Finite Group Databasebeta

← C3×#10591642 #13158659

Subgroups — C3×#10591642 #13158659

Order 1728.

620 conjugacy classes of subgroups.

The subgroup lattice has 574 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C32×#942C211yesmaximal, characteristic, complemented
2C2×C32×#215C211yesmaximal, characteristic, complemented
3C3×#63823C211yesmaximal, characteristic
4C3×#63814C211yesmaximal, characteristic, complemented
5C3×#63663C211yesmaximal, characteristic, complemented
6C3×#63577C211yesmaximal, characteristic, complemented
7C3×#63558C211yesmaximal, characteristic, complemented
8C3×#5203—31nomaximal
9C3×#5177—31nomaximal
10#10591642C311yesmaximal, characteristic, direct product factor
11C2×C33×Q8C2211yescharacteristic
12C32×#235C2211yescharacteristic
13C2×C32×D12C2211yescharacteristic
14C3×#3748C2211yescharacteristic
15C3×#3748C2211yescharacteristic
16C32×#216—21no—
17C32×#215—21no—
18C32×#215—21no—
19C32×#215—21no—
20C32×#208C2211yescharacteristic
21C2×C32×C3⋊C8C2211yescharacteristic
22C3×#3715—21no—
23C3×#3713—21no—
24C3×#3711—21no—
25C3×D3×C3⋊C8—21no—
26C32×#137—31no—
27C2×C32×SD16—31no—
28C3×#949—31no—
29C3×#948—31no—
30C3×#942—21no—
31C3×#942D311yescharacteristic, complemented
32C3×#942C2×C311yescharacteristic, complemented
33C2×C3×#215—21no—
34C2×C3×#215D311yescharacteristic, complemented
35C2×C3×#215C2×C311yescharacteristic, complemented
36C3×#932—31no—
37C3×#901—31no—
38#63823C2×C311yescharacteristic
39#63814C2×C311yescharacteristic, complemented
40#63663C2×C311yescharacteristic, complemented
41C3×#836—31no—
42C3×#836—31no—
43C3×#836—61no—
44C3×#833—31no—
45C3×#823—31no—
46C3×#822—31no—
47C3×#819—31no—
48#63577C2×C311yescharacteristic, complemented
49#63558C2×C311yescharacteristic, complemented
50C22×C32×D3—21no—
51C33×Q8—21no—
52C33×Q8—21no—
53C2×C33×C4—21no—
54C2×C33×C4C2311yescharacteristic
55C32×#67—21no—
56C32×D12—21no—
57C32×D12—21no—
58C32×D12D411yescharacteristic
59C32×C4×D3—21no—
60C32×Dic6D411yescharacteristic
61C3×C3⋊(C3⋊C8)—21no—
62C3×C3⋊(C3⋊C8)—21no—
63C32×C3⋊C8—21no—
64C32×C3⋊C8D411yescharacteristic
65C32×C3⋊C8D411yescharacteristic
66C3×#471—91no—
67#5203—31no—
68#5177—31no—
69C32×#41—31no—
70C2×C32×Q8—21no—
71C2×C32×Q8—21no—
72C2×C32×Q8—21no—
73C2×C32×Q8—41no—
74C2×C32×Q8C2×D311yescharacteristic
75C2×C32×Q8C2×D311yescharacteristic
76C2×C32×Q8C22×C311yescharacteristic
77C2×C32×D4—31no—
78C3×#235—21no—
79C3×#235C2×D311yescharacteristic
80C3×#235C22×C311yescharacteristic
81C2×C3×D12—21no—
82C2×C3×D12C2×D311yescharacteristic
83C2×C3×D12C22×C311yescharacteristic
84C32×Q16—61no—
85C32×SD16—61no—
86C32×SD16—61no—
87C32×SD16—61no—
88C32×M16—31no—
