Finite Group Databasebeta

← #13150986

Subgroups — #13150986

Order 1728.

1068 conjugacy classes of subgroups.

The subgroup lattice has 574 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C9×#237C211yesmaximal, characteristic, complemented
2C22×Dic3×D9C211yesmaximal, characteristic, complemented
3#11721835C211yesmaximal, characteristic, complemented
4C3×#63499C211yesmaximal, characteristic, complemented
5#11721285C211yesmaximal, characteristic, complemented
6Dic3×#611C212yesmaximal, complemented
7#11721270C212yesmaximal, complemented
8#11721266C211yesmaximal, characteristic, complemented
9#11721244C211yesmaximal, characteristic, complemented
10#11720984C212yesmaximal, complemented
11#11720976C212yesmaximal, complemented
12#10595154—31nomaximal
13#10589930—31nomaximal
14C23×C3×D9C2211yescharacteristic
15C2×C3×D4×C9C2211yescharacteristic
16C22×C3⋊Dic9C2211yescharacteristic
17C2×C9×#67C2211yescharacteristic
18C2×C9×#67C2211yescharacteristic, complemented
19C2×C3×#611C2212yes—
20C22×C9×Dic3C2211yescharacteristic
21C22×C9×Dic3C2211yescharacteristic, complemented
22C9×#237—24no—
23C2×C4×D3×C9C2211yescharacteristic, complemented
24C2×C9×Dic6C2211yescharacteristic, complemented
25C2×#6194C2212yes—
26C2×Dic3×D9—24no—
27C2×Dic3×D9C2212yes—
28C2×Dic3×D9C2212yes—
29#88551C2211yescharacteristic
30#88551C2212yescomplemented
31#88534C2212yescomplemented
32C3×#3676C2211yescharacteristic
33C3×(C2×D9)⋊C4C2212yes—
34(C2×D3)⋊Dic9C2212yescomplemented
35(C2×D9)⋊Dic3C2212yes—
36Dic3⋊Dic9C2212yescomplemented
37Dic9⋊Dic3C2212yescomplemented
38Dic3×Dic9C2212yescomplemented
39C2×C3×#237D311yescharacteristic
40C22×D3×Dic3—31no—
41#64144—31no—
42C3×#937—31no—
43#63976—31no—
44Dic3×#67—32no—
45#63970—32no—
46#63962—31no—
47#63955—31no—
48#63887—32no—
49#63879—32no—
50C2×C9×#41—31no—
51C22×C4×D9—31no—
52#63508—32no—
53#63500—32no—
54#63499—31no—
55#63499D311yescharacteristic, complemented
56#63497—31no—
57#63497—31no—
58#63497—61no—
59C4×#611—32no—
60C4×#611—32no—
61#63488—31no—
62C23×C3×C9C2311yescharacteristic
63C23×C3×C9C2311yescharacteristic
64C22×D3×C9C2311yescharacteristic
65C22×C3×D9—22no—
66C22×C3×D9—24no—
67C22×C3×D9C2312yes—
68C3×D4×C9—22no—
69C2×C3×C4×C9C2311yescharacteristic
70C2×C3⋊Dic9—22no—
71C2×C3⋊Dic9C2312yes—
72C2×C3⋊Dic9C2312yes—
73C9×#67—22no—
74C9×#67—22no—
75C3×#611—24no—
76C2×C9×Dic3—22no—
77C2×C9×Dic3D414yescomplemented
78C2×C9×Dic3C2311yescharacteristic
79C2×C9×Dic3C2311yescharacteristic
80C2×C9×Dic3C2311yescharacteristic
81C2×C3×Dic9C2312yes—
82C4×D3×C9—22no—
83C9×Dic6—22no—
84#6194—24no—
85Dic3×D9—24no—
86Dic3×D9—24no—
87#5828—91no—
88#5586—91no—
