Finite Group Databasebeta

← #13147703

Subgroups — #13147703

Order 1728.

1090 conjugacy classes of subgroups.

The subgroup lattice has 566 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#88646C211yesmaximal, characteristic
2#11721709C211yesmaximal, characteristic, complemented
3C4×C9×D12C211yesmaximal, characteristic, complemented
4C3×C4×D36C211yesmaximal, characteristic, complemented
5D3×(C2×D9)⋊C4C212yesmaximal, complemented
6D9×(C2×D3)⋊C4C212yesmaximal, complemented
7#11721025C211yesmaximal, characteristic, complemented
8#11721012C212yesmaximal, complemented
9#11720998C211yesmaximal, characteristic, complemented
10#11720976C212yesmaximal, complemented
11#11720922C211yesmaximal, characteristic, complemented
12#10593996—31nomaximal
13#10589647—31nomaximal
14C22×D3×D9C2212yes—
15C2×C4×C3⋊D9C2211yescharacteristic
16C2×C9×D12C2211yescharacteristic
17C2×C4×D3×C9C2212yescomplemented
18C2×C3×D36C2211yescharacteristic
19C2×C3×C4×D9C2212yescomplemented
20C2×#6194C2212yes—
21C2×D3×Dic9C2212yescomplemented
22C2×Dic3×D9C2212yescomplemented
23#88646C418yescomplemented
24C3×C42×C9C2211yescharacteristic, complemented
25#88536C2211yescharacteristic
26#88536C2211yescharacteristic
27#88534C2211yescharacteristic, complemented
28#88534C2211yescharacteristic, complemented
29C4×C3⋊Dic9C2211yescharacteristic, complemented
30C9×(C2×D3)⋊C4C2212yescomplemented
31C3×(C2×D9)⋊C4C2212yescomplemented
32C9×C4⋊Dic3C2211yescharacteristic, complemented
33C3×C4⋊Dic9C2211yescharacteristic, complemented
34(C2×D3)⋊Dic9C2212yescomplemented
35#88445C2212yes—
36(C2×D9)⋊Dic3C2212yescomplemented
37Dic3×Dic9C2212yescomplemented
38C2×#3802—31no—
39#64081—31no—
40C3×C4×D12—31no—
41C3×C4×D12D311yescharacteristic
42D3×(C2×D3)⋊C4—32no—
43D3×(C2×D3)⋊C4—32no—
44#63898—31no—
45#63898—31no—
46#63879—32no—
47#63879—32no—
48#63849—31no—
49C2×D4×D9—31no—
50C4×D4×C9—31no—
51D4×Dic9—31no—
52C4×#611—32no—
53C2×(C2×D9)⋊C4—32no—
54#63455—31no—
55#63454—31no—
56#611⋊C4—32no—
57C22⋊C4×D9—32no—
58C4×D36D311yescharacteristic, complemented
59#63434—31no—
60#63434—61no—
61C22×C3⋊D9C2311yescharacteristic
62C22×D3×C9C2312yes—
63C22×C3×D9C2312yes—
64C2×D3×D9—42no—
65C2×D3×D9—42no—
66C2×D3×D9C2×C418yescomplemented
67C2×C3×C4×C9C2311yescharacteristic
68C2×C3×C4×C9C2311yescharacteristic
69C2×C3×C4×C9C2311yescharacteristic
70C2×C3⋊Dic9C2311yescharacteristic
71C2×C3⋊Dic9C2311yescharacteristic
72C2×C3⋊Dic9C2311yescharacteristic
73C4×C3⋊D9C2×C414yescomplemented
74C2×C9×Dic3C2312yes—
75C2×C3×Dic9C2312yes—
76C9×D12C2×C414yescomplemented
77C4×D3×C9—42no—
78C3×D36C2×C414yescomplemented
79C3×C4×D9—42no—
80#6194C2×C418yescomplemented
81D3×Dic9—42no—
82Dic3×D9—42no—
83#5545—91no—
84#5545—91no—
85C22×D32—32no—
86C2×C4×C3⋊D3—31no—
87C2×C3×D12—31no—
88C2×C3×D12C2×D311yescharacteristic
89C2×C3×C4×D3—32no—
90C2×C3×C4×D3C2×D312yes—
91C2×#625—32no—
92C2×D3×Dic3—32no—
93C2×D3×Dic3—32no—
94#3802—38no—
95C23×D9—32no—
96C32×C42C2×D311yescharacteristic
97#3752—31no—
98#3752—31no—
99#3750—31no—
100#3750—31no—
101C4×C3⋊Dic3—31no—
102C3×(C2×D3)⋊C4—32no—
103C3×(C2×D3)⋊C4C2×D312yes—
