Finite Group Databasebeta

← C3×#10591406 #13158520

Subgroups — C3×#10591406 #13158520

Order 1728.

676 conjugacy classes of subgroups.

The subgroup lattice has 606 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#64141C211yesmaximal, characteristic, complemented
2C2×C32×#213C211yesmaximal, characteristic, complemented
3C3×#63830C211yesmaximal, characteristic, complemented
4C3×#63817C211yesmaximal, characteristic, complemented
5C32×#833C211yesmaximal, characteristic, complemented
6C3×#63577C211yesmaximal, characteristic, complemented
7C3×#63558C211yesmaximal, characteristic, complemented
8C3×#5160—31nomaximal
9C3×#4749—31nomaximal
10#10591406C311yesmaximal, characteristic, direct product factor
11C2×C33×D4C2211yescharacteristic
12C3×#3828C2211yescharacteristic
13C2×C32×D12C2211yescharacteristic
14C3×#3754—21no—
15C3×#3753—21no—
16C3×#3748C2211yescharacteristic
17C32×#213—21no—
18C32×#213—21no—
19C32×#208C2211yescharacteristic
20C32×#208C2211yescharacteristic
21C2×C32×C3⋊C8C2211yescharacteristic
22C3×#3716—21no—
23C3×#3714—21no—
24C3×#3712—21no—
25C3×#3710—21no—
26C2×C32×D8—31no—
27#64141C2×C311yescharacteristic, complemented
28C3×#932—31no—
29C3×#932—31no—
30C3×#932—31no—
31C3×#932—61no—
32C2×C3×#213—21no—
33C2×C3×#213D311yescharacteristic, complemented
34C2×C3×#213C2×C311yescharacteristic, complemented
35C3×#908—31no—
36C3×#907—31no—
37C3×#901—31no—
38#63830C2×C311yescharacteristic, complemented
39#63817C2×C311yescharacteristic, complemented
40C32×#100—31no—
41C3×#833—21no—
42C3×#833—31no—
43C3×#833D311yescharacteristic, complemented
44C3×#833C2×C311yescharacteristic, complemented
45C3×#823—31no—
46C3×#822—31no—
47C3×#819—31no—
48#63577C2×C311yescharacteristic, complemented
49#63558C2×C311yescharacteristic, complemented
50C23×C33—21no—
51C22×C32×D3—21no—
52C33×D4—21no—
53C33×D4—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×#472—91no—
67#5160—31no—
68#4749—31no—
69C2×C32×D4—21no—
70C2×C32×D4—21no—
71C2×C32×D4—21no—
72C2×C32×D4—31no—
73C2×C32×D4—41no—
74C2×C32×D4C2×D311yescharacteristic
75C2×C32×D4C2×D311yescharacteristic
76C2×C32×D4C22×C311yescharacteristic
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×D8—61no—
85C32×D8—61no—
86C32×M16—31no—
87C32×M16—31no—
88C2×C32×C8—31no—
89#3754—21no—
90#3753—21no—
91#3748C22×C311yescharacteristic
92C3×#214—61no—
93C3×#214—61no—
94C3×#214—61no—
95C3×#214—62no—
96C3×#213—22no—
97C3×#213—22no—
98C3×#213—21no—
99C3×#213—21no—
100C3×#213—21no—
101C3×#213—21no—
102C3×#213—61no—
103C3×#213—61no—
104C3×#213—61no—
