Finite Group Databasebeta

← D4×Dic3×D9 #13150957

Subgroups — D4×Dic3×D9 #13150957

Order 1728.

1372 conjugacy classes of subgroups.

The subgroup lattice has 580 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×D4×D9C211yesmaximal, characteristic
2C22×Dic3×D9C212yesmaximal, complemented
3D4×C3⋊Dic9C211yesmaximal, characteristic, complemented
4D4×C9×Dic3C211yesmaximal, characteristic, complemented
5#611⋊Dic3C212yesmaximal, complemented
6Dic3×#611C212yesmaximal, complemented
7D9×#217C212yesmaximal, complemented
8#11720998C211yesmaximal, characteristic, complemented
9Dic3×D36C211yesmaximal, characteristic, complemented
10D9×C4⋊Dic3C211yesmaximal, characteristic, complemented
11C4×Dic3×D9C211yesmaximal, characteristic, complemented
12D3×D4×Dic3—31nomaximal
13C4×D4×D9—31nomaximal
14C23×C3×D9C2212yes—
15C2×C3×D4×C9C2211yescharacteristic
16C22×C3⋊Dic9C2212yescomplemented
17C2×C3×#611C2212yes—
18C22×C9×Dic3C2212yescomplemented
19C3×D4×D9C418yescomplemented
20C2×C3×D36C2211yescharacteristic
21C2×C3×C4×D9C2211yescharacteristic
22C2×Dic3×D9—28no—
23C2×Dic3×D9C2211yescharacteristic
24C2×Dic3×D9C2212yes—
25C2×Dic3×D9C2211yescharacteristic
26C2×Dic3×D9C2212yescomplemented
27C2×Dic3×D9C2212yescomplemented
28#88551C2212yescomplemented
29C4⋊(C3⋊Dic9)C2211yescharacteristic, complemented
30C4×C3⋊Dic9C2211yescharacteristic, complemented
31C9×#217C2212yescomplemented
32C9×C4⋊Dic3C2211yescharacteristic, complemented
33C4×C9×Dic3C2211yescharacteristic, complemented
34(C2×D9)⋊Dic3C2212yescomplemented
35(C2×D9)⋊Dic3C2212yescomplemented
36Dic9⋊Dic3C2212yescomplemented
37Dic3×Dic9C2211yescharacteristic, complemented
38Dic3×Dic9C2211yescharacteristic, complemented
39C2×C3×D3×D4—31no—
40C22×D3×Dic3—32no—
41D4×C3⋊Dic3—31no—
42C3×D4×Dic3D311yescharacteristic
43#67⋊Dic3—32no—
44Dic3×#67—32no—
45D3×#217—32no—
46#63898—31no—
47Dic3×D12—31no—
48D3×C4⋊Dic3—31no—
49C4×D3×Dic3—31no—
50C2×D4×D9D311yescharacteristic
51C22×C4×D9—32no—
52C4×D4×C9—31no—
53D4×Dic9—31no—
54D4×Dic9—61no—
55C4×#611—32no—
56#63455—31no—
57C4⋊C4×D9—31no—
58#611⋊C4—32no—
59C22⋊C4×D9—32no—
60C4×D36—31no—
61C42×D9—31no—
62C23×C3×C9C2312yes—
63C22×C3×D9—24no—
64C22×C3×D9—24no—
65C22×C3×D9C2×C418yescomplemented
66C22×C3×D9C2312yes—
67C22×C3×D9C2311yescharacteristic
68C3×D4×C9C2×C414yescomplemented
69C2×C3×C4×C9C2311yescharacteristic
70C2×C3⋊Dic9—24no—
71C2×C3⋊Dic9C2312yes—
72C2×C3⋊Dic9C2311yescharacteristic
73C2×C3⋊Dic9C2311yescharacteristic
74C3×#611C2×C418yescomplemented
75C2×C9×Dic3—24no—
