Finite Group Databasebeta

← C3×#10591239 #13158173

Subgroups — C3×#10591239 #13158173

Order 1728.

1026 conjugacy classes of subgroups.

The subgroup lattice has 518 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C32×C4×Dic3C211yesmaximal, characteristic
2C2×C3×Dic32C212yesmaximal
3C3×#63658C211yesmaximal, characteristic
4C32×C3⋊#95C211yesmaximal, characteristic
5C3×#63579C212yesmaximal
6C3×#4932—31nomaximal
7C3×#4648—31nomaximal
8#10591239C311yesmaximal, characteristic, direct product factor
9C22×C33×C4C2211yescharacteristic
10C22×C3×C3⋊Dic3C2212yes—
11C22×C32×Dic3C2211yescharacteristic
12C22×C32×Dic3C2212yes—
13C22×C32×Dic3C2211yescharacteristic
14C32×C4×Dic3C414yescomplemented
15C3×Dic32C418yescomplemented
16C2×C32×C42—31no—
17C2×C3×C4×Dic3—21no—
18C2×C3×C4×Dic3—32no—
19C2×C3×C4×Dic3—32no—
20C2×C3×C4×Dic3D311yescharacteristic
21C2×C3×C4×Dic3C2×C311yescharacteristic
22C2×Dic32C2×C312yes—
23C32×#95—31no—
24#63658C2×C311yescharacteristic
25C3×C3⋊#95—21no—
26C3×C3⋊#95—31no—
27C3×C3⋊#95—31no—
28C3×C3⋊#95—32no—
29C3×C3⋊#95—32no—
30C3×C3⋊#95—61no—
31C3×C3⋊#95D311yescharacteristic
32C3×C3⋊#95C2×C311yescharacteristic
33#63579C2×C312yes—
34C23×C33C2311yescharacteristic
35C2×C33×C4—22no—
36C2×C33×C4C2×C412yes—
37C2×C3×C3⋊Dic3—22no—
38C2×C3×C3⋊Dic3—22no—
39C2×C3×C3⋊Dic3C2×C414yes—
40C2×C32×Dic3—21no—
41C2×C32×Dic3—21no—
42C2×C32×Dic3—22no—
43C2×C32×Dic3—22no—
44C2×C32×Dic3C2×C414yes—
45C2×C32×Dic3C2×C412yes—
46C2×C32×Dic3C2×C414yes—
47C2×C32×Dic3C2×C412yes—
48C3×#376—91no—
49#4932—31no—
50#4648—31no—
51C22×C32×C4—21no—
52C22×C32×C4—21no—
53C22×C32×C4—21no—
54C22×C32×C4—31no—
55C22×C32×C4—32no—
56C22×C32×C4—31no—
57C22×C32×C4—41no—
58C22×C32×C4C2×D311yescharacteristic
59C22×C32×C4C2×D311yescharacteristic
60C22×C32×C4C22×C311yescharacteristic
61C22×C3⋊Dic3C22×C312yes—
62C22×C3×Dic3—21no—
63C22×C3×Dic3—21no—
64C22×C3×Dic3—22no—
65C22×C3×Dic3—32no—
66C22×C3×Dic3—32no—
67C22×C3×Dic3—62no—
68C22×C3×Dic3C2×D312yes—
69C22×C3×Dic3C2×D311yescharacteristic
70C22×C3×Dic3C2×D311yescharacteristic
71C22×C3×Dic3C22×C311yescharacteristic
72C22×C3×Dic3C22×C312yes—
73C22×C3×Dic3C22×C311yescharacteristic
74C32×C42—34no—
75C3×C4×Dic3—24no—
76C3×C4×Dic3—38no—
77C3×C4×Dic3—38no—
78C3×C4×Dic3Dic314yescomplemented
79C3×C4×Dic3C3×C414yescomplemented
80Dic32C3×C418yescomplemented
81C22×C33C22×C412yes—
82C22×C33C22×C411yescharacteristic
83C22×C33#4111yescharacteristic
84C22×C33#4112yes—
85C22×C33#4111yescharacteristic
86C33×C4—22no—
