Finite Group Databasebeta

← #35422147

Subgroups — #35422147

Order 21248.

810 conjugacy classes of subgroups.

The subgroup lattice has 588 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19127743C211yesmaximal, characteristic
2#24910354C211yesmaximal, characteristic
3#24910330C211yesmaximal, characteristic
4#24908201C211yesmaximal, characteristic, complemented
5#24908196C211yesmaximal, characteristic, complemented
6#24908056C211yesmaximal, characteristic, complemented
7#24908049C211yesmaximal, characteristic, complemented
8#24908042C211yesmaximal, characteristic, complemented
9#24908035C211yesmaximal, characteristic, complemented
10#24907559C211yesmaximal, characteristic
11#24907554C211yesmaximal, characteristic
12#24907496C211yesmaximal, characteristic
13#24907360C211yesmaximal, characteristic, complemented
14C4×#14581724C211yesmaximal, characteristic, complemented
15C83×#1753C211yesmaximal, characteristic
16C2×#14581751C2211yescharacteristic
17C2×#14581748—21no—
18C22×C4⋊Dic83C2211yescharacteristic
19#19127743C418yescomplemented
20C2×#14581726C2211yescharacteristic
21C2×#14581726C2211yescharacteristic
22C2×#14581726C2211yescharacteristic
23C2×#14581724C2211yescharacteristic
24C2×#14581724C2211yescharacteristic
25C2×#14581724C2211yescharacteristic
26#19127599C2211yescharacteristic
27#19127599C2211yescharacteristic
28#19127599C2211yescharacteristic
29#19127599C2211yescharacteristic
30#19127595—21no—
31#19127595—21no—
32#19127586—21no—
33#19127586—21no—
34#19127568—21no—
35#19127562—21no—
36#19127561C2211yescharacteristic
37#19127561C2211yescharacteristic
38C4×C4⋊Dic83C2211yescharacteristic
39C4×C4⋊Dic83C2211yescharacteristic
40C4×C4⋊Dic83C2211yescharacteristic
41C4×C4⋊Dic83C2211yescharacteristic
42#19127552—21no—
43#19127464C2211yescharacteristic
44#19127464C2211yescharacteristic
45#19127464C2211yescharacteristic
46#19127464C2211yescharacteristic
47#19127464C2211yescharacteristic
48#19127464C2211yescharacteristic
49#19127464C2211yescharacteristic
50#19127464C2211yescharacteristic
51#19127464C2211yescharacteristic
52#19127464C2211yescharacteristic
53C22×M16×C83C2211yescharacteristic
54C2×C22⋊C8×C83C2211yescharacteristic
55C2×#97×C83C2211yescharacteristic
56C2×C4×C8×C83C2211yescharacteristic
57C4×C22⋊C4×C83C2211yescharacteristic
58#336×C83C2211yescharacteristic
59#336×C83C2211yescharacteristic
60C23×Dic83—21no—
61#14581751—22no—
62#14581751C2×C414yes—
63#14581748—44no—
64C2×C4⋊Dic83—21no—
65C2×C4⋊Dic83—22no—
66C2×C4⋊Dic83C2×C414yes—
67C2×C4⋊Dic83C2311yescharacteristic
68C2×C4⋊Dic83C2311yescharacteristic
69C2×C4⋊Dic83C2311yescharacteristic
70C2×C4⋊Dic83C2311yescharacteristic
71C2×C4⋊Dic83C2311yescharacteristic
72C2×C4⋊Dic83C2311yescharacteristic
73C2×C4⋊Dic83C2311yescharacteristic
74C2×C4⋊Dic83C2311yescharacteristic
75C2×Dic83⋊C4—21no—
76C2×Dic83⋊C4—21no—
77C2×Dic83⋊C4—21no—
