Finite Group Databasebeta

← C3×#10591131 #13158160

Subgroups — C3×#10591131 #13158160

Order 1728.

762 conjugacy classes of subgroups.

The subgroup lattice has 512 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×C4⋊(C3⋊Dic3)C211yesmaximal, characteristic
2C32×#926C211yesmaximal, characteristic
3C2×C32×#208C211yesmaximal, characteristic
4C3×#63575C212yesmaximal, complemented
5C3×#63573C212yesmaximal, complemented
6C3×#5121—31nomaximal
7C3×#4995—31nomaximal
8#10591131C311yesmaximal, characteristic, direct product factor
9C22×C33×C4C2211yescharacteristic
10C22×C3×C3⋊Dic3—21no—
11C3×C4⋊(C3⋊Dic3)—21no—
12C3×C4⋊(C3⋊Dic3)C2212yes—
13C32×#217—21no—
14C32×C4⋊Dic3C2211yescharacteristic
15C32×C4⋊Dic3C2211yescharacteristic
16C32×C4×Dic3—21no—
17C32×#208C414yescomplemented
18C2×C32×C3⋊C8C2212yes—
19C2×C32×M16—31no—
20C32×#117—31no—
21C2×C4⋊(C3⋊Dic3)C2×C311yescharacteristic
22C3×#926—21no—
23C3×#926D311yescharacteristic
24C3×#926C2×C311yescharacteristic
25C2×C3×C4⋊Dic3—31no—
26C2×C3×C4⋊Dic3—31no—
27C2×C3×C4⋊Dic3—61no—
28C2×C3×#208—21no—
29C2×C3×#208D311yescharacteristic
30C2×C3×#208C2×C311yescharacteristic
31C3×#818—32no—
32C3×#817—32no—
33C3×#808—32no—
34C3×#807—32no—
35#63575C2×C312yescomplemented
36#63573C2×C312yescomplemented
37C23×C33D411yescharacteristic
38C2×C33×C4C2×C412yes—
39C2×C33×C4D411yescharacteristic
40C2×C33×C4Q812yes—
41C2×C33×C4C2311yescharacteristic
42C2×C3×C3⋊Dic3—22no—
43C2×C3×C3⋊Dic3—42no—
44C2×C32×Dic3—21no—
45C2×C32×Dic3—21no—
46C32×C3⋊C8C2×C414yescomplemented
47C3×#428—91no—
48#5121—31no—
49#4995—31no—
50C22×C32×C4—21no—
51C22×C32×C4—21no—
52C22×C32×C4—21no—
53C22×C32×C4—41no—
54C22×C32×C4C2×D311yescharacteristic
55C22×C32×C4C2×D311yescharacteristic
56C22×C32×C4C22×C311yescharacteristic
57C22×C3⋊Dic3—21no—
58C22×C3×Dic3—62no—
59C22×C3×Dic3—61no—
60C22×C3×Dic3—61no—
61C32×M16—34no—
62C2×C32×C8—32no—
63C32×C4⋊C4—31no—
64C32×C4⋊C4—31no—
65C32×C22⋊C4—61no—
66C32×C42—61no—
67C4⋊(C3⋊Dic3)—21no—
68C4⋊(C3⋊Dic3)C22×C312yes—
69C3×#217—22no—
70C3×#217—21no—
71C3×#217—21no—
72C3×C4⋊Dic3—21no—
73C3×C4⋊Dic3—21no—
74C3×C4⋊Dic3—32no—
75C3×C4⋊Dic3—32no—
76C3×C4⋊Dic3—62no—
77C3×C4⋊Dic3—62no—
78C3×C4⋊Dic3—61no—
79C3×C4⋊Dic3—61no—
80C3×C4⋊Dic3C2×D311yescharacteristic
81C3×C4⋊Dic3C2×D311yescharacteristic
82C3×C4⋊Dic3C22×C311yescharacteristic
83C3×C4⋊Dic3C22×C311yescharacteristic
84C3×C4×Dic3—22no—
85C3×C4×Dic3—21no—
86C3×C4×Dic3—21no—
87C3×#208—24no—
88C3×#208Dic314yescomplemented
89C3×#208C3×C414yescomplemented
90C2×C3×C3⋊C8—22no—
91C2×C3×C3⋊C8C2×D312yes—
92C2×C3×C3⋊C8C22×C312yes—
