Finite Group Databasebeta

← C3×#10590991 #13158000

Subgroups — C3×#10590991 #13158000

Order 1728.

582 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
1C3×#64116C211yesmaximal, characteristic, complemented
2C3×#63833C211yesmaximal, characteristic, complemented
3C3×#63830C211yesmaximal, characteristic
4C2×C3×#3715C211yesmaximal, characteristic, complemented
5C32×#824C211yesmaximal, characteristic, complemented
6C32×#823C211yesmaximal, characteristic, complemented
7C3×#63578C211yesmaximal, characteristic, complemented
8C3×#4754—31nomaximal
9C3×#4748—31nomaximal
10#10590991C311yesmaximal, characteristic, direct product factor
11C3×#3828C2211yescharacteristic
12C2×C32×D12C2211yescharacteristic
13C2×C32×Dic6C2211yescharacteristic
14C33×M16C2211yescharacteristic
15C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
16C3×#3743—21no—
17C3×C8×C3⋊D3—21no—
18C32×#208C2211yescharacteristic
19C32×#208C2211yescharacteristic
20C3×#3719—21no—
21C3×#3717—21no—
22C3×#3716—21no—
23C3×#3715—21no—
24C3×#3715—21no—
25C3×#3714—21no—
26#64116C2×C311yescharacteristic, complemented
27C3×#942—31no—
28C2×C3×#215—31no—
29C2×C3×#214—31no—
30C3×#932—31no—
31C3×#909—31no—
32C3×#908—31no—
33C3×#907—31no—
34C3×#907—31no—
35C3×#907—61no—
36#63833C2×C311yescharacteristic, complemented
37#63830C2×C311yescharacteristic
38C2×#3715C2×C311yescharacteristic, complemented
39C32×#101—31no—
40C32×#100—31no—
41C3×#824—21no—
42C3×#824D311yescharacteristic, complemented
43C3×#824C2×C311yescharacteristic, complemented
44C3×#823—21no—
45C3×#823D311yescharacteristic, complemented
46C3×#823C2×C311yescharacteristic, complemented
47C3×#822—31no—
48C3×#822—31no—
49#63578C2×C311yescharacteristic, complemented
50C22×C32×D3—21no—
51C2×C33×C4C2311yescharacteristic
52C3×#638—21no—
53C3×C4⋊(C3⋊D3)D411yescharacteristic
54C3×C4×C3⋊D3—21no—
55C3×#634D411yescharacteristic
56C2×C32×Dic3—21no—
57C32×D12—21no—
58C32×D12—21no—
59C32×Dic6—21no—
60C32×Dic6—21no—
61C33×C8—21no—
62C3×C3⋊(C3⋊C8)D411yescharacteristic
63C3×C3⋊(C3⋊C8)D411yescharacteristic
64C32×C3⋊C8—21no—
65C32×C3⋊C8—21no—
66C3×#471—91no—
67#4754—31no—
68#4748—31no—
69C2×C32×Q8—31no—
70C2×C32×D4—31no—
71#3828C22×C311yescharacteristic
72C3×#235—31no—
73C3×#235—31no—
74C3×#235—61no—
75C2×C3×D12—21no—
76C2×C3×D12C2×D311yescharacteristic
77C2×C3×D12C22×C311yescharacteristic
78C2×C3×Dic6—21no—
79C2×C3×Dic6C2×D311yescharacteristic
80C2×C3×Dic6C22×C311yescharacteristic
81C32×M16—21no—
82C32×M16—21no—
83C32×M16—21no—
84C32×M16—31no—