89C2×C32×C8—31no—
90#3748C22×C311yescharacteristic
91#3748C22×C311yescharacteristic
92C3×#216—22no—
93C3×#216—21no—
94C3×#216—21no—
95C3×#215—22no—
96C3×#215—22no—
97C3×#215—22no—
98C3×#215—21no—
99C3×#215—21no—
100C3×#215—21no—
101C3×#215—21no—
102C3×#215—21no—
103C3×#215—21no—
104C3×#215—61no—
105C3×#214—61no—
106C3×#213—61no—
107C3×#213—61no—
108C3×#208—21no—
109C3×#208—31no—
110C3×#208—31no—
111C3×#208—31no—
112C3×#208—31no—
113C3×#208—61no—
114C3×#208—61no—
115C3×#208—61no—
116C3×#208C2×D311yescharacteristic
117C3×#208C22×C311yescharacteristic
118C2×C3×C3⋊C8—21no—
119C2×C3×C3⋊C8—61no—
120C2×C3×C3⋊C8C2×D311yescharacteristic
121C2×C3×C3⋊C8C22×C311yescharacteristic
122C3×ΓL(1,𝔽25)—61no—
123C3×D3×C8—61no—
124#3715—21no—
125#3713—21no—
126#3711—21no—
127D3×C3⋊C8—21no—
128C22×C33#4111yescharacteristic
129C2×C32×D3—21no—
130C2×C32×D3—41no—
131C2×C32×D3—81no—
132C33×C4—41no—
133C33×C4C2×D411yescharacteristic
134C33×C4C2×D411yescharacteristic
135C32×Dic3—21no—
136C3×#137—31no—
137C3×#137—31no—
138C3×#137—61no—
139C3×AGL(1,ℤ/8)—91no—
140C2×C3×SD16—31no—
141C2×C3×SD16—31no—
142C2×C3×SD16—61no—
143C3×#131—91no—
144#949—31no—
145#948—31no—
146#942C3×D311yescharacteristic, complemented
147C2×#215C3×D311yescharacteristic, complemented
148#932—31no—
149#901—31no—
150C3×#108—91no—
151C3×#101—91no—
152C3×#100—91no—
153#836—31no—
154#836—31no—
155#836—61no—
156#833—31no—
157#823—31no—
158#822—31no—
159#819—31no—
160C23×C32—61no—
161C22×C3×D3—21no—
162C22×C3×D3—21no—
163C22×C3×D3—41no—
164C32×Q8—21no—
165C32×Q8—21no—
166C32×Q8—22no—
167C32×Q8—22no—
168C32×Q8—21no—
169C32×Q8—21no—
170C32×Q8—21no—
171C32×Q8—21no—
172C32×Q8—31no—
173C32×Q8—41no—
174C32×Q8—41no—
175C32×Q8—41no—
176C32×Q8—41no—
177C32×Q8—42no—
178C32×Q8—42no—
179C32×D4—31no—
180C32×D4—61no—
181C32×D4—61no—
182C32×D4—61no—
183C2×C32×C4—21no—
184C2×C32×C4—22no—
185C2×C32×C4—21no—
186C2×C32×C4—21no—
187C2×C32×C4—21no—
188C2×C32×C4—21no—
189C2×C32×C4—21no—
190C2×C32×C4—41no—
191C2×C32×C4—41no—
192C2×C32×C4—41no—
193C2×C32×C4—42no—
194C2×C32×C4—61no—
195C2×C32×C4C22×D311yescharacteristic
196C2×C32×C4C22×D311yescharacteristic
197C2×C32×C4C23×C311yescharacteristic
198C3×#67—21no—
199C3×#67—21no—
200C3×#67—41no—
201C3×D12—22no—
202C3×D12—21no—
203C3×D12—21no—
204C3×D12—21no—
205C3×D12—21no—
206C3×D12—21no—
207C3×D12—41no—
208C3×D12#6711yescharacteristic
209C3×D12C3×D411yescharacteristic
210C3×C4×D3—21no—
211C3×C4×D3—21no—
212C3×C4×D3—41no—
213C3×Dic6—21no—
214C3×Dic6#6711yescharacteristic
215C3×Dic6C3×D411yescharacteristic
216C32×C8—31no—
217C32×C8—31no—
218C32×C8—61no—
219C3⋊(C3⋊C8)—21no—
220C3⋊(C3⋊C8)—21no—
221C3×C3⋊C8—22no—
