Finite Group Databasebeta

← C2×C3×#63869 #13174400

Subgroups — C2×C3×#63869 #13174400

Order 1728.

1250 conjugacy classes of subgroups.

The subgroup lattice has 492 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×#3724C212yesmaximal
5C2×C3×Dic3⋊Dic3C212yesmaximal
6C3×#63869C218yesmaximal, direct product factor
7C2×C3×#890—31nomaximal
8C2×C3×#870—31nomaximal
9C2×#63869C311yesmaximal, characteristic, direct product factor
10C22×C33×C4C2211yescharacteristic
11C22×C3×C3⋊Dic3C2212yes—
12C22×C32×Dic3C2211yescharacteristic
13C22×C32×Dic3C2211yescharacteristic
14C22×C32×Dic3C2212yes—
15C3×C4⋊(C3⋊Dic3)C2214yes—
16C32×C4⋊Dic3C2214yes—
17C32×C4×Dic3C2214yes—
18C3×#3724C2218yes—
19C3×Dic3⋊Dic3C2218yes—
20C2×C32×C4⋊C4—31no—
21C2×C32×C42—31no—
22C2×C4⋊(C3⋊Dic3)C2×C311yescharacteristic
23C2×C3×C4⋊Dic3—21no—
24C2×C3×C4⋊Dic3—31no—
25C2×C3×C4⋊Dic3—31no—
26C2×C3×C4⋊Dic3—32no—
27C2×C3×C4⋊Dic3—61no—
28C2×C3×C4⋊Dic3D311yescharacteristic
29C2×C3×C4⋊Dic3C2×C311yescharacteristic
30C2×C3×Dic3⋊C4—32no—
31C2×C3×Dic3⋊C4—32no—
32C2×C3×Dic3⋊C4—32no—
33C2×C3×C4×Dic3—21no—
34C2×C3×C4×Dic3D311yescharacteristic
35C2×C3×C4×Dic3C2×C311yescharacteristic
36C3×#890—38no—
37C3×#870—38no—
38C2×#3724C2×C312yes—
39C2×Dic3⋊Dic3C2×C312yes—
40#63869C2×C318yesdirect product factor
41C23×C33C2311yescharacteristic
42C2×C33×C4Q814yes—
43C2×C33×C4C2312yes—
44C2×C3×C3⋊Dic3—24no—
45C2×C3×C3⋊Dic3C2314yes—
46C2×C32×Dic3—22no—
47C2×C32×Dic3—22no—
48C2×C32×Dic3—24no—
49C2×C32×Dic3C2312yes—
50C2×C32×Dic3C2314yes—
51C2×C32×Dic3C2312yes—
52C2×C3×#125—91no—
53C2×#890—31no—
54C2×#870—31no—
55C22×C32×C4—21no—
56C22×C32×C4—21no—
57C22×C32×C4—21no—
58C22×C32×C4—31no—
59C22×C32×C4—31no—
60C22×C32×C4—32no—
61C22×C32×C4—41no—
62C22×C32×C4C2×D311yescharacteristic
63C22×C32×C4C2×D311yescharacteristic
64C22×C32×C4C22×C311yescharacteristic
65C22×C3⋊Dic3C22×C312yes—
66C22×C3×Dic3—21no—
67C22×C3×Dic3—21no—
68C22×C3×Dic3—22no—
69C22×C3×Dic3—32no—
70C22×C3×Dic3—32no—
71C22×C3×Dic3—62no—
72C22×C3×Dic3C2×D311yescharacteristic
73C22×C3×Dic3C2×D311yescharacteristic
74C22×C3×Dic3C2×D312yes—
75C22×C3×Dic3C22×C311yescharacteristic
76C22×C3×Dic3C22×C311yescharacteristic
77C22×C3×Dic3C22×C312yes—
78C32×C4⋊C4—34no—
79C32×C42—34no—
80C4⋊(C3⋊Dic3)C22×C314yes—
81C3×C4⋊Dic3—24no—
82C3×C4⋊Dic3—34no—
83C3×C4⋊Dic3—34no—
84C3×C4⋊Dic3—38no—
85C3×C4⋊Dic3—64no—
86C3×C4⋊Dic3C2×D314yes—
87C3×C4⋊Dic3C22×C314yes—
88C3×Dic3⋊C4—38no—
