Finite Group Databasebeta

← C2×C3×#64125 #13179512

Subgroups — C2×C3×#64125 #13179512

Order 1728.

1868 conjugacy classes of subgroups.

The subgroup lattice has 466 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#3832C211yesmaximal, characteristic, complemented
2C2×C3×#3830C211yesmaximal, characteristic, complemented
3C2×C33×SD16C211yesmaximal, characteristic, complemented
4C2×C3×#3756C211yesmaximal, characteristic, complemented
5C2×C3×#3753C211yesmaximal, characteristic, complemented
6C3×#64125C218yesmaximal, direct product factor
7C2×C3×#3744C211yesmaximal, characteristic, complemented
8C2×C3×C8×C3⋊D3C211yesmaximal, characteristic, complemented
9C2×#64125C311yesmaximal, characteristic, direct product factor
10C2×C3×#916—34nomaximal
11C2×C33×Q8C2211yescharacteristic
12C2×C33×D4C2211yescharacteristic
13C2×C3×#638—21no—
14C22×C3×C3⋊Dic3—21no—
15C3×#3832—22no—
16C3×#3832C2214yescomplemented
17C3×#3830—22no—
18C3×#3830C2214yescomplemented
19C2×C3×C4⋊(C3⋊D3)—21no—
20C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
21C2×C3×C4×C3⋊D3—21no—
22C2×C3×C4×C3⋊D3C2211yescharacteristic
23C2×C3×#634C2211yescharacteristic
24C33×SD16C2214yescomplemented
25C2×C33×C8C2211yescharacteristic
26C3×#3756C2214yescomplemented
27C3×#3753C2214yescomplemented
28C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
29C3×#3744C2214yescomplemented
30C3×C8×C3⋊D3C2214yescomplemented
31C2×#3832C2×C311yescharacteristic, complemented
32C2×#3830C2×C311yescharacteristic, complemented
33C2×C3×#239—34no—
34C2×C3×#237—34no—
35C2×C32×SD16—24no—
36C2×C32×SD16D314yescomplemented
37C2×C32×SD16C2×C311yescharacteristic, complemented
38C2×#3756C2×C311yescharacteristic, complemented
39C2×#3753C2×C311yescharacteristic, complemented
40#64125C2×C318yesdirect product factor
41C2×#3744C2×C311yescharacteristic, complemented
42C2×C8×C3⋊D3C2×C311yescharacteristic, complemented
43C2×C3×#216—34no—
44C2×C3×#213—34no—
45C3×#916—332no—
46C2×C3×#204—34no—
47C2×C3×D3×C8—34no—
48C23×C33—21no—
49C22×C3×C3⋊D3—21no—
50C22×C3×C3⋊D3D411yescharacteristic
51C33×Q8—21no—
52C33×Q8C2312yes—
53C33×D4—21no—
54C33×D4C2312yes—
55C2×C33×C4—21no—
56C2×C33×C4C2311yescharacteristic
57C3×#638—24no—
58C2×C3×C3⋊Dic3—21no—
59C2×C3×C3⋊Dic3—24no—
60C2×C3×C3⋊Dic3D411yescharacteristic
61C3×C4⋊(C3⋊D3)—24no—
62C3×C4⋊(C3⋊D3)—21no—
63C3×C4⋊(C3⋊D3)C2312yes—
64C3×C4×C3⋊D3—24no—
65C3×C4×C3⋊D3D412yes—
66C3×C4×C3⋊D3C2312yes—
67C3×#634—21no—
68C3×#634C2312yes—
69C33×C8C2312yes—
70C3×C3⋊(C3⋊C8)C2312yes—
71C2×C3×#135—91no—
