Finite Group Databasebeta

← #35462472

Subgroups — #35462472

Order 21248.

696 conjugacy classes of subgroups.

The subgroup lattice has 584 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#24911181C211yesmaximal, characteristic
2#24910348C211yesmaximal, characteristic
3#24909034C211yesmaximal, characteristic, complemented
4#24908967C211yesmaximal, characteristic, complemented
5#24908963C211yesmaximal, characteristic, complemented
6#24908804C211yesmaximal, characteristic, complemented
7#24908417C211yesmaximal, characteristic, complemented
8#24908234C211yesmaximal, characteristic, complemented
9#24908044C211yesmaximal, characteristic, complemented
10#24908040C211yesmaximal, characteristic, complemented
11#24907656C211yesmaximal, characteristic
12#24907646C211yesmaximal, characteristic
13#24907558C211yesmaximal, characteristic
14#24907496C211yesmaximal, characteristic
15C83×#1736C211yesmaximal, characteristic
16C2×#14581783C2211yescharacteristic
17C2×#14581726C2211yescharacteristic
18C2×#14581724C2211yescharacteristic
19C2×#14581723C2211yescharacteristic
20C2×#14581723C2211yescharacteristic
21C2×Dic83⋊C8C2211yescharacteristic
22C2×Dic83⋊C8C2211yescharacteristic
23C2×C22⋊(C83⋊C8)C2211yescharacteristic
24#19127640—21no—
25#19127634—21no—
26#19127600C2211yescharacteristic
27#19127600C2211yescharacteristic
28#19127600C2211yescharacteristic
29#19127597C2211yescharacteristic
30#19127594—21no—
31#19127594—21no—
32#19127574—21no—
33#19127567—21no—
34#19127566C2211yescharacteristic
35#19127561—21no—
36#19127561C2211yescharacteristic
37#19127561C2211yescharacteristic
38C4⋊C4×Dic83C2211yescharacteristic
39C4⋊C4×Dic83C2211yescharacteristic
40C4×C22⋊Dic83C2211yescharacteristic
41#19127547—21no—
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—
63C2×C4⋊Dic83—21no—
64C2×C4⋊Dic83—21no—
65C2×C4⋊Dic83—21no—
66C2×C4⋊Dic83—21no—
67C2×C4⋊Dic83C2311yescharacteristic
68C2×C4⋊Dic83C2311yescharacteristic
69C2×C4⋊Dic83C2311yescharacteristic
70C2×C4⋊Dic83C2311yescharacteristic
71C2×Dic83⋊C4—21no—
72C2×Dic83⋊C4—21no—
73C2×C22⋊Dic83—21no—
74C2×C22⋊Dic83—21no—
75C2×C22⋊Dic83D411yescharacteristic
76C2×C22⋊Dic83D411yescharacteristic
77C2×C4×Dic83—21no—
78C2×C4×Dic83—21no—
79C2×C4×Dic83C2311yescharacteristic
80C2×C4×Dic83C2311yescharacteristic
81#14581726—41no—
82#14581724—22no—
83#14581723—22no—
84#14581723—41no—
85Dic83⋊C8—22no—
86Dic83⋊C8—41no—
87C22⋊(C83⋊C8)—22no—
88C83⋊#95—21no—
89C83⋊#95—21no—
90C83⋊#95—21no—
91C83⋊#95—21no—
92C83⋊#95—21no—
93C83⋊#95—21no—
94C83⋊#95—21no—
95C83⋊#95—21no—
96C83⋊#95—21no—
97C83⋊#95—21no—
98C83⋊#95D411yescharacteristic
99C83⋊#95D411yescharacteristic
100C23×C4×C83—21no—
101C23×C4×C83C2311yescharacteristic
102C2×M16×C83—22no—
103C2×M16×C83—22no—
104C2×M16×C83—41no—
105C22×C8×C83C2311yescharacteristic
106C22×C8×C83C2311yescharacteristic
107#117×C83—22no—
108#117×C83Q814yes—
109C2×C4⋊C4×C83—22no—
110C2×C4⋊C4×C83—21no—
111C2×C4⋊C4×C83—41no—
112C2×C4⋊C4×C83C2311yescharacteristic
113C2×C4⋊C4×C83C2311yescharacteristic
114C2×C4⋊C4×C83C2311yescharacteristic
115C2×C4⋊C4×C83C2311yescharacteristic
116C2×C22⋊C4×C83—21no—
117C2×C42×C83—21no—
118C22×Dic83—21no—
119C22×Dic83—21no—
120C22×Dic83—21no—