87C3×C3⋊Dic3—24no—
88C32×Dic3—22no—
89C32×Dic3—24no—
90C32×Dic3C4214yes—
91C2×C3×C42—31no—
92C2×C3×C42—31no—
93C2×C3×C42—61no—
94C2×C3×C42—92no—
95C2×C4×Dic3—32no—
96C2×C4×Dic3—32no—
97C2×C4×Dic3C3×D311yescharacteristic
98C3×#95—31no—
99C3×#95—31no—
100C3×#95—61no—
101C3×#95—92no—
102C3×#95—91no—
103C3⋊#95—31no—
104C3⋊#95—31no—
105C3⋊#95—32no—
106C3⋊#95—32no—
107C3⋊#95—61no—
108C3⋊#95C3×D311yescharacteristic
109C23×C32—21no—
110C23×C32—21no—
111C23×C32—21no—
112C23×C32—41no—
113C23×C32C22×D311yescharacteristic
114C23×C32C22×D311yescharacteristic
115C23×C32C23×C311yescharacteristic
116C2×C32×C4—22no—
117C2×C32×C4—24no—
118C2×C32×C4—22no—
119C2×C32×C4—22no—
120C2×C32×C4—22no—
121C2×C32×C4—22no—
122C2×C32×C4—22no—
123C2×C32×C4—34no—
124C2×C32×C4—32no—
125C2×C32×C4—34no—
126C2×C32×C4—32no—
127C2×C32×C4—42no—
128C2×C32×C4—42no—
129C2×C32×C4—42no—
130C2×C32×C4—44no—
131C2×C32×C4—61no—
132C2×C32×C4—61no—
133C2×C32×C4—62no—
134C2×C32×C4—62no—
135C2×C32×C4C4×D312yes—
136C2×C32×C4C2×Dic312yes—
137C2×C32×C4C2×C3×C412yes—
138C2×C3⋊Dic3—22no—
139C2×C3⋊Dic3—22no—
140C2×C3⋊Dic3C2×C3×C414yes—
141C2×C3×Dic3—22no—
142C2×C3×Dic3—22no—
143C2×C3×Dic3—21no—
144C2×C3×Dic3—21no—
145C2×C3×Dic3—22no—
146C2×C3×Dic3—22no—
147C2×C3×Dic3—21no—
148C2×C3×Dic3—21no—
149C2×C3×Dic3—22no—
150C2×C3×Dic3—22no—
151C2×C3×Dic3—24no—
152C2×C3×Dic3—24no—
153C2×C3×Dic3—22no—
154C2×C3×Dic3—22no—
155C2×C3×Dic3—34no—
156C2×C3×Dic3—34no—
157C2×C3×Dic3—42no—
158C2×C3×Dic3—42no—
159C2×C3×Dic3—64no—
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×Dic3C4×D314yes—
167C2×C3×Dic3C4×D314yes—
168C2×C3×Dic3C2×Dic312yes—
169C2×C3×Dic3C2×Dic312yes—
170C2×C3×Dic3C2×C3×C414yes—
171C2×C3×Dic3C2×C3×C414yes—
172C2×C3×Dic3C2×C3×C412yes—
173C2×C3×Dic3C2×C3×C412yes—
174#376—91noSylow
175C2×C33C2×C4211yescharacteristic
176C2×C33#11712yes—
177C2×C33#11712yes—
178C2×C33#11711yescharacteristic
179C2×C33#11711yescharacteristic
180C22×C3×C4—21no—
181C22×C3×C4—21no—
182C22×C3×C4—21no—
183C22×C3×C4—31no—
184C22×C3×C4—31no—
185C22×C3×C4—32no—
186C22×C3×C4—31no—
187C22×C3×C4—31no—
188C22×C3×C4—32no—
189C22×C3×C4—41no—
190C22×C3×C4—61no—
191C22×C3×C4—61no—
192C22×C3×C4—62no—
193C22×C3×C4—92no—
194C22×C3×C4D3211yescharacteristic
195C22×C3×C4C2×C3×D311yescharacteristic
196C22×C3×C4C2×C3×D311yescharacteristic
197C22×Dic3—32no—
198C22×Dic3—32no—
199C22×Dic3—62no—
200C22×Dic3C2×C3×D312yes—
201C22×Dic3C2×C3×D311yescharacteristic
202C22×Dic3C2×C3×D311yescharacteristic
203C3×C42—34no—