85C32×M16—31no—
86C32×M16—41no—
87C32×M16C2×D311yescharacteristic
88C32×M16C2×D311yescharacteristic
89C32×M16C22×C311yescharacteristic
90C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
91#3743—21no—
92C8×C3⋊D3—21no—
93C3×#216—61no—
94C3×#215—61no—
95C3×#215—61no—
96C3×#215—61no—
97C3×#214—61no—
98C3×#214—61no—
99C3×#214—61no—
100C3×#213—61no—
101C3×#208—21no—
102C3×#208—21no—
103C3×#208C2×D311yescharacteristic
104C3×#208C2×D311yescharacteristic
105C3×#208C22×C311yescharacteristic
106C3×#208C22×C311yescharacteristic
107C2×C3×C3⋊C8—31no—
108C2×C3×C3⋊C8—31no—
109C2×C3×C3⋊C8—61no—
110C3×Dic12—61no—
111C3×D24—61no—
112C3×#204—61no—
113C3×#204—61no—
114C3×ΓL(1,𝔽25)—61no—
115C3×ΓL(1,𝔽25)—61no—
116C3×ΓL(1,𝔽25)—62no—
117C3×D3×C8—61no—
118C3×D3×C8—61no—
119C3×D3×C8—62no—
120#3719—21no—
121#3717—21no—
122#3716—21no—
123#3715—21no—
124#3715—21no—
125#3714—21no—
126C22×C33#4111yescharacteristic
127C2×C3×C3⋊D3—21no—
128C2×C32×D3—41no—
129C2×C32×D3—81no—
130C33×C4C2×D411yescharacteristic
131C33×C4C2×D411yescharacteristic
132C3×C3⋊Dic3—21no—
133C32×Dic3—41no—
134C3×#137—91no—
135C3×AGL(1,ℤ/8)—91no—
136C2×C3×SD16—91no—
137C3×#131—91no—
138#942—31no—
139C2×#215—31no—
140C2×#214—31no—
141#932—31no—
142#909—31no—
143#908—31no—
144#907—31no—
145#907—31no—
146#907—61no—
147C3×#108—91no—
148C3×#101—31no—
149C3×#101—31no—
150C3×#101—61no—
151C3×#100—31no—
152C3×#100—31no—
153C3×#100—61no—
154#824C3×D311yescharacteristic, complemented
155#823C3×D311yescharacteristic, complemented
156#822—31no—
157#822—31no—
158C23×C32—61no—
159C22×C3×D3—22no—
160C22×C3×D3—21no—
161C22×C3×D3—21no—
162C32×Q8—61no—
163C32×Q8—61no—
164C32×D4—61no—
165C32×D4—61no—
166C2×C32×C4—21no—
167C2×C32×C4—21no—
168C2×C32×C4—21no—
169C2×C32×C4—41no—
170C2×C32×C4—61no—
171C2×C32×C4C22×D311yescharacteristic
172C2×C32×C4C22×D311yescharacteristic
173C2×C32×C4C23×C311yescharacteristic
174#638—21no—
175C4⋊(C3⋊D3)C3×D411yescharacteristic
176C4×C3⋊D3—21no—
177#634C3×D411yescharacteristic
178C3×#67—62no—
179C3×#67—61no—
180C3×#67—61no—
181C2×C3×Dic3—22no—
182C2×C3×Dic3—21no—
183C2×C3×Dic3—21no—
184C3×D12—21no—
185C3×D12—21no—
186C3×D12—21no—
187C3×D12—21no—
188C3×D12—31no—
189C3×D12—31no—
190C3×D12—41no—
191C3×D12—41no—
192C3×D12—61no—
193C3×C4×D3—62no—
194C3×C4×D3—61no—
195C3×C4×D3—61no—
196C3×Dic6—21no—
197C3×Dic6—21no—
198C3×Dic6—21no—
199C3×Dic6—21no—
200C3×Dic6—31no—
201C3×Dic6—31no—
202C3×Dic6—41no—
203C3×Dic6—41no—
204C3×Dic6—61no—
205C32×C8—21no—
206C32×C8—22no—
207C32×C8—21no—
208C32×C8—21no—
