Finite Group Databasebeta

← #36711513

Subgroups — #36711513

Order 22784.

570 conjugacy classes of subgroups.

The subgroup lattice has 526 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26031939C211yesmaximal, characteristic
2#26022491C211yesmaximal, characteristic, complemented
3#26022454C211yesmaximal, characteristic, complemented
4#26022432C211yesmaximal, characteristic, complemented
5#26022386C211yesmaximal, characteristic, complemented
6#26022367C211yesmaximal, characteristic, complemented
7#26022341C211yesmaximal, characteristic, complemented
8#26022314C211yesmaximal, characteristic, complemented
9#26022190C211yesmaximal, characteristic, complemented
10#26022180C211yesmaximal, characteristic, complemented
11#26022114C211yesmaximal, characteristic, complemented
12#26022078C211yesmaximal, characteristic, complemented
13#26021962C211yesmaximal, characteristic, complemented
14C2×#19151398C211yesmaximal, characteristic, complemented
15C89×#3338C211yesmaximal, characteristic, complemented
16C2×#15668034C2211yescharacteristic
17C22×D712C2211yescharacteristic
18#19152244—21no—
19C2×#15667958C2211yescharacteristic, complemented
20C2×#15667955C2211yescharacteristic
21C2×#15667955C2211yescharacteristic
22C2×#15667952C2211yescharacteristic, complemented
23C22×#13083549C2211yescharacteristic
24#19151982C2211yescharacteristic
25#19151839C2211yescharacteristic
26#19151834C2211yescharacteristic
27#19151814—21no—
28#19151751—21no—
29#19151745—21no—
30C4×D712C2211yescharacteristic
31#19151423C2211yescharacteristic, complemented
32#19151423C2211yescharacteristic, 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
47#19151374C2211yescharacteristic, complemented
48#554×C89—21no—
49#500×C89C2211yescharacteristic, complemented
50#484×C89—21no—
51#484×C89—21no—
52#477×C89—21no—
53#467×C89C2211yescharacteristic
54#467×C89C2211yescharacteristic, complemented
55C4×Q16×C89C2211yescharacteristic, complemented
56C8×D4×C89C2211yescharacteristic
57C2×#107×C89C2211yescharacteristic
58C2×#107×C89C2211yescharacteristic, complemented
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—
82#15667952—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
95C4⋊Q8×C89—21no—
96#125×C89—41no—
97C22⋊Q8×C89—21no—
98C22⋊Q8×C89—21no—
99C22⋊Q8×C89—41no—
100C22⋊Q8×C89—41no—
101C4×Q8×C89—21no—
102C4×D4×C89—41no—
103C4×D4×C89D411yescharacteristic
104C2×C4⋊C4×C89—21no—
105C2×C4⋊C4×C89—21no—
106#107×C89—21no—
107#107×C89—21no—
108#107×C89—21no—
109#107×C89C2311yescharacteristic
110#107×C89C2311yescharacteristic
111#107×C89C2311yescharacteristic
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×Q8×C89—41no—
154C2×D4×C89D811yescharacteristic
155C22×C4×C89—41no—
156C22×C4×C89—41no—
157C22×C4×C89C2×D411yescharacteristic
158C22×C4×C89C2×D411yescharacteristic
159Q16×C89—21no—
160Q16×C89—21no—
161C2×C8×C89—21no—
162C2×C8×C89—21no—
163C2×C8×C89—21no—
164C2×C8×C89—21no—
165C2×C8×C89C2×D411yescharacteristic
166C2×C8×C89C2×D411yescharacteristic
167C2×C8×C89C2×D411yescharacteristic
168C2×C8×C89C2411yescharacteristic
169C4⋊C4×C89—21no—
170C4⋊C4×C89—21no—
171C4⋊C4×C89—21no—
172C4⋊C4×C89—41no—
173C4⋊C4×C89—41no—
174C4⋊C4×C89—41no—
175C4⋊C4×C89—41no—
176C4⋊C4×C89—41no—
177C4⋊C4×C89—41no—
178C4⋊C4×C89D811yescharacteristic
179C22⋊C4×C89—41no—
180C22⋊C4×C89—41no—
181C22⋊C4×C89D811yescharacteristic
182C22⋊C4×C89D811yescharacteristic
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—
