Finite Group Databasebeta

← C2×C3×#63953 #13175117

Subgroups — C2×C3×#63953 #13175117

Order 1728.

1954 conjugacy classes of subgroups.

The subgroup lattice has 504 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×#638C211yesmaximal, characteristic
2C22×C3×#627C211yesmaximal, characteristic, complemented
3C2×C32×#217C212yesmaximal, complemented
4C2×C3×#3722C212yesmaximal, complemented
5C2×C3×Dic32C211yesmaximal, characteristic, complemented
6C3×#63953C218yesmaximal, direct product factor
7C2×C3×#885—32nomaximal
8C2×#63953C311yesmaximal, characteristic, direct product factor
9C24×C33C2211yescharacteristic
10C23×C3×C3⋊D3C2211yescharacteristic
11C2×C3×#638—24no—
12C2×C3×#638C2214yes—
13C22×C3×C3⋊Dic3C2211yescharacteristic
14C22×C32×Dic3C2212yes—
15C22×C32×Dic3C2212yes—
16C2×C3×#627—24no—
17C2×C3×#627C2214yescomplemented
18C32×#217C2218yescomplemented
19C3×#3722C2218yescomplemented
20C3×Dic32C2214yescomplemented
21C22×#638C2×C311yescharacteristic
22C22×C3×#67—32no—
23C22×C3×#67—61no—
24C22×C3×Dic6—32no—
25C22×#627C2×C311yescharacteristic, complemented
26C2×C32×C22⋊C4—32no—
27C2×C3×#217—22no—
28C2×C3×#217D312yescomplemented
29C2×C3×#217C2×C312yescomplemented
30C2×C3×(C2×D3)⋊C4—32no—
31C2×C3×(C2×D3)⋊C4—32no—
32C2×C3×C4×Dic3—32no—
33C3×#885—316no—
34C2×#3722C2×C312yescomplemented
35C2×Dic32C2×C311yescharacteristic, complemented
36#63953C2×C318yesdirect product factor
37C23×C33—22no—
38C23×C33—24no—
39C23×C33C2312yes—
40C23×C33C2311yescharacteristic
41C22×C3×C3⋊D3—22no—
42C22×C3×C3⋊D3—24no—
43C22×C3×C3⋊D3C2312yes—
44C3×#638—28no—
45C2×C3×C3⋊Dic3D414yes—
46C2×C3×C3⋊Dic3C2312yes—
47C2×C32×Dic3—24no—
48C2×C32×Dic3—24no—
49C2×C32×Dic3C2314yes—
50C2×C32×Dic3C2314yes—
51C3×#627—28no—
52C2×C3×#124—91no—
53C2×#885—32no—
54C24×C32—21no—
55C24×C32—22no—
56C24×C32—41no—
57C24×C32C2×D312yes—
58C24×C32C22×C311yescharacteristic
59C23×C3⋊D3C22×C311yescharacteristic
60C23×C3×D3—32no—
61C23×C3×D3—61no—
62C22×C32×C4—32no—
63C22×C32×C4—32no—
64C2×#638—24no—
65C2×#638C22×C314yes—
66C22×C3⋊Dic3C22×C311yescharacteristic
67C2×C3×#67—38no—
68C2×C3×#67—68no—
69C2×C3×#67—64no—
70C2×C3×#67—68no—
71C22×C3×Dic3—22no—
72C22×C3×Dic3—22no—
73C22×C3×Dic3—32no—
74C22×C3×Dic3—61no—
75C22×C3×Dic3C2×D312yes—
76C22×C3×Dic3C2×D312yes—
77C22×C3×Dic3C22×C312yes—
78C22×C3×Dic3C22×C312yes—
79C2×C3×Dic6—38no—
80C2×C3×Dic6—68no—
81C2×#627—24no—
