Finite Group Databasebeta

← #35559229

Subgroups — #35559229

Order 21248.

632 conjugacy classes of subgroups.

The subgroup lattice has 534 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#466×D83C211yesmaximal, characteristic
2#427×D83C211yesmaximal, characteristic, complemented
3#24909655C211yesmaximal, characteristic, complemented
4#24909654C211yesmaximal, characteristic, complemented
5#24909651C211yesmaximal, characteristic, complemented
6#24909650C211yesmaximal, characteristic, complemented
7#24909647C211yesmaximal, characteristic, complemented
8#24909645C211yesmaximal, characteristic, complemented
9#24909642C211yesmaximal, characteristic, complemented
10#24909250C211yesmaximal, characteristic, complemented
11#24909248C211yesmaximal, characteristic, complemented
12#24909177C211yesmaximal, characteristic, complemented
13#24909177C211yesmaximal, characteristic, complemented
14#24909072C211yesmaximal, characteristic, complemented
15C83×#2234C211yesmaximal, characteristic, complemented
16C2×D8×D83C2211yescharacteristic
17C22×C8×D83C2211yescharacteristic
18#121×D83—21no—
19#117×D83—21no—
20C2×#14581731C2211yescharacteristic
21C2×#14581731C2211yescharacteristic
22#19127938C2211yescharacteristic
23#19127937C2211yescharacteristic
24#19127936—21no—
25#19127935—21no—
26#19127934C2211yescharacteristic
27#19127739C2211yescharacteristic, complemented
28#19127738C2211yescharacteristic
29#19127738C2211yescharacteristic
30#19127737—21no—
31#19127736—21no—
32#107×D83C2211yescharacteristic
33#107×D83C2211yescharacteristic, complemented
34#107×D83C2211yescharacteristic, complemented
35#106×D83—21no—
36#102×D83—21no—
37#19127529C2211yescharacteristic, complemented
38#19127528C2211yescharacteristic, complemented
39#19127528C2211yescharacteristic, complemented
40#19127527C2211yescharacteristic, complemented
41#19127526C2211yescharacteristic, complemented
42#19127526C2211yescharacteristic, complemented
43#19127503C2211yescharacteristic, complemented
44#19127503C2211yescharacteristic, complemented
45#19127502C2211yescharacteristic, complemented
46#19127502C2211yescharacteristic, complemented
47#19127501C2211yescharacteristic, complemented
48#19127501C2211yescharacteristic, complemented
49C22⋊(C83⋊C16)C2211yescharacteristic, complemented
50(C2×D83)⋊C16C2211yescharacteristic, complemented
51(C2×D83)⋊C16C2211yescharacteristic, complemented
52#466×C83C2211yescharacteristic
53#427×C83C2211yescharacteristic, complemented
54#367×C83C2211yescharacteristic, complemented
55#366×C83C2211yescharacteristic, complemented
56#357×C83C2211yescharacteristic, complemented
57#357×C83C2211yescharacteristic, complemented
58C22⋊C16×C83C2211yescharacteristic, complemented
59C2×D4×D83—21no—
60C2×D4×D83—41no—
61C22×C4×D83D411yescharacteristic
62C2×D664C2311yescharacteristic
63C2×#13050575—21no—
64D8×D83—22no—
65D8×D83—22no—
66C22×C83⋊C8D411yescharacteristic
67C2×C8×D83—22no—
68C2×C8×D83—22no—
69C2×C8×D83D411yescharacteristic
70C2×C8×D83D411yescharacteristic
71C2×C8×D83C2311yescharacteristic
72C2×C8×D83C2311yescharacteristic
73#14581767—21no—
