Finite Group Databasebeta

← Dic9×#237 #13151168

Subgroups — Dic9×#237 #13151168

Order 1728.

1024 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×C9×#237C211yesmaximal, characteristic
2D4×C3⋊Dic9C211yesmaximal, characteristic, complemented
3C3×D4×Dic9C211yesmaximal, characteristic, complemented
4#67⋊Dic9C212yesmaximal, complemented
5#67×Dic9C212yesmaximal, complemented
6#11721244C212yesmaximal, complemented
7C2×Dic3×Dic9C212yesmaximal, complemented
8C4×D3×Dic9C211yesmaximal, characteristic, complemented
9#11720911C211yesmaximal, characteristic, complemented
10Dic6×Dic9C211yesmaximal, characteristic, complemented
11#11720906C211yesmaximal, characteristic, complemented
12Dic3×#237—31nomaximal
13#41×Dic9—31nomaximal
14C2×C3×D4×C9C2211yescharacteristic
15C22×C3⋊Dic9C2212yes—
16C2×C9×#67C2212yes—
17C22×C9×Dic3C2212yes—
18C22×C3×Dic9C2212yes—
19C9×#237C418yescomplemented
20C2×C4×D3×C9C2211yescharacteristic
21C2×C9×Dic6C2211yescharacteristic
22C2×D3×Dic9C2211yescharacteristic
23C2×D3×Dic9C2211yescharacteristic
24#88551C2212yes—
25C4⋊(C3⋊Dic9)C2211yescharacteristic
26C4×C3⋊Dic9C2211yescharacteristic
27C3×#3676C2212yes—
28C3×C4⋊Dic9C2211yescharacteristic
29C3×C4×Dic9C2211yescharacteristic
30(C2×D3)⋊Dic9C2212yes—
31Dic3⋊Dic9C2212yes—
32Dic3⋊Dic9C2212yes—
33Dic3×Dic9C2211yescharacteristic
34Dic3×Dic9C2212yes—
35Dic3×Dic9C2212yes—
36Dic3×Dic9C2212yes—
37Dic3×Dic9C2211yescharacteristic
38C2×C3×#237D311yescharacteristic
39D4×C3⋊Dic3—31no—
40C3×D4×Dic3—31no—
41#67⋊Dic3—32no—
42Dic3×#67—32no—
43#63955—32no—
44C2×Dic32—32no—
45C4×D3×Dic3—31no—
46#63843—31no—
47Dic3×Dic6—31no—
48#63841—31no—
49C2×C9×#41—31no—
50Q8×Dic9—31no—
51Q8×Dic9—31no—
52D4×Dic9—31no—
53D4×Dic9—32no—
54D4×Dic9—32no—
55D4×Dic9—61no—
56D4×Dic9D311yescharacteristic, complemented
57#63488—32no—
58#63488—31no—
59C2×C4×Dic9—32no—
60C2×C4×Dic9—31no—
61C23×C3×C9C2312yes—
62C22×D3×C9C2311yescharacteristic
63C3×D4×C9C2×C414yescomplemented
64C2×C3×C4×C9C2311yescharacteristic
65C2×C3⋊Dic9—24no—
66C2×C3⋊Dic9C2311yescharacteristic
67C2×C3⋊Dic9C2312yes—
68C2×C3⋊Dic9C2311yescharacteristic
69C9×#67C2×C418yescomplemented
70C2×C9×Dic3C2×C418yescomplemented
71C2×C9×Dic3C2312yes—
72C2×C9×Dic3C2311yescharacteristic
73C2×C3×Dic9—24no—
74C2×C3×Dic9C2311yescharacteristic
75C2×C3×Dic9C2312yes—
76C2×C3×Dic9C2311yescharacteristic
77C4×D3×C9C2×C414yescomplemented
78C9×Dic6C2×C414yescomplemented
79D3×Dic9—22no—
80D3×Dic9—22no—
81Dic3×#41—91no—
82C4×#237—91no—
83C2×C32×D4C2×D311yescharacteristic
