Finite Group Databasebeta

← C3×#10590999 #13158001

Subgroups — C3×#10590999 #13158001

Order 1728.

558 conjugacy classes of subgroups.

The subgroup lattice has 532 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C32×#907C211yesmaximal, characteristic
2C2×C3×#3719C211yesmaximal, characteristic, complemented
3C3×#63831C211yesmaximal, characteristic
4C3×#63827C211yesmaximal, characteristic
5C3×#63651C211yesmaximal, characteristic, complemented
6C32×#824C211yesmaximal, characteristic, complemented
7C3×#63577C211yesmaximal, characteristic, complemented
8C3×#5143—31nomaximal
9C3×#4755—31nomaximal
10#10590999C311yesmaximal, characteristic, direct product factor
11C2×C3×#634C2211yescharacteristic
12C32×#235C2211yescharacteristic
13C2×C32×Dic6C2211yescharacteristic
14C33×M16C2211yescharacteristic
15C3×#3748C2211yescharacteristic
16C32×#208C2211yescharacteristic
17C2×C32×C3⋊C8C2211yescharacteristic
18C32×ΓL(1,𝔽25)—21no—
19C32×D3×C8—21no—
20C3×#3719—21no—
21C3×#3719—21no—
22C3×#3719—21no—
23C3×#3718—21no—
24C3×#3716—21no—
25C3×#3715—21no—
26C32×#131—31no—
27C3×#951—31no—
28C3×#951—31no—
29C2×C3×#216—31no—
30C3×#942—31no—
31C3×#909—31no—
32C3×#907—21no—
33C3×#907D311yescharacteristic
34C3×#907C2×C311yescharacteristic
35C2×C3×Dic12—31no—
36C2×#3719C2×C311yescharacteristic, complemented
37#63831C2×C311yescharacteristic
38#63827C2×C311yescharacteristic
39C32×#101—31no—
40#63651C2×C311yescharacteristic, complemented
41C3×#824—21no—
42C3×#824—31no—
43C3×#824—31no—
44C3×#824—61no—
45C3×#824D311yescharacteristic, complemented
46C3×#824C2×C311yescharacteristic, complemented
47C3×#822—31no—
48C3×#819—31no—
49#63577C2×C311yescharacteristic, complemented
50C2×C33×C4C2311yescharacteristic
51C2×C3×C3⋊Dic3—21no—
52C3×#634—21no—
53C3×#634—21no—
54C32×#67—21no—
55C2×C32×Dic3—21no—
56C32×D12D411yescharacteristic
57C32×C4×D3—21no—
58C32×Dic6—21no—
59C32×Dic6—21no—
60C32×Dic6D411yescharacteristic
61C33×C8—21no—
62C3×C3⋊(C3⋊C8)—21no—
63C32×C3⋊C8—21no—
64C32×C3⋊C8D411yescharacteristic
65C32×C3⋊C8D411yescharacteristic
66C3×#473—91no—
67#5143—31no—
68#4755—31no—
69C32×#41—31no—
70C2×C32×Q8—31no—
71C2×#634C22×C311yescharacteristic
72C3×#235—21no—
73C3×#235C2×D311yescharacteristic
74C3×#235C22×C311yescharacteristic
75C2×C3×Dic6—21no—
76C2×C3×Dic6—31no—
77C2×C3×Dic6—31no—
78C2×C3×Dic6—61no—
79C2×C3×Dic6C2×D311yescharacteristic
80C2×C3×Dic6C22×C311yescharacteristic
81C32×M16—21no—
82C32×M16—21no—
83C32×M16—21no—
