Finite Group Databasebeta

← C3×#10591507 #13158540

Subgroups — C3×#10591507 #13158540

Order 1728.

652 conjugacy classes of subgroups.

The subgroup lattice has 586 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#3754C211yesmaximal, characteristic, complemented
2C32×#932C211yesmaximal, characteristic
3C3×#63831C211yesmaximal, characteristic, complemented
4C3×#63815C211yesmaximal, characteristic, complemented
5C32×#833C211yesmaximal, characteristic, complemented
6C3×#63578C211yesmaximal, characteristic, complemented
7C3×#63561C211yesmaximal, characteristic, complemented
8C3×#5200—31nomaximal
9C3×#4748—31nomaximal
10#10591507C311yesmaximal, characteristic, direct product factor
11C2×C33×D4C2211yescharacteristic
12C2×C3×#634C2211yescharacteristic
13C32×#235C2211yescharacteristic
14C3×#3754—21no—
15C3×#3754—21no—
16C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
17C32×#214—21no—
18C32×#213—21no—
19C32×#208C2211yescharacteristic
20C32×#208C2211yescharacteristic
21C32×#208C2211yescharacteristic
22C3×#3719—21no—
23C3×#3716—21no—
24C3×#3711—21no—
25C3×D3×C3⋊C8—21no—
26C32×AGL(1,ℤ/8)—31no—
27C2×#3754C2×C311yescharacteristic, complemented
28C3×#951—31no—
29C3×#948—31no—
30C2×C3×#214—31no—
31C2×C3×#214—31no—
32C2×C3×#214—61no—
33C3×#932—21no—
34C3×#932D311yescharacteristic
35C3×#932C2×C311yescharacteristic
36C3×#909—31no—
37C3×#907—31no—
38#63831C2×C311yescharacteristic, complemented
39#63815C2×C311yescharacteristic, complemented
40C32×#100—31no—
41C3×#833—21no—
42C3×#833D311yescharacteristic, complemented
43C3×#833C2×C311yescharacteristic, complemented
44C3×#824—31no—
45C3×#824—31no—
46C3×#822—31no—
47C3×#822—31no—
48#63578C2×C311yescharacteristic, complemented
49#63561C2×C311yescharacteristic, complemented
50C23×C33—21no—
51C33×D4—21no—
52C33×D4—21no—
53C2×C33×C4C2311yescharacteristic
54C2×C3×C3⋊Dic3—21no—
55C3×#634—21no—
56C3×#634—21no—
57C32×#67—21no—
58C32×D12D411yescharacteristic
59C32×C4×D3—21no—
60C32×Dic6D411yescharacteristic
61C3×C3⋊(C3⋊C8)D411yescharacteristic
62C3×C3⋊(C3⋊C8)D411yescharacteristic
63C32×C3⋊C8—21no—
64C32×C3⋊C8—21no—
65C32×C3⋊C8—21no—
66C3×#471—91no—
67#5200—31no—
68#4748—31no—
69C32×#41—31no—
70C2×C32×D4—21no—
71C2×C32×D4—21no—
72C2×C32×D4—21no—
73C2×C32×D4—41no—
74C2×C32×D4C2×D311yescharacteristic
75C2×C32×D4C2×D311yescharacteristic
76C2×C32×D4C22×C311yescharacteristic
77C2×#634C22×C311yescharacteristic
78C3×#235—21no—
79C3×#235C2×D311yescharacteristic
80C3×#235C22×C311yescharacteristic
81C2×C3×Dic6—31no—
82C2×C3×Dic6—31no—
83C2×C3×Dic6—61no—
84C32×SD16—61no—
85C32×D8—61no—
86C32×M16—31no—
87C32×M16—31no—
