Finite Group Databasebeta

← D9×#924 #13150175

Subgroups — D9×#924 #13150175

Order 1728.

1040 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
1C22×C3×C4×D9C211yesmaximal, characteristic
2C2×D9×Dic6C211yesmaximal, characteristic, complemented
3#11721813C211yesmaximal, characteristic, complemented
4C9×#924C211yesmaximal, characteristic, complemented
5D9×#217C212yesmaximal
6#11721253C212yesmaximal, complemented
7D9×C4⋊Dic3C211yesmaximal, characteristic, complemented
8D9×Dic3⋊C4C212yesmaximal, complemented
9#11720945C211yesmaximal, characteristic, complemented
10#11720944C211yesmaximal, characteristic, complemented
11#11720939C212yesmaximal, complemented
12D3×#924—31nomaximal
13D9×C22⋊Q8—31nomaximal
14C23×C3×D9C2211yescharacteristic
15C22×C3×C4×C9C2211yescharacteristic
16C2×#6226C2211yescharacteristic, complemented
17C22×C3×Dic9C2211yescharacteristic
18C2×C3×C4×D9—24no—
19C2×C3×C4×D9C2211yescharacteristic
20C2×C3×C4×D9C2211yescharacteristic
21C2×C3×C4×D9C2211yescharacteristic
22C2×C3×C4×D9C2211yescharacteristic
23C2×C9×Dic6C2211yescharacteristic, complemented
24C2×Dic3×D9C2212yes—
25C2×Dic3×D9C2212yes—
26C2×#6189C2212yescomplemented
27D9×Dic6—24no—
28#88551C2212yes—
29C4⋊(C3⋊Dic9)C2211yescharacteristic, complemented
30#88534C2212yescomplemented
31C9×#217C2212yes—
32C9×C4⋊Dic3C2211yescharacteristic, complemented
33C9×Dic3⋊C4C2212yescomplemented
34(C2×D9)⋊Dic3C2212yes—
35(C2×D9)⋊Dic3C2212yes—
36Dic3⋊Dic9C2212yescomplemented
37#88442C2212yescomplemented
38Dic9⋊Dic3C2212yescomplemented
39C22×C3×C4×D3—31no—
40C2×D3×Dic6—31no—
41#64133—31no—
42C3×#924D311yescharacteristic
43D3×#217—32no—
44#63964—32no—
45D3×C4⋊Dic3—31no—
46D3×Dic3⋊C4—32no—
47#63857—31no—
48#63856—31no—
49#63851—32no—
50C2×Q8×D9—31no—
51C22×C4×D9D311yescharacteristic
52C9×C22⋊Q8—31no—
53#63508—31no—
54#63486—31no—
55#63486—61no—
56#63459—31no—
57#63458—32no—
58C4⋊C4×D9—32no—
59C4⋊C4×D9—31no—
60C22⋊C4×D9—32no—
61#63440—32no—
62C23×C3×C9C2311yescharacteristic
63C22×C3×D9—22no—
64C22×C3×D9—22no—
65C22×C3×D9Q814yescomplemented
66C22×C3×D9C2311yescharacteristic
67C22×C3×D9C2311yescharacteristic
68C2×C3×C4×C9—22no—
69C2×C3×C4×C9C2311yescharacteristic
70C2×C3×C4×C9C2311yescharacteristic
71C2×C3⋊Dic9C2312yes—
72C2×C3⋊Dic9C2312yes—
73#6226—22no—
74C2×C9×Dic3C2312yes—
75C2×C9×Dic3C2312yes—