76C2×C9×Dic3C2312yes—
77C2×C9×Dic3C2311yescharacteristic
78C2×C9×Dic3C2311yescharacteristic
79C2×C3×Dic9C2311yescharacteristic
80C3×D36C2×C414yescomplemented
81C3×C4×D9C2×C414yescomplemented
82Dic3×D9—24no—
83Dic3×D9—22no—
84Dic3×D9—24no—
85Dic3×D9—24no—
86Dic3×D9D414yesdirect product factor
87C2×D4×Dic3—91no—
88C4×D3×D4—91no—
89C23×C3×D3—32no—
90C2×C32×D4C2×D311yescharacteristic
91C22×C3⋊Dic3—32no—
92C2×C3×#67—32no—
93C22×C3×Dic3C2×D312yes—
94C3×D3×D4—38no—
95C2×C3×D12—31no—
96C2×C3×C4×D3—31no—
97C2×D3×Dic3—31no—
98C2×D3×Dic3—32no—
99C2×D3×Dic3—32no—
100C2×D3×Dic3—32no—
101C2×D3×Dic3—31no—
102C2×D3×Dic3—68no—
103C23×D9C2×D312yes—
104#3757—32no—
105C4⋊(C3⋊Dic3)—31no—
106C4×C3⋊Dic3—31no—
107C3×#217C2×D312yes—
108C3×C4⋊Dic3C2×D311yescharacteristic
109C3×C4×Dic3C2×D311yescharacteristic
110Dic3⋊Dic3—32no—
111(C2×D3)⋊Dic3—32no—
112(C2×D3)⋊Dic3—32no—
113Dic32—31no—
114Dic32—31no—
115C2×D4×C9—21no—
116C2×D4×C9C2×D311yescharacteristic
117C22×C4×C9—32no—
118C2×#611C2×D312yes—
119C22×Dic9—32no—
120C22×Dic9—62no—
121D4×D9Dic318yesdirect product factor
122C2×D36C2×D311yescharacteristic
123C2×C4×D9—31no—
124C2×C4×D9—32no—
125C2×C4×D9—31no—
126C2×C4×D9—32no—
127C2×C4×D9—32no—
128C2×C4×D9—68no—
129C2×C4×D9C2×D311yescharacteristic
130C9×C4⋊C4—31no—
131C9×C22⋊C4—32no—
132C42×C9—31no—
133#3676—32no—
134#3676—62no—
135(C2×D9)⋊C4—32no—
136(C2×D9)⋊C4—32no—
137C4⋊Dic9—31no—
138C4⋊Dic9—61no—
139Dic9⋊C4—32no—
140C4×Dic9—31no—
141C4×Dic9—31no—
142C4×Dic9—31no—
143C4×Dic9—61no—
144C22×C3×C9—22no—
145C22×C3×C9—22no—
146C22×C3×C9C22×C414yescomplemented
147C22×C3×C9C2411yescharacteristic
148C2×C3×D9—22no—
149C2×C3×D9—22no—
150C2×C3×D9—216no—
151C2×C3×D9C22×C412yes—
152C2×C3×D9C22×C414yescomplemented
153C2×C3×D9C2×D412yes—
154C2×C3×D9#4112yes—
155C3×C4×C9C22×C412yescomplemented
156C3⋊Dic9—22no—
157C3⋊Dic9—21no—
158C3⋊Dic9C2×D412yescomplemented
159C9×Dic3—22no—
160C9×Dic3—21no—
161C9×Dic3C2×D412yescomplemented
162C3×Dic9C22×C412yescomplemented
163C22×C3×D4—91no—
164C23×Dic3—92no—
165C2×D3×D4—31no—
166C22×C4×D3—92no—
167C3×C4×D4—31no—
168C2×#217—92no—
169D4×Dic3—91no—
170D4×Dic3—91no—
171D4×Dic3—92no—
172D4×Dic3—92no—
173D4×Dic3—91no—
174D4×Dic3—91no—
175D4×Dic3—181no—
176D4×Dic3D911yescharacteristic, direct product factor
177C4×#67—92no—
178C2×C4⋊Dic3—91no—
179C2×C4×Dic3—91no—
180#893—91no—
181D3×C4⋊C4—91no—
182#67⋊C4—92no—
183D3×C22⋊C4—92no—
184C4×D12—91no—
185C42×D3—91no—
186C23×C32C22×D312yes—
187C22×C3×D3—31no—
188C22×C3×D3—38no—
189C22×C3×D3—32no—
190C22×C3×D3—64no—