72C2×#916—34no—
73C2×C32×Q8—24no—
74C2×C32×Q8C2×D314yes—
75C2×C32×Q8C22×C311yescharacteristic
76C2×C32×D4—24no—
77C2×C32×D4C2×D314yes—
78C2×C32×D4C22×C311yescharacteristic
79C2×#638—21no—
80C22×C3⋊Dic3—21no—
81#3832—22no—
82#3832C22×C314yescomplemented
83#3830—22no—
84#3830C22×C314yescomplemented
85C2×C4⋊(C3⋊D3)—21no—
86C2×C4⋊(C3⋊D3)C22×C311yescharacteristic
87C2×C4×C3⋊D3—21no—
88C2×C4×C3⋊D3C22×C311yescharacteristic
89C2×#634C22×C311yescharacteristic
90C2×C3×#67—64no—
91C22×C3×Dic3—64no—
92C3×#239—316no—
93C3×#239—68no—
94C3×#237—316no—
95C3×#237—68no—
96C2×C3×D12—34no—
97C2×C3×D12—64no—
98C2×C3×C4×D3—34no—
99C2×C3×C4×D3—64no—
100C2×C3×Dic6—34no—
101C32×SD16—216no—
102C32×SD16C2×D3116yescomplemented
103C32×SD16C22×C314yescomplemented
104C2×C32×C8—24no—
105C2×C32×C8C2×D314yes—
106C2×C32×C8C22×C311yescharacteristic
107#3756C22×C314yescomplemented
108#3753C22×C314yescomplemented
109C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
110#3744C22×C314yescomplemented
111C8×C3⋊D3C22×C314yescomplemented
112C3×#216—316no—
113C3×#213—316no—
114C2×C3×C3⋊C8—34no—
115C3×#204—316no—
116C3×D3×C8—316no—
117C22×C33—22no—
118C22×C33—42no—
119C22×C33C2×D411yescharacteristic
120C2×C3×C3⋊D3—22no—
121C2×C3×C3⋊D3—22no—
122C2×C3×C3⋊D3—42no—
123C2×C3×C3⋊D3C2×D412yes—
124C33×C4—22no—
125C33×C4C2×D411yescharacteristic
126C33×C4C2411yescharacteristic
127C3×C3⋊Dic3—22no—
128C3×C3⋊Dic3C2×D412yes—
129C2×C3×#41—91no—
130C2×C3×#41—91no—
131C2×#239—34no—
132C2×#237—34no—
133C3×#135—98no—
134C2×C3×Q16—91no—
135C2×C3×SD16—24no—
136C2×C3×SD16—91no—
137C2×C3×SD16C3×D314yescomplemented
138C2×C3×SD16C3⋊D311yescharacteristic, complemented
139C2×C3×D8—91no—
140C22×C3×C8—91no—
141C2×#216—34no—
142C2×#213—34no—
143#916—332no—
144C2×#204—34no—
145C2×D3×C8—34no—
146C23×C32—21no—
147C23×C32—24no—
148C23×C32—44no—
149C22×C3⋊D3—21no—
150C22×C3⋊D3C3×D411yescharacteristic
151C22×C3×D3—34no—
152C22×C3×D3—64no—
153C32×Q8—21no—
154C32×Q8—28no—
155C32×Q8—24no—
156C32×Q8—44no—
157C32×Q8C22×D318yes—
158C32×Q8C23×C312yes—
159C32×D4—21no—
160C32×D4—28no—
161C32×D4—24no—
162C32×D4—44no—
163C32×D4C22×D318yes—
164C32×D4C23×C312yes—
165C2×C32×C4—21no—
166C2×C32×C4—24no—
167C2×C32×C4—24no—
168C2×C32×C4—44no—
169C2×C32×C4C22×D314yes—
170C2×C32×C4C23×C311yescharacteristic
171#638—24no—
172C2×C3⋊Dic3—21no—
173C2×C3⋊Dic3—24no—
174C2×C3⋊Dic3C3×D411yescharacteristic
175C4⋊(C3⋊D3)—24no—
176C4⋊(C3⋊D3)—21no—
177C4⋊(C3⋊D3)C23×C312yes—
178C4×C3⋊D3—24no—
179C4×C3⋊D3C3×D412yes—
180C4×C3⋊D3C23×C312yes—
181#634—21no—
