Finite Group Databasebeta

← C3×#10589632 #13152491

Subgroups — C3×#10589632 #13152491

Order 1728.

790 conjugacy classes of subgroups.

The subgroup lattice has 524 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×C9×#117C211yesmaximal, characteristic, complemented
2C3×#63492C211yesmaximal, characteristic, complemented
3C3×C4×#611C211yesmaximal, characteristic, complemented
4C3×#63457C211yesmaximal, characteristic, complemented
5C3×#63455C211yesmaximal, characteristic, complemented
6C3×C4⋊C4×D9C211yesmaximal, characteristic, complemented
7C3×#63452C211yesmaximal, characteristic, complemented
8C3×#63446C212yesmaximal, complemented
9C3×#63445C212yesmaximal, complemented
10C3×#63438C212yesmaximal, complemented
11C3×C4×D36C212yesmaximal, complemented
12C3×#5530—31nomaximal
13#10589632C311yesmaximal, characteristic, direct product factor
14C22×C3×C4×C9C2211yescharacteristic
15C2×C3×#611C2211yescharacteristic
16C2×C3×#611C2212yes—
17C2×C3×D36C2211yescharacteristic
18C2×C3×D36C2212yes—
19C2×C3×C4×D9C2211yescharacteristic
20C2×C3×C4×D9C2212yes—
21C2×C3×C4×D9C2212yes—
22C3×C9×C4⋊C4C2211yescharacteristic
23C3×C9×C4⋊C4C2211yescharacteristic
24C3×C9×C22⋊C4C2212yes—
25C3×C42×C9C2212yes—
26C3×#3676C2211yescharacteristic
27C3×(C2×D9)⋊C4C2211yescharacteristic
28C3×(C2×D9)⋊C4C2212yes—
29C3×(C2×D9)⋊C4C2212yes—
30C3×(C2×D9)⋊C4C2212yes—
31C3×C4⋊Dic9C2211yescharacteristic
32C3×C4⋊Dic9C2212yes—
33C3×Dic9⋊C4C2211yescharacteristic
34C3×Dic9⋊C4C2212yes—
35C3×Dic9⋊C4C2212yes—
36C3×C4×Dic9C2211yescharacteristic
37C32×#117D311yescharacteristic
38C3×#930—31no—
39C3×C4×#67—31no—
40C3×#895—31no—
41C3×#893—31no—
42C3×D3×C4⋊C4—31no—
43C3×#890—31no—
44C3×#884—32no—
45C3×#883—32no—
46C3×#876—32no—
47C3×C4×D12—32no—
48C9×#117—21no—
49C9×#117C2×C311yescharacteristic, complemented
50#63492C2×C311yescharacteristic, complemented
51C4×#611C2×C311yescharacteristic, complemented
52#63457C2×C311yescharacteristic, complemented
53#63455C2×C311yescharacteristic, complemented
54C4⋊C4×D9C2×C311yescharacteristic, complemented
55#63452C2×C311yescharacteristic, complemented
56#63446C2×C312yescomplemented
57#63445C2×C312yescomplemented
58#63438C2×C312yescomplemented
59C4×D36C2×C312yescomplemented
60C23×C3×C9C2311yescharacteristic
61C22×C3×D9C2312yes—
62C22×C3×D9C2311yescharacteristic
63C2×C3×C4×C9—22no—
64C2×C3×C4×C9C2312yes—
65C2×C3×C4×C9C2312yes—
66C2×C3×C4×C9C2311yescharacteristic
67C2×C3×C4×C9C2311yescharacteristic
68C3×#611—22no—
69C3×#611—42no—
70C2×C3×Dic9C2311yescharacteristic
71C2×C3×Dic9C2312yes—
72C2×C3×Dic9C2311yescharacteristic
73C2×C3×Dic9C2311yescharacteristic
74C3×D36—24no—
75C3×D36—22no—
76C3×C4×D9—22no—
77C3×C4×D9—42no—
78C3×C4×D9—42no—
79C3×#553—91no—
80#5530—31no—
81C22×C32×C4C2×D311yescharacteristic
82C2×C3×#67—31no—
83C2×C3×#67—32no—
84C2×C3×D12—32no—
85C2×C3×D12—31no—
86C2×C3×C4×D3—31no—
87C2×C3×C4×D3—32no—
88C2×C3×C4×D3—32no—
89C32×C4⋊C4C2×D311yescharacteristic
90C32×C4⋊C4C2×D311yescharacteristic
91C32×C22⋊C4C2×D312yes—
92C32×C42C2×D312yes—
