Finite Group Databasebeta

← #13148269

Subgroups — #13148269

Order 1728.

1090 conjugacy classes of subgroups.

The subgroup lattice has 568 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#6194C211yesmaximal, characteristic
2#11721723C211yesmaximal, characteristic, complemented
3D3×C9×C22⋊C4C211yesmaximal, characteristic, complemented
4C3×C22⋊C4×D9C211yesmaximal, characteristic, complemented
5D3×#3676C211yesmaximal, characteristic, complemented
6D9×#217C211yesmaximal, characteristic, complemented
7#11721248C211yesmaximal, characteristic, complemented
8#11721028C212yesmaximal, complemented
9C4×#6194C212yesmaximal, complemented
10#11721012C212yesmaximal, complemented
11#11720976C212yesmaximal, complemented
12#10594181—31nomaximal
13#10589588—31nomaximal
14C23×D3×C9C2211yescharacteristic
15C23×C3×D9C2211yescharacteristic
16C22×C3⋊Dic9C2211yescharacteristic
17C2×C4×D3×C9C2212yescomplemented
18C2×C3×C4×D9C2212yescomplemented
19C2×#6194C418yescomplemented
20C2×#6194C2211yescharacteristic
21C2×#6194C2211yescharacteristic
22C2×#6194C2211yescharacteristic
23C2×#6194C2211yescharacteristic
24C2×D3×Dic9C2212yescomplemented
25C2×Dic3×D9C2212yescomplemented
26C3×C9×C22⋊C4C2211yescharacteristic
27#88551C2211yescharacteristic
28#88534C2212yescomplemented
29C4×C3⋊Dic9C2212yescomplemented
30C9×#217C2211yescharacteristic
31C3×#3676C2211yescharacteristic
32C9×(C2×D3)⋊C4C2212yescomplemented
33C3×(C2×D9)⋊C4C2212yescomplemented
34(C2×D3)⋊Dic9C2212yescomplemented
35(C2×D9)⋊Dic3C2212yescomplemented
36#88442C2212yescomplemented
37Dic3×Dic9C2212yescomplemented
38C22×#625—31no—
39#64086—31no—
40C3×D3×C22⋊C4—31no—
41C3×D3×C22⋊C4D311yescharacteristic
42D3×#217—31no—
43D3×#217—31no—
44#63957—31no—
45#63902—32no—
46C4×#625—32no—
47#63879—32no—
48#63879—32no—
49C22×#611—31no—
50C2×C9×C22⋊C4—31no—
51C2×#3676—31no—
52C4×#611—32no—
53C4×#611—32no—
54#611⋊C4—32no—
55#611⋊C4—32no—
56C22⋊C4×D9—31no—
57C22⋊C4×D9D311yescharacteristic, complemented
58#63439—31no—
59#63439—31no—
60#63439—61no—
61C23×C3×C9C2311yescharacteristic
62C22×D3×C9—42no—
63C22×D3×C9C2×C414yescomplemented
64C22×D3×C9C2311yescharacteristic
65C22×D3×C9C2311yescharacteristic
66C22×C3×D9—42no—
67C22×C3×D9C2×C414yescomplemented
68C22×C3×D9C2311yescharacteristic
69C22×C3×D9C2311yescharacteristic
70C2×C3×C4×C9C2312yes—
71C2×C3⋊Dic9C2×C414yescomplemented
72C2×C3⋊Dic9C2312yes—
73C2×C3⋊Dic9C2311yescharacteristic
74C2×C3⋊Dic9C2311yescharacteristic
75C2×C9×Dic3C2312yes—
76C2×C3×Dic9C2312yes—
77C4×D3×C9—42no—
78C3×C4×D9—42no—
79#6194C2×C414yescomplemented
80#6194C2×C414yescomplemented
81#6194C2×C414yescomplemented
82#6194C2×C414yescomplemented
83D3×Dic9—42no—
84Dic3×D9—42no—
85#5486—91no—
86#5486—91no—
87C23×C3×D3—31no—
88C23×C3×D3C2×D311yescharacteristic
89C22×C3⋊Dic3—31no—
90C2×C3×C4×D3—32no—
91C2×C3×C4×D3C2×D312yes—
92C2×#625—38no—
93C2×#625—31no—
94C2×#625—31no—
95C2×#625—31no—
96C2×#625—31no—
97C2×D3×Dic3—32no—
98C2×D3×Dic3—32no—
99C24×C9—31no—
100C23×D9C2×D311yescharacteristic
