Finite Group Databasebeta

← #13150917

Subgroups — #13150917

Order 1728.

842 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
1C9×#1004C211yesmaximal, characteristic, complemented
2#11722464C211yesmaximal, characteristic, complemented
3#11721833C211yesmaximal, characteristic, complemented
4C2×C3×#3672C211yesmaximal, characteristic, complemented
5#11721171C212yesmaximal, complemented
6#11721170C212yesmaximal, complemented
7#11721165C212yesmaximal, complemented
8D3×#3672C212yesmaximal, complemented
9#11720889C211yesmaximal, characteristic, complemented
10#11720887C211yesmaximal, characteristic, complemented
11#11720877C211yesmaximal, characteristic, complemented
12#10595105—31nomaximal
13#10589938—31nomaximal
14C2×C3×D4×C9C2211yescharacteristic
15#88736C2211yescharacteristic, complemented
16C2×C9×#67—22no—
17C9×#237—21no—
18C9×#237C2212yes—
19D3×D4×C9—21no—
20D3×D4×C9C2212yes—
21C9×#235C2211yescharacteristic
22C9×#235C2211yescharacteristic
23C2×C3×Dic18C2211yescharacteristic, complemented
24#88662—22no—
25#88642C2212yescomplemented
26#88639—21no—
27D3×Dic18C2212yescomplemented
28(C3⋊D9)⋊Q8—21no—
29#88544C2212yescomplemented
30#88543C2212yescomplemented
31#88532C2211yescharacteristic, complemented
32C3×#3672C2212yescomplemented
33C3×#3672C2212yescomplemented
34C2×C3×C9⋊C8C2211yescharacteristic, complemented
35#88428C2211yescharacteristic, complemented
36#88427C2212yescomplemented
37#88426C2211yescharacteristic, complemented
38#88424C2211yescharacteristic, complemented
39#88423C2212yescomplemented
40#88422C2211yescharacteristic, complemented
41#88420C2212yescomplemented
42D3×C9⋊C8C2212yescomplemented
43C3×#1004D311yescharacteristic
44#64297—31no—
45#64141—31no—
46C2×C3×#214—31no—
47#63935—32no—
48#63934—32no—
49#63929—32no—
50D3×#214—32no—
51#63829—31no—
52#63827—31no—
53#63814—31no—
54C9×H(5,𝔽2)—31no—
55#63716—31no—
56#63513—32no—
57#63513—31no—
58#63511—31no—
59#63511—32no—
60#63510—31no—
61#63495—31no—
62#63495—32no—
63#63495—61no—
64C2×#3672—32no—
65C2×#3672D311yescharacteristic, complemented
66C23×C3×C9—21no—
67C22×D3×C9—22no—
68C3×D4×C9—21no—
69C3×D4×C9C2312yes—
70C2×C3×C4×C9C2311yescharacteristic
71#6233—21no—
72C4⋊(C3⋊D9)C2311yescharacteristic
73C4×C3⋊D9—21no—
74#6226C2311yescharacteristic
75C9×#67—21no—
76C9×#67—22no—
77C9×#67—22no—
78C9×#67D412yes—
79C2×C9×Dic3—22no—
80C2×C3×Dic9—21no—
81C9×D12D411yescharacteristic
82C9×D12C2311yescharacteristic
83C4×D3×C9—21no—
84C4×D3×C9C2312yes—
85C9×Dic6D411yescharacteristic
86C9×Dic6C2311yescharacteristic
87C3×Dic18—21no—
88C3×Dic18C2312yes—
89#6195—22no—
90D3×Dic9—22no—
91(C3⋊D9)⋊C4—22no—
92#6189—22no—
93C3⋊(C9⋊C8)C2312yes—
94C3×C9⋊C8C2311yescharacteristic
95C3×C9⋊C8C2311yescharacteristic
96#5836—91no—
97#5762—91no—
98C2×C32×D4C2×D311yescharacteristic
99#3828—31no—
100C2×C3×#67—22no—
101C3×#237—21no—
102C3×#237C2×D312yes—
103C3×D3×D4—21no—
104C3×D3×D4C2×D312yes—
105C3×#235C2×D311yescharacteristic
106C3×#235C2×D311yescharacteristic
107C2×C3×Dic6—31no—
108#3804—62no—
109#3799—32no—