76C2×C3×Dic9—22no—
77C2×C3×Dic9C2311yescharacteristic
78C2×C3×Dic9C2311yescharacteristic
79C3×C4×D9—22no—
80C3×C4×D9—22no—
81C3×C4×D9—22no—
82C3×C4×D9D414yes—
83C9×Dic6—22no—
84Dic3×D9—24no—
85Dic3×D9—24no—
86#6189—24no—
87C2×#924—91no—
88D3×C22⋊Q8—91no—
89C23×C3×D3—31no—
90C22×C32×C4C2×D311yescharacteristic
91C2×#634—31no—
92C22×C3×Dic3—31no—
93C2×C3×C4×D3—31no—
94C2×C3×C4×D3—31no—
95C2×C3×C4×D3—31no—
96C2×C3×C4×D3—31no—
97C2×C3×C4×D3—64no—
98C2×C3×Dic6C2×D311yescharacteristic
99C2×#627—32no—
100C2×D3×Dic3—32no—
101C2×D3×Dic3—32no—
102D3×Dic6—64no—
103C23×D9C2×D311yescharacteristic
104#3757—32no—
105C4⋊(C3⋊Dic3)—31no—
106#3750—32no—
107C3×#217C2×D312yes—
108C3×C4⋊Dic3C2×D311yescharacteristic
109C3×Dic3⋊C4C2×D312yes—
110#3724—32no—
111Dic3⋊Dic3—32no—
112Dic3⋊Dic3—32no—
113(C2×D3)⋊Dic3—32no—
114(C2×D3)⋊Dic3—32no—
115C2×Q8×C9—31no—
116C22×C4×C9—21no—
117C22×C4×C9C2×D311yescharacteristic
118C22×Dic9C2×D311yescharacteristic
119Q8×D9—64no—
120C2×C4×D9—24no—
121C2×C4×D9—32no—
122C2×C4×D9—32no—
123C2×C4×D9C2×D311yescharacteristic
124C2×C4×D9C2×D311yescharacteristic
125C2×C4×D9C2×D311yescharacteristic
126C2×C4×D9C2×D311yescharacteristic
127C2×Dic18—31no—
128C2×Dic18—32no—
129C2×Dic18—61no—
130C9×C4⋊C4—32no—
131C9×C4⋊C4—31no—
132C9×C22⋊C4—32no—
133#3676—32no—
134#3676—62no—
135(C2×D9)⋊C4—32no—
136(C2×D9)⋊C4—32no—
137C4⋊Dic9—31no—
138C4⋊Dic9—32no—
139C4⋊Dic9—61no—
140Dic9⋊C4—32no—
141Dic9⋊C4—32no—
142Dic9⋊C4—32no—
143Dic9⋊C4—62no—
144C22×C3×C9—21no—
145C22×C3×C9—21no—
146C22×C3×C9C2×Q812yescomplemented
147C22×C3×C9C2411yescharacteristic
148C2×C3×D9—21no—
149C2×C3×D9—21no—
150C2×C3×D9—28no—
151C2×C3×D9C2×D412yes—
152C2×C3×D9C2×Q812yes—
153C2×C3×D9C2×Q812yescomplemented
154C2×C3×D9#4112yes—
155C3×C4×C9—21no—
156C3×C4×C9C2×D412yes—
157C3⋊Dic9—22no—
158C3⋊Dic9—22no—
159C9×Dic3—22no—
160C9×Dic3—22no—
161C3×Dic9—21no—
162C3×Dic9C2×D412yes—
163C23×C3×C4—91no—
164C2×D3×Q8—91no—
165C22×C4×D3—31no—
166C22×Dic6—91no—
167C3×C22⋊Q8—31no—
168C2×#217—92no—
169#946—91no—
170C2×C4⋊Dic3—91no—
171#924—91no—
172#924—91no—
173#924—91no—
174#924—92no—
175#924—91no—
176#924—92no—
177#924—181no—
178#924D911yescharacteristic, direct product factor
179C2×Dic3⋊C4—92no—
180#897—91no—
181#896—92no—
182D3×C4⋊C4—92no—
183D3×C4⋊C4—91no—
184D3×C22⋊C4—92no—
185#878—92no—