74#14581766—21no—
75#14581765—41no—
76#14581764—41no—
77#14581763—21no—
78#14581751—21no—
79#14581750—21no—
80#14581750—21no—
81#14581749—41no—
82#14581748—41no—
83C4⋊C4×D83—21no—
84C4⋊C4×D83—21no—
85C4⋊C4×D83—21no—
86C22⋊C4×D83—41no—
87C22⋊C4×D83—41no—
88C42×D83—41no—
89#14581731—22no—
90#14581731—22no—
91C2×C83⋊C16C2311yescharacteristic
92C2×C83⋊C16C2311yescharacteristic
93#14581726C2311yescharacteristic
94#14581726C2311yescharacteristic
95#14581726C2311yescharacteristic
96#14581725—21no—
97#14581725—21no—
98#14581725—21no—
99#14581724—21no—
100#14581723—21no—
101#14581723—21no—
102#14581714—21no—
103#14581713—21no—
104#14581712—21no—
105C2×D8×C83C2311yescharacteristic
106C22×C8×C83C2311yescharacteristic
107#121×C83—21no—
108#117×C83—21no—
109C2×C16×C83C2311yescharacteristic
110C2×C16×C83C2311yescharacteristic
111#107×C83C2311yescharacteristic
112#107×C83C2311yescharacteristic
113#107×C83C2311yescharacteristic
114#106×C83—21no—
115#102×C83—21no—
116C23×D83—41no—
117C23×D83D811yescharacteristic
118C2×D332—21no—
119C2×D332—41no—
120C2×C22⋊D83—41no—
121C2×C22⋊D83—41no—
122C2×C22⋊D83—41no—
123D4×D83—42no—
124D4×D83—42no—
125D4×D83—84no—
126C22×Dic83D811yescharacteristic
127C2×C4×D83—22no—
128C2×C4×D83—22no—
129C2×C4×D83—41no—
130C2×C4×D83—41no—
131C2×C4×D83—41no—
132C2×C4×D83D811yescharacteristic
133C2×C4×D83D811yescharacteristic
134C2×C4×D83C2×D411yescharacteristic
135C2×C4×D83C2×D411yescharacteristic
136D664—21no—
137D664—21no—
138#13050575—42no—
139C2×C83⋊C8—22no—
140C2×C83⋊C8C2×D411yescharacteristic
141C2×C83⋊C8C2×D411yescharacteristic
142C8×D83—22no—
143C8×D83—21no—
144C8×D83—21no—
145C8×D83—22no—
146C8×D83#4112yes—
147C8×D83#4112yes—
148C4⋊Dic83—21no—
149C4⋊Dic83—21no—
150C4⋊Dic83—21no—
151Dic83⋊C4—41no—
152Dic83⋊C4—41no—
153Dic83⋊C4—41no—
154C22⋊Dic83—41no—
155C22⋊Dic83—41no—
156(C2×D83)⋊C4—41no—
157(C2×D83)⋊C4—41no—
158(C2×D83)⋊C4—41no—
159(C2×D83)⋊C4—41no—
160C4×Dic83—41no—
161C4×Dic83—41no—
162C4×Dic83—41no—
163C83⋊C16—21no—
164C83⋊C16—21no—
165C2×D4×C83—21no—
166C2×D4×C83—41no—
167C22×C4×C83C2×D411yescharacteristic
168D8×C83—21no—
169D8×C83—21no—
170C2×C8×C83—21no—
171C2×C8×C83—21no—
172C2×C8×C83C2×D411yescharacteristic
173C2×C8×C83C2411yescharacteristic
174C4⋊C4×C83—21no—
175C4⋊C4×C83—21no—
176C4⋊C4×C83—21no—
177C22⋊C4×C83—41no—
178C22⋊C4×C83—41no—
179C42×C83—41no—
180C16×C83—21no—
181C16×C83—21no—
182C23×C83—41no—
183C23×C83C2×D811yescharacteristic
184C22×D83—22no—
185C22×D83—22no—
186C22×D83—41no—
187C22×D83—41no—
188C22×D83—82no—
189C22×D83—82no—
190C22×D83—82no—
191C22×D83C2×D811yescharacteristic
192C22×D83C2×D811yescharacteristic
193D4×C83—41no—
194D4×C83—41no—
195D4×C83—82no—
196C2×C4×C83—21no—
197C2×C4×C83—21no—
198C2×C4×C83—41no—
199C2×C4×C83—41no—
200C2×C4×C83—41no—
201C2×C4×C83C2×D811yescharacteristic
202C2×C4×C83C22×D411yescharacteristic
203C22⋊D83—82no—
204C22⋊D83—82no—
205C22⋊D83—82no—
206C2×Dic83—22no—
207C2×Dic83—41no—
208C2×Dic83—41no—
209C2×Dic83—41no—
210C2×Dic83C2×D811yescharacteristic
211C2×Dic83C2×D811yescharacteristic
212D332—41no—
213D332—41no—
214D332—82no—
215C4×D83—22no—
