Finite Group Databasebeta

← #36711267

Subgroups — #36711267

Order 22784.

568 conjugacy classes of subgroups.

The subgroup lattice has 522 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26031939C211yesmaximal, characteristic
2#26022486C211yesmaximal, characteristic, complemented
3#26022449C211yesmaximal, characteristic, complemented
4#26022443C211yesmaximal, characteristic, complemented
5#26022406C211yesmaximal, characteristic, complemented
6#26022350C211yesmaximal, characteristic, complemented
7#26022341C211yesmaximal, characteristic, complemented
8#26022314C211yesmaximal, characteristic, complemented
9#26022190C211yesmaximal, characteristic, complemented
10#26022180C211yesmaximal, characteristic, complemented
11C4×#15667955C211yesmaximal, characteristic, complemented
12D4×#13083549C211yesmaximal, characteristic, complemented
13#26021957C211yesmaximal, characteristic, complemented
14C2×#19151398C211yesmaximal, characteristic, complemented
15C89×#3336C211yesmaximal, characteristic, complemented
16C2×#15668034C2211yescharacteristic
17C22×D712C2211yescharacteristic
18#19152244—21no—
19C2×#15667958C2211yescharacteristic, complemented
20C2×#15667955C2211yescharacteristic
21C2×#15667955C2211yescharacteristic
22C22×#13083549C2211yescharacteristic
23C22×#13083549C2211yescharacteristic, complemented
24#19151982C2211yescharacteristic
25#19151839C2211yescharacteristic
26#19151834C2211yescharacteristic
27#19151814—21no—
28#19151751—21no—
29#19151745—21no—
30C4×D712C2211yescharacteristic
31#19151425C2211yescharacteristic, complemented
32#19151425C2211yescharacteristic, complemented
33#19151423C2211yescharacteristic, complemented
34C4⋊#13083549C2211yescharacteristic, complemented
35#19151402C2211yescharacteristic, complemented
36#19151401C2211yescharacteristic, complemented
37#19151401C2211yescharacteristic, complemented
38#19151401C2211yescharacteristic, complemented
39#19151398C2211yescharacteristic
40#19151398C2211yescharacteristic
41#19151398C2211yescharacteristic
42#19151398C2211yescharacteristic
43#19151398C2211yescharacteristic
44#19151398C2211yescharacteristic
45C22⋊#13083549C2211yescharacteristic
46C22⋊#13083549C2211yescharacteristic, complemented
47C4×#13083549C2211yescharacteristic, complemented
48#552×C89—21no—
49#499×C89C2211yescharacteristic, complemented
50#485×C89—21no—
51#483×C89—21no—
52#479×C89—21no—
53#467×C89C2211yescharacteristic
54#467×C89C2211yescharacteristic, complemented
55C4×Q16×C89C2211yescharacteristic, complemented
56C8×D4×C89C2211yescharacteristic
57#427×C89C2211yescharacteristic, complemented
58C2×#107×C89C2211yescharacteristic
59C2×#13083574—21no—
60C22×D356—21no—
61#15668034—22no—
62#15668034—41no—
63C2×Dic356C2311yescharacteristic
64C2×#13083563—21no—
65C2×#13083563—21no—
66C2×D712—22no—
67C2×D712—21no—
68C2×D712—41no—
69C2×D712C2311yescharacteristic
70C2×D712C2311yescharacteristic
71#15668000—21no—
72#15667997—41no—
73#15667991—21no—
74#15667986—21no—
75#15667985—41no—
76#15667980—41no—
77C4×D356—21no—
78C22⋊C4×D89—41no—
79#15667958—41no—
80#15667955—22no—
81#15667955—41no—
