Finite Group Databasebeta

← D9×#886 #13148412

Subgroups — D9×#886 #13148412

Order 1728.

1104 conjugacy classes of subgroups.

The subgroup lattice has 586 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×D9×#67C211yesmaximal, characteristic
2C22×Dic3×D9C211yesmaximal, characteristic, complemented
3#11721732C211yesmaximal, characteristic, complemented
4C9×#886C211yesmaximal, characteristic, complemented
5C3×C22⋊C4×D9C211yesmaximal, characteristic, complemented
6#11721266C211yesmaximal, characteristic, complemented
7#11721251C211yesmaximal, characteristic, complemented
8D9×(C2×D3)⋊C4C212yesmaximal, complemented
9#11720989C212yesmaximal, complemented
10#11720988C212yesmaximal, complemented
11D9×C4⋊Dic3C212yesmaximal, complemented
12D3×#886—31nomaximal
13D9×#123—31nomaximal
14C23×C3×D9C2211yescharacteristic
15C22×D3×D9C2211yescharacteristic
16C2×#6233C2211yescharacteristic
17C22×C3⋊Dic9C2211yescharacteristic, complemented
18C2×C9×#67C2211yescharacteristic
19C22×C9×Dic3C2211yescharacteristic, complemented
20C2×C3×C4×D9C2212yes—
21D9×#67—24no—
22C2×#6194C2211yescharacteristic
23C2×#6192C2211yescharacteristic
24C2×Dic3×D9—24no—
25C2×Dic3×D9C2211yescharacteristic
26C2×Dic3×D9C2212yes—
27C2×Dic3×D9C2212yescomplemented
28C3×C9×C22⋊C4C2211yescharacteristic, complemented
29#88536C2212yescomplemented
30C4⋊(C3⋊Dic9)C2212yescomplemented
31C3×#3676C2211yescharacteristic, complemented
32C9×(C2×D3)⋊C4C2212yescomplemented
33C3×(C2×D9)⋊C4C2212yescomplemented
34C9×C4⋊Dic3C2212yescomplemented
35(C2×D3)⋊Dic9C2212yescomplemented
36#88445C2212yescomplemented
37Dic9⋊Dic3C2212yescomplemented
38Dic9⋊Dic3C2212yescomplemented
39C2×D3×#67—31no—
40C22×D3×Dic3—31no—
41#64095—31no—
42C3×#886D311yescharacteristic
43C3×D3×C22⋊C4—31no—
44#63966—31no—
45#63962—31no—
46D3×(C2×D3)⋊C4—32no—
47#63891—32no—
48#63890—32no—
49D3×C4⋊Dic3—32no—
50C2×D4×D9—31no—
51C22×C4×D9—31no—
52C9×#123—31no—
53#63497—31no—
54#63491—31no—
55#63456—32no—
56C4⋊C4×D9—32no—
57#63448—31no—
58#63448—61no—
59#63445—32no—
60C22⋊C4×D9—32no—
61C22⋊C4×D9D311yescharacteristic, complemented
62C23×C3×C9C2311yescharacteristic
63C22×C3⋊D9C2311yescharacteristic
64C22×D3×C9C2311yescharacteristic
65C22×C3×D9—24no—
66C22×C3×D9D414yes—
67C22×C3×D9C2311yescharacteristic
68C22×C3×D9C2311yescharacteristic
69C2×D3×D9—22no—
70C2×D3×D9—24no—
71C2×C3×C4×C9C2312yes—
72#6233—22no—
73C2×C3⋊Dic9—22no—
74C2×C3⋊Dic9C2312yes—
75C2×C3⋊Dic9C2311yescharacteristic
76C9×#67—22no—
77C2×C9×Dic3—22no—
78C2×C9×Dic3C2312yes—
79C2×C9×Dic3C2311yescharacteristic
80C2×C3×Dic9C2312yes—
81C3×C4×D9—24no—
82#6194—22no—
83#6192—22no—
84Dic3×D9—24no—
85Dic3×D9—24no—
86Dic3×D9—22no—
87D3×#123—91no—
88C2×#886—91no—
89C23×C3×D3—31no—
90C22×D32—31no—
