Finite Group Databasebeta

← C3×#10591325 #13158244

Subgroups — C3×#10591325 #13158244

Order 1728.

1410 conjugacy classes of subgroups.

The subgroup lattice has 478 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×C4×C3⋊D3C211yesmaximal, characteristic
2C2×C32×C4⋊Dic3C212yesmaximal, complemented
3C2×C3×#3722C212yesmaximal
4C3×#63579C212yesmaximal, complemented
5C3×#4987—32nomaximal
6#10591325C311yesmaximal, characteristic, direct product factor
7C23×C3×C3⋊D3C2211yescharacteristic
8C22×C33×C4C2211yescharacteristic
9C22×C3×C3⋊Dic3C2211yescharacteristic
10C2×C3×C4×C3⋊D3—24no—
11C2×C3×C4×C3⋊D3C414yescomplemented
12C22×C32×Dic3C2214yes—
13C32×C4⋊Dic3—24no—
14C3×#3722—24no—
15C22×C4×C3⋊D3C2×C311yescharacteristic
16C22×C3×C4×D3—32no—
17C22×C3×C4×D3—61no—
18C2×C32×C4⋊C4—32no—
19C2×C3×(C2×D3)⋊C4—34no—
20C2×C3×C4⋊Dic3—22no—
21C2×C3×C4⋊Dic3D312yescomplemented
22C2×C3×C4⋊Dic3C2×C312yescomplemented
23C2×#3722C2×C312yes—
24C3×C3⋊#95—34no—
25#63579C2×C312yescomplemented
26C23×C33C2311yescharacteristic
27C22×C3×C3⋊D3—22no—
28C22×C3×C3⋊D3—22no—
29C22×C3×C3⋊D3C2×C412yes—
30C22×C3×C3⋊D3D412yes—
31C22×C3×C3⋊D3Q812yes—
32C2×C33×C4C2×C412yes—
33C2×C33×C4D414yes—
34C2×C3×C3⋊Dic3—22no—
35C2×C3×C3⋊Dic3C2×C412yes—
36C3×C4×C3⋊D3—28no—
37C2×C32×Dic3—24no—
38C2×C32×Dic3—24no—
39C2×C32×Dic3—24no—
40C3×#380—91no—
41#4987—32no—
42C23×C3⋊D3C22×C311yescharacteristic
43C23×C3×D3—32no—
44C23×C3×D3—61no—
45C22×C32×C4—21no—
46C22×C32×C4—22no—
47C22×C32×C4—34no—
48C22×C32×C4—41no—
49C22×C32×C4C2×D312yes—
50C22×C32×C4C22×C311yescharacteristic
51C22×C3⋊Dic3C22×C311yescharacteristic
52C2×C4×C3⋊D3—24no—
53C2×C4×C3⋊D3C3×C414yescomplemented
54C22×C3×Dic3—24no—
55C22×C3×Dic3—32no—
56C22×C3×Dic3—61no—
57C22×C3×Dic3C2×D314yes—
58C22×C3×Dic3C22×C314yes—
59C2×C3×C4×D3—38no—
60C2×C3×C4×D3—64no—
61C2×C3×C4×D3—68no—
62C2×C3×C4×D3—64no—
63C2×C3×C4×D3—64no—
64C32×C4⋊C4—64no—
65C3×(C2×D3)⋊C4—68no—
66C3×C4⋊Dic3—28no—
67C3×C4⋊Dic3—24no—
68C3×C4⋊Dic3—24no—
69#3722—24no—
70C22×C33C22×C411yescharacteristic
71C22×C33C2×D412yes—
72C22×C33C2×D411yescharacteristic
73C22×C33C2×Q811yescharacteristic
74C22×C33#4112yes—
75C2×C3×C3⋊D3—22no—
76C2×C3×C3⋊D3—22no—
77C2×C3×C3⋊D3—24no—
78C2×C3×C3⋊D3—24no—
79C2×C3×C3⋊D3C4⋊C414yescomplemented
80C33×C4C22⋊C414yescomplemented
81C3×C3⋊Dic3—22no—
82C32×Dic3—44no—
83C23×C3×C4—91no—
84C22×C4×D3—32no—
85C22×C4×D3—61no—
86C2×C3×C4⋊C4—32no—
87C2×C3×C4⋊C4—32no—
88C2×C3×C4⋊C4—62no—
89C2×C3×C22⋊C4—92no—
90C2×(C2×D3)⋊C4—34no—
91C2×C4⋊Dic3C3×D312yescomplemented
92C3×#95—92no—
93C3⋊#95—34no—
94C23×C32—21no—
95C23×C32—22no—
96C23×C32—41no—
97C23×C32C22×D312yes—
