Finite Group Databasebeta

← #13147701

Subgroups — #13147701

Order 1728.

1108 conjugacy classes of subgroups.

The subgroup lattice has 556 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×D3×D36C211yesmaximal, characteristic
2C4×C4⋊(C3⋊D9)C211yesmaximal, characteristic, complemented
3C9×#872C211yesmaximal, characteristic, complemented
4C3×C4×D36C211yesmaximal, characteristic, complemented
5#11721048C212yesmaximal, complemented
6D9×(C2×D3)⋊C4C212yesmaximal, complemented
7#11721002C212yesmaximal, complemented
8#11720993C211yesmaximal, characteristic, complemented
9Dic3×D36C211yesmaximal, characteristic, complemented
10#11720978C212yesmaximal, complemented
11#11720906C211yesmaximal, characteristic, complemented
12#10593994—31nomaximal
13#10589623—31nomaximal
14C22×D3×D9C2212yes—
15C2×C4⋊(C3⋊D9)C2211yescharacteristic
16C2×C4×C3⋊D9C2212yescomplemented
17C2×C4×D3×C9C2211yescharacteristic
18C2×C3×D36C2211yescharacteristic
19C2×C3×C4×D9C2212yescomplemented
20C2×#6192C2212yes—
21C2×(C3⋊D9)⋊C4C2212yescomplemented
22C2×Dic3×D9C2212yescomplemented
23D3×D36C418yescomplemented
24C3×C42×C9C2211yescharacteristic, complemented
25#88536C2212yescomplemented
26C4⋊(C3⋊Dic9)C2211yescharacteristic, complemented
27C9×(C2×D3)⋊C4C2211yescharacteristic
28C9×(C2×D3)⋊C4C2211yescharacteristic
29C3×(C2×D9)⋊C4C2212yescomplemented
30C9×Dic3⋊C4C2211yescharacteristic, complemented
31C9×Dic3⋊C4C2211yescharacteristic, complemented
32C4×C9×Dic3C2211yescharacteristic, complemented
33C3×C4⋊Dic9C2211yescharacteristic, complemented
34(C2×D3)⋊Dic9C2212yes—
35#88445C2212yescomplemented
36(C2×D9)⋊Dic3C2212yescomplemented
37Dic3×Dic9C2212yescomplemented
38C2×D3×D12—31no—
39C4×C4⋊(C3⋊D3)—31no—
40C3×C4×D12—31no—
41C3×#872D311yescharacteristic
42#63921—32no—
43D3×(C2×D3)⋊C4—32no—
44#63904—32no—
45#63894—31no—
46Dic3×D12—31no—
47#63881—32no—
48#63841—31no—
49C22×D36—31no—
50C9×#117—31no—
51#63488—31no—
52#63455—32no—
53#63455—32no—
54C22⋊C4×D9—32no—
55C22⋊C4×D9—32no—
56C4×D36—31no—
57C4×D36—31no—
58C4×D36—31no—
59C4×D36—61no—
60C4×D36D311yescharacteristic, complemented
61C22×C3⋊D9C2312yes—
62C22×D3×C9C2311yescharacteristic
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⋊Dic9C2312yes—
71C4⋊(C3⋊D9)C2×C414yescomplemented
72C4×C3⋊D9—42no—
73C2×C9×Dic3C2311yescharacteristic
74C2×C9×Dic3C2311yescharacteristic
75C2×C9×Dic3C2311yescharacteristic
76C2×C3×Dic9C2312yes—
77C4×D3×C9C2×C414yescomplemented
78C3×D36C2×C414yescomplemented
79C3×C4×D9—42no—
80#6192C2×C418yescomplemented
81(C3⋊D9)⋊C4—42no—
82Dic3×D9—42no—
83#5545—91no—
84#5521—91no—
85C22×D32—32no—
86C2×C4⋊(C3⋊D3)—31no—
87C2×C4×C3⋊D3—32no—
88C2×C3×D12—31no—
89C2×C3×C4×D3—32no—
90C2×C3×C4×D3C2×D311yescharacteristic
91C2×#626—32no—
92C2×#624—32no—
93C2×D3×Dic3—32no—
