Finite Group Databasebeta

← #13147700

Subgroups — #13147700

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×D9×D12C211yesmaximal, characteristic
2C4×C4⋊(C3⋊D9)C211yesmaximal, characteristic, complemented
3C4×C9×D12C211yesmaximal, characteristic, complemented
4C3×#63434C211yesmaximal, characteristic, complemented
5#11721048C212yesmaximal, complemented
6D3×(C2×D9)⋊C4C212yesmaximal, complemented
7#11721030C212yesmaximal, complemented
8D12×Dic9C211yesmaximal, characteristic, complemented
9#11721014C212yesmaximal, complemented
10#11720992C211yesmaximal, characteristic, complemented
11#11720907C211yesmaximal, characteristic, complemented
12#10593994—31nomaximal
13#10589647—31nomaximal
14C22×D3×D9C2212yes—
15C2×C4⋊(C3⋊D9)C2211yescharacteristic
16C2×C4×C3⋊D9C2212yescomplemented
17C2×C9×D12C2211yescharacteristic
18C2×C4×D3×C9C2212yescomplemented
19C2×C3×C4×D9C2211yescharacteristic
20C2×#6195C2212yes—
21C2×D3×Dic9C2212yescomplemented
22C2×(C3⋊D9)⋊C4C2212yescomplemented
23D9×D12C418yescomplemented
24C3×C42×C9C2211yescharacteristic, complemented
25#88536C2212yescomplemented
26C4⋊(C3⋊Dic9)C2211yescharacteristic, complemented
27C9×(C2×D3)⋊C4C2212yescomplemented
28C3×(C2×D9)⋊C4C2211yescharacteristic
29C3×(C2×D9)⋊C4C2211yescharacteristic
30C9×C4⋊Dic3C2211yescharacteristic, complemented
31C3×Dic9⋊C4C2211yescharacteristic, complemented
32C3×Dic9⋊C4C2211yescharacteristic, complemented
33C3×C4×Dic9C2211yescharacteristic, complemented
34(C2×D3)⋊Dic9C2212yescomplemented
35#88445C2212yescomplemented
36(C2×D9)⋊Dic3C2212yes—
37Dic3×Dic9C2212yescomplemented
38C2×D3×D12—31no—
39C4×C4⋊(C3⋊D3)—31no—
40C3×C4×D12D311yescharacteristic
41C3×#872—31no—
42#63921—32no—
43D3×(C2×D3)⋊C4—32no—
44#63904—32no—
45#63894—31no—
46Dic3×D12—31no—
47#63881—32no—
48#63841—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×D36—31no—
59C4×D36—61no—
60#63434D311yescharacteristic, complemented
61C22×C3⋊D9C2312yes—
62C22×D3×C9C2312yes—
63C22×C3×D9C2311yescharacteristic
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×Dic3C2312yes—
74C2×C3×Dic9C2311yescharacteristic
75C2×C3×Dic9C2311yescharacteristic
76C2×C3×Dic9C2311yescharacteristic
77C9×D12C2×C414yescomplemented
78C4×D3×C9—42no—
79C3×C4×D9C2×C414yescomplemented
80#6195C2×C418yescomplemented
81D3×Dic9—42no—
82(C3⋊D9)⋊C4—42no—
83#5545—91no—
84#5521—91no—
85C22×D32—32no—
86C2×C4⋊(C3⋊D3)—31no—
87C2×C4×C3⋊D3—32no—
88C2×C3×D12C2×D311yescharacteristic
89C2×C3×C4×D3—31no—
90C2×C3×C4×D3C2×D312yes—
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—31no—
100C3×(C2×D3)⋊C4—31no—
101C3×(C2×D3)⋊C4C2×D312yes—
102C3×C4⋊Dic3C2×D311yescharacteristic
103C3×Dic3⋊C4—31no—
104C3×Dic3⋊C4—31no—
105C3×C4×Dic3—31no—
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×D36—31no—
116C2×D36—61no—
117C2×C4×D9—32no—
118C2×C4×D9—32no—
119C2×C4×D9—62no—
120C2×C4×D9C2×D311yescharacteristic
121C9×C4⋊C4—31no—
122C9×C22⋊C4—32no—
123C42×C9—21no—
124C42×C9C2×D311yescharacteristic, complemented
125#3676—32no—
126(C2×D9)⋊C4—32no—
127(C2×D9)⋊C4—32no—
128(C2×D9)⋊C4—32no—
129(C2×D9)⋊C4—62no—
