Finite Group Databasebeta

← C3×#10595190 #13174747

Subgroups — C3×#10595190 #13174747

Order 1728.

1646 conjugacy classes of subgroups.

The subgroup lattice has 520 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#3830C211yesmaximal, characteristic
2C32×D4×Dic3C212yesmaximal, complemented
3C3×#63957C212yesmaximal, complemented
4C2×C3×Dic32C212yesmaximal, complemented
5C3×#63952C214yesmaximal, complemented
6C3×C4×#624C211yesmaximal, characteristic, complemented
7C3×#63849C211yesmaximal, characteristic, complemented
8C3×#63844C212yesmaximal, complemented
9C3×C4×#237—32nomaximal
10#10595190C311yesmaximal, characteristic, direct product factor
11C2×C33×D4C2211yescharacteristic
12C2×C3×#638C2212yes—
13C22×C3×C3⋊Dic3C2212yes—
14C3×#3830C418yescomplemented
15C2×C3×C4×C3⋊D3C2211yescharacteristic
16C2×C3×#634C2211yescharacteristic
17C22×C32×Dic3C2214yes—
18C2×C3×#624C2212yes—
19C32×#217C2214yes—
20C32×C4⋊Dic3C2212yes—
21C32×C4×Dic3C2212yes—
22C3×#3724C2214yes—
23C3×#3722C2212yes—
24C3×Dic32C2214yes—
25C3×Dic32C2211yescharacteristic
26C3×Dic32C2212yes—
27C3×Dic32C2211yescharacteristic
28C2×#3830C2×C311yescharacteristic
29C2×C3×#237—32no—
30C2×C3×#237—61no—
31C32×C4×D4—32no—
32C3×D4×Dic3—22no—
33C3×D4×Dic3D312yescomplemented
34C3×D4×Dic3C2×C312yescomplemented
35C3×C4×#67—34no—
36C2×C3×C4×Dic3—34no—
37C3×#892—32no—
38C3×#887—32no—
39C3×#67⋊C4—34no—
40C3×#877—34no—
41C3×C42×D3—32no—
42C3×C4×Dic6—32no—
43#63957C2×C312yescomplemented
44C2×Dic32C2×C312yescomplemented
45#63952C2×C314yescomplemented
46C4×#624C2×C311yescharacteristic, complemented
47#63849C2×C311yescharacteristic, complemented
48#63844C2×C312yescomplemented
49C23×C33C2312yes—
50C22×C3×C3⋊D3C2311yescharacteristic
51C33×D4C2×C414yescomplemented
52C2×C33×C4C2311yescharacteristic
53C3×#638C2×C418yescomplemented
54C2×C3×C3⋊Dic3C2×C418yescomplemented
55C2×C3×C3⋊Dic3C2311yescharacteristic
56C2×C3×C3⋊Dic3C2312yes—
57C3×C4×C3⋊D3C2×C414yescomplemented
58C3×#634C2×C414yescomplemented
59C2×C32×Dic3—28no—
60C2×C32×Dic3C2312yes—
61C2×C32×Dic3C2314yes—
62C2×C32×Dic3C2312yes—
63C3×#624—24no—
64C3×C4×#41—91no—
65C4×#237—32no—
66C2×C32×D4—21no—
67C2×C32×D4—22no—
68C2×C32×D4—41no—
69C2×C32×D4C2×D312yes—
70C2×C32×D4C22×C311yescharacteristic
71C22×C32×C4—34no—
72C2×#638C22×C312yes—
73C22×C3⋊Dic3C22×C312yes—
74#3830C3×C418yescomplemented
75C2×C4×C3⋊D3C22×C311yescharacteristic
76C2×#634C22×C311yescharacteristic
77C2×C3×#67—34no—
78C2×C3×#67—62no—
79C22×C3×Dic3—24no—
80C22×C3×Dic3—34no—
81C22×C3×Dic3—62no—
82C22×C3×Dic3C2×D314yes—