89C3×Dic3⋊C4—38no—
90C3×Dic3⋊C4—38no—
91C3×C4×Dic3—24no—
92C3×C4×Dic3C2×D314yes—
93C3×C4×Dic3C22×C314yes—
94#3724C22×C318yes—
95Dic3⋊Dic3C22×C318yes—
96C22×C33C2×Q812yes—
97C22×C33#4112yes—
98C22×C33#4112yes—
99C22×C33C2411yescharacteristic
100C33×C4C2×Q814yes—
101C3×C3⋊Dic3—24no—
102C32×Dic3—22no—
103C32×Dic3—22no—
104C32×Dic3—24no—
105C3×#125—98no—
106C2×C3×C4⋊C4—31no—
107C2×C3×C4⋊C4—31no—
108C2×C3×C4⋊C4—61no—
109C2×C3×C4⋊C4—92no—
110C2×C3×C4⋊C4—92no—
111C2×C3×C4⋊C4—91no—
112C2×C3×C42—31no—
113C2×C3×C42—31no—
114C2×C3×C42—61no—
115C2×C4⋊Dic3—31no—
116C2×C4⋊Dic3—31no—
117C2×C4⋊Dic3—32no—
118C2×C4⋊Dic3—61no—
119C2×C4⋊Dic3C3×D311yescharacteristic
120C2×Dic3⋊C4—32no—
121C2×Dic3⋊C4—32no—
122C2×Dic3⋊C4—32no—
123C2×C4×Dic3C3×D311yescharacteristic
124#890—38no—
125#870—38no—
126C23×C32—21no—
127C23×C32—21no—
128C23×C32—21no—
129C23×C32—41no—
130C23×C32C22×D311yescharacteristic
131C23×C32C22×D311yescharacteristic
132C23×C32C23×C311yescharacteristic
133C2×C32×C4—24no—
134C2×C32×C4—24no—
135C2×C32×C4—24no—
136C2×C32×C4—22no—
137C2×C32×C4—22no—
138C2×C32×C4—22no—
139C2×C32×C4—32no—
140C2×C32×C4—32no—
141C2×C32×C4—34no—
142C2×C32×C4—44no—
143C2×C32×C4—42no—
144C2×C32×C4—62no—
145C2×C32×C4—62no—
146C2×C32×C4—64no—
147C2×C32×C4Dic614yes—
148C2×C32×C4Dic614yes—
149C2×C32×C4C3×Q814yes—
150C2×C32×C4C22×D312yes—
151C2×C32×C4C22×D312yes—
152C2×C32×C4C23×C312yes—
153C2×C3⋊Dic3—24no—
154C2×C3⋊Dic3C23×C314yes—
155C2×C3×Dic3—24no—
156C2×C3×Dic3—24no—
157C2×C3×Dic3—22no—
158C2×C3×Dic3—22no—
159C2×C3×Dic3—24no—
160C2×C3×Dic3—22no—
161C2×C3×Dic3—22no—
162C2×C3×Dic3—24no—
163C2×C3×Dic3—22no—
164C2×C3×Dic3—22no—
165C2×C3×Dic3—24no—
166C2×C3×Dic3—34no—
167C2×C3×Dic3—34no—
168C2×C3×Dic3—44no—
169C2×C3×Dic3—64no—
170C2×C3×Dic3—64no—
171C2×C3×Dic3—64no—
172C2×C3×Dic3—124no—
173C2×C3×Dic3C22×D312yes—
174C2×C3×Dic3C22×D312yes—
175C2×C3×Dic3C22×D314yes—
176C2×C3×Dic3C23×C312yes—
177C2×C3×Dic3C23×C314yes—
178C2×C3×Dic3C23×C312yes—
179C2×#125—91noSylow
180C2×C33#12514yescomplemented
181C2×C33C22×Q811yescharacteristic
182C2×C33C2×#4111yescharacteristic
183C2×C33C2×#4111yescharacteristic
184C22×C3×C4—21no—
185C22×C3×C4—21no—
186C22×C3×C4—21no—
187C22×C3×C4—31no—
188C22×C3×C4—31no—
189C22×C3×C4—32no—
190C22×C3×C4—31no—
191C22×C3×C4—31no—
192C22×C3×C4—32no—
193C22×C3×C4—41no—
194C22×C3×C4—61no—
195C22×C3×C4—61no—
196C22×C3×C4—62no—
197C22×C3×C4—92no—
198C22×C3×C4D3211yescharacteristic
