Finite Group Databasebeta

← C2×C3×Dic3×Dic6 #13174369

Subgroups — C2×C3×Dic3×Dic6 #13174369

Order 1728.

1602 conjugacy classes of subgroups.

The subgroup lattice has 512 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C32×Dic6C211yesmaximal, characteristic
2C2×C3×C4⋊(C3⋊Dic3)C211yesmaximal, characteristic
3C2×C32×C4×Dic3C211yesmaximal, characteristic
4C2×C3×Dic3⋊Dic3C212yesmaximal
5C2×C3×Dic32C212yesmaximal
6C3×Dic3×Dic6C218yesmaximal, direct product factor
7C2×C3×Q8×Dic3—31nomaximal
8C2×C3×C4×Dic6—31nomaximal
9C2×Dic3×Dic6C311yesmaximal, characteristic, direct product factor
10C22×C33×C4C2211yescharacteristic
11C22×C3×C3⋊Dic3C2212yes—
12C22×C32×Dic3C2211yescharacteristic
13C22×C32×Dic3C2212yes—
14C22×C32×Dic3C2211yescharacteristic
15C2×C32×Dic6C418yescomplemented
16C2×C32×Dic6C2214yes—
17C3×C4⋊(C3⋊Dic3)C2214yes—
18C32×C4×Dic3C2214yes—
19C3×Dic3⋊Dic3C2218yes—
20C3×Dic32C2218yes—
21C22×C32×Q8—31no—
22C22×C3×Dic6—21no—
23C22×C3×Dic6D311yescharacteristic
24C22×C3×Dic6C2×C311yescharacteristic
25C2×C32×C42—31no—
26C2×C4⋊(C3⋊Dic3)C2×C311yescharacteristic
27C3×Q8×Dic3—38no—
28C2×C3×C4⋊Dic3—31no—
29C2×C3×C4⋊Dic3—31no—
30C2×C3×C4⋊Dic3—32no—
31C2×C3×C4⋊Dic3—61no—
32C2×C3×Dic3⋊C4—32no—
33C2×C3×C4×Dic3—21no—
34C2×C3×C4×Dic3—32no—
35C2×C3×C4×Dic3—32no—
36C2×C3×C4×Dic3D311yescharacteristic
37C2×C3×C4×Dic3C2×C311yescharacteristic
38C3×C4×Dic6—38no—
39C2×Dic3⋊Dic3C2×C312yes—
40C2×Dic32C2×C312yes—
41Dic3×Dic6C2×C318yesdirect product factor
42C23×C33C2311yescharacteristic
43C2×C33×C4C2×C414yes—
44C2×C33×C4C2312yes—
45C2×C3×C3⋊Dic3—24no—
46C2×C3×C3⋊Dic3C2314yes—
47C2×C32×Dic3—22no—
48C2×C32×Dic3C2×C418yes—
49C2×C32×Dic3Q814yescomplemented
50C2×C32×Dic3C2312yes—
51C2×C32×Dic3C2314yes—
52C2×C32×Dic3C2312yes—
53C32×Dic6C2×C4116yescomplemented
54C2×C3×C4×Q8—91no—
55C2×Q8×Dic3—31no—
56C2×C4×Dic6—31no—
57C2×C32×Q8—38no—
58C2×C32×Q8—34no—
59C22×C32×C4—21no—
60C22×C32×C4—21no—
61C22×C32×C4—21no—
62C22×C32×C4—31no—
63C22×C32×C4—32no—
64C22×C32×C4—31no—
65C22×C32×C4—41no—
66C22×C32×C4C2×D311yescharacteristic
67C22×C32×C4C2×D311yescharacteristic
68C22×C32×C4C22×C311yescharacteristic
69C22×C3⋊Dic3C22×C312yes—
70C22×C3×Dic3—21no—
71C22×C3×Dic3—21no—
72C22×C3×Dic3—22no—
73C22×C3×Dic3—32no—
74C22×C3×Dic3—32no—
75C22×C3×Dic3—62no—