84C32×M16—31no—
85C32×M16—41no—
86C32×M16—61no—
87C32×M16C2×D311yescharacteristic
88C32×M16C2×D311yescharacteristic
89C32×M16C22×C311yescharacteristic
90C2×C32×C8—31no—
91C2×C32×C8—61no—
92#3748C22×C311yescharacteristic
93C3×#216—61no—
94C3×#216—61no—
95C3×#216—61no—
96C3×#216—61no—
97C3×#216—61no—
98C3×#215—61no—
99C3×#214—61no—
100C3×#214—61no—
101C3×#208—21no—
102C3×#208—31no—
103C3×#208—31no—
104C3×#208—61no—
105C3×#208C2×D311yescharacteristic
106C3×#208C22×C311yescharacteristic
107C2×C3×C3⋊C8—21no—
108C2×C3×C3⋊C8C2×D311yescharacteristic
109C2×C3×C3⋊C8C22×C311yescharacteristic
110C3×Dic12—61no—
111C3×Dic12—61no—
112C3×Dic12—61no—
113C3×#204—61no—
114C3×ΓL(1,𝔽25)—22no—
115C3×ΓL(1,𝔽25)—21no—
116C3×ΓL(1,𝔽25)—21no—
117C3×D3×C8—22no—
118C3×D3×C8—21no—
119C3×D3×C8—21no—
120#3719—21no—
121#3719—21no—
122#3719—21no—
123#3718—21no—
124#3716—21no—
125#3715—21no—
126C22×C33#4111yescharacteristic
127C2×C32×D3—21no—
128C33×C4C2×D411yescharacteristic
129C33×C4C2×D411yescharacteristic
130C3×C3⋊Dic3—41no—
131C32×Dic3—21no—
132C32×Dic3—41no—
133C3×#137—91no—
134C3×#137—91no—
135C2×C3×Q16—91no—
136C3×#131—31no—
137C3×#131—31no—
138C3×#131—61no—
139#951—31no—
140#951—31no—
141C2×#216—31no—
142#942—31no—
143#909—31no—
144#907C3×D311yescharacteristic
145C2×Dic12—31no—
146C3×#108—91no—
147C3×#101—31no—
148C3×#101—31no—
149C3×#101—61no—
150C3×#101—91no—
151#824—31no—
152#824—31no—
153#824—61no—
154#824C3×D311yescharacteristic, complemented
155#822—31no—
156#819—31no—
157C32×Q8—31no—
158C32×Q8—61no—
159C32×Q8—61no—
160C32×D4—31no—
161C32×D4—61no—
162C2×C32×C4—21no—
163C2×C32×C4—21no—
164C2×C32×C4—21no—
165C2×C32×C4—41no—
166C2×C32×C4—61no—
167C2×C32×C4—61no—
168C2×C32×C4C22×D311yescharacteristic
169C2×C32×C4C22×D311yescharacteristic
170C2×C32×C4C23×C311yescharacteristic
171C2×C3⋊Dic3—21no—
172#634—21no—
173#634—21no—
174C3×#67—22no—
175C3×#67—21no—
176C3×#67—21no—
177C2×C3×Dic3—22no—
178C2×C3×Dic3—21no—
179C2×C3×Dic3—21no—
180C2×C3×Dic3—62no—
181C2×C3×Dic3—61no—
182C2×C3×Dic3—61no—
183C3×D12—21no—
184C3×D12D1211yescharacteristic
185C3×D12C3×D411yescharacteristic
186C3×C4×D3—22no—
187C3×C4×D3—21no—
188C3×C4×D3—21no—
189C3×Dic6—21no—
190C3×Dic6—21no—
191C3×Dic6—21no—
192C3×Dic6—21no—
193C3×Dic6—21no—
194C3×Dic6—41no—
195C3×Dic6—41no—
196C3×Dic6—61no—
197C3×Dic6—61no—
198C3×Dic6—61no—
199C3×Dic6—61no—
200C3×Dic6—121no—
201C3×Dic6—121no—
202C3×Dic6D1211yescharacteristic
203C3×Dic6C3×D411yescharacteristic
