Finite Group Databasebeta

← #36707749

Subgroups — #36707749

Order 22784.

738 conjugacy classes of subgroups.

The subgroup lattice has 530 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19152032C211yesmaximal, characteristic
2C4×#15668031C211yesmaximal, characteristic, complemented
3C4×D8×D89C211yesmaximal, characteristic, complemented
4#26022126C211yesmaximal, characteristic, complemented
5#26022118C211yesmaximal, characteristic, complemented
6#26022116C211yesmaximal, characteristic, complemented
7#26022107C211yesmaximal, characteristic, complemented
8#26022103C211yesmaximal, characteristic, complemented
9D16×Dic89C211yesmaximal, characteristic, complemented
10#26022096C211yesmaximal, characteristic, complemented
11#26022039C211yesmaximal, characteristic, complemented
12#26021966C211yesmaximal, characteristic, complemented
13#26021961C211yesmaximal, characteristic, complemented
14#26021845C211yesmaximal, characteristic, complemented
15C89×#2177C211yesmaximal, characteristic, complemented
16C2×#15668031C2211yescharacteristic
17C2×D8×D89C2211yescharacteristic
18C4×#13083573—21no—
19C4×D4×D89—21no—
20#19152032C418yescomplemented
21C2×#15667957C2211yescharacteristic
22C2×#15667956C2211yescharacteristic
23C2×#15667953C2211yescharacteristic
24C2×#15667951C2211yescharacteristic
25C2×#15667951C2211yescharacteristic
26C4×Dic356C2211yescharacteristic, complemented
27#19151692C2211yescharacteristic, complemented
28#19151689—21no—
29C4×#13083562—21no—
30C4×D712C2211yescharacteristic
31D8×Dic89C2211yescharacteristic
32D8×Dic89C2211yescharacteristic, complemented
33#19151680C2211yescharacteristic
34C4×#13083559—21no—
35#19151678—21no—
36#19151635C2211yescharacteristic
37#107×D89C2211yescharacteristic
38#19151591—21no—
39#102×D89—21no—
40C4×C8×D89C2211yescharacteristic
41#19151425C2211yescharacteristic, complemented
42#19151424C2211yescharacteristic, complemented
43#19151423C2211yescharacteristic, complemented
44#19151403C2211yescharacteristic, complemented
45#19151401C2211yescharacteristic, complemented
46#19151400C2211yescharacteristic
47#19151399C2211yescharacteristic
48#19151399C2211yescharacteristic, complemented
49#19151398C2211yescharacteristic
50#19151374C2211yescharacteristic, complemented
51C16×Dic89C2211yescharacteristic, complemented
52C16×Dic89C2211yescharacteristic, complemented
53C2×D16×C89C2211yescharacteristic
54C4×D8×C89C2211yescharacteristic
55C4×D8×C89C2211yescharacteristic, complemented
56#367×C89C2211yescharacteristic, complemented
57#357×C89C2211yescharacteristic
58#357×C89C2211yescharacteristic, complemented
59#346×C89C2211yescharacteristic, complemented
60C2×#13083573—21no—
61C2×D4×D89—21no—
62C22×C4×D89—41no—
63#15668031—22no—
64#15668031C2×C414yescomplemented
65C2×Dic356C2311yescharacteristic
66C2×#13083562—21no—
67C2×D712C2311yescharacteristic
68C2×#13083559—21no—
69D8×D89—22no—
70D8×D89C2×C414yescomplemented
71C2×C8×D89D411yescharacteristic
72C2×C8×D89C2311yescharacteristic
73C4×Dic178—21no—
74#15667977—21no—
75C4×D356—21no—
76D4×Dic89—21no—
77D4×Dic89—21no—
78#15667974—21no—
79C4×C22⋊D89—41no—
80C4×C22⋊D89—41no—
81#15667972—41no—
