Finite Group Databasebeta

← C3×#10591349 #13158289

Subgroups — C3×#10591349 #13158289

Order 1728.

1090 conjugacy classes of subgroups.

The subgroup lattice has 486 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×C4⋊(C3⋊Dic3)C211yesmaximal, characteristic
2C2×C32×C4⋊Dic3C211yesmaximal, characteristic
3C2×C32×C4×Dic3C211yesmaximal, characteristic
4C2×C3×Dic3⋊Dic3C214yesmaximal
5C3×#4975—31nomaximal
6C3×#4935—31nomaximal
7#10591349C311yesmaximal, characteristic, direct product factor
8C22×C33×C4C2211yescharacteristic
9C22×C3×C3⋊Dic3C2212yes—
10C22×C32×Dic3C2212yes—
11C22×C32×Dic3C2212yes—
12C3×C4⋊(C3⋊Dic3)—22no—
13C32×C4⋊Dic3—22no—
14C32×C4×Dic3C414yescomplemented
15C3×Dic3⋊Dic3—28no—
16C2×C32×C4⋊C4—31no—
17C2×C32×C42—31no—
18C2×C4⋊(C3⋊Dic3)C2×C311yescharacteristic
19C2×C3×C4⋊Dic3—21no—
20C2×C3×C4⋊Dic3—31no—
21C2×C3×C4⋊Dic3—31no—
22C2×C3×C4⋊Dic3—34no—
23C2×C3×C4⋊Dic3—61no—
24C2×C3×C4⋊Dic3D311yescharacteristic
25C2×C3×C4⋊Dic3C2×C311yescharacteristic
26C2×C3×Dic3⋊C4—34no—
27C2×C3×C4×Dic3—21no—
28C2×C3×C4×Dic3D311yescharacteristic
29C2×C3×C4×Dic3C2×C311yescharacteristic
30C2×Dic3⋊Dic3C2×C314yes—
31C23×C33C2311yescharacteristic
32C2×C33×C4C2×C412yes—
33C2×C33×C4D412yes—
34C2×C33×C4Q812yes—
35C2×C3×C3⋊Dic3—22no—
36C2×C3×C3⋊Dic3—24no—
37C2×C32×Dic3—22no—
38C2×C32×Dic3—24no—
39C2×C32×Dic3C2×C414yes—
40C2×C32×Dic3D414yes—
41C2×C32×Dic3Q814yes—
42C3×#384—91no—
43#4975—31no—
44#4935—31no—
45C22×C32×C4—21no—
46C22×C32×C4—21no—
47C22×C32×C4—21no—
48C22×C32×C4—32no—
49C22×C32×C4—32no—
50C22×C32×C4—41no—
51C22×C32×C4C2×D311yescharacteristic
52C22×C32×C4C2×D311yescharacteristic
53C22×C32×C4C22×C311yescharacteristic
54C22×C3⋊Dic3C22×C312yes—
55C22×C3×Dic3—22no—
56C22×C3×Dic3—22no—
57C22×C3×Dic3—32no—
58C22×C3×Dic3—32no—
59C22×C3×Dic3—62no—
60C22×C3×Dic3C2×D312yes—
61C22×C3×Dic3C2×D312yes—
62C22×C3×Dic3C22×C312yes—
63C22×C3×Dic3C22×C312yes—
64C32×C4⋊C4—62no—
65C32×C42—34no—
66C4⋊(C3⋊Dic3)—22no—
67C3×C4⋊Dic3—24no—
68C3×C4⋊Dic3—22no—
69C3×C4⋊Dic3—22no—
70C3×C4⋊Dic3—68no—
71C3×C4⋊Dic3—64no—
72C3×C4⋊Dic3—62no—
73C3×C4⋊Dic3—62no—
74C3×Dic3⋊C4—68no—
75C3×C4×Dic3—24no—
76C3×C4×Dic3Dic314yescomplemented
77C3×C4×Dic3C3×C414yescomplemented
78Dic3⋊Dic3—28no—
79C22×C33C22×C411yescharacteristic
80C22×C33C2×D412yes—
81C22×C33C2×D411yescharacteristic
82C22×C33C2×Q811yescharacteristic
83C22×C33C2×Q812yes—
84C33×C4C4⋊C414yescomplemented
85C3×C3⋊Dic3—42no—
86C32×Dic3—42no—
87C32×Dic3C4⋊C418yescomplemented
88C2×C3×C4⋊C4—31no—
89C2×C3×C4⋊C4—31no—
90C2×C3×C4⋊C4—61no—
91C2×C3×C4⋊C4—94no—
92C2×C3×C4⋊C4—91no—
93C2×C3×C42—31no—
94C2×C3×C42—31no—
95C2×C3×C42—61no—
96C2×C4⋊Dic3—31no—
97C2×C4⋊Dic3—31no—
98C2×C4⋊Dic3—34no—
99C2×C4⋊Dic3—61no—
100C2×C4⋊Dic3C3×D311yescharacteristic
101C2×Dic3⋊C4—34no—
102C2×C4×Dic3C3×D311yescharacteristic
103C23×C32—21no—
104C23×C32—21no—
105C23×C32—21no—
106C23×C32—41no—