78C2×Dic83⋊C4—21no—
79C2×C22⋊Dic83—21no—
80C2×C22⋊Dic83—21no—
81C2×C22⋊Dic83—21no—
82C2×C4×Dic83—21no—
83C2×C4×Dic83—21no—
84C2×C4×Dic83—21no—
85C2×C4×Dic83—21no—
86C2×C4×Dic83—21no—
87#14581726—22no—
88#14581726C2×C414yes—
89#14581726C2×C414yes—
90#14581724—22no—
91#14581724C2×C414yes—
92#14581724C2×C414yes—
93C83⋊#95—21no—
94C83⋊#95—21no—
95C83⋊#95—21no—
96C83⋊#95—21no—
97C83⋊#95—21no—
98C83⋊#95—21no—
99C83⋊#95—21no—
100C83⋊#95—21no—
101C83⋊#95—21no—
102C83⋊#95—21no—
103C23×C4×C83C2311yescharacteristic
104C2×M16×C83—22no—
105C2×M16×C83—22no—
106C2×M16×C83C2×C414yes—
107C22×C8×C83C2311yescharacteristic
108C22×C8×C83C2311yescharacteristic
109C22×C8×C83C2311yescharacteristic
110C22×C8×C83C2311yescharacteristic
111C2×C22⋊C4×C83D411yescharacteristic
112C2×C22⋊C4×C83D411yescharacteristic
113C2×C42×C83C2311yescharacteristic
114C2×C42×C83C2311yescharacteristic
115C22⋊C8×C83—22no—
116#97×C83—22no—
117C4×C8×C83—22no—
118#95×C83D411yescharacteristic
119#95×C83D411yescharacteristic
120C22×Dic83—24no—
121C22×Dic83—21no—
122C22×Dic83—21no—
123C22×Dic83—21no—
124C22×Dic83—21no—
125C22×Dic83—21no—
126C22×Dic83—21no—
127C22×Dic83—21no—
128C22×Dic83—21no—
129C22×Dic83—42no—
130C22×Dic83—42no—
131C4⋊Dic83—24no—
132C4⋊Dic83—21no—
133C4⋊Dic83—21no—
134C4⋊Dic83—21no—
135C4⋊Dic83—21no—
136C4⋊Dic83—21no—
137C4⋊Dic83—21no—
138C4⋊Dic83—21no—
139C4⋊Dic83—21no—
140C4⋊Dic83—21no—
141C4⋊Dic83—21no—
142C4⋊Dic83—21no—
143C4⋊Dic83—21no—
144C4⋊Dic83C22×C412yes—
145C4⋊Dic83C22×C412yes—
146C4⋊Dic83C22×C412yes—
147C4⋊Dic83C22×C412yes—
148Dic83⋊C4—42no—
149Dic83⋊C4—42no—
150Dic83⋊C4—42no—
151Dic83⋊C4—42no—
152C22⋊Dic83—24no—
153C22⋊Dic83—42no—
154C22⋊Dic83—81no—
155C4×Dic83—24no—
156C4×Dic83—42no—
157C4×Dic83—42no—
158C4×Dic83—81no—
159C4×Dic83—81no—
160C24×C83C2×D411yescharacteristic
161C22×C4×C83—21no—
162C22×C4×C83—21no—
163C22×C4×C83—21no—
164C22×C4×C83—21no—
165C22×C4×C83C22×C412yes—
166C22×C4×C83C2×D411yescharacteristic
167C22×C4×C83C2×D411yescharacteristic
168C22×C4×C83C2×D411yescharacteristic
169C22×C4×C83C2×D411yescharacteristic
170C22×C4×C83C2×D411yescharacteristic
171C22×C4×C83#4112yes—
172C22×C4×C83C2411yescharacteristic
173M16×C83—28no—
174C2×C8×C83—21no—
175C2×C8×C83—21no—
176C2×C8×C83—21no—
177C2×C8×C83—21no—
178C2×C8×C83—21no—
179C2×C8×C83—21no—
180C2×C8×C83—21no—
181C2×C8×C83—21no—
182C2×C8×C83C22×C412yes—
183C2×C8×C83C22×C412yes—
184C2×C8×C83#4112yes—
185C2×C8×C83#4112yes—
186C22⋊C4×C83—22no—
187C22⋊C4×C83—41no—
188C42×C83—21no—
189C42×C83—21no—
190C42×C83—21no—
191C42×C83—21no—
192C23×C83—21no—
193C23×C83—21no—
194C23×C83—21no—
195C23×C83—21no—
196C23×C83—21no—
197C23×C83—21no—
198C23×C83C4×D412yes—
199C23×C83C22×D411yescharacteristic
200C2×C4×C83—21no—
201C2×C4×C83—21no—
202C2×C4×C83—21no—
203C2×C4×C83—21no—
204C2×C4×C83—22no—
205C2×C4×C83—22no—
206C2×C4×C83—22no—
207C2×C4×C83—22no—
208C2×C4×C83—21no—