84C22×C3⋊Dic3—32no—
85C2×C3×#67C2×D312yes—
86C22×C3×Dic3—32no—
87C22×C3×Dic3C2×D312yes—
88C3×#237Dic318yes—
89C2×C3×C4×D3C2×D311yescharacteristic
90C2×C3×Dic6C2×D311yescharacteristic
91C2×D3×Dic3—31no—
92C2×D3×Dic3—31no—
93#3757—32no—
94C4⋊(C3⋊Dic3)—31no—
95C4×C3⋊Dic3—31no—
96C3×#217—32no—
97C3×C4⋊Dic3—31no—
98C3×C4×Dic3—31no—
99Dic3⋊Dic3—32no—
100Dic3⋊Dic3—32no—
101(C2×D3)⋊Dic3—32no—
102Dic32—32no—
103Dic32—32no—
104Dic32—32no—
105Dic32—31no—
106Dic32—31no—
107C9×#41—38no—
108C2×Q8×C9—31no—
109C2×D4×C9—21no—
110C2×D4×C9—32no—
111C2×D4×C9C2×D311yescharacteristic
112C22×C4×C9—32no—
113C22×C4×C9—31no—
114C22×Dic9—32no—
115C22×Dic9—31no—
116C22×Dic9—31no—
117C22×Dic9—62no—
118C22×Dic9C2×D312yes—
119#3676—32no—
120#3676—32no—
121#3676—62no—
122#3676C2×D312yes—
123C4⋊Dic9—31no—
124C4⋊Dic9—32no—
125C4⋊Dic9—32no—
126C4⋊Dic9—61no—
127C4⋊Dic9C2×D311yescharacteristic
128C4×Dic9—31no—
129C4×Dic9—32no—
130C4×Dic9—31no—
131C4×Dic9—32no—
132C4×Dic9—32no—
133C4×Dic9—31no—
134C4×Dic9—61no—
135C4×Dic9C2×D311yescharacteristic
136C22×C3×C9—22no—
137C22×C3×C9—22no—
138C22×C3×C9C22×C414yes—
139C22×C3×C9C2411yescharacteristic
140C2×D3×C9—21no—
141C2×D3×C9—21no—
142C2×D3×C9C22×C412yes—
143C3×C4×C9C22×C412yes—
144C3⋊Dic9—21no—
145C3⋊Dic9—22no—
146C3⋊Dic9#4112yescomplemented
147C9×Dic3C22×C414yes—
148C9×Dic3C22×C412yes—
149C3×Dic9—21no—
150C3×Dic9—22no—
151C3×Dic9#4112yescomplemented
152C2×C3×#41—31no—
153C2×#237D911yescharacteristic
154C3×C4×D4—91no—
155Q8×Dic3—91no—
156Q8×Dic3—91no—
157D4×Dic3—31no—
158D4×Dic3—92no—
159D4×Dic3—92no—
160D4×Dic3—91no—
161D4×Dic3—91no—
162D4×Dic3—181no—
163C4×#67—92no—
164#926—91no—
165#926—92no—
166C2×C4×Dic3—91no—
167C2×C4×Dic3—92no—
168C2×C4×Dic3—92no—
169#892—91no—
170#887—91no—
171#67⋊C4—92no—
172#877—92no—
173C42×D3—91no—
174C4×Dic6—91no—
175C23×C32C22×D312yes—
176C22×C3×D3C22×D311yescharacteristic
177C32×D4C2×Dic314yes—
178C2×C32×C4C22×D311yescharacteristic
179C2×C3⋊Dic3—31no—
180C2×C3⋊Dic3—32no—
181C2×C3⋊Dic3—31no—
182C2×C3⋊Dic3—64no—
183C3×#67C2×Dic318yes—
184C2×C3×Dic3—31no—
185C2×C3×Dic3—32no—
186C2×C3×Dic3—31no—
187C2×C3×Dic3—64no—
188C2×C3×Dic3C2×Dic318yes—
189C2×C3×Dic3C22×D311yescharacteristic
190C2×C3×Dic3C22×D312yes—
191C3×C4×D3C2×Dic314yes—
192C3×Dic6C2×Dic314yes—
193D3×Dic3—62no—
194D3×Dic3—62no—
195C23×C9—22no—
196C23×C9—31no—
197C23×C9C22×D312yes—
198Q8×C9—34no—
199D4×C9—24no—
200D4×C9—38no—
201D4×C9C4×D314yescomplemented
202C2×C4×C9—21no—
203C2×C4×C9—38no—
204C2×C4×C9—31no—
205C2×C4×C9—32no—
206C2×C4×C9—34no—
