Finite Group Databasebeta

← C2×C3×#64298 #13181533

Subgroups — C2×C3×#64298 #13181533

Order 1728.

3714 conjugacy classes of subgroups.

The subgroup lattice has 476 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×#634C211yesmaximal, characteristic, complemented
2C2×C32×#235C212yesmaximal, complemented
3C2×C3×#3805C214yesmaximal, complemented
4C3×#64298C2116yesmaximal, direct product factor
5C2×C3×#3795C214yesmaximal, complemented
6C2×C3×D3×Dic6C214yesmaximal, complemented
7C2×C3×#1010—32nomaximal
8C2×#64298C311yesmaximal, characteristic, direct product factor
9C22×C33×C4C2211yescharacteristic
10C22×C3×C3⋊Dic3C2212yes—
11C2×C3×#634C2214yes—
12C2×C3×#634C2218yescomplemented
13C2×C32×#67C2214yes—
14C32×#235C22116yescomplemented
15C2×C32×D12C2212yes—
16C2×C32×C4×D3C2214yes—
17C2×C32×Dic6C2212yes—
18C2×C3×#627C2214yes—
19C2×C3×#625C2214yes—
20C3×#3805C22132yescomplemented
21C2×C3×D3×Dic3C2218yes—
22C3×#3795C22132yescomplemented
23C3×D3×Dic6C22132yescomplemented
24C2×C32×#41—32no—
25C22×#634C2×C311yescharacteristic, complemented
26C3×#1010—332no—
27C2×C3×D3×Q8—34no—
28C2×C3×#237—38no—
29C2×C3×#237—34no—
30C2×C3×#235—22no—
31C2×C3×#235—34no—
32C2×C3×#235D312yescomplemented
33C2×C3×#235C2×C312yescomplemented
34C22×C3×Dic6—32no—
35C22×C3×Dic6—34no—
36C22×C3×Dic6—61no—
37C2×#3805C2×C314yescomplemented
38#64298C2×C3116yesdirect product factor
39C2×#3795C2×C314yescomplemented
40C2×D3×Dic6C2×C314yescomplemented
41C23×C33C2311yescharacteristic
42C22×C32×D3C2314yes—
43C2×C33×C4C2312yes—
44C2×C33×C4C2314yes—
45C2×C3×C3⋊Dic3C2314yes—
46C2×C3×C3⋊Dic3C2318yes—
47C3×#634C23116yes—
48C32×#67C23116yes—
49C2×C32×Dic3C2314yes—
50C32×D12C2318yes—
51C32×C4×D3C23116yes—
52C32×Dic6C2318yes—
53C3×#627C23116yes—
54C3×#625C23116yes—
55C3×D3×Dic3C23132yes—
56C2×C3×2−1+4—91no—
57C2×#1010—32no—
58C32×#41—316no—
59C2×C32×Q8—32no—
60C2×C32×D4—34no—
61C2×C32×D4—32no—
62C22×C32×C4—21no—
63C22×C32×C4—22no—
64C22×C32×C4—34no—
65C22×C32×C4—41no—
66C22×C32×C4C2×D312yes—
67C22×C32×C4C22×C311yescharacteristic
68C22×C3⋊Dic3C22×C312yes—
69C2×#634C22×C314yes—
70C2×#634C22×C318yescomplemented
71C2×C3×#67—24no—
72C2×C3×#67—38no—
73C2×C3×#67C2×D314yes—
74C2×C3×#67C22×C314yes—
75C22×C3×Dic3—34no—
76C22×C3×Dic3—38no—
77C22×C3×Dic3—62no—
78C3×D3×Q8—332no—
79C3×#237—364no—
80C3×#237—332no—
81C3×#235—216no—
