Finite Group Databasebeta

← #13150554

Subgroups — #13150554

Order 1728.

1132 conjugacy classes of subgroups.

The subgroup lattice has 570 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×D3×#611C212yesmaximal, complemented
2#11721819C211yesmaximal, characteristic, complemented
3C2×C9×(C2×D3)⋊C4C211yesmaximal, characteristic, complemented
4C3×#63492C211yesmaximal, characteristic, complemented
5#11721277C212yesmaximal, complemented
6#11721051C212yesmaximal, complemented
7#11721049C212yesmaximal, complemented
8#11721021C212yesmaximal, complemented
9#11721020C212yesmaximal, complemented
10#10595072—31nomaximal
11#10589595—31nomaximal
12C23×D3×C9C2211yescharacteristic
13C22×D3×D9C2212yescomplemented
14C22×C3×C4×C9C2211yescharacteristic, complemented
15C2×#6233C2212yescomplemented
16C2×C4⋊(C3⋊D9)C2211yescharacteristic, complemented
17C2×C3×#611C2212yescomplemented
18C22×C9×Dic3C2211yescharacteristic, complemented
19C2×C3×D36C2211yescharacteristic, complemented
20D3×#611—28no—
21C2×#6195C2212yes—
22C2×#6194C2212yes—
23C2×D3×Dic9C2212yescomplemented
24C2×#6192C2212yescomplemented
25#88536C2212yescomplemented
26C4⋊(C3⋊Dic9)C2211yescharacteristic, complemented
27C9×(C2×D3)⋊C4C2212yescomplemented
28C9×(C2×D3)⋊C4C2212yescomplemented
29C3×(C2×D9)⋊C4C2212yescomplemented
30C3×C4⋊Dic9C2211yescharacteristic, complemented
31#88445C2212yescomplemented
32(C2×D9)⋊Dic3C2212yescomplemented
33Dic3⋊Dic9C2212yescomplemented
34C2×D3×#67—32no—
35#64139—31no—
36C3×#930—31no—
37C2×C3×(C2×D3)⋊C4D311yescharacteristic
38#63978—32no—
39#63923—32no—
40#63922—32no—
41#63891—32no—
42#63890—32no—
43C22×#611—32no—
44C2×C9×C22⋊C4—31no—
45#63492—31no—
46#63492—32no—
47#63492—61no—
48#63492D311yescharacteristic, complemented
49#63448—32no—
50#63448—32no—
51#63444—32no—
52#63444—32no—
53C23×C3×C9C2311yescharacteristic
54C22×C3⋊D9C2312yes—
55C22×D3×C9—41no—
56C22×D3×C9—41no—
57C22×D3×C9D414yescomplemented
58C22×D3×C9C2312yes—
59C22×C3×D9C2312yes—
60C2×D3×D9—24no—
61C2×D3×D9—42no—
62C2×D3×D9—42no—
63C2×C3×C4×C9—41no—
64C2×C3×C4×C9C2311yescharacteristic
65C2×C3×C4×C9C2311yescharacteristic
66#6233—24no—
67C2×C3⋊Dic9C2312yes—
68C4⋊(C3⋊D9)—41no—
69C3×#611—24no—
70C2×C9×Dic3—41no—
71C2×C9×Dic3C2312yes—
72C2×C3×Dic9C2312yes—
73C3×D36—41no—
74#6195—24no—
75#6194—24no—
76D3×Dic9—24no—
77#6192—42no—
78#5605—91no—
79#5493—91no—
80C23×C3×D3C2×D311yescharacteristic
81C22×D32—32no—
82C22×C32×C4C2×D311yescharacteristic
83C2×#638—32no—
84C2×C4⋊(C3⋊D3)—31no—
85C2×C3×#67—32no—
86C22×C3×Dic3C2×D311yescharacteristic
87C2×C3×D12—31no—
88D3×#67—68no—
89C2×#626—32no—
90C2×#626—32no—
91C2×#625—32no—
92C2×D3×Dic3—32no—
93C24×C9—31no—
94C23×D9—32no—
95#3752—32no—
