Finite Group Databasebeta

← #13177658

Subgroups — #13177658

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
1C3×#64361C211yesmaximal, characteristic, complemented
2#11724563C211yesmaximal, characteristic, complemented
3C2×#88861C211yesmaximal, characteristic, complemented
4C32×#932C211yesmaximal, characteristic, complemented
5#11723560C212yesmaximal, complemented
6#11723559C212yesmaximal, complemented
7#11723552C212yesmaximal, complemented
8C3⋊D3×#214C212yesmaximal, complemented
9#11723388C211yesmaximal, characteristic, complemented
10#11723363C211yesmaximal, characteristic, complemented
11#11723354C211yesmaximal, characteristic, complemented
12#10596246—31nomaximal
13#10595106—34nomaximal
14C2×C33×D4C2211yescharacteristic
15C2×#6308C2211yescharacteristic, complemented
16C2×C3×#638—22no—
17C3×#3830—21no—
18C3×#3830C2212yes—
19C3×D4×C3⋊D3—21no—
20C3×D4×C3⋊D3C2212yes—
21C3×#3828C2211yescharacteristic
22C3×#3828C2211yescharacteristic
23C32×#235C2211yescharacteristic, complemented
24#89029—22no—
25#89021—21no—
26C3⋊D3×Dic6C2212yescomplemented
27#89014C2212yescomplemented
28D3×#634—21no—
29#88861C2212yescomplemented
30#88861C2212yescomplemented
31C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic, complemented
32C32×#214C2212yescomplemented
33C32×#213C2212yescomplemented
34C32×#208C2211yescharacteristic, complemented
35#88800C2212yescomplemented
36#88798C2211yescharacteristic, complemented
37#88796C2212yescomplemented
38#88793C2211yescharacteristic, complemented
39#88792C2211yescharacteristic, complemented
40#88789C2211yescharacteristic, complemented
41#88787C2212yescomplemented
42C3⋊D3×C3⋊C8C2212yescomplemented
43#64367—31no—
44#64361D311yescharacteristic, complemented
45C3×#1004—34no—
46#64298—34no—
47C32×AGL(1,ℤ/8)—31no—
48#64159—31no—
49C2×#3754—31no—
50C2×#3754—34no—
51C2×#3754—64no—
52#64124—32no—
53SD16×C3⋊D3—32no—
54#64121—32no—
55#64120—32no—
56#64118—31no—
57#64116—31no—
58C3×#932D314yescomplemented
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×C3×C3⋊D3—22no—
68C33×D4—21no—
69C33×D4C2312yes—
70C2×C33×C4C2311yescharacteristic
71C2×C3⋊(C3⋊Dic3)—21no—
72#6308—21no—
73#6308C2312yes—
74C3×#638—21no—
75C3×#638—22no—
76C3×#638—22no—
77C3×#638D412yes—
78C2×C3×C3⋊Dic3—22no—
79C3×C4⋊(C3⋊D3)D411yescharacteristic
80C3×C4⋊(C3⋊D3)C2311yescharacteristic
81C3×C4×C3⋊D3—21no—
82C3×C4×C3⋊D3C2312yes—
83C3×#634D411yescharacteristic
84C3×#634C2311yescharacteristic
85C32×#67—21no—
86C32×D12C2311yescharacteristic
87C32×C4×D3—21no—
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