204C3×C42—34no—
205C3×C42—64no—
206C3×C42—98no—
207C4×Dic3—38no—
208C4×Dic3—38no—
209C4×Dic3C3×Dic314yescomplemented
210C22×C32—21no—
211C22×C32—21no—
212C22×C32—22no—
213C22×C32—21no—
214C22×C32—22no—
215C22×C32—22no—
216C22×C32—22no—
217C22×C32—21no—
218C22×C32—21no—
219C22×C32—21no—
220C22×C32—22no—
221C22×C32—21no—
222C22×C32—22no—
223C22×C32—21no—
224C22×C32—21no—
225C22×C32—41no—
226C22×C32—42no—
227C22×C32—42no—
228C22×C32—41no—
229C22×C32—41no—
230C22×C32C2×C4×D312yes—
231C22×C32C2×C4×D312yes—
232C22×C32C2×C4×D311yescharacteristic
233C22×C32#23511yescharacteristic
234C22×C32#23511yescharacteristic
235C22×C32#23512yes—
236C22×C32#23711yescharacteristic
237C22×C32#23712yes—
238C22×C32#23911yescharacteristic
239C22×C32C22×Dic311yescharacteristic
240C22×C32C22×C3×C412yes—
241C22×C32C22×C3×C411yescharacteristic
242C22×C32C3×#4111yescharacteristic
243C22×C32C3×#4112yes—
244C22×C32C3×#4111yescharacteristic
245C32×C4—22no—
246C32×C4—24no—
247C32×C4—22no—
248C32×C4—22no—
249C32×C4—34no—
250C32×C4—42no—
251C32×C4—42no—
252C32×C4—44no—
253C32×C4—62no—
254C32×C4—64no—
255C3⋊Dic3—24no—
256C3×Dic3—24no—
257C3×Dic3—24no—
258C3×Dic3—22no—
259C3×Dic3—24no—
260C3×Dic3—22no—
261C3×Dic3—24no—
262C3×Dic3—44no—
263C3×Dic3—64no—
264C3×Dic3—64no—
265C3×Dic3—124no—
266C3×Dic3C4×Dic314yes—
267C3×Dic3C3×C4214yes—
268C2×C42—31no—
269C2×C42—92no—
270#95—31no—
271#95—92no—
272#95—91no—
273C33#37611yescharacteristic, complemented, Sylow
274C23×C3—21no—
275C23×C3—21no—
276C23×C3—21no—
277C23×C3—41no—
278C23×C3C2×D3211yescharacteristic
279C23×C3C22×C3×D311yescharacteristic
280C23×C3C22×C3×D311yescharacteristic
281C2×C3×C4—24no—
282C2×C3×C4—22no—
283C2×C3×C4—22no—
284C2×C3×C4—22no—
285C2×C3×C4—22no—
286C2×C3×C4—22no—
287C2×C3×C4—22no—
288C2×C3×C4—34no—
289C2×C3×C4—32no—
290C2×C3×C4—34no—
291C2×C3×C4—32no—
292C2×C3×C4—34no—
293C2×C3×C4—32no—
294C2×C3×C4—34no—
295C2×C3×C4—32no—
296C2×C3×C4—42no—
297C2×C3×C4—42no—
298C2×C3×C4—44no—
299C2×C3×C4—42no—
300C2×C3×C4—62no—
301C2×C3×C4—62no—
302C2×C3×C4—64no—
303C2×C3×C4—64no—
304C2×C3×C4—62no—
305C2×C3×C4—62no—
306C2×C3×C4—61no—
307C2×C3×C4—62no—
308C2×C3×C4—61no—
309C2×C3×C4—62no—
310C2×C3×C4—62no—
311C2×C3×C4—62no—
312C2×C3×C4—61no—
313C2×C3×C4—61no—
314C2×C3×C4—94no—
315C2×C3×C4—122no—
316C2×C3×C4—122no—
317C2×C3×C4—182no—
318C2×C3×C4—182no—
319C2×C3×C4D3×Dic312yes—
320C2×C3×C4C3×C4×D312yes—
321C2×C3×C4C2×C3×Dic312yes—
322C2×Dic3—21no—
323C2×Dic3—21no—
324C2×Dic3—22no—
325C2×Dic3—22no—
326C2×Dic3—34no—
