Finite Group Databasebeta

← C2×C3×#63915 #13174595

Subgroups — C2×C3×#63915 #13174595

Order 1728.

1746 conjugacy classes of subgroups.

The subgroup lattice has 484 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×#627C211yesmaximal, characteristic, complemented
2C22×C3×#625C211yesmaximal, characteristic
3C2×C3×C4×C3⋊Dic3C211yesmaximal, characteristic, complemented
4C2×C32×(C2×D3)⋊C4C212yesmaximal, complemented
5C2×C3×(C2×D3)⋊Dic3C212yesmaximal, complemented
6C3×#63915C218yesmaximal, direct product factor
7C2×C3×#885—32nomaximal
8C2×#63915C311yesmaximal, characteristic, direct product factor
9C23×C32×D3C2212yes—
10C22×C33×C4C2211yescharacteristic
11C22×C3×C3⋊Dic3C2211yescharacteristic
12C22×C3×C3⋊Dic3C2211yescharacteristic
13C22×C32×Dic3C2212yes—
14C2×C3×#627—24no—
15C2×C3×#627C2214yescomplemented
16C2×C3×#625—24no—
17C2×C3×#625C2214yes—
18C3×C4×C3⋊Dic3C2214yescomplemented
19C32×(C2×D3)⋊C4C2218yescomplemented
20C3×(C2×D3)⋊Dic3C2218yescomplemented
21C22×C3×#67—32no—
22C22×C3×Dic6—32no—
23C22×#627C2×C311yescharacteristic, complemented
24C22×#625C2×C311yescharacteristic
25C2×C32×C22⋊C4—32no—
26C2×C4×C3⋊Dic3C2×C311yescharacteristic, complemented
27C2×C3×#217—32no—
28C2×C3×(C2×D3)⋊C4—22no—
29C2×C3×(C2×D3)⋊C4—32no—
30C2×C3×(C2×D3)⋊C4D312yescomplemented
31C2×C3×(C2×D3)⋊C4C2×C312yescomplemented
32C2×C3×C4×Dic3—32no—
33C2×C3×C4×Dic3—61no—
34C3×#885—316no—
35C2×(C2×D3)⋊Dic3C2×C312yescomplemented
36#63915C2×C318yesdirect product factor
37C23×C33C2311yescharacteristic
38C22×C32×D3—24no—
39C22×C32×D3—24no—
40C22×C32×D3—24no—
41C22×C32×D3C2314yes—
42C2×C33×C4—22no—
43C2×C33×C4C2312yes—
44C2×C3×C3⋊Dic3—22no—
45C2×C3×C3⋊Dic3D414yes—
46C2×C3×C3⋊Dic3C2312yes—
47C2×C3×C3⋊Dic3C2312yes—
48C2×C32×Dic3—24no—
49C2×C32×Dic3C2314yes—
50C3×#627—28no—
51C3×#625—28no—
52C2×C3×#124—91no—
53C2×#885—32no—
54C24×C32—32no—
55C23×C3×D3—22no—
56C23×C3×D3C2×D312yes—
57C23×C3×D3C22×C312yes—
58C22×C32×C4—21no—
59C22×C32×C4—22no—
60C22×C32×C4—32no—
61C22×C32×C4—41no—
62C22×C32×C4C2×D312yes—
63C22×C32×C4C22×C311yescharacteristic
64C22×C3⋊Dic3C22×C311yescharacteristic
65C22×C3⋊Dic3C22×C311yescharacteristic
66C2×C3×#67—38no—
67C2×C3×#67—68no—
68C22×C3×Dic3—22no—
69C22×C3×Dic3—32no—
70C22×C3×Dic3—32no—
71C22×C3×Dic3—61no—
72C22×C3×Dic3—61no—
73C22×C3×Dic3C2×D312yes—
74C22×C3×Dic3C22×C312yes—
75C2×C3×Dic6—38no—
76C2×C3×Dic6—68no—
77C2×#627—24no—