199C22×C3×C4C2×C3×D311yescharacteristic
200C22×C3×C4C2×C3×D311yescharacteristic
201C22×Dic3—32no—
202C22×Dic3—32no—
203C22×Dic3—62no—
204C22×Dic3C2×C3×D311yescharacteristic
205C22×Dic3C2×C3×D311yescharacteristic
206C22×Dic3C2×C3×D312yes—
207C3×C4⋊C4—34no—
208C3×C4⋊C4—34no—
209C3×C4⋊C4—64no—
210C3×C4⋊C4—98no—
211C3×C4⋊C4—98no—
212C3×C4⋊C4—94no—
213C3×C42—34no—
214C3×C42—34no—
215C3×C42—64no—
216C4⋊Dic3—34no—
217C4⋊Dic3—34no—
218C4⋊Dic3—38no—
219C4⋊Dic3—64no—
220C4⋊Dic3C2×C3×D314yes—
221Dic3⋊C4—38no—
222Dic3⋊C4—38no—
223Dic3⋊C4—38no—
224C4×Dic3C2×C3×D314yes—
225C22×C32—21no—
226C22×C32—21no—
227C22×C32—21no—
228C22×C32—22no—
229C22×C32—22no—
230C22×C32—22no—
231C22×C32—22no—
232C22×C32—22no—
233C22×C32—22no—
234C22×C32—22no—
235C22×C32—22no—
236C22×C32—22no—
237C22×C32—42no—
238C22×C32—42no—
239C22×C32—42no—
240C22×C32—41no—
241C22×C32C2×Dic612yes—
242C22×C32C2×Dic612yes—
243C22×C32#23512yes—
244C22×C32#23512yes—
245C22×C32#23712yes—
246C22×C32#23912yes—
247C22×C32C2×C3×Q812yes—
248C22×C32C3×#4112yes—
249C22×C32C3×#4112yes—
250C22×C32C23×D311yescharacteristic
251C22×C32C23×D311yescharacteristic
252C22×C32C24×C311yescharacteristic
253C32×C4—24no—
254C32×C4—24no—
255C32×C4—24no—
256C32×C4—44no—
257C32×C4—62no—
258C32×C4—62no—
259C32×C4—64no—
260C32×C4C2×Dic614yes—
261C32×C4C2×Dic614yes—
262C32×C4C2×C3×Q814yes—
263C3⋊Dic3—24no—
264C3×Dic3—24no—
265C3×Dic3—24no—
266C3×Dic3—22no—
267C3×Dic3—22no—
268C3×Dic3—24no—
269C3×Dic3—22no—
270C3×Dic3—22no—
271C3×Dic3—24no—
272C3×Dic3—44no—
273C3×Dic3—64no—
274C3×Dic3—64no—
275C3×Dic3—124no—
276#125—98no—
277C2×C4⋊C4—31no—
278C2×C4⋊C4—92no—
279C2×C4⋊C4—92no—
280C2×C4⋊C4—91no—
281C2×C42—31no—
282C33C2×#12511yescharacteristic, complemented, Sylow
283C23×C3—21no—
284C23×C3—21no—
285C23×C3—21no—
286C23×C3—41no—
287C23×C3C2×D3211yescharacteristic
288C23×C3C22×C3×D311yescharacteristic
289C23×C3C22×C3×D311yescharacteristic
290C2×C3×C4—24no—
291C2×C3×C4—24no—
292C2×C3×C4—24no—
293C2×C3×C4—22no—
294C2×C3×C4—22no—
295C2×C3×C4—22no—
296C2×C3×C4—32no—
297C2×C3×C4—32no—
298C2×C3×C4—34no—
299C2×C3×C4—32no—
300C2×C3×C4—32no—
301C2×C3×C4—34no—
302C2×C3×C4—44no—
303C2×C3×C4—42no—
304C2×C3×C4—62no—
305C2×C3×C4—62no—
306C2×C3×C4—64no—
307C2×C3×C4—64no—
308C2×C3×C4—64no—
309C2×C3×C4—62no—
310C2×C3×C4—64no—
311C2×C3×C4—62no—
312C2×C3×C4—64no—
313C2×C3×C4—62no—
314C2×C3×C4—62no—
315C2×C3×C4—94no—
316C2×C3×C4—124no—
317C2×C3×C4—184no—
