Finite Group Databasebeta

← C3×#10591223 #13158071

Subgroups — C3×#10591223 #13158071

Order 1728.

866 conjugacy classes of subgroups.

The subgroup lattice has 510 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C32×C4×Dic3C211yesmaximal, characteristic
2C3×#63658C211yesmaximal, characteristic
3C32×C3⋊#95C211yesmaximal, characteristic
4C3×#63579C212yesmaximal
5C3×#63579C212yesmaximal
6C3×#4937—31nomaximal
7C3×#4651—31nomaximal
8#10591223C311yesmaximal, characteristic, direct product factor
9C22×C33×C4C2211yescharacteristic
10C22×C3×C3⋊Dic3C2212yes—
11C22×C32×Dic3C2212yes—
12C22×C32×Dic3C2211yescharacteristic
13C22×C32×Dic3C2211yescharacteristic
14C32×C4×Dic3C414yescomplemented
15C2×C32×C42—31no—
16C2×C3×C4×Dic3—21no—
17C2×C3×C4×Dic3D311yescharacteristic
18C2×C3×C4×Dic3C2×C311yescharacteristic
19C32×#95—31no—
20#63658C2×C311yescharacteristic
21C3×C3⋊#95—21no—
22C3×C3⋊#95—31no—
23C3×C3⋊#95—31no—
24C3×C3⋊#95—32no—
25C3×C3⋊#95—32no—
26C3×C3⋊#95—32no—
27C3×C3⋊#95—32no—
28C3×C3⋊#95—61no—
29C3×C3⋊#95D311yescharacteristic
30C3×C3⋊#95C2×C311yescharacteristic
31#63579C2×C312yes—
32#63579C2×C312yes—
33C23×C33C2311yescharacteristic
34C2×C33×C4—22no—
35C2×C33×C4C2×C412yes—
36C2×C3×C3⋊Dic3—22no—
37C2×C3×C3⋊Dic3—22no—
38C2×C3×C3⋊Dic3—22no—
39C2×C32×Dic3—22no—
40C2×C32×Dic3—21no—
41C2×C32×Dic3—21no—
42C2×C32×Dic3—22no—
43C2×C32×Dic3—22no—
44C2×C32×Dic3—22no—
45C2×C32×Dic3C2×C412yes—
46C2×C32×Dic3C2×C412yes—
47C3×#383—91no—
48#4937—31no—
49#4651—31no—
50C22×C32×C4—21no—
51C22×C32×C4—21no—
52C22×C32×C4—21no—
53C22×C32×C4—31no—
54C22×C32×C4—31no—
55C22×C32×C4—32no—
56C22×C32×C4—41no—
57C22×C32×C4C2×D311yescharacteristic
58C22×C32×C4C2×D311yescharacteristic
59C22×C32×C4C22×C311yescharacteristic
60C22×C3⋊Dic3C22×C312yes—
61C22×C3×Dic3—21no—
62C22×C3×Dic3—21no—
63C22×C3×Dic3—22no—
64C22×C3×Dic3—32no—
65C22×C3×Dic3—32no—
66C22×C3×Dic3—62no—
67C22×C3×Dic3C2×D312yes—
68C22×C3×Dic3C2×D311yescharacteristic
69C22×C3×Dic3C2×D311yescharacteristic
70C22×C3×Dic3C22×C312yes—
71C22×C3×Dic3C22×C311yescharacteristic
72C22×C3×Dic3C22×C311yescharacteristic
73C32×C42—34no—
74C3×C4×Dic3—24no—
75C3×C4×Dic3Dic314yescomplemented
76C3×C4×Dic3C3×C414yescomplemented
77C22×C33C22×C411yescharacteristic
78C22×C33#4112yes—
79C22×C33#4111yescharacteristic
80C22×C33#4112yes—
81C22×C33#4111yescharacteristic
82C33×C4—22no—
83C3×C3⋊Dic3—42no—
84C32×Dic3—22no—
85C32×Dic3—22no—
86C32×Dic3—42no—
