Finite Group Databasebeta

← #13150918

Subgroups — #13150918

Order 1728.

842 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
1C9×#1004C211yesmaximal, characteristic, complemented
2#11722465C211yesmaximal, characteristic, complemented
3C2×#88543C211yesmaximal, characteristic, complemented
4C3×#63495C211yesmaximal, characteristic, complemented
5#11721169C212yesmaximal, complemented
6#11721168C212yesmaximal, complemented
7#11721165C212yesmaximal, complemented
8D3×#3672C212yesmaximal, complemented
9#11720891C211yesmaximal, characteristic, complemented
10#11720881C211yesmaximal, characteristic, complemented
11#11720878C211yesmaximal, characteristic, complemented
12#10595106—31nomaximal
13#10589938—31nomaximal
14C2×C3×D4×C9C2211yescharacteristic
15C2×#6226C2211yescharacteristic, complemented
16C2×C9×#67—22no—
17C9×#237—21no—
18C9×#237C2212yes—
19D3×D4×C9—21no—
20D3×D4×C9C2212yes—
21C9×#235C2211yescharacteristic
22C9×#235C2211yescharacteristic
23C3×#3697C2211yescharacteristic, complemented
24#88663—22no—
25#88641C2212yescomplemented
26#88638—21no—
27D3×Dic18C2212yescomplemented
28D9×Dic6—21no—
29#88543C2212yescomplemented
30#88543C2212yescomplemented
31C2×C3⋊(C9⋊C8)C2211yescharacteristic, complemented
32C3×#3673C2212yescomplemented
33C3×#3672C2212yescomplemented
34C3×#3667C2211yescharacteristic, complemented
35#88428C2212yescomplemented
36#88427C2211yescharacteristic, complemented
37#88425C2211yescharacteristic, complemented
38#88424C2212yescomplemented
39#88423C2211yescharacteristic, complemented
40#88421C2211yescharacteristic, complemented
41#88420C2212yescomplemented
42D3×C9⋊C8C2212yescomplemented
43C3×#1004D311yescharacteristic
44#64298—31no—
45C2×#3754—31no—
46C3×#932—31no—
47#63933—32no—
48#63932—32no—
49#63929—32no—
50D3×#214—32no—
51#63831—31no—
52#63822—31no—
53#63815—31no—
54C9×H(5,𝔽2)—31no—
55#63716—31no—
56#63513—31no—
57#63513—32no—
58#63511—32no—
59#63511—31no—
60#63510—31no—
61#63495—32no—
62#63495D311yescharacteristic, complemented
63C2×#3672—31no—
64C2×#3672—32no—
65C2×#3672—61no—
66C23×C3×C9—21no—
67C22×D3×C9—22no—
68C3×D4×C9—21no—
69C3×D4×C9C2312yes—
70C2×C3×C4×C9C2311yescharacteristic
71C2×C3⋊Dic9—21no—
72#6226—21no—
73#6226C2312yes—
74C9×#67—21no—
75C9×#67—22no—
76C9×#67—22no—
77C9×#67D412yes—
78C3×#611—21no—
79C2×C9×Dic3—22no—
80C9×D12D411yescharacteristic
81C9×D12C2311yescharacteristic
82C4×D3×C9—21no—
83C4×D3×C9C2312yes—
84C3×D36C2311yescharacteristic
85C3×C4×D9—21no—
86C9×Dic6D411yescharacteristic
87C9×Dic6C2311yescharacteristic
