Finite Group Databasebeta

← C3×#10591328 #13158269

Subgroups — C3×#10591328 #13158269

Order 1728.

994 conjugacy classes of subgroups.

The subgroup lattice has 504 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×C4×C3⋊Dic3C211yesmaximal, characteristic
2C2×C3×#3724C211yesmaximal, characteristic
3C2×C3×#3724C211yesmaximal, characteristic
4C2×C3×Dic3⋊Dic3C212yesmaximal
5C32×C3⋊#95C212yesmaximal
6C3×#4647—32nomaximal
7#10591328C311yesmaximal, characteristic, direct product factor
8C22×C33×C4C2211yescharacteristic
9C22×C3×C3⋊Dic3C2211yescharacteristic
10C22×C3×C3⋊Dic3C2211yescharacteristic
11C22×C32×Dic3C2212yes—
12C22×C32×Dic3C2212yes—
13C3×C4×C3⋊Dic3—22no—
14C3×#3724—22no—
15C3×#3724C414yescomplemented
16C3×Dic3⋊Dic3—24no—
17C2×C4×C3⋊Dic3C2×C311yescharacteristic
18C2×C3×C4⋊Dic3—32no—
19C2×C3×Dic3⋊C4—32no—
20C2×C3×Dic3⋊C4—32no—
21C2×C3×Dic3⋊C4—32no—
22C2×C3×C4×Dic3—32no—
23C2×C3×C4×Dic3—61no—
24C2×#3724C2×C311yescharacteristic
25C2×#3724C2×C311yescharacteristic
26C2×Dic3⋊Dic3C2×C312yes—
27C32×#95—32no—
28C3×C3⋊#95—22no—
29C3×C3⋊#95D312yes—
30C3×C3⋊#95C2×C312yes—
31C23×C33C2311yescharacteristic
32C2×C33×C4—21no—
33C2×C33×C4—21no—
34C2×C33×C4Q812yes—
35C2×C3×C3⋊Dic3—21no—
36C2×C3×C3⋊Dic3—21no—
37C2×C3×C3⋊Dic3C2×C412yes—
38C2×C3×C3⋊Dic3D412yes—
39C2×C3×C3⋊Dic3D412yes—
40C2×C3×C3⋊Dic3Q812yes—
41C2×C32×Dic3—22no—
42C2×C32×Dic3—22no—
43C2×C32×Dic3—22no—
44C2×C32×Dic3—22no—
45C2×C32×Dic3—22no—
46C2×C32×Dic3C2×C414yes—
47C3×#389—91no—
48#4647—32no—
49C22×C32×C4—21no—
50C22×C32×C4—22no—
51C22×C32×C4—32no—
52C22×C32×C4—32no—
53C22×C32×C4—41no—
54C22×C32×C4C2×D312yes—
55C22×C32×C4C22×C311yescharacteristic
56C22×C3⋊Dic3C22×C311yescharacteristic
57C22×C3⋊Dic3C22×C311yescharacteristic
58C22×C3×Dic3—22no—
59C22×C3×Dic3—22no—
60C22×C3×Dic3—32no—
61C22×C3×Dic3—32no—
62C22×C3×Dic3—61no—
63C22×C3×Dic3—61no—
64C22×C3×Dic3C2×D312yes—
65C22×C3×Dic3C2×D312yes—
66C22×C3×Dic3C22×C312yes—
67C22×C3×Dic3C22×C312yes—
68C4×C3⋊Dic3—22no—
69C3×C4⋊Dic3—64no—
70C3×Dic3⋊C4—38no—
71C3×Dic3⋊C4—64no—
72C3×Dic3⋊C4—64no—
73C3×C4×Dic3—64no—
74C3×C4×Dic3—64no—
75#3724—22no—
76#3724C3×C414yescomplemented
77Dic3⋊Dic3—24no—
78C22×C33C22×C411yescharacteristic
79C22×C33C2×D411yescharacteristic
80C22×C33C2×D411yescharacteristic
81C22×C33C2×Q811yescharacteristic
82C22×C33C2×Q811yescharacteristic
83C22×C33#4111yescharacteristic