209C2×C4×C83—21no—
210C2×C4×C83—21no—
211C2×C4×C83—21no—
212C2×C4×C83—21no—
213C2×C4×C83—21no—
214C2×C4×C83—21no—
215C2×C4×C83—21no—
216C2×C4×C83—21no—
217C2×C4×C83C4×D412yes—
218C2×C4×C83AGL(1,ℤ/8)12yes—
219C2×C4×C83C23×C411yescharacteristic
220C2×C4×C83C2×#4111yescharacteristic
221C2×C4×C83C2×#4111yescharacteristic
222C2×C4×C83C2×#4111yescharacteristic
223C2×C4×C832−1+411yescharacteristic
224C2×C4×C832−1+411yescharacteristic
225C2×Dic83—22no—
226C2×Dic83—22no—
227C2×Dic83—22no—
228C2×Dic83—22no—
229C2×Dic83—48no—
230C2×Dic83—41no—
231C2×Dic83—41no—
232C2×Dic83—41no—
233C2×Dic83—41no—
234C2×Dic83—41no—
235C2×Dic83—41no—
236C2×Dic83—41no—
237C2×Dic83—41no—
238C2×Dic83—41no—
239C2×Dic83—41no—
240C2×Dic83—41no—
241C2×Dic83—41no—
242C2×Dic83—41no—
243C2×Dic83—41no—
244C2×Dic83—41no—
245C2×Dic83—41no—
246C2×Dic83—41no—
247C2×Dic83—41no—
248C2×Dic83—41no—
249C2×Dic83—41no—
250C8×C83—22no—
251C8×C83—41no—
252C8×C83—41no—
253C8×C83#11714yescomplemented
254C22×C83—22no—
255C22×C83—22no—
256C22×C83—22no—
257C22×C83—41no—
258C22×C83—41no—
259C22×C83—41no—
260C22×C83—41no—
261C22×C83C2×C4×D411yescharacteristic
262C22×C83#54711yescharacteristic
263C22×C83#54711yescharacteristic
264C22×C83C2×AGL(1,ℤ/8)11yescharacteristic
265C22×C83#57511yescharacteristic
266C22×C83#57711yescharacteristic
267C22×C83#57811yescharacteristic
268C4×C83—22no—
269C4×C83—22no—
270C4×C83—41no—
271C4×C83—41no—
272C4×C83—41no—
273C4×C83C2×#11711yescharacteristic
274C4×C83#51911yescharacteristic
275C4×C83#55211yescharacteristic
276C4×C83#55211yescharacteristic
277Dic83—42no—
278Dic83—42no—
279Dic83—42no—
280Dic83—42no—
281Dic83—81no—
282Dic83—81no—
283Dic83—81no—
284Dic83—81no—
285#22622—831nomaximal, Sylow
286C2×C83—42no—
287C2×C83#234411yescharacteristic
288C2×C83C4×AGL(1,ℤ/8)11yescharacteristic
289C2×C83#294711yescharacteristic
290C2×C83#334511yescharacteristic
291C2×C83#334911yescharacteristic
292C2×C83#335111yescharacteristic
293C2×C83#335311yescharacteristic
294C2×#428—831no—
295#2347—831no—
296#2344—831no—
297#1943—831no—
298#1942—831no—
299#1921—831no—
300#1920—831no—
301#1919—831no—
302#1918—831no—
303#1825—831no—
304#1824—831no—
305#1809—831no—
306#1776—831no—
307C4×#106—831no—
308#1753D8311yescharacteristic
309C83#2262211yescharacteristic, complemented, Sylow
310C22×M16C2×D8311yescharacteristic
311C2×#117—831no—
312C2×#117—1661no—
313C22×C4⋊C4—831no—
314#428—838no—
315C2×#107—831no—
316C2×#107—831no—
317C2×#107—831no—
318C2×#106—831no—
319C2×#106—831no—
320C2×#106—831no—
321C2×C22⋊C8C2×D8311yescharacteristic
322C2×#97C2×D8311yescharacteristic
323C2×C4×C8C2×D8311yescharacteristic
324#389—831no—
325#389—831no—
326#389—831no—
327#389—831no—
328#388—1661no—
329#388—1661no—
330#387—1661no—
331#387—1661no—
332#383—1661no—
333#381—1661no—
334#380—831no—
335#380—831no—
336C4×C4⋊C4—831no—
337C4×C4⋊C4—831no—
338C4×C4⋊C4—831no—