121C22×Dic83—21no—
122C22×Dic83C2×D411yescharacteristic
123C22×Dic83C2×D411yescharacteristic
124C22×Dic83C2×D411yescharacteristic
125C22×Dic83C2×D411yescharacteristic
126C2×C83⋊C8—21no—
127C2×C83⋊C8—21no—
128C2×C83⋊C8—21no—
129C2×C83⋊C8—21no—
130C2×C83⋊C8—21no—
131C2×C83⋊C8—21no—
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—42no—
141C4⋊Dic83—81no—
142C4⋊Dic83—81no—
143C4⋊Dic83—81no—
144Dic83⋊C4—42no—
145Dic83⋊C4—81no—
146C22⋊Dic83—41no—
147C22⋊Dic83—41no—
148C22⋊Dic83—42no—
149C22⋊Dic83—81no—
150C4×Dic83—21no—
151C4×Dic83—21no—
152C4×Dic83—21no—
153C4×Dic83—21no—
154C4×Dic83—42no—
155C4×Dic83—42no—
156C24×C83C2×D411yescharacteristic
157C22×C4×C83—21no—
158C22×C4×C83—21no—
159C22×C4×C83—21no—
160C22×C4×C83—21no—
161C22×C4×C83—21no—
162C22×C4×C83—21no—
163C22×C4×C83—21no—
164C22×C4×C83—21no—
165C22×C4×C83—42no—
166C22×C4×C83—42no—
167C22×C4×C83—81no—
168C22×C4×C83C2×D411yescharacteristic
169C22×C4×C83C2×Q812yes—
170C22×C4×C83#4112yes—
171C22×C4×C83C2411yescharacteristic
172M16×C83—44no—
173C2×C8×C83—21no—
174C2×C8×C83—21no—
175C2×C8×C83—21no—
176C2×C8×C83—21no—
177C2×C8×C83—21no—
178C2×C8×C83#4112yes—
179C4⋊C4×C83—21no—
180C4⋊C4×C83—21no—
181C4⋊C4×C83—21no—
182C4⋊C4×C83—21no—
183C4⋊C4×C83—21no—
184C4⋊C4×C83—42no—
185C4⋊C4×C83C2×Q812yes—
186C4⋊C4×C83C2×Q812yes—
187C4⋊C4×C83#4112yes—
188C22⋊C4×C83—24no—
189C42×C83—24no—
190C23×C83—21no—
191C23×C83—21no—
192C23×C83—21no—
193C23×C83—21no—
194C23×C83—21no—
195C23×C83—21no—
196C23×C83C22⋊Q812yes—
197C23×C83C22×D411yescharacteristic
198C2×C4×C83—21no—
199C2×C4×C83—21no—
200C2×C4×C83—22no—
201C2×C4×C83—22no—
202C2×C4×C83—22no—
203C2×C4×C83—22no—
204C2×C4×C83—22no—
205C2×C4×C83—22no—
206C2×C4×C83—22no—
207C2×C4×C83—41no—
208C2×C4×C83—41no—
209C2×C4×C83—41no—
210C2×C4×C83—41no—
211C2×C4×C83—41no—
212C2×C4×C83—41no—
213C2×C4×C83—41no—
214C2×C4×C83—41no—
215C2×C4×C83—41no—
216C2×C4×C83—84no—
217C2×C4×C83C22⋊Q812yes—
218C2×C4×C83C22×Q811yescharacteristic
219C2×C4×C83C2×#4111yescharacteristic
220C2×C4×C83C2×#4111yescharacteristic
221C2×C4×C83C2×#4111yescharacteristic
222C2×C4×C83H(5,𝔽2)11yescharacteristic
223C2×C4×C832−1+411yescharacteristic
224C2×Dic83—21no—
225C2×Dic83—21no—
226C2×Dic83—21no—
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—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—
248C8×C83—22no—
249C8×C83—41no—
250C83⋊C8—41no—
251C83⋊C8—41no—
252C22×C83—22no—
253C22×C83—22no—
254C22×C83—22no—
255C22×C83—41no—
256C22×C83—41no—
257C22×C83—41no—
258C22×C83—41no—
259C22×C83C2×C22⋊Q811yescharacteristic
260C22×C83#54611yescharacteristic
261C22×C83#54711yescharacteristic
262C22×C83#57511yescharacteristic
263C22×C83#57511yescharacteristic
264C22×C83#57711yescharacteristic
265C22×C83#57711yescharacteristic
266C4×C83—22no—
267C4×C83—42no—
268C4×C83—42no—
269C4×C83—42no—
270C4×C83—81no—
271C4×C83#53311yescharacteristic
272C4×C83#54411yescharacteristic
273C4×C83#55111yescharacteristic
274C4×C83#55211yescharacteristic
275Dic83—41no—
276Dic83—41no—
277Dic83—41no—
278Dic83—41no—
279Dic83—81no—
280Dic83—81no—
281Dic83—81no—
282Dic83—81no—
283#26019—831nomaximal, Sylow
284C2×C83—42no—
285C2×C83#243811yescharacteristic
286C2×C83#308011yescharacteristic