182#634C23×C312yes—
183C3×#67—616no—
184C2×C3×Dic3—34no—
185C2×C3×Dic3—616no—
186C2×C3×Dic3—64no—
187C3×D12—38no—
188C3×D12—616no—
189C3×D12—64no—
190C3×C4×D3—38no—
191C3×C4×D3—38no—
192C3×C4×D3—616no—
193C3×Dic6—38no—
194C3×Dic6—64no—
195C32×C8—28no—
196C32×C8C22×D318yes—
197C32×C8C23×C312yes—
198C3⋊(C3⋊C8)C23×C312yes—
199C3×C3⋊C8—38no—
200C2×#135—91noSylow
201C2×C33—42no—
202C2×C33#13512yescomplemented
203C2×C33C22×D411yescharacteristic
204C3×C3⋊D3—22no—
205C3×C3⋊D3—42no—
206C3×#41—94no—
207C3×#41—94no—
208C3×#41—182no—
209C3×#41—182no—
210C2×C3×Q8—24no—
211C2×C3×Q8—91no—
212C2×C3×Q8C2×C3×D314yes—
213C2×C3×Q8C2×C3⋊D311yescharacteristic
214C2×C3×D4—24no—
215C2×C3×D4—91no—
216C2×C3×D4—181no—
217C2×C3×D4—181no—
218C2×C3×D4C2×C3×D314yes—
219C2×C3×D4C2×C3⋊D311yescharacteristic
220C22×C3×C4—91no—
221C22×C3×C4—181no—
222C22×C3×C4—181no—
223C2×#67—64no—
224C22×Dic3—64no—
225#239—316no—
226#239—68no—
227#237—316no—
228#237—68no—
229C2×D12—34no—
230C2×D12—64no—
231C2×C4×D3—34no—
232C2×C4×D3—64no—
233C2×Dic6—34no—
234C3×Q16—94no—
235C3×SD16—216no—
236C3×SD16—94no—
237C3×SD16C2×C3×D3116yescomplemented
238C3×SD16C2×C3⋊D314yescomplemented
239C3×D8—94no—
240C2×C3×C8—24no—
241C2×C3×C8—91no—
242C2×C3×C8—94no—
243C2×C3×C8C2×C3×D314yes—
244C2×C3×C8C2×C3⋊D311yescharacteristic
245#216—316no—
246#213—316no—
247C2×C3⋊C8—34no—
248#204—316no—
249D3×C8—316no—
250C22×C32—22no—
251C22×C32—24no—
252C22×C32—28no—
253C22×C32—48no—
254C22×C32—42no—
255C22×C32—48no—
256C22×C32—88no—
257C22×C32D3×D414yes—
258C22×C32C2×C3×D411yescharacteristic
259C2×C3⋊D3—22no—
260C2×C3⋊D3—22no—
261C2×C3⋊D3—42no—
262C2×C3⋊D3C2×C3×D412yes—
263C2×C3×D3—38no—
264C2×C3×D3—68no—
265C2×C3×D3—68no—
266C2×C3×D3—128no—
267C32×C4—22no—
268C32×C4—24no—
269C32×C4—24no—
270C32×C4—28no—
271C32×C4—48no—
272C32×C4D3×D414yes—
273C32×C4C2×C3×D411yescharacteristic
274C32×C4C23×D314yes—
275C32×C4C24×C311yescharacteristic
276C3⋊Dic3—22no—
277C3⋊Dic3C2×C3×D412yes—
278C3×Dic3—38no—
279C3×Dic3—68no—
280C2×#41—91no—
281C2×#41—91no—
282#135—98no—
283C2×Q16—91no—
284C2×SD16—91no—
285C2×SD16C3×C3⋊D311yescharacteristic, complemented
286C2×D8—91no—
287C22×C8—91no—
288C33C2×#13511yescharacteristic, complemented, Sylow
289C23×C3—24no—
290C23×C3—21no—
291C23×C3—44no—
292C23×C3—91no—
293C23×C3—181no—
294C22×D3—34no—
295C22×D3—64no—
296C3×Q8—24no—
297C3×Q8—21no—
298C3×Q8—28no—
299C3×Q8—44no—
300C3×Q8—92no—
301C3×Q8—181no—
302C3×Q8C22×C3×D318yes—
303C3×Q8C22×C3⋊D312yes—
304C3×D4—24no—
305C3×D4—21no—
306C3×D4—28no—