78C2×#627C22×C314yescomplemented
79C2×#625—24no—
80C2×#625C22×C314yes—
81C32×C22⋊C4—38no—
82C4×C3⋊Dic3C22×C314yescomplemented
83C3×#217—38no—
84C3×(C2×D3)⋊C4—28no—
85C3×(C2×D3)⋊C4—38no—
86C3×(C2×D3)⋊C4C2×D318yescomplemented
87C3×(C2×D3)⋊C4C22×C318yescomplemented
88C3×C4×Dic3—38no—
89C3×C4×Dic3—64no—
90(C2×D3)⋊Dic3C22×C318yescomplemented
91C22×C33C2×D412yes—
92C22×C33#4112yes—
93C22×C33#4112yes—
94C22×C33C2411yescharacteristic
95C2×C32×D3—24no—
96C2×C32×D3—24no—
97C2×C32×D3—24no—
98C2×C32×D3—48no—
99C33×C4—22no—
100C3×C3⋊Dic3—22no—
101C3×C3⋊Dic3C2×D414yes—
102C32×Dic3—24no—
103C22×C3×Q8—91no—
104C22×C3×D4—91no—
105C22×#67—32no—
106C22×Dic6—32no—
107C3×#124—98no—
108C2×C3×C22⋊C4—32no—
109C2×C3×C22⋊C4—32no—
110C2×C3×C22⋊C4—62no—
111C2×C3×C22⋊C4—92no—
112C2×C3×C42—91no—
113C2×#217—32no—
114C2×(C2×D3)⋊C4—32no—
115C2×(C2×D3)⋊C4C3×D312yescomplemented
116C2×C4×Dic3—32no—
117C2×C4×Dic3—61no—
118#885—316no—
119C23×C32—21no—
120C23×C32—22no—
121C23×C32—34no—
122C23×C32—41no—
123C23×C32—64no—
124C23×C32—64no—
125C23×C32—64no—
126C23×C32C22×D312yes—
127C23×C32C23×C311yescharacteristic
128C22×C3×D3—28no—
129C22×C3×D3—24no—
130C22×C3×D3—24no—
131C22×C3×D3—24no—
132C22×C3×D3—24no—
133C22×C3×D3—24no—
134C22×C3×D3—24no—
135C22×C3×D3—24no—
136C22×C3×D3—44no—
137C22×C3×D3—44no—
138C22×C3×D3C22×D314yes—
139C22×C3×D3C23×C314yes—
140C2×C32×C4—22no—
141C2×C32×C4—24no—
142C2×C32×C4—24no—
143C2×C32×C4—22no—
144C2×C32×C4—24no—
145C2×C32×C4—34no—
146C2×C32×C4—42no—
147C2×C32×C4—44no—
148C2×C32×C4—44no—
149C2×C32×C4—64no—
150C2×C32×C4C22×D314yes—
151C2×C32×C4C23×C312yes—
152C2×C3⋊Dic3—22no—
153C2×C3⋊Dic3C3×D414yes—
154C2×C3⋊Dic3C23×C312yes—
155C2×C3⋊Dic3C23×C312yes—
156C3×#67—616no—
157C2×C3×Dic3—24no—
158C2×C3×Dic3—24no—
159C2×C3×Dic3—24no—
160C2×C3×Dic3—38no—
161C2×C3×Dic3—34no—
162C2×C3×Dic3—34no—
163C2×C3×Dic3—44no—
164C2×C3×Dic3—62no—
165C2×C3×Dic3—62no—
166C2×C3×Dic3—64no—
167C2×C3×Dic3—64no—
168C2×C3×Dic3—64no—
169C2×C3×Dic3C22×D314yes—
170C2×C3×Dic3C23×C314yes—
171C3×Dic6—616no—
172#627—28no—
173#625—28no—
174C2×#124—91noSylow
175C2×C33#12414yescomplemented
176C2×C33C22×D411yescharacteristic
177C2×C33C2×#4111yescharacteristic
178C2×C33C2×#4111yescharacteristic
179C32×D3—44no—
180C24×C3—32no—
181C24×C3—32no—
182C24×C3—62no—
183C23×D3C2×C3×D312yes—
184C2×C3×Q8—94no—
185C2×C3×Q8—184no—
186C2×C3×D4—94no—
187C2×C3×D4—184no—
188C22×C3×C4—21no—
189C22×C3×C4—22no—