82C2×#13083549—41no—
83C2×#13083549—41no—
84C2×#13083549C2311yescharacteristic
85C2×#13083549C2311yescharacteristic
86C2×#13083549C2311yescharacteristic
87C2×#13083549C2311yescharacteristic
88#15667946C2311yescharacteristic
89#15667937—21no—
90#15667934—21no—
91#15667934—21no—
92C2×Q16×C89C2311yescharacteristic
93C22×C8×C89C2311yescharacteristic
94C22×C8×C89C2311yescharacteristic
95#126×C89—41no—
96#125×C89—21no—
97#125×C89—41no—
98#123×C89—41no—
99C22⋊Q8×C89—21no—
100C22⋊Q8×C89—21no—
101C4×Q8×C89—21no—
102C4×D4×C89D411yescharacteristic
103#117×C89—21no—
104#117×C89—41no—
105C2×C4⋊C4×C89—21no—
106#107×C89—21no—
107#107×C89C2311yescharacteristic
108#107×C89C2311yescharacteristic
109#107×C89C2311yescharacteristic
110#106×C89—21no—
111#106×C89—21no—
112C4⋊C8×C89D411yescharacteristic
113#103×C89—21no—
114#103×C89—21no—
115#103×C89—21no—
116C22⋊C8×C89D411yescharacteristic
117C22⋊C8×C89D411yescharacteristic
118C4×C8×C89C2311yescharacteristic
119C23×D89—41no—
120#13083574—42no—
121#13083574—81no—
122C2×Dic178—21no—
123C2×D356—21no—
124C2×D356—21no—
125C2×D356—42no—
126C2×D356—41no—
127C2×D356—81no—
128C2×C22⋊D89—41no—
129C2×C22⋊D89—41no—
130C2×C4×D89—41no—
131C2×C4×D89—41no—
132Dic356—21no—
133Dic356—21no—
134#13083563—42no—
135#13083563—81no—
136D712—21no—
137D712—21no—
138D712—21no—
139D712—42no—
140D712#4112yes—
141C4⋊Dic89—21no—
142#13083553—41no—
143C22⋊Dic89—41no—
144#13083551—41no—
145#13083551—41no—
146#13083551—41no—
147#13083551—41no—
148#13083549—21no—
149#13083549—21no—
150#13083549—21no—
151#13083549—21no—
152C2×Q8×C89—21no—
153C2×D4×C89D811yescharacteristic
154C22×C4×C89—41no—
155C22×C4×C89C2×D411yescharacteristic
156C22×C4×C89C2×D411yescharacteristic
157Q16×C89—21no—
158Q16×C89—21no—
159C2×C8×C89—21no—
160C2×C8×C89—21no—
161C2×C8×C89—21no—
162C2×C8×C89—21no—
163C2×C8×C89C2×D411yescharacteristic
164C2×C8×C89C2×D411yescharacteristic
165C2×C8×C89C2×D411yescharacteristic
166C2×C8×C89C2411yescharacteristic
167C4⋊C4×C89—21no—
168C4⋊C4×C89—21no—
169C4⋊C4×C89—21no—
170C4⋊C4×C89—41no—
171C4⋊C4×C89—41no—
172C4⋊C4×C89—41no—
173C4⋊C4×C89—41no—
174C4⋊C4×C89—41no—
175C4⋊C4×C89—41no—
176C4⋊C4×C89D811yescharacteristic
177C22⋊C4×C89—41no—
178C22⋊C4×C89—41no—
179C22⋊C4×C89—41no—
180C22⋊C4×C89D811yescharacteristic
181C22⋊C4×C89D811yescharacteristic
182C42×C89—41no—
183C42×C89—41no—
184C42×C89C2×D411yescharacteristic
185C23×C89C2×D811yescharacteristic
186C23×C89C2×D811yescharacteristic
187C22×D89—41no—
188C22×D89—41no—
189C22×D89—82no—
190C22×D89—82no—
191C22×D89—161no—
192Q8×C89—41no—
193Q8×C89—41no—
194D4×C89—22no—
195C2×C4×C89—21no—
196C2×C4×C89—21no—
197C2×C4×C89—21no—
198C2×C4×C89—21no—
199C2×C4×C89—41no—
200C2×C4×C89—41no—
201C2×C4×C89—41no—
202C2×C4×C89—41no—
203C2×C4×C89—82no—
204C2×C4×C89C2×D811yescharacteristic
205C2×C4×C89C2×D811yescharacteristic
206C2×C4×C89C2×D811yescharacteristic
207C2×C4×C89C2×D811yescharacteristic
208C2×C4×C89C22×D411yescharacteristic
209C22⋊D89—82no—
210C22⋊D89—161no—
211C2×Dic89—41no—
212C2×Dic89—41no—
213D356—22no—
214D356—41no—
215D356—41no—
216D356—41no—
217D356—82no—
218C4×D89—82no—
219C4×D89—161no—