82C2×#627C22×C314yescomplemented
83C32×C22⋊C4—38no—
84C3×#217—28no—
85C3×#217C2×D318yescomplemented
86C3×#217C22×C318yescomplemented
87C3×(C2×D3)⋊C4—38no—
88C3×(C2×D3)⋊C4—38no—
89C3×C4×Dic3—38no—
90#3722C22×C318yescomplemented
91Dic32C22×C314yescomplemented
92C22×C33—22no—
93C22×C33—24no—
94C22×C33—44no—
95C22×C33C2×D412yes—
96C22×C33#4114yes—
97C22×C33C2411yescharacteristic
98C2×C3×C3⋊D3—22no—
99C2×C3×C3⋊D3—24no—
100C2×C3×C3⋊D3—44no—
101C3×C3⋊Dic3C2×D414yes—
102C32×Dic3—24no—
103C32×Dic3—24no—
104C22×C3×Q8—91no—
105C22×C3×D4—91no—
106C22×#67—32no—
107C22×#67—61no—
108C22×Dic6—32no—
109C3×#124—98no—
110C2×C3×C22⋊C4—32no—
111C2×C3×C22⋊C4—32no—
112C2×C3×C22⋊C4—62no—
113C2×C3×C22⋊C4—92no—
114C2×C3×C42—91no—
115C2×#217C3×D312yescomplemented
116C2×(C2×D3)⋊C4—32no—
117C2×(C2×D3)⋊C4—32no—
118C2×C4×Dic3—32no—
119#885—316no—
120C23×C32—24no—
121C23×C32—22no—
122C23×C32—28no—
123C23×C32—24no—
124C23×C32—24no—
125C23×C32—24no—
126C23×C32—24no—
127C23×C32—22no—
128C23×C32—21no—
129C23×C32—22no—
130C23×C32—24no—
131C23×C32—41no—
132C23×C32—42no—
133C23×C32—44no—
134C23×C32—44no—
135C23×C32—44no—
136C23×C32—44no—
137C23×C32—48no—
138C23×C32C22×D314yes—
139C23×C32C22×D312yes—
140C23×C32C23×C312yes—
141C23×C32C23×C311yescharacteristic
142C22×C3⋊D3—22no—
143C22×C3⋊D3—24no—
144C22×C3⋊D3C23×C312yes—
145C22×C3×D3—34no—
146C22×C3×D3—62no—
147C22×C3×D3—64no—
148C22×C3×D3—64no—
149C22×C3×D3—64no—
150C22×C3×D3—64no—
151C22×C3×D3—124no—
152C2×C32×C4—34no—
153C2×C32×C4—34no—
154C2×C32×C4—64no—
155C2×C32×C4—64no—
156#638—28no—
157C2×C3⋊Dic3C3×D414yes—
158C2×C3⋊Dic3C23×C312yes—
159C3×#67—616no—
160C3×#67—616no—
161C2×C3×Dic3—24no—
162C2×C3×Dic3—24no—
163C2×C3×Dic3—24no—
164C2×C3×Dic3—24no—
165C2×C3×Dic3—24no—
166C2×C3×Dic3—24no—
167C2×C3×Dic3—38no—
168C2×C3×Dic3—34no—
169C2×C3×Dic3—44no—
170C2×C3×Dic3—44no—
171C2×C3×Dic3—64no—
172C2×C3×Dic3—62no—
173C2×C3×Dic3C22×D314yes—
174C2×C3×Dic3C22×D314yes—
175C2×C3×Dic3C23×C314yes—
176C2×C3×Dic3C23×C314yes—
177C3×Dic6—616no—
178#627—28no—
179C2×#124—91noSylow
180C2×C33—42no—
181C2×C33#12414yescomplemented
182C2×C33C22×D411yescharacteristic
183C2×C33C2×#4112yes—
184C3×C3⋊D3—42no—
185C24×C3—22no—
186C24×C3—21no—
187C24×C3—41no—
188C24×C3—91no—
189C24×C3D3211yescharacteristic
190C24×C3C2×C3×D312yes—
191C23×D3—32no—
192C23×D3—61no—
193C2×C3×Q8—94no—
194C2×C3×Q8—184no—
195C2×C3×D4—94no—