83C22×C3×Dic3C22×C314yes—
84C3×#237—316no—
85C3×#237—68no—
86C2×C3×C4×D3—32no—
87C2×C3×C4×D3—32no—
88C2×C3×C4×D3—32no—
89C2×C3×C4×D3—61no—
90C2×C3×Dic6—32no—
91C2×C3×Dic6—61no—
92C2×#624C22×C312yes—
93C32×C4⋊C4—32no—
94C32×C22⋊C4—34no—
95C32×C42—32no—
96C3×#217—24no—
97C3×#217C2×D314yes—
98C3×#217C22×C314yes—
99C3×(C2×D3)⋊C4—34no—
100C3×C4⋊Dic3—22no—
101C3×C4⋊Dic3C2×D312yes—
102C3×C4⋊Dic3C22×C312yes—
103C3×Dic3⋊C4—34no—
104C3×Dic3⋊C4—34no—
105C3×C4×Dic3—22no—
106C3×C4×Dic3—32no—
107C3×C4×Dic3—34no—
108C3×C4×Dic3—34no—
109C3×C4×Dic3—32no—
110C3×C4×Dic3—34no—
111C3×C4×Dic3C2×D312yes—
112C3×C4×Dic3C22×C312yes—
113#3724C22×C314yes—
114#3722C22×C312yes—
115Dic32C22×C314yes—
116Dic32C22×C312yes—
117Dic32C22×C311yescharacteristic
118Dic32C22×C311yescharacteristic
119C22×C33—24no—
120C22×C33C22×C414yes—
121C22×C33C2411yescharacteristic
122C2×C3×C3⋊D3—22no—
123C2×C3×C3⋊D3C22×C412yes—
124C33×C4C22×C412yes—
125C3×C3⋊Dic3C22×C414yes—
126C3×C3⋊Dic3C22×C412yes—
127C32×Dic3—22no—
128C32×Dic3—24no—
129C32×Dic3#4114yescomplemented
130C2×C3×#41—91no—
131C2×#237—32no—
132C2×#237—61no—
133C3×C4×Q8—92no—
134C3×C4×D4—32no—
135C3×C4×D4—32no—
136C3×C4×D4—62no—
137C3×C4×D4—94no—
138C3×#117—92no—
139C3×#117—91no—
140C2×C3×C42—91no—
141C2×C3×C42—92no—
142D4×Dic3C3×D312yescomplemented
143C4×#67—34no—
144C2×C4×Dic3—34no—
145#892—32no—
146#887—32no—
147#67⋊C4—34no—
148#877—34no—
149C42×D3—32no—
150C4×Dic6—32no—
151C23×C32—22no—
152C23×C32—24no—
153C23×C32—42no—
154C23×C32C22×D314yes—
155C23×C32C23×C312yes—
156C22×C3⋊D3C23×C311yescharacteristic
157C22×C3×D3—32no—
158C22×C3×D3—61no—
159C32×D4—24no—
160C32×D4—28no—
161C32×D4—44no—
162C32×D4C4×D318yescomplemented
163C32×D4C2×C3×C414yescomplemented
164C2×C32×C4—21no—
165C2×C32×C4—22no—
166C2×C32×C4—32no—
167C2×C32×C4—34no—
168C2×C32×C4—32no—
169C2×C32×C4—41no—
170C2×C32×C4—68no—
171C2×C32×C4C22×D312yes—
172C2×C32×C4C23×C311yescharacteristic
173#638C2×C3×C418yescomplemented
174C2×C3⋊Dic3C2×C3×C418yescomplemented
175C2×C3⋊Dic3C23×C311yescharacteristic
176C2×C3⋊Dic3C23×C312yes—
177C4×C3⋊D3C2×C3×C414yescomplemented
178#634C2×C3×C414yescomplemented
179C3×#67—316no—
180C3×#67—68no—
181C2×C3×Dic3—28no—
182C2×C3×Dic3—28no—
183C2×C3×Dic3—22no—
184C2×C3×Dic3—22no—
185C2×C3×Dic3—24no—
186C2×C3×Dic3—316no—
187C2×C3×Dic3—32no—
188C2×C3×Dic3—34no—
189C2×C3×Dic3—48no—
190C2×C3×Dic3—68no—
191C2×C3×Dic3—62no—
192C2×C3×Dic3—61no—
193C2×C3×Dic3C22×D314yes—
194C2×C3×Dic3C22×D312yes—
195C2×C3×Dic3C22×D312yes—
196C2×C3×Dic3C23×C314yes—
197C2×C3×Dic3C23×C312yes—
198C2×C3×Dic3C23×C312yes—
