Finite Group Databasebeta

← #35462397

Subgroups — #35462397

Order 21248.

732 conjugacy classes of subgroups.

The subgroup lattice has 590 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#24911170C211yesmaximal, characteristic
2#24910332C211yesmaximal, characteristic
3#24909029C211yesmaximal, characteristic, complemented
4#24908966C211yesmaximal, characteristic, complemented
5#24908953C211yesmaximal, characteristic, complemented
6#24908800C211yesmaximal, characteristic, complemented
7#24908417C211yesmaximal, characteristic, complemented
8#24908225C211yesmaximal, characteristic, complemented
9#24908054C211yesmaximal, characteristic, complemented
10#24908037C211yesmaximal, characteristic, complemented
11#24907662C211yesmaximal, characteristic
12#24907646C211yesmaximal, characteristic
13#24907553C211yesmaximal, characteristic
14#24907496C211yesmaximal, characteristic
15C83×#1736C211yesmaximal, characteristic
16C2×#14581783C2211yescharacteristic
17C2×#14581751—21no—
18C2×#14581726C2211yescharacteristic
19C2×#14581725C2211yescharacteristic
20C2×#14581724C2211yescharacteristic
21C2×#14581723C2211yescharacteristic
22C2×Dic83⋊C8C2211yescharacteristic
23C2×Dic83⋊C8C2211yescharacteristic
24C2×C22⋊(C83⋊C8)C2211yescharacteristic
25#19127637—21no—
26#19127624—21no—
27#19127600C2211yescharacteristic
28#19127600C2211yescharacteristic
29#19127600C2211yescharacteristic
30#19127599—21no—
31#19127597C2211yescharacteristic
32#19127587—21no—
33#19127587—21no—
34#19127585—21no—
35#19127570C2211yescharacteristic
36#19127564—21no—
37#19127561C2211yescharacteristic
38#19127561C2211yescharacteristic
39C4⋊C4×Dic83C2211yescharacteristic
40C4⋊C4×Dic83C2211yescharacteristic
41C4×C22⋊Dic83C2211yescharacteristic
42#19127464C2211yescharacteristic
43#19127464C2211yescharacteristic
44#19127463C2211yescharacteristic
45#19127463C2211yescharacteristic
46#19127463C2211yescharacteristic
47#19127463C2211yescharacteristic
48#19127463C2211yescharacteristic
49#19127463C2211yescharacteristic
50#19127455C2211yescharacteristic
51#19127455C2211yescharacteristic
52C22×M16×C83C2211yescharacteristic
53C2×#117×C83C2211yescharacteristic
54C22×C4⋊C4×C83C2211yescharacteristic
55#340×C83C2211yescharacteristic
56#340×C83C2211yescharacteristic
57#340×C83C2211yescharacteristic
58#340×C83C2211yescharacteristic
59C22×C83⋊C8C2311yescharacteristic
60C22×C83⋊C8C2311yescharacteristic
61#14581783—22no—
62#14581783—41no—
63#14581751—44no—
64C2×C4⋊Dic83—21no—
65C2×C4⋊Dic83—21no—
66C2×C4⋊Dic83—21no—
67C2×C4⋊Dic83—21no—
68C2×C4⋊Dic83C2311yescharacteristic
69C2×C4⋊Dic83C2311yescharacteristic
70C2×C4⋊Dic83C2311yescharacteristic
71C2×C4⋊Dic83C2311yescharacteristic
72C2×Dic83⋊C4—21no—
73C2×Dic83⋊C4—21no—
