Finite Group Databasebeta

← D9×#877 #13147983

Subgroups — D9×#877 #13147983

Order 1728.

1066 conjugacy classes of subgroups.

The subgroup lattice has 520 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×Dic3×D9C211yesmaximal, characteristic
2#11721723C211yesmaximal, characteristic, complemented
3C3×C22⋊C4×D9C211yesmaximal, characteristic
4C9×#877C211yesmaximal, characteristic, complemented
5D9×#217C211yesmaximal, characteristic
6#11721248C211yesmaximal, characteristic, complemented
7#11721244C211yesmaximal, characteristic, complemented
8#11720975C212yesmaximal, complemented
9#11720974C212yesmaximal, complemented
10D9×Dic3⋊C4C212yesmaximal, complemented
11C4×Dic3×D9C212yesmaximal, complemented
12D3×#877—31nomaximal
13D9×#117—31nomaximal
14C23×C3×D9C2211yescharacteristic
15C22×C3⋊Dic9C2211yescharacteristic
16C22×C9×Dic3C2211yescharacteristic
17C2×C3×C4×D9C2212yes—
18C2×Dic3×D9C418yescomplemented
19C2×Dic3×D9C2212yes—
20C2×Dic3×D9C2211yescharacteristic
21C2×Dic3×D9C2211yescharacteristic
22C2×Dic3×D9C2211yescharacteristic
23C2×Dic3×D9C2211yescharacteristic
24C3×C9×C22⋊C4C2211yescharacteristic
25#88551C2211yescharacteristic
26#88534C2212yescomplemented
27C4×C3⋊Dic9C2212yescomplemented
28C9×#217C2211yescharacteristic
29C3×#3676C2211yescharacteristic
30C3×(C2×D9)⋊C4C2212yes—
31C9×Dic3⋊C4C2212yescomplemented
32C4×C9×Dic3C2212yescomplemented
33(C2×D9)⋊Dic3C2212yes—
34Dic3⋊Dic9C2212yescomplemented
35#88442C2212yescomplemented
36Dic3×Dic9C2212yescomplemented
37Dic3×Dic9C2212yescomplemented
38C22×D3×Dic3—31no—
39#64086—31no—
40C3×D3×C22⋊C4—31no—
41C3×#877D311yescharacteristic
42D3×#217—31no—
43#63957—31no—
44#63955—31no—
45#63878—32no—
46#63877—32no—
47D3×Dic3⋊C4—32no—
48C4×D3×Dic3—32no—
49C22×C4×D9—31no—
50C9×#117—31no—
51#63488—31no—
52#63454—32no—
53C4⋊C4×D9—32no—
54C22⋊C4×D9—31no—
55C22⋊C4×D9D311yescharacteristic
56#63439—31no—
57#63439—31no—
58#63439—61no—
59#63434—32no—
60C42×D9—32no—
61C23×C3×C9C2311yescharacteristic
62C22×C3×D9—24no—
63C22×C3×D9C2×C414yes—
64C22×C3×D9C2311yescharacteristic
65C22×C3×D9C2311yescharacteristic
66C2×C3×C4×C9C2312yes—
67C2×C3⋊Dic9C2×C414yescomplemented
68C2×C3⋊Dic9C2312yes—
69C2×C3⋊Dic9C2311yescharacteristic
70C2×C3⋊Dic9C2311yescharacteristic
71C2×C9×Dic3C2×C414yescomplemented
72C2×C9×Dic3C2312yes—
73C2×C9×Dic3C2311yescharacteristic
74C2×C9×Dic3C2311yescharacteristic
75C2×C3×Dic9C2312yes—
76C3×C4×D9—24no—
77Dic3×D9—24no—
78Dic3×D9C2×C414yes—
79Dic3×D9C2×C414yes—
80Dic3×D9C2×C414yescomplemented
81Dic3×D9C2×C414yescomplemented
82D3×#117—91no—
83C2×#877—91no—
84C23×C3×D3—31no—
85C22×C3⋊Dic3—31no—
86C22×C3×Dic3C2×D311yescharacteristic
87C2×C3×C4×D3—32no—
88C2×D3×Dic3—38no—
89C2×D3×Dic3—31no—
90C2×D3×Dic3—31no—
91C2×D3×Dic3—31no—
92C2×D3×Dic3—31no—
93C2×D3×Dic3—32no—
94C23×D9C2×D311yescharacteristic
