Finite Group Databasebeta

← #13150982

Subgroups — #13150982

Order 1728.

1078 conjugacy classes of subgroups.

The subgroup lattice has 598 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×D4×D9C211yesmaximal, characteristic, complemented
2C2×#88641C211yesmaximal, characteristic, complemented
3#11721836C211yesmaximal, characteristic, complemented
4C9×#938C211yesmaximal, characteristic, complemented
5#11721285C212yesmaximal, complemented
6#11721270C212yesmaximal, complemented
7#11721269C212yesmaximal, complemented
8D9×#217C212yesmaximal, complemented
9C4×#6194C211yesmaximal, characteristic, complemented
10#11720998C211yesmaximal, characteristic, complemented
11#11720945C211yesmaximal, characteristic, complemented
12#10595228—31nomaximal
13#10589714—31nomaximal
14C23×C3×D9C2212yes—
15C2×C3×D4×C9C2211yescharacteristic
16C2×#6226C2211yescharacteristic, complemented
17C2×C9×#67C2212yescomplemented
18C2×C3×#611C2212yes—
19C3×D4×D9—24no—
20C2×C4×D3×C9C2211yescharacteristic, complemented
21C2×C3×D36C2211yescharacteristic
22C2×C3×C4×D9C2211yescharacteristic
23C2×#6194C2211yescharacteristic
24C2×#6194C2212yescomplemented
25C2×Dic3×D9C2212yescomplemented
26C2×Dic3×D9C2212yescomplemented
27#88641—24no—
28#88551C2212yescomplemented
29#88551C2212yescomplemented
30C4×C3⋊Dic9C2211yescharacteristic, complemented
31C9×#217C2212yescomplemented
32C9×C4⋊Dic3C2211yescharacteristic, complemented
33(C2×D3)⋊Dic9C2211yescharacteristic, complemented
34(C2×D9)⋊Dic3C2211yescharacteristic
35(C2×D9)⋊Dic3C2212yes—
36(C2×D9)⋊Dic3C2212yescomplemented
37Dic3⋊Dic9C2212yescomplemented
38#88442C2211yescharacteristic, complemented
39C2×C3×D3×D4—31no—
40C2×#3795—31no—
41#64145—31no—
42C3×#938D311yescharacteristic
43#63976—32no—
44#63970—32no—
45#63969—32no—
46D3×#217—32no—
47C4×#625—31no—
48#63898—31no—
49#63857—31no—
50C2×D4×D9D311yescharacteristic, complemented
51C2×#3697—31no—
52C9×#121—31no—
53#63500—32no—
54#63499—32no—
55#63498—31no—
56#63498—61no—
57C4×#611—31no—
58#63459—31no—
59#63455—31no—
60#63445—32no—
61C22⋊C4×D9—32no—
62C23×C3×C9C2312yes—
63C22×D3×C9C2311yescharacteristic
64C22×C3×D9—24no—
65C22×C3×D9—24no—
66C22×C3×D9—42no—
67C22×C3×D9C2312yes—
68C22×C3×D9C2311yescharacteristic
69C3×D4×C9—22no—
70C2×C3×C4×C9C2311yescharacteristic
71C2×C3⋊Dic9C2311yescharacteristic
72C2×C3⋊Dic9C2312yes—
73C2×C3⋊Dic9C2311yescharacteristic
74#6226—22no—
75C9×#67—42no—
76C3×#611—24no—
77C2×C9×Dic3C2311yescharacteristic
78C2×C9×Dic3C2312yes—
79C2×C3×Dic9C2311yescharacteristic
80C4×D3×C9—22no—
81C3×D36D414yescomplemented
82C3×C4×D9—22no—
83#6194—24no—
84#6194—22no—
85Dic3×D9—24no—
86Dic3×D9—42no—
87#5806—91no—
88#5612—91no—
89C23×C3×D3—32no—
90C2×C32×D4C2×D311yescharacteristic