220Dic178—41no—
221Dic178—41no—
222C8×C89—21no—
223C8×C89—21no—
224C8×C89—21no—
225C8×C89C2×#4111yescharacteristic
226C8×C89H(5,𝔽2)11yescharacteristic
227C22×C89—21no—
228C22×C89—21no—
229C22×C89—21no—
230C22×C89—21no—
231C22×C89—21no—
232C22×C89#50812yes—
233C22×C89C22×D811yescharacteristic
234C2×D89—42no—
235C2×D89—81no—
236C2×D89—81no—
237C2×D89—81no—
238C2×D89—81no—
239C2×D89—81no—
240C2×D89—164no—
241C4×C89—21no—
242C4×C89—21no—
243C4×C89—21no—
244C4×C89—21no—
245C4×C89—81no—
246C4×C89—81no—
247C4×C89—81no—
248C4×C89—81no—
249C4×C89#54611yescharacteristic
250C4×C89#57711yescharacteristic
251Dic89—81no—
252Dic89—81no—
253#33863—891nomaximal, Sylow
254C2×C89—22no—
255C2×C89—41no—
256C2×C89#329411yescharacteristic
257C2×C89C2×#50811yescharacteristic
258C2×C89#341611yescharacteristic
259D89—82no—
260D89—161no—
261#3336D8911yescharacteristic, complemented
262#3280—891no—
263#2242—891no—
264#2235—891no—
265#2234—891no—
266#2228—891no—
267#2217—891no—
268#2216—891no—
269#2211—891no—
270#2189—891no—
271#2187—891no—
272C4×SD32—891no—
273D4×C16—891no—
274#2138—891no—
275C2×#357—891no—
276C89#3386311yescharacteristic, complemented, Sylow
277C2×#135—891no—
278C22×D8—891no—
279#552—21no—
280#546—1781no—
281C2×Q32—891no—
282C2×SD32—891no—
283C2×SD32—891no—
284C22×C16—891no—
285C22×C16—891no—
286#499C2×D8911yescharacteristic, complemented
287#495—891no—
288#485—21no—
289#483—21no—
290#479—21no—
291#467—891no—
292#467C2×D8911yescharacteristic
293#467C2×D8911yescharacteristic, complemented
294#466—891no—
295#463—1781no—
296#450—1781no—
297#449—1781no—
298C4×Q16C2×D8911yescharacteristic, complemented
299C4×D8—891no—
300C8×D4C2×D8911yescharacteristic
301#427C2×D8911yescharacteristic, complemented
302C2×#107C2×D8911yescharacteristic
303#367—891no—
304#367—891no—
305#366—891no—
306C4⋊C16—891no—
307#358—891no—
308#358—891no—
309#358—891no—
310#358—891no—
311#357—891no—
312#357—891no—
313#357—891no—
314#357—891no—
315#357—891no—
316#357—891no—
317C22⋊C16—891no—
318C22⋊C16—891no—
319C4×C16—891no—
320C2×#41—1781no—
321C22×D4—1781no—
322#135—3561no—
323#135—1782no—
324C2×Q16—891no—
325C2×Q16C22×D8911yescharacteristic
326C2×SD16—1781no—
327C2×SD16—1781no—
328C2×D8—891no—
329C2×D8—891no—
330C2×D8—3561no—
331C2×D8—1782no—
332C2×D8—1781no—
333C22×C8C22×D8911yescharacteristic
334C22×C8C22×D8911yescharacteristic
335#126—41no—
336#125—21no—
337#125—41no—
338#124—1781no—
339#123—41no—
340#123—3561no—
341C22⋊Q8—21no—
342C22⋊Q8—21no—
343C22⋊Q8—1781no—
344#121—3561no—
345#121—1781no—
346#120—3561no—
347C4×Q8—21no—
348C4×D4—1781no—
349C4×D4C22⋊D8911yescharacteristic
350#117—21no—
351#117—41no—
352C2×C4⋊C4—21no—
353C2×C22⋊C4—3561no—
354Q32—3561no—
355SD32—3561no—
356SD32—1782no—
357C2×C16—891no—
358C2×C16—891no—
359C2×C16—891no—
360C2×C16—891no—
361C2×C16—3561no—
362C2×C16—3561no—
363#107—21no—
364#107—891no—
365#107C22×D8911yescharacteristic
366#107C22×D8911yescharacteristic
367#107C22×D8911yescharacteristic
368#106—21no—
369#106—21no—
370C4⋊C8C22⋊D8911yescharacteristic
371#103—21no—
372#103—21no—
373#103—21no—
374#103—1781no—
375#102—1781no—