216C4×D83—21no—
217C4×D83—21no—
218C4×D83—22no—
219C4×D83—82no—
220C4×D83—82no—
221C4×D83—82no—
222C4×D83#12312yes—
223C4×D83#13712yes—
224C8×C83—21no—
225C8×C83C2×#4111yescharacteristic
226C8×C83C2×#4111yescharacteristic
227C83⋊C8—21no—
228C83⋊C8#12312yes—
229C22×C83—21no—
230C22×C83—21no—
231C22×C83—21no—
232C22×C83—81no—
233C22×C83—81no—
234C22×C83—81no—
235C22×C83C22×D811yescharacteristic
236C2×D83—21no—
237C2×D83—21no—
238C2×D83—21no—
239C2×D83—21no—
240C2×D83—44no—
241C2×D83—81no—
242C2×D83—81no—
243C2×D83—81no—
244C2×D83—164no—
245C2×D83#48012yes—
246C2×D83#50912yes—
247C4×C83—21no—
248C4×C83—81no—
249C4×C83—81no—
250C4×C83—81no—
251C4×C83C2×#12311yescharacteristic
252C4×C83C2×#13711yescharacteristic
253Dic83—21no—
254Dic83—81no—
255Dic83—81no—
256Dic83—81no—
257Dic83#48012yes—
258#33648—831nomaximal, Sylow
259C2×C83—41no—
260C2×C83—161no—
261C2×C83C2×#48011yescharacteristic
262C2×C83C2×#50911yescharacteristic
263C2×C83#341411yescharacteristic
264D83—22no—
265D83—22no—
266D83—161no—
267C2×#466—831no—
268C2×#427—831no—
269#2234—831no—
270#2234—831no—
271#2234—831no—
272#2234D8311yescharacteristic, complemented
273#2233—831no—
274#2233—831no—
275#2232—831no—
276#2232—831no—
277#2159—831no—
278#2159—831no—
279#2140—831no—
280#2140—831no—
281#2112—831no—
282C83#3364811yescharacteristic, complemented, Sylow
283C22×D8—831no—
284C23×C8—831no—
285C2×#121—1661no—
286C2×#117—1661no—
287C2×M32—831no—
288C2×M32—831no—
289#466—831no—
290#466—831no—
291#466—831no—
292#466—1661no—
293#466—1661no—
294#466C2×D8311yescharacteristic
295#427—831no—
296#427—831no—
297#427—831no—
298#427—1661no—
299#427—1661no—
300#427C2×D8311yescharacteristic, complemented
301C2×#107—831no—
302C2×#107—831no—
303C2×#107—831no—
304C2×#106—1661no—
305C2×#102—1661no—
306#367—831no—
307#367—831no—
308#367—831no—
309#367C2×D8311yescharacteristic, complemented
310#366—831no—
311#366—831no—
312#366—831no—
313#366C2×D8311yescharacteristic, complemented
314#357—831no—
315#357—831no—
316#357—831no—
317#357—831no—
318#357—831no—
319#357—831no—
320#357C2×D8311yescharacteristic, complemented
321#357C2×D8311yescharacteristic, complemented
322C22⋊C16—831no—
323C22⋊C16—831no—
324C22⋊C16—831no—
325C22⋊C16C2×D8311yescharacteristic, complemented
326C22×D4—3321no—
327C22×D4—1661no—
328C23×C4—831no—
329C2×D8—831no—
330C2×D8—1662no—
331C2×D8—1662no—
332C2×D8—1661no—
333C2×D8C22×D8311yescharacteristic
334C22×C8—831no—
335C22×C8—831no—
336C22×C8—831no—
337C22×C8—831no—
338C22×C8—831no—
339C22×C8—1662no—
340C22×C8—1662no—
341C22×C8C22×D8311yescharacteristic
342#121—21no—
343#121—3321no—
344#121—3321no—
345#121—1661no—
346#121—1661no—
347#121—1661no—
348#117—21no—
349#117—3321no—
350#117—3321no—
351#117—1661no—
352#117—1661no—
353#117—1661no—
354C2×C4⋊C4—1661no—
355C2×C4⋊C4—1661no—
356C2×C4⋊C4—1661no—
357C2×C22⋊C4—3321no—
358C2×C22⋊C4—3321no—
359C2×C42—3321no—
360M32—1662no—
361M32—1662no—
362C2×C16—831no—
363C2×C16—831no—
364C2×C16C22×D8311yescharacteristic
365C2×C16C22×D8311yescharacteristic
366#107—831no—
367#107—831no—
368#107—831no—
369#107—1661no—
370#107—1661no—
