Finite Group Databasebeta

← C2×C3×#63918 #13174635

Subgroups — C2×C3×#63918 #13174635

Order 1728.

1506 conjugacy classes of subgroups.

The subgroup lattice has 508 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×#627C212yesmaximal
2C2×C3×C4⋊(C3⋊Dic3)C211yesmaximal, characteristic
3C2×C32×C4⋊Dic3C211yesmaximal, characteristic
4C2×C32×C4×Dic3C211yesmaximal, characteristic
5C2×C3×Dic3⋊Dic3C212yesmaximal
6C3×#63918C218yesmaximal, direct product factor
7C2×C3×#888—31nomaximal
8C2×C3×#869—31nomaximal
9C2×#63918C311yesmaximal, characteristic, direct product factor
10C22×C33×C4C2211yescharacteristic
11C22×C3×C3⋊Dic3C2212yes—
12C22×C32×Dic3C2212yes—
13C22×C32×Dic3C2211yescharacteristic
14C22×C32×Dic3C2211yescharacteristic
15C2×C3×#627—28no—
16C2×C3×#627C2218yes—
17C3×C4⋊(C3⋊Dic3)C2214yes—
18C32×C4⋊Dic3C2214yes—
19C32×C4×Dic3C2214yes—
20C3×Dic3⋊Dic3C2218yes—
21C22×C3×Dic6—32no—
22C22×C3×Dic6—32no—
23C22×#627C2×C312yes—
24C2×C32×C4⋊C4—31no—
25C2×C32×C42—31no—
26C2×C4⋊(C3⋊Dic3)C2×C311yescharacteristic
27C2×C3×C4⋊Dic3—21no—
28C2×C3×C4⋊Dic3—31no—
29C2×C3×C4⋊Dic3—31no—
30C2×C3×C4⋊Dic3—32no—
31C2×C3×C4⋊Dic3—61no—
32C2×C3×C4⋊Dic3D311yescharacteristic
33C2×C3×C4⋊Dic3C2×C311yescharacteristic
34C2×C3×Dic3⋊C4—32no—
35C2×C3×C4×Dic3—21no—
36C2×C3×C4×Dic3D311yescharacteristic
37C2×C3×C4×Dic3C2×C311yescharacteristic
38C3×#888—38no—
39C3×#869—38no—
40C2×Dic3⋊Dic3C2×C312yes—
41#63918C2×C318yesdirect product factor
42C23×C33C2311yescharacteristic
43C2×C33×C4Q814yescomplemented
44C2×C33×C4C2312yes—
45C2×C3×C3⋊Dic3—24no—
46C2×C3×C3⋊Dic3C2314yes—
47C2×C32×Dic3—24no—
48C2×C32×Dic3D414yes—
49C2×C32×Dic3Q814yescomplemented
50C2×C32×Dic3C2314yes—
51C2×C32×Dic3C2312yes—
52C2×C32×Dic3C2312yes—
53C3×#627—216no—
54C2×C3×C4⋊Q8—91no—
55C2×#888—31no—
56C2×#869—31no—
57C22×C32×C4—21no—
58C22×C32×C4—21no—
59C22×C32×C4—21no—
60C22×C32×C4—31no—
61C22×C32×C4—31no—
62C22×C32×C4—32no—
63C22×C32×C4—41no—
64C22×C32×C4C2×D311yescharacteristic
65C22×C32×C4C2×D311yescharacteristic
66C22×C32×C4C22×C311yescharacteristic
67C22×C3⋊Dic3C22×C312yes—
68C22×C3×Dic3—21no—
69C22×C3×Dic3—21no—
70C22×C3×Dic3—22no—
71C22×C3×Dic3—32no—
72C22×C3×Dic3—32no—
73C22×C3×Dic3—62no—
74C22×C3×Dic3C2×D312yes—
75C22×C3×Dic3C2×D311yescharacteristic
76C22×C3×Dic3C2×D311yescharacteristic
77C22×C3×Dic3C22×C312yes—
78C22×C3×Dic3C22×C311yescharacteristic
79C22×C3×Dic3C22×C311yescharacteristic
80C2×C3×Dic6—38no—
81C2×C3×Dic6—38no—
82C2×C3×Dic6—68no—
83C2×C3×Dic6—68no—
84C2×#627—28no—
85C2×#627C22×C318yes—
86C32×C4⋊C4—34no—
87C32×C42—34no—
88C4⋊(C3⋊Dic3)C22×C314yes—
89C3×C4⋊Dic3—24no—
90C3×C4⋊Dic3—34no—
91C3×C4⋊Dic3—34no—
92C3×C4⋊Dic3—38no—
93C3×C4⋊Dic3—64no—
94C3×C4⋊Dic3C2×D314yes—