98C23×C32C23×C311yescharacteristic
99C22×C3⋊D3—22no—
100C22×C3⋊D3—22no—
101C22×C3⋊D3C2×C3×C412yes—
102C22×C3⋊D3C3×D412yes—
103C22×C3⋊D3C3×Q812yes—
104C22×C3×D3—34no—
105C22×C3×D3—34no—
106C22×C3×D3—34no—
107C22×C3×D3—62no—
108C22×C3×D3—62no—
109C22×C3×D3—62no—
110C22×C3×D3—64no—
111C22×C3×D3—64no—
112C22×C3×D3—62no—
113C22×C3×D3—62no—
114C22×C3×D3—62no—
115C22×C3×D3—62no—
116C2×C32×C4—22no—
117C2×C32×C4—24no—
118C2×C32×C4—24no—
119C2×C32×C4—24no—
120C2×C32×C4—24no—
121C2×C32×C4—42no—
122C2×C32×C4—44no—
123C2×C32×C4—64no—
124C2×C32×C4—64no—
125C2×C32×C4—64no—
126C2×C32×C4C4×D314yes—
127C2×C32×C4D1214yes—
128C2×C32×C4#6714yes—
129C2×C32×C4C2×C3×C412yes—
130C2×C32×C4C3×D414yes—
131C2×C3⋊Dic3—22no—
132C2×C3⋊Dic3C2×C3×C412yes—
133C4×C3⋊D3—28no—
134C2×C3×Dic3—28no—
135C2×C3×Dic3—24no—
136C2×C3×Dic3—24no—
137C2×C3×Dic3—24no—
138C2×C3×Dic3—24no—
139C2×C3×Dic3—24no—
140C2×C3×Dic3—24no—
141C2×C3×Dic3—34no—
142C2×C3×Dic3—44no—
143C2×C3×Dic3—44no—
144C2×C3×Dic3—62no—
145C2×C3×Dic3—64no—
146C2×C3×Dic3—62no—
147C2×C3×Dic3—62no—
148C3×C4×D3—616no—
149C3×C4×D3—616no—
150#380—91noSylow
151C2×C33C2×C22⋊C411yescharacteristic
152C2×C33C2×C4⋊C411yescharacteristic
153C2×C33#11711yescharacteristic
154C2×C33#12112yes—
155C2×C33C22⋊Q812yes—
156C3×C3⋊D3—24no—
157C24×C3—91no—
158C23×D3—32no—
159C23×D3—61no—
160C22×C3×C4—22no—
161C22×C3×C4—21no—
162C22×C3×C4—34no—
163C22×C3×C4—34no—
164C22×C3×C4—41no—
165C22×C3×C4—64no—
166C22×C3×C4—91no—
167C22×C3×C4—94no—
168C22×C3×C4—184no—
169C22×C3×C4D3211yescharacteristic
170C22×C3×C4C2×C3×D312yes—
171C22×Dic3—32no—
172C22×Dic3—61no—
173C22×Dic3C2×C3×D314yes—
174C2×C4×D3—38no—
175C2×C4×D3—64no—
176C2×C4×D3—68no—
177C2×C4×D3—64no—
178C2×C4×D3—64no—
179C3×C4⋊C4—68no—
180C3×C4⋊C4—64no—
181C3×C4⋊C4—64no—
182C3×C22⋊C4—184no—
183(C2×D3)⋊C4—68no—
184C4⋊Dic3—24no—
185C22×C32—22no—
186C22×C32—21no—
187C22×C32—21no—
188C22×C32—21no—
189C22×C32—22no—
190C22×C32—22no—
191C22×C32—22no—
192C22×C32—22no—
193C22×C32—22no—
194C22×C32—22no—
195C22×C32—22no—
196C22×C32—22no—
197C22×C32—41no—
198C22×C32—42no—
199C22×C32—41no—
200C22×C32—41no—
201C22×C32—42no—
202C22×C32C2×C4×D312yes—
203C22×C32C2×D1212yes—
204C22×C32D3×D412yes—
205C22×C32#23712yes—
206C22×C32D3×Q812yes—
207C22×C32#23912yes—
208C22×C32C2×#6712yes—
209C22×C32C22×C3×C411yescharacteristic
210C22×C32C2×C3×D412yes—
211C22×C32C2×C3×D411yescharacteristic
212C22×C32C2×C3×Q811yescharacteristic
213C22×C32C3×#4112yes—
214C2×C3⋊D3—22no—
215C2×C3⋊D3—22no—
216C2×C3⋊D3—24no—
217C2×C3⋊D3—24no—
218C2×C3⋊D3C3×C4⋊C414yescomplemented
219C2×C3×D3—38no—
220C2×C3×D3—64no—
221C2×C3×D3—64no—
222C2×C3×D3—64no—