101C32×C22⋊C4C2×D311yescharacteristic
102#3757—31no—
103#3750—32no—
104C4×C3⋊Dic3—32no—
105C3×#217—31no—
106C3×#217C2×D311yescharacteristic
107C3×(C2×D3)⋊C4—32no—
108C3×(C2×D3)⋊C4C2×D312yes—
109#3724—32no—
110(C2×D3)⋊Dic3—32no—
111(C2×D3)⋊Dic3—32no—
112Dic32—32no—
113C22×C4×C9—32no—
114C2×#611—38no—
115C2×#611—31no—
116C2×#611—31no—
117C2×#611—31no—
118C2×#611—31no—
119C22×Dic9—31no—
120C22×Dic9—32no—
121C22×Dic9—61no—
122C2×C4×D9—32no—
123C2×C4×D9C2×D312yescomplemented
124C9×C22⋊C4—21no—
125C9×C22⋊C4—31no—
126C9×C22⋊C4—32no—
127C9×C22⋊C4C2×D311yescharacteristic
128#3676—31no—
129#3676—32no—
130#3676—61no—
131#3676C2×D311yescharacteristic
132(C2×D9)⋊C4—32no—
133(C2×D9)⋊C4C2×D312yescomplemented
134Dic9⋊C4—32no—
135Dic9⋊C4—32no—
136Dic9⋊C4—62no—
137C4×Dic9—32no—
138C4×Dic9—32no—
139C4×Dic9—62no—
140C22×C3×C9—22no—
141C22×C3×C9C22×C412yes—
142C22×C3×C9C2411yescharacteristic
143C2×D3×C9—22no—
144C2×D3×C9—22no—
145C2×D3×C9—44no—
146C2×D3×C9C22×C412yes—
147C2×D3×C9C22×C412yes—
148C2×C3×D9—22no—
149C2×C3×D9—22no—
150C2×C3×D9—44no—
151C2×C3×D9C22×C412yes—
152C2×C3×D9C22×C412yes—
153C3×C4×C9—22no—
154C3⋊Dic9—22no—
155C3⋊Dic9C22×C412yes—
156C3⋊Dic9C22×C412yes—
157C9×Dic3—22no—
158C3×Dic9—22no—
159C22×#67—91no—
160C22×#67—91no—
161C2×C3×C22⋊C4—31no—
162C2×C3×C22⋊C4—91no—
163C2×#217—91no—
164C2×#217—91no—
165C4×#67—92no—
166C4×#67—92no—
167C4×#67—92no—
168C4×#67—92no—
169#67⋊C4—92no—
170#67⋊C4—92no—
171#67⋊C4—92no—
172#67⋊C4—92no—
173D3×C22⋊C4—31no—
174D3×C22⋊C4—91no—
175D3×C22⋊C4—91no—
176D3×C22⋊C4D911yescharacteristic, complemented
177#877—91no—
178#877—91no—
179#877—91no—
180#877—91no—
181#877—181no—
182C23×C32C22×D311yescharacteristic
183C22×C3×D3—31no—
184C22×C3×D3—31no—
185C22×C3×D3—34no—
186C22×C3×D3—42no—
187C22×C3×D3—122no—
188C22×C3×D3C4×D314yes—
189C22×C3×D3C22×D311yescharacteristic
190C22×C3×D3C22×D311yescharacteristic
191C2×C32×C4C22×D312yes—
192C2×C3⋊Dic3—34no—
193C2×C3⋊Dic3—31no—
194C2×C3⋊Dic3—31no—
195C2×C3⋊Dic3—32no—
196C2×C3×Dic3—32no—
197C2×C3×Dic3C22×D312yes—
198C3×C4×D3—42no—
199C3×C4×D3—122no—
200#625—34no—
201#625—34no—
202#625—34no—
203#625—34no—
204D3×Dic3—122no—
205D3×Dic3—122no—
206C23×C9—21no—
207C23×C9—31no—
208C23×C9—31no—
209C23×C9—34no—
210C23×C9—122no—
211C23×C9C22×D311yescharacteristic
212C22×D9—42no—
213C22×D9C4×D314yescomplemented
214C22×D9C22×D311yescharacteristic
215C22×D9C22×D311yescharacteristic
216C2×C4×C9—22no—
217C2×C4×C9—32no—
218C2×C4×C9—122no—
219C2×C4×C9C22×D312yes—
220#611—34no—
221#611—34no—
222#611—34no—
223#611—34no—
224C2×Dic9—34no—
225C2×Dic9—31no—
226C2×Dic9—31no—
227C2×Dic9—32no—
228C2×Dic9—61no—
229C2×Dic9—61no—
230C2×Dic9—64no—
231C2×Dic9—62no—
232C2×Dic9—122no—
233C2×Dic9C22×D312yes—
234C4×D9—42no—
235C4×D9—122no—
236#518—271noSylow
237C2×C3×C9—22no—
238C2×C3×C9C23×C411yescharacteristic
239C2×C3×C9H(5,𝔽2)12yes—
240D3×C9—22no—
241D3×C9—22no—
242C3×D9—22no—
243C3×D9—22no—
244C24×C3—31no—
245C24×C3—91no—
246C23×D3—31no—
247C23×D3C2×D911yescharacteristic
248C22×C3×C4—32no—
249C22×C3×C4—92no—
