Finite Group Databasebeta

← C3×#10591273 #13158191

Subgroups — C3×#10591273 #13158191

Order 1728.

1106 conjugacy classes of subgroups.

The subgroup lattice has 488 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×#634C211yesmaximal, characteristic
2C2×C3×#3724C211yesmaximal, characteristic
3C2×C3×Dic32C211yesmaximal, characteristic
4C32×C3⋊#95C212yesmaximal
5C3×#63579C212yesmaximal
6C3×#4646—32nomaximal
7#10591273C311yesmaximal, characteristic, direct product factor
8C22×C33×C4C2211yescharacteristic
9C22×C3×C3⋊Dic3C2211yescharacteristic
10C22×C3×C3⋊Dic3C2211yescharacteristic
11C2×C3×#634—22no—
12C2×C3×#634—22no—
13C2×C3×#634C414yescomplemented
14C22×C32×Dic3C2212yes—
15C22×C32×Dic3C2212yes—
16C3×#3724—22no—
17C3×Dic32—22no—
18C22×#634C2×C311yescharacteristic
19C22×C3×Dic6—32no—
20C22×C3×Dic6—61no—
21C2×C3×Dic3⋊C4—32no—
22C2×C3×C4×Dic3—32no—
23C2×#3724C2×C311yescharacteristic
24C2×Dic32C2×C311yescharacteristic
25C32×#95—32no—
26C3×C3⋊#95—22no—
27C3×C3⋊#95—32no—
28C3×C3⋊#95—32no—
29C3×C3⋊#95D312yes—
30C3×C3⋊#95C2×C312yes—
31#63579C2×C312yes—
32C23×C33C2311yescharacteristic
33C2×C33×C4—21no—
34C2×C33×C4—21no—
35C2×C33×C4C2×C412yes—
36C2×C3×C3⋊Dic3—21no—
37C2×C3×C3⋊Dic3—21no—
38C2×C3×C3⋊Dic3C2×C412yes—
39C2×C3×C3⋊Dic3C2×C412yes—
40C2×C3×C3⋊Dic3D412yes—
41C2×C3×C3⋊Dic3D412yes—
42C3×#634—28no—
43C2×C32×Dic3—22no—
44C2×C32×Dic3—22no—
45C2×C32×Dic3—22no—
46C2×C32×Dic3—22no—
47C2×C32×Dic3—22no—
48C2×C32×Dic3Q814yescomplemented
49C3×#391—91no—
50#4646—32no—
51C22×C32×C4—21no—
52C22×C32×C4—22no—
53C22×C32×C4—32no—
54C22×C32×C4—32no—
55C22×C32×C4—41no—
56C22×C32×C4C2×D312yes—
57C22×C32×C4C22×C311yescharacteristic
58C22×C3⋊Dic3C22×C311yescharacteristic
59C22×C3⋊Dic3C22×C311yescharacteristic
60C2×#634—22no—
61C2×#634—22no—
62C2×#634C3×C414yescomplemented
63C22×C3×Dic3—22no—
64C22×C3×Dic3—22no—
65C22×C3×Dic3—32no—
66C22×C3×Dic3—32no—
67C22×C3×Dic3—61no—
68C22×C3×Dic3—61no—
69C22×C3×Dic3C2×D312yes—
70C22×C3×Dic3C2×D312yes—
71C22×C3×Dic3C22×C312yes—
72C22×C3×Dic3C22×C312yes—
73C2×C3×Dic6—38no—
74C2×C3×Dic6—64no—
75C2×C3×Dic6—64no—
76C2×C3×Dic6—64no—
77C2×C3×Dic6—64no—
78C2×C3×Dic6—64no—
79C3×Dic3⋊C4—64no—
80C3×C4×Dic3—64no—
81#3724—22no—
82Dic32—22no—
83C22×C33C22×C411yescharacteristic
84C22×C33C2×D411yescharacteristic
85C22×C33C2×D411yescharacteristic
86C22×C33C2×Q812yes—
87C22×C33#4112yes—
88C33×C4—22no—
89C3×C3⋊Dic3—22no—
90C3×C3⋊Dic3C22⋊C414yes—
91C32×Dic3—24no—
92C32×Dic3—42no—
93C22×C3×Q8—91no—
94C22×Dic6—32no—
95C22×Dic6—61no—
96C2×C3×C4⋊C4—91no—
97C2×C3×C42—91no—
98C2×Dic3⋊C4—32no—
99C2×C4×Dic3—32no—
100C3×#95—32no—
101C3×#95—32no—
102C3×#95—62no—
103C3×#95—92no—
104C3⋊#95—32no—
105C3⋊#95—32no—
106C3⋊#95C3×D312yes—
107C23×C32—21no—
