Finite Group Databasebeta

← C2×C3×#63844 #13174373

Subgroups — C2×C3×#63844 #13174373

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×C3×#634C211yesmaximal, characteristic
2C2×C32×C4⋊Dic3C211yesmaximal, characteristic
3C2×C32×C4×Dic3C211yesmaximal, characteristic
4C2×C3×#3724C212yesmaximal
5C2×C3×Dic32C212yesmaximal
6C3×#63844C218yesmaximal, direct product factor
7C2×C3×#887—31nomaximal
8C2×C3×C4×Dic6—31nomaximal
9C2×#63844C311yesmaximal, characteristic, direct product factor
10C22×C33×C4C2211yescharacteristic
11C22×C3×C3⋊Dic3C2212yes—
12C2×C3×#634C418yescomplemented
13C2×C3×#634C2214yes—
14C22×C32×Dic3C2212yes—
15C22×C32×Dic3C2211yescharacteristic
16C22×C32×Dic3C2211yescharacteristic
17C32×C4⋊Dic3C2214yes—
18C32×C4×Dic3C2214yes—
19C3×#3724C2218yes—
20C3×Dic32C2218yes—
21C22×#634C2×C311yescharacteristic
22C22×C3×Dic6—31no—
23C22×C3×Dic6—31no—
24C22×C3×Dic6—61no—
25C2×C32×C4⋊C4—31no—
26C2×C32×C42—31no—
27C2×C3×C4⋊Dic3—21no—
28C2×C3×C4⋊Dic3D311yescharacteristic
29C2×C3×C4⋊Dic3C2×C311yescharacteristic
30C2×C3×Dic3⋊C4—32no—
31C2×C3×Dic3⋊C4—32no—
32C2×C3×C4×Dic3—21no—
33C2×C3×C4×Dic3—32no—
34C2×C3×C4×Dic3—32no—
35C2×C3×C4×Dic3D311yescharacteristic
36C2×C3×C4×Dic3C2×C311yescharacteristic
37C3×#887—38no—
38C3×C4×Dic6—38no—
39C2×#3724C2×C312yes—
40C2×Dic32C2×C312yes—
41#63844C2×C318yesdirect product factor
42C23×C33C2311yescharacteristic
43C2×C33×C4C2×C414yes—
44C2×C33×C4C2312yes—
45C2×C3×C3⋊Dic3C2×C418yes—
46C2×C3×C3⋊Dic3C2314yes—
47C3×#634C2×C4116yescomplemented
48C2×C32×Dic3—22no—
49C2×C32×Dic3—24no—
50C2×C32×Dic3Q814yescomplemented
51C2×C32×Dic3C2314yes—
52C2×C32×Dic3C2312yes—
53C2×C32×Dic3C2312yes—
54C2×C3×C4×Q8—91no—
55C2×#887—31no—
56C2×C4×Dic6—31no—
57C22×C32×C4—21no—
58C22×C32×C4—21no—
59C22×C32×C4—21no—
60C22×C32×C4—31no—
61C22×C32×C4—32no—
62C22×C32×C4—31no—
63C22×C32×C4—41no—
64C22×C32×C4C2×D311yescharacteristic
65C22×C32×C4C2×D311yescharacteristic
66C22×C32×C4C22×C311yescharacteristic
67C22×C3⋊Dic3C22×C312yes—
68C2×#634C3×C418yescomplemented
69C2×#634C22×C314yes—
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×D312yes—
77C22×C3×Dic3C2×D311yescharacteristic
78C22×C3×Dic3C2×D311yescharacteristic
79C22×C3×Dic3C22×C311yescharacteristic
80C22×C3×Dic3C22×C312yes—
81C22×C3×Dic3C22×C311yescharacteristic
82C2×C3×Dic6—38no—
83C2×C3×Dic6—38no—
84C2×C3×Dic6—34no—
85C2×C3×Dic6—34no—
86C2×C3×Dic6—68no—
87C2×C3×Dic6—64no—
88C32×C4⋊C4—34no—
89C32×C42—34no—
90C3×C4⋊Dic3—24no—
91C3×C4⋊Dic3C2×D314yes—
92C3×C4⋊Dic3C22×C314yes—
93C3×Dic3⋊C4—38no—
94C3×Dic3⋊C4—38no—
95C3×C4×Dic3—24no—
96C3×C4×Dic3—38no—
97C3×C4×Dic3—38no—
98C3×C4×Dic3C2×D314yes—
99C3×C4×Dic3C22×C314yes—
100#3724C22×C318yes—
101Dic32C22×C318yes—
102C22×C33C22×C412yes—
103C22×C33C2×Q812yes—
104C22×C33#4112yes—
105C22×C33C2411yescharacteristic
