Finite Group Databasebeta

← #13149903

Subgroups — #13149903

Order 1728.

874 conjugacy classes of subgroups.

The subgroup lattice has 564 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11722498C212yesmaximal, complemented
2D8×C3⋊D9C211yesmaximal, characteristic, complemented
3C9×#912C211yesmaximal, characteristic, complemented
4C3×#63474C211yesmaximal, characteristic, complemented
5#11721185C212yesmaximal, complemented
6#11721182C212yesmaximal, complemented
7#11721171C212yesmaximal, complemented
8#11720830C211yesmaximal, characteristic, complemented
9#11720829C211yesmaximal, characteristic, complemented
10#11720812C211yesmaximal, characteristic, complemented
11D3×C8×D9C211yesmaximal, characteristic, complemented
12#10594730—31nomaximal
13D9×#135—31nomaximal
14D4×C3⋊D9C2212yes—
15C9×#237C2212yescomplemented
16C3×#3699C2212yescomplemented
17#88668—22no—
18#88662—22no—
19C2×(C3⋊D9)⋊C4—22no—
20#88658—22no—
21C4×D3×D9C2211yescharacteristic
22#88642C2212yescomplemented
23(C3⋊D9)⋊Q8C2212yes—
24#88635C2212yescomplemented
25C3×C9×D8C2211yescharacteristic, complemented
26#88544C2212yescomplemented
27#88525C2211yescharacteristic, complemented
28C8×C3⋊D9C2211yescharacteristic
29C9×#214C2212yescomplemented
30C3×#3672C2212yescomplemented
31C9×Dic12C2211yescharacteristic, complemented
32D3×C8×C9C2211yescharacteristic, complemented
33C3×C8×D9C2211yescharacteristic, complemented
34C3×Dic36C2211yescharacteristic, complemented
35#88427C2212yescomplemented
36#88426C2212yescomplemented
37D3×C9⋊C8C2211yescharacteristic, complemented
38#88416C2212yescomplemented
39(C3⋊D9)⋊C8C2211yescharacteristic
40D9×C3⋊C8C2211yescharacteristic, complemented
41#64312—32no—
42D8×C3⋊D3—31no—
43C3×#912—31no—
44C3×#912D311yescharacteristic
45#63935—32no—
46#63935—32no—
47#63930—32no—
48#63808—31no—
49#63808—31no—
50#63802—31no—
51D32×C8—31no—
52#41×D9—32no—
53C9×#135—31no—
54#63513—32no—
55#63481—31no—
56Q16×D9—31no—
57#63478—32no—
58SD16×D9—32no—
59#63474D311yescharacteristic, complemented
60D8×D9—31no—
61D8×D9—61no—
62#63467—31no—
63C2×C8×D9—31no—
64C22×C3⋊D9—22no—
65C2×D3×D9D411yescharacteristic
66C3×D4×C9C2312yes—
67#6233—22no—
68C4⋊(C3⋊D9)C2312yes—
69C4×C3⋊D9C2311yescharacteristic
70C9×#67—22no—
71C3×#611—22no—
72C2×C9×Dic3—22no—
73C2×C3×Dic9—22no—
74C4×D3×C9C2311yescharacteristic
75C3×C4×D9C2311yescharacteristic
76C9×Dic6C2312yes—
77C3×Dic18C2312yes—
78#6195—22no—
79D3×Dic9D411yescharacteristic
80#6192—22no—
81(C3⋊D9)⋊C4—22no—
82(C3⋊D9)⋊C4—22no—
83(C3⋊D9)⋊C4—22no—
84(C3⋊D9)⋊C4D411yescharacteristic