88C32×M16—31no—
89#3754—21no—
90#3754—21no—
91C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
92C3×#216—61no—
93C3×#214—22no—
94C3×#214—21no—
95C3×#214—21no—
96C3×#214—61no—
97C3×#214—61no—
98C3×#214—61no—
99C3×#214—61no—
100C3×#214—61no—
101C3×#214—62no—
102C3×#214—62no—
103C3×#213—22no—
104C3×#213—21no—
105C3×#213—21no—
106C3×#208—21no—
107C3×#208—21no—
108C3×#208—21no—
109C3×#208—61no—
110C3×#208C2×D311yescharacteristic
111C3×#208C2×D311yescharacteristic
112C3×#208C2×D311yescharacteristic
113C3×#208C22×C311yescharacteristic
114C3×#208C22×C311yescharacteristic
115C3×#208C22×C311yescharacteristic
116C2×C3×C3⋊C8—31no—
117C2×C3×C3⋊C8—31no—
118C2×C3×C3⋊C8—61no—
119C2×C3×C3⋊C8—61no—
120C3×Dic12—61no—
121C3×#204—61no—
122C3×ΓL(1,𝔽25)—61no—
123C3×D3×C8—61no—
124#3719—21no—
125#3716—21no—
126#3711—21no—
127D3×C3⋊C8—21no—
128C22×C33—41no—
129C22×C33—81no—
130C22×C33#4111yescharacteristic
131C2×C32×D3—21no—
132C33×C4C2×D411yescharacteristic
133C33×C4C2×D411yescharacteristic
134C3×C3⋊Dic3—41no—
135C32×Dic3—21no—
136C3×#137—91no—
137C3×AGL(1,ℤ/8)—31no—
138C3×AGL(1,ℤ/8)—31no—
139C3×AGL(1,ℤ/8)—61no—
140C2×C3×SD16—91no—
141C3×#131—91no—
142#951—31no—
143#948—31no—
144C2×#214—31no—
145C2×#214—31no—
146C2×#214—61no—
147#932C3×D311yescharacteristic
148#909—31no—
149#907—31no—
150C3×#108—91no—
151C3×#101—91no—
152C3×#100—31no—
153C3×#100—31no—
154C3×#100—61no—
155#833C3×D311yescharacteristic, complemented
156#824—31no—
157#824—31no—
158#822—31no—
159#822—31no—
160C23×C32—21no—
161C23×C32—22no—
162C23×C32—21no—
163C23×C32—21no—
164C23×C32—41no—
165C23×C32—41no—
166C23×C32—42no—
167C32×Q8—31no—
168C32×D4—21no—
169C32×D4—21no—
170C32×D4—22no—
171C32×D4—22no—
172C32×D4—21no—
173C32×D4—21no—
174C32×D4—21no—
175C32×D4—21no—
176C32×D4—31no—
177C32×D4—41no—
178C32×D4—41no—
179C32×D4—41no—
180C32×D4—41no—
181C32×D4—42no—
182C32×D4—42no—
183C32×D4—61no—
184C2×C32×C4—21no—
185C2×C32×C4—21no—
186C2×C32×C4—21no—
187C2×C32×C4—41no—
188C2×C32×C4—61no—
189C2×C32×C4C22×D311yescharacteristic
190C2×C32×C4C22×D311yescharacteristic
191C2×C32×C4C23×C311yescharacteristic
192C2×C3⋊Dic3—21no—
193#634—21no—
194#634—21no—
195C3×#67—21no—
196C3×#67—21no—
197C3×#67—41no—
198C2×C3×Dic3—61no—
199C2×C3×Dic3—61no—
200C2×C3×Dic3—121no—
201C3×D12—21no—
202C3×D12#6711yescharacteristic
203C3×D12C3×D411yescharacteristic
204C3×C4×D3—21no—
205C3×C4×D3—21no—
206C3×C4×D3—41no—
207C3×Dic6—21no—
208C3×Dic6—62no—
209C3×Dic6—61no—
210C3×Dic6—61no—
211C3×Dic6—61no—
212C3×Dic6—61no—
213C3×Dic6—121no—
214C3×Dic6#6711yescharacteristic
215C3×Dic6C3×D411yescharacteristic
216C32×C8—61no—
