Finite Group Databasebeta

← #35463161

Subgroups — #35463161

Order 21248.

838 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×#19128111C211yesmaximal, characteristic, complemented
2C22×#14581725C211yesmaximal, characteristic
3#24911165C211yesmaximal, characteristic
4#24908945C211yesmaximal, characteristic, complemented
5#24908934C211yesmaximal, characteristic, complemented
6#24908798C211yesmaximal, characteristic, complemented
7#24908791C211yesmaximal, characteristic, complemented
8#24908203C211yesmaximal, characteristic, complemented
9#24908198C211yesmaximal, characteristic, complemented
10#24907655C211yesmaximal, characteristic
11#24907651C211yesmaximal, characteristic
12#24907632C211yesmaximal, characteristic, complemented
13#24907486C211yesmaximal, characteristic
14#24907241C211yesmaximal, characteristic
15C83×#2442C211yesmaximal, characteristic
16C23×C83⋊C8C2211yescharacteristic
17C2×#14581792C2211yescharacteristic
18C2×#14581769—21no—
19C22×C4⋊Dic83C2211yescharacteristic
20#19128111—24no—
21C2×#14581725—24no—
22C2×#14581725C2211yescharacteristic
23C2×#14581725C2211yescharacteristic
24C2×#14581725C2211yescharacteristic
25C2×#14581725C2211yescharacteristic
26C2×#14581725C2211yescharacteristic
27C2×#14581725C2211yescharacteristic
28C2×#14581723C2211yescharacteristic
29C2×#14581723C2211yescharacteristic
30C2×#14581723C2211yescharacteristic
31C2×#14581723C2211yescharacteristic
32C2×C22⋊(C83⋊C8)C2211yescharacteristic
33C2×#14581703C2211yescharacteristic
34C2×#14581703C2211yescharacteristic
35#19127633—21no—
36#19127622—21no—
37#19127599C2211yescharacteristic
38#19127599C2211yescharacteristic
39#19127576—21no—
40#19127571C2211yescharacteristic
41#19127561C2211yescharacteristic
42#19127463C2211yescharacteristic
43#19127463C2211yescharacteristic
44#19127463C2211yescharacteristic
45#19127463C2211yescharacteristic
46#19127463C2211yescharacteristic
47#19127463C2211yescharacteristic
48#19127456C2211yescharacteristic
49#19127456C2211yescharacteristic
50C2×#125×C83C2211yescharacteristic
51C22×C4⋊C4×C83C2211yescharacteristic
52#400×C83—21no—
53#398×C83—21no—
54#389×C83C2211yescharacteristic
55#389×C83C2211yescharacteristic
56#385×C83—21no—
57#382×C83C2211yescharacteristic
58#380×C83—21no—
59#380×C83C2211yescharacteristic
60C4×C22⋊C4×C83C2211yescharacteristic
61C22×Dic166—21no—
62C23×Dic83—21no—
63C22×C83⋊C8—22no—
64C22×C83⋊C8—22no—
65C22×C83⋊C8Q814yescomplemented
66C22×C83⋊C8C2311yescharacteristic
67C22×C83⋊C8C2311yescharacteristic
68C22×C83⋊C8C2311yescharacteristic
69C22×C83⋊C8C2311yescharacteristic
70#14581792—22no—
71#14581792—41no—
72#14581769—44no—
73C2×C4⋊Dic83—22no—
74C2×C4⋊Dic83—21no—
75C2×C4⋊Dic83—21no—
76C2×C4⋊Dic83—21no—
77C2×C4⋊Dic83—21no—
78C2×C4⋊Dic83—41no—
79C2×C4⋊Dic83C2311yescharacteristic
80C2×C4⋊Dic83C2311yescharacteristic
81C2×C4⋊Dic83C2311yescharacteristic
82C2×C4⋊Dic83C2311yescharacteristic