186C23×C32C22×D311yescharacteristic
187C22×C3×D3—31no—
188C22×C3×D3—34no—
189C22×C3×D3—31no—
190C22×C3×D3—62no—
191C22×C3×D3—62no—
192C2×C32×C4—22no—
193C2×C32×C4C22×D311yescharacteristic
194C2×C32×C4C22×D311yescharacteristic
195C2×C3⋊Dic3—32no—
196C2×C3⋊Dic3—32no—
197#634—62no—
198C2×C3×Dic3—31no—
199C2×C3×Dic3—31no—
200C2×C3×Dic3—62no—
201C2×C3×Dic3C22×D312yes—
202C2×C3×Dic3C22×D312yes—
203C3×C4×D3—34no—
204C3×C4×D3—62no—
205C3×C4×D3—62no—
206C3×C4×D3—62no—
207C3×Dic6—22no—
208#627—64no—
209D3×Dic3—64no—
210D3×Dic3—64no—
211C23×C9—21no—
212C23×C9C22×D311yescharacteristic
213C22×D9—22no—
214C22×D9—22no—
215C22×D9Dic614yescomplemented
216C22×D9C22×D311yescharacteristic
217C22×D9C22×D311yescharacteristic
218Q8×C9—62no—
219C2×C4×C9—22no—
220C2×C4×C9—21no—
221C2×C4×C9—21no—
222C2×C4×C9—32no—
223C2×C4×C9—32no—
224C2×C4×C9—42no—
225C2×C4×C9C22×D311yescharacteristic
226C2×C4×C9C22×D311yescharacteristic
227C2×Dic9—22no—
228C2×Dic9—32no—
229C2×Dic9—32no—
230C2×Dic9—62no—
231C2×Dic9—62no—
232C2×Dic9C22×D311yescharacteristic
233C2×Dic9C22×D311yescharacteristic
234C4×D9—22no—
235C4×D9—22no—
236C4×D9—22no—
237C4×D9—64no—
238C4×D9—64no—
239C4×D9#6714yes—
240Dic18—62no—
241Dic18—64no—
242Dic18—122no—
243C2×C22⋊Q8—271noSylow
244C2×C3×C9—22no—
245C2×C3×C9C22×D411yescharacteristic
246C2×C3×C9C22×Q811yescharacteristic
247C2×C3×C9C2×#4111yescharacteristic
248C3×D9—22no—
249C3×D9C22⋊Q814yescomplemented
250C24×C3—91no—
251C23×D3—31no—
252C2×C3×Q8—31no—
253C22×C3×C4—21no—
254C22×C3×C4—91no—
255C22×C3×C4—91no—
256C22×C3×C4—91no—
257C22×C3×C4—91no—
258C22×C3×C4—91no—
259C22×C3×C4—184no—
260C22×C3×C4C2×D911yescharacteristic
261C22×C3×C4D3211yescharacteristic
262C22×Dic3—31no—
263C22×Dic3—92no—
264C22×Dic3—92no—
265D3×Q8—184no—
266C2×C4×D3—31no—
267C2×C4×D3—31no—
268C2×C4×D3—31no—
269C2×C4×D3—31no—
270C2×C4×D3—64no—
271C2×C4×D3—92no—
272C2×C4×D3—92no—
273C2×Dic6—92no—
274C2×Dic6—91no—
275C2×Dic6—91no—
276C2×Dic6—92no—
277C2×Dic6—181no—
278C2×Dic6—184no—
279C2×Dic6C2×D911yescharacteristic, complemented
280C3×C4⋊C4—32no—
281C3×C4⋊C4—31no—
282C3×C22⋊C4—32no—
283#217—92no—
284#217—92no—
285#217—92no—
286#217—92no—
287#217—182no—
288#217C2×D912yes—
289(C2×D3)⋊C4—92no—
290(C2×D3)⋊C4—92no—
291C4⋊Dic3—92no—
292C4⋊Dic3—91no—
293C4⋊Dic3—91no—
294C4⋊Dic3—92no—
295C4⋊Dic3—181no—