85Dic3×D9D411yescharacteristic
86#6189—22no—
87C3×C8×C9C2311yescharacteristic
88C3⋊(C9⋊C8)C2311yescharacteristic
89C9×C3⋊C8C2311yescharacteristic
90C3×C9⋊C8C2311yescharacteristic
91D3×#135—91no—
92D3×#135—91no—
93D4×C3⋊D3—32no—
94C3×#237—32no—
95C3×#237C2×D312yes—
96#3811—62no—
97C2×#624—62no—
98#3804—62no—
99#3804—62no—
100C4×D32—31no—
101#3799—32no—
102#3799—32no—
103#3797—32no—
104C32×D8C2×D311yescharacteristic
105#3753—32no—
106#3745—31no—
107C8×C3⋊D3—31no—
108C3×#214—32no—
109C3×#214C2×D312yes—
110C3×Dic12—31no—
111C3×Dic12C2×D311yescharacteristic
112C3×D3×C8—31no—
113C3×D3×C8C2×D311yescharacteristic
114#3718—32no—
115#3717—32no—
116#3717—32no—
117#3710—31no—
118D3×C3⋊C8—31no—
119D3×C3⋊C8—31no—
120C9×#41—32no—
121#3701—32no—
122Q8×D9—32no—
123#3699—62no—
124#3699C2×D312yescomplemented
125D4×D9—32no—
126D4×D9—62no—
127D4×D9—62no—
128#3697—32no—
129#3697—62no—
130C2×C4×D9—31no—
131C2×C4×D9—62no—
132C9×Q16—31no—
133C9×SD16—32no—
134C9×D8—21no—
135C9×D8C2×D311yescharacteristic, complemented
136C2×C8×C9—31no—
137#3675—32no—
138#3674—32no—
139#3673—32no—
140#3673—62no—
141#3672C2×D312yescomplemented
142C2×C9⋊C8—31no—
143D72—31no—
144D72—61no—
145#3664—32no—
146C8×D9—31no—
147C8×D9—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—44no—
155C2×C3⋊D9C2×D411yescharacteristic
156C2×D3×C9C2×D411yescharacteristic
157C2×C3×D9C2×D411yescharacteristic
158D3×D9—22no—
159C3×C4×C9C2411yescharacteristic
160C3⋊Dic9C2×D411yescharacteristic
161C9×Dic3—22no—
162C9×Dic3C2×D411yescharacteristic
163C3×Dic9—22no—
164C3×Dic9C2×D411yescharacteristic
165D3×#41—92no—
166D3×#41—92no—
167C3×#135—31no—
168C3×#135—91no—
169#950—92no—
170#950—92no—
171#919—91no—
172#919—91no—
173D3×Q16—91no—
174D3×Q16—91no—
175#916—92no—
176#916—92no—
177D3×SD16—92no—
178D3×SD16—92no—
179#912—31no—
180#912D911yescharacteristic, complemented
181D3×D8—91no—
182D3×D8—91no—
183D3×D8—181no—
184#904—91no—
185#904—91no—
186C2×D3×C8—91no—
187C2×D3×C8—91no—
188C22×C3⋊D3—62no—
189C2×D32—31no—
190C32×D4C22×D312yes—
191#638—62no—
192C4⋊(C3⋊D3)—32no—
193C4×C3⋊D3—31no—
194C3×#67—22no—
195C3×#67—62no—
196C2×C3×Dic3—22no—
197C2×C3×Dic3—62no—
198C3×C4×D3—31no—
199C3×C4×D3C22×D311yescharacteristic
200C3×Dic6—32no—
201C3×Dic6C22×D312yes—
202#627—62no—
203#626—62no—
204#626—62no—
205#624—31no—
206#624—62no—
207#624—62no—
208#624—62no—
209D3×Dic3—31no—
210D3×Dic3—31no—
211C22×D9—31no—
212C22×D9—62no—
213C22×D9—122no—
214C32×C8C22×D311yescharacteristic
215C3⋊(C3⋊C8)—31no—
216C3×C3⋊C8—31no—
217C3×C3⋊C8C22×D311yescharacteristic
218Q8×C9—32no—
219D4×C9—22no—
