Finite Group Databasebeta

← #13150991

Subgroups — #13150991

Order 1728.

1120 conjugacy classes of subgroups.

The subgroup lattice has 586 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#3699C211yesmaximal, characteristic, complemented
2C22×(C3⋊D9)⋊C4C211yesmaximal, characteristic, complemented
3#11721837C211yesmaximal, characteristic, complemented
4C9×#935C211yesmaximal, characteristic, complemented
5#11721278C211yesmaximal, characteristic, complemented
6#11721259C211yesmaximal, characteristic, complemented
7#11721256C212yesmaximal, complemented
8#11721245C211yesmaximal, characteristic, complemented
9#11721241C212yesmaximal, complemented
10#11721002C212yesmaximal, complemented
11#11720985C212yesmaximal, complemented
12#10595159—31nomaximal
13#10589758—31nomaximal
14C23×C3⋊D9C2211yescharacteristic
15C2×C3×D4×C9C2211yescharacteristic
16C2×#6233C2212yes—
17C2×C3×#611C2211yescharacteristic
18C2×C3×#611C2211yescharacteristic, complemented
19C22×C9×Dic3C2211yescharacteristic
20C22×C3×Dic9C2211yescharacteristic
21C22×C3×Dic9C2211yescharacteristic, complemented
22C3×#3699—24no—
23C2×C3×C4×D9C2211yescharacteristic, complemented
24C2×C3×Dic18C2211yescharacteristic, complemented
25C2×#6192C2212yes—
26C2×(C3⋊D9)⋊C4—24no—
27C2×(C3⋊D9)⋊C4C2212yes—
28C2×(C3⋊D9)⋊C4C2212yes—
29#88551C2211yescharacteristic
30#88536C2212yes—
31C9×#217C2211yescharacteristic
32C9×#217C2212yescomplemented
33C9×Dic3⋊C4C2212yescomplemented
34#88445C2212yes—
35(C2×D9)⋊Dic3C2212yescomplemented
36#88442C2212yescomplemented
37Dic9⋊Dic3C2212yescomplemented
38Dic3×Dic9C2212yescomplemented
39C2×C3×#237—31no—
40C22×#624—31no—
41#64146—31no—
42C3×#935D311yescharacteristic
43#63981—31no—
44#63961—31no—
45#63958—32no—
46#63955—31no—
47#63952—32no—
48#63904—32no—
49#63888—32no—
50C2×#3699D311yescharacteristic, complemented
51C22×C4×D9—31no—
52C9×#123—31no—
53#63499—31no—
54#63499—61no—
55#63491—31no—
56#63458—32no—
57#63455—32no—
58#63446—32no—
59#63444—31no—
60#611⋊C4—32no—
61#63439—31no—
62C23×C3×C9C2311yescharacteristic
63C23×C3×C9C2311yescharacteristic
64C22×C3⋊D9—22no—
65C22×C3⋊D9—24no—
66C22×C3⋊D9C2312yes—
67C22×C3×D9C2311yescharacteristic
68C3×D4×C9—22no—
69C2×C3×C4×C9C2311yescharacteristic
70#6233—24no—
71C2×C3⋊Dic9C2312yes—
72C3×#611—22no—
73C3×#611—22no—
74C2×C9×Dic3—22no—
75C2×C9×Dic3C2312yes—
76C2×C9×Dic3C2312yes—
77C2×C3×Dic9—22no—
78C2×C3×Dic9D414yescomplemented
79C2×C3×Dic9C2311yescharacteristic
80C2×C3×Dic9C2311yescharacteristic
81C2×C3×Dic9C2311yescharacteristic
82C3×C4×D9—22no—
83C3×Dic18—22no—
84#6192—24no—
85(C3⋊D9)⋊C4—24no—
86(C3⋊D9)⋊C4—24no—
87#5828—91no—
88#5656—91no—
89C23×C3⋊D3—31no—
90C2×C32×D4C2×D311yescharacteristic