223C2×C3×D3—68no—
224C2×C3×D3—68no—
225C2×C3×D3—64no—
226C2×C3×D3—64no—
227C2×C3×D3—64no—
228C2×C3×D3—64no—
229C2×C3×D3—64no—
230C2×C3×D3—64no—
231C32×C4—24no—
232C32×C4—28no—
233C32×C4—44no—
234C32×C4—124no—
235C32×C4(C2×D3)⋊C418yescomplemented
236C32×C4C3×C22⋊C414yescomplemented
237C3⋊Dic3—22no—
238C3×Dic3—48no—
239C3×Dic3—44no—
240C3×Dic3—44no—
241C3×Dic3—64no—
242C3×Dic3—64no—
243C23×C4—91no—
244C2×C4⋊C4—32no—
245C2×C22⋊C4—92no—
246#95—92no—
247C33#38011yescharacteristic, complemented, Sylow
248C23×C3—22no—
249C23×C3—21no—
250C23×C3—41no—
251C23×C3—92no—
252C23×C3—92no—
253C23×C3—92no—
254C23×C3—182no—
255C23×C3—182no—
256C23×C3C2×D3211yescharacteristic
257C23×C3C22×C3×D312yes—
258C22×D3—34no—
259C22×D3—34no—
260C22×D3—34no—
261C22×D3—62no—
262C22×D3—62no—
263C22×D3—62no—
264C22×D3—64no—
265C22×D3—64no—
266C22×D3—62no—
267C22×D3—62no—
268C22×D3—62no—
269C22×D3—62no—
270C2×C3×C4—24no—
271C2×C3×C4—22no—
272C2×C3×C4—24no—
273C2×C3×C4—24no—
274C2×C3×C4—24no—
275C2×C3×C4—42no—
276C2×C3×C4—44no—
277C2×C3×C4—68no—
278C2×C3×C4—64no—
279C2×C3×C4—64no—
280C2×C3×C4—64no—
281C2×C3×C4—64no—
282C2×C3×C4—64no—
283C2×C3×C4—64no—
284C2×C3×C4—92no—
285C2×C3×C4—124no—
286C2×C3×C4—124no—
287C2×C3×C4—182no—
288C2×C3×C4—188no—
289C2×C3×C4#62412yes—
290C2×C3×C4#62614yes—
291C2×C3×C4C3×C4×D314yes—
292C2×C3×C4C3×D1214yes—
293C2×C3×C4C3×#6714yes—
294C2×Dic3—24no—
295C2×Dic3—24no—
296C2×Dic3—24no—
297C2×Dic3—34no—
298C2×Dic3—62no—
299C2×Dic3—64no—
300C2×Dic3—62no—
301C2×Dic3—62no—
302C4×D3—616no—
303C4×D3—616no—
304C2×C32—22no—
305C2×C32—22no—
306C2×C32—22no—
307C2×C32—22no—
308C2×C32—22no—
309C2×C32—22no—
310C2×C32—22no—
311C2×C32—21no—
312C2×C32—22no—
313C2×C32—21no—
314C2×C32—22no—
315C2×C32—21no—
316C2×C32—41no—
317C2×C32—41no—
318C2×C32—41no—
319C2×C32—42no—
320C2×C32—42no—
321C2×C32D3×C4⋊C412yes—
322C2×C32#89212yes—
323C2×C32#89512yes—
324C2×C32#89712yes—
325C2×C32C2×(C2×D3)⋊C412yes—
326C2×C32#93812yes—
327C2×C32#94612yes—
328C2×C32C2×C3×C22⋊C411yescharacteristic
329C2×C32C2×C3×C4⋊C411yescharacteristic
330C2×C32C3×#11711yescharacteristic
331C2×C32C3×#12112yes—
332C2×C32C3×C22⋊Q812yes—
333C3⋊D3—24no—
334C3×D3—68no—
335C3×D3—68no—
336C24—91no—
337C22×C4—34no—
338C22×C4—91no—
339C22×C4—94no—
340C22×C4—184no—
341C22×C4C3×D3211yescharacteristic
342C4⋊C4—64no—
343C22⋊C4—184no—
344C22×C3—22no—
345C22×C3—22no—
346C22×C3—22no—
347C22×C3—21no—
348C22×C3—22no—
349C22×C3—22no—
350C22×C3—21no—
351C22×C3—21no—
352C22×C3—22no—
353C22×C3—22no—
354C22×C3—22no—
355C22×C3—22no—
356C22×C3—42no—
357C22×C3—41no—
358C22×C3—41no—
359C22×C3—41no—
360C22×C3—42no—
361C22×C3—94no—