250C2×#67—91no—
251C2×#67—91no—
252C2×#67—91no—
253C2×#67—91no—
254C2×#67—98no—
255C2×#67—98no—
256C2×#67—91no—
257C2×#67—91no—
258C2×#67—91no—
259C2×#67—91no—
260C22×Dic3—92no—
261C22×Dic3—91no—
262C22×Dic3—91no—
263C22×Dic3—92no—
264C22×Dic3—181no—
265C2×C4×D3—32no—
266C2×C4×D3—92no—
267C2×C4×D3—92no—
268C2×C4×D3C2×D912yescomplemented
269C3×C22⋊C4—21no—
270C3×C22⋊C4—31no—
271C3×C22⋊C4—32no—
272C3×C22⋊C4—91no—
273C3×C22⋊C4—92no—
274C3×C22⋊C4C2×D911yescharacteristic
275C3×C22⋊C4D3211yescharacteristic
276#217—31no—
277#217—92no—
278#217—91no—
279#217—91no—
280#217—92no—
281#217—181no—
282#217C2×D911yescharacteristic
283(C2×D3)⋊C4—32no—
284(C2×D3)⋊C4—92no—
285(C2×D3)⋊C4—92no—
286(C2×D3)⋊C4C2×D912yescomplemented
287Dic3⋊C4—92no—
288Dic3⋊C4—92no—
289Dic3⋊C4—92no—
290Dic3⋊C4—92no—
291Dic3⋊C4—182no—
292C4×Dic3—92no—
293C4×Dic3—92no—
294C4×Dic3—92no—
295C4×Dic3—92no—
296C4×Dic3—182no—
297C22×C32—22no—
298C22×C32C2×C4×D312yes—
299C22×C32C23×D311yescharacteristic
300C2×C3×D3—22no—
301C2×C3×D3—22no—
302C2×C3×D3—32no—
303C2×C3×D3—32no—
304C2×C3×D3—44no—
305C2×C3×D3—62no—
306C2×C3×D3—62no—
307C2×C3×D3—124no—
308C2×C3×D3C2×C4×D312yes—
309C2×C3×D3C2×C4×D312yes—
310C32×C4—22no—
311C3⋊Dic3—32no—
312C3⋊Dic3—32no—
313C3⋊Dic3—62no—
314C3×Dic3—22no—
315C3×Dic3—62no—
316C22×C9—22no—
317C22×C9—21no—
318C22×C9—22no—
319C22×C9—32no—
320C22×C9—32no—
321C22×C9—42no—
322C22×C9—62no—
323C22×C9—62no—
324C22×C9—124no—
325C22×C9C2×C4×D312yes—
326C22×C9C23×D311yescharacteristic
327C2×D9—22no—
328C2×D9—22no—
329C2×D9—44no—
330C2×D9C2×C4×D312yes—
331C2×D9C2×C4×D312yes—
332C4×C9—22no—
333C4×C9—42no—
334C4×C9—62no—
335Dic9—22no—
336Dic9—32no—
337Dic9—32no—
338Dic9—62no—
339Dic9—62no—
340Dic9—62no—
341Dic9—122no—
342C22×D4—271no—
343C4×D4—272no—
344C4×D4—272no—
345C4×D4—272no—
346C4×D4—272no—
347#117—271no—
348#117—271no—
349C2×C22⋊C4—31no—
350C2×C22⋊C4—91no—
351C2×C22⋊C4—271no—
352C2×C22⋊C4—271no—
353C3×C9#51811yescharacteristic, complemented, Sylow
354C23×C3—21no—
355C23×C3—31no—
356C23×C3—31no—
357C23×C3—34no—
358C23×C3—91no—
359C23×C3—91no—
360C23×C3—94no—
361C23×C3—122no—
362C23×C3—362no—
363C23×C3C22×D911yescharacteristic
364C23×C3C2×D3211yescharacteristic
365C22×D3—31no—
366C22×D3—31no—
367C22×D3—34no—
368C22×D3—42no—
369C22×D3—122no—
370C22×D3C4×D914yescomplemented
371C22×D3C22×D911yescharacteristic
372C22×D3C22×D911yescharacteristic
373C2×C3×C4—22no—
374C2×C3×C4—32no—
375C2×C3×C4—92no—
376C2×C3×C4—122no—
377C2×C3×C4—362no—
378C2×C3×C4C22×D912yes—
379C2×C3×C4C2×D3212yes—
380#67—94no—
381#67—94no—
382#67—94no—
383#67—94no—
384#67—94no—
385#67—94no—
386#67—94no—
387#67—94no—
388C2×Dic3—32no—
389C2×Dic3—91no—
390C2×Dic3—91no—
391C2×Dic3—94no—
392C2×Dic3—91no—
393C2×Dic3—91no—
394C2×Dic3—94no—
395C2×Dic3—92no—
396C2×Dic3—92no—
397C2×Dic3—181no—
398C2×Dic3—181no—
399C2×Dic3—184no—
400C2×Dic3—182no—
401C2×Dic3—362no—
402C2×Dic3—362no—
403C2×Dic3C22×D912yes—
404C4×D3—42no—
405C4×D3—122no—
406C4×D3—362no—
407C4×D3—362no—
408C2×C32—22no—
