Finite Group Databasebeta

← C3×#10591236 #13158409

Subgroups — C3×#10591236 #13158409

Order 1728.

1042 conjugacy classes of subgroups.

The subgroup lattice has 468 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×#627C211yesmaximal, characteristic
2C2×C3×#3750C211yesmaximal, characteristic
3C2×C3×Dic32C211yesmaximal, characteristic
4C32×C3⋊#95C212yesmaximal
5C3×#63579C212yesmaximal
6C3×#4646—32nomaximal
7#10591236C311yesmaximal, characteristic, direct product factor
8C22×C33×C4C2211yescharacteristic
9C22×C3×C3⋊Dic3C2211yescharacteristic
10C22×C3×C3⋊Dic3C2211yescharacteristic
11C22×C32×Dic3C2212yes—
12C22×C32×Dic3C2212yes—
13C2×C3×#627—22no—
14C2×C3×#627—22no—
15C2×C3×#627C414yescomplemented
16C3×#3750—22no—
17C3×Dic32—22no—
18C22×C3×Dic6—32no—
19C22×#627C2×C311yescharacteristic
20C2×#3750C2×C311yescharacteristic
21C2×C3×Dic3⋊C4—32no—
22C2×C3×Dic3⋊C4—61no—
23C2×C3×C4×Dic3—32no—
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—22no—
35C2×C3×C3⋊Dic3—21no—
36C2×C3×C3⋊Dic3—22no—
37C2×C3×C3⋊Dic3C2×C412yes—
38C2×C3×C3⋊Dic3D412yes—
39C2×C3×C3⋊Dic3D412yes—
40C2×C32×Dic3—22no—
41C2×C32×Dic3—22no—
42C2×C32×Dic3—22no—
43C2×C32×Dic3—22no—
44C2×C32×Dic3C2×C414yes—
45C2×C32×Dic3Q814yescomplemented
46C3×#627—28no—
47C3×#391—91no—
48#4646—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—
68C2×C3×Dic6—38no—
69C2×C3×Dic6—64no—
70C2×C3×Dic6—64no—
71C2×#627—22no—
72C2×#627—22no—
73C2×#627C3×C414yescomplemented
74#3750—22no—
75C3×Dic3⋊C4—64no—
76C3×Dic3⋊C4—64no—
77C3×C4×Dic3—64no—
78Dic32—22no—
79C22×C33C22×C411yescharacteristic
80C22×C33C2×D411yescharacteristic
81C22×C33C2×D411yescharacteristic
82C22×C33C2×Q812yes—
83C22×C33#4112yes—
84C33×C4—41no—
85C3×C3⋊Dic3—41no—
86C3×C3⋊Dic3C22⋊C414yes—
87C32×Dic3—24no—
88C32×Dic3—24no—
89C22×C3×Q8—91no—
90C22×Dic6—32no—
91C2×C3×C4⋊C4—91no—
92C2×C3×C42—91no—
93C2×Dic3⋊C4—32no—
94C2×Dic3⋊C4—61no—
95C2×C4×Dic3—32no—
96C3×#95—32no—
97C3×#95—32no—
98C3×#95—62no—
99C3×#95—92no—
100C3⋊#95—32no—
101C3⋊#95—32no—
102C3⋊#95C3×D312yes—
103C23×C32—21no—
104C23×C32—22no—
105C23×C32—41no—
106C23×C32C22×D312yes—
107C23×C32C23×C311yescharacteristic
108C2×C32×C4—22no—
109C2×C32×C4—21no—
110C2×C32×C4—22no—
111C2×C32×C4—24no—
112C2×C32×C4—22no—
113C2×C32×C4—22no—
114C2×C32×C4—22no—
115C2×C32×C4—34no—
116C2×C32×C4—34no—
117C2×C32×C4—42no—
118C2×C32×C4—42no—
119C2×C32×C4—42no—
120C2×C32×C4—42no—
121C2×C32×C4—44no—
122C2×C32×C4—62no—
123C2×C32×C4—62no—
124C2×C32×C4—62no—
