Finite Group Databasebeta

← #36707792

Subgroups — #36707792

Order 22784.

682 conjugacy classes of subgroups.

The subgroup lattice has 512 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×SD32×D89C211yesmaximal, characteristic
2#26029115C211yesmaximal, characteristic, complemented
3#26029099C211yesmaximal, characteristic
4#26022119C211yesmaximal, characteristic, complemented
5#26022116C211yesmaximal, characteristic, complemented
6#26022114C211yesmaximal, characteristic, complemented
7#26022113C211yesmaximal, characteristic, complemented
8C4×#15667957C211yesmaximal, characteristic, complemented
9SD32×Dic89C211yesmaximal, characteristic, complemented
10#26022103C211yesmaximal, characteristic, complemented
11#26022035C211yesmaximal, characteristic, complemented
12#26021970C211yesmaximal, characteristic, complemented
13#26021966C211yesmaximal, characteristic
14#26021847C211yesmaximal, characteristic, complemented
15C4×SD32×C89C211yesmaximal, characteristic, complemented
16C2×Q16×D89C2211yescharacteristic
17C2×D8×D89C2211yescharacteristic
18#19152074—21no—
19#19152070—21no—
20C2×#15667957C2211yescharacteristic
21C2×#15667956C2211yescharacteristic
22C2×#15667955C2211yescharacteristic
23SD32×D89C418yescomplemented
24C2×#15667951C2211yescharacteristic
25C2×C16×D89C2211yescharacteristic
26#19151708—21no—
27#19151707—21no—
28#19151703—21no—
29#19151702—21no—
30C4×Dic356C2211yescharacteristic, complemented
31Q16×Dic89C2211yescharacteristic, complemented
32#19151692C2211yescharacteristic, complemented
33C4×D712C2211yescharacteristic
34D8×Dic89C2211yescharacteristic
35#19151680C2211yescharacteristic
36#19151635C2211yescharacteristic
37#19151635C2211yescharacteristic
38#19151605—21no—
39#19151601—21no—
40#19151552C2211yescharacteristic
41#19151426C2211yescharacteristic, complemented
42#19151423C2211yescharacteristic, complemented
43#19151423C2211yescharacteristic, complemented
44#19151403C2211yescharacteristic, complemented
45#19151402C2211yescharacteristic, complemented
46#19151401C2211yescharacteristic, complemented
47#19151400C2211yescharacteristic
48#19151399C2211yescharacteristic
49#19151398C2211yescharacteristic
50#19151374C2211yescharacteristic, complemented
51#19151373C2211yescharacteristic, complemented
52C16×Dic89C2211yescharacteristic, complemented
53C2×SD32×C89C2211yescharacteristic
54C4×Q16×C89C2211yescharacteristic, complemented
55C4×D8×C89C2211yescharacteristic
56#367×C89C2211yescharacteristic, complemented
57#358×C89C2211yescharacteristic, complemented
58#357×C89C2211yescharacteristic
59C4×C16×C89C2211yescharacteristic, complemented
60C2×Q8×D89—21no—
61C2×D4×D89—21no—
62C2×Dic356C2311yescharacteristic
63C2×#13083564—21no—
64Q16×D89—22no—
65Q16×D89C2×C414yescomplemented
66C2×D712C2311yescharacteristic
67C2×#13083559—21no—
68D8×D89—22no—
69D8×D89C2×C414yes—
70C2×#13083557D411yescharacteristic
71C2×C8×D89C2311yescharacteristic
72Q8×Dic89—21no—
73C4×Dic178—21no—
74C4×Dic178—41no—
75#15667977—21no—
76#15667977—41no—
77C4×D356—21no—
