Finite Group Databasebeta

← C3×#10590593 #13157610

Subgroups — C3×#10590593 #13157610

Order 1728.

1026 conjugacy classes of subgroups.

The subgroup lattice has 464 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#3802C211yesmaximal, characteristic
2C3×C4⋊C4×C3⋊D3C211yesmaximal, characteristic, complemented
3C2×C3×#3710C211yesmaximal, characteristic, complemented
4C32×#809C212yesmaximal, complemented
5C3×#63563C212yesmaximal, complemented
6C3×D3×#102—32nomaximal
7#10590593C311yesmaximal, characteristic, direct product factor
8C22×C3×D32—21no—
9C2×C3×C4×C3⋊D3—21no—
10C2×C3×C4×C3⋊D3C2211yescharacteristic
11C2×C32×D12C2212yes—
12C2×C3×#625—21no—
13C3×#3802—22no—
14C3×#3802C414yescomplemented
15C33×C4⋊C4C2211yescharacteristic, complemented
16C3×C4⋊(C3⋊Dic3)C2211yescharacteristic, complemented
17C3×#3750—21no—
18C2×C32×C3⋊C8C2212yescomplemented
19C3×#3710—22no—
20C2×C3×D3×D4—32no—
21C2×#3802C2×C311yescharacteristic
22C4⋊C4×C3⋊D3C2×C311yescharacteristic, complemented
23C2×C3×D3×C8—32no—
24C3×D3×C4⋊C4—32no—
25C3×D3×C4⋊C4—61no—
26C2×#3710C2×C311yescharacteristic, complemented
27C32×#102—32no—
28C3×#832—32no—
29C3×#821—32no—
30C3×#809—22no—
31C3×#809D312yescomplemented
32C3×#809C2×C312yescomplemented
33#63563C2×C312yescomplemented
34C22×C3×C3⋊D3D411yescharacteristic
35C22×C32×D3—22no—
36C2×C3×D32—24no—
37C2×C3×D32—42no—
38C2×C3×D32—42no—
39C2×C33×C4—21no—
40C2×C33×C4C2311yescharacteristic
41C2×C3×C3⋊Dic3—21no—
42C2×C3×C3⋊Dic3D411yescharacteristic
43C3×C4×C3⋊D3—42no—
44C3×C4×C3⋊D3C2×C412yes—
45C3×C4×C3⋊D3D412yes—
46C32×D12—22no—
47C32×D12C2×C414yescomplemented
48C3×#625—24no—
49C32×C3⋊C8—22no—
50C2×C3×#102—91no—
51D3×#102—32no—
52C23×C3×D3—62no—
53C22×D32—21no—
54C2×C32×D4—32no—
55C2×C4×C3⋊D3—21no—
56C2×C4×C3⋊D3C22×C311yescharacteristic
57C2×C3×#67—62no—
58C3×D3×D4—38no—
59C3×D3×D4—64no—
60C2×C3×D12—22no—
61C2×C3×D12C2×D312yes—
62C2×C3×D12C22×C312yes—
63C2×C3×C4×D3—32no—
64C2×C3×C4×D3—61no—
65C2×C3×C4×D3—62no—
66C2×C3×C4×D3—62no—
67C2×#625—21no—
68#3802—22no—
69#3802C3×C414yescomplemented
70C2×C32×C8—32no—
71C32×C4⋊C4—21no—
72C32×C4⋊C4—22no—
73C32×C4⋊C4—41no—
74C32×C4⋊C4C2×D312yescomplemented
75C32×C4⋊C4C22×C311yescharacteristic, complemented
76C4⋊(C3⋊Dic3)C22×C311yescharacteristic, complemented
77#3750—21no—
78C3×C4⋊Dic3—32no—
79C3×C4⋊Dic3—61no—
80C3×Dic3⋊C4—62no—
81C3×Dic3⋊C4—62no—
82C2×C3×C3⋊C8—22no—
83C2×C3×C3⋊C8C2×D312yescomplemented
84C2×C3×C3⋊C8C22×C312yescomplemented
85C3×D3×C8—64no—
86#3710—22no—
87C22×C33C2×D411yescharacteristic
88C2×C3×C3⋊D3C22⋊C412yes—
89C2×C3×C3⋊D3D812yes—
90C2×C3×C3⋊D3SD1612yes—
91C2×C32×D3—24no—
92C2×C32×D3—42no—
93C2×C32×D3—42no—
94C3×D32—48no—
95C33×C4—41no—
96C33×C4C22×C411yescharacteristic
97C33×C4C2×D411yescharacteristic
98C3×C3⋊Dic3—41no—
99C3×C3⋊Dic3C22⋊C412yes—
100C22×C3×D4—91no—
101C2×D3×D4—32no—
102C22×C3×C8—91no—
103C2×C3×C4⋊C4—91no—
104C2×D3×C8—32no—
105D3×C4⋊C4—32no—
106D3×C4⋊C4—61no—
107C3×#102—32no—
108C3×#102—32no—
109C3×#102—62no—
110C3×#102—92no—
111#832—32no—
112#821—32no—
113#809C3×D312yescomplemented
114C23×C32—62no—
115C22×C3⋊D3C3×D411yescharacteristic
116C22×C3×D3—22no—
117C22×C3×D3—22no—
118C22×C3×D3—32no—
119C22×C3×D3—42no—
120C22×C3×D3—61no—
121C22×C3×D3—68no—
122C22×C3×D3—124no—
123C22×C3×D3—124no—
124C2×D32—24no—
125C2×D32—42no—
126C2×D32—42no—
127C32×D4—34no—
128C32×D4—62no—
129C2×C32×C4—21no—
130C2×C32×C4—22no—
