Finite Group Databasebeta

← C3×#10595041 #13174624

Subgroups — C3×#10595041 #13174624

Order 1728.

1278 conjugacy classes of subgroups.

The subgroup lattice has 490 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#64135C211yesmaximal, characteristic, complemented
2C32×#930C212yesmaximal, complemented
3C3×#63970C214yesmaximal, complemented
4C3×#63914C211yesmaximal, characteristic, complemented
5C3×#63898C212yesmaximal, complemented
6C3×#63891C214yesmaximal, complemented
7C3×#63872C211yesmaximal, characteristic, complemented
8C3×#5619—32nomaximal
9#10595041C311yesmaximal, characteristic, direct product factor
10C22×C33×C4C2211yescharacteristic
11C2×C32×#67C2214yes—
12C2×C32×D12C2212yes—
13C2×C3×#625C2214yes—
14C2×C3×D3×Dic3C2214yes—
15C3×#3757C2212yes—
16C3×C4⋊(C3⋊Dic3)C2212yes—
17C3×C4×C3⋊Dic3C2211yescharacteristic
18C3×C4×C3⋊Dic3C2211yescharacteristic
19C32×(C2×D3)⋊C4C2214yes—
20C32×C4⋊Dic3C2212yes—
21C3×Dic3⋊Dic3C2214yes—
22C3×(C2×D3)⋊Dic3C2214yes—
23C32×#121—32no—
24#64135C2×C311yescharacteristic, complemented
25C3×#940—32no—
26C3×#939—34no—
27C3×#938—34no—
28C3×D4×Dic3—32no—
29C3×#930—22no—
30C3×#930D312yescomplemented
31C3×#930C2×C312yescomplemented
32C3×#926—32no—
33C3×#926—61no—
34C3×#893—32no—
35C3×#890—32no—
36C3×#886—34no—
37C3×#883—34no—
38#63970C2×C314yescomplemented
39#63914C2×C311yescharacteristic, complemented
40#63898C2×C312yescomplemented
41#63891C2×C314yescomplemented
42#63872C2×C311yescharacteristic, complemented
43C23×C33C2311yescharacteristic
44C22×C32×D3C2314yes—
45C2×C33×C4—22no—
46C2×C33×C4C2311yescharacteristic
47C2×C33×C4C2311yescharacteristic
48C2×C3×C3⋊Dic3C2312yes—
49C2×C3×C3⋊Dic3C2312yes—
50C32×#67—44no—
51C2×C32×Dic3C2314yes—
52C32×D12—24no—
53C3×#625—44no—
54C3×D3×Dic3—44no—
55C3×#540—91no—
56#5619—32no—
57C2×C32×D4—34no—
58C2×C32×D4—32no—
59C22×C32×C4—21no—
60C22×C32×C4—22no—
61C22×C32×C4—41no—
62C22×C32×C4C2×D312yes—
63C22×C32×C4C22×C311yescharacteristic
64C2×C3×#67—24no—
65C2×C3×#67—34no—
66C2×C3×#67—34no—
67C2×C3×#67C2×D314yes—
68C2×C3×#67C22×C314yes—
69C22×C3×Dic3—34no—
70C2×C3×D12—22no—
71C2×C3×D12C2×D312yes—
72C2×C3×D12C22×C312yes—
73C2×C3×C4×D3—34no—
74C2×#625C22×C314yes—
75C2×D3×Dic3C22×C314yes—
76C32×C4⋊C4—32no—
77C32×C22⋊C4—34no—
78#3757C22×C312yes—
79C4⋊(C3⋊Dic3)C22×C312yes—
80C4×C3⋊Dic3C22×C311yescharacteristic
81C4×C3⋊Dic3C22×C311yescharacteristic
82C3×#217—32no—
83C3×#217—32no—
84C3×#217—34no—
85C3×#217—62no—
86C3×(C2×D3)⋊C4—24no—
87C3×(C2×D3)⋊C4—34no—
88C3×(C2×D3)⋊C4C2×D314yes—
89C3×(C2×D3)⋊C4C22×C314yes—
90C3×C4⋊Dic3—22no—
91C3×C4⋊Dic3—32no—
92C3×C4⋊Dic3—32no—
93C3×C4⋊Dic3—34no—
94C3×C4⋊Dic3—62no—
95C3×C4⋊Dic3C2×D312yes—
96C3×C4⋊Dic3C22×C312yes—
97C3×Dic3⋊C4—34no—
98C3×C4×Dic3—32no—
99C3×C4×Dic3—32no—
100C3×C4×Dic3—61no—