106C33×C4C22×C414yes—
107C3×C3⋊Dic3C22×C418yes—
108C32×Dic3—22no—
109C32×Dic3—24no—
110C32×Dic3C2×Q814yescomplemented
111C22×C3×Q8—91no—
112C22×Dic6—31no—
113C22×Dic6—31no—
114C22×Dic6—61no—
115C3×C4×Q8—98no—
116C2×C3×C4⋊C4—31no—
117C2×C3×C4⋊C4—31no—
118C2×C3×C4⋊C4—61no—
119C2×C3×C4⋊C4—92no—
120C2×C3×C42—31no—
121C2×C3×C42—31no—
122C2×C3×C42—61no—
123C2×C3×C42—92no—
124C2×C4⋊Dic3C3×D311yescharacteristic
125C2×Dic3⋊C4—32no—
126C2×Dic3⋊C4—32no—
127C2×C4×Dic3—32no—
128C2×C4×Dic3—32no—
129C2×C4×Dic3C3×D311yescharacteristic
130#887—38no—
131C4×Dic6—38no—
132C23×C32—21no—
133C23×C32—21no—
134C23×C32—21no—
135C23×C32—41no—
136C23×C32C22×D311yescharacteristic
137C23×C32C22×D311yescharacteristic
138C23×C32C23×C311yescharacteristic
139C2×C32×C4—24no—
140C2×C32×C4—24no—
141C2×C32×C4—24no—
142C2×C32×C4—22no—
143C2×C32×C4—22no—
144C2×C32×C4—22no—
145C2×C32×C4—34no—
146C2×C32×C4—32no—
147C2×C32×C4—34no—
148C2×C32×C4—32no—
149C2×C32×C4—44no—
150C2×C32×C4—42no—
151C2×C32×C4—62no—
152C2×C32×C4—64no—
153C2×C32×C4C4×D314yes—
154C2×C32×C4C4×D314yes—
155C2×C32×C4C2×C3×C414yes—
156C2×C32×C4C22×D312yes—
157C2×C32×C4C22×D312yes—
158C2×C32×C4C23×C312yes—
159C2×C3⋊Dic3C2×C3×C418yes—
160C2×C3⋊Dic3C23×C314yes—
161#634C2×C3×C4116yescomplemented
162C2×C3×Dic3—24no—
163C2×C3×Dic3—22no—
164C2×C3×Dic3—24no—
165C2×C3×Dic3—22no—
166C2×C3×Dic3—24no—
167C2×C3×Dic3—24no—
168C2×C3×Dic3—22no—
169C2×C3×Dic3—22no—
170C2×C3×Dic3—24no—
171C2×C3×Dic3—38no—
172C2×C3×Dic3—38no—
173C2×C3×Dic3—34no—
174C2×C3×Dic3—34no—
175C2×C3×Dic3—44no—
176C2×C3×Dic3—68no—
177C2×C3×Dic3—64no—
178C2×C3×Dic3Dic614yescomplemented
179C2×C3×Dic3C3×Q814yescomplemented
180C2×C3×Dic3C22×D314yes—
181C2×C3×Dic3C22×D312yes—
182C2×C3×Dic3C22×D312yes—
183C2×C3×Dic3C23×C314yes—
184C2×C3×Dic3C23×C312yes—
185C2×C3×Dic3C23×C312yes—
186C3×Dic6—316no—
187C3×Dic6—316no—
188C3×Dic6—616no—
189C2×C4×Q8—91noSylow
190C2×C33C4×Q814yescomplemented
191C2×C33C23×C411yescharacteristic
192C2×C33C22×Q811yescharacteristic
193C2×C33C2×#4111yescharacteristic
194C2×C3×Q8—98no—
195C2×C3×Q8—94no—
196C22×C3×C4—21no—
197C22×C3×C4—21no—
198C22×C3×C4—21no—
199C22×C3×C4—31no—
200C22×C3×C4—31no—
201C22×C3×C4—31no—
202C22×C3×C4—32no—
203C22×C3×C4—32no—
204C22×C3×C4—31no—
205C22×C3×C4—41no—
206C22×C3×C4—61no—
207C22×C3×C4—61no—
208C22×C3×C4—62no—
209C22×C3×C4—92no—
210C22×C3×C4D3211yescharacteristic
211C22×C3×C4C2×C3×D311yescharacteristic
212C22×C3×C4C2×C3×D311yescharacteristic
213C22×Dic3—32no—
214C22×Dic3—32no—
215C22×Dic3—62no—
216C22×Dic3C2×C3×D312yes—
217C22×Dic3C2×C3×D311yescharacteristic
218C22×Dic3C2×C3×D311yescharacteristic
219C2×Dic6—38no—
220C2×Dic6—38no—
221C2×Dic6—34no—
222C2×Dic6—34no—
223C2×Dic6—68no—
224C2×Dic6—64no—
225C3×C4⋊C4—34no—
226C3×C4⋊C4—34no—
227C3×C4⋊C4—64no—
228C3×C4⋊C4—98no—
229C3×C42—34no—
230C3×C42—34no—