296C4⋊Dic3C2×D911yescharacteristic, complemented
297Dic3⋊C4—92no—
298Dic3⋊C4—92no—
299Dic3⋊C4—92no—
300Dic3⋊C4—92no—
301Dic3⋊C4—92no—
302Dic3⋊C4—92no—
303Dic3⋊C4—182no—
304Dic3⋊C4C2×D912yescomplemented
305C22×C32—21no—
306C22×C32—21no—
307C22×C32D3×Q812yes—
308C22×C32C23×D311yescharacteristic
309C2×C3×D3—32no—
310C2×C3×D3—32no—
311C2×C3×D3—32no—
312C2×C3×D3—32no—
313C2×C3×D3—68no—
314C2×C3×D3—61no—
315C2×C3×D3—61no—
316C32×C4—21no—
317C32×C4D3×D412yes—
318C3⋊Dic3—62no—
319C3⋊Dic3—62no—
320C3×Dic3—22no—
321C3×Dic3—22no—
322C3×Dic3—32no—
323C3×Dic3—61no—
324C22×C9—21no—
325C22×C9—21no—
326C22×C9—21no—
327C22×C9—22no—
328C22×C9—41no—
329C22×C9—41no—
330C22×C9C2×Dic612yescomplemented
331C22×C9C23×D311yescharacteristic
332C2×D9—21no—
333C2×D9—21no—
334C2×D9—28no—
335C2×D9C2×Dic612yes—
336C2×D9C2×Dic612yescomplemented
337C2×D9#23512yes—
338C2×D9C2×#6712yes—
339C4×C9—21no—
340C4×C9—22no—
341C4×C9—41no—
342C4×C9—62no—
343C4×C9—62no—
344C4×C9C2×#6712yes—
345Dic9—21no—
346Dic9—62no—
347Dic9—62no—
348Dic9—122no—
349Dic9—122no—
350Dic9C2×#6712yes—
351C22×Q8—271no—
352C23×C4—91no—
353C22⋊Q8—31no—
354C22⋊Q8—272no—
355C22⋊Q8—272no—
356C22⋊Q8—271no—
357C22⋊Q8—271no—
358C22⋊Q8—271no—
359C2×C4⋊C4—272no—
360C2×C4⋊C4—271no—
361C2×C22⋊C4—272no—
362C3×C9C2×C22⋊Q811yescharacteristic, complemented, Sylow
363C23×C3—21no—
364C23×C3—91no—
365C23×C3—94no—
366C23×C3—91no—
367C23×C3—182no—
368C23×C3—182no—
369C23×C3C22×D911yescharacteristic
370C23×C3C2×D3211yescharacteristic
371C22×D3—31no—
372C22×D3—31no—
373C22×D3—34no—
374C22×D3—62no—
375C22×D3—62no—
376C3×Q8—62no—
377C2×C3×C4—22no—
378C2×C3×C4—22no—
379C2×C3×C4—21no—
380C2×C3×C4—21no—
381C2×C3×C4—32no—
382C2×C3×C4—32no—
383C2×C3×C4—42no—
384C2×C3×C4—91no—
385C2×C3×C4—94no—
386C2×C3×C4—91no—
387C2×C3×C4—182no—
388C2×C3×C4—182no—
389C2×C3×C4—182no—
390C2×C3×C4—182no—
391C2×C3×C4C22×D911yescharacteristic
392C2×C3×C4C22×D911yescharacteristic
393C2×C3×C4C2×D3211yescharacteristic
394C2×C3×C4C2×D3211yescharacteristic
395C2×Dic3—31no—
396C2×Dic3—31no—
397C2×Dic3—62no—
398C2×Dic3—92no—
399C2×Dic3—92no—
400C2×Dic3—92no—
401C2×Dic3—92no—
402C2×Dic3—182no—
403C2×Dic3—182no—
404C2×Dic3—184no—
405C2×Dic3—184no—
406C2×Dic3C22×D912yes—
407C2×Dic3C22×D912yes—
408C4×D3—34no—
409C4×D3—62no—
410C4×D3—62no—
411C4×D3—62no—
412C4×D3—184no—
