Finite Group Databasebeta

← D9×#872 #13147695

Subgroups — D9×#872 #13147695

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
1C2×C4×D3×D9C211yesmaximal, characteristic
2#11721709C211yesmaximal, characteristic, complemented
3C9×#872C211yesmaximal, characteristic, complemented
4C3×C42×D9C211yesmaximal, characteristic, complemented
5D9×(C2×D3)⋊C4C212yesmaximal
6#11721010C212yesmaximal, complemented
7D9×Dic3⋊C4C212yesmaximal, complemented
8C4×Dic3×D9C211yesmaximal, characteristic, complemented
9#11720959C211yesmaximal, characteristic, complemented
10#11720931C211yesmaximal, characteristic, complemented
11#11720899C212yesmaximal, complemented
12D3×#872—31nomaximal
13D9×#117—31nomaximal
14C22×D3×D9C2211yescharacteristic
15C2×C4×C3⋊D9C2211yescharacteristic
16C2×C4×D3×C9C2211yescharacteristic
17C2×C3×C4×D9C2212yes—
18C2×C3×C4×D9C2211yescharacteristic
19C2×D3×Dic9C2211yescharacteristic
20C2×(C3⋊D9)⋊C4C2211yescharacteristic
21C2×Dic3×D9C2212yes—
22C2×Dic3×D9C2211yescharacteristic
23C4×D3×D9C418yescomplemented
24C3×C42×C9C2211yescharacteristic, complemented
25#88536C2212yes—
26#88534C2212yescomplemented
27C4×C3⋊Dic9C2211yescharacteristic, complemented
28C9×(C2×D3)⋊C4C2212yes—
29C9×Dic3⋊C4C2212yescomplemented
30C4×C9×Dic3C2211yescharacteristic, complemented
31C3×C4×Dic9C2212yescomplemented
32C3×C4×Dic9C2211yescharacteristic, complemented
33(C2×D3)⋊Dic9C2212yes—
34#88445C2212yes—
35Dic3⋊Dic9C2212yescomplemented
36#88442C2212yescomplemented
37Dic3×Dic9C2212yescomplemented
38C2×C4×D32—31no—
39#64081—31no—
40C3×#872D311yescharacteristic
41C3×C42×D3—31no—
42D3×(C2×D3)⋊C4—32no—
43#63877—32no—
44D3×Dic3⋊C4—32no—
45C4×D3×Dic3—31no—
46#63874—31no—
47#63855—31no—
48#63836—32no—
49C22×C4×D9—31no—
50C9×#117—31no—
51#63488—31no—
52#63454—32no—
53C4⋊C4×D9—32no—
54C22⋊C4×D9—32no—
55#63439—32no—
56#63434—31no—
57#63434—31no—
58#63434—61no—
59C42×D9—31no—
60C42×D9D311yescharacteristic, complemented
61C22×C3⋊D9C2311yescharacteristic
62C22×D3×C9C2311yescharacteristic
63C22×C3×D9C2311yescharacteristic
64C2×D3×D9—24no—
65C2×D3×D9C2×C414yes—
66C2×C3×C4×C9C2312yes—
67C2×C3×C4×C9C2311yescharacteristic
68C2×C3⋊Dic9C2312yes—
69C2×C3⋊Dic9C2311yescharacteristic
70C4×C3⋊D9C2×C414yescomplemented
71C2×C9×Dic3C2312yes—
72C2×C9×Dic3C2311yescharacteristic
73C2×C3×Dic9C2312yes—
74C2×C3×Dic9C2311yescharacteristic
75C4×D3×C9C2×C414yescomplemented
76C3×C4×D9—24no—
77C3×C4×D9C2×C414yes—
78D3×Dic9C2×C414yescomplemented
79(C3⋊D9)⋊C4C2×C414yescomplemented
80Dic3×D9—24no—
81Dic3×D9C2×C414yes—
82D3×#117—91no—
83C2×#872—91no—
84C22×D32—31no—
85C2×C4×C3⋊D3—31no—
86C2×C3×C4×D3—31no—
87C2×C3×C4×D3—32no—
88C2×C3×C4×D3C2×D311yescharacteristic
89C2×#624—31no—
90C2×D3×Dic3—31no—
91C2×D3×Dic3—31no—
92C2×D3×Dic3—32no—
93C4×D32—38no—
94C23×D9—31no—