74C2×Dic83⋊C4—21no—
75C2×Dic83⋊C4—21no—
76C2×C22⋊Dic83—21no—
77C2×C22⋊Dic83—21no—
78C2×C22⋊Dic83D411yescharacteristic
79C2×C22⋊Dic83D411yescharacteristic
80C2×C4×Dic83—21no—
81C2×C4×Dic83—21no—
82C2×C4×Dic83C2311yescharacteristic
83C2×C4×Dic83C2311yescharacteristic
84#14581726—22no—
85#14581725—22no—
86#14581724—41no—
87#14581723—41no—
88Dic83⋊C8—22no—
89Dic83⋊C8—41no—
90C22⋊(C83⋊C8)—22no—
91C83⋊#95—21no—
92C83⋊#95—21no—
93C83⋊#95—21no—
94C83⋊#95—21no—
95C83⋊#95—21no—
96C83⋊#95—21no—
97C83⋊#95—21no—
98C83⋊#95—21no—
99C83⋊#95D411yescharacteristic
100C83⋊#95D411yescharacteristic
101C23×C4×C83—21no—
102C23×C4×C83C2311yescharacteristic
103C2×M16×C83—22no—
104C2×M16×C83—22no—
105C2×M16×C83—41no—
106C22×C8×C83C2311yescharacteristic
107C22×C8×C83C2311yescharacteristic
108#117×C83—22no—
109#117×C83—41no—
110C2×C4⋊C4×C83—21no—
111C2×C4⋊C4×C83—22no—
112C2×C4⋊C4×C83Q814yes—
113C2×C4⋊C4×C83C2311yescharacteristic
114C2×C4⋊C4×C83C2311yescharacteristic
115C2×C4⋊C4×C83C2311yescharacteristic
116C2×C4⋊C4×C83C2311yescharacteristic
117C2×C22⋊C4×C83—21no—
118C2×C42×C83—21no—
119C22×Dic83—21no—
120C22×Dic83—21no—
121C22×Dic83—21no—
122C22×Dic83—21no—
123C22×Dic83C2×D411yescharacteristic
124C22×Dic83C2×D411yescharacteristic
125C22×Dic83C2×D411yescharacteristic
126C22×Dic83C2×D411yescharacteristic
127C2×C83⋊C8—21no—
128C2×C83⋊C8—21no—
129C2×C83⋊C8—21no—
130C2×C83⋊C8—21no—
131C2×C83⋊C8—21no—
132C2×C83⋊C8—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—42no—
142C4⋊Dic83—42no—
143C4⋊Dic83—42no—
144C4⋊Dic83—42no—
145Dic83⋊C4—42no—
146Dic83⋊C4—42no—
147Dic83⋊C4—81no—
148Dic83⋊C4—81no—
149C22⋊Dic83—22no—
150C22⋊Dic83—41no—
151C22⋊Dic83—42no—
152C22⋊Dic83—81no—
153C4×Dic83—21no—
154C4×Dic83—21no—
155C4×Dic83—21no—
156C4×Dic83—21no—
157C4×Dic83—42no—
158C4×Dic83—42no—
159C24×C83C2×D411yescharacteristic
160C22×C4×C83—21no—
161C22×C4×C83—21no—
162C22×C4×C83—21no—
163C22×C4×C83—21no—
164C22×C4×C83—24no—
165C22×C4×C83—21no—
166C22×C4×C83—21no—
167C22×C4×C83—21no—
168C22×C4×C83—21no—
169C22×C4×C83—42no—
170C22×C4×C83—42no—
171C22×C4×C83C2×D411yescharacteristic
172C22×C4×C83C2×Q812yes—
173C22×C4×C83#4112yes—
174C22×C4×C83C2411yescharacteristic
175M16×C83—44no—
176C2×C8×C83—21no—
177C2×C8×C83—21no—
178C2×C8×C83—21no—
179C2×C8×C83—21no—
180C2×C8×C83—21no—
181C2×C8×C83#4112yes—
182C4⋊C4×C83—24no—
183C4⋊C4×C83—21no—
184C4⋊C4×C83—21no—
185C4⋊C4×C83—21no—
186C4⋊C4×C83—21no—
187C4⋊C4×C83—21no—
188C4⋊C4×C83C2×Q812yes—
189C4⋊C4×C83C2×Q812yes—
190C4⋊C4×C83#4112yes—
191C22⋊C4×C83—42no—
192C42×C83—42no—
193C23×C83—21no—
194C23×C83—21no—
195C23×C83—21no—
196C23×C83—21no—
197C23×C83—21no—
198C23×C83—21no—
199C23×C83C22⋊Q812yes—
200C23×C83C22×D411yescharacteristic
201C2×C4×C83—21no—
202C2×C4×C83—21no—
