Finite Group Databasebeta

← #13148018

Subgroups — #13148018

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×#88658C211yesmaximal, characteristic, complemented
2C22×Dic3×D9C211yesmaximal, characteristic, complemented
3#11721732C211yesmaximal, characteristic, complemented
4C3×#63444C211yesmaximal, characteristic, complemented
5C9×#877C211yesmaximal, characteristic, complemented
6#11721276C211yesmaximal, characteristic, complemented
7#11721260C211yesmaximal, characteristic, complemented
8#11720991C212yesmaximal, complemented
9Dic3×D36C212yesmaximal, complemented
10#11720983C212yesmaximal, complemented
11#11720977C212yesmaximal, complemented
12#10594078—31nomaximal
13#10589626—31nomaximal
14C23×C3×D9C2211yescharacteristic
15C2×#6233C2211yescharacteristic
16C22×C3⋊Dic9C2211yescharacteristic
17C2×C3×#611C2211yescharacteristic
18C22×C9×Dic3C2211yescharacteristic
19C2×C3×D36C2212yes—
20C2×#6192C2212yes—
21C2×#6192C2211yescharacteristic, complemented
22C2×(C3⋊D9)⋊C4C2211yescharacteristic, complemented
23#88658—24no—
24C2×Dic3×D9—24no—
25C2×Dic3×D9C2212yes—
26C2×Dic3×D9C2212yes—
27C2×Dic3×D9C2211yescharacteristic, complemented
28C2×#6189C2211yescharacteristic, complemented
29C3×C9×C22⋊C4C2211yescharacteristic
30#88536C2212yescomplemented
31C4⋊(C3⋊Dic9)C2212yescomplemented
32C9×#217C2211yescharacteristic
33C3×(C2×D9)⋊C4C2212yes—
34C9×Dic3⋊C4C2212yescomplemented
35C4×C9×Dic3C2212yescomplemented
36(C2×D9)⋊Dic3C2212yes—
37(C2×D9)⋊Dic3C2212yescomplemented
38Dic3⋊Dic9C2212yescomplemented
39C2×#3804—31no—
40C22×D3×Dic3—31no—
41#64095—31no—
42C3×#882—31no—
43C3×#877D311yescharacteristic
44#63977—31no—
45#63962—31no—
46#63893—32no—
47Dic3×D12—32no—
48#63886—32no—
49#63880—32no—
50C2×#3697—31no—
51C22×C4×D9—31no—
52C9×#117—31no—
53#63459—32no—
54#63457—32no—
55#63448—31no—
56#63448—31no—
57#63448—61no—
58#63444—31no—
59#63444D311yescharacteristic, complemented
60C4×D36—32no—
61C4×D36—32no—
62C23×C3×C9C2311yescharacteristic
63C22×C3⋊D9C2311yescharacteristic
64C22×C3×D9—22no—
65C22×C3×D9—24no—
66C22×C3×D9C2311yescharacteristic
67C22×C3×D9C2312yes—
68C2×C3×C4×C9C2312yes—
69#6233—22no—
70C2×C3⋊Dic9—22no—
71C2×C3⋊Dic9C2311yescharacteristic
72C2×C3⋊Dic9C2312yes—
73C3×#611—22no—
74C2×C9×Dic3D414yescomplemented
75C2×C9×Dic3C2312yes—
76C2×C9×Dic3C2311yescharacteristic
77C2×C9×Dic3C2311yescharacteristic
78C2×C3×Dic9C2311yescharacteristic
79C3×D36—24no—
80#6192—24no—
81#6192—22no—
82(C3⋊D9)⋊C4—22no—
83Dic3×D9—24no—
84Dic3×D9—24no—
85Dic3×D9—22no—
86#6189—22no—
87#5586—91no—
88#5524—91no—
89C23×C3×D3—31no—