82C3×#235—332no—
83C3×#235C2×D3116yescomplemented
84C3×#235C22×C3116yescomplemented
85C2×C3×D12—22no—
86C2×C3×D12C2×D312yes—
87C2×C3×D12C22×C312yes—
88C2×C3×C4×D3—24no—
89C2×C3×C4×D3—38no—
90C2×C3×C4×D3C2×D314yes—
91C2×C3×C4×D3C22×C314yes—
92C2×C3×Dic6—22no—
93C2×C3×Dic6—34no—
94C2×C3×Dic6—34no—
95C2×C3×Dic6—316no—
96C2×C3×Dic6—38no—
97C2×C3×Dic6—332no—
98C2×C3×Dic6—64no—
99C2×C3×Dic6—68no—
100C2×C3×Dic6C2×D312yes—
101C2×C3×Dic6C22×C312yes—
102C2×#627C22×C314yes—
103C2×#625C22×C314yes—
104#3805C22×C3132yescomplemented
105C2×D3×Dic3C22×C318yes—
106#3795C22×C3132yescomplemented
107D3×Dic6C22×C3132yescomplemented
108C22×C33—22no—
109C22×C33C2412yes—
110C22×C33C2411yescharacteristic
111C2×C32×D3—28no—
112C2×C32×D3C2418yes—
113C33×C4C2414yes—
114C3×C3⋊Dic3C2418yes—
115C32×Dic3C2418yes—
116C3×2−1+4—916no—
117C2×C3×#41—32no—
118C2×C3×#41—32no—
119C2×C3×#41—62no—
120C2×C3×#41—94no—
121C2×C3×#41—94no—
122C22×C3×Q8—94no—
123C22×C3×Q8—91no—
124#1010—332no—
125C2×D3×Q8—34no—
126C2×#237—38no—
127C2×#237—34no—
128C2×#235—34no—
129C2×#235C3×D312yescomplemented
130C22×Dic6—32no—
131C22×Dic6—34no—
132C22×Dic6—61no—
133C23×C32—21no—
134C23×C32—22no—
135C23×C32—34no—
136C23×C32—41no—
137C23×C32C22×D312yes—
138C23×C32C23×C311yescharacteristic
139C22×C3×D3—24no—
140C22×C3×D3C22×D314yes—
141C22×C3×D3C23×C314yes—
142C32×Q8—38no—
143C32×D4—316no—
144C32×D4—38no—
145C2×C32×C4—24no—
146C2×C32×C4—22no—
147C2×C32×C4—22no—
148C2×C32×C4—22no—
149C2×C32×C4—28no—
150C2×C32×C4—34no—
151C2×C32×C4—316no—
152C2×C32×C4—42no—
153C2×C32×C4—44no—
154C2×C32×C4C22×D312yes—
155C2×C32×C4C22×D312yes—
156C2×C32×C4C22×D318yes—
157C2×C32×C4C23×C312yes—
158C2×C32×C4C23×C314yes—
159C2×C3⋊Dic3C23×C314yes—
160C2×C3⋊Dic3C23×C318yes—
161#634C23×C3116yes—
162C3×#67—216no—
163C3×#67—332no—
164C3×#67C22×D3116yes—
165C3×#67C23×C3116yes—
166C2×C3×Dic3—24no—
167C2×C3×Dic3—38no—
168C2×C3×Dic3—316no—
169C2×C3×Dic3—332no—
170C2×C3×Dic3—68no—
171C2×C3×Dic3—64no—
172C2×C3×Dic3C22×D314yes—
173C2×C3×Dic3C23×C314yes—
174C3×D12—28no—
175C3×D12C22×D318yes—
176C3×D12C23×C318yes—
177C3×C4×D3—216no—
178C3×C4×D3—332no—
179C3×C4×D3C22×D3116yes—
180C3×C4×D3C23×C3116yes—
181C3×Dic6—28no—
182C3×Dic6—316no—
183C3×Dic6—316no—
184C3×Dic6—332no—
185C3×Dic6—616no—
186C3×Dic6C22×D318yes—
187C3×Dic6C23×C318yes—
188#627C23×C3116yes—
189#625C23×C3116yes—