318C2×C3×C4#62714yes—
319C2×C3×C4C3×Dic614yes—
320C2×C3×C4C3×Dic614yes—
321C2×C3×C4C2×D3212yes—
322C2×C3×C4C22×C3×D312yes—
323C2×C3×C4C22×C3×D312yes—
324C2×Dic3—22no—
325C2×Dic3—22no—
326C2×Dic3—24no—
327C2×Dic3—34no—
328C2×Dic3—34no—
329C2×Dic3—64no—
330C2×Dic3—64no—
331C2×Dic3—64no—
332C2×Dic3—124no—
333C2×Dic3C22×C3×D312yes—
334C2×Dic3C22×C3×D312yes—
335C2×Dic3C22×C3×D314yes—
336C2×C32—24no—
337C2×C32—21no—
338C2×C32—21no—
339C2×C32—21no—
340C2×C32—21no—
341C2×C32—21no—
342C2×C32—21no—
343C2×C32—21no—
344C2×C32—21no—
345C2×C32—21no—
346C2×C32—24no—
347C2×C32—24no—
348C2×C32—44no—
349C2×C32—41no—
350C2×C32—41no—
351C2×C32—41no—
352C2×C32#87014yescomplemented
353C2×C32#89014yescomplemented
354C2×C32C3×#12514yescomplemented
355C2×C32C22×Dic611yescharacteristic
356C2×C32C22×Dic611yescharacteristic
357C2×C32C2×#23511yescharacteristic
358C2×C32C2×#23511yescharacteristic
359C2×C32C2×#23711yescharacteristic
360C2×C32C2×#23911yescharacteristic
361C2×C32C22×C3×Q811yescharacteristic
362C2×C32C2×C3×#4111yescharacteristic
363C2×C32C2×C3×#4111yescharacteristic
364C22×C4—31no—
365C22×C4—31no—
366C22×C4—32no—
367C22×C4—92no—
368C22×C4C3×D3211yescharacteristic
369C4⋊C4—34no—
370C4⋊C4—98no—
371C4⋊C4—98no—
372C4⋊C4—94no—
373C42—34no—
374C22×C3—21no—
375C22×C3—21no—
376C22×C3—21no—
377C22×C3—22no—
378C22×C3—22no—
379C22×C3—22no—
380C22×C3—22no—
381C22×C3—22no—
382C22×C3—22no—
383C22×C3—22no—
384C22×C3—22no—
385C22×C3—22no—
386C22×C3—41no—
387C22×C3—42no—
388C22×C3—42no—
389C22×C3—42no—
390C22×C3#379512yes—
391C22×C3#379912yes—
392C22×C3C2×#62712yes—
393C22×C3C2×C3×Dic612yes—
394C22×C3C2×C3×Dic612yes—
395C22×C3C3×#23512yes—
396C22×C3C3×#23512yes—
397C22×C3C3×#23712yes—
398C22×C3C3×#23912yes—
399C22×C3C22×D3211yescharacteristic
400C22×C3C23×C3×D311yescharacteristic
401C22×C3C23×C3×D311yescharacteristic
402C3×C4—24no—
403C3×C4—24no—
404C3×C4—24no—
405C3×C4—44no—
406C3×C4—64no—
407C3×C4—64no—
408C3×C4—62no—
409C3×C4—64no—
410C3×C4—62no—
411C3×C4—64no—
412C3×C4—62no—
413C3×C4—62no—
414C3×C4—124no—
415C3×C4—184no—
416C3×C4C2×#62714yes—
417C3×C4C2×C3×Dic614yes—
418C3×C4C2×C3×Dic614yes—
419Dic3—22no—
420Dic3—22no—
421Dic3—24no—
422Dic3—64no—
423Dic3—64no—
424Dic3—124no—
425C32—21no—
426C32—21no—
427C32—21no—
428C32—41no—
429C32C2×#87011yescharacteristic, complemented
430C32C2×#89011yescharacteristic, complemented
431C32C2×C3×#12511yescharacteristic, complemented