87C2×C3×C42—31no—
88C2×C3×C42—31no—
89C2×C3×C42—61no—
90C2×C4×Dic3C3×D311yescharacteristic
91C3×#95—31no—
92C3×#95—31no—
93C3×#95—61no—
94C3×#95—92no—
95C3×#95—92no—
96C3×#95—91no—
97C3⋊#95—31no—
98C3⋊#95—31no—
99C3⋊#95—32no—
100C3⋊#95—32no—
101C3⋊#95—32no—
102C3⋊#95—32no—
103C3⋊#95—61no—
104C3⋊#95C3×D311yescharacteristic
105C23×C32—21no—
106C23×C32—21no—
107C23×C32—21no—
108C23×C32—41no—
109C23×C32C22×D311yescharacteristic
110C23×C32C22×D311yescharacteristic
111C23×C32C23×C311yescharacteristic
112C2×C32×C4—22no—
113C2×C32×C4—24no—
114C2×C32×C4—22no—
115C2×C32×C4—22no—
116C2×C32×C4—22no—
117C2×C32×C4—22no—
118C2×C32×C4—22no—
119C2×C32×C4—32no—
120C2×C32×C4—32no—
121C2×C32×C4—42no—
122C2×C32×C4—42no—
123C2×C32×C4—42no—
124C2×C32×C4—44no—
125C2×C32×C4—62no—
126C2×C32×C4—61no—
127C2×C32×C4—61no—
128C2×C32×C4—62no—
129C2×C32×C4—62no—
130C2×C32×C4—62no—
131C2×C32×C4C4×D312yes—
132C2×C32×C4C2×Dic312yes—
133C2×C32×C4C2×C3×C412yes—
134C2×C3⋊Dic3—22no—
135C2×C3⋊Dic3—22no—
136C2×C3⋊Dic3—22no—
137C2×C3×Dic3—22no—
138C2×C3×Dic3—24no—
139C2×C3×Dic3—22no—
140C2×C3×Dic3—21no—
141C2×C3×Dic3—22no—
142C2×C3×Dic3—21no—
143C2×C3×Dic3—22no—
144C2×C3×Dic3—21no—
145C2×C3×Dic3—21no—
146C2×C3×Dic3—24no—
147C2×C3×Dic3—22no—
148C2×C3×Dic3—22no—
149C2×C3×Dic3—22no—
150C2×C3×Dic3—22no—
151C2×C3×Dic3—22no—
152C2×C3×Dic3—22no—
153C2×C3×Dic3—22no—
154C2×C3×Dic3—22no—
155C2×C3×Dic3—42no—
156C2×C3×Dic3—42no—
157C2×C3×Dic3—64no—
158C2×C3×Dic3—62no—
159C2×C3×Dic3—62no—
160C2×C3×Dic3—62no—
161C2×C3×Dic3—62no—
162C2×C3×Dic3—62no—
163C2×C3×Dic3—62no—
164C2×C3×Dic3—122no—
165C2×C3×Dic3—122no—
166C2×C3×Dic3C2×Dic312yes—
167C2×C3×Dic3C2×Dic312yes—
168C2×C3×Dic3C2×C3×C412yes—
169C2×C3×Dic3C2×C3×C412yes—
170#383—91noSylow
171C2×C33#11711yescharacteristic
172C2×C33#11711yescharacteristic
173C2×C33#11711yescharacteristic
174C2×C33#12612yes—
175C2×C33#12612yes—
176C22×C3×C4—21no—
177C22×C3×C4—21no—
178C22×C3×C4—21no—
179C22×C3×C4—31no—
180C22×C3×C4—31no—
181C22×C3×C4—31no—
182C22×C3×C4—31no—
183C22×C3×C4—32no—
184C22×C3×C4—32no—
185C22×C3×C4—41no—
186C22×C3×C4—61no—
187C22×C3×C4—61no—
188C22×C3×C4—62no—
189C22×C3×C4—92no—
190C22×C3×C4D3211yescharacteristic
191C22×C3×C4C2×C3×D311yescharacteristic
192C22×C3×C4C2×C3×D311yescharacteristic
193C22×Dic3—32no—
194C22×Dic3—32no—
195C22×Dic3—62no—
196C22×Dic3C2×C3×D312yes—
197C22×Dic3C2×C3×D311yescharacteristic
198C22×Dic3C2×C3×D311yescharacteristic
199C3×C42—34no—