190D3×Dic3C23×C3132yes—
191C2×2−1+4—91noSylow
192C2×C33—22no—
193C2×C332−1+412yescomplemented
194C2×C33C2511yescharacteristic
195C32×D3—28no—
196C3×#41—316no—
197C3×#41—316no—
198C3×#41—616no—
199C3×#41—932no—
200C3×#41—932no—
201C2×C3×Q8—32no—
202C2×C3×Q8—32no—
203C2×C3×Q8—62no—
204C2×C3×Q8—94no—
205C2×C3×Q8—932no—
206C2×C3×Q8—94no—
207C2×C3×Q8—98no—
208C2×C3×D4—34no—
209C2×C3×D4—34no—
210C2×C3×D4—32no—
211C2×C3×D4—32no—
212C2×C3×D4—62no—
213C2×C3×D4—64no—
214C2×C3×D4—94no—
215C22×C3×C4—21no—
216C22×C3×C4—22no—
217C22×C3×C4—34no—
218C22×C3×C4—34no—
219C22×C3×C4—41no—
220C22×C3×C4—64no—
221C22×C3×C4—98no—
222C22×C3×C4—92no—
223C22×C3×C4D3211yescharacteristic
224C22×C3×C4C2×C3×D312yes—
225C2×#67—38no—
226C2×#67C2×C3×D314yes—
227C22×Dic3—34no—
228C22×Dic3—38no—
229C22×Dic3—62no—
230D3×Q8—332no—
231#237—364no—
232#237—332no—
233#235—332no—
234#235C2×C3×D3116yescomplemented
235C2×D12C2×C3×D312yes—
236C2×C4×D3—38no—
237C2×C4×D3C2×C3×D314yes—
238C2×Dic6—34no—
239C2×Dic6—34no—
240C2×Dic6—316no—
241C2×Dic6—38no—
242C2×Dic6—332no—
243C2×Dic6—64no—
244C2×Dic6—68no—
245C2×Dic6C2×C3×D312yes—
246C22×C32—22no—
247C22×C32—24no—
248C22×C32—24no—
249C22×C32—22no—
250C22×C32—21no—
251C22×C32—22no—
252C22×C32—24no—
253C22×C32—38no—
254C22×C32—41no—
255C22×C32—42no—
256C22×C32—44no—
257C22×C32—44no—
258C22×C32—68no—
259C22×C32C23×D314yes—
260C22×C32C23×D312yes—
261C22×C32C24×C312yes—
262C22×C32C24×C311yescharacteristic
263C2×C3×D3—28no—
264C2×C3×D3—28no—
265C2×C3×D3—28no—
266C2×C3×D3—48no—
267C2×C3×D3C23×D318yes—
268C2×C3×D3C24×C318yes—
269C32×C4—24no—
270C32×C4—24no—
271C32×C4—24no—
272C32×C4—38no—
273C32×C4—44no—
274C32×C4C23×D314yes—
275C32×C4C23×D314yes—
276C32×C4C24×C314yes—
277C3⋊Dic3C24×C318yes—
278C3×Dic3—28no—
279C3×Dic3—316no—
280C3×Dic3—68no—
281C3×Dic3C23×D318yes—
282C3×Dic3C24×C318yes—
2832−1+4—916no—
284C2×#41—32no—
285C2×#41—94no—
286C2×#41—94no—
287C22×Q8—94no—
288C22×Q8—91no—
289C33C2×2−1+411yescharacteristic, complemented, Sylow
290C23×C3—21no—
291C23×C3—22no—
292C23×C3—34no—
293C23×C3—34no—
294C23×C3—41no—
295C23×C3—64no—
296C23×C3C2×D3211yescharacteristic
297C23×C3C22×C3×D312yes—
298C22×D3C22×C3×D314yes—
299C3×Q8—38no—
300C3×Q8—38no—
301C3×Q8—68no—
302C3×Q8—916no—
303C3×Q8—916no—
304C3×D4—316no—
305C3×D4—316no—
306C3×D4—38no—
307C3×D4—38no—
308C3×D4—68no—
309C3×D4—616no—
