Finite Group Databasebeta

← C3×#10591397 #13158413

Subgroups — C3×#10591397 #13158413

Order 1728.

1026 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×C4⋊(C3⋊Dic3)C211yesmaximal, characteristic
3C2×C3×Dic3⋊Dic3C212yesmaximal
4C32×C3⋊#95C212yesmaximal
5C3×#63579C211yesmaximal, characteristic
6C3×#4656—32nomaximal
7#10591397C311yesmaximal, 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—24no—
15C3×C4⋊(C3⋊Dic3)—22no—
16C3×Dic3⋊Dic3—24no—
17C22×C3×Dic6—32no—
18C22×#627C2×C311yescharacteristic
19C2×C4⋊(C3⋊Dic3)C2×C311yescharacteristic
20C2×C3×C4⋊Dic3—32no—
21C2×C3×C4⋊Dic3—32no—
22C2×C3×C4⋊Dic3—61no—
23C2×C3×Dic3⋊C4—32no—
24C2×Dic3⋊Dic3C2×C312yes—
25C32×#95—32no—
26C3×C3⋊#95—22no—
27C3×C3⋊#95—32no—
28C3×C3⋊#95D312yes—
29C3×C3⋊#95C2×C312yes—
30#63579C2×C311yescharacteristic
31C23×C33C2311yescharacteristic
32C2×C33×C4—21no—
33C2×C33×C4—21no—
34C2×C33×C4Q812yescomplemented
35C2×C3×C3⋊Dic3—22no—
36C2×C3×C3⋊Dic3—21no—
37C2×C3×C3⋊Dic3—22no—
38C2×C3×C3⋊Dic3D412yes—
39C2×C32×Dic3—22no—
40C2×C32×Dic3—22no—
41C2×C32×Dic3—22no—
42C2×C32×Dic3—22no—
43C2×C32×Dic3D414yes—
44C2×C32×Dic3Q814yescomplemented
45C3×#627—28no—
46C3×#395—91no—
47#4656—32no—
48C22×C32×C4—21no—
49C22×C32×C4—22no—
50C22×C32×C4—32no—
51C22×C32×C4—32no—
52C22×C32×C4—41no—
53C22×C32×C4C2×D312yes—
54C22×C32×C4C22×C311yescharacteristic
55C22×C3⋊Dic3C22×C311yescharacteristic
56C22×C3⋊Dic3C22×C311yescharacteristic
57C22×C3×Dic3—22no—
58C22×C3×Dic3—22no—
59C22×C3×Dic3—32no—
60C22×C3×Dic3—32no—
61C22×C3×Dic3—61no—
62C22×C3×Dic3—61no—
63C22×C3×Dic3C2×D312yes—
64C22×C3×Dic3C2×D312yes—
65C22×C3×Dic3C22×C312yes—
66C22×C3×Dic3C22×C312yes—
67C2×C3×Dic6—64no—
68C2×C3×Dic6—64no—
69C2×C3×Dic6—64no—
70C2×#627—22no—
71C2×#627—24no—
72C4⋊(C3⋊Dic3)—22no—
73C3×C4⋊Dic3—64no—
74C3×C4⋊Dic3—64no—
75C3×C4⋊Dic3—64no—
76C3×Dic3⋊C4—64no—
77Dic3⋊Dic3—24no—
78C22×C33C2×D412yes—
79C22×C33C2×D411yescharacteristic
80C22×C33C2×Q811yescharacteristic
81C22×C33C2×Q812yes—
82C22×C33#4111yescharacteristic
83C33×C4—22no—
84C3×C3⋊Dic3—22no—
85C3×C3⋊Dic3—41no—
86C32×Dic3—24no—
87C32×Dic3—24no—
88C22×C3×Q8—91no—
89C22×Dic6—32no—
90C2×C3×C4⋊C4—92no—
91C2×C3×C4⋊C4—91no—
92C2×C4⋊Dic3—32no—
93C2×C4⋊Dic3—32no—
94C2×C4⋊Dic3—61no—
95C2×Dic3⋊C4—32no—
96C3×#95—32no—
97C3×#95—32no—
98C3×#95—62no—
99C3×#95—91no—
100C3⋊#95—32no—
101C3⋊#95C3×D312yes—
102C23×C32—21no—
103C23×C32—22no—
104C23×C32—41no—
105C23×C32C22×D312yes—
106C23×C32C23×C311yescharacteristic