95C32×C22⋊C4C2×D311yescharacteristic
96#3757—31no—
97#3750—32no—
98C4×C3⋊Dic3—32no—
99C3×#217—31no—
100C3×#217C2×D311yescharacteristic
101C3×(C2×D3)⋊C4—32no—
102C3×Dic3⋊C4C2×D312yes—
103C3×C4×Dic3C2×D312yes—
104#3724—32no—
105Dic3⋊Dic3—32no—
106(C2×D3)⋊Dic3—32no—
107Dic32—32no—
108Dic32—32no—
109C22×C4×C9—31no—
110C22×Dic9—31no—
111C22×Dic9—61no—
112C2×C4×D9—38no—
113C2×C4×D9—31no—
114C2×C4×D9—31no—
115C2×C4×D9—31no—
116C2×C4×D9—31no—
117C2×C4×D9—32no—
118C2×C4×D9C2×D312yes—
119C9×C4⋊C4—32no—
120C9×C22⋊C4—21no—
121C9×C22⋊C4—31no—
122C9×C22⋊C4C2×D311yescharacteristic
123C42×C9—32no—
124#3676—31no—
125#3676—61no—
126#3676C2×D311yescharacteristic
127(C2×D9)⋊C4—32no—
128(C2×D9)⋊C4C2×D312yes—
129C4⋊Dic9—32no—
130Dic9⋊C4—32no—
131Dic9⋊C4—32no—
132Dic9⋊C4—62no—
133C4×Dic9—32no—
134C4×Dic9—32no—
135C4×Dic9—32no—
136C4×Dic9—62no—
137C22×C3×C9—22no—
138C22×C3×C9C22×C412yes—
139C22×C3×C9C2411yescharacteristic
140C2×C3×D9—22no—
141C2×C3×D9—28no—
142C2×C3×D9C22×C412yes—
143C2×C3×D9C22×C412yes—
144C2×C3×D9#4114yes—
145C3×C4×C9—22no—
146C3⋊Dic9—22no—
147C3⋊Dic9C22×C412yes—
148C3⋊Dic9C22×C412yes—
149C9×Dic3—22no—
150C9×Dic3C22×C412yes—
151C9×Dic3C22×C412yes—
152C3×Dic9—22no—
153C23×Dic3—91no—
154C22×C4×D3—91no—
155C3×#117—31no—
156C2×C3×C22⋊C4—91no—
157C2×#217—91no—
158#926—91no—
159C2×Dic3⋊C4—92no—
160C2×C4×Dic3—92no—
161#892—92no—
162D3×C4⋊C4—92no—
163D3×C22⋊C4—31no—
164D3×C22⋊C4—91no—
165#877—91no—
166#877—91no—
167#877—91no—
168#877—91no—
169#877—91no—
170#877—92no—
171#877—92no—
172#877—181no—
173#877D911yescharacteristic, direct product factor
174#872—92no—
175C42×D3—92no—
176C23×C32C22×D311yescharacteristic
177C22×C3×D3—31no—
178C22×C3×D3—34no—
179C22×C3×D3—31no—
180C22×C3×D3—64no—
181C2×C32×C4C22×D312yes—
182C2×C3⋊Dic3—34no—
183C2×C3⋊Dic3—31no—
184C2×C3⋊Dic3—31no—
185C2×C3⋊Dic3—32no—
186C2×C3×Dic3—32no—
187C2×C3×Dic3C4×D314yes—
188C2×C3×Dic3C22×D312yes—
189C2×C3×Dic3C22×D311yescharacteristic
190C2×C3×Dic3C22×D311yescharacteristic
191C3×C4×D3—64no—
192D3×Dic3—34no—
193D3×Dic3—34no—
194D3×Dic3—34no—
195D3×Dic3—34no—
196D3×Dic3—64no—
197C23×C9—21no—
198C23×C9C22×D311yescharacteristic
199C22×D9—24no—
200C22×D9C4×D314yes—
201C22×D9C22×D311yescharacteristic
202C22×D9C22×D311yescharacteristic
203C2×C4×C9—22no—
204C2×C4×C9—34no—
205C2×C4×C9—31no—
206C2×C4×C9—31no—
207C2×C4×C9—32no—
208C2×C4×C9C22×D312yes—
209C2×Dic9—34no—
210C2×Dic9—31no—
211C2×Dic9—31no—
212C2×Dic9—32no—
213C2×Dic9—61no—
214C2×Dic9—61no—
215C2×Dic9—64no—
216C2×Dic9—62no—
217C2×Dic9C22×D312yes—
218C4×D9—24no—
219C4×D9—34no—
220C4×D9—34no—
221C4×D9—34no—
222C4×D9—34no—
223C4×D9—64no—
224C2×#117—271noSylow
225C2×C3×C9—22no—
226C2×C3×C9C23×C411yescharacteristic
227C2×C3×C9C2×#4112yes—
228C3×D9—22no—
229C3×D9#11714yescomplemented
230C24×C3—91no—
231C23×D3—31no—
232C22×C3×C4—31no—