362C22×C3—182no—
363C22×C3—184no—
364C22×C3—184no—
365C22×C3—182no—
366C22×C3#379612yes—
367C22×C3#379711yescharacteristic
368C22×C3#380211yescharacteristic
369C22×C3C2×#62411yescharacteristic
370C22×C3C2×#62612yes—
371C22×C3C2×C3×C4×D312yes—
372C22×C3C2×C3×D1212yes—
373C22×C3C3×D3×D412yes—
374C22×C3C3×#23712yes—
375C22×C3C3×D3×Q812yes—
376C22×C3C3×#23912yes—
377C22×C3C2×C3×#6712yes—
378C2×D3—38no—
379C2×D3—64no—
380C2×D3—64no—
381C2×D3—64no—
382C2×D3—68no—
383C2×D3—68no—
384C2×D3—64no—
385C2×D3—64no—
386C2×D3—64no—
387C2×D3—64no—
388C2×D3—64no—
389C2×D3—64no—
390C3×C4—24no—
391C3×C4—28no—
392C3×C4—44no—
393C3×C4—128no—
394C3×C4—124no—
395C3×C4—124no—
396C3×C4—182no—
397C3×C4#372214yescomplemented
398C3×C4C3×(C2×D3)⋊C418yescomplemented
399Dic3—44no—
400Dic3—64no—
401Dic3—64no—
402C32—22no—
403C32—21no—
404C32—41no—
405C32#498712yescomplemented
406C32C3×#38011yescharacteristic, complemented
407C23—92no—
408C23—92no—
409C23—92no—
410C23—182no—
411C23—182no—
412C23C2×C3×D3211yescharacteristic
413C2×C4—64no—
414C2×C4—64no—
415C2×C4—64no—
416C2×C4—92no—
417C2×C4—182no—
418C2×C4—188no—
419C2×C4C3×#62412yes—
420C2×C4C3×#62614yes—
421C2×C3—21no—
422C2×C3—22no—
423C2×C3—21no—
424C2×C3—22no—
425C2×C3—22no—
426C2×C3—22no—
427C2×C3—22no—
428C2×C3—21no—
429C2×C3—22no—
430C2×C3—22no—
431C2×C3—22no—
432C2×C3—22no—
433C2×C3—41no—
434C2×C3—41no—
435C2×C3—42no—
436C2×C3—41no—
437C2×C3—42no—
438C2×C3—184no—
439C2×C3#6384911yescharacteristic
440C2×C3#6387112yes—
441C2×C3#6390011yescharacteristic
442C2×C3#6391612yes—
443C2×C3C2×#372211yescharacteristic
444C2×C3C3×D3×C4⋊C412yes—
445C2×C3C3×#89212yes—
446C2×C3C3×#89512yes—
447C2×C3C3×#89712yes—
448C2×C3C2×C3×(C2×D3)⋊C412yes—
449C2×C3C3×#93812yes—
450C2×C3C3×#94612yes—
451D3—68no—
452D3—68no—
453C22—94no—
454C22—182no—
455C22—182no—
456C22—184no—
457C22—184no—
458C22C3×#379612yes—
459C22C3×#379711yescharacteristic
460C22C3×#380211yescharacteristic
461C22C2×C3×#62411yescharacteristic
462C22C2×C3×#62612yes—
463C4—124no—
464C4—182no—
465C4C3×#372214yescomplemented
466C3—21no—
467C3—22no—
468C3—41no—
469C3#1059132511yescharacteristic, direct product factor
470C3C3×#498712yescomplemented
471C2—184no—
472C2C3×#6384911yescharacteristic
473C2C3×#6387112yes—
474C2C3×#6390011yescharacteristic
475C2C3×#6391612yes—
476C2C2×C3×#372211yescharacteristic
subgroupG/H
centerC23×C3C2×D32
commutatorC22×C32C22×C3×C4
FrattiniC23C2×C3×D32
FittingC22×C33×C4C22
socleC23×C33C23
prime radicalC23×C3×C3⋊D3C22
cosocleC23×C32C23×C3
hypercenterC22×C3×C4D32
hypocenterC32C3×#380
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C23×C3 ◁ C22×C3×C4

