Finite Group Databasebeta

← #13177709

Subgroups — #13177709

Order 1728.

1820 conjugacy classes of subgroups.

The subgroup lattice has 596 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C32×#1004C211yesmaximal, characteristic, complemented
2#11724563C211yesmaximal, characteristic, complemented
3C2×#88861C211yesmaximal, characteristic, complemented
4C3×#64141C211yesmaximal, characteristic, complemented
5#11723556C212yesmaximal, complemented
6#11723555C212yesmaximal, complemented
7#11723550C212yesmaximal, complemented
8D3×#3754C212yesmaximal, complemented
9#11723379C211yesmaximal, characteristic, complemented
10#11723363C211yesmaximal, characteristic, complemented
11#11723350C211yesmaximal, characteristic, complemented
12#10596307—31nomaximal
13#10595106—34nomaximal
14C2×C33×D4C2211yescharacteristic
15C2×#6308C2211yescharacteristic, complemented
16C3×#3828C2211yescharacteristic, complemented
17C2×C32×#67—22no—
18C32×#237—21no—
19C32×#237C2212yes—
20C32×D3×D4—21no—
21C32×D3×D4C2212yes—
22C32×#235C2211yescharacteristic
23C32×#235C2211yescharacteristic
24#89029—22no—
25#89021C2212yescomplemented
26C3⋊D3×Dic6—21no—
27#89014—21no—
28D3×#634C2212yescomplemented
29#88861C2212yescomplemented
30#88861C2212yescomplemented
31C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic, complemented
32C3×#3754C2212yescomplemented
33C3×#3753C2212yescomplemented
34C3×#3748C2211yescharacteristic, complemented
35#88799C2212yescomplemented
36#88798C2211yescharacteristic, complemented
37#88794C2212yescomplemented
38#88793C2211yescharacteristic, complemented
39#88792C2211yescharacteristic, complemented
40#88789C2211yescharacteristic, complemented
41#88786C2212yescomplemented
42D3×C3⋊(C3⋊C8)C2212yescomplemented
43C32×H(5,𝔽2)—31no—
44#64367—31no—
45C3×#1004D314yescomplemented
46#64298—34no—
47#64160—32no—
48#64160—31no—
49#64159—31no—
50#64159—32no—
51#64157—31no—
52C2×#3754—31no—
53C2×#3754—32no—
54C2×#3754—34no—
55C2×#3754—64no—
56#64141—32no—
57#64141D311yescharacteristic, complemented
58C3×#932—34no—
59#63933—38no—
60#63932—38no—
61#63929—38no—
62D3×#214—38no—
63#63831—34no—
64#63822—34no—
65#63815—34no—
66C23×C33—21no—
67C22×C32×D3—22no—
68C33×D4—21no—
69C33×D4C2312yes—
70C2×C33×C4C2311yescharacteristic
71C2×C3⋊(C3⋊Dic3)—21no—
72#6308—21no—
73#6308C2312yes—
74C3×#638—21no—
75C3×C4⋊(C3⋊D3)C2311yescharacteristic
76C3×C4×C3⋊D3—21no—
77C3×#634C2311yescharacteristic
78C32×#67—21no—
79C32×#67—22no—
80C32×#67—22no—
81C32×#67D412yes—
82C2×C32×Dic3—22no—
83C32×D12D411yescharacteristic
84C32×D12C2311yescharacteristic
85C32×C4×D3—21no—
86C32×C4×D3C2312yes—
87C32×Dic6D411yescharacteristic
88C32×Dic6C2311yescharacteristic