84C22×C33#4111yescharacteristic
85C33×C4—22no—
86C3×C3⋊Dic3—22no—
87C3×C3⋊Dic3C4⋊C414yescomplemented
88C32×Dic3—24no—
89C32×Dic3—42no—
90C2×C3×C4⋊C4—91no—
91C2×C3×C4⋊C4—91no—
92C2×C3×C4⋊C4—92no—
93C2×C3×C42—91no—
94C2×C4⋊Dic3—32no—
95C2×Dic3⋊C4—32no—
96C2×Dic3⋊C4—32no—
97C2×Dic3⋊C4—32no—
98C2×C4×Dic3—32no—
99C2×C4×Dic3—61no—
100C3×#95—32no—
101C3×#95—32no—
102C3×#95—62no—
103C3⋊#95C3×D312yes—
104C23×C32—21no—
105C23×C32—22no—
106C23×C32—41no—
107C23×C32C22×D312yes—
108C23×C32C23×C311yescharacteristic
109C2×C32×C4—21no—
110C2×C32×C4—21no—
111C2×C32×C4—22no—
112C2×C32×C4—22no—
113C2×C32×C4—22no—
114C2×C32×C4—22no—
115C2×C32×C4—22no—
116C2×C32×C4—24no—
117C2×C32×C4—34no—
118C2×C32×C4—42no—
119C2×C32×C4—42no—
120C2×C32×C4—42no—
121C2×C32×C4—42no—
122C2×C32×C4—42no—
123C2×C32×C4—62no—
124C2×C32×C4—62no—
125C2×C32×C4—62no—
126C2×C32×C4—62no—
127C2×C32×C4—62no—
128C2×C32×C4Dic614yes—
129C2×C32×C4C3×Q812yes—
130C2×C3⋊Dic3—21no—
131C2×C3⋊Dic3—21no—
132C2×C3⋊Dic3C2×C3×C412yes—
133C2×C3⋊Dic3C3×D412yes—
134C2×C3⋊Dic3C3×D412yes—
135C2×C3⋊Dic3C3×Q812yes—
136C2×C3×Dic3—22no—
137C2×C3×Dic3—22no—
138C2×C3×Dic3—22no—
139C2×C3×Dic3—22no—
140C2×C3×Dic3—24no—
141C2×C3×Dic3—22no—
142C2×C3×Dic3—22no—
143C2×C3×Dic3—22no—
144C2×C3×Dic3—22no—
145C2×C3×Dic3—22no—
146C2×C3×Dic3—22no—
147C2×C3×Dic3—24no—
148C2×C3×Dic3—34no—
149C2×C3×Dic3—34no—
150C2×C3×Dic3—34no—
151C2×C3×Dic3—34no—
152C2×C3×Dic3—42no—
153C2×C3×Dic3—42no—
154C2×C3×Dic3—42no—
155C2×C3×Dic3—42no—
156C2×C3×Dic3—62no—
157C2×C3×Dic3—62no—
158C2×C3×Dic3—62no—
159C2×C3×Dic3—62no—
160C2×C3×Dic3—62no—
161C2×C3×Dic3—62no—
162C2×C3×Dic3—62no—
163C2×C3×Dic3—62no—
164C2×C3×Dic3C4×D314yes—
165C2×C3×Dic3C2×C3×C414yes—
166#389—91noSylow
167C2×C33C2×C4⋊C411yescharacteristic
168C2×C33C4×D411yescharacteristic
169C2×C33C4×Q811yescharacteristic
170C2×C33#12111yescharacteristic
171C2×C33C22⋊Q811yescharacteristic
172C2×C33#12511yescharacteristic
173C2×C33C4⋊Q811yescharacteristic
174C22×C3×C4—21no—
175C22×C3×C4—22no—
176C22×C3×C4—32no—
177C22×C3×C4—32no—
178C22×C3×C4—32no—
179C22×C3×C4—32no—
180C22×C3×C4—41no—
181C22×C3×C4—62no—
182C22×C3×C4—62no—
183C22×C3×C4—91no—
184C22×C3×C4—91no—
185C22×C3×C4D3211yescharacteristic
186C22×C3×C4C2×C3×D312yes—
187C22×Dic3—32no—
188C22×Dic3—32no—
189C22×Dic3—61no—
190C22×Dic3—61no—
191C22×Dic3C2×C3×D312yes—
192C22×Dic3C2×C3×D312yes—
193C3×C4⋊C4—94no—
194C3×C4⋊C4—182no—