204C32×C8—21no—
205C32×C8—22no—
206C32×C8—21no—
207C32×C8—21no—
208C32×C8—31no—
209C32×C8—31no—
210C32×C8—41no—
211C32×C8—41no—
212C32×C8—42no—
213C32×C8—61no—
214C3⋊(C3⋊C8)—21no—
215C3×C3⋊C8—21no—
216C3×C3⋊C8—21no—
217C3×C3⋊C8—21no—
218C3×C3⋊C8—21no—
219C3×C3⋊C8—41no—
220C3×C3⋊C8—61no—
221C3×C3⋊C8—61no—
222C3×C3⋊C8—121no—
223C3×C3⋊C8#6711yescharacteristic
224C3×C3⋊C8#6711yescharacteristic
225C3×C3⋊C8C3×D411yescharacteristic
226C3×C3⋊C8C3×D411yescharacteristic
227#473—91noSylow
228C2×C33—21no—
229C2×C33#12111yescharacteristic
230C32×D3—41no—
231C3×#41—31no—
232C3×#41—31no—
233C3×#41—61no—
234C2×C3×Q8—31no—
235C2×C3×Q8—31no—
236C2×C3×Q8—61no—
237C2×C3×Q8—91no—
238#235C2×C3×D311yescharacteristic
239C2×Dic6—31no—
240C2×Dic6—31no—
241C2×Dic6—61no—
242C2×Dic6C2×C3×D311yescharacteristic
243C3×Q16—181no—
244C3×Q16—181no—
245C3×Q16—181no—
246C3×Q16—181no—
247C3×SD16—181no—
248C3×SD16—181no—
249C3×M16—21no—
250C3×M16—21no—
251C3×M16—21no—
252C3×M16—31no—
253C3×M16—31no—
254C3×M16—41no—
255C3×M16—61no—
256C3×M16—61no—
257C3×M16—61no—
258C3×M16—62no—
259C3×M16—91no—
260C3×M16D3211yescharacteristic
261C3×M16C2×C3×D311yescharacteristic
262C3×M16C2×C3×D311yescharacteristic
263C2×C3×C8—31no—
264C2×C3×C8—31no—
265C2×C3×C8—61no—
266C2×C3×C8—61no—
267C2×C3×C8—61no—
268C2×C3×C8—62no—
269#216—61no—
270#216—61no—
271#216—61no—
272#216—61no—
273#216—61no—
274#215—61no—
275#214—61no—
276#214—61no—
277#208—31no—
278#208—31no—
279#208—61no—
280#208C2×C3×D311yescharacteristic
281C2×C3⋊C8C2×C3×D311yescharacteristic
282Dic12—61no—
283Dic12—61no—
284Dic12—61no—
285#204—61no—
286ΓL(1,𝔽25)—21no—
287D3×C8—21no—
288C22×C32—21no—
289C22×C32—21no—
290C22×C32—21no—
291C22×C32—41no—
292C22×C32—61no—
293C22×C32#23511yescharacteristic
294C22×C32#23511yescharacteristic
295C22×C32C3×#4111yescharacteristic
296C2×C3×D3—22no—
297C2×C3×D3—21no—
298C2×C3×D3—21no—
299C32×C4—21no—
300C32×C4—21no—
301C32×C4—21no—
302C32×C4—21no—
303C32×C4—21no—
304C32×C4—21no—
305C32×C4—41no—
306C32×C4—41no—
307C32×C4—61no—
308C32×C4—121no—
309C32×C4C2×D1211yescharacteristic
310C32×C4D3×D411yescharacteristic
311C32×C4D3×D411yescharacteristic
312C32×C4C2×#6711yescharacteristic
313C32×C4C2×C3×D411yescharacteristic
314C32×C4C2×C3×D411yescharacteristic
315C3⋊Dic3—41no—
316C3×Dic3—22no—
317C3×Dic3—21no—
318C3×Dic3—21no—
319C3×Dic3—42no—
320C3×Dic3—41no—
321C3×Dic3—41no—
322C3×Dic3—122no—
323C3×Dic3—121no—
324C3×Dic3—121no—
325#137—91no—
326#137—91no—