91C2×#634—31no—
92C2×C3×#67—32no—
93C2×C3×#67C2×D312yes—
94C3×D3×D4—64no—
95C2×C3×D12—31no—
96C2×C3×C4×D3—31no—
97C2×C3×C4×D3C2×D311yescharacteristic
98C2×#625—32no—
99C2×#625—31no—
100C2×D3×Dic3—32no—
101C2×D3×Dic3—32no—
102#3795—64no—
103C23×D9C2×D312yes—
104#3757—32no—
105#3757—32no—
106C4×C3⋊Dic3—31no—
107C3×#217C2×D312yes—
108C3×C4⋊Dic3C2×D311yescharacteristic
109#3724—31no—
110Dic3⋊Dic3—32no—
111(C2×D3)⋊Dic3—31no—
112(C2×D3)⋊Dic3—32no—
113(C2×D3)⋊Dic3—31no—
114(C2×D3)⋊Dic3—32no—
115C2×D4×C9—21no—
116C2×D4×C9—32no—
117C2×D4×C9C2×D311yescharacteristic
118C22×C4×C9—31no—
119C2×#611—31no—
120C2×#611—32no—
121C2×#611C2×D312yes—
122D4×D9—24no—
123#3697—64no—
124C2×D36C2×D311yescharacteristic
125C2×C4×D9—32no—
126C2×C4×D9—32no—
127C2×C4×D9C2×D311yescharacteristic
128C2×Dic18—31no—
129C2×Dic18—61no—
130C9×C4⋊C4—31no—
131C9×C22⋊C4—32no—
132#3676—32no—
133#3676—32no—
134#3676—31no—
135#3676—62no—
136#3676—62no—
137(C2×D9)⋊C4—31no—
138(C2×D9)⋊C4—32no—
139(C2×D9)⋊C4—32no—
140C4⋊Dic9—32no—
141Dic9⋊C4—31no—
142C4×Dic9—31no—
143C4×Dic9—61no—
144C22×C3×C9—22no—
145C22×C3×C9—22no—
146C22×C3×C9—22no—
147C22×C3×C9C2411yescharacteristic
148C2×D3×C9—21no—
149C2×D3×C9—21no—
150C2×D3×C9—21no—
151C2×C3×D9—21no—
152C2×C3×D9—22no—
153C2×C3×D9—22no—
154C2×C3×D9—21no—
155C2×C3×D9—48no—
156C2×C3×D9C2×D414yes—
157C2×C3×D9#4112yescomplemented
158C3×C4×C9C2×D412yes—
159C3⋊Dic9—22no—
160C3⋊Dic9—21no—
161C3⋊Dic9—21no—
162C9×Dic3—21no—
163C9×Dic3—22no—
164C3×Dic9—21no—
165C22×C3×D4—91no—
166C2×#237—91no—
167C2×D3×D4—31no—
168C2×#235—91no—
169C3×#121—31no—
170#953—92no—
171C2×#217—92no—
172#939—92no—
173#938—92no—
174#938D911yescharacteristic, complemented
175#937—92no—
176#936—91no—
177#936—91no—
178#936—181no—
179#935—92no—
180D4×Dic3—91no—
181C4×#67—91no—
182C4×#67—91no—
183#924—91no—
184#897—91no—
185#893—91no—
186#883—92no—
187D3×C22⋊C4—92no—
188C23×C32C22×D312yes—
189C22×C3×D3—31no—
190C22×C3×D3—32no—
191C22×C3×D3—64no—
192C22×C3×D3—64no—
193C22×C3×D3—122no—
194C22×C3×D3C22×D311yescharacteristic
195C32×D4—22no—
196C2×C32×C4C22×D311yescharacteristic
197C2×C3⋊Dic3—31no—
198C2×C3⋊Dic3—31no—
199C2×C3⋊Dic3—32no—
200#634—62no—
201C3×#67—42no—
202C3×#67—64no—
203C2×C3×Dic3—31no—
204C2×C3×Dic3C22×D312yes—
205C2×C3×Dic3C22×D311yescharacteristic
206C3×D12—34no—
207C3×C4×D3—22no—
208C3×C4×D3—62no—
209#625—64no—
210#625—62no—
211D3×Dic3—64no—
212D3×Dic3—122no—
213C23×C9—22no—
214C23×C9—31no—
215C23×C9C22×D312yes—
216C22×D9—24no—
217C22×D9—24no—
218C22×D9—42no—
219C22×D9C22×D311yescharacteristic
220C22×D9C22×D312yes—
221D4×C9—22no—
222D4×C9—42no—
223D4×C9—122no—
224C2×C4×C9—21no—
225C2×C4×C9—31no—
226C2×C4×C9—32no—
227C2×C4×C9—62no—