96C4⋊(C3⋊Dic3)—31no—
97C3×(C2×D3)⋊C4—32no—
98C3×(C2×D3)⋊C4C2×D312yes—
99C3×(C2×D3)⋊C4C2×D312yes—
100C3×C4⋊Dic3—31no—
101Dic3⋊Dic3—32no—
102#3722—32no—
103(C2×D3)⋊Dic3—32no—
104C22×C4×C9—21no—
105C22×C4×C9—31no—
106C22×C4×C9C2×D311yescharacteristic, complemented
107C2×#611—32no—
108C2×#611—32no—
109C2×#611—32no—
110C2×#611—68no—
111C2×#611—62no—
112C2×#611C2×D312yescomplemented
113C22×Dic9—32no—
114C2×D36—32no—
115C2×D36—31no—
116C2×D36—61no—
117C2×D36C2×D311yescharacteristic, complemented
118C9×C22⋊C4—32no—
119C9×C22⋊C4—32no—
120(C2×D9)⋊C4—32no—
121(C2×D9)⋊C4—32no—
122(C2×D9)⋊C4—32no—
123(C2×D9)⋊C4—62no—
124(C2×D9)⋊C4C2×D312yescomplemented
125C4⋊Dic9—31no—
126C4⋊Dic9—32no—
127C4⋊Dic9—61no—
128C4⋊Dic9C2×D311yescharacteristic, complemented
129C22×C3×C9—21no—
130C22×C3×C9—21no—
131C22×C3×C9C2×D412yes—
132C22×C3×C9C2411yescharacteristic
133C2×C3⋊D9—22no—
134C2×C3⋊D9—22no—
135C2×C3⋊D9—22no—
136C2×D3×C9—22no—
137C2×D3×C9—22no—
138C2×D3×C9—44no—
139C2×D3×C9C2×D414yes—
140C2×C3×D9—22no—
141C2×C3×D9—22no—
142C2×C3×D9—22no—
143D3×D9—48no—
144C3×C4×C9—21no—
145C3×C4×C9—21no—
146C3⋊Dic9—22no—
147C9×Dic3—22no—
148C3×Dic9—22no—
149C22×#67—92no—
150C2×D3×D4—92no—
151C3×#121—91no—
152C2×C3×C22⋊C4—31no—
153#939—92no—
154#937—92no—
155#930—31no—
156#930—92no—
157#930—91no—
158#930—91no—
159#930—181no—
160C2×(C2×D3)⋊C4D911yescharacteristic, complemented
161#894—92no—
162#886—92no—
163#886—92no—
164#883—92no—
165#882—92no—
166#882—92no—
167#882—92no—
168C23×C32C22×D311yescharacteristic
169C22×C3⋊D3—32no—
170C22×C3×D3—32no—
171C22×C3×D3—41no—
172C22×C3×D3—41no—
173C22×C3×D3D1214yes—
174C22×C3×D3C22×D312yes—
175C2×D32—64no—
176C2×D32—122no—
177C2×D32—122no—
178C2×C32×C4—41no—
179C2×C32×C4C22×D311yescharacteristic
180C2×C32×C4C22×D311yescharacteristic
181#638—64no—
182C2×C3⋊Dic3—32no—
183C4⋊(C3⋊D3)—121no—
184C3×#67—64no—
185C2×C3×Dic3—32no—
186C2×C3×Dic3—41no—
187C2×C3×Dic3C22×D312yes—
188C3×D12—121no—
189#626—64no—
190#626—122no—
191#625—64no—
192D3×Dic3—64no—
193C23×C9—21no—
194C23×C9—32no—
195C23×C9—34no—
196C23×C9—121no—
197C23×C9—121no—
198C23×C9C22×D311yescharacteristic
199C22×D9—32no—
200C22×D9—62no—
201C22×D9—64no—
202C22×D9—122no—
203C22×D9—122no—
204C22×D9C22×D312yes—
205C2×C4×C9—21no—
206C2×C4×C9—21no—
207C2×C4×C9—32no—
208C2×C4×C9—41no—
209C2×C4×C9—42no—
210C2×C4×C9—121no—
211C2×C4×C9C22×D311yescharacteristic
212C2×C4×C9C22×D311yescharacteristic
213#611—24no—
214#611—64no—
215#611—64no—
216#611—64no—
217#611—124no—
218C2×Dic9—32no—
219C2×Dic9—62no—
220C2×Dic9—64no—
221C2×Dic9C22×D312yes—
222D36—41no—
223D36—121no—
224D36—122no—
225D36—122no—
226#534—271noSylow
227C2×C3×C9—22no—
228C2×C3×C9C22×D411yescharacteristic
