Finite Group Databasebeta

← #13176692

Subgroups — #13176692

Order 1728.

1820 conjugacy classes of subgroups.

The subgroup lattice has 600 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11724581C211yesmaximal, characteristic, complemented
2#11724566C211yesmaximal, characteristic, complemented
3#11723949C211yesmaximal, characteristic, complemented
4C3×#64116C211yesmaximal, characteristic, complemented
5C32×#909C211yesmaximal, characteristic, complemented
6C2×#88795C211yesmaximal, characteristic, complemented
7#11723377C211yesmaximal, characteristic, complemented
8#11723341C212yesmaximal, complemented
9#11723329C212yesmaximal, complemented
10#11723319C212yesmaximal, complemented
11C3⋊D3×#204C212yesmaximal, complemented
12#10596249—31nomaximal
13#10594715—34nomaximal
14#89076C2211yescharacteristic, complemented
15C2×C4⋊(C3⋊(C3⋊D3))C2211yescharacteristic
16C3×#3828C2211yescharacteristic
17C32×#235C2211yescharacteristic
18C2×C32×Dic6C2211yescharacteristic, complemented
19#89039—22no—
20#89031—22no—
21C3⋊D3×D12C2212yes—
22D3×C4⋊(C3⋊D3)—21no—
23#89022—21no—
24#89019—21no—
25#89018C2212yes—
26#89018C2212yescomplemented
27#89017—21no—
28C3⋊D3×Dic6C2212yescomplemented
29C33×M16C2211yescharacteristic, complemented
30#88852C2212yescomplemented
31#88851C2212yescomplemented
32C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic, complemented
33C3×#3743C2212yescomplemented
34C3×C8×C3⋊D3C2212yescomplemented
35C32×Dic12C2212yescomplemented
36C32×#204C2212yescomplemented
37#88799C2211yescharacteristic, complemented
38#88795C2212yescomplemented
39#88795C2212yescomplemented
40#88795C2211yescharacteristic, complemented
41#88794C2211yescharacteristic, complemented
42#88790C2211yescharacteristic, complemented
43#64366—31no—
44#64364—31no—
45#64308—34no—
46#64297—34no—
47C32×#137—31no—
48#64159—31no—
49C2×#3755—31no—
50#64128—32no—
51#64127—32no—
52#64123—32no—
53SD16×C3⋊D3—32no—
54#64117—31no—
55#64117—34no—
56#64117—64no—
57#64116D311yescharacteristic, complemented
58C3×#909D314yescomplemented
59C3×#907—34no—
60C2×#3717—34no—
61#63829—34no—
62#63808—38no—
63#63801—38no—
64#63794—38no—
65D3×#204—38no—
66C22×C3⋊(C3⋊D3)—21no—
67C2×D3×C3⋊D3—22no—
68C2×C33×C4C2311yescharacteristic
69#6312—21no—
70C4⋊(C3⋊(C3⋊D3))—21no—
71C4⋊(C3⋊(C3⋊D3))C2312yes—
72C4⋊(C3⋊(C3⋊D3))C2311yescharacteristic
73C4×C3⋊(C3⋊D3)—21no—
74#6308C2311yescharacteristic
75C3×#638D412yes—
76C3×C4⋊(C3⋊D3)D411yescharacteristic
77C3×C4×C3⋊D3C2312yes—
78C3×#634D411yescharacteristic
79C32×#67—21no—
80C2×C32×Dic3—21no—
81C32×D12C2311yescharacteristic
82C32×C4×D3—21no—
83C32×Dic6—21no—
84C32×Dic6C2312yes—
85C32×Dic6C2311yescharacteristic
86#6293—22no—
87#6292—22no—
88#6292—22no—
89#6291—22no—
90#6289—22no—
91#6289—22no—
92Dic3×C3⋊D3—22no—
93C33×C8C2312yes—
94C3×C3⋊(C3⋊C8)C2311yescharacteristic
95C3×C3⋊(C3⋊C8)C2311yescharacteristic
96#5778—94no—
97#5751—91no—
98C32×#41—31no—
99C2×C32×Q8—31no—
100#3832—32no—
101#3832—32no—
102#3832—61no—
103Q8×C3⋊D3—32no—
104Q8×C3⋊D3—61no—
105D4×C3⋊D3—32no—
106D4×C3⋊D3—62no—
107#3828—31no—
108#3828—34no—
