Finite Group Databasebeta

← #31762363

Subgroups — #31762363

Order 17152.

2038 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
1#22574978C212yesmaximal, complemented
2#22573361C214yesmaximal, complemented
3#22573355C214yesmaximal, complemented
4#22573354C214yesmaximal, complemented
5#22573349C214yesmaximal, complemented
6#22573345C211yesmaximal, characteristic, complemented
7#22573342C214yesmaximal, complemented
8C2×#16863826C212yesmaximal, complemented
9#508×D67C214yesmaximal, complemented
10C2×#16863814C211yesmaximal, characteristic, complemented
11C67×#3414C211yesmaximal, characteristic, complemented
12#16864232—22no—
13#16864210C2214yescomplemented
14#16864209C2214yescomplemented
15#16864203—24no—
16#16864202—24no—
17#16864198C2214yescomplemented
18#16864197C2214yescomplemented
19#16864194—22no—
20#16864193—24no—
21C2×#14524065C2212yescomplemented
22#135×D67C2218yescomplemented
23C2×#14524049C2211yescharacteristic
24#16863839C2214yescomplemented
25#16863838C2218yescomplemented
26#16863835C2218yescomplemented
27#16863834C2214yescomplemented
28#16863833C2218yescomplemented
29#16863830C2218yescomplemented
30#16863829C2214yescomplemented
31#16863828C2214yescomplemented
32#16863827C2218yescomplemented
33#16863826C2214yescomplemented
34#16863826C2214yescomplemented
35#16863826C2214yescomplemented
36#16863826C2214yescomplemented
37#16863825C2214yescomplemented
38C2×#14523989C2212yescomplemented
39Q32×D67C2214yescomplemented
40C2×#14523987C2212yescomplemented
41C2×#14523986C2212yescomplemented
42SD32×D67C2218yescomplemented
43C2×#14523984C2212yescomplemented
44D16×D67C2214yescomplemented
45#16863814C2214yescomplemented
46#16863814C2214yescomplemented
47C2×#14523982C2212yescomplemented
48C22×C67⋊C16C2211yescharacteristic, complemented
49C2×C16×D67C2212yescomplemented
50C2×#135×C67C2212yescomplemented
51#510×C67C2214yescomplemented
52#509×C67C2214yescomplemented
53#508×C67C2214yescomplemented
54C2×M32×C67C2211yescharacteristic, complemented
55#14524091—24no—
56#14524090—24no—
57#14524090—42no—
58#14524089—24no—
59#14524089—42no—
60#14524088—24no—
61#14524088—44no—
62C2×#11761628—22no—
63C2×#11761628—44no—
64C2×#11761628D411yescharacteristic
65#41×D67—28no—
66#41×D67—44no—
67#14524075—28no—
68#14524074—24no—
69#14524073—24no—
70#14524072—28no—
71#14524070—28no—
72#14524069—24no—
73#14524068—24no—
74#14524067—28no—
75#14524066C2318yes—
76#14524065—22no—
77#14524065—22no—
78#14524065C2314yes—
79#14524065C2314yes—
80#14524064—28no—
81#14524063—28no—
82#14524062C2318yes—
83C2×Dic268C2312yes—
84Q16×D67C2318yes—
85C2×#11761617—22no—
86SD16×D67—28no—
87C2×D536C2312yes—
88D8×D67C2318yes—
89#14524049D414yes—
90#14524049C2314yes—
91C2×#11761612D411yescharacteristic
92C2×#11761611D412yes—
93C2×C8×D67C2312yes—
94Dic536C2314yes—
95#14523989C2314yes—
96#14523989C2314yes—
97#14523988C2318yes—
98#14523987C2314yes—
99#14523987C2314yes—
100#14523986C2314yes—
101#14523986C2314yes—
102D1072C2314yes—
103#14523984C2314yes—
104#14523984C2314yes—
105#14523982C2318yes—
106C2×C67⋊C16C2311yescharacteristic
107C2×C67⋊C16C2311yescharacteristic
108C2×C67⋊C16C2312yes—
109C2×C67⋊C16C2312yes—
110C16×D67C2314yes—
111C16×D67C2314yes—
112C2×#41×C67—22no—
113#135×C67—22no—
114#135×C67—22no—
115#135×C67C2314yes—
116#135×C67C2314yes—
117C2×Q16×C67C2312yes—
118C2×SD16×C67—22no—
119C2×D8×C67C2312yes—
120C22×C8×C67C2311yescharacteristic
121Q32×C67C2314yes—
122SD32×C67C2318yes—
123D16×C67C2314yes—
124M32×C67C2314yes—
125C2×C16×C67C2312yes—
126#11761629—28no—
127#11761629—44no—
128#11761628—24no—
129#11761628—24no—
130#11761628—42no—
131#11761628—42no—
132#11761628—48no—
133#11761628—44no—
134#11761628—48no—
135#11761628—44no—
136#11761628D814yes—
137#11761628C2×D414yes—
138#11761627—28no—
139#11761627—48no—