287C2×C83#308411yescharacteristic
288C2×C83#332111yescharacteristic
289C2×C83#332311yescharacteristic
290C2×C83#334811yescharacteristic
291C2×C83#335111yescharacteristic
292#2442—831no—
293#2346—831no—
294#2101—831no—
295#2088—831no—
296#2088—831no—
297#2059—831no—
298#1989—831no—
299#1949—831no—
300#1919—831no—
301#1919—831no—
302#1853—831no—
303#1851—831no—
304#1825—831no—
305#1809—831no—
306#1736D8311yescharacteristic
307C83#2601911yescharacteristic, complemented, Sylow
308C22×M16C2×D8311yescharacteristic
309C2×#125—831no—
310C2×#117C2×D8311yescharacteristic
311C22×C4⋊C4C2×D8311yescharacteristic
312C2×#107—831no—
313C2×#106—831no—
314C2×#106—831no—
315C2×#106—831no—
316C2×C4⋊C8—831no—
317C2×C4⋊C8—831no—
318C2×C22⋊C8—831no—
319#400—1661no—
320#398—1661no—
321#389—831no—
322#389—831no—
323#389—831no—
324#389—831no—
325#388—1661no—
326#388—1661no—
327#385—1661no—
328#382—831no—
329#382—1661no—
330#380—831no—
331#380—831no—
332#380—1661no—
333C4×C4⋊C4—831no—
334C4×C4⋊C4—831no—
335C4×C22⋊C4—831no—
336#376—1661no—
337#340—831no—
338#340—831no—
339#340—831no—
340#340—831no—
341#340—831no—
342#340—831no—
343#340—831no—
344#340—831no—
345#340C2×D8311yescharacteristic
346#340C2×D8311yescharacteristic
347#340C2×D8311yescharacteristic
348#340C2×D8311yescharacteristic
349#336—831no—
350#336—831no—
351C23×C4—21no—
352C23×C4C22×D8311yescharacteristic
353C2×M16—22no—
354C2×M16—22no—
355C2×M16—41no—
356C22×C8—831no—
357C22×C8—831no—
358C22×C8C22×D8311yescharacteristic
359C22×C8C22×D8311yescharacteristic
360#125—3321no—
361#125—1662no—
362#117—22no—
363#117Dic16614yes—
364C2×C4⋊C4—22no—
365C2×C4⋊C4—21no—
366C2×C4⋊C4—41no—
367C2×C4⋊C4—831no—
368C2×C4⋊C4—831no—
369C2×C4⋊C4—831no—
370C2×C4⋊C4—831no—
371C2×C4⋊C4—1661no—
372C2×C4⋊C4—1661no—
373C2×C4⋊C4—1661no—
374C2×C4⋊C4—1661no—
375C2×C4⋊C4—1661no—
376C2×C4⋊C4—1661no—
377C2×C4⋊C4C22×D8311yescharacteristic
378C2×C4⋊C4C22×D8311yescharacteristic
379C2×C4⋊C4C22×D8311yescharacteristic
380C2×C4⋊C4C22×D8311yescharacteristic
381C2×C22⋊C4—21no—
382C2×C22⋊C4—831no—
383C2×C22⋊C4—831no—
384C2×C22⋊C4—1661no—
385C2×C22⋊C4—1661no—
386C2×C42—21no—
387C2×C42—831no—
388C2×C42—831no—
389C2×C42—1661no—
390C2×C42—1661no—
391#107—3321no—
392#106—3321no—
393#106—1662no—
394#106—1662no—
395C4⋊C8—3321no—
396C4⋊C8—1662no—
397C22⋊C8—1662no—
398#95—831no—
399#95—831no—
400#95—1661no—
401#95—1661no—
402#95—1661no—
403#95—1661no—
404#95—1661no—
405#95—1661no—
406#95—1661no—
407#95—1661no—
408#95—1661no—
409#95—1661no—
410C24D4×D8311yescharacteristic
411C22×C4—21no—
412C22×C4—21no—
413C22×C4—21no—
414C22×C4—21no—
415C22×C4—21no—
416C22×C4—21no—
417C22×C4—21no—
418C22×C4—21no—
419C22×C4—42no—
420C22×C4—42no—
421C22×C4—81no—
422C22×C4—831no—
423C22×C4—831no—
424C22×C4—831no—
425C22×C4—831no—
426C22×C4—1661no—
427C22×C4—1661no—
428C22×C4—1661no—
429C22×C4—1661no—
430C22×C4D4×D8311yescharacteristic
431C22×C4C2×Dic16612yes—
432C22×C4#1305058912yes—
433C22×C4C23×D8311yescharacteristic
434M16—44no—
435C2×C8—21no—
436C2×C8—21no—
437C2×C8—21no—
438C2×C8—21no—
439C2×C8—21no—
440C2×C8—1661no—
441C2×C8—1661no—
442C2×C8—1661no—
443C2×C8—1661no—
444C2×C8—1661no—
445C2×C8—1661no—
446C2×C8#1305059012yes—
447C4⋊C4—21no—