107C2×C32×C4—21no—
108C2×C32×C4—21no—
109C2×C32×C4—22no—
110C2×C32×C4—22no—
111C2×C32×C4—22no—
112C2×C32×C4—22no—
113C2×C32×C4—22no—
114C2×C32×C4—24no—
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—42no—
122C2×C32×C4—62no—
123C2×C32×C4—62no—
124C2×C32×C4—62no—
125C2×C32×C4—62no—
126C2×C32×C4Dic614yescomplemented
127C2×C32×C4C3×Q812yescomplemented
128C2×C3⋊Dic3—21no—
129C2×C3⋊Dic3—22no—
130C2×C3⋊Dic3—22no—
131C2×C3⋊Dic3C3×D412yes—
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—22no—
140C2×C3×Dic3—24no—
141C2×C3×Dic3—24no—
142C2×C3×Dic3—34no—
143C2×C3×Dic3—42no—
144C2×C3×Dic3—42no—
145C2×C3×Dic3—42no—
146C2×C3×Dic3—42no—
147C2×C3×Dic3—62no—
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—122no—
155C2×C3×Dic3—122no—
156C2×C3×Dic3Dic614yescomplemented
157C2×C3×Dic3D1214yes—
158C2×C3×Dic3C3×D414yes—
159C2×C3×Dic3C3×Q814yescomplemented
160C3×Dic6—616no—
161#627—28no—
162#395—91noSylow
163C2×C33#12011yescharacteristic
164C2×C33C22⋊Q811yescharacteristic
165C2×C33C22⋊Q812yes—
166C2×C33C4⋊Q812yes—
167C2×C33C4⋊Q811yescharacteristic
168C2×C3×Q8—184no—
169C2×C3×Q8—182no—
170C22×C3×C4—21no—
171C22×C3×C4—22no—
172C22×C3×C4—32no—
173C22×C3×C4—32no—
174C22×C3×C4—32no—
175C22×C3×C4—32no—
176C22×C3×C4—41no—
177C22×C3×C4—62no—
178C22×C3×C4—62no—
179C22×C3×C4—91no—
180C22×C3×C4—91no—
181C22×C3×C4D3211yescharacteristic
182C22×C3×C4C2×C3×D312yes—
183C22×Dic3—32no—
184C22×Dic3—32no—
185C22×Dic3—61no—
186C22×Dic3—61no—
187C22×Dic3C2×C3×D312yes—
188C22×Dic3C2×C3×D312yes—
189C2×Dic6—64no—
190C2×Dic6—64no—
191C2×Dic6—64no—
192C3×C4⋊C4—182no—
193C3×C4⋊C4—184no—
194C4⋊Dic3—64no—
195C4⋊Dic3—64no—
196C4⋊Dic3—64no—
197Dic3⋊C4—64no—
198C22×C32—22no—
199C22×C32—22no—
200C22×C32—21no—
201C22×C32—21no—
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—42no—
212C22×C32—41no—
213C22×C32—42no—
214C22×C32—41no—
215C22×C32C2×Dic612yes—
216C22×C32C2×Dic612yes—
217C22×C32C2×D1212yes—
218C22×C32D3×D412yes—
219C22×C32D3×D412yes—
220C22×C32#23712yes—
221C22×C32D3×Q812yes—
222C22×C32C2×C3×D412yes—
223C22×C32C2×C3×D411yescharacteristic
224C22×C32C2×C3×Q811yescharacteristic
225C22×C32C2×C3×Q812yes—
226C22×C32C3×#4111yescharacteristic
227C32×C4—22no—
228C32×C4—24no—
229C32×C4—24no—
230C32×C4—44no—
231C32×C4—44no—
232C32×C4—64no—
233C32×C4—64no—
234C3⋊Dic3—22no—
235C3⋊Dic3—41no—
236C3×Dic3—24no—
237C3×Dic3—24no—
238C3×Dic3—24no—
239C3×Dic3—24no—
240C3×Dic3—44no—
