23—21no—
524C23#1458179112yes—
525C23C2×D4×D8311yescharacteristic
526Q8—6644no—
527Q8—6644no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—3321no—
547C2×C4—3321no—
548C2×C4—3321no—
549C2×C4—3321no—
550C2×C4—3321no—
551C2×C4—3321no—
552C2×C4—3321no—
553C2×C4—3321no—
554C2×C4—3321no—
555C2×C4—3321no—
556C2×C4—832no—
557C2×C4—832no—
558C2×C4—6644no—
559C2×C4—1662no—
560C2×C4—1662no—
561C2×C4—1661no—
562C2×C4—1661no—
563C2×C4—1661no—
564C2×C4—1661no—
565C2×C4—1661no—
566C2×C4—1661no—
567C2×C4—1661no—
568C2×C4—1661no—
569C2×C4—1661no—
570C2×C4—1661no—
571C2×C4#1458179112yes—
572C2×C4C2×D4×D8311yescharacteristic
573C2×C4C2×Q8×D8311yescharacteristic
574C2×C4C22×Dic16611yescharacteristic
575C2×C4C2×#1305058911yescharacteristic
576C2×C4#1458183811yescharacteristic
577C2×C4#1458183911yescharacteristic
578C8—22no—
579C8—1662no—
580C8—1662no—
581C8#1458176914yescomplemented
582C22—22no—
583C22—22no—
584C22—22no—
585C22—41no—
586C22—41no—
587C22—41no—
588C22—41no—
589C22C2×#1458179111yescharacteristic
590C22#1912833511yescharacteristic
591C22#1912835111yescharacteristic
592C22AGL(1,ℤ/8)×D8311yescharacteristic
593C22#137×D8311yescharacteristic
594C22#1912850511yescharacteristic
595C22#1912851711yescharacteristic
596C4—22no—
597C4—81no—
598C4—81no—
599C4—81no—
600C4—81no—
601C4—3321no—
602C4—3322no—
603C4—3322no—
604C4—3321no—
605C4—6641no—
606C4—6641no—
607C4—1662no—
608C4—1662no—
609C4C2×#1458176911yescharacteristic
610C4#1912832911yescharacteristic
611C4#1912836411yescharacteristic
612C4#1912840911yescharacteristic
613C2—42no—
614C2#2491116611yescharacteristic
615C2#2491800111yescharacteristic
616C2#2491800711yescharacteristic
617C2#2491895011yescharacteristic
618C2#2491904011yescharacteristic
619C2#2492004211yescharacteristic
620C2#2492008411yescharacteristic
subgroupG/H
centerC23C2×D4×D83
commutator, cosocleC22×C4×C83C24
FrattiniC22×C4C23×D83
FittingC83×#1809C2
socleC23×C83C22×D4
prime radicalC24×C83C2×D4
hypercenter#1809D83
hypocenterC83#26074
LCSG ▷ C22×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1809

Complements

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

NH ≅ G/Ntype
1. C2×#19128105C2semidirect
5. #24908949C2semidirect
6. #24908938C2semidirect
7. #24908795C2semidirect
8. #24908790C2semidirect
9. #24908199C2semidirect
10. #24908192C2semidirect
13. #24907630C2semidirect
102. C22⋊(C83⋊C8)Q8semidirect
116. C2×M16×C83Q8semidirect
268. C8×C83C22⋊Q8semidirect
326. C83#26074semidirect
378. C2×M16Dic166semidirect
581. C8#14581769semidirect

Maximal subgroups

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

subgroupindex
C2×#191281052
C2×#191277432
#249111662
#249111552
#249089492
#249089382
#249087952
#249087902
#249081992
#249081922
#249076602
#249076502
#249076302
#249072442
C83×#18092
#2607483