220D4×C9—62no—
221D4×C9C22×D312yes—
222C2×C4×C9—31no—
223C2×C4×C9—62no—
224#611—22no—
225#611—62no—
226#611—62no—
227#611—122no—
228C2×Dic9—22no—
229C2×Dic9—31no—
230D36—32no—
231D36—62no—
232D36—62no—
233C4×D9—31no—
234C4×D9—31no—
235C4×D9—31no—
236C4×D9—61no—
237C4×D9—62no—
238C4×D9—62no—
239C4×D9—62no—
240C4×D9C22×D311yescharacteristic
241Dic18—62no—
242Dic18C22×D312yes—
243C8×C9—21no—
244C8×C9—31no—
245C8×C9C22×D311yescharacteristic
246C9⋊C8—31no—
247C9⋊C8—61no—
248C9⋊C8C22×D311yescharacteristic
249C2×#135—271noSylow
250C2×C3×C9—42no—
251C2×C3×C9C22×D411yescharacteristic
252C3⋊D9—42no—
253C3⋊D9#13512yescomplemented
254D3×C9—21no—
255C3×D9—21no—
256C3×#41—32no—
257C3×#41—92no—
258#239—92no—
259#239—92no—
260D3×Q8—92no—
261D3×Q8—92no—
262#237—32no—
263#237—182no—
264#237—182no—
265#237C2×D912yescomplemented
266D3×D4—92no—
267D3×D4—92no—
268D3×D4—182no—
269D3×D4—182no—
270D3×D4—182no—
271#235—92no—
272#235—92no—
273#235—182no—
274#235—182no—
275C2×C4×D3—91no—
276C2×C4×D3—91no—
277C2×C4×D3—182no—
278C2×C4×D3—182no—
279C3×Q16—31no—
280C3×Q16—91no—
281C3×SD16—32no—
282C3×SD16—92no—
283C3×D8—21no—
284C3×D8C2×D911yescharacteristic, complemented
285C3×D8D3211yescharacteristic
286C2×C3×C8—31no—
287C2×C3×C8—91no—
288#216—92no—
289#216—92no—
290#215—92no—
291#215—92no—
292#214—32no—
293#214C2×D912yescomplemented
294#213—92no—
295#213—92no—
296#213—182no—
297C2×C3⋊C8—91no—
298C2×C3⋊C8—91no—
299Dic12—31no—
300Dic12C2×D911yescharacteristic, complemented
301D24—91no—
302D24—91no—
303D24—181no—
304#204—92no—
305#204—92no—
306D3×C8—31no—
307D3×C8—91no—
308D3×C8—91no—
309D3×C8—91no—
310D3×C8—91no—
311D3×C8—91no—
312D3×C8—91no—
313D3×C8—181no—
314D3×C8C2×D911yescharacteristic, complemented
315C22×C32—22no—
316C2×C3⋊D3—31no—
317C2×C3⋊D3—62no—
318C2×C3⋊D3—124no—
319C2×C3×D3—31no—
320C2×C3×D3D3×D411yescharacteristic
321D32—62no—
322C32×C4C23×D311yescharacteristic
323C3⋊Dic3—31no—
324C3×Dic3—22no—
325C3×Dic3—31no—
326C3×Dic3—62no—
327C3×Dic3D3×D411yescharacteristic
328C22×C9—22no—
329C22×C9—31no—
330C22×C9—42no—
331C2×D9—31no—
332C2×D9—61no—
333C2×D9—62no—
334C2×D9—62no—
335C2×D9—122no—
336C2×D9—124no—
337C2×D9—244no—
338C2×D9D3×D411yescharacteristic
339C4×C9—21no—
340C4×C9—31no—
341C4×C9—62no—
342C4×C9C23×D311yescharacteristic
343Dic9—22no—
344Dic9—31no—
345Dic9—61no—
346Dic9D3×D411yescharacteristic
347C2×#41—272no—
348#135—31no—
349#135—91no—
350#135—271no—
351#135—271no—
352#135—272no—
353#135—272no—
354C2×Q16—271no—
355C2×SD16—272no—
356C2×D8—271no—
357C22×C8—271no—
358C3×C9C2×#13511yescharacteristic, complemented, Sylow