107C23×C32C22×D311yescharacteristic
108C23×C32C22×D311yescharacteristic
109C23×C32C23×C311yescharacteristic
110C2×C32×C4—22no—
111C2×C32×C4—22no—
112C2×C32×C4—22no—
113C2×C32×C4—22no—
114C2×C32×C4—22no—
115C2×C32×C4—22no—
116C2×C32×C4—22no—
117C2×C32×C4—22no—
118C2×C32×C4—22no—
119C2×C32×C4—34no—
120C2×C32×C4—34no—
121C2×C32×C4—34no—
122C2×C32×C4—42no—
123C2×C32×C4—42no—
124C2×C32×C4—42no—
125C2×C32×C4—62no—
126C2×C32×C4—64no—
127C2×C32×C4Dic612yes—
128C2×C32×C4Dic612yes—
129C2×C32×C4C4×D312yes—
130C2×C32×C4D1212yes—
131C2×C32×C4C2×Dic312yes—
132C2×C32×C4#6712yes—
133C2×C32×C4C2×C3×C412yes—
134C2×C32×C4C3×D412yes—
135C2×C32×C4C3×Q812yes—
136C2×C3⋊Dic3—22no—
137C2×C3⋊Dic3—24no—
138C2×C3×Dic3—24no—
139C2×C3×Dic3—24no—
140C2×C3×Dic3—22no—
141C2×C3×Dic3—24no—
142C2×C3×Dic3—22no—
143C2×C3×Dic3—24no—
144C2×C3×Dic3—24no—
145C2×C3×Dic3—24no—
146C2×C3×Dic3—44no—
147C2×C3×Dic3—64no—
148C2×C3×Dic3—64no—
149C2×C3×Dic3—62no—
150C2×C3×Dic3—64no—
151C2×C3×Dic3—62no—
152C2×C3×Dic3—124no—
153C2×C3×Dic3Dic614yes—
154C2×C3×Dic3D1214yes—
155C2×C3×Dic3C2×Dic314yes—
156C2×C3×Dic3C2×C3×C414yes—
157C2×C3×Dic3C3×D414yes—
158C2×C3×Dic3C3×Q814yes—
159#384—91noSylow
160C2×C33C2×C4⋊C412yes—
161C2×C33C2×C4⋊C411yescharacteristic
162C2×C33#12711yescharacteristic
163C2×C33C4⋊Q812yes—
164C2×C33C4⋊Q811yescharacteristic
165C22×C3×C4—21no—
166C22×C3×C4—21no—
167C22×C3×C4—21no—
168C22×C3×C4—32no—
169C22×C3×C4—32no—
170C22×C3×C4—32no—
171C22×C3×C4—32no—
172C22×C3×C4—41no—
173C22×C3×C4—62no—
174C22×C3×C4—62no—
175C22×C3×C4—92no—
176C22×C3×C4D3211yescharacteristic
177C22×C3×C4C2×C3×D311yescharacteristic
178C22×C3×C4C2×C3×D311yescharacteristic
179C22×Dic3—32no—
180C22×Dic3—32no—
181C22×Dic3—62no—
182C22×Dic3C2×C3×D312yes—
183C22×Dic3C2×C3×D312yes—
184C3×C4⋊C4—64no—
185C3×C4⋊C4—62no—
186C3×C4⋊C4—62no—
187C3×C4⋊C4—188no—
188C3×C4⋊C4—182no—
189C3×C42—34no—
190C3×C42—34no—
191C3×C42—64no—
192C4⋊Dic3—22no—
193C4⋊Dic3—68no—
194C4⋊Dic3—64no—
195C4⋊Dic3—62no—
196C4⋊Dic3—62no—
197Dic3⋊C4—68no—
198C4×Dic3C3×Dic314yescomplemented
199C22×C32—22no—
200C22×C32—22no—
201C22×C32—21no—
202C22×C32—21no—
203C22×C32—21no—
204C22×C32—22no—
205C22×C32—21no—
206C22×C32—21no—
207C22×C32—21no—
208C22×C32—22no—
209C22×C32—21no—
210C22×C32—22no—
211C22×C32—22no—
212C22×C32—21no—
213C22×C32—21no—
214C22×C32—41no—
215C22×C32—41no—
216C22×C32—42no—
217C22×C32—41no—
218C22×C32—42no—
219C22×C32C2×Dic612yes—
220C22×C32C2×Dic611yescharacteristic
221C22×C32C2×Dic611yescharacteristic
222C22×C32C2×C4×D311yescharacteristic
223C22×C32C2×D1212yes—
224C22×C32C2×D1211yescharacteristic
225C22×C32D3×D412yes—
226C22×C32D3×Q812yes—
227C22×C32C22×Dic311yescharacteristic
228C22×C32C2×#6711yescharacteristic
229C22×C32C22×C3×C411yescharacteristic
230C22×C32C2×C3×D412yes—
231C22×C32C2×C3×D411yescharacteristic
232C22×C32C2×C3×Q811yescharacteristic
233C22×C32C2×C3×Q812yes—