95C3×C4⋊Dic3C22×C314yes—
96C3×Dic3⋊C4—38no—
97C3×C4×Dic3—24no—
98C3×C4×Dic3C2×D314yes—
99C3×C4×Dic3C22×C314yes—
100Dic3⋊Dic3C22×C318yes—
101C22×C33C2×D412yes—
102C22×C33C2×Q812yes—
103C22×C33C2×Q812yes—
104C22×C33C2411yescharacteristic
105C33×C4C2×Q814yescomplemented
106C3×C3⋊Dic3—24no—
107C32×Dic3—24no—
108C32×Dic3C2×D414yes—
109C32×Dic3C2×Q814yescomplemented
110C22×C3×Q8—92no—
111C22×Dic6—32no—
112C22×Dic6—32no—
113C3×C4⋊Q8—98no—
114C2×C3×C4⋊C4—31no—
115C2×C3×C4⋊C4—31no—
116C2×C3×C4⋊C4—61no—
117C2×C3×C4⋊C4—92no—
118C2×C3×C4⋊C4—91no—
119C2×C3×C42—31no—
120C2×C3×C42—31no—
121C2×C3×C42—61no—
122C2×C4⋊Dic3—31no—
123C2×C4⋊Dic3—31no—
124C2×C4⋊Dic3—32no—
125C2×C4⋊Dic3—61no—
126C2×C4⋊Dic3C3×D311yescharacteristic
127C2×Dic3⋊C4—32no—
128C2×C4×Dic3C3×D311yescharacteristic
129#888—38no—
130#869—38no—
131C23×C32—21no—
132C23×C32—21no—
133C23×C32—21no—
134C23×C32—41no—
135C23×C32C22×D311yescharacteristic
136C23×C32C22×D311yescharacteristic
137C23×C32C23×C311yescharacteristic
138C2×C32×C4—24no—
139C2×C32×C4—24no—
140C2×C32×C4—24no—
141C2×C32×C4—22no—
142C2×C32×C4—22no—
143C2×C32×C4—22no—
144C2×C32×C4—34no—
145C2×C32×C4—34no—
146C2×C32×C4—32no—
147C2×C32×C4—32no—
148C2×C32×C4—34no—
149C2×C32×C4—44no—
150C2×C32×C4—42no—
151C2×C32×C4—64no—
152C2×C32×C4Dic614yescomplemented
153C2×C32×C4Dic614yescomplemented
154C2×C32×C4C3×Q814yescomplemented
155C2×C32×C4C22×D312yes—
156C2×C32×C4C22×D312yes—
157C2×C32×C4C23×C312yes—
158C2×C3⋊Dic3—24no—
159C2×C3⋊Dic3C23×C314yes—
160C2×C3×Dic3—24no—
161C2×C3×Dic3—24no—
162C2×C3×Dic3—24no—
163C2×C3×Dic3—24no—
164C2×C3×Dic3—22no—
165C2×C3×Dic3—22no—
166C2×C3×Dic3—24no—
167C2×C3×Dic3—34no—
168C2×C3×Dic3—34no—
169C2×C3×Dic3—44no—
170C2×C3×Dic3—64no—
171C2×C3×Dic3—64no—
172C2×C3×Dic3—64no—
173C2×C3×Dic3—124no—
174C2×C3×Dic3Dic614yescomplemented
175C2×C3×Dic3D1214yes—
176C2×C3×Dic3C3×D414yes—
177C2×C3×Dic3C3×Q814yescomplemented
178C2×C3×Dic3C22×D314yes—
179C2×C3×Dic3C22×D312yes—
180C2×C3×Dic3C22×D312yes—
181C2×C3×Dic3C23×C314yes—
182C2×C3×Dic3C23×C312yes—
183C2×C3×Dic3C23×C312yes—
184C3×Dic6—616no—
185C3×Dic6—616no—
186#627—216no—
187C2×C4⋊Q8—91noSylow
188C2×C33C4⋊Q814yescomplemented
189C2×C33C22×D411yescharacteristic
190C2×C33C22×Q811yescharacteristic
191C2×C33C22×Q811yescharacteristic
192C2×C3×Q8—98no—
193C2×C3×Q8—188no—
194C22×C3×C4—21no—
195C22×C3×C4—21no—
196C22×C3×C4—21no—
197C22×C3×C4—31no—
198C22×C3×C4—31no—
199C22×C3×C4—31no—
200C22×C3×C4—31no—
201C22×C3×C4—32no—
202C22×C3×C4—32no—
203C22×C3×C4—41no—
204C22×C3×C4—61no—
205C22×C3×C4—61no—
206C22×C3×C4—62no—
207C22×C3×C4—92no—
208C22×C3×C4D3211yescharacteristic
209C22×C3×C4C2×C3×D311yescharacteristic
210C22×C3×C4C2×C3×D311yescharacteristic