209C32×C8—41no—
210C32×C8—41no—
211C32×C8—42no—
212C32×C8—61no—
213C32×C8—61no—
214C3⋊(C3⋊C8)C3×D411yescharacteristic
215C3⋊(C3⋊C8)C3×D411yescharacteristic
216C3×C3⋊C8—21no—
217C3×C3⋊C8—21no—
218C3×C3⋊C8—21no—
219C3×C3⋊C8—21no—
220C3×C3⋊C8—31no—
221C3×C3⋊C8—31no—
222C3×C3⋊C8—31no—
223C3×C3⋊C8—31no—
224C3×C3⋊C8—41no—
225C3×C3⋊C8—41no—
226C3×C3⋊C8—61no—
227C3×C3⋊C8—61no—
228#471—91noSylow
229C2×C33—21no—
230C2×C33#12111yescharacteristic
231C3×C3⋊D3—41no—
232C32×D3—81no—
233C3×#41—91no—
234C2×C3×Q8—31no—
235C2×C3×Q8—31no—
236C2×C3×Q8—61no—
237C2×C3×D4—31no—
238C2×C3×D4—31no—
239C2×C3×D4—61no—
240#235—31no—
241#235—31no—
242#235—61no—
243C2×D12C2×C3×D311yescharacteristic
244C2×Dic6C2×C3×D311yescharacteristic
245C3×Q16—181no—
246C3×SD16—181no—
247C3×SD16—181no—
248C3×SD16—181no—
249C3×SD16—181no—
250C3×D8—181no—
251C3×M16—21no—
252C3×M16—21no—
253C3×M16—21no—
254C3×M16—31no—
255C3×M16—31no—
256C3×M16—31no—
257C3×M16—31no—
258C3×M16—41no—
259C3×M16—61no—
260C3×M16—61no—
261C3×M16—181no—
262C3×M16D3211yescharacteristic
263C3×M16C2×C3×D311yescharacteristic
264C3×M16C2×C3×D311yescharacteristic
265C2×C3×C8—91no—
266C2×C3×C8—181no—
267#216—61no—
268#215—61no—
269#215—61no—
270#215—61no—
271#214—61no—
272#214—61no—
273#214—61no—
274#213—61no—
275#208C2×C3×D311yescharacteristic
276#208C2×C3×D311yescharacteristic
277C2×C3⋊C8—31no—
278C2×C3⋊C8—31no—
279C2×C3⋊C8—61no—
280Dic12—61no—
281D24—61no—
282#204—61no—
283#204—61no—
284ΓL(1,𝔽25)—61no—
285ΓL(1,𝔽25)—61no—
286ΓL(1,𝔽25)—62no—
287D3×C8—61no—
288D3×C8—61no—
289D3×C8—62no—
290C22×C32—21no—
291C22×C32—21no—
292C22×C32—21no—
293C22×C32—41no—
294C22×C32—121no—
295C22×C32—241no—
296C22×C32#23511yescharacteristic
297C22×C32#23511yescharacteristic
298C22×C32C3×#4111yescharacteristic
299C2×C3⋊D3—21no—
300C2×C3×D3—42no—
301C2×C3×D3—41no—
302C2×C3×D3—41no—
303C2×C3×D3—62no—
304C2×C3×D3—61no—
305C2×C3×D3—61no—
306C2×C3×D3—81no—
307C2×C3×D3—81no—
308C2×C3×D3—82no—
309C32×C4—21no—
310C32×C4—21no—
311C32×C4—21no—
312C32×C4—21no—
313C32×C4—21no—
314C32×C4—21no—
315C32×C4—41no—
316C32×C4—41no—
317C32×C4—121no—
318C32×C4D3×D411yescharacteristic
319C32×C4D3×D411yescharacteristic
320C32×C4D3×D411yescharacteristic
321C32×C4D3×D411yescharacteristic
322C32×C4C2×C3×D411yescharacteristic
323C32×C4C2×C3×D411yescharacteristic
324C3⋊Dic3—21no—
325C3×Dic3—42no—
326C3×Dic3—41no—
327C3×Dic3—41no—
328C3×Dic3—62no—
329C3×Dic3—61no—
330C3×Dic3—61no—
331#137—91no—
332AGL(1,ℤ/8)—91no—
333C2×SD16—91no—
334#131—91no—
335#108—91no—
336#101—31no—