234C32×C4—24no—
235C32×C4—24no—
236C32×C4—24no—
237C32×C4—38no—
238C32×C4—44no—
239C32×C4—122no—
240C32×C4Dic3⋊C414yescomplemented
241C32×C4C4⋊Dic314yescomplemented
242C32×C4C3×C4⋊C414yescomplemented
243C3⋊Dic3—42no—
244C3×Dic3—28no—
245C3×Dic3—44no—
246C3×Dic3—42no—
247C3×Dic3—42no—
248C3×Dic3—124no—
249C3×Dic3—122no—
250C3×Dic3—122no—
251C3×Dic3C4⋊Dic318yescomplemented
252C3×Dic3C3×C4⋊C418yescomplemented
253C2×C4⋊C4—31no—
254C2×C4⋊C4—94no—
255C2×C4⋊C4—91no—
256C2×C42—31no—
257C33#38411yescharacteristic, complemented, Sylow
258C23×C3—21no—
259C23×C3—21no—
260C23×C3—21no—
261C23×C3—41no—
262C23×C3C2×D3211yescharacteristic
263C23×C3C22×C3×D311yescharacteristic
264C23×C3C22×C3×D311yescharacteristic
265C2×C3×C4—22no—
266C2×C3×C4—22no—
267C2×C3×C4—22no—
268C2×C3×C4—22no—
269C2×C3×C4—22no—
270C2×C3×C4—22no—
271C2×C3×C4—22no—
272C2×C3×C4—22no—
273C2×C3×C4—22no—
274C2×C3×C4—34no—
275C2×C3×C4—34no—
276C2×C3×C4—34no—
277C2×C3×C4—34no—
278C2×C3×C4—34no—
279C2×C3×C4—34no—
280C2×C3×C4—42no—
281C2×C3×C4—42no—
282C2×C3×C4—42no—
283C2×C3×C4—64no—
284C2×C3×C4—64no—
285C2×C3×C4—64no—
286C2×C3×C4—64no—
287C2×C3×C4—64no—
288C2×C3×C4—62no—
289C2×C3×C4—64no—
290C2×C3×C4—62no—
291C2×C3×C4—124no—
292C2×C3×C4—184no—
293C2×C3×C4—182no—
294C2×C3×C4D3×Dic312yes—
295C2×C3×C4#62612yes—
296C2×C3×C4#62712yes—
297C2×C3×C4C3×Dic612yes—
298C2×C3×C4C3×Dic612yes—
299C2×C3×C4C3×C4×D312yes—
300C2×C3×C4C3×D1212yes—
301C2×C3×C4C2×C3×Dic312yes—
302C2×C3×C4C3×#6712yes—
303C2×Dic3—24no—
304C2×Dic3—22no—
305C2×Dic3—64no—
306C2×Dic3—64no—
307C2×Dic3—62no—
308C2×Dic3—64no—
309C2×Dic3—62no—
310C2×Dic3—124no—
311C2×Dic3C3×Dic614yes—
312C2×Dic3C3×D1214yes—
313C2×Dic3C2×C3×Dic314yes—
314C2×C32—22no—
315C2×C32—22no—
316C2×C32—21no—
317C2×C32—22no—
318C2×C32—21no—
319C2×C32—21no—
320C2×C32—21no—
321C2×C32—21no—
322C2×C32—22no—
323C2×C32—21no—
324C2×C32—22no—
325C2×C32—21no—
326C2×C32—21no—
327C2×C32—22no—
328C2×C32—21no—
329C2×C32—41no—
330C2×C32—42no—
331C2×C32—41no—
332C2×C32—41no—
333C2×C32—42no—
334C2×C32#86911yescharacteristic
335C2×C32#86912yes—
336C2×C32#87411yescharacteristic
337C2×C32#88812yes—
338C2×C32D3×C4⋊C412yes—
339C2×C32C2×Dic3⋊C411yescharacteristic
340C2×C32C2×C4⋊Dic312yes—
341C2×C32C2×C4⋊Dic311yescharacteristic
342C2×C32#94011yescharacteristic
343C2×C32#94411yescharacteristic
344C2×C32C2×C3×C4⋊C412yes—
345C2×C32C2×C3×C4⋊C411yescharacteristic
346C2×C32C3×#12711yescharacteristic
347C2×C32C3×C4⋊Q812yes—
348C2×C32C3×C4⋊Q811yescharacteristic
349C22×C4—32no—
350C22×C4—32no—
351C22×C4—92no—
352C22×C4C3×D3211yescharacteristic
353C4⋊C4—62no—
354C4⋊C4—188no—
355C4⋊C4—182no—
356C42—34no—
357C22×C3—22no—
358C22×C3—22no—
359C22×C3—22no—
360C22×C3—22no—
361C22×C3—22no—
362C22×C3—22no—
363C22×C3—21no—
364C22×C3—21no—
365C22×C3—21no—
366C22×C3—21no—
367C22×C3—21no—
368C22×C3—21no—
369C22×C3—21no—
370C22×C3—21no—
371C22×C3—21no—