222C3×C3⋊C8—21no—
223C3×C3⋊C8—21no—
224C3×C3⋊C8—21no—
225C3×C3⋊C8—21no—
226C3×C3⋊C8—61no—
227C3×C3⋊C8—61no—
228C3×C3⋊C8—61no—
229C3×C3⋊C8—61no—
230C3×C3⋊C8—121no—
231C3×C3⋊C8—121no—
232C3×C3⋊C8#6711yescharacteristic
233C3×C3⋊C8#6711yescharacteristic
234C3×C3⋊C8C3×D411yescharacteristic
235C3×C3⋊C8C3×D411yescharacteristic
236#471—91noSylow
237C2×C33—21no—
238C2×C33#12111yescharacteristic
239C32×D3—41no—
240C32×D3—81no—
241C3×#41—31no—
242C3×#41—31no—
243C3×#41—61no—
244C2×C3×Q8—21no—
245C2×C3×Q8—21no—
246C2×C3×Q8—21no—
247C2×C3×Q8—41no—
248C2×C3×Q8D3211yescharacteristic
249C2×C3×Q8C2×C3×D311yescharacteristic
250C2×C3×Q8C2×C3×D311yescharacteristic
251C2×C3×D4—31no—
252C2×C3×D4—31no—
253C2×C3×D4—61no—
254#235C2×C3×D311yescharacteristic
255C2×D12C2×C3×D311yescharacteristic
256C3×Q16—61no—
257C3×Q16—61no—
258C3×Q16—62no—
259C3×SD16—61no—
260C3×SD16—61no—
261C3×SD16—61no—
262C3×SD16—61no—
263C3×SD16—61no—
264C3×SD16—61no—
265C3×SD16—62no—
266C3×SD16—62no—
267C3×SD16—62no—
268C3×SD16—181no—
269C3×D8—181no—
270C3×M16—31no—
271C3×M16—31no—
272C3×M16—61no—
273C3×M16—91no—
274C3×M16—91no—
275C3×M16—181no—
276C2×C3×C8—31no—
277C2×C3×C8—31no—
278C2×C3×C8—61no—
279C2×C3×C8—181no—
280#216—21no—
281#215—21no—
282#215—21no—
283#215—21no—
284#215—61no—
285#214—61no—
286#213—61no—
287#213—61no—
288#208—31no—
289#208—31no—
290#208—31no—
291#208—31no—
292#208—61no—
293#208—61no—
294#208—61no—
295#208C2×C3×D311yescharacteristic
296C2×C3⋊C8—61no—
297C2×C3⋊C8C2×C3×D311yescharacteristic
298ΓL(1,𝔽25)—61no—
299D3×C8—61no—
300C22×C32—21no—
301C22×C32—21no—
302C22×C32—21no—
303C22×C32—41no—
304C22×C32—61no—
305C22×C32—121no—
306C22×C32—241no—
307C22×C32#23711yescharacteristic
308C22×C32#23711yescharacteristic
309C22×C32C3×#4111yescharacteristic
310C2×C3×D3—21no—
311C2×C3×D3—21no—
312C2×C3×D3—41no—
313C2×C3×D3—42no—
314C2×C3×D3—41no—
315C2×C3×D3—41no—
316C2×C3×D3—82no—
317C2×C3×D3—81no—
318C2×C3×D3—81no—
319C32×C4—21no—
320C32×C4—21no—
321C32×C4—21no—
322C32×C4—21no—
323C32×C4—21no—
324C32×C4—21no—
325C32×C4—41no—
326C32×C4—41no—
327C32×C4—42no—
328C32×C4—42no—
329C32×C4—42no—
330C32×C4—44no—
331C32×C4—41no—
332C32×C4—41no—
333C32×C4—41no—
334C32×C4—61no—
335C32×C4D3×D411yescharacteristic
336C32×C4D3×D411yescharacteristic
337C32×C4C2×#6711yescharacteristic
338C32×C4C2×#6711yescharacteristic
339C32×C4C2×C3×D411yescharacteristic
340C32×C4C2×C3×D411yescharacteristic
341C3×Dic3—21no—
342C3×Dic3—21no—
343C3×Dic3—41no—
344#137—31no—
345AGL(1,ℤ/8)—91no—
346C2×SD16—31no—
347#131—91no—
348#108—91no—
349#101—91no—
350#100—91no—
351C33#47111yescharacteristic, complemented, Sylow
352C23×C3—61no—
353C23×C3—61no—
