Finite Group Databasebeta

← C3×#10591542 #13158647

Subgroups — C3×#10591542 #13158647

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
1C3×#64151C211yesmaximal, characteristic, complemented
2C2×C32×#215C211yesmaximal, characteristic, complemented
3C3×#63830C211yesmaximal, characteristic
4C3×#63817C211yesmaximal, characteristic, complemented
5C32×#836C211yesmaximal, characteristic, complemented
6C3×#63577C211yesmaximal, characteristic, complemented
7C3×#63558C211yesmaximal, characteristic, complemented
8C3×#5177—31nomaximal
9C3×#4754—31nomaximal
10#10591542C311yesmaximal, characteristic, direct product factor
11C2×C33×Q8C2211yescharacteristic
12C3×#3828C2211yescharacteristic
13C2×C32×D12C2211yescharacteristic
14C3×#3756—21no—
15C3×#3755—21no—
16C3×#3748C2211yescharacteristic
17C32×#215—21no—
18C32×#215—21no—
19C32×#208C2211yescharacteristic
20C32×#208C2211yescharacteristic
21C2×C32×C3⋊C8C2211yescharacteristic
22C3×#3716—21no—
23C3×#3714—21no—
24C3×#3712—21no—
25C3×#3710—21no—
26C2×C32×SD16—31no—
27#64151C2×C311yescharacteristic, complemented
28C3×#942—31no—
29C3×#942—31no—
30C3×#942—61no—
31C2×C3×#215—21no—
32C2×C3×#215D311yescharacteristic, complemented
33C2×C3×#215C2×C311yescharacteristic, complemented
34C3×#932—31no—
35C3×#908—31no—
36C3×#907—31no—
37C3×#901—31no—
38#63830C2×C311yescharacteristic
39#63817C2×C311yescharacteristic, complemented
40C32×#101—31no—
41C3×#836—21no—
42C3×#836D311yescharacteristic, complemented
43C3×#836C2×C311yescharacteristic, complemented
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
55C3×#638—21no—
56C3×C4⋊(C3⋊D3)D411yescharacteristic
57C3×C4×C3⋊D3—21no—
58C3×#634D411yescharacteristic
59C32×D12—21no—
60C32×D12—21no—
61C3×C3⋊(C3⋊C8)—21no—
62C32×C3⋊C8—21no—
63C32×C3⋊C8—21no—
64C32×C3⋊C8D411yescharacteristic
65C32×C3⋊C8D411yescharacteristic
66C3×#471—91no—
67#5177—31no—
68#4754—31no—
69C2×C32×Q8—21no—
70C2×C32×Q8—21no—
71C2×C32×Q8—21no—
72C2×C32×Q8—41no—
73C2×C32×Q8C2×D311yescharacteristic
74C2×C32×Q8C2×D311yescharacteristic
75C2×C32×Q8C22×C311yescharacteristic
76C2×C32×D4—31no—
77#3828C22×C311yescharacteristic
78C3×#235—31no—
79C3×#235—31no—
80C3×#235—61no—
81C2×C3×D12—21no—
82C2×C3×D12C2×D311yescharacteristic
83C2×C3×D12C22×C311yescharacteristic
84C32×SD16—61no—
85C32×SD16—61no—
86C32×M16—31no—
87C32×M16—31no—
88C2×C32×C8—31no—
89#3756—21no—
90#3755—21no—
91#3748C22×C311yescharacteristic
92C3×#216—61no—
93C3×#216—61no—
94C3×#216—62no—
95C3×#215—22no—
96C3×#215—22no—
97C3×#215—21no—
98C3×#215—21no—
99C3×#215—21no—
100C3×#215—21no—
101C3×#215—61no—
102C3×#215—61no—
103C3×#215—62no—
104C3×#214—61no—
105C3×#213—61no—
106C3×#208—21no—
107C3×#208—21no—
108C3×#208—31no—
109C3×#208—31no—
110C3×#208—61no—
111C3×#208C2×D311yescharacteristic
112C3×#208C2×D311yescharacteristic
113C3×#208C22×C311yescharacteristic