88C3×Dic18C2311yescharacteristic
89#6194—22no—
90D3×Dic9—22no—
91Dic3×D9—22no—
92#6189—22no—
93C3⋊(C9⋊C8)C2311yescharacteristic
94C3⋊(C9⋊C8)C2311yescharacteristic
95C3×C9⋊C8C2312yes—
96#5836—91no—
97#5775—91no—
98C2×C32×D4C2×D311yescharacteristic
99C2×#634—31no—
100C2×C3×#67—22no—
101C3×#237—21no—
102C3×#237C2×D312yes—
103C3×D3×D4—21no—
104C3×D3×D4C2×D312yes—
105C3×#235—31no—
106C3×#235C2×D311yescharacteristic
107C3×#235C2×D311yescharacteristic
108#3805—62no—
109#3795—32no—
110#3795—61no—
111D3×Dic6—32no—
112D3×Dic6—61no—
113#3754—32no—
114#3754—32no—
115C2×C3⋊(C3⋊C8)—31no—
116C3×#214—32no—
117C3×#213—32no—
118C3×#208—31no—
119#3719—32no—
120#3718—31no—
121#3716—32no—
122#3715—31no—
123#3715—31no—
124#3713—31no—
125#3711—32no—
126D3×C3⋊C8—32no—
127C9×#41—31no—
128C9×#41—32no—
129C9×#41—31no—
130C9×#41—61no—
131C2×D4×C9—21no—
132C2×D4×C9—32no—
133C2×D4×C9—61no—
134C2×D4×C9—62no—
135C2×D4×C9C2×D311yescharacteristic
136Q8×D9—61no—
137#3699—61no—
138#3699—62no—
139#3697—32no—
140#3697C2×D311yescharacteristic, complemented
141C2×Dic18—31no—
142C2×Dic18—32no—
143C2×Dic18—61no—
144#3675—31no—
145#3674—31no—
146#3674—32no—
147#3673—31no—
148#3673C2×D312yescomplemented
149#3672—32no—
150#3672—32no—
151#3672—31no—
152#3672—32no—
153#3672—62no—
154#3672—62no—
155#3672C2×D312yescomplemented
156#3667—32no—
157#3667C2×D311yescharacteristic, complemented
158C2×C9⋊C8—31no—
159C2×C9⋊C8—32no—
160C2×C9⋊C8—61no—
161C22×C3×C9—22no—
162C22×C3×C9—81no—
163C22×C3×C9C2×D411yescharacteristic
164C2×D3×C9—21no—
165C2×D3×C9—82no—
166C2×D3×C9C2×D412yes—
167C2×C3×D9—21no—
168C3×C4×C9C2×D411yescharacteristic
169C3×C4×C9C2411yescharacteristic
170C3⋊Dic9—22no—
171C9×Dic3—21no—
172C9×Dic3C2×D412yes—
173C3×Dic9—21no—
174C3×H(5,𝔽2)—31no—
175#1010—91no—
176#1010—91no—
177#1004D911yescharacteristic, complemented
178C3×AGL(1,ℤ/8)—91no—
179#951—91no—
180#951—92no—
181#950—92no—
182#950—91no—
183#950—91no—
184#948—91no—
185C2×#214—92no—
186C2×#214—91no—
187C2×#214—91no—
188C2×#214—181no—
189#932—31no—
190#932—92no—
191#915—92no—
192D3×SD16—92no—
193#912—92no—
194#911—92no—
195#909—91no—
196#907—91no—
197C23×C32—21no—
198C22×C3×D3—22no—
199C32×D4—21no—
200C32×D4C22×D312yes—
201C2×C32×C4C22×D311yescharacteristic
202C2×C3⋊Dic3—61no—
203#634—32no—
204#634—61no—
205C3×#67—21no—
206C3×#67—22no—
207C3×#67—22no—
208C3×#67—61no—
209C3×#67#6712yes—
210C2×C3×Dic3—22no—
211C3×D12—31no—
212C3×D12#6711yescharacteristic
213C3×D12C22×D311yescharacteristic
214C3×C4×D3—21no—
215C3×C4×D3—61no—
216C3×C4×D3C22×D312yes—