78D4×Dic89—21no—
79#15667974—21no—
80C4×C22⋊D89—41no—
81#15667972—41no—
82#15667970—21no—
83#15667970—21no—
84#15667970—41no—
85#15667969—41no—
86#15667969D411yescharacteristic
87C2×#13083551—41no—
88C22⋊C4×D89—41no—
89#15667957C2×C414yescomplemented
90#15667956C2×C414yescomplemented
91#15667955C2×C414yescomplemented
92#15667951—22no—
93C2×#13083549C2311yescharacteristic
94C2×#13083549C2311yescharacteristic
95C16×D89C2×C414yescomplemented
96#15667946C2311yescharacteristic
97#15667946C2311yescharacteristic
98#15667943—21no—
99#15667943—21no—
100#15667937—21no—
101#15667936—21no—
102#15667935—21no—
103#15667934—21no—
104#15667933—21no—
105#15667932—21no—
106#15667923D411yescharacteristic
107#15667922D411yescharacteristic
108C8×Dic89C2311yescharacteristic
109C2×Q16×C89C2311yescharacteristic
110C2×D8×C89C2311yescharacteristic
111C4×Q8×C89—21no—
112C4×D4×C89—21no—
113SD32×C89C2×C414yescomplemented
114C2×C16×C89C2311yescharacteristic
115C2×C16×C89C2311yescharacteristic
116#107×C89C2311yescharacteristic
117#107×C89C2311yescharacteristic
118#103×C89—21no—
119#102×C89—21no—
120C4×C8×C89C2311yescharacteristic
121C23×D89—41no—
122C2×Dic178—21no—
123C2×Dic178—41no—
124Q8×D89—24no—
125Q8×D89—42no—
126C2×D356—21no—
127C2×C22⋊D89—41no—
128D4×D89—24no—
129D4×D89—42no—
130C22×Dic89—41no—
131C2×C4×D89—41no—
132C2×C4×D89—41no—
133C2×C4×D89—41no—
134C2×C4×D89C2×D411yescharacteristic
135Dic356—21no—
136Dic356C22×C412yes—
137#13083564—24no—
138D712—21no—
139D712C22×C412yes—
140#13083559—24no—
141#13083557—22no—
142C2×#10623494C2×D411yescharacteristic
143C2×#10623494C2×D411yescharacteristic
144C8×D89C22×C412yes—
145C8×D89#4112yes—
146C4⋊Dic89—21no—
147C4⋊Dic89—21no—
148C4⋊Dic89—41no—
149#13083553—41no—
150#13083553—41no—
151#13083553D811yescharacteristic
152#13083553D811yescharacteristic
153C22⋊Dic89—41no—
154#13083551—41no—
155#13083551—41no—
156#13083551—41no—
157#13083551—41no—
158#13083551D811yescharacteristic
159#13083551D811yescharacteristic
160C4×Dic89—41no—
161C4×Dic89—41no—
162C4×Dic89—41no—
163C4×Dic89—41no—
164C4×Dic89—41no—
165C4×Dic89C2×D411yescharacteristic
166#13083549—21no—
167#13083549C22×C412yes—
168C2×Q8×C89—21no—
169C2×D4×C89—21no—
170C22×C4×C89—41no—
171Q16×C89—21no—
172Q16×C89C22×C412yes—
173D8×C89—21no—
174D8×C89C22×C412yes—
175C2×C8×C89C2×D411yescharacteristic
176C2×C8×C89C2411yescharacteristic
177C4⋊C4×C89—21no—
178C4⋊C4×C89—21no—
179C4⋊C4×C89—41no—
180C22⋊C4×C89—41no—
181C42×C89—41no—
182C42×C89C2×D411yescharacteristic
183C16×C89—21no—
184C16×C89C22×C412yes—
185C23×C89—41no—
186C22×D89—41no—
187C22×D89—44no—
188C22×D89—161no—
189C22×D89—161no—
190C22×D89C2×D811yescharacteristic
191Q8×C89—22no—
192Q8×C89—41no—
193D4×C89—22no—
194D4×C89—41no—
195C2×C4×C89—41no—
196C2×C4×C89—41no—
197C2×C4×C89—41no—
198C2×C4×C89—161no—
199C2×C4×C89C2×D811yescharacteristic
200C2×C4×C89C2×D811yescharacteristic
201C2×C4×C89C22×D411yescharacteristic
202C22⋊D89—44no—
203C2×Dic89—41no—
204C2×Dic89—41no—
205C2×Dic89—41no—
206C2×Dic89—161no—
207C2×Dic89C2×D811yescharacteristic