413C4×D3—184no—
414Dic6—22no—
415Dic6—184no—
416Dic6—184no—
417Dic6—182no—
418Dic6—182no—
419Dic6—362no—
420C2×C32—22no—
421C2×C32C2×D3×D411yescharacteristic
422C2×C32C2×D3×Q811yescharacteristic
423C2×C32D3×#4111yescharacteristic
424C3×D3—34no—
425C3×D3—62no—
426C2×C9—22no—
427C2×C9—21no—
428C2×C9—21no—
429C2×C9—21no—
430C2×C9—42no—
431C2×C9C22×Dic611yescharacteristic
432C2×C9C2×#23511yescharacteristic
433C2×C9C22×#6711yescharacteristic
434D9—22no—
435D9#92414yesdirect product factor
436C24—91no—
437C2×Q8—31no—
438C2×Q8—272no—
439C2×Q8—271no—
440C2×Q8—544no—
441C22×C4—91no—
442C22×C4—91no—
443C22×C4—91no—
444C22×C4—91no—
445C22×C4—91no—
446C22×C4—184no—
447C22×C4—272no—
448C22×C4—272no—
449C22×C4D3×D911yescharacteristic
450C4⋊C4—32no—
451C4⋊C4—31no—
452C4⋊C4—272no—
453C4⋊C4—272no—
454C4⋊C4—272no—
455C4⋊C4—272no—
456C4⋊C4—271no—
457C22⋊C4—32no—
458C22⋊C4—272no—
459C22⋊C4—272no—
460C22⋊C4—272no—
461C22×C3—21no—
462C22×C3—21no—
463C22×C3—21no—
464C22×C3—21no—
465C22×C3—22no—
466C22×C3—21no—
467C22×C3—41no—
468C22×C3—41no—
469C22×C3—92no—
470C22×C3—92no—
471C22×C3—92no—
472C22×C3—92no—
473C22×C3—188no—
474C22×C3—181no—
475C22×C3—181no—
476C22×C3Q8×D912yescomplemented
477C22×C3C23×D911yescharacteristic
478C22×C3D3×Dic612yes—
479C22×C3C22×D3211yescharacteristic
480C2×D3—32no—
481C2×D3—32no—
482C2×D3—32no—
483C2×D3—32no—
484C2×D3—68no—
485C2×D3—61no—
486C2×D3—61no—
487C3×C4—21no—
488C3×C4—21no—
489C3×C4—22no—
490C3×C4—41no—
491C3×C4—62no—
492C3×C4—62no—
493C3×C4—92no—
494C3×C4—181no—
495C3×C4D4×D912yes—
496C3×C4D3×#6712yes—
497Dic3—22no—
498Dic3—22no—
499Dic3—32no—
500Dic3—61no—
501Dic3—182no—
502Dic3—182no—
503Dic3—182no—
504Dic3—182no—
505Dic3—362no—
506Dic3—362no—
507C32D3×C22⋊Q811yescharacteristic
508C9—21no—
509C9C2×#92411yescharacteristic, complemented
510C23—91no—
511C23—94no—
512C23—91no—
513C23—182no—
514C23—182no—
515C23C2×D3×D911yescharacteristic
516Q8—62no—
517Q8—544no—
518Q8—542no—
519C2×C4—22no—
520C2×C4—32no—
521C2×C4—32no—
522C2×C4—91no—
523C2×C4—94no—
524C2×C4—91no—
525C2×C4—182no—
526C2×C4—182no—
527C2×C4—182no—
528C2×C4—182no—
529C2×C4—272no—
530C2×C4—272no—
531C2×C4—544no—
532C2×C4—544no—
533C2×C4C2×D3×D911yescharacteristic
534C2×C4C2×D3×D911yescharacteristic
535C2×C3—22no—
536C2×C3—22no—
537C2×C3—21no—
538C2×C3—21no—