228C2×C4×C9C22×D311yescharacteristic
229#611—24no—
230#611—62no—
231#611—64no—
232C2×Dic9—31no—
233C2×Dic9—31no—
234C2×Dic9—32no—
235C2×Dic9—61no—
236C2×Dic9—61no—
237C2×Dic9—62no—
238C2×Dic9C22×D311yescharacteristic
239D36#6714yescomplemented
240C4×D9—22no—
241C4×D9—64no—
242C4×D9—122no—
243Dic18—62no—
244Dic18—122no—
245#546—271noSylow
246C2×C3×C9—42no—
247C2×C3×C9C22×D411yescharacteristic
248C2×C3×C9C2×#4111yescharacteristic
249C2×C3×C9H(5,𝔽2)11yescharacteristic
250D3×C9—41no—
251C3×D9—22no—
252C3×D9—24no—
253C24×C3—92no—
254C23×D3—32no—
255C2×C3×D4—21no—
256C2×C3×D4—32no—
257C2×C3×D4—92no—
258C2×C3×D4—91no—
259C2×C3×D4—184no—
260C2×C3×D4C2×D911yescharacteristic
261C2×C3×D4D3211yescharacteristic
262C22×C3×C4—31no—
263C22×C3×C4—91no—
264C2×#67—32no—
265C2×#67—91no—
266C2×#67—92no—
267C2×#67—91no—
268C2×#67—92no—
269C2×#67C2×D912yescomplemented
270C22×Dic3—92no—
271C22×Dic3—92no—
272#237—184no—
273D3×D4—64no—
274#235—184no—
275C2×D12—31no—
276C2×C4×D3—31no—
277C2×C4×D3—92no—
278C2×C4×D3—92no—
279C2×C4×D3C2×D911yescharacteristic, complemented
280C2×Dic6—91no—
281C2×Dic6—91no—
282C2×Dic6—181no—
283C3×C4⋊C4—31no—
284C3×C22⋊C4—32no—
285#217—91no—
286#217—92no—
287#217—92no—
288#217—92no—
289#217—92no—
290#217—92no—
291#217—91no—
292#217—92no—
293#217—182no—
294#217—182no—
295#217C2×D912yescomplemented
296(C2×D3)⋊C4—91no—
297(C2×D3)⋊C4—92no—
298(C2×D3)⋊C4—92no—
299(C2×D3)⋊C4—91no—
300C4⋊Dic3—92no—
301C4⋊Dic3C2×D911yescharacteristic, complemented
302Dic3⋊C4—91no—
303Dic3⋊C4—92no—
304Dic3⋊C4—91no—
305C4×Dic3—91no—
306C4×Dic3—91no—
307C4×Dic3—181no—
308C22×C32—22no—
309C22×C32—22no—
310C22×C32—22no—
311C22×C32C23×D311yescharacteristic
312C2×C3×D3—21no—
313C2×C3×D3—21no—
314C2×C3×D3—21no—
315C2×C3×D3—34no—
316C2×C3×D3—32no—
317C2×C3×D3—61no—
318C2×C3×D3—62no—
319C2×C3×D3—61no—
320C2×C3×D3—62no—
321C2×C3×D3—128no—
322C32×C4D3×D412yes—
323C3⋊Dic3—62no—
324C3⋊Dic3—61no—
325C3⋊Dic3—61no—
326C3×Dic3—21no—
327C3×Dic3—22no—
328C3×Dic3—61no—
329C22×C9—22no—
330C22×C9—22no—
331C22×C9—22no—
332C22×C9—21no—
333C22×C9—42no—
334C22×C9—42no—
335C22×C9—42no—
336C22×C9—61no—
337C22×C9—61no—
338C22×C9—61no—
339C22×C9C23×D311yescharacteristic
340C2×D9—22no—
341C2×D9—22no—
342C2×D9—21no—
343C2×D9—21no—
344C2×D9—48no—
345C2×D9#23712yescomplemented
346C2×D9C2×#6714yes—
347C4×C9—22no—
348C4×C9—62no—
349C4×C9—61no—
350C4×C9C2×#6712yes—
351Dic9—21no—
352Dic9—61no—
353Dic9—61no—
354Dic9—62no—
355Dic9—122no—
356Dic9—121no—
357Dic9—121no—
358C2×#41—271no—
359C22×D4—91no—
360#124—271no—
361#123—272no—
362C22⋊Q8—271no—
363#121—31no—
364#121—272no—
365#120—272no—
366C4×D4—271no—
367C4×D4—271no—
368C2×C22⋊C4—272no—
369C3×C9#54611yescharacteristic, complemented, Sylow
370C23×C3—22no—