95C32×C3⋊C8C2312yes—
96#5836—91no—
97#5775—94no—
98C32×#41—31no—
99C2×C32×D4—24no—
100C2×C32×D4C2×D314yes—
101C2×C32×D4C2×D311yescharacteristic
102C2×#638—22no—
103Q8×C3⋊D3—32no—
104#3830—21no—
105#3830—32no—
106#3830—62no—
107#3830C2×D312yes—
108D4×C3⋊D3—21no—
109D4×C3⋊D3C2×D312yes—
110#3828—61no—
111#3828C2×D311yescharacteristic
112#3828C2×D311yescharacteristic
113C2×#634—31no—
114C2×#634—34no—
115C2×#634—61no—
116C2×#634—64no—
117C2×C3×#67—68no—
118C3×#237—38no—
119C3×#237—64no—
120C3×D3×D4—38no—
121C3×D3×D4—64no—
122C3×#235—34no—
123C3×#235—34no—
124C3×#235C2×D314yescomplemented
125#3805—68no—
126#3795—38no—
127#3795—64no—
128D3×Dic6—38no—
129D3×Dic6—64no—
130C32×SD16—32no—
131C32×D8—32no—
132C32×M16—31no—
133#3756—31no—
134#3755—31no—
135#3754—32no—
136#3754—32no—
137#3754—38no—
138#3754—38no—
139#3754—31no—
140#3754—68no—
141#3754—68no—
142#3753—31no—
143C2×C3⋊(C3⋊C8)—31no—
144C2×C3⋊(C3⋊C8)—34no—
145C2×C3⋊(C3⋊C8)—64no—
146#3746—32no—
147#3744—32no—
148#3743—32no—
149C8×C3⋊D3—32no—
150C3×#214C2×D318yescomplemented
151C3×#213C2×D318yescomplemented
152C3×#208C2×D314yescomplemented
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×C3×C3⋊D3—82no—
166C2×C3×C3⋊D3C2×D412yes—
167C2×C32×D3—21no—
168C33×C4C2×D411yescharacteristic
169C33×C4C2411yescharacteristic
170C3⋊(C3⋊Dic3)—22no—
171C3×C3⋊Dic3—21no—
172C3×C3⋊Dic3C2×D412yes—
173C32×Dic3—21no—
174C3×H(5,𝔽2)—91no—
175#1010—91no—
176#1010—94no—
177#1004—34no—
178C3×AGL(1,ℤ/8)—34no—
179#951—92no—
180#951—91no—
181#950—94no—
182#950—91no—
183#950—92no—
184#948—91no—
185C2×#214—94no—
186C2×#214—91no—
187C2×#214—92no—
188C2×#214—184no—
189#932—92no—
190#932C3⋊D311yescharacteristic, complemented
191#915—98no—
192D3×SD16—98no—
193#912—98no—
194#911—98no—
195#909—94no—
196#907—94no—
197C23×C32—21no—
198C23×C32—24no—
199C23×C32—44no—
200C22×C3⋊D3—22no—
201C22×C3×D3—68no—
202C32×Q8—31no—
203C32×D4—21no—
204C32×D4—28no—
205C32×D4—24no—
206C32×D4—31no—
207C32×D4—44no—
208C32×D4—61no—
209C32×D4C22×D318yes—
210C32×D4C22×D312yes—
211C2×C32×C4—24no—
212C2×C32×C4—61no—
213C2×C32×C4C22×D314yes—
214C2×C32×C4C22×D311yescharacteristic
215#638—21no—
216#638—22no—
217#638—22no—
218#638—62no—
219#638#6712yes—
220C2×C3⋊Dic3—22no—
221C2×C3⋊Dic3—61no—
222C2×C3⋊Dic3—62no—
223C2×C3⋊Dic3—64no—
224C2×C3⋊Dic3—124no—
225C4⋊(C3⋊D3)#6711yescharacteristic
226C4⋊(C3⋊D3)C22×D311yescharacteristic
227C4×C3⋊D3—21no—
228C4×C3⋊D3—62no—
229C4×C3⋊D3C22×D312yes—
230#634—32no—
231#634—38no—
232#634—68no—
233#634—61no—
234#634—62no—
235#634—64no—
236#634—124no—
237#634#6711yescharacteristic