91C2×#638—32no—
92C2×C3×#67—31no—
93C2×C3×#67—31no—
94C22×C3×Dic3—31no—
95C22×C3×Dic3—31no—
96C22×C3×Dic3C2×D311yescharacteristic
97C3×#237—64no—
98C2×C3×C4×D3—31no—
99C2×C3×Dic6—31no—
100C2×#626—32no—
101C2×#624—32no—
102C2×#624—32no—
103C2×#624—64no—
104C23×D9—31no—
105C23×D9—61no—
106#3757—31no—
107#3752—32no—
108C3×#217C2×D311yescharacteristic
109C3×#217C2×D312yes—
110C3×Dic3⋊C4C2×D312yes—
111#3724—32no—
112Dic3⋊Dic3—32no—
113#3722—32no—
114(C2×D3)⋊Dic3—32no—
115Dic32—32no—
116C2×D4×C9—21no—
117C2×D4×C9C2×D311yescharacteristic
118C22×C4×C9—31no—
119C2×#611—32no—
120C2×#611—62no—
121C2×#611C2×D311yescharacteristic
122C2×#611C2×D311yescharacteristic, complemented
123C22×Dic9C2×D311yescharacteristic
124C22×Dic9C2×D311yescharacteristic, complemented
125#3699—24no—
126C2×D36—32no—
127C2×C4×D9—32no—
128C2×C4×D9—32no—
129C2×C4×D9—64no—
130C2×C4×D9C2×D311yescharacteristic, complemented
131C2×Dic18C2×D311yescharacteristic, complemented
132C9×C4⋊C4—32no—
133C9×C22⋊C4—31no—
134C9×C22⋊C4—32no—
135#3676—31no—
136#3676—61no—
137(C2×D9)⋊C4—32no—
138(C2×D9)⋊C4—32no—
139(C2×D9)⋊C4—32no—
140(C2×D9)⋊C4—62no—
141Dic9⋊C4—32no—
142Dic9⋊C4—32no—
143C4×Dic9—32no—
144C22×C3×C9—22no—
145C22×C3×C9—21no—
146C22×C3×C9—22no—
147C22×C3×C9C2×D412yes—
148C22×C3×C9C2411yescharacteristic
149C2×C3⋊D9—28no—
150C2×C3⋊D9—22no—
151C2×C3⋊D9—22no—
152C2×C3⋊D9#4114yescomplemented
153C2×C3×D9—21no—
154C2×C3×D9—22no—
155C3×C4×C9—21no—
156C3⋊Dic9—22no—
157C9×Dic3—22no—
158C9×Dic3—22no—
159C3×Dic9—21no—
160C3×Dic9C2×D412yes—
161C3×Dic9C2×D412yes—
162C2×C3×#41—91no—
163C2×#237—31no—
164C22×C4×D3—91no—
165C22×C4×D3—91no—
166C3×#123—31no—
167#946—92no—
168#938—92no—
169#937—91no—
170#937—91no—
171#937—91no—
172#937—181no—
173#935—91no—
174#935D911yescharacteristic, complemented
175#929—91no—
176C4×#67—92no—
177C4×#67—92no—
178#926—91no—
179#896—92no—
180#893—92no—
181#884—92no—
182#882—91no—
183#67⋊C4—92no—
184#877—91no—
185C23×C32C22×D311yescharacteristic
186C23×C32C22×D311yescharacteristic
187C22×C3⋊D3—32no—
188C22×C3⋊D3—64no—
189C22×C3⋊D3—62no—
190C22×C3×D3—31no—
191C32×D4—22no—
192C2×C32×C4C22×D311yescharacteristic
193#638—64no—
194C2×C3⋊Dic3—32no—
195C3×#67—62no—
196C3×#67—62no—
197C2×C3×Dic3—22no—
198C2×C3×Dic3—34no—
199C2×C3×Dic3—31no—
200C2×C3×Dic3—31no—
201C2×C3×Dic3—31no—
202C2×C3×Dic3—62no—
203C2×C3×Dic3C22×D312yes—
204C2×C3×Dic3C22×D312yes—
205C3×C4×D3—62no—
206C3×Dic6—62no—
207#626—64no—
208#624—64no—
209#624—64no—
210C23×C9—21no—
211C23×C9—21no—
212C23×C9C22×D311yescharacteristic
213C23×C9C22×D311yescharacteristic
214C22×D9—32no—
215C22×D9—62no—
216C22×D9—64no—
217C22×D9—62no—
218C22×D9—124no—
219C22×D9—122no—