196C2×C3×D4—184no—
197C22×C3×C4—32no—
198C22×C3×C4—32no—
199C22×C3×C4—32no—
200C22×C3×C4—32no—
201C22×C3×C4—62no—
202C22×C3×C4—62no—
203C22×C3×C4—91no—
204C2×#67—38no—
205C2×#67—64no—
206C2×#67—68no—
207C2×#67—68no—
208C22×Dic3—32no—
209C22×Dic3—61no—
210C22×Dic3C2×C3×D312yes—
211C22×Dic3C2×C3×D312yes—
212C2×Dic6—38no—
213C2×Dic6—68no—
214C3×C22⋊C4—38no—
215C3×C22⋊C4—38no—
216C3×C22⋊C4—68no—
217C3×C22⋊C4—98no—
218C3×C42—94no—
219#217C2×C3×D318yescomplemented
220(C2×D3)⋊C4—38no—
221(C2×D3)⋊C4—38no—
222C4×Dic3—38no—
223C22×C32—24no—
224C22×C32—22no—
225C22×C32—24no—
226C22×C32—28no—
227C22×C32—24no—
228C22×C32—24no—
229C22×C32—24no—
230C22×C32—21no—
231C22×C32—22no—
232C22×C32—24no—
233C22×C32—22no—
234C22×C32—24no—
235C22×C32—24no—
236C22×C32—24no—
237C22×C32—42no—
238C22×C32—44no—
239C22×C32—41no—
240C22×C32—44no—
241C22×C32—44no—
242C22×C32—44no—
243C22×C32—44no—
244C22×C32—48no—
245C22×C32—416no—
246C22×C32—48no—
247C22×C32—416no—
248C22×C32—48no—
249C22×C32—44no—
250C22×C32#23514yes—
251C22×C32D3×D414yes—
252C22×C32#23714yes—
253C22×C32C2×C3×D412yes—
254C22×C32C3×#4114yes—
255C22×C32C23×D312yes—
256C22×C32C24×C311yescharacteristic
257C2×C3⋊D3—22no—
258C2×C3⋊D3—24no—
259C2×C3⋊D3—44no—
260C2×C3×D3—64no—
261C2×C3×D3—64no—
262C2×C3×D3—64no—
263C2×C3×D3—64no—
264C2×C3×D3—124no—
265C2×C3×D3—128no—
266C2×C3×D3—128no—
267C32×C4—64no—
268C32×C4—64no—
269C3⋊Dic3C2×C3×D414yes—
270C3×Dic3—24no—
271C3×Dic3—24no—
272C3×Dic3—24no—
273C3×Dic3—24no—
274C3×Dic3—38no—
275C3×Dic3—44no—
276C3×Dic3—44no—
277C3×Dic3—64no—
278C22×Q8—91no—
279C22×D4—91no—
280#124—98no—
281C2×C22⋊C4—32no—
282C2×C22⋊C4—92no—
283C2×C42—91no—
284C33C2×#12411yescharacteristic, complemented, Sylow
285C23×C3—24no—
286C23×C3—28no—
287C23×C3—24no—
288C23×C3—22no—
289C23×C3—24no—
290C23×C3—24no—
291C23×C3—24no—
292C23×C3—22no—
293C23×C3—21no—
294C23×C3—24no—
295C23×C3—22no—
296C23×C3—44no—
297C23×C3—44no—
298C23×C3—44no—
299C23×C3—48no—
300C23×C3—44no—
301C23×C3—42no—
302C23×C3—41no—
303C23×C3—92no—
304C23×C3—184no—
305C23×C3—182no—
306C23×C3C2×D3212yes—
307C23×C3C2×D3211yescharacteristic
308C23×C3C22×C3×D314yes—
309C23×C3C22×C3×D312yes—
310C22×D3—34no—
311C22×D3—62no—
312C22×D3—64no—
313C22×D3—64no—
314C22×D3—64no—
315C22×D3—64no—
316C22×D3—124no—
317C3×Q8—188no—
318C3×D4—188no—
319C2×C3×C4—34no—
320C2×C3×C4—34no—
321C2×C3×C4—34no—
322C2×C3×C4—34no—