91C2×#638—31no—
92C22×C3⋊Dic3—31no—
93C2×C3×#67C2×D311yescharacteristic
94C22×C3×Dic3C2×D311yescharacteristic
95C2×C3×C4×D3—32no—
96D3×#67—64no—
97C2×#626—31no—
98C2×#625—31no—
99C2×D3×Dic3—32no—
100C2×D3×Dic3—32no—
101C2×D3×Dic3—31no—
102C2×D3×Dic3—64no—
103C23×D9—31no—
104C23×D9C2×D311yescharacteristic
105C32×C22⋊C4C2×D311yescharacteristic
106#3752—32no—
107C4⋊(C3⋊Dic3)—32no—
108C3×#217—31no—
109C3×(C2×D3)⋊C4—32no—
110C3×(C2×D3)⋊C4C2×D312yes—
111C3×C4⋊Dic3C2×D312yes—
112Dic3⋊Dic3—32no—
113Dic3⋊Dic3—32no—
114#3722—32no—
115(C2×D3)⋊Dic3—32no—
116C2×D4×C9—31no—
117C22×C4×C9—31no—
118C2×#611—31no—
119C2×#611—31no—
120C2×#611—61no—
121C22×Dic9—31no—
122C22×Dic9—61no—
123D4×D9—64no—
124C2×D36—31no—
125C2×C4×D9—32no—
126C2×C4×D9—31no—
127C2×C4×D9—32no—
128C2×C4×D9—64no—
129C2×C4×D9C2×D312yes—
130C9×C4⋊C4—32no—
131C9×C22⋊C4—21no—
132C9×C22⋊C4—32no—
133C9×C22⋊C4C2×D311yescharacteristic, complemented
134#3676—32no—
135#3676C2×D311yescharacteristic, complemented
136(C2×D9)⋊C4—32no—
137(C2×D9)⋊C4—32no—
138(C2×D9)⋊C4—62no—
139(C2×D9)⋊C4C2×D312yescomplemented
140C4⋊Dic9—32no—
141C4⋊Dic9—62no—
142Dic9⋊C4—32no—
143Dic9⋊C4—32no—
144C22×C3×C9—22no—
145C22×C3×C9C2×D412yes—
146C22×C3×C9C2411yescharacteristic
147C2×C3⋊D9—21no—
148C2×C3⋊D9—22no—
149C2×D3×C9—21no—
150C2×D3×C9—22no—
151C2×C3×D9—22no—
152C2×C3×D9—28no—
153C2×C3×D9C2×D412yes—
154C2×C3×D9C2×D412yes—
155C2×C3×D9#4114yes—
156D3×D9—44no—
157C3×C4×C9—22no—
158C3⋊Dic9—22no—
159C3⋊Dic9—21no—
160C9×Dic3—22no—
161C9×Dic3—21no—
162C3×Dic9—22no—
163C22×#67—91no—
164C23×Dic3—91no—
165C2×D3×D4—91no—
166C22×C4×D3—91no—
167C3×#123—31no—
168C2×C3×C22⋊C4—91no—
169#935—91no—
170#929—91no—
171C2×(C2×D3)⋊C4—92no—
172C2×C4⋊Dic3—92no—
173#894—92no—
174D3×C4⋊C4—92no—
175#886—91no—
176#886—91no—
177#886—91no—
178#886—91no—
179#886—92no—
180#886—92no—
181#886—181no—
182#886D911yescharacteristic, direct product factor
183#883—92no—
184D3×C22⋊C4—31no—
185D3×C22⋊C4—92no—
186C23×C32C22×D311yescharacteristic
187C22×C3⋊D3—31no—
188C22×C3×D3—31no—
189C22×C3×D3—34no—
190C22×C3×D3—31no—
191C22×C3×D3—64no—
192C22×C3×D3C22×D311yescharacteristic
193C2×D32—64no—
194C2×D32—62no—
195C2×C32×C4C22×D312yes—
196#638—62no—
197C2×C3⋊Dic3—32no—
198C2×C3⋊Dic3—31no—
199C2×C3⋊Dic3—62no—
200C3×#67—22no—
201C2×C3×Dic3—22no—
202C2×C3×Dic3—32no—
203C2×C3×Dic3C22×D312yes—
204C2×C3×Dic3C22×D311yescharacteristic
205C3×C4×D3—64no—
206#626—62no—
207#625—62no—
208D3×Dic3—64no—
209D3×Dic3—64no—
210D3×Dic3—62no—
211C23×C9—21no—
212C23×C9—31no—
213C23×C9C22×D311yescharacteristic
214C22×D9—24no—
215C22×D9—31no—
216C22×D9—61no—
217C22×D9—62no—
218C22×D9—64no—
219C22×D9D1214yes—
220C22×D9C22×D311yescharacteristic
221C22×D9C22×D311yescharacteristic
222D4×C9—62no—