208C2×Dic89C2×D811yescharacteristic
209C2×Dic89C2×D811yescharacteristic
210D356—22no—
211D356—41no—
212C4×D89—44no—
213C4×D89—161no—
214C4×D89—161no—
215C4×D89C4×D412yes—
216C4×D89#13512yes—
217Dic178—22no—
218Dic178—41no—
219Dic178—44no—
220C8×C89—21no—
221C8×C89C23×C411yescharacteristic
222C8×C89C2×#4111yescharacteristic
223#10623494—21no—
224#10623494C4×D412yes—
225C22×C89—42no—
226C22×C89—81no—
227C22×C89—81no—
228C22×C89C22×D811yescharacteristic
229C2×D89—21no—
230C2×D89—21no—
231C2×D89—42no—
232C2×D89—81no—
233C2×D89—81no—
234C2×D89—164no—
235C2×D89C4×D812yes—
236C4×C89—21no—
237C4×C89—21no—
238C4×C89—42no—
239C4×C89—81no—
240C4×C89—81no—
241C4×C89C2×C4×D411yescharacteristic
242C4×C89C2×#13511yescharacteristic
243Dic89—21no—
244Dic89—21no—
245Dic89—42no—
246Dic89—81no—
247Dic89—81no—
248Dic89C4×D812yes—
249#33457—891nomaximal, Sylow
250C2×C89—82no—
251C2×C89C2×C4×D811yescharacteristic
252C2×C89#341611yescharacteristic
253C2×C89#341611yescharacteristic
254D89—22no—
255D89—82no—
256C22×SD32—891no—
257#2950—891no—
258#2948—891no—
259#2175—891no—
260#2175—891no—
261#2175—891no—
262#2175—891no—
263C4×SD32—891no—
264C4×SD32—891no—
265C4×SD32—891no—
266C4×SD32D8911yescharacteristic, complemented
267#2158—891no—
268#2141—891no—
269#2140—891no—
270#2108—891no—
271C89#3345711yescharacteristic, complemented, Sylow
272C22×Q16—891no—
273C22×D8—891no—
274#519—1781no—
275#518—1781no—
276C2×SD32—891no—
277C2×SD32—891no—
278C2×SD32—898no—
279C2×SD32—891no—
280C2×SD32C2×D8911yescharacteristic
281C2×M32—891no—
282C22×C16—891no—
283#442—1781no—
284#442—1781no—
285#441—1781no—
286#441—1781no—
287C4×Q16—891no—
288C4×Q16—891no—
289C4×Q16—891no—
290C4×Q16C2×D8911yescharacteristic, complemented
291C4×D8—891no—
292C4×D8—891no—
293C4×D8—891no—
294C4×D8C2×D8911yescharacteristic
295#427—891no—
296#427—891no—
297#419—1781no—
298#418—1781no—
299#405—891no—
300#367—891no—
301#367C2×D8911yescharacteristic, complemented
302#366—891no—
303#366—891no—
304#358—891no—
305#358—891no—
306#358—891no—
307#358C2×D8911yescharacteristic, complemented
308#357—891no—
309#357—891no—
310#357—891no—
311#357C2×D8911yescharacteristic
312#346—891no—
313#346—891no—
314C4×C16—891no—
315C4×C16C2×D8911yescharacteristic, complemented
316C22×Q8—1781no—
317C22×D4—1781no—
318C2×Q16—894no—
319C2×Q16—891no—
320C2×Q16—1782no—
321C2×Q16—1781no—
322C2×Q16C22×D8911yescharacteristic
323C2×D8—894no—
324C2×D8—891no—
325C2×D8—1782no—
326C2×D8—1781no—
327C2×D8C22×D8911yescharacteristic
328C2×M16—891no—
329C22×C8—891no—
330C4×Q8—21no—
331C4×Q8—3561no—
332C4×Q8—3561no—
333C4×Q8—1781no—
334C4×Q8—1781no—
335C4×Q8—1781no—
336C4×D4—21no—
337C4×D4—3561no—
338C4×D4—3561no—
339C4×D4—1781no—
340C4×D4—1781no—
341C4×D4—1781no—
342#117—891no—
343#117—3561no—
344#117—3561no—
345#117—1781no—
346#117—1781no—
347C2×C22⋊C4—3561no—
348C2×C22⋊C4—3561no—
349SD32—894no—
350SD32—894no—
351SD32—894no—
352SD32C4×D8914yescomplemented
353M32—1782no—
354C2×C16—891no—
355C2×C16—894no—
356C2×C16—891no—
357C2×C16C22×D8911yescharacteristic