94D3×D12—38no—
95C23×D9—32no—
96C32×C42C2×D311yescharacteristic
97#3752—32no—
98C4⋊(C3⋊Dic3)—31no—
99C3×(C2×D3)⋊C4—32no—
100C3×(C2×D3)⋊C4C2×D311yescharacteristic
101C3×(C2×D3)⋊C4C2×D311yescharacteristic
102C3×C4⋊Dic3—31no—
103C3×Dic3⋊C4C2×D311yescharacteristic
104C3×Dic3⋊C4C2×D311yescharacteristic
105C3×C4×Dic3C2×D311yescharacteristic
106#3722—32no—
107(C2×D3)⋊Dic3—32no—
108(C2×D3)⋊Dic3—32no—
109Dic32—32no—
110C22×C4×C9—31no—
111C2×D36—32no—
112C2×D36—31no—
113C2×D36—38no—
114C2×D36—61no—
115C2×D36C2×D311yescharacteristic
116C2×C4×D9—32no—
117C2×C4×D9—32no—
118C2×C4×D9—32no—
119C2×C4×D9—62no—
120C2×C4×D9C2×D312yescomplemented
121C9×C4⋊C4—31no—
122C9×C4⋊C4—31no—
123C9×C22⋊C4—31no—
124C9×C22⋊C4—31no—
125C42×C9—21no—
126C42×C9—31no—
127C42×C9C2×D311yescharacteristic, complemented
128#3676—32no—
129(C2×D9)⋊C4—32no—
130(C2×D9)⋊C4—32no—
131(C2×D9)⋊C4—32no—
132(C2×D9)⋊C4—62no—
133(C2×D9)⋊C4C2×D312yescomplemented
134C4⋊Dic9—31no—
135C4⋊Dic9—61no—
136C4⋊Dic9C2×D311yescharacteristic, complemented
137C4×Dic9—32no—
138C22×C3×C9C2411yescharacteristic
139C2×C3⋊D9—22no—
140C2×C3⋊D9—22no—
141C2×C3⋊D9C22×C414yes—
142C2×D3×C9—21no—
143C2×D3×C9—21no—
144C2×D3×C9C22×C412yes—
145C2×C3×D9—22no—
146C2×C3×D9—22no—
147C2×C3×D9C22×C414yes—
148D3×D9—48no—
149C3×C4×C9—21no—
150C3×C4×C9—21no—
151C3×C4×C9C22×C412yes—
152C3⋊Dic9—22no—
153C9×Dic3—21no—
154C9×Dic3—21no—
155C9×Dic3C22×C412yes—
156C3×Dic9—22no—
157C2×D3×D4—91no—
158C22×D12—91no—
159C3×C4×D4—91no—
160C3×#117—31no—
161D4×Dic3—91no—
162C4×#67—92no—
163C2×(C2×D3)⋊C4—92no—
164#926—91no—
165#893—92no—
166#893—92no—
167#893—91no—
168#892—91no—
169#67⋊C4—92no—
170D3×C22⋊C4—92no—
171D3×C22⋊C4—92no—
172D3×C22⋊C4—92no—
173C4×D12—31no—
174C4×D12—91no—
175C4×D12—91no—
176C4×D12—91no—
177C4×D12—91no—
178C4×D12—181no—
179#872D911yescharacteristic, complemented
180C22×C3⋊D3—32no—
181C22×C3×D3—32no—
182C22×C3×D3C22×D311yescharacteristic
183C2×D32—38no—
184C2×D32—122no—
185C2×D32—122no—
186C2×C32×C4C22×D311yescharacteristic
187C2×C32×C4C22×D311yescharacteristic
188C2×C32×C4C22×D311yescharacteristic
189C2×C3⋊Dic3—32no—
190C4⋊(C3⋊D3)—34no—
191C4×C3⋊D3—122no—
192C2×C3×Dic3—32no—
193C2×C3×Dic3C22×D311yescharacteristic
194C2×C3×Dic3C22×D311yescharacteristic
195C2×C3×Dic3C22×D311yescharacteristic
196C3×D12—34no—
197C3×C4×D3—122no—
198C3×C4×D3C4×D314yes—
199#626—38no—
200#624—122no—
201D3×Dic3—122no—
202C23×C9—31no—
203C22×D9—32no—
204C22×D9—38no—
205C22×D9—62no—
206C22×D9—122no—
207C22×D9—122no—
208C22×D9C22×D312yes—
209C2×C4×C9—21no—
210C2×C4×C9—21no—
211C2×C4×C9—21no—
212C2×C4×C9—31no—
213C2×C4×C9—34no—
214C2×C4×C9—31no—
215C2×C4×C9—31no—
216C2×C4×C9C22×D311yescharacteristic
217C2×C4×C9C22×D311yescharacteristic
218C2×C4×C9C22×D311yescharacteristic
219C2×Dic9—32no—