223C2×C4×C9—22no—
224C2×C4×C9—31no—
225C2×C4×C9—32no—
226C2×C4×C9—62no—
227C2×C4×C9C22×D312yes—
228#611—62no—
229#611—62no—
230#611—122no—
231C2×Dic9—31no—
232C2×Dic9—32no—
233C2×Dic9—61no—
234C2×Dic9—62no—
235C2×Dic9—62no—
236C2×Dic9—122no—
237C2×Dic9C22×D312yes—
238D36—62no—
239C4×D9—24no—
240C4×D9—64no—
241C4×D9—64no—
242C4×D9—62no—
243C2×#123—271noSylow
244C2×C3×C9—22no—
245C2×C3×C9C22×D411yescharacteristic
246C2×C3×C9C2×#4112yes—
247C3⋊D9—41no—
248D3×C9—41no—
249C3×D9—22no—
250C3×D9#12314yescomplemented
251C24×C3—91no—
252C23×D3—31no—
253C23×D3—91no—
254C23×D3—91no—
255C2×C3×D4—31no—
256C22×C3×C4—31no—
257C22×C3×C4—92no—
258C2×#67—91no—
259C2×#67—91no—
260C2×#67—91no—
261C2×#67—91no—
262C2×#67—91no—
263C2×#67—181no—
264C2×#67—184no—
265C2×#67C2×D911yescharacteristic
266C22×Dic3—91no—
267C22×Dic3—91no—
268C22×Dic3—92no—
269C22×Dic3—91no—
270C22×Dic3—92no—
271C22×Dic3—181no—
272C22×Dic3—184no—
273C22×Dic3C2×D911yescharacteristic, complemented
274D3×D4—184no—
275C2×D12—91no—
276C2×C4×D3—32no—
277C2×C4×D3—92no—
278C2×C4×D3—92no—
279C2×C4×D3—91no—
280C2×C4×D3—184no—
281C3×C4⋊C4—32no—
282C3×C22⋊C4—21no—
283C3×C22⋊C4—32no—
284C3×C22⋊C4—91no—
285C3×C22⋊C4—92no—
286C3×C22⋊C4C2×D911yescharacteristic, complemented
287C3×C22⋊C4D3211yescharacteristic
288#217—31no—
289#217—92no—
290(C2×D3)⋊C4—32no—
291(C2×D3)⋊C4—92no—
292(C2×D3)⋊C4—92no—
293(C2×D3)⋊C4—92no—
294(C2×D3)⋊C4—92no—
295(C2×D3)⋊C4—92no—
296(C2×D3)⋊C4—182no—
297(C2×D3)⋊C4C2×D912yescomplemented
298C4⋊Dic3—92no—
299C4⋊Dic3—92no—
300C4⋊Dic3—92no—
301C4⋊Dic3—92no—
302C4⋊Dic3—182no—
303C4⋊Dic3C2×D912yescomplemented
304Dic3⋊C4—92no—
305Dic3⋊C4—92no—
306C22×C32—22no—
307C22×C32D3×D412yes—
308C22×C32C23×D311yescharacteristic
309C2×C3⋊D3—61no—
310C2×C3⋊D3—62no—
311C2×C3×D3—21no—
312C2×C3×D3—22no—
313C2×C3×D3—32no—
314C2×C3×D3—34no—
315C2×C3×D3—32no—
316C2×C3×D3—68no—
317C2×C3×D3—62no—
318D32—124no—
319C32×C4—22no—
320C3⋊Dic3—62no—
321C3⋊Dic3—61no—
322C3×Dic3—22no—
323C3×Dic3—21no—
324C3×Dic3—62no—
325C22×C9—22no—
326C22×C9—21no—
327C22×C9—22no—
328C22×C9—42no—
329C22×C9—61no—
330C22×C9—62no—
331C22×C9C2×D1212yes—
332C22×C9C23×D311yescharacteristic
333C2×D9—22no—
334C2×D9—28no—
335C2×D9—61no—
336C2×D9—62no—
337C2×D9—121no—
338C2×D9—122no—
339C2×D9—124no—
340C2×D9C2×D1212yes—
341C2×D9C2×D1212yes—
342C2×D9#23714yes—
343C4×C9—22no—
344C4×C9—42no—
345C4×C9—62no—
346C4×C9—61no—
347Dic9—22no—
348Dic9—62no—
349Dic9—61no—
350Dic9—122no—
351Dic9—121no—
352C22×D4—271no—
353C23×C4—271no—
354#123—31no—
355#123—272no—
356#123—272no—
357#123—271no—
358#123—271no—
359#123—271no—
360C2×C4⋊C4—272no—
361C2×C22⋊C4—91no—
362C2×C22⋊C4—272no—
363C3×C9C2×#12311yescharacteristic, complemented, Sylow
364C23×C3—21no—
365C23×C3—31no—