233C22×C3×C4—92no—
234C22×Dic3—91no—
235C22×Dic3—91no—
236C22×Dic3—98no—
237C22×Dic3—91no—
238C22×Dic3—91no—
239C22×Dic3—91no—
240C22×Dic3—91no—
241C22×Dic3—92no—
242C22×Dic3—181no—
243C22×Dic3C2×D911yescharacteristic
244C2×C4×D3—32no—
245C2×C4×D3—91no—
246C2×C4×D3—91no—
247C2×C4×D3—91no—
248C2×C4×D3—91no—
249C2×C4×D3—98no—
250C2×C4×D3—92no—
251C3×C4⋊C4—32no—
252C3×C22⋊C4—21no—
253C3×C22⋊C4—31no—
254C3×C22⋊C4—91no—
255C3×C22⋊C4—92no—
256C3×C22⋊C4C2×D911yescharacteristic
257C3×C22⋊C4D3211yescharacteristic
258C3×C42—32no—
259#217—31no—
260#217—91no—
261#217—91no—
262#217—92no—
263#217—181no—
264#217C2×D911yescharacteristic
265(C2×D3)⋊C4—32no—
266(C2×D3)⋊C4—92no—
267C4⋊Dic3—92no—
268Dic3⋊C4—92no—
269Dic3⋊C4—92no—
270Dic3⋊C4—92no—
271Dic3⋊C4—92no—
272Dic3⋊C4—92no—
273Dic3⋊C4—182no—
274Dic3⋊C4C2×D912yescomplemented
275C4×Dic3—92no—
276C4×Dic3—92no—
277C4×Dic3—92no—
278C4×Dic3—92no—
279C4×Dic3—92no—
280C4×Dic3—92no—
281C4×Dic3—182no—
282C4×Dic3C2×D912yescomplemented
283C22×C32—22no—
284C22×C32C2×C4×D312yes—
285C22×C32C23×D311yescharacteristic
286C2×C3×D3—32no—
287C2×C3×D3—34no—
288C2×C3×D3—32no—
289C2×C3×D3—68no—
290C2×C3×D3—62no—
291C32×C4—22no—
292C3⋊Dic3—32no—
293C3⋊Dic3—32no—
294C3⋊Dic3—62no—
295C3×Dic3—22no—
296C3×Dic3—62no—
297C3×Dic3C2×C4×D312yes—
298C3×Dic3C2×C4×D312yes—
299C22×C9—22no—
300C22×C9—21no—
301C22×C9—22no—
302C22×C9—42no—
303C22×C9C2×C4×D312yes—
304C22×C9C23×D311yescharacteristic
305C2×D9—22no—
306C2×D9—28no—
307C2×D9C2×C4×D312yes—
308C2×D9C2×C4×D312yes—
309C2×D9#23714yes—
310C4×C9—22no—
311C4×C9—32no—
312C4×C9—32no—
313C4×C9—42no—
314C4×C9—62no—
315Dic9—22no—
316Dic9—32no—
317Dic9—32no—
318Dic9—62no—
319Dic9—62no—
320Dic9—62no—
321Dic9—122no—
322C23×C4—271no—
323#117—31no—
324#117—272no—
325#117—271no—
326#117—272no—
327#117—271no—
328#117—271no—
329C2×C4⋊C4—272no—
330C2×C22⋊C4—91no—
331C2×C22⋊C4—271no—
332C2×C42—272no—
333C3×C9C2×#11711yescharacteristic, complemented, Sylow
334C23×C3—21no—
335C23×C3—91no—
336C23×C3—94no—
337C23×C3—91no—
338C23×C3—184no—
339C23×C3C22×D911yescharacteristic
340C23×C3C2×D3211yescharacteristic
341C22×D3—31no—
342C22×D3—31no—
343C22×D3—34no—
344C22×D3—64no—
345C2×C3×C4—22no—
346C2×C3×C4—31no—
347C2×C3×C4—31no—
348C2×C3×C4—34no—
349C2×C3×C4—32no—
350C2×C3×C4—92no—
351C2×C3×C4—184no—
352C2×C3×C4C22×D912yes—
353C2×C3×C4C2×D3212yes—
354C2×Dic3—32no—
355C2×Dic3—91no—
356C2×Dic3—91no—
357C2×Dic3—94no—
358C2×Dic3—91no—
359C2×Dic3—91no—
360C2×Dic3—94no—
361C2×Dic3—94no—
362C2×Dic3—94no—
363C2×Dic3—94no—
364C2×Dic3—94no—
365C2×Dic3—92no—
366C2×Dic3—92no—
367C2×Dic3—181no—
368C2×Dic3—181no—
369C2×Dic3—184no—
370C2×Dic3—182no—
371C2×Dic3—184no—
372C2×Dic3C4×D914yescomplemented
373C2×Dic3C22×D912yes—
374C2×Dic3C22×D911yescharacteristic
375C2×Dic3C22×D911yescharacteristic
376C4×D3—64no—
377C4×D3—94no—