207C2×C4×C9C22×D311yescharacteristic
208C2×Dic9—24no—
209C2×Dic9—31no—
210C2×Dic9—31no—
211C2×Dic9—32no—
212C2×Dic9—61no—
213C2×Dic9—61no—
214C2×Dic9—62no—
215C2×Dic9—62no—
216C2×Dic9—64no—
217C2×Dic9—62no—
218C2×Dic9—124no—
219C2×Dic9C22×D312yes—
220C2×Dic9C22×D311yescharacteristic
221C2×Dic9C22×D311yescharacteristic
222C4×#41—271noSylow
223C2×C3×C9—24no—
224C2×C3×C9C23×C411yescharacteristic
225C2×C3×C9C2×#4111yescharacteristic
226C2×C3×C9C2×#4111yescharacteristic
227D3×C9—22no—
228C3×#41—38no—
229C2×C3×Q8—31no—
230C2×C3×D4—21no—
231C2×C3×D4—32no—
232C2×C3×D4C2×D911yescharacteristic
233C2×C3×D4D3211yescharacteristic
234C22×C3×C4—32no—
235C22×C3×C4—31no—
236C22×C3×C4—92no—
237C2×#67C2×D912yes—
238C22×Dic3—32no—
239C22×Dic3—91no—
240C22×Dic3—91no—
241C22×Dic3—92no—
242C22×Dic3—92no—
243C22×Dic3—182no—
244C22×Dic3C2×D912yes—
245#237Dic918yesdirect product factor
246C2×C4×D3—91no—
247C2×C4×D3—91no—
248C2×C4×D3C2×D911yescharacteristic
249C2×Dic6C2×D911yescharacteristic
250C3×C4⋊C4—91no—
251C3×C22⋊C4—92no—
252C3×C42—91no—
253#217—32no—
254#217—92no—
255#217—92no—
256#217—92no—
257#217—182no—
258(C2×D3)⋊C4—92no—
259C4⋊Dic3—31no—
260C4⋊Dic3—92no—
261C4⋊Dic3—92no—
262C4⋊Dic3—91no—
263C4⋊Dic3—91no—
264C4⋊Dic3—181no—
265Dic3⋊C4—92no—
266Dic3⋊C4—92no—
267C4×Dic3—31no—
268C4×Dic3—91no—
269C4×Dic3—91no—
270C4×Dic3—92no—
271C4×Dic3—92no—
272C4×Dic3—92no—
273C4×Dic3—91no—
274C4×Dic3—91no—
275C4×Dic3—92no—
276C4×Dic3—91no—
277C4×Dic3—91no—
278C4×Dic3—92no—
279C4×Dic3—92no—
280C4×Dic3—181no—
281C22×C32—22no—
282C22×C32—22no—
283C22×C32C22×Dic314yes—
284C22×C32C23×D311yescharacteristic
285C2×C3×D3—21no—
286C2×C3×D3—21no—
287C2×C3×D3C22×Dic312yes—
288C32×C4C22×Dic312yes—
289C3⋊Dic3—32no—
290C3⋊Dic3—61no—
291C3⋊Dic3—62no—
292C3×Dic3—32no—
293C3×Dic3—61no—
294C3×Dic3—62no—
295C3×Dic3C22×Dic314yes—
296C3×Dic3C22×Dic312yes—
297C22×C9—22no—
298C22×C9—22no—
299C22×C9—21no—
300C22×C9—24no—
301C22×C9—32no—
302C22×C9—42no—
303C22×C9—42no—
304C22×C9—61no—
305C22×C9—61no—
306C22×C9C2×C4×D314yes—
307C22×C9C23×D311yescharacteristic
308C4×C9—22no—
309C4×C9—32no—
310C4×C9—34no—
311C4×C9C2×C4×D312yes—
312Dic9—22no—
313Dic9—21no—
314Dic9—32no—
315Dic9—62no—
316Dic9—61no—
317Dic9—62no—
318Dic9—121no—
319Dic9—122no—
320Dic9#23712yesdirect product factor
321C2×#41—31no—
322C4×Q8—271no—
323C4×Q8—271no—
324C4×D4—91no—
325C4×D4—272no—
326C4×D4—272no—
327C4×D4—271no—
328#117—272no—
329#117—271no—
330C2×C42—271no—
331C2×C42—272no—
332C3×C9C4×#4111yescharacteristic, complemented, Sylow