432C23C2×C3×D3211yescharacteristic
433C2×C4—32no—
434C2×C4—32no—
435C2×C4—34no—
436C2×C4—62no—
437C2×C4—64no—
438C2×C4—62no—
439C2×C4—94no—
440C2×C4—184no—
441C2×C4C3×#62714yes—
442C2×C4C2×C3×D3212yes—
443C2×C3—21no—
444C2×C3—21no—
445C2×C3—21no—
446C2×C3—24no—
447C2×C3—24no—
448C2×C3—24no—
449C2×C3—21no—
450C2×C3—21no—
451C2×C3—21no—
452C2×C3—21no—
453C2×C3—21no—
454C2×C3—21no—
455C2×C3—41no—
456C2×C3—44no—
457C2×C3—41no—
458C2×C3—41no—
459C2×C3#6386914yesdirect product factor
460C2×C3C3×#87014yescomplemented
461C2×C3C3×#89014yescomplemented
462C2×C3C2×#379511yescharacteristic
463C2×C3C2×#379911yescharacteristic
464C2×C3C22×#62711yescharacteristic
465C2×C3C22×C3×Dic611yescharacteristic
466C2×C3C22×C3×Dic611yescharacteristic
467C2×C3C2×C3×#23511yescharacteristic
468C2×C3C2×C3×#23511yescharacteristic
469C2×C3C2×C3×#23711yescharacteristic
470C2×C3C2×C3×#23911yescharacteristic
471C22C3×#379512yes—
472C22C3×#379912yes—
473C22C2×C3×#62712yes—
474C22C22×C3×D3211yescharacteristic
475C4—62no—
476C4—64no—
477C4—62no—
478C4—184no—
479C4C2×C3×#62714yes—
480C3—21no—
481C3—21no—
482C3—21no—
483C3—41no—
484C3C2×#6386911yescharacteristic, direct product factor
485C3C2×C3×#87011yescharacteristic, complemented
486C3C2×C3×#89011yescharacteristic, complemented
487C2C3×#6386914yesdirect product factor
488C2C2×C3×#379511yescharacteristic
489C2C2×C3×#379911yescharacteristic
490C2C22×C3×#62711yescharacteristic
subgroupG/H
centerC23×C3C2×D32
commutator, cosocleC22×C32C24×C3
FrattiniC22C22×C3×D32
FittingC22×C33×C4C22
socle, prime radicalC23×C33C23
hypercenterC22×C3×C4D32
hypocenterC32C2×C3×#125
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C23×C3 ◁ C22×C3×C4

Complements

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

NH ≅ G/Ntype
6. C3×#63869C2direct (N × H)
9. C2×#63869C3direct (N × H)
40. #63869C2×C3direct (N × H)
180. C2×C33#125semidirect
282. C33C2×#125semidirect
352. C2×C32#870semidirect
353. C2×C32#890semidirect
354. C2×C32C3×#125semidirect
429. C32C2×#870semidirect
430. C32C2×#890semidirect
431. C32C2×C3×#125semidirect
459. C2×C3#63869direct (N × H)
460. C2×C3C3×#870semidirect
461. C2×C3C3×#890semidirect
484. C3C2×#63869direct (N × H)
485. C3C2×C3×#870semidirect
486. C3C2×C3×#890semidirect
487. C2C3×#63869direct (N × H)

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×#37242
C2×C3×Dic3⋊Dic32
C3×#638692
C2×C3×#8903
C2×C3×#8703
C2×#638693