Finite Group Databasebeta

← #13148817

Subgroups — #13148817

Order 1728.

1024 conjugacy classes of subgroups.

The subgroup lattice has 556 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#88634C212yesmaximal, complemented
2#11721749C211yesmaximal, characteristic, complemented
3C3×#63457C211yesmaximal, characteristic, complemented
4C9×#888C211yesmaximal, characteristic, complemented
5#11721006C211yesmaximal, characteristic, complemented
6#11721004C211yesmaximal, characteristic, complemented
7#11721003C212yesmaximal, complemented
8#11720998C211yesmaximal, characteristic, complemented
9#11720983C212yesmaximal, complemented
10#11720978C212yesmaximal, complemented
11C4×Dic3×D9C211yesmaximal, characteristic, complemented
12#10594369—31nomaximal
13#10589827—31nomaximal
14C2×C4⋊(C3⋊D9)C2211yescharacteristic
15C2×C4×C3⋊D9C2212yes—
16C2×C3×D36C2212yes—
17C2×C3×D36C2211yescharacteristic
18C2×C3×C4×D9C2211yescharacteristic
19C2×C9×Dic6C2212yes—
20C2×#6192C2212yes—
21C2×#6192C2212yes—
22C2×#6192C2212yes—
23C2×Dic3×D9C2211yescharacteristic
24C2×Dic3×D9C2212yes—
25C2×Dic3×D9C2211yescharacteristic
26#88634—28no—
27C3×C9×C4⋊C4C2211yescharacteristic
28#88536C2212yes—
29C4×C3⋊Dic9C2211yescharacteristic
30C3×(C2×D9)⋊C4C2212yes—
31C9×C4⋊Dic3C2211yescharacteristic
32C9×Dic3⋊C4C2212yes—
33C4×C9×Dic3C2211yescharacteristic
34#88445C2212yes—
35(C2×D9)⋊Dic3C2212yes—
36Dic3×Dic9C2211yescharacteristic
37Dic3×Dic9C2211yescharacteristic
38C2×#3796—32no—
39#64102—31no—
40C3×#895—31no—
41C3×#888D311yescharacteristic
42#63911—31no—
43#63909—31no—
44#63907—32no—
45#63898—31no—
46#63886—32no—
47#63881—32no—
48C4×D3×Dic3—31no—
49C2×#3701—32no—
50C9×C4⋊Q8—31no—
51#63509—32no—
52#63457—32no—
53#63457—31no—
54#63457D311yescharacteristic, complemented
55#63455—31no—
56#63455—31no—
57#63455—32no—
58#63455—61no—
59#63436—31no—
60C42×D9—31no—
61C22×C3⋊D9C2312yes—
62C22×C3×D9C2312yes—
63C22×C3×D9C2311yescharacteristic
64C2×C3×C4×C9C2312yes—
65C2×C3×C4×C9C2311yescharacteristic
66C2×C3⋊Dic9C2311yescharacteristic
67C2×C3⋊Dic9C2311yescharacteristic
68C4⋊(C3⋊D9)—22no—
69C4×C3⋊D9—24no—
70C2×C9×Dic3C2312yes—
71C2×C9×Dic3C2311yescharacteristic
72C2×C9×Dic3C2311yescharacteristic
73C2×C3×Dic9C2311yescharacteristic
74C3×D36—24no—
75C3×D36—22no—
76C3×C4×D9—22no—
77C9×Dic6—24no—
78#6192—24no—
79#6192—24no—
80#6192—24no—
81Dic3×D9—24no—
82Dic3×D9—22no—
83Dic3×D9D414yescomplemented
84#5725—91no—
85#5596—91no—
86C2×C4⋊(C3⋊D3)—31no—
87C2×C4×C3⋊D3—32no—