95C32×C42C2×D311yescharacteristic
96#3752—32no—
97#3750—32no—
98C4×C3⋊Dic3—31no—
99C3×(C2×D3)⋊C4C2×D312yes—
100C3×Dic3⋊C4C2×D312yes—
101C3×C4×Dic3—32no—
102C3×C4×Dic3—31no—
103C3×C4×Dic3C2×D311yescharacteristic
104#3724—32no—
105Dic3⋊Dic3—32no—
106#3722—32no—
107(C2×D3)⋊Dic3—32no—
108Dic32—32no—
109C22×C4×C9—31no—
110C22×Dic9—31no—
111C2×C4×D9—31no—
112C2×C4×D9—31no—
113C2×C4×D9—31no—
114C2×C4×D9—38no—
115C2×C4×D9—32no—
116C2×C4×D9—61no—
117C2×C4×D9C2×D312yes—
118C2×C4×D9C2×D311yescharacteristic
119C9×C4⋊C4—32no—
120C9×C22⋊C4—32no—
121C42×C9—21no—
122C42×C9—31no—
123C42×C9C2×D311yescharacteristic, complemented
124#3676—32no—
125(C2×D9)⋊C4—32no—
126(C2×D9)⋊C4—32no—
127(C2×D9)⋊C4—62no—
128C4⋊Dic9—32no—
129Dic9⋊C4—32no—
130Dic9⋊C4—32no—
131Dic9⋊C4—62no—
132C4×Dic9—31no—
133C4×Dic9—32no—
134C4×Dic9—61no—
135C4×Dic9C2×D311yescharacteristic, complemented
136C4×Dic9C2×D312yescomplemented
137C22×C3×C9C2411yescharacteristic
138C2×C3⋊D9—22no—
139C2×C3⋊D9C22×C412yes—
140C2×D3×C9—22no—
141C2×D3×C9C22×C412yes—
142C2×C3×D9C22×C412yes—
143C2×C3×D9#4114yes—
144D3×D9—28no—
145C3×C4×C9—22no—
146C3×C4×C9C22×C412yes—
147C3⋊Dic9—22no—
148C3⋊Dic9C22×C412yes—
149C9×Dic3—22no—
150C9×Dic3C22×C412yes—
151C3×Dic9—22no—
152C3×Dic9C22×C412yes—
153C22×C4×D3—91no—
154C22×C4×D3—91no—
155C3×#117—31no—
156C2×C3×C42—91no—
157C2×(C2×D3)⋊C4—92no—
158#926—91no—
159C2×Dic3⋊C4—92no—
160C2×C4×Dic3—91no—
161#892—92no—
162D3×C4⋊C4—92no—
163D3×C22⋊C4—92no—
164#877—92no—
165#872—91no—
166#872—91no—
167#872—91no—
168#872—91no—
169#872—92no—
170#872—92no—
171#872—91no—
172#872—181no—
173#872D911yescharacteristic, direct product factor
174C42×D3—31no—
175C42×D3—91no—
176C22×C3⋊D3—31no—
177C22×C3×D3—31no—
178C22×C3×D3C22×D311yescharacteristic
179C2×D32—34no—
180C2×D32—64no—
181C2×C32×C4C22×D312yes—
182C2×C32×C4C22×D311yescharacteristic
183C2×C3⋊Dic3—31no—
184C2×C3⋊Dic3—32no—
185C4×C3⋊D3—34no—
186C2×C3×Dic3—31no—
187C2×C3×Dic3—32no—
188C2×C3×Dic3C22×D312yes—
189C2×C3×Dic3C22×D311yescharacteristic
190C3×C4×D3—34no—
191C3×C4×D3—64no—
192C3×C4×D3C4×D314yes—
193#624—34no—
194D3×Dic3—34no—
195D3×Dic3—34no—
196D3×Dic3—64no—
197C23×C9—31no—
198C22×D9—31no—
199C22×D9—34no—
200C22×D9—61no—
201C22×D9—64no—
202C22×D9C22×D311yescharacteristic
203C2×C4×C9—21no—
204C2×C4×C9—22no—
205C2×C4×C9—31no—
206C2×C4×C9—34no—
207C2×C4×C9—32no—
208C2×C4×C9C22×D312yes—
209C2×C4×C9C22×D311yescharacteristic
210C2×Dic9—31no—
211C2×Dic9—34no—
212C2×Dic9—32no—
213C2×Dic9—61no—
214C2×Dic9—62no—
215C2×Dic9C22×D312yes—
216C2×Dic9C22×D311yescharacteristic
217C4×D9—24no—
218C4×D9—34no—
219C4×D9—34no—
220C4×D9—34no—
221C4×D9—64no—
222C4×D9—64no—
223C4×D9C4×D314yes—
224C2×#117—271noSylow
225C2×C3×C9C23×C411yescharacteristic
226C2×C3×C9C2×#4112yes—
227C3⋊D9—22no—
228D3×C9—22no—
229C3×D9#11714yescomplemented