310C3×D4—916no—
311C2×C3×C4—22no—
312C2×C3×C4—22no—
313C2×C3×C4—28no—
314C2×C3×C4—22no—
315C2×C3×C4—24no—
316C2×C3×C4—34no—
317C2×C3×C4—34no—
318C2×C3×C4—316no—
319C2×C3×C4—316no—
320C2×C3×C4—42no—
321C2×C3×C4—44no—
322C2×C3×C4—616no—
323C2×C3×C4—64no—
324C2×C3×C4—932no—
325C2×C3×C4—98no—
326C2×C3×C4—94no—
327C2×C3×C4C2×D3214yes—
328C2×C3×C4C2×D3212yes—
329C2×C3×C4C22×C3×D312yes—
330C2×C3×C4C22×C3×D312yes—
331C2×C3×C4C22×C3×D318yes—
332#67—332no—
333#67C22×C3×D3116yes—
334C2×Dic3—38no—
335C2×Dic3—316no—
336C2×Dic3—332no—
337C2×Dic3—64no—
338C2×Dic3—68no—
339C2×Dic3C22×C3×D314yes—
340D12C22×C3×D318yes—
341C4×D3—332no—
342C4×D3C22×C3×D3116yes—
343Dic6—316no—
344Dic6—316no—
345Dic6—332no—
346Dic6—616no—
347Dic6C22×C3×D318yes—
348C2×C32—22no—
349C2×C32—24no—
350C2×C32—24no—
351C2×C32—22no—
352C2×C32—24no—
353C2×C32—21no—
354C2×C32—22no—
355C2×C32—42no—
356C2×C32—41no—
357C2×C32—44no—
358C2×C32—44no—
359C2×C32—68no—
360C2×C32#101014yescomplemented
361C2×C32C3×2−1+412yescomplemented
362C2×C32C24×D312yes—
363C2×C32C25×C311yescharacteristic
364C3×D3—28no—
365C3×D3—28no—
366C3×D3—48no—
367#41—316no—
368#41—932no—
369#41—932no—
370C2×Q8—32no—
371C2×Q8—94no—
372C2×Q8—932no—
373C2×Q8—94no—
374C2×Q8—98no—
375C2×D4—34no—
376C2×D4—32no—
377C2×D4—94no—
378C22×C4—34no—
379C22×C4—98no—
380C22×C4—92no—
381C22×C4C3×D3211yescharacteristic
382C22×C3—24no—
383C22×C3—22no—
384C22×C3—24no—
385C22×C3—22no—
386C22×C3—21no—
387C22×C3—22no—
388C22×C3—24no—
389C22×C3—38no—
390C22×C3—38no—
391C22×C3—44no—
392C22×C3—44no—
393C22×C3—42no—
394C22×C3—41no—
395C22×C3—68no—
396C22×C3—68no—
397C22×C3—68no—
398C22×C3—128no—
399C22×C3C22×D3212yes—
400C22×C3C22×D3211yescharacteristic
401C22×C3C23×C3×D314yes—
402C22×C3C23×C3×D312yes—
403C2×D3—28no—
404C2×D3C23×C3×D318yes—
405C3×C4—24no—
406C3×C4—24no—
407C3×C4—24no—
408C3×C4—38no—
409C3×C4—38no—
410C3×C4—44no—
411C3×C4—68no—
412C3×C4—98no—
413C3×C4C22×D3214yes—
414C3×C4C23×C3×D314yes—
415C3×C4C23×C3×D314yes—
416Dic3—316no—
417Dic3—68no—
418Dic3C23×C3×D318yes—
419C32—21no—
420C32—22no—
421C32—41no—
422C32C2×#101012yescomplemented
423C32C2×C3×2−1+411yescharacteristic, complemented
424C23—34no—
425C23C2×C3×D3211yescharacteristic
426Q8—38no—
427Q8—916no—
428Q8—916no—
429D4—316no—
430D4—38no—
431D4—916no—
432C2×C4—34no—
433C2×C4—316no—
434C2×C4—932no—
435C2×C4—94no—