82#15667972—41no—
83#15667970—21no—
84#15667968—41no—
85C4⋊C4×D89—21no—
86C22⋊C4×D89—41no—
87C2×C4×Dic89—41no—
88C42×D89D411yescharacteristic
89#15667957C2×C414yescomplemented
90#15667956C2×C414yescomplemented
91#15667953C2×C414yescomplemented
92#15667951—22no—
93#15667951C2×C414yescomplemented
94C2×#13083549C2311yescharacteristic
95C2×#13083549C2311yescharacteristic
96#15667946C2311yescharacteristic
97#15667946C2311yescharacteristic
98#15667945—21no—
99#15667943—21no—
100#15667937—21no—
101#15667935—21no—
102#15667934—21no—
103#15667933—21no—
104#15667933—21no—
105#15667932—21no—
106C4×#10623494D411yescharacteristic
107C8×Dic89D411yescharacteristic
108C8×Dic89C2311yescharacteristic
109C2×D8×C89C2311yescharacteristic
110C2×D8×C89C2311yescharacteristic
111C4×D4×C89—21no—
112C4×D4×C89—21no—
113D16×C89C2×C414yescomplemented
114C2×C16×C89C2311yescharacteristic
115C2×C16×C89C2311yescharacteristic
116#107×C89C2311yescharacteristic
117#107×C89C2311yescharacteristic
118#102×C89—21no—
119#102×C89—21no—
120C4×C8×C89C2311yescharacteristic
121C23×D89—41no—
122#13083573—24no—
123#13083573—42no—
124C2×Dic178—21no—
125C2×D356—21no—
126C2×C22⋊D89—41no—
127C2×C22⋊D89—41no—
128D4×D89—24no—
129D4×D89—42no—
130C22×Dic89—41no—
131C22×Dic89—41no—
132C22×Dic89—41no—
133C2×C4×D89—41no—
134C2×C4×D89—41no—
135C2×C4×D89—41no—
136C2×C4×D89—84no—
137C2×C4×D89D811yescharacteristic
138C2×C4×D89D811yescharacteristic
139C2×C4×D89C2×D411yescharacteristic
140Dic356—21no—
141Dic356C22×C412yescomplemented
142#13083562—24no—
143D712—21no—
144D712C22×C412yes—
145#13083559—24no—
146C2×#10623494C2×D411yescharacteristic
147C2×#10623494C2×D411yescharacteristic
148C8×D89—22no—
149C8×D89C22×C412yes—
150C8×D89#4112yes—
151C4⋊Dic89—21no—
152C4⋊Dic89—21no—
153#13083553—41no—
154#13083553—41no—
155#13083553—41no—
156C22⋊Dic89—41no—
157C22⋊Dic89—41no—
158#13083551—41no—
159#13083551—41no—
160#13083551—41no—
161C4×Dic89—41no—
162C4×Dic89—41no—
163C4×Dic89—41no—
164C4×Dic89D811yescharacteristic
165C4×Dic89D811yescharacteristic
166C4×Dic89C2×D411yescharacteristic
167#13083549—21no—
168#13083549C22×C412yescomplemented
169C2×D4×C89—21no—
170C2×D4×C89—21no—
171C22×C4×C89—41no—
172C22×C4×C89—41no—
173D8×C89—21no—
174D8×C89—21no—
175D8×C89C22×C412yes—
176D8×C89C22×C412yescomplemented
177C2×C8×C89C2×D411yescharacteristic
178C2×C8×C89C2411yescharacteristic
179C4⋊C4×C89—21no—
180C4⋊C4×C89—21no—
181C22⋊C4×C89—41no—
182C22⋊C4×C89—41no—
183C42×C89C2×D411yescharacteristic
184C16×C89—21no—
185C16×C89C22×C412yescomplemented
186C23×C89—41no—
187C23×C89—41no—
188C22×D89—41no—
189C22×D89—44no—
190C22×D89—82no—
191C22×D89—82no—
192C22×D89C2×D811yescharacteristic
193D4×C89—22no—
194D4×C89—22no—
195D4×C89—41no—
196D4×C89—41no—
197C2×C4×C89—41no—
198C2×C4×C89—41no—
199C2×C4×C89—82no—
200C2×C4×C89—82no—
201C2×C4×C89C2×D811yescharacteristic
202C2×C4×C89C2×D811yescharacteristic
203C2×C4×C89C22×D411yescharacteristic
204C22⋊D89—44no—
205C22⋊D89—44no—
206C2×Dic89—41no—
207C2×Dic89—44no—
208C2×Dic89—41no—
209C2×Dic89—41no—
210C2×Dic89—82no—
211C2×Dic89—82no—
212C2×Dic89C2×D811yescharacteristic
213C2×Dic89C2×D811yescharacteristic