130(C2×D9)⋊C4C2×D311yescharacteristic
131(C2×D9)⋊C4C2×D311yescharacteristic
132C4⋊Dic9—31no—
133C4⋊Dic9—61no—
134Dic9⋊C4C2×D311yescharacteristic, complemented
135Dic9⋊C4C2×D311yescharacteristic, complemented
136C4×Dic9—32no—
137C4×Dic9C2×D311yescharacteristic, complemented
138C22×C3×C9C2411yescharacteristic
139C2×C3⋊D9—22no—
140C2×C3⋊D9—22no—
141C2×C3⋊D9C22×C414yes—
142C2×D3×C9—22no—
143C2×D3×C9—22no—
144C2×D3×C9C22×C414yes—
145C2×C3×D9—21no—
146C2×C3×D9—21no—
147C2×C3×D9C22×C412yes—
148D3×D9—48no—
149C3×C4×C9—21no—
150C3×C4×C9—21no—
151C3×C4×C9C22×C412yes—
152C3⋊Dic9—22no—
153C9×Dic3—22no—
154C3×Dic9—21no—
155C3×Dic9—21no—
156C3×Dic9C22×C412yes—
157C2×D3×D4—91no—
158C22×D12—91no—
159C3×C4×D4—31no—
160C3×#117—91no—
161D4×Dic3—91no—
162C4×#67—92no—
163C2×(C2×D3)⋊C4—92no—
164#926—91no—
165#893—91no—
166#893—92no—
167#893—92no—
168#892—91no—
169#67⋊C4—92no—
170D3×C22⋊C4—92no—
171D3×C22⋊C4—92no—
172D3×C22⋊C4—92no—
173C4×D12—91no—
174C4×D12—91no—
175C4×D12—91no—
176C4×D12—91no—
177C4×D12—181no—
178C4×D12D911yescharacteristic, complemented
179#872—31no—
180C22×C3⋊D3—32no—
181C22×C3×D3—31no—
182C22×C3×D3C22×D312yes—
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—31no—
193C2×C3×Dic3—31no—
194C2×C3×Dic3—31no—
195C2×C3×Dic3C22×D312yes—
196C3×D12C4×D314yes—
197C3×C4×D3—34no—
198C3×C4×D3—42no—
199#626—38no—
200#624—122no—
201D3×Dic3—122no—
202C23×C9—32no—
203C22×D9—32no—
204C22×D9—38no—
205C22×D9—62no—
206C22×D9—122no—
207C22×D9—122no—
208C22×D9C22×D311yescharacteristic
209D4×C9—34no—
210C2×C4×C9—21no—
211C2×C4×C9—21no—
212C2×C4×C9—21no—
213C2×C4×C9—32no—
214C2×C4×C9—122no—
215C2×C4×C9C22×D311yescharacteristic
216C2×C4×C9C22×D311yescharacteristic
217C2×C4×C9C22×D311yescharacteristic
218#611—38no—
219C2×Dic9—32no—
220C2×Dic9—62no—
221C2×Dic9—122no—
222C2×Dic9C22×D311yescharacteristic
223C2×Dic9C22×D311yescharacteristic
224C2×Dic9C22×D311yescharacteristic
225D36—34no—
226D36—64no—
227C4×D9—122no—
228C4×D9—122no—
229C4×D9—124no—
230C4×D9C4×D314yescomplemented
231#518—271noSylow
232C2×C3×C9C23×C411yescharacteristic
233C2×C3×C9H(5,𝔽2)11yescharacteristic
234C2×C3×C9H(5,𝔽2)11yescharacteristic
235C3⋊D9—24no—
236D3×C9—24no—
237C3×D9—22no—
238C23×D3—92no—
239C23×D3—92no—
240C2×C3×D4—31no—
241C22×C3×C4—32no—
242C22×C3×C4—91no—
243C2×#67—92no—
244C22×Dic3—92no—
245D3×D4—98no—
246C2×D12—92no—
247C2×D12—91no—
248C2×D12—91no—
249C2×D12—98no—
250C2×D12—181no—
251C2×D12C2×D911yescharacteristic
252C2×C4×D3—31no—
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×D912yescomplemented
260C3×C4⋊C4—31no—
261C3×C4⋊C4—91no—
262C3×C4⋊C4—91no—
263C3×C22⋊C4—32no—
264C3×C22⋊C4—91no—
265C3×C22⋊C4—91no—
266C3×C42—21no—
267C3×C42—91no—
268C3×C42C2×D911yescharacteristic, complemented
269C3×C42D3211yescharacteristic
270#217—92no—
271#217—92no—
272(C2×D3)⋊C4—31no—
273(C2×D3)⋊C4—31no—
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—92no—
280(C2×D3)⋊C4—182no—
281(C2×D3)⋊C4C2×D912yescomplemented
282C4⋊Dic3—91no—
283C4⋊Dic3—91no—
284C4⋊Dic3—181no—
285C4⋊Dic3C2×D911yescharacteristic, complemented
