Finite Group Databasebeta

← C3×#10595032 #13174608

Subgroups — C3×#10595032 #13174608

Order 1728.

1730 conjugacy classes of subgroups.

The subgroup lattice has 472 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C32×#235C212yesmaximal, complemented
2C2×C3×C4×C3⋊Dic3C211yesmaximal, characteristic, complemented
3C3×#63980C214yesmaximal, complemented
4C3×#63914C211yesmaximal, characteristic, complemented
5C3×C4×#625C214yesmaximal, complemented
6C3×#63873C211yesmaximal, characteristic, complemented
7C3×#63863C212yesmaximal, complemented
8C3×#5832—32nomaximal
9#10595032C311yesmaximal, characteristic, direct product factor
10C22×C33×C4C2211yescharacteristic
11C22×C3×C3⋊Dic3C2212yes—
12C2×C32×#67C2214yes—
13C32×#235—28no—
14C2×C32×D12C2212yes—
15C2×C32×C4×D3C2214yes—
16C2×C32×Dic6C2212yes—
17C2×C3×#625C2214yes—
18C3×C4×C3⋊Dic3C2211yescharacteristic
19C3×C4×C3⋊Dic3C2211yescharacteristic
20C3×C4×C3⋊Dic3C2212yes—
21C3×#3724C2214yes—
22C3×(C2×D3)⋊Dic3C2218yes—
23C2×C32×#41—32no—
24C2×C3×#235—22no—
25C2×C3×#235D312yescomplemented
26C2×C3×#235C2×C312yescomplemented
27C2×C4×C3⋊Dic3C2×C311yescharacteristic, complemented
28C3×#947—32no—
29C3×#944—32no—
30C3×#940—32no—
31C3×#939—38no—
32C3×#936—32no—
33C3×C4×#67—38no—
34C2×C3×C4×Dic3—32no—
35C2×C3×C4×Dic3—61no—
36#63980C2×C314yescomplemented
37#63914C2×C311yescharacteristic, complemented
38C4×#625C2×C314yescomplemented
39#63873C2×C311yescharacteristic, complemented
40#63863C2×C312yescomplemented
41C23×C33C2311yescharacteristic
42C22×C32×D3C2314yes—
43C2×C33×C4D414yescomplemented
44C2×C33×C4C2311yescharacteristic
45C2×C33×C4C2311yescharacteristic
46C2×C3×C3⋊Dic3—24no—
47C2×C3×C3⋊Dic3C2312yes—
48C2×C3×C3⋊Dic3C2312yes—
49C32×#67—28no—
50C2×C32×Dic3C2314yes—
51C32×D12—24no—
52C32×C4×D3—28no—
53C32×Dic6—24no—
54C3×#625—28no—
55C3×#532—91no—
56#5832—32no—
57C32×#41—68no—
58C2×C32×Q8—32no—
59C2×C32×D4—34no—
60C2×C32×D4—32no—
61C22×C32×C4—21no—
62C22×C32×C4—22no—
63C22×C32×C4—34no—
64C22×C32×C4—41no—
65C22×C32×C4C2×D312yes—
66C22×C32×C4C22×C311yescharacteristic
67C22×C3⋊Dic3C22×C312yes—
68C2×C3×#67—24no—
69C2×C3×#67—38no—
70C2×C3×#67C2×D314yes—
71C2×C3×#67C22×C314yes—
72C22×C3×Dic3—34no—
73C22×C3×Dic3—62no—
74C3×#235—216no—
75C3×#235—28no—
76C3×#235—28no—
77C2×C3×D12—22no—
78C2×C3×D12C2×D312yes—
79C2×C3×D12C22×C312yes—
80C2×C3×C4×D3—24no—
81C2×C3×C4×D3C2×D314yes—
82C2×C3×C4×D3C22×C314yes—
83C2×C3×Dic6—22no—
84C2×C3×Dic6C2×D312yes—
85C2×C3×Dic6C22×C312yes—
86C2×#625C22×C314yes—
87C4×C3⋊Dic3C22×C311yescharacteristic
88C4×C3⋊Dic3C22×C311yescharacteristic
89C4×C3⋊Dic3C22×C312yes—
90C3×#217—38no—
91C3×(C2×D3)⋊C4—38no—
92C3×Dic3⋊C4—38no—
93C3×C4×Dic3—34no—
94C3×C4×Dic3—32no—
95C3×C4×Dic3—32no—
96C3×C4×Dic3—61no—
97C3×C4×Dic3—61no—
98C3×C4×Dic3—62no—
99#3724C22×C314yes—
100(C2×D3)⋊Dic3C22×C318yes—
101C22×C33—22no—
102C22×C33C2×D412yes—
103C22×C33C2411yescharacteristic
104C2×C32×D3—24no—
105C2×C32×D3—28no—
106C33×C4C2×D412yes—
107C33×C4C2×D412yes—
108C3×C3⋊Dic3—22no—
109C3×C3⋊Dic3#4114yescomplemented
110C32×Dic3—24no—
111C2×C3×#41—32no—
112C2×C3×#41—32no—
113C2×C3×#41—62no—
114C2×#235C3×D312yescomplemented
115C3×C4⋊Q8—91no—
116C3×#127—91no—
117C3×#124—92no—
118C3×#121—94no—
119C3×C4×D4—94no—
120C2×C3×C42—91no—
121#947—32no—
122#944—32no—
123#940—32no—
124#939—38no—
125#936—32no—
126C4×#67—38no—
127C2×C4×Dic3—32no—
128C2×C4×Dic3—61no—
129C23×C32—21no—
130C23×C32—22no—