76C22×C3×Dic3C2×D311yescharacteristic
77C22×C3×Dic3C2×D311yescharacteristic
78C22×C3×Dic3C2×D312yes—
79C22×C3×Dic3C22×C311yescharacteristic
80C22×C3×Dic3C22×C312yes—
81C22×C3×Dic3C22×C311yescharacteristic
82C2×C3×Dic6—28no—
83C2×C3×Dic6—24no—
84C2×C3×Dic6Dic318yesdirect product factor
85C2×C3×Dic6C3×C418yescomplemented
86C2×C3×Dic6C2×D314yes—
87C2×C3×Dic6C22×C314yes—
88C32×C42—34no—
89C4⋊(C3⋊Dic3)C22×C314yes—
90C3×C4⋊Dic3—34no—
91C3×C4⋊Dic3—34no—
92C3×C4⋊Dic3—38no—
93C3×C4⋊Dic3—64no—
94C3×Dic3⋊C4—38no—
95C3×C4×Dic3—24no—
96C3×C4×Dic3—38no—
97C3×C4×Dic3—38no—
98C3×C4×Dic3C2×D314yes—
99C3×C4×Dic3C22×C314yes—
100Dic3⋊Dic3C22×C318yes—
101Dic32C22×C318yes—
102C22×C33C22×C412yes—
103C22×C33C2×Q812yes—
104C22×C33#4112yes—
105C22×C33C2411yescharacteristic
106C33×C4C22×C414yes—
107C3×C3⋊Dic3—24no—
108C32×Dic3—22no—
109C32×Dic3C22×C418yes—
110C32×Dic3C2×Q814yescomplemented
111C22×C3×Q8—31no—
112C22×C3×Q8—31no—
113C22×C3×Q8—61no—
114C22×Dic6C3×D311yescharacteristic
115C3×C4×Q8—98no—
116C2×C3×C4⋊C4—92no—
117C2×C3×C4⋊C4—91no—
118C2×C3×C42—31no—
119C2×C3×C42—31no—
120C2×C3×C42—61no—
121C2×C3×C42—92no—
122Q8×Dic3—38no—
123C2×C4⋊Dic3—31no—
124C2×C4⋊Dic3—31no—
125C2×C4⋊Dic3—32no—
126C2×C4⋊Dic3—61no—
127C2×Dic3⋊C4—32no—
128C2×C4×Dic3—32no—
129C2×C4×Dic3—32no—
130C2×C4×Dic3C3×D311yescharacteristic
131C4×Dic6—38no—
132C23×C32—21no—
133C23×C32—21no—
134C23×C32—21no—
135C23×C32—41no—
136C23×C32C22×D311yescharacteristic
137C23×C32C22×D311yescharacteristic
138C23×C32C23×C311yescharacteristic
139C32×Q8—316no—
140C2×C32×C4—24no—
141C2×C32×C4—24no—
142C2×C32×C4—24no—
143C2×C32×C4—22no—
144C2×C32×C4—22no—
145C2×C32×C4—22no—
146C2×C32×C4—38no—
147C2×C32×C4—34no—
148C2×C32×C4—32no—
149C2×C32×C4—34no—
150C2×C32×C4—32no—
151C2×C32×C4—44no—
152C2×C32×C4—42no—
153C2×C32×C4—62no—
154C2×C32×C4C4×D314yes—
155C2×C32×C4C2×Dic314yes—
156C2×C32×C4C2×C3×C414yes—
157C2×C32×C4C22×D312yes—
158C2×C32×C4C22×D312yes—
159C2×C32×C4C23×C312yes—
160C2×C3⋊Dic3—24no—
161C2×C3⋊Dic3C23×C314yes—
162C2×C3×Dic3—24no—
163C2×C3×Dic3—22no—
164C2×C3×Dic3—22no—
165C2×C3×Dic3—24no—
166C2×C3×Dic3—28no—
167C2×C3×Dic3—22no—
168C2×C3×Dic3—22no—
169C2×C3×Dic3—24no—
170C2×C3×Dic3—34no—
171C2×C3×Dic3—34no—
172C2×C3×Dic3—64no—
173C2×C3×Dic3—64no—
174C2×C3×Dic3—64no—
175C2×C3×Dic3—124no—