539C2×C3—21no—
540C2×C3—42no—
541C2×C3—94no—
542C2×C3—182no—
543C2×C3C2×D4×D911yescharacteristic
544C2×C3C2×Q8×D911yescharacteristic
545C2×C3#41×D911yescharacteristic
546C2×C3C2×D3×Dic611yescharacteristic
547C2×C3D3×#23511yescharacteristic
548C2×C3C2×D3×#6711yescharacteristic
549D3—34no—
550D3—62no—
551C22—21no—
552C22—21no—
553C22—92no—
554C22—92no—
555C22—92no—
556C22—92no—
557C22—188no—
558C22—181no—
559C22—181no—
560C22D9×Dic612yescomplemented
561C22C22×D3×D911yescharacteristic
562C4—21no—
563C4—62no—
564C4—62no—
565C4—92no—
566C4—181no—
567C4—542no—
568C4—542no—
569C4D9×#6712yes—
570C3—21no—
571C3D9×C22⋊Q811yescharacteristic, complemented
572C3D3×#92411yescharacteristic
573C2—22no—
574C2—94no—
575C2—182no—
576C2C2×D9×Dic611yescharacteristic
577C2D9×#23511yescharacteristic
578C2C2×D9×#6711yescharacteristic
subgroupG/H
centerC22C22×D3×D9
commutator, cosocleC22×C3×C9C24
FrattiniC22×C3C22×D32
FittingC22×C3×C4×C9C22
socleC22×C32C23×D3
prime radicalC23×C3×D9C22
hypercenterC22×C4D3×D9
hypocenterC3×C9C2×C22⋊Q8
LCSG ▷ C22×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
2. C2×D9×Dic6C2semidirect
3. #11721813C2semidirect
4. C9×#924C2semidirect
6. #11721253C2semidirect
7. D9×C4⋊Dic3C2semidirect
8. D9×Dic3⋊C4C2semidirect
9. #11720945C2semidirect
10. #11720944C2semidirect
11. #11720939C2semidirect
16. C2×#6226C22semidirect
23. C2×C9×Dic6C22semidirect
26. C2×#6189C22semidirect
29. C4⋊(C3⋊Dic9)C22semidirect
30. #88534C22semidirect
32. C9×C4⋊Dic3C22semidirect
33. C9×Dic3⋊C4C22semidirect
36. Dic3⋊Dic9C22semidirect
37. #88442C22semidirect
38. Dic9⋊Dic3C22semidirect
65. C22×C3×D9Q8semidirect
146. C22×C3×C9C2×Q8semidirect
153. C2×C3×D9C2×Q8semidirect
178. #924D9direct (N × H)
215. C22×D9Dic6semidirect
249. C3×D9C22⋊Q8semidirect
279. C2×Dic6C2×D9semidirect
296. C4⋊Dic3C2×D9semidirect
304. Dic3⋊C4C2×D9semidirect
330. C22×C9C2×Dic6semidirect
336. C2×D9C2×Dic6semidirect
362. C3×C9C2×C22⋊Q8semidirect
435. D9#924direct (N × H)
476. C22×C3Q8×D9semidirect
509. C9C2×#924semidirect
560. C22D9×Dic6semidirect
571. C3D9×C22⋊Q8semidirect

Maximal subgroups

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

subgroupindex
C22×C3×C4×D92
C2×D9×Dic62
#117218132
C9×#9242
D9×#2172
#117212532
D9×C4⋊Dic32
D9×Dic3⋊C42
#117209452
#117209442
#117209392
D3×#9243
D9×C22⋊Q83