114C3×#208C22×C311yescharacteristic
115C2×C3×C3⋊C8—21no—
116C2×C3×C3⋊C8C2×D311yescharacteristic
117C2×C3×C3⋊C8C22×C311yescharacteristic
118C3×D24—61no—
119C3×#204—61no—
120C3×ΓL(1,𝔽25)—61no—
121C3×ΓL(1,𝔽25)—61no—
122C3×D3×C8—61no—
123C3×D3×C8—61no—
124#3716—21no—
125#3714—21no—
126#3712—21no—
127#3710—21no—
128C22×C33#4111yescharacteristic
129C2×C3×C3⋊D3—21no—
130C2×C32×D3—41no—
131C2×C32×D3—81no—
132C33×C4—41no—
133C33×C4C2×D411yescharacteristic
134C33×C4C2×D411yescharacteristic
135C3×C3⋊Dic3—21no—
136C3×#137—91no—
137C3×AGL(1,ℤ/8)—91no—
138C2×C3×SD16—31no—
139C2×C3×SD16—31no—
140C2×C3×SD16—61no—
141C3×#131—91no—
142#942—31no—
143#942—31no—
144#942—61no—
145C2×#215C3×D311yescharacteristic, complemented
146#932—31no—
147#908—31no—
148#907—31no—
149#901—31no—
150C3×#108—91no—
151C3×#101—31no—
152C3×#101—31no—
153C3×#101—61no—
154C3×#100—91no—
155#836C3×D311yescharacteristic, complemented
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—41no—
173C32×Q8—41no—
174C32×Q8—41no—
175C32×Q8—41no—
176C32×Q8—42no—
177C32×Q8—42no—
178C32×D4—61no—
179C32×D4—61no—
180C2×C32×C4—21no—
181C2×C32×C4—22no—
182C2×C32×C4—21no—
183C2×C32×C4—21no—
184C2×C32×C4—21no—
185C2×C32×C4—21no—
186C2×C32×C4—21no—
187C2×C32×C4—41no—
188C2×C32×C4—41no—
189C2×C32×C4—41no—
190C2×C32×C4—42no—
191C2×C32×C4C22×D311yescharacteristic
192C2×C32×C4C22×D311yescharacteristic
193C2×C32×C4C23×C311yescharacteristic
194#638—21no—
195C4⋊(C3⋊D3)C3×D411yescharacteristic
196C4×C3⋊D3—21no—
197#634C3×D411yescharacteristic
198C3×#67—61no—
199C3×#67—61no—
200C3×#67—121no—
201C3×D12—22no—
202C3×D12—21no—
203C3×D12—21no—
204C3×D12—21no—
205C3×D12—21no—
206C3×D12—31no—
207C3×D12—31no—
208C3×D12—41no—
209C3×D12—61no—
210C3×C4×D3—61no—
211C3×C4×D3—61no—
212C3×C4×D3—121no—
213C3×Dic6—31no—
214C3×Dic6—31no—
215C3×Dic6—61no—
216C32×C8—31no—
217C32×C8—31no—
218C32×C8—61no—
219C32×C8—61no—
220C3⋊(C3⋊C8)—21no—
221C3×C3⋊C8—21no—
222C3×C3⋊C8—21no—
223C3×C3⋊C8—21no—
224C3×C3⋊C8—21no—
225C3×C3⋊C8—21no—
226C3×C3⋊C8—21no—
227C3×C3⋊C8—41no—
228C3×C3⋊C8—41no—
229C3×C3⋊C8—62no—
230C3×C3⋊C8—61no—
231C3×C3⋊C8—61no—
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
239C3×C3⋊D3—41no—
240C32×D3—81no—
241C3×#41—91no—
242C2×C3×Q8—21no—
243C2×C3×Q8—21no—
244C2×C3×Q8—21no—
245C2×C3×Q8—41no—
246C2×C3×Q8D3211yescharacteristic
247C2×C3×Q8C2×C3×D311yescharacteristic
248C2×C3×Q8C2×C3×D311yescharacteristic
249C2×C3×D4—31no—
250C2×C3×D4—31no—
251C2×C3×D4—61no—
252#235—31no—
253#235—31no—
254#235—61no—
255C2×D12C2×C3×D311yescharacteristic
256C3×Q16—181no—
257C3×SD16—61no—
258C3×SD16—61no—
259C3×SD16—61no—
260C3×SD16—61no—
261C3×SD16—62no—
262C3×SD16—62no—
263C3×SD16—181no—
264C3×SD16—181no—
265C3×D8—181no—
266C3×M16—31no—
267C3×M16—31no—
268C3×M16—31no—
269C3×M16—31no—
270C3×M16—61no—
271C3×M16—61no—
272C3×M16—91no—
273C3×M16—181no—
274C2×C3×C8—31no—
275C2×C3×C8—31no—