131C23×C32—34no—
132C23×C32—41no—
133C23×C32C22×D312yes—
134C23×C32C23×C311yescharacteristic
135C22×C3×D3—24no—
136C22×C3×D3C22×D314yes—
137C22×C3×D3C23×C314yes—
138C32×Q8—64no—
139C32×D4—68no—
140C32×D4—64no—
141C2×C32×C4—24no—
142C2×C32×C4—28no—
143C2×C32×C4—21no—
144C2×C32×C4—21no—
145C2×C32×C4—22no—
146C2×C32×C4—22no—
147C2×C32×C4—34no—
148C2×C32×C4—44no—
149C2×C32×C4—41no—
150C2×C32×C4—41no—
151C2×C32×C4—68no—
152C2×C32×C4#6718yescomplemented
153C2×C32×C4C3×D414yescomplemented
154C2×C32×C4C22×D312yes—
155C2×C32×C4C22×D312yes—
156C2×C32×C4C23×C311yescharacteristic
157C2×C32×C4C23×C311yescharacteristic
158C2×C3⋊Dic3—24no—
159C2×C3⋊Dic3C23×C312yes—
160C2×C3⋊Dic3C23×C312yes—
161C3×#67—216no—
162C3×#67—28no—
163C3×#67—28no—
164C3×#67—616no—
165C2×C3×Dic3—24no—
166C2×C3×Dic3—34no—
167C2×C3×Dic3—34no—
168C2×C3×Dic3—62no—
169C2×C3×Dic3—62no—
170C2×C3×Dic3—68no—
171C2×C3×Dic3—68no—
172C2×C3×Dic3C22×D314yes—
173C2×C3×Dic3C23×C314yes—
174C3×D12—28no—
175C3×D12—24no—
176C3×D12—24no—
177C3×C4×D3—216no—
178C3×C4×D3—28no—
179C3×C4×D3—28no—
180C3×Dic6—28no—
181C3×Dic6—24no—
182C3×Dic6—24no—
183#625—28no—
184#532—91noSylow
185C2×C33—22no—
186C2×C33C22×D411yescharacteristic
187C2×C33C2×#4112yes—
188C32×D3—44no—
189C3×#41—616no—
190C3×#41—68no—
191C3×#41—68no—
192C2×C3×Q8—32no—
193C2×C3×Q8—32no—
194C2×C3×Q8—62no—
195C2×C3×D4—34no—
196C2×C3×D4—34no—
197C2×C3×D4—32no—
198C2×C3×D4—32no—
199C2×C3×D4—62no—
200C2×C3×D4—64no—
201C2×C3×D4—94no—
202C22×C3×C4—21no—
203C22×C3×C4—22no—
204C22×C3×C4—34no—
205C22×C3×C4—34no—
206C22×C3×C4—41no—
207C22×C3×C4—64no—
208C22×C3×C4—92no—
209C22×C3×C4D3211yescharacteristic
210C22×C3×C4C2×C3×D312yes—
211C2×#67—38no—
212C2×#67C2×C3×D314yes—
213C22×Dic3—34no—
214C22×Dic3—62no—
215#235—28no—
216C2×D12C2×C3×D312yes—
217C2×C4×D3C2×C3×D314yes—
218C2×Dic6C2×C3×D312yes—
219C3×C4⋊C4—94no—
220C3×C22⋊C4—98no—
221C3×C42—91no—
222C3×C42—91no—
223C3×C42—92no—
224#217—38no—
225(C2×D3)⋊C4—38no—
226Dic3⋊C4—38no—
227C4×Dic3—34no—
228C4×Dic3—32no—
229C4×Dic3—32no—
230C4×Dic3—62no—
231C4×Dic3—61no—
232C4×Dic3—61no—
233C22×C32—22no—
234C22×C32—24no—
235C22×C32—24no—
236C22×C32—21no—
237C22×C32—22no—
238C22×C32—22no—
239C22×C32—24no—
240C22×C32—42no—
241C22×C32—41no—
242C22×C32—44no—
243C22×C32—44no—
244C22×C32—64no—
245C22×C32—68no—
246C22×C32C2×#6714yes—
247C22×C32C2×C3×D412yes—
248C22×C32C23×D312yes—
249C22×C32C24×C311yescharacteristic
250C2×C3×D3—28no—
251C2×C3×D3—28no—
252C2×C3×D3—24no—
253C2×C3×D3—24no—
254C2×C3×D3—28no—
255C2×C3×D3—48no—
256C32×C4—22no—
257C32×C4—22no—
258C32×C4—24no—
259C32×C4—24no—
260C32×C4—42no—
261C32×C4—42no—
262C32×C4—64no—
263C32×C4C2×#6714yes—
264C32×C4C2×#6714yes—
265C32×C4C2×C3×D412yes—
266C32×C4C2×C3×D412yes—
267C3⋊Dic3—22no—
268C3⋊Dic3C3×#4114yescomplemented
269C3×Dic3—28no—
270C3×Dic3—24no—
271C3×Dic3—24no—
272C3×Dic3—38no—
273C3×Dic3—64no—
274C3×Dic3—64no—
275C3×Dic3—64no—
276C2×#41—32no—
277C4⋊Q8—91no—
278#127—91no—
279#124—92no—
280#121—94no—
281C4×D4—94no—
282C2×C42—91no—
283C33#53211yescharacteristic, complemented, Sylow
284C23×C3—21no—
285C23×C3—22no—
286C23×C3—34no—
287C23×C3—34no—
288C23×C3—41no—
289C23×C3—64no—
290C23×C3C2×D3211yescharacteristic
291C23×C3C22×C3×D312yes—
292C22×D3C22×C3×D314yes—
293C3×Q8—68no—
294C3×Q8—64no—
295C3×Q8—64no—
296C3×D4—68no—
297C3×D4—616no—
298C3×D4—68no—
299C3×D4—68no—
300C3×D4—64no—
301C3×D4—64no—
302C3×D4—188no—
303C2×C3×C4—28no—