323C2×C3×C4—64no—
324C2×C3×C4—64no—
325C2×C3×C4—64no—
326C2×C3×C4—64no—
327C2×C3×C4—64no—
328C2×C3×C4—64no—
329C2×C3×C4—92no—
330C2×C3×C4—94no—
331C2×C3×C4—124no—
332C2×C3×C4—124no—
333#67—616no—
334#67—616no—
335C2×Dic3—24no—
336C2×Dic3—24no—
337C2×Dic3—34no—
338C2×Dic3—38no—
339C2×Dic3—64no—
340C2×Dic3—62no—
341C2×Dic3C22×C3×D314yes—
342C2×Dic3C22×C3×D314yes—
343Dic6—616no—
344C2×C32—28no—
345C2×C32—22no—
346C2×C32—21no—
347C2×C32—22no—
348C2×C32—22no—
349C2×C32—22no—
350C2×C32—24no—
351C2×C32—44no—
352C2×C32—41no—
353C2×C32—42no—
354C2×C32—48no—
355C2×C32—44no—
356C2×C32—48no—
357C2×C32—44no—
358C2×C32—42no—
359C2×C32#88518yescomplemented
360C2×C32C3×#12414yescomplemented
361C2×C32C2×#23512yes—
362C2×C32C2×D3×D412yes—
363C2×C32C2×#23712yes—
364C2×C32C22×C3×D411yescharacteristic
365C2×C32C2×C3×#4112yes—
366C3⋊D3—42no—
367C3×D3—124no—
368C3×D3—124no—
369C24—91no—
370C24C3×D3211yescharacteristic
371C2×Q8—94no—
372C2×Q8—184no—
373C2×D4—94no—
374C2×D4—184no—
375C22×C4—32no—
376C22×C4—32no—
377C22×C4—91no—
378C22⋊C4—38no—
379C22⋊C4—98no—
380C42—94no—
381C22×C3—24no—
382C22×C3—28no—
383C22×C3—24no—
384C22×C3—22no—
385C22×C3—24no—
386C22×C3—24no—
387C22×C3—24no—
388C22×C3—21no—
389C22×C3—22no—
390C22×C3—24no—
391C22×C3—24no—
392C22×C3—24no—
393C22×C3—22no—
394C22×C3—24no—
395C22×C3—44no—
396C22×C3—44no—
397C22×C3—44no—
398C22×C3—48no—
399C22×C3—44no—
400C22×C3—48no—
401C22×C3—416no—
402C22×C3—416no—
403C22×C3—48no—
404C22×C3—44no—
405C22×C3—41no—
406C22×C3—44no—
407C22×C3—42no—
408C22×C3—184no—
409C22×C3—182no—
410C22×C3—364no—
411C22×C3#380414yes—
412C22×C3#381112yes—
413C22×C3C3×#23514yes—
414C22×C3C3×D3×D414yes—
415C22×C3C3×#23714yes—
416C22×C3C22×D3211yescharacteristic
417C22×C3C23×C3×D312yes—
418C2×D3—64no—
419C2×D3—64no—
420C2×D3—64no—
421C2×D3—64no—
422C2×D3—124no—
423C2×D3—128no—
424C2×D3—128no—
425C3×C4—64no—
426C3×C4—64no—
427C3×C4—64no—
428C3×C4—64no—
429C3×C4—94no—
430C3×C4—124no—
431C3×C4—124no—
432Dic3—24no—
433Dic3—24no—
434Dic3—38no—
435Dic3—64no—
436C32—22no—
437C32—21no—
438C32—41no—
439C32C2×#88512yescomplemented
440C32C2×C3×#12411yescharacteristic, complemented
441C23—24no—
442C23—22no—
443C23—92no—
444C23—184no—
445C23—182no—
446C23C2×C3×D3212yes—
447C23C2×C3×D3211yescharacteristic
448Q8—188no—
449D4—188no—
450C2×C4—34no—
451C2×C4—34no—
452C2×C4—64no—
453C2×C4—64no—
454C2×C4—92no—
455C2×C4—94no—