104C3×C4⋊Dic3—31no—
105C3×C4⋊Dic3C2×D311yescharacteristic
106#3722—32no—
107(C2×D3)⋊Dic3—32no—
108(C2×D3)⋊Dic3—32no—
109Dic32—32no—
110C2×D4×C9—31no—
111C22×C4×C9—32no—
112C2×#611—32no—
113C22×Dic9—32no—
114D4×D9—38no—
115C2×D36C2×D311yescharacteristic
116C2×C4×D9—31no—
117C2×C4×D9—32no—
118C2×C4×D9—61no—
119C2×C4×D9C2×D312yescomplemented
120C9×C4⋊C4—31no—
121C9×C22⋊C4—32no—
122C42×C9—21no—
123C42×C9C2×D311yescharacteristic, complemented
124#3676—32no—
125(C2×D9)⋊C4—32no—
126(C2×D9)⋊C4—31no—
127(C2×D9)⋊C4—31no—
128(C2×D9)⋊C4—32no—
129(C2×D9)⋊C4—61no—
130(C2×D9)⋊C4—61no—
131(C2×D9)⋊C4C2×D312yescomplemented
132C4⋊Dic9C2×D311yescharacteristic, complemented
133Dic9⋊C4—31no—
134Dic9⋊C4—31no—
135Dic9⋊C4—61no—
136Dic9⋊C4—61no—
137C4×Dic9—31no—
138C4×Dic9—32no—
139C4×Dic9—61no—
140C22×C3×C9C2411yescharacteristic
141C2×C3⋊D9—21no—
142C2×C3⋊D9—21no—
143C2×C3⋊D9C22×C412yes—
144C2×D3×C9—22no—
145C2×D3×C9—22no—
146C2×D3×C9C22×C414yes—
147C2×C3×D9—22no—
148C2×C3×D9—22no—
149C2×C3×D9C22×C414yes—
150D3×D9—48no—
151C3×C4×C9—21no—
152C3×C4×C9—21no—
153C3×C4×C9C22×C412yes—
154C3⋊Dic9—21no—
155C3⋊Dic9—21no—
156C3⋊Dic9C22×C412yes—
157C9×Dic3—22no—
158C3×Dic9—22no—
159C2×D3×D4—91no—
160C2×D3×D4—91no—
161C3×C4×D4—31no—
162C3×C4×D4—91no—
163D4×Dic3—91no—
164D4×Dic3—91no—
165C4×#67—92no—
166C4×#67—92no—
167C2×(C2×D3)⋊C4—92no—
168C2×(C2×D3)⋊C4—92no—
169#893—91no—
170#893—91no—
171#892—91no—
172#892—91no—
173#67⋊C4—92no—
174#67⋊C4—92no—
175D3×C22⋊C4—92no—
176D3×C22⋊C4—92no—
177C4×D12—31no—
178C4×D12D911yescharacteristic, complemented
179#872—91no—
180#872—91no—
181#872—181no—
182C22×C3⋊D3—31no—
183C22×C3×D3—32no—
184C22×C3×D3C22×D312yes—
185C2×D32—38no—
186C2×D32—122no—
187C2×D32—122no—
188C2×C32×C4C22×D311yescharacteristic
189C2×C32×C4C22×D311yescharacteristic
190C2×C32×C4C22×D311yescharacteristic
191C2×C3⋊Dic3—31no—
192C2×C3⋊Dic3—31no—
193C2×C3⋊Dic3—31no—
194C4×C3⋊D3—34no—
195C2×C3×Dic3—32no—
196C2×C3×Dic3C22×D312yes—
197C3×D12—34no—
198C3×D12C4×D314yes—
199C3×C4×D3—42no—
200C3×C4×D3—122no—
201#625—38no—
202D3×Dic3—122no—
203D3×Dic3—122no—
204C23×C9—32no—
205C22×D9—31no—
206C22×D9—38no—
207C22×D9—61no—
208C22×D9—122no—
209C22×D9—122no—
210C22×D9C22×D312yes—
211D4×C9—34no—
212C2×C4×C9—21no—
213C2×C4×C9—21no—
214C2×C4×C9—21no—
215C2×C4×C9—32no—
216C2×C4×C9—122no—
217C2×C4×C9C22×D311yescharacteristic
218C2×C4×C9C22×D311yescharacteristic
219C2×C4×C9C22×D311yescharacteristic
220#611—38no—
221C2×Dic9—31no—
222C2×Dic9—31no—
223C2×Dic9—31no—
224C2×Dic9—61no—
225C2×Dic9—61no—
226C2×Dic9—61no—
227C2×Dic9—122no—
228C2×Dic9C22×D312yes—
229D36C4×D314yescomplemented
230C4×D9—34no—
231C4×D9—42no—
232C4×D9—64no—
233C4×D9—122no—
234#518—271noSylow
235C2×C3×C9C23×C411yescharacteristic
236C2×C3×C9H(5,𝔽2)11yescharacteristic
237C2×C3×C9H(5,𝔽2)11yescharacteristic
238C3⋊D9—22no—
239D3×C9—24no—
240C3×D9—24no—
241C23×D3—92no—
242C23×D3—92no—
243C2×C3×D4—31no—
244C2×C3×D4—91no—
245C22×C3×C4—32no—
246C22×C3×C4—92no—
247C2×#67—92no—
248C2×#67—92no—
249C22×Dic3—92no—
250C22×Dic3—92no—
251D3×D4—98no—
252D3×D4—98no—
253C2×D12—31no—
254C2×D12C2×D911yescharacteristic