211C22×Dic3—32no—
212C22×Dic3—32no—
213C22×Dic3—62no—
214C22×Dic3C2×C3×D312yes—
215C22×Dic3C2×C3×D311yescharacteristic
216C22×Dic3C2×C3×D311yescharacteristic
217C2×Dic6—38no—
218C2×Dic6—38no—
219C2×Dic6—68no—
220C2×Dic6—68no—
221C3×C4⋊C4—34no—
222C3×C4⋊C4—34no—
223C3×C4⋊C4—64no—
224C3×C4⋊C4—98no—
225C3×C4⋊C4—94no—
226C3×C42—34no—
227C3×C42—34no—
228C3×C42—64no—
229C4⋊Dic3—34no—
230C4⋊Dic3—34no—
231C4⋊Dic3—38no—
232C4⋊Dic3—64no—
233C4⋊Dic3C2×C3×D314yes—
234Dic3⋊C4—38no—
235C4×Dic3C2×C3×D314yes—
236C22×C32—21no—
237C22×C32—21no—
238C22×C32—21no—
239C22×C32—22no—
240C22×C32—22no—
241C22×C32—22no—
242C22×C32—22no—
243C22×C32—22no—
244C22×C32—22no—
245C22×C32—22no—
246C22×C32—22no—
247C22×C32—22no—
248C22×C32—42no—
249C22×C32—42no—
250C22×C32—42no—
251C22×C32—41no—
252C22×C32C2×Dic612yes—
253C22×C32C2×Dic612yes—
254C22×C32C2×Dic612yes—
255C22×C32C2×D1212yes—
256C22×C32D3×D412yes—
257C22×C32D3×Q812yes—
258C22×C32C2×C3×D412yes—
259C22×C32C2×C3×Q812yes—
260C22×C32C2×C3×Q812yes—
261C22×C32C23×D311yescharacteristic
262C22×C32C23×D311yescharacteristic
263C22×C32C24×C311yescharacteristic
264C32×C4—24no—
265C32×C4—24no—
266C32×C4—24no—
267C32×C4—34no—
268C32×C4—34no—
269C32×C4—44no—
270C32×C4—64no—
271C32×C4C2×Dic614yescomplemented
272C32×C4C2×Dic614yescomplemented
273C32×C4C2×C3×Q814yescomplemented
274C3⋊Dic3—24no—
275C3×Dic3—24no—
276C3×Dic3—24no—
277C3×Dic3—24no—
278C3×Dic3—24no—
279C3×Dic3—44no—
280C3×Dic3—64no—
281C3×Dic3—64no—
282C3×Dic3—124no—
283C3×Dic3C2×Dic614yescomplemented
284C3×Dic3C2×D1214yes—
285C3×Dic3C2×C3×D414yes—
286C3×Dic3C2×C3×Q814yescomplemented
287C22×Q8—92no—
288C4⋊Q8—98no—
289C2×C4⋊C4—31no—
290C2×C4⋊C4—92no—
291C2×C4⋊C4—91no—
292C2×C42—31no—
293C33C2×C4⋊Q811yescharacteristic, complemented, Sylow
294C23×C3—21no—
295C23×C3—21no—
296C23×C3—21no—
297C23×C3—41no—
298C23×C3C2×D3211yescharacteristic
299C23×C3C22×C3×D311yescharacteristic
300C23×C3C22×C3×D311yescharacteristic
301C3×Q8—1816no—
302C2×C3×C4—24no—
303C2×C3×C4—24no—
304C2×C3×C4—24no—
305C2×C3×C4—22no—
306C2×C3×C4—22no—
307C2×C3×C4—22no—
308C2×C3×C4—32no—
309C2×C3×C4—32no—
310C2×C3×C4—32no—
311C2×C3×C4—32no—
312C2×C3×C4—34no—
313C2×C3×C4—34no—
314C2×C3×C4—34no—
315C2×C3×C4—34no—
316C2×C3×C4—34no—
317C2×C3×C4—34no—
318C2×C3×C4—44no—
319C2×C3×C4—42no—
320C2×C3×C4—62no—
321C2×C3×C4—62no—
322C2×C3×C4—64no—
323C2×C3×C4—64no—
324C2×C3×C4—64no—
325C2×C3×C4—64no—
326C2×C3×C4—64no—
327C2×C3×C4—94no—
328C2×C3×C4—124no—
329C2×C3×C4—184no—
330C2×C3×C4#62714yescomplemented
331C2×C3×C4C3×Dic614yescomplemented
332C2×C3×C4C3×Dic614yescomplemented
333C2×C3×C4C2×D3212yes—
334C2×C3×C4C22×C3×D312yes—
335C2×C3×C4C22×C3×D312yes—
