Finite Group Databasebeta

← C3×#10593867 #13173452

Subgroups — C3×#10593867 #13173452

Order 1728.

998 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
1C3×#64081C211yesmaximal, characteristic
2C32×#870C211yesmaximal, characteristic, complemented
3C32×#869C211yesmaximal, characteristic, complemented
4C3×#63871C212yesmaximal
5C3×#63848C214yesmaximal
6C3×#63846C214yesmaximal, complemented
7C3×#63843C212yesmaximal, complemented
8C3×#5731—31nomaximal
9C3×#5698—31nomaximal
10#10593867C311yesmaximal, characteristic, direct product factor
11C2×C3×C4×C3⋊D3C2211yescharacteristic
12C2×C32×Dic6C2212yes—
13C33×C42C2211yescharacteristic
14C3×#3752C2212yes—
15C3×#3750C2212yes—
16C3×C4×C3⋊Dic3C2211yescharacteristic
17C32×C4⋊Dic3C2212yes—
18C32×C4⋊Dic3C2214yes—
19C32×Dic3⋊C4C2214yes—
20C3×Dic3⋊Dic3C2214yes—
21C3×Dic3⋊Dic3C2214yes—
22C3×#3722C2214yes—
23C3×Dic32C2214yes—
24C32×C4⋊Q8—31no—
25C32×#125—31no—
26#64081C2×C311yescharacteristic
27C3×#946—32no—
28C3×Q8×Dic3—32no—
29C3×#898—34no—
30C3×#898—34no—
31C3×#897—32no—
32C3×#890—34no—
33C3×#889—34no—
34C3×#887—32no—
35C3×#872—31no—
36C3×#872—31no—
37C3×#872—61no—
38C3×#870—21no—
39C3×#870D311yescharacteristic, complemented
40C3×#870C2×C311yescharacteristic, complemented
41C3×#869—21no—
42C3×#869D311yescharacteristic, complemented
43C3×#869C2×C311yescharacteristic, complemented
44#63871C2×C312yes—
45#63848C2×C314yes—
46#63846C2×C314yescomplemented
47#63843C2×C312yescomplemented
48C22×C3×C3⋊D3C2311yescharacteristic
49C2×C33×C4C2312yes—
50C2×C33×C4C2311yescharacteristic
51C2×C3×C3⋊Dic3C2312yes—
52C2×C3×C3⋊Dic3C2311yescharacteristic
53C3×C4×C3⋊D3—22no—
54C2×C32×Dic3C2314yes—
55C2×C32×Dic3C2314yes—
56C32×Dic6—24no—
57C3×#541—91no—
58#5731—31no—
59#5698—31no—
60C2×C32×Q8—32no—
61C2×C4×C3⋊D3C22×C311yescharacteristic
62C2×C3×C4×D3—31no—
63C2×C3×C4×D3—31no—
64C2×C3×C4×D3—61no—
65C2×C3×Dic6—22no—
66C2×C3×Dic6C2×D312yes—
67C2×C3×Dic6C22×C312yes—
68C32×C4⋊C4—32no—
69C32×C4⋊C4—34no—
70C32×C4⋊C4—34no—
71C32×C42—21no—
72C32×C42—21no—
73C32×C42—21no—
74C32×C42—41no—
75C32×C42C2×D311yescharacteristic
76C32×C42C2×D311yescharacteristic
77C32×C42C22×C311yescharacteristic
78#3752C22×C312yes—
79#3750C22×C312yes—
80C4×C3⋊Dic3C22×C311yescharacteristic
81C3×(C2×D3)⋊C4—32no—
82C3×(C2×D3)⋊C4—32no—
83C3×(C2×D3)⋊C4—34no—
84C3×(C2×D3)⋊C4—34no—
85C3×(C2×D3)⋊C4—62no—
86C3×C4⋊Dic3—24no—
87C3×C4⋊Dic3—22no—
88C3×C4⋊Dic3—34no—
89C3×C4⋊Dic3—34no—
90C3×C4⋊Dic3C2×D314yes—
91C3×C4⋊Dic3C2×D312yes—
92C3×C4⋊Dic3C22×C312yes—
93C3×C4⋊Dic3C22×C314yes—
94C3×Dic3⋊C4—24no—
95C3×Dic3⋊C4—32no—
96C3×Dic3⋊C4—32no—
97C3×Dic3⋊C4—34no—
98C3×Dic3⋊C4—34no—
99C3×Dic3⋊C4—62no—
100C3×Dic3⋊C4C2×D314yes—
101C3×Dic3⋊C4C22×C314yes—
102C3×C4×Dic3—31no—
103C3×C4×Dic3—31no—