101C3×C4×Dic3—61no—
102Dic3⋊Dic3C22×C314yes—
103(C2×D3)⋊Dic3C22×C314yes—
104C22×C33—21no—
105C22×C33—22no—
106C22×C33C2411yescharacteristic
107C2×C32×D3—24no—
108C2×C32×D3—24no—
109C2×C32×D3—24no—
110C33×C4—21no—
111C33×C4#4112yes—
112C3×C3⋊Dic3—22no—
113C3×C3⋊Dic3—22no—
114C32×Dic3—24no—
115C3×#127—91no—
116C3×#125—91no—
117C3×#123—94no—
118C3×#121—32no—
119C3×#121—32no—
120C3×#121—62no—
121C3×#121—94no—
122C3×C4×D4—92no—
123C3×#117—91no—
124#940—32no—
125#939—34no—
126#938—34no—
127D4×Dic3—32no—
128#930C3×D312yescomplemented
129#926—32no—
130#926—61no—
131#893—32no—
132#890—32no—
133#886—34no—
134#883—34no—
135C23×C32—21no—
136C23×C32—22no—
137C23×C32—34no—
138C23×C32—41no—
139C23×C32C22×D312yes—
140C23×C32C23×C311yescharacteristic
141C22×C3×D3—24no—
142C22×C3×D3C22×D314yes—
143C22×C3×D3C23×C314yes—
144C32×D4—64no—
145C32×D4—124no—
146C2×C32×C4—22no—
147C2×C32×C4—24no—
148C2×C32×C4—24no—
149C2×C32×C4—21no—
150C2×C32×C4—21no—
151C2×C32×C4—22no—
152C2×C32×C4—22no—
153C2×C32×C4—34no—
154C2×C32×C4—41no—
155C2×C32×C4—41no—
156C2×C32×C4—44no—
157C2×C32×C4—44no—
158C2×C32×C4C22×D312yes—
159C2×C32×C4C22×D312yes—
160C2×C32×C4C23×C311yescharacteristic
161C2×C32×C4C23×C311yescharacteristic
162C2×C3⋊Dic3C23×C312yes—
163C2×C3⋊Dic3C23×C312yes—
164C3×#67—44no—
165C3×#67—44no—
166C3×#67—48no—
167C3×#67—124no—
168C3×#67—124no—
169C2×C3×Dic3—24no—
170C2×C3×Dic3—32no—
171C2×C3×Dic3—32no—
172C2×C3×Dic3—32no—
173C2×C3×Dic3—32no—
174C2×C3×Dic3—62no—
175C2×C3×Dic3—62no—
176C2×C3×Dic3—124no—
177C2×C3×Dic3C22×D314yes—
178C2×C3×Dic3C23×C314yes—
179C3×D12—24no—
180C3×D12—24no—
181C3×D12—44no—
182C3×C4×D3—124no—
183#625—44no—
184D3×Dic3—44no—
185#540—91noSylow
186C2×C33—41no—
187C2×C33C2×#4111yescharacteristic
188C2×C33H(5,𝔽2)12yes—
189C32×D3—44no—
190C2×C3×D4—34no—
191C2×C3×D4—34no—
192C2×C3×D4—32no—
193C2×C3×D4—32no—
194C2×C3×D4—62no—
195C2×C3×D4—64no—
196C2×C3×D4—94no—
197C22×C3×C4—21no—
198C22×C3×C4—22no—
199C22×C3×C4—41no—
200C22×C3×C4—94no—
201C22×C3×C4D3211yescharacteristic
202C22×C3×C4C2×C3×D312yes—
203C2×#67—34no—
204C2×#67—34no—
205C2×#67C2×C3×D314yes—
206C22×Dic3—34no—
207C2×D12C2×C3×D312yes—
208C2×C4×D3—34no—
209C3×C4⋊C4—32no—
210C3×C4⋊C4—32no—
211C3×C4⋊C4—62no—
212C3×C4⋊C4—94no—
213C3×C4⋊C4—92no—
214C3×C22⋊C4—34no—
215C3×C22⋊C4—34no—
216C3×C22⋊C4—64no—
217C3×C22⋊C4—94no—
218C3×C22⋊C4—92no—
219C3×C42—91no—
220C3×C42—91no—
221#217—32no—
222#217—32no—
223#217—34no—
224#217—62no—
225(C2×D3)⋊C4—34no—
226(C2×D3)⋊C4C2×C3×D314yes—
227C4⋊Dic3—32no—
228C4⋊Dic3—32no—
229C4⋊Dic3—34no—
230C4⋊Dic3—62no—
231C4⋊Dic3C2×C3×D312yes—
232Dic3⋊C4—34no—
233C4×Dic3—32no—
234C4×Dic3—32no—
235C4×Dic3—61no—
236C4×Dic3—61no—
237C22×C32—22no—
238C22×C32—21no—
239C22×C32—24no—
240C22×C32—22no—
241C22×C32—22no—
242C22×C32—22no—
243C22×C32—22no—