93C3×#217—31no—
94C3×(C2×D3)⋊C4—32no—
95C3×(C2×D3)⋊C4—32no—
96C3×(C2×D3)⋊C4—31no—
97C3×(C2×D3)⋊C4—32no—
98C3×C4⋊Dic3—32no—
99C3×C4⋊Dic3—31no—
100C3×Dic3⋊C4—32no—
101C3×Dic3⋊C4—32no—
102C3×Dic3⋊C4—31no—
103C3×C4×Dic3—31no—
104C22×C4×C9—21no—
105C22×C4×C9C22×C311yescharacteristic
106C2×#611C22×C311yescharacteristic
107C2×#611C22×C312yes—
108C2×D36C22×C311yescharacteristic
109C2×D36C22×C312yes—
110C2×C4×D9C22×C311yescharacteristic
111C2×C4×D9C22×C312yes—
112C2×C4×D9C22×C312yes—
113C9×C4⋊C4—21no—
114C9×C4⋊C4—21no—
115C9×C4⋊C4C22×C311yescharacteristic
116C9×C4⋊C4C22×C311yescharacteristic
117C9×C22⋊C4—22no—
118C9×C22⋊C4C22×C312yes—
119C42×C9—22no—
120C42×C9C22×C312yes—
121#3676C22×C311yescharacteristic
122(C2×D9)⋊C4C22×C311yescharacteristic
123(C2×D9)⋊C4C22×C312yes—
124(C2×D9)⋊C4C22×C312yes—
125(C2×D9)⋊C4C22×C312yes—
126C4⋊Dic9C22×C311yescharacteristic
127C4⋊Dic9C22×C312yes—
128Dic9⋊C4C22×C311yescharacteristic
129Dic9⋊C4C22×C312yes—
130Dic9⋊C4C22×C312yes—
131C4×Dic9C22×C311yescharacteristic
132C22×C3×C9—21no—
133C22×C3×C9—21no—
134C22×C3×C9—21no—
135C22×C3×C9C2411yescharacteristic
136C2×C3×D9—21no—
137C2×C3×D9—21no—
138C2×C3×D9—22no—
139C2×C3×D9—22no—
140C2×C3×D9—22no—
141C2×C3×D9#4112yes—
142C3×C4×C9—21no—
143C3×C4×C9—22no—
144C3×C4×C9—22no—
145C3×C4×C9#4112yes—
146C3×Dic9—21no—
147C3×Dic9—21no—
148C3×Dic9—21no—
149C3×Dic9—22no—
150C3×#126—92no—
151C3×#125—91no—
152C3×#123—92no—
153C3×#121—91no—
154C3×#121—91no—
155C3×#121—92no—
156C3×C4×D4—91no—
157C3×C4×D4—91no—
158C3×C4×D4—92no—
159C3×#117—21no—
160C3×#117D911yescharacteristic, complemented
161C3×#117C3×D311yescharacteristic
162C2×C3×C4⋊C4—91no—
163#930—31no—
164C4×#67—31no—
165#895—31no—
166#893—31no—
167D3×C4⋊C4—31no—
168#890—31no—
169#884—32no—
170#883—32no—
171#876—32no—
172C4×D12—32no—
173C23×C32C22×D311yescharacteristic
174C22×C3×D3—31no—
175C22×C3×D3—32no—
176C2×C32×C4—22no—
177C2×C32×C4C22×D312yes—
178C2×C32×C4C22×D311yescharacteristic
179C2×C32×C4C22×D312yes—
180C2×C32×C4C22×D311yescharacteristic
181C3×#67—62no—
182C3×#67—122no—
183C2×C3×Dic3—31no—
184C2×C3×Dic3—31no—
185C2×C3×Dic3—31no—
186C2×C3×Dic3—32no—
187C3×D12—64no—
188C3×D12—62no—
189C3×C4×D3—62no—
190C3×C4×D3—122no—
191C3×C4×D3—122no—
192C23×C9—21no—
193C23×C9C23×C311yescharacteristic
194C22×D9C23×C312yes—
195C22×D9C23×C311yescharacteristic
196C2×C4×C9—24no—
197C2×C4×C9—22no—
198C2×C4×C9—21no—
199C2×C4×C9—21no—
200C2×C4×C9—22no—
201C2×C4×C9—22no—
202C2×C4×C9C23×C312yes—
203C2×C4×C9C23×C311yescharacteristic
204C2×C4×C9C23×C312yes—
205C2×C4×C9C23×C311yescharacteristic
206#611—22no—
207#611—42no—
208C2×Dic9C23×C311yescharacteristic
209C2×Dic9C23×C311yescharacteristic
210C2×Dic9C23×C311yescharacteristic
211C2×Dic9C23×C312yes—
212D36—24no—
213D36—22no—
214C4×D9—22no—
215C4×D9—42no—
216C4×D9—42no—
217#553—91noSylow
218C2×C3×C9—41no—
219C2×C3×C9C2×#4111yescharacteristic
220C2×C3×C9C2×#4111yescharacteristic
221C2×C3×C9H(5,𝔽2)11yescharacteristic
222C3×D9—22no—
223C3×D9—42no—
224C2×C3×D4—91no—
225C2×C3×D4—92no—
226C2×C3×D4—92no—