378C4×D3—94no—
379C4×D3—94no—
380C4×D3—94no—
381C4×D3—184no—
382C2×C32—22no—
383C2×C32C22×C4×D311yescharacteristic
384C2×C32D3×#4112yes—
385C3×D3—34no—
386C3×D3—62no—
387C2×C9—22no—
388C2×C9—21no—
389C2×C9—22no—
390C2×C9—42no—
391C2×C9C22×C4×D311yescharacteristic
392C2×C9C2×#23712yes—
393D9—22no—
394D9#87714yesdirect product factor
395C24—91no—
396C22×C4—31no—
397C22×C4—92no—
398C22×C4—271no—
399C22×C4—271no—
400C22×C4—271no—
401C22×C4—271no—
402C22×C4—278no—
403C22×C4—272no—
404C22×C4—271no—
405C4⋊C4—32no—
406C4⋊C4—272no—
407C4⋊C4—272no—
408C4⋊C4—272no—
409C22⋊C4—31no—
410C22⋊C4—91no—
411C22⋊C4—92no—
412C22⋊C4—272no—
413C22⋊C4—271no—
414C22⋊C4D3×D911yescharacteristic
415C42—32no—
416C42—272no—
417C42—272no—
418C42—272no—
419C22×C3—22no—
420C22×C3—22no—
421C22×C3—21no—
422C22×C3—22no—
423C22×C3—42no—
424C22×C3—92no—
425C22×C3—94no—
426C22×C3—92no—
427C22×C3—188no—
428C22×C3—182no—
429C22×C3C2×C4×D912yes—
430C22×C3C23×D911yescharacteristic
431C22×C3C4×D3212yes—
432C22×C3C22×D3211yescharacteristic
433C2×D3—32no—
434C2×D3—34no—
435C2×D3—32no—
436C2×D3—68no—
437C2×D3—62no—
438C3×C4—22no—
439C3×C4—22no—
440C3×C4—32no—
441C3×C4—32no—
442C3×C4—42no—
443C3×C4—62no—
444C3×C4—182no—
445Dic3—22no—
446Dic3—62no—
447Dic3—92no—
448Dic3—92no—
449Dic3—92no—
450Dic3—92no—
451Dic3—182no—
452Dic3—182no—
453Dic3—182no—
454Dic3—182no—
455Dic3—362no—
456Dic3C2×C4×D912yes—
457Dic3C2×C4×D912yes—
458C32D3×#11711yescharacteristic
459C9—21no—
460C9C2×#87711yescharacteristic, complemented
461C23—91no—
462C23—91no—
463C23—94no—
464C23—184no—
465C23C2×D3×D911yescharacteristic
466C2×C4—31no—
467C2×C4—31no—
468C2×C4—34no—
469C2×C4—32no—
470C2×C4—92no—
471C2×C4—184no—
472C2×C4—274no—
473C2×C4—274no—
474C2×C4—274no—
475C2×C4—274no—
476C2×C4—271no—
477C2×C4—271no—
478C2×C4—274no—
479C2×C4—272no—
480C2×C4—544no—
481C2×C4C2×D3×D912yes—
482C2×C3—22no—
483C2×C3—22no—
484C2×C3—22no—
485C2×C3—21no—
486C2×C3—42no—
487C2×C3—94no—
488C2×C3—182no—
489C2×C3C22×C4×D911yescharacteristic
490C2×C3#41×D912yes—
491C2×C3C2×C4×D3211yescharacteristic
492C2×C3D3×#23712yes—
493D3—34no—
494D3—62no—
495C22—22no—
496C22—94no—
497C22—92no—
498C22—92no—
499C22—188no—
500C22—182no—
501C22C4×D3×D912yes—
502C22C22×D3×D911yescharacteristic
503C4—22no—
504C4—32no—
505C4—32no—
506C4—62no—
507C4—182no—
508C4—272no—
509C4—272no—
510C4—542no—
511C3—21no—
512C3D9×#11711yescharacteristic, complemented
513C3D3×#87711yescharacteristic
514C2—22no—
515C2—94no—
516C2—182no—
517C2C2×C4×D3×D911yescharacteristic
518C2D9×#23712yes—
subgroupG/H
centerC22C22×D3×D9
commutatorC2×C3×C9C23×C4
FrattiniC22×C3C22×D32
FittingC3×C9×C22⋊C4C22
socleC22×C32C23×D3
prime radicalC23×C3×D9C22
cosocleC22×C3×C9C24
hypercenterC22⋊C4D3×D9
hypocenterC3×C9C2×#117
LCSG ▷ C2×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C22⋊C4