131C2×C32×C4—22no—
132C2×C32×C4—21no—
133C2×C32×C4—22no—
134C2×C32×C4—41no—
135C2×C32×C4—42no—
136C2×C32×C4—42no—
137C2×C32×C4C22×D312yes—
138C2×C32×C4C23×C311yescharacteristic
139C2×C3⋊Dic3—21no—
140C2×C3⋊Dic3C3×D411yescharacteristic
141C4×C3⋊D3—42no—
142C4×C3⋊D3C2×C3×C412yes—
143C4×C3⋊D3C3×D412yes—
144C3×#67—68no—
145C2×C3×Dic3—32no—
146C2×C3×Dic3—61no—
147C2×C3×Dic3—62no—
148C2×C3×Dic3—62no—
149C3×D12—22no—
150C3×D12—22no—
151C3×D12—24no—
152C3×D12—42no—
153C3×D12C4×D314yescomplemented
154C3×D12C2×C3×C414yescomplemented
155C3×C4×D3—34no—
156C3×C4×D3—34no—
157C3×C4×D3—62no—
158C3×C4×D3—62no—
159C3×C4×D3—124no—
160C3×C4×D3—124no—
161#625—24no—
162C32×C8—62no—
163C3×C3⋊C8—22no—
164C3×C3⋊C8—22no—
165C3×C3⋊C8—42no—
166C2×#102—91noSylow
167C2×C33C2×C22⋊C411yescharacteristic
168C2×C33C2×D811yescharacteristic
169C2×C33C2×SD1611yescharacteristic
170C3×C3⋊D3#10214yescomplemented
171C32×D3—44no—
172C24×C3—181no—
173C23×D3—62no—
174C2×C3×D4—32no—
175C2×C3×D4—32no—
176C2×C3×D4—62no—
177C2×C3×D4—94no—
178C2×C3×D4—182no—
179C2×C3×D4—181no—
180C22×C3×C4—91no—
181C22×C3×C4—181no—
182C2×#67—62no—
183D3×D4—38no—
184D3×D4—64no—
185C2×D12C2×C3×D312yes—
186C2×C4×D3—32no—
187C2×C4×D3—61no—
188C2×C4×D3—62no—
189C2×C4×D3—62no—
190C2×C3×C8—32no—
191C2×C3×C8—32no—
192C2×C3×C8—62no—
193C2×C3×C8—182no—
194C3×C4⋊C4—21no—
195C3×C4⋊C4—22no—
196C3×C4⋊C4—41no—
197C3×C4⋊C4—91no—
198C3×C4⋊C4—181no—
199C3×C4⋊C4D3211yescharacteristic, complemented
200C3×C4⋊C4C2×C3×D312yescomplemented
201C4⋊Dic3—32no—
202C4⋊Dic3—61no—
203Dic3⋊C4—62no—
204Dic3⋊C4—62no—
205C2×C3⋊C8C2×C3×D312yescomplemented
206D3×C8—64no—
207C22×C32—21no—
208C22×C32—22no—
209C22×C32—41no—
210C22×C32—64no—
211C22×C32—122no—
212C22×C32—122no—
213C22×C32D3×D412yes—
214C22×C32C2×C3×D411yescharacteristic
215C2×C3⋊D3C3×C22⋊C412yes—
216C2×C3⋊D3C3×D812yes—
217C2×C3⋊D3C3×SD1612yes—
218C2×C3×D3—24no—
219C2×C3×D3—24no—
220C2×C3×D3—34no—
221C2×C3×D3—34no—
222C2×C3×D3—34no—
223C2×C3×D3—44no—
224C2×C3×D3—42no—
225C2×C3×D3—42no—
226C2×C3×D3—42no—
227C2×C3×D3—42no—
228C2×C3×D3—62no—
229C2×C3×D3—62no—
230C2×C3×D3—62no—
231C2×C3×D3—82no—
232C2×C3×D3—82no—
233C2×C3×D3—1216no—
234D32—48no—
235C32×C4—21no—
236C32×C4—21no—
237C32×C4—22no—
238C32×C4—22no—
239C32×C4—41no—
240C32×C4—41no—
241C32×C4—41no—
242C32×C4—42no—
243C32×C4—42no—
244C32×C4—82no—
245C32×C4—82no—
246C32×C4C2×C4×D312yes—
247C32×C4D3×D412yes—
248C32×C4C22×C3×C411yescharacteristic
249C32×C4C2×C3×D411yescharacteristic
250C3⋊Dic3—41no—
251C3⋊Dic3C3×C22⋊C412yes—
252C3×Dic3—34no—
253C3×Dic3—62no—
254C3×Dic3—122no—
255C3×Dic3—122no—
256C22×D4—91no—
257C22×C8—91no—
258C2×C4⋊C4—91no—
259#102—32no—
260#102—92no—
261C33C2×#10211yescharacteristic, complemented, Sylow
262C23×C3—62no—
263C23×C3—62no—
264C23×C3—91no—
265C23×C3—122no—
266C23×C3—184no—
267C23×C3—362no—
268C23×C3—362no—
269C22×D3—22no—
270C22×D3—32no—
271C22×D3—61no—
272C22×D3—68no—
273C22×D3—124no—
274C22×D3—124no—
275C3×D4—34no—
276C3×D4—34no—
277C3×D4—64no—
278C3×D4—62no—
279C3×D4—62no—
280C3×D4—122no—
281C3×D4—184no—
282C2×C3×C4—22no—
283C2×C3×C4—21no—
284C2×C3×C4—22no—
285C2×C3×C4—22no—
286C2×C3×C4—21no—
287C2×C3×C4—42no—
288C2×C3×C4—42no—
289C2×C3×C4—41no—
290C2×C3×C4—91no—
291C2×C3×C4—92no—
292C2×C3×C4—92no—
293C2×C3×C4—181no—
294C2×C3×C4—362no—
295C2×C3×C4C2×D3211yescharacteristic
296C2×C3×C4C22×C3×D312yes—
297#67—68no—
298C2×Dic3—32no—