238#634C22×D311yescharacteristic
239C3×#67—24no—
240C3×#67—38no—
241C3×#67—68no—
242C3×#67—68no—
243C3×#67—64no—
244C2×C3×Dic3—68no—
245C3×D12—34no—
246C3×D12—34no—
247C3×D12C22×D314yes—
248C3×C4×D3—24no—
249C3×C4×D3—38no—
250C3×C4×D3—64no—
251C3×Dic6—34no—
252C3×Dic6—34no—
253C3×Dic6C22×D314yes—
254#627—68no—
255#625—68no—
256D3×Dic3—68no—
257D3×Dic3—68no—
258C32×C8—32no—
259C3⋊(C3⋊C8)—31no—
260C3⋊(C3⋊C8)—31no—
261C3⋊(C3⋊C8)—34no—
262C3⋊(C3⋊C8)—34no—
263C3⋊(C3⋊C8)—64no—
264C3⋊(C3⋊C8)—64no—
265C3×C3⋊C8C22×D318yes—
266#577—271noSylow
267C2×C33—21no—
268C2×C33—42no—
269C2×C33C22×D411yescharacteristic
270C3×C3⋊D3—22no—
271C3×C3⋊D3—41no—
272C32×D3—41no—
273C3×#41—34no—
274C3×#41—91no—
275C3×#41—92no—
276C3×#41—91no—
277C3×#41—181no—
278C2×C3×D4—24no—
279C2×C3×D4—92no—
280C2×C3×D4—181no—
281C2×C3×D4—182no—
282C2×C3×D4D3214yes—
283C2×C3×D4C2×C3⋊D311yescharacteristic
284C2×#67—68no—
285D3×Q8—98no—
286D3×Q8—181no—
287#237—38no—
288#237—64no—
289#237—98no—
290#237—181no—
291#237—182no—
292#237—188no—
293D3×D4—38no—
294D3×D4—64no—
295#235—34no—
296#235—34no—
297#235—92no—
298#235—184no—
299#235C2×C3⋊D311yescharacteristic, complemented
300C2×Dic6—94no—
301C2×Dic6—91no—
302C2×Dic6—92no—
303C2×Dic6—184no—
304C2×Dic6—184no—
305C3×SD16—38no—
306C3×D8—38no—
307C3×M16—34no—
308#216—94no—
309#216—91no—
310#216—92no—
311#215—94no—
312#215—91no—
313#214—94no—
314#214—98no—
315#214—98no—
316#214—92no—
317#214—92no—
318#214—91no—
319#214—92no—
320#214—188no—
321#214—188no—
322#214C2×C3⋊D312yescomplemented
323#213—94no—
324#213—91no—
325#213C2×C3⋊D312yescomplemented
326#208—92no—
327#208C2×C3⋊D311yescharacteristic, complemented
328C2×C3⋊C8—94no—
329C2×C3⋊C8—91no—
330C2×C3⋊C8—92no—
331C2×C3⋊C8—184no—
332Dic12—98no—
333#204—98no—
334ΓL(1,𝔽25)—98no—
335D3×C8—98no—
336C22×C32—22no—
337C22×C32—24no—
338C22×C32—28no—
339C22×C32—48no—
340C22×C32—61no—
341C22×C32—81no—
342C22×C32—84no—
343C22×C32—88no—
344C22×C32D3×D414yes—
345C22×C32C2×#6711yescharacteristic
346C2×C3⋊D3—21no—
347C2×C3⋊D3—82no—
348C2×C3⋊D3C2×#6712yes—
349C2×C3×D3—24no—
350C2×C3×D3—38no—
351C2×C3×D3—64no—
352C2×C3×D3—248no—
353C32×C4—24no—
354C32×C4—24no—
355C32×C4—61no—
356C32×C4D3×D414yes—
357C32×C4C2×#6711yescharacteristic
358C32×C4C23×D314yes—
359C32×C4C23×D311yescharacteristic
360C3⋊Dic3—21no—
361C3⋊Dic3—62no—
362C3⋊Dic3—68no—
363C3⋊Dic3—128no—
364C3⋊Dic3C2×#6712yes—
365C3×Dic3—24no—
366C3×Dic3—38no—
367C3×Dic3—64no—
3682−1+4—271no—
369H(5,𝔽2)—91no—
370#137—271no—
371#137—272no—
372AGL(1,ℤ/8)—31no—
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—182no—