227C2×C3×D4—91no—
228C22×C3×C4—21no—
229C22×C3×C4—91no—
230C22×C3×C4—92no—
231C22×C3×C4—92no—
232C22×C3×C4C2×D911yescharacteristic
233C22×C3×C4C2×C3×D311yescharacteristic
234C2×#67—31no—
235C2×#67—32no—
236C2×D12—32no—
237C2×D12—31no—
238C2×C4×D3—31no—
239C2×C4×D3—32no—
240C2×C4×D3—32no—
241C3×C4⋊C4—21no—
242C3×C4⋊C4—21no—
243C3×C4⋊C4—92no—
244C3×C4⋊C4—91no—
245C3×C4⋊C4—92no—
246C3×C4⋊C4—91no—
247C3×C4⋊C4—92no—
248C3×C4⋊C4C2×D911yescharacteristic
249C3×C4⋊C4C2×D911yescharacteristic
250C3×C4⋊C4C2×C3×D311yescharacteristic
251C3×C4⋊C4C2×C3×D311yescharacteristic
252C3×C22⋊C4—22no—
253C3×C22⋊C4—91no—
254C3×C22⋊C4—91no—
255C3×C22⋊C4—92no—
256C3×C22⋊C4—92no—
257C3×C22⋊C4—92no—
258C3×C22⋊C4C2×D912yes—
259C3×C22⋊C4C2×C3×D312yes—
260C3×C42—22no—
261C3×C42—91no—
262C3×C42C2×D912yes—
263C3×C42C2×C3×D312yes—
264#217—31no—
265(C2×D3)⋊C4—31no—
266(C2×D3)⋊C4—32no—
267(C2×D3)⋊C4—32no—
268(C2×D3)⋊C4—32no—
269C4⋊Dic3—32no—
270C4⋊Dic3—31no—
271Dic3⋊C4—31no—
272Dic3⋊C4—32no—
273Dic3⋊C4—32no—
274C4×Dic3—31no—
275C22×C32—21no—
276C22×C32—21no—
277C22×C32—21no—
278C22×C32C23×D311yescharacteristic
279C2×C3×D3—32no—
280C2×C3×D3—61no—
281C2×C3×D3—61no—
282C2×C3×D3—62no—
283C2×C3×D3—62no—
284C2×C3×D3—62no—
285C32×C4—21no—
286C32×C4—22no—
287C32×C4—22no—
288C32×C4#23512yes—
289C3×Dic3—61no—
290C3×Dic3—61no—
291C3×Dic3—61no—
292C3×Dic3—62no—
293C22×C9—22no—
294C22×C9—21no—
295C22×C9—21no—
296C22×C9—21no—
297C22×C9—21no—
298C22×C9—41no—
299C22×C9—41no—
300C22×C9C24×C311yescharacteristic
301C2×D9—21no—
302C2×D9—21no—
303C2×D9—22no—
304C2×D9—22no—
305C2×D9—22no—
306C2×D9C3×#4112yes—
307C4×C9—22no—
308C4×C9—22no—
309C4×C9—21no—
310C4×C9—22no—
311C4×C9—22no—
312C4×C9—42no—
313C4×C9—42no—
314C4×C9C3×#4112yes—
315Dic9—21no—
316Dic9—21no—
317Dic9—21no—
318Dic9—22no—
319#126—92no—
320#125—91no—
321#123—92no—
322#121—91no—
323#121—91no—
324#121—92no—
325C4×D4—91no—
326C4×D4—91no—
327C4×D4—92no—
328#117C3×D911yescharacteristic, complemented
329C2×C4⋊C4—91no—
330C3×C9#55311yescharacteristic, complemented, Sylow
331C23×C3—21no—
332C23×C3—91no—
333C23×C3—92no—
334C23×C3C22×D911yescharacteristic
335C23×C3C22×C3×D311yescharacteristic
336C22×D3—31no—
337C22×D3—32no—
338C3×D4—184no—
339C3×D4—182no—
340C3×D4—182no—
341C3×D4—362no—
342C2×C3×C4—24no—
343C2×C3×C4—22no—
344C2×C3×C4—22no—
345C2×C3×C4—21no—
346C2×C3×C4—21no—
347C2×C3×C4—22no—
348C2×C3×C4—22no—
349C2×C3×C4—91no—
350C2×C3×C4—91no—
351C2×C3×C4—91no—
352C2×C3×C4—92no—
353C2×C3×C4—182no—
354C2×C3×C4—362no—
355C2×C3×C4—362no—
356C2×C3×C4C22×D911yescharacteristic
357C2×C3×C4C22×D912yes—
358C2×C3×C4C22×D912yes—
359C2×C3×C4C22×D911yescharacteristic
360C2×C3×C4C22×C3×D312yes—
361C2×C3×C4C22×C3×D311yescharacteristic
362C2×C3×C4C22×C3×D312yes—
363C2×C3×C4C22×C3×D311yescharacteristic
364#67—62no—
365#67—122no—
366C2×Dic3—31no—
367C2×Dic3—31no—
368C2×Dic3—31no—
369C2×Dic3—32no—
370D12—64no—
371D12—62no—
372C4×D3—62no—
373C4×D3—122no—
374C4×D3—122no—
375C2×C32—41no—
376C2×C32C2×#23511yescharacteristic
377C2×C32D3×#4111yescharacteristic