203C2×C4×C83—22no—
204C2×C4×C83—22no—
205C2×C4×C83—22no—
206C2×C4×C83—22no—
207C2×C4×C83—22no—
208C2×C4×C83—22no—
209C2×C4×C83—22no—
210C2×C4×C83—48no—
211C2×C4×C83—41no—
212C2×C4×C83—41no—
213C2×C4×C83—41no—
214C2×C4×C83—41no—
215C2×C4×C83—41no—
216C2×C4×C83—41no—
217C2×C4×C83—41no—
218C2×C4×C83—41no—
219C2×C4×C83—41no—
220C2×C4×C83C22⋊Q812yes—
221C2×C4×C83C22×Q811yescharacteristic
222C2×C4×C83C2×#4111yescharacteristic
223C2×C4×C83C2×#4111yescharacteristic
224C2×C4×C83C2×#4111yescharacteristic
225C2×C4×C83H(5,𝔽2)11yescharacteristic
226C2×C4×C832−1+411yescharacteristic
227C2×Dic83—21no—
228C2×Dic83—21no—
229C2×Dic83—21no—
230C2×Dic83—21no—
231C2×Dic83—21no—
232C2×Dic83—21no—
233C2×Dic83—21no—
234C2×Dic83—21no—
235C2×Dic83—21no—
236C2×Dic83—21no—
237C2×Dic83—21no—
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—
250C2×Dic83AGL(1,ℤ/8)12yes—
251C8×C83—22no—
252C8×C83—41no—
253C83⋊C8—41no—
254C83⋊C8—41no—
255C22×C83—22no—
256C22×C83—22no—
257C22×C83—22no—
258C22×C83—41no—
259C22×C83—41no—
260C22×C83—41no—
261C22×C83—41no—
262C22×C83C2×C22⋊Q811yescharacteristic
263C22×C83#54611yescharacteristic
264C22×C83#54711yescharacteristic
265C22×C83C2×AGL(1,ℤ/8)11yescharacteristic
266C22×C83#57511yescharacteristic
267C22×C83#57711yescharacteristic
268C22×C83#57811yescharacteristic
269C4×C83—22no—
270C4×C83—42no—
271C4×C83—42no—
272C4×C83—42no—
273C4×C83—81no—
274C4×C83#53311yescharacteristic
275C4×C83#54411yescharacteristic
276C4×C83#55111yescharacteristic
277C4×C83#55211yescharacteristic
278Dic83—22no—
279Dic83—41no—
280Dic83—41no—
281Dic83—41no—
282Dic83—81no—
283Dic83—81no—
284Dic83—81no—
285Dic83—81no—
286#26014—831nomaximal, Sylow
287C2×C83—42no—
288C2×C83#243811yescharacteristic
289C2×C83#307711yescharacteristic
290C2×C83#308211yescharacteristic
291C2×C83#331711yescharacteristic
292C2×C83#333011yescharacteristic
293C2×C83#334511yescharacteristic
294C2×C83#335211yescharacteristic
295#2441—831no—
296#2344—831no—
297#2100—831no—
298#2088—831no—
299#2086—831no—
300#2059—831no—
301#1989—831no—
302#1948—831no—
303#1921—831no—
304#1918—831no—
305#1854—831no—
306#1851—831no—
307#1824—831no—
308#1809—831no—
309#1736D8311yescharacteristic
310C83#2601411yescharacteristic, complemented, Sylow
311C22×M16C2×D8311yescharacteristic
312C2×#125—831no—
313C2×#117—1661no—
314C2×#117C2×D8311yescharacteristic
315C22×C4⋊C4C2×D8311yescharacteristic
316C2×#107—831no—
317C2×#107—831no—
318C2×#106—831no—
319C2×#106—831no—
320C2×C4⋊C8—831no—
321C2×C4⋊C8—831no—
322C2×C22⋊C8—831no—
323#399—1661no—
324#396—1661no—
325#389—831no—
326#389—831no—
327#389—831no—
328#389—831no—
329#389—1661no—
330#387—1661no—
331#387—1661no—
332#387—1661no—
333#384—831no—
334#381—1661no—
335#380—831no—
