Finite Group Databasebeta

← #13149909

Subgroups — #13149909

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#11722500C212yesmaximal, complemented
2D8×C3⋊D9C211yesmaximal, characteristic, complemented
3D3×C9×D8C211yesmaximal, characteristic, complemented
4C3×#63474C211yesmaximal, characteristic, complemented
5#11721179C212yesmaximal, complemented
6#11721173C212yesmaximal, complemented
7#11721170C212yesmaximal, complemented
8#11720824C211yesmaximal, characteristic, complemented
9D9×D24C211yesmaximal, characteristic, complemented
10#11720812C211yesmaximal, characteristic, complemented
11#11720806C211yesmaximal, characteristic, complemented
12#10594734—31nomaximal
13#10589859—31nomaximal
14D4×C3⋊D9C2212yescomplemented
15D3×D4×C9C2212yescomplemented
16C3×#3699C2212yescomplemented
17#88668—22no—
18D3×#611—22no—
19C2×#6195—22no—
20#88662—22no—
21D9×D12C2212yescomplemented
22#88642C2212yescomplemented
23#88640C2211yescharacteristic
24#88639C2212yescomplemented
25C3×C9×D8C2211yescharacteristic, complemented
26#88544C2212yescomplemented
27#88525C2211yescharacteristic, complemented
28C8×C3⋊D9C2211yescharacteristic, complemented
29C9×#213C2212yescomplemented
30C3×#3672C2212yescomplemented
31C9×D24C2211yescharacteristic, complemented
32D3×C8×C9C2211yescharacteristic, complemented
33C3×C8×D9C2211yescharacteristic, complemented
34C3×Dic36C2211yescharacteristic, complemented
35#88423C2212yescomplemented
36#88422C2212yescomplemented
37#88420C2211yescharacteristic, complemented
38#88416C2212yescomplemented
39#88414C2211yescharacteristic, complemented
40#88413C2211yescharacteristic, complemented
41#64314—32no—
42D8×C3⋊D3—31no—
43C3×#912—31no—
44C3×D3×D8D311yescharacteristic
45#63937—32no—
46#63934—32no—
47#63925—32no—
48#63810—31no—
49#63808—31no—
50#63803—31no—
51D3×D24—31no—
52#63710—32no—
53C2×C9×D8—31no—
54#63495—32no—
55#63474—31no—
56#63474D311yescharacteristic, complemented
57#63473—32no—
58#63473—32no—
59D8×D9—31no—
60D8×D9—31no—
61D8×D9—61no—
62#63467—31no—
63#63464—31no—
64C22×C3⋊D9—22no—
65C22×D3×C9—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×C3×Dic9—22no—
74C9×D12C2312yes—
75C4×D3×C9C2311yescharacteristic
76C3×C4×D9C2311yescharacteristic
77C3×Dic18C2312yes—
78#6195—22no—
79#6195—22no—
80#6195—22no—
81#6195D411yescharacteristic
82#6194D411yescharacteristic
83D3×Dic9—22no—
84#6192D411yescharacteristic
85(C3⋊D9)⋊C4—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×#237—32no—
95C3×D3×D4C2×D312yes—
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×#214—32no—
109C3×#213C2×D312yes—
110C3×Dic12—31no—
111C3×D24C2×D311yescharacteristic
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—
120C2×D4×C9—32no—
121C2×#611—62no—
122C2×#611—62no—
123#3699—32no—
124#3699—62no—
125#3699C2×D312yescomplemented
126D4×D9—32no—
127D4×D9—32no—
128D4×D9—62no—
129D4×D9—62no—
130#3697—31no—
131#3697—32no—
132C9×D8—21no—
133C9×D8—32no—
134C9×D8—31no—
135C9×D8C2×D311yescharacteristic, complemented
136C2×C8×C9—31no—
137#3673—32no—
138#3673—32no—
139#3673—62no—
140#3672—32no—
141#3672C2×D312yescomplemented
142#3667—31no—
143D72—31no—
144D72—61no—
145#3664—32no—
146#3663—31no—
147#3663—31no—
148C8×D9—31no—
149C8×D9—61no—
150C8×D9C2×D311yescharacteristic, complemented
151Dic36C2×D311yescharacteristic, complemented
152C22×C3×C9—22no—
153C2×C3⋊D9—22no—
154C2×C3⋊D9—82no—
155C2×C3⋊D9C2×D411yescharacteristic
156C2×D3×C9—22no—
157C2×D3×C9—82no—
158C2×D3×C9C2×D411yescharacteristic
159C2×C3×D9C2×D411yescharacteristic
160D3×D9—82no—