110#3797—61no—
111#3796—61no—
112D3×Dic6—32no—
113#3754—32no—
114#3753—32no—
115#3748—31no—
116C3×#214—32no—
117C3×#214—32no—
118C2×C3×C3⋊C8—31no—
119#3719—31no—
120#3718—32no—
121#3717—31no—
122#3716—31no—
123#3715—32no—
124#3714—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—31no—
140#3697—32no—
141#3697—61no—
142C2×Dic18—32no—
143C2×Dic18C2×D311yescharacteristic, complemented
144#3675—31no—
145#3674—32no—
146#3674—31no—
147#3673—32no—
148#3673—31no—
149#3673—62no—
150#3672—32no—
151#3672—32no—
152#3672—31no—
153#3672—62no—
154#3672C2×D312yescomplemented
155#3672C2×D312yescomplemented
156#3667—31no—
157#3667—32no—
158#3667—61no—
159C2×C9⋊C8—32no—
160C2×C9⋊C8C2×D311yescharacteristic, complemented
161C22×C3×C9—22no—
162C22×C3×C9—81no—
163C22×C3×C9C2×D411yescharacteristic
164C2×C3⋊D9—21no—
165C2×D3×C9—21no—
166C2×D3×C9—82no—
167C2×D3×C9C2×D412yes—
168C3×C4×C9C2×D411yescharacteristic
169C3×C4×C9C2411yescharacteristic
170C3⋊Dic9—21no—
171C9×Dic3—21no—
172C9×Dic3C2×D412yes—
173C3×Dic9—22no—
174C3×H(5,𝔽2)—31no—
175#1010—91no—
176#1007—91no—
177#1004D911yescharacteristic, complemented
178C2×C3×SD16—91no—
179#951—92no—
180#951—91no—
181#950—91no—
182#950—92no—
183#948—91no—
184#942—91no—
185C2×#214—31no—
186C2×#214—92no—
187#932—92no—
188#932—91no—
189#932—91no—
190#932—181no—
191#916—92no—
192#915—92no—
193#914—92no—
194D3×SD16—92no—
195#904—91no—
196#901—91no—
197C23×C32—21no—
198C22×C3×D3—22no—
199C32×D4—21no—
200C32×D4C22×D312yes—
201C2×C32×C4C22×D311yescharacteristic
202#638—61no—
203C4⋊(C3⋊D3)—31no—
204C4×C3⋊D3—61no—
205#634—31no—
206C3×#67—21no—
207C3×#67—22no—
208C3×#67—22no—
209C3×#67#6712yes—
210C2×C3×Dic3—22no—
211C2×C3×Dic3—61no—
212C3×D12#6711yescharacteristic
213C3×D12C22×D311yescharacteristic
214C3×C4×D3—21no—
215C3×C4×D3C22×D312yes—
216C3×Dic6—32no—
217C3×Dic6—61no—
218C3×Dic6#6711yescharacteristic
219C3×Dic6C22×D311yescharacteristic
220#627—62no—
221#626—62no—
222#624—62no—
223D3×Dic3—62no—
224C23×C9—21no—
225C23×C9—41no—
226C23×C9—62no—
227C3⋊(C3⋊C8)—32no—
228C3×C3⋊C8—31no—
229C3×C3⋊C8—31no—
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—62no—
248#611—61no—
249#611—121no—
250C2×Dic9—21no—
251C2×Dic9—62no—
252D36—31no—
253D36—61no—
254C4×D9—62no—
255C4×D9—61no—
256C4×D9—121no—
257Dic18—21no—
258Dic18—31no—
259Dic18—61no—
260Dic18—62no—
261Dic18C22×D312yes—
262C9⋊C8—32no—
263C9⋊C8—62no—
264C9⋊C8C22×D311yescharacteristic
265C9⋊C8C22×D311yescharacteristic
266#577—271noSylow
267C2×C3×C9—21no—
268C2×C3×C9—42no—
269C2×C3×C9C22×D411yescharacteristic
270C3⋊D9—41no—
271D3×C9—22no—
272D3×C9—41no—
273C3×#41—31no—
274C3×#41—32no—
275C3×#41—31no—
276C3×#41—61no—
277C2×C3×Q8—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—
285#239—92no—
286#239—181no—
287D3×Q8—92no—
288D3×Q8—181no—
289D3×Q8—181no—
290#237—21no—
291#237—181no—
292#237—182no—
293#237C2×D912yes—
294D3×D4—21no—
295D3×D4C2×D912yes—
296#235—92no—
297#235—91no—
298#235—91no—
299#235—181no—
300#235—182no—
301#235C2×D911yescharacteristic
302#235C2×D911yescharacteristic