105C3×#213—62no—
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—41no—
129C22×C33—81no—
130C22×C33#4111yescharacteristic
131C2×C3×C3⋊D3—21no—
132C2×C32×D3—41no—
133C2×C32×D3—81no—
134C33×C4C2×D411yescharacteristic
135C33×C4C2×D411yescharacteristic
136C3×C3⋊Dic3—21no—
137C3×AGL(1,ℤ/8)—91no—
138C3×AGL(1,ℤ/8)—91no—
139C2×C3×D8—31no—
140C2×C3×D8—31no—
141C2×C3×D8—61no—
142C3×#131—91no—
143#932—31no—
144#932—31no—
145#932—31no—
146#932—61no—
147C2×#213C3×D311yescharacteristic, complemented
148#908—31no—
149#907—31no—
150#901—31no—
151C3×#108—91no—
152C3×#100—31no—
153C3×#100—31no—
154C3×#100—61no—
155C3×#100—91no—
156#833—31no—
157#833C3×D311yescharacteristic, complemented
158#823—31no—
159#822—31no—
160#819—31no—
161C23×C32—21no—
162C23×C32—22no—
163C23×C32—21no—
164C23×C32—21no—
165C23×C32—41no—
166C23×C32—41no—
167C23×C32—42no—
168C23×C32—61no—
169C22×C3×D3—21no—
170C22×C3×D3—21no—
171C22×C3×D3—41no—
172C32×D4—21no—
173C32×D4—21no—
174C32×D4—22no—
175C32×D4—22no—
176C32×D4—21no—
177C32×D4—21no—
178C32×D4—21no—
179C32×D4—21no—
180C32×D4—41no—
181C32×D4—41no—
182C32×D4—41no—
183C32×D4—41no—
184C32×D4—42no—
185C32×D4—42no—
186C32×D4—61no—
187C32×D4—61no—
188C2×C32×C4—21no—
189C2×C32×C4—21no—
190C2×C32×C4—21no—
191C2×C32×C4—41no—
192C2×C32×C4C22×D311yescharacteristic
193C2×C32×C4C22×D311yescharacteristic
194C2×C32×C4C23×C311yescharacteristic
195#638—21no—
196C4⋊(C3⋊D3)C3×D411yescharacteristic
197C4×C3⋊D3—21no—
198#634C3×D411yescharacteristic
199C3×#67—61no—
200C3×#67—61no—
201C3×#67—121no—
202C3×D12—22no—
203C3×D12—21no—
204C3×D12—21no—
205C3×D12—21no—
206C3×D12—21no—
207C3×D12—31no—
208C3×D12—31no—
209C3×D12—41no—
210C3×D12—61no—
211C3×C4×D3—61no—
212C3×C4×D3—61no—
213C3×C4×D3—121no—
214C3×Dic6—31no—
215C3×Dic6—31no—
216C3×Dic6—61no—
217C32×C8—31no—
218C32×C8—31no—
219C32×C8—61no—
220C32×C8—61no—
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—21no—
228C3×C3⋊C8—41no—
229C3×C3⋊C8—41no—
230C3×C3⋊C8—62no—
231C3×C3⋊C8—61no—
232C3×C3⋊C8—61no—
233C3×C3⋊C8#6711yescharacteristic
234C3×C3⋊C8#6711yescharacteristic
235C3×C3⋊C8C3×D411yescharacteristic
236C3×C3⋊C8C3×D411yescharacteristic
237#472—91noSylow
238C2×C33—21no—
239C2×C33—81no—
240C2×C33#12111yescharacteristic
241C3×C3⋊D3—41no—
242C32×D3—81no—
243C3×#41—91no—
244C2×C3×D4—21no—
245C2×C3×D4—21no—
246C2×C3×D4—21no—
247C2×C3×D4—31no—
248C2×C3×D4—31no—
249C2×C3×D4—41no—
250C2×C3×D4—61no—
251C2×C3×D4D3211yescharacteristic
252C2×C3×D4C2×C3×D311yescharacteristic
253C2×C3×D4C2×C3×D311yescharacteristic
254#235—31no—
255#235—31no—