304C2×C3×C4—24no—
305C2×C3×C4—22no—
306C2×C3×C4—22no—
307C2×C3×C4—21no—
308C2×C3×C4—21no—
309C2×C3×C4—34no—
310C2×C3×C4—34no—
311C2×C3×C4—44no—
312C2×C3×C4—41no—
313C2×C3×C4—41no—
314C2×C3×C4—64no—
315C2×C3×C4—616no—
316C2×C3×C4—68no—
317C2×C3×C4—68no—
318C2×C3×C4—92no—
319C2×C3×C4—92no—
320C2×C3×C4—184no—
321C2×C3×C4#62514yescomplemented
322C2×C3×C4C3×#6718yescomplemented
323C2×C3×C4C2×D3211yescharacteristic
324C2×C3×C4C2×D3211yescharacteristic
325C2×C3×C4C22×C3×D312yes—
326C2×C3×C4C22×C3×D312yes—
327#67—28no—
328#67—616no—
329C2×Dic3—34no—
330C2×Dic3—34no—
331C2×Dic3—62no—
332C2×Dic3—62no—
333C2×Dic3—68no—
334C2×Dic3—68no—
335C2×Dic3C22×C3×D314yes—
336D12—24no—
337C4×D3—28no—
338Dic6—24no—
339C2×C32—22no—
340C2×C32—24no—
341C2×C32—24no—
342C2×C32—22no—
343C2×C32—21no—
344C2×C32—24no—
345C2×C32—22no—
346C2×C32—41no—
347C2×C32—42no—
348C2×C32—44no—
349C2×C32—44no—
350C2×C32—124no—
351C2×C32D3×#4114yes—
352C2×C32C22×#6712yes—
353C2×C32C22×C3×D411yescharacteristic
354C2×C32C2×C3×#4112yes—
355C3×D3—48no—
356C3×D3—44no—
357C3×D3—44no—
358#41—68no—
359C2×Q8—32no—
360C2×D4—34no—
361C2×D4—32no—
362C2×D4—94no—
363C22×C4—34no—
364C22×C4—92no—
365C22×C4C3×D3211yescharacteristic
366C4⋊C4—94no—
367C22⋊C4—98no—
368C42—92no—
369C42—91no—
370C42—91no—
371C22×C3—24no—
372C22×C3—22no—
373C22×C3—24no—
374C22×C3—21no—
375C22×C3—22no—
376C22×C3—22no—
377C22×C3—24no—
378C22×C3—44no—
379C22×C3—44no—
380C22×C3—41no—
381C22×C3—42no—
382C22×C3—68no—
383C22×C3—68no—
384C22×C3—64no—
385C22×C3—68no—
386C22×C3—64no—
387C22×C3—128no—
388C22×C3C2×#62512yes—
389C22×C3C2×C3×#6714yes—
390C22×C3C22×D3211yescharacteristic
391C22×C3C23×C3×D312yes—
392C2×D3—28no—
393C2×D3—24no—
394C3×C4—24no—
395C3×C4—24no—
396C3×C4—22no—
397C3×C4—22no—
398C3×C4—42no—
399C3×C4—42no—
400C3×C4—68no—
401C3×C4—64no—
402C3×C4—64no—
403C3×C4—94no—
404C3×C4—182no—
405C3×C4C2×#62512yes—
406C3×C4C2×#62512yes—
407C3×C4C2×C3×#6714yes—
408C3×C4C2×C3×#6714yes—
409Dic3—24no—
410Dic3—38no—
411Dic3—64no—
412Dic3—64no—
413Dic3—64no—
414C32—22no—
415C32—21no—
416C32—41no—
417C32#583212yescomplemented
418C32C3×#53211yescharacteristic, complemented
419C23—34no—
420C23C2×C3×D3211yescharacteristic
421Q8—64no—
422D4—68no—
423D4—64no—
424D4—188no—
425C2×C4—34no—
426C2×C4—68no—
427C2×C4—92no—
428C2×C4—92no—
429C2×C4—184no—
430C2×C4C3×#62514yescomplemented
431C2×C4C2×C3×D3211yescharacteristic
432C2×C4C2×C3×D3211yescharacteristic
433C2×C3—24no—
434C2×C3—24no—
435C2×C3—22no—
436C2×C3—22no—
437C2×C3—21no—
438C2×C3—22no—
439C2×C3—24no—
440C2×C3—44no—
441C2×C3—44no—
442C2×C3—41no—
443C2×C3—42no—
444C2×C3—128no—
445C2×C3—124no—
446C2×C3—124no—
447C2×C3#6430612yes—
448C2×C3C22×#62511yescharacteristic
449C2×C3C3×D3×#4114yes—
450C2×C3C22×C3×#6712yes—
451D3—44no—
452C22—22no—
453C22—64no—
454C22—68no—
455C22C2×C3×#62512yes—
456C22C22×C3×D3211yescharacteristic
457C4—64no—
458C4—94no—
459C4—182no—
460C4C2×C3×#62512yes—
461C4C2×C3×#62512yes—
462C3—21no—
463C3—22no—
464C3—41no—
465C3#1059503211yescharacteristic, direct product factor
466C3C3×#583212yescomplemented
467C2—22no—
468C2—124no—
469C2C3×#6430612yes—
470C2C22×C3×#62511yescharacteristic
subgroupG/H
centerC2×C3×C4C2×D32
commutator, cosocleC22×C32C24×C3
FrattiniC22C22×C3×D32
FittingC22×C33×C4C22
socleC22×C33C24
hypercenterC22×C3×C4D32
hypocenterC32C3×#532
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C2×C3×C4 ◁ C22×C3×C4