217C32×C8—61no—
218C32×C8—61no—
219C3⋊(C3⋊C8)C3×D411yescharacteristic
220C3⋊(C3⋊C8)C3×D411yescharacteristic
221C3×C3⋊C8—22no—
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—31no—
229C3×C3⋊C8—31no—
230C3×C3⋊C8—31no—
231C3×C3⋊C8—31no—
232C3×C3⋊C8—41no—
233C3×C3⋊C8—41no—
234C3×C3⋊C8—61no—
235C3×C3⋊C8—61no—
236#471—91noSylow
237C2×C33—21no—
238C2×C33—81no—
239C2×C33#12111yescharacteristic
240C32×D3—41no—
241C3×#41—31no—
242C3×#41—31no—
243C3×#41—61no—
244C2×C3×Q8—91no—
245C2×C3×D4—21no—
246C2×C3×D4—21no—
247C2×C3×D4—21no—
248C2×C3×D4—41no—
249C2×C3×D4D3211yescharacteristic
250C2×C3×D4C2×C3×D311yescharacteristic
251C2×C3×D4C2×C3×D311yescharacteristic
252#235C2×C3×D311yescharacteristic
253C2×Dic6—31no—
254C2×Dic6—31no—
255C2×Dic6—61no—
256C3×Q16—181no—
257C3×SD16—61no—
258C3×SD16—61no—
259C3×SD16—62no—
260C3×SD16—181no—
261C3×SD16—181no—
262C3×SD16—181no—
263C3×D8—61no—
264C3×D8—61no—
265C3×D8—62no—
266C3×M16—31no—
267C3×M16—31no—
268C3×M16—31no—
269C3×M16—31no—
270C3×M16—31no—
271C3×M16—31no—
272C3×M16—61no—
273C3×M16—61no—
274C3×M16—61no—
275C3×M16—181no—
276C2×C3×C8—91no—
277C2×C3×C8—181no—
278#216—61no—
279#214—21no—
280#214—61no—
281#214—61no—
282#214—61no—
283#214—61no—
284#214—61no—
285#214—62no—
286#214—62no—
287#213—21no—
288#208—61no—
289#208C2×C3×D311yescharacteristic
290#208C2×C3×D311yescharacteristic
291#208C2×C3×D311yescharacteristic
292C2×C3⋊C8—31no—
293C2×C3⋊C8—31no—
294C2×C3⋊C8—61no—
295C2×C3⋊C8—61no—
296Dic12—61no—
297#204—61no—
298ΓL(1,𝔽25)—61no—
299D3×C8—61no—
300C22×C32—21no—
301C22×C32—21no—
302C22×C32—21no—
303C22×C32—41no—
304C22×C32—42no—
305C22×C32—42no—
306C22×C32—42no—
307C22×C32—44no—
308C22×C32—41no—
309C22×C32—41no—
310C22×C32—41no—
311C22×C32—61no—
312C22×C32—81no—
313C22×C32—81no—
314C22×C32—81no—
315C22×C32—82no—
316C22×C32—82no—
317C22×C32—82no—
318C22×C32—84no—
319C22×C32#23511yescharacteristic
320C22×C32#23711yescharacteristic
321C22×C32C3×#4111yescharacteristic
322C2×C3×D3—21no—
323C2×C3×D3—21no—
324C2×C3×D3—41no—
325C32×C4—21no—
326C32×C4—21no—
327C32×C4—21no—
328C32×C4—21no—
329C32×C4—21no—
330C32×C4—21no—
331C32×C4—41no—
332C32×C4—41no—
333C32×C4—61no—
334C32×C4D3×D411yescharacteristic
335C32×C4D3×D411yescharacteristic
336C32×C4D3×D411yescharacteristic
337C32×C4C2×#6711yescharacteristic
338C32×C4C2×C3×D411yescharacteristic
339C32×C4C2×C3×D411yescharacteristic
340C3⋊Dic3—41no—
341C3×Dic3—21no—
342C3×Dic3—21no—
343C3×Dic3—41no—
344C3×Dic3—122no—
345C3×Dic3—121no—
346C3×Dic3—121no—
347#137—91no—
348AGL(1,ℤ/8)—31no—
349C2×SD16—91no—
350#131—91no—
351#108—91no—
352#101—91no—
353#100—31no—