333C23×C3—22no—
334C23×C3—31no—
335C23×C3C22×D912yes—
336C23×C3C2×D3212yes—
337C22×D3C22×D911yescharacteristic
338C3×Q8—34no—
339C3×D4—24no—
340C3×D4—38no—
341C3×D4C2×Dic914yescomplemented
342C3×D4D3×Dic314yes—
343C2×C3×C4—21no—
344C2×C3×C4—31no—
345C2×C3×C4—32no—
346C2×C3×C4—38no—
347C2×C3×C4—34no—
348C2×C3×C4—91no—
349C2×C3×C4—92no—
350C2×C3×C4—91no—
351C2×C3×C4—184no—
352C2×C3×C4C22×D911yescharacteristic
353C2×C3×C4C2×D3211yescharacteristic
354#67C2×Dic918yescomplemented
355C2×Dic3—31no—
356C2×Dic3—31no—
357C2×Dic3—32no—
358C2×Dic3—64no—
359C2×Dic3—91no—
360C2×Dic3—91no—
361C2×Dic3—91no—
362C2×Dic3—92no—
363C2×Dic3—91no—
364C2×Dic3—92no—
365C2×Dic3—181no—
366C2×Dic3—181no—
367C2×Dic3—182no—
368C2×Dic3—182no—
369C2×Dic3—182no—
370C2×Dic3—184no—
371C2×Dic3—184no—
372C2×Dic3—364no—
373C2×Dic3C2×Dic918yescomplemented
374C2×Dic3C22×D911yescharacteristic
375C2×Dic3C22×D912yes—
376C4×D3—182no—
377C4×D3—182no—
378C4×D3C2×Dic914yescomplemented
379Dic6C2×Dic914yescomplemented
380C2×C32—24no—
381C2×C32D3×#4111yescharacteristic
382C2×C32D3×#4111yescharacteristic
383C2×C32C23×Dic311yescharacteristic
384C3×D3—22no—
385C2×C9—24no—
386C2×C9—21no—
387C2×C9—21no—
388C2×C9—21no—
389C2×C9—44no—
390C2×C9—62no—
391C2×C9C22×C4×D311yescharacteristic
392C2×C9C2×#23711yescharacteristic
393C2×C9D3×#4111yescharacteristic
394#41—38no—
395C2×Q8—31no—
396C2×D4—32no—
397C2×D4D3×D911yescharacteristic
398C22×C4—32no—
399C22×C4—31no—
400C22×C4—92no—
401C22×C4—271no—
402C22×C4—271no—
403C22×C4—272no—
404C4⋊C4—91no—
405C4⋊C4—272no—
406C4⋊C4—272no—
407C4⋊C4—271no—
408C22⋊C4—92no—
409C22⋊C4—272no—
410C22⋊C4—272no—
411C42—91no—
412C42—271no—
413C42—271no—
414C42—272no—
415C42—272no—
416C42—272no—
417C42—271no—
418C22×C3—22no—
419C22×C3—22no—
420C22×C3—22no—
421C22×C3—22no—
422C22×C3—21no—
423C22×C3—24no—
424C22×C3—32no—
425C22×C3—42no—
426C22×C3—42no—
427C22×C3—61no—
428C22×C3—61no—
429C22×C3C22×Dic914yes—
430C22×C3C23×D911yescharacteristic
431C22×C3C2×D3×Dic314yes—
432C22×C3C22×D3211yescharacteristic
433C2×D3—21no—
434C2×D3—21no—
435C2×D3C22×Dic912yes—
436C3×C4—22no—
437C3×C4—32no—
438C3×C4—34no—
439C3×C4—92no—
440C3×C4—181no—
441C3×C4—182no—
442C3×C4C22×Dic912yes—
443C3×C4C2×D3×Dic312yes—
444Dic3—32no—
445Dic3—62no—
446Dic3—61no—
447Dic3—92no—
448Dic3—92no—
449Dic3—182no—
450Dic3—181no—
451Dic3—182no—
452Dic3—181no—
453Dic3—182no—
454Dic3—362no—
455Dic3—361no—
456Dic3C22×Dic914yes—
457Dic3C22×Dic912yes—
458C32Dic3×#4111yescharacteristic
459C9—21no—
460C9C4×#23711yescharacteristic, complemented