89C23×C3×D3—31no—
90C2×C32×D4C2×D311yescharacteristic
91C22×C3⋊Dic3—31no—
92C2×C3×#67—32no—
93C2×C3×#67C2×D311yescharacteristic
94C2×C3×#67C2×D311yescharacteristic
95C22×C3×Dic3C2×D311yescharacteristic
96C22×C3×Dic3C2×D311yescharacteristic
97C3×#237—24no—
98C2×C3×C4×D3C2×D311yescharacteristic
99C2×C3×Dic6C2×D311yescharacteristic
100C2×#625—32no—
101C2×D3×Dic3—32no—
102C2×D3×Dic3—32no—
103C2×D3×Dic3—64no—
104C23×D9C2×D311yescharacteristic
105#3757—31no—
106#3757—32no—
107#3750—32no—
108C3×#217—31no—
109C3×(C2×D3)⋊C4—32no—
110Dic3⋊Dic3—32no—
111Dic3⋊Dic3—32no—
112(C2×D3)⋊Dic3—32no—
113(C2×D3)⋊Dic3—32no—
114Dic32—32no—
115C9×#41—64no—
116C2×Q8×C9—31no—
117C2×D4×C9—21no—
118C2×D4×C9—31no—
119C2×D4×C9—31no—
120C2×D4×C9C2×D311yescharacteristic
121C22×C4×C9—31no—
122C22×C4×C9—31no—
123C22×C4×C9—31no—
124C2×#611—32no—
125C2×#611C2×D312yes—
126C22×Dic9—31no—
127C22×Dic9—61no—
128C2×C4×D9—32no—
129C2×C4×D9—32no—
130C2×C4×D9—64no—
131#3676—31no—
132#3676—32no—
133#3676—32no—
134#3676—61no—
135#3676—62no—
136#3676C2×D311yescharacteristic
137(C2×D9)⋊C4—32no—
138(C2×D9)⋊C4C2×D312yes—
139C4⋊Dic9—32no—
140Dic9⋊C4—32no—
141Dic9⋊C4—32no—
142Dic9⋊C4—62no—
143C4×Dic9—32no—
144C22×C3×C9—22no—
145C22×C3×C9—21no—
146C22×C3×C9—22no—
147C22×C3×C9C2×D412yes—
148C22×C3×C9C2411yescharacteristic
149C2×D3×C9—21no—
150C2×D3×C9—22no—
151C2×C3×D9—28no—
152C2×C3×D9—22no—
153C2×C3×D9—22no—
154C2×C3×D9#4114yescomplemented
155C3×C4×C9—21no—
156C3⋊Dic9—22no—
157C3⋊Dic9—22no—
158C9×Dic3—21no—
159C9×Dic3C2×D412yes—
160C9×Dic3C2×D412yes—
161C3×Dic9—22no—
162C2×C3×#41—31no—
163C23×Dic3—91no—
164C2×#237D911yescharacteristic, complemented
165C22×C4×D3—91no—
166C3×#120—91no—
167#953—91no—
168#946—92no—
169#939—92no—
170#938—92no—
171#937—31no—
172#937—91no—
173#935—91no—
174#935—91no—
175#935—91no—
176#935—181no—
177D4×Dic3—92no—
178C4×#67—92no—
179C4×#67—92no—
180#926—91no—
181#886—91no—
182#67⋊C4—92no—
183#878—92no—
184#877—91no—
185C23×C32C22×D311yescharacteristic
186C23×C32C22×D311yescharacteristic
187C22×C3×D3—32no—
188C22×C3×D3—64no—
189C22×C3×D3—62no—
190C22×C3×D3C22×D311yescharacteristic
191C32×D4—22no—
192C2×C32×C4C22×D311yescharacteristic
193C2×C3⋊Dic3—32no—
194C2×C3⋊Dic3—32no—
195C2×C3⋊Dic3—62no—
196C3×#67—22no—
197C3×#67—22no—
198C3×#67—64no—
199C2×C3×Dic3—22no—
200C2×C3×Dic3—32no—
201C2×C3×Dic3#6714yes—
202C2×C3×Dic3C22×D311yescharacteristic
203C2×C3×Dic3C22×D311yescharacteristic
204C2×C3×Dic3C22×D311yescharacteristic
205C3×C4×D3—22no—
206C3×Dic6—22no—
207#625—64no—
208D3×Dic3—64no—
209D3×Dic3—64no—
210C23×C9—21no—
211C23×C9—21no—
212C23×C9—31no—
213C23×C9C22×D311yescharacteristic
214C23×C9C22×D311yescharacteristic
