Finite Group Databasebeta

← #13149908

Subgroups — #13149908

Order 1728.

880 conjugacy classes of subgroups.

The subgroup lattice has 576 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11722501C212yesmaximal, complemented
2D8×C3⋊D9C211yesmaximal, characteristic, complemented
3C9×#912C211yesmaximal, characteristic, complemented
4C3×D8×D9C211yesmaximal, characteristic, complemented
5#11721187C212yesmaximal, complemented
6#11721184C212yesmaximal, complemented
7#11721167C212yesmaximal, complemented
8#11720831C211yesmaximal, characteristic, complemented
9#11720830C211yesmaximal, characteristic, complemented
10#11720806C211yesmaximal, characteristic, complemented
11D3×D72C211yesmaximal, characteristic, complemented
12#10594734—31nomaximal
13#10589871—31nomaximal
14D4×C3⋊D9C2212yescomplemented
15C9×#237C2212yescomplemented
16C3×D4×D9C2212yescomplemented
17#88668—22no—
18D9×#67—22no—
19C2×#6192—22no—
20#88658—22no—
21D3×D36C2212yescomplemented
22#88640C2211yescharacteristic
23#88635C2212yescomplemented
24#88634C2212yescomplemented
25C3×C9×D8C2211yescharacteristic, complemented
26#88544C2212yescomplemented
27#88525C2211yescharacteristic, complemented
28C8×C3⋊D9C2211yescharacteristic, complemented
29C9×#214C2212yescomplemented
30C3×#3673C2212yescomplemented
31C9×Dic12C2211yescharacteristic, complemented
32D3×C8×C9C2211yescharacteristic, complemented
33C3×D72C2211yescharacteristic, complemented
34C3×C8×D9C2211yescharacteristic, complemented
35#88426C2212yescomplemented
36#88425C2212yescomplemented
37#88420C2211yescharacteristic, complemented
38#88417C2212yescomplemented
39#88414C2211yescharacteristic, complemented
40#88413C2211yescharacteristic, complemented
41#64314—32no—
42D8×C3⋊D3—31no—
43C3×#912D311yescharacteristic
44C3×D3×D8—31no—
45#63937—32no—
46#63934—32no—
47#63925—32no—
48#63810—31no—
49#63808—31no—
50#63803—31no—
51D3×D24—31no—
52#63715—32no—
53C9×#135—31no—
54#63512—32no—
55#63481—31no—
56#63481—31no—
57#63476—32no—
58#63476—32no—
59D8×D9—31no—
60D8×D9—61no—
61D8×D9D311yescharacteristic, complemented
62C2×D72—31no—
63#63464—31no—
64C22×C3⋊D9—22no—
65C22×C3×D9—22no—
66C2×D3×D9—22no—
67C3×D4×C9C2312yes—
68#6233—22no—
69C4⋊(C3⋊D9)C2312yes—
70C4×C3⋊D9C2311yescharacteristic
71C9×#67—22no—
72C3×#611—22no—
73C2×C9×Dic3—22no—
74C4×D3×C9C2311yescharacteristic
75C3×D36C2312yes—
76C3×C4×D9C2311yescharacteristic
77C9×Dic6C2312yes—
78#6195D411yescharacteristic
79#6194D411yescharacteristic
80#6192—22no—
81#6192—22no—
82#6192—22no—
83#6192D411yescharacteristic
84(C3⋊D9)⋊C4—22no—
85Dic3×D9—22no—
86#6189D411yescharacteristic
87C3×C8×C9C2311yescharacteristic
88C3⋊(C9⋊C8)C2311yescharacteristic
89C9×C3⋊C8C2311yescharacteristic
90C3×C9⋊C8C2311yescharacteristic
91#5769—91no—
92#5757—91no—
93D4×C3⋊D3—32no—
94C3×#237C2×D312yes—
95C3×D3×D4—32no—
96#3811—62no—
97D3×#67—62no—
98C2×#626—62no—
99#3804—62no—
100D3×D12—32no—