286Dic3⋊C4—31no—
287Dic3⋊C4—31no—
288C4×Dic3—31no—
289C4×Dic3—92no—
290C4×Dic3—92no—
291C22×C32C23×D311yescharacteristic
292C2×C3⋊D3—34no—
293C2×C3⋊D3—62no—
294C2×C3⋊D3—62no—
295C2×C3×D3—22no—
296C2×C3×D3—22no—
297C2×C3×D3—32no—
298C2×C3×D3—61no—
299C2×C3×D3—61no—
300C2×C3×D3C2×C4×D314yes—
301D32—128no—
302C32×C4—21no—
303C32×C4—21no—
304C32×C4C2×C4×D312yes—
305C3⋊Dic3—62no—
306C3×Dic3—22no—
307C3×Dic3—32no—
308C3×Dic3—61no—
309C3×Dic3—61no—
310C22×C9—21no—
311C22×C9—34no—
312C22×C9—62no—
313C22×C9—62no—
314C22×C9C23×D311yescharacteristic
315C2×D9—21no—
316C2×D9—21no—
317C2×D9—34no—
318C2×D9—64no—
319C2×D9—62no—
320C2×D9—62no—
321C2×D9—122no—
322C2×D9—122no—
323C2×D9—128no—
324C2×D9C2×C4×D312yes—
325C4×C9—22no—
326C4×C9—21no—
327C4×C9—21no—
328C4×C9—22no—
329C4×C9—41no—
330C4×C9—62no—
331C4×C9C2×C4×D312yes—
332Dic9—21no—
333Dic9—21no—
334Dic9—62no—
335Dic9—122no—
336Dic9C2×C4×D312yes—
337C22×D4—271no—
338C4×D4—31no—
339C4×D4—271no—
340C4×D4—271no—
341C4×D4—272no—
342C4×D4—272no—
343C4×D4—271no—
344#117—91no—
345#117—271no—
346C2×C22⋊C4—272no—
347C2×C22⋊C4—272no—
348C3×C9#51811yescharacteristic, complemented, Sylow
349C23×C3—32no—
350C23×C3—91no—
351C22×D3—31no—
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×D912yes—
362C3×D4—34no—
363C2×C3×C4—21no—
364C2×C3×C4—21no—
365C2×C3×C4—21no—
366C2×C3×C4—32no—
367C2×C3×C4—91no—
368C2×C3×C4—94no—
369C2×C3×C4—91no—
370C2×C3×C4—91no—
371C2×C3×C4—122no—
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—31no—
380C2×Dic3—31no—
381C2×Dic3—31no—
382C2×Dic3—92no—
383C2×Dic3—92no—
384C2×Dic3—182no—
385C2×Dic3—362no—
386C2×Dic3C22×D912yes—
387D12—98no—
388D12—94no—
389D12—94no—
390D12—184no—
391D12C4×D914yescomplemented
392C4×D3—34no—
393C4×D3—42no—
394C4×D3—362no—
395C4×D3—362no—
396C4×D3—362no—
397C4×D3—362no—
398C4×D3—362no—
399C4×D3—364no—
400C2×C32C22×C4×D311yescharacteristic
401C2×C32#100411yescharacteristic
402C2×C32#100911yescharacteristic
403C3⋊D3—64no—
404C3×D3—24no—
405C3×D3—62no—
406C2×C9—21no—
407C2×C9—21no—
408C2×C9—21no—
409C2×C9—64no—
410C2×C9C22×C4×D311yescharacteristic
411C2×C9#100911yescharacteristic
412C2×C9#100911yescharacteristic
413D9—22no—
414D9—64no—
415D9—124no—
416C24—272no—
417C2×D4—31no—
418C2×D4—278no—
419C2×D4—272no—
420C2×D4—271no—
421C22×C4—32no—
422C22×C4—91no—
423C22×C4—272no—
424C22×C4—272no—
425C22×C4—272no—
426C4⋊C4—31no—
427C4⋊C4—91no—
428C4⋊C4—91no—
429C4⋊C4—271no—
430C22⋊C4—32no—
431C22⋊C4—91no—
432C22⋊C4—91no—
433C22⋊C4—272no—
434C22⋊C4—272no—
435C22⋊C4—272no—
436C22⋊C4—272no—
437C42—91no—
438C42—272no—
439C42D3×D911yescharacteristic, complemented
440C22×C3—21no—
441C22×C3—34no—
442C22×C3—62no—
443C22×C3—62no—
444C22×C3—92no—
445C22×C3—181no—
446C22×C3—181no—
447C22×C3C23×D911yescharacteristic
448C22×C3C22×D3211yescharacteristic
449C2×D3—22no—
450C2×D3—22no—
451C2×D3—32no—
452C2×D3—61no—
453C2×D3—61no—
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×D914yes—
466C3×C4—22no—
467C3×C4—21no—
468C3×C4—21no—
469C3×C4—21no—
470C3×C4—21no—
471C3×C4—22no—
472C3×C4—41no—
473C3×C4—62no—
474C3×C4—92no—
