Finite Group Databasebeta

← C3×#10594527 #13173945

Subgroups — C3×#10594527 #13173945

Order 1728.

1532 conjugacy classes of subgroups.

The subgroup lattice has 552 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#3795C212yesmaximal, complemented
2C3×#64096C211yesmaximal, characteristic, complemented
3C32×#895C212yesmaximal, complemented
4C3×#63914C211yesmaximal, characteristic, complemented
5C3×#63914C212yesmaximal, complemented
6C3×#63902C212yesmaximal, complemented
7C3×C4×#625C211yesmaximal, characteristic, complemented
8C3×#63885C214yesmaximal, complemented
9C3×#5599—32nomaximal
10#10594527C311yesmaximal, characteristic, direct product factor
11C2×C3×#634C2211yescharacteristic
12C2×C32×D12C2212yes—
13C2×C32×D12C2214yes—
14C2×C32×C4×D3C2212yes—
15C2×C3×#625C2214yes—
16C2×C3×#625C2211yescharacteristic
17C2×C3×#625C2212yes—
18C2×C3×#625C2212yes—
19C2×C3×D3×Dic3C2214yes—
20C3×#3795—28no—
21C33×C4⋊C4C2211yescharacteristic
22C3×#3750C2212yes—
23C3×C4×C3⋊Dic3C2211yescharacteristic
24C3×C4×C3⋊Dic3C2212yes—
25C32×(C2×D3)⋊C4C2214yes—
26C3×#3724C2211yescharacteristic
27C3×(C2×D3)⋊Dic3C2212yes—
28C2×C3×#237—32no—
29C2×C3×#235—32no—
30C2×#3795C2×C312yescomplemented
31C32×#121—32no—
32#64096C2×C311yescharacteristic, complemented
33C3×#940—34no—
34C3×#940—32no—
35C3×#939—34no—
36C3×C4×#67—32no—
37C3×#895—22no—
38C3×#895D312yescomplemented
39C3×#895C2×C312yescomplemented
40C3×#887—32no—
41C3×#887—61no—
42C3×#884—34no—
43C3×#67⋊C4—34no—
44#63914C2×C311yescharacteristic, complemented
45#63914C2×C312yescomplemented
46#63902C2×C312yescomplemented
47C4×#625C2×C311yescharacteristic, complemented
48#63885C2×C314yescomplemented
49C22×C32×D3C2314yes—
50C22×C32×D3C2312yes—
51C2×C33×C4C2312yes—
52C2×C33×C4C2311yescharacteristic
53C2×C3×C3⋊Dic3C2311yescharacteristic
54C2×C3×C3⋊Dic3C2312yes—
55C2×C3×C3⋊Dic3C2311yescharacteristic
56C3×#634D414yescomplemented
57C2×C32×Dic3C2312yes—
58C32×D12—24no—
59C32×D12—44no—
60C32×C4×D3—24no—
61C3×#625—28no—
62C3×#625—22no—
63C3×#625—24no—
64C3×#625—42no—
65C3×D3×Dic3—28no—
66C3×#550—91no—
67#5599—32no—
68C2×C32×D4—32no—
69C2×C32×D4—34no—
70C22×C32×C4—32no—
71C2×#634C22×C311yescharacteristic
72C2×C3×#67—34no—
73C2×C3×#67—34no—
74C2×C3×#67—32no—
75C2×C3×#67—34no—
76C2×C3×#67—34no—
77C22×C3×Dic3—34no—
78C3×#237—68no—
79C3×#235—68no—
80C2×C3×D12—22no—
81C2×C3×D12—24no—
82C2×C3×D12C2×D312yes—
83C2×C3×D12C2×D314yes—
84C2×C3×D12C22×C312yes—
85C2×C3×D12C22×C314yes—
86C2×C3×C4×D3—22no—
87C2×C3×C4×D3—34no—