199C3×C4×D3—38no—
200C3×C4×D3—64no—
201C3×C4×D3—64no—
202C3×C4×D3—64no—
203C3×Dic6—38no—
204C3×Dic6—64no—
205#624—24no—
206C4×#41—91noSylow
207C2×C33—24no—
208C2×C33C23×C411yescharacteristic
209C2×C33C2×#4112yes—
210C3×C3⋊D3—22no—
211C3×#41—98no—
212C2×C3×Q8—91no—
213C2×C3×D4—22no—
214C2×C3×D4—21no—
215C2×C3×D4—41no—
216C2×C3×D4—92no—
217C2×C3×D4D3211yescharacteristic
218C2×C3×D4C2×C3×D312yes—
219C22×C3×C4—34no—
220C22×C3×C4—34no—
221C22×C3×C4—64no—
222C22×C3×C4—92no—
223C22×C3×C4—92no—
224C22×C3×C4—91no—
225C2×#67—34no—
226C2×#67—62no—
227C22×Dic3—34no—
228C22×Dic3—62no—
229C22×Dic3C2×C3×D314yes—
230#237—316no—
231#237—68no—
232C2×C4×D3—32no—
233C2×C4×D3—32no—
234C2×C4×D3—32no—
235C2×C4×D3—61no—
236C2×Dic6—32no—
237C2×Dic6—61no—
238C3×C4⋊C4—32no—
239C3×C4⋊C4—32no—
240C3×C4⋊C4—62no—
241C3×C4⋊C4—94no—
242C3×C22⋊C4—34no—
243C3×C22⋊C4—34no—
244C3×C22⋊C4—64no—
245C3×C22⋊C4—92no—
246C3×C42—32no—
247C3×C42—32no—
248C3×C42—62no—
249C3×C42—91no—
250C3×C42—91no—
251C3×C42—92no—
252C3×C42—94no—
253#217C2×C3×D314yes—
254(C2×D3)⋊C4—34no—
255C4⋊Dic3C2×C3×D312yes—
256Dic3⋊C4—34no—
257Dic3⋊C4—34no—
258C4×Dic3—34no—
259C4×Dic3—32no—
260C4×Dic3—32no—
261C4×Dic3—34no—
262C4×Dic3—34no—
263C4×Dic3C2×C3×D312yes—
264C22×C32—24no—
265C22×C32—24no—
266C22×C32—24no—
267C22×C32—21no—
268C22×C32—22no—
269C22×C32—24no—
270C22×C32—28no—
271C22×C32—41no—
272C22×C32—44no—
273C22×C32—44no—
274C22×C32—44no—
275C22×C32—44no—
276C22×C32—84no—
277C22×C32C2×C4×D318yes—
278C22×C32C22×C3×C414yes—
279C22×C32C23×D312yes—
280C22×C32C24×C311yescharacteristic
281C2×C3⋊D3—22no—
282C2×C3⋊D3C22×C3×C412yes—
283C2×C3×D3—34no—
284C2×C3×D3—62no—
285C2×C3×D3—62no—
286C2×C3×D3—62no—
287C2×C3×D3—122no—
288C32×C4—22no—
289C32×C4—24no—
290C32×C4—34no—
291C32×C4—42no—
292C32×C4—62no—
293C32×C4—64no—
294C32×C4C2×C4×D314yes—
295C32×C4C22×C3×C412yes—
296C3⋊Dic3C22×C3×C414yes—
297C3⋊Dic3C22×C3×C412yes—
298C3×Dic3—24no—
299C3×Dic3—22no—
300C3×Dic3—22no—
301C3×Dic3—24no—
302C3×Dic3—24no—
303C3×Dic3—38no—
304C3×Dic3—34no—
305C3×Dic3—42no—
306C3×Dic3—44no—
307C3×Dic3—64no—
308C3×Dic3—62no—
309C3×Dic3#23714yescomplemented
310C3×Dic3C3×#4114yescomplemented
311C2×#41—91no—
312C4×Q8—92no—
313C4×D4—32no—
314C4×D4—94no—
315#117—92no—
316#117—91no—
317C2×C42—91no—
318C2×C42—92no—
319C33C4×#4111yescharacteristic, complemented, Sylow
320C23×C3—24no—
321C23×C3—22no—
322C23×C3—42no—
323C23×C3—91no—
324C23×C3C2×D3212yes—
325C23×C3C22×C3×D314yes—
326C22×D3—32no—
327C22×D3—61no—
328C3×Q8—94no—
329C3×D4—28no—
330C3×D4—24no—
331C3×D4—44no—
332C3×D4—98no—