371C23×C3—31no—
372C23×C3—91no—
373C23×C3—92no—
374C23×C3—184no—
375C23×C3—184no—
376C23×C3—362no—
377C23×C3C22×D912yes—
378C23×C3C2×D3212yes—
379C22×D3—31no—
380C22×D3—32no—
381C22×D3—64no—
382C22×D3—64no—
383C22×D3—122no—
384C22×D3C22×D911yescharacteristic
385C3×D4—22no—
386C3×D4—22no—
387C3×D4—42no—
388C3×D4—94no—
389C3×D4—122no—
390C3×D4—184no—
391C2×C3×C4—21no—
392C2×C3×C4—31no—
393C2×C3×C4—32no—
394C2×C3×C4—62no—
395C2×C3×C4—91no—
396C2×C3×C4—182no—
397C2×C3×C4C22×D911yescharacteristic
398C2×C3×C4C2×D3211yescharacteristic
399#67—42no—
400#67—64no—
401#67—184no—
402#67—182no—
403#67—184no—
404#67—182no—
405C2×Dic3—31no—
406C2×Dic3—91no—
407C2×Dic3—91no—
408C2×Dic3—92no—
409C2×Dic3—91no—
410C2×Dic3—91no—
411C2×Dic3—92no—
412C2×Dic3—181no—
413C2×Dic3—181no—
414C2×Dic3—182no—
415C2×Dic3—184no—
416C2×Dic3—362no—
417C2×Dic3C22×D912yes—
418C2×Dic3C22×D911yescharacteristic
419D12—34no—
420C4×D3—22no—
421C4×D3—62no—
422C4×D3—184no—
423C4×D3—362no—
424Dic6—182no—
425Dic6—182no—
426Dic6—362no—
427C2×C32—42no—
428C2×C32C2×D3×D411yescharacteristic
429C2×C32#100411yescharacteristic
430C2×C32D3×#4111yescharacteristic
431C3×D3—41no—
432C3×D3—64no—
433C3×D3—62no—
434C2×C9—21no—
435C2×C9—21no—
436C2×C9—21no—
437C2×C9—42no—
438C2×C9—82no—
439C2×C9—121no—
440C2×C9C2×#23711yescharacteristic
441C2×C9#100411yescharacteristic
442C2×C9C22×#6711yescharacteristic
443D9—22no—
444D9—24no—
445C24—92no—
446#41—544no—
447C2×Q8—271no—
448C2×D4—32no—
449C2×D4—92no—
450C2×D4—91no—
451C2×D4—184no—
452C2×D4—271no—
453C2×D4—272no—
454C2×D4D3×D911yescharacteristic
455C22×C4—31no—
456C22×C4—91no—
457C22×C4—272no—
458C22×C4—272no—
459C4⋊C4—31no—
460C4⋊C4—271no—
461C4⋊C4—272no—
462C22⋊C4—32no—
463C22⋊C4—271no—
464C22⋊C4—272no—
465C22⋊C4—272no—
466C22⋊C4—271no—
467C22⋊C4—272no—
468C22⋊C4—272no—
469C42—271no—
470C22×C3—22no—
471C22×C3—22no—
472C22×C3—22no—
473C22×C3—22no—
474C22×C3—22no—
475C22×C3—22no—
476C22×C3—21no—
477C22×C3—42no—
478C22×C3—42no—
479C22×C3—42no—
480C22×C3—61no—
481C22×C3—61no—
482C22×C3—61no—
483C22×C3—94no—
484C22×C3—92no—
485C22×C3—181no—
486C22×C3—182no—
487C22×C3—181no—
488C22×C3—182no—
489C22×C3—368no—
490C22×C3C23×D911yescharacteristic
491C22×C3C22×D3211yescharacteristic
492C2×D3—21no—
493C2×D3—21no—
494C2×D3—21no—
495C2×D3—32no—
496C2×D3—34no—
497C2×D3—62no—
498C2×D3—62no—
499C2×D3—61no—
500C2×D3—61no—
501C2×D3—128no—
502C3×C4—22no—
503C3×C4—62no—
504C3×C4—61no—
505C3×C4—181no—
506C3×C4D4×D912yes—
507C3×C4D3×#6712yes—
508Dic3—21no—
509Dic3—22no—
510Dic3—61no—
511Dic3—181no—
512Dic3—181no—
513Dic3—182no—
514Dic3—182no—
515Dic3—181no—
516Dic3—181no—
517Dic3—362no—
518Dic3—361no—
519Dic3—361no—
520C32#561211yescharacteristic
521C9—21no—