220C22×D9C22×D311yescharacteristic
221D4×C9—22no—
222D4×C9—42no—
223C2×C4×C9—21no—
224C2×C4×C9—32no—
225C2×C4×C9—32no—
226C2×C4×C9—62no—
227C2×C4×C9C22×D311yescharacteristic
228#611—22no—
229#611—22no—
230#611—64no—
231#611—124no—
232C2×Dic9—22no—
233C2×Dic9—32no—
234C2×Dic9—62no—
235C2×Dic9#6714yescomplemented
236C2×Dic9C22×D311yescharacteristic
237C2×Dic9C22×D311yescharacteristic
238C2×Dic9C22×D311yescharacteristic
239D36—64no—
240C4×D9—22no—
241C4×D9—64no—
242C4×D9—64no—
243Dic18—22no—
244#525—271noSylow
245C2×C3×C9—22no—
246C2×C3×C9—41no—
247C2×C3×C9C22×D411yescharacteristic
248C2×C3×C9C2×#4112yes—
249C3⋊D9—24no—
250C3×D9—41no—
251C23×D3—91no—
252C23×D3—91no—
253C23×D3—181no—
254C3×#41—184no—
255C2×C3×Q8—91no—
256C2×C3×D4—21no—
257C2×C3×D4—91no—
258C2×C3×D4—91no—
259C2×C3×D4C2×D911yescharacteristic
260C2×C3×D4D3211yescharacteristic
261C22×C3×C4—31no—
262C22×C3×C4—91no—
263C22×C3×C4—91no—
264C22×C3×C4—91no—
265C2×#67—31no—
266C2×#67—31no—
267C2×#67—92no—
268C2×#67—92no—
269C2×#67—92no—
270C2×#67—182no—
271C22×Dic3—31no—
272C22×Dic3—31no—
273C22×Dic3C2×D911yescharacteristic
274#237—64no—
275C2×D12—92no—
276C2×C4×D3—31no—
277C2×C4×D3—92no—
278C2×C4×D3—92no—
279C2×C4×D3—92no—
280C2×C4×D3—92no—
281C2×C4×D3—184no—
282C2×C4×D3—184no—
283C2×Dic6—31no—
284C3×C4⋊C4—32no—
285C3×C22⋊C4—31no—
286C3×C22⋊C4—32no—
287#217—91no—
288#217—91no—
289#217—92no—
290#217—181no—
291#217C2×D911yescharacteristic
292#217C2×D912yescomplemented
293(C2×D3)⋊C4—92no—
294(C2×D3)⋊C4—92no—
295(C2×D3)⋊C4—92no—
296(C2×D3)⋊C4—92no—
297(C2×D3)⋊C4—92no—
298(C2×D3)⋊C4—182no—
299C4⋊Dic3—92no—
300Dic3⋊C4—92no—
301Dic3⋊C4—92no—
302Dic3⋊C4—92no—
303Dic3⋊C4C2×D912yescomplemented
304C4×Dic3—92no—
305C4×Dic3—92no—
306C22×C32—22no—
307C22×C32—21no—
308C22×C32—22no—
309C22×C32D3×D412yes—
310C22×C32C23×D311yescharacteristic
311C2×C3⋊D3—34no—
312C2×C3⋊D3—68no—
313C2×C3⋊D3—62no—
314C2×C3⋊D3—62no—
315C2×C3×D3—61no—
316C2×C3×D3—62no—
317C32×C4—21no—
318C3⋊Dic3—62no—
319C3×Dic3—22no—
320C3×Dic3—22no—
321C3×Dic3—32no—
322C3×Dic3—32no—
323C3×Dic3—61no—
324C22×C9—22no—
325C22×C9—22no—
326C22×C9—21no—
327C22×C9—21no—
328C22×C9—22no—
329C22×C9—42no—
330C22×C9—42no—
331C22×C9—41no—
332C22×C9C2×#6712yes—
333C22×C9C23×D311yescharacteristic
334C2×D9—22no—
335C2×D9—21no—
336C2×D9—34no—
337C2×D9—64no—
338C2×D9—68no—
339C2×D9—62no—
340C2×D9—62no—
341C2×D9—122no—
342C2×D9—122no—
343C2×D9—128no—
344C4×C9—21no—
345C4×C9—41no—
346C4×C9—62no—
347C4×C9—62no—
348Dic9—21no—
349Dic9—62no—
350Dic9—122no—
351Dic9C2×#6712yes—
352Dic9C2×#6712yes—
353C2×#41—91no—
354C23×C4—271no—
355#123—31no—
356#123—271no—
357C22⋊Q8—272no—
358#121—272no—
359#120—271no—
360#120—271no—
361C4×D4—272no—
