Finite Group Databasebeta

← C3×D3×#833 #13158513

Subgroups — C3×D3×#833 #13158513

Order 1728.

1126 conjugacy classes of subgroups.

The subgroup lattice has 572 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C32×D3×D4C211yesmaximal, characteristic
2C3×D3×#208C212yesmaximal, complemented
3C3×#63660C211yesmaximal, characteristic, complemented
4C32×#833C211yesmaximal, characteristic, complemented
5C3×#63558C212yesmaximal, complemented
6C2×C3×#833—31nomaximal
7C3×D3×#100—31nomaximal
8D3×#833C311yesmaximal, characteristic, direct product factor
9C23×C32×D3C412yes—
10C2×C33×D4C2211yescharacteristic
11C2×C32×#67C412yes—
12C32×D3×D4—22no—
13C32×D3×D4—22no—
14C2×C32×D12C2211yescharacteristic
15C2×C32×C4×D3C2211yescharacteristic
16C3×#3748C2212yescomplemented
17C32×#208C2212yescomplemented
18C3×#3711—22no—
19C3×D3×C3⋊C8—22no—
20C22×C32×D4—31no—
21C2×C3×D3×D4—21no—
22C2×C3×D3×D4D311yescharacteristic
23C2×C3×D3×D4C2×C311yescharacteristic
24C2×C3×#208—32no—
25C3×D3×M16—32no—
26D3×#208C2×C312yescomplemented
27C32×#100—31no—
28#63660C2×C311yescharacteristic, complemented
29C3×#833—21no—
30C3×#833—31no—
31C3×#833—31no—
32C3×#833—32no—
33C3×#833—61no—
34C3×#833D311yescharacteristic, direct product factor
35C3×#833C2×C311yescharacteristic, complemented
36C3×#823—32no—
37#63558C2×C312yescomplemented
38C23×C33C2×C412yes—
39C22×C32×D3—22no—
40C22×C32×D3—22no—
41C22×C32×D3—44no—
42C22×C32×D3C2×C411yescharacteristic
43C22×C32×D3C2×C412yes—
44C33×D4—22no—
45C2×C33×C4C2311yescharacteristic
46C32×#67—24no—
47C2×C32×Dic3C2×C411yescharacteristic
48C32×D12—22no—
49C32×C4×D3D412yes—
50C32×C4×D3D412yes—
51C3×C3⋊(C3⋊C8)—22no—
52C32×C3⋊C8—22no—
53C2×C3×#100—91no—
54C2×#833—31no—
55D3×#100—31no—
56C24×C32—32no—
57C23×C3×D3—22no—
58C23×C3×D3Dic312yes—
59C23×C3×D3C3×C412yes—
60C2×C32×D4—21no—
61C2×C32×D4—21no—
62C2×C32×D4—21no—
63C2×C32×D4—32no—
64C2×C32×D4—31no—
65C2×C32×D4—41no—
66C2×C32×D4—62no—
67C2×C32×D4—62no—
68C2×C32×D4C2×D311yescharacteristic
69C2×C32×D4C2×D311yescharacteristic
70C2×C32×D4C22×C311yescharacteristic
71C22×C32×C4—31no—
72C2×C3×#67—22no—
73C2×C3×#67Dic312yes—
74C2×C3×#67C3×C412yes—
75C3×D3×D4—24no—
76C3×D3×D4—24no—
77C3×D3×D4—22no—
78C3×D3×D4—22no—
79C3×D3×D4—22no—
80C3×D3×D4—22no—
81C2×C3×D12—21no—
82C2×C3×D12C2×D311yescharacteristic
83C2×C3×D12C22×C311yescharacteristic
84C2×C3×C4×D3—21no—
85C2×C3×C4×D3C2×D311yescharacteristic
86C2×C3×C4×D3C22×C311yescharacteristic
87C32×M16—32no—
88#3748C22×C312yescomplemented