230C23×D3—91no—
231C23×D3—91no—
232C22×C3×C4—31no—
233C22×C3×C4—91no—
234C22×C3×C4—92no—
235C22×Dic3—91no—
236C22×Dic3—91no—
237C22×Dic3—92no—
238C2×C4×D3—31no—
239C2×C4×D3—32no—
240C2×C4×D3—98no—
241C2×C4×D3—91no—
242C2×C4×D3—91no—
243C2×C4×D3—91no—
244C2×C4×D3—91no—
245C2×C4×D3—91no—
246C2×C4×D3—91no—
247C2×C4×D3—98no—
248C2×C4×D3—92no—
249C2×C4×D3—181no—
250C2×C4×D3C2×D911yescharacteristic
251C3×C4⋊C4—32no—
252C3×C22⋊C4—32no—
253C3×C42—21no—
254C3×C42—31no—
255C3×C42—92no—
256C3×C42—91no—
257C3×C42C2×D911yescharacteristic, complemented
258C3×C42D3211yescharacteristic
259#217—92no—
260(C2×D3)⋊C4—92no—
261(C2×D3)⋊C4—92no—
262(C2×D3)⋊C4—92no—
263(C2×D3)⋊C4—92no—
264(C2×D3)⋊C4—92no—
265(C2×D3)⋊C4—182no—
266(C2×D3)⋊C4C2×D912yes—
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—31no—
276C4×Dic3—32no—
277C4×Dic3—92no—
278C4×Dic3—91no—
279C4×Dic3—91no—
280C4×Dic3—92no—
281C4×Dic3—181no—
282C4×Dic3C2×D911yescharacteristic, complemented
283C22×C32C23×D311yescharacteristic
284C2×C3⋊D3—32no—
285C2×C3⋊D3—62no—
286C2×C3×D3—22no—
287C2×C3×D3—32no—
288C2×C3×D3—34no—
289C2×C3×D3C2×C4×D312yes—
290D32—68no—
291C32×C4—22no—
292C32×C4C2×C4×D312yes—
293C3⋊Dic3—32no—
294C3⋊Dic3—62no—
295C3×Dic3—22no—
296C3×Dic3—32no—
297C3×Dic3—62no—
298C3×Dic3C2×C4×D312yes—
299C22×C9—21no—
300C22×C9—32no—
301C22×C9—62no—
302C22×C9C23×D311yescharacteristic
303C2×D9—32no—
304C2×D9—62no—
305C2×D9—62no—
306C2×D9—68no—
307C2×D9—122no—
308C2×D9C2×C4×D312yes—
309C2×D9#23514yes—
310C4×C9—22no—
311C4×C9—22no—
312C4×C9—32no—
313C4×C9—42no—
314C4×C9—62no—
315C4×C9C2×C4×D312yes—
316Dic9—22no—
317Dic9—32no—
318Dic9—62no—
319Dic9—62no—
320Dic9—122no—
321Dic9C2×C4×D312yes—
322C23×C4—271no—
323#117—31no—
324#117—271no—
325#117—272no—
326#117—272no—
327#117—271no—
328#117—271no—
329C2×C4⋊C4—272no—
330C2×C22⋊C4—272no—
331C2×C42—91no—
332C2×C42—271no—
333C3×C9C2×#11711yescharacteristic, complemented, Sylow
334C23×C3—31no—
335C23×C3—91no—
336C22×D3—31no—
337C22×D3—94no—
338C22×D3—91no—
339C22×D3—91no—
340C22×D3—94no—
341C22×D3—181no—
342C22×D3—184no—
343C22×D3—184no—
344C22×D3C22×D911yescharacteristic
345C2×C3×C4—22no—
346C2×C3×C4—21no—
347C2×C3×C4—31no—
348C2×C3×C4—34no—
349C2×C3×C4—32no—
350C2×C3×C4—91no—
351C2×C3×C4—94no—
352C2×C3×C4—92no—
353C2×C3×C4—184no—
354C2×C3×C4C22×D912yes—
355C2×C3×C4C22×D911yescharacteristic
356C2×C3×C4C2×D3212yes—
357C2×C3×C4C2×D3211yescharacteristic
358C2×Dic3—31no—
359C2×Dic3—32no—
360C2×Dic3—94no—
361C2×Dic3—91no—
362C2×Dic3—91no—
363C2×Dic3—94no—
364C2×Dic3—92no—
365C2×Dic3—92no—
366C2×Dic3—181no—
367C2×Dic3—182no—
368C2×Dic3—184no—
369C2×Dic3C22×D912yes—
370C2×Dic3C22×D911yescharacteristic
371C4×D3—34no—
372C4×D3—64no—
373C4×D3—94no—
374C4×D3—94no—
375C4×D3—94no—
376C4×D3—94no—
377C4×D3—94no—
378C4×D3—94no—