88C2×C3×C4×D3C2×D312yes—
89C2×C3×C4×D3C22×C312yes—
90C2×C3×Dic6—32no—
91C2×C3×Dic6—61no—
92C2×#625C22×C314yes—
93C2×#625C22×C311yescharacteristic
94C2×#625C22×C312yes—
95C2×#625C22×C312yes—
96C2×D3×Dic3C22×C314yes—
97#3795—28no—
98C32×C4⋊C4—21no—
99C32×C4⋊C4—22no—
100C32×C4⋊C4—41no—
101C32×C4⋊C4C2×D312yes—
102C32×C4⋊C4C22×C311yescharacteristic
103C32×C22⋊C4—34no—
104#3750C22×C312yes—
105C4×C3⋊Dic3C22×C311yescharacteristic
106C4×C3⋊Dic3C22×C312yes—
107C3×#217—32no—
108C3×(C2×D3)⋊C4—24no—
109C3×(C2×D3)⋊C4—32no—
110C3×(C2×D3)⋊C4C2×D314yes—
111C3×(C2×D3)⋊C4C22×C314yes—
112C3×Dic3⋊C4—34no—
113C3×Dic3⋊C4—32no—
114C3×Dic3⋊C4—62no—
115C3×C4×Dic3—34no—
116C3×C4×Dic3—32no—
117C3×C4×Dic3—62no—
118C3×C4×Dic3—61no—
119#3724C22×C311yescharacteristic
120(C2×D3)⋊Dic3C22×C312yes—
121C22×C33C2411yescharacteristic
122C2×C32×D3—24no—
123C2×C32×D3—22no—
124C2×C32×D3—22no—
125C2×C32×D3—24no—
126C2×C32×D3—22no—
127C2×C32×D3—24no—
128C33×C4—22no—
129C33×C4C2×D412yes—
130C3×C3⋊Dic3—21no—
131C3×C3⋊Dic3C2×D414yes—
132C3×C3⋊Dic3#4112yescomplemented
133C32×Dic3—22no—
134C2×C3×#41—92no—
135C2×#237—32no—
136C2×#235—32no—
137C3×#127—91no—
138C3×#127—92no—
139C3×#121—32no—
140C3×#121—32no—
141C3×#121—62no—
142C3×#121—94no—
143C3×C4×Q8—91no—
144C3×C4×D4—92no—
145C3×C4×D4—91no—
146#940—34no—
147#940—32no—
148#939—34no—
149C4×#67—32no—
150#895C3×D312yescomplemented
151#887—32no—
152#887—61no—
153#884—34no—
154#67⋊C4—34no—
155C23×C32—32no—
156C23×C32—34no—
157C22×C3×D3—22no—
158C22×C3×D3—24no—
159C22×C3×D3C22×D312yes—
160C22×C3×D3C22×D314yes—
161C22×C3×D3C23×C312yes—
162C22×C3×D3C23×C314yes—
163C32×D4—64no—
164C32×D4—124no—
165C2×C32×C4—21no—
166C2×C32×C4—22no—
167C2×C32×C4—22no—
168C2×C32×C4—24no—
169C2×C32×C4—32no—
170C2×C32×C4—41no—
171C2×C32×C4—42no—
172C2×C32×C4—64no—
173C2×C32×C4C22×D314yes—
174C2×C32×C4C22×D312yes—
175C2×C32×C4C23×C312yes—
176C2×C32×C4C23×C311yescharacteristic
177C2×C3⋊Dic3C23×C311yescharacteristic
178C2×C3⋊Dic3C23×C312yes—
179C2×C3⋊Dic3C23×C311yescharacteristic
180#634C3×D414yescomplemented
181C3×#67—68no—
182C3×#67—68no—
183C3×#67—68no—
184C3×#67—64no—
185C3×#67—124no—
186C2×C3×Dic3—22no—
187C2×C3×Dic3—32no—
188C2×C3×Dic3—34no—
189C2×C3×Dic3—32no—
190C2×C3×Dic3—61no—
191C2×C3×Dic3—62no—
192C2×C3×Dic3—61no—
193C2×C3×Dic3—68no—
194C2×C3×Dic3C22×D312yes—
195C2×C3×Dic3C23×C312yes—
196C3×D12—24no—
197C3×D12—24no—
198C3×D12—44no—
199C3×D12—44no—
200C3×D12—44no—
201C3×D12—48no—
202C3×C4×D3—24no—
203C3×C4×D3—24no—
204C3×C4×D3—44no—
205C3×C4×D3—68no—