93C22×C33—21no—
94C22×C33—21no—
95C22×C33C4⋊C412yes—
96C22×C33C2×D411yescharacteristic
97C33×C4C4⋊C412yes—
98C33×C4C22×C411yescharacteristic
99C33×C4C2×Q811yescharacteristic
100C3×C3⋊Dic3—42no—
101C32×Dic3—41no—
102C32×Dic3—41no—
103C2×C3×M16—31no—
104C2×C3×M16—31no—
105C2×C3×M16—61no—
106C3×#117—31no—
107C3×#117—31no—
108C3×#117—61no—
109C2×C3×C4⋊C4—91no—
110#926C3×D311yescharacteristic
111C2×C4⋊Dic3—31no—
112C2×C4⋊Dic3—31no—
113C2×C4⋊Dic3—61no—
114C2×#208C3×D311yescharacteristic
115C3×#107—92no—
116C3×#106—92no—
117#818—32no—
118#817—32no—
119#808—32no—
120#807—32no—
121C23×C32—21no—
122C23×C32—21no—
123C23×C32—21no—
124C23×C32—41no—
125C23×C32D1211yescharacteristic
126C23×C32#6711yescharacteristic
127C23×C32C3×D411yescharacteristic
128C2×C32×C4—22no—
129C2×C32×C4—21no—
130C2×C32×C4—22no—
131C2×C32×C4—22no—
132C2×C32×C4—22no—
133C2×C32×C4—22no—
134C2×C32×C4—21no—
135C2×C32×C4—22no—
136C2×C32×C4—21no—
137C2×C32×C4—21no—
138C2×C32×C4—21no—
139C2×C32×C4—21no—
140C2×C32×C4—42no—
141C2×C32×C4—41no—
142C2×C32×C4—42no—
143C2×C32×C4—41no—
144C2×C32×C4—61no—
145C2×C32×C4—61no—
146C2×C32×C4Dic612yes—
147C2×C32×C4Dic612yes—
148C2×C32×C4C4×D312yes—
149C2×C32×C4D1211yescharacteristic
150C2×C32×C4C2×Dic312yes—
151C2×C32×C4#6711yescharacteristic
152C2×C32×C4C2×C3×C412yes—
153C2×C32×C4C3×D411yescharacteristic
154C2×C32×C4C3×Q812yes—
155C2×C32×C4C22×D311yescharacteristic
156C2×C32×C4C22×D311yescharacteristic
157C2×C32×C4C23×C311yescharacteristic
158C2×C3⋊Dic3—22no—
159C2×C3⋊Dic3—42no—
160C2×C3×Dic3—22no—
161C2×C3×Dic3—22no—
162C2×C3×Dic3—21no—
163C2×C3×Dic3—21no—
164C2×C3×Dic3—21no—
165C2×C3×Dic3—21no—
166C2×C3×Dic3—64no—
167C2×C3×Dic3—62no—
168C2×C3×Dic3—62no—
169C2×C3×Dic3—124no—
170C2×C3×Dic3—122no—
171C2×C3×Dic3—122no—
172C32×C8—34no—
173C3×C3⋊C8—24no—
174C3×C3⋊C8C2×Dic314yescomplemented
175C3×C3⋊C8C2×C3×C414yescomplemented
176#428—91noSylow
177C2×C33—22no—
178C2×C33C2×C4⋊C411yescharacteristic
179C2×C33AGL(1,ℤ/8)11yescharacteristic
180C2×C33#13711yescharacteristic
181C22×C3×C4—21no—
182C22×C3×C4—21no—
183C22×C3×C4—21no—
184C22×C3×C4—41no—
185C22×C3×C4—181no—
186C22×C3×C4D3211yescharacteristic
187C22×C3×C4C2×C3×D311yescharacteristic
188C22×C3×C4C2×C3×D311yescharacteristic
189C22×Dic3—62no—
190C22×Dic3—61no—
191C22×Dic3—61no—
192C3×M16—34no—
193C3×M16—34no—
194C3×M16—64no—
195C2×C3×C8—32no—
196C2×C3×C8—32no—
197C2×C3×C8—62no—
198C3×C4⋊C4—31no—
199C3×C4⋊C4—31no—
200C3×C4⋊C4—31no—
201C3×C4⋊C4—31no—
202C3×C4⋊C4—61no—
203C3×C4⋊C4—61no—
204C3×C4⋊C4—92no—
205C3×C4⋊C4—181no—
206C3×C22⋊C4—62no—
207C3×C22⋊C4—61no—