101#3799—32no—
102#3798—31no—
103#3796—32no—
104C32×D8C2×D311yescharacteristic
105#3753—32no—
106#3745—31no—
107C8×C3⋊D3—31no—
108C3×#214C2×D312yes—
109C3×#213—32no—
110C3×Dic12C2×D311yescharacteristic
111C3×D24—31no—
112C3×D3×C8—31no—
113C3×D3×C8C2×D311yescharacteristic
114#3717—32no—
115#3715—32no—
116#3714—32no—
117#3712—31no—
118#3711—31no—
119#3711—31no—
120C9×#41—32no—
121#3701—32no—
122#3701—32no—
123D4×D9—32no—
124D4×D9—62no—
125D4×D9—62no—
126D4×D9—62no—
127D4×D9C2×D312yescomplemented
128#3697—31no—
129#3697—62no—
130C2×D36—32no—
131C2×D36—62no—
132C9×Q16—31no—
133C9×SD16—32no—
134C9×D8—21no—
135C9×D8C2×D311yescharacteristic, complemented
136C2×C8×C9—31no—
137#3675—32no—
138#3675—32no—
139#3673—32no—
140#3673—62no—
141#3673C2×D312yescomplemented
142#3667—31no—
143D72—32no—
144D72—31no—
145D72—61no—
146D72C2×D311yescharacteristic, complemented
147#3663—31no—
148#3663—31no—
149C8×D9—31no—
150C8×D9—61no—
151C8×D9C2×D311yescharacteristic, complemented
152C22×C3×C9—22no—
153C2×C3⋊D9—22no—
154C2×C3⋊D9—82no—
155C2×C3⋊D9C2×D411yescharacteristic
156C2×D3×C9C2×D411yescharacteristic
157C2×C3×D9—22no—
158C2×C3×D9—82no—
159C2×C3×D9C2×D411yescharacteristic
160D3×D9—82no—
161C3×C4×C9C2411yescharacteristic
162C3⋊Dic9C2×D411yescharacteristic
163C9×Dic3—22no—
164C9×Dic3C2×D411yescharacteristic
165C3×Dic9C2×D411yescharacteristic
166#1009—92no—
167#1004—92no—
168C3×#135—31no—
169C2×C3×D8—91no—
170#949—92no—
171#932—92no—
172#919—91no—
173#919—91no—
174#914—92no—
175#914—92no—
176#912—91no—
177#912D911yescharacteristic, complemented
178#911—92no—
179#911—92no—
180D3×D8—31no—
181D3×D8—91no—
182D3×D8—91no—
183D3×D8—91no—
184D3×D8—181no—
185#904—91no—
186C2×D24—91no—
187#901—91no—
188#901—91no—
189C22×C3⋊D3—62no—
190C22×C3×D3—62no—
191C2×D32—62no—
192C32×D4C22×D312yes—
193#638—62no—
194C4⋊(C3⋊D3)—32no—
195C4×C3⋊D3—31no—
196C3×#67—22no—
197C3×#67—62no—
198C2×C3×Dic3—22no—
199C3×D12—32no—
200C3×C4×D3—31no—
201C3×C4×D3C22×D311yescharacteristic
202C3×Dic6C22×D312yes—
203#627—31no—
204#626—31no—
205#626—31no—
206#626—62no—
207#626—62no—
208#626—62no—
209#625—31no—
210#624—62no—
211D3×Dic3—62no—
212C22×D9—22no—
213C22×D9—62no—
214C22×D9—62no—
215C22×D9—122no—
216C32×C8C22×D311yescharacteristic
217C3⋊(C3⋊C8)—31no—
218C3×C3⋊C8—31no—
219C3×C3⋊C8C22×D311yescharacteristic
220Q8×C9—32no—
221D4×C9—22no—
222D4×C9—62no—
223D4×C9C22×D312yes—
224C2×C4×C9—31no—
225C2×C4×C9—62no—
226#611—22no—
227#611—31no—
228#611—31no—
229#611—62no—
230#611—122no—
231D36—31no—
232D36—32no—
233D36—62no—
234D36—62no—
235D36—62no—
236D36—62no—
237D36C22×D312yes—
238C4×D9—31no—
239C4×D9—61no—
240C4×D9—62no—
241C4×D9—62no—
242C4×D9C22×D311yescharacteristic
243Dic18—31no—
244C8×C9—21no—
245C8×C9—31no—
246C8×C9C22×D311yescharacteristic
247C9⋊C8—31no—
248C9⋊C8—61no—
249C9⋊C8C22×D311yescharacteristic
250#576—271noSylow
251C2×C3×C9—42no—
252C2×C3×C9C22×D411yescharacteristic