303C2×Dic6—31no—
304C2×Dic6—92no—
305C3×SD16—92no—
306C3×SD16—92no—
307C2×C3×C8—91no—
308#216—91no—
309#216—92no—
310#216—92no—
311#215—91no—
312#215—92no—
313#214—32no—
314#214—32no—
315#214—91no—
316#214—92no—
317#214—92no—
318#214—92no—
319#214—182no—
320#213—91no—
321#213—92no—
322#213—92no—
323#213—182no—
324#208—92no—
325#208—91no—
326#208—91no—
327#208—181no—
328C2×C3⋊C8—31no—
329C2×C3⋊C8—92no—
330Dic12—91no—
331D24—91no—
332#204—91no—
333#204—91no—
334ΓL(1,𝔽25)—92no—
335D3×C8—92no—
336C22×C32—22no—
337C22×C32—81no—
338C22×C32C2×#6711yescharacteristic
339C2×C3⋊D3—61no—
340C2×C3×D3—21no—
341C2×C3×D3—82no—
342C2×C3×D3C2×#6712yes—
343C32×C4C2×#6711yescharacteristic
344C32×C4C23×D311yescharacteristic
345C3⋊Dic3—61no—
346C3×Dic3—21no—
347C3×Dic3—62no—
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—61no—
359C2×D9—121no—
360C4×C9—21no—
361C4×C9—21no—
362C4×C9—32no—
363C4×C9—61no—
364C4×C9D3×D411yescharacteristic
365C4×C9C23×D311yescharacteristic
366Dic9—22no—
367Dic9—61no—
368Dic9—121no—
3692−1+4—271no—
370H(5,𝔽2)—31no—
371#137—272no—
372#137—271no—
373AGL(1,ℤ/8)—272no—
374AGL(1,ℤ/8)—271no—
375#135—271no—
376#135—272no—
377C2×SD16—91no—
378C2×SD16—272no—
379#131—271no—
380C3×C9#57711yescharacteristic, complemented, Sylow
381C23×C3—21no—
382C23×C3—21no—
383C23×C3—41no—
384C23×C3—62no—
385C22×D3—22no—
386C3×Q8—31no—
387C3×Q8—31no—
388C3×Q8—92no—
389C3×Q8—181no—
390C3×D4—21no—
391C3×D4—21no—
392C3×D4—22no—
393C3×D4—32no—
394C3×D4—31no—
395C3×D4—31no—
396C3×D4—41no—
397C3×D4—62no—
398C3×D4—62no—
399C3×D4—61no—
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—182no—
413#67—181no—
414#67—181no—
415#67—361no—
416#67#61112yes—
417C2×Dic3—22no—
418C2×Dic3—61no—
419C2×Dic3—182no—
420D12—91no—
421D12—91no—
422D12—181no—
423D12—182no—
424D12#61111yescharacteristic
425D12C22×D911yescharacteristic
426C4×D3—21no—
427C4×D3—182no—
428C4×D3—182no—
429C4×D3—181no—
430C4×D3—181no—
431C4×D3—182no—
432C4×D3—361no—
433C4×D3C22×D912yes—
434Dic6—32no—
435Dic6—61no—
436Dic6—91no—
437Dic6—91no—
438Dic6—181no—
439Dic6—182no—
440Dic6—182no—
441Dic6#61111yescharacteristic
442Dic6C22×D911yescharacteristic
443C3×C8—91no—
444C3×C8—91no—
445C3⋊C8—31no—
446C3⋊C8—31no—
447C3⋊C8—92no—
448C3⋊C8—92no—
449C3⋊C8—182no—
450C2×C32—21no—
451C2×C32—42no—
452C2×C32C22×#6711yescharacteristic
453C3⋊D3—121no—
454C3×D3—22no—
455C3×D3—41no—
456C2×C9—21no—
457C2×C9—21no—
458C2×C9—41no—
459C2×C9—42no—
460C2×C9—62no—
461C2×C9—82no—
462C2×C9—121no—
463C2×C9C2×D3×D411yescharacteristic
464D9—121no—
465D9—241no—
466#41—31no—
467#41—32no—
468#41—31no—
469#41—61no—
470#41—272no—
471#41—271no—
472#41—542no—
473#41—541no—
474C2×Q8—91no—
475C2×Q8—272no—
476C2×Q8—541no—
477C2×D4—32no—
478C2×D4—61no—
479C2×D4—62no—
480C2×D4D3×D911yescharacteristic
481Q16—271no—
482Q16—272no—
483SD16—92no—
484SD16—92no—
485SD16—271no—
486SD16—271no—
487SD16—272no—
488SD16—272no—
489D8—271no—
490D8—272no—
491M16—272no—
492M16—271no—
493C2×C8—91no—
494C2×C8—272no—
495C22×C3—22no—
496C22×C3—22no—
497C22×C3—21no—