140#11761627—48no—
141#11761627—44no—
142C2×Dic134—22no—
143C2×Dic134—44no—
144C2×Dic134D811yescharacteristic
145Q8×D67—28no—
146Q8×D67—44no—
147C2×D268—22no—
148C2×D268—44no—
149C2×D268D811yescharacteristic
150C2×C22⋊D67—42no—
151C2×C22⋊D67—48no—
152C2×C22⋊D67D812yes—
153D4×D67—28no—
154D4×D67—48no—
155D4×D67—48no—
156D4×D67—44no—
157C2×C4×D67—42no—
158C2×C4×D67—48no—
159C2×C4×D67C2×D412yes—
160Dic268—22no—
161Dic268C2414yes—
162#11761618—28no—
163#11761617—24no—
164#11761617—24no—
165#11761616—28no—
166#11761615—28no—
167D536—22no—
168D536C2414yes—
169#11761613—28no—
170#11761612C2×D414yes—
171#11761611C2×D418yes—
172C2×C67⋊C8C2×D412yes—
173C8×D67C2×D414yes—
174C8×D67C2414yes—
175C67⋊C16C2411yescharacteristic
176C67⋊C16C2411yescharacteristic
177C67⋊C16C2412yes—
178#41×C67—24no—
179#41×C67—24no—
180#41×C67—42no—
181#41×C67—42no—
182C2×Q8×C67—22no—
183C2×D4×C67—22no—
184C2×D4×C67—42no—
185C22×C4×C67—42no—
186C22×C4×C67C2×D411yescharacteristic
187Q16×C67—22no—
188Q16×C67C2414yes—
189SD16×C67—24no—
190SD16×C67—24no—
191D8×C67—22no—
192D8×C67C2414yes—
193C2×C8×C67C2×D412yes—
194C2×C8×C67C2×D411yescharacteristic
195C2×C8×C67C2411yescharacteristic
196C2×C8×C67C2412yes—
197C16×C67C2414yes—
198C23×C67—42no—
199C23×C67C2×D811yescharacteristic
200C22×D67—42no—
201C22×D67—48no—
202C22×D67C2×D812yes—
203Q8×C67—24no—
204Q8×C67—42no—
205D4×C67—24no—
206D4×C67—44no—
207D4×C67—44no—
208D4×C67—42no—
209C2×C4×C67—42no—
210C2×C4×C67—44no—
211C2×C4×C67—44no—
212C2×C4×C67C2×D812yes—
213C2×C4×C67C2×D811yescharacteristic
214C2×C4×C67C22×D412yes—
215C2×C4×C67C22×D411yescharacteristic
216C22⋊D67—44no—
217C22⋊D67—44no—
218C22⋊D67—48no—
219C22⋊D67—48no—
220C22⋊D67C2×D818yes—
221C2×Dic67—42no—
222C2×Dic67—48no—
223C2×Dic67C2×D812yes—
224D268—24no—
225D268—48no—
226D268—42no—
227D268C2×D814yes—
228C4×D67—48no—
229C4×D67—44no—
230C4×D67—44no—
231C4×D67—48no—
232C4×D67C2×D814yes—
233C4×D67C22×D414yes—
234Dic134—24no—
235Dic134—42no—
236Dic134—48no—
237Dic134C2×D814yes—
238C8×C67C22×D412yes—
239C8×C67C22×D411yescharacteristic
240C8×C67C2511yescharacteristic
241C67⋊C8C22×D414yes—
242C22×C67—41no—
243C22×C67—44no—
244C22×C67—162no—
245C22×C67C22×D812yes—
246C22×C67C22×D811yescharacteristic
247C2×D67—42no—
248C2×D67—44no—
249C2×D67—168no—
250C2×D67—162no—
251C2×D67C22×D814yes—
252C4×C67—44no—
253C4×C67C22×D812yes—
254C4×C67C22×D811yescharacteristic
255C4×C67C23×D411yescharacteristic
256Dic67—44no—
257Dic67C22×D814yes—
258#62635—671nomaximal, Sylow
259C2×C67—21no—
260C2×C67—22no—
261C2×C67—84no—
262C2×C67C23×D811yescharacteristic
263D67—24no—
264D67—84no—
265#3584—672no—
266#3417—674no—
267#3416—674no—
268#3416—674no—
269#3415—674no—
270#3414—674no—
271#3414—671no—
272#3414D6711yescharacteristic, complemented
273C2×#508—674no—
274C2×#508—672no—
275C2×#504—671no—
276C67#6263511yescharacteristic, complemented, Sylow
277#585—1342no—
278#578—674no—
279#578—674no—
280#577—1344no—
281#577—1344no—
282#576—674no—
283#576—674no—
284#575—1344no—
285#575—1342no—
286C2×#135—678no—
287C2×#135—672no—
288C2×#135C2×D6712yescomplemented
289C2×#131—671no—
290#510—678no—
291#510—678no—
292#510—674no—
293#510C2×D6714yescomplemented
294#509—678no—
295#509—678no—
296#509—674no—
297#509C2×D6714yescomplemented
298#508—674no—
299#508—674no—
300#508—674no—
301#508—674no—
302#508—674no—
303#508—678no—
304#508—674no—
305#508—674no—
306#508C2×D6714yescomplemented
307C2×Q32—672no—
308C2×Q32—674no—
309C2×SD32—672no—
310C2×SD32—672no—
311C2×SD32—678no—
312C2×D16—672no—
313C2×D16—674no—
314#504—674no—
315#504—674no—
316C2×M32—672no—
317C2×M32C2×D6711yescharacteristic, complemented
318C22×C16—671no—
319C22×C16—672no—
3202−1+4—2682no—
3212−1+4—1344no—
3222−1+4—1344no—
323H(5,𝔽2)—2684no—
324H(5,𝔽2)—2682no—
325H(5,𝔽2)—1344no—