379C4×D3—184no—
380C4×D3—184no—
381C4×D3C4×D914yescomplemented
382C2×C32C22×C4×D311yescharacteristic
383C2×C32D3×#4112yes—
384C3⋊D3—62no—
385C3×D3—22no—
386C3×D3—34no—
387C2×C9—21no—
388C2×C9—22no—
389C2×C9—62no—
390C2×C9C22×C4×D311yescharacteristic
391C2×C9C2×#23512yes—
392D9—62no—
393D9—122no—
394D9#87214yesdirect product factor
395C24—271no—
396C22×C4—31no—
397C22×C4—91no—
398C22×C4—92no—
399C22×C4—271no—
400C22×C4—278no—
401C22×C4—271no—
402C22×C4—272no—
403C22×C4—271no—
404C22×C4—271no—
405C4⋊C4—32no—
406C4⋊C4—272no—
407C4⋊C4—272no—
408C4⋊C4—272no—
409C22⋊C4—32no—
410C22⋊C4—272no—
411C22⋊C4—272no—
412C22⋊C4—272no—
413C42—31no—
414C42—92no—
415C42—91no—
416C42—272no—
417C42—271no—
418C42D3×D911yescharacteristic, complemented
419C22×C3—21no—
420C22×C3—32no—
421C22×C3—62no—
422C22×C3—92no—
423C22×C3—94no—
424C22×C3C23×D911yescharacteristic
425C22×C3C22×D3211yescharacteristic
426C2×D3—22no—
427C2×D3—32no—
428C2×D3—34no—
429C2×D3—92no—
430C2×D3—92no—
431C2×D3—182no—
432C2×D3—188no—
433C2×D3—188no—
434C2×D3—182no—
435C2×D3—182no—
436C2×D3—362no—
437C2×D3C2×C4×D912yes—
438C3×C4—22no—
439C3×C4—22no—
440C3×C4—22no—
441C3×C4—32no—
442C3×C4—42no—
443C3×C4—62no—
444C3×C4—92no—
445C3×C4—182no—
446C3×C4C2×C4×D912yes—
447C3×C4C4×D3212yes—
448Dic3—22no—
449Dic3—32no—
450Dic3—62no—
451Dic3—92no—
452Dic3—92no—
453Dic3—182no—
454Dic3—182no—
455Dic3—182no—
456Dic3—362no—
457Dic3C2×C4×D912yes—
458C32D3×#11711yescharacteristic
459C9—21no—
460C9C2×#87211yescharacteristic, complemented
461C23—31no—
462C23—91no—
463C23—274no—
464C23—271no—
465C23—544no—
466C2×C4—31no—
467C2×C4—34no—
468C2×C4—32no—
469C2×C4—91no—
470C2×C4—94no—
471C2×C4—92no—
472C2×C4—184no—
473C2×C4—274no—
474C2×C4—274no—
475C2×C4—274no—
476C2×C4—271no—
477C2×C4—274no—
478C2×C4—272no—
479C2×C4—544no—
480C2×C4C2×D3×D912yes—
481C2×C4C2×D3×D911yescharacteristic
482C2×C3—21no—
483C2×C3—22no—
484C2×C3—62no—
485C2×C3—94no—
486C2×C3C22×C4×D911yescharacteristic
487C2×C3#41×D912yes—
488C2×C3C2×C4×D3211yescharacteristic
489C2×C3D3×#23512yes—
490D3—22no—
491D3—34no—
492D3—182no—
493D3—182no—
494D3—362no—
495C22—32no—
496C22—62no—
497C22—92no—
498C22—94no—
499C22—272no—
500C22—548no—
501C22—542no—
502C22C22×D3×D911yescharacteristic
503C4—22no—
504C4—32no—
505C4—62no—
506C4—92no—
507C4—182no—
508C4—272no—
509C4—542no—
510C4C4×D3×D912yes—
511C3—21no—
512C3D9×#11711yescharacteristic, complemented
513C3D3×#87211yescharacteristic
514C2—62no—
515C2—94no—
516C2—542no—
517C2C2×C4×D3×D911yescharacteristic
518C2D9×#23512yes—
subgroupG/H
centerC2×C4C2×D3×D9
commutatorC2×C3×C9C23×C4
FrattiniC22×C3C22×D32
FittingC3×C42×C9C22
socleC22×C32C23×D3
prime radicalC22×D3×D9C22
cosocleC22×C3×C9C24
hypercenterC42D3×D9
hypocenterC3×C9C2×#117
LCSG ▷ C2×C3×C9 ▷ C3×C9
UCSC1 ◁ C2×C4 ◁ C42