89#6293—22no—
90#6290—22no—
91Dic3×C3⋊D3—22no—
92D3×C3⋊Dic3—22no—
93C3⋊(C3⋊(C3⋊C8))C2311yescharacteristic
94C3⋊(C3⋊(C3⋊C8))C2311yescharacteristic
95C3×C3⋊(C3⋊C8)C2312yes—
96#5836—94no—
97#5775—91no—
98C32×#41—31no—
99C32×#41—32no—
100C32×#41—31no—
101C32×#41—61no—
102C2×C32×D4—24no—
103C2×C32×D4—32no—
104C2×C32×D4—61no—
105C2×C32×D4—62no—
106C2×C32×D4C2×D314yes—
107C2×C32×D4C2×D311yescharacteristic
108Q8×C3⋊D3—61no—
109#3830—61no—
110#3830—62no—
111#3828—32no—
112#3828C2×D311yescharacteristic, complemented
113C2×#634—31no—
114C2×#634—34no—
115C2×#634—32no—
116C2×#634—64no—
117C2×C3×#67—28no—
118C3×#237—24no—
119C3×#237C2×D318yes—
120C3×D3×D4—24no—
121C3×D3×D4C2×D318yes—
122C3×#235—34no—
123C3×#235C2×D314yes—
124C3×#235C2×D314yes—
125#3805—68no—
126#3795—38no—
127#3795—64no—
128D3×Dic6—38no—
129D3×Dic6—64no—
130#3756—31no—
131#3756—32no—
132#3755—31no—
133#3754—32no—
134#3754—32no—
135#3754—38no—
136#3754—38no—
137#3754—31no—
138#3754—32no—
139#3754—68no—
140#3754—68no—
141#3754C2×D312yescomplemented
142#3753—31no—
143#3753C2×D312yescomplemented
144#3748—32no—
145#3748C2×D311yescharacteristic, complemented
146C2×C3⋊(C3⋊C8)—31no—
147C2×C3⋊(C3⋊C8)—34no—
148C2×C3⋊(C3⋊C8)—32no—
149C2×C3⋊(C3⋊C8)—64no—
150C3×#214—38no—
151C3×#213—38no—
152C3×#208—34no—
153#3719—38no—
154#3718—34no—
155#3716—38no—
156#3715—34no—
157#3715—34no—
158#3713—34no—
159#3711—38no—
160D3×C3⋊C8—38no—
161C22×C33—22no—
162C22×C33—81no—
163C22×C33C2×D411yescharacteristic
164C2×C3×C3⋊D3—21no—
165C2×C32×D3—21no—
166C2×C32×D3—82no—
167C2×C32×D3C2×D412yes—
168C33×C4C2×D411yescharacteristic
169C33×C4C2411yescharacteristic
170C3⋊(C3⋊Dic3)—22no—
171C3×C3⋊Dic3—21no—
172C32×Dic3—21no—
173C32×Dic3C2×D412yes—
174C3×H(5,𝔽2)—34no—
175#1010—94no—
176#1010—91no—
177#1004C3⋊D311yescharacteristic, complemented
178C3×AGL(1,ℤ/8)—91no—
179#951—94no—
180#951—98no—
181#950—98no—
182#950—94no—
183#950—91no—
184#948—94no—
185C2×#214—98no—
186C2×#214—94no—
187C2×#214—91no—
188C2×#214—184no—
189#932—34no—
190#932—98no—
191#915—92no—
192D3×SD16—92no—
193#912—92no—
194#911—92no—
195#909—91no—
196#907—91no—
197C23×C32—21no—
198C23×C32—24no—
199C23×C32—44no—
200C23×C32—62no—
201C22×C3×D3—28no—
202C32×Q8—31no—
203C32×Q8—31no—
204C32×D4—21no—
205C32×D4—28no—
206C32×D4—24no—
207C32×D4—32no—
208C32×D4—31no—
209C32×D4—31no—
210C32×D4—44no—
211C32×D4—62no—
212C32×D4—62no—
213C32×D4—61no—
214C32×D4C22×D318yes—
215C32×D4C22×D312yes—
216C2×C32×C4—24no—
217C2×C32×C4—32no—
218C2×C32×C4—62no—
219C2×C32×C4—61no—
220C2×C32×C4C22×D314yes—
221C2×C32×C4C22×D311yescharacteristic
222#638—21no—
223#638—62no—
224C2×C3⋊Dic3—62no—
225C2×C3⋊Dic3—61no—
226C2×C3⋊Dic3—64no—
227C2×C3⋊Dic3—124no—