89C3×#208—22no—
90C3×#208—32no—
91C3×#208—32no—
92C3×#208—62no—
93C3×#208—62no—
94C3×#208C2×D312yescomplemented
95C3×#208C22×C312yescomplemented
96C2×C3×C3⋊C8—62no—
97C3×ΓL(1,𝔽25)—62no—
98C3×D3×C8—62no—
99#3711—22no—
100D3×C3⋊C8—22no—
101C22×C33—22no—
102C22×C33—42no—
103C22×C33C22×C411yescharacteristic
104C2×C32×D3—22no—
105C2×C32×D3—22no—
106C2×C32×D3—44no—
107C2×C32×D3—44no—
108C2×C32×D3—42no—
109C2×C32×D3C22⋊C411yescharacteristic
110C2×C32×D3C22⋊C411yescharacteristic
111C33×C4C2×D412yes—
112C32×Dic3C22⋊C412yes—
113C22×C3×D4—31no—
114C22×C3×D4—31no—
115C22×C3×D4—61no—
116C2×D3×D4C3×D311yescharacteristic
117C2×C3×M16—92no—
118C2×#208—32no—
119D3×M16—32no—
120C3×#100—31no—
121C3×#100—31no—
122C3×#100—61no—
123C3×#100—92no—
124C3×#100—91no—
125#833—31no—
126#833—31no—
127#833—32no—
128#833—61no—
129#833C3×D311yescharacteristic, direct product factor
130#823—32no—
131C23×C32—22no—
132C23×C32—22no—
133C23×C32—22no—
134C23×C32—31no—
135C23×C32—32no—
136C23×C32—42no—
137C23×C32—62no—
138C23×C32—62no—
139C23×C32—124no—
140C23×C32C4×D312yes—
141C23×C32C2×Dic312yes—
142C23×C32C2×C3×C412yes—
143C22×C3×D3—24no—
144C22×C3×D3—24no—
145C22×C3×D3—22no—
146C22×C3×D3—22no—
147C22×C3×D3—22no—
148C22×C3×D3—22no—
149C22×C3×D3—21no—
150C22×C3×D3—22no—
151C22×C3×D3—48no—
152C22×C3×D3—44no—
153C22×C3×D3—44no—
154C22×C3×D3C2×Dic311yescharacteristic
155C22×C3×D3C2×Dic312yes—
156C22×C3×D3C2×C3×C411yescharacteristic
157C22×C3×D3C2×C3×C412yes—
158C32×D4—22no—
159C32×D4—24no—
160C32×D4—22no—
161C32×D4—22no—
162C32×D4—42no—
163C32×D4—42no—
164C32×D4—44no—
165C32×D4—64no—
166C32×D4—62no—
167C2×C32×C4—21no—
168C2×C32×C4—21no—
169C2×C32×C4—21no—
170C2×C32×C4—31no—
171C2×C32×C4—32no—
172C2×C32×C4—32no—
173C2×C32×C4—41no—
174C2×C32×C4C22×D311yescharacteristic
175C2×C32×C4C22×D311yescharacteristic
176C2×C32×C4C23×C311yescharacteristic
177C3×#67—28no—
178C3×#67—24no—
179C3×#67—24no—
180C2×C3×Dic3—21no—
181C2×C3×Dic3C2×Dic311yescharacteristic
182C2×C3×Dic3C2×C3×C411yescharacteristic
183C3×D12—24no—
184C3×D12—22no—
185C3×D12—22no—
186C3×C4×D3—22no—
187C3×C4×D3—22no—
188C3×C4×D3#6712yes—
189C3×C4×D3#6712yes—
190C3×C4×D3C3×D412yes—
191C3×C4×D3C3×D412yes—
192C32×C8—62no—
193C3⋊(C3⋊C8)—22no—
194C3×C3⋊C8—22no—
195C3×C3⋊C8—22no—
196C3×C3⋊C8—42no—
197C3×C3⋊C8—62no—
198C3×C3⋊C8—62no—
199C3×C3⋊C8—122no—
200C2×#100—91noSylow
201C2×C33—21no—
202C2×C33—42no—
203C2×C33C2×C22⋊C411yescharacteristic
204C32×D3—21no—
205C32×D3—42no—
206C32×D3#10012yescomplemented