354C23×C3—121no—
355C22×D3—21no—
356C3×Q8—22no—
357C3×Q8—22no—
358C3×Q8—21no—
359C3×Q8—21no—
360C3×Q8—21no—
361C3×Q8—21no—
362C3×Q8—21no—
363C3×Q8—21no—
364C3×Q8—31no—
365C3×Q8—31no—
366C3×Q8—41no—
367C3×Q8—41no—
368C3×Q8—41no—
369C3×Q8—41no—
370C3×Q8—42no—
371C3×Q8—42no—
372C3×Q8—61no—
373C3×D4—31no—
374C3×D4—31no—
375C3×D4—61no—
376C3×D4—62no—
377C3×D4—61no—
378C3×D4—61no—
379C3×D4—61no—
380C3×D4—61no—
381C3×D4—61no—
382C3×D4—61no—
383C3×D4—121no—
384C3×D4—121no—
385C2×C3×C4—22no—
386C2×C3×C4—21no—
387C2×C3×C4—21no—
388C2×C3×C4—21no—
389C2×C3×C4—21no—
390C2×C3×C4—21no—
391C2×C3×C4—21no—
392C2×C3×C4—41no—
393C2×C3×C4—41no—
394C2×C3×C4—42no—
395C2×C3×C4—41no—
396C2×C3×C4—61no—
397C2×C3×C4—61no—
398C2×C3×C4—121no—
399C2×C3×C4C2×D3211yescharacteristic
400C2×C3×C4C22×C3×D311yescharacteristic
401C2×C3×C4C22×C3×D311yescharacteristic
402#67—21no—
403D12—21no—
404D12—21no—
405D12C3×#6711yescharacteristic
406C4×D3—21no—
407Dic6C3×#6711yescharacteristic
408C3×C8—31no—
409C3×C8—31no—
410C3×C8—31no—
411C3×C8—31no—
412C3×C8—61no—
413C3×C8—61no—
414C3×C8—62no—
415C3×C8—61no—
416C3×C8—61no—
417C3×C8—181no—
418C3×C8—181no—
419C3⋊C8—21no—
420C3⋊C8—61no—
421C3⋊C8—61no—
422C3⋊C8—61no—
423C3⋊C8—61no—
424C3⋊C8—121no—
425C3⋊C8—121no—
426C3⋊C8C3×#6711yescharacteristic
427C3⋊C8C3×#6711yescharacteristic
428C2×C32—21no—
429C2×C32—22no—
430C2×C32—21no—
431C2×C32—21no—
432C2×C32—21no—
433C2×C32—21no—
434C2×C32—21no—
435C2×C32—41no—
436C2×C32—41no—
437C2×C32—41no—
438C2×C32—42no—
439C2×C32—121no—
440C2×C32—241no—
441C2×C32#93811yescharacteristic
442C2×C32#93911yescharacteristic
443C2×C32C3×#12111yescharacteristic
444C3×D3—41no—
445C3×D3—41no—
446C3×D3—81no—
447C3×D3—82no—
448C3×D3—81no—
449C3×D3—81no—
450#41—31no—
451C2×Q8C3×D3211yescharacteristic
452C2×D4—31no—
453Q16—61no—
454SD16—61no—
455SD16—61no—
456SD16—61no—
457SD16—181no—
458D8—181no—
459M16—31no—
460M16—91no—
461M16—91no—
462M16—181no—
463C2×C8—31no—
464C2×C8—181no—
465C22×C3—21no—
466C22×C3—21no—
467C22×C3—21no—
468C22×C3—41no—
469C22×C3—61no—
470C22×C3—61no—
471C22×C3—121no—
472C22×C3—122no—
473C22×C3—121no—
474C22×C3—121no—
475C22×C3—241no—
476C22×C3—241no—
477C22×C3—242no—
478C22×C3#380511yescharacteristic
479C22×C3C3×#23711yescharacteristic
480C22×C3C3×#23711yescharacteristic
481C2×D3—21no—
482C2×D3—41no—
483C2×D3—81no—
484C3×C4—21no—
485C3×C4—21no—
486C3×C4—21no—
487C3×C4—21no—
488C3×C4—21no—
489C3×C4—21no—
490C3×C4—42no—
491C3×C4—42no—
492C3×C4—42no—
493C3×C4—44no—
494C3×C4—41no—
495C3×C4—41no—
496C3×C4—41no—
497C3×C4—41no—
498C3×C4—41no—
499C3×C4—61no—
500C3×C4—61no—
501C3×C4—121no—
502C3×C4D3×#6711yescharacteristic