194Q8×C89—161no—
195D4×C89—22no—
196C2×C4×C89—21no—
197C2×C4×C89—21no—
198C2×C4×C89—21no—
199C2×C4×C89—21no—
200C2×C4×C89—41no—
201C2×C4×C89—41no—
202C2×C4×C89—41no—
203C2×C4×C89—41no—
204C2×C4×C89—161no—
205C2×C4×C89—161no—
206C2×C4×C89C2×D811yescharacteristic
207C2×C4×C89C2×D811yescharacteristic
208C2×C4×C89C2×D811yescharacteristic
209C2×C4×C89C2×D811yescharacteristic
210C2×C4×C89C22×D411yescharacteristic
211C22⋊D89—82no—
212C22⋊D89—161no—
213C2×Dic89—41no—
214C2×Dic89—41no—
215D356—22no—
216D356—41no—
217D356—41no—
218D356—41no—
219D356—82no—
220C4×D89—82no—
221C4×D89—161no—
222Dic178—41no—
223Dic178—41no—
224C8×C89—21no—
225C8×C89—21no—
226C8×C89—21no—
227C8×C89C2×#4111yescharacteristic
228C8×C89H(5,𝔽2)11yescharacteristic
229C22×C89—21no—
230C22×C89—21no—
231C22×C89—21no—
232C22×C89—21no—
233C22×C89—21no—
234C22×C89#51012yes—
235C22×C89C22×D811yescharacteristic
236C2×D89—42no—
237C2×D89—81no—
238C2×D89—81no—
239C2×D89—81no—
240C2×D89—81no—
241C2×D89—81no—
242C2×D89—164no—
243C4×C89—21no—
244C4×C89—21no—
245C4×C89—21no—
246C4×C89—21no—
247C4×C89—81no—
248C4×C89—81no—
249C4×C89—81no—
250C4×C89—81no—
251C4×C89#54611yescharacteristic
252C4×C89#57711yescharacteristic
253Dic89—81no—
254Dic89—81no—
255#33881—891nomaximal, Sylow
256C2×C89—22no—
257C2×C89—41no—
258C2×C89#329411yescharacteristic
259C2×C89C2×#51011yescharacteristic
260C2×C89#341511yescharacteristic
261D89—82no—
262D89—161no—
263#3338D8911yescharacteristic, complemented
264#3280—891no—
265#2243—891no—
266#2236—891no—
267#2232—891no—
268#2224—891no—
269#2220—891no—
270#2216—891no—
271#2211—891no—
272#2189—891no—
273#2187—891no—
274#2175—891no—
275#2168—891no—
276#2139—891no—
277C2×#357—891no—
278C89#3388111yescharacteristic, complemented, Sylow
279C2×#135—891no—
280C22×D8—891no—
281#554—21no—
282#546—1781no—
283C2×Q32—891no—
284C2×SD32—891no—
285C2×SD32—891no—
286C2×M32—891no—
287C22×C16—891no—
288#500C2×D8911yescharacteristic, complemented
289#495—891no—
290#484—21no—
291#484—21no—
292#477—21no—
293#467—891no—
294#467C2×D8911yescharacteristic
295#467C2×D8911yescharacteristic, complemented
296#466—891no—
297#463—1781no—
298#450—1781no—
299#449—1781no—
300C4×Q16C2×D8911yescharacteristic, complemented
301C4×D8—891no—
302C8×D4C2×D8911yescharacteristic
303C2×#107C2×D8911yescharacteristic
304C2×#107C2×D8911yescharacteristic, complemented
305#366—891no—
306#366—891no—
307#366—891no—
308C4⋊C16—891no—
309#358—891no—
310#358—891no—
311#358—891no—
312#358—891no—
313#357—891no—
314#357—891no—
315#357—891no—
316#357—891no—
317#357—891no—
318#357—891no—
319C22⋊C16—891no—
320C22⋊C16—891no—
321#346—891no—
322C2×#41—1781no—
323C22×D4—1781no—
324#135—3561no—
325#135—1782no—
326C2×Q16—891no—
327C2×Q16C22×D8911yescharacteristic
328C2×SD16—1781no—
329C2×SD16—1781no—
330C2×D8—891no—
331C2×D8—891no—
332C2×D8—3561no—
333C2×D8—1782no—
334C2×D8—1781no—
335C22×C8C22×D8911yescharacteristic
336C22×C8C22×D8911yescharacteristic
337C4⋊Q8—21no—
338#125—41no—
339#124—1781no—
340#123—3561no—
341C22⋊Q8—21no—
342C22⋊Q8—21no—
343C22⋊Q8—41no—
344C22⋊Q8—41no—
345C22⋊Q8—1781no—
346#121—3561no—
347#121—1781no—
348#120—3561no—
349C4×Q8—21no—
350C4×D4—41no—
351C4×D4—1781no—
352C4×D4C22⋊D8911yescharacteristic
353C2×C4⋊C4—21no—
354C2×C4⋊C4—21no—
355C2×C22⋊C4—3561no—
356Q32—3561no—
357SD32—3561no—
358SD32—1782no—
359M32—3561no—
360C2×C16—891no—
361C2×C16—891no—
362C2×C16—891no—
363C2×C16—891no—