109#3828—62no—
110#3828—64no—
111#3828—61no—
112#3828C2×D311yescharacteristic
113C2×C4⋊(C3⋊D3)—31no—
114C2×C4⋊(C3⋊D3)—34no—
115C2×C4⋊(C3⋊D3)—61no—
116C2×C4⋊(C3⋊D3)—64no—
117C3×#235—34no—
118C3×#235C2×D314yes—
119C2×C3×Dic6C2×D314yescomplemented
120#3811—68no—
121#3804—68no—
122D3×D12—38no—
123D3×D12—64no—
124#3799—38no—
125#3799—38no—
126#3799—64no—
127#3797—64no—
128#3796—64no—
129D3×Dic6—38no—
130C32×Q16—32no—
131C32×SD16—32no—
132C32×M16—24no—
133C32×M16C2×D314yescomplemented
134C32×M16C2×D311yescharacteristic, complemented
135#3756—31no—
136#3755—32no—
137#3755—32no—
138#3755—31no—
139#3754—31no—
140#3753—31no—
141C2×C3⋊(C3⋊C8)C2×D311yescharacteristic, complemented
142#3745—32no—
143#3745—38no—
144#3745—68no—
145#3744—32no—
146#3744—38no—
147#3744—68no—
148#3743C2×D312yescomplemented
149C8×C3⋊D3C2×D312yescomplemented
150C2×C3×C3⋊C8—34no—
151C3×Dic12C2×D318yescomplemented
152C3×#204C2×D318yescomplemented
153C3×ΓL(1,𝔽25)—38no—
154C3×D3×C8—38no—
155#3719—34no—
156#3717—38no—
157#3717—38no—
158#3717—34no—
159#3716—34no—
160#3714—34no—
161C22×C33C2×D411yescharacteristic
162C2×C3⋊(C3⋊D3)—21no—
163C2×C3⋊(C3⋊D3)—22no—
164C2×C3⋊(C3⋊D3)—81no—
165C2×C3×C3⋊D3C2×D412yes—
166C2×C32×D3—21no—
167D3×C3⋊D3—82no—
168C33×C4C2×D411yescharacteristic
169C33×C4C2411yescharacteristic
170C3⋊(C3⋊Dic3)—21no—
171C3×C3⋊Dic3C2×D412yes—
172C32×Dic3—22no—
173C32×Dic3—21no—
174#1010—91no—
175#1009—91no—
176#1009—94no—
177#1007—94no—
178C3×#137—34no—
179C3×#131—91no—
180#950—94no—
181C2×#215—94no—
182#919—98no—
183#918—98no—
184#914—98no—
185D3×SD16—98no—
186#909—92no—
187#909C3⋊D311yescharacteristic, complemented
188#908—94no—
189#908—91no—
190#908—92no—
191#908—184no—
192#907—34no—
193#904—91no—
194#904—92no—
195C2×#204—91no—
196C2×#204—92no—
197C22×C3⋊D3—61no—
198C22×C3⋊D3—62no—
199C22×C3⋊D3—64no—
200C22×C3⋊D3—124no—
201C2×D32—68no—
202C32×Q8—32no—
203C32×Q8—31no—
204C32×Q8—61no—
205C32×D4—31no—
206C32×D4—61no—
207C2×C32×C4—24no—
208C2×C32×C4—61no—
209C2×C32×C4—61no—
210C2×C32×C4C22×D314yes—
211C2×C32×C4C22×D311yescharacteristic
212#638—61no—
213#638—62no—
214#638—64no—
215#638—124no—
216#638D1212yes—
217C4⋊(C3⋊D3)—32no—
218C4⋊(C3⋊D3)—31no—
219C4⋊(C3⋊D3)—34no—
220C4⋊(C3⋊D3)—38no—
221C4⋊(C3⋊D3)—68no—
222C4⋊(C3⋊D3)—64no—
223C4⋊(C3⋊D3)—62no—
224C4⋊(C3⋊D3)—62no—
225C4⋊(C3⋊D3)—61no—
226C4⋊(C3⋊D3)—64no—
227C4⋊(C3⋊D3)—124no—
228C4⋊(C3⋊D3)D1211yescharacteristic
229C4×C3⋊D3—62no—
230C4×C3⋊D3—61no—
231C4×C3⋊D3—62no—
232C4×C3⋊D3—62no—
233C4×C3⋊D3—64no—
234C4×C3⋊D3—124no—
235C4×C3⋊D3C22×D312yes—
236#634—31no—
237#634—34no—
238#634—64no—
239#634—62no—
240#634D1211yescharacteristic
241C3×#67—24no—
242C3×#67—38no—
243C2×C3×Dic3—24no—
244C3×D12—34no—
245C3×D12C22×D314yes—
246C3×C4×D3—24no—
247C3×C4×D3—38no—
248C3×Dic6—24no—
249C3×Dic6—34no—
250C3×Dic6C22×D318yes—
251C3×Dic6C22×D314yes—
252#627—68no—
253#626—68no—
254#626—68no—
255#626—68no—
256#624—68no—
257#624—68no—
258D3×Dic3—68no—
259C32×C8—28no—
260C32×C8C22×D318yes—
261C32×C8C22×D312yes—
262C3⋊(C3⋊C8)C22×D311yescharacteristic
263C3⋊(C3⋊C8)C22×D311yescharacteristic
264C3×C3⋊C8—34no—