215C22×D9—22no—
216C22×D9—24no—
217C22×D9C22×D312yes—
218Q8×C9—62no—
219D4×C9—22no—
220D4×C9—42no—
221D4×C9—62no—
222D4×C9—62no—
223C2×C4×C9—21no—
224C2×C4×C9—34no—
225C2×C4×C9—31no—
226C2×C4×C9—31no—
227C2×C4×C9—31no—
228C2×C4×C9—62no—
229C2×C4×C9—62no—
230C2×C4×C9C22×D311yescharacteristic
231#611—24no—
232#611—64no—
233C2×Dic9—32no—
234C2×Dic9—32no—
235C2×Dic9—62no—
236C2×Dic9—62no—
237C2×Dic9—62no—
238C2×Dic9—122no—
239C2×Dic9C22×D312yes—
240C4×D9—64no—
241C4×D9—64no—
242#525—271noSylow
243C2×C3×C9—22no—
244C2×C3×C9—41no—
245C2×C3×C9C22×D411yescharacteristic
246C2×C3×C9C2×#4112yes—
247D3×C9—41no—
248C3×D9—24no—
249C24×C3—91no—
250C23×D3—31no—
251C3×#41—64no—
252C2×C3×Q8—31no—
253C2×C3×D4—21no—
254C2×C3×D4—31no—
255C2×C3×D4—31no—
256C2×C3×D4—92no—
257C2×C3×D4C2×D911yescharacteristic
258C2×C3×D4D3211yescharacteristic
259C22×C3×C4—31no—
260C22×C3×C4—31no—
261C22×C3×C4—31no—
262C2×#67—32no—
263C2×#67—92no—
264C2×#67—92no—
265C2×#67C2×D911yescharacteristic
266C2×#67C2×D911yescharacteristic, complemented
267C22×Dic3—91no—
268C22×Dic3—91no—
269C22×Dic3—92no—
270C22×Dic3—92no—
271C22×Dic3—181no—
272C22×Dic3—184no—
273C22×Dic3C2×D911yescharacteristic
274C22×Dic3C2×D911yescharacteristic, complemented
275#237—24no—
276C2×C4×D3—92no—
277C2×C4×D3—92no—
278C2×C4×D3—184no—
279C2×C4×D3C2×D911yescharacteristic, complemented
280C2×Dic6C2×D911yescharacteristic, complemented
281C3×C22⋊C4—91no—
282C3×C22⋊C4—92no—
283#217—31no—
284#217—92no—
285#217—91no—
286#217—92no—
287#217—91no—
288#217—92no—
289#217—92no—
290#217—181no—
291#217—182no—
292(C2×D3)⋊C4—32no—
293(C2×D3)⋊C4—92no—
294(C2×D3)⋊C4—92no—
295C4⋊Dic3—92no—
296C4⋊Dic3—92no—
297Dic3⋊C4—92no—
298Dic3⋊C4—92no—
299Dic3⋊C4—92no—
300Dic3⋊C4—92no—
301Dic3⋊C4—182no—
302C4×Dic3—92no—
303C4×Dic3—92no—
304C22×C32—22no—
305C22×C32—22no—
306C22×C32—21no—
307C22×C32C2×#6712yes—
308C22×C32C23×D311yescharacteristic
309C2×C3×D3—22no—
310C2×C3×D3—21no—
311C2×C3×D3—34no—
312C2×C3×D3—68no—
313C2×C3×D3—62no—
314C2×C3×D3—62no—
315C32×C4—21no—
316C3⋊Dic3—62no—
317C3⋊Dic3—62no—
318C3×Dic3—21no—
319C3×Dic3—62no—
320C3×Dic3C2×#6712yes—
321C3×Dic3C2×#6712yes—
322C22×C9—22no—
323C22×C9—21no—
324C22×C9—22no—
325C22×C9—21no—
326C22×C9—22no—
327C22×C9—42no—
328C22×C9—41no—
329C22×C9—42no—
330C22×C9—61no—
331C22×C9—62no—
332C22×C9D3×D412yes—
333C22×C9C23×D311yescharacteristic
334C2×D9—28no—
335C2×D9—22no—
336C2×D9—22no—
337C2×D9#23714yescomplemented
338C4×C9—21no—
339C4×C9—32no—
340C4×C9—32no—
341C4×C9—41no—
342C4×C9—61no—
343Dic9—22no—
344Dic9—62no—
345Dic9—62no—
346Dic9—122no—
347Dic9—122no—
348C2×#41—31no—
349C23×C4—271no—
350#123—271no—