125C2×C32×C4—62no—
126C2×C3⋊Dic3—21no—
127C2×C3⋊Dic3—22no—
128C2×C3⋊Dic3C2×C3×C412yes—
129C2×C3⋊Dic3C3×D412yes—
130C2×C3⋊Dic3C3×D412yes—
131C2×C3×Dic3—22no—
132C2×C3×Dic3—22no—
133C2×C3×Dic3—22no—
134C2×C3×Dic3—22no—
135C2×C3×Dic3—22no—
136C2×C3×Dic3—22no—
137C2×C3×Dic3—22no—
138C2×C3×Dic3—22no—
139C2×C3×Dic3—24no—
140C2×C3×Dic3—24no—
141C2×C3×Dic3—34no—
142C2×C3×Dic3—34no—
143C2×C3×Dic3—34no—
144C2×C3×Dic3—42no—
145C2×C3×Dic3—42no—
146C2×C3×Dic3—42no—
147C2×C3×Dic3—42no—
148C2×C3×Dic3—62no—
149C2×C3×Dic3—62no—
150C2×C3×Dic3—62no—
151C2×C3×Dic3—62no—
152C2×C3×Dic3—62no—
153C2×C3×Dic3—62no—
154C2×C3×Dic3—62no—
155C2×C3×Dic3—122no—
156C2×C3×Dic3Dic614yescomplemented
157C2×C3×Dic3C4×D314yes—
158C2×C3×Dic3C2×C3×C414yes—
159C2×C3×Dic3C3×Q814yescomplemented
160C3×Dic6—616no—
161#627—28no—
162#391—91noSylow
163C2×C33C2×C22⋊C411yescharacteristic
164C2×C33C4×Q812yes—
165C2×C33C22⋊Q812yes—
166C2×C33#12411yescharacteristic
167C2×C33C4⋊Q811yescharacteristic
168C2×C3×Q8—94no—
169C2×C3×Q8—182no—
170C2×C3×Q8—182no—
171C22×C3×C4—22no—
172C22×C3×C4—21no—
173C22×C3×C4—32no—
174C22×C3×C4—32no—
175C22×C3×C4—32no—
176C22×C3×C4—32no—
177C22×C3×C4—41no—
178C22×C3×C4—62no—
179C22×C3×C4—62no—
180C22×C3×C4—91no—
181C22×C3×C4—91no—
182C22×C3×C4D3211yescharacteristic
183C22×C3×C4C2×C3×D312yes—
184C22×Dic3—32no—
185C22×Dic3—32no—
186C22×Dic3—61no—
187C22×Dic3—61no—
188C22×Dic3C2×C3×D312yes—
189C22×Dic3C2×C3×D312yes—
190C2×Dic6—38no—
191C2×Dic6—64no—
192C2×Dic6—64no—
193C3×C4⋊C4—182no—
194C3×C42—182no—
195Dic3⋊C4—64no—
196Dic3⋊C4—64no—
197C4×Dic3—64no—
198C22×C32—22no—
199C22×C32—21no—
200C22×C32—21no—
201C22×C32—22no—
202C22×C32—21no—
203C22×C32—22no—
204C22×C32—22no—
205C22×C32—22no—
206C22×C32—22no—
207C22×C32—22no—
208C22×C32—22no—
209C22×C32—22no—
210C22×C32—41no—
211C22×C32—41no—
212C22×C32—41no—
213C22×C32—42no—
214C22×C32—42no—
215C22×C32C2×Dic612yes—
216C22×C32C2×C4×D312yes—
217C22×C32#23512yes—
218C22×C32D3×D412yes—
219C22×C32D3×D412yes—
220C22×C32#23712yes—
221C22×C32D3×Q812yes—
222C22×C32C22×C3×C411yescharacteristic
223C22×C32C2×C3×D411yescharacteristic
224C22×C32C2×C3×D411yescharacteristic
225C22×C32C2×C3×Q812yes—
226C22×C32C3×#4112yes—
227C32×C4—44no—
228C32×C4—42no—
229C32×C4—44no—
230C32×C4—42no—
231C32×C4—41no—
232C32×C4—64no—
233C32×C4—64no—
234C3⋊Dic3—41no—
235C3⋊Dic3C3×C22⋊C414yes—
236C3×Dic3—24no—
237C3×Dic3—24no—
238C3×Dic3—24no—
239C3×Dic3—24no—
240C3×Dic3—38no—
241C3×Dic3—44no—
242C3×Dic3—44no—