366C23×C3—91no—
367C23×C3—94no—
368C23×C3—91no—
369C23×C3—184no—
370C23×C3C22×D911yescharacteristic
371C23×C3C2×D3211yescharacteristic
372C22×D3—31no—
373C22×D3—31no—
374C22×D3—34no—
375C22×D3—64no—
376C22×D3—91no—
377C22×D3—91no—
378C22×D3—181no—
379C22×D3—182no—
380C22×D3—184no—
381C22×D3—182no—
382C22×D3—184no—
383C22×D3C22×D911yescharacteristic
384C3×D4—62no—
385C2×C3×C4—22no—
386C2×C3×C4—31no—
387C2×C3×C4—32no—
388C2×C3×C4—62no—
389C2×C3×C4—92no—
390C2×C3×C4—184no—
391C2×C3×C4C22×D912yes—
392C2×C3×C4C2×D3212yes—
393#67—22no—
394#67—182no—
395#67—182no—
396#67—182no—
397#67—182no—
398#67—182no—
399#67—362no—
400C2×Dic3—22no—
401C2×Dic3—32no—
402C2×Dic3—91no—
403C2×Dic3—92no—
404C2×Dic3—91no—
405C2×Dic3—92no—
406C2×Dic3—181no—
407C2×Dic3—182no—
408C2×Dic3—184no—
409C2×Dic3—184no—
410C2×Dic3—182no—
411C2×Dic3—182no—
412C2×Dic3—182no—
413C2×Dic3—362no—
414C2×Dic3C22×D911yescharacteristic
415C2×Dic3C22×D912yes—
416D12—182no—
417C4×D3—64no—
418C4×D3—184no—
419C4×D3—184no—
420C4×D3—182no—
421C2×C32—22no—
422C2×C32C2×D3×D411yescharacteristic
423C2×C32D3×#4112yes—
424C3⋊D3—121no—
425C3×D3—34no—
426C3×D3—41no—
427C3×D3—62no—
428C2×C9—22no—
429C2×C9—21no—
430C2×C9—22no—
431C2×C9—42no—
432C2×C9—121no—
433C2×C9C22×D1211yescharacteristic
434C2×C9C2×#23712yes—
435D9—22no—
436D9—121no—
437D9—241no—
438D9#88614yesdirect product factor
439C24—91no—
440C24—271no—
441C2×D4—31no—
442C2×D4—271no—
443C2×D4—271no—
444C2×D4—271no—
445C2×D4—544no—
446C22×C4—31no—
447C22×C4—92no—
448C22×C4—272no—
449C22×C4—271no—
450C22×C4—272no—
451C22×C4—271no—
452C22×C4—544no—
453C4⋊C4—32no—
454C4⋊C4—272no—
455C4⋊C4—272no—
456C4⋊C4—272no—
457C22⋊C4—32no—
458C22⋊C4—91no—
459C22⋊C4—92no—
460C22⋊C4—272no—
461C22⋊C4—272no—
462C22⋊C4—272no—
463C22⋊C4D3×D911yescharacteristic, complemented
464C22×C3—22no—
465C22×C3—22no—
466C22×C3—21no—
467C22×C3—22no—
468C22×C3—42no—
469C22×C3—61no—
470C22×C3—62no—
471C22×C3—92no—
472C22×C3—94no—
473C22×C3—92no—
474C22×C3—188no—
475C22×C3—182no—
476C22×C3D4×D912yes—
477C22×C3C23×D911yescharacteristic
478C22×C3D3×D1212yes—
479C22×C3C22×D3211yescharacteristic
480C2×D3—21no—
481C2×D3—22no—
482C2×D3—32no—
483C2×D3—34no—
484C2×D3—32no—
485C2×D3—68no—
486C2×D3—62no—
487C2×D3—181no—
488C2×D3—182no—
489C2×D3—181no—
490C2×D3—182no—
491C2×D3—361no—
492C2×D3—362no—
493C2×D3—364no—
494C2×D3—364no—
495C3×C4—22no—
496C3×C4—22no—
497C3×C4—42no—
498C3×C4—62no—
499C3×C4—61no—
500C3×C4—182no—
501Dic3—22no—
502Dic3—21no—
503Dic3—62no—
504Dic3—182no—
505Dic3—181no—
506Dic3—182no—
507Dic3—181no—
508Dic3—362no—
509Dic3—361no—
510C32D3×#12311yescharacteristic
511C9—21no—
512C9C2×#88611yescharacteristic, complemented
513C23—31no—
514C23—91no—
515C23—91no—
516C23—94no—