255C2×C4×D3—32no—
256C2×C4×D3—91no—
257C2×C4×D3—91no—
258C2×C4×D3—92no—
259C2×C4×D3—92no—
260C2×C4×D3—181no—
261C2×C4×D3C2×D912yescomplemented
262C3×C4⋊C4—31no—
263C3×C4⋊C4—91no—
264C3×C22⋊C4—32no—
265C3×C22⋊C4—92no—
266C3×C42—21no—
267C3×C42C2×D911yescharacteristic, complemented
268C3×C42D3211yescharacteristic
269#217—92no—
270#217—92no—
271(C2×D3)⋊C4—32no—
272(C2×D3)⋊C4—92no—
273(C2×D3)⋊C4—92no—
274(C2×D3)⋊C4—92no—
275(C2×D3)⋊C4—91no—
276(C2×D3)⋊C4—91no—
277(C2×D3)⋊C4—91no—
278(C2×D3)⋊C4—91no—
279(C2×D3)⋊C4—92no—
280(C2×D3)⋊C4—181no—
281(C2×D3)⋊C4—181no—
282(C2×D3)⋊C4C2×D912yescomplemented
283C4⋊Dic3—31no—
284C4⋊Dic3C2×D911yescharacteristic, complemented
285Dic3⋊C4—91no—
286Dic3⋊C4—91no—
287Dic3⋊C4—91no—
288Dic3⋊C4—91no—
289Dic3⋊C4—181no—
290Dic3⋊C4—181no—
291C4×Dic3—92no—
292C4×Dic3—91no—
293C4×Dic3—91no—
294C4×Dic3—92no—
295C4×Dic3—181no—
296C22×C32C23×D311yescharacteristic
297C2×C3⋊D3—32no—
298C2×C3⋊D3—61no—
299C2×C3⋊D3—61no—
300C2×C3×D3—22no—
301C2×C3×D3—22no—
302C2×C3×D3—34no—
303C2×C3×D3—62no—
304C2×C3×D3—62no—
305C2×C3×D3C2×C4×D314yes—
306D32—128no—
307C32×C4—21no—
308C32×C4—21no—
309C32×C4C2×C4×D312yes—
310C3⋊Dic3—32no—
311C3⋊Dic3—61no—
312C3⋊Dic3—61no—
313C3×Dic3—22no—
314C3×Dic3—62no—
315C22×C9—21no—
316C22×C9—34no—
317C22×C9—62no—
318C22×C9—62no—
319C22×C9C23×D311yescharacteristic
320C2×D9—22no—
321C2×D9—22no—
322C2×D9—32no—
323C2×D9—62no—
324C2×D9—61no—
325C2×D9—61no—
326C2×D9—121no—
327C2×D9—121no—
328C2×D9—128no—
329C2×D9C2×C4×D314yes—
330C4×C9—21no—
331C4×C9—21no—
332C4×C9—22no—
333C4×C9—41no—
334C4×C9—41no—
335C4×C9—62no—
336C4×C9C2×C4×D312yes—
337Dic9—22no—
338Dic9—32no—
339Dic9—62no—
340Dic9—61no—
341Dic9—61no—
342Dic9—121no—
343Dic9—121no—
344C22×D4—271no—
345C4×D4—31no—
346C4×D4—91no—
347C4×D4—272no—
348C4×D4—272no—
349C4×D4—271no—
350C4×D4—271no—
351#117—271no—
352#117—271no—
353C2×C22⋊C4—272no—
354C2×C22⋊C4—272no—
355C3×C9#51811yescharacteristic, complemented, Sylow
356C23×C3—32no—
357C23×C3—92no—
358C22×D3—32no—
359C22×D3—98no—
360C22×D3—91no—
361C22×D3—91no—
362C22×D3—98no—
363C22×D3—181no—
364C22×D3—362no—
365C22×D3—362no—
366C22×D3—362no—
367C22×D3—362no—
368C22×D3C22×D912yes—
369C3×D4—34no—
370C3×D4—94no—
371C2×C3×C4—21no—
372C2×C3×C4—21no—
373C2×C3×C4—21no—
374C2×C3×C4—32no—
375C2×C3×C4—92no—
376C2×C3×C4—122no—
377C2×C3×C4—362no—
378C2×C3×C4C22×D911yescharacteristic
379C2×C3×C4C22×D911yescharacteristic
380C2×C3×C4C22×D911yescharacteristic
381C2×C3×C4C2×D3211yescharacteristic
382C2×C3×C4C2×D3211yescharacteristic
383C2×C3×C4C2×D3211yescharacteristic
384#67—98no—
385#67—98no—
386C2×Dic3—32no—
387C2×Dic3—91no—
388C2×Dic3—91no—
389C2×Dic3—91no—
390C2×Dic3—91no—
391C2×Dic3—91no—
392C2×Dic3—91no—
393C2×Dic3—181no—
394C2×Dic3—181no—
395C2×Dic3—181no—
396C2×Dic3—362no—
397C2×Dic3—362no—
398C2×Dic3C22×D912yes—
399D12—34no—
400D12C4×D914yescomplemented
401C4×D3—42no—
402C4×D3—94no—
403C4×D3—94no—
404C4×D3—122no—
405C4×D3—184no—
406C4×D3—362no—
407C4×D3—362no—
408C2×C32C22×C4×D311yescharacteristic
409C2×C32#100411yescharacteristic
410C2×C32#100911yescharacteristic
411C3⋊D3—62no—