243C3×Dic3—64no—
244C3×Dic3—122no—
245C3×Dic3—122no—
246C22×Q8—91no—
247C2×C4⋊C4—91no—
248C2×C42—91no—
249#95—32no—
250#95—92no—
251C33#39111yescharacteristic, complemented, Sylow
252C23×C3—22no—
253C23×C3—21no—
254C23×C3—41no—
255C23×C3C2×D3211yescharacteristic
256C23×C3C22×C3×D312yes—
257C3×Q8—188no—
258C2×C3×C4—22no—
259C2×C3×C4—24no—
260C2×C3×C4—22no—
261C2×C3×C4—21no—
262C2×C3×C4—22no—
263C2×C3×C4—22no—
264C2×C3×C4—22no—
265C2×C3×C4—34no—
266C2×C3×C4—34no—
267C2×C3×C4—34no—
268C2×C3×C4—34no—
269C2×C3×C4—42no—
270C2×C3×C4—42no—
271C2×C3×C4—42no—
272C2×C3×C4—44no—
273C2×C3×C4—42no—
274C2×C3×C4—64no—
275C2×C3×C4—64no—
276C2×C3×C4—62no—
277C2×C3×C4—62no—
278C2×C3×C4—62no—
279C2×C3×C4—62no—
280C2×C3×C4—62no—
281C2×C3×C4—62no—
282C2×C3×C4—62no—
283C2×C3×C4—62no—
284C2×C3×C4—92no—
285C2×C3×C4—92no—
286C2×C3×C4—92no—
287C2×C3×C4—122no—
288C2×C3×C4—122no—
289C2×C3×C4—122no—
290C2×C3×C4—122no—
291C2×C3×C4—182no—
292C2×C3×C4—181no—
293C2×Dic3—22no—
294C2×Dic3—22no—
295C2×Dic3—22no—
296C2×Dic3—22no—
297C2×Dic3—34no—
298C2×Dic3—34no—
299C2×Dic3—34no—
300C2×Dic3—62no—
301C2×Dic3—62no—
302C2×Dic3—62no—
303C2×Dic3—62no—
304C2×Dic3—62no—
305C2×Dic3—62no—
306C2×Dic3—62no—
307C2×Dic3—122no—
308C2×Dic3C3×Dic614yescomplemented
309C2×Dic3C3×C4×D314yes—
310Dic6—616no—
311C2×C32—22no—
312C2×C32—22no—
313C2×C32—22no—
314C2×C32—22no—
315C2×C32—22no—
316C2×C32—22no—
317C2×C32—22no—
318C2×C32—21no—
319C2×C32—22no—
320C2×C32—21no—
321C2×C32—21no—
322C2×C32—22no—
323C2×C32—41no—
324C2×C32—42no—
325C2×C32—42no—
326C2×C32—41no—
327C2×C32—41no—
328C2×C32C4×Dic612yes—
329C2×C32#87812yes—
330C2×C32D3×C22⋊C412yes—
331C2×C32#88512yes—
332C2×C32#88712yes—
333C2×C32#88812yes—
334C2×C32#89612yes—
335C2×C32C2×C3×C22⋊C411yescharacteristic
336C2×C32C3×C4×Q812yes—
337C2×C32C3×C22⋊Q812yes—
338C2×C32C3×#12411yescharacteristic
339C2×C32C3×C4⋊Q811yescharacteristic
340C2×Q8—94no—
341C2×Q8—182no—
342C2×Q8—182no—
343C22×C4—32no—
344C22×C4—32no—
345C22×C4—91no—
346C22×C4—91no—
347C22×C4C3×D3211yescharacteristic
348C4⋊C4—182no—
349C42—182no—
350C22×C3—22no—
351C22×C3—22no—
352C22×C3—22no—
353C22×C3—21no—
354C22×C3—22no—
355C22×C3—21no—
356C22×C3—22no—
357C22×C3—22no—
358C22×C3—22no—
359C22×C3—22no—
360C22×C3—21no—
361C22×C3—22no—
362C22×C3—42no—
363C22×C3—41no—
364C22×C3—41no—
365C22×C3—42no—
366C22×C3—41no—
367C22×C3D3×Dic612yes—
368C22×C3C4×D3211yescharacteristic
369C22×C3#380211yescharacteristic
370C22×C3#380412yes—