333C3×D4#62414yescomplemented
334C3×D4C3×C4×D318yescomplemented
335C2×C3×C4—22no—
336C2×C3×C4—21no—
337C2×C3×C4—32no—
338C2×C3×C4—32no—
339C2×C3×C4—32no—
340C2×C3×C4—34no—
341C2×C3×C4—34no—
342C2×C3×C4—32no—
343C2×C3×C4—41no—
344C2×C3×C4—62no—
345C2×C3×C4—62no—
346C2×C3×C4—64no—
347C2×C3×C4—68no—
348C2×C3×C4—68no—
349C2×C3×C4—91no—
350C2×C3×C4—92no—
351C2×C3×C4—98no—
352C2×C3×C4—94no—
353C2×C3×C4—128no—
354C2×C3×C4—184no—
355C2×C3×C4C2×D3211yescharacteristic
356C2×C3×C4C22×C3×D312yes—
357#67—316no—
358#67—68no—
359C2×Dic3—28no—
360C2×Dic3—32no—
361C2×Dic3—34no—
362C2×Dic3—316no—
363C2×Dic3—68no—
364C2×Dic3—62no—
365C2×Dic3—61no—
366C2×Dic3C22×C3×D314yes—
367C2×Dic3C22×C3×D312yes—
368C2×Dic3C22×C3×D312yes—
369C4×D3—38no—
370C4×D3—64no—
371C4×D3—64no—
372C4×D3—64no—
373Dic6—38no—
374Dic6—64no—
375C2×C32—28no—
376C2×C32—24no—
377C2×C32—22no—
378C2×C32—21no—
379C2×C32—22no—
380C2×C32—22no—
381C2×C32—22no—
382C2×C32—41no—
383C2×C32—42no—
384C2×C32—48no—
385C2×C32—44no—
386C2×C32—84no—
387C2×C32C22×C4×D312yes—
388C2×C32C2×#23712yes—
389C2×C32D3×#4112yes—
390C2×C32C23×C3×C411yescharacteristic
391C2×C32C2×C3×#4112yes—
392C3⋊D3—22no—
393C3×D3—64no—
394C3×D3—122no—
395#41—98no—
396C2×Q8—91no—
397C2×D4—92no—
398C2×D4C3×D3211yescharacteristic
399C22×C4—34no—
400C22×C4—92no—
401C22×C4—92no—
402C22×C4—91no—
403C4⋊C4—32no—
404C4⋊C4—94no—
405C22⋊C4—34no—
406C22⋊C4—92no—
407C42—32no—
408C42—91no—
409C42—91no—
410C42—92no—
411C42—94no—
412C22×C3—24no—
413C22×C3—24no—
414C22×C3—24no—
415C22×C3—21no—
416C22×C3—22no—
417C22×C3—24no—
418C22×C3—28no—
419C22×C3—44no—
420C22×C3—44no—
421C22×C3—44no—
422C22×C3—41no—
423C22×C3—44no—
424C22×C3—84no—
425C22×C3—92no—
426C22×C3—182no—
427C22×C3C2×#62414yes—
428C22×C3C2×C3×C4×D318yes—
429C22×C3C22×D3211yescharacteristic
430C22×C3C23×C3×D312yes—
431C2×D3—34no—
432C2×D3—62no—
433C2×D3—62no—
434C2×D3—62no—
435C2×D3—122no—
436C3×C4—22no—
437C3×C4—24no—
438C3×C4—34no—
439C3×C4—34no—
440C3×C4—42no—
441C3×C4—64no—
442C3×C4—64no—
443C3×C4—62no—
444C3×C4—62no—
445C3×C4—64no—
446C3×C4—92no—
447C3×C4—94no—
448C3×C4—124no—
449C3×C4—122no—
450C3×C4C2×#62412yes—
451C3×C4C2×C3×C4×D314yes—
452Dic3—24no—
453Dic3—22no—
454Dic3—38no—
455Dic3—34no—
456Dic3—64no—
457Dic3—62no—
458Dic3C3×#23714yescomplemented
459C32—22no—
460C32—21no—
461C32—41no—
462C32C4×#23712yescomplemented
463C32C3×C4×#4111yescharacteristic, complemented
464C23—91no—
465C23C2×C3×D3212yes—
466Q8—94no—
467D4—98no—
468D4C3×#62414yescomplemented
469C2×C4—32no—
470C2×C4—32no—
471C2×C4—34no—
472C2×C4—68no—
473C2×C4—91no—