214C2×Dic89C2×D811yescharacteristic
215D356—22no—
216D356—41no—
217C4×D89—22no—
218C4×D89—41no—
219C4×D89—82no—
220C4×D89—82no—
221C4×D89—82no—
222C4×D89C4×D412yes—
223C4×D89#13512yes—
224Dic178—22no—
225Dic178—41no—
226C8×C89—21no—
227C8×C89C23×C411yescharacteristic
228C8×C89C2×#4111yescharacteristic
229#10623494—21no—
230#10623494C4×D412yes—
231C22×C89—42no—
232C22×C89—42no—
233C22×C89—81no—
234C22×C89—81no—
235C22×C89—81no—
236C22×C89—81no—
237C22×C89C22×D811yescharacteristic
238C2×D89—21no—
239C2×D89—21no—
240C2×D89—42no—
241C2×D89—88no—
242C2×D89—81no—
243C2×D89—81no—
244C2×D89C4×D812yes—
245C4×C89—21no—
246C4×C89—21no—
247C4×C89—81no—
248C4×C89—81no—
249C4×C89C2×C4×D411yescharacteristic
250C4×C89C2×#13511yescharacteristic
251Dic89—21no—
252Dic89—42no—
253Dic89—81no—
254Dic89—81no—
255Dic89C4×D812yes—
256Dic89#50912yescomplemented
257C4×#509—891nomaximal, Sylow
258C2×C89—82no—
259C2×C89—82no—
260C2×C89C2×C4×D811yescharacteristic
261C2×C89C2×#50911yescharacteristic
262C2×C89#341411yescharacteristic
263D89—22no—
264D89—82no—
265C2×#509—891no—
266C4×#135—891no—
267C2×C4×D8—891no—
268#2177—891no—
269#2177D8911yescharacteristic, complemented
270#2175—891no—
271#2175—891no—
272C4×SD32—891no—
273C4×SD32—891no—
274C4×D16—891no—
275C4×D16—891no—
276#2159—891no—
277#2140—891no—
278#2139—891no—
279C4×M32—891no—
280C89C4×#50911yescharacteristic, complemented, Sylow
281C2×#135—891no—
282C22×D8—891no—
283C4×#41—1781no—
284C2×C4×D4—1781no—
285#509—898no—
286C2×SD32—891no—
287C2×SD32—891no—
288C2×D16—891no—
289C2×D16C2×D8911yescharacteristic
290C2×M32—891no—
291C2×M32—891no—
292C4×Q16—891no—
293C4×Q16—891no—
294C4×SD16—1781no—
295C4×SD16—1781no—
296C4×D8—891no—
297C4×D8—891no—
298C4×D8—891no—
299C4×D8—891no—
300C4×D8—1781no—
301C4×D8—1781no—
302C4×D8C2×D8911yescharacteristic
303C4×D8C2×D8911yescharacteristic, complemented
304#427—891no—
305C2×#107—891no—
306#416—1781no—
307C2×#102—1781no—
308C2×C4×C8—891no—
309#367—891no—
310#367C2×D8911yescharacteristic, complemented
311#366—891no—
312#366—891no—
313#358—891no—
314#358—891no—
315#357—891no—
316#357—891no—
317#357—891no—
318#357—891no—
319#357C2×D8911yescharacteristic
320#357C2×D8911yescharacteristic, complemented
321#346—891no—
322#346C2×D8911yescharacteristic, complemented
323C4×C16—891no—
324C4×C16—891no—
325C2×#41—1781no—
326C22×D4—1781no—
327C23×C4—3561no—
328#135—894no—
329#135—1782no—
330C2×Q16—891no—
331C2×SD16—1781no—
332C2×D8—894no—
333C2×D8—891no—
334C2×D8—1782no—
335C2×D8—1781no—
336C2×D8C22×D8911yescharacteristic
337C2×D8C22×D8911yescharacteristic
338C22×C8—891no—
339C22×C8—891no—
340C4×Q8—1781no—
341C4×Q8—1781no—
342C4×D4—21no—
343C4×D4—21no—
344C4×D4—3561no—
345C4×D4—3561no—
346C4×D4—3561no—
347C4×D4—3561no—
348C4×D4—1781no—
349C4×D4—1781no—
350C4×D4—1781no—
351C4×D4—1781no—
352#117—3561no—
353#117—1781no—
354C2×C4⋊C4—1781no—
355C2×C22⋊C4—3561no—
356C2×C42—891no—
357C2×C42—3561no—
358SD32—894no—
359SD32—894no—
360D16—894no—
361D16C4×D8914yescomplemented
362M32—894no—
363M32—1782no—
364C2×C16—891no—
365C2×C16—891no—
366C2×C16C22×D8911yescharacteristic
367C2×C16C22×D8911yescharacteristic