339C4×C4⋊C4—831no—
340C4×C4⋊C4—1661no—
341C4×C22⋊C4C2×D8311yescharacteristic
342#340—831no—
343#340—831no—
344#340—831no—
345#340—831no—
346#340—831no—
347#340—831no—
348#340—831no—
349#340—831no—
350#340—831no—
351#340—831no—
352#336C2×D8311yescharacteristic
353#336C2×D8311yescharacteristic
354C23×C4—1661no—
355C23×C4C22×D8311yescharacteristic
356C2×M16—22no—
357C2×M16—22no—
358C2×M16C4×D8314yes—
359C22×C8C22×D8311yescharacteristic
360C22×C8C22×D8311yescharacteristic
361C22×C8C22×D8311yescharacteristic
362C22×C8C22×D8311yescharacteristic
363#117—3324no—
364#117—834no—
365#117—1662no—
366C2×C4⋊C4—834no—
367C2×C4⋊C4—831no—
368C2×C4⋊C4—831no—
369C2×C4⋊C4—831no—
370C2×C4⋊C4—831no—
371C2×C4⋊C4—831no—
372C2×C4⋊C4—831no—
373C2×C4⋊C4—831no—
374C2×C4⋊C4—831no—
375C2×C4⋊C4—1661no—
376C2×C4⋊C4—1661no—
377C2×C4⋊C4—1661no—
378C2×C4⋊C4—1661no—
379C2×C4⋊C4—1661no—
380C2×C4⋊C4—1662no—
381C2×C22⋊C4—1661no—
382C2×C22⋊C4—1661no—
383C2×C22⋊C4—1661no—
384C2×C22⋊C4D33211yescharacteristic
385C2×C22⋊C4D33211yescharacteristic
386C2×C42—1661no—
387C2×C42—1661no—
388C2×C42—1661no—
389C2×C42—1661no—
390C2×C42—1661no—
391C2×C42C22×D8311yescharacteristic
392C2×C42C22×D8311yescharacteristic
393#107—834no—
394#107—834no—
395#107—1662no—
396#106—834no—
397#106—834no—
398#106—1662no—
399C22⋊C8—22no—
400#97—22no—
401C4×C8—22no—
402#95—1661no—
403#95—1661no—
404#95—1661no—
405#95—1661no—
406#95—1661no—
407#95—1661no—
408#95—1661no—
409#95—1661no—
410#95—1661no—
411#95—1661no—
412#95D33211yescharacteristic
413#95D33211yescharacteristic
414C24C2×D33211yescharacteristic
415C22×C4—21no—
416C22×C4—21no—
417C22×C4—21no—
418C22×C4—21no—
419C22×C4—3322no—
420C22×C4—3322no—
421C22×C4—1661no—
422C22×C4—1661no—
423C22×C4—1661no—
424C22×C4—1661no—
425C22×C4—1661no—
426C22×C4—1661no—
427C22×C4—1661no—
428C22×C4—1661no—
429C22×C4—1664no—
430C22×C4C2×C4×D8312yes—
431C22×C4C2×D33211yescharacteristic
432C22×C4C2×D33211yescharacteristic
433C22×C4C2×D33211yescharacteristic
434C22×C4C2×D33211yescharacteristic
435C22×C4C2×D33211yescharacteristic
436C22×C4#1305059012yes—
437C22×C4C23×D8311yescharacteristic
438M16—28no—
439C2×C8—21no—
440C2×C8—21no—
441C2×C8—21no—
442C2×C8—21no—
443C2×C8—21no—
444C2×C8—21no—
445C2×C8—21no—
446C2×C8—21no—
447C2×C8C2×C4×D8312yes—
448C2×C8C2×C4×D8312yes—
449C2×C8#1305058912yes—
450C2×C8#1305058912yes—
451C4⋊C4—3322no—
452C4⋊C4—3322no—
453C4⋊C4—3322no—
454C4⋊C4—3322no—
455C4⋊C4—832no—
456C4⋊C4—832no—
457C4⋊C4—832no—
458C4⋊C4—832no—
459C4⋊C4—1661no—
460C4⋊C4—1661no—
461C4⋊C4—1661no—
462C4⋊C4—1661no—
463C4⋊C4—1661no—
464C4⋊C4—1661no—
465C4⋊C4—1661no—
466C4⋊C4—1661no—
467C4⋊C4—1661no—
468C4⋊C4—1661no—
469C4⋊C4—1661no—
470C4⋊C4—1661no—
471C4⋊C4—1664no—
472C22⋊C4—22no—
473C22⋊C4—41no—
474C22⋊C4—3322no—
475C22⋊C4—6641no—
476C22⋊C4—1664no—
477C42—21no—
478C42—21no—
479C42—21no—
480C42—21no—
481C42—3322no—
482C42—3322no—