362C4×D4—272no—
363#117—271no—
364C3×C9#52511yescharacteristic, complemented, Sylow
365C23×C3—21no—
366C23×C3—21no—
367C23×C3—91no—
368C23×C3C22×D911yescharacteristic
369C23×C3C22×D911yescharacteristic
370C23×C3C2×D3211yescharacteristic
371C23×C3C2×D3211yescharacteristic
372C22×D3—31no—
373C22×D3—92no—
374C22×D3—92no—
375C22×D3—182no—
376C22×D3—184no—
377C22×D3—184no—
378C22×D3—182no—
379C22×D3—182no—
380C22×D3—364no—
381C22×D3—362no—
382C3×Q8—182no—
383C3×D4—22no—
384C3×D4—22no—
385C3×D4—42no—
386C3×D4—182no—
387C3×D4—182no—
388C2×C3×C4—21no—
389C2×C3×C4—32no—
390C2×C3×C4—32no—
391C2×C3×C4—62no—
392C2×C3×C4—94no—
393C2×C3×C4—91no—
394C2×C3×C4—91no—
395C2×C3×C4—91no—
396C2×C3×C4—182no—
397C2×C3×C4—182no—
398C2×C3×C4C22×D911yescharacteristic
399C2×C3×C4C2×D3211yescharacteristic
400#67—62no—
401#67—62no—
402#67—184no—
403#67—184no—
404#67—184no—
405#67—364no—
406C2×Dic3—22no—
407C2×Dic3—31no—
408C2×Dic3—31no—
409C2×Dic3—34no—
410C2×Dic3—31no—
411C2×Dic3—62no—
412C2×Dic3—92no—
413C2×Dic3—92no—
414C2×Dic3—182no—
415C2×Dic3C22×D912yes—
416C2×Dic3C22×D912yes—
417D12—184no—
418C4×D3—62no—
419C4×D3—184no—
420C4×D3—184no—
421C4×D3—184no—
422C4×D3—184no—
423Dic6—62no—
424C2×C32—22no—
425C2×C32—41no—
426C2×C32C2×D3×D411yescharacteristic
427C2×C32D3×#4112yes—
428C3⋊D3—64no—
429C3×D3—121no—
430C2×C9—22no—
431C2×C9—22no—
432C2×C9—21no—
433C2×C9—42no—
434C2×C9—41no—
435C2×C9—81no—
436C2×C9D3×#4112yes—
437C2×C9C22×#6711yescharacteristic
438D9—41no—
439D9—64no—
440D9—124no—
441C24—271no—
442#41—184no—
443C2×Q8—91no—
444C2×D4—91no—
445C2×D4—91no—
446C2×D4—272no—
447C2×D4—272no—
448C2×D4D3×D911yescharacteristic
449C22×C4—31no—
450C22×C4—91no—
451C22×C4—91no—
452C22×C4—91no—
453C22×C4—272no—
454C22×C4—272no—
455C22×C4—544no—
456C4⋊C4—32no—
457C4⋊C4—272no—
458C4⋊C4—272no—
459C22⋊C4—31no—
460C22⋊C4—32no—
461C22⋊C4—272no—
462C22⋊C4—272no—
463C22⋊C4—271no—
464C22⋊C4—272no—
465C42—272no—
466C22×C3—22no—
467C22×C3—22no—
468C22×C3—22no—
469C22×C3—21no—
470C22×C3—21no—
471C22×C3—22no—
472C22×C3—21no—
473C22×C3—22no—
474C22×C3—42no—
475C22×C3—42no—
476C22×C3—41no—
477C22×C3—181no—
478C22×C3—182no—
479C22×C3D4×D912yes—
480C22×C3C23×D911yescharacteristic
481C22×C3D3×#6712yes—
482C22×C3C22×D3211yescharacteristic
483C2×D3—62no—
484C2×D3—61no—
485C2×D3—94no—
486C2×D3—94no—
487C2×D3—184no—
488C2×D3—188no—
489C2×D3—188no—
490C2×D3—182no—
491C2×D3—182no—
492C2×D3—182no—
493C2×D3—182no—
494C2×D3—362no—
495C2×D3—362no—
496C2×D3—368no—
497C3×C4—21no—
498C3×C4—21no—
499C3×C4—41no—
500C3×C4—62no—
501C3×C4—62no—
502C3×C4—92no—
503C3×C4—92no—
504C3×C4—181no—
505Dic3—22no—
506Dic3—22no—
507Dic3—32no—
508Dic3—32no—
509Dic3—61no—
510Dic3—182no—