412C3×D3—24no—
413C3×D3—64no—
414C2×C9—21no—
415C2×C9—21no—
416C2×C9—21no—
417C2×C9—64no—
418C2×C9C22×C4×D311yescharacteristic
419C2×C9#100411yescharacteristic
420C2×C9#100911yescharacteristic
421D9—24no—
422D9—62no—
423D9—122no—
424C24—272no—
425C2×D4—31no—
426C2×D4—91no—
427C2×D4—278no—
428C2×D4—272no—
429C22×C4—32no—
430C22×C4—92no—
431C22×C4—272no—
432C22×C4—272no—
433C22×C4—271no—
434C4⋊C4—31no—
435C4⋊C4—91no—
436C4⋊C4—271no—
437C4⋊C4—271no—
438C22⋊C4—32no—
439C22⋊C4—92no—
440C22⋊C4—272no—
441C22⋊C4—272no—
442C22⋊C4—272no—
443C22⋊C4—271no—
444C22⋊C4—271no—
445C42—272no—
446C42—271no—
447C42D3×D911yescharacteristic, complemented
448C22×C3—21no—
449C22×C3—34no—
450C22×C3—62no—
451C22×C3—62no—
452C22×C3—94no—
453C22×C3—182no—
454C22×C3—182no—
455C22×C3C23×D911yescharacteristic
456C22×C3C22×D3211yescharacteristic
457C2×D3—22no—
458C2×D3—22no—
459C2×D3—34no—
460C2×D3—62no—
461C2×D3—62no—
462C2×D3—92no—
463C2×D3—92no—
464C2×D3—182no—
465C2×D3—181no—
466C2×D3—181no—
467C2×D3—181no—
468C2×D3—181no—
469C2×D3—361no—
470C2×D3—361no—
471C2×D3—368no—
472C2×D3—368no—
473C2×D3C2×C4×D914yes—
474C3×C4—21no—
475C3×C4—21no—
476C3×C4—21no—
477C3×C4—21no—
478C3×C4—22no—
479C3×C4—41no—
480C3×C4—41no—
481C3×C4—62no—
482C3×C4—182no—
483C3×C4C2×C4×D912yes—
484C3×C4C4×D3212yes—
485Dic3—22no—
486Dic3—62no—
487Dic3—92no—
488Dic3—92no—
489Dic3—182no—
490Dic3—181no—
491Dic3—181no—
492Dic3—181no—
493Dic3—181no—
494Dic3—361no—
495Dic3—361no—
496C32#554511yescharacteristic
497C9—21no—
498C9#554511yescharacteristic, complemented
499C23—32no—
500C23—92no—
501C23—278no—
502C23—271no—
503C23—1082no—
504C23—1082no—
505D4—34no—
506D4—94no—
507D4—278no—
508C2×C4—32no—
509C2×C4—92no—
510C2×C4—122no—
511C2×C4—271no—
512C2×C4—274no—
513C2×C4—271no—
514C2×C4—271no—
515C2×C4—362no—
516C2×C4—1082no—
517C2×C4—1082no—
518C2×C4C2×D3×D911yescharacteristic
519C2×C4C2×D3×D911yescharacteristic
520C2×C4C2×D3×D911yescharacteristic
521C2×C3—21no—
522C2×C3—21no—
523C2×C3—21no—
524C2×C3—64no—
525C2×C3—184no—
526C2×C3C22×C4×D911yescharacteristic
527C2×C3#6371011yescharacteristic
528C2×C3#6371511yescharacteristic
529C2×C3C2×C4×D3211yescharacteristic
530C2×C3#6430711yescharacteristic
531C2×C3#6430711yescharacteristic
532D3—24no—
533D3—64no—
534D3—182no—
535D3—182no—
536D3—362no—
537C22—34no—
538C22—62no—
539C22—62no—
540C22—94no—
541C22—182no—
542C22—182no—
543C22—272no—
544C22—541no—
545C22—541no—
546C22—1088no—
547C22C22×D3×D911yescharacteristic
548C4—21no—
549C4—21no—
550C4—62no—
551C4—182no—
552C4—272no—
553C4—541no—
554C4—541no—
555C4C4×D3×D912yes—
556C3—21no—
557C3#1058964711yescharacteristic, complemented
558C3#1059399611yescharacteristic
559C2—64no—
560C2—184no—
561C2—542no—
562C2C2×C4×D3×D911yescharacteristic
563C2#1172247611yescharacteristic
564C2#1172247711yescharacteristic
subgroupG/H
centerC22C22×D3×D9
commutatorC2×C3×C9C23×C4
FrattiniC22×C3C22×D32
FittingC3×C42×C9C22
socleC22×C32C23×D3
prime radicalC2×#88646C2
cosocleC22×C3×C9C24
hypercenterC42D3×D9
hypocenterC3×C9#518
LCSG ▷ C2×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C42