378C2×C32#100911yescharacteristic
379C3×D3—62no—
380C3×D3—122no—
381C2×C9—21no—
382C2×C9—21no—
383C2×C9—21no—
384C2×C9—42no—
385C2×C9—41no—
386C2×C9C2×C3×#4111yescharacteristic
387C2×C9C2×C3×#4111yescharacteristic
388C2×C9C3×H(5,𝔽2)11yescharacteristic
389D9—22no—
390D9—42no—
391C2×D4—91no—
392C2×D4—92no—
393C2×D4—92no—
394C2×D4—91no—
395C22×C4—91no—
396C22×C4—92no—
397C22×C4—92no—
398C22×C4C2×C3×D911yescharacteristic
399C4⋊C4—91no—
400C4⋊C4—92no—
401C4⋊C4—92no—
402C4⋊C4—91no—
403C4⋊C4—92no—
404C4⋊C4C2×C3×D911yescharacteristic
405C4⋊C4C2×C3×D911yescharacteristic
406C22⋊C4—91no—
407C22⋊C4—91no—
408C22⋊C4—92no—
409C22⋊C4—92no—
410C22⋊C4—92no—
411C22⋊C4C2×C3×D912yes—
412C42—91no—
413C42C2×C3×D912yes—
414C22×C3—22no—
415C22×C3—21no—
416C22×C3—21no—
417C22×C3—21no—
418C22×C3—21no—
419C22×C3—21no—
420C22×C3—21no—
421C22×C3—21no—
422C22×C3—41no—
423C22×C3—41no—
424C22×C3—92no—
425C22×C3—181no—
426C22×C3—181no—
427C22×C3—182no—
428C22×C3—182no—
429C22×C3—182no—
430C22×C3C23×D911yescharacteristic
431C22×C3C23×C3×D311yescharacteristic
432C2×D3—32no—
433C2×D3—61no—
434C2×D3—61no—
435C2×D3—62no—
436C2×D3—62no—
437C2×D3—62no—
438C3×C4—22no—
439C3×C4—21no—
440C3×C4—22no—
441C3×C4—22no—
442C3×C4—21no—
443C3×C4—22no—
444C3×C4—22no—
445C3×C4—22no—
446C3×C4—42no—
447C3×C4—42no—
448C3×C4—181no—
449C3×C4—181no—
450C3×C4—181no—
451C3×C4—182no—
452C3×C4#369712yes—
453C3×C4C3×#23512yes—
454Dic3—61no—
455Dic3—61no—
456Dic3—61no—
457Dic3—62no—
458C32#553011yescharacteristic
459C9—21no—
460C9C3×#55311yescharacteristic, complemented
461C23—91no—
462C23—92no—
463C23C22×C3×D911yescharacteristic
464D4—184no—
465D4—182no—
466D4—182no—
467D4—362no—
468C2×C4—22no—
469C2×C4—91no—
470C2×C4—91no—
471C2×C4—91no—
472C2×C4—92no—
473C2×C4—182no—
474C2×C4—362no—
475C2×C4—362no—
476C2×C4C22×C3×D911yescharacteristic
477C2×C4C22×C3×D912yes—
478C2×C4C22×C3×D912yes—
479C2×C4C22×C3×D911yescharacteristic
480C2×C3—21no—
481C2×C3—21no—
482C2×C3—21no—
483C2×C3—42no—
484C2×C3—41no—
485C2×C3—41no—
486C2×C3—182no—
487C2×C3—362no—
488C2×C3C2×#369711yescharacteristic
489C2×C3#41×D911yescharacteristic
490C2×C3#6371511yescharacteristic
491C2×C3C2×C3×#23511yescharacteristic
492C2×C3C3×D3×#4111yescharacteristic
493C2×C3C3×#100911yescharacteristic
494D3—62no—
495D3—122no—
496C22—21no—
497C22—21no—
498C22—21no—
499C22—92no—
500C22—181no—
501C22—181no—
502C22—182no—
503C22—182no—
504C22—182no—
505C22C23×C3×D911yescharacteristic
506C4—21no—
507C4—22no—
508C4—22no—
509C4—181no—
510C4—181no—
511C4—181no—
512C4—182no—
513C4C3×#369712yes—
514C3—21no—
515C3#1058963211yescharacteristic, direct product factor
516C3C3×#553011yescharacteristic
517C2—41no—
518C2—182no—
519C2—362no—
520C2C2×C3×#369711yescharacteristic
521C2C3×#41×D911yescharacteristic
522C2C3×#6371511yescharacteristic
subgroupG/H
centerC22×C3C23×D9
commutator, cosocleC22×C9C24×C3
FrattiniC22×C3C23×C3×D3
FittingC3×C9×#117C2
socleC22×C32C23×D3
hypercenterC3×#117D9
hypocenterC9C3×#553
LCSG ▷ C22×C9 ▷ C9
UCSC1 ◁ C22×C3 ◁ C3×#117