108C23×C32—22no—
109C23×C32—41no—
110C23×C32C22×D312yes—
111C23×C32C23×C311yescharacteristic
112C2×C32×C4—21no—
113C2×C32×C4—21no—
114C2×C32×C4—22no—
115C2×C32×C4—22no—
116C2×C32×C4—22no—
117C2×C32×C4—22no—
118C2×C32×C4—22no—
119C2×C32×C4—24no—
120C2×C32×C4—34no—
121C2×C32×C4—42no—
122C2×C32×C4—42no—
123C2×C32×C4—42no—
124C2×C32×C4—42no—
125C2×C32×C4—42no—
126C2×C32×C4—62no—
127C2×C32×C4—62no—
128C2×C32×C4—62no—
129C2×C32×C4—62no—
130C2×C32×C4—62no—
131C2×C32×C4C4×D314yes—
132C2×C32×C4C2×C3×C412yes—
133C2×C3⋊Dic3—21no—
134C2×C3⋊Dic3—21no—
135C2×C3⋊Dic3C2×C3×C412yes—
136C2×C3⋊Dic3C2×C3×C412yes—
137C2×C3⋊Dic3C3×D412yes—
138C2×C3⋊Dic3C3×D412yes—
139#634—28no—
140C2×C3×Dic3—22no—
141C2×C3×Dic3—22no—
142C2×C3×Dic3—22no—
143C2×C3×Dic3—22no—
144C2×C3×Dic3—24no—
145C2×C3×Dic3—22no—
146C2×C3×Dic3—22no—
147C2×C3×Dic3—22no—
148C2×C3×Dic3—22no—
149C2×C3×Dic3—22no—
150C2×C3×Dic3—22no—
151C2×C3×Dic3—24no—
152C2×C3×Dic3—34no—
153C2×C3×Dic3—34no—
154C2×C3×Dic3—34no—
155C2×C3×Dic3—34no—
156C2×C3×Dic3—42no—
157C2×C3×Dic3—42no—
158C2×C3×Dic3—42no—
159C2×C3×Dic3—42no—
160C2×C3×Dic3—62no—
161C2×C3×Dic3—62no—
162C2×C3×Dic3—62no—
163C2×C3×Dic3—62no—
164C2×C3×Dic3—62no—
165C2×C3×Dic3—62no—
166C2×C3×Dic3—62no—
167C2×C3×Dic3—62no—
168C2×C3×Dic3Dic614yescomplemented
169C2×C3×Dic3C3×Q814yescomplemented
170C3×Dic6—616no—
171C3×Dic6—616no—
172#391—91noSylow
173C2×C33C2×C22⋊C411yescharacteristic
174C2×C33C4×Q812yes—
175C2×C33C22⋊Q812yes—
176C2×C33#12411yescharacteristic
177C2×C33C4⋊Q811yescharacteristic
178C2×C3×Q8—94no—
179C2×C3×Q8—182no—
180C2×C3×Q8—182no—
181C22×C3×C4—21no—
182C22×C3×C4—22no—
183C22×C3×C4—32no—
184C22×C3×C4—32no—
185C22×C3×C4—32no—
186C22×C3×C4—32no—
187C22×C3×C4—41no—
188C22×C3×C4—62no—
189C22×C3×C4—62no—
190C22×C3×C4—91no—
191C22×C3×C4—91no—
192C22×C3×C4D3211yescharacteristic
193C22×C3×C4C2×C3×D312yes—
194C22×Dic3—32no—
195C22×Dic3—32no—
196C22×Dic3—61no—
197C22×Dic3—61no—
198C22×Dic3C2×C3×D312yes—
199C22×Dic3C2×C3×D312yes—
200C2×Dic6—38no—
201C2×Dic6—64no—
202C2×Dic6—64no—
203C2×Dic6—64no—
204C2×Dic6—64no—
205C2×Dic6—64no—
206C3×C4⋊C4—182no—
207C3×C42—182no—
208Dic3⋊C4—64no—
209C4×Dic3—64no—
210C22×C32—22no—
211C22×C32—22no—
212C22×C32—21no—
213C22×C32—21no—
214C22×C32—21no—
215C22×C32—22no—
216C22×C32—22no—
217C22×C32—22no—
218C22×C32—22no—
219C22×C32—22no—
220C22×C32—22no—
221C22×C32—22no—
222C22×C32—41no—
223C22×C32—41no—
224C22×C32—41no—
225C22×C32—42no—
226C22×C32—42no—
227C22×C32C2×Dic612yes—
228C22×C32C2×C4×D312yes—
229C22×C32#23512yes—
230C22×C32D3×D412yes—
231C22×C32D3×D412yes—
232C22×C32#23712yes—
233C22×C32D3×Q812yes—
234C22×C32C22×C3×C411yescharacteristic
235C22×C32C2×C3×D411yescharacteristic
236C22×C32C2×C3×D411yescharacteristic