307C3×D4—44no—
308C3×D4—92no—
309C3×D4—184no—
310C3×D4—184no—
311C3×D4—181no—
312C3×D4C22×C3×D318yes—
313C3×D4C22×C3⋊D312yes—
314C2×C3×C4—24no—
315C2×C3×C4—21no—
316C2×C3×C4—24no—
317C2×C3×C4—44no—
318C2×C3×C4—91no—
319C2×C3×C4—92no—
320C2×C3×C4—92no—
321C2×C3×C4—184no—
322C2×C3×C4—184no—
323C2×C3×C4—181no—
324C2×C3×C4C22×C3×D314yes—
325C2×C3×C4C22×C3⋊D311yescharacteristic
326#67—616no—
327C2×Dic3—34no—
328C2×Dic3—616no—
329C2×Dic3—64no—
330D12—38no—
331D12—616no—
332D12—64no—
333C4×D3—38no—
334C4×D3—38no—
335C4×D3—616no—
336Dic6—38no—
337Dic6—64no—
338C3×C8—28no—
339C3×C8—92no—
340C3×C8C22×C3×D318yes—
341C3×C8C22×C3⋊D312yes—
342C3⋊C8—38no—
343C2×C32—28no—
344C2×C32—24no—
345C2×C32—42no—
346C2×C32—48no—
347C2×C32—88no—
348C2×C32#91618yescomplemented
349C2×C32C3×#13512yescomplemented
350C2×C32C2×D3×D414yes—
351C2×C32C22×C3×D411yescharacteristic
352C3⋊D3—22no—
353C3⋊D3—42no—
354C3×D3—68no—
355C3×D3—128no—
356#41—94no—
357#41—94no—
358#41—182no—
359#41—182no—
360C2×Q8—91no—
361C2×Q8C2×C3×C3⋊D311yescharacteristic
362C2×D4—91no—
363C2×D4—181no—
364C2×D4—181no—
365C2×D4C2×C3×C3⋊D311yescharacteristic
366C22×C4—91no—
367C22×C4—181no—
368C22×C4—181no—
369Q16—94no—
370SD16—94no—
371SD16C2×C3×C3⋊D314yescomplemented
372D8—94no—
373C2×C8—91no—
374C2×C8—94no—
375C2×C8C2×C3×C3⋊D311yescharacteristic
376C22×C3—28no—
377C22×C3—22no—
378C22×C3—24no—
379C22×C3—48no—
380C22×C3—48no—
381C22×C3—42no—
382C22×C3—88no—
383C22×C3—92no—
384C22×C3—182no—
385C22×C3—182no—
386C22×C3—362no—
387C22×C3C3×D3×D414yes—
388C22×C3D4×C3⋊D311yescharacteristic
389C2×D3—38no—
390C2×D3—68no—
391C2×D3—68no—
392C2×D3—128no—
393C3×C4—28no—
394C3×C4—22no—
395C3×C4—24no—
396C3×C4—24no—
397C3×C4—48no—
398C3×C4—92no—
399C3×C4—182no—
400C3×C4C3×D3×D414yes—
401C3×C4D4×C3⋊D311yescharacteristic
402C3×C4C23×C3×D314yes—
403C3×C4C23×C3⋊D311yescharacteristic
404Dic3—38no—
405Dic3—68no—
406C32—24no—
407C32C2×#91614yescomplemented
408C32C2×C3×#13511yescharacteristic, complemented
409C23—21no—
410C23—91no—
411C23—181no—
412Q8—21no—
413Q8—92no—
414Q8—181no—
415Q8C22×C3×C3⋊D312yes—
416D4—21no—
417D4—92no—
418D4—184no—
419D4—184no—
420D4—181no—
421D4C22×C3×C3⋊D312yes—
422C2×C4—21no—
423C2×C4—91no—
424C2×C4—92no—
425C2×C4—92no—
426C2×C4—184no—
427C2×C4—184no—
428C2×C4—181no—
429C2×C4C22×C3×C3⋊D311yescharacteristic
430C8—92no—
431C8C22×C3×C3⋊D312yes—
432C2×C3—28no—
433C2×C3—24no—
434C2×C3—48no—
435C2×C3—42no—
436C2×C3—88no—
437C2×C3—182no—
438C2×C3—362no—
439C2×C3C3×#91618yescomplemented