208C3×C22⋊C4—61no—
209C3×C42—62no—
210C3×C42—61no—
211C3×C42—61no—
212#217—21no—
213C4⋊Dic3—32no—
214C4⋊Dic3—32no—
215C4⋊Dic3—62no—
216C4⋊Dic3—62no—
217C4⋊Dic3—61no—
218C4⋊Dic3—61no—
219C4⋊Dic3C2×C3×D311yescharacteristic
220C4⋊Dic3C2×C3×D311yescharacteristic
221C4×Dic3—21no—
222#208C3×Dic314yescomplemented
223C2×C3⋊C8C2×C3×D312yes—
224C22×C32—21no—
225C22×C32—21no—
226C22×C32—22no—
227C22×C32—22no—
228C22×C32—21no—
229C22×C32—21no—
230C22×C32—21no—
231C22×C32—21no—
232C22×C32—22no—
233C22×C32—22no—
234C22×C32—22no—
235C22×C32—21no—
236C22×C32—21no—
237C22×C32—21no—
238C22×C32—42no—
239C22×C32—41no—
240C22×C32—41no—
241C22×C32—41no—
242C22×C32—41no—
243C22×C32—41no—
244C22×C32—42no—
245C22×C32—42no—
246C22×C32Dic3⋊C412yes—
247C22×C32C4⋊Dic312yes—
248C22×C32C3×C4⋊C412yes—
249C22×C32C2×D1211yescharacteristic
250C22×C32C2×#6711yescharacteristic
251C22×C32C2×C3×D411yescharacteristic
252C32×C4—22no—
253C32×C4—22no—
254C32×C4—22no—
255C32×C4—21no—
256C32×C4—21no—
257C32×C4—21no—
258C32×C4—21no—
259C32×C4—21no—
260C32×C4—21no—
261C32×C4—42no—
262C32×C4—41no—
263C32×C4—41no—
264C32×C4—121no—
265C32×C4—121no—
266C32×C4Dic3⋊C412yes—
267C32×C4C4⋊Dic312yes—
268C32×C4C3×C4⋊C412yes—
269C32×C4C2×Dic611yescharacteristic
270C32×C4C2×Dic611yescharacteristic
271C32×C4C2×C4×D311yescharacteristic
272C32×C4C22×Dic311yescharacteristic
273C32×C4C22×C3×C411yescharacteristic
274C32×C4C2×C3×Q811yescharacteristic
275C3⋊Dic3—42no—
276C3×Dic3—42no—
277C3×Dic3—42no—
278C3×Dic3—41no—
279C3×Dic3—41no—
280C3×Dic3—41no—
281C3×Dic3—41no—
282C3×Dic3—124no—
283C3×Dic3—122no—
284C3×Dic3—122no—
285C2×M16—31no—
286#117—31no—
287C2×C4⋊C4—91no—
288#107—92no—
289#106—92no—
290C33#42811yescharacteristic, complemented, Sylow
291C23×C3—21no—
292C23×C3—21no—
293C23×C3—21no—
294C23×C3—41no—
295C23×C3#62611yescharacteristic
296C23×C3C3×D1211yescharacteristic
297C23×C3C3×#6711yescharacteristic
298C2×C3×C4—22no—
299C2×C3×C4—22no—
300C2×C3×C4—22no—
301C2×C3×C4—21no—
302C2×C3×C4—21no—
303C2×C3×C4—21no—
304C2×C3×C4—22no—
305C2×C3×C4—22no—
306C2×C3×C4—22no—
307C2×C3×C4—21no—
308C2×C3×C4—21no—
309C2×C3×C4—21no—
310C2×C3×C4—42no—
311C2×C3×C4—41no—
312C2×C3×C4—42no—
313C2×C3×C4—41no—
314C2×C3×C4—62no—
315C2×C3×C4—62no—
316C2×C3×C4—61no—
317C2×C3×C4—61no—
318C2×C3×C4—61no—
319C2×C3×C4—61no—
320C2×C3×C4—182no—
321C2×C3×C4—362no—
322C2×C3×C4D3×Dic312yes—
323C2×C3×C4#62611yescharacteristic
324C2×C3×C4#62712yes—
325C2×C3×C4C3×Dic612yes—
326C2×C3×C4C3×Dic612yes—
327C2×C3×C4C3×C4×D312yes—
328C2×C3×C4C3×D1211yescharacteristic
329C2×C3×C4C2×C3×Dic312yes—