483C42—6641no—
484C42—6641no—
485C42—1664no—
486C23—21no—
487C23—21no—
488C23—21no—
489C23—21no—
490C23—21no—
491C23—21no—
492C23C4×D33212yes—
493C23C22×D33211yescharacteristic
494C2×C4—21no—
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—3328no—
512C2×C4—3321no—
513C2×C4—3321no—
514C2×C4—3321no—
515C2×C4—3321no—
516C2×C4—3321no—
517C2×C4—3321no—
518C2×C4—3321no—
519C2×C4—3321no—
520C2×C4—3321no—
521C2×C4—3321no—
522C2×C4—3321no—
523C2×C4—3321no—
524C2×C4—3321no—
525C2×C4—3321no—
526C2×C4—3321no—
527C2×C4—3321no—
528C2×C4—3321no—
529C2×C4—3321no—
530C2×C4—3321no—
531C2×C4—3321no—
532C2×C4—1662no—
533C2×C4—1662no—
534C2×C4—1662no—
535C2×C4—1662no—
536C2×C4C4×D33212yes—
537C2×C4#1458182012yes—
538C2×C4C22×C4×D8311yescharacteristic
539C2×C4C2×#1305058911yescharacteristic
540C2×C4C2×#1305058911yescharacteristic
541C2×C4C2×#1305059011yescharacteristic
542C2×C4#1458183911yescharacteristic
543C2×C4#1458183911yescharacteristic
544C8—22no—
545C8—41no—
546C8—41no—
547C8#1458174814yescomplemented
548C22—22no—
549C22—22no—
550C22—22no—
551C22—41no—
552C22—41no—
553C22—41no—
554C22—41no—
555C22C2×C4×D33211yescharacteristic
556C22#1912835511yescharacteristic
557C22#1912835511yescharacteristic
558C22C2×#1458182011yescharacteristic
559C22#1912850111yescharacteristic
560C22#1912851311yescharacteristic
561C22#1912851811yescharacteristic
562C4—22no—
563C4—22no—
564C4—41no—
565C4—41no—
566C4—41no—
567C4—3322no—
568C4—3322no—
569C4—3322no—
570C4—3322no—
571C4—6641no—
572C4—6641no—
573C4—6641no—
574C4—6641no—
575C4C2×#1458174811yescharacteristic
576C4#1912817811yescharacteristic
577C4#1912839111yescharacteristic
578C4#1912839111yescharacteristic
579C2—42no—
580C2#2491033011yescharacteristic
581C2C4×#1458182011yescharacteristic
582C2#2491630811yescharacteristic
583C2#2491993511yescharacteristic
584C2#2491999511yescharacteristic
585C2#2492002511yescharacteristic
586C2#2492004811yescharacteristic
subgroupG/H
centerC23C22×D332
commutatorC2×C4×C83C23×C4
FrattiniC22×C4C23×D83
FittingC83×#1753C2
socleC23×C83C22×D4
prime radicalC24×C83C2×D4
cosocleC22×C4×C83C24
hypercenter#1753D83
hypocenterC83#22622
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #1753

Complements

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

NH ≅ G/Ntype
4. #24908201C2semidirect
5. #24908196C2semidirect
6. #24908056C2semidirect
7. #24908049C2semidirect
8. #24908042C2semidirect
9. #24908035C2semidirect
13. #24907360C2semidirect
14. C4×#14581724C2semidirect
19. #19127743C4semidirect
253. C8×C83#117semidirect
309. C83#22622semidirect
547. C8#14581748semidirect

Maximal subgroups

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

subgroupindex
C2×#191277432
#249103542
#249103302
#249082012
#249081962
#249080562
#249080492
#249080422
#249080352
#249075592
#249075542
#249074962
#249073602
C4×#145817242
C83×#17532
#2262283