200C3×C42—34no—
201C3×C42—64no—
202C4×Dic3C3×Dic314yescomplemented
203C22×C32—21no—
204C22×C32—22no—
205C22×C32—21no—
206C22×C32—21no—
207C22×C32—22no—
208C22×C32—21no—
209C22×C32—21no—
210C22×C32—22no—
211C22×C32—22no—
212C22×C32—21no—
213C22×C32—22no—
214C22×C32—21no—
215C22×C32—22no—
216C22×C32—21no—
217C22×C32—21no—
218C22×C32—41no—
219C22×C32—42no—
220C22×C32—41no—
221C22×C32—42no—
222C22×C32—41no—
223C22×C32C2×C4×D311yescharacteristic
224C22×C32#23511yescharacteristic
225C22×C32#23512yes—
226C22×C32#23512yes—
227C22×C32#23511yescharacteristic
228C22×C32#23512yes—
229C22×C32#23711yescharacteristic
230C22×C32#23712yes—
231C22×C32#23911yescharacteristic
232C22×C32C22×Dic311yescharacteristic
233C22×C32C22×C3×C411yescharacteristic
234C22×C32C3×#4112yes—
235C22×C32C3×#4111yescharacteristic
236C22×C32C3×#4112yes—
237C22×C32C3×#4111yescharacteristic
238C32×C4—22no—
239C32×C4—24no—
240C32×C4—22no—
241C32×C4—22no—
242C32×C4—42no—
243C32×C4—42no—
244C32×C4—44no—
245C32×C4—62no—
246C32×C4—62no—
247C32×C4—122no—
248C3⋊Dic3—42no—
249C3×Dic3—24no—
250C3×Dic3—24no—
251C3×Dic3—22no—
252C3×Dic3—22no—
253C3×Dic3—22no—
254C3×Dic3—22no—
255C3×Dic3—44no—
256C3×Dic3—42no—
257C3×Dic3—42no—
258C3×Dic3—124no—
259C3×Dic3—122no—
260C3×Dic3—122no—
261C2×C42—31no—
262#95—31no—
263#95—92no—
264#95—92no—
265#95—91no—
266C33#38311yescharacteristic, complemented, Sylow
267C23×C3—21no—
268C23×C3—21no—
269C23×C3—21no—
270C23×C3—41no—
271C23×C3C2×D3211yescharacteristic
272C23×C3C22×C3×D311yescharacteristic
273C23×C3C22×C3×D311yescharacteristic
274C2×C3×C4—24no—
275C2×C3×C4—22no—
276C2×C3×C4—22no—
277C2×C3×C4—22no—
278C2×C3×C4—22no—
279C2×C3×C4—22no—
280C2×C3×C4—22no—
281C2×C3×C4—32no—
282C2×C3×C4—32no—
283C2×C3×C4—32no—
284C2×C3×C4—32no—
285C2×C3×C4—42no—
286C2×C3×C4—42no—
287C2×C3×C4—44no—
288C2×C3×C4—42no—
289C2×C3×C4—62no—
290C2×C3×C4—62no—
291C2×C3×C4—62no—
292C2×C3×C4—64no—
293C2×C3×C4—62no—
294C2×C3×C4—61no—
295C2×C3×C4—62no—
296C2×C3×C4—61no—
297C2×C3×C4—62no—
298C2×C3×C4—61no—
299C2×C3×C4—61no—
300C2×C3×C4—64no—
301C2×C3×C4—62no—
302C2×C3×C4—62no—
303C2×C3×C4—62no—
304C2×C3×C4—62no—
305C2×C3×C4—62no—
306C2×C3×C4—62no—
307C2×C3×C4—122no—
308C2×C3×C4—122no—
309C2×C3×C4—182no—
310C2×C3×C4—182no—
311C2×C3×C4—182no—
312C2×C3×C4D3×Dic312yes—
313C2×C3×C4C3×C4×D312yes—
314C2×C3×C4C2×C3×Dic312yes—
315C2×Dic3—21no—
316C2×Dic3—22no—
317C2×Dic3—21no—
318C2×Dic3—22no—
319C2×Dic3—22no—