195C3×C4⋊C4—184no—
196C3×C42—182no—
197C4⋊Dic3—64no—
198Dic3⋊C4—38no—
199Dic3⋊C4—64no—
200Dic3⋊C4—64no—
201C4×Dic3—64no—
202C4×Dic3—64no—
203C22×C32—21no—
204C22×C32—21no—
205C22×C32—21no—
206C22×C32—21no—
207C22×C32—21no—
208C22×C32—21no—
209C22×C32—21no—
210C22×C32—22no—
211C22×C32—22no—
212C22×C32—22no—
213C22×C32—22no—
214C22×C32—22no—
215C22×C32—22no—
216C22×C32—22no—
217C22×C32—41no—
218C22×C32—41no—
219C22×C32—41no—
220C22×C32—41no—
221C22×C32—41no—
222C22×C32—41no—
223C22×C32—41no—
224C22×C32C2×Dic612yes—
225C22×C32C2×C4×D312yes—
226C22×C32#23512yes—
227C22×C32D3×D412yes—
228C22×C32D3×D412yes—
229C22×C32#23712yes—
230C22×C32D3×Q812yes—
231C22×C32C22×C3×C411yescharacteristic
232C22×C32C2×C3×D411yescharacteristic
233C22×C32C2×C3×D411yescharacteristic
234C22×C32C2×C3×Q811yescharacteristic
235C22×C32C2×C3×Q811yescharacteristic
236C22×C32C3×#4111yescharacteristic
237C22×C32C3×#4111yescharacteristic
238C32×C4—22no—
239C32×C4—24no—
240C32×C4—24no—
241C32×C4—44no—
242C32×C4—44no—
243C32×C4—64no—
244C32×C4—122no—
245C3⋊Dic3—22no—
246C3⋊Dic3C3×C4⋊C414yescomplemented
247C3×Dic3—24no—
248C3×Dic3—24no—
249C3×Dic3—38no—
250C3×Dic3—44no—
251C3×Dic3—44no—
252C3×Dic3—42no—
253C3×Dic3—42no—
254C3×Dic3—64no—
255C3×Dic3—64no—
256C3×Dic3—64no—
257C2×C4⋊C4—91no—
258C2×C4⋊C4—91no—
259C2×C4⋊C4—92no—
260C2×C42—91no—
261#95—32no—
262C33#38911yescharacteristic, complemented, Sylow
263C23×C3—22no—
264C23×C3—21no—
265C23×C3—41no—
266C23×C3C2×D3211yescharacteristic
267C23×C3C22×C3×D312yes—
268C2×C3×C4—22no—
269C2×C3×C4—22no—
270C2×C3×C4—21no—
271C2×C3×C4—21no—
272C2×C3×C4—22no—
273C2×C3×C4—22no—
274C2×C3×C4—22no—
275C2×C3×C4—24no—
276C2×C3×C4—34no—
277C2×C3×C4—34no—
278C2×C3×C4—42no—
279C2×C3×C4—42no—
280C2×C3×C4—42no—
281C2×C3×C4—42no—
282C2×C3×C4—42no—
283C2×C3×C4—64no—
284C2×C3×C4—62no—
285C2×C3×C4—62no—
286C2×C3×C4—62no—
287C2×C3×C4—62no—
288C2×C3×C4—64no—
289C2×C3×C4—62no—
290C2×C3×C4—62no—
291C2×C3×C4—62no—
292C2×C3×C4—62no—
293C2×C3×C4—62no—
294C2×C3×C4—62no—
295C2×C3×C4—92no—
296C2×C3×C4—92no—
297C2×C3×C4—92no—
298C2×C3×C4—92no—
299C2×C3×C4—122no—
300C2×C3×C4—122no—
301C2×C3×C4—122no—
302C2×C3×C4—122no—
303C2×C3×C4—181no—
304C2×C3×C4—181no—
305C2×C3×C4#62712yes—
306C2×C3×C4C3×Dic614yes—
307C2×Dic3—22no—
308C2×Dic3—22no—
309C2×Dic3—22no—
310C2×Dic3—22no—
311C2×Dic3—22no—
312C2×Dic3—34no—
313C2×Dic3—34no—