Complements

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

NH ≅ G/Ntype
2. #11721723C2semidirect
4. C9×#877C2semidirect
6. #11721248C2semidirect
7. #11721244C2semidirect
8. #11720975C2semidirect
9. #11720974C2semidirect
10. D9×Dic3⋊C4C2semidirect
11. C4×Dic3×D9C2semidirect
18. C2×Dic3×D9C4semidirect
26. #88534C22semidirect
27. C4×C3⋊Dic9C22semidirect
31. C9×Dic3⋊C4C22semidirect
32. C4×C9×Dic3C22semidirect
34. Dic3⋊Dic9C22semidirect
35. #88442C22semidirect
36. Dic3×Dic9C22semidirect
37. Dic3×Dic9C22semidirect
67. C2×C3⋊Dic9C2×C4semidirect
71. C2×C9×Dic3C2×C4semidirect
80. Dic3×D9C2×C4semidirect
81. Dic3×D9C2×C4semidirect
173. #877D9direct (N × H)
229. C3×D9#117semidirect
274. Dic3⋊C4C2×D9semidirect
282. C4×Dic3C2×D9semidirect
333. C3×C9C2×#117semidirect
372. C2×Dic3C4×D9semidirect
394. D9#877direct (N × H)
460. C9C2×#877semidirect
512. C3D9×#117semidirect

Maximal subgroups

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

subgroupindex
C22×Dic3×D92
#117217232
C3×C22⋊C4×D92
C9×#8772
D9×#2172
#117212482
#117212442
#117209752
#117209742
D9×Dic3⋊C42
C4×Dic3×D92
D3×#8773
D9×#1173