354C33#47111yescharacteristic, complemented, Sylow
355C23×C3—22no—
356C23×C3—21no—
357C23×C3—21no—
358C23×C3—21no—
359C23×C3—41no—
360C23×C3—41no—
361C23×C3—42no—
362C3×Q8—31no—
363C3×Q8—31no—
364C3×Q8—61no—
365C3×Q8—181no—
366C3×Q8—181no—
367C3×D4—22no—
368C3×D4—22no—
369C3×D4—21no—
370C3×D4—21no—
371C3×D4—21no—
372C3×D4—21no—
373C3×D4—21no—
374C3×D4—21no—
375C3×D4—31no—
376C3×D4—31no—
377C3×D4—41no—
378C3×D4—41no—
379C3×D4—41no—
380C3×D4—41no—
381C3×D4—42no—
382C3×D4—42no—
383C3×D4—61no—
384C3×D4—61no—
385C3×D4—61no—
386C3×D4—121no—
387C2×C3×C4—21no—
388C2×C3×C4—21no—
389C2×C3×C4—21no—
390C2×C3×C4—41no—
391C2×C3×C4—61no—
392C2×C3×C4—61no—
393C2×C3×C4—121no—
394C2×C3×C4—181no—
395C2×C3×C4C2×D3211yescharacteristic
396C2×C3×C4C22×C3×D311yescharacteristic
397C2×C3×C4C22×C3×D311yescharacteristic
398#67—21no—
399C2×Dic3—61no—
400C2×Dic3—61no—
401C2×Dic3—121no—
402D12C3×#6711yescharacteristic
403C4×D3—21no—
404Dic6—62no—
405Dic6—61no—
406Dic6—61no—
407Dic6—61no—
408Dic6—61no—
409Dic6—121no—
410Dic6C3×#6711yescharacteristic
411C3×C8—62no—
412C3×C8—61no—
413C3×C8—61no—
414C3×C8—61no—
415C3×C8—61no—
416C3×C8—61no—
417C3×C8—61no—
418C3×C8—91no—
419C3×C8—91no—
420C3×C8—121no—
421C3×C8—121no—
422C3⋊C8—21no—
423C3⋊C8—21no—
424C3⋊C8—21no—
425C3⋊C8—31no—
426C3⋊C8—31no—
427C3⋊C8—31no—
428C3⋊C8—31no—
429C3⋊C8—61no—
430C3⋊C8—61no—
431C2×C32—21no—
432C2×C32—22no—
433C2×C32—21no—
434C2×C32—21no—
435C2×C32—21no—
436C2×C32—21no—
437C2×C32—21no—
438C2×C32—41no—
439C2×C32—41no—
440C2×C32—41no—
441C2×C32—42no—
442C2×C32—82no—
443C2×C32—82no—
444C2×C32—82no—
445C2×C32—84no—
446C2×C32—81no—
447C2×C32—81no—
448C2×C32—81no—
449C2×C32—121no—
450C2×C32#88411yescharacteristic
451C2×C32#93911yescharacteristic
452C2×C32C3×#12111yescharacteristic
453C3×D3—41no—
454C3×D3—41no—
455C3×D3—81no—
456#41—31no—
457C2×Q8—91no—
458C2×D4C3×D3211yescharacteristic
459Q16—181no—
460SD16—61no—
461SD16—181no—
462SD16—181no—
463SD16—181no—
464D8—61no—
465M16—31no—
466M16—31no—
467M16—31no—
468M16—181no—
469C2×C8—91no—
470C2×C8—181no—
471C22×C3—21no—
472C22×C3—21no—
473C22×C3—21no—
474C22×C3—42no—
475C22×C3—42no—
476C22×C3—42no—
477C22×C3—44no—
478C22×C3—41no—
479C22×C3—41no—
480C22×C3—41no—
481C22×C3—41no—
482C22×C3—61no—
483C22×C3—61no—
484C22×C3—81no—
485C22×C3—81no—
486C22×C3—81no—
487C22×C3—82no—
488C22×C3—82no—
489C22×C3—82no—
490C22×C3—84no—
491C22×C3—121no—
492C22×C3#380411yescharacteristic
493C22×C3C3×#23511yescharacteristic
494C22×C3C3×#23711yescharacteristic
495C2×D3—21no—
496C3×C4—21no—
497C3×C4—21no—
498C3×C4—21no—
499C3×C4—21no—
500C3×C4—21no—
501C3×C4—21no—
502C3×C4—41no—
503C3×C4—41no—