359C22×D3—91no—
360C22×D3—91no—
361C22×D3—182no—
362C22×D3—182no—
363C22×D3—362no—
364C3×Q8—32no—
365C3×Q8—92no—
366C3×D4—22no—
367C3×D4—62no—
368C3×D4—182no—
369C3×D4C22×D912yes—
370C3×D4C2×D3212yes—
371C2×C3×C4—31no—
372C2×C3×C4—62no—
373C2×C3×C4—91no—
374C2×C3×C4—182no—
375#67—22no—
376#67—62no—
377#67—182no—
378#67—182no—
379#67—182no—
380#67—182no—
381#67—362no—
382C2×Dic3—22no—
383C2×Dic3—62no—
384C2×Dic3—91no—
385C2×Dic3—91no—
386D12—92no—
387D12—92no—
388D12—182no—
389D12—182no—
390D12—182no—
391C4×D3—31no—
392C4×D3—91no—
393C4×D3—91no—
394C4×D3—91no—
395C4×D3—91no—
396C4×D3—91no—
397C4×D3—91no—
398C4×D3—181no—
399C4×D3—182no—
400C4×D3—182no—
401C4×D3—182no—
402C4×D3—182no—
403C4×D3—182no—
404C4×D3—182no—
405C4×D3C22×D911yescharacteristic
406Dic6—32no—
407Dic6—182no—
408Dic6—182no—
409Dic6C22×D912yes—
410C3×C8—21no—
411C3×C8—31no—
412C3×C8—91no—
413C3×C8C22×D911yescharacteristic
414C3×C8C2×D3211yescharacteristic
415C3⋊C8—31no—
416C3⋊C8—91no—
417C3⋊C8—91no—
418C3⋊C8—181no—
419C3⋊C8C22×D911yescharacteristic
420C2×C32—42no—
421C2×C32C2×D3×D411yescharacteristic
422C3⋊D3—32no—
423C3⋊D3—122no—
424C3×D3—21no—
425C3×D3—61no—
426C2×C9—21no—
427C2×C9—42no—
428C2×C9—61no—
429C2×C9—82no—
430C2×C9C2×D3×D411yescharacteristic
431D9—21no—
432D9—32no—
433D9—62no—
434D9—122no—
435D9—242no—
436#41—32no—
437#41—92no—
438#41—272no—
439#41—272no—
440#41—542no—
441#41—542no—
442C2×Q8—272no—
443C2×D4—272no—
444C2×D4—542no—
445C22×C4—271no—
446C22×C4—542no—
447Q16—31no—
448Q16—91no—
449Q16—272no—
450SD16—32no—
451SD16—92no—
452SD16—272no—
453SD16—272no—
454D8—272no—
455D8—271no—
456D8D3×D911yescharacteristic, complemented
457C2×C8—31no—
458C2×C8—91no—
459C2×C8—271no—
460C2×C8—271no—
461C2×C8—271no—
462C2×C8—271no—
463C22×C3—22no—
464C22×C3—22no—
465C22×C3—31no—
466C22×C3—42no—
467C22×C3—91no—
468C2×D3—31no—
469C2×D3—91no—
470C2×D3—91no—
471C2×D3—181no—
472C2×D3—182no—
473C2×D3—182no—
474C2×D3—182no—
475C2×D3—182no—
476C2×D3—362no—
477C2×D3—364no—
478C2×D3—364no—
479C2×D3—724no—
480C2×D3D4×D911yescharacteristic
481C3×C4—21no—
482C3×C4—31no—
483C3×C4—62no—
484C3×C4—91no—
485C3×C4—182no—
486C3×C4C23×D911yescharacteristic
487C3×C4C22×D3211yescharacteristic
488Dic3—22no—
489Dic3—31no—
490Dic3—62no—
491Dic3—91no—
492Dic3—91no—
493Dic3—181no—
494Dic3D4×D911yescharacteristic
495C32D3×#13511yescharacteristic
496C9—21no—
497C9D3×#13511yescharacteristic, complemented
498C23—271no—
499C23—542no—
500Q8—32no—
501Q8—92no—
502Q8—542no—
503D4—62no—
504D4—182no—
505D4—272no—
506D4—542no—