461C23—31no—
462C23C2×D3×D912yes—
463Q8—34no—
464D4—38no—
465D4D3×Dic914yescomplemented
466C2×C4—31no—
467C2×C4—32no—
468C2×C4—38no—
469C2×C4—34no—
470C2×C4—91no—
471C2×C4—91no—
472C2×C4—92no—
473C2×C4—184no—
474C2×C4—271no—
475C2×C4—271no—
476C2×C4—272no—
477C2×C4—542no—
478C2×C4—542no—
479C2×C4—544no—
480C2×C4C2×D3×D911yescharacteristic
481C2×C3—24no—
482C2×C3—24no—
483C2×C3—21no—
484C2×C3—21no—
485C2×C3—21no—
486C2×C3—44no—
487C2×C3—62no—
488C2×C3#41×D911yescharacteristic
489C2×C3#41×D911yescharacteristic
490C2×C3C23×Dic911yescharacteristic
491C2×C3D3×#23711yescharacteristic
492C2×C3#6431211yescharacteristic
493C2×C3C22×D3×Dic311yescharacteristic
494D3—22no—
495C22—22no—
496C22—22no—
497C22—32no—
498C22—61no—
499C22—61no—
500C22C2×D3×Dic914yes—
501C22C22×D3×D911yescharacteristic
502C4—32no—
503C4—34no—
504C4—92no—
505C4—182no—
506C4—181no—
507C4—272no—
508C4—541no—
509C4—542no—
510C4C2×D3×Dic912yes—
511C3—21no—
512C3#41×Dic911yescharacteristic, complemented
513C3Dic3×#23711yescharacteristic
514C2—24no—
515C2—62no—
516C2D9×#23711yescharacteristic
517C2#1172249811yescharacteristic
518C2C22×D3×Dic911yescharacteristic
subgroupG/H
centerC22C22×D3×D9
commutatorC2×C3×C9C23×C4
FrattiniC22×C3C22×D32
FittingC2×C3×D4×C9C22
socleC22×C32C23×D3
prime radicalC2×C3×#237D3
cosocleC22×C3×C9C24
hypercenterC2×D4D3×D9
hypocenterC3×C9C4×#41
LCSG ▷ C2×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
2. D4×C3⋊Dic9C2semidirect
3. C3×D4×Dic9C2semidirect
4. #67⋊Dic9C2semidirect
5. #67×Dic9C2semidirect
6. #11721244C2semidirect
7. C2×Dic3×Dic9C2semidirect
8. C4×D3×Dic9C2semidirect
9. #11720911C2semidirect
10. Dic6×Dic9C2semidirect
11. #11720906C2semidirect
19. C9×#237C4semidirect
56. D4×Dic9D3semidirect
63. C3×D4×C9C2×C4semidirect
69. C9×#67C2×C4semidirect
70. C2×C9×Dic3C2×C4semidirect
77. C4×D3×C9C2×C4semidirect
78. C9×Dic6C2×C4semidirect
146. C3⋊Dic9#41semidirect
151. C3×Dic9#41semidirect
201. D4×C9C4×D3semidirect
245. #237Dic9direct (N × H)
320. Dic9#237direct (N × H)
332. C3×C9C4×#41semidirect
341. C3×D4C2×Dic9semidirect
354. #67C2×Dic9semidirect
373. C2×Dic3C2×Dic9semidirect
378. C4×D3C2×Dic9semidirect
379. Dic6C2×Dic9semidirect
460. C9C4×#237semidirect
465. D4D3×Dic9semidirect
512. C3#41×Dic9semidirect

Maximal subgroups

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

subgroupindex
C2×C9×#2372
D4×C3⋊Dic92
C3×D4×Dic92
#67⋊Dic92
#67×Dic92
#117212442
C2×Dic3×Dic92
C4×D3×Dic92
#117209112
Dic6×Dic92
#117209062
Dic3×#2373
#41×Dic93