237C22×C32C2×C3×Q812yes—
238C22×C32C3×#4112yes—
239C32×C4—22no—
240C32×C4—24no—
241C32×C4—24no—
242C32×C4—44no—
243C32×C4—44no—
244C32×C4—64no—
245C32×C4—122no—
246C3⋊Dic3—22no—
247C3⋊Dic3C3×C22⋊C414yes—
248C3×Dic3—24no—
249C3×Dic3—24no—
250C3×Dic3—38no—
251C3×Dic3—44no—
252C3×Dic3—44no—
253C3×Dic3—42no—
254C3×Dic3—42no—
255C3×Dic3—64no—
256C3×Dic3—64no—
257C3×Dic3—64no—
258C22×Q8—91no—
259C2×C4⋊C4—91no—
260C2×C42—91no—
261#95—32no—
262#95—92no—
263C33#39111yescharacteristic, complemented, Sylow
264C23×C3—22no—
265C23×C3—21no—
266C23×C3—41no—
267C23×C3C2×D3211yescharacteristic
268C23×C3C22×C3×D312yes—
269C3×Q8—188no—
270C2×C3×C4—22no—
271C2×C3×C4—22no—
272C2×C3×C4—21no—
273C2×C3×C4—21no—
274C2×C3×C4—22no—
275C2×C3×C4—22no—
276C2×C3×C4—22no—
277C2×C3×C4—24no—
278C2×C3×C4—34no—
279C2×C3×C4—34no—
280C2×C3×C4—42no—
281C2×C3×C4—42no—
282C2×C3×C4—42no—
283C2×C3×C4—42no—
284C2×C3×C4—42no—
285C2×C3×C4—64no—
286C2×C3×C4—62no—
287C2×C3×C4—62no—
288C2×C3×C4—62no—
289C2×C3×C4—62no—
290C2×C3×C4—64no—
291C2×C3×C4—62no—
292C2×C3×C4—62no—
293C2×C3×C4—62no—
294C2×C3×C4—62no—
295C2×C3×C4—62no—
296C2×C3×C4—62no—
297C2×C3×C4—92no—
298C2×C3×C4—92no—
299C2×C3×C4—92no—
300C2×C3×C4—92no—
301C2×C3×C4—122no—
302C2×C3×C4—122no—
303C2×C3×C4—122no—
304C2×C3×C4—122no—
305C2×C3×C4—181no—
306C2×C3×C4—181no—
307C2×C3×C4#62412yes—
308C2×C3×C4C3×C4×D314yes—
309C2×Dic3—22no—
310C2×Dic3—22no—
311C2×Dic3—22no—
312C2×Dic3—22no—
313C2×Dic3—22no—
314C2×Dic3—34no—
315C2×Dic3—34no—
316C2×Dic3—34no—
317C2×Dic3—34no—
318C2×Dic3—62no—
319C2×Dic3—62no—
320C2×Dic3—62no—
321C2×Dic3—62no—
322C2×Dic3—62no—
323C2×Dic3—62no—
324C2×Dic3—62no—
325C2×Dic3—62no—
326C2×Dic3C3×Dic614yescomplemented
327Dic6—616no—
328Dic6—616no—
329C2×C32—22no—
330C2×C32—22no—
331C2×C32—22no—
332C2×C32—22no—
333C2×C32—22no—
334C2×C32—22no—
335C2×C32—21no—
336C2×C32—22no—
337C2×C32—22no—
338C2×C32—21no—
339C2×C32—21no—
340C2×C32—22no—
341C2×C32—41no—
342C2×C32—42no—
343C2×C32—42no—
344C2×C32—41no—
345C2×C32—41no—
346C2×C32C4×Dic612yes—
347C2×C32#87812yes—
348C2×C32D3×C22⋊C412yes—
349C2×C32#88512yes—
350C2×C32#88712yes—
351C2×C32#88812yes—
352C2×C32#89612yes—
353C2×C32C2×C3×C22⋊C411yescharacteristic
354C2×C32C3×C4×Q812yes—
355C2×C32C3×C22⋊Q812yes—
356C2×C32C3×#12411yescharacteristic
357C2×C32C3×C4⋊Q811yescharacteristic
358C2×Q8—94no—
359C2×Q8—182no—
360C2×Q8—182no—
361C22×C4—32no—
362C22×C4—32no—
363C22×C4—91no—
364C22×C4—91no—
365C22×C4C3×D3211yescharacteristic
366C4⋊C4—182no—
367C42—182no—
368C22×C3—22no—
369C22×C3—22no—
370C22×C3—22no—
371C22×C3—22no—
372C22×C3—22no—
373C22×C3—22no—
374C22×C3—21no—
375C22×C3—22no—
376C22×C3—21no—
377C22×C3—21no—
378C22×C3—22no—
379C22×C3—22no—
380C22×C3—42no—
381C22×C3—42no—
382C22×C3—41no—