206C3×Dic6—38no—
207C3×Dic6—64no—
208#625—28no—
209#625—22no—
210#625—24no—
211#625—42no—
212D3×Dic3—28no—
213#550—91noSylow
214C2×C33C22×D411yescharacteristic
215C2×C33C2×#4111yescharacteristic
216C2×C33H(5,𝔽2)11yescharacteristic
217C32×D3—42no—
218C32×D3—44no—
219C3×#41—188no—
220C2×C3×Q8—91no—
221C2×C3×D4—32no—
222C2×C3×D4—34no—
223C2×C3×D4—32no—
224C2×C3×D4—34no—
225C2×C3×D4—62no—
226C2×C3×D4—64no—
227C2×C3×D4—92no—
228C2×C3×D4—91no—
229C2×C3×D4—94no—
230C2×C3×D4—92no—
231C22×C3×C4—32no—
232C22×C3×C4—32no—
233C22×C3×C4—62no—
234C22×C3×C4—94no—
235C2×#67—34no—
236C2×#67—34no—
237C2×#67—32no—
238C2×#67—34no—
239C2×#67—34no—
240C22×Dic3—34no—
241#237—68no—
242#235—68no—
243C2×D12C2×C3×D312yes—
244C2×D12C2×C3×D314yes—
245C2×C4×D3—34no—
246C2×C4×D3C2×C3×D312yes—
247C2×Dic6—32no—
248C2×Dic6—61no—
249C3×C4⋊C4—21no—
250C3×C4⋊C4—22no—
251C3×C4⋊C4—41no—
252C3×C4⋊C4—91no—
253C3×C4⋊C4—92no—
254C3×C4⋊C4D3211yescharacteristic
255C3×C4⋊C4C2×C3×D312yes—
256C3×C22⋊C4—34no—
257C3×C22⋊C4—34no—
258C3×C22⋊C4—64no—
259C3×C22⋊C4—92no—
260C3×C42—92no—
261C3×C42—91no—
262#217—32no—
263(C2×D3)⋊C4—32no—
264(C2×D3)⋊C4C2×C3×D314yes—
265Dic3⋊C4—34no—
266Dic3⋊C4—32no—
267Dic3⋊C4—62no—
268C4×Dic3—34no—
269C4×Dic3—32no—
270C4×Dic3—62no—
271C4×Dic3—61no—
272C22×C32—21no—
273C22×C32—22no—
274C22×C32—41no—
275C22×C32—62no—
276C22×C32—64no—
277C22×C32—62no—
278C22×C32—64no—
279C22×C32—62no—
280C22×C32—64no—
281C22×C32C23×D312yes—
282C22×C32C24×C311yescharacteristic
283C2×C3×D3—24no—
284C2×C3×D3—22no—
285C2×C3×D3—24no—
286C2×C3×D3—22no—
287C2×C3×D3—24no—
288C2×C3×D3—22no—
289C2×C3×D3—24no—
290C2×C3×D3—24no—
291C2×C3×D3—22no—
292C2×C3×D3—22no—
293C2×C3×D3—24no—
294C2×C3×D3—24no—
295C2×C3×D3—22no—
296C2×C3×D3—42no—
297C2×C3×D3—42no—
298C2×C3×D3—44no—
299C2×C3×D3—44no—
300C2×C3×D3—44no—
301C32×C4—24no—
302C32×C4—22no—
303C32×C4—22no—
304C32×C4—24no—
305C32×C4—42no—
306C32×C4—44no—
307C32×C4—42no—
308C32×C4—62no—
309C32×C4—82no—
310C32×C4D3×D414yes—
311C32×C4C2×C3×D412yes—
312C3⋊Dic3—21no—
313C3⋊Dic3C2×C3×D414yes—
314C3⋊Dic3C3×#4112yescomplemented
315C3×Dic3—22no—
316C3×Dic3—22no—
317C3×Dic3—38no—
318C3×Dic3—34no—
319C3×Dic3—42no—
320C3×Dic3—64no—
321C3×Dic3—62no—
322C3×Dic3—62no—
323C3×Dic3—121no—
324C2×#41—92no—
325#127—91no—
326#127—92no—
327#121—32no—
328#121—94no—
329C4×Q8—91no—
330C4×D4—92no—
331C4×D4—91no—
332C33#55011yescharacteristic, complemented, Sylow
333C23×C3—34no—
334C23×C3—32no—
335C23×C3—32no—
336C23×C3—34no—
337C23×C3—62no—
338C23×C3—64no—