522C9#580611yescharacteristic, complemented
523C23—31no—
524C23—91no—
525C23—92no—
526C23—184no—
527C23—184no—
528C23—362no—
529C23C2×D3×D912yes—
530Q8—542no—
531D4—22no—
532D4—94no—
533D4—122no—
534D4—184no—
535D4—544no—
536D4—542no—
537C2×C4—31no—
538C2×C4—32no—
539C2×C4—62no—
540C2×C4—91no—
541C2×C4—182no—
542C2×C4—271no—
543C2×C4—271no—
544C2×C4—272no—
545C2×C4—544no—
546C2×C4—1082no—
547C2×C4C2×D3×D911yescharacteristic
548C2×C3—21no—
549C2×C3—21no—
550C2×C3—21no—
551C2×C3—42no—
552C2×C3—42no—
553C2×C3—82no—
554C2×C3—121no—
555C2×C3—182no—
556C2×C3—184no—
557C2×C3C2×D4×D911yescharacteristic
558C2×C3#6371011yescharacteristic
559C2×C3#41×D911yescharacteristic
560C2×C3D3×#23711yescharacteristic
561C2×C3#6431311yescharacteristic
562C2×C3C2×D3×#6711yescharacteristic
563D3—41no—
564D3—62no—
565D3—64no—
566C22—22no—
567C22—22no—
568C22—22no—
569C22—61no—
570C22—61no—
571C22—61no—
572C22—92no—
573C22—94no—
574C22—182no—
575C22—182no—
576C22—181no—
577C22—181no—
578C22—368no—
579C22C22×D3×D911yescharacteristic
580C4—62no—
581C4—61no—
582C4—181no—
583C4—541no—
584C4—541no—
585C4—542no—
586C4D9×#6712yes—
587C3—21no—
588C3#1058971411yescharacteristic, complemented
589C3#1059522811yescharacteristic
590C2—42no—
591C2—121no—
592C2—182no—
593C2—184no—
594C2D9×#23711yescharacteristic
595C2#1172249911yescharacteristic
596C2C2×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#546
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×D4×D9C2semidirect
2. C2×#88641C2semidirect
3. #11721836C2semidirect
4. C9×#938C2semidirect
5. #11721285C2semidirect
6. #11721270C2semidirect
7. #11721269C2semidirect
8. D9×#217C2semidirect
9. C4×#6194C2semidirect
10. #11720998C2semidirect
11. #11720945C2semidirect
16. C2×#6226C22semidirect
17. C2×C9×#67C22semidirect
20. C2×C4×D3×C9C22semidirect
24. C2×#6194C22semidirect
25. C2×Dic3×D9C22semidirect
26. C2×Dic3×D9C22semidirect
28. #88551C22semidirect
29. #88551C22semidirect
30. C4×C3⋊Dic9C22semidirect
31. C9×#217C22semidirect
32. C9×C4⋊Dic3C22semidirect
33. (C2×D3)⋊Dic9C22semidirect
36. (C2×D9)⋊Dic3C22semidirect
37. Dic3⋊Dic9C22semidirect
38. #88442C22semidirect
50. C2×D4×D9D3semidirect
81. C3×D36D4semidirect
157. C2×C3×D9#41semidirect
174. #938D9semidirect
239. D36#67semidirect
269. C2×#67C2×D9semidirect
279. C2×C4×D3C2×D9semidirect
295. #217C2×D9semidirect
301. C4⋊Dic3C2×D9semidirect
345. C2×D9#237semidirect
369. C3×C9#546semidirect
522. C9#5806semidirect
588. C3#10589714semidirect

Maximal subgroups

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

subgroupindex
C2×C3×D4×D92
C2×#886412
#117218362
C9×#9382
#117212852
#117212702
#117212692
D9×#2172
C4×#61942
#117209982
#117209452
#105952283
#105897143