358C2×C16C22×D8911yescharacteristic
359#107—891no—
360#107—891no—
361#107C22×D8911yescharacteristic
362#107C22×D8911yescharacteristic
363#106—1781no—
364#106—1781no—
365#103—21no—
366#103—1781no—
367#103—1781no—
368#103—1781no—
369#102—21no—
370#102—1781no—
371#102—1781no—
372#102—1781no—
373#97—891no—
374#97—891no—
375C4×C8—891no—
376C4×C8C22×D8911yescharacteristic
377C24—3561no—
378C2×Q8—21no—
379C2×Q8—3561no—
380C2×Q8—3562no—
381C2×Q8—1784no—
382C2×Q8—1781no—
383C2×D4—21no—
384C2×D4—3562no—
385C2×D4—3561no—
386C2×D4—1784no—
387C2×D4—1781no—
388C22×C4—41no—
389C22×C4—891no—
390C22×C4—3561no—
391C22×C4—3561no—
392C22×C4—3561no—
393C22×C4—3561no—
394Q16—21no—
395Q16—892no—
396Q16—1784no—
397Q16—1781no—
398Q16C2×C4×D8912yes—
399D8—21no—
400D8—892no—
401D8—1784no—
402D8—1781no—
403D8C2×C4×D8912yes—
404M16—1782no—
405C2×C8—891no—
406C2×C8—892no—
407C2×C8—892no—
408C2×C8—891no—
409C2×C8D4×D8911yescharacteristic
410C2×C8C23×D8911yescharacteristic
411C4⋊C4—21no—
412C4⋊C4—21no—
413C4⋊C4—41no—
414C4⋊C4—891no—
415C4⋊C4—891no—
416C4⋊C4—3561no—
417C4⋊C4—3561no—
418C4⋊C4—3561no—
419C4⋊C4—1781no—
420C4⋊C4—1781no—
421C22⋊C4—41no—
422C22⋊C4—891no—
423C22⋊C4—891no—
424C22⋊C4—3561no—
425C22⋊C4—3561no—
426C22⋊C4—3561no—
427C22⋊C4—3561no—
428C22⋊C4—3561no—
429C42—41no—
430C42—891no—
431C42—3561no—
432C42—3561no—
433C42—3561no—
434C42—3561no—
435C42—3561no—
436C42D4×D8911yescharacteristic
437C16—21no—
438C16—892no—
439C16—1781no—
440C16C2×C4×D8912yes—
441C23—41no—
442C23—891no—
443C23—3564no—
444C23—3561no—
445C23—14241no—
446C23—14241no—
447Q8—22no—
448Q8—41no—
449Q8—3564no—
450Q8—3561no—
451Q8—1782no—
452D4—22no—
453D4—41no—
454D4—3564no—
455D4—3561no—
456D4—1782no—
457C2×C4—41no—
458C2×C4—41no—
459C2×C4—41no—
460C2×C4—161no—
461C2×C4—891no—
462C2×C4—892no—
463C2×C4—892no—
464C2×C4—891no—
465C2×C4—891no—
466C2×C4—3564no—
467C2×C4—3561no—
468C2×C4—3561no—
469C2×C4—3561no—
470C2×C4—14241no—
471C2×C4—14241no—
472C2×C4—14241no—
473C2×C4D8×D8911yescharacteristic
474C2×C4D8×D8911yescharacteristic
475C2×C4C2×D4×D8911yescharacteristic
476C8—21no—
477C8—892no—
478C8—1781no—
479C8C22×C4×D8911yescharacteristic
480C8#41×D8911yescharacteristic
481C22—42no—
482C22—81no—
483C22—81no—
484C22—892no—
485C22—3562no—
486C22—14244no—
487C22—1781no—
488C22—1781no—
489C22—7121no—
490C22—7121no—
491C22C2×D8×D8911yescharacteristic
492C4—21no—
493C4—21no—
494C4—42no—
495C4—81no—
496C4—81no—
497C4—892no—
498C4—3562no—
499C4—1781no—
500C4—1781no—
501C4—7121no—
502C4—7121no—
503C4C4×D4×D8911yescharacteristic
504C4#135×D8911yescharacteristic
505C2—82no—
506C2—1782no—
507C2—7122no—
508C2C4×D8×D8911yescharacteristic
509C2#2603339411yescharacteristic
510C2#2603339511yescharacteristic
subgroupG/H
centerC22C2×D8×D89
commutatorC8×C89C23×C4
FrattiniC2×C8C23×D89
FittingC4×SD32×C89C2
socleC22×C89C22×D8
prime radicalC2×D8×D89C22
cosocleC2×C8×C89C24
hypercenterC4×SD32D89
hypocenterC89#33457
LCSG ▷ C8×C89 ▷ C4×C89 ▷ C2×C89 ▷ C89
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ C4×SD32