190C22×C3×C4—32no—
191C22×C3×C4—32no—
192C22×C3×C4—41no—
193C22×C3×C4—62no—
194C22×C3×C4—91no—
195C22×C3×C4—91no—
196C22×C3×C4D3211yescharacteristic
197C22×C3×C4C2×C3×D312yes—
198C2×#67—38no—
199C2×#67—68no—
200C22×Dic3—32no—
201C22×Dic3—32no—
202C22×Dic3—61no—
203C22×Dic3—61no—
204C22×Dic3C2×C3×D312yes—
205C2×Dic6—38no—
206C2×Dic6—68no—
207C3×C22⋊C4—38no—
208C3×C22⋊C4—38no—
209C3×C22⋊C4—68no—
210C3×C22⋊C4—98no—
211C3×C42—94no—
212#217—38no—
213(C2×D3)⋊C4—38no—
214(C2×D3)⋊C4C2×C3×D318yescomplemented
215C4×Dic3—38no—
216C4×Dic3—64no—
217C22×C32—21no—
218C22×C32—22no—
219C22×C32—22no—
220C22×C32—22no—
221C22×C32—22no—
222C22×C32—24no—
223C22×C32—24no—
224C22×C32—24no—
225C22×C32—42no—
226C22×C32—42no—
227C22×C32—42no—
228C22×C32—41no—
229C22×C32—64no—
230C22×C32—64no—
231C22×C32—64no—
232C22×C32—128no—
233C22×C32#23514yes—
234C22×C32D3×D414yes—
235C22×C32#23714yes—
236C22×C32C2×C3×D412yes—
237C22×C32C3×#4112yes—
238C22×C32C3×#4112yes—
239C22×C32C23×D312yes—
240C22×C32C24×C311yescharacteristic
241C2×C3×D3—28no—
242C2×C3×D3—24no—
243C2×C3×D3—24no—
244C2×C3×D3—24no—
245C2×C3×D3—24no—
246C2×C3×D3—24no—
247C2×C3×D3—24no—
248C2×C3×D3—44no—
249C2×C3×D3—44no—
250C2×C3×D3—416no—
251C2×C3×D3—48no—
252C2×C3×D3—48no—
253C32×C4—22no—
254C32×C4—24no—
255C32×C4—24no—
256C32×C4—44no—
257C32×C4—44no—
258C32×C4—64no—
259C3⋊Dic3—22no—
260C3⋊Dic3C2×C3×D414yes—
261C3×Dic3—24no—
262C3×Dic3—24no—
263C3×Dic3—38no—
264C3×Dic3—44no—
265C3×Dic3—64no—
266C3×Dic3—64no—
267C3×Dic3—64no—
268C22×Q8—91no—
269C22×D4—91no—
270#124—98no—
271C2×C22⋊C4—32no—
272C2×C22⋊C4—92no—
273C2×C42—91no—
274C33C2×#12411yescharacteristic, complemented, Sylow
275C23×C3—22no—
276C23×C3—21no—
277C23×C3—34no—
278C23×C3—34no—
279C23×C3—41no—
280C23×C3—64no—
281C23×C3—68no—
282C23×C3—64no—
283C23×C3—64no—
284C23×C3—64no—
285C23×C3—64no—
286C23×C3—64no—
287C23×C3—64no—
288C23×C3—124no—
289C23×C3—124no—
290C23×C3C2×D3211yescharacteristic
291C23×C3C22×C3×D312yes—
292C22×D3—24no—
293C22×D3—24no—
294C22×D3—24no—
295C22×D3C22×C3×D314yes—
296C3×Q8—188no—
297C3×D4—188no—
298C2×C3×C4—24no—
299C2×C3×C4—22no—
300C2×C3×C4—24no—
301C2×C3×C4—22no—
302C2×C3×C4—24no—
303C2×C3×C4—34no—
304C2×C3×C4—34no—
305C2×C3×C4—44no—
306C2×C3×C4—44no—
307C2×C3×C4—42no—
308C2×C3×C4—64no—
309C2×C3×C4—64no—
310C2×C3×C4—64no—
311C2×C3×C4—92no—
312C2×C3×C4—94no—
313C2×C3×C4—92no—
314C2×C3×C4—124no—
315C2×C3×C4—182no—