504C3×C4—61no—
505C3×C4—61no—
506C3×C4—121no—
507C3×C4—361no—
508C3×C4D3×#6711yescharacteristic
509C3×C4#381111yescharacteristic
510C3×C4C3×D3×D411yescharacteristic
511C3×C4C3×D3×D411yescharacteristic
512C3×C4C3×D3×D411yescharacteristic
513C3×C4C2×C3×#6711yescharacteristic
514Dic3—21no—
515Dic3—122no—
516Dic3—121no—
517Dic3—121no—
518C32—21no—
519C32—21no—
520C32—21no—
521C32—41no—
522C32#474811yescharacteristic, complemented
523C32#520011yescharacteristic, complemented
524C32C3×#47111yescharacteristic, complemented
525C23—21no—
526Q8—31no—
527Q8—181no—
528Q8—181no—
529D4—21no—
530D4—21no—
531D4—31no—
532D4—61no—
533C2×C4—61no—
534C2×C4—181no—
535C2×C4C2×C3×D3211yescharacteristic
536C8—61no—
537C8—61no—
538C8—61no—
539C8—91no—
540C8—91no—
541C2×C3—22no—
542C2×C3—21no—
543C2×C3—21no—
544C2×C3—21no—
545C2×C3—21no—
546C2×C3—21no—
547C2×C3—21no—
548C2×C3—41no—
549C2×C3—41no—
550C2×C3—42no—
551C2×C3—41no—
552C2×C3—82no—
553C2×C3—82no—
554C2×C3—82no—
555C2×C3—84no—
556C2×C3—81no—
557C2×C3—81no—
558C2×C3—81no—
559C2×C3—121no—
560C2×C3—121no—
561C2×C3—241no—
562C2×C3#6397111yescharacteristic
563C2×C3C3×#88411yescharacteristic
564C2×C3C3×#93911yescharacteristic
565D3—41no—
566C22—41no—
567C22—61no—
568C22—81no—
569C22C3×#380411yescharacteristic
570C4—61no—
571C4—361no—
572C4C3×D3×#6711yescharacteristic
573C4C3×#381111yescharacteristic
574C3—21no—
575C3—21no—
576C3—21no—
577C3—41no—
578C3#1059150711yescharacteristic, direct product factor
579C3C3×#474811yescharacteristic, complemented
580C3C3×#520011yescharacteristic, complemented
581C2—21no—
582C2—81no—
583C2—121no—
584C2C3×#6397111yescharacteristic
subgroupG/H
centerC2×C3#63971
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C4C2×C3×D32
FittingC2×C33×D4C22
socleC2×C33#121
prime radicalC32×#932C2
hypercenterC2×C3×D4D32
hypocenterC32C3×#471
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. C2×C3×#3754C2semidirect
3. C3×#63831C2semidirect
4. C3×#63815C2semidirect
5. C32×#833C2semidirect
6. C3×#63578C2semidirect
7. C3×#63561C2semidirect
10. #10591507C3direct (N × H)
27. C2×#3754C2×C3semidirect
38. #63831C2×C3semidirect
39. #63815C2×C3semidirect
42. C3×#833D3semidirect
43. C3×#833C2×C3semidirect
48. #63578C2×C3semidirect
49. #63561C2×C3semidirect
155. #833C3×D3semidirect
354. C33#471semidirect
522. C32#4748semidirect
523. C32#5200semidirect
524. C32C3×#471semidirect
578. C3#10591507direct (N × H)
579. C3C3×#4748semidirect
580. C3C3×#5200semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#37542
C32×#9322
C3×#638312
C3×#638152
C32×#8332
C3×#635782
C3×#635612
C3×#52003
C3×#47483
#105915073