104C3×C4×Dic3—34no—
105C3×C4×Dic3—34no—
106C3×C4×Dic3—61no—
107Dic3⋊Dic3C22×C314yes—
108Dic3⋊Dic3C22×C314yes—
109#3722C22×C314yes—
110Dic32C22×C314yes—
111C22×C33C2411yescharacteristic
112C2×C3×C3⋊D3—21no—
113C2×C3×C3⋊D3—22no—
114C33×C4—22no—
115C33×C4#4112yes—
116C3×C3⋊Dic3—21no—
117C3×C3⋊Dic3—22no—
118C32×Dic3—24no—
119C32×Dic3—24no—
120C3×C4⋊Q8—31no—
121C3×C4⋊Q8—31no—
122C3×C4⋊Q8—61no—
123C3×#126—94no—
124C3×#125—31no—
125C3×#125—31no—
126C3×#125—61no—
127C3×#125—94no—
128C3×C22⋊Q8—92no—
129C3×C4×Q8—92no—
130C3×#117—91no—
131#946—32no—
132Q8×Dic3—32no—
133#898—34no—
134#898—34no—
135#897—32no—
136#890—34no—
137#889—34no—
138#887—32no—
139#872—31no—
140#872—31no—
141#872—61no—
142#870C3×D311yescharacteristic, complemented
143#869C3×D311yescharacteristic, complemented
144C22×C3⋊D3C23×C311yescharacteristic
145C22×C3×D3—31no—
146C22×C3×D3—31no—
147C22×C3×D3—61no—
148C32×Q8—64no—
149C2×C32×C4—21no—
150C2×C32×C4—22no—
151C2×C32×C4—22no—
152C2×C32×C4—21no—
153C2×C32×C4—21no—
154C2×C32×C4—22no—
155C2×C32×C4—34no—
156C2×C32×C4—34no—
157C2×C32×C4—41no—
158C2×C32×C4—42no—
159C2×C32×C4C22×D312yes—
160C2×C32×C4C22×D312yes—
161C2×C32×C4C22×D311yescharacteristic
162C2×C32×C4C22×D311yescharacteristic
163C2×C32×C4C23×C312yes—
164C2×C32×C4C23×C311yescharacteristic
165C2×C3⋊Dic3C23×C312yes—
166C2×C3⋊Dic3C23×C311yescharacteristic
167C4×C3⋊D3—22no—
168C2×C3×Dic3—24no—
169C2×C3×Dic3—24no—
170C2×C3×Dic3—32no—
171C2×C3×Dic3—32no—
172C2×C3×Dic3—31no—
173C2×C3×Dic3—31no—
174C2×C3×Dic3—61no—
175C2×C3×Dic3—62no—
176C2×C3×Dic3C22×D314yes—
177C2×C3×Dic3C22×D314yes—
178C2×C3×Dic3C23×C314yes—
179C2×C3×Dic3C23×C314yes—
180C3×C4×D3—64no—
181C3×C4×D3—62no—
182C3×C4×D3—62no—
183C3×Dic6—24no—
184C3×Dic6—24no—
185C3×Dic6—44no—
186#541—91noSylow
187C2×C33C2×#4111yescharacteristic
188C2×C332−1+412yes—
189C3×C3⋊D3—41no—
190C2×C3×Q8—32no—
191C2×C3×Q8—32no—
192C2×C3×Q8—62no—
193C22×C3×C4—91no—
194C2×C4×D3—31no—
195C2×C4×D3—31no—
196C2×C4×D3—61no—
197C2×Dic6C2×C3×D312yes—
198C3×C4⋊C4—34no—
199C3×C4⋊C4—32no—
200C3×C4⋊C4—34no—
201C3×C4⋊C4—32no—
202C3×C4⋊C4—34no—
203C3×C4⋊C4—34no—
204C3×C4⋊C4—64no—
205C3×C4⋊C4—62no—
206C3×C4⋊C4—64no—
207C3×C4⋊C4—94no—
208C3×C4⋊C4—94no—
209C3×C4⋊C4—92no—
210C3×C22⋊C4—94no—
211C3×C22⋊C4—92no—
212C3×C42—21no—
213C3×C42—21no—
214C3×C42—21no—
215C3×C42—41no—
216C3×C42—94no—
217C3×C42—91no—
218C3×C42D3211yescharacteristic
219C3×C42C2×C3×D311yescharacteristic
220C3×C42C2×C3×D311yescharacteristic
221(C2×D3)⋊C4—32no—
222(C2×D3)⋊C4—32no—
223(C2×D3)⋊C4—34no—
224(C2×D3)⋊C4—34no—
225(C2×D3)⋊C4—62no—
226C4⋊Dic3—34no—
227C4⋊Dic3—34no—
228C4⋊Dic3C2×C3×D314yes—
229C4⋊Dic3C2×C3×D312yes—
230Dic3⋊C4—32no—
231Dic3⋊C4—32no—
232Dic3⋊C4—34no—
233Dic3⋊C4—34no—
234Dic3⋊C4—62no—
235Dic3⋊C4C2×C3×D314yes—