253C3⋊D9—21no—
254C3⋊D9—42no—
255D3×C9—21no—
256C3×D9—21no—
257C3×D9—42no—
258C3×#41—32no—
259C2×C3×D4—92no—
260C2×#67—182no—
261C2×#67—182no—
262#239—92no—
263#239—92no—
264#237—92no—
265#237—182no—
266#237C2×D912yescomplemented
267D3×D4—32no—
268D3×D4—92no—
269D3×D4—92no—
270D3×D4—92no—
271D3×D4—182no—
272D3×D4—182no—
273D3×D4—182no—
274D3×D4—182no—
275#235—91no—
276#235—91no—
277#235—92no—
278#235—182no—
279C2×D12—92no—
280C2×D12—182no—
281C3×Q16—31no—
282C3×SD16—32no—
283C3×D8—21no—
284C3×D8—92no—
285C3×D8—91no—
286C3×D8C2×D911yescharacteristic, complemented
287C3×D8D3211yescharacteristic
288C2×C3×C8—31no—
289C2×C3×C8—91no—
290#215—92no—
291#215—92no—
292#214—92no—
293#214C2×D912yescomplemented
294#213—32no—
295#213—92no—
296#213—92no—
297#213—92no—
298#213—182no—
299#208—91no—
300#208—91no—
301Dic12C2×D911yescharacteristic, complemented
302D24—31no—
303D24—92no—
304D24—91no—
305D24—91no—
306D24—181no—
307#204—92no—
308ΓL(1,𝔽25)—91no—
309ΓL(1,𝔽25)—91no—
310ΓL(1,𝔽25)—91no—
311ΓL(1,𝔽25)—91no—
312D3×C8—31no—
313D3×C8—91no—
314D3×C8—91no—
315D3×C8—181no—
316D3×C8C2×D911yescharacteristic, complemented
317C22×C32—22no—
318C2×C3⋊D3—31no—
319C2×C3⋊D3—62no—
320C2×C3⋊D3—242no—
321C2×C3×D3—31no—
322C2×C3×D3—62no—
323C2×C3×D3—242no—
324C2×C3×D3D3×D411yescharacteristic
325D32—242no—
326C32×C4C23×D311yescharacteristic
327C3⋊Dic3—31no—
328C3×Dic3—22no—
329C3×Dic3—31no—
330C3×Dic3D3×D411yescharacteristic
331C22×C9—22no—
332C22×C9—31no—
333C22×C9—42no—
334C2×D9—22no—
335C2×D9—31no—
336C2×D9—61no—
337C2×D9—62no—
338C2×D9—82no—
339C2×D9—122no—
340C2×D9—242no—
341C2×D9—242no—
342C2×D9—244no—
343C2×D9D3×D411yescharacteristic
344C4×C9—21no—
345C4×C9—31no—
346C4×C9—62no—
347C4×C9C23×D311yescharacteristic
348Dic9—31no—
349Dic9—61no—
350Dic9D3×D411yescharacteristic
351H(5,𝔽2)—272no—
352AGL(1,ℤ/8)—272no—
353AGL(1,ℤ/8)—272no—
354AGL(1,ℤ/8)—272no—
355#135—31no—
356#135—271no—
357#135—271no—
358C2×D8—91no—
359C2×D8—271no—
360C2×D8—271no—
361#131—271no—
362C3×C9#57611yescharacteristic, complemented, Sylow
363C23×C3—182no—
364C22×D3—62no—
365C22×D3—182no—
366C22×D3—182no—
367C22×D3—182no—
368C22×D3—182no—
369C22×D3—362no—
370C3×Q8—32no—
371C3×D4—22no—
372C3×D4—62no—
373C3×D4—92no—
374C3×D4—182no—
375C3×D4C22×D912yes—
376C3×D4C2×D3212yes—
377C2×C3×C4—31no—
378C2×C3×C4—62no—
379C2×C3×C4—91no—
380#67—22no—
381#67—62no—
382#67—91no—
383#67—91no—
384#67—91no—
385#67—91no—
386#67—182no—
387#67—182no—
388#67—182no—
389#67—182no—
390#67—182no—
391#67—362no—
392C2×Dic3—22no—
393C2×Dic3—182no—
394D12—32no—
395D12—91no—
396D12—91no—
397D12—92no—
398D12—92no—
399D12—182no—
400D12—182no—
401D12—182no—
402D12—182no—
403C4×D3—31no—
404C4×D3—91no—
405C4×D3—91no—
406C4×D3—181no—
407C4×D3—182no—
408C4×D3—182no—
409C4×D3—182no—
410C4×D3C22×D911yescharacteristic
411Dic6—91no—
412Dic6—91no—
413Dic6C22×D912yes—
414C3×C8—21no—
415C3×C8—31no—
416C3×C8—91no—