217C3×Dic6—31no—
218C3×Dic6#6711yescharacteristic
219C3×Dic6C22×D311yescharacteristic
220#627—62no—
221#625—62no—
222D3×Dic3—62no—
223D3×Dic3—62no—
224C23×C9—21no—
225C23×C9—41no—
226C23×C9—62no—
227C3⋊(C3⋊C8)—31no—
228C3⋊(C3⋊C8)—31no—
229C3×C3⋊C8—32no—
230Q8×C9—31no—
231Q8×C9—31no—
232D4×C9—21no—
233D4×C9—22no—
234D4×C9—32no—
235D4×C9—31no—
236D4×C9—31no—
237D4×C9—41no—
238D4×C9—62no—
239D4×C9—62no—
240D4×C9—61no—
241D4×C9C22×D312yes—
242C2×C4×C9—21no—
243C2×C4×C9—32no—
244C2×C4×C9—62no—
245C2×C4×C9—61no—
246C2×C4×C9C22×D311yescharacteristic
247#611—21no—
248#611—62no—
249C2×Dic9—62no—
250C2×Dic9—61no—
251C2×Dic9—121no—
252D36C22×D311yescharacteristic
253C4×D9—21no—
254C4×D9—62no—
255Dic18—32no—
256Dic18—62no—
257Dic18—62no—
258Dic18—61no—
259Dic18—121no—
260Dic18C22×D311yescharacteristic
261C9⋊C8—31no—
262C9⋊C8—31no—
263C9⋊C8—61no—
264C9⋊C8—61no—
265C9⋊C8C22×D312yes—
266#577—271noSylow
267C2×C3×C9—21no—
268C2×C3×C9—42no—
269C2×C3×C9C22×D411yescharacteristic
270D3×C9—22no—
271D3×C9—41no—
272C3×D9—41no—
273C3×#41—31no—
274C3×#41—32no—
275C3×#41—31no—
276C3×#41—61no—
277C3×#41—91no—
278C2×C3×D4—21no—
279C2×C3×D4—32no—
280C2×C3×D4—61no—
281C2×C3×D4—62no—
282C2×C3×D4C2×D911yescharacteristic
283C2×C3×D4D3211yescharacteristic
284C2×#67—22no—
285D3×Q8—92no—
286D3×Q8—181no—
287#237—21no—
288#237—92no—
289#237—181no—
290#237—182no—
291#237—182no—
292#237C2×D912yes—
293D3×D4—21no—
294D3×D4C2×D912yes—
295#235—31no—
296#235—92no—
297#235—181no—
298#235C2×D911yescharacteristic
299#235C2×D911yescharacteristic
300C2×Dic6—92no—
301C2×Dic6—91no—
302C2×Dic6—91no—
303C2×Dic6—181no—
304C2×Dic6—181no—
305C3×SD16—92no—
306C3×D8—92no—
307C3×M16—91no—
308#216—92no—
309#216—91no—
310#216—91no—
311#215—91no—
312#215—91no—
313#214—32no—
314#214—92no—
315#214—91no—
316#214—92no—
317#214—92no—
318#214—92no—
319#214—92no—
320#214—91no—
321#214—182no—
322#214—182no—
323#213—32no—
324#213—91no—
325#213—91no—
326#208—31no—
327#208—92no—
328C2×C3⋊C8—92no—
329C2×C3⋊C8—91no—
330C2×C3⋊C8—91no—
331C2×C3⋊C8—181no—
332Dic12—92no—
333#204—92no—
334ΓL(1,𝔽25)—92no—
335D3×C8—92no—
336C22×C32—22no—
337C22×C32—81no—
338C22×C32C2×#6711yescharacteristic
339C2×C3×D3—21no—
340C2×C3×D3—61no—
341C2×C3×D3—82no—
342C2×C3×D3C2×#6712yes—
343C32×C4C2×#6711yescharacteristic
344C32×C4C23×D311yescharacteristic
345C3⋊Dic3—62no—
346C3×Dic3—21no—
347C3×Dic3—61no—
348C3×Dic3C2×#6712yes—
349C22×C9—22no—
350C22×C9—21no—
351C22×C9—32no—
352C22×C9—42no—
353C22×C9—61no—
354C22×C9—81no—
355C22×C9—82no—