88C2×C3×D12—32no—
89C2×C3×D12—31no—
90C2×C3×C4×D3—31no—
91C2×C3×Dic6C2×D312yes—
92C2×#626—32no—
93C2×#626—32no—
94C2×#626—32no—
95C2×D3×Dic3—31no—
96C2×D3×Dic3—32no—
97C2×D3×Dic3—31no—
98#3796—68no—
99C32×C4⋊C4C2×D311yescharacteristic
100#3752—32no—
101C4×C3⋊Dic3—31no—
102C3×(C2×D3)⋊C4—32no—
103C3×C4⋊Dic3C2×D311yescharacteristic
104C3×Dic3⋊C4C2×D312yes—
105C3×C4×Dic3C2×D311yescharacteristic
106#3722—32no—
107(C2×D3)⋊Dic3—32no—
108Dic32—31no—
109Dic32—31no—
110C2×Q8×C9—32no—
111#3701—68no—
112C2×D36—32no—
113C2×D36—32no—
114C2×D36—32no—
115C2×D36—31no—
116C2×D36—61no—
117C2×D36C2×D311yescharacteristic
118C2×D36C2×D312yes—
119C2×C4×D9—31no—
120C2×C4×D9—31no—
121C2×C4×D9—32no—
122C2×C4×D9—32no—
123C2×C4×D9—62no—
124C2×C4×D9C2×D311yescharacteristic
125C9×C4⋊C4—21no—
126C9×C4⋊C4—32no—
127C9×C4⋊C4—31no—
128C9×C4⋊C4C2×D311yescharacteristic
129C42×C9—31no—
130(C2×D9)⋊C4—32no—
131(C2×D9)⋊C4—32no—
132(C2×D9)⋊C4—32no—
133(C2×D9)⋊C4—62no—
134(C2×D9)⋊C4C2×D312yes—
135C4×Dic9—31no—
136C4×Dic9—31no—
137C4×Dic9—31no—
138C4×Dic9—61no—
139C22×C3×C9C2411yescharacteristic
140C2×C3⋊D9—22no—
141C2×C3⋊D9—22no—
142C2×C3⋊D9—22no—
143C2×C3×D9—21no—
144C2×C3×D9—21no—
145C2×C3×D9—22no—
146C2×C3×D9—22no—
147C2×C3×D9—22no—
148C2×C3×D9C2×D412yes—
149C3×C4×C9—22no—
150C3×C4×C9#4112yescomplemented
151C3⋊Dic9—21no—
152C3⋊Dic9C2×D412yes—
153C9×Dic3—22no—
154C9×Dic3C2×D412yes—
155C9×Dic3#4112yescomplemented
156C3×Dic9—21no—
157C2×#239—92no—
158C2×#237—92no—
159C3×C4⋊Q8—31no—
160C3×#121—91no—
161#947—92no—
162#940—91no—
163#939—92no—
164D4×Dic3—91no—
165#930—91no—
166C2×C4×Dic3—91no—
167#895—31no—
168#895—91no—
169#895—92no—
170#893—92no—
171#893—91no—
172#893—91no—
173#893—91no—
174#893—181no—
175#888D911yescharacteristic, complemented
176#885—92no—
177#67⋊C4—92no—
178#874—91no—
179C42×D3—91no—
180C22×C3⋊D3—32no—
181C22×C3×D3—31no—
182C22×C3×D3—32no—
183C2×C32×C4C22×D312yes—
184C2×C32×C4C22×D311yescharacteristic
185C2×C3⋊Dic3—31no—
186C2×C3⋊Dic3—31no—
187C4⋊(C3⋊D3)—62no—
188C4×C3⋊D3—64no—
189C2×C3×Dic3—31no—
190C2×C3×Dic3C22×D312yes—
191C2×C3×Dic3C22×D311yescharacteristic
192C2×C3×Dic3C22×D311yescharacteristic
193C3×D12—64no—
194C3×D12—62no—
195C3×C4×D3—62no—
196C3×Dic6—24no—
197#626—64no—
198#626—64no—
199#626—64no—
200D3×Dic3—34no—
201D3×Dic3—64no—
202D3×Dic3—62no—
203C22×D9—32no—
204C22×D9—62no—
205C22×D9C22×D312yes—
206C22×D9C22×D311yescharacteristic
207Q8×C9—64no—
208C2×C4×C9—21no—
209C2×C4×C9—22no—
210C2×C4×C9—31no—
211C2×C4×C9—31no—