Complements

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

NH ≅ G/Ntype
1. C3×C9×#117C2semidirect
2. C3×#63492C2semidirect
3. C3×C4×#611C2semidirect
4. C3×#63457C2semidirect
5. C3×#63455C2semidirect
6. C3×C4⋊C4×D9C2semidirect
7. C3×#63452C2semidirect
8. C3×#63446C2semidirect
9. C3×#63445C2semidirect
10. C3×#63438C2semidirect
11. C3×C4×D36C2semidirect
13. #10589632C3direct (N × H)
49. C9×#117C2×C3semidirect
50. #63492C2×C3semidirect
51. C4×#611C2×C3semidirect
52. #63457C2×C3semidirect
53. #63455C2×C3semidirect
54. C4⋊C4×D9C2×C3semidirect
55. #63452C2×C3semidirect
56. #63446C2×C3semidirect
57. #63445C2×C3semidirect
58. #63438C2×C3semidirect
59. C4×D36C2×C3semidirect
160. C3×#117D9semidirect
328. #117C3×D9semidirect
330. C3×C9#553semidirect
460. C9C3×#553semidirect
515. C3#10589632direct (N × H)

Maximal subgroups

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

subgroupindex
C3×C9×#1172
C3×#634922
C3×C4×#6112
C3×#634572
C3×#634552
C3×C4⋊C4×D92
C3×#634522
C3×#634462
C3×#634452
C3×#634382
C3×C4×D362
C3×#55303
#105896323