383C22×C3—41no—
384C22×C3—41no—
385C22×C3D3×Dic612yes—
386C22×C3#379512yes—
387C22×C3C2×#62411yescharacteristic
388C22×C3#381111yescharacteristic
389C22×C3#381111yescharacteristic
390C22×C3C2×C3×Dic612yes—
391C22×C3C2×C3×C4×D312yes—
392C22×C3C3×#23512yes—
393C22×C3C3×D3×D412yes—
394C22×C3C3×D3×D412yes—
395C22×C3C3×#23712yes—
396C22×C3C3×D3×Q812yes—
397C3×C4—24no—
398C3×C4—24no—
399C3×C4—22no—
400C3×C4—44no—
401C3×C4—44no—
402C3×C4—64no—
403C3×C4—64no—
404C3×C4—94no—
405C3×C4—124no—
406C3×C4—124no—
407C3×C4—122no—
408C3×C4—122no—
409C3×C4—182no—
410Dic3—24no—
411Dic3—38no—
412Dic3—42no—
413Dic3—64no—
414Dic3—64no—
415Dic3—64no—
416C32—21no—
417C32—22no—
418C32—41no—
419C32#464612yescomplemented
420C32C3×#39111yescharacteristic, complemented
421C23C2×C3×D3211yescharacteristic
422Q8—188no—
423C2×C4—21no—
424C2×C4—21no—
425C2×C4—34no—
426C2×C4—62no—
427C2×C4—62no—
428C2×C4—62no—
429C2×C4—62no—
430C2×C4—62no—
431C2×C4—92no—
432C2×C4—92no—
433C2×C4—92no—
434C2×C4—92no—
435C2×C4—181no—
436C2×C4—181no—
437C2×C4C3×#62412yes—
438C2×C3—21no—
439C2×C3—22no—
440C2×C3—22no—
441C2×C3—21no—
442C2×C3—22no—
443C2×C3—22no—
444C2×C3—22no—
445C2×C3—22no—
446C2×C3—22no—
447C2×C3—22no—
448C2×C3—22no—
449C2×C3—21no—
450C2×C3—41no—
451C2×C3—41no—
452C2×C3—42no—
453C2×C3—42no—
454C2×C3—41no—
455C2×C3#6384011yescharacteristic
456C2×C3#6384412yes—
457C2×C3#6389612yes—
458C2×C3#6391311yescharacteristic
459C2×C3#6397411yescharacteristic
460C2×C3C3×C4×Dic612yes—
461C2×C3C3×#87812yes—
462C2×C3C3×D3×C22⋊C412yes—
463C2×C3C3×#88512yes—
464C2×C3C3×#88712yes—
465C2×C3C3×#88812yes—
466C2×C3C3×#89612yes—
467C22C3×D3×Dic612yes—
468C22C3×#379512yes—
469C22C2×C3×#62411yescharacteristic
470C22C3×#381111yescharacteristic
471C22C3×#381111yescharacteristic
472C4—22no—
473C4—64no—
474C4—94no—
475C4—122no—
476C4—182no—
477C3—22no—
478C3—21no—
479C3—41no—
480C3#1059127311yescharacteristic, direct product factor
481C3C3×#464612yescomplemented
482C2C3×#6384011yescharacteristic
483C2C3×#6384412yes—
484C2C3×#6389612yes—
485C2C3×#6391311yescharacteristic
486C2C3×#6397411yescharacteristic
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×#391
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. #10591273C3direct (N × H)
13. C2×C3×#634C4semidirect
48. C2×C32×Dic3Q8semidirect
62. C2×#634C3×C4semidirect
168. C2×C3×Dic3Dic6semidirect
169. C2×C3×Dic3C3×Q8semidirect
263. C33#391semidirect
326. C2×Dic3C3×Dic6semidirect
419. C32#4646semidirect
420. C32C3×#391semidirect
480. C3#10591273direct (N × H)
481. C3C3×#4646semidirect

Maximal subgroups

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

subgroupindex
C22×C3×#6342
C2×C3×#37242
C2×C3×Dic322
C32×C3⋊#952
C3×#635792
C3×#46463
#105912733