336C2×Dic3—24no—
337C2×Dic3—34no—
338C2×Dic3—34no—
339C2×Dic3—64no—
340C2×Dic3—64no—
341C2×Dic3—64no—
342C2×Dic3—124no—
343C2×Dic3C3×Dic614yescomplemented
344C2×Dic3C3×D1214yes—
345C2×Dic3C22×C3×D314yes—
346C2×Dic3C22×C3×D312yes—
347C2×Dic3C22×C3×D312yes—
348Dic6—616no—
349Dic6—616no—
350C2×C32—24no—
351C2×C32—21no—
352C2×C32—21no—
353C2×C32—21no—
354C2×C32—21no—
355C2×C32—21no—
356C2×C32—21no—
357C2×C32—21no—
358C2×C32—21no—
359C2×C32—21no—
360C2×C32—24no—
361C2×C32—24no—
362C2×C32—44no—
363C2×C32—41no—
364C2×C32—41no—
365C2×C32—41no—
366C2×C32#86914yescomplemented
367C2×C32#88814yescomplemented
368C2×C32C3×C4⋊Q814yescomplemented
369C2×C32C22×Dic611yescharacteristic
370C2×C32C22×Dic611yescharacteristic
371C2×C32C22×Dic611yescharacteristic
372C2×C32C22×D1211yescharacteristic
373C2×C32C2×D3×D411yescharacteristic
374C2×C32C2×D3×Q811yescharacteristic
375C2×C32C22×C3×D411yescharacteristic
376C2×C32C22×C3×Q811yescharacteristic
377C2×C32C22×C3×Q811yescharacteristic
378C2×Q8—98no—
379C2×Q8—188no—
380C22×C4—31no—
381C22×C4—31no—
382C22×C4—32no—
383C22×C4—92no—
384C22×C4C3×D3211yescharacteristic
385C4⋊C4—34no—
386C4⋊C4—98no—
387C4⋊C4—94no—
388C42—34no—
389C22×C3—21no—
390C22×C3—21no—
391C22×C3—21no—
392C22×C3—22no—
393C22×C3—22no—
394C22×C3—22no—
395C22×C3—22no—
396C22×C3—22no—
397C22×C3—22no—
398C22×C3—22no—
399C22×C3—22no—
400C22×C3—22no—
401C22×C3—41no—
402C22×C3—42no—
403C22×C3—42no—
404C22×C3—42no—
405C22×C3D3×Dic612yes—
406C22×C3D3×D1212yes—
407C22×C3C2×#62712yes—
408C22×C3C2×C3×Dic612yes—
409C22×C3C2×C3×Dic612yes—
410C22×C3C2×C3×Dic612yes—
411C22×C3C2×C3×D1212yes—
412C22×C3C3×D3×D412yes—
413C22×C3C3×D3×Q812yes—
414C22×C3C22×D3211yescharacteristic
415C22×C3C23×C3×D311yescharacteristic
416C22×C3C23×C3×D311yescharacteristic
417C3×C4—24no—
418C3×C4—24no—
419C3×C4—24no—
420C3×C4—34no—
421C3×C4—34no—
422C3×C4—34no—
423C3×C4—34no—
424C3×C4—44no—
425C3×C4—64no—
426C3×C4—64no—
427C3×C4—64no—
428C3×C4—64no—
429C3×C4—124no—
430C3×C4—184no—
431C3×C4C2×#62714yescomplemented
432C3×C4C2×C3×Dic614yescomplemented
433C3×C4C2×C3×Dic614yescomplemented
434Dic3—24no—
435Dic3—64no—
436Dic3—64no—
437Dic3—124no—
438Dic3C2×C3×Dic614yescomplemented
439Dic3C2×C3×D1214yes—
440C32—21no—
441C32—21no—
442C32—21no—
443C32—41no—
444C32C2×#86911yescharacteristic, complemented
445C32C2×#88811yescharacteristic, complemented
446C32C2×C3×C4⋊Q811yescharacteristic, complemented
447C23C2×C3×D3211yescharacteristic
448Q8—1816no—
449C2×C4—32no—
450C2×C4—32no—
451C2×C4—34no—
452C2×C4—34no—
453C2×C4—34no—
454C2×C4—64no—
455C2×C4—94no—
456C2×C4—184no—
457C2×C4C3×#62714yescomplemented
458C2×C4C2×C3×D3212yes—
459C2×C3—21no—
460C2×C3—21no—
461C2×C3—21no—