229C2×C3×C9H(5,𝔽2)11yescharacteristic
230C2×C3×C9H(5,𝔽2)11yescharacteristic
231C3⋊D9—42no—
232D3×C9—24no—
233C3×D9—42no—
234C24×C3—31no—
235C23×D3—92no—
236C23×D3—92no—
237C23×D3C2×D911yescharacteristic
238C2×C3×D4—92no—
239C2×C3×D4—91no—
240C22×C3×C4—21no—
241C22×C3×C4—31no—
242C22×C3×C4C2×D911yescharacteristic, complemented
243C22×C3×C4D3211yescharacteristic
244C2×#67—32no—
245C2×#67—92no—
246C2×#67—92no—
247C2×#67—92no—
248C2×#67—92no—
249C2×#67—92no—
250C2×#67—92no—
251C2×#67—182no—
252C2×#67—188no—
253C22×Dic3—92no—
254C22×Dic3C2×D911yescharacteristic, complemented
255D3×D4—188no—
256C2×D12—31no—
257C2×D12—92no—
258C2×D12—92no—
259C2×D12—91no—
260C2×D12—91no—
261C2×D12—181no—
262C2×C4×D3—92no—
263C3×C4⋊C4—91no—
264C3×C22⋊C4—32no—
265C3×C22⋊C4—32no—
266C3×C22⋊C4—92no—
267#217—92no—
268(C2×D3)⋊C4—32no—
269(C2×D3)⋊C4—92no—
270(C2×D3)⋊C4—92no—
271(C2×D3)⋊C4—92no—
272(C2×D3)⋊C4—92no—
273(C2×D3)⋊C4—92no—
274(C2×D3)⋊C4—182no—
275(C2×D3)⋊C4C2×D912yescomplemented
276(C2×D3)⋊C4C2×D912yescomplemented
277C4⋊Dic3—31no—
278C4⋊Dic3—92no—
279C4⋊Dic3—91no—
280C4⋊Dic3—91no—
281C4⋊Dic3—181no—
282Dic3⋊C4—92no—
283C22×C32—21no—
284C22×C32—21no—
285C22×C32C2×D1212yes—
286C22×C32C23×D311yescharacteristic
287C2×C3⋊D3—62no—
288C2×C3⋊D3—62no—
289C2×C3⋊D3—62no—
290C2×C3×D3—22no—
291C2×C3×D3—22no—
292C2×C3×D3—44no—
293C2×C3×D3—62no—
294C2×C3×D3—62no—
295C2×C3×D3—62no—
296C2×C3×D3C2×D1214yes—
297D32—128no—
298C32×C4—21no—
299C32×C4—21no—
300C3⋊Dic3—62no—
301C3×Dic3—22no—
302C3×Dic3—62no—
303C22×C9—21no—
304C22×C9—21no—
305C22×C9—21no—
306C22×C9—22no—
307C22×C9—34no—
308C22×C9—41no—
309C22×C9—41no—
310C22×C9—62no—
311C22×C9—62no—
312C22×C9—124no—
313C22×C9D3×D412yes—
314C22×C9C23×D311yescharacteristic
315C2×D9—22no—
316C2×D9—22no—
317C2×D9—22no—
318C2×D9—62no—
319C2×D9—62no—
320C2×D9—62no—
321C2×D9—122no—
322C2×D9—122no—
323C2×D9—122no—
324C2×D9—128no—
325C4×C9—22no—
326C4×C9—21no—
327C4×C9—21no—
328C4×C9—41no—
329C4×C9—62no—
330Dic9—22no—
331Dic9—62no—
332Dic9—122no—
333C22×D4—272no—
334#123—272no—
335#123—272no—
336#121—91no—
337#121—272no—
338#121—271no—
339#120—272no—
340#120—272no—
341C2×C22⋊C4—31no—
342C3×C9#53411yescharacteristic, complemented, Sylow
343C23×C3—21no—
344C23×C3—32no—
345C23×C3—34no—
346C23×C3—92no—
347C23×C3—121no—
348C23×C3—121no—
349C23×C3C22×D911yescharacteristic
350C23×C3C2×D3211yescharacteristic
351C22×D3—32no—
352C22×D3—41no—
353C22×D3—41no—
354C22×D3—92no—
355C22×D3—92no—
356C22×D3—182no—
357C22×D3—184no—
358C22×D3—184no—
359C22×D3—362no—
360C22×D3—362no—
361C22×D3—362no—
362C22×D3—362no—
363C22×D3D3614yescomplemented
364C22×D3C22×D912yes—
365C3×D4—184no—
366C3×D4—361no—
367C2×C3×C4—21no—
368C2×C3×C4—21no—