327C2×Q16—91no—
328#131—31no—
329#108—91no—
330#101—31no—
331#101—91no—
332C33#47311yescharacteristic, complemented, Sylow
333C3×Q8—31no—
334C3×Q8—31no—
335C3×Q8—61no—
336C3×Q8—61no—
337C3×Q8—61no—
338C3×Q8—61no—
339C3×Q8—61no—
340C3×Q8—121no—
341C3×Q8—121no—
342C3×Q8—181no—
343C3×Q8—181no—
344C3×D4—31no—
345C3×D4—31no—
346C3×D4—61no—
347C3×D4—62no—
348C3×D4—61no—
349C3×D4—61no—
350C2×C3×C4—21no—
351C2×C3×C4—21no—
352C2×C3×C4—21no—
353C2×C3×C4—41no—
354C2×C3×C4—62no—
355C2×C3×C4—61no—
356C2×C3×C4—61no—
357C2×C3×C4—62no—
358C2×C3×C4—61no—
359C2×C3×C4—61no—
360C2×C3×C4—181no—
361C2×C3×C4C2×D3211yescharacteristic
362C2×C3×C4C22×C3×D311yescharacteristic
363C2×C3×C4C22×C3×D311yescharacteristic
364#67—21no—
365C2×Dic3—21no—
366C2×Dic3—62no—
367C2×Dic3—61no—
368C2×Dic3—61no—
369D12C3×D1211yescharacteristic
370C4×D3—21no—
371Dic6—21no—
372Dic6—21no—
373Dic6—61no—
374Dic6—61no—
375Dic6—61no—
376Dic6—61no—
377Dic6—121no—
378Dic6—121no—
379Dic6C3×D1211yescharacteristic
380C3×C8—22no—
381C3×C8—21no—
382C3×C8—21no—
383C3×C8—21no—
384C3×C8—31no—
385C3×C8—31no—
386C3×C8—31no—
387C3×C8—31no—
388C3×C8—41no—
389C3×C8—41no—
390C3×C8—42no—
391C3×C8—61no—
392C3×C8—61no—
393C3×C8—61no—
394C3×C8—61no—
395C3×C8—121no—
396C3×C8—181no—
397C3⋊C8—21no—
398C3⋊C8—61no—
399C3⋊C8—61no—
400C3⋊C8—121no—
401C3⋊C8C3×#6711yescharacteristic
402C3⋊C8C3×#6711yescharacteristic
403C2×C32—21no—
404C2×C32—21no—
405C2×C32—21no—
406C2×C32—21no—
407C2×C32—21no—
408C2×C32—21no—
409C2×C32—41no—
410C2×C32—41no—
411C2×C32—41no—
412C2×C32—41no—
413C2×C32—81no—
414C2×C32—121no—
415C2×C32#88411yescharacteristic
416C2×C32#93011yescharacteristic
417C2×C32C3×#12111yescharacteristic
418C3×D3—42no—
419C3×D3—41no—
420C3×D3—41no—
421#41—31no—
422C2×Q8—31no—
423C2×Q8—91no—
424Q16—181no—
425Q16—181no—
426Q16—181no—
427Q16—181no—
428SD16—181no—
429SD16—181no—
430M16—31no—
431M16—61no—
432M16—91no—
433M16C3×D3211yescharacteristic
434C2×C8—31no—
435C2×C8—61no—
436C22×C3—21no—
437C22×C3—21no—
438C22×C3—21no—
439C22×C3—41no—
440C22×C3—62no—
441C22×C3—61no—
442C22×C3—61no—
443C22×C3#379811yescharacteristic
444C22×C3C3×#23511yescharacteristic
445C22×C3C3×#23511yescharacteristic
446C2×D3—21no—
447C3×C4—21no—
448C3×C4—21no—
449C3×C4—21no—
450C3×C4—21no—
451C3×C4—21no—
452C3×C4—21no—
453C3×C4—41no—
454C3×C4—41no—
455C3×C4—62no—
456C3×C4—61no—
457C3×C4—61no—
458C3×C4—122no—
459C3×C4—121no—
460C3×C4—121no—
461C3×C4—361no—
462C3×C4D3×D1211yescharacteristic