326H(5,𝔽2)—1344no—
327C2×#41—22no—
328C2×#41—2684no—
329C2×#41—2684no—
330C2×#41—671no—
331C2×#41—1348no—
332C2×#41—1342no—
333#137—1348no—
334#137—1348no—
335#137—1344no—
336#137—1344no—
337AGL(1,ℤ/8)—1348no—
338AGL(1,ℤ/8)—1348no—
339AGL(1,ℤ/8)—1344no—
340AGL(1,ℤ/8)—1344no—
341#135—22no—
342#135—22no—
343#135—678no—
344#135—678no—
345#135—674no—
346#135—674no—
347#135—1348no—
348#135—1348no—
349#135—1342no—
350#135—1342no—
351#135C22×D6714yes—
352#135C22×D6714yes—
353C2×Q16—678no—
354C2×Q16—672no—
355C2×Q16C22×D6712yes—
356C2×SD16—22no—
357C2×SD16—1348no—
358C2×SD16—1342no—
359C2×D8—678no—
360C2×D8—672no—
361C2×D8C22×D6712yes—
362#131—674no—
363#131—674no—
364C2×M16—671no—
365C2×M16—672no—
366C22×C8—672no—
367C22×C8C22×D6711yescharacteristic
368Q32—674no—
369Q32—674no—
370Q32—674no—
371Q32C22×D6714yes—
372SD32—674no—
373SD32—674no—
374SD32—674no—
375SD32—674no—
376SD32—678no—
377SD32C22×D6718yes—
378D16—674no—
379D16—674no—
380D16—674no—
381D16C22×D6714yes—
382M32—678no—
383M32C22×D6714yes—
384C2×C16—671no—
385C2×C16—671no—
386C2×C16—672no—
387C2×C16—672no—
388C2×C16—674no—
389C2×C16—674no—
390C2×C16C22×D6712yes—
391#41—24no—
392#41—24no—
393#41—42no—
394#41—42no—
395#41—2684no—
396#41—2688no—
397#41—2684no—
398#41—2688no—
399#41—2684no—
400#41—2688no—
401#41—2684no—
402#41—2688no—
403#41—2682no—
404#41—2682no—
405#41—674no—
406#41—674no—
407#41—1348no—
408#41—1348no—
409#41—1344no—
410#41—1344no—
411C2×Q8—22no—
412C2×Q8—2684no—
413C2×Q8—2684no—
414C2×Q8—671no—
415C2×Q8—1348no—
416C2×Q8—1342no—
417C2×D4—22no—
418C2×D4—42no—
419C2×D4—2684no—
420C2×D4—2688no—
421C2×D4—2688no—
422C2×D4—2684no—
423C2×D4—2682no—
424C2×D4—2688no—
425C2×D4—672no—
426C2×D4—671no—
427C2×D4—1348no—
428C2×D4—1342no—
429C22×C4—42no—
430C22×C4—2688no—
431C22×C4—2682no—
432C22×C4—672no—
433C22×C4D4×D6711yescharacteristic
434Q16—22no—
435Q16—674no—
436Q16—1348no—
437Q16—1342no—
438Q16C23×D6714yes—
439SD16—24no—
440SD16—24no—
441SD16—1348no—
442SD16—1348no—
443SD16—1344no—
444SD16—1344no—
445D8—22no—
446D8—674no—
447D8—1348no—
448D8—1342no—
449D8C23×D6714yes—
450M16—674no—
451M16—678no—
452C2×C8—672no—
453C2×C8—674no—
454C2×C8—674no—
455C2×C8D4×D6711yescharacteristic
456C2×C8D4×D6712yes—
457C2×C8C23×D6711yescharacteristic
458C2×C8C23×D6712yes—
459C16—671no—
460C16—671no—
461C16—672no—
462C16C23×D6714yes—
463C23—42no—
464C23—2688no—
465C23—2682no—
466C23—672no—
467C23D8×D6711yescharacteristic
468Q8—24no—
469Q8—42no—
470Q8—2688no—
471Q8—2682no—
472Q8—674no—
473Q8—1344no—
474D4—24no—
475D4—44no—
476D4—44no—
477D4—42no—
478D4—2688no—
479D4—2688no—
480D4—2684no—
481D4—2684no—
482D4—2688no—
483D4—2682no—
484D4—678no—
485D4—674no—
486D4—1344no—
487C2×C4—42no—
488C2×C4—44no—
489C2×C4—44no—
490C2×C4—2688no—
491C2×C4—2688no—
492C2×C4—2682no—
493C2×C4—2688no—
494C2×C4—2684no—
495C2×C4—2684no—
496C2×C4—672no—
497C2×C4—674no—
498C2×C4—674no—
499C2×C4D8×D6711yescharacteristic
500C2×C4D8×D6712yes—
501C2×C4C2×D4×D6712yes—
502C2×C4C2×D4×D6711yescharacteristic
503C8—674no—
504C8C2×D4×D6711yescharacteristic
505C8C2×D4×D6712yes—
506C8C24×D6711yescharacteristic
507C22—41no—
508C22—44no—
509C22—2682no—
510C22—2684no—
511C22—162no—
512C22—10722no—
513C22—10728no—
514C22—674no—
515C22C2×D8×D6712yes—
516C22C2×D8×D6711yescharacteristic
517C4—44no—
518C4—2684no—
519C4—674no—
520C4C2×D8×D6712yes—
521C4C2×D8×D6711yescharacteristic
522C4C22×D4×D6711yescharacteristic
523C2—21no—
524C2—22no—
525C2—84no—
526C2—5364no—
527C2—1344no—
528C2C22×D8×D6711yescharacteristic
subgroupG/H
centerC4C2×D8×D67
commutator, cosocleC8×C67C25
FrattiniC8C24×D67
FittingC67×#3414C2
socleC2×C67C23×D8
hypercenter#3414D67
hypocenterC67#62635
LCSG ▷ C8×C67 ▷ C4×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C4 ◁ C22×C4 ◁ C22×C8 ◁ #3414