191C22×C3×D3—64no—
192C32×D4C4×D314yes—
193C2×C32×C4C22×D311yescharacteristic
194C2×C3⋊Dic3—31no—
195C2×C3⋊Dic3—31no—
196C2×C3⋊Dic3—32no—
197C2×C3⋊Dic3—64no—
198C3×#67—38no—
199C2×C3×Dic3—24no—
200C2×C3×Dic3—31no—
201C2×C3×Dic3C22×D311yescharacteristic
202C2×C3×Dic3C22×D312yes—
203C2×C3×Dic3C22×D311yescharacteristic
204C3×D12—34no—
205C3×C4×D3—34no—
206D3×Dic3—34no—
207D3×Dic3—64no—
208D3×Dic3—62no—
209D3×Dic3—64no—
210D3×Dic3—64no—
211C23×C9—22no—
212C23×C9C22×D312yes—
213C22×D9—24no—
214C22×D9—24no—
215C22×D9C2×Dic318yescomplemented
216C22×D9C22×D312yes—
217C22×D9C22×D311yescharacteristic
218D4×C9—24no—
219D4×C9C2×Dic314yescomplemented
220C2×C4×C9—21no—
221C2×C4×C9—31no—
222C2×C4×C9—31no—
223C2×C4×C9—32no—
224C2×C4×C9—64no—
225C2×C4×C9C22×D311yescharacteristic
226#611C2×Dic318yescomplemented
227C2×Dic9—31no—
228C2×Dic9—31no—
229C2×Dic9—32no—
230C2×Dic9—61no—
231C2×Dic9—61no—
232C2×Dic9—62no—
233C2×Dic9—64no—
234C2×Dic9—124no—
235C2×Dic9C22×D311yescharacteristic
236D36C2×Dic314yescomplemented
237C4×D9—34no—
238C4×D9—64no—
239C4×D9—64no—
240C4×D9—64no—
241C4×D9—62no—
242C4×D9C2×Dic314yescomplemented
243C2×C4×D4—271noSylow
244C2×C3×C9—24no—
245C2×C3×C9C23×C411yescharacteristic
246C2×C3×C9C22×D411yescharacteristic
247C2×C3×C9C2×#4111yescharacteristic
248C3×D9—24no—
249C3×D9C4×D414yescomplemented
250C24×C3—92no—
251C23×D3—32no—
252C2×C3×D4—21no—
253C2×C3×D4—92no—
254C2×C3×D4—98no—
255C2×C3×D4—91no—
256C2×C3×D4C2×D911yescharacteristic
257C2×C3×D4D3211yescharacteristic
258C22×C3×C4—32no—
259C22×C3×C4—91no—
260C2×#67—32no—
261C22×Dic3—91no—
262C22×Dic3—92no—
263C22×Dic3—92no—
264C22×Dic3—91no—
265C22×Dic3—92no—
266C22×Dic3—92no—
267C22×Dic3—92no—
268C22×Dic3—182no—
269C22×Dic3—188no—
270C22×Dic3C2×D912yescomplemented
271D3×D4—38no—
272C2×D12—31no—
273C2×C4×D3—31no—
274C2×C4×D3—91no—
275C2×C4×D3—91no—
276C2×C4×D3—92no—
277C2×C4×D3—92no—
278C2×C4×D3—92no—
279C2×C4×D3—188no—
280C3×C4⋊C4—31no—
281C3×C22⋊C4—32no—
282C3×C42—31no—
283#217—92no—
284#217—92no—
285#217—92no—
286#217—92no—
287#217—182no—
288#217C2×D912yescomplemented
289(C2×D3)⋊C4—92no—
290(C2×D3)⋊C4—92no—
291C4⋊Dic3—91no—
292C4⋊Dic3—91no—
293C4⋊Dic3—92no—
294C4⋊Dic3—181no—
295C4⋊Dic3C2×D911yescharacteristic, complemented
296Dic3⋊C4—92no—
297C4×Dic3—91no—
298C4×Dic3—91no—
299C4×Dic3—91no—
300C4×Dic3—91no—
301C4×Dic3—91no—
302C4×Dic3—91no—
303C4×Dic3—181no—
304C4×Dic3C2×D911yescharacteristic, complemented
305C22×C32—22no—
306C22×C32—22no—
307C22×C32C2×C4×D314yes—
308C22×C32C23×D311yescharacteristic