83C2×Dic83⋊C4—21no—
84C2×Dic83⋊C4—21no—
85C2×C22⋊Dic83—21no—
86#14581725—22no—
87#14581725—22no—
88#14581725—22no—
89#14581725—22no—
90#14581725—41no—
91#14581725D414yes—
92#14581723—22no—
93#14581723—22no—
94#14581723—22no—
95#14581723—41no—
96C22⋊(C83⋊C8)Q814yescomplemented
97#14581703—22no—
98#14581703—22no—
99C83⋊#95—21no—
100C83⋊#95—21no—
101C83⋊#95—21no—
102C23×C4×C83—21no—
103C23×C4×C83C2311yescharacteristic
104#125×C83—22no—
105#125×C83—41no—
106C2×C4⋊C4×C83—22no—
107C2×C4⋊C4×C83—21no—
108C2×C4⋊C4×C83—21no—
109C2×C4⋊C4×C83—21no—
110C2×C4⋊C4×C83—21no—
111C2×C4⋊C4×C83—21no—
112C2×C4⋊C4×C83—41no—
113C2×C4⋊C4×C83C2311yescharacteristic
114C2×C4⋊C4×C83C2311yescharacteristic
115C2×C4⋊C4×C83C2311yescharacteristic
116C2×C4⋊C4×C83C2311yescharacteristic
117C2×C22⋊C4×C83—21no—
118C2×C22⋊C4×C83D411yescharacteristic
119C2×C22⋊C4×C83D411yescharacteristic
120C2×C42×C83C2311yescharacteristic
121C2×C42×C83C2311yescharacteristic
122#95×C83—21no—
123#95×C83—21no—
124#95×C83—21no—
125#95×C83—21no—
126#95×C83—21no—
127#95×C83D411yescharacteristic
128#95×C83D411yescharacteristic
129C2×Dic166—42no—
130C2×Dic166—42no—
131C2×Dic166—81no—
132C22×Dic83—21no—
133C22×Dic83—21no—
134C22×Dic83—21no—
135C22×Dic83—21no—
136C22×Dic83—42no—
137C22×Dic83—42no—
138C22×Dic83—81no—
139C2×C83⋊C8—21no—
140C2×C83⋊C8—21no—
141C2×C83⋊C8—21no—
142C2×C83⋊C8—21no—
143C2×C83⋊C8—21no—
144C2×C83⋊C8—21no—
145C2×C83⋊C8—28no—
146C2×C83⋊C8C2×D412yes—
147C2×C83⋊C8C2×Q812yes—
148C2×C83⋊C8C2×Q812yes—
149C2×C83⋊C8C2×Q812yes—
150C2×C83⋊C8C2×Q812yes—
151C2×C83⋊C8#4112yes—
152C4⋊Dic83—21no—
153C4⋊Dic83—21no—
154C4⋊Dic83—21no—
155C4⋊Dic83—21no—
156C4⋊Dic83—21no—
157C4⋊Dic83—21no—
158C4⋊Dic83—21no—
159C4⋊Dic83—42no—
160C4⋊Dic83—42no—
161C4⋊Dic83—42no—
162C4⋊Dic83—81no—
163C4⋊Dic83C2×D412yes—
164Dic83⋊C4—42no—
165Dic83⋊C4—42no—
166C22⋊Dic83—42no—
167C24×C83C2×D411yescharacteristic
168C22×C4×C83—21no—
169C22×C4×C83—21no—
170C22×C4×C83—21no—
171C22×C4×C83—21no—
172C22×C4×C83—21no—
173C22×C4×C83—21no—
174C22×C4×C83—21no—
175C22×C4×C83—21no—
176C22×C4×C83—42no—
177C22×C4×C83—42no—
178C22×C4×C83—81no—
179C22×C4×C83C2×D411yescharacteristic
180C22×C4×C83C2×D411yescharacteristic
181C22×C4×C83C2×D411yescharacteristic
182C22×C4×C83C2×D411yescharacteristic
183C22×C4×C83C2×D411yescharacteristic
184C22×C4×C83C2×Q812yes—
185C22×C4×C83C2×Q812yes—
186C22×C4×C83C2411yescharacteristic
187C4⋊C4×C83—21no—
188C4⋊C4×C83—21no—
189C4⋊C4×C83—21no—
190C4⋊C4×C83—21no—
191C4⋊C4×C83—21no—
192C4⋊C4×C83—21no—
193C4⋊C4×C83—21no—
194C4⋊C4×C83—42no—
195C4⋊C4×C83—42no—
196C4⋊C4×C83—42no—
197C4⋊C4×C83—81no—
198C4⋊C4×C83—81no—
199C4⋊C4×C83C2×D412yes—
200C22⋊C4×C83—22no—
201C22⋊C4×C83—22no—
202C22⋊C4×C83—42no—
203C42×C83—21no—
204C42×C83—21no—
205C42×C83—21no—
206C42×C83—21no—
207C23×C83—21no—
208C23×C83—21no—
209C23×C83—21no—
210C23×C83—21no—
211C23×C83C4⋊Q812yes—