212C2×C4×C9—32no—
213C2×C4×C9C22×D312yes—
214C2×C4×C9C22×D311yescharacteristic
215C2×Dic9—31no—
216C2×Dic9—31no—
217C2×Dic9—61no—
218C2×Dic9—61no—
219C2×Dic9C22×D311yescharacteristic
220D36—24no—
221D36—22no—
222D36—64no—
223D36—64no—
224D36—64no—
225D36—64no—
226D36—62no—
227C4×D9—22no—
228C4×D9—34no—
229C4×D9—64no—
230C4×D9—64no—
231C4×D9—62no—
232C4×D9—124no—
233#532—271noSylow
234C2×C3×C9C22×D411yescharacteristic
235C2×C3×C9C2×#4111yescharacteristic
236C2×C3×C9C2×#4111yescharacteristic
237C3⋊D9—42no—
238C3×D9—22no—
239C3×D9—42no—
240C2×C3×Q8—32no—
241C2×C3×D4—91no—
242C2×C3×D4—92no—
243C22×C3×C4—91no—
244C2×#67—92no—
245C2×#67—92no—
246C2×#67—92no—
247C22×Dic3—91no—
248C22×Dic3—92no—
249C22×Dic3—91no—
250#239—188no—
251#237—188no—
252C2×D12—31no—
253C2×D12—32no—
254C2×D12—92no—
255C2×D12—92no—
256C2×D12—92no—
257C2×D12—91no—
258C2×D12—91no—
259C2×D12—181no—
260C2×C4×D3—31no—
261C2×C4×D3—91no—
262C2×C4×D3—91no—
263C2×C4×D3—92no—
264C2×C4×D3—92no—
265C2×C4×D3—92no—
266C2×C4×D3—182no—
267C2×Dic6C2×D912yes—
268C3×C4⋊C4—21no—
269C3×C4⋊C4—32no—
270C3×C4⋊C4—31no—
271C3×C4⋊C4C2×D911yescharacteristic
272C3×C4⋊C4D3211yescharacteristic
273C3×C22⋊C4—92no—
274C3×C42—31no—
275#217—92no—
276(C2×D3)⋊C4—32no—
277(C2×D3)⋊C4—92no—
278(C2×D3)⋊C4—92no—
279(C2×D3)⋊C4—92no—
280(C2×D3)⋊C4—92no—
281(C2×D3)⋊C4—92no—
282(C2×D3)⋊C4—182no—
283C4⋊Dic3C2×D911yescharacteristic
284Dic3⋊C4C2×D912yes—
285C4×Dic3—91no—
286C4×Dic3—91no—
287C4×Dic3—91no—
288C4×Dic3—91no—
289C4×Dic3—91no—
290C4×Dic3—91no—
291C4×Dic3—181no—
292C4×Dic3C2×D911yescharacteristic
293C22×C32C23×D311yescharacteristic
294C2×C3⋊D3—62no—
295C2×C3⋊D3—62no—
296C2×C3⋊D3—62no—
297C2×C3×D3—32no—
298C2×C3×D3—61no—
299C2×C3×D3—61no—
300C2×C3×D3—62no—
301C2×C3×D3—62no—
302C2×C3×D3—62no—
303C32×C4—22no—
304C32×C4#23912yes—
305C3⋊Dic3—32no—
306C3⋊Dic3—61no—
307C3×Dic3—22no—
308C3×Dic3—61no—
309C3×Dic3D3×D412yes—
310C3×Dic3#23912yes—
311C22×C9—21no—
312C22×C9C23×D311yescharacteristic
313C2×D9—21no—
314C2×D9—21no—
315C2×D9—22no—
316C2×D9—22no—
317C2×D9—22no—
318C2×D9—64no—
319C2×D9—62no—
320C2×D9—62no—
321C2×D9—62no—
322C2×D9—122no—
323C2×D9—122no—
324C2×D9D3×D412yes—
325C4×C9—22no—
326C4×C9—22no—
327C4×C9—32no—
328C4×C9—32no—
329C4×C9—42no—
330C4×C9—62no—
331C4×C9#23712yescomplemented
332Dic9—21no—
333Dic9—32no—
334Dic9—62no—
335Dic9—61no—
336Dic9—121no—
337C2×#41—272no—
338C4⋊Q8—31no—
339#127—271no—
340#124—272no—
341#121—91no—
342#121—271no—
343#121—272no—
344C4×D4—272no—
345C4×D4—271no—