207C24×C3—32no—
208C24×C3—32no—
209C24×C3—62no—
210C23×D3C3×Dic312yes—
211C2×C3×D4—21no—
212C2×C3×D4—21no—
213C2×C3×D4—21no—
214C2×C3×D4—32no—
215C2×C3×D4—32no—
216C2×C3×D4—31no—
217C2×C3×D4—31no—
218C2×C3×D4—41no—
219C2×C3×D4—61no—
220C2×C3×D4—62no—
221C2×C3×D4—62no—
222C2×C3×D4—62no—
223C2×C3×D4—62no—
224C2×C3×D4—62no—
225C2×C3×D4—64no—
226C2×C3×D4—64no—
227C2×C3×D4D3211yescharacteristic
228C2×C3×D4C2×C3×D311yescharacteristic
229C2×C3×D4C2×C3×D311yescharacteristic
230C22×C3×C4—31no—
231C22×C3×C4—31no—
232C22×C3×C4—61no—
233C2×#67C3×Dic312yes—
234D3×D4—22no—
235D3×D4—22no—
236C2×D12C2×C3×D311yescharacteristic
237C2×C4×D3C2×C3×D311yescharacteristic
238C3×M16—32no—
239C3×M16—32no—
240C3×M16—62no—
241C3×M16—92no—
242C3×M16—182no—
243C2×C3×C8—182no—
244#208—32no—
245#208—32no—
246#208—62no—
247#208—62no—
248#208C2×C3×D312yescomplemented
249C2×C3⋊C8—62no—
250ΓL(1,𝔽25)—62no—
251D3×C8—62no—
252C22×C32—22no—
253C22×C32—24no—
254C22×C32—22no—
255C22×C32—22no—
256C22×C32—21no—
257C22×C32—21no—
258C22×C32—21no—
259C22×C32—31no—
260C22×C32—31no—
261C22×C32—41no—
262C22×C32—42no—
263C22×C32—42no—
264C22×C32—44no—
265C22×C32—42no—
266C22×C32—42no—
267C22×C32—42no—
268C22×C32—44no—
269C22×C32—62no—
270C22×C32—62no—
271C22×C32—82no—
272C22×C32—82no—
273C22×C32—84no—
274C22×C32—122no—
275C22×C32—124no—
276C22×C32—124no—
277C22×C32C2×C4×D311yescharacteristic
278C22×C32C22×Dic311yescharacteristic
279C22×C32C22×C3×C411yescharacteristic
280C2×C3×D3—24no—
281C2×C3×D3—22no—
282C2×C3×D3—22no—
283C2×C3×D3—24no—
284C2×C3×D3—22no—
285C2×C3×D3—22no—
286C2×C3×D3—21no—
287C2×C3×D3—21no—
288C2×C3×D3—48no—
289C2×C3×D3—44no—
290C2×C3×D3—44no—
291C2×C3×D3—48no—
292C2×C3×D3—44no—
293C2×C3×D3—44no—
294C2×C3×D3—44no—
295C2×C3×D3—42no—
296C2×C3×D3—42no—
297C2×C3×D3#21711yescharacteristic
298C2×C3×D3#21711yescharacteristic
299C2×C3×D3C3×C22⋊C411yescharacteristic
300C2×C3×D3C3×C22⋊C411yescharacteristic
301C32×C4—22no—
302C32×C4—22no—
303C32×C4—22no—
304C32×C4—32no—
305C32×C4—42no—
306C32×C4D3×D412yes—
307C32×C4C2×#6712yes—
308C32×C4C2×C3×D412yes—
309C3×Dic3—22no—
310C3×Dic3#21712yes—
311C3×Dic3C3×C22⋊C412yes—
312C22×D4—31no—
313C2×M16—92no—
314#100—31no—
315#100—92no—
316#100—91no—
317C33C2×#10011yescharacteristic, complemented, Sylow
318C23×C3—22no—
319C23×C3—22no—
320C23×C3—22no—
321C23×C3—31no—
322C23×C3—31no—
323C23×C3—32no—
324C23×C3—32no—
325C23×C3—42no—
326C23×C3—61no—
327C23×C3—62no—
328C23×C3—62no—
329C23×C3—62no—
330C23×C3—62no—
331C23×C3—62no—