90C2×#638—31no—
91C22×C3⋊Dic3—31no—
92C2×C3×#67—31no—
93C22×C3×Dic3C2×D311yescharacteristic
94C2×C3×D12—32no—
95C2×#627—31no—
96C2×#626—31no—
97C2×#626—32no—
98C2×#624—31no—
99#3804—64no—
100C2×D3×Dic3—32no—
101C2×D3×Dic3—32no—
102C2×D3×Dic3—31no—
103C2×D3×Dic3—64no—
104C23×D9C2×D311yescharacteristic
105C32×C22⋊C4C2×D311yescharacteristic
106#3752—32no—
107C4⋊(C3⋊Dic3)—32no—
108C3×#217C2×D311yescharacteristic
109C3×(C2×D3)⋊C4—32no—
110C3×Dic3⋊C4C2×D312yes—
111C3×C4×Dic3C2×D312yes—
112Dic3⋊Dic3—32no—
113(C2×D3)⋊Dic3—32no—
114(C2×D3)⋊Dic3—32no—
115C22×C4×C9—31no—
116C2×#611—31no—
117C2×#611—61no—
118C2×#611C2×D311yescharacteristic
119C22×Dic9—31no—
120C22×Dic9—61no—
121#3697—64no—
122C2×D36—32no—
123C2×D36—31no—
124C2×D36C2×D312yes—
125C2×C4×D9—32no—
126C2×C4×D9—32no—
127C2×C4×D9—31no—
128C2×C4×D9—31no—
129C2×C4×D9—64no—
130C2×Dic18—31no—
131C9×C4⋊C4—32no—
132C9×C22⋊C4—21no—
133C9×C22⋊C4—31no—
134C9×C22⋊C4C2×D311yescharacteristic
135C42×C9—32no—
136(C2×D9)⋊C4—32no—
137(C2×D9)⋊C4—32no—
138(C2×D9)⋊C4—32no—
139(C2×D9)⋊C4—62no—
140(C2×D9)⋊C4C2×D312yes—
141C4⋊Dic9—32no—
142C4⋊Dic9—32no—
143C4⋊Dic9—62no—
144C22×C3×C9—22no—
145C22×C3×C9C2×D412yes—
146C22×C3×C9C2411yescharacteristic
147C2×C3⋊D9—21no—
148C2×C3⋊D9—22no—
149C2×C3×D9—28no—
150C2×C3×D9—22no—
151C2×C3×D9—22no—
152C2×C3×D9—21no—
153C2×C3×D9—22no—
154C2×C3×D9#4114yescomplemented
155C3×C4×C9—22no—
156C3⋊Dic9—21no—
157C3⋊Dic9—22no—
158C9×Dic3—22no—
159C9×Dic3C2×D412yes—
160C9×Dic3C2×D412yes—
161C3×Dic9—21no—
162C23×Dic3—91no—
163C2×#237—91no—
164C2×#235—91no—
165C22×C4×D3—91no—
166C3×#120—91no—
167C3×#117—31no—
168#953—91no—
169#939—92no—
170#935—91no—
171D4×Dic3—92no—
172#897—92no—
173#895—92no—
174#886—91no—
175#886—91no—
176#886—91no—
177#886—181no—
178#882—31no—
179#882—91no—
180#67⋊C4—92no—
181#878—92no—
182#877D911yescharacteristic, complemented
183C4×D12—92no—
184C4×D12—92no—
185C23×C32C22×D311yescharacteristic
186C22×C3⋊D3—31no—
187C22×C3×D3—32no—
188C22×C3×D3—31no—
189C22×C3×D3—64no—
190C22×C3×D3—62no—
191C2×C32×C4C22×D312yes—
192#638—62no—
193C2×C3⋊Dic3—31no—
194C2×C3⋊Dic3—32no—
195C2×C3⋊Dic3—62no—
196C3×#67—62no—
197C2×C3×Dic3—31no—
198C2×C3×Dic3D1214yes—
199C2×C3×Dic3C22×D312yes—
200C2×C3×Dic3C22×D311yescharacteristic
201C2×C3×Dic3C22×D311yescharacteristic
202C3×D12—64no—
203#627—62no—
204#626—64no—
205#626—62no—
206#624—62no—
207D3×Dic3—64no—
208D3×Dic3—64no—
209D3×Dic3—62no—
210C23×C9—21no—
211C23×C9C22×D311yescharacteristic
212C22×D9—22no—
213C22×D9—24no—
214C22×D9—31no—
215C22×D9—61no—
216C22×D9C22×D311yescharacteristic