176C2×C3×Dic3Dic614yesdirect product factor
177C2×C3×Dic3C2×Dic318yes—
178C2×C3×Dic3C2×C3×C418yes—
179C2×C3×Dic3C3×Q814yescomplemented
180C2×C3×Dic3C22×D312yes—
181C2×C3×Dic3C22×D312yes—
182C2×C3×Dic3C22×D314yes—
183C2×C3×Dic3C23×C312yes—
184C2×C3×Dic3C23×C314yes—
185C2×C3×Dic3C23×C312yes—
186C3×Dic6—216no—
187C3×Dic6C2×Dic3116yesdirect product factor
188C3×Dic6C2×C3×C4116yescomplemented
189C2×C4×Q8—91noSylow
190C2×C33C4×Q814yescomplemented
191C2×C33C23×C411yescharacteristic
192C2×C33C22×Q811yescharacteristic
193C2×C33C2×#4111yescharacteristic
194C2×C3×Q8—38no—
195C2×C3×Q8—38no—
196C2×C3×Q8—34no—
197C2×C3×Q8—34no—
198C2×C3×Q8—68no—
199C2×C3×Q8—64no—
200C22×C3×C4—21no—
201C22×C3×C4—21no—
202C22×C3×C4—21no—
203C22×C3×C4—31no—
204C22×C3×C4—32no—
205C22×C3×C4—31no—
206C22×C3×C4—32no—
207C22×C3×C4—31no—
208C22×C3×C4—31no—
209C22×C3×C4—41no—
210C22×C3×C4—61no—
211C22×C3×C4—61no—
212C22×C3×C4—62no—
213C22×C3×C4—92no—
214C22×C3×C4D3211yescharacteristic
215C22×C3×C4C2×C3×D311yescharacteristic
216C22×C3×C4C2×C3×D311yescharacteristic
217C22×Dic3—32no—
218C22×Dic3—32no—
219C22×Dic3—62no—
220C22×Dic3C2×C3×D311yescharacteristic
221C22×Dic3C2×C3×D311yescharacteristic
222C22×Dic3C2×C3×D312yes—
223C2×Dic6C3×Dic318yesdirect product factor
224C2×Dic6C2×C3×D314yes—
225C3×C4⋊C4—98no—
226C3×C4⋊C4—94no—
227C3×C42—34no—
228C3×C42—34no—
229C3×C42—64no—
230C3×C42—98no—
231C4⋊Dic3—34no—
232C4⋊Dic3—34no—
233C4⋊Dic3—38no—
234C4⋊Dic3—64no—
235Dic3⋊C4—38no—
236C4×Dic3—38no—
237C4×Dic3—38no—
238C4×Dic3C2×C3×D314yes—
239C22×C32—21no—
240C22×C32—21no—
241C22×C32—21no—
242C22×C32—22no—
243C22×C32—22no—
244C22×C32—22no—
245C22×C32—22no—
246C22×C32—22no—
247C22×C32—22no—
248C22×C32—22no—
249C22×C32—22no—
250C22×C32—22no—
251C22×C32—42no—
252C22×C32—42no—
253C22×C32—42no—
254C22×C32—41no—
255C22×C32C2×Dic612yes—
256C22×C32C2×C4×D312yes—
257C22×C32#23512yes—
258C22×C32D3×Q812yes—
259C22×C32#23912yes—
260C22×C32C22×Dic312yes—
261C22×C32C22×C3×C412yes—
262C22×C32C2×C3×Q812yes—
263C22×C32C3×#4112yes—
264C22×C32C23×D311yescharacteristic
265C22×C32C23×D311yescharacteristic
266C22×C32C24×C311yescharacteristic
267C32×C4—24no—
268C32×C4—24no—
269C32×C4—24no—
270C32×C4—38no—
271C32×C4—34no—
272C32×C4—44no—
273C32×C4—62no—
274C32×C4C2×C4×D314yes—
275C32×C4C22×Dic314yes—