364C2×C16—3561no—
365#107—21no—
366#107—21no—
367#107—21no—
368#107—891no—
369#107C22×D8911yescharacteristic
370#107C22×D8911yescharacteristic
371#107C22×D8911yescharacteristic
372C4⋊C8C22⋊D8911yescharacteristic
373#103—21no—
374#103—21no—
375#103—21no—
376#103—1781no—
377#102—1781no—
378#102—1781no—
379C22⋊C8C22⋊D8911yescharacteristic
380C22⋊C8C22⋊D8911yescharacteristic
381C4×C8C22×D8911yescharacteristic
382C24—3561no—
383#41—3562no—
384#41—7121no—
385C2×Q8—21no—
386C2×Q8—41no—
387C2×Q8—1781no—
388C2×D4—3562no—
389C2×D4—3561no—
390C2×D4—3561no—
391C2×D4—3561no—
392C2×D4—1781no—
393C2×D4—1781no—
394C2×D4—7121no—
395C2×D4#1308355911yescharacteristic
396C22×C4—41no—
397C22×C4—41no—
398C22×C4—3561no—
399C22×C4—3561no—
400C22×C4C2×C22⋊D8911yescharacteristic
401C22×C4C2×C22⋊D8911yescharacteristic
402Q16—21no—
403Q16—21no—
404Q16—1781no—
405Q16—1781no—
406SD16—3562no—
407SD16—7121no—
408D8—892no—
409D8—3562no—
410D8—1781no—
411D8—1781no—
412D8—1781no—
413C2×C8—21no—
414C2×C8—21no—
415C2×C8—21no—
416C2×C8—21no—
417C2×C8C2×C22⋊D8911yescharacteristic
418C2×C8C2×C22⋊D8911yescharacteristic
419C2×C8C2×C22⋊D8911yescharacteristic
420C2×C8C23×D8911yescharacteristic
421C4⋊C4—21no—
422C4⋊C4—21no—
423C4⋊C4—21no—
424C4⋊C4—41no—
425C4⋊C4—41no—
426C4⋊C4—41no—
427C4⋊C4—41no—
428C4⋊C4—41no—
429C4⋊C4—41no—
430C4⋊C4—3561no—
431C4⋊C4—1781no—
432C4⋊C4#1308355911yescharacteristic
433C22⋊C4—41no—
434C22⋊C4—41no—
435C22⋊C4—3561no—
436C22⋊C4—3561no—
437C22⋊C4—3561no—
438C22⋊C4—3561no—
439C22⋊C4—3561no—
440C22⋊C4#1308355911yescharacteristic
441C22⋊C4#1308355911yescharacteristic
442C42—41no—
443C42C2×C22⋊D8911yescharacteristic
444C16—1781no—
445C16—1781no—
446C16—1781no—
447C16—1781no—
448C23—3561no—
449C23—3561no—
450C23—14241no—
451C23—7122no—
452C23—7122no—
453C23C2×#1308355911yescharacteristic
454C23C2×#1308355911yescharacteristic
455Q8—41no—
456Q8—41no—
457Q8—161no—
458Q8—3561no—
459Q8—3561no—
460D4—22no—
461D4—3561no—
462D4—3561no—
463D4—3561no—
464D4—14241no—
465D4—1782no—
466D4—7122no—
467D4—7122no—
468C2×C4—21no—
469C2×C4—21no—
470C2×C4—21no—
471C2×C4—21no—
472C2×C4—41no—
473C2×C4—41no—
474C2×C4—41no—
475C2×C4—41no—
476C2×C4—161no—
477C2×C4—161no—
478C2×C4—3561no—
479C2×C4—3561no—
480C2×C4—14241no—
481C2×C4—7122no—
482C2×C4C2×#1308355911yescharacteristic
483C2×C4C2×#1308355911yescharacteristic
484C2×C4C2×#1308355911yescharacteristic
485C2×C4C2×#1308355911yescharacteristic
486C2×C4C22×C22⋊D8911yescharacteristic
487C8—21no—
488C8—21no—
489C8—21no—
490C8#41×D8911yescharacteristic
491C8#1566805811yescharacteristic
492C22—21no—
493C22—21no—
494C22—21no—
495C22—21no—
496C22—21no—
497C22—3562no—
498C22—14244no—
499C22—7121no—
500C22—7121no—
501C22—7121no—
502C22—7121no—
503C22—7121no—
504C22#1915203912yes—
505C22C22×#1308355911yescharacteristic
506C4—21no—
507C4—21no—
508C4—21no—
509C4—21no—
510C4—81no—
511C4—81no—
512C4—81no—
513C4—81no—
514C4—7121no—
515C4—7121no—
516C4#1915223611yescharacteristic
517C4#1915240611yescharacteristic
518C2—22no—
519C2—41no—
520C2—14241no—
521C2—7122no—
522C2#2603205911yescharacteristic
523C2C2×#1915203911yescharacteristic
524C2#2603339111yescharacteristic
subgroupG/H
centerC22C22×#13083559
commutator, cosocleC2×C8×C89C24
FrattiniC2×C8C23×D89
FittingC89×#3338C2
socleC22×C89C22×D8
prime radical#26031939C2
hypercenter#3338D89
hypocenterC89#33881
LCSG ▷ C2×C8×C89 ▷ C4×C89 ▷ C2×C89 ▷ C89
UCSC1 ◁ C22 ◁ C4×D4 ◁ C8×D4 ◁ #3338