217C22×D9C22×D312yes—
218C2×C4×C9—22no—
219C2×C4×C9—34no—
220C2×C4×C9—31no—
221C2×C4×C9—31no—
222C2×C4×C9—32no—
223C2×C4×C9C22×D312yes—
224#611—22no—
225#611—62no—
226#611—122no—
227C2×Dic9—31no—
228C2×Dic9—32no—
229C2×Dic9—61no—
230C2×Dic9—62no—
231C2×Dic9—62no—
232C2×Dic9—122no—
233C2×Dic9C22×D311yescharacteristic
234D36—24no—
235D36—62no—
236D36—64no—
237C4×D9—62no—
238C4×D9—64no—
239C4×D9—64no—
240C4×D9—62no—
241Dic18—62no—
242#525—271noSylow
243C2×C3×C9—22no—
244C2×C3×C9C22×D411yescharacteristic
245C2×C3×C9C2×#4112yes—
246C3⋊D9—41no—
247C3×D9—24no—
248C3×D9—41no—
249C24×C3—91no—
250C23×D3—31no—
251C2×C3×D4—91no—
252C2×C3×D4—92no—
253C22×C3×C4—31no—
254C2×#67—31no—
255C2×#67—91no—
256C2×#67—91no—
257C2×#67—92no—
258C2×#67—91no—
259C2×#67—181no—
260C22×Dic3—91no—
261C22×Dic3—91no—
262C22×Dic3—92no—
263C22×Dic3—92no—
264C22×Dic3—91no—
265C22×Dic3—181no—
266C22×Dic3—184no—
267C22×Dic3C2×D911yescharacteristic
268#237—184no—
269#235—184no—
270C2×D12—32no—
271C2×D12—92no—
272C2×D12—91no—
273C2×C4×D3—91no—
274C2×C4×D3—92no—
275C2×C4×D3—91no—
276C2×C4×D3—92no—
277C2×C4×D3—91no—
278C2×C4×D3—184no—
279C2×Dic6—91no—
280C2×Dic6—91no—
281C3×C4⋊C4—32no—
282C3×C22⋊C4—21no—
283C3×C22⋊C4—31no—
284C3×C22⋊C4—92no—
285C3×C22⋊C4C2×D911yescharacteristic
286C3×C22⋊C4D3211yescharacteristic
287C3×C42—32no—
288#217—92no—
289#217—92no—
290#217C2×D911yescharacteristic
291(C2×D3)⋊C4—32no—
292(C2×D3)⋊C4—92no—
293(C2×D3)⋊C4—92no—
294(C2×D3)⋊C4—92no—
295(C2×D3)⋊C4—92no—
296(C2×D3)⋊C4—182no—
297C4⋊Dic3—92no—
298C4⋊Dic3—92no—
299C4⋊Dic3—92no—
300C4⋊Dic3—182no—
301Dic3⋊C4—92no—
302Dic3⋊C4C2×D912yescomplemented
303C4×Dic3C2×D912yescomplemented
304C22×C32—22no—
305C22×C32C2×D1212yes—
306C22×C32C23×D311yescharacteristic
307C2×C3⋊D3—61no—
308C2×C3⋊D3—62no—
309C2×C3×D3—34no—
310C2×C3×D3—68no—
311C2×C3×D3—62no—
312C2×C3×D3—62no—
313C2×C3×D3—61no—
314C2×C3×D3—62no—
315C32×C4—22no—
316C3⋊Dic3—62no—
317C3⋊Dic3—61no—
318C3×Dic3—22no—
319C3×Dic3—61no—
320C3×Dic3C2×D1212yes—
321C3×Dic3C2×D1212yes—
322C22×C9—22no—
323C22×C9—21no—
324C22×C9—22no—
325C22×C9—42no—
326C22×C9D3×D412yes—
327C22×C9C23×D311yescharacteristic
328C2×D9—28no—
329C2×D9—22no—
330C2×D9—22no—
331C2×D9—21no—
332C2×D9—22no—
333C2×D9—61no—
334C2×D9—62no—
335C2×D9—121no—
336C2×D9—122no—
337C2×D9#23714yescomplemented
338C4×C9—22no—
339C4×C9—32no—
340C4×C9—32no—
341C4×C9—42no—
342C4×C9—62no—
343Dic9—21no—
344Dic9—61no—
345Dic9—62no—
346Dic9—122no—
347Dic9—121no—
348C2×#41—271no—
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—31no—