Complements

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

NH ≅ G/Ntype
2. #26029115C2semidirect
4. #26022119C2semidirect
5. #26022116C2semidirect
6. #26022114C2semidirect
7. #26022113C2semidirect
8. C4×#15667957C2semidirect
9. SD32×Dic89C2semidirect
10. #26022103C2semidirect
11. #26022035C2semidirect
12. #26021970C2semidirect
14. #26021847C2semidirect
15. C4×SD32×C89C2semidirect
23. SD32×D89C4semidirect
30. C4×Dic356C22semidirect
31. Q16×Dic89C22semidirect
32. #19151692C22semidirect
41. #19151426C22semidirect
42. #19151423C22semidirect
43. #19151423C22semidirect
44. #19151403C22semidirect
45. #19151402C22semidirect
46. #19151401C22semidirect
50. #19151374C22semidirect
51. #19151373C22semidirect
52. C16×Dic89C22semidirect
54. C4×Q16×C89C22semidirect
56. #367×C89C22semidirect
57. #358×C89C22semidirect
59. C4×C16×C89C22semidirect
65. Q16×D89C2×C4semidirect
89. #15667957C2×C4semidirect
90. #15667956C2×C4semidirect
91. #15667955C2×C4semidirect
95. C16×D89C2×C4semidirect
113. SD32×C89C2×C4semidirect
266. C4×SD32D89semidirect
271. C89#33457semidirect
290. C4×Q16C2×D89semidirect
301. #367C2×D89semidirect
307. #358C2×D89semidirect
315. C4×C16C2×D89semidirect
352. SD32C4×D89semidirect

Maximal subgroups

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

subgroupindex
C2×SD32×D892
#260291152
#260290992
#260221192
#260221162
#260221142
#260221132
C4×#156679572
SD32×Dic892
#260221032
#260220352
#260219702
#260219662
#260218472
C4×SD32×C892
#3345789