265C3×C3⋊C8—34no—
266#577—271noSylow
267C2×C33—21no—
268C2×C33C22×D411yescharacteristic
269C3⋊(C3⋊D3)—42no—
270C3⋊(C3⋊D3)—41no—
271C3×C3⋊D3—22no—
272C32×D3—41no—
273C3×#41—34no—
274C3×#41—91no—
275C2×C3×Q8—34no—
276#239—98no—
277#239—98no—
278#239—181no—
279#239—184no—
280D3×Q8—98no—
281D3×Q8—181no—
282D3×Q8—184no—
283#237—181no—
284#237—182no—
285D3×D4—98no—
286D3×D4—181no—
287D3×D4—182no—
288D3×D4—188no—
289#235—34no—
290#235—94no—
291#235—91no—
292#235—92no—
293#235—92no—
294#235—184no—
295#235—184no—
296#235—188no—
297#235C2×C3⋊D311yescharacteristic
298C2×D12—94no—
299C2×D12—91no—
300C2×D12—92no—
301C2×D12—184no—
302C2×D12—184no—
303C2×Dic6—92no—
304C2×Dic6C2×C3⋊D311yescharacteristic, complemented
305C3×Q16—38no—
306C3×SD16—38no—
307C3×M16—24no—
308C3×M16—92no—
309C3×M16D3214yescomplemented
310C3×M16C2×C3⋊D311yescharacteristic, complemented
311C2×C3×C8—91no—
312C2×C3×C8—92no—
313#216—94no—
314#215—98no—
315#215—98no—
316#215—94no—
317#214—94no—
318#213—94no—
319C2×C3⋊C8—34no—
320Dic12—91no—
321Dic12C2×C3⋊D312yescomplemented
322D24—91no—
323D24—98no—
324D24—92no—
325D24—188no—
326#204—91no—
327#204—92no—
328#204—92no—
329#204—91no—
330#204—98no—
331#204—92no—
332#204—188no—
333#204C2×C3⋊D312yescomplemented
334ΓL(1,𝔽25)—38no—
335D3×C8—38no—
336C22×C32—24no—
337C22×C32—61no—
338C22×C32C2×D1211yescharacteristic
339C22×C32D3×D414yes—
340C2×C3⋊D3—61no—
341C2×C3⋊D3—62no—
342C2×C3⋊D3—64no—
343C2×C3⋊D3—68no—
344C2×C3⋊D3—124no—
345C2×C3⋊D3—128no—
346C2×C3⋊D3—242no—
347C2×C3⋊D3—241no—
348C2×C3⋊D3—244no—
349C2×C3⋊D3—248no—
350C2×C3⋊D3C2×D1212yes—
351C2×C3×D3—24no—
352C2×C3×D3—38no—
353D32—248no—
354C32×C4—24no—
355C32×C4—24no—
356C32×C4—62no—
357C32×C4—61no—
358C32×C4C2×D1211yescharacteristic
359C32×C4D3×D414yes—
360C32×C4C23×D314yes—
361C32×C4C23×D311yescharacteristic
362C3⋊Dic3—61no—
363C3⋊Dic3—64no—
364C3⋊Dic3—124no—
365C3⋊Dic3C2×D1212yes—
366C3×Dic3—28no—
367C3×Dic3—24no—
368C3×Dic3—38no—
3692−1+4—271no—
370H(5,𝔽2)—271no—
371#137—31no—
372#137—272no—
373AGL(1,ℤ/8)—272no—
374AGL(1,ℤ/8)—271no—
375#135—272no—
376#135—271no—
377C2×SD16—272no—
378C2×SD16—271no—
379#131—91no—
380C33#57711yescharacteristic, complemented, Sylow
381C22×D3—182no—
382C22×D3—188no—
383C22×D3—184no—
384C22×D3—181no—
385C22×D3—364no—
386C3×Q8—38no—
387C3×Q8—34no—
388C3×Q8—64no—
389C3×Q8—91no—
390C3×D4—34no—
391C3×D4—64no—
392C3×D4—92no—
393C3×D4—91no—
394C2×C3×C4—24no—
395C2×C3×C4—64no—
396C2×C3×C4—64no—
397C2×C3×C4—92no—
398C2×C3×C4C2×D3214yes—
399C2×C3×C4C22×C3⋊D311yescharacteristic
400#67—21no—
401#67—38no—
402#67—182no—
403#67—182no—
404#67—188no—
405#67—184no—
406#67—181no—
407#67—364no—
408C2×Dic3—21no—
409C2×Dic3—182no—
410D12—34no—
411D12—98no—
412D12—94no—
413D12—92no—
414D12—91no—
415D12—188no—
416D12—184no—
417D12—188no—
418D12—188no—
419D12—182no—
420D12—184no—
421D12—181no—
422D12—364no—
423D12C22×C3⋊D311yescharacteristic
424C4×D3—21no—
425C4×D3—38no—
426C4×D3—182no—
427C4×D3—182no—
428C4×D3—188no—
429C4×D3—188no—
430C4×D3—188no—
431C4×D3—184no—
432C4×D3—181no—
433C4×D3—364no—
434Dic6—21no—
435Dic6—34no—