336#380—831no—
337C4×C4⋊C4—831no—
338C4×C4⋊C4—831no—
339C4×C22⋊C4—831no—
340#340—831no—
341#340—831no—
342#340—831no—
343#340—831no—
344#340—831no—
345#340—831no—
346#340—831no—
347#340—831no—
348#340C2×D8311yescharacteristic
349#340C2×D8311yescharacteristic
350#340C2×D8311yescharacteristic
351#340C2×D8311yescharacteristic
352#336—831no—
353#336—831no—
354C23×C4—21no—
355C23×C4C22×D8311yescharacteristic
356C2×M16—22no—
357C2×M16—22no—
358C2×M16—41no—
359C22×C8—831no—
360C22×C8—831no—
361C22×C8C22×D8311yescharacteristic
362C22×C8C22×D8311yescharacteristic
363#125—3321no—
364#125—1662no—
365#117—22no—
366#117—41no—
367#117—3324no—
368C2×C4⋊C4—21no—
369C2×C4⋊C4—22no—
370C2×C4⋊C4—831no—
371C2×C4⋊C4—831no—
372C2×C4⋊C4—831no—
373C2×C4⋊C4—831no—
374C2×C4⋊C4—1661no—
375C2×C4⋊C4—1661no—
376C2×C4⋊C4—1661no—
377C2×C4⋊C4—1661no—
378C2×C4⋊C4—1661no—
379C2×C4⋊C4—1661no—
380C2×C4⋊C4—1661no—
381C2×C4⋊C4—1661no—
382C2×C4⋊C4Dic16614yes—
383C2×C4⋊C4C22×D8311yescharacteristic
384C2×C4⋊C4C22×D8311yescharacteristic
385C2×C4⋊C4C22×D8311yescharacteristic
386C2×C4⋊C4C22×D8311yescharacteristic
387C2×C22⋊C4—21no—
388C2×C22⋊C4—831no—
389C2×C22⋊C4—831no—
390C2×C22⋊C4—1661no—
391C2×C22⋊C4—1661no—
392C2×C42—21no—
393C2×C42—831no—
394C2×C42—831no—
395C2×C42—1661no—
396C2×C42—1661no—
397#107—1662no—
398#107—1662no—
399#106—3321no—
400#106—3321no—
401C4⋊C8—3321no—
402C4⋊C8—1662no—
403C22⋊C8—1662no—
404#95—831no—
405#95—831no—
406#95—1661no—
407#95—1661no—
408#95—1661no—
409#95—1661no—
410#95—1661no—
411#95—1661no—
412#95—1661no—
413#95—1661no—
414C24D4×D8311yescharacteristic
415C22×C4—21no—
416C22×C4—21no—
417C22×C4—21no—
418C22×C4—21no—
419C22×C4—24no—
420C22×C4—21no—
421C22×C4—21no—
422C22×C4—21no—
423C22×C4—21no—
424C22×C4—42no—
425C22×C4—42no—
426C22×C4—831no—
427C22×C4—831no—
428C22×C4—831no—
429C22×C4—831no—
430C22×C4—1661no—
431C22×C4—1661no—
432C22×C4—1661no—
433C22×C4—1661no—
434C22×C4D4×D8311yescharacteristic
435C22×C4C2×Dic16612yes—
436C22×C4#1305058912yes—
437C22×C4C23×D8311yescharacteristic
438M16—44no—
439C2×C8—21no—
440C2×C8—21no—
441C2×C8—21no—
442C2×C8—21no—
443C2×C8—21no—
444C2×C8—1661no—
445C2×C8—1661no—
446C2×C8—1661no—
447C2×C8—1661no—
448C2×C8—1661no—
449C2×C8—1661no—
450C2×C8#1305059012yes—
451C4⋊C4—24no—
452C4⋊C4—21no—
453C4⋊C4—21no—
454C4⋊C4—21no—
455C4⋊C4—21no—
456C4⋊C4—21no—
457C4⋊C4—3322no—
458C4⋊C4—3322no—
459C4⋊C4—3322no—
460C4⋊C4—3322no—
461C4⋊C4—3322no—
462C4⋊C4—3322no—
463C4⋊C4—6641no—
464C4⋊C4—6641no—
465C4⋊C4—1661no—
466C4⋊C4—1661no—
467C4⋊C4—1661no—
468C4⋊C4—1661no—
469C4⋊C4—1661no—
470C4⋊C4—1661no—
471C4⋊C4—1661no—
472C4⋊C4—1661no—
473C4⋊C4C2×Dic16612yes—
474C4⋊C4C2×Dic16612yes—
475C4⋊C4#1305059012yes—
476C22⋊C4—42no—
477C22⋊C4—3322no—