Complements

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

NH ≅ G/Ntype
1. #22574978C2semidirect
2. #22573361C2semidirect
3. #22573355C2semidirect
4. #22573354C2semidirect
5. #22573349C2semidirect
6. #22573345C2semidirect
7. #22573342C2semidirect
8. C2×#16863826C2semidirect
9. #508×D67C2semidirect
10. C2×#16863814C2semidirect
11. C67×#3414C2semidirect
13. #16864210C22semidirect
14. #16864209C22semidirect
17. #16864198C22semidirect
18. #16864197C22semidirect
21. C2×#14524065C22semidirect
22. #135×D67C22semidirect
24. #16863839C22semidirect
25. #16863838C22semidirect
26. #16863835C22semidirect
27. #16863834C22semidirect
28. #16863833C22semidirect
29. #16863830C22semidirect
30. #16863829C22semidirect
31. #16863828C22semidirect
32. #16863827C22semidirect
33. #16863826C22semidirect
34. #16863826C22semidirect
35. #16863826C22semidirect
36. #16863826C22semidirect
37. #16863825C22semidirect
38. C2×#14523989C22semidirect
39. Q32×D67C22semidirect
40. C2×#14523987C22semidirect
41. C2×#14523986C22semidirect
42. SD32×D67C22semidirect
43. C2×#14523984C22semidirect
44. D16×D67C22semidirect
45. #16863814C22semidirect
46. #16863814C22semidirect
47. C2×#14523982C22semidirect
48. C22×C67⋊C16C22semidirect
49. C2×C16×D67C22semidirect
50. C2×#135×C67C22semidirect
51. #510×C67C22semidirect
52. #509×C67C22semidirect
53. #508×C67C22semidirect
54. C2×M32×C67C22semidirect
272. #3414D67semidirect
276. C67#62635semidirect
288. C2×#135C2×D67semidirect
293. #510C2×D67semidirect
297. #509C2×D67semidirect
306. #508C2×D67semidirect
317. C2×M32C2×D67semidirect

Maximal subgroups

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

subgroupindex
#225749782
#225733612
#225733552
#225733542
#225733492
#225733452
#225733422
C2×#168638262
#508×D672
C2×#168638142
C67×#34142
#6263567