244C22×C32—21no—
245C22×C32—22no—
246C22×C32—41no—
247C22×C32—42no—
248C22×C32—42no—
249C22×C32—42no—
250C22×C32—42no—
251C22×C32—44no—
252C22×C32—64no—
253C22×C32—64no—
254C22×C32—64no—
255C22×C32C23×D312yes—
256C22×C32C24×C311yescharacteristic
257C2×C3×D3—24no—
258C2×C3×D3—24no—
259C2×C3×D3—24no—
260C2×C3×D3—24no—
261C2×C3×D3—24no—
262C2×C3×D3—24no—
263C2×C3×D3—44no—
264C2×C3×D3—44no—
265C2×C3×D3—44no—
266C32×C4—21no—
267C32×C4—22no—
268C32×C4—22no—
269C32×C4—22no—
270C32×C4—24no—
271C32×C4—42no—
272C32×C4—42no—
273C32×C4—42no—
274C32×C4—64no—
275C32×C4#23714yes—
276C32×C4C3×#4112yes—
277C3⋊Dic3—22no—
278C3⋊Dic3—22no—
279C3×Dic3—24no—
280C3×Dic3—24no—
281C3×Dic3—44no—
282C3×Dic3—62no—
283C3×Dic3—62no—
284C3×Dic3—62no—
285C3×Dic3—62no—
286C3×Dic3—122no—
287C3×Dic3—122no—
288#127—91no—
289#125—91no—
290#123—94no—
291#121—32no—
292#121—94no—
293C4×D4—92no—
294#117—91no—
295C33#54011yescharacteristic, complemented, Sylow
296C23×C3—22no—
297C23×C3—21no—
298C23×C3—34no—
299C23×C3—34no—
300C23×C3—41no—
301C23×C3—64no—
302C23×C3C2×D3211yescharacteristic
303C23×C3C22×C3×D312yes—
304C22×D3C22×C3×D314yes—
305C3×D4—64no—
306C3×D4—64no—
307C3×D4—124no—
308C3×D4—124no—
309C3×D4—124no—
310C3×D4—128no—
311C3×D4—364no—
312C2×C3×C4—24no—
313C2×C3×C4—22no—
314C2×C3×C4—24no—
315C2×C3×C4—22no—
316C2×C3×C4—21no—
317C2×C3×C4—21no—
318C2×C3×C4—22no—
319C2×C3×C4—34no—
320C2×C3×C4—34no—
321C2×C3×C4—44no—
322C2×C3×C4—44no—
323C2×C3×C4—41no—
324C2×C3×C4—41no—
325C2×C3×C4—64no—
326C2×C3×C4—92no—
327C2×C3×C4—92no—
328C2×C3×C4—364no—
329C2×C3×C4C2×D3211yescharacteristic
330C2×C3×C4C2×D3211yescharacteristic
331C2×C3×C4C22×C3×D312yes—
332C2×C3×C4C22×C3×D312yes—
333#67—44no—
334#67—124no—
335#67—124no—
336C2×Dic3—32no—
337C2×Dic3—32no—
338C2×Dic3—32no—
339C2×Dic3—32no—
340C2×Dic3—62no—
341C2×Dic3—62no—
342C2×Dic3—124no—
343C2×Dic3C22×C3×D314yes—
344D12—24no—
345C4×D3—124no—
346C2×C32—22no—
347C2×C32—21no—
348C2×C32—22no—
349C2×C32—22no—
350C2×C32—22no—
351C2×C32—41no—
352C2×C32—42no—
353C2×C32—44no—
354C2×C32—42no—
355C2×C32—44no—
356C2×C32—41no—
357C2×C32—42no—
358C2×C32—124no—
359C2×C32C2×#23712yes—
360C2×C32#100412yes—
361C2×C32#100912yes—
362C2×C32C2×C3×#4111yescharacteristic
363C2×C32C3×H(5,𝔽2)12yes—
364C3×D3—44no—
365C3×D3—44no—
366C3×D3—84no—
367C2×D4—34no—
368C2×D4—32no—
369C2×D4—94no—
370C22×C4—94no—
371C22×C4C3×D3211yescharacteristic
372C4⋊C4—32no—
373C4⋊C4—94no—
374C4⋊C4—92no—
375C22⋊C4—34no—
376C22⋊C4—94no—
377C22⋊C4—92no—
378C42—91no—
379C42—91no—
380C22×C3—22no—
381C22×C3—24no—
382C22×C3—22no—
383C22×C3—21no—
384C22×C3—22no—
385C22×C3—22no—
386C22×C3—22no—
387C22×C3—21no—
388C22×C3—22no—
389C22×C3—42no—
390C22×C3—42no—
391C22×C3—42no—
392C22×C3—42no—
393C22×C3—44no—
394C22×C3—41no—
395C22×C3—64no—