351#123—271no—
352C22⋊Q8—272no—
353#121—272no—
354#120—91no—
355#120—271no—
356C4×D4—272no—
357C4×D4—272no—
358#117—271no—
359C3×C9#52511yescharacteristic, complemented, Sylow
360C23×C3—21no—
361C23×C3—21no—
362C23×C3—31no—
363C23×C3—92no—
364C23×C3—184no—
365C23×C3—182no—
366C23×C3C22×D911yescharacteristic
367C23×C3C22×D911yescharacteristic
368C23×C3C2×D3211yescharacteristic
369C23×C3C2×D3211yescharacteristic
370C22×D3—32no—
371C22×D3—64no—
372C22×D3—62no—
373C22×D3C22×D911yescharacteristic
374C3×Q8—62no—
375C3×D4—22no—
376C3×D4—22no—
377C3×D4—42no—
378C3×D4—62no—
379C3×D4—62no—
380C3×D4—184no—
381C2×C3×C4—21no—
382C2×C3×C4—31no—
383C2×C3×C4—31no—
384C2×C3×C4—34no—
385C2×C3×C4—31no—
386C2×C3×C4—62no—
387C2×C3×C4—62no—
388C2×C3×C4—92no—
389C2×C3×C4C22×D911yescharacteristic
390C2×C3×C4C2×D3211yescharacteristic
391#67—22no—
392#67—22no—
393#67—64no—
394#67—184no—
395#67—184no—
396C2×Dic3—22no—
397C2×Dic3—32no—
398C2×Dic3—92no—
399C2×Dic3—92no—
400C2×Dic3—92no—
401C2×Dic3—92no—
402C2×Dic3—182no—
403C2×Dic3—182no—
404C2×Dic3—182no—
405C2×Dic3—182no—
406C2×Dic3—184no—
407C2×Dic3—184no—
408C2×Dic3—362no—
409C2×Dic3#61114yescomplemented
410C2×Dic3C22×D911yescharacteristic
411C2×Dic3C22×D911yescharacteristic
412C2×Dic3C22×D911yescharacteristic
413C4×D3—22no—
414C4×D3—184no—
415C4×D3—184no—
416Dic6—22no—
417C2×C32—22no—
418C2×C32—41no—
419C2×C32D3×#4112yes—
420C2×C32C22×#6711yescharacteristic
421C3×D3—41no—
422C3×D3—64no—
423C2×C9—22no—
424C2×C9—22no—
425C2×C9—21no—
426C2×C9—42no—
427C2×C9—41no—
428C2×C9—81no—
429C2×C9—121no—
430C2×C9C2×D3×D411yescharacteristic
431C2×C9C2×#23712yes—
432D9—24no—
433C24—91no—
434#41—64no—
435C2×Q8—31no—
436C2×D4—31no—
437C2×D4—31no—
438C2×D4—92no—
439C2×D4—272no—
440C2×D4D3×D911yescharacteristic
441C22×C4—31no—
442C22×C4—31no—
443C22×C4—31no—
444C22×C4—272no—
445C22×C4—272no—
446C22×C4—271no—
447C22×C4—544no—
448C4⋊C4—272no—
449C4⋊C4—272no—
450C4⋊C4—272no—
451C22⋊C4—91no—
452C22⋊C4—92no—
453C22⋊C4—272no—
454C22⋊C4—272no—
455C22⋊C4—271no—
456C22⋊C4—272no—
457C42—272no—
458C22×C3—22no—
459C22×C3—22no—
460C22×C3—22no—
461C22×C3—21no—
462C22×C3—22no—
463C22×C3—21no—
464C22×C3—21no—
465C22×C3—22no—
466C22×C3—42no—
467C22×C3—41no—
468C22×C3—42no—
469C22×C3—62no—
470C22×C3—61no—
471C22×C3—94no—
472C22×C3—188no—
473C22×C3—182no—
474C22×C3—182no—
475C22×C3C2×#61112yes—
476C22×C3C23×D911yescharacteristic
477C22×C3D3×#6712yes—
478C22×C3C22×D3211yescharacteristic
479C2×D3—22no—
480C2×D3—21no—
481C2×D3—34no—
482C2×D3—68no—
483C2×D3—62no—
484C2×D3—62no—
485C3×C4—21no—
486C3×C4—21no—
487C3×C4—32no—
488C3×C4—32no—
489C3×C4—41no—
490C3×C4—61no—