314C2×Dic3—34no—
315C2×Dic3—34no—
316C2×Dic3—62no—
317C2×Dic3—62no—
318C2×Dic3—62no—
319C2×Dic3—62no—
320C2×Dic3—62no—
321C2×Dic3—62no—
322C2×Dic3—62no—
323C2×Dic3—62no—
324C2×Dic3C3×C4×D314yes—
325C2×C32—22no—
326C2×C32—22no—
327C2×C32—22no—
328C2×C32—22no—
329C2×C32—22no—
330C2×C32—22no—
331C2×C32—21no—
332C2×C32—21no—
333C2×C32—21no—
334C2×C32—21no—
335C2×C32—21no—
336C2×C32—21no—
337C2×C32—21no—
338C2×C32—22no—
339C2×C32—41no—
340C2×C32—41no—
341C2×C32—41no—
342C2×C32—41no—
343C2×C32—41no—
344C2×C32—41no—
345C2×C32—41no—
346C2×C32C4×Dic612yes—
347C2×C32#87812yes—
348C2×C32#67⋊C412yes—
349C2×C32#88412yes—
350C2×C32#88812yes—
351C2×C32#88912yes—
352C2×C32D3×C4⋊C412yes—
353C2×C32C2×C3×C4⋊C411yescharacteristic
354C2×C32C3×C4×D411yescharacteristic
355C2×C32C3×C4×Q811yescharacteristic
356C2×C32C3×#12111yescharacteristic
357C2×C32C3×C22⋊Q811yescharacteristic
358C2×C32C3×#12511yescharacteristic
359C2×C32C3×C4⋊Q811yescharacteristic
360C22×C4—32no—
361C22×C4—32no—
362C22×C4—91no—
363C22×C4—91no—
364C22×C4C3×D3211yescharacteristic
365C4⋊C4—94no—
366C4⋊C4—182no—
367C4⋊C4—184no—
368C42—182no—
369C22×C3—21no—
370C22×C3—22no—
371C22×C3—21no—
372C22×C3—22no—
373C22×C3—21no—
374C22×C3—22no—
375C22×C3—21no—
376C22×C3—22no—
377C22×C3—21no—
378C22×C3—22no—
379C22×C3—22no—
380C22×C3—21no—
381C22×C3—21no—
382C22×C3—22no—
383C22×C3—41no—
384C22×C3—41no—
385C22×C3—41no—
386C22×C3—41no—
387C22×C3—41no—
388C22×C3—41no—
389C22×C3—41no—
390C22×C3#379711yescharacteristic
391C22×C3#379811yescharacteristic
392C22×C3C4×D3211yescharacteristic
393C22×C3#380211yescharacteristic
394C22×C3#380511yescharacteristic
395C22×C3C2×#62711yescharacteristic
396C22×C3#381111yescharacteristic
397C22×C3C2×C3×Dic612yes—
398C22×C3C2×C3×C4×D312yes—
399C22×C3C3×#23512yes—
400C22×C3C3×D3×D412yes—
401C22×C3C3×D3×D412yes—
402C22×C3C3×#23712yes—
403C22×C3C3×D3×Q812yes—
404C3×C4—24no—
405C3×C4—24no—
406C3×C4—22no—
407C3×C4—44no—
408C3×C4—44no—
409C3×C4—64no—
410C3×C4—64no—
411C3×C4—94no—
412C3×C4—124no—
413C3×C4—124no—
414C3×C4—122no—
415C3×C4—122no—
416C3×C4—182no—
417Dic3—24no—
418Dic3—38no—
419Dic3—42no—
420Dic3—64no—
421Dic3—64no—
422Dic3—64no—
423C32—22no—
424C32—21no—
425C32—41no—
426C32#464712yescomplemented
427C32C3×#38911yescharacteristic, complemented
428C23C2×C3×D3211yescharacteristic
429C2×C4—21no—
430C2×C4—21no—
431C2×C4—34no—
432C2×C4—62no—