327C2×Dic3—34no—
328C2×Dic3—64no—
329C2×Dic3—62no—
330C2×Dic3—62no—
331C2×Dic3—62no—
332C2×Dic3—62no—
333C2×Dic3—122no—
334C2×Dic3—122no—
335C2×Dic3C3×C4×D314yes—
336C2×Dic3C3×C4×D314yes—
337C2×Dic3C2×C3×Dic312yes—
338C2×Dic3C2×C3×Dic312yes—
339C2×C32—22no—
340C2×C32—21no—
341C2×C32—22no—
342C2×C32—21no—
343C2×C32—21no—
344C2×C32—21no—
345C2×C32—21no—
346C2×C32—22no—
347C2×C32—21no—
348C2×C32—21no—
349C2×C32—22no—
350C2×C32—21no—
351C2×C32—21no—
352C2×C32—22no—
353C2×C32—22no—
354C2×C32—41no—
355C2×C32—41no—
356C2×C32—42no—
357C2×C32—41no—
358C2×C32—42no—
359C2×C32C42×D311yescharacteristic
360C2×C32#87212yes—
361C2×C32#87212yes—
362C2×C32#87712yes—
363C2×C32#87711yescharacteristic
364C2×C32#89212yes—
365C2×C32#89211yescharacteristic
366C2×C32C2×C4×Dic311yescharacteristic
367C2×C32#92611yescharacteristic
368C2×C32#92611yescharacteristic
369C2×C32C2×C3×C4211yescharacteristic
370C2×C32C3×#11711yescharacteristic
371C2×C32C3×#11712yes—
372C2×C32C3×#11711yescharacteristic
373C2×C32C3×#11712yes—
374C22×C4—31no—
375C22×C4—32no—
376C22×C4—31no—
377C22×C4—92no—
378C22×C4C3×D3211yescharacteristic
379C42—34no—
380C42—98no—
381C22×C3—21no—
382C22×C3—21no—
383C22×C3—21no—
384C22×C3—21no—
385C22×C3—21no—
386C22×C3—21no—
387C22×C3—22no—
388C22×C3—22no—
389C22×C3—22no—
390C22×C3—21no—
391C22×C3—21no—
392C22×C3—21no—
393C22×C3—22no—
394C22×C3—22no—
395C22×C3—22no—
396C22×C3—41no—
397C22×C3—41no—
398C22×C3—42no—
399C22×C3—41no—
400C22×C3—42no—
401C22×C3#379511yescharacteristic
402C22×C3#379911yescharacteristic
403C22×C3C4×D3212yes—
404C22×C3C2×D3×Dic311yescharacteristic
405C22×C3#380412yes—
406C22×C3C2×C3×C4×D312yes—
407C22×C3C2×C3×C4×D312yes—
408C22×C3C2×C3×C4×D311yescharacteristic
409C22×C3C3×#23511yescharacteristic
410C22×C3C3×#23511yescharacteristic
411C22×C3C3×#23512yes—
412C22×C3C3×#23711yescharacteristic
413C22×C3C3×#23712yes—
414C22×C3C3×#23911yescharacteristic
415C22×C3C22×C3×Dic311yescharacteristic
416C3×C4—24no—
417C3×C4—22no—
418C3×C4—22no—
419C3×C4—22no—
420C3×C4—34no—
421C3×C4—34no—
422C3×C4—42no—
423C3×C4—42no—
424C3×C4—44no—
425C3×C4—64no—
426C3×C4—64no—
427C3×C4—62no—
428C3×C4—62no—
429C3×C4—64no—
430C3×C4—64no—
431C3×C4—124no—
432C3×C4—184no—
433Dic3—24no—
434Dic3—22no—
435Dic3—64no—
436Dic3—64no—
437Dic3—124no—
438Dic3C3×C4×Dic314yes—
439C32—21no—
440C32—21no—
441C32—21no—
442C32—41no—
443C32#464811yescharacteristic, complemented
444C32#493211yescharacteristic, complemented
445C32C3×#37611yescharacteristic, complemented