359C3×C9#52511yescharacteristic, complemented, Sylow
360C23×C3—21no—
361C23×C3—92no—
362C23×C3—91no—
363C23×C3—184no—
364C23×C3—182no—
365C23×C3C22×D911yescharacteristic
366C23×C3C2×D3211yescharacteristic
367C22×D3—32no—
368C22×D3—31no—
369C22×D3—64no—
370C22×D3—62no—
371C22×D3—91no—
372C22×D3—91no—
373C22×D3—181no—
374C3×D4—184no—
375C3×D4—182no—
376C2×C3×C4—22no—
377C2×C3×C4—31no—
378C2×C3×C4—31no—
379C2×C3×C4—34no—
380C2×C3×C4—32no—
381C2×C3×C4—91no—
382C2×C3×C4C22×D912yes—
383C2×C3×C4C2×D3212yes—
384#67—62no—
385#67—184no—
386#67—182no—
387#67—182no—
388#67—182no—
389#67—362no—
390C2×Dic3—31no—
391C2×Dic3—91no—
392C2×Dic3—92no—
393C2×Dic3—92no—
394C2×Dic3—91no—
395C2×Dic3—181no—
396C2×Dic3—182no—
397C2×Dic3—182no—
398C2×Dic3—182no—
399C2×Dic3—182no—
400C2×Dic3—184no—
401C2×Dic3—184no—
402C2×Dic3—362no—
403C2×Dic3D3614yescomplemented
404C2×Dic3C22×D912yes—
405C2×Dic3C22×D911yescharacteristic
406C2×Dic3C22×D911yescharacteristic
407D12—64no—
408D12—182no—
409D12—184no—
410C4×D3—182no—
411C4×D3—184no—
412C4×D3—184no—
413C4×D3—182no—
414C4×D3—182no—
415Dic6—182no—
416Dic6—182no—
417C2×C32—22no—
418C2×C32C22×D1211yescharacteristic
419C2×C32D3×#4112yes—
420C3⋊D3—121no—
421C3×D3—64no—
422C3×D3—121no—
423C2×C9—22no—
424C2×C9—21no—
425C2×C9—22no—
426C2×C9—42no—
427C2×C9C2×D3×D411yescharacteristic
428C2×C9C2×#23712yes—
429D9—24no—
430D9—41no—
431D9—121no—
432D9—241no—
433C24—91no—
434#41—544no—
435C2×Q8—271no—
436C2×D4—91no—
437C2×D4—92no—
438C2×D4—272no—
439C2×D4—271no—
440C2×D4—271no—
441C22×C4—31no—
442C22×C4—271no—
443C22×C4—272no—
444C22×C4—271no—
445C22×C4—272no—
446C22×C4—271no—
447C22×C4—544no—
448C4⋊C4—32no—
449C4⋊C4—272no—
450C4⋊C4—272no—
451C22⋊C4—31no—
452C22⋊C4—92no—
453C22⋊C4—272no—
454C22⋊C4—272no—
455C22⋊C4—272no—
456C22⋊C4D3×D911yescharacteristic
457C42—32no—
458C22×C3—22no—
459C22×C3—22no—
460C22×C3—21no—
461C22×C3—22no—
462C22×C3—42no—
463C22×C3—94no—
464C22×C3—188no—
465C22×C3—182no—
466C22×C3—182no—
467C22×C3—181no—
468C22×C3—182no—
469C22×C3C2×D3612yes—
470C22×C3C23×D911yescharacteristic
471C22×C3D3×D1212yes—
472C22×C3C22×D3211yescharacteristic
473C2×D3—34no—
474C2×D3—68no—
475C2×D3—62no—
476C2×D3—62no—
477C2×D3—61no—
478C2×D3—62no—
479C2×D3—181no—
480C2×D3—182no—
481C2×D3—181no—
482C2×D3—182no—
483C2×D3—361no—
484C2×D3—362no—
485C3×C4—22no—
486C3×C4—22no—
487C3×C4—32no—
488C3×C4—32no—
489C3×C4—42no—
490C3×C4—62no—
491C3×C4—181no—
492Dic3—22no—
493Dic3—61no—
494Dic3—181no—
495Dic3—182no—
496Dic3—181no—
497Dic3—182no—
498Dic3—362no—
499Dic3—361no—
500Dic3C2×D3612yes—