409C2×C32C22×C4×D311yescharacteristic
410C2×C32#100412yes—
411C3×D3—22no—
412C3×D3—22no—
413C3×D3—62no—
414C3×D3—62no—
415C2×C9—22no—
416C2×C9—21no—
417C2×C9—22no—
418C2×C9—42no—
419C2×C9—62no—
420C2×C9—62no—
421C2×C9C22×C4×D311yescharacteristic
422C2×C9#100412yes—
423D9—22no—
424D9—22no—
425C24—31no—
426C24—91no—
427C2×D4—271no—
428C2×D4—271no—
429C2×D4—271no—
430C2×D4—271no—
431C2×D4—278no—
432C22×C4—32no—
433C22×C4—92no—
434C22×C4—272no—
435C22×C4—272no—
436C22×C4—271no—
437C4⋊C4—272no—
438C4⋊C4—272no—
439C22⋊C4—31no—
440C22⋊C4—32no—
441C22⋊C4—91no—
442C22⋊C4—92no—
443C22⋊C4—272no—
444C22⋊C4—272no—
445C22⋊C4—271no—
446C22⋊C4D3×D911yescharacteristic
447C42—272no—
448C42—272no—
449C22×C3—22no—
450C22×C3—22no—
451C22×C3—21no—
452C22×C3—22no—
453C22×C3—32no—
454C22×C3—32no—
455C22×C3—42no—
456C22×C3—62no—
457C22×C3—62no—
458C22×C3—92no—
459C22×C3—92no—
460C22×C3—124no—
461C22×C3—182no—
462C22×C3—182no—
463C22×C3—364no—
464C22×C3C2×C4×D912yes—
465C22×C3C23×D911yescharacteristic
466C22×C3C4×D3212yes—
467C22×C3C22×D3211yescharacteristic
468C2×D3—22no—
469C2×D3—22no—
470C2×D3—32no—
471C2×D3—32no—
472C2×D3—44no—
473C2×D3—62no—
474C2×D3—62no—
475C2×D3—124no—
476C2×D3C2×C4×D912yes—
477C2×D3C2×C4×D912yes—
478C3×C4—22no—
479C3×C4—22no—
480C3×C4—42no—
481C3×C4—62no—
482C3×C4—182no—
483Dic3—22no—
484Dic3—62no—
485Dic3—92no—
486Dic3—92no—
487Dic3—92no—
488Dic3—92no—
489Dic3—182no—
490Dic3—182no—
491Dic3—182no—
492Dic3—182no—
493Dic3—362no—
494C32#548611yescharacteristic
495C9—21no—
496C9#548611yescharacteristic, complemented
497C23—31no—
498C23—31no—
499C23—34no—
500C23—91no—
501C23—91no—
502C23—94no—
503C23—122no—
504C23—362no—
505C23C2×D3×D911yescharacteristic
506D4—274no—
507D4—274no—
508D4—274no—
509D4—274no—
510C2×C4—32no—
511C2×C4—92no—
512C2×C4—122no—
513C2×C4—271no—
514C2×C4—271no—
515C2×C4—274no—
516C2×C4—272no—
517C2×C4—362no—
518C2×C4—1082no—
519C2×C4—1082no—
520C2×C4C2×D3×D912yes—
521C2×C3—22no—
522C2×C3—22no—
523C2×C3—22no—
524C2×C3—21no—
525C2×C3—42no—
526C2×C3—62no—
527C2×C3—62no—
528C2×C3—182no—
529C2×C3—182no—
530C2×C3C22×C4×D911yescharacteristic
531C2×C3#6371012yes—
532C2×C3C2×C4×D3211yescharacteristic
533C2×C3#6431312yes—
534D3—22no—
535D3—22no—
536D3—62no—
537D3—62no—
538C22—22no—
539C22—32no—
540C22—32no—
541C22—62no—
542C22—62no—
543C22—92no—
544C22—92no—
545C22—124no—
546C22—182no—
547C22—182no—
548C22—364no—
549C22C4×D3×D912yes—
550C22C22×D3×D911yescharacteristic
551C4—22no—
552C4—62no—
553C4—182no—
554C4—272no—
555C4—272no—
556C4—542no—
557C3—21no—
558C3#1058958811yescharacteristic, complemented
559C3#1059418111yescharacteristic
560C2—22no—
561C2—62no—
562C2—62no—
563C2—182no—
564C2—182no—
565C2C2×C4×D3×D911yescharacteristic
566C2#1172249912yes—
subgroupG/H
centerC22C22×D3×D9
commutatorC2×C3×C9C23×C4
FrattiniC22×C3C22×D32
FittingC3×C9×C22⋊C4C22
socleC22×C32C23×D3
prime radicalC22×#6194C2
cosocleC22×C3×C9C24
hypercenterC22⋊C4D3×D9
hypocenterC3×C9#518
LCSG ▷ C2×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C22⋊C4