346C4×D4—271no—
347C2×C42—271no—
348C3×C9#53211yescharacteristic, complemented, Sylow
349C23×C3—91no—
350C23×C3—92no—
351C22×D3—31no—
352C22×D3—32no—
353C22×D3—92no—
354C22×D3—92no—
355C22×D3—182no—
356C3×Q8—64no—
357C3×D4—184no—
358C3×D4—182no—
359C2×C3×C4—21no—
360C2×C3×C4—22no—
361C2×C3×C4—31no—
362C2×C3×C4—31no—
363C2×C3×C4—32no—
364C2×C3×C4—91no—
365C2×C3×C4—182no—
366C2×C3×C4C22×D912yes—
367C2×C3×C4C22×D911yescharacteristic
368C2×C3×C4C2×D3212yes—
369C2×C3×C4C2×D3211yescharacteristic
370#67—184no—
371#67—184no—
372#67—184no—
373C2×Dic3—31no—
374C2×Dic3—91no—
375C2×Dic3—91no—
376C2×Dic3—94no—
377C2×Dic3—91no—
378C2×Dic3—91no—
379C2×Dic3—181no—
380C2×Dic3—181no—
381C2×Dic3—184no—
382C2×Dic3—182no—
383C2×Dic3C22×D912yes—
384C2×Dic3C22×D911yescharacteristic
385C2×Dic3C22×D911yescharacteristic
386D12—64no—
387D12—62no—
388D12—184no—
389D12—184no—
390D12—184no—
391D12—184no—
392D12—182no—
393D12—182no—
394C4×D3—62no—
395C4×D3—94no—
396C4×D3—184no—
397C4×D3—184no—
398C4×D3—184no—
399C4×D3—182no—
400C4×D3—364no—
401Dic6—24no—
402C2×C32C2×D3×D411yescharacteristic
403C2×C32C2×#23911yescharacteristic
404C2×C32C2×#23911yescharacteristic
405C3⋊D3—122no—
406C3×D3—62no—
407C3×D3—122no—
408C2×C9—21no—
409C2×C9—21no—
410C2×C9—21no—
411C2×C9C2×D3×D411yescharacteristic
412C2×C9C2×#23711yescharacteristic
413C2×C9D3×#4111yescharacteristic
414D9—22no—
415D9—42no—
416D9—124no—
417D9—122no—
418#41—548no—
419C2×Q8—32no—
420C2×D4—91no—
421C2×D4—92no—
422C2×D4—272no—
423C2×D4—272no—
424C2×D4—272no—
425C2×D4—271no—
426C22×C4—91no—
427C22×C4—271no—
428C22×C4—271no—
429C22×C4—272no—
430C22×C4—272no—
431C4⋊C4—32no—
432C4⋊C4—31no—
433C4⋊C4D3×D911yescharacteristic
434C22⋊C4—92no—
435C22⋊C4—272no—
436C22⋊C4—272no—
437C22⋊C4—272no—
438C42—31no—
439C42—271no—
440C42—271no—
441C42—271no—
442C22×C3—21no—
443C22×C3—92no—
444C22×C3—181no—
445C22×C3—181no—
446C22×C3—182no—
447C22×C3—182no—
448C22×C3—182no—
449C22×C3C23×D911yescharacteristic
450C22×C3C22×D3211yescharacteristic
451C2×D3—32no—
452C2×D3—61no—
453C2×D3—61no—
454C2×D3—62no—
455C2×D3—62no—
456C2×D3—62no—
457C2×D3—184no—
458C2×D3—182no—
459C2×D3—182no—
460C2×D3—182no—
461C2×D3—182no—
462C2×D3—182no—
463C2×D3—182no—
464C2×D3—362no—
465C2×D3—362no—
466C3×C4—22no—
467C3×C4—22no—
468C3×C4—22no—
469C3×C4—32no—
470C3×C4—32no—
471C3×C4—42no—
472C3×C4—62no—
473C3×C4—181no—
474C3×C4#370112yescomplemented
475C3×C4#379612yes—
476Dic3—22no—
477Dic3—61no—
478Dic3—92no—
479Dic3—92no—