507D4—542no—
508D4—542no—
509D4C2×D3×D912yes—
510C2×C4—31no—
511C2×C4—62no—
512C2×C4—91no—
513C2×C4—182no—
514C2×C4—271no—
515C2×C4—271no—
516C2×C4—271no—
517C2×C4—271no—
518C2×C4—542no—
519C2×C4—542no—
520C2×C4—542no—
521C8—31no—
522C8—91no—
523C8—271no—
524C8C2×D3×D911yescharacteristic
525C2×C3—21no—
526C2×C3—42no—
527C2×C3—42no—
528C2×C3—61no—
529C2×C3—82no—
530C2×C3—181no—
531C2×C3C2×D4×D911yescharacteristic
532C2×C3D32×D411yescharacteristic
533D3—21no—
534D3—61no—
535D3—92no—
536D3—92no—
537D3—182no—
538D3—362no—
539D3—362no—
540D3—722no—
541C22—22no—
542C22—31no—
543C22—91no—
544C22—271no—
545C22—542no—
546C22—542no—
547C22—1084no—
548C4—31no—
549C4—62no—
550C4—91no—
551C4—182no—
552C4—271no—
553C4C22×D3×D911yescharacteristic
554C3—21no—
555C3D9×#13511yescharacteristic, complemented
556C3#1059473011yescharacteristic
557C2—42no—
558C2—61no—
559C2—181no—
560C2—272no—
561C2—1082no—
562C2D3×D4×D911yescharacteristic
subgroupG/H
centerC2D3×D4×D9
commutator, cosocleC3×C4×C9C24
FrattiniC3×C4C22×D32
FittingC3×C9×D8C22
socleC2×C32C2×D3×D4
hypercenterD8D3×D9
hypocenterC3×C9C2×#135
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. #11722498C2semidirect
2. D8×C3⋊D9C2semidirect
3. C9×#912C2semidirect
4. C3×#63474C2semidirect
5. #11721185C2semidirect
6. #11721182C2semidirect
7. #11721171C2semidirect
8. #11720830C2semidirect
9. #11720829C2semidirect
10. #11720812C2semidirect
11. D3×C8×D9C2semidirect
15. C9×#237C22semidirect
16. C3×#3699C22semidirect
22. #88642C22semidirect
24. #88635C22semidirect
25. C3×C9×D8C22semidirect
26. #88544C22semidirect
27. #88525C22semidirect
29. C9×#214C22semidirect
30. C3×#3672C22semidirect
31. C9×Dic12C22semidirect
32. D3×C8×C9C22semidirect
33. C3×C8×D9C22semidirect
34. C3×Dic36C22semidirect
35. #88427C22semidirect
36. #88426C22semidirect
37. D3×C9⋊C8C22semidirect
38. #88416C22semidirect
40. D9×C3⋊C8C22semidirect
59. #63474D3semidirect
124. #3699C2×D3semidirect
135. C9×D8C2×D3semidirect
141. #3672C2×D3semidirect
150. C8×D9C2×D3semidirect
151. Dic36C2×D3semidirect
180. #912D9semidirect
253. C3⋊D9#135semidirect
265. #237C2×D9semidirect
284. C3×D8C2×D9semidirect
293. #214C2×D9semidirect
300. Dic12C2×D9semidirect
314. D3×C8C2×D9semidirect
358. C3×C9C2×#135semidirect
456. D8D3×D9semidirect
497. C9D3×#135semidirect
555. C3D9×#135semidirect

Maximal subgroups

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

subgroupindex
#117224982
D8×C3⋊D92
C9×#9122
C3×#634742
#117211852
#117211822
#117211712
#117208302
#117208292
#117208122
D3×C8×D92
#105947303
D9×#1353