517C23—184no—
518C23—271no—
519C23—542no—
520C23—544no—
521C23C2×D3×D911yescharacteristic
522D4—62no—
523D4—542no—
524D4—542no—
525D4—542no—
526C2×C4—32no—
527C2×C4—31no—
528C2×C4—62no—
529C2×C4—92no—
530C2×C4—184no—
531C2×C4—272no—
532C2×C4—271no—
533C2×C4—544no—
534C2×C4—544no—
535C2×C4—542no—
536C2×C4—542no—
537C2×C4C2×D3×D912yes—
538C2×C3—22no—
539C2×C3—22no—
540C2×C3—22no—
541C2×C3—21no—
542C2×C3—42no—
543C2×C3—94no—
544C2×C3—121no—
545C2×C3—182no—
546C2×C3C2×D4×D911yescharacteristic
547C2×C3#41×D912yes—
548C2×C3C2×D3×D1211yescharacteristic
549C2×C3D3×#23712yes—
550D3—34no—
551D3—41no—
552D3—62no—
553D3—361no—
554D3—361no—
555D3—721no—
556C22—22no—
557C22—62no—
558C22—61no—
559C22—94no—
560C22—92no—
561C22—92no—
562C22—188no—
563C22—182no—
564C22—542no—
565C22—541no—
566C22—1084no—
567C22D9×D1212yes—
568C22C22×D3×D911yescharacteristic
569C4—22no—
570C4—62no—
571C4—61no—
572C4—182no—
573C4—542no—
574C4—541no—
575C3—21no—
576C3D9×#12311yescharacteristic, complemented
577C3D3×#88611yescharacteristic
578C2—22no—
579C2—94no—
580C2—121no—
581C2—182no—
582C2—1081no—
583C2C2×D9×D1211yescharacteristic
584C2D9×#23712yes—
subgroupG/H
centerC22C22×D3×D9
commutator, cosocleC22×C3×C9C24
FrattiniC22×C3C22×D32
FittingC3×C9×C22⋊C4C22
socleC22×C32C23×D3
prime radicalC2×D9×#67C2
hypercenterC22⋊C4D3×D9
hypocenterC3×C9C2×#123
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. C22×Dic3×D9C2semidirect
3. #11721732C2semidirect
4. C9×#886C2semidirect
5. C3×C22⋊C4×D9C2semidirect
6. #11721266C2semidirect
7. #11721251C2semidirect
8. D9×(C2×D3)⋊C4C2semidirect
9. #11720989C2semidirect
10. #11720988C2semidirect
11. D9×C4⋊Dic3C2semidirect
17. C22×C3⋊Dic9C22semidirect
19. C22×C9×Dic3C22semidirect
27. C2×Dic3×D9C22semidirect
28. C3×C9×C22⋊C4C22semidirect
29. #88536C22semidirect
30. C4⋊(C3⋊Dic9)C22semidirect
31. C3×#3676C22semidirect
32. C9×(C2×D3)⋊C4C22semidirect
33. C3×(C2×D9)⋊C4C22semidirect
34. C9×C4⋊Dic3C22semidirect
35. (C2×D3)⋊Dic9C22semidirect
36. #88445C22semidirect
37. Dic9⋊Dic3C22semidirect
38. Dic9⋊Dic3C22semidirect
61. C22⋊C4×D9D3semidirect
133. C9×C22⋊C4C2×D3semidirect
135. #3676C2×D3semidirect
139. (C2×D9)⋊C4C2×D3semidirect
182. #886D9direct (N × H)
250. C3×D9#123semidirect
273. C22×Dic3C2×D9semidirect
286. C3×C22⋊C4C2×D9semidirect
297. (C2×D3)⋊C4C2×D9semidirect
303. C4⋊Dic3C2×D9semidirect
363. C3×C9C2×#123semidirect
438. D9#886direct (N × H)
463. C22⋊C4D3×D9semidirect
512. C9C2×#886semidirect
576. C3D9×#123semidirect

Maximal subgroups

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

subgroupindex
C2×D9×#672
C22×Dic3×D92
#117217322
C9×#8862
C3×C22⋊C4×D92
#117212662
#117212512
D9×(C2×D3)⋊C42
#117209892
#117209882
D9×C4⋊Dic32
D3×#8863
D9×#1233