309C2×C3×D3—32no—
310C2×C3×D3—32no—
311C2×C3×D3—32no—
312C2×C3×D3—34no—
313C2×C3×D3—616no—
314C2×C3×D3—62no—
315C2×C3×D3—62no—
316C32×C4C2×C4×D312yes—
317C3⋊Dic3—32no—
318C3⋊Dic3—62no—
319C3⋊Dic3—61no—
320C3×Dic3—22no—
321C3×Dic3—21no—
322C3×Dic3—32no—
323C3×Dic3D3×D412yes—
324C22×C9—22no—
325C22×C9—22no—
326C22×C9—21no—
327C22×C9—24no—
328C22×C9—42no—
329C22×C9—42no—
330C22×C9C22×Dic314yescomplemented
331C22×C9C23×D311yescharacteristic
332C2×D9—22no—
333C2×D9—22no—
334C2×D9—216no—
335C2×D9D3×D412yes—
336C2×D9#23712yes—
337C2×D9C22×Dic312yes—
338C2×D9C22×Dic314yescomplemented
339C4×C9—22no—
340C4×C9—32no—
341C4×C9—62no—
342C4×C9—61no—
343C4×C9C22×Dic312yescomplemented
344Dic9—32no—
345Dic9—62no—
346Dic9—62no—
347Dic9—61no—
348Dic9—121no—
349Dic9—122no—
350Dic9C22×Dic312yescomplemented
351C22×D4—91no—
352C23×C4—272no—
353C4×D4—31no—
354C4×D4—271no—
355C4×D4—272no—
356C4×D4—271no—
357C4×D4—272no—
358C4×D4—271no—
359C2×C4⋊C4—271no—
360C2×C22⋊C4—272no—
361C2×C42—271no—
362C3×C9C2×C4×D411yescharacteristic, complemented, Sylow
363C23×C3—22no—
364C23×C3—91no—
365C23×C3—98no—
366C23×C3—92no—
367C23×C3—184no—
368C23×C3—184no—
369C23×C3C22×D912yes—
370C23×C3C2×D3212yes—
371C22×D3—31no—
372C22×D3—32no—
373C22×D3—38no—
374C22×D3—64no—
375C22×D3—64no—
376C3×D4—24no—
377C3×D4—98no—
378C3×D4—94no—
379C3×D4C4×D914yescomplemented
380C3×D4D3×Dic314yes—
381C2×C3×C4—21no—
382C2×C3×C4—31no—
383C2×C3×C4—31no—
384C2×C3×C4—32no—
385C2×C3×C4—64no—
386C2×C3×C4—91no—
387C2×C3×C4—94no—
388C2×C3×C4C22×D911yescharacteristic
389C2×C3×C4C2×D3211yescharacteristic
390#67—38no—
391C2×Dic3—24no—
392C2×Dic3—31no—
393C2×Dic3—91no—
394C2×Dic3—91no—
395C2×Dic3—94no—
396C2×Dic3—91no—
397C2×Dic3—92no—
398C2×Dic3—91no—
399C2×Dic3—92no—
400C2×Dic3—181no—
401C2×Dic3—181no—
402C2×Dic3—182no—
403C2×Dic3—184no—
404C2×Dic3—182no—
405C2×Dic3—184no—
406C2×Dic3—184no—
407C2×Dic3—184no—
408C2×Dic3—184no—
409C2×Dic3—364no—
410C2×Dic3C22×D911yescharacteristic
411C2×Dic3C22×D912yes—
412C2×Dic3C22×D911yescharacteristic
413D12—34no—
414C4×D3—34no—
415C4×D3—94no—
416C4×D3—184no—
417C4×D3—184no—
418C4×D3—182no—
419C4×D3—184no—
420C2×C32—24no—
421C2×C32C22×C4×D311yescharacteristic
422C2×C32C2×D3×D411yescharacteristic
423C2×C32D3×#4111yescharacteristic
424C3×D3—34no—
425C3×D3—64no—
426C2×C9—24no—
427C2×C9—21no—
428C2×C9—21no—
429C2×C9—21no—
430C2×C9—44no—
431C2×C9C2×D3×D411yescharacteristic
432C2×C9C2×#23711yescharacteristic
433C2×C9C23×Dic311yescharacteristic
434D9—24no—
435D9D4×Dic314yesdirect product factor