256#235—61no—
257C2×D12C2×C3×D311yescharacteristic
258C3×SD16—181no—
259C3×SD16—181no—
260C3×D8—61no—
261C3×D8—61no—
262C3×D8—61no—
263C3×D8—61no—
264C3×D8—62no—
265C3×D8—62no—
266C3×D8—181no—
267C3×D8—181no—
268C3×M16—31no—
269C3×M16—31no—
270C3×M16—31no—
271C3×M16—31no—
272C3×M16—61no—
273C3×M16—61no—
274C3×M16—91no—
275C3×M16—181no—
276C2×C3×C8—31no—
277C2×C3×C8—31no—
278C2×C3×C8—61no—
279C2×C3×C8—181no—
280#214—61no—
281#214—61no—
282#214—61no—
283#214—62no—
284#213—21no—
285#213—21no—
286#213—61no—
287#213—61no—
288#213—61no—
289#213—62no—
290#208—31no—
291#208—31no—
292#208—61no—
293#208C2×C3×D311yescharacteristic
294#208C2×C3×D311yescharacteristic
295C2×C3⋊C8C2×C3×D311yescharacteristic
296D24—61no—
297#204—61no—
298ΓL(1,𝔽25)—61no—
299ΓL(1,𝔽25)—61no—
300D3×C8—61no—
301D3×C8—61no—
302C22×C32—21no—
303C22×C32—21no—
304C22×C32—21no—
305C22×C32—41no—
306C22×C32—42no—
307C22×C32—42no—
308C22×C32—42no—
309C22×C32—44no—
310C22×C32—41no—
311C22×C32—41no—
312C22×C32—41no—
313C22×C32—81no—
314C22×C32—81no—
315C22×C32—81no—
316C22×C32—82no—
317C22×C32—82no—
318C22×C32—82no—
319C22×C32—84no—
320C22×C32—121no—
321C22×C32—241no—
322C22×C32#23511yescharacteristic
323C22×C32#23711yescharacteristic
324C22×C32C3×#4111yescharacteristic
325C2×C3⋊D3—21no—
326C2×C3×D3—42no—
327C2×C3×D3—41no—
328C2×C3×D3—41no—
329C2×C3×D3—61no—
330C2×C3×D3—61no—
331C2×C3×D3—82no—
332C2×C3×D3—81no—
333C2×C3×D3—81no—
334C2×C3×D3—121no—
335C32×C4—21no—
336C32×C4—21no—
337C32×C4—21no—
338C32×C4—21no—
339C32×C4—21no—
340C32×C4—21no—
341C32×C4—41no—
342C32×C4—41no—
343C32×C4D3×D411yescharacteristic
344C32×C4D3×D411yescharacteristic
345C32×C4D3×D411yescharacteristic
346C32×C4C2×#6711yescharacteristic
347C32×C4C2×C3×D411yescharacteristic
348C32×C4C2×C3×D411yescharacteristic
349C3⋊Dic3—21no—
350C3×Dic3—61no—
351C3×Dic3—61no—
352C3×Dic3—121no—
353AGL(1,ℤ/8)—91no—
354AGL(1,ℤ/8)—91no—
355C2×D8—31no—
356#131—91no—
357#108—91no—
358#100—31no—
359#100—91no—
360C33#47211yescharacteristic, complemented, Sylow
361C23×C3—22no—
362C23×C3—21no—
363C23×C3—21no—
364C23×C3—21no—
365C23×C3—41no—
366C23×C3—41no—
367C23×C3—42no—
368C23×C3—61no—
369C23×C3—61no—
370C23×C3—121no—
371C22×D3—21no—
372C3×Q8—91no—
373C3×D4—22no—
374C3×D4—22no—
375C3×D4—21no—
376C3×D4—21no—
377C3×D4—21no—
378C3×D4—21no—
379C3×D4—21no—
380C3×D4—21no—
381C3×D4—41no—
382C3×D4—41no—
383C3×D4—41no—
384C3×D4—41no—
385C3×D4—42no—
386C3×D4—42no—
387C3×D4—62no—
388C3×D4—61no—
389C3×D4—61no—
390C3×D4—61no—
391C3×D4—61no—
392C3×D4—91no—
393C3×D4—121no—
394C3×D4—181no—