228C4⋊(C3⋊D3)C22×D311yescharacteristic
229C4×C3⋊D3—21no—
230C4×C3⋊D3—62no—
231#634—32no—
232#634—38no—
233#634—68no—
234#634—62no—
235#634—61no—
236#634—64no—
237#634—124no—
238#634C22×D311yescharacteristic
239C3×#67—24no—
240C3×#67—28no—
241C3×#67—28no—
242C3×#67—64no—
243C3×#67#6718yes—
244C2×C3×Dic3—28no—
245C3×D12—34no—
246C3×D12#6714yes—
247C3×D12C22×D314yes—
248C3×C4×D3—24no—
249C3×C4×D3—64no—
250C3×C4×D3C22×D318yes—
251C3×Dic6—34no—
252C3×Dic6#6714yes—
253C3×Dic6C22×D314yes—
254#627—68no—
255#625—68no—
256D3×Dic3—68no—
257D3×Dic3—68no—
258C3⋊(C3⋊C8)—31no—
259C3⋊(C3⋊C8)—31no—
260C3⋊(C3⋊C8)—34no—
261C3⋊(C3⋊C8)—34no—
262C3⋊(C3⋊C8)—64no—
263C3⋊(C3⋊C8)—64no—
264C3⋊(C3⋊C8)C22×D312yes—
265C3×C3⋊C8—38no—
266#577—271noSylow
267C2×C33—21no—
268C2×C33—42no—
269C2×C33C22×D411yescharacteristic
270C3×C3⋊D3—41no—
271C32×D3—22no—
272C32×D3—41no—
273C3×#41—34no—
274C3×#41—38no—
275C3×#41—34no—
276C3×#41—64no—
277C3×#41—91no—
278C2×C3×D4—24no—
279C2×C3×D4—38no—
280C2×C3×D4—64no—
281C2×C3×D4—68no—
282C2×C3×D4D3214yes—
283C2×C3×D4C2×C3⋊D311yescharacteristic
284C2×#67—22no—
285D3×Q8—92no—
286D3×Q8—184no—
287#237—21no—
288#237—92no—
289#237—184no—
290#237—188no—
291#237—182no—
292#237C2×C3⋊D312yes—
293D3×D4—21no—
294D3×D4C2×C3⋊D312yes—
295#235—34no—
296#235—98no—
297#235—181no—
298#235C2×C3⋊D311yescharacteristic
299#235C2×C3⋊D311yescharacteristic
300C2×Dic6—98no—
301C2×Dic6—94no—
302C2×Dic6—91no—
303C2×Dic6—184no—
304C2×Dic6—181no—
305C3×SD16—92no—
306C3×D8—92no—
307C3×M16—91no—
308#216—98no—
309#216—94no—
310#216—91no—
311#215—94no—
312#215—91no—
313#214—38no—
314#214—98no—
315#214—94no—
316#214—98no—
317#214—98no—
318#214—92no—
319#214—92no—
320#214—91no—
321#214—188no—
322#214—188no—
323#213—38no—
324#213—94no—
325#213—91no—
326#208—34no—
327#208—98no—
328C2×C3⋊C8—98no—
329C2×C3⋊C8—94no—
330C2×C3⋊C8—91no—
331C2×C3⋊C8—184no—
332Dic12—92no—
333#204—92no—
334ΓL(1,𝔽25)—92no—
335D3×C8—92no—
336C22×C32—22no—
337C22×C32—24no—
338C22×C32—28no—
339C22×C32—32no—
340C22×C32—48no—
341C22×C32—61no—
342C22×C32—81no—
343C22×C32—84no—
344C22×C32—88no—
345C22×C32—242no—
346C22×C32D3×D411yescharacteristic
347C22×C32C2×#6714yes—
348C2×C3⋊D3—21no—
349C2×C3×D3—24no—
350C2×C3×D3—64no—
351C2×C3×D3—88no—
352C2×C3×D3C2×#6718yes—
353C32×C4—24no—
354C32×C4—24no—
355C32×C4—32no—
356C32×C4—61no—
357C32×C4D3×D411yescharacteristic
358C32×C4C2×#6714yes—
359C32×C4C23×D314yes—
360C32×C4C23×D311yescharacteristic
361C3⋊Dic3—21no—
362C3⋊Dic3—62no—
363C3⋊Dic3—68no—
364C3⋊Dic3—128no—
365C3×Dic3—24no—
366C3×Dic3—64no—
367C3×Dic3C2×#6718yes—
3682−1+4—271no—
369H(5,𝔽2)—31no—
370#137—271no—
371#137—272no—
372AGL(1,ℤ/8)—91no—
373AGL(1,ℤ/8)—272no—
374#135—272no—