Complements

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

NH ≅ G/Ntype
2. #11721709C2semidirect
3. C9×#872C2semidirect
4. C3×C42×D9C2semidirect
6. #11721010C2semidirect
7. D9×Dic3⋊C4C2semidirect
8. C4×Dic3×D9C2semidirect
9. #11720959C2semidirect
10. #11720931C2semidirect
11. #11720899C2semidirect
23. C4×D3×D9C4semidirect
24. C3×C42×C9C22semidirect
26. #88534C22semidirect
27. C4×C3⋊Dic9C22semidirect
29. C9×Dic3⋊C4C22semidirect
30. C4×C9×Dic3C22semidirect
31. C3×C4×Dic9C22semidirect
32. C3×C4×Dic9C22semidirect
35. Dic3⋊Dic9C22semidirect
36. #88442C22semidirect
37. Dic3×Dic9C22semidirect
60. C42×D9D3semidirect
70. C4×C3⋊D9C2×C4semidirect
75. C4×D3×C9C2×C4semidirect
78. D3×Dic9C2×C4semidirect
79. (C3⋊D9)⋊C4C2×C4semidirect
123. C42×C9C2×D3semidirect
135. C4×Dic9C2×D3semidirect
136. C4×Dic9C2×D3semidirect
173. #872D9direct (N × H)
229. C3×D9#117semidirect
257. C3×C42C2×D9semidirect
274. Dic3⋊C4C2×D9semidirect
282. C4×Dic3C2×D9semidirect
333. C3×C9C2×#117semidirect
381. C4×D3C4×D9semidirect
394. D9#872direct (N × H)
418. C42D3×D9semidirect
460. C9C2×#872semidirect
512. C3D9×#117semidirect

Maximal subgroups

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

subgroupindex
C2×C4×D3×D92
#117217092
C9×#8722
C3×C42×D92
D9×(C2×D3)⋊C42
#117210102
D9×Dic3⋊C42
C4×Dic3×D92
#117209592
#117209312
#117208992
D3×#8723
D9×#1173