446C23C2×C3×D3211yescharacteristic
447C2×C4—22no—
448C2×C4—34no—
449C2×C4—32no—
450C2×C4—34no—
451C2×C4—32no—
452C2×C4—61no—
453C2×C4—62no—
454C2×C4—61no—
455C2×C4—62no—
456C2×C4—94no—
457C2×C4—182no—
458C2×C4—182no—
459C2×C4C3×D3×Dic312yes—
460C2×C3—21no—
461C2×C3—21no—
462C2×C3—21no—
463C2×C3—22no—
464C2×C3—22no—
465C2×C3—22no—
466C2×C3—21no—
467C2×C3—21no—
468C2×C3—21no—
469C2×C3—21no—
470C2×C3—21no—
471C2×C3—21no—
472C2×C3—22no—
473C2×C3—22no—
474C2×C3—22no—
475C2×C3—41no—
476C2×C3—42no—
477C2×C3—41no—
478C2×C3—41no—
479C2×C3—42no—
480C2×C3#6383612yes—
481C2×C3#6384111yescharacteristic
482C2×C3C4×D3×Dic311yescharacteristic
483C2×C3#6387712yes—
484C2×C3#6395511yescharacteristic
485C2×C3C3×C42×D311yescharacteristic
486C2×C3C3×#87212yes—
487C2×C3C3×#87212yes—
488C2×C3C3×#87711yescharacteristic
489C2×C3C3×#87712yes—
490C2×C3C3×#89212yes—
491C2×C3C3×#89211yescharacteristic
492C2×C3C2×C3×C4×Dic311yescharacteristic
493C2×C3C3×#92611yescharacteristic
494C2×C3C3×#92611yescharacteristic
495C22C3×#379511yescharacteristic
496C22C3×#379911yescharacteristic
497C22C3×C4×D3212yes—
498C22C2×C3×D3×Dic311yescharacteristic
499C22C3×#380412yes—
500C4—22no—
501C4—34no—
502C4—62no—
503C4—64no—
504C4—184no—
505C3—21no—
506C3—21no—
507C3—21no—
508C3—41no—
509C3#1059123911yescharacteristic, direct product factor
510C3C3×#464811yescharacteristic, complemented
511C3C3×#493211yescharacteristic, complemented
512C2C3×#6383612yes—
513C2C3×#6384111yescharacteristic
514C2C3×C4×D3×Dic311yescharacteristic
515C2C3×#6387712yes—
516C2C3×#6395511yescharacteristic
subgroupG/H
centerC23×C3C2×D32
commutatorC2×C32C2×C3×C42
FrattiniC23C2×C3×D32
FittingC22×C33×C4C22
socle, prime radicalC23×C33C23
cosocleC23×C32C23×C3
hypercenterC22×C3×C4D32
hypocenterC32C3×#376
LCSG ▷ C2×C32 ▷ C32
UCSC1 ◁ C23×C3 ◁ C22×C3×C4

Complements

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

NH ≅ G/Ntype
8. #10591239C3direct (N × H)
14. C32×C4×Dic3C4semidirect
15. C3×Dic32C4semidirect
78. C3×C4×Dic3Dic3semidirect
79. C3×C4×Dic3C3×C4semidirect
80. Dic32C3×C4semidirect
209. C4×Dic3C3×Dic3semidirect
273. C33#376semidirect
443. C32#4648semidirect
444. C32#4932semidirect
445. C32C3×#376semidirect
509. C3#10591239direct (N × H)
510. C3C3×#4648semidirect
511. C3C3×#4932semidirect

Maximal subgroups

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

subgroupindex
C2×C32×C4×Dic32
C2×C3×Dic322
C3×#636582
C32×C3⋊#952
C3×#635792
C3×#49323
C3×#46483
#105912393