503C3×C4D3×#6711yescharacteristic
504C3×C4C3×D3×D411yescharacteristic
505C3×C4C3×D3×D411yescharacteristic
506C3×C4C2×C3×#6711yescharacteristic
507C3×C4C2×C3×#6711yescharacteristic
508Dic3—21no—
509C32—21no—
510C32—21no—
511C32—21no—
512C32—41no—
513C32#517711yescharacteristic, complemented
514C32#520311yescharacteristic, complemented
515C32C3×#47111yescharacteristic, complemented
516C23—61no—
517Q8—21no—
518Q8—21no—
519Q8—31no—
520D4—31no—
521D4—61no—
522D4—61no—
523D4—61no—
524C2×C4—21no—
525C2×C4—61no—
526C2×C4C2×C3×D3211yescharacteristic
527C8—31no—
528C8—31no—
529C8—61no—
530C8—181no—
531C8—181no—
532C2×C3—22no—
533C2×C3—21no—
534C2×C3—21no—
535C2×C3—21no—
536C2×C3—21no—
537C2×C3—21no—
538C2×C3—21no—
539C2×C3—41no—
540C2×C3—41no—
541C2×C3—42no—
542C2×C3—41no—
543C2×C3—121no—
544C2×C3—121no—
545C2×C3—241no—
546C2×C3—242no—
547C2×C3—241no—
548C2×C3—241no—
549C2×C3#6397011yescharacteristic
550C2×C3C3×#93811yescharacteristic
551C2×C3C3×#93911yescharacteristic
552D3—41no—
553D3—81no—
554C22—61no—
555C22—121no—
556C22—241no—
557C22C3×#380511yescharacteristic
558C4—41no—
559C4—61no—
560C4C3×D3×#6711yescharacteristic
561C4C3×D3×#6711yescharacteristic
562C3—21no—
563C3—21no—
564C3—21no—
565C3—41no—
566C3#1059164211yescharacteristic, direct product factor
567C3C3×#517711yescharacteristic, complemented
568C3C3×#520311yescharacteristic, complemented
569C2—21no—
570C2—121no—
571C2—241no—
572C2C3×#6397011yescharacteristic
subgroupG/H
centerC2×C3#63970
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C4C2×C3×D32
FittingC2×C33×Q8C22
socleC2×C33#121
prime radicalC3×#63823C2
hypercenterC2×C3×Q8D32
hypocenterC32C3×#471
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C2×C3 ◁ C2×C3×C4 ◁ C2×C3×Q8

Complements

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

NH ≅ G/Ntype
1. C32×#942C2semidirect
2. C2×C32×#215C2semidirect
4. C3×#63814C2semidirect
5. C3×#63663C2semidirect
6. C3×#63577C2semidirect
7. C3×#63558C2semidirect
10. #10591642C3direct (N × H)
31. C3×#942D3semidirect
32. C3×#942C2×C3semidirect
34. C2×C3×#215D3semidirect
35. C2×C3×#215C2×C3semidirect
39. #63814C2×C3semidirect
40. #63663C2×C3semidirect
48. #63577C2×C3semidirect
49. #63558C2×C3semidirect
146. #942C3×D3semidirect
147. C2×#215C3×D3semidirect
351. C33#471semidirect
513. C32#5177semidirect
514. C32#5203semidirect
515. C32C3×#471semidirect
566. C3#10591642direct (N × H)
567. C3C3×#5177semidirect
568. C3C3×#5203semidirect

Maximal subgroups

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

subgroupindex
C32×#9422
C2×C32×#2152
C3×#638232
C3×#638142
C3×#636632
C3×#635772
C3×#635582
C3×#52033
C3×#51773
#105916423