384C22×D3—68no—
385C3×Q8—34no—
386C3×Q8—91no—
387C3×Q8—91no—
388C3×D4—24no—
389C3×D4—21no—
390C3×D4—28no—
391C3×D4—34no—
392C3×D4—44no—
393C3×D4—64no—
394C3×D4—92no—
395C3×D4—91no—
396C3×D4—91no—
397C3×D4—182no—
398C3×D4—182no—
399C3×D4—181no—
400C3×D4C2×D3218yes—
401C3×D4C22×C3⋊D312yes—
402C2×C3×C4—24no—
403C2×C3×C4—64no—
404C2×C3×C4—92no—
405C2×C3×C4—182no—
406C2×C3×C4—181no—
407C2×C3×C4C2×D3214yes—
408C2×C3×C4C22×C3⋊D311yescharacteristic
409#67—21no—
410#67—38no—
411#67—68no—
412#67—68no—
413#67—64no—
414#67—182no—
415#67—188no—
416C2×Dic3—68no—
417C2×Dic3—182no—
418C2×Dic3—188no—
419C2×Dic3—184no—
420C2×Dic3—181no—
421C2×Dic3—364no—
422D12—34no—
423D12—34no—
424D12C22×C3⋊D311yescharacteristic
425C4×D3—21no—
426C4×D3—38no—
427C4×D3—64no—
428C4×D3—188no—
429C4×D3—182no—
430Dic6—34no—
431Dic6—34no—
432Dic6—98no—
433Dic6—92no—
434Dic6—188no—
435Dic6—182no—
436Dic6—188no—
437Dic6—184no—
438Dic6—181no—
439Dic6—364no—
440Dic6C22×C3⋊D311yescharacteristic
441C3×C8—38no—
442C3⋊C8—94no—
443C3⋊C8—94no—
444C3⋊C8—91no—
445C3⋊C8—91no—
446C3⋊C8—184no—
447C3⋊C8—184no—
448C3⋊C8C22×C3⋊D312yes—
449C2×C32—28no—
450C2×C32—24no—
451C2×C32—21no—
452C2×C32—24no—
453C2×C32—42no—
454C2×C32—48no—
455C2×C32—88no—
456C2×C32—121no—
457C2×C32C2×D3×D414yes—
458C2×C32C22×#6711yescharacteristic
459C3⋊D3—22no—
460C3⋊D3—41no—
461C3×D3—44no—
462C3×D3—68no—
463C3×D3—124no—
464#41—31no—
465#41—91no—
466#41—92no—
467#41—91no—
468#41—181no—
469#41—272no—
470#41—541no—
471#41—542no—
472C2×Q8—272no—
473C2×Q8—271no—
474C2×Q8—541no—
475C2×D4—92no—
476C2×D4—181no—
477C2×D4—182no—
478C2×D4D3×C3⋊D311yescharacteristic
479Q16—272no—
480Q16—271no—
481SD16—32no—
482SD16—272no—
483SD16—271no—
484SD16—271no—
485SD16—272no—
486SD16—272no—
487D8—32no—
488D8—271no—
489M16—31no—
490M16—272no—
491C2×C8—272no—
492C2×C8—271no—
493C22×C3—28no—
494C22×C3—22no—
495C22×C3—24no—
496C22×C3—48no—
497C22×C3—64no—
498C22×C3—84no—
499C22×C3—81no—
500C22×C3—88no—
501C22×C3—92no—
502C22×C3—181no—
503C22×C3—722no—
504C22×C3D3×#6714yes—
505C22×C3D4×C3⋊D311yescharacteristic
506C2×D3—21no—
507C2×D3—38no—
508C2×D3—64no—
509C2×D3—248no—
510C3×C4—24no—
511C3×C4—24no—
512C3×C4—64no—
513C3×C4—92no—
514C3×C4—181no—
515C3×C4D3×#6714yes—
516C3×C4D4×C3⋊D311yescharacteristic
517C3×C4C22×D3214yes—
518C3×C4C23×C3⋊D311yescharacteristic
519Dic3—21no—
520Dic3—38no—
521Dic3—64no—
522Dic3—188no—
523Dic3—182no—
524Dic3—368no—
525C32—24no—
526C32#577514yescomplemented
527C32#583611yescharacteristic, complemented
528C23—21no—
529C23—182no—
530Q8—31no—
531Q8—91no—
532Q8—91no—