220C2×Dic9—62no—
221C2×Dic9C22×D312yes—
222D36—38no—
223D36—34no—
224D36—64no—
225D36C4×D314yescomplemented
226C4×D9—42no—
227C4×D9—122no—
228C4×D9—122no—
229C4×D9—122no—
230C4×D9—124no—
231#518—271noSylow
232C2×C3×C9C23×C411yescharacteristic
233C2×C3×C9H(5,𝔽2)11yescharacteristic
234C2×C3×C9H(5,𝔽2)11yescharacteristic
235C3⋊D9—24no—
236D3×C9—22no—
237C3×D9—24no—
238C23×D3—92no—
239C23×D3—92no—
240C2×C3×D4—91no—
241C22×C3×C4—31no—
242C22×C3×C4—92no—
243C2×#67—92no—
244C22×Dic3—92no—
245D3×D4—98no—
246C2×D12—31no—
247C2×D12—98no—
248C2×D12—92no—
249C2×D12—91no—
250C2×D12—91no—
251C2×D12—181no—
252C2×C4×D3—32no—
253C2×C4×D3—92no—
254C2×C4×D3—92no—
255C2×C4×D3—92no—
256C2×C4×D3—92no—
257C2×C4×D3—92no—
258C2×C4×D3—182no—
259C2×C4×D3C2×D911yescharacteristic
260C3×C4⋊C4—31no—
261C3×C4⋊C4—31no—
262C3×C4⋊C4—91no—
263C3×C22⋊C4—31no—
264C3×C22⋊C4—31no—
265C3×C22⋊C4—92no—
266C3×C42—21no—
267C3×C42—31no—
268C3×C42C2×D911yescharacteristic, complemented
269C3×C42D3211yescharacteristic
270#217—92no—
271#217—92no—
272(C2×D3)⋊C4—32no—
273(C2×D3)⋊C4—92no—
274(C2×D3)⋊C4—92no—
275(C2×D3)⋊C4—92no—
276(C2×D3)⋊C4—92no—
277(C2×D3)⋊C4—92no—
278(C2×D3)⋊C4—92no—
279(C2×D3)⋊C4—182no—
280(C2×D3)⋊C4C2×D911yescharacteristic
281(C2×D3)⋊C4C2×D911yescharacteristic
282C4⋊Dic3—31no—
283C4⋊Dic3—91no—
284C4⋊Dic3—91no—
285C4⋊Dic3—181no—
286Dic3⋊C4C2×D911yescharacteristic, complemented
287Dic3⋊C4C2×D911yescharacteristic, complemented
288C4×Dic3—92no—
289C4×Dic3—92no—
290C4×Dic3C2×D911yescharacteristic, complemented
291C22×C32C23×D311yescharacteristic
292C2×C3⋊D3—34no—
293C2×C3⋊D3—62no—
294C2×C3⋊D3—62no—
295C2×C3×D3—21no—
296C2×C3×D3—21no—
297C2×C3×D3—34no—
298C2×C3×D3—62no—
299C2×C3×D3—62no—
300C2×C3×D3C2×C4×D312yes—
301D32—128no—
302C32×C4—21no—
303C32×C4—21no—
304C32×C4C2×C4×D312yes—
305C3⋊Dic3—62no—
306C3×Dic3—21no—
307C3×Dic3—21no—
308C3×Dic3—62no—
309C3×Dic3C2×C4×D312yes—
310C22×C9—21no—
311C22×C9—32no—
312C22×C9—61no—
313C22×C9—61no—
314C22×C9C23×D311yescharacteristic
315C2×D9—22no—
316C2×D9—22no—
317C2×D9—34no—
318C2×D9—64no—
319C2×D9—62no—
320C2×D9—62no—
321C2×D9—122no—
322C2×D9—122no—
323C2×D9—128no—
324C2×D9C2×C4×D314yes—
325C4×C9—22no—
326C4×C9—21no—
327C4×C9—21no—
328C4×C9—22no—
329C4×C9—32no—
330C4×C9—41no—
331C4×C9—61no—
332C4×C9—61no—
333C4×C9C2×C4×D312yes—
334Dic9—22no—
335Dic9—62no—
336Dic9—122no—
337C22×D4—271no—
338C4×D4—91no—
339C4×D4—271no—
340C4×D4—271no—
341C4×D4—272no—
342C4×D4—272no—
343C4×D4—271no—
344#117—31no—
345#117—271no—
346C2×C22⋊C4—272no—
347C2×C22⋊C4—272no—
348C3×C9#51811yescharacteristic, complemented, Sylow
349C23×C3—31no—
350C23×C3—92no—
351C22×D3—32no—
352C22×D3—98no—
353C22×D3—92no—
354C22×D3—92no—
355C22×D3—98no—
356C22×D3—182no—
357C22×D3—362no—
358C22×D3—362no—