Complements

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

NH ≅ G/Ntype
2. #26022491C2semidirect
3. #26022454C2semidirect
4. #26022432C2semidirect
5. #26022386C2semidirect
6. #26022367C2semidirect
7. #26022341C2semidirect
8. #26022314C2semidirect
9. #26022190C2semidirect
10. #26022180C2semidirect
11. #26022114C2semidirect
12. #26022078C2semidirect
13. #26021962C2semidirect
14. C2×#19151398C2semidirect
15. C89×#3338C2semidirect
19. C2×#15667958C22semidirect
22. C2×#15667952C22semidirect
31. #19151423C22semidirect
32. #19151423C22semidirect
33. #19151423C22semidirect
34. C4⋊#13083549C22semidirect
35. #19151402C22semidirect
36. #19151401C22semidirect
37. #19151401C22semidirect
38. #19151401C22semidirect
46. C22⋊#13083549C22semidirect
47. #19151374C22semidirect
49. #500×C89C22semidirect
54. #467×C89C22semidirect
55. C4×Q16×C89C22semidirect
58. C2×#107×C89C22semidirect
263. #3338D89semidirect
278. C89#33881semidirect
288. #500C2×D89semidirect
295. #467C2×D89semidirect
300. C4×Q16C2×D89semidirect
304. C2×#107C2×D89semidirect

Maximal subgroups

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

subgroupindex
#260319392
#260224912
#260224542
#260224322
#260223862
#260223672
#260223412
#260223142
#260221902
#260221802
#260221142
#260220782
#260219622
C2×#191513982
C89×#33382
#3388189