316C2×C3×C4C2×D3212yes—
317C2×C3×C4C22×C3×D314yes—
318#67—616no—
319C2×Dic3—24no—
320C2×Dic3—34no—
321C2×Dic3—38no—
322C2×Dic3—34no—
323C2×Dic3—64no—
324C2×Dic3—62no—
325C2×Dic3—62no—
326C2×Dic3—64no—
327C2×Dic3—64no—
328C2×Dic3C22×C3×D314yes—
329Dic6—616no—
330C2×C32—28no—
331C2×C32—21no—
332C2×C32—21no—
333C2×C32—21no—
334C2×C32—22no—
335C2×C32—22no—
336C2×C32—22no—
337C2×C32—24no—
338C2×C32—44no—
339C2×C32—41no—
340C2×C32—41no—
341C2×C32—41no—
342C2×C32—124no—
343C2×C32#88518yescomplemented
344C2×C32C3×#12414yescomplemented
345C2×C32C2×#23512yes—
346C2×C32C2×D3×D412yes—
347C2×C32C2×#23712yes—
348C2×C32C22×C3×D411yescharacteristic
349C2×C32C2×C3×#4111yescharacteristic
350C2×C32C2×C3×#4111yescharacteristic
351C3×D3—48no—
352C3×D3—44no—
353C3×D3—44no—
354C24—32no—
355C2×Q8—94no—
356C2×Q8—184no—
357C2×D4—94no—
358C2×D4—184no—
359C22×C4—32no—
360C22×C4—91no—
361C22×C4—91no—
362C22×C4C3×D3211yescharacteristic
363C22⋊C4—38no—
364C22⋊C4—98no—
365C42—94no—
366C22×C3—21no—
367C22×C3—22no—
368C22×C3—22no—
369C22×C3—24no—
370C22×C3—22no—
371C22×C3—24no—
372C22×C3—22no—
373C22×C3—24no—
374C22×C3—41no—
375C22×C3—42no—
376C22×C3—42no—
377C22×C3—42no—
378C22×C3—68no—
379C22×C3—64no—
380C22×C3—64no—
381C22×C3—64no—
382C22×C3—64no—
383C22×C3—64no—
384C22×C3—64no—
385C22×C3—124no—
386C22×C3—124no—
387C22×C3—1216no—
388C22×C3—128no—
389C22×C3—128no—
390C22×C3#379812yes—
391C22×C3#380212yes—
392C22×C3#380512yes—
393C22×C3C3×#23514yes—
394C22×C3C3×D3×D414yes—
395C22×C3C3×#23714yes—
396C22×C3C22×D3211yescharacteristic
397C22×C3C23×C3×D312yes—
398C2×D3—24no—
399C2×D3—24no—
400C2×D3—24no—
401C2×D3—48no—
402C3×C4—24no—
403C3×C4—22no—
404C3×C4—24no—
405C3×C4—44no—
406C3×C4—44no—
407C3×C4—64no—
408C3×C4—64no—
409C3×C4—94no—
410C3×C4—124no—
411C3×C4—182no—
412Dic3—24no—
413Dic3—38no—
414Dic3—64no—
415Dic3—64no—
416Dic3—64no—
417C32—22no—
418C32—21no—
419C32—41no—
420C32C2×#88512yescomplemented
421C32C2×C3×#12411yescharacteristic, complemented
422C23—34no—
423C23—64no—
424C23—64no—
425C23—64no—
426C23C2×C3×D3211yescharacteristic
427Q8—188no—
428D4—188no—
429C2×C4—22no—
430C2×C4—34no—
431C2×C4—64no—
432C2×C4—92no—
433C2×C4—94no—
434C2×C4—92no—
435C2×C4—182no—
436C2×C4C2×C3×D3212yes—
437C2×C3—21no—
438C2×C3—22no—
439C2×C3—28no—
440C2×C3—24no—
441C2×C3—21no—
442C2×C3—22no—
443C2×C3—21no—
444C2×C3—22no—
445C2×C3—41no—
446C2×C3—44no—
447C2×C3—41no—