475C3×C4—181no—
476C3×C4—181no—
477C3×C4C2×C4×D912yes—
478C3×C4C4×D3212yes—
479Dic3—22no—
480Dic3—32no—
481Dic3—61no—
482Dic3—61no—
483Dic3—182no—
484Dic3—182no—
485Dic3—362no—
486C32#554511yescharacteristic
487C9—21no—
488C9#552111yescharacteristic, complemented
489C23—32no—
490C23—91no—
491C23—278no—
492C23—272no—
493C23—1082no—
494C23—1082no—
495D4—34no—
496D4—278no—
497D4—274no—
498C2×C4—32no—
499C2×C4—91no—
500C2×C4—94no—
501C2×C4—91no—
502C2×C4—91no—
503C2×C4—122no—
504C2×C4—272no—
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—64no—
515C2×C3—182no—
516C2×C3C22×C4×D911yescharacteristic
517C2×C3#6371011yescharacteristic
518C2×C3#6371511yescharacteristic
519C2×C3C2×C4×D3211yescharacteristic
520C2×C3#6430711yescharacteristic
521C2×C3#6430811yescharacteristic
522D3—24no—
523D3—62no—
524D3—184no—
525D3—184no—
526D3—364no—
527C22—34no—
528C22—62no—
529C22—62no—
530C22—92no—
531C22—181no—
532C22—181no—
533C22—274no—
534C22—542no—
535C22—542no—
536C22—1088no—
537C22C22×D3×D911yescharacteristic
538C4—21no—
539C4—21no—
540C4—62no—
541C4—92no—
542C4—181no—
543C4—181no—
544C4—542no—
545C4C4×D3×D912yes—
546C3—21no—
547C3#1058964711yescharacteristic, complemented
548C3#1059399411yescharacteristic
549C2—64no—
550C2—182no—
551C2—544no—
552C2C2×C4×D3×D911yescharacteristic
553C2#1172247611yescharacteristic
554C2#1172247811yescharacteristic
subgroupG/H
centerC22C22×D3×D9
commutatorC2×C3×C9C23×C4
FrattiniC22×C3C22×D32
FittingC3×C42×C9C22
socleC22×C32C23×D3
prime radicalC2×D9×D12C2
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. C4×C9×D12C2semidirect
4. C3×#63434C2semidirect
5. #11721048C2semidirect
6. D3×(C2×D9)⋊C4C2semidirect
7. #11721030C2semidirect
8. D12×Dic9C2semidirect
9. #11721014C2semidirect
10. #11720992C2semidirect
11. #11720907C2semidirect
16. C2×C4×C3⋊D9C22semidirect
18. C2×C4×D3×C9C22semidirect
21. C2×D3×Dic9C22semidirect
22. C2×(C3⋊D9)⋊C4C22semidirect
23. D9×D12C4semidirect
24. C3×C42×C9C22semidirect
25. #88536C22semidirect
26. C4⋊(C3⋊Dic9)C22semidirect
27. C9×(C2×D3)⋊C4C22semidirect
30. C9×C4⋊Dic3C22semidirect
31. C3×Dic9⋊C4C22semidirect
32. C3×Dic9⋊C4C22semidirect
33. C3×C4×Dic9C22semidirect
34. (C2×D3)⋊Dic9C22semidirect
35. #88445C22semidirect
37. Dic3×Dic9C22semidirect
60. #63434D3semidirect
66. C2×D3×D9C2×C4semidirect
71. C4⋊(C3⋊D9)C2×C4semidirect
77. C9×D12C2×C4semidirect
79. C3×C4×D9C2×C4semidirect
80. #6195C2×C4semidirect
124. C42×C9C2×D3semidirect
134. Dic9⋊C4C2×D3semidirect
135. Dic9⋊C4C2×D3semidirect
137. C4×Dic9C2×D3semidirect
178. C4×D12D9semidirect
230. C4×D9C4×D3semidirect
259. C2×C4×D3C2×D9semidirect
268. C3×C42C2×D9semidirect
281. (C2×D3)⋊C4C2×D9semidirect
285. C4⋊Dic3C2×D9semidirect
348. C3×C9#518semidirect
391. D12C4×D9semidirect
439. C42D3×D9semidirect
488. C9#5521semidirect
547. C3#10589647semidirect

Maximal subgroups

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

subgroupindex
C2×D9×D122
C4×C4⋊(C3⋊D9)2
C4×C9×D122
C3×#634342
#117210482
D3×(C2×D9)⋊C42
#117210302
D12×Dic92
#117210142
#117209922
#117209072
#105939943
#105896473