478C22⋊C4—3321no—
479C22⋊C4—6641no—
480C22⋊C4—1662no—
481C42—42no—
482C42—3322no—
483C42—3322no—
484C42—1661no—
485C42—1661no—
486C42—1661no—
487C42—1661no—
488C23—21no—
489C23—21no—
490C23—21no—
491C23—21no—
492C23—21no—
493C23—21no—
494C23#1458176912yes—
495C23C2×D4×D8311yescharacteristic
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—48no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—41no—
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—832no—
528C2×C4—1661no—
529C2×C4—1661no—
530C2×C4—1661no—
531C2×C4—1661no—
532C2×C4—1661no—
533C2×C4—1661no—
534C2×C4—1661no—
535C2×C4—1661no—
536C2×C4—1661no—
537C2×C4—1661no—
538C2×C4—1661no—
539C2×C4#1458176912yes—
540C2×C4C22×Dic16611yescharacteristic
541C2×C4C2×#1305058911yescharacteristic
542C2×C4C2×#1305059011yescharacteristic
543C2×C4C2×#1305059011yescharacteristic
544C2×C4#1458183711yescharacteristic
545C2×C4#1458184011yescharacteristic
546C8—22no—
547C8—41no—
548C8—3321no—
549C8—3321no—
550C22—22no—
551C22—22no—
552C22—22no—
553C22—41no—
554C22—41no—
555C22—41no—
556C22—41no—
557C22C2×#1458176911yescharacteristic
558C22#1912834111yescharacteristic
559C22#1912835211yescharacteristic
560C22AGL(1,ℤ/8)×D8311yescharacteristic
561C22#1912849711yescharacteristic
562C22#1912851011yescharacteristic
563C22#1912851711yescharacteristic
564C4—22no—
565C4—42no—
566C4—42no—
567C4—42no—
568C4—81no—
569C4—3321no—
570C4—3321no—
571C4—3321no—
572C4—6641no—
573C4—6641no—
574C4—6641no—
575C4—6641no—
576C4—1662no—
577C4#1912825811yescharacteristic
578C4#1912833211yescharacteristic
579C4#1912837711yescharacteristic
580C4#1912838611yescharacteristic
581C2—42no—
582C2#2491113911yescharacteristic
583C2#2491725711yescharacteristic
584C2#2491730711yescharacteristic
585C2#2491959011yescharacteristic
586C2#2491976411yescharacteristic
587C2#2491993411yescharacteristic
588C2#2492003411yescharacteristic
subgroupG/H
centerC23C2×D4×D83
commutator, cosocleC22×C4×C83C24
FrattiniC22×C4C23×D83
FittingC83×#1736C2
socleC23×C83C22×D4
prime radicalC24×C83C2×D4
hypercenter#1736D83
hypocenterC83#26014
LCSG ▷ C22×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1736

Complements

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

NH ≅ G/Ntype
3. #24909029C2semidirect
4. #24908966C2semidirect
5. #24908953C2semidirect
6. #24908800C2semidirect
7. #24908417C2semidirect
8. #24908225C2semidirect
9. #24908054C2semidirect
10. #24908037C2semidirect
310. C83#26014semidirect

Maximal subgroups

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

subgroupindex
#249111702
#249103322
#249090292
#249089662
#249089532
#249088002
#249084172
#249082252
#249080542
#249080372
#249076622
#249076462
#249075532
#249074962
C83×#17362
#2601483