480Dic3—182no—
481Dic3—181no—
482Dic3—181no—
483Dic3—361no—
484Dic3D4×D912yes—
485Dic3#370112yescomplemented
486C32#572511yescharacteristic
487C9—21no—
488C9#559611yescharacteristic, complemented
489C23—91no—
490C23—92no—
491C23—272no—
492Q8—64no—
493D4—184no—
494D4—182no—
495D4—544no—
496D4—544no—
497D4—544no—
498D4—542no—
499C2×C4—31no—
500C2×C4—31no—
501C2×C4—32no—
502C2×C4—91no—
503C2×C4—182no—
504C2×C4—274no—
505C2×C4—271no—
506C2×C4—271no—
507C2×C4—544no—
508C2×C4—542no—
509C2×C4—544no—
510C2×C4C2×D3×D912yes—
511C2×C4C2×D3×D911yescharacteristic
512C2×C3—21no—
513C2×C3—21no—
514C2×C3—21no—
515C2×C3—182no—
516C2×C3—362no—
517C2×C3C2×D4×D911yescharacteristic
518C2×C3C2×#370111yescharacteristic
519C2×C3C2×#370111yescharacteristic
520C2×C3C2×#379611yescharacteristic
521C2×C3D32×D411yescharacteristic
522C2×C3D3×#23911yescharacteristic
523D3—62no—
524D3—122no—
525D3—364no—
526D3—362no—
527D3—362no—
528C22—92no—
529C22—181no—
530C22—181no—
531C22—182no—
532C22—182no—
533C22—182no—
534C22—542no—
535C22—542no—
536C22—542no—
537C22C22×D3×D911yescharacteristic
538C4—22no—
539C4—32no—
540C4—32no—
541C4—62no—
542C4—181no—
543C4—272no—
544C4—541no—
545C4#8863412yescomplemented
546C3—21no—
547C3#1058982711yescharacteristic, complemented
548C3#1059436911yescharacteristic
549C2—182no—
550C2—362no—
551C2—1082no—
552C2C2×#8863411yescharacteristic
553C2D3×D4×D911yescharacteristic
554C2D3×#370111yescharacteristic
subgroupG/H
centerC22C22×D3×D9
commutator, cosocleC22×C3×C9C24
FrattiniC22×C3C22×D32
FittingC3×C9×C4⋊C4C22
socleC22×C32C23×D3
hypercenterC4⋊C4D3×D9
hypocenterC3×C9#532
LCSG ▷ C22×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
1. C2×#88634C2semidirect
2. #11721749C2semidirect
3. C3×#63457C2semidirect
4. C9×#888C2semidirect
5. #11721006C2semidirect
6. #11721004C2semidirect
7. #11721003C2semidirect
8. #11720998C2semidirect
9. #11720983C2semidirect
10. #11720978C2semidirect
11. C4×Dic3×D9C2semidirect
54. #63457D3semidirect
83. Dic3×D9D4semidirect
150. C3×C4×C9#41semidirect
155. C9×Dic3#41semidirect
175. #888D9semidirect
331. C4×C9#237semidirect
348. C3×C9#532semidirect
474. C3×C4#3701semidirect
485. Dic3#3701semidirect
488. C9#5596semidirect
545. C4#88634semidirect
547. C3#10589827semidirect

Maximal subgroups

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

subgroupindex
C2×#886342
#117217492
C3×#634572
C9×#8882
#117210062
#117210042
#117210032
#117209982
#117209832
#117209782
C4×Dic3×D92
#105943693
#105898273