375#135—271no—
376C2×SD16—272no—
377C2×SD16—271no—
378#131—271no—
379C33#57711yescharacteristic, complemented, Sylow
380C23×C3—24no—
381C23×C3—21no—
382C23×C3—44no—
383C23×C3—68no—
384C22×D3—22no—
385C3×Q8—34no—
386C3×Q8—34no—
387C3×Q8—91no—
388C3×D4—24no—
389C3×D4—21no—
390C3×D4—28no—
391C3×D4—38no—
392C3×D4—34no—
393C3×D4—34no—
394C3×D4—44no—
395C3×D4—68no—
396C3×D4—68no—
397C3×D4—64no—
398C3×D4—91no—
399C3×D4—181no—
400C3×D4C2×D3218yes—
401C3×D4C22×C3⋊D312yes—
402C2×C3×C4—24no—
403C2×C3×C4—38no—
404C2×C3×C4—68no—
405C2×C3×C4—64no—
406C2×C3×C4—181no—
407C2×C3×C4C2×D3214yes—
408C2×C3×C4C22×C3⋊D311yescharacteristic
409#67—21no—
410#67—22no—
411#67—22no—
412#67—64no—
413#67—188no—
414#67—182no—
415#67#63812yes—
416C2×Dic3—22no—
417C2×Dic3—188no—
418C2×Dic3—184no—
419C2×Dic3—181no—
420C2×Dic3—182no—
421C2×Dic3—364no—
422D12—34no—
423D12#63811yescharacteristic
424D12C22×C3⋊D311yescharacteristic
425C4×D3—21no—
426C4×D3—64no—
427C4×D3—188no—
428C4×D3—182no—
429C4×D3C22×C3⋊D312yes—
430Dic6—34no—
431Dic6—98no—
432Dic6—92no—
433Dic6—188no—
434Dic6—188no—
435Dic6—184no—
436Dic6—181no—
437Dic6—182no—
438Dic6—364no—
439Dic6#63811yescharacteristic
440Dic6C22×C3⋊D311yescharacteristic
441C3×C8—92no—
442C3⋊C8—38no—
443C3⋊C8—94no—
444C3⋊C8—94no—
445C3⋊C8—91no—
446C3⋊C8—91no—
447C3⋊C8—184no—
448C3⋊C8—184no—
449C2×C32—28no—
450C2×C32—21no—
451C2×C32—24no—
452C2×C32—24no—
453C2×C32—42no—
454C2×C32—48no—
455C2×C32—62no—
456C2×C32—88no—
457C2×C32—121no—
458C2×C32C2×D3×D411yescharacteristic
459C2×C32C22×#6714yes—
460C3⋊D3—41no—
461C3×D3—28no—
462C3×D3—44no—
463C3×D3—124no—
464#41—31no—
465#41—32no—
466#41—31no—
467#41—61no—
468#41—91no—
469#41—272no—
470#41—541no—
471#41—542no—
472C2×Q8—272no—
473C2×Q8—271no—
474C2×Q8—541no—
475C2×D4—32no—
476C2×D4—61no—
477C2×D4—62no—
478C2×D4D3×C3⋊D311yescharacteristic
479Q16—272no—
480Q16—271no—
481SD16—92no—
482SD16—272no—
483SD16—271no—
484SD16—271no—
485SD16—272no—
486SD16—272no—
487D8—92no—
488D8—271no—
489M16—91no—
490M16—272no—
491C2×C8—272no—
492C2×C8—271no—
493C22×C3—28no—
494C22×C3—22no—
495C22×C3—24no—
496C22×C3—38no—
497C22×C3—48no—
498C22×C3—64no—
499C22×C3—84no—
500C22×C3—81no—
501C22×C3—88no—
502C22×C3—181no—
503C22×C3—248no—
504C22×C3D3×#6714yes—
505C22×C3C2×#63811yescharacteristic
506C2×D3—21no—
507C2×D3—64no—
508C2×D3—82no—
509C2×D3C2×#63812yes—
510C3×C4—24no—
511C3×C4—24no—
512C3×C4—38no—
513C3×C4—64no—
514C3×C4—181no—
515C3×C4D3×#6714yes—
516C3×C4C2×#63811yescharacteristic
517C3×C4C22×D3214yes—
518C3×C4C23×C3⋊D311yescharacteristic
519Dic3—21no—
520Dic3—64no—
521Dic3—188no—
522Dic3—182no—
523Dic3—368no—
524Dic3C2×#63812yes—
525C32—24no—
526C32#577511yescharacteristic, complemented