339C22×D3C22×C3×D312yes—
340C22×D3C22×C3×D314yes—
341C3×Q8—94no—
342C3×D4—64no—
343C3×D4—64no—
344C3×D4—124no—
345C3×D4—124no—
346C3×D4—124no—
347C3×D4—128no—
348C3×D4—184no—
349C3×D4—182no—
350C3×D4—188no—
351C3×D4—362no—
352C2×C3×C4—22no—
353C2×C3×C4—21no—
354C2×C3×C4—24no—
355C2×C3×C4—22no—
356C2×C3×C4—32no—
357C2×C3×C4—32no—
358C2×C3×C4—41no—
359C2×C3×C4—42no—
360C2×C3×C4—62no—
361C2×C3×C4—64no—
362C2×C3×C4—64no—
363C2×C3×C4—91no—
364C2×C3×C4—92no—
365C2×C3×C4—91no—
366C2×C3×C4—124no—
367C2×C3×C4—188no—
368C2×C3×C4C2×D3212yes—
369C2×C3×C4C2×D3211yescharacteristic
370C2×C3×C4C22×C3×D314yes—
371C2×C3×C4C22×C3×D312yes—
372#67—68no—
373#67—68no—
374#67—68no—
375#67—64no—
376#67—124no—
377C2×Dic3—32no—
378C2×Dic3—34no—
379C2×Dic3—32no—
380C2×Dic3—61no—
381C2×Dic3—62no—
382C2×Dic3—61no—
383C2×Dic3—68no—
384C2×Dic3C22×C3×D312yes—
385D12—24no—
386D12—44no—
387C4×D3—24no—
388C4×D3—68no—
389Dic6—38no—
390Dic6—64no—
391C2×C32—21no—
392C2×C32—21no—
393C2×C32—21no—
394C2×C32—22no—
395C2×C32—22no—
396C2×C32—22no—
397C2×C32—41no—
398C2×C32—41no—
399C2×C32—41no—
400C2×C32—122no—
401C2×C32—124no—
402C2×C32C2×D3×D412yes—
403C2×C32#100412yes—
404C2×C32D3×#4112yes—
405C2×C32C22×C3×D411yescharacteristic
406C2×C32C2×C3×#4111yescharacteristic
407C2×C32C3×H(5,𝔽2)11yescharacteristic
408C3×D3—44no—
409C3×D3—42no—
410C3×D3—44no—
411C3×D3—42no—
412C3×D3—44no—
413C3×D3—84no—
414#41—188no—
415C2×Q8—91no—
416C2×D4—32no—
417C2×D4—34no—
418C2×D4—92no—
419C2×D4—91no—
420C2×D4—94no—
421C2×D4—92no—
422C22×C4—32no—
423C22×C4—94no—
424C4⋊C4—91no—
425C4⋊C4—92no—
426C4⋊C4C3×D3211yescharacteristic
427C22⋊C4—34no—
428C22⋊C4—92no—
429C42—92no—
430C42—91no—
431C22×C3—21no—
432C22×C3—22no—
433C22×C3—41no—
434C22×C3—64no—
435C22×C3—62no—
436C22×C3—64no—
437C22×C3—62no—
438C22×C3—64no—
439C22×C3—64no—
440C22×C3—62no—
441C22×C3—64no—
442C22×C3—62no—
443C22×C3—64no—
444C22×C3—62no—
445C22×C3—62no—
446C22×C3—64no—
447C22×C3—122no—
448C22×C3—122no—
449C22×C3—124no—
450C22×C3—124no—
451C22×C3—124no—
452C22×C3C22×D3211yescharacteristic
453C22×C3C23×C3×D312yes—
454C2×D3—22no—
455C2×D3—24no—
456C2×D3—22no—
457C2×D3—24no—
458C2×D3—22no—
459C2×D3—24no—
460C3×C4—22no—
461C3×C4—24no—
462C3×C4—22no—
463C3×C4—24no—
464C3×C4—44no—
465C3×C4—42no—
466C3×C4—42no—
467C3×C4—62no—
468C3×C4—62no—
469C3×C4—82no—
470C3×C4—94no—
471C3×C4—92no—
472C3×C4—122no—
473C3×C4—181no—
474C3×C4#380212yes—
475C3×C4C3×D3×D414yes—
476Dic3—22no—
477Dic3—38no—
478Dic3—34no—
479Dic3—64no—
480Dic3—62no—