436C24—92no—
437C2×D4—92no—
438C2×D4—98no—
439C2×D4—91no—
440C2×D4D3×D911yescharacteristic
441C22×C4—32no—
442C22×C4—91no—
443C22×C4—271no—
444C22×C4—271no—
445C22×C4—272no—
446C22×C4—272no—
447C22×C4—272no—
448C22×C4—272no—
449C22×C4—548no—
450C4⋊C4—31no—
451C4⋊C4—272no—
452C4⋊C4—271no—
453C22⋊C4—32no—
454C22⋊C4—272no—
455C22⋊C4—272no—
456C22⋊C4—272no—
457C42—31no—
458C42—271no—
459C42—271no—
460C42—271no—
461C22×C3—22no—
462C22×C3—22no—
463C22×C3—22no—
464C22×C3—22no—
465C22×C3—21no—
466C22×C3—24no—
467C22×C3—42no—
468C22×C3—42no—
469C22×C3—92no—
470C22×C3—92no—
471C22×C3—92no—
472C22×C3—94no—
473C22×C3—1816no—
474C22×C3—182no—
475C22×C3—182no—
476C22×C3C2×C4×D914yescomplemented
477C22×C3C23×D911yescharacteristic
478C22×C3C2×D3×Dic314yes—
479C22×C3C22×D3211yescharacteristic
480C2×D3—32no—
481C2×D3—32no—
482C2×D3—32no—
483C2×D3—34no—
484C2×D3—616no—
485C2×D3—62no—
486C2×D3—62no—
487C3×C4—22no—
488C3×C4—32no—
489C3×C4—62no—
490C3×C4—61no—
491C3×C4—92no—
492C3×C4C2×C4×D912yescomplemented
493C3×C4C2×D3×Dic312yes—
494Dic3—22no—
495Dic3—21no—
496Dic3—32no—
497Dic3—92no—
498Dic3—92no—
499Dic3—182no—
500Dic3—182no—
501Dic3—181no—
502Dic3—182no—
503Dic3—181no—
504Dic3—362no—
505Dic3—361no—
506Dic3D4×D912yesdirect product factor
507C32C4×D3×D411yescharacteristic
508C9—21no—
509C9C2×D4×Dic311yescharacteristic, complemented
510C23—91no—
511C23—92no—
512C23—98no—
513C23—184no—
514C23—184no—
515C23C2×D3×D912yes—
516D4—98no—
517D4—94no—
518D4Dic3×D914yesdirect product factor
519C2×C4—31no—
520C2×C4—31no—
521C2×C4—32no—
522C2×C4—64no—
523C2×C4—91no—
524C2×C4—94no—
525C2×C4—274no—
526C2×C4—271no—
527C2×C4—271no—
528C2×C4—272no—
529C2×C4—544no—
530C2×C4—544no—
531C2×C4—544no—
532C2×C4—542no—
533C2×C4—544no—
534C2×C4C2×D3×D911yescharacteristic
535C2×C3—24no—
536C2×C3—24no—
537C2×C3—21no—
538C2×C3—21no—
539C2×C3—21no—
540C2×C3—44no—
541C2×C3—94no—
542C2×C3—184no—
543C2×C3C22×C4×D911yescharacteristic
544C2×C3C2×D4×D911yescharacteristic
545C2×C3#41×D911yescharacteristic
546C2×C3D32×D411yescharacteristic
547C2×C3D3×#23711yescharacteristic
548C2×C3C22×D3×Dic311yescharacteristic
549D3—34no—
550D3—64no—
551C22—22no—
552C22—22no—
553C22—92no—
554C22—92no—
555C22—92no—
556C22—94no—
557C22—1816no—
558C22—182no—
559C22—182no—
560C22C2×Dic3×D914yescomplemented
561C22C22×D3×D911yescharacteristic
562C4—32no—
563C4—61no—
564C4—62no—
565C4—92no—
566C4—272no—
567C4—541no—
568C4—542no—
569C4C2×Dic3×D912yescomplemented
570C3—21no—
571C3C4×D4×D911yescharacteristic, complemented