236C4×Dic3—31no—
237C4×Dic3—31no—
238C4×Dic3—34no—
239C4×Dic3—34no—
240C4×Dic3—61no—
241C22×C32—21no—
242C22×C32—21no—
243C22×C32—21no—
244C22×C32—41no—
245C22×C32C23×D311yescharacteristic
246C22×C32C23×D311yescharacteristic
247C22×C32C24×C311yescharacteristic
248C2×C3⋊D3—21no—
249C2×C3⋊D3—22no—
250C2×C3×D3—62no—
251C2×C3×D3—61no—
252C2×C3×D3—61no—
253C2×C3×D3—62no—
254C2×C3×D3—62no—
255C2×C3×D3—122no—
256C32×C4—22no—
257C32×C4—24no—
258C32×C4—22no—
259C32×C4—22no—
260C32×C4—22no—
261C32×C4—22no—
262C32×C4—22no—
263C32×C4—42no—
264C32×C4—42no—
265C32×C4—42no—
266C32×C4—44no—
267C32×C4—64no—
268C32×C4—64no—
269C32×C4#23712yes—
270C32×C4#23912yes—
271C32×C4C3×#4112yes—
272C3⋊Dic3—21no—
273C3⋊Dic3—22no—
274C3×Dic3—24no—
275C3×Dic3—24no—
276C3×Dic3—24no—
277C3×Dic3—24no—
278C3×Dic3—44no—
279C3×Dic3—44no—
280C3×Dic3—62no—
281C3×Dic3—61no—
282C3×Dic3—61no—
283C3×Dic3—62no—
284C3×Dic3—62no—
285C3×Dic3—122no—
286C4⋊Q8—31no—
287#126—94no—
288#125—31no—
289#125—94no—
290C22⋊Q8—92no—
291C4×Q8—92no—
292#117—91no—
293C33#54111yescharacteristic, complemented, Sylow
294C23×C3—91no—
295C22×D3—31no—
296C22×D3—31no—
297C22×D3—61no—
298C3×Q8—64no—
299C3×Q8—64no—
300C3×Q8—124no—
301C2×C3×C4—22no—
302C2×C3×C4—21no—
303C2×C3×C4—21no—
304C2×C3×C4—21no—
305C2×C3×C4—22no—
306C2×C3×C4—22no—
307C2×C3×C4—34no—
308C2×C3×C4—34no—
309C2×C3×C4—34no—
310C2×C3×C4—34no—
311C2×C3×C4—42no—
312C2×C3×C4—41no—
313C2×C3×C4—64no—
314C2×C3×C4—64no—
315C2×C3×C4—92no—
316C2×C3×C4—91no—
317C2×C3×C4—182no—
318C2×C3×C4C2×D3212yes—
319C2×C3×C4C2×D3211yescharacteristic
320C2×C3×C4C22×C3×D312yes—
321C2×C3×C4C22×C3×D312yes—
322C2×C3×C4C22×C3×D311yescharacteristic
323C2×C3×C4C22×C3×D311yescharacteristic
324C2×Dic3—32no—
325C2×Dic3—32no—
326C2×Dic3—31no—
327C2×Dic3—31no—
328C2×Dic3—61no—
329C2×Dic3—62no—
330C2×Dic3C22×C3×D314yes—
331C2×Dic3C22×C3×D314yes—
332C4×D3—64no—
333C4×D3—62no—
334C4×D3—62no—
335Dic6—24no—
336C2×C32—21no—
337C2×C32—22no—
338C2×C32—21no—
339C2×C32—21no—
340C2×C32—22no—
341C2×C32—22no—
342C2×C32—41no—
343C2×C32—42no—
344C2×C32C2×#23711yescharacteristic
345C2×C32C2×#23911yescharacteristic
346C2×C32#100712yes—
347C2×C32#101012yes—
348C2×C32C2×C3×#4111yescharacteristic
349C2×C32C3×2−1+412yes—
350C3⋊D3—41no—
351C3×D3—122no—
352C3×D3—121no—
353C3×D3—121no—
354C2×Q8—32no—
355C22×C4—91no—
356C4⋊C4—32no—
357C4⋊C4—34no—
358C4⋊C4—34no—
359C4⋊C4—94no—
360C4⋊C4—94no—
361C4⋊C4—92no—
362C22⋊C4—94no—
363C22⋊C4—92no—
364C42—94no—
365C42—91no—
366C42C3×D3211yescharacteristic
367C22×C3—21no—
368C22×C3—21no—
369C22×C3—21no—
370C22×C3—41no—
371C22×C3—181no—
372C22×C3—182no—
373C22×C3C22×D3211yescharacteristic
374C22×C3C23×C3×D311yescharacteristic
375C22×C3C23×C3×D311yescharacteristic
376C2×D3—62no—