276C32×C4C22×C3×C414yes—
277C3⋊Dic3—24no—
278C3×Dic3—24no—
279C3×Dic3—22no—
280C3×Dic3—22no—
281C3×Dic3—24no—
282C3×Dic3—28no—
283C3×Dic3—64no—
284C3×Dic3—64no—
285C3×Dic3—124no—
286C3×Dic3C2×Dic614yesdirect product factor
287C3×Dic3C22×Dic318yes—
288C3×Dic3C22×C3×C418yes—
289C3×Dic3C2×C3×Q814yescomplemented
290C22×Q8—31no—
291C4×Q8—98no—
292C2×C4⋊C4—92no—
293C2×C4⋊C4—91no—
294C2×C42—31no—
295C2×C42—92no—
296C33C2×C4×Q811yescharacteristic, complemented, Sylow
297C23×C3—21no—
298C23×C3—21no—
299C23×C3—21no—
300C23×C3—41no—
301C23×C3C2×D3211yescharacteristic
302C23×C3C22×C3×D311yescharacteristic
303C23×C3C22×C3×D311yescharacteristic
304C3×Q8—316no—
305C3×Q8—316no—
306C3×Q8—616no—
307C2×C3×C4—24no—
308C2×C3×C4—24no—
309C2×C3×C4—24no—
310C2×C3×C4—22no—
311C2×C3×C4—22no—
312C2×C3×C4—22no—
313C2×C3×C4—32no—
314C2×C3×C4—34no—
315C2×C3×C4—32no—
316C2×C3×C4—34no—
317C2×C3×C4—34no—
318C2×C3×C4—38no—
319C2×C3×C4—38no—
320C2×C3×C4—34no—
321C2×C3×C4—32no—
322C2×C3×C4—32no—
323C2×C3×C4—44no—
324C2×C3×C4—42no—
325C2×C3×C4—62no—
326C2×C3×C4—62no—
327C2×C3×C4—64no—
328C2×C3×C4—64no—
329C2×C3×C4—68no—
330C2×C3×C4—64no—
331C2×C3×C4—62no—
332C2×C3×C4—62no—
333C2×C3×C4—94no—
334C2×C3×C4—184no—
335C2×C3×C4D3×Dic314yes—
336C2×C3×C4C3×C4×D314yes—
337C2×C3×C4C2×C3×Dic314yes—
338C2×C3×C4C2×D3212yes—
339C2×C3×C4C22×C3×D312yes—
340C2×C3×C4C22×C3×D312yes—
341C2×Dic3—22no—
342C2×Dic3—34no—
343C2×Dic3—34no—
344C2×Dic3—64no—
345C2×Dic3—64no—
346C2×Dic3—64no—
347C2×Dic3—124no—
348C2×Dic3C3×Dic614yesdirect product factor
349C2×Dic3C2×C3×Dic318yes—
350C2×Dic3C22×C3×D312yes—
351C2×Dic3C22×C3×D312yes—
352C2×Dic3C22×C3×D314yes—
353Dic6C2×C3×Dic3116yesdirect product factor
354C2×C32—21no—
355C2×C32—21no—
356C2×C32—21no—
357C2×C32—24no—
358C2×C32—21no—
359C2×C32—21no—
360C2×C32—21no—
361C2×C32—21no—
362C2×C32—21no—
363C2×C32—21no—
364C2×C32—24no—
365C2×C32—24no—
366C2×C32—44no—
367C2×C32—41no—
368C2×C32—41no—
369C2×C32—41no—
370C2×C32C4×Dic614yescomplemented
371C2×C32Q8×Dic314yescomplemented
372C2×C32C3×C4×Q814yescomplemented
373C2×C32C22×Dic611yescharacteristic
374C2×C32C22×C4×D311yescharacteristic
375C2×C32C2×#23511yescharacteristic
376C2×C32C2×D3×Q811yescharacteristic
377C2×C32C2×#23911yescharacteristic
378C2×C32C23×Dic311yescharacteristic
379C2×C32C23×C3×C411yescharacteristic