320C2×Dic3—22no—
321C2×Dic3—64no—
322C2×Dic3—62no—
323C2×Dic3—62no—
324C2×Dic3—62no—
325C2×Dic3—62no—
326C2×Dic3—62no—
327C2×Dic3—62no—
328C2×Dic3—122no—
329C2×Dic3—122no—
330C2×Dic3C2×C3×Dic312yes—
331C2×Dic3C2×C3×Dic312yes—
332C2×C32—21no—
333C2×C32—22no—
334C2×C32—21no—
335C2×C32—22no—
336C2×C32—21no—
337C2×C32—21no—
338C2×C32—21no—
339C2×C32—21no—
340C2×C32—21no—
341C2×C32—22no—
342C2×C32—22no—
343C2×C32—21no—
344C2×C32—21no—
345C2×C32—22no—
346C2×C32—22no—
347C2×C32—41no—
348C2×C32—41no—
349C2×C32—41no—
350C2×C32—42no—
351C2×C32—42no—
352C2×C32#87211yescharacteristic
353C2×C32#87612yes—
354C2×C32#87612yes—
355C2×C32#87711yescharacteristic
356C2×C32#87912yes—
357C2×C32#89211yescharacteristic
358C2×C32#89812yes—
359C2×C32#92611yescharacteristic
360C2×C32#92611yescharacteristic
361C2×C32#92611yescharacteristic
362C2×C32C3×#11711yescharacteristic
363C2×C32C3×#11711yescharacteristic
364C2×C32C3×#11711yescharacteristic
365C2×C32C3×#12612yes—
366C2×C32C3×#12612yes—
367C22×C4—31no—
368C22×C4—31no—
369C22×C4—32no—
370C22×C4—92no—
371C22×C4C3×D3211yescharacteristic
372C42—34no—
373C22×C3—21no—
374C22×C3—21no—
375C22×C3—21no—
376C22×C3—22no—
377C22×C3—22no—
378C22×C3—22no—
379C22×C3—21no—
380C22×C3—21no—
381C22×C3—21no—
382C22×C3—21no—
383C22×C3—21no—
384C22×C3—21no—
385C22×C3—22no—
386C22×C3—22no—
387C22×C3—22no—
388C22×C3—41no—
389C22×C3—42no—
390C22×C3—41no—
391C22×C3—41no—
392C22×C3—42no—
393C22×C3#379511yescharacteristic
394C22×C3#379812yes—
395C22×C3#379911yescharacteristic
396C22×C3C2×D3×Dic311yescharacteristic
397C22×C3#380412yes—
398C22×C3C2×C3×C4×D311yescharacteristic
399C22×C3C3×#23511yescharacteristic
400C22×C3C3×#23512yes—
401C22×C3C3×#23512yes—
402C22×C3C3×#23511yescharacteristic
403C22×C3C3×#23512yes—
404C22×C3C3×#23711yescharacteristic
405C22×C3C3×#23712yes—
406C22×C3C3×#23911yescharacteristic
407C22×C3C22×C3×Dic311yescharacteristic
408C3×C4—24no—
409C3×C4—22no—
410C3×C4—22no—
411C3×C4—22no—
412C3×C4—42no—
413C3×C4—42no—
414C3×C4—44no—
415C3×C4—64no—
416C3×C4—64no—
417C3×C4—62no—
418C3×C4—62no—
419C3×C4—62no—
420C3×C4—62no—
421C3×C4—124no—
422C3×C4—122no—
423C3×C4—122no—
424C3×C4—362no—
425Dic3—22no—
426Dic3—22no—
427Dic3—42no—
428Dic3—124no—
429Dic3—122no—
430Dic3—122no—
431C32—21no—
432C32—21no—
433C32—21no—
434C32—41no—
435C32#465111yescharacteristic, complemented
436C32#493711yescharacteristic, complemented
437C32C3×#38311yescharacteristic, complemented