498C22×C3—32no—
499C22×C3—42no—
500C22×C3—61no—
501C22×C3—81no—
502C22×C3—81no—
503C22×C3—82no—
504C22×C3—242no—
505C22×C3C2×#61111yescharacteristic
506C22×C3D3×#6711yescharacteristic
507C2×D3—21no—
508C2×D3—82no—
509C2×D3—181no—
510C2×D3—181no—
511C2×D3—361no—
512C2×D3C2×#61112yes—
513C3×C4—21no—
514C3×C4—21no—
515C3×C4—32no—
516C3×C4—61no—
517C3×C4—182no—
518C3×C4C2×#61111yescharacteristic
519C3×C4C23×D911yescharacteristic
520C3×C4D3×#6711yescharacteristic
521C3×C4C22×D3211yescharacteristic
522Dic3—21no—
523Dic3—62no—
524Dic3—181no—
525Dic3—181no—
526Dic3—361no—
527Dic3C2×#61112yes—
528C32#583611yescharacteristic
529C9—21no—
530C9#576211yescharacteristic, complemented
531C23—21no—
532C23—62no—
533Q8—31no—
534Q8—31no—
535Q8—92no—
536Q8—181no—
537Q8—271no—
538Q8—542no—
539D4—21no—
540D4—32no—
541D4—31no—
542D4—31no—
543D4—62no—
544D4—62no—
545D4—61no—
546D4—271no—
547D4—542no—
548D4—541no—
549D4C2×D3×D912yes—
550C2×C4—32no—
551C2×C4—62no—
552C2×C4—61no—
553C2×C4—181no—
554C2×C4—542no—
555C2×C4—542no—
556C2×C4—541no—
557C2×C4C2×D3×D911yescharacteristic
558C8—91no—
559C8—91no—
560C8—272no—
561C2×C3—21no—
562C2×C3—21no—
563C2×C3—21no—
564C2×C3—41no—
565C2×C3—42no—
566C2×C3—42no—
567C2×C3—62no—
568C2×C3—82no—
569C2×C3—121no—
570C2×C3C22×#61111yescharacteristic
571C2×C3C2×D3×#6711yescharacteristic
572D3—22no—
573D3—41no—
574D3—361no—
575D3—361no—
576D3—721no—
577C22—22no—
578C22—32no—
579C22—61no—
580C22—81no—
581C22—242no—
582C22—541no—
583C22D3×#61111yescharacteristic
584C4—32no—
585C4—61no—
586C4—182no—
587C4—541no—
588C4D3×#61111yescharacteristic
589C4C22×D3×D911yescharacteristic
590C3—21no—
591C3#1058993811yescharacteristic, complemented
592C3#1059510511yescharacteristic
593C2—21no—
594C2—42no—
595C2—62no—
596C2—121no—
597C2—1081no—
598C2C2×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. #11722464C2semidirect
3. #11721833C2semidirect
4. C2×C3×#3672C2semidirect
5. #11721171C2semidirect
6. #11721170C2semidirect
7. #11721165C2semidirect
8. D3×#3672C2semidirect
9. #11720889C2semidirect
10. #11720887C2semidirect
11. #11720877C2semidirect
15. #88736C22semidirect
23. C2×C3×Dic18C22semidirect
25. #88642C22semidirect
27. D3×Dic18C22semidirect
29. #88544C22semidirect
30. #88543C22semidirect
31. #88532C22semidirect
32. C3×#3672C22semidirect
33. C3×#3672C22semidirect
34. C2×C3×C9⋊C8C22semidirect
35. #88428C22semidirect
36. #88427C22semidirect
37. #88426C22semidirect
38. #88424C22semidirect
39. #88423C22semidirect
40. #88422C22semidirect
41. #88420C22semidirect
42. D3×C9⋊C8C22semidirect
65. C2×#3672D3semidirect
143. C2×Dic18C2×D3semidirect
154. #3672C2×D3semidirect
155. #3672C2×D3semidirect
160. C2×C9⋊C8C2×D3semidirect
177. #1004D9semidirect
380. C3×C9#577semidirect
530. C9#5762semidirect
591. C3#10589938semidirect

Maximal subgroups

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

subgroupindex
C9×#10042
#117224642
#117218332
C2×C3×#36722
#117211712
#117211702
#117211652
D3×#36722
#117208892
#117208872
#117208772
#105951053
#105899383