231C3×C42—64no—
232C3×C42—98no—
233C4⋊Dic3C2×C3×D314yes—
234Dic3⋊C4—38no—
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×C32C2×C4×D312yes—
258C22×C32#23512yes—
259C22×C32#23712yes—
260C22×C32D3×Q812yes—
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—34no—
271C32×C4—44no—
272C32×C4—62no—
273C32×C4—64no—
274C32×C4C2×C4×D314yes—
275C32×C4C2×C4×D314yes—
276C32×C4C22×C3×C414yes—
277C3⋊Dic3C22×C3×C418yes—
278C3×Dic3—24no—
279C3×Dic3—22no—
280C3×Dic3—24no—
281C3×Dic3—22no—
282C3×Dic3—24no—
283C3×Dic3—24no—
284C3×Dic3—38no—
285C3×Dic3—38no—
286C3×Dic3—44no—
287C3×Dic3—68no—
288C3×Dic3C2×Dic614yescomplemented
289C3×Dic3C2×C3×Q814yescomplemented
290C22×Q8—91no—
291C4×Q8—98no—
292C2×C4⋊C4—31no—
293C2×C4⋊C4—92no—
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—916no—
305C2×C3×C4—24no—
306C2×C3×C4—24no—
307C2×C3×C4—24no—
308C2×C3×C4—22no—
309C2×C3×C4—22no—
310C2×C3×C4—22no—
311C2×C3×C4—32no—
312C2×C3×C4—32no—
313C2×C3×C4—34no—
314C2×C3×C4—34no—
315C2×C3×C4—32no—
316C2×C3×C4—34no—
317C2×C3×C4—34no—
318C2×C3×C4—32no—
319C2×C3×C4—44no—
320C2×C3×C4—42no—
321C2×C3×C4—62no—
322C2×C3×C4—62no—
323C2×C3×C4—64no—
324C2×C3×C4—64no—
325C2×C3×C4—64no—
326C2×C3×C4—62no—
327C2×C3×C4—64no—
328C2×C3×C4—64no—
329C2×C3×C4—62no—
330C2×C3×C4—94no—
331C2×C3×C4—98no—
332C2×C3×C4—124no—
333C2×C3×C4#62414yes—
334C2×C3×C4C3×C4×D314yes—
335C2×C3×C4C3×C4×D314yes—
336C2×C3×C4C2×D3212yes—
337C2×C3×C4C22×C3×D312yes—
338C2×C3×C4C22×C3×D312yes—
339C2×Dic3—22no—
340C2×Dic3—24no—
341C2×Dic3—34no—
342C2×Dic3—34no—
343C2×Dic3—38no—
344C2×Dic3—38no—
345C2×Dic3—68no—
346C2×Dic3—64no—
347C2×Dic3C3×Dic614yescomplemented
348C2×Dic3C22×C3×D314yes—
349C2×Dic3C22×C3×D312yes—
350C2×Dic3C22×C3×D312yes—
351Dic6—316no—
352Dic6—316no—
353Dic6—616no—
354C2×C32—24no—
355C2×C32—21no—
356C2×C32—21no—
357C2×C32—21no—
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×C32#88714yescomplemented
372C2×C32C3×C4×Q814yescomplemented
373C2×C32C22×Dic611yescharacteristic
374C2×C32C22×C4×D311yescharacteristic
375C2×C32C22×C4×D311yescharacteristic
376C2×C32C2×#23511yescharacteristic
377C2×C32C2×#23711yescharacteristic
378C2×C32C2×D3×Q811yescharacteristic
379C2×C32C23×C3×C411yescharacteristic
380C2×C32C22×C3×Q811yescharacteristic
381C2×C32C2×C3×#4111yescharacteristic
382C2×Q8—98no—
383C2×Q8—94no—
384C22×C4—31no—
385C22×C4—31no—
386C22×C4—32no—
387C22×C4—92no—
388C22×C4C3×D3211yescharacteristic
389C4⋊C4—34no—
390C4⋊C4—98no—
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#379512yes—
411C22×C3C2×#62412yes—
412C22×C3C2×C3×Dic612yes—
413C22×C3C2×C3×C4×D312yes—
414C22×C3C2×C3×C4×D312yes—
415C22×C3C3×#23512yes—
416C22×C3C3×#23712yes—
417C22×C3C3×D3×Q812yes—
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—34no—
426C3×C4—44no—
427C3×C4—64no—
428C3×C4—64no—