368#107—891no—
369#107—891no—
370#107—1781no—
371#107C22×D8911yescharacteristic
372#107C22×D8911yescharacteristic
373#106—1781no—
374#103—1781no—
375#103—1781no—
376#102—21no—
377#102—21no—
378#102—1781no—
379#102—1781no—
380#102—1781no—
381#102—1781no—
382C4×C8—891no—
383C4×C8—891no—
384C4×C8—891no—
385C4×C8C22×D8911yescharacteristic
386C24—3561no—
387#41—3562no—
388#41—1784no—
389C2×Q8—1781no—
390C2×D4—21no—
391C2×D4—21no—
392C2×D4—3562no—
393C2×D4—3561no—
394C2×D4—3561no—
395C2×D4—1784no—
396C2×D4—1781no—
397C22×C4—41no—
398C22×C4—41no—
399C22×C4—891no—
400C22×C4—891no—
401C22×C4—891no—
402C22×C4—3561no—
403C22×C4—3561no—
404C22×C4—3561no—
405C22×C4—3561no—
406C22×C4—3561no—
407C22×C4—3561no—
408C22×C4—7124no—
409Q16—892no—
410Q16—1781no—
411SD16—1784no—
412D8—21no—
413D8—21no—
414D8—892no—
415D8—1784no—
416D8—1781no—
417D8C2×C4×D8912yes—
418D8C2×C4×D8912yescomplemented
419C2×C8—891no—
420C2×C8—892no—
421C2×C8—892no—
422C2×C8—891no—
423C2×C8—1782no—
424C2×C8D4×D8911yescharacteristic
425C2×C8C23×D8911yescharacteristic
426C4⋊C4—21no—
427C4⋊C4—21no—
428C4⋊C4—3561no—
429C4⋊C4—3561no—
430C4⋊C4—3561no—
431C4⋊C4—1781no—
432C4⋊C4—1781no—
433C22⋊C4—41no—
434C22⋊C4—41no—
435C22⋊C4—3561no—
436C22⋊C4—3561no—
437C22⋊C4—3561no—
438C22⋊C4—3561no—
439C22⋊C4—3561no—
440C42—891no—
441C42—891no—
442C42—891no—
443C42—3561no—
444C42—3561no—
445C42—3561no—
446C42D4×D8911yescharacteristic
447C16—21no—
448C16—892no—
449C16—1781no—
450C16C2×C4×D8912yescomplemented
451C23—41no—
452C23—41no—
453C23—891no—
454C23—3564no—
455C23—3561no—
456C23—7122no—
457C23—7122no—
458Q8—3561no—
459Q8—1782no—
460D4—22no—
461D4—22no—
462D4—41no—
463D4—41no—
464D4—3564no—
465D4—3564no—
466D4—3561no—
467D4—1782no—
468C2×C4—41no—
469C2×C4—41no—
470C2×C4—82no—
471C2×C4—82no—
472C2×C4—891no—
473C2×C4—891no—
474C2×C4—892no—
475C2×C4—892no—
476C2×C4—891no—
477C2×C4—3561no—
478C2×C4—3564no—
479C2×C4—3561no—
480C2×C4—3561no—
481C2×C4—3561no—
482C2×C4—1782no—
483C2×C4—7122no—
484C2×C4—7122no—
485C2×C4—7122no—
486C2×C4—7122no—
487C2×C4—7122no—
488C2×C4D8×D8911yescharacteristic
489C2×C4D8×D8911yescharacteristic
490C2×C4C2×D4×D8911yescharacteristic
491C8—21no—
492C8—892no—
493C8—1781no—
494C8C22×C4×D8911yescharacteristic
495C8#41×D8911yescharacteristic
496C22—42no—
497C22—42no—
498C22—81no—
499C22—81no—
500C22—81no—
501C22—81no—
502C22—892no—
503C22—3562no—
504C22—1781no—
505C22—1781no—
506C22—7128no—
507C22—7121no—
508C22—7121no—
509C22C2×D8×D8911yescharacteristic
510C4—21no—
511C4—21no—
512C4—81no—
513C4—81no—
514C4—892no—
515C4—892no—
516C4—3562no—
517C4—1781no—
518C4—7121no—
519C4—7121no—
520C4C4×D4×D8911yescharacteristic
521C4#135×D8911yescharacteristic
522C2—82no—
523C2—82no—
524C2—1782no—
525C2—7122no—
526C2C4×D8×D8911yescharacteristic
527C2#509×D8911yescharacteristic
528C2#2603338411yescharacteristic
subgroupG/H
centerC22C2×D8×D89
commutatorC8×C89C23×C4
FrattiniC2×C8C23×D89
FittingC89×#2177C2
socleC22×C89C22×D8
prime radicalC2×#19152032C2
cosocleC2×C8×C89C24
hypercenter#2177D89
hypocenterC89C4×#509
LCSG ▷ C8×C89 ▷ C4×C89 ▷ C2×C89 ▷ C89
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ #2177