Complements

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

NH ≅ G/Ntype
1. C2×C32×#235C2semidirect
2. C2×C3×C4×C3⋊Dic3C2semidirect
3. C3×#63980C2semidirect
4. C3×#63914C2semidirect
5. C3×C4×#625C2semidirect
6. C3×#63873C2semidirect
7. C3×#63863C2semidirect
9. #10595032C3direct (N × H)
25. C2×C3×#235D3semidirect
26. C2×C3×#235C2×C3semidirect
27. C2×C4×C3⋊Dic3C2×C3semidirect
36. #63980C2×C3semidirect
37. #63914C2×C3semidirect
38. C4×#625C2×C3semidirect
39. #63873C2×C3semidirect
40. #63863C2×C3semidirect
43. C2×C33×C4D4semidirect
109. C3×C3⋊Dic3#41semidirect
114. C2×#235C3×D3semidirect
152. C2×C32×C4#67semidirect
153. C2×C32×C4C3×D4semidirect
268. C3⋊Dic3C3×#41semidirect
283. C33#532semidirect
321. C2×C3×C4#625semidirect
322. C2×C3×C4C3×#67semidirect
417. C32#5832semidirect
418. C32C3×#532semidirect
430. C2×C4C3×#625semidirect
465. C3#10595032direct (N × H)
466. C3C3×#5832semidirect

Maximal subgroups

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

subgroupindex
C2×C32×#2352
C2×C3×C4×C3⋊Dic32
C3×#639802
C3×#639142
C3×C4×#6252
C3×#638732
C3×#638632
C3×#58323
#105950323