448C4⋊C4—21no—
449C4⋊C4—21no—
450C4⋊C4—21no—
451C4⋊C4—21no—
452C4⋊C4—42no—
453C4⋊C4—3322no—
454C4⋊C4—3322no—
455C4⋊C4—6641no—
456C4⋊C4—6641no—
457C4⋊C4—6641no—
458C4⋊C4—6641no—
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⋊C4C2×Dic16612yes—
468C4⋊C4C2×Dic16612yes—
469C4⋊C4#1305059012yes—
470C22⋊C4—24no—
471C22⋊C4—3322no—
472C22⋊C4—3321no—
473C22⋊C4—3321no—
474C22⋊C4—6641no—
475C42—24no—
476C42—3322no—
477C42—3322no—
478C42—1661no—
479C42—1661no—
480C42—1661no—
481C42—1661no—
482C23—21no—
483C23—21no—
484C23—21no—
485C23—21no—
486C23—21no—
487C23—21no—
488C23#1458176912yes—
489C23C2×D4×D8311yescharacteristic
490C2×C4—21no—
491C2×C4—21no—
492C2×C4—22no—
493C2×C4—22no—
494C2×C4—22no—
495C2×C4—22no—
496C2×C4—22no—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—41no—
500C2×C4—41no—
501C2×C4—41no—
502C2×C4—41no—
503C2×C4—41no—
504C2×C4—41no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—84no—
509C2×C4—3321no—
510C2×C4—3321no—
511C2×C4—3321no—
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—1661no—
522C2×C4—1661no—
523C2×C4—1661no—
524C2×C4—1661no—
525C2×C4—1661no—
526C2×C4—1661no—
527C2×C4—1661no—
528C2×C4—1661no—
529C2×C4—1661no—
530C2×C4—1661no—
531C2×C4—1661no—
532C2×C4—1661no—
533C2×C4#1458176912yes—
534C2×C4C22×Dic16611yescharacteristic
535C2×C4C2×#1305058911yescharacteristic
536C2×C4C2×#1305059011yescharacteristic
537C2×C4C2×#1305059011yescharacteristic
538C2×C4#1458183711yescharacteristic
539C2×C4#1458184011yescharacteristic
540C8—22no—
541C8—41no—
542C8—3321no—
543C8—3321no—
544C22—22no—
545C22—22no—
546C22—22no—
547C22—41no—
548C22—41no—
549C22—41no—
550C22—41no—
551C22C2×#1458176911yescharacteristic
552C22#1912834111yescharacteristic
553C22#1912835211yescharacteristic
554C22#1912849711yescharacteristic
555C22#1912850011yescharacteristic
556C22#1912851011yescharacteristic
557C22#1912851111yescharacteristic
558C4—22no—
559C4—42no—
560C4—42no—
561C4—42no—
562C4—81no—
563C4—3321no—
564C4—3321no—
565C4—3321no—
566C4—3321no—
567C4—6641no—
568C4—6641no—
569C4—6641no—
570C4—6641no—
571C4#1912825811yescharacteristic
572C4#1912833211yescharacteristic
573C4#1912837711yescharacteristic
574C4#1912838611yescharacteristic
575C2—42no—
576C2#2491113911yescharacteristic
577C2#2491728311yescharacteristic
578C2#2491732911yescharacteristic
579C2#2491965011yescharacteristic
580C2#2491967511yescharacteristic
581C2#2491997911yescharacteristic
582C2#2492002411yescharacteristic
subgroupG/H
centerC23C2×D4×D83
commutator, cosocleC22×C4×C83C24
FrattiniC22×C4C23×D83
FittingC83×#1736C2
socleC23×C83C22×D4
prime radicalC24×C83C2×D4
hypercenter#1736D83
hypocenterC83#26019
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. #24909034C2semidirect
4. #24908967C2semidirect
5. #24908963C2semidirect
6. #24908804C2semidirect
7. #24908417C2semidirect
8. #24908234C2semidirect
9. #24908044C2semidirect
10. #24908040C2semidirect
307. C83#26019semidirect

Maximal subgroups

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

subgroupindex
#249111812
#249103482
#249090342
#249089672
#249089632
#249088042
#249084172
#249082342
#249080442
#249080402
#249076562
#249076462
#249075582
#249074962
C83×#17362
#2601983