241C3×Dic3—44no—
242C3×Dic3—64no—
243C3×Dic3—122no—
244C3×Dic3—122no—
245C3×Dic3—122no—
246C22×Q8—91no—
247C2×C4⋊C4—92no—
248C2×C4⋊C4—91no—
249#95—32no—
250#95—91no—
251C33#39511yescharacteristic, 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—22no—
260C2×C3×C4—21no—
261C2×C3×C4—21no—
262C2×C3×C4—22no—
263C2×C3×C4—22no—
264C2×C3×C4—22no—
265C2×C3×C4—24no—
266C2×C3×C4—34no—
267C2×C3×C4—34no—
268C2×C3×C4—34no—
269C2×C3×C4—34no—
270C2×C3×C4—42no—
271C2×C3×C4—42no—
272C2×C3×C4—42no—
273C2×C3×C4—42no—
274C2×C3×C4—42no—
275C2×C3×C4—64no—
276C2×C3×C4—64no—
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—62no—
285C2×C3×C4—92no—
286C2×C3×C4—122no—
287C2×C3×C4—122no—
288C2×C3×C4—122no—
289C2×C3×C4—122no—
290C2×C3×C4—182no—
291C2×C3×C4—182no—
292C2×C3×C4—181no—
293C2×C3×C4#62712yescomplemented
294C2×C3×C4C3×Dic614yescomplemented
295C2×Dic3—22no—
296C2×Dic3—22no—
297C2×Dic3—22no—
298C2×Dic3—22no—
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×Dic3—122no—
309C2×Dic3C3×Dic614yescomplemented
310C2×Dic3C3×D1214yes—
311Dic6—616no—
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—22no—
321C2×C32—21no—
322C2×C32—21no—
323C2×C32—22no—
324C2×C32—41no—
325C2×C32—41no—
326C2×C32—42no—
327C2×C32—41no—
328C2×C32—42no—
329C2×C32#86912yes—
330C2×C32#87812yes—
331C2×C32#87812yes—
332C2×C32#88212yes—
333C2×C32#88812yes—
334C2×C32#88812yes—
335C2×C32#89712yes—
336C2×C32C3×#12011yescharacteristic
337C2×C32C3×C22⋊Q811yescharacteristic
338C2×C32C3×C22⋊Q812yes—
339C2×C32C3×C4⋊Q811yescharacteristic
340C2×C32C3×C4⋊Q812yes—
341C2×Q8—184no—
342C2×Q8—182no—
343C22×C4—32no—
344C22×C4—32no—
345C22×C4—91no—
346C22×C4—91no—
347C22×C4C3×D3211yescharacteristic
348C4⋊C4—182no—
349C4⋊C4—184no—
350C22×C3—22no—
351C22×C3—22no—
352C22×C3—22no—
353C22×C3—22no—
354C22×C3—22no—
355C22×C3—22no—
356C22×C3—21no—
357C22×C3—22no—
358C22×C3—22no—
359C22×C3—21no—
360C22×C3—21no—
361C22×C3—22no—
362C22×C3—42no—
363C22×C3—42no—
364C22×C3—41no—
365C22×C3—41no—
366C22×C3—41no—
367C22×C3D3×Dic612yes—
368C22×C3D3×D1212yes—
369C22×C3#380511yescharacteristic
370C22×C3C2×#62711yescharacteristic
371C22×C3#381111yescharacteristic
372C22×C3C2×C3×Dic612yes—
373C22×C3C2×C3×Dic612yes—
374C22×C3C2×C3×D1212yes—
375C22×C3C3×D3×D412yes—
376C22×C3C3×D3×D412yes—
377C22×C3C3×#23712yes—
378C22×C3C3×D3×Q812yes—
379C3×C4—24no—
380C3×C4—24no—
381C3×C4—22no—
382C3×C4—44no—
383C3×C4—44no—
384C3×C4—64no—
385C3×C4—64no—
386C3×C4—64no—