161C3×C4×C9C2411yescharacteristic
162C3⋊Dic9C2×D411yescharacteristic
163C9×Dic3C2×D411yescharacteristic
164C3×Dic9—22no—
165C3×Dic9C2×D411yescharacteristic
166#1009—92no—
167#1004—92no—
168C3×#135—91no—
169C2×C3×D8—31no—
170#949—92no—
171#932—92no—
172#919—91no—
173#919—91no—
174#914—92no—
175#914—92no—
176#912—31no—
177#912—91no—
178#911—92no—
179#911—92no—
180D3×D8—91no—
181D3×D8—91no—
182D3×D8—91no—
183D3×D8—181no—
184D3×D8D911yescharacteristic, complemented
185#904—91no—
186C2×D24—91no—
187#901—91no—
188#901—91no—
189C22×C3⋊D3—62no—
190C22×C3×D3—22no—
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—62no—
199C3×D12C22×D312yes—
200C3×C4×D3—31no—
201C3×C4×D3C22×D311yescharacteristic
202C3×Dic6—32no—
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—
212C23×C9—62no—
213C22×D9—62no—
214C22×D9—62no—
215C22×D9—122no—
216C32×C8C22×D311yescharacteristic
217C3⋊(C3⋊C8)—31no—
218C3×C3⋊C8—31no—
219C3×C3⋊C8C22×D311yescharacteristic
220D4×C9—22no—
221D4×C9—32no—
222D4×C9—62no—
223D4×C9C22×D312yes—
224C2×C4×C9—31no—
225#611—22no—
226#611—31no—
227#611—31no—
228#611—62no—
229#611—62no—
230#611—62no—
231#611—62no—
232#611—122no—
233C2×Dic9—22no—
234C2×Dic9—62no—
235D36—31no—
236D36—32no—
237D36—62no—
238C4×D9—31no—
239C4×D9—61no—
240C4×D9—62no—
241C4×D9C22×D311yescharacteristic
242Dic18—31no—
243Dic18C22×D312yes—
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—
256D3×C9—42no—
257C3×D9—21no—
258C3×#41—92no—
259C2×C3×D4—32no—
260C2×#67—182no—
261C2×#67—182no—
262#239—92no—
263#239—92no—
264#237—32no—
265#237—92no—
266#237—182no—
267D3×D4—92no—
268D3×D4—92no—
269D3×D4—92no—
270D3×D4—182no—
271D3×D4—182no—
272D3×D4—182no—
273D3×D4—182no—
274D3×D4C2×D912yescomplemented
275#235—92no—
276#235—91no—
277#235—91no—
278#235—182no—
279C2×D12—92no—
280C2×D12—182no—
281C3×Q16—91no—
282C3×SD16—92no—
283C3×D8—21no—
284C3×D8—32no—
285C3×D8—31no—
286C3×D8C2×D911yescharacteristic, complemented
287C3×D8D3211yescharacteristic
288C2×C3×C8—31no—
289C2×C3×C8—91no—
290#215—92no—
291#215—92no—
292#214—32no—
293#214—92no—
294#213—92no—
295#213—92no—
296#213—92no—
297#213—182no—
298#213C2×D912yescomplemented
299#208—91no—
300#208—91no—
301Dic12—31no—
302D24—92no—
303D24—91no—
304D24—91no—
305D24—181no—
306D24C2×D911yescharacteristic, complemented
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—22no—
322C2×C3×D3—31no—
323C2×C3×D3—82no—
324C2×C3×D3D3×D411yescharacteristic
325D32—242no—
326C32×C4C23×D311yescharacteristic
327C3⋊Dic3—31no—
328C3×Dic3—31no—
329C3×Dic3—62no—
330C3×Dic3D3×D411yescharacteristic
331C22×C9—22no—
332C22×C9—31no—
333C22×C9—42no—
334C22×C9—62no—
335C22×C9—242no—
336C2×D9—31no—
337C2×D9—61no—
338C2×D9—62no—
339C2×D9—122no—
340C2×D9—242no—
341C2×D9—242no—
342C2×D9—244no—
343C2×D9D3×D411yescharacteristic
344C4×C9—21no—
345C4×C9—31no—
346C4×C9C23×D311yescharacteristic
347Dic9—22no—
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—91no—
356#135—271no—
357#135—271no—
358C2×D8—31no—
359C2×D8—271no—
360C2×D8—271no—
361#131—271no—
362C3×C9#57611yescharacteristic, complemented, Sylow
363C23×C3—62no—
364C22×D3—22no—
365C22×D3—182no—
366C22×D3—182no—
367C22×D3—182no—
368C22×D3—182no—
369C22×D3—362no—
370C3×Q8—92no—
371C3×D4—22no—
372C3×D4—32no—
373C3×D4—62no—
374C3×D4—182no—
375C3×D4C22×D912yes—
376C3×D4C2×D3212yes—
377C2×C3×C4—31no—
378C2×C3×C4—91no—
379C2×C3×C4—182no—
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—62no—
393C2×Dic3—182no—
394D12—91no—
395D12—91no—
396D12—92no—
397D12—92no—
398D12—182no—