299C2×Dic3—61no—
300C2×Dic3—62no—
301C2×Dic3—62no—
302D12—22no—
303D12C3×C4×D314yescomplemented
304C4×D3—34no—
305C4×D3—34no—
306C4×D3—62no—
307C4×D3—62no—
308C4×D3—124no—
309C4×D3—124no—
310C3×C8—62no—
311C3×C8—62no—
312C3×C8—122no—
313C3⋊C8—22no—
314C2×C32—22no—
315C2×C32—22no—
316C2×C32—22no—
317C2×C32—21no—
318C2×C32—21no—
319C2×C32—21no—
320C2×C32—41no—
321C2×C32—41no—
322C2×C32—41no—
323C2×C32—124no—
324C2×C32D3×C22⋊C412yes—
325C2×C32D3×D812yes—
326C2×C32D3×SD1612yes—
327C2×C32C2×C3×C22⋊C411yescharacteristic
328C2×C32C2×C3×D811yescharacteristic
329C2×C32C2×C3×SD1611yescharacteristic
330C3⋊D3C3×#10214yescomplemented
331C3×D3—38no—
332C3×D3—44no—
333C3×D3—44no—
334C3×D3—64no—
335C3×D3—84no—
336C24—181no—
337C2×D4—32no—
338C2×D4—94no—
339C2×D4—182no—
340C2×D4—181no—
341C22×C4—91no—
342C22×C4—181no—
343C2×C8—32no—
344C2×C8—182no—
345C4⋊C4—91no—
346C4⋊C4—181no—
347C4⋊C4C3×D3211yescharacteristic, complemented
348C22×C3—21no—
349C22×C3—22no—
350C22×C3—41no—
351C22×C3—64no—
352C22×C3—64no—
353C22×C3—92no—
354C22×C3—92no—
355C22×C3—92no—
356C22×C3—124no—
357C22×C3—122no—
358C22×C3—122no—
359C22×C3—122no—
360C22×C3—122no—
361C22×C3—242no—
362C22×C3—242no—
363C22×C3—368no—
364C22×C3#381111yescharacteristic
365C22×C3C3×D3×D412yes—
366C2×D3—24no—
367C2×D3—34no—
368C2×D3—34no—
369C2×D3—34no—
370C2×D3—42no—
371C2×D3—42no—
372C2×D3—62no—
373C2×D3—62no—
374C2×D3—62no—
375C2×D3—1216no—
376C3×C4—21no—
377C3×C4—22no—
378C3×C4—22no—
379C3×C4—21no—
380C3×C4—42no—
381C3×C4—42no—
382C3×C4—41no—
383C3×C4—41no—
384C3×C4—41no—
385C3×C4—82no—
386C3×C4—82no—
387C3×C4—92no—
388C3×C4—361no—
389C3×C4C4×D3211yescharacteristic
390C3×C4#380211yescharacteristic
391C3×C4C2×C3×C4×D312yes—
392C3×C4C3×D3×D412yes—
393Dic3—34no—
394Dic3—62no—
395Dic3—122no—
396Dic3—122no—
397C32—22no—
398C32—21no—
399C32—41no—
400C32D3×#10212yescomplemented
401C32C2×C3×#10211yescharacteristic, complemented
402C23—62no—
403C23—91no—
404C23—184no—
405C23—362no—
406C23—362no—
407D4—34no—
408D4—62no—
409D4—184no—
410C2×C4—21no—
411C2×C4—91no—
412C2×C4—92no—
413C2×C4—92no—
414C2×C4—181no—
415C2×C4—362no—
416C2×C4C2×C3×D3211yescharacteristic
417C8—62no—
418C2×C3—21no—
419C2×C3—22no—
420C2×C3—21no—
421C2×C3—22no—
422C2×C3—21no—
423C2×C3—22no—
424C2×C3—41no—
425C2×C3—41no—
426C2×C3—41no—
427C2×C3—94no—
428C2×C3—124no—
429C2×C3—124no—
430C2×C3—244no—
431C2×C3#6392111yescharacteristic
432C2×C3#6392611yescharacteristic
433C2×C3#6394011yescharacteristic
434C2×C3C3×D3×C22⋊C412yes—
435C2×C3C3×D3×D812yes—
436C2×C3C3×D3×SD1612yes—
437D3—38no—
438D3—44no—
439D3—64no—
440C22—64no—
441C22—92no—
442C22—92no—
443C22—92no—
444C22—122no—
445C22—122no—
446C22—368no—
447C22C3×#381111yescharacteristic
448C4—41no—
449C4—92no—
450C4—361no—
451C4C3×C4×D3211yescharacteristic
452C4C3×#380211yescharacteristic
453C3—21no—
454C3—22no—
455C3—41no—
456C3#1059059311yescharacteristic, direct product factor
457C3C3×D3×#10212yescomplemented
458C2—94no—
459C2—124no—
460C2C3×#6392111yescharacteristic
461C2C3×#6392611yescharacteristic
462C2C3×#6394011yescharacteristic
subgroupG/H
centerC22×C3#3811
commutatorC32×C4C22×C3×C4
FrattiniC2×C4C2×C3×D32
FittingC33×C4⋊C4C22
socleC22×C33C2×D4
prime radicalC2×C3×#3802C2
cosocleC2×C32×C4C23×C3
hypercenterC3×C4⋊C4D32
hypocenterC32C2×C3×#102
LCSG ▷ C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C2×C3×C4 ◁ C3×C4⋊C4