527C32#583614yescomplemented
528C23—21no—
529C23—62no—
530Q8—31no—
531Q8—31no—
532Q8—91no—
533Q8—272no—
534Q8—542no—
535Q8—541no—
536D4—21no—
537D4—32no—
538D4—31no—
539D4—31no—
540D4—62no—
541D4—62no—
542D4—61no—
543D4—91no—
544D4—181no—
545D4—542no—
546D4C2×D3×C3⋊D312yes—
547C2×C4—32no—
548C2×C4—62no—
549C2×C4—61no—
550C2×C4—181no—
551C2×C4—542no—
552C2×C4—542no—
553C2×C4—541no—
554C2×C4C2×D3×C3⋊D311yescharacteristic
555C8—92no—
556C8—271no—
557C8—271no—
558C2×C3—28no—
559C2×C3—24no—
560C2×C3—21no—
561C2×C3—24no—
562C2×C3—48no—
563C2×C3—42no—
564C2×C3—68no—
565C2×C3—88no—
566C2×C3—124no—
567C2×C3—361no—
568C2×C3C2×D3×#6714yes—
569C2×C3C22×#63811yescharacteristic
570D3—22no—
571D3—41no—
572D3—124no—
573C22—22no—
574C22—32no—
575C22—61no—
576C22—81no—
577C22—181no—
578C22—242no—
579C22D3×#63811yescharacteristic
580C4—32no—
581C4—61no—
582C4—181no—
583C4—542no—
584C4D3×#63811yescharacteristic
585C4C22×D3×C3⋊D311yescharacteristic
586C3—24no—
587C3#1059510614yescomplemented
588C3#1059630711yescharacteristic, complemented
589C2—21no—
590C2—42no—
591C2—62no—
592C2—121no—
593C2—361no—
594C2C2×D3×#63811yescharacteristic
subgroupG/H
centerC2C2×D3×#638
commutator, cosocleC33×C4C24
FrattiniC4C22×D3×C3⋊D3
FittingC2×C33×D4C22
socleC2×C33C22×D4
hypercenterC2×D4D3×C3⋊D3
hypocenterC33#577
LCSG ▷ C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
1. C32×#1004C2semidirect
2. #11724563C2semidirect
3. C2×#88861C2semidirect
4. C3×#64141C2semidirect
5. #11723556C2semidirect
6. #11723555C2semidirect
7. #11723550C2semidirect
8. D3×#3754C2semidirect
9. #11723379C2semidirect
10. #11723363C2semidirect
11. #11723350C2semidirect
15. C2×#6308C22semidirect
16. C3×#3828C22semidirect
25. #89021C22semidirect
28. D3×#634C22semidirect
29. #88861C22semidirect
30. #88861C22semidirect
31. C2×C3⋊(C3⋊(C3⋊C8))C22semidirect
32. C3×#3754C22semidirect
33. C3×#3753C22semidirect
34. C3×#3748C22semidirect
35. #88799C22semidirect
36. #88798C22semidirect
37. #88794C22semidirect
38. #88793C22semidirect
39. #88792C22semidirect
40. #88789C22semidirect
41. #88786C22semidirect
42. D3×C3⋊(C3⋊C8)C22semidirect
45. C3×#1004D3semidirect
57. #64141D3semidirect
112. #3828C2×D3semidirect
141. #3754C2×D3semidirect
143. #3753C2×D3semidirect
145. #3748C2×D3semidirect
177. #1004C3⋊D3semidirect
379. C33#577semidirect
526. C32#5775semidirect
527. C32#5836semidirect
587. C3#10595106semidirect
588. C3#10596307semidirect

Maximal subgroups

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

subgroupindex
C32×#10042
#117245632
C2×#888612
C3×#641412
#117235562
#117235552
#117235502
D3×#37542
#117233792
#117233632
#117233502
#105963073
#105951063