572C3D3×D4×Dic311yescharacteristic
573C2—24no—
574C2—94no—
575C2—184no—
576C2D3×D4×D911yescharacteristic
577C2D9×#23711yescharacteristic
578C2C22×Dic3×D911yescharacteristic
subgroupG/H
centerC22C22×D3×D9
commutatorC2×C3×C9C23×C4
FrattiniC22×C3C22×D32
FittingC2×C3×D4×C9C22
socleC22×C32C23×D3
prime radicalC2×C3×D4×D9C2
cosocleC22×C3×C9C24
hypercenterC2×D4D3×D9
hypocenterC3×C9C2×C4×D4
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. C22×Dic3×D9C2semidirect
3. D4×C3⋊Dic9C2semidirect
4. D4×C9×Dic3C2semidirect
5. #611⋊Dic3C2semidirect
6. Dic3×#611C2semidirect
7. D9×#217C2semidirect
8. #11720998C2semidirect
9. Dic3×D36C2semidirect
10. D9×C4⋊Dic3C2semidirect
11. C4×Dic3×D9C2semidirect
16. C22×C3⋊Dic9C22semidirect
18. C22×C9×Dic3C22semidirect
19. C3×D4×D9C4semidirect
26. C2×Dic3×D9C22semidirect
27. C2×Dic3×D9C22semidirect
28. #88551C22semidirect
29. C4⋊(C3⋊Dic9)C22semidirect
30. C4×C3⋊Dic9C22semidirect
31. C9×#217C22semidirect
32. C9×C4⋊Dic3C22semidirect
33. C4×C9×Dic3C22semidirect
34. (C2×D9)⋊Dic3C22semidirect
35. (C2×D9)⋊Dic3C22semidirect
36. Dic9⋊Dic3C22semidirect
37. Dic3×Dic9C22semidirect
38. Dic3×Dic9C22semidirect
65. C22×C3×D9C2×C4semidirect
68. C3×D4×C9C2×C4semidirect
74. C3×#611C2×C4semidirect
80. C3×D36C2×C4semidirect
81. C3×C4×D9C2×C4semidirect
86. Dic3×D9D4direct (N × H)
121. D4×D9Dic3direct (N × H)
146. C22×C3×C9C22×C4semidirect
152. C2×C3×D9C22×C4semidirect
155. C3×C4×C9C22×C4semidirect
158. C3⋊Dic9C2×D4semidirect
161. C9×Dic3C2×D4semidirect
162. C3×Dic9C22×C4semidirect
176. D4×Dic3D9direct (N × H)
215. C22×D9C2×Dic3semidirect
219. D4×C9C2×Dic3semidirect
226. #611C2×Dic3semidirect
236. D36C2×Dic3semidirect
242. C4×D9C2×Dic3semidirect
249. C3×D9C4×D4semidirect
270. C22×Dic3C2×D9semidirect
288. #217C2×D9semidirect
295. C4⋊Dic3C2×D9semidirect
304. C4×Dic3C2×D9semidirect
330. C22×C9C22×Dic3semidirect
338. C2×D9C22×Dic3semidirect
343. C4×C9C22×Dic3semidirect
350. Dic9C22×Dic3semidirect
362. C3×C9C2×C4×D4semidirect
379. C3×D4C4×D9semidirect
435. D9D4×Dic3direct (N × H)
476. C22×C3C2×C4×D9semidirect
492. C3×C4C2×C4×D9semidirect
506. Dic3D4×D9direct (N × H)
509. C9C2×D4×Dic3semidirect
518. D4Dic3×D9direct (N × H)
560. C22C2×Dic3×D9semidirect
569. C4C2×Dic3×D9semidirect
571. C3C4×D4×D9semidirect

Maximal subgroups

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

subgroupindex
C2×C3×D4×D92
C22×Dic3×D92
D4×C3⋊Dic92
D4×C9×Dic32
#611⋊Dic32
Dic3×#6112
D9×#2172
#117209982
Dic3×D362
D9×C4⋊Dic32
C4×Dic3×D92
D3×D4×Dic33
C4×D4×D93