501Dic3C2×D3612yes—
502C32#552411yescharacteristic
503C9—21no—
504C9#558611yescharacteristic, complemented
505C23—92no—
506C23—91no—
507C23—184no—
508C23—182no—
509C23—271no—
510C23C2×D3×D911yescharacteristic
511Q8—542no—
512D4—184no—
513D4—182no—
514D4—542no—
515D4—544no—
516D4—542no—
517C2×C4—31no—
518C2×C4—31no—
519C2×C4—34no—
520C2×C4—32no—
521C2×C4—91no—
522C2×C4—272no—
523C2×C4—271no—
524C2×C4—542no—
525C2×C4—544no—
526C2×C4—544no—
527C2×C4—542no—
528C2×C4—542no—
529C2×C4C2×D3×D912yes—
530C2×C3—22no—
531C2×C3—22no—
532C2×C3—22no—
533C2×C3—21no—
534C2×C3—42no—
535C2×C3—184no—
536C2×C3—361no—
537C2×C3C22×D3611yescharacteristic
538C2×C3#41×D912yes—
539C2×C3C2×D3×D1211yescharacteristic
540C2×C3D3×#23712yes—
541D3—64no—
542D3—121no—
543D3—361no—
544D3—361no—
545D3—721no—
546C22—22no—
547C22—94no—
548C22—188no—
549C22—182no—
550C22—182no—
551C22—182no—
552C22—181no—
553C22—542no—
554C22—541no—
555C22D3×D3612yes—
556C22C22×D3×D911yescharacteristic
557C4—22no—
558C4—32no—
559C4—32no—
560C4—62no—
561C4—181no—
562C4—542no—
563C4—541no—
564C3—21no—
565C3#1058962611yescharacteristic, complemented
566C3#1059407811yescharacteristic
567C2—22no—
568C2—184no—
569C2—361no—
570C2—1081no—
571C2C2×D3×D3611yescharacteristic
572C2D9×#23712yes—
subgroupG/H
centerC22C22×D3×D9
commutator, cosocleC22×C3×C9C24
FrattiniC22×C3C22×D32
FittingC3×C9×C22⋊C4C22
socleC22×C32C23×D3
hypercenterC22⋊C4D3×D9
hypocenterC3×C9#525
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×#88658C2semidirect
2. C22×Dic3×D9C2semidirect
3. #11721732C2semidirect
4. C3×#63444C2semidirect
5. C9×#877C2semidirect
6. #11721276C2semidirect
7. #11721260C2semidirect
8. #11720991C2semidirect
9. Dic3×D36C2semidirect
10. #11720983C2semidirect
11. #11720977C2semidirect
21. C2×#6192C22semidirect
22. C2×(C3⋊D9)⋊C4C22semidirect
27. C2×Dic3×D9C22semidirect
28. C2×#6189C22semidirect
30. #88536C22semidirect
31. C4⋊(C3⋊Dic9)C22semidirect
34. C9×Dic3⋊C4C22semidirect
35. C4×C9×Dic3C22semidirect
37. (C2×D9)⋊Dic3C22semidirect
38. Dic3⋊Dic9C22semidirect
59. #63444D3semidirect
74. C2×C9×Dic3D4semidirect
154. C2×C3×D9#41semidirect
182. #877D9semidirect
302. Dic3⋊C4C2×D9semidirect
303. C4×Dic3C2×D9semidirect
337. C2×D9#237semidirect
359. C3×C9#525semidirect
403. C2×Dic3D36semidirect
504. C9#5586semidirect
565. C3#10589626semidirect

Maximal subgroups

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

subgroupindex
C2×#886582
C22×Dic3×D92
#117217322
C3×#634442
C9×#8772
#117212762
#117212602
#117209912
Dic3×D362
#117209832
#117209772
#105940783
#105896263