337#100—31no—
338C33#47111yescharacteristic, complemented, Sylow
339C23×C3—62no—
340C23×C3—61no—
341C23×C3—61no—
342C22×D3—21no—
343C3×Q8—61no—
344C3×Q8—61no—
345C3×Q8—61no—
346C3×Q8—61no—
347C3×Q8—91no—
348C3×Q8—121no—
349C3×Q8—121no—
350C3×D4—61no—
351C3×D4—61no—
352C3×D4—61no—
353C3×D4—61no—
354C3×D4—91no—
355C3×D4—121no—
356C3×D4—121no—
357C3×D4—181no—
358C2×C3×C4—21no—
359C2×C3×C4—21no—
360C2×C3×C4—21no—
361C2×C3×C4—41no—
362C2×C3×C4—62no—
363C2×C3×C4—61no—
364C2×C3×C4—61no—
365C2×C3×C4—181no—
366C2×C3×C4C2×D3211yescharacteristic
367C2×C3×C4C22×C3×D311yescharacteristic
368C2×C3×C4C22×C3×D311yescharacteristic
369#67—62no—
370#67—61no—
371#67—61no—
372C2×Dic3—21no—
373D12—21no—
374D12—21no—
375D12—31no—
376D12—31no—
377D12—61no—
378C4×D3—62no—
379C4×D3—61no—
380C4×D3—61no—
381Dic6—21no—
382Dic6—21no—
383Dic6—31no—
384Dic6—31no—
385Dic6—61no—
386C3×C8—22no—
387C3×C8—21no—
388C3×C8—21no—
389C3×C8—21no—
390C3×C8—41no—
391C3×C8—41no—
392C3×C8—42no—
393C3×C8—61no—
394C3×C8—61no—
395C3×C8—61no—
396C3×C8—61no—
397C3×C8—91no—
398C3×C8—91no—
399C3×C8—121no—
400C3×C8—121no—
401C3⋊C8—21no—
402C3⋊C8—21no—
403C3⋊C8—31no—
404C3⋊C8—31no—
405C3⋊C8—31no—
406C3⋊C8—31no—
407C3⋊C8—61no—
408C3⋊C8—61no—
409C2×C32—21no—
410C2×C32—21no—
411C2×C32—21no—
412C2×C32—21no—
413C2×C32—21no—
414C2×C32—21no—
415C2×C32—41no—
416C2×C32—41no—
417C2×C32—41no—
418C2×C32—41no—
419C2×C32—81no—
420C2×C32—241no—
421C2×C32#88411yescharacteristic
422C2×C32#88411yescharacteristic
423C2×C32C3×#12111yescharacteristic
424C3⋊D3—41no—
425C3×D3—82no—
426C3×D3—81no—
427C3×D3—81no—
428C3×D3—122no—
429C3×D3—121no—
430C3×D3—121no—
431#41—91no—
432C2×Q8—31no—
433C2×D4—31no—
434Q16—181no—
435SD16—181no—
436SD16—181no—
437SD16—181no—
438SD16—181no—
439D8—181no—
440M16—31no—
441M16—31no—
442M16—181no—
443M16C3×D3211yescharacteristic
444C2×C8—91no—
445C2×C8—181no—
446C22×C3—21no—
447C22×C3—21no—
448C22×C3—21no—
449C22×C3—41no—
450C22×C3—122no—
451C22×C3—121no—
452C22×C3—121no—
453C22×C3—181no—
454C22×C3—241no—
455C22×C3—241no—
456C22×C3—242no—
457C22×C3#379811yescharacteristic
458C22×C3C3×#23511yescharacteristic
459C22×C3C3×#23511yescharacteristic
460C2×D3—41no—
461C2×D3—62no—
462C2×D3—61no—
463C2×D3—61no—
464C2×D3—81no—
465C3×C4—21no—
466C3×C4—21no—
467C3×C4—21no—
468C3×C4—21no—
469C3×C4—21no—
470C3×C4—21no—
471C3×C4—41no—
472C3×C4—41no—
473C3×C4—122no—
474C3×C4—121no—
475C3×C4—121no—
476C3×C4—181no—
477C3×C4#380211yescharacteristic
478C3×C4#381111yescharacteristic
479C3×C4C3×D3×D411yescharacteristic