417C3×C8C22×D911yescharacteristic
418C3×C8C2×D3211yescharacteristic
419C3⋊C8—31no—
420C3⋊C8—91no—
421C3⋊C8—91no—
422C3⋊C8—181no—
423C3⋊C8C22×D911yescharacteristic
424C2×C32—42no—
425C2×C32C2×D3×D411yescharacteristic
426C3⋊D3—61no—
427C3⋊D3—122no—
428C3×D3—21no—
429C3×D3—61no—
430C3×D3—122no—
431C2×C9—21no—
432C2×C9—42no—
433C2×C9—61no—
434C2×C9—82no—
435C2×C9C2×D3×D411yescharacteristic
436D9—21no—
437D9—42no—
438D9—62no—
439D9—61no—
440D9—122no—
441D9—242no—
442#41—32no—
443#41—271no—
444#41—272no—
445#41—272no—
446#41—542no—
447C2×D4—92no—
448C2×D4—272no—
449C2×D4—272no—
450C2×D4—542no—
451C2×D4—542no—
452C2×D4—542no—
453Q16—31no—
454SD16—32no—
455SD16—272no—
456SD16—272no—
457D8—92no—
458D8—91no—
459D8—272no—
460D8—272no—
461D8—271no—
462D8D3×D911yescharacteristic, complemented
463M16—271no—
464M16—271no—
465M16—271no—
466C2×C8—31no—
467C2×C8—91no—
468C2×C8—271no—
469C22×C3—22no—
470C22×C3—22no—
471C22×C3—31no—
472C22×C3—42no—
473C22×C3—91no—
474C22×C3—182no—
475C22×C3—722no—
476C2×D3—31no—
477C2×D3—62no—
478C2×D3—91no—
479C2×D3—91no—
480C2×D3—181no—
481C2×D3—182no—
482C2×D3—182no—
483C2×D3—242no—
484C2×D3—362no—
485C2×D3—722no—
486C2×D3—722no—
487C2×D3—722no—
488C2×D3—722no—
489C2×D3—724no—
490C2×D3D4×D911yescharacteristic
491C3×C4—21no—
492C3×C4—31no—
493C3×C4—62no—
494C3×C4—91no—
495C3×C4C23×D911yescharacteristic
496C3×C4C22×D3211yescharacteristic
497Dic3—22no—
498Dic3—31no—
499Dic3—91no—
500Dic3—91no—
501Dic3—181no—
502Dic3D4×D911yescharacteristic
503C32#576911yescharacteristic
504C9—21no—
505C9#575711yescharacteristic, complemented
506C23—182no—
507C23—542no—
508C23—542no—
509Q8—32no—
510Q8—271no—
511D4—62no—
512D4—92no—
513D4—182no—
514D4—271no—
515D4—271no—
516D4—271no—
517D4—272no—
518D4—542no—
519D4—542no—
520D4—542no—
521D4—542no—
522D4C2×D3×D912yes—
523C2×C4—31no—
524C2×C4—62no—
525C2×C4—91no—
526C2×C4—271no—
527C2×C4—542no—
528C2×C4—542no—
529C8—31no—
530C8—91no—
531C8—271no—
532C8C2×D3×D911yescharacteristic
533C2×C3—21no—
534C2×C3—42no—
535C2×C3—42no—
536C2×C3—61no—
537C2×C3—82no—
538C2×C3—181no—
539C2×C3—362no—
540C2×C3C2×D4×D911yescharacteristic
541C2×C3D32×D411yescharacteristic
542D3—21no—
543D3—61no—
544D3—122no—
545D3—182no—
546D3—181no—
547D3—181no—
548D3—362no—
549D3—362no—
550D3—722no—
551C22—22no—
552C22—31no—
553C22—91no—
554C22—182no—
555C22—271no—
556C22—542no—
557C22—722no—
558C22—2162no—
559C22—2162no—
560C4—31no—
561C4—62no—
562C4—91no—
563C4—271no—
564C4C22×D3×D911yescharacteristic
565C3—21no—
566C3#1058987111yescharacteristic, complemented
567C3#1059473411yescharacteristic
568C2—42no—
569C2—61no—
570C2—181no—
571C2—362no—
572C2—541no—
573C2—1082no—
574C2D3×D4×D911yescharacteristic
subgroupG/H
centerC2D3×D4×D9
commutator, cosocleC3×C4×C9C24
FrattiniC3×C4C22×D32
FittingC3×C9×D8C22
socleC2×C32C2×D3×D4
hypercenterD8D3×D9
hypocenterC3×C9#576
LCSG ▷ C3×C4×C9 ▷ C2×C3×C9 ▷ C3×C9
UCSC1 ◁ C2 ◁ C4 ◁ D8