448C2×C3—41no—
449C2×C3—128no—
450C2×C3—124no—
451C2×C3—124no—
452C2×C3#6391514yesdirect product factor
453C2×C3C3×#88518yescomplemented
454C2×C3C2×#379811yescharacteristic
455C2×C3C2×#380211yescharacteristic
456C2×C3C2×#380511yescharacteristic
457C2×C3C2×C3×#23512yes—
458C2×C3C2×C3×D3×D412yes—
459C2×C3C2×C3×#23712yes—
460D3—44no—
461C22—64no—
462C22—64no—
463C22—64no—
464C22—128no—
465C22C3×#379812yes—
466C22C3×#380212yes—
467C22C3×#380512yes—
468C22C22×C3×D3211yescharacteristic
469C4—22no—
470C4—64no—
471C4—94no—
472C4—182no—
473C3—22no—
474C3—21no—
475C3—41no—
476C3C2×#6391511yescharacteristic, direct product factor
477C3C2×C3×#88512yescomplemented
478C2—124no—
479C2C3×#6391514yesdirect product factor
480C2C2×C3×#379811yescharacteristic
481C2C2×C3×#380211yescharacteristic
482C2C2×C3×#380511yescharacteristic
subgroupG/H
centerC23×C3C2×D32
commutator, cosocleC22×C32C24×C3
FrattiniC22C22×C3×D32
FittingC22×C33×C4C22
socleC23×C33C23
prime radicalC22×C3×#625C2
hypercenterC22×C3×C4D32
hypocenterC32C2×C3×#124
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C23×C3 ◁ C22×C3×C4

Complements

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

NH ≅ G/Ntype
1. C22×C3×#627C2semidirect
3. C2×C3×C4×C3⋊Dic3C2semidirect
4. C2×C32×(C2×D3)⋊C4C2semidirect
5. C2×C3×(C2×D3)⋊Dic3C2semidirect
6. C3×#63915C2direct (N × H)
8. C2×#63915C3direct (N × H)
15. C2×C3×#627C22semidirect
18. C3×C4×C3⋊Dic3C22semidirect
19. C32×(C2×D3)⋊C4C22semidirect
20. C3×(C2×D3)⋊Dic3C22semidirect
23. C22×#627C2×C3semidirect
26. C2×C4×C3⋊Dic3C2×C3semidirect
30. C2×C3×(C2×D3)⋊C4D3semidirect
31. C2×C3×(C2×D3)⋊C4C2×C3semidirect
35. C2×(C2×D3)⋊Dic3C2×C3semidirect
36. #63915C2×C3direct (N × H)
78. C2×#627C22×C3semidirect
82. C4×C3⋊Dic3C22×C3semidirect
86. C3×(C2×D3)⋊C4C2×D3semidirect
87. C3×(C2×D3)⋊C4C22×C3semidirect
90. (C2×D3)⋊Dic3C22×C3semidirect
115. C2×(C2×D3)⋊C4C3×D3semidirect
175. C2×C33#124semidirect
214. (C2×D3)⋊C4C2×C3×D3semidirect
274. C33C2×#124semidirect
343. C2×C32#885semidirect
344. C2×C32C3×#124semidirect
420. C32C2×#885semidirect
421. C32C2×C3×#124semidirect
452. C2×C3#63915direct (N × H)
453. C2×C3C3×#885semidirect
476. C3C2×#63915direct (N × H)
477. C3C2×C3×#885semidirect
479. C2C3×#63915direct (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
C22×C3×#6252
C2×C3×C4×C3⋊Dic32
C2×C32×(C2×D3)⋊C42
C2×C3×(C2×D3)⋊Dic32
C3×#639152
C2×C3×#8853
C2×#639153