212C23×C83AGL(1,ℤ/8)12yes—
213C23×C83#13712yes—
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—21no—
222C2×C4×C83—21no—
223C2×C4×C83—22no—
224C2×C4×C83—21no—
225C2×C4×C83—21no—
226C2×C4×C83—21no—
227C2×C4×C83—21no—
228C2×C4×C83—21no—
229C2×C4×C83—21no—
230C2×C4×C83—21no—
231C2×C4×C83—21no—
232C2×C4×C83—21no—
233C2×C4×C83—21no—
234C2×C4×C83—41no—
235C2×C4×C83—41no—
236C2×C4×C83—41no—
237C2×C4×C83—41no—
238C2×C4×C83—41no—
239C2×C4×C83—41no—
240C2×C4×C83—41no—
241C2×C4×C83—41no—
242C2×C4×C83—41no—
243C2×C4×C83—41no—
244C2×C4×C83—41no—
245C2×C4×C83—84no—
246C2×C4×C83C4⋊Q812yes—
247C2×C4×C83C22×D411yescharacteristic
248C2×C4×C83C22×Q811yescharacteristic
249C2×C4×C83C22×Q811yescharacteristic
250C2×C4×C83C2×#4111yescharacteristic
251C2×C4×C83H(5,𝔽2)11yescharacteristic
252C2×C4×C832−1+411yescharacteristic
253C2×Dic83—22no—
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—41no—
263C2×Dic83—41no—
264C2×Dic83—41no—
265C2×Dic83—84no—
266Dic166—84no—
267C83⋊C8—22no—
268C83⋊C8—22no—
269C83⋊C8—22no—
270C83⋊C8C22⋊Q814yescomplemented
271C22×C83—22no—
272C22×C83—22no—
273C22×C83—22no—
274C22×C83—22no—
275C22×C83—22no—
276C22×C83—22no—
277C22×C83#50114yescomplemented
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—41no—
287C4×C83—41no—
288C4×C83—41no—
289C4×C83—42no—
290C4×C83—41no—
291C4×C83—81no—
292C4×C83—81no—
293C4×C83—81no—
294C4×C83C2×C22⋊Q811yescharacteristic
295C4×C83#54411yescharacteristic
296C4×C83D4×Q811yescharacteristic
297C4×C83#55411yescharacteristic
298Dic83—42no—
299Dic83—81no—
300Dic83—81no—
301Dic83—81no—
302#26071—831nomaximal, Sylow
303C2×C83—24no—
304C2×C83#244011yescharacteristic
305C2×C83C2×#50111yescharacteristic
306C2×C83#316411yescharacteristic
307C2×C83#326411yescharacteristic
308C2×C83#327011yescharacteristic
309C2×C83#334311yescharacteristic
310C2×C83#334611yescharacteristic
311C2×#501—831no—
312C22×#107—831no—
313#2442D8311yescharacteristic
314#2440—831no—
315#2085—831no—
316#2083—831no—
317#2059—831no—
318#2057—831no—
319#1944—831no—
320#1943—831no—
321#1853—831no—
322#1852—831no—
323#1847—831no—
324#1807—831no—
325#1752—831no—
326C83#2607111yescharacteristic, complemented, Sylow
327C23×C8—831no—
328C2×C4⋊Q8—831no—
329C2×#125C2×D8311yescharacteristic
330C2×C22⋊Q8—1661no—
331C22×C4⋊C4—831no—
332C22×C4⋊C4C2×D8311yescharacteristic
333#501—1664no—
334C2×#107—831no—
335C2×#107—831no—
336C2×#107—831no—
337C2×#107—831no—
338C2×#107—831no—
339C2×#107—831no—
340C2×#107—1664no—
341C2×#106—831no—
342C2×#106—831no—
343C2×#106—831no—
344C2×#106—831no—
345C2×C22⋊C8—831no—
346C2×#97—831no—
347C2×#97—831no—
348#400—21no—
349#398—21no—
350#398—1661no—
351#395—1661no—
352#389—831no—
353#389—831no—
354#389C2×D8311yescharacteristic
355#389C2×D8311yescharacteristic
356#385—21no—
357#385—1661no—
358#384—831no—
359#382C2×D8311yescharacteristic
360#380—21no—
361#380—831no—
362#380C2×D8311yescharacteristic