380C2×C32C22×C3×Q811yescharacteristic
381C2×C32C2×C3×#4111yescharacteristic
382C2×Q8—38no—
383C2×Q8—34no—
384C22×C4—31no—
385C22×C4—32no—
386C22×C4—31no—
387C22×C4—92no—
388C22×C4C3×D3211yescharacteristic
389C4⋊C4—98no—
390C4⋊C4—94no—
391C42—34no—
392C42—98no—
393C22×C3—21no—
394C22×C3—21no—
395C22×C3—21no—
396C22×C3—22no—
397C22×C3—22no—
398C22×C3—22no—
399C22×C3—22no—
400C22×C3—22no—
401C22×C3—22no—
402C22×C3—22no—
403C22×C3—22no—
404C22×C3—22no—
405C22×C3—41no—
406C22×C3—42no—
407C22×C3—42no—
408C22×C3—42no—
409C22×C3D3×Dic612yes—
410C22×C3#379912yes—
411C22×C3C2×D3×Dic312yes—
412C22×C3C2×C3×Dic612yes—
413C22×C3C2×C3×C4×D312yes—
414C22×C3C3×#23512yes—
415C22×C3C3×D3×Q812yes—
416C22×C3C3×#23912yes—
417C22×C3C22×C3×Dic312yes—
418C22×C3C22×D3211yescharacteristic
419C22×C3C23×C3×D311yescharacteristic
420C22×C3C23×C3×D311yescharacteristic
421C3×C4—24no—
422C3×C4—24no—
423C3×C4—24no—
424C3×C4—34no—
425C3×C4—38no—
426C3×C4—38no—
427C3×C4—34no—
428C3×C4—44no—
429C3×C4—68no—
430C3×C4—64no—
431C3×C4—64no—
432C3×C4—62no—
433C3×C4—62no—
434C3×C4—184no—
435C3×C4C2×D3×Dic314yes—
436C3×C4C2×C3×C4×D314yes—
437C3×C4C22×C3×Dic314yes—
438Dic3—22no—
439Dic3—64no—
440Dic3—64no—
441Dic3—124no—
442Dic3C2×C3×Dic614yesdirect product factor
443Dic3C22×C3×Dic318yes—
444C32—21no—
445C32—21no—
446C32—21no—
447C32—41no—
448C32C2×C4×Dic611yescharacteristic, complemented
449C32C2×Q8×Dic311yescharacteristic, complemented
450C32C2×C3×C4×Q811yescharacteristic, complemented
451C23C2×C3×D3211yescharacteristic
452Q8—316no—
453C2×C4—32no—
454C2×C4—34no—
455C2×C4—34no—
456C2×C4—38no—
457C2×C4—32no—
458C2×C4—62no—
459C2×C4—94no—
460C2×C4—184no—
461C2×C4C3×D3×Dic314yes—
462C2×C4C2×C3×D3212yes—
463C2×C3—21no—
464C2×C3—21no—
465C2×C3—21no—
466C2×C3—24no—
467C2×C3—24no—
468C2×C3—24no—
469C2×C3—21no—
470C2×C3—21no—
471C2×C3—21no—
472C2×C3—21no—
473C2×C3—21no—
474C2×C3—21no—
475C2×C3—41no—
476C2×C3—44no—
477C2×C3—41no—
478C2×C3—41no—
479C2×C3Dic3×Dic614yesdirect product factor
480C2×C3C3×C4×Dic614yescomplemented
481C2×C3C3×Q8×Dic314yescomplemented
482C2×C3C2×D3×Dic611yescharacteristic
483C2×C3C2×#379911yescharacteristic
484C2×C3C22×D3×Dic311yescharacteristic
485C2×C3C22×C3×Dic611yescharacteristic
486C2×C3C22×C3×C4×D311yescharacteristic