Complements

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

NH ≅ G/Ntype
2. C2×C32×C4⋊Dic3C2semidirect
4. C3×#63579C2semidirect
6. #10591325C3direct (N × H)
11. C2×C3×C4×C3⋊D3C4semidirect
21. C2×C3×C4⋊Dic3D3semidirect
22. C2×C3×C4⋊Dic3C2×C3semidirect
25. #63579C2×C3semidirect
53. C2×C4×C3⋊D3C3×C4semidirect
79. C2×C3×C3⋊D3C4⋊C4semidirect
80. C33×C4C22⋊C4semidirect
91. C2×C4⋊Dic3C3×D3semidirect
218. C2×C3⋊D3C3×C4⋊C4semidirect
235. C32×C4(C2×D3)⋊C4semidirect
236. C32×C4C3×C22⋊C4semidirect
247. C33#380semidirect
397. C3×C4#3722semidirect
398. C3×C4C3×(C2×D3)⋊C4semidirect
405. C32#4987semidirect
406. C32C3×#380semidirect
465. C4C3×#3722semidirect
469. C3#10591325direct (N × H)
470. C3C3×#4987semidirect

Maximal subgroups

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

subgroupindex
C22×C3×C4×C3⋊D32
C2×C32×C4⋊Dic32
C2×C3×#37222
C3×#635792
C3×#49873
#105913253