377C2×D3—61no—
378C2×D3—61no—
379C2×D3—62no—
380C2×D3—62no—
381C2×D3—122no—
382C3×C4—24no—
383C3×C4—22no—
384C3×C4—22no—
385C3×C4—22no—
386C3×C4—22no—
387C3×C4—22no—
388C3×C4—22no—
389C3×C4—42no—
390C3×C4—42no—
391C3×C4—44no—
392C3×C4—42no—
393C3×C4—64no—
394C3×C4—64no—
395C3×C4—64no—
396C3×C4—64no—
397C3×C4—124no—
398C3×C4—124no—
399C3×C4—181no—
400C3×C4—182no—
401C3×C4#379612yes—
402C3×C4C3×#23712yes—
403C3×C4C3×#23912yes—
404Dic3—24no—
405Dic3—24no—
406Dic3—62no—
407Dic3—61no—
408Dic3—61no—
409Dic3—62no—
410Dic3—62no—
411Dic3—122no—
412C32—21no—
413C32—21no—
414C32—21no—
415C32—41no—
416C32#569811yescharacteristic, complemented
417C32#573111yescharacteristic, complemented
418C32C3×#54111yescharacteristic, complemented
419C23—91no—
420Q8—64no—
421C2×C4—34no—
422C2×C4—34no—
423C2×C4—92no—
424C2×C4—91no—
425C2×C4—182no—
426C2×C4C2×C3×D3212yes—
427C2×C4C2×C3×D3211yescharacteristic
428C2×C3—21no—
429C2×C3—21no—
430C2×C3—21no—
431C2×C3—22no—
432C2×C3—22no—
433C2×C3—22no—
434C2×C3—41no—
435C2×C3—42no—
436C2×C3—361no—
437C2×C3C2×#379611yescharacteristic
438C2×C3#6429712yes—
439C2×C3C2×C3×#23711yescharacteristic
440C2×C3C2×C3×#23911yescharacteristic
441C2×C3C3×#100712yes—
442C2×C3C3×#101012yes—
443D3—122no—
444D3—121no—
445D3—121no—
446C22—181no—
447C22—182no—
448C22C22×C3×D3211yescharacteristic
449C4—22no—
450C4—64no—
451C4—64no—
452C4—181no—
453C4—182no—
454C4C3×#379612yes—
455C3—21no—
456C3—21no—
457C3—21no—
458C3—41no—
459C3#1059386711yescharacteristic, direct product factor
460C3C3×#569811yescharacteristic, complemented
461C3C3×#573111yescharacteristic, complemented
462C2—361no—
463C2C2×C3×#379611yescharacteristic
464C2C3×#6429712yes—
subgroupG/H
centerC22×C3C22×D32
commutator, cosocleC22×C32C24×C3
FrattiniC22C22×C3×D32
FittingC33×C42C22
socleC22×C33C24
prime radicalC22×C3×C3⋊D3C23
hypercenterC3×C42D32
hypocenterC32C3×#541
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C3×C42

Complements

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

NH ≅ G/Ntype
2. C32×#870C2semidirect
3. C32×#869C2semidirect
6. C3×#63846C2semidirect
7. C3×#63843C2semidirect
10. #10593867C3direct (N × H)
39. C3×#870D3semidirect
40. C3×#870C2×C3semidirect
42. C3×#869D3semidirect
43. C3×#869C2×C3semidirect
46. #63846C2×C3semidirect
47. #63843C2×C3semidirect
142. #870C3×D3semidirect
143. #869C3×D3semidirect
293. C33#541semidirect
416. C32#5698semidirect
417. C32#5731semidirect
418. C32C3×#541semidirect
459. C3#10593867direct (N × H)
460. C3C3×#5698semidirect
461. C3C3×#5731semidirect

Maximal subgroups

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

subgroupindex
C3×#640812
C32×#8702
C32×#8692
C3×#638712
C3×#638482
C3×#638462
C3×#638432
C3×#57313
C3×#56983
#105938673