462C2×C3—21no—
463C2×C3—21no—
464C2×C3—21no—
465C2×C3—24no—
466C2×C3—24no—
467C2×C3—24no—
468C2×C3—21no—
469C2×C3—21no—
470C2×C3—21no—
471C2×C3—41no—
472C2×C3—41no—
473C2×C3—44no—
474C2×C3—41no—
475C2×C3#6391814yesdirect product factor
476C2×C3C3×#86914yescomplemented
477C2×C3C3×#88814yescomplemented
478C2×C3C2×D3×Dic611yescharacteristic
479C2×C3C2×D3×D1211yescharacteristic
480C2×C3C22×#62711yescharacteristic
481C2×C3C22×C3×Dic611yescharacteristic
482C2×C3C22×C3×Dic611yescharacteristic
483C2×C3C22×C3×Dic611yescharacteristic
484C2×C3C22×C3×D1211yescharacteristic
485C2×C3C2×C3×D3×D411yescharacteristic
486C2×C3C2×C3×D3×Q811yescharacteristic
487C22C3×D3×Dic612yes—
488C22C3×D3×D1212yes—
489C22C2×C3×#62712yes—
490C22C22×C3×D3211yescharacteristic
491C4—34no—
492C4—34no—
493C4—64no—
494C4—184no—
495C4C2×C3×#62714yescomplemented
496C3—21no—
497C3—21no—
498C3—21no—
499C3—41no—
500C3C2×#6391811yescharacteristic, direct product factor
501C3C2×C3×#86911yescharacteristic, complemented
502C3C2×C3×#88811yescharacteristic, complemented
503C2C3×#6391814yesdirect product factor
504C2C2×C3×D3×Dic611yescharacteristic
505C2C2×C3×D3×D1211yescharacteristic
506C2C22×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×C4⋊Q8
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×#63918C2direct (N × H)
9. C2×#63918C3direct (N × H)
41. #63918C2×C3direct (N × H)
43. C2×C33×C4Q8semidirect
49. C2×C32×Dic3Q8semidirect
105. C33×C4C2×Q8semidirect
109. C32×Dic3C2×Q8semidirect
152. C2×C32×C4Dic6semidirect
153. C2×C32×C4Dic6semidirect
154. C2×C32×C4C3×Q8semidirect
174. C2×C3×Dic3Dic6semidirect
177. C2×C3×Dic3C3×Q8semidirect
188. C2×C33C4⋊Q8semidirect
271. C32×C4C2×Dic6semidirect
272. C32×C4C2×Dic6semidirect
273. C32×C4C2×C3×Q8semidirect
283. C3×Dic3C2×Dic6semidirect
286. C3×Dic3C2×C3×Q8semidirect
293. C33C2×C4⋊Q8semidirect
330. C2×C3×C4#627semidirect
331. C2×C3×C4C3×Dic6semidirect
332. C2×C3×C4C3×Dic6semidirect
343. C2×Dic3C3×Dic6semidirect
366. C2×C32#869semidirect
367. C2×C32#888semidirect
368. C2×C32C3×C4⋊Q8semidirect
431. C3×C4C2×#627semidirect
432. C3×C4C2×C3×Dic6semidirect
433. C3×C4C2×C3×Dic6semidirect
438. Dic3C2×C3×Dic6semidirect
444. C32C2×#869semidirect
445. C32C2×#888semidirect
446. C32C2×C3×C4⋊Q8semidirect
457. C2×C4C3×#627semidirect
475. C2×C3#63918direct (N × H)
476. C2×C3C3×#869semidirect
477. C2×C3C3×#888semidirect
495. C4C2×C3×#627semidirect
500. C3C2×#63918direct (N × H)
501. C3C2×C3×#869semidirect
502. C3C2×C3×#888semidirect
503. C2C3×#63918direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×#6272
C2×C3×C4⋊(C3⋊Dic3)2
C2×C32×C4⋊Dic32
C2×C32×C4×Dic32
C2×C3×Dic3⋊Dic32
C3×#639182
C2×C3×#8883
C2×C3×#8693
C2×#639183