371#107—1661no—
372#107C22×D8311yescharacteristic
373#107C22×D8311yescharacteristic
374#107C22×D8311yescharacteristic
375#106—21no—
376#106—1661no—
377#106—1661no—
378#106—1661no—
379#102—21no—
380#102—1661no—
381#102—1661no—
382#102—1661no—
383C24—3321no—
384C24—831no—
385C2×D4—21no—
386C2×D4—41no—
387C2×D4—3321no—
388C2×D4—3321no—
389C2×D4—3321no—
390C2×D4—3322no—
391C2×D4—3322no—
392C2×D4—3321no—
393C2×D4—6644no—
394C2×D4—1661no—
395C22×C4—3321no—
396C22×C4—3321no—
397C22×C4—3321no—
398C22×C4—831no—
399C22×C4—831no—
400C22×C4—831no—
401C22×C4—831no—
402C22×C4—831no—
403C22×C4—1662no—
404C22×C4—1662no—
405C22×C4D4×D8311yescharacteristic
406D8—21no—
407D8—21no—
408D8—3322no—
409D8—1661no—
410D8—1661no—
411C2×C8—21no—
412C2×C8—21no—
413C2×C8—831no—
414C2×C8—832no—
415C2×C8—832no—
416C2×C8—831no—
417C2×C8—1662no—
418C2×C8—1662no—
419C2×C8—1662no—
420C2×C8—1661no—
421C2×C8—1661no—
422C2×C8D4×D8311yescharacteristic
423C2×C8C23×D8311yescharacteristic
424C4⋊C4—21no—
425C4⋊C4—21no—
426C4⋊C4—21no—
427C4⋊C4—3321no—
428C4⋊C4—3321no—
429C4⋊C4—3321no—
430C4⋊C4—1661no—
431C4⋊C4—1661no—
432C4⋊C4—1661no—
433C22⋊C4—41no—
434C22⋊C4—41no—
435C22⋊C4—3321no—
436C22⋊C4—3321no—
437C22⋊C4—3321no—
438C22⋊C4—3321no—
439C22⋊C4—3321no—
440C22⋊C4—3321no—
441C42—41no—
442C42—3321no—
443C42—3321no—
444C42—3321no—
445C16—21no—
446C16—21no—
447C16—1661no—
448C16—1661no—
449C23—41no—
450C23—3321no—
451C23—3321no—
452C23—831no—
453C23—831no—
454C23—6642no—
455C23—6642no—
456C23—6642no—
457C23—1662no—
458C23—1662no—
459C23D8×D8311yescharacteristic
460D4—41no—
461D4—41no—
462D4—82no—
463D4—3321no—
464D4—3321no—
465D4—6642no—
466D4—6642no—
467D4—6642no—
468D4—6642no—
469C2×C4—21no—
470C2×C4—21no—
471C2×C4—41no—
472C2×C4—41no—
473C2×C4—41no—
474C2×C4—3321no—
475C2×C4—3321no—
476C2×C4—3321no—
477C2×C4—831no—
478C2×C4—832no—
479C2×C4—832no—
480C2×C4—831no—
481C2×C4—6642no—
482C2×C4—6642no—
483C2×C4—6642no—
484C2×C4—1662no—
485C2×C4—1662no—
486C2×C4—1662no—
487C2×C4—1661no—
488C2×C4—1661no—
489C2×C4D8×D8311yescharacteristic
490C2×C4C2×D4×D8311yescharacteristic
491C8—21no—
492C8—832no—
493C8—1661no—
494C8#41×D8311yescharacteristic
495C8#41×D8311yescharacteristic
496C22—21no—
497C22—21no—
498C22—21no—
499C22—81no—
500C22—81no—
501C22—81no—
502C22—13284no—
503C22—3324no—
504C22—832no—
505C22—832no—
506C22—6641no—
507C22—6641no—
508C22—6641no—
509C22—1661no—
510C22—1661no—
511C22—1661no—
512C22—1661no—
513C22C2×D8×D8311yescharacteristic
514C4—21no—
515C4—81no—
516C4—81no—
517C4—81no—
518C4—832no—
519C4—6641no—
520C4—6641no—
521C4—6641no—
522C4—1661no—
523C4#123×D8311yescharacteristic
524C4#137×D8311yescharacteristic
525C2—41no—
526C2—161no—
527C2—13281no—
528C2—1662no—
529C2—1662no—
530C2#480×D8311yescharacteristic
531C2#509×D8311yescharacteristic
532C2#2492059611yescharacteristic
subgroupG/H
centerC22C2×D8×D83
commutator, cosocleC2×C8×C83C24
FrattiniC2×C8C23×D83
FittingC83×#2234C2
socleC22×C83C22×D8
prime radical#466×D83C2
hypercenter#2234D83
hypocenterC83#33648
LCSG ▷ C2×C8×C83 ▷ C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ #2234