276C2×C3×C8—61no—
277C2×C3×C8—181no—
278#216—61no—
279#216—61no—
280#216—62no—
281#215—21no—
282#215—21no—
283#215—61no—
284#215—61no—
285#215—62no—
286#214—61no—
287#213—61no—
288#208—31no—
289#208—31no—
290#208—61no—
291#208C2×C3×D311yescharacteristic
292#208C2×C3×D311yescharacteristic
293C2×C3⋊C8C2×C3×D311yescharacteristic
294D24—61no—
295#204—61no—
296ΓL(1,𝔽25)—61no—
297ΓL(1,𝔽25)—61no—
298D3×C8—61no—
299D3×C8—61no—
300C22×C32—21no—
301C22×C32—21no—
302C22×C32—21no—
303C22×C32—41no—
304C22×C32—121no—
305C22×C32—241no—
306C22×C32#23511yescharacteristic
307C22×C32#23711yescharacteristic
308C22×C32C3×#4111yescharacteristic
309C2×C3⋊D3—21no—
310C2×C3×D3—42no—
311C2×C3×D3—41no—
312C2×C3×D3—41no—
313C2×C3×D3—61no—
314C2×C3×D3—61no—
315C2×C3×D3—82no—
316C2×C3×D3—81no—
317C2×C3×D3—81no—
318C2×C3×D3—121no—
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×C4D3×D411yescharacteristic
335C32×C4D3×D411yescharacteristic
336C32×C4D3×D411yescharacteristic
337C32×C4C2×#6711yescharacteristic
338C32×C4C2×C3×D411yescharacteristic
339C32×C4C2×C3×D411yescharacteristic
340C3⋊Dic3—21no—
341C3×Dic3—61no—
342C3×Dic3—61no—
343C3×Dic3—121no—
344#137—91no—
345AGL(1,ℤ/8)—91no—
346C2×SD16—31no—
347#131—91no—
348#108—91no—
349#101—31no—
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—41no—
365C3×Q8—41no—
366C3×Q8—41no—
367C3×Q8—41no—
368C3×Q8—42no—
369C3×Q8—42no—
370C3×Q8—91no—
371C3×D4—62no—
372C3×D4—61no—
373C3×D4—61no—
374C3×D4—61no—
375C3×D4—61no—
376C3×D4—91no—
377C3×D4—121no—
378C3×D4—181no—
379C2×C3×C4—22no—
380C2×C3×C4—21no—
381C2×C3×C4—21no—
382C2×C3×C4—21no—
383C2×C3×C4—21no—
384C2×C3×C4—21no—
385C2×C3×C4—21no—
386C2×C3×C4—41no—
387C2×C3×C4—41no—
388C2×C3×C4—42no—
389C2×C3×C4—41no—
390C2×C3×C4—181no—
391C2×C3×C4C2×D3211yescharacteristic
392C2×C3×C4C22×C3×D311yescharacteristic
393C2×C3×C4C22×C3×D311yescharacteristic
394#67—61no—
395#67—61no—
396#67—121no—
397D12—21no—
398D12—21no—
399D12—31no—
400D12—31no—
401D12—61no—
402C4×D3—61no—
403C4×D3—61no—
404C4×D3—121no—
405Dic6—31no—
406Dic6—31no—
407Dic6—61no—
408C3×C8—31no—
409C3×C8—31no—
410C3×C8—31no—
411C3×C8—31no—
412C3×C8—61no—
413C3×C8—61no—
414C3×C8—61no—
415C3×C8—61no—
416C3×C8—61no—
417C3×C8—61no—
418C3×C8—121no—
419C3×C8—121no—
420C3×C8—181no—
421C3⋊C8—21no—
422C3⋊C8—21no—
423C3⋊C8—62no—
424C3⋊C8—61no—
425C3⋊C8—61no—
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—241no—
440C2×C32#88411yescharacteristic
441C2×C32#93811yescharacteristic
442C2×C32C3×#12111yescharacteristic
443C3⋊D3—41no—
444C3×D3—82no—
445C3×D3—81no—
446C3×D3—81no—
447C3×D3—121no—
448C3×D3—121no—
449C3×D3—241no—
450#41—91no—
451C2×Q8C3×D3211yescharacteristic
452C2×D4—31no—
453Q16—181no—
454SD16—61no—
455SD16—61no—
456SD16—181no—
457SD16—181no—
458D8—181no—
459M16—31no—
460M16—31no—
461M16—91no—
462M16—181no—
463C2×C8—31no—
464C2×C8—181no—