436C2×C4—98no—
437C2×C4C2×C3×D3212yes—
438C2×C4C2×C3×D3214yes—
439C2×C3—24no—
440C2×C3—22no—
441C2×C3—24no—
442C2×C3—21no—
443C2×C3—22no—
444C2×C3—22no—
445C2×C3—24no—
446C2×C3—44no—
447C2×C3—44no—
448C2×C3—41no—
449C2×C3—42no—
450C2×C3—68no—
451C2×C3—68no—
452C2×C3—128no—
453C2×C3#6429812yesdirect product factor
454C2×C3C3×#101014yescomplemented
455C2×C3C23×D3211yescharacteristic
456C2×C3C24×C3×D312yes—
457D3—28no—
458C22—22no—
459C22—38no—
460C22—68no—
461C22C22×C3×D3212yes—
462C22C22×C3×D3211yescharacteristic
463C4—38no—
464C4—98no—
465C4C22×C3×D3214yes—
466C3—21no—
467C3—22no—
468C3—41no—
469C3C2×#6429811yescharacteristic, direct product factor
470C3C2×C3×#101012yescomplemented
471C2—22no—
472C2—68no—
473C2C3×#6429812yesdirect product factor
474C2C23×C3×D3211yescharacteristic
subgroupG/H
centerC22×C3C22×D32
commutator, cosocleC2×C32C25×C3
FrattiniC2C23×C3×D32
FittingC22×C33×C4C22
socleC22×C33C24
hypercenterC22×C3×C4D32
hypocenterC32C2×C3×2−1+4
LCSG ▷ C2×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C22×C3×C4

Complements

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

NH ≅ G/Ntype
1. C22×C3×#634C2semidirect
2. C2×C32×#235C2semidirect
3. C2×C3×#3805C2semidirect
4. C3×#64298C2direct (N × H)
5. C2×C3×#3795C2semidirect
6. C2×C3×D3×Dic6C2semidirect
8. C2×#64298C3direct (N × H)
12. C2×C3×#634C22semidirect
14. C32×#235C22semidirect
20. C3×#3805C22semidirect
22. C3×#3795C22semidirect
23. C3×D3×Dic6C22semidirect
25. C22×#634C2×C3semidirect
32. C2×C3×#235D3semidirect
33. C2×C3×#235C2×C3semidirect
37. C2×#3805C2×C3semidirect
38. #64298C2×C3direct (N × H)
39. C2×#3795C2×C3semidirect
40. C2×D3×Dic6C2×C3semidirect
70. C2×#634C22×C3semidirect
83. C3×#235C2×D3semidirect
84. C3×#235C22×C3semidirect
104. #3805C22×C3semidirect
106. #3795C22×C3semidirect
107. D3×Dic6C22×C3semidirect
129. C2×#235C3×D3semidirect
193. C2×C332−1+4semidirect
234. #235C2×C3×D3semidirect
289. C33C2×2−1+4semidirect
360. C2×C32#1010semidirect
361. C2×C32C3×2−1+4semidirect
422. C32C2×#1010semidirect
423. C32C2×C3×2−1+4semidirect
453. C2×C3#64298direct (N × H)
454. C2×C3C3×#1010semidirect
469. C3C2×#64298direct (N × H)
470. C3C2×C3×#1010semidirect
473. C2C3×#64298direct (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×#2352
C2×C3×#38052
C3×#642982
C2×C3×#37952
C2×C3×D3×Dic62
C2×C3×#10103
C2×#642983