511Dic3—182no—
512Dic3—362no—
513C32#565611yescharacteristic
514C9—21no—
515C9#582811yescharacteristic, complemented
516C23—91no—
517C23—272no—
518C23—544no—
519C23—542no—
520C23C2×D3×D911yescharacteristic
521C23C2×D3×D911yescharacteristic
522Q8—182no—
523D4—22no—
524D4—182no—
525D4—182no—
526D4—544no—
527D4—544no—
528C2×C4—32no—
529C2×C4—32no—
530C2×C4—62no—
531C2×C4—91no—
532C2×C4—91no—
533C2×C4—94no—
534C2×C4—91no—
535C2×C4—182no—
536C2×C4—182no—
537C2×C4—272no—
538C2×C4—544no—
539C2×C4—544no—
540C2×C4C2×D3×D911yescharacteristic
541C2×C3—22no—
542C2×C3—22no—
543C2×C3—22no—
544C2×C3—21no—
545C2×C3—42no—
546C2×C3—41no—
547C2×C3—41no—
548C2×C3—81no—
549C2×C3—361no—
550C2×C3C2×D4×D911yescharacteristic
551C2×C3#41×D912yes—
552C2×C3#6431212yes—
553C2×C3C2×D3×#6711yescharacteristic
554D3—121no—
555D3—184no—
556D3—184no—
557D3—364no—
558C22—22no—
559C22—22no—
560C22—21no—
561C22—182no—
562C22—181no—
563C22—274no—
564C22—548no—
565C22—542no—
566C22—542no—
567C22D9×#6712yes—
568C22C22×D3×D911yescharacteristic
569C4—21no—
570C4—62no—
571C4—62no—
572C4—92no—
573C4—92no—
574C4—181no—
575C4—542no—
576C3—21no—
577C3#1058975811yescharacteristic, complemented
578C3#1059515911yescharacteristic
579C2—22no—
580C2—41no—
581C2—361no—
582C2—544no—
583C2#1172249812yes—
584C2C2×D9×#6711yescharacteristic
subgroupG/H
centerC22C22×D3×D9
commutator, cosocleC22×C3×C9C24
FrattiniC22×C3C22×D32
FittingC2×C3×D4×C9C22
socleC22×C32C23×D3
hypercenterC2×D4D3×D9
hypocenterC3×C9#525
LCSG ▷ C22×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
1. C2×C3×#3699C2semidirect
2. C22×(C3⋊D9)⋊C4C2semidirect
3. #11721837C2semidirect
4. C9×#935C2semidirect
5. #11721278C2semidirect
6. #11721259C2semidirect
7. #11721256C2semidirect
8. #11721245C2semidirect
9. #11721241C2semidirect
10. #11721002C2semidirect
11. #11720985C2semidirect
18. C2×C3×#611C22semidirect
21. C22×C3×Dic9C22semidirect
23. C2×C3×C4×D9C22semidirect
24. C2×C3×Dic18C22semidirect
32. C9×#217C22semidirect
33. C9×Dic3⋊C4C22semidirect
35. (C2×D9)⋊Dic3C22semidirect
36. #88442C22semidirect
37. Dic9⋊Dic3C22semidirect
38. Dic3×Dic9C22semidirect
50. C2×#3699D3semidirect
78. C2×C3×Dic9D4semidirect
122. C2×#611C2×D3semidirect
124. C22×Dic9C2×D3semidirect
130. C2×C4×D9C2×D3semidirect
131. C2×Dic18C2×D3semidirect
152. C2×C3⋊D9#41semidirect
174. #935D9semidirect
235. C2×Dic9#67semidirect
292. #217C2×D9semidirect
303. Dic3⋊C4C2×D9semidirect
364. C3×C9#525semidirect
515. C9#5828semidirect
577. C3#10589758semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#36992
C22×(C3⋊D9)⋊C42
#117218372
C9×#9352
#117212782
#117212592
#117212562
#117212452
#117212412
#117210022
#117209852
#105951593
#105897583