438C23C2×C3×D3211yescharacteristic
439C2×C4—22no—
440C2×C4—32no—
441C2×C4—32no—
442C2×C4—61no—
443C2×C4—62no—
444C2×C4—61no—
445C2×C4—62no—
446C2×C4—62no—
447C2×C4—62no—
448C2×C4—182no—
449C2×C4—182no—
450C2×C4—182no—
451C2×C4C3×D3×Dic312yes—
452C2×C3—21no—
453C2×C3—21no—
454C2×C3—21no—
455C2×C3—21no—
456C2×C3—21no—
457C2×C3—21no—
458C2×C3—21no—
459C2×C3—21no—
460C2×C3—21no—
461C2×C3—22no—
462C2×C3—22no—
463C2×C3—22no—
464C2×C3—22no—
465C2×C3—22no—
466C2×C3—22no—
467C2×C3—41no—
468C2×C3—41no—
469C2×C3—41no—
470C2×C3—42no—
471C2×C3—42no—
472C2×C3#6384111yescharacteristic
473C2×C3#6385511yescharacteristic
474C2×C3#6385912yes—
475C2×C3#6386812yes—
476C2×C3#6395511yescharacteristic
477C2×C3C3×#87211yescharacteristic
478C2×C3C3×#87612yes—
479C2×C3C3×#87612yes—
480C2×C3C3×#87711yescharacteristic
481C2×C3C3×#87912yes—
482C2×C3C3×#89211yescharacteristic
483C2×C3C3×#89812yes—
484C2×C3C3×#92611yescharacteristic
485C2×C3C3×#92611yescharacteristic
486C2×C3C3×#92611yescharacteristic
487C22C3×#379511yescharacteristic
488C22C3×#379812yes—
489C22C3×#379911yescharacteristic
490C22C2×C3×D3×Dic311yescharacteristic
491C22C3×#380412yes—
492C4—22no—
493C4—62no—
494C4—62no—
495C4—122no—
496C4—362no—
497C3—21no—
498C3—21no—
499C3—21no—
500C3—41no—
501C3#1059122311yescharacteristic, direct product factor
502C3C3×#465111yescharacteristic, complemented
503C3C3×#493711yescharacteristic, complemented
504C2C3×#6384111yescharacteristic
505C2C3×#6385511yescharacteristic
506C2C3×#6385912yes—
507C2C3×#6386812yes—
508C2C3×#6395511yescharacteristic
subgroupG/H
centerC23×C3C2×D32
commutatorC22×C32C22×C3×C4
FrattiniC23C2×C3×D32
FittingC22×C33×C4C22
socle, prime radicalC23×C33C23
cosocleC23×C32C23×C3
hypercenterC22×C3×C4D32
hypocenterC32C3×#383
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C23×C3 ◁ C22×C3×C4

Complements

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

NH ≅ G/Ntype
8. #10591223C3direct (N × H)
14. C32×C4×Dic3C4semidirect
75. C3×C4×Dic3Dic3semidirect
76. C3×C4×Dic3C3×C4semidirect
202. C4×Dic3C3×Dic3semidirect
266. C33#383semidirect
435. C32#4651semidirect
436. C32#4937semidirect
437. C32C3×#383semidirect
501. C3#10591223direct (N × H)
502. C3C3×#4651semidirect
503. C3C3×#4937semidirect

Maximal subgroups

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

subgroupindex
C2×C32×C4×Dic32
C3×#636582
C32×C3⋊#952
C3×#635792
C3×#49373
C3×#46513
#105912233