440C2×C3#6412512yesdirect product factor
441C2×C3C2×C3×D3×D414yes—
442C2×C3C2×D4×C3⋊D311yescharacteristic
443D3—68no—
444D3—128no—
445C22—22no—
446C22—42no—
447C22—92no—
448C22—182no—
449C22—182no—
450C22—362no—
451C22C3×D4×C3⋊D311yescharacteristic
452C4—22no—
453C4—92no—
454C4—182no—
455C4C3×D4×C3⋊D311yescharacteristic
456C4C23×C3×C3⋊D311yescharacteristic
457C3—24no—
458C3C2×C3×#91614yescomplemented
459C3C2×#6412511yescharacteristic, direct product factor
460C2—42no—
461C2—182no—
462C2—362no—
463C2C3×#6412512yesdirect product factor
464C2C2×C3×D4×C3⋊D311yescharacteristic
subgroupG/H
centerC22×C3D4×C3⋊D3
commutator, cosocleC32×C4C24×C3
FrattiniC4C23×C3×C3⋊D3
FittingC2×C33×SD16C2
socleC22×C33C2×D4
hypercenterC2×C3×SD16C3⋊D3
hypocenterC32C2×C3×#135
LCSG ▷ C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C2×C3×C4 ◁ C2×C3×SD16

Complements

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

NH ≅ G/Ntype
1. C2×C3×#3832C2semidirect
2. C2×C3×#3830C2semidirect
3. C2×C33×SD16C2semidirect
4. C2×C3×#3756C2semidirect
5. C2×C3×#3753C2semidirect
6. C3×#64125C2direct (N × H)
7. C2×C3×#3744C2semidirect
8. C2×C3×C8×C3⋊D3C2semidirect
9. C2×#64125C3direct (N × H)
16. C3×#3832C22semidirect
18. C3×#3830C22semidirect
24. C33×SD16C22semidirect
26. C3×#3756C22semidirect
27. C3×#3753C22semidirect
29. C3×#3744C22semidirect
30. C3×C8×C3⋊D3C22semidirect
31. C2×#3832C2×C3semidirect
32. C2×#3830C2×C3semidirect
36. C2×C32×SD16D3semidirect
37. C2×C32×SD16C2×C3semidirect
38. C2×#3756C2×C3semidirect
39. C2×#3753C2×C3semidirect
40. #64125C2×C3direct (N × H)
41. C2×#3744C2×C3semidirect
42. C2×C8×C3⋊D3C2×C3semidirect
82. #3832C22×C3semidirect
84. #3830C22×C3semidirect
102. C32×SD16C2×D3semidirect
103. C32×SD16C22×C3semidirect
107. #3756C22×C3semidirect
108. #3753C22×C3semidirect
110. #3744C22×C3semidirect
111. C8×C3⋊D3C22×C3semidirect
137. C2×C3×SD16C3×D3semidirect
138. C2×C3×SD16C3⋊D3semidirect
202. C2×C33#135semidirect
237. C3×SD16C2×C3×D3semidirect
238. C3×SD16C2×C3⋊D3semidirect
285. C2×SD16C3×C3⋊D3semidirect
288. C33C2×#135semidirect
348. C2×C32#916semidirect
349. C2×C32C3×#135semidirect
371. SD16C2×C3×C3⋊D3semidirect
407. C32C2×#916semidirect
408. C32C2×C3×#135semidirect
439. C2×C3C3×#916semidirect
440. C2×C3#64125direct (N × H)
458. C3C2×C3×#916semidirect
459. C3C2×#64125direct (N × H)
463. C2C3×#64125direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#38322
C2×C3×#38302
C2×C33×SD162
C2×C3×#37562
C2×C3×#37532
C3×#641252
C2×C3×#37442
C2×C3×C8×C3⋊D32
C2×#641253
C2×C3×#9163