Complements

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

NH ≅ G/Ntype
1. #11722501C2semidirect
2. D8×C3⋊D9C2semidirect
3. C9×#912C2semidirect
4. C3×D8×D9C2semidirect
5. #11721187C2semidirect
6. #11721184C2semidirect
7. #11721167C2semidirect
8. #11720831C2semidirect
9. #11720830C2semidirect
10. #11720806C2semidirect
11. D3×D72C2semidirect
14. D4×C3⋊D9C22semidirect
15. C9×#237C22semidirect
16. C3×D4×D9C22semidirect
21. D3×D36C22semidirect
23. #88635C22semidirect
24. #88634C22semidirect
25. C3×C9×D8C22semidirect
26. #88544C22semidirect
27. #88525C22semidirect
28. C8×C3⋊D9C22semidirect
29. C9×#214C22semidirect
30. C3×#3673C22semidirect
31. C9×Dic12C22semidirect
32. D3×C8×C9C22semidirect
33. C3×D72C22semidirect
34. C3×C8×D9C22semidirect
35. #88426C22semidirect
36. #88425C22semidirect
37. #88420C22semidirect
38. #88417C22semidirect
39. #88414C22semidirect
40. #88413C22semidirect
61. D8×D9D3semidirect
127. D4×D9C2×D3semidirect
135. C9×D8C2×D3semidirect
141. #3673C2×D3semidirect
146. D72C2×D3semidirect
151. C8×D9C2×D3semidirect
177. #912D9semidirect
266. #237C2×D9semidirect
286. C3×D8C2×D9semidirect
293. #214C2×D9semidirect
301. Dic12C2×D9semidirect
316. D3×C8C2×D9semidirect
362. C3×C9#576semidirect
462. D8D3×D9semidirect
505. C9#5757semidirect
566. C3#10589871semidirect

Maximal subgroups

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

subgroupindex
#117225012
D8×C3⋊D92
C9×#9122
C3×D8×D92
#117211872
#117211842
#117211672
#117208312
#117208302
#117208062
D3×D722
#105947343
#105898713