372C22×C3—42no—
373C22×C3—42no—
374C22×C3—41no—
375C22×C3—41no—
376C22×C3—41no—
377C22×C3D3×Dic612yes—
378C22×C3D3×D1212yes—
379C22×C3C2×D3×Dic311yescharacteristic
380C22×C3C2×#62611yescharacteristic
381C22×C3C2×#62711yescharacteristic
382C22×C3C2×C3×Dic612yes—
383C22×C3C2×C3×Dic611yescharacteristic
384C22×C3C2×C3×Dic611yescharacteristic
385C22×C3C2×C3×C4×D311yescharacteristic
386C22×C3C2×C3×D1212yes—
387C22×C3C2×C3×D1211yescharacteristic
388C22×C3C3×D3×D412yes—
389C22×C3C3×D3×Q812yes—
390C22×C3C22×C3×Dic311yescharacteristic
391C22×C3C2×C3×#6711yescharacteristic
392C3×C4—24no—
393C3×C4—24no—
394C3×C4—24no—
395C3×C4—38no—
396C3×C4—38no—
397C3×C4—44no—
398C3×C4—68no—
399C3×C4—124no—
400C3×C4—122no—
401C3×C4—122no—
402C3×C4—362no—
403C3×C4Dic3⋊Dic314yescomplemented
404C3×C4C3×Dic3⋊C414yescomplemented
405C3×C4C3×C4⋊Dic314yescomplemented
406Dic3—42no—
407Dic3—124no—
408Dic3—122no—
409Dic3—122no—
410Dic3C3×C4⋊Dic318yescomplemented
411C32—21no—
412C32—21no—
413C32—21no—
414C32—41no—
415C32#493511yescharacteristic, complemented
416C32#497511yescharacteristic, complemented
417C32C3×#38411yescharacteristic, complemented
418C23C2×C3×D3211yescharacteristic
419C2×C4—34no—
420C2×C4—34no—
421C2×C4—34no—
422C2×C4—64no—
423C2×C4—62no—
424C2×C4—184no—
425C2×C4—182no—
426C2×C4C3×D3×Dic312yes—
427C2×C4C3×#62612yes—
428C2×C4C3×#62712yes—
429C2×C3—21no—
430C2×C3—21no—
431C2×C3—21no—
432C2×C3—21no—
433C2×C3—21no—
434C2×C3—21no—
435C2×C3—22no—
436C2×C3—22no—
437C2×C3—22no—
438C2×C3—21no—
439C2×C3—21no—
440C2×C3—21no—
441C2×C3—22no—
442C2×C3—22no—
443C2×C3—22no—
444C2×C3—41no—
445C2×C3—41no—
446C2×C3—42no—
447C2×C3—41no—
448C2×C3—42no—
449C2×C3#6386611yescharacteristic
450C2×C3D3×C4⋊Dic312yes—
451C2×C3#6391111yescharacteristic
452C2×C3#6391812yes—
453C2×C3C2×Dic3⋊Dic311yescharacteristic
454C2×C3C3×#86911yescharacteristic
455C2×C3C3×#86912yes—
456C2×C3C3×#87411yescharacteristic
457C2×C3C3×#88812yes—
458C2×C3C3×D3×C4⋊C412yes—
459C2×C3C2×C3×Dic3⋊C411yescharacteristic
460C2×C3C2×C3×C4⋊Dic311yescharacteristic
461C2×C3C2×C3×C4⋊Dic312yes—
462C2×C3C3×#94011yescharacteristic
463C2×C3C3×#94411yescharacteristic
464C22C3×D3×Dic612yes—
465C22C3×D3×D1212yes—
466C22C2×C3×D3×Dic311yescharacteristic
467C22C2×C3×#62611yescharacteristic
468C22C2×C3×#62711yescharacteristic
469C4—38no—
470C4—122no—
471C4—362no—
472C4C3×Dic3⋊Dic314yescomplemented
473C3—21no—
474C3—21no—
475C3—21no—
476C3—41no—
477C3#1059134911yescharacteristic, direct product factor
478C3C3×#493511yescharacteristic, complemented
479C3C3×#497511yescharacteristic, complemented
480C2C3×#6386611yescharacteristic
481C2C3×D3×C4⋊Dic312yes—
482C2C3×#6391111yescharacteristic
483C2C3×#6391812yes—
484C2C2×C3×Dic3⋊Dic311yescharacteristic
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×#384
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C23×C3 ◁ C22×C3×C4