429C3×C4—62no—
430C3×C4—64no—
431C3×C4—64no—
432C3×C4—62no—
433C3×C4—98no—
434C3×C4—124no—
435C3×C4C2×#62414yes—
436C3×C4C2×C3×C4×D314yes—
437C3×C4C2×C3×C4×D314yes—
438Dic3—22no—
439Dic3—24no—
440Dic3—38no—
441Dic3—38no—
442Dic3—68no—
443Dic3C2×C3×Dic614yescomplemented
444C32—21no—
445C32—21no—
446C32—21no—
447C32—41no—
448C32C2×C4×Dic611yescharacteristic, complemented
449C32C2×#88711yescharacteristic, complemented
450C32C2×C3×C4×Q811yescharacteristic, complemented
451C23C2×C3×D3211yescharacteristic
452Q8—916no—
453C2×C4—32no—
454C2×C4—34no—
455C2×C4—32no—
456C2×C4—34no—
457C2×C4—62no—
458C2×C4—64no—
459C2×C4—94no—
460C2×C4—98no—
461C2×C4C3×#62414yes—
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×C3#6384414yesdirect product factor
480C2×C3C3×C4×Dic614yescomplemented
481C2×C3C3×#88714yescomplemented
482C2×C3C2×D3×Dic611yescharacteristic
483C2×C3C2×#379511yescharacteristic
484C2×C3C22×#62411yescharacteristic
485C2×C3C22×C3×Dic611yescharacteristic
486C2×C3C22×C3×C4×D311yescharacteristic
487C2×C3C22×C3×C4×D311yescharacteristic
488C2×C3C2×C3×#23511yescharacteristic
489C2×C3C2×C3×#23711yescharacteristic
490C2×C3C2×C3×D3×Q811yescharacteristic
491C22C3×D3×Dic612yes—
492C22C3×#379512yes—
493C22C2×C3×#62412yes—
494C22C22×C3×D3211yescharacteristic
495C4—34no—
496C4—62no—
497C4—64no—
498C4—98no—
499C4C2×C3×#62414yes—
500C3—21no—
501C3—21no—
502C3—21no—
503C3—41no—
504C3C2×#6384411yescharacteristic, direct product factor
505C3C2×C3×C4×Dic611yescharacteristic, complemented
506C3C2×C3×#88711yescharacteristic, complemented
507C2C3×#6384414yesdirect product factor
508C2C2×C3×D3×Dic611yescharacteristic
509C2C2×C3×#379511yescharacteristic
510C2C22×C3×#62411yescharacteristic
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×#63844C2direct (N × H)
9. C2×#63844C3direct (N × H)
12. C2×C3×#634C4semidirect
41. #63844C2×C3direct (N × H)
47. C3×#634C2×C4semidirect
50. C2×C32×Dic3Q8semidirect
68. C2×#634C3×C4semidirect
110. C32×Dic3C2×Q8semidirect
161. #634C2×C3×C4semidirect
178. C2×C3×Dic3Dic6semidirect
179. C2×C3×Dic3C3×Q8semidirect
190. C2×C33C4×Q8semidirect
288. C3×Dic3C2×Dic6semidirect
289. C3×Dic3C2×C3×Q8semidirect
296. C33C2×C4×Q8semidirect
347. C2×Dic3C3×Dic6semidirect
370. C2×C32C4×Dic6semidirect
371. C2×C32#887semidirect
372. C2×C32C3×C4×Q8semidirect
443. Dic3C2×C3×Dic6semidirect
448. C32C2×C4×Dic6semidirect
449. C32C2×#887semidirect
450. C32C2×C3×C4×Q8semidirect
479. C2×C3#63844direct (N × H)
480. C2×C3C3×C4×Dic6semidirect
481. C2×C3C3×#887semidirect
504. C3C2×#63844direct (N × H)
505. C3C2×C3×C4×Dic6semidirect
506. C3C2×C3×#887semidirect
507. C2C3×#63844direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×#6342
C2×C32×C4⋊Dic32
C2×C32×C4×Dic32
C2×C3×#37242
C2×C3×Dic322
C3×#638442
C2×C3×#8873
C2×C3×C4×Dic63
C2×#638443