463C3×C4D3×#6711yescharacteristic
464C3×C4C2×C3×D1211yescharacteristic
465C3×C4C3×D3×D411yescharacteristic
466C3×C4C3×D3×D411yescharacteristic
467C3×C4C2×C3×#6711yescharacteristic
468Dic3—21no—
469Dic3—41no—
470Dic3—122no—
471Dic3—121no—
472Dic3—121no—
473C32—21no—
474C32—21no—
475C32—21no—
476C32—41no—
477C32#475511yescharacteristic, complemented
478C32#514311yescharacteristic, complemented
479C32C3×#47311yescharacteristic, complemented
480Q8—31no—
481Q8—61no—
482Q8—61no—
483Q8—181no—
484Q8—181no—
485D4—31no—
486D4—61no—
487C2×C4—61no—
488C2×C4—61no—
489C2×C4—181no—
490C2×C4C2×C3×D3211yescharacteristic
491C8—21no—
492C8—31no—
493C8—31no—
494C8—61no—
495C8—181no—
496C2×C3—21no—
497C2×C3—21no—
498C2×C3—21no—
499C2×C3—21no—
500C2×C3—21no—
501C2×C3—21no—
502C2×C3—41no—
503C2×C3—41no—
504C2×C3—41no—
505C2×C3—41no—
506C2×C3—81no—
507C2×C3—122no—
508C2×C3—121no—
509C2×C3—121no—
510C2×C3#6390611yescharacteristic
511C2×C3C3×#88411yescharacteristic
512C2×C3C3×#93011yescharacteristic
513D3—41no—
514C22—61no—
515C22C3×#379811yescharacteristic
516C4—61no—
517C4—121no—
518C4—361no—
519C4C3×D3×D1211yescharacteristic
520C4C3×D3×#6711yescharacteristic
521C3—21no—
522C3—21no—
523C3—21no—
524C3—41no—
525C3#1059099911yescharacteristic, direct product factor
526C3C3×#475511yescharacteristic, complemented
527C3C3×#514311yescharacteristic, complemented
528C2—21no—
529C2—121no—
530C2C3×#6390611yescharacteristic
subgroupG/H
centerC2×C3#63906
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C4C2×C3×D32
FittingC33×M16C22
socleC2×C33#121
prime radicalC32×#235C22
hypercenterC3×M16D32
hypocenterC32C3×#473
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C2×C3 ◁ C2×C3×C4 ◁ C3×M16

Complements

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

NH ≅ G/Ntype
2. C2×C3×#3719C2semidirect
5. C3×#63651C2semidirect
6. C32×#824C2semidirect
7. C3×#63577C2semidirect
10. #10590999C3direct (N × H)
36. C2×#3719C2×C3semidirect
40. #63651C2×C3semidirect
45. C3×#824D3semidirect
46. C3×#824C2×C3semidirect
49. #63577C2×C3semidirect
154. #824C3×D3semidirect
332. C33#473semidirect
477. C32#4755semidirect
478. C32#5143semidirect
479. C32C3×#473semidirect
525. C3#10590999direct (N × H)
526. C3C3×#4755semidirect
527. C3C3×#5143semidirect

Maximal subgroups

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

subgroupindex
C32×#9072
C2×C3×#37192
C3×#638312
C3×#638272
C3×#636512
C32×#8242
C3×#635772
C3×#51433
C3×#47553
#105909993