359C22×D3—362no—
360C22×D3—362no—
361C22×D3C22×D911yescharacteristic
362C3×D4—94no—
363C2×C3×C4—21no—
364C2×C3×C4—21no—
365C2×C3×C4—21no—
366C2×C3×C4—31no—
367C2×C3×C4—34no—
368C2×C3×C4—31no—
369C2×C3×C4—31no—
370C2×C3×C4—92no—
371C2×C3×C4—362no—
372C2×C3×C4C22×D911yescharacteristic
373C2×C3×C4C22×D911yescharacteristic
374C2×C3×C4C22×D911yescharacteristic
375C2×C3×C4C2×D3211yescharacteristic
376C2×C3×C4C2×D3211yescharacteristic
377C2×C3×C4C2×D3211yescharacteristic
378#67—98no—
379C2×Dic3—32no—
380C2×Dic3—92no—
381C2×Dic3—92no—
382C2×Dic3—182no—
383C2×Dic3—362no—
384C2×Dic3C22×D911yescharacteristic
385C2×Dic3C22×D911yescharacteristic
386C2×Dic3C22×D911yescharacteristic
387D12—34no—
388D12—98no—
389D12—94no—
390D12—94no—
391D12—184no—
392C4×D3—122no—
393C4×D3—362no—
394C4×D3—362no—
395C4×D3—362no—
396C4×D3—362no—
397C4×D3—362no—
398C4×D3—364no—
399C4×D3C4×D914yescomplemented
400C2×C32C22×C4×D311yescharacteristic
401C2×C32#100911yescharacteristic
402C2×C32#100911yescharacteristic
403C3⋊D3—64no—
404C3×D3—22no—
405C3×D3—64no—
406C2×C9—21no—
407C2×C9—21no—
408C2×C9—21no—
409C2×C9—62no—
410C2×C9C22×C4×D311yescharacteristic
411C2×C9#100411yescharacteristic
412C2×C9#100911yescharacteristic
413D9—24no—
414D9—64no—
415D9—124no—
416C24—272no—
417C2×D4—91no—
418C2×D4—278no—
419C2×D4—272no—
420C2×D4—271no—
421C22×C4—31no—
422C22×C4—92no—
423C22×C4—272no—
424C22×C4—272no—
425C22×C4—272no—
426C4⋊C4—31no—
427C4⋊C4—31no—
428C4⋊C4—91no—
429C4⋊C4—271no—
430C22⋊C4—31no—
431C22⋊C4—31no—
432C22⋊C4—92no—
433C22⋊C4—272no—
434C22⋊C4—272no—
435C22⋊C4—272no—
436C22⋊C4—272no—
437C42—31no—
438C42—272no—
439C42D3×D911yescharacteristic, complemented
440C22×C3—21no—
441C22×C3—32no—
442C22×C3—61no—
443C22×C3—61no—
444C22×C3—94no—
445C22×C3—182no—
446C22×C3—182no—
447C22×C3C23×D911yescharacteristic
448C22×C3C22×D3211yescharacteristic
449C2×D3—21no—
450C2×D3—21no—
451C2×D3—34no—
452C2×D3—62no—
453C2×D3—62no—
454C2×D3—94no—
455C2×D3—94no—
456C2×D3—184no—
457C2×D3—182no—
458C2×D3—182no—
459C2×D3—182no—
460C2×D3—182no—
461C2×D3—362no—
462C2×D3—362no—
463C2×D3—368no—
464C2×D3—368no—
465C2×D3C2×C4×D912yes—
466C3×C4—22no—
467C3×C4—21no—
468C3×C4—21no—
469C3×C4—21no—
470C3×C4—21no—
471C3×C4—22no—
472C3×C4—32no—
473C3×C4—41no—
474C3×C4—61no—
475C3×C4—61no—
476C3×C4—182no—
477C3×C4C2×C4×D912yes—
478C3×C4C4×D3212yes—
479Dic3—21no—
480Dic3—21no—
481Dic3—62no—
482Dic3—182no—
483Dic3—182no—
484Dic3—362no—
485Dic3C2×C4×D912yes—
486C32#552111yescharacteristic
487C9—21no—
488C9#554511yescharacteristic, complemented
489C23—31no—
490C23—92no—
491C23—278no—
492C23—272no—
493C23—1082no—
494C23—1082no—
495D4—94no—
496D4—278no—
497D4—274no—
498C2×C4—31no—
499C2×C4—34no—
500C2×C4—31no—
501C2×C4—31no—
502C2×C4—92no—
503C2×C4—272no—