487C2×C3C2×C3×#23511yescharacteristic
488C2×C3C2×C3×D3×Q811yescharacteristic
489C2×C3C2×C3×#23911yescharacteristic
490C2×C3C23×C3×Dic311yescharacteristic
491C22C3×D3×Dic612yes—
492C22C3×#379912yes—
493C22C2×C3×D3×Dic312yes—
494C22C22×C3×D3211yescharacteristic
495C4—38no—
496C4—34no—
497C4—62no—
498C4—184no—
499C4C2×C3×D3×Dic314yes—
500C3—21no—
501C3—21no—
502C3—21no—
503C3—41no—
504C3C2×Dic3×Dic611yescharacteristic, direct product factor
505C3C2×C3×C4×Dic611yescharacteristic, complemented
506C3C2×C3×Q8×Dic311yescharacteristic, complemented
507C2C3×Dic3×Dic614yesdirect product factor
508C2C2×C3×D3×Dic611yescharacteristic
509C2C2×C3×#379911yescharacteristic
510C2C22×C3×D3×Dic311yescharacteristic
subgroupG/H
centerC23×C3C2×D32
commutatorC2×C32C23×C3×C4
FrattiniC22C22×C3×D32
FittingC22×C33×C4C22
socle, prime radicalC23×C33C23
cosocleC22×C32C24×C3
hypercenterC22×C3×C4D32
hypocenterC32C2×C3×C4×Q8
LCSG ▷ C2×C32 ▷ C32
UCSC1 ◁ C23×C3 ◁ C22×C3×C4

Complements

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

NH ≅ G/Ntype
6. C3×Dic3×Dic6C2direct (N × H)
9. C2×Dic3×Dic6C3direct (N × H)
15. C2×C32×Dic6C4semidirect
41. Dic3×Dic6C2×C3direct (N × H)
49. C2×C32×Dic3Q8semidirect
53. C32×Dic6C2×C4semidirect
84. C2×C3×Dic6Dic3direct (N × H)
85. C2×C3×Dic6C3×C4semidirect
110. C32×Dic3C2×Q8semidirect
176. C2×C3×Dic3Dic6direct (N × H)
179. C2×C3×Dic3C3×Q8semidirect
187. C3×Dic6C2×Dic3direct (N × H)
188. C3×Dic6C2×C3×C4semidirect
190. C2×C33C4×Q8semidirect
223. C2×Dic6C3×Dic3direct (N × H)
286. C3×Dic3C2×Dic6direct (N × H)
289. C3×Dic3C2×C3×Q8semidirect
296. C33C2×C4×Q8semidirect
348. C2×Dic3C3×Dic6direct (N × H)
353. Dic6C2×C3×Dic3direct (N × H)
370. C2×C32C4×Dic6semidirect
371. C2×C32Q8×Dic3semidirect
372. C2×C32C3×C4×Q8semidirect
442. Dic3C2×C3×Dic6direct (N × H)
448. C32C2×C4×Dic6semidirect
449. C32C2×Q8×Dic3semidirect
450. C32C2×C3×C4×Q8semidirect
479. C2×C3Dic3×Dic6direct (N × H)
480. C2×C3C3×C4×Dic6semidirect
481. C2×C3C3×Q8×Dic3semidirect
504. C3C2×Dic3×Dic6direct (N × H)
505. C3C2×C3×C4×Dic6semidirect
506. C3C2×C3×Q8×Dic3semidirect
507. C2C3×Dic3×Dic6direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C32×Dic62
C2×C3×C4⋊(C3⋊Dic3)2
C2×C32×C4×Dic32
C2×C3×Dic3⋊Dic32
C2×C3×Dic322
C3×Dic3×Dic62
C2×C3×Q8×Dic33
C2×C3×C4×Dic63
C2×Dic3×Dic63