Complements

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

NH ≅ G/Ntype
7. #10591349C3direct (N × H)
14. C32×C4×Dic3C4semidirect
76. C3×C4×Dic3Dic3semidirect
77. C3×C4×Dic3C3×C4semidirect
84. C33×C4C4⋊C4semidirect
87. C32×Dic3C4⋊C4semidirect
198. C4×Dic3C3×Dic3semidirect
240. C32×C4Dic3⋊C4semidirect
241. C32×C4C4⋊Dic3semidirect
242. C32×C4C3×C4⋊C4semidirect
251. C3×Dic3C4⋊Dic3semidirect
252. C3×Dic3C3×C4⋊C4semidirect
257. C33#384semidirect
403. C3×C4Dic3⋊Dic3semidirect
404. C3×C4C3×Dic3⋊C4semidirect
405. C3×C4C3×C4⋊Dic3semidirect
410. Dic3C3×C4⋊Dic3semidirect
415. C32#4935semidirect
416. C32#4975semidirect
417. C32C3×#384semidirect
472. C4C3×Dic3⋊Dic3semidirect
477. C3#10591349direct (N × H)
478. C3C3×#4935semidirect
479. C3C3×#4975semidirect

Maximal subgroups

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

subgroupindex
C2×C3×C4⋊(C3⋊Dic3)2
C2×C32×C4⋊Dic32
C2×C32×C4×Dic32
C2×C3×Dic3⋊Dic32
C3×#49753
C3×#49353
#105913493