395C2×C3×C4—21no—
396C2×C3×C4—21no—
397C2×C3×C4—21no—
398C2×C3×C4—41no—
399C2×C3×C4—181no—
400C2×C3×C4C2×D3211yescharacteristic
401C2×C3×C4C22×C3×D311yescharacteristic
402C2×C3×C4C22×C3×D311yescharacteristic
403#67—61no—
404#67—61no—
405#67—121no—
406D12—21no—
407D12—21no—
408D12—31no—
409D12—31no—
410D12—61no—
411C4×D3—61no—
412C4×D3—61no—
413C4×D3—121no—
414Dic6—31no—
415Dic6—31no—
416Dic6—61no—
417C3×C8—31no—
418C3×C8—31no—
419C3×C8—31no—
420C3×C8—31no—
421C3×C8—61no—
422C3×C8—61no—
423C3×C8—61no—
424C3×C8—61no—
425C3×C8—61no—
426C3×C8—61no—
427C3×C8—121no—
428C3×C8—121no—
429C3×C8—181no—
430C3⋊C8—21no—
431C3⋊C8—21no—
432C3⋊C8—62no—
433C3⋊C8—61no—
434C3⋊C8—61no—
435C3⋊C8C3×#6711yescharacteristic
436C3⋊C8C3×#6711yescharacteristic
437C2×C32—21no—
438C2×C32—22no—
439C2×C32—21no—
440C2×C32—21no—
441C2×C32—21no—
442C2×C32—21no—
443C2×C32—21no—
444C2×C32—41no—
445C2×C32—41no—
446C2×C32—41no—
447C2×C32—42no—
448C2×C32—82no—
449C2×C32—82no—
450C2×C32—82no—
451C2×C32—84no—
452C2×C32—81no—
453C2×C32—81no—
454C2×C32—81no—
455C2×C32—241no—
456C2×C32#88411yescharacteristic
457C2×C32#93811yescharacteristic
458C2×C32C3×#12111yescharacteristic
459C3⋊D3—41no—
460C3×D3—82no—
461C3×D3—81no—
462C3×D3—81no—
463C3×D3—121no—
464C3×D3—121no—
465C3×D3—241no—
466#41—91no—
467C2×D4—31no—
468C2×D4C3×D3211yescharacteristic
469SD16—181no—
470SD16—181no—
471D8—61no—
472D8—61no—
473D8—181no—
474D8—181no—
475M16—31no—
476M16—31no—
477M16—91no—
478M16—181no—
479C2×C8—31no—
480C2×C8—181no—
481C22×C3—21no—
482C22×C3—21no—
483C22×C3—21no—
484C22×C3—42no—
485C22×C3—42no—
486C22×C3—42no—
487C22×C3—44no—
488C22×C3—41no—
489C22×C3—41no—
490C22×C3—41no—
491C22×C3—41no—
492C22×C3—81no—
493C22×C3—81no—
494C22×C3—81no—
495C22×C3—82no—
496C22×C3—82no—
497C22×C3—82no—
498C22×C3—84no—
499C22×C3—122no—
500C22×C3—121no—
501C22×C3—121no—
502C22×C3—181no—
503C22×C3—241no—
504C22×C3—241no—
505C22×C3—242no—
506C22×C3#380411yescharacteristic
507C22×C3C3×#23511yescharacteristic
508C22×C3C3×#23711yescharacteristic
509C2×D3—41no—
510C2×D3—61no—
511C2×D3—61no—
512C2×D3—81no—
513C2×D3—121no—
514C3×C4—21no—
515C3×C4—21no—
516C3×C4—21no—
517C3×C4—21no—
518C3×C4—21no—
519C3×C4—21no—
520C3×C4—41no—
521C3×C4—41no—
522C3×C4—181no—
523C3×C4D3×#6711yescharacteristic
524C3×C4#381111yescharacteristic
525C3×C4C3×D3×D411yescharacteristic
526C3×C4C3×D3×D411yescharacteristic
527C3×C4C3×D3×D411yescharacteristic
528C3×C4C2×C3×#6711yescharacteristic
529Dic3—61no—
530Dic3—61no—
531Dic3—121no—
532C32—21no—
533C32—21no—
534C32—21no—
535C32—41no—