504C2×C4—362no—
505C2×C4—1082no—
506C2×C4—1082no—
507C2×C4—1082no—
508C2×C4C2×D3×D911yescharacteristic
509C2×C4C2×D3×D911yescharacteristic
510C2×C4C2×D3×D911yescharacteristic
511C2×C3—21no—
512C2×C3—21no—
513C2×C3—21no—
514C2×C3—62no—
515C2×C3—184no—
516C2×C3C22×C4×D911yescharacteristic
517C2×C3#6371511yescharacteristic
518C2×C3#6371511yescharacteristic
519C2×C3C2×C4×D3211yescharacteristic
520C2×C3#6430711yescharacteristic
521C2×C3#6430811yescharacteristic
522D3—22no—
523D3—64no—
524D3—184no—
525D3—184no—
526D3—364no—
527C22—32no—
528C22—61no—
529C22—61no—
530C22—94no—
531C22—182no—
532C22—182no—
533C22—274no—
534C22—542no—
535C22—542no—
536C22—1088no—
537C22C22×D3×D911yescharacteristic
538C4—21no—
539C4—21no—
540C4—32no—
541C4—61no—
542C4—61no—
543C4—182no—
544C4—542no—
545C4C4×D3×D912yes—
546C3—21no—
547C3#1058962311yescharacteristic, complemented
548C3#1059399411yescharacteristic
549C2—62no—
550C2—184no—
551C2—544no—
552C2C2×C4×D3×D911yescharacteristic
553C2#1172247711yescharacteristic
554C2#1172247811yescharacteristic
subgroupG/H
centerC22C22×D3×D9
commutatorC2×C3×C9C23×C4
FrattiniC22×C3C22×D32
FittingC3×C42×C9C22
socleC22×C32C23×D3
prime radicalC2×D3×D36C2
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. C4×C4⋊(C3⋊D9)C2semidirect
3. C9×#872C2semidirect
4. C3×C4×D36C2semidirect
5. #11721048C2semidirect
6. D9×(C2×D3)⋊C4C2semidirect
7. #11721002C2semidirect
8. #11720993C2semidirect
9. Dic3×D36C2semidirect
10. #11720978C2semidirect
11. #11720906C2semidirect
16. C2×C4×C3⋊D9C22semidirect
19. C2×C3×C4×D9C22semidirect
21. C2×(C3⋊D9)⋊C4C22semidirect
22. C2×Dic3×D9C22semidirect
23. D3×D36C4semidirect
24. C3×C42×C9C22semidirect
25. #88536C22semidirect
26. C4⋊(C3⋊Dic9)C22semidirect
29. C3×(C2×D9)⋊C4C22semidirect
30. C9×Dic3⋊C4C22semidirect
31. C9×Dic3⋊C4C22semidirect
32. C4×C9×Dic3C22semidirect
33. C3×C4⋊Dic9C22semidirect
35. #88445C22semidirect
36. (C2×D9)⋊Dic3C22semidirect
37. Dic3×Dic9C22semidirect
60. C4×D36D3semidirect
66. C2×D3×D9C2×C4semidirect
71. C4⋊(C3⋊D9)C2×C4semidirect
77. C4×D3×C9C2×C4semidirect
78. C3×D36C2×C4semidirect
80. #6192C2×C4semidirect
120. C2×C4×D9C2×D3semidirect
127. C42×C9C2×D3semidirect
133. (C2×D9)⋊C4C2×D3semidirect
136. C4⋊Dic9C2×D3semidirect
179. #872D9semidirect
225. D36C4×D3semidirect
268. C3×C42C2×D9semidirect
286. Dic3⋊C4C2×D9semidirect
287. Dic3⋊C4C2×D9semidirect
290. C4×Dic3C2×D9semidirect
348. C3×C9#518semidirect
399. C4×D3C4×D9semidirect
439. C42D3×D9semidirect
488. C9#5545semidirect
547. C3#10589623semidirect

Maximal subgroups

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

subgroupindex
C2×D3×D362
C4×C4⋊(C3⋊D9)2
C9×#8722
C3×C4×D362
#117210482
D9×(C2×D3)⋊C42
#117210022
#117209932
Dic3×D362
#117209782
#117209062
#105939943
#105896233