Complements

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

NH ≅ G/Ntype
2. C3×C4⋊C4×C3⋊D3C2semidirect
3. C2×C3×#3710C2semidirect
4. C32×#809C2semidirect
5. C3×#63563C2semidirect
7. #10590593C3direct (N × H)
14. C3×#3802C4semidirect
15. C33×C4⋊C4C22semidirect
16. C3×C4⋊(C3⋊Dic3)C22semidirect
18. C2×C32×C3⋊C8C22semidirect
22. C4⋊C4×C3⋊D3C2×C3semidirect
26. C2×#3710C2×C3semidirect
31. C3×#809D3semidirect
32. C3×#809C2×C3semidirect
33. #63563C2×C3semidirect
47. C32×D12C2×C4semidirect
69. #3802C3×C4semidirect
74. C32×C4⋊C4C2×D3semidirect
75. C32×C4⋊C4C22×C3semidirect
76. C4⋊(C3⋊Dic3)C22×C3semidirect
83. C2×C3×C3⋊C8C2×D3semidirect
84. C2×C3×C3⋊C8C22×C3semidirect
113. #809C3×D3semidirect
153. C3×D12C4×D3semidirect
154. C3×D12C2×C3×C4semidirect
170. C3×C3⋊D3#102semidirect
199. C3×C4⋊C4D32semidirect
200. C3×C4⋊C4C2×C3×D3semidirect
205. C2×C3⋊C8C2×C3×D3semidirect
261. C33C2×#102semidirect
303. D12C3×C4×D3semidirect
330. C3⋊D3C3×#102semidirect
347. C4⋊C4C3×D32semidirect
400. C32D3×#102semidirect
401. C32C2×C3×#102semidirect
456. C3#10590593direct (N × H)
457. C3C3×D3×#102semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#38022
C3×C4⋊C4×C3⋊D32
C2×C3×#37102
C32×#8092
C3×#635632
C3×D3×#1023
#105905933