536C32#474911yescharacteristic, complemented
537C32#516011yescharacteristic, complemented
538C32C3×#47211yescharacteristic, complemented
539C23—21no—
540C23—61no—
541Q8—91no—
542D4—21no—
543D4—21no—
544D4—61no—
545D4—61no—
546D4—91no—
547D4—181no—
548C2×C4—181no—
549C2×C4C2×C3×D3211yescharacteristic
550C8—31no—
551C8—31no—
552C8—61no—
553C8—61no—
554C8—181no—
555C2×C3—22no—
556C2×C3—21no—
557C2×C3—21no—
558C2×C3—21no—
559C2×C3—21no—
560C2×C3—21no—
561C2×C3—21no—
562C2×C3—41no—
563C2×C3—41no—
564C2×C3—42no—
565C2×C3—41no—
566C2×C3—82no—
567C2×C3—82no—
568C2×C3—82no—
569C2×C3—84no—
570C2×C3—81no—
571C2×C3—81no—
572C2×C3—81no—
573C2×C3—242no—
574C2×C3—241no—
575C2×C3—241no—
576C2×C3—361no—
577C2×C3#6395811yescharacteristic
578C2×C3C3×#88411yescharacteristic
579C2×C3C3×#93811yescharacteristic
580D3—81no—
581D3—121no—
582D3—121no—
583D3—241no—
584C22—41no—
585C22—81no—
586C22—121no—
587C22—181no—
588C22—241no—
589C22C3×#380411yescharacteristic
590C4—181no—
591C4C3×D3×#6711yescharacteristic
592C4C3×#381111yescharacteristic
593C3—21no—
594C3—21no—
595C3—21no—
596C3—41no—
597C3#1059140611yescharacteristic, direct product factor
598C3C3×#474911yescharacteristic, complemented
599C3C3×#516011yescharacteristic, complemented
600C2—21no—
601C2—81no—
602C2—241no—
603C2—361no—
604C2C3×#6395811yescharacteristic
subgroupG/H
centerC2×C3#63958
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C4C2×C3×D32
FittingC2×C33×D4C22
socleC2×C33#121
hypercenterC2×C3×D4D32
hypocenterC32C3×#472
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C2×C3 ◁ C2×C3×C4 ◁ C2×C3×D4

Complements

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

NH ≅ G/Ntype
1. C3×#64141C2semidirect
2. C2×C32×#213C2semidirect
3. C3×#63830C2semidirect
4. C3×#63817C2semidirect
5. C32×#833C2semidirect
6. C3×#63577C2semidirect
7. C3×#63558C2semidirect
10. #10591406C3direct (N × H)
27. #64141C2×C3semidirect
33. C2×C3×#213D3semidirect
34. C2×C3×#213C2×C3semidirect
38. #63830C2×C3semidirect
39. #63817C2×C3semidirect
43. C3×#833D3semidirect
44. C3×#833C2×C3semidirect
48. #63577C2×C3semidirect
49. #63558C2×C3semidirect
147. C2×#213C3×D3semidirect
157. #833C3×D3semidirect
360. C33#472semidirect
536. C32#4749semidirect
537. C32#5160semidirect
538. C32C3×#472semidirect
597. C3#10591406direct (N × H)
598. C3C3×#4749semidirect
599. C3C3×#5160semidirect

Maximal subgroups

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

subgroupindex
C3×#641412
C2×C32×#2132
C3×#638302
C3×#638172
C32×#8332
C3×#635772
C3×#635582
C3×#51603
C3×#47493
#105914063