387C3×C4—64no—
388C3×C4—124no—
389C3×C4—124no—
390C3×C4—182no—
391C3×C4—361no—
392Dic3—24no—
393Dic3—24no—
394Dic3—64no—
395Dic3—122no—
396Dic3—122no—
397Dic3—122no—
398C32—22no—
399C32—21no—
400C32—41no—
401C32#465612yescomplemented
402C32C3×#39511yescharacteristic, complemented
403C23C2×C3×D3211yescharacteristic
404Q8—188no—
405C2×C4—21no—
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—182no—
415C2×C4—182no—
416C2×C4—181no—
417C2×C4C3×#62712yescomplemented
418C2×C3—21no—
419C2×C3—22no—
420C2×C3—22no—
421C2×C3—21no—
422C2×C3—22no—
423C2×C3—22no—
424C2×C3—22no—
425C2×C3—22no—
426C2×C3—22no—
427C2×C3—22no—
428C2×C3—21no—
429C2×C3—22no—
430C2×C3—41no—
431C2×C3—41no—
432C2×C3—42no—
433C2×C3—42no—
434C2×C3—41no—
435C2×C3#6384011yescharacteristic
436C2×C3#6389312yes—
437C2×C3#6391812yes—
438C2×C3#6392311yescharacteristic
439C2×C3#6398211yescharacteristic
440C2×C3C3×#86912yes—
441C2×C3C3×#87812yes—
442C2×C3C3×#87812yes—
443C2×C3C3×#88212yes—
444C2×C3C3×#88812yes—
445C2×C3C3×#88812yes—
446C2×C3C3×#89712yes—
447C22C3×D3×Dic612yes—
448C22C3×D3×D1212yes—
449C22C3×#380511yescharacteristic
450C22C2×C3×#62711yescharacteristic
451C22C3×#381111yescharacteristic
452C4—22no—
453C4—64no—
454C4—64no—
455C4—182no—
456C4—361no—
457C3—22no—
458C3—21no—
459C3—41no—
460C3#1059139711yescharacteristic, direct product factor
461C3C3×#465612yescomplemented
462C2C3×#6384011yescharacteristic
463C2C3×#6389312yes—
464C2C3×#6391812yes—
465C2C3×#6392311yescharacteristic
466C2C3×#6398211yescharacteristic
subgroupG/H
centerC23×C3C2×D32
commutator, cosocleC23×C32C23×C3
FrattiniC23C2×C3×D32
FittingC22×C33×C4C22
socle, prime radicalC23×C33C23
hypercenterC22×C3×C4D32
hypocenterC32C3×#395
LCSG ▷ C23×C32 ▷ C32
UCSC1 ◁ C23×C3 ◁ C22×C3×C4

Complements

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

NH ≅ G/Ntype
7. #10591397C3direct (N × H)
34. C2×C33×C4Q8semidirect
44. C2×C32×Dic3Q8semidirect
126. C2×C32×C4Dic6semidirect
127. C2×C32×C4C3×Q8semidirect
156. C2×C3×Dic3Dic6semidirect
159. C2×C3×Dic3C3×Q8semidirect
251. C33#395semidirect
293. C2×C3×C4#627semidirect
294. C2×C3×C4C3×Dic6semidirect
309. C2×Dic3C3×Dic6semidirect
401. C32#4656semidirect
402. C32C3×#395semidirect
417. C2×C4C3×#627semidirect
460. C3#10591397direct (N × H)
461. C3C3×#4656semidirect

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×C4⋊(C3⋊Dic3)2
C2×C3×Dic3⋊Dic32
C32×C3⋊#952
C3×#635792
C3×#46563
#105913973