356C22×C9—242no—
357C22×C9D3×D411yescharacteristic
358C2×D9—21no—
359C4×C9—21no—
360C4×C9—21no—
361C4×C9—32no—
362C4×C9—61no—
363C4×C9D3×D411yescharacteristic
364C4×C9C23×D311yescharacteristic
365Dic9—21no—
366Dic9—62no—
367Dic9—122no—
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—
379C3×C9#57711yescharacteristic, complemented, Sylow
380C23×C3—21no—
381C23×C3—21no—
382C23×C3—41no—
383C23×C3—62no—
384C22×D3—22no—
385C3×Q8—31no—
386C3×Q8—31no—
387C3×Q8—91no—
388C3×D4—21no—
389C3×D4—21no—
390C3×D4—22no—
391C3×D4—32no—
392C3×D4—31no—
393C3×D4—31no—
394C3×D4—41no—
395C3×D4—62no—
396C3×D4—62no—
397C3×D4—61no—
398C3×D4—91no—
399C3×D4—181no—
400C3×D4C22×D912yes—
401C3×D4C2×D3212yes—
402C2×C3×C4—21no—
403C2×C3×C4—32no—
404C2×C3×C4—62no—
405C2×C3×C4—61no—
406C2×C3×C4—181no—
407C2×C3×C4C22×D911yescharacteristic
408C2×C3×C4C2×D3211yescharacteristic
409#67—21no—
410#67—22no—
411#67—22no—
412#67—61no—
413#67—182no—
414#67—182no—
415#67#61112yes—
416C2×Dic3—22no—
417C2×Dic3—182no—
418C2×Dic3—181no—
419C2×Dic3—181no—
420C2×Dic3—182no—
421C2×Dic3—361no—
422D12—31no—
423D12#61111yescharacteristic
424D12C22×D911yescharacteristic
425C4×D3—21no—
426C4×D3—61no—
427C4×D3—182no—
428C4×D3—182no—
429C4×D3C22×D912yes—
430Dic6—31no—
431Dic6—92no—
432Dic6—92no—
433Dic6—182no—
434Dic6—182no—
435Dic6—181no—
436Dic6—181no—
437Dic6—182no—
438Dic6—361no—
439Dic6#61111yescharacteristic
440Dic6C22×D911yescharacteristic
441C3×C8—92no—
442C3⋊C8—32no—
443C3⋊C8—91no—
444C3⋊C8—91no—
445C3⋊C8—91no—
446C3⋊C8—91no—
447C3⋊C8—181no—
448C3⋊C8—181no—
449C2×C32—21no—
450C2×C32—42no—
451C2×C32C22×#6711yescharacteristic
452C3×D3—22no—
453C3×D3—41no—
454C3×D3—121no—
455C2×C9—22no—
456C2×C9—21no—
457C2×C9—21no—
458C2×C9—42no—
459C2×C9—62no—
460C2×C9—82no—
461C2×C9—121no—
462C2×C9C2×D3×D411yescharacteristic
463D9—41no—
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×D911yescharacteristic
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—22no—
494C22×C3—22no—
495C22×C3—21no—
496C22×C3—32no—
497C22×C3—42no—
498C22×C3—61no—
499C22×C3—81no—
500C22×C3—81no—
501C22×C3—82no—
502C22×C3—181no—
503C22×C3—242no—
504C22×C3C2×#61111yescharacteristic
505C22×C3D3×#6711yescharacteristic
506C2×D3—21no—
507C2×D3—61no—
508C2×D3—82no—
509C2×D3C2×#61112yes—
510C3×C4—21no—
511C3×C4—21no—
512C3×C4—32no—
513C3×C4—61no—
514C3×C4—181no—
515C3×C4C2×#61111yescharacteristic
516C3×C4C23×D911yescharacteristic
517C3×C4D3×#6711yescharacteristic
518C3×C4C22×D3211yescharacteristic
519Dic3—21no—
520Dic3—61no—
521Dic3—182no—