480C3×C4C3×D3×D411yescharacteristic
481C3×C4C3×D3×D411yescharacteristic
482C3×C4C3×D3×D411yescharacteristic
483Dic3—41no—
484Dic3—62no—
485Dic3—61no—
486Dic3—61no—
487C32—21no—
488C32—21no—
489C32—21no—
490C32—41no—
491C32#474811yescharacteristic, complemented
492C32#475411yescharacteristic, complemented
493C32C3×#47111yescharacteristic, complemented
494C23—61no—
495Q8—61no—
496Q8—61no—
497Q8—91no—
498D4—61no—
499D4—61no—
500D4—91no—
501D4—181no—
502C2×C4—61no—
503C2×C4—181no—
504C2×C4C2×C3×D3211yescharacteristic
505C8—21no—
506C8—61no—
507C8—61no—
508C8—91no—
509C8—91no—
510C2×C3—21no—
511C2×C3—21no—
512C2×C3—21no—
513C2×C3—21no—
514C2×C3—21no—
515C2×C3—21no—
516C2×C3—41no—
517C2×C3—41no—
518C2×C3—41no—
519C2×C3—41no—
520C2×C3—81no—
521C2×C3—242no—
522C2×C3—241no—
523C2×C3—241no—
524C2×C3—361no—
525C2×C3#6391211yescharacteristic
526C2×C3C3×#88411yescharacteristic
527C2×C3C3×#88411yescharacteristic
528D3—81no—
529D3—122no—
530D3—121no—
531D3—121no—
532C22—121no—
533C22—181no—
534C22—241no—
535C22C3×#379811yescharacteristic
536C4—121no—
537C4—181no—
538C4C3×#380211yescharacteristic
539C4C3×#381111yescharacteristic
540C3—21no—
541C3—21no—
542C3—21no—
543C3—41no—
544C3#1059099111yescharacteristic, direct product factor
545C3C3×#474811yescharacteristic, complemented
546C3C3×#475411yescharacteristic, complemented
547C2—21no—
548C2—241no—
549C2—361no—
550C2C3×#6391211yescharacteristic
subgroupG/H
centerC2×C3#63912
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C4C2×C3×D32
FittingC33×M16C22
socleC2×C33#121
prime radicalC3×#63830C2
hypercenterC3×M16D32
hypocenterC32C3×#471
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
1. C3×#64116C2semidirect
2. C3×#63833C2semidirect
4. C2×C3×#3715C2semidirect
5. C32×#824C2semidirect
6. C32×#823C2semidirect
7. C3×#63578C2semidirect
10. #10590991C3direct (N × H)
26. #64116C2×C3semidirect
36. #63833C2×C3semidirect
38. C2×#3715C2×C3semidirect
42. C3×#824D3semidirect
43. C3×#824C2×C3semidirect
45. C3×#823D3semidirect
46. C3×#823C2×C3semidirect
49. #63578C2×C3semidirect
154. #824C3×D3semidirect
155. #823C3×D3semidirect
338. C33#471semidirect
491. C32#4748semidirect
492. C32#4754semidirect
493. C32C3×#471semidirect
544. C3#10590991direct (N × H)
545. C3C3×#4748semidirect
546. C3C3×#4754semidirect

Maximal subgroups

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

subgroupindex
C3×#641162
C3×#638332
C3×#638302
C2×C3×#37152
C32×#8242
C32×#8232
C3×#635782
C3×#47543
C3×#47483
#105909913