399D12—182no—
400D12—182no—
401D12—182no—
402D12C22×D912yes—
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—32no—
412Dic6—91no—
413Dic6—91no—
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—42no—
430C3×D3—61no—
431C2×C9—21no—
432C2×C9—42no—
433C2×C9—61no—
434C2×C9—82no—
435C2×C9—122no—
436C2×C9C2×D3×D411yescharacteristic
437D9—21no—
438D9—62no—
439D9—61no—
440D9—122no—
441D9—242no—
442#41—92no—
443#41—271no—
444#41—272no—
445#41—272no—
446#41—542no—
447C2×D4—32no—
448C2×D4—272no—
449C2×D4—272no—
450C2×D4—542no—
451C2×D4—542no—
452C2×D4—542no—
453Q16—91no—
454SD16—92no—
455SD16—272no—
456SD16—272no—
457D8—32no—
458D8—31no—
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—62no—
474C22×C3—91no—
475C22×C3—242no—
476C2×D3—22no—
477C2×D3—31no—
478C2×D3—82no—
479C2×D3—91no—
480C2×D3—91no—
481C2×D3—181no—
482C2×D3—182no—
483C2×D3—182no—
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—91no—
494C3×C4—182no—
495C3×C4C23×D911yescharacteristic
496C3×C4C22×D3211yescharacteristic
497Dic3—31no—
498Dic3—62no—
499Dic3—91no—
500Dic3—91no—
501Dic3—181no—
502Dic3D4×D911yescharacteristic
503C32#575711yescharacteristic
504C9—21no—
505C9#576911yescharacteristic, complemented
506C23—62no—
507C23—542no—
508C23—542no—
509Q8—92no—
510Q8—271no—
511D4—32no—
512D4—62no—
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—91no—
525C2×C4—182no—
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—122no—
539C2×C3—181no—
540C2×C3C2×D4×D911yescharacteristic
541C2×C3D32×D411yescharacteristic
542D3—21no—
543D3—42no—
544D3—61no—
545D3—182no—
546D3—181no—
547D3—181no—
548D3—362no—
549D3—362no—
550D3—722no—
551C22—22no—
552C22—31no—
553C22—62no—
554C22—91no—
555C22—242no—
556C22—271no—
557C22—542no—
558C22—2162no—
559C22—2162no—
560C4—31no—
561C4—91no—
562C4—182no—
563C4—271no—
564C4C22×D3×D911yescharacteristic
565C3—21no—
566C3#1058985911yescharacteristic, complemented
567C3#1059473411yescharacteristic
568C2—42no—
569C2—61no—
570C2—122no—
571C2—181no—
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. #11722500C2semidirect
2. D8×C3⋊D9C2semidirect
3. D3×C9×D8C2semidirect
4. C3×#63474C2semidirect
5. #11721179C2semidirect
6. #11721173C2semidirect
7. #11721170C2semidirect
8. #11720824C2semidirect
9. D9×D24C2semidirect
10. #11720812C2semidirect
11. #11720806C2semidirect
14. D4×C3⋊D9C22semidirect
15. D3×D4×C9C22semidirect
16. C3×#3699C22semidirect
21. D9×D12C22semidirect
22. #88642C22semidirect
24. #88639C22semidirect
25. C3×C9×D8C22semidirect
26. #88544C22semidirect
27. #88525C22semidirect
28. C8×C3⋊D9C22semidirect
29. C9×#213C22semidirect
30. C3×#3672C22semidirect
31. C9×D24C22semidirect
32. D3×C8×C9C22semidirect
33. C3×C8×D9C22semidirect
34. C3×Dic36C22semidirect
35. #88423C22semidirect
36. #88422C22semidirect
37. #88420C22semidirect
38. #88416C22semidirect
39. #88414C22semidirect
40. #88413C22semidirect
56. #63474D3semidirect
125. #3699C2×D3semidirect
135. C9×D8C2×D3semidirect
141. #3672C2×D3semidirect
150. C8×D9C2×D3semidirect
151. Dic36C2×D3semidirect
184. D3×D8D9semidirect
274. D3×D4C2×D9semidirect
286. C3×D8C2×D9semidirect
298. #213C2×D9semidirect
306. D24C2×D9semidirect
316. D3×C8C2×D9semidirect
362. C3×C9#576semidirect
462. D8D3×D9semidirect
505. C9#5769semidirect
566. C3#10589859semidirect

Maximal subgroups

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

subgroupindex
#117225002
D8×C3⋊D92
D3×C9×D82
C3×#634742
#117211792
#117211732
#117211702
#117208242
D9×D242
#117208122
#117208062
#105947343
#105898593