330C2×C3×C4C3×#6711yescharacteristic
331C2×C3×C4C2×D3211yescharacteristic
332C2×C3×C4C22×C3×D311yescharacteristic
333C2×C3×C4C22×C3×D311yescharacteristic
334C2×Dic3—21no—
335C2×Dic3—21no—
336C2×Dic3—64no—
337C2×Dic3—62no—
338C2×Dic3—62no—
339C2×Dic3—124no—
340C2×Dic3—122no—
341C2×Dic3—122no—
342C3×C8—34no—
343C3×C8—34no—
344C3×C8—64no—
345C3⋊C8C2×C3×Dic314yescomplemented
346C2×C32—22no—
347C2×C32—24no—
348C2×C32—22no—
349C2×C32—22no—
350C2×C32—21no—
351C2×C32—21no—
352C2×C32—21no—
353C2×C32—21no—
354C2×C32—21no—
355C2×C32—21no—
356C2×C32—21no—
357C2×C32—21no—
358C2×C32—21no—
359C2×C32—41no—
360C2×C32—41no—
361C2×C32—41no—
362C2×C32—42no—
363C2×C32—42no—
364C2×C32—44no—
365C2×C32#90811yescharacteristic
366C2×C32#90911yescharacteristic
367C2×C32C2×Dic3⋊C411yescharacteristic
368C2×C32C2×C4⋊Dic311yescharacteristic
369C2×C32#94911yescharacteristic
370C2×C32#95111yescharacteristic
371C2×C32C2×C3×C4⋊C411yescharacteristic
372C2×C32C3×AGL(1,ℤ/8)11yescharacteristic
373C2×C32C3×#13711yescharacteristic
374C22×C4—181no—
375C22×C4C3×D3211yescharacteristic
376M16—34no—
377C2×C8—32no—
378C4⋊C4—31no—
379C4⋊C4—31no—
380C4⋊C4—92no—
381C4⋊C4—181no—
382C22⋊C4—61no—
383C42—61no—
384C22×C3—22no—
385C22×C3—22no—
386C22×C3—21no—
387C22×C3—21no—
388C22×C3—21no—
389C22×C3—21no—
390C22×C3—21no—
391C22×C3—21no—
392C22×C3—21no—
393C22×C3—21no—
394C22×C3—21no—
395C22×C3—22no—
396C22×C3—22no—
397C22×C3—22no—
398C22×C3—41no—
399C22×C3—41no—
400C22×C3—41no—
401C22×C3—41no—
402C22×C3—42no—
403C22×C3—42no—
404C22×C3—41no—
405C22×C3—42no—
406C22×C3Dic3⋊Dic312yes—
407C22×C3C3×Dic3⋊C412yes—
408C22×C3C3×C4⋊Dic312yes—
409C22×C3C2×#62611yescharacteristic
410C22×C3C2×C3×D1211yescharacteristic
411C22×C3C2×C3×#6711yescharacteristic
412C3×C4—21no—
413C3×C4—21no—
414C3×C4—21no—
415C3×C4—21no—
416C3×C4—21no—
417C3×C4—21no—
418C3×C4—22no—
419C3×C4—22no—
420C3×C4—22no—
421C3×C4—41no—
422C3×C4—41no—
423C3×C4—42no—
424C3×C4—122no—
425C3×C4—122no—
426C3×C4—121no—
427C3×C4—121no—
428C3×C4—121no—
429C3×C4—121no—
430C3×C4—362no—
431C3×C4Dic3⋊Dic312yes—
432C3×C4C3×Dic3⋊C412yes—
433C3×C4C3×C4⋊Dic312yes—
434C3×C4C2×D3×Dic311yescharacteristic
435C3×C4C2×#62711yescharacteristic
436C3×C4C2×C3×Dic611yescharacteristic
437C3×C4C2×C3×Dic611yescharacteristic
438C3×C4C2×C3×C4×D311yescharacteristic
439C3×C4C22×C3×Dic311yescharacteristic
440Dic3—41no—
441Dic3—41no—
442Dic3—124no—
443Dic3—122no—
444Dic3—122no—
445C32—21no—
446C32—21no—
447C32—21no—
448C32—41no—
449C32#499511yescharacteristic, complemented
450C32#512111yescharacteristic, complemented