491C3×C4—182no—
492Dic3—21no—
493Dic3—62no—
494Dic3—182no—
495Dic3—182no—
496Dic3—182no—
497Dic3—182no—
498Dic3—362no—
499Dic3—362no—
500Dic3C2×#61112yes—
501Dic3C2×#61112yes—
502C32#582811yescharacteristic
503C9—21no—
504C9#558611yescharacteristic, complemented
505C23—31no—
506C23—92no—
507C23—184no—
508C23—182no—
509C23C2×D3×D911yescharacteristic
510C23C2×D3×D911yescharacteristic
511Q8—62no—
512D4—22no—
513D4—62no—
514D4—62no—
515D4—184no—
516D4—544no—
517C2×C4—31no—
518C2×C4—31no—
519C2×C4—34no—
520C2×C4—31no—
521C2×C4—62no—
522C2×C4—62no—
523C2×C4—92no—
524C2×C4—272no—
525C2×C4—272no—
526C2×C4—544no—
527C2×C4—544no—
528C2×C4—542no—
529C2×C4C2×D3×D911yescharacteristic
530C2×C3—22no—
531C2×C3—22no—
532C2×C3—22no—
533C2×C3—21no—
534C2×C3—42no—
535C2×C3—41no—
536C2×C3—41no—
537C2×C3—81no—
538C2×C3—121no—
539C2×C3—184no—
540C2×C3#41×D912yes—
541C2×C3C22×#61111yescharacteristic
542C2×C3D3×#23712yes—
543C2×C3C2×D3×#6711yescharacteristic
544D3—41no—
545D3—64no—
546C22—22no—
547C22—22no—
548C22—21no—
549C22—62no—
550C22—61no—
551C22—94no—
552C22—188no—
553C22—182no—
554C22—182no—
555C22D3×#61112yes—
556C22C22×D3×D911yescharacteristic
557C4—21no—
558C4—32no—
559C4—32no—
560C4—61no—
561C4—182no—
562C4—542no—
563C4—542no—
564C3—21no—
565C3#1058993011yescharacteristic, complemented
566C3#1059515411yescharacteristic
567C2—22no—
568C2—41no—
569C2—121no—
570C2—184no—
571C2D9×#23712yes—
572C2C2×D3×#61111yescharacteristic
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×C9×#237C2semidirect
2. C22×Dic3×D9C2semidirect
3. #11721835C2semidirect
4. C3×#63499C2semidirect
5. #11721285C2semidirect
6. Dic3×#611C2semidirect
7. #11721270C2semidirect
8. #11721266C2semidirect
9. #11721244C2semidirect
10. #11720984C2semidirect
11. #11720976C2semidirect
18. C2×C9×#67C22semidirect
21. C22×C9×Dic3C22semidirect
23. C2×C4×D3×C9C22semidirect
24. C2×C9×Dic6C22semidirect
30. #88551C22semidirect
31. #88534C22semidirect
34. (C2×D3)⋊Dic9C22semidirect
36. Dic3⋊Dic9C22semidirect
37. Dic9⋊Dic3C22semidirect
38. Dic3×Dic9C22semidirect
55. #63499D3semidirect
77. C2×C9×Dic3D4semidirect
154. C2×C3×D9#41semidirect
164. C2×#237D9semidirect
266. C2×#67C2×D9semidirect
274. C22×Dic3C2×D9semidirect
279. C2×C4×D3C2×D9semidirect
280. C2×Dic6C2×D9semidirect
337. C2×D9#237semidirect
359. C3×C9#525semidirect
409. C2×Dic3#611semidirect
504. C9#5586semidirect
565. C3#10589930semidirect

Maximal subgroups

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

subgroupindex
C2×C9×#2372
C22×Dic3×D92
#117218352
C3×#634992
#117212852
Dic3×#6112
#117212702
#117212662
#117212442
#117209842
#117209762
#105951543
#105899303