396C22×C3—64no—
397C22×C3—64no—
398C22×C3—64no—
399C22×C3—64no—
400C22×C3—64no—
401C22×C3—124no—
402C22×C3—124no—
403C22×C3—124no—
404C22×C3C22×D3211yescharacteristic
405C22×C3C23×C3×D312yes—
406C2×D3—24no—
407C2×D3—24no—
408C2×D3—24no—
409C3×C4—22no—
410C3×C4—21no—
411C3×C4—22no—
412C3×C4—22no—
413C3×C4—24no—
414C3×C4—42no—
415C3×C4—42no—
416C3×C4—42no—
417C3×C4—64no—
418C3×C4—64no—
419C3×C4—124no—
420C3×C4—182no—
421C3×C4—182no—
422C3×C4#380512yes—
423C3×C4C3×#23714yes—
424Dic3—24no—
425Dic3—62no—
426Dic3—62no—
427Dic3—62no—
428Dic3—62no—
429Dic3—122no—
430Dic3—122no—
431C32—22no—
432C32—21no—
433C32—41no—
434C32#561912yescomplemented
435C32C3×#54011yescharacteristic, complemented
436C23—34no—
437C23C2×C3×D3211yescharacteristic
438D4—64no—
439D4—124no—
440D4—364no—
441C2×C4—22no—
442C2×C4—34no—
443C2×C4—92no—
444C2×C4—92no—
445C2×C4—364no—
446C2×C4C2×C3×D3211yescharacteristic
447C2×C4C2×C3×D3211yescharacteristic
448C2×C3—21no—
449C2×C3—22no—
450C2×C3—22no—
451C2×C3—22no—
452C2×C3—22no—
453C2×C3—44no—
454C2×C3—42no—
455C2×C3—44no—
456C2×C3—42no—
457C2×C3—41no—
458C2×C3—41no—
459C2×C3—42no—
460C2×C3—124no—
461C2×C3—124no—
462C2×C3—244no—
463C2×C3#6430712yes—
464C2×C3C2×#380511yescharacteristic
465C2×C3C2×C3×#23712yes—
466C2×C3C3×#100412yes—
467C2×C3C3×#100912yes—
468D3—44no—
469C22—21no—
470C22—22no—
471C22—64no—
472C22—64no—
473C22—64no—
474C22C22×C3×D3211yescharacteristic
475C4—21no—
476C4—64no—
477C4—182no—
478C4—182no—
479C4C3×#380512yes—
480C3—22no—
481C3—21no—
482C3—41no—
483C3#1059504111yescharacteristic, direct product factor
484C3C3×#561912yescomplemented
485C2—41no—
486C2—124no—
487C2C3×#6430712yes—
488C2C2×C3×#380511yescharacteristic
subgroupG/H
centerC22×C3C22×D32
commutator, cosocleC22×C32C24×C3
FrattiniC22C22×C3×D32
FittingC22×C33×C4C22
socleC22×C33C24
hypercenterC22×C3×C4D32
hypocenterC32C3×#540
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C22×C3×C4

Complements

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

NH ≅ G/Ntype
1. C3×#64135C2semidirect
2. C32×#930C2semidirect
3. C3×#63970C2semidirect
4. C3×#63914C2semidirect
5. C3×#63898C2semidirect
6. C3×#63891C2semidirect
7. C3×#63872C2semidirect
9. #10595041C3direct (N × H)
24. #64135C2×C3semidirect
30. C3×#930D3semidirect
31. C3×#930C2×C3semidirect
38. #63970C2×C3semidirect
39. #63914C2×C3semidirect
40. #63898C2×C3semidirect
41. #63891C2×C3semidirect
42. #63872C2×C3semidirect
128. #930C3×D3semidirect
295. C33#540semidirect
434. C32#5619semidirect
435. C32C3×#540semidirect
483. C3#10595041direct (N × H)
484. C3C3×#5619semidirect

Maximal subgroups

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

subgroupindex
C3×#641352
C32×#9302
C3×#639702
C3×#639142
C3×#638982
C3×#638912
C3×#638722
C3×#56193
#105950413