436Dic6—94no—
437Dic6—91no—
438Dic6—184no—
439Dic6—182no—
440Dic6—188no—
441Dic6C22×C3⋊D312yes—
442Dic6C22×C3⋊D311yescharacteristic
443C3×C8—28no—
444C3×C8—91no—
445C3×C8—91no—
446C3×C8C2×D3218yes—
447C3×C8C22×C3⋊D312yes—
448C3⋊C8—34no—
449C3⋊C8—34no—
450C2×C32—21no—
451C2×C32—24no—
452C2×C32—24no—
453C2×C32—44no—
454C2×C32—121no—
455C2×C32C22×D1211yescharacteristic
456C2×C32C2×D3×D414yes—
457C3⋊D3—22no—
458C3⋊D3—121no—
459C3⋊D3—122no—
460C3⋊D3—128no—
461C3⋊D3—124no—
462C3⋊D3—244no—
463C3⋊D3—248no—
464C3×D3—44no—
465C3×D3—68no—
466#41—31no—
467#41—91no—
468#41—272no—
469#41—272no—
470#41—271no—
471#41—542no—
472#41—541no—
473#41—541no—
474C2×Q8—31no—
475C2×Q8—272no—
476C2×Q8—541no—
477C2×D4—272no—
478C2×D4—271no—
479C2×D4—541no—
480C2×D4—542no—
481Q16—32no—
482Q16—271no—
483SD16—32no—
484SD16—271no—
485SD16—272no—
486SD16—272no—
487SD16—271no—
488SD16—272no—
489D8—271no—
490D8—272no—
491M16—92no—
492M16D3×C3⋊D311yescharacteristic, complemented
493C2×C8—91no—
494C2×C8—92no—
495C22×C3—24no—
496C22×C3—64no—
497C22×C3—92no—
498C22×C3D3×D1214yes—
499C22×C3D4×C3⋊D311yescharacteristic
500C2×D3—21no—
501C2×D3—38no—
502C2×D3—184no—
503C2×D3—188no—
504C2×D3—181no—
505C2×D3—182no—
506C2×D3—364no—
507C2×D3—368no—
508C2×D3—722no—
509C2×D3—728no—
510C2×D3—721no—
511C2×D3—724no—
512C2×D3—728no—
513C3×C4—24no—
514C3×C4—24no—
515C3×C4—68no—
516C3×C4—64no—
517C3×C4—92no—
518C3×C4D3×D1214yes—
519C3×C4D4×C3⋊D311yescharacteristic
520C3×C4C22×D3214yes—
521C3×C4C23×C3⋊D311yescharacteristic
522Dic3—22no—
523Dic3—21no—
524Dic3—38no—
525Dic3—184no—
526Dic3—181no—
527Dic3—364no—
528C32—24no—
529C32#575111yescharacteristic, complemented
530C32#577814yescomplemented
531C23—542no—
532C23—541no—
533Q8—32no—
534Q8—31no—
535Q8—61no—
536Q8—91no—
537Q8—271no—
538Q8—542no—
539D4—31no—
540D4—61no—
541D4—92no—
542D4—91no—
543D4—271no—
544D4—272no—
545D4—542no—
546D4—542no—
547D4—542no—
548D4—541no—
549D4—541no—
550C2×C4—61no—
551C2×C4—61no—
552C2×C4—92no—
553C2×C4—542no—
554C2×C4—542no—
555C2×C4—542no—
556C2×C4—541no—
557C2×C4C2×D3×C3⋊D311yescharacteristic
558C8—91no—
559C8—91no—
560C8C2×D3×C3⋊D312yes—
561C2×C3—21no—
562C2×C3—24no—
563C2×C3—24no—
564C2×C3—44no—
565C2×C3—124no—
566C2×C3—182no—
567C2×C3C2×D3×D1214yes—
568C2×C3C2×D4×C3⋊D311yescharacteristic
569D3—41no—
570D3—68no—
571D3—362no—
572D3—361no—
573D3—368no—
574D3—364no—
575D3—728no—
576D3—724no—
577C22—61no—
578C22—92no—
579C22—541no—
580C22—542no—
581C22—2161no—
582C22—2162no—
583C22C3⋊D3×D1211yescharacteristic
584C4—61no—
585C4—62no—
586C4—92no—
587C4—541no—
588C4C3⋊D3×D1211yescharacteristic
589C4C22×D3×C3⋊D311yescharacteristic
590C3—24no—
591C3#1059471514yescomplemented
592C3#1059624911yescharacteristic, complemented
593C2—21no—
594C2—121no—
595C2—182no—
596C2—1082no—
597C2—1081no—
598C2C2×C3⋊D3×D1211yescharacteristic
subgroupG/H
centerC2C2×C3⋊D3×D12
commutator, cosocleC33×C4C24
FrattiniC4C22×D3×C3⋊D3
FittingC33×M16C22
socleC2×C33C22×D4
hypercenterM16D3×C3⋊D3
hypocenterC33#577
LCSG ▷ C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C2 ◁ C2×C4 ◁ M16