533Q8—272no—
534Q8—542no—
535Q8—541no—
536D4—21no—
537D4—31no—
538D4—61no—
539D4—92no—
540D4—91no—
541D4—91no—
542D4—182no—
543D4—182no—
544D4—181no—
545D4—542no—
546D4C2×D3×C3⋊D312yes—
547C2×C4—61no—
548C2×C4—92no—
549C2×C4—182no—
550C2×C4—181no—
551C2×C4—542no—
552C2×C4—542no—
553C2×C4—541no—
554C2×C4C2×D3×C3⋊D311yescharacteristic
555C8—32no—
556C8—271no—
557C8—271no—
558C2×C3—28no—
559C2×C3—24no—
560C2×C3—21no—
561C2×C3—24no—
562C2×C3—48no—
563C2×C3—42no—
564C2×C3—88no—
565C2×C3—124no—
566C2×C3—182no—
567C2×C3—361no—
568C2×C3C2×D3×#6714yes—
569C2×C3C2×D4×C3⋊D311yescharacteristic
570D3—41no—
571D3—68no—
572D3—124no—
573C22—22no—
574C22—61no—
575C22—81no—
576C22—92no—
577C22—181no—
578C22—722no—
579C22C3⋊D3×#6711yescharacteristic
580C4—61no—
581C4—92no—
582C4—181no—
583C4—542no—
584C4C3⋊D3×#6711yescharacteristic
585C4C22×D3×C3⋊D311yescharacteristic
586C3—24no—
587C3#1059510614yescomplemented
588C3#1059624611yescharacteristic, complemented
589C2—21no—
590C2—42no—
591C2—121no—
592C2—182no—
593C2—361no—
594C2C2×C3⋊D3×#6711yescharacteristic
subgroupG/H
centerC2C2×C3⋊D3×#67
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. C3×#64361C2semidirect
2. #11724563C2semidirect
3. C2×#88861C2semidirect
4. C32×#932C2semidirect
5. #11723560C2semidirect
6. #11723559C2semidirect
7. #11723552C2semidirect
8. C3⋊D3×#214C2semidirect
9. #11723388C2semidirect
10. #11723363C2semidirect
11. #11723354C2semidirect
15. C2×#6308C22semidirect
23. C32×#235C22semidirect
26. C3⋊D3×Dic6C22semidirect
27. #89014C22semidirect
29. #88861C22semidirect
30. #88861C22semidirect
31. C2×C3⋊(C3⋊(C3⋊C8))C22semidirect
32. C32×#214C22semidirect
33. C32×#213C22semidirect
34. C32×#208C22semidirect
35. #88800C22semidirect
36. #88798C22semidirect
37. #88796C22semidirect
38. #88793C22semidirect
39. #88792C22semidirect
40. #88789C22semidirect
41. #88787C22semidirect
42. C3⋊D3×C3⋊C8C22semidirect
44. #64361D3semidirect
58. C3×#932D3semidirect
124. C3×#235C2×D3semidirect
150. C3×#214C2×D3semidirect
151. C3×#213C2×D3semidirect
152. C3×#208C2×D3semidirect
190. #932C3⋊D3semidirect
299. #235C2×C3⋊D3semidirect
322. #214C2×C3⋊D3semidirect
325. #213C2×C3⋊D3semidirect
327. #208C2×C3⋊D3semidirect
379. C33#577semidirect
526. C32#5775semidirect
527. C32#5836semidirect
587. C3#10595106semidirect
588. C3#10596246semidirect

Maximal subgroups

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

subgroupindex
C3×#643612
#117245632
C2×#888612
C32×#9322
#117235602
#117235592
#117235522
C3⋊D3×#2142
#117233882
#117233632
#117233542
#105962463
#105951063