433C2×C4—62no—
434C2×C4—62no—
435C2×C4—62no—
436C2×C4—62no—
437C2×C4—92no—
438C2×C4—92no—
439C2×C4—92no—
440C2×C4—92no—
441C2×C4—181no—
442C2×C4—181no—
443C2×C4C3×#62712yes—
444C2×C3—21no—
445C2×C3—22no—
446C2×C3—22no—
447C2×C3—21no—
448C2×C3—21no—
449C2×C3—22no—
450C2×C3—22no—
451C2×C3—21no—
452C2×C3—21no—
453C2×C3—22no—
454C2×C3—22no—
455C2×C3—21no—
456C2×C3—22no—
457C2×C3—21no—
458C2×C3—41no—
459C2×C3—41no—
460C2×C3—41no—
461C2×C3—41no—
462C2×C3—41no—
463C2×C3—41no—
464C2×C3—41no—
465C2×C3#6387011yescharacteristic
466C2×C3#6388311yescharacteristic
467C2×C3#6390211yescharacteristic
468C2×C3#6391211yescharacteristic
469C2×C3C4×#62711yescharacteristic
470C2×C3#6391911yescharacteristic
471C2×C3#6398211yescharacteristic
472C2×C3C3×C4×Dic612yes—
473C2×C3C3×#87812yes—
474C2×C3C3×#67⋊C412yes—
475C2×C3C3×#88412yes—
476C2×C3C3×#88812yes—
477C2×C3C3×#88912yes—
478C2×C3C3×D3×C4⋊C412yes—
479C22C3×#379711yescharacteristic
480C22C3×#379811yescharacteristic
481C22C3×C4×D3211yescharacteristic
482C22C3×#380211yescharacteristic
483C22C3×#380511yescharacteristic
484C22C2×C3×#62711yescharacteristic
485C22C3×#381111yescharacteristic
486C4—22no—
487C4—64no—
488C4—94no—
489C4—122no—
490C4—182no—
491C3—22no—
492C3—21no—
493C3—41no—
494C3#1059132811yescharacteristic, direct product factor
495C3C3×#464712yescomplemented
496C2C3×#6387011yescharacteristic
497C2C3×#6388311yescharacteristic
498C2C3×#6390211yescharacteristic
499C2C3×#6391211yescharacteristic
500C2C3×C4×#62711yescharacteristic
501C2C3×#6391911yescharacteristic
502C2C3×#6398211yescharacteristic
subgroupG/H
centerC23×C3C2×D32
commutatorC22×C32C22×C3×C4
FrattiniC23C2×C3×D32
FittingC22×C33×C4C22
socle, prime radicalC23×C33C23
cosocleC23×C32C23×C3
hypercenterC22×C3×C4D32
hypocenterC32C3×#389
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C23×C3 ◁ C22×C3×C4

Complements

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

NH ≅ G/Ntype
7. #10591328C3direct (N × H)
15. C3×#3724C4semidirect
76. #3724C3×C4semidirect
87. C3×C3⋊Dic3C4⋊C4semidirect
246. C3⋊Dic3C3×C4⋊C4semidirect
262. C33#389semidirect
426. C32#4647semidirect
427. C32C3×#389semidirect
494. C3#10591328direct (N × H)
495. C3C3×#4647semidirect

Maximal subgroups

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

subgroupindex
C2×C3×C4×C3⋊Dic32
C2×C3×#37242
C2×C3×Dic3⋊Dic32
C32×C3⋊#952
C3×#46473
#105913283