376#102—1781no—
377C22⋊C8C22⋊D8911yescharacteristic
378C22⋊C8C22⋊D8911yescharacteristic
379C4×C8C22×D8911yescharacteristic
380C24—3561no—
381#41—3562no—
382#41—7121no—
383C2×Q8—21no—
384C2×Q8—1781no—
385C2×D4—3562no—
386C2×D4—3561no—
387C2×D4—3561no—
388C2×D4—3561no—
389C2×D4—1781no—
390C2×D4—1781no—
391C2×D4—7121no—
392C2×D4#1308355911yescharacteristic
393C22×C4—41no—
394C22×C4—3561no—
395C22×C4—3561no—
396C22×C4C2×C22⋊D8911yescharacteristic
397C22×C4C2×C22⋊D8911yescharacteristic
398Q16—21no—
399Q16—21no—
400Q16—1781no—
401Q16—1781no—
402SD16—3562no—
403SD16—7121no—
404D8—892no—
405D8—3562no—
406D8—1781no—
407D8—1781no—
408D8—1781no—
409C2×C8—21no—
410C2×C8—21no—
411C2×C8—21no—
412C2×C8—21no—
413C2×C8C2×C22⋊D8911yescharacteristic
414C2×C8C2×C22⋊D8911yescharacteristic
415C2×C8C2×C22⋊D8911yescharacteristic
416C2×C8C23×D8911yescharacteristic
417C4⋊C4—21no—
418C4⋊C4—21no—
419C4⋊C4—21no—
420C4⋊C4—41no—
421C4⋊C4—41no—
422C4⋊C4—41no—
423C4⋊C4—41no—
424C4⋊C4—41no—
425C4⋊C4—41no—
426C4⋊C4—3561no—
427C4⋊C4—1781no—
428C4⋊C4#1308355911yescharacteristic
429C22⋊C4—41no—
430C22⋊C4—41no—
431C22⋊C4—41no—
432C22⋊C4—3561no—
433C22⋊C4—3561no—
434C22⋊C4—3561no—
435C22⋊C4—3561no—
436C22⋊C4—3561no—
437C22⋊C4#1308355911yescharacteristic
438C22⋊C4#1308355911yescharacteristic
439C42—41no—
440C42—41no—
441C42C2×C22⋊D8911yescharacteristic
442C16—1781no—
443C16—1781no—
444C16—1781no—
445C16—1781no—
446C23—3561no—
447C23—3561no—
448C23—14241no—
449C23—7122no—
450C23—7122no—
451C23C2×#1308355911yescharacteristic
452C23C2×#1308355911yescharacteristic
453Q8—41no—
454Q8—41no—
455Q8—3561no—
456Q8—3561no—
457D4—22no—
458D4—3561no—
459D4—3561no—
460D4—3561no—
461D4—14241no—
462D4—1782no—
463D4—7122no—
464D4—7122no—
465C2×C4—21no—
466C2×C4—21no—
467C2×C4—21no—
468C2×C4—21no—
469C2×C4—41no—
470C2×C4—41no—
471C2×C4—41no—
472C2×C4—41no—
473C2×C4—82no—
474C2×C4—3561no—
475C2×C4—3561no—
476C2×C4—14241no—
477C2×C4—7122no—
478C2×C4C2×#1308355911yescharacteristic
479C2×C4C2×#1308355911yescharacteristic
480C2×C4C2×#1308355911yescharacteristic
481C2×C4C2×#1308355911yescharacteristic
482C2×C4C22×C22⋊D8911yescharacteristic
483C8—21no—
484C8—21no—
485C8—21no—
486C8#41×D8911yescharacteristic
487C8#1566805811yescharacteristic
488C22—21no—
489C22—21no—
490C22—21no—
491C22—21no—
492C22—21no—
493C22—3562no—
494C22—14244no—
495C22—7121no—
496C22—7121no—
497C22—7121no—
498C22—7121no—
499C22—7121no—
500C22#1915202812yes—
501C22C22×#1308355911yescharacteristic
502C4—21no—
503C4—21no—
504C4—21no—
505C4—21no—
506C4—81no—
507C4—81no—
508C4—81no—
509C4—81no—
510C4—7121no—
511C4—7121no—
512C4#1915223611yescharacteristic
513C4#1915240611yescharacteristic
514C2—22no—
515C2—41no—
516C2—14241no—
517C2—7122no—
518C2#2603205911yescharacteristic
519C2C2×#1915202811yescharacteristic
520C2#2603339611yescharacteristic
subgroupG/H
centerC22C22×#13083559
commutator, cosocleC2×C8×C89C24
FrattiniC2×C8C23×D89
FittingC89×#3336C2
socleC22×C89C22×D8
prime radical#26031939C2
hypercenter#3336D89
hypocenterC89#33863
LCSG ▷ C2×C8×C89 ▷ C4×C89 ▷ C2×C89 ▷ C89
UCSC1 ◁ C22 ◁ C4×D4 ◁ C8×D4 ◁ #3336