481Dic3—62no—
482Dic3—121no—
483C32—22no—
484C32—21no—
485C32—41no—
486C32#559912yescomplemented
487C32C3×#55011yescharacteristic, complemented
488C23—34no—
489C23—32no—
490Q8—94no—
491D4—64no—
492D4—124no—
493D4—188no—
494D4—182no—
495D4—184no—
496D4—362no—
497C2×C4—32no—
498C2×C4—64no—
499C2×C4—91no—
500C2×C4—92no—
501C2×C4—91no—
502C2×C4—188no—
503C2×C4C2×C3×D3212yes—
504C2×C4C2×C3×D3211yescharacteristic
505C2×C3—21no—
506C2×C3—22no—
507C2×C3—21no—
508C2×C3—22no—
509C2×C3—21no—
510C2×C3—22no—
511C2×C3—41no—
512C2×C3—41no—
513C2×C3—41no—
514C2×C3—124no—
515C2×C3—122no—
516C2×C3—124no—
517C2×C3—122no—
518C2×C3—124no—
519C2×C3—244no—
520C2×C3C2×#380211yescharacteristic
521C2×C3#6431311yescharacteristic
522C2×C3#6431911yescharacteristic
523C2×C3C2×C3×D3×D412yes—
524C2×C3C3×#100412yes—
525C2×C3C3×D3×#4112yes—
526D3—42no—
527D3—44no—
528C22—62no—
529C22—64no—
530C22—62no—
531C22—64no—
532C22—62no—
533C22—64no—
534C22C22×C3×D3211yescharacteristic
535C4—22no—
536C4—62no—
537C4—92no—
538C4—94no—
539C4—181no—
540C4C3×#380212yes—
541C3—22no—
542C3—21no—
543C3—41no—
544C3#1059452711yescharacteristic, direct product factor
545C3C3×#559912yescomplemented
546C2—122no—
547C2—124no—
548C2C2×C3×#380211yescharacteristic
549C2C3×#6431311yescharacteristic
550C2C3×#6431911yescharacteristic
subgroupG/H
centerC22×C3C22×D32
commutator, cosocleC22×C32C24×C3
FrattiniC22C22×C3×D32
FittingC33×C4⋊C4C22
socleC22×C33C24
hypercenterC3×C4⋊C4D32
hypocenterC32C3×#550
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C3×C4⋊C4

Complements

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

NH ≅ G/Ntype
1. C2×C3×#3795C2semidirect
2. C3×#64096C2semidirect
3. C32×#895C2semidirect
4. C3×#63914C2semidirect
5. C3×#63914C2semidirect
6. C3×#63902C2semidirect
7. C3×C4×#625C2semidirect
8. C3×#63885C2semidirect
10. #10594527C3direct (N × H)
30. C2×#3795C2×C3semidirect
32. #64096C2×C3semidirect
38. C3×#895D3semidirect
39. C3×#895C2×C3semidirect
44. #63914C2×C3semidirect
45. #63914C2×C3semidirect
46. #63902C2×C3semidirect
47. C4×#625C2×C3semidirect
48. #63885C2×C3semidirect
56. C3×#634D4semidirect
132. C3×C3⋊Dic3#41semidirect
150. #895C3×D3semidirect
180. #634C3×D4semidirect
314. C3⋊Dic3C3×#41semidirect
332. C33#550semidirect
486. C32#5599semidirect
487. C32C3×#550semidirect
544. C3#10594527direct (N × H)
545. C3C3×#5599semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#37952
C3×#640962
C32×#8952
C3×#639142
C3×#639022
C3×C4×#6252
C3×#638852
C3×#55993
#105945273