465C22×C3—21no—
466C22×C3—21no—
467C22×C3—21no—
468C22×C3—41no—
469C22×C3—122no—
470C22×C3—121no—
471C22×C3—121no—
472C22×C3—181no—
473C22×C3—241no—
474C22×C3—241no—
475C22×C3—242no—
476C22×C3#380411yescharacteristic
477C22×C3C3×#23511yescharacteristic
478C22×C3C3×#23711yescharacteristic
479C2×D3—41no—
480C2×D3—61no—
481C2×D3—61no—
482C2×D3—81no—
483C2×D3—121no—
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—181no—
500C3×C4D3×#6711yescharacteristic
501C3×C4#381111yescharacteristic
502C3×C4C3×D3×D411yescharacteristic
503C3×C4C3×D3×D411yescharacteristic
504C3×C4C3×D3×D411yescharacteristic
505C3×C4C2×C3×#6711yescharacteristic
506Dic3—61no—
507Dic3—61no—
508Dic3—121no—
509C32—21no—
510C32—21no—
511C32—21no—
512C32—41no—
513C32#475411yescharacteristic, complemented
514C32#517711yescharacteristic, complemented
515C32C3×#47111yescharacteristic, complemented
516C23—61no—
517Q8—21no—
518Q8—21no—
519Q8—91no—
520D4—61no—
521D4—61no—
522D4—91no—
523D4—181no—
524C2×C4—21no—
525C2×C4—181no—
526C2×C4C2×C3×D3211yescharacteristic
527C8—31no—
528C8—31no—
529C8—61no—
530C8—61no—
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—242no—
544C2×C3—241no—
545C2×C3—241no—
546C2×C3—361no—
547C2×C3#6395811yescharacteristic
548C2×C3C3×#88411yescharacteristic
549C2×C3C3×#93811yescharacteristic
550D3—81no—
551D3—121no—
552D3—121no—
553D3—241no—
554C22—121no—
555C22—181no—
556C22—241no—
557C22C3×#380411yescharacteristic
558C4—41no—
559C4—181no—
560C4C3×D3×#6711yescharacteristic
561C4C3×#381111yescharacteristic
562C3—21no—
563C3—21no—
564C3—21no—
565C3—41no—
566C3#1059154211yescharacteristic, direct product factor
567C3C3×#475411yescharacteristic, complemented
568C3C3×#517711yescharacteristic, complemented
569C2—21no—
570C2—241no—
571C2—361no—
572C2C3×#6395811yescharacteristic
subgroupG/H
centerC2×C3#63958
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C4C2×C3×D32
FittingC2×C33×Q8C22
socleC2×C33#121
prime radicalC3×#63830C2
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. C3×#64151C2semidirect
2. C2×C32×#215C2semidirect
4. C3×#63817C2semidirect
5. C32×#836C2semidirect
6. C3×#63577C2semidirect
7. C3×#63558C2semidirect
10. #10591542C3direct (N × H)
27. #64151C2×C3semidirect
32. C2×C3×#215D3semidirect
33. C2×C3×#215C2×C3semidirect
39. #63817C2×C3semidirect
42. C3×#836D3semidirect
43. C3×#836C2×C3semidirect
48. #63577C2×C3semidirect
49. #63558C2×C3semidirect
145. C2×#215C3×D3semidirect
155. #836C3×D3semidirect
351. C33#471semidirect
513. C32#4754semidirect
514. C32#5177semidirect
515. C32C3×#471semidirect
566. C3#10591542direct (N × H)
567. C3C3×#4754semidirect
568. C3C3×#5177semidirect

Maximal subgroups

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

subgroupindex
C3×#641512
C2×C32×#2152
C3×#638302
C3×#638172
C32×#8362
C3×#635772
C3×#635582
C3×#51773
C3×#47543
#105915423