Complements

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

NH ≅ G/Ntype
2. #26022486C2semidirect
3. #26022449C2semidirect
4. #26022443C2semidirect
5. #26022406C2semidirect
6. #26022350C2semidirect
7. #26022341C2semidirect
8. #26022314C2semidirect
9. #26022190C2semidirect
10. #26022180C2semidirect
11. C4×#15667955C2semidirect
12. D4×#13083549C2semidirect
13. #26021957C2semidirect
14. C2×#19151398C2semidirect
15. C89×#3336C2semidirect
19. C2×#15667958C22semidirect
23. C22×#13083549C22semidirect
31. #19151425C22semidirect
32. #19151425C22semidirect
33. #19151423C22semidirect
34. C4⋊#13083549C22semidirect
35. #19151402C22semidirect
36. #19151401C22semidirect
37. #19151401C22semidirect
38. #19151401C22semidirect
46. C22⋊#13083549C22semidirect
47. C4×#13083549C22semidirect
49. #499×C89C22semidirect
54. #467×C89C22semidirect
55. C4×Q16×C89C22semidirect
57. #427×C89C22semidirect
261. #3336D89semidirect
276. C89#33863semidirect
286. #499C2×D89semidirect
293. #467C2×D89semidirect
298. C4×Q16C2×D89semidirect
301. #427C2×D89semidirect

Maximal subgroups

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

subgroupindex
#260319392
#260224862
#260224492
#260224432
#260224062
#260223502
#260223412
#260223142
#260221902
#260221802
C4×#156679552
D4×#130835492
#260219572
C2×#191513982
C89×#33362
#3386389