522Dic3—182no—
523Dic3—362no—
524Dic3C2×#61112yes—
525C32#583611yescharacteristic
526C9—21no—
527C9#577511yescharacteristic, complemented
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×D912yes—
547C2×C4—32no—
548C2×C4—62no—
549C2×C4—61no—
550C2×C4—181no—
551C2×C4—542no—
552C2×C4—542no—
553C2×C4—541no—
554C2×C4C2×D3×D911yescharacteristic
555C8—92no—
556C8—271no—
557C8—271no—
558C2×C3—22no—
559C2×C3—21no—
560C2×C3—21no—
561C2×C3—21no—
562C2×C3—42no—
563C2×C3—42no—
564C2×C3—62no—
565C2×C3—82no—
566C2×C3—121no—
567C2×C3—361no—
568C2×C3C22×#61111yescharacteristic
569C2×C3C2×D3×#6711yescharacteristic
570D3—22no—
571D3—41no—
572D3—121no—
573C22—22no—
574C22—32no—
575C22—61no—
576C22—81no—
577C22—181no—
578C22—242no—
579C22D3×#61111yescharacteristic
580C4—32no—
581C4—61no—
582C4—181no—
583C4—542no—
584C4D3×#61111yescharacteristic
585C4C22×D3×D911yescharacteristic
586C3—21no—
587C3#1058993811yescharacteristic, complemented
588C3#1059510611yescharacteristic
589C2—21no—
590C2—42no—
591C2—62no—
592C2—121no—
593C2—361no—
594C2C2×D3×#61111yescharacteristic
subgroupG/H
centerC2C2×D3×#611
commutator, cosocleC3×C4×C9C24
FrattiniC3×C4C22×D32
FittingC2×C3×D4×C9C22
socleC2×C32C22×#67
hypercenterC2×D4D3×D9
hypocenterC3×C9#577
LCSG ▷ C3×C4×C9 ▷ C2×C3×C9 ▷ C3×C9
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
1. C9×#1004C2semidirect
2. #11722465C2semidirect
3. C2×#88543C2semidirect
4. C3×#63495C2semidirect
5. #11721169C2semidirect
6. #11721168C2semidirect
7. #11721165C2semidirect
8. D3×#3672C2semidirect
9. #11720891C2semidirect
10. #11720881C2semidirect
11. #11720878C2semidirect
15. C2×#6226C22semidirect
23. C3×#3697C22semidirect
25. #88641C22semidirect
27. D3×Dic18C22semidirect
29. #88543C22semidirect
30. #88543C22semidirect
31. C2×C3⋊(C9⋊C8)C22semidirect
32. C3×#3673C22semidirect
33. C3×#3672C22semidirect
34. C3×#3667C22semidirect
35. #88428C22semidirect
36. #88427C22semidirect
37. #88425C22semidirect
38. #88424C22semidirect
39. #88423C22semidirect
40. #88421C22semidirect
41. #88420C22semidirect
42. D3×C9⋊C8C22semidirect
62. #63495D3semidirect
140. #3697C2×D3semidirect
148. #3673C2×D3semidirect
155. #3672C2×D3semidirect
157. #3667C2×D3semidirect
177. #1004D9semidirect
379. C3×C9#577semidirect
527. C9#5775semidirect
587. C3#10589938semidirect

Maximal subgroups

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

subgroupindex
C9×#10042
#117224652
C2×#885432
C3×#634952
#117211692
#117211682
#117211652
D3×#36722
#117208912
#117208812
#117208782
#105951063
#105899383