Complements

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

NH ≅ G/Ntype
2. #427×D83C2semidirect
3. #24909655C2semidirect
4. #24909654C2semidirect
5. #24909651C2semidirect
6. #24909650C2semidirect
7. #24909647C2semidirect
8. #24909645C2semidirect
9. #24909642C2semidirect
10. #24909250C2semidirect
11. #24909248C2semidirect
12. #24909177C2semidirect
13. #24909177C2semidirect
14. #24909072C2semidirect
15. C83×#2234C2semidirect
27. #19127739C22semidirect
33. #107×D83C22semidirect
34. #107×D83C22semidirect
37. #19127529C22semidirect
38. #19127528C22semidirect
39. #19127528C22semidirect
40. #19127527C22semidirect
41. #19127526C22semidirect
42. #19127526C22semidirect
43. #19127503C22semidirect
44. #19127503C22semidirect
45. #19127502C22semidirect
46. #19127502C22semidirect
47. #19127501C22semidirect
48. #19127501C22semidirect
49. C22⋊(C83⋊C16)C22semidirect
50. (C2×D83)⋊C16C22semidirect
51. (C2×D83)⋊C16C22semidirect
53. #427×C83C22semidirect
54. #367×C83C22semidirect
55. #366×C83C22semidirect
56. #357×C83C22semidirect
57. #357×C83C22semidirect
58. C22⋊C16×C83C22semidirect
272. #2234D83semidirect
282. C83#33648semidirect
300. #427C2×D83semidirect
309. #367C2×D83semidirect
313. #366C2×D83semidirect
320. #357C2×D83semidirect
321. #357C2×D83semidirect
325. C22⋊C16C2×D83semidirect

Maximal subgroups

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

subgroupindex
#466×D832
#427×D832
#249096552
#249096542
#249096512
#249096502
#249096472
#249096452
#249096422
#249092502
#249092482
#249091772
#249090722
C83×#22342
#3364883