369C2×C3×C4—32no—
370C2×C3×C4—41no—
371C2×C3×C4—41no—
372C2×C3×C4—42no—
373C2×C3×C4—92no—
374C2×C3×C4—121no—
375C2×C3×C4C22×D911yescharacteristic
376C2×C3×C4C22×D911yescharacteristic
377C2×C3×C4C2×D3211yescharacteristic
378C2×C3×C4C2×D3211yescharacteristic
379#67—64no—
380#67—184no—
381#67—184no—
382#67—184no—
383#67—184no—
384#67—184no—
385#67—364no—
386#67—362no—
387C2×Dic3—32no—
388C2×Dic3—41no—
389C2×Dic3—92no—
390C2×Dic3—92no—
391C2×Dic3—182no—
392C2×Dic3—184no—
393C2×Dic3C22×D912yes—
394D12—121no—
395D12—184no—
396D12—361no—
397D12—361no—
398D12—362no—
399D12—362no—
400C4×D3—184no—
401C2×C32—22no—
402C2×C32C22×D1211yescharacteristic
403C2×C32#100411yescharacteristic
404C2×C32#100411yescharacteristic
405C3⋊D3—122no—
406C3×D3—24no—
407C3×D3—122no—
408C2×C9—22no—
409C2×C9—21no—
410C2×C9—21no—
411C2×C9—21no—
412C2×C9—42no—
413C2×C9—64no—
414C2×C9C2×D3×D411yescharacteristic
415C2×C9#100411yescharacteristic
416C2×C9#100911yescharacteristic
417D9—42no—
418D9—122no—
419D9—242no—
420C24—31no—
421C24—272no—
422C2×D4—92no—
423C2×D4—91no—
424C2×D4—272no—
425C2×D4—272no—
426C2×D4—272no—
427C2×D4—272no—
428C2×D4—271no—
429C2×D4—548no—
430C22×C4—31no—
431C22×C4—272no—
432C22×C4D3×D911yescharacteristic, complemented
433C4⋊C4—91no—
434C4⋊C4—272no—
435C4⋊C4—271no—
436C22⋊C4—32no—
437C22⋊C4—32no—
438C22⋊C4—92no—
439C22⋊C4—272no—
440C22⋊C4—272no—
441C22⋊C4—272no—
442C22×C3—21no—
443C22×C3—21no—
444C22×C3—21no—
445C22×C3—21no—
446C22×C3—21no—
447C22×C3—22no—
448C22×C3—34no—
449C22×C3—41no—
450C22×C3—41no—
451C22×C3—62no—
452C22×C3—62no—
453C22×C3—124no—
454C22×C3—182no—
455C22×C3—182no—
456C22×C3—182no—
457C22×C3C2×D3612yes—
458C22×C3C23×D911yescharacteristic
459C22×C3D3×D1212yes—
460C22×C3C22×D3211yescharacteristic
461C2×D3—22no—
462C2×D3—22no—
463C2×D3—44no—
464C2×D3—62no—
465C2×D3—62no—
466C2×D3—62no—
467C2×D3—182no—
468C2×D3—182no—
469C2×D3—182no—
470C2×D3—182no—
471C2×D3—182no—
472C2×D3—182no—
473C2×D3—362no—
474C2×D3—362no—
475C2×D3—362no—
476C2×D3—368no—
477C2×D3—368no—
478C2×D3C2×D3614yes—
479C3×C4—22no—
480C3×C4—21no—
481C3×C4—21no—
482C3×C4—21no—
483C3×C4—21no—
484C3×C4—41no—
485C3×C4—62no—
486C3×C4—182no—
487Dic3—22no—
488Dic3—62no—
489Dic3—182no—
490Dic3—182no—
491Dic3—362no—
492C32#549311yescharacteristic
493C9—21no—
494C9#560511yescharacteristic, complemented
495C23—32no—
496C23—34no—
497C23—92no—
498C23—121no—
499C23—121no—
500C23—272no—
501C23—544no—
502C23—1082no—
503C23—1082no—
504C23C2×D3×D911yescharacteristic
505D4—184no—
506D4—361no—
507D4—544no—
508D4—544no—
509D4—544no—
510D4—1081no—
511D4—1082no—
512C2×C4—32no—
513C2×C4—41no—
514C2×C4—92no—
515C2×C4—121no—
516C2×C4—272no—
517C2×C4—544no—