Complements

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

NH ≅ G/Ntype
2. #11721709C2semidirect
3. C4×C9×D12C2semidirect
4. C3×C4×D36C2semidirect
5. D3×(C2×D9)⋊C4C2semidirect
6. D9×(C2×D3)⋊C4C2semidirect
7. #11721025C2semidirect
8. #11721012C2semidirect
9. #11720998C2semidirect
10. #11720976C2semidirect
11. #11720922C2semidirect
17. C2×C4×D3×C9C22semidirect
19. C2×C3×C4×D9C22semidirect
21. C2×D3×Dic9C22semidirect
22. C2×Dic3×D9C22semidirect
23. #88646C4semidirect
24. C3×C42×C9C22semidirect
27. #88534C22semidirect
28. #88534C22semidirect
29. C4×C3⋊Dic9C22semidirect
30. C9×(C2×D3)⋊C4C22semidirect
31. C3×(C2×D9)⋊C4C22semidirect
32. C9×C4⋊Dic3C22semidirect
33. C3×C4⋊Dic9C22semidirect
34. (C2×D3)⋊Dic9C22semidirect
36. (C2×D9)⋊Dic3C22semidirect
37. Dic3×Dic9C22semidirect
58. C4×D36D3semidirect
66. C2×D3×D9C2×C4semidirect
73. C4×C3⋊D9C2×C4semidirect
76. C9×D12C2×C4semidirect
78. C3×D36C2×C4semidirect
80. #6194C2×C4semidirect
119. C2×C4×D9C2×D3semidirect
123. C42×C9C2×D3semidirect
131. (C2×D9)⋊C4C2×D3semidirect
132. C4⋊Dic9C2×D3semidirect
178. C4×D12D9semidirect
229. D36C4×D3semidirect
261. C2×C4×D3C2×D9semidirect
267. C3×C42C2×D9semidirect
282. (C2×D3)⋊C4C2×D9semidirect
284. C4⋊Dic3C2×D9semidirect
355. C3×C9#518semidirect
400. D12C4×D9semidirect
447. C42D3×D9semidirect
498. C9#5545semidirect
557. C3#10589647semidirect

Maximal subgroups

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

subgroupindex
C2×#886462
#117217092
C4×C9×D122
C3×C4×D362
D3×(C2×D9)⋊C42
D9×(C2×D3)⋊C42
#117210252
#117210122
#117209982
#117209762
#117209222
#105939963
#105896473