363C4×C22⋊C4C2×D8311yescharacteristic
364#340—831no—
365#340—831no—
366#340—831no—
367#340—831no—
368#340—831no—
369#340—831no—
370#336—831no—
371#336—831no—
372C22×Q8—1661no—
373C23×C4—21no—
374C23×C4—1661no—
375C23×C4C22×D8311yescharacteristic
376C22×C8—831no—
377C22×C8—834no—
378C22×C8—831no—
379C22×C8—831no—
380C22×C8—831no—
381C22×C8—1662no—
382C22×C8—1662no—
383C4⋊Q8—3321no—
384C4⋊Q8—1662no—
385#125—22no—
386#125—41no—
387C22⋊Q8—3324no—
388C2×C4⋊C4—22no—
389C2×C4⋊C4—21no—
390C2×C4⋊C4—21no—
391C2×C4⋊C4—21no—
392C2×C4⋊C4—21no—
393C2×C4⋊C4—21no—
394C2×C4⋊C4—41no—
395C2×C4⋊C4—3321no—
396C2×C4⋊C4—831no—
397C2×C4⋊C4—831no—
398C2×C4⋊C4—831no—
399C2×C4⋊C4—831no—
400C2×C4⋊C4—1661no—
401C2×C4⋊C4—1661no—
402C2×C4⋊C4—1661no—
403C2×C4⋊C4—1661no—
404C2×C4⋊C4—1661no—
405C2×C4⋊C4—1662no—
406C2×C4⋊C4—1661no—
407C2×C4⋊C4C22×D8311yescharacteristic
408C2×C4⋊C4C22×D8311yescharacteristic
409C2×C4⋊C4C22×D8311yescharacteristic
410C2×C4⋊C4C22×D8311yescharacteristic
411C2×C22⋊C4—21no—
412C2×C22⋊C4—1661no—
413C2×C22⋊C4C22⋊D8311yescharacteristic
414C2×C22⋊C4C22⋊D8311yescharacteristic
415C2×C42C22×D8311yescharacteristic
416C2×C42C22×D8311yescharacteristic
417#107—3321no—
418#107—834no—
419#107—1662no—
420#107—1662no—
421#107—1662no—
422#107—1662no—
423#106—3321no—
424#106—1662no—
425#106—1662no—
426#106—1662no—
427C22⋊C8—834no—
428#97—1662no—
429#97—1662no—
430#95—21no—
431#95—21no—
432#95—21no—
433#95—21no—
434#95—21no—
435#95—1661no—
436#95—1661no—
437#95—1661no—
438#95C22⋊D8311yescharacteristic
439#95C22⋊D8311yescharacteristic
440C24C2×C22⋊D8311yescharacteristic
441C2×Q8—3322no—
442C2×Q8—3322no—
443C2×Q8—6641no—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—21no—
447C22×C4—21no—
448C22×C4—21no—
449C22×C4—21no—
450C22×C4—21no—
451C22×C4—21no—
452C22×C4—42no—
453C22×C4—42no—
454C22×C4—81no—
455C22×C4—3322no—
456C22×C4—3322no—
457C22×C4—6641no—
458C22×C4—1661no—
459C22×C4—1661no—
460C22×C4—1661no—
461C22×C4—1661no—
462C22×C4C2×C22⋊D8311yescharacteristic
463C22×C4C2×C22⋊D8311yescharacteristic
464C22×C4C2×C22⋊D8311yescharacteristic
465C22×C4C2×C22⋊D8311yescharacteristic
466C22×C4C2×C22⋊D8311yescharacteristic
467C22×C4Q8×D8312yes—
468C22×C4Q8×D8312yes—
469C22×C4C23×D8311yescharacteristic
470C2×C8—832no—
471C2×C8—832no—
472C2×C8—832no—
473C2×C8—832no—
474C2×C8—832no—
475C2×C8—832no—
476C2×C8—1668no—
477C2×C8—1661no—
478C2×C8—1661no—
479C2×C8—1661no—
480C2×C8—1661no—
481C2×C8—1661no—
482C2×C8—1661no—
483C4⋊C4—21no—
484C4⋊C4—21no—
485C4⋊C4—21no—
486C4⋊C4—21no—
487C4⋊C4—21no—
488C4⋊C4—21no—
489C4⋊C4—21no—
490C4⋊C4—42no—
491C4⋊C4—42no—
492C4⋊C4—42no—
493C4⋊C4—81no—
494C4⋊C4—81no—
495C4⋊C4—3322no—
496C4⋊C4—3322no—
497C4⋊C4—3322no—
498C4⋊C4—3322no—
499C4⋊C4—3322no—
500C4⋊C4—832no—
501C4⋊C4—6641no—
502C4⋊C4—1661no—
503C4⋊C4—1661no—
504C4⋊C4—1661no—
505C4⋊C4—1661no—
506C4⋊C4—1661no—
507C4⋊C4—1661no—
508C4⋊C4—1661no—