518C2×C4C2×D3×D911yescharacteristic
519C2×C4C2×D3×D911yescharacteristic
520C2×C3—22no—
521C2×C3—22no—
522C2×C3—21no—
523C2×C3—21no—
524C2×C3—21no—
525C2×C3—42no—
526C2×C3—64no—
527C2×C3—362no—
528C2×C3C22×D3611yescharacteristic
529C2×C3#6371011yescharacteristic
530C2×C3#6371011yescharacteristic
531C2×C3C2×D3×D1211yescharacteristic
532C2×C3#6430711yescharacteristic
533C2×C3#6433011yescharacteristic
534D3—24no—
535D3—122no—
536D3—362no—
537D3—362no—
538D3—722no—
539C22—21no—
540C22—21no—
541C22—34no—
542C22—62no—
543C22—62no—
544C22—124no—
545C22—182no—
546C22—182no—
547C22—182no—
548C22—542no—
549C22—542no—
550C22—542no—
551C22—1088no—
552C22D3×D3612yes—
553C22C22×D3×D911yescharacteristic
554C4—21no—
555C4—21no—
556C4—62no—
557C4—182no—
558C4—542no—
559C3—21no—
560C3#1058959511yescharacteristic, complemented
561C3#1059507211yescharacteristic
562C2—22no—
563C2—64no—
564C2—362no—
565C2—1082no—
566C2C2×D3×D3611yescharacteristic
567C2#1172247611yescharacteristic
568C2#1172253511yescharacteristic
subgroupG/H
centerC22C22×D3×D9
commutator, cosocleC22×C3×C9C24
FrattiniC22×C3C22×D32
FittingC22×C3×C4×C9C22
socleC22×C32C23×D3
hypercenterC22×C4D3×D9
hypocenterC3×C9#534
LCSG ▷ C22×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
1. C2×D3×#611C2semidirect
2. #11721819C2semidirect
3. C2×C9×(C2×D3)⋊C4C2semidirect
4. C3×#63492C2semidirect
5. #11721277C2semidirect
6. #11721051C2semidirect
7. #11721049C2semidirect
8. #11721021C2semidirect
9. #11721020C2semidirect
13. C22×D3×D9C22semidirect
14. C22×C3×C4×C9C22semidirect
15. C2×#6233C22semidirect
16. C2×C4⋊(C3⋊D9)C22semidirect
17. C2×C3×#611C22semidirect
18. C22×C9×Dic3C22semidirect
19. C2×C3×D36C22semidirect
23. C2×D3×Dic9C22semidirect
24. C2×#6192C22semidirect
25. #88536C22semidirect
26. C4⋊(C3⋊Dic9)C22semidirect
27. C9×(C2×D3)⋊C4C22semidirect
28. C9×(C2×D3)⋊C4C22semidirect
29. C3×(C2×D9)⋊C4C22semidirect
30. C3×C4⋊Dic9C22semidirect
31. #88445C22semidirect
32. (C2×D9)⋊Dic3C22semidirect
33. Dic3⋊Dic9C22semidirect
48. #63492D3semidirect
57. C22×D3×C9D4semidirect
106. C22×C4×C9C2×D3semidirect
112. C2×#611C2×D3semidirect
117. C2×D36C2×D3semidirect
124. (C2×D9)⋊C4C2×D3semidirect
128. C4⋊Dic9C2×D3semidirect
160. C2×(C2×D3)⋊C4D9semidirect
242. C22×C3×C4C2×D9semidirect
254. C22×Dic3C2×D9semidirect
275. (C2×D3)⋊C4C2×D9semidirect
276. (C2×D3)⋊C4C2×D9semidirect
342. C3×C9#534semidirect
363. C22×D3D36semidirect
432. C22×C4D3×D9semidirect
494. C9#5605semidirect
560. C3#10589595semidirect

Maximal subgroups

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

subgroupindex
C2×D3×#6112
#117218192
C2×C9×(C2×D3)⋊C42
C3×#634922
#117212772
#117210512
#117210492
#117210212
#117210202
#105950723
#105895953