Complements

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

NH ≅ G/Ntype
1. #11724581C2semidirect
2. #11724566C2semidirect
3. #11723949C2semidirect
4. C3×#64116C2semidirect
5. C32×#909C2semidirect
6. C2×#88795C2semidirect
7. #11723377C2semidirect
8. #11723341C2semidirect
9. #11723329C2semidirect
10. #11723319C2semidirect
11. C3⋊D3×#204C2semidirect
14. #89076C22semidirect
18. C2×C32×Dic6C22semidirect
26. #89018C22semidirect
28. C3⋊D3×Dic6C22semidirect
29. C33×M16C22semidirect
30. #88852C22semidirect
31. #88851C22semidirect
32. C2×C3×C3⋊(C3⋊C8)C22semidirect
33. C3×#3743C22semidirect
34. C3×C8×C3⋊D3C22semidirect
35. C32×Dic12C22semidirect
36. C32×#204C22semidirect
37. #88799C22semidirect
38. #88795C22semidirect
39. #88795C22semidirect
40. #88795C22semidirect
41. #88794C22semidirect
42. #88790C22semidirect
57. #64116D3semidirect
58. C3×#909D3semidirect
119. C2×C3×Dic6C2×D3semidirect
133. C32×M16C2×D3semidirect
134. C32×M16C2×D3semidirect
141. C2×C3⋊(C3⋊C8)C2×D3semidirect
148. #3743C2×D3semidirect
149. C8×C3⋊D3C2×D3semidirect
151. C3×Dic12C2×D3semidirect
152. C3×#204C2×D3semidirect
187. #909C3⋊D3semidirect
304. C2×Dic6C2×C3⋊D3semidirect
309. C3×M16D32semidirect
310. C3×M16C2×C3⋊D3semidirect
321. Dic12C2×C3⋊D3semidirect
333. #204C2×C3⋊D3semidirect
380. C33#577semidirect
492. M16D3×C3⋊D3semidirect
529. C32#5751semidirect
530. C32#5778semidirect
591. C3#10594715semidirect
592. C3#10596249semidirect

Maximal subgroups

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

subgroupindex
#117245812
#117245662
#117239492
C3×#641162
C32×#9092
C2×#887952
#117233772
#117233412
#117233292
#117233192
C3⋊D3×#2042
#105962493
#105947153