509C4⋊C4D4×D8312yes—
510C22⋊C4—22no—
511C22⋊C4—22no—
512C22⋊C4—42no—
513C22⋊C4—3322no—
514C42—21no—
515C42—21no—
516C42—21no—
517C42—21no—
518C23—21no—
519C23—21no—
520C23—21no—
521C23—21no—
522C23#1458179012yes—
523C23#1458181712yes—
524C23#1458182212yes—
525C23C22×C22⋊D8311yescharacteristic
526Q8—6644no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—22no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—84no—
558C2×C4—3321no—
559C2×C4—3321no—
560C2×C4—3321no—
561C2×C4—3321no—
562C2×C4—3321no—
563C2×C4—3321no—
564C2×C4—3321no—
565C2×C4—3321no—
566C2×C4—3321no—
567C2×C4—3321no—
568C2×C4—3321no—
569C2×C4—6644no—
570C2×C4—1662no—
571C2×C4#1458179012yes—
572C2×C4C2×D4×D8311yescharacteristic
573C2×C4C2×Q8×D8311yescharacteristic
574C2×C4C2×Q8×D8311yescharacteristic
575C2×C4#41×D8311yescharacteristic
576C2×C4#1458183711yescharacteristic
577C2×C4#1458183911yescharacteristic
578C8—834no—
579C8—1662no—
580C8—1662no—
581C8—1662no—
582C22—22no—
583C22—22no—
584C22—22no—
585C22—22no—
586C22—22no—
587C22—22no—
588C22#1912811114yescomplemented
589C22C2×#1458179011yescharacteristic
590C22D4×C22⋊D8311yescharacteristic
591C22#1912835011yescharacteristic
592C22C2×#1458181711yescharacteristic
593C22C2×#1458182211yescharacteristic
594C22#1912850411yescharacteristic
595C22#1912851611yescharacteristic
596C4—22no—
597C4—41no—
598C4—41no—
599C4—41no—
600C4—42no—
601C4—41no—
602C4—81no—
603C4—81no—
604C4—81no—
605C4—3322no—
606C4—6641no—
607C4—6641no—
608C4—6641no—
609C4C22⋊Q8×D8311yescharacteristic
610C4#1912833011yescharacteristic
611C4#1912836411yescharacteristic
612C4#1912840511yescharacteristic
613C2—24no—
614C2#2491116211yescharacteristic
615C2C2×#1912811111yescharacteristic
616C2#2491799711yescharacteristic
617C2#2491894411yescharacteristic
618C2#2491902911yescharacteristic
619C2#2491990311yescharacteristic
620C2#2491994811yescharacteristic
subgroupG/H
centerC23C22×C22⋊D83
commutator, cosocleC22×C4×C83C24
FrattiniC22×C4C23×D83
FittingC83×#2442C2
socleC23×C83C22×D4
prime radicalC24×C83C2×D4
hypercenter#2442D83
hypocenterC83#26071
LCSG ▷ C22×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #2442

Complements

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

NH ≅ G/Ntype
1. C2×#19128111C2semidirect
4. #24908945C2semidirect
5. #24908934C2semidirect
6. #24908798C2semidirect
7. #24908791C2semidirect
8. #24908203C2semidirect
9. #24908198C2semidirect
12. #24907632C2semidirect
65. C22×C83⋊C8Q8semidirect
96. C22⋊(C83⋊C8)Q8semidirect
270. C83⋊C8C22⋊Q8semidirect
277. C22×C83#501semidirect
326. C83#26071semidirect
588. C22#19128111semidirect

Maximal subgroups

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

subgroupindex
C2×#191281112
C22×#145817252
#249111652
#249089452
#249089342
#249087982
#249087912
#249082032
#249081982
#249076552
#249076512
#249076322
#249074862
#249072412
C83×#24422
#2607183