Complements

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

NH ≅ G/Ntype
2. C4×#15668031C2semidirect
3. C4×D8×D89C2semidirect
4. #26022126C2semidirect
5. #26022118C2semidirect
6. #26022116C2semidirect
7. #26022107C2semidirect
8. #26022103C2semidirect
9. D16×Dic89C2semidirect
10. #26022096C2semidirect
11. #26022039C2semidirect
12. #26021966C2semidirect
13. #26021961C2semidirect
14. #26021845C2semidirect
15. C89×#2177C2semidirect
20. #19152032C4semidirect
26. C4×Dic356C22semidirect
27. #19151692C22semidirect
32. D8×Dic89C22semidirect
41. #19151425C22semidirect
42. #19151424C22semidirect
43. #19151423C22semidirect
44. #19151403C22semidirect
45. #19151401C22semidirect
48. #19151399C22semidirect
50. #19151374C22semidirect
51. C16×Dic89C22semidirect
52. C16×Dic89C22semidirect
55. C4×D8×C89C22semidirect
56. #367×C89C22semidirect
58. #357×C89C22semidirect
59. #346×C89C22semidirect
64. #15668031C2×C4semidirect
70. D8×D89C2×C4semidirect
89. #15667957C2×C4semidirect
90. #15667956C2×C4semidirect
91. #15667953C2×C4semidirect
93. #15667951C2×C4semidirect
113. D16×C89C2×C4semidirect
141. Dic356C22×C4semidirect
168. #13083549C22×C4semidirect
176. D8×C89C22×C4semidirect
185. C16×C89C22×C4semidirect
256. Dic89#509semidirect
269. #2177D89semidirect
280. C89C4×#509semidirect
303. C4×D8C2×D89semidirect
310. #367C2×D89semidirect
320. #357C2×D89semidirect
322. #346C2×D89semidirect
361. D16C4×D89semidirect
418. D8C2×C4×D89semidirect
450. C16C2×C4×D89semidirect

Maximal subgroups

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

subgroupindex
C2×#191520322
C4×#156680312
C4×D8×D892
#260221262
#260221182
#260221162
#260221072
#260221032
D16×Dic892
#260220962
#260220392
#260219662
#260219612
#260218452
C89×#21772
C4×#50989