451C32C3×#42811yescharacteristic, complemented
452C23C3×#62611yescharacteristic
453C2×C4—61no—
454C2×C4—61no—
455C2×C4—182no—
456C2×C4—362no—
457C2×C4C3×D3×Dic312yes—
458C2×C4C3×#62611yescharacteristic
459C2×C4C3×#62712yes—
460C2×C4C2×C3×D3211yescharacteristic
461C8—34no—
462C2×C3—24no—
463C2×C3—22no—
464C2×C3—22no—
465C2×C3—22no—
466C2×C3—21no—
467C2×C3—21no—
468C2×C3—21no—
469C2×C3—21no—
470C2×C3—21no—
471C2×C3—21no—
472C2×C3—21no—
473C2×C3—21no—
474C2×C3—21no—
475C2×C3—42no—
476C2×C3—42no—
477C2×C3—44no—
478C2×C3—41no—
479C2×C3—41no—
480C2×C3—41no—
481C2×C3#6382511yescharacteristic
482C2×C3#6383111yescharacteristic
483C2×C3C2×Dic3⋊Dic311yescharacteristic
484C2×C3C3×#90811yescharacteristic
485C2×C3C3×#90911yescharacteristic
486C2×C3C2×C3×Dic3⋊C411yescharacteristic
487C2×C3C2×C3×C4⋊Dic311yescharacteristic
488C2×C3C3×#94911yescharacteristic
489C2×C3C3×#95111yescharacteristic
490C22—21no—
491C22—21no—
492C22C3×Dic3⋊Dic312yes—
493C22C2×C3×#62611yescharacteristic
494C4—121no—
495C4—121no—
496C4—362no—
497C4C3×Dic3⋊Dic312yes—
498C4C2×C3×D3×Dic311yescharacteristic
499C4C2×C3×#62711yescharacteristic
500C3—21no—
501C3—21no—
502C3—21no—
503C3—41no—
504C3#1059113111yescharacteristic, direct product factor
505C3C3×#499511yescharacteristic, complemented
506C3C3×#512111yescharacteristic, complemented
507C2—22no—
508C2C3×#6382511yescharacteristic
509C2C3×#6383111yescharacteristic
510C2C2×C3×Dic3⋊Dic311yescharacteristic
subgroupG/H
centerC22×C3C2×#626
commutatorC32×C4C22×C3×C4
FrattiniC2×C4C2×C3×D32
FittingC22×C33×C4C22
socleC22×C33C2×D4
prime radicalC23×C33D4
cosocleC2×C32×C4C23×C3
hypercenterC22×C3×C4D32
hypocenterC32C3×#428
LCSG ▷ C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C22×C3×C4

Complements

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

NH ≅ G/Ntype
4. C3×#63575C2semidirect
5. C3×#63573C2semidirect
8. #10591131C3direct (N × H)
17. C32×#208C4semidirect
35. #63575C2×C3semidirect
36. #63573C2×C3semidirect
46. C32×C3⋊C8C2×C4semidirect
88. C3×#208Dic3semidirect
89. C3×#208C3×C4semidirect
174. C3×C3⋊C8C2×Dic3semidirect
175. C3×C3⋊C8C2×C3×C4semidirect
222. #208C3×Dic3semidirect
290. C33#428semidirect
345. C3⋊C8C2×C3×Dic3semidirect
449. C32#4995semidirect
450. C32#5121semidirect
451. C32C3×#428semidirect
504. C3#10591131direct (N × H)
505. C3C3×#4995semidirect
506. C3C3×#5121semidirect

Maximal subgroups

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

subgroupindex
C2×C3×C4⋊(C3⋊Dic3)2
C32×#9262
C2×C32×#2082
C3×#635752
C3×#635732
C3×#51213
C3×#49953
#105911313