Complements

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

NH ≅ G/Ntype
2. #11721723C2semidirect
3. D3×C9×C22⋊C4C2semidirect
4. C3×C22⋊C4×D9C2semidirect
5. D3×#3676C2semidirect
6. D9×#217C2semidirect
7. #11721248C2semidirect
8. #11721028C2semidirect
9. C4×#6194C2semidirect
10. #11721012C2semidirect
11. #11720976C2semidirect
17. C2×C4×D3×C9C22semidirect
18. C2×C3×C4×D9C22semidirect
19. C2×#6194C4semidirect
24. C2×D3×Dic9C22semidirect
25. C2×Dic3×D9C22semidirect
28. #88534C22semidirect
29. C4×C3⋊Dic9C22semidirect
32. C9×(C2×D3)⋊C4C22semidirect
33. C3×(C2×D9)⋊C4C22semidirect
34. (C2×D3)⋊Dic9C22semidirect
35. (C2×D9)⋊Dic3C22semidirect
36. #88442C22semidirect
37. Dic3×Dic9C22semidirect
57. C22⋊C4×D9D3semidirect
63. C22×D3×C9C2×C4semidirect
67. C22×C3×D9C2×C4semidirect
71. C2×C3⋊Dic9C2×C4semidirect
79. #6194C2×C4semidirect
80. #6194C2×C4semidirect
81. #6194C2×C4semidirect
82. #6194C2×C4semidirect
123. C2×C4×D9C2×D3semidirect
133. (C2×D9)⋊C4C2×D3semidirect
176. D3×C22⋊C4D9semidirect
213. C22×D9C4×D3semidirect
268. C2×C4×D3C2×D9semidirect
286. (C2×D3)⋊C4C2×D9semidirect
353. C3×C9#518semidirect
370. C22×D3C4×D9semidirect
496. C9#5486semidirect
558. C3#10589588semidirect

Maximal subgroups

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

subgroupindex
C22×#61942
#117217232
D3×C9×C22⋊C42
C3×C22⋊C4×D92
D3×#36762
D9×#2172
#117212482
#117210282
C4×#61942
#117210122
#117209762
#105941813
#105895883