474C2×C4—92no—
475C2×C4—98no—
476C2×C4—94no—
477C2×C4—184no—
478C2×C4C2×C3×D3211yescharacteristic
479C2×C3—28no—
480C2×C3—24no—
481C2×C3—21no—
482C2×C3—22no—
483C2×C3—22no—
484C2×C3—22no—
485C2×C3—22no—
486C2×C3—44no—
487C2×C3—48no—
488C2×C3—41no—
489C2×C3—42no—
490C2×C3—84no—
491C2×C3—182no—
492C2×C3D3×#23712yes—
493C2×C3C22×#62411yescharacteristic
494C2×C3C22×C3×C4×D312yes—
495C2×C3C2×C3×#23712yes—
496C2×C3C3×D3×#4112yes—
497D3—64no—
498D3—122no—
499C22—24no—
500C22—92no—
501C22—182no—
502C22C2×C3×#62414yes—
503C22C22×C3×D3211yescharacteristic
504C4—34no—
505C4—64no—
506C4—62no—
507C4—92no—
508C4—94no—
509C4C2×C3×#62412yes—
510C3—21no—
511C3—22no—
512C3—41no—
513C3#1059519011yescharacteristic, direct product factor
514C3C3×C4×#23712yescomplemented
515C2—24no—
516C2—182no—
517C2C3×D3×#23712yes—
518C2C22×C3×#62411yescharacteristic
subgroupG/H
centerC22×C3C22×D32
commutatorC2×C32C23×C3×C4
FrattiniC22C22×C3×D32
FittingC2×C33×D4C22
socleC22×C33C24
prime radicalC2×C3×#3830C2
cosocleC22×C32C24×C3
hypercenterC2×C3×D4D32
hypocenterC32C3×C4×#41
LCSG ▷ C2×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C2×C3×D4

Complements

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

NH ≅ G/Ntype
2. C32×D4×Dic3C2semidirect
3. C3×#63957C2semidirect
4. C2×C3×Dic32C2semidirect
5. C3×#63952C2semidirect
6. C3×C4×#624C2semidirect
7. C3×#63849C2semidirect
8. C3×#63844C2semidirect
10. #10595190C3direct (N × H)
14. C3×#3830C4semidirect
33. C3×D4×Dic3D3semidirect
34. C3×D4×Dic3C2×C3semidirect
43. #63957C2×C3semidirect
44. C2×Dic32C2×C3semidirect
45. #63952C2×C3semidirect
46. C4×#624C2×C3semidirect
47. #63849C2×C3semidirect
48. #63844C2×C3semidirect
51. C33×D4C2×C4semidirect
53. C3×#638C2×C4semidirect
54. C2×C3×C3⋊Dic3C2×C4semidirect
57. C3×C4×C3⋊D3C2×C4semidirect
58. C3×#634C2×C4semidirect
74. #3830C3×C4semidirect
129. C32×Dic3#41semidirect
142. D4×Dic3C3×D3semidirect
162. C32×D4C4×D3semidirect
163. C32×D4C2×C3×C4semidirect
173. #638C2×C3×C4semidirect
174. C2×C3⋊Dic3C2×C3×C4semidirect
177. C4×C3⋊D3C2×C3×C4semidirect
178. #634C2×C3×C4semidirect
309. C3×Dic3#237semidirect
310. C3×Dic3C3×#41semidirect
319. C33C4×#41semidirect
333. C3×D4#624semidirect
334. C3×D4C3×C4×D3semidirect
458. Dic3C3×#237semidirect
462. C32C4×#237semidirect
463. C32C3×C4×#41semidirect
468. D4C3×#624semidirect
513. C3#10595190direct (N × H)
514. C3C3×C4×#237semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#38302
C32×D4×Dic32
C3×#639572
C2×C3×Dic322
C3×#639522
C3×C4×#6242
C3×#638492
C3×#638442
C3×C4×#2373
#105951903