371C22×C3#381111yescharacteristic
372C22×C3C2×C3×Dic612yes—
373C22×C3C2×C3×C4×D312yes—
374C22×C3C3×#23512yes—
375C22×C3C3×D3×D412yes—
376C22×C3C3×D3×D412yes—
377C22×C3C3×#23712yes—
378C22×C3C3×D3×Q812yes—
379C3×C4—42no—
380C3×C4—44no—
381C3×C4—44no—
382C3×C4—42no—
383C3×C4—41no—
384C3×C4—64no—
385C3×C4—64no—
386C3×C4—64no—
387C3×C4—64no—
388C3×C4—94no—
389C3×C4—124no—
390C3×C4—124no—
391C3×C4—361no—
392Dic3—24no—
393Dic3—24no—
394Dic3—38no—
395Dic3—64no—
396Dic3—122no—
397Dic3—122no—
398C32—21no—
399C32—22no—
400C32—41no—
401C32#464612yescomplemented
402C32C3×#39111yescharacteristic, complemented
403C23C2×C3×D3211yescharacteristic
404Q8—188no—
405C2×C4—22no—
406C2×C4—21no—
407C2×C4—34no—
408C2×C4—34no—
409C2×C4—62no—
410C2×C4—62no—
411C2×C4—62no—
412C2×C4—62no—
413C2×C4—92no—
414C2×C4—92no—
415C2×C4—92no—
416C2×C4—182no—
417C2×C4—181no—
418C2×C3—21no—
419C2×C3—22no—
420C2×C3—22no—
421C2×C3—22no—
422C2×C3—22no—
423C2×C3—22no—
424C2×C3—21no—
425C2×C3—22no—
426C2×C3—22no—
427C2×C3—22no—
428C2×C3—21no—
429C2×C3—22no—
430C2×C3—41no—
431C2×C3—42no—
432C2×C3—41no—
433C2×C3—42no—
434C2×C3—41no—
435C2×C3#6383712yes—
436C2×C3#6384011yescharacteristic
437C2×C3#6389912yes—
438C2×C3#6392111yescharacteristic
439C2×C3#6395311yescharacteristic
440C2×C3C3×C4×Dic612yes—
441C2×C3C3×#87812yes—
442C2×C3C3×D3×C22⋊C412yes—
443C2×C3C3×#88512yes—
444C2×C3C3×#88712yes—
445C2×C3C3×#88812yes—
446C2×C3C3×#89612yes—
447C22C3×D3×Dic612yes—
448C22C3×C4×D3211yescharacteristic
449C22C3×#380211yescharacteristic
450C22C3×#380412yes—
451C22C3×#381111yescharacteristic
452C4—41no—
453C4—64no—
454C4—64no—
455C4—94no—
456C4—361no—
457C3—22no—
458C3—21no—
459C3—41no—
460C3#1059123611yescharacteristic, direct product factor
461C3C3×#464612yescomplemented
462C2C3×#6383712yes—
463C2C3×#6384011yescharacteristic
464C2C3×#6389912yes—
465C2C3×#6392111yescharacteristic
466C2C3×#6395311yescharacteristic
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. #10591236C3direct (N × H)
15. C2×C3×#627C4semidirect
45. C2×C32×Dic3Q8semidirect
73. C2×#627C3×C4semidirect
156. C2×C3×Dic3Dic6semidirect
159. C2×C3×Dic3C3×Q8semidirect
251. C33#391semidirect
308. C2×Dic3C3×Dic6semidirect
401. C32#4646semidirect
402. C32C3×#391semidirect
460. C3#10591236direct (N × H)
461. C3C3×#4646semidirect

Maximal subgroups

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

subgroupindex
C22×C3×#6272
C2×C3×#37502
C2×C3×Dic322
C32×C3⋊#952
C3×#635792
C3×#46463
#105912363