456C2×C3—22no—
457C2×C3—22no—
458C2×C3—22no—
459C2×C3—21no—
460C2×C3—22no—
461C2×C3—24no—
462C2×C3—28no—
463C2×C3—44no—
464C2×C3—48no—
465C2×C3—48no—
466C2×C3—44no—
467C2×C3—42no—
468C2×C3—42no—
469C2×C3—41no—
470C2×C3—44no—
471C2×C3—362no—
472C2×C3#6395314yesdirect product factor
473C2×C3C3×#88518yescomplemented
474C2×C3C2×#380412yes—
475C2×C3C2×#381111yescharacteristic
476C2×C3C2×C3×#23512yes—
477C2×C3C2×C3×D3×D412yes—
478C2×C3C2×C3×#23712yes—
479D3—124no—
480D3—124no—
481C22—24no—
482C22—22no—
483C22—44no—
484C22—184no—
485C22—182no—
486C22—364no—
487C22C3×#380414yes—
488C22C3×#381112yes—
489C22C22×C3×D3211yescharacteristic
490C4—64no—
491C4—64no—
492C4—94no—
493C3—21no—
494C3—22no—
495C3—41no—
496C3C2×#6395311yescharacteristic, direct product factor
497C3C2×C3×#88512yescomplemented
498C2—42no—
499C2—362no—
500C2C3×#6395314yesdirect product factor
501C2C2×C3×#380412yes—
502C2C2×C3×#381111yescharacteristic
subgroupG/H
centerC23×C3C2×D32
commutator, cosocleC22×C32C24×C3
FrattiniC22C22×C3×D32
FittingC24×C33C22
socleC23×C33C23
prime radicalC22×C3×#638C2
hypercenterC24×C3D32
hypocenterC32C2×C3×#124
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C23×C3 ◁ C24×C3

Complements

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

NH ≅ G/Ntype
2. C22×C3×#627C2semidirect
3. C2×C32×#217C2semidirect
4. C2×C3×#3722C2semidirect
5. C2×C3×Dic32C2semidirect
6. C3×#63953C2direct (N × H)
8. C2×#63953C3direct (N × H)
17. C2×C3×#627C22semidirect
18. C32×#217C22semidirect
19. C3×#3722C22semidirect
20. C3×Dic32C22semidirect
25. C22×#627C2×C3semidirect
28. C2×C3×#217D3semidirect
29. C2×C3×#217C2×C3semidirect
34. C2×#3722C2×C3semidirect
35. C2×Dic32C2×C3semidirect
36. #63953C2×C3direct (N × H)
82. C2×#627C22×C3semidirect
85. C3×#217C2×D3semidirect
86. C3×#217C22×C3semidirect
90. #3722C22×C3semidirect
91. Dic32C22×C3semidirect
115. C2×#217C3×D3semidirect
181. C2×C33#124semidirect
219. #217C2×C3×D3semidirect
284. C33C2×#124semidirect
359. C2×C32#885semidirect
360. C2×C32C3×#124semidirect
439. C32C2×#885semidirect
440. C32C2×C3×#124semidirect
472. C2×C3#63953direct (N × H)
473. C2×C3C3×#885semidirect
496. C3C2×#63953direct (N × H)
497. C3C2×C3×#885semidirect
500. C2C3×#63953direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×#6382
C22×C3×#6272
C2×C32×#2172
C2×C3×#37222
C2×C3×Dic322
C3×#639532
C2×C3×#8853
C2×#639533