332C23×C3—64no—
333C23×C3—64no—
334C23×C3—124no—
335C23×C3—124no—
336C23×C3—128no—
337C23×C3D3×Dic312yes—
338C23×C3C3×C4×D312yes—
339C23×C3C2×C3×Dic312yes—
340C22×D3—22no—
341C22×D3—22no—
342C22×D3—44no—
343C22×D3C2×C3×Dic311yescharacteristic
344C22×D3C2×C3×Dic312yes—
345C3×D4—24no—
346C3×D4—22no—
347C3×D4—22no—
348C3×D4—22no—
349C3×D4—42no—
350C3×D4—42no—
351C3×D4—44no—
352C3×D4—64no—
353C3×D4—64no—
354C3×D4—64no—
355C3×D4—68no—
356C3×D4—62no—
357C3×D4—62no—
358C2×C3×C4—21no—
359C2×C3×C4—21no—
360C2×C3×C4—21no—
361C2×C3×C4—31no—
362C2×C3×C4—31no—
363C2×C3×C4—32no—
364C2×C3×C4—32no—
365C2×C3×C4—32no—
366C2×C3×C4—32no—
367C2×C3×C4—41no—
368C2×C3×C4—62no—
369C2×C3×C4—62no—
370C2×C3×C4—61no—
371C2×C3×C4C2×D3211yescharacteristic
372C2×C3×C4C22×C3×D311yescharacteristic
373C2×C3×C4C22×C3×D311yescharacteristic
374#67—24no—
375C2×Dic3C2×C3×Dic311yescharacteristic
376D12—22no—
377C4×D3C3×#6712yes—
378C4×D3C3×#6712yes—
379C3×C8—62no—
380C3×C8—62no—
381C3×C8—122no—
382C3×C8—182no—
383C3⋊C8—22no—
384C3⋊C8—62no—
385C3⋊C8—62no—
386C3⋊C8—122no—
387C2×C32—21no—
388C2×C32—22no—
389C2×C32—21no—
390C2×C32—21no—
391C2×C32—21no—
392C2×C32—21no—
393C2×C32—21no—
394C2×C32—32no—
395C2×C32—41no—
396C2×C32—41no—
397C2×C32—41no—
398C2×C32—42no—
399C2×C32—42no—
400C2×C32—44no—
401C2×C32—42no—
402C2×C32—42no—
403C2×C32—61no—
404C2×C32—82no—
405C2×C32—82no—
406C2×C32—84no—
407C2×C32—122no—
408C2×C32D3×C22⋊C411yescharacteristic
409C2×C32C2×#21711yescharacteristic
410C2×C32C2×C3×C22⋊C411yescharacteristic
411C3×D3—22no—
412C3×D3—21no—
413C3×D3—21no—
414C3×D3—22no—
415C3×D3—44no—
416C3×D3—42no—
417C3×D3—42no—
418C3×D3#83312yesdirect product factor
419C3×D3C3×#10012yescomplemented
420C24—32no—
421C2×D4—32no—
422C2×D4—31no—
423C2×D4—62no—
424C2×D4—62no—
425C2×D4C3×D3211yescharacteristic
426C22×C4—31no—
427M16—32no—
428M16—92no—
429M16—182no—
430C2×C8—182no—
431C22×C3—24no—
432C22×C3—22no—
433C22×C3—22no—
434C22×C3—22no—
435C22×C3—21no—
436C22×C3—21no—
437C22×C3—21no—
438C22×C3—31no—
439C22×C3—31no—
440C22×C3—31no—
441C22×C3—31no—
442C22×C3—42no—
443C22×C3—42no—
444C22×C3—44no—
445C22×C3—42no—
446C22×C3—42no—
447C22×C3—42no—
448C22×C3—44no—
449C22×C3—41no—
450C22×C3—61no—
451C22×C3—61no—
452C22×C3—62no—
453C22×C3—62no—
454C22×C3—64no—
455C22×C3—64no—
456C22×C3—62no—
457C22×C3—62no—
458C22×C3—82no—
459C22×C3—82no—
460C22×C3—84no—
461C22×C3—122no—
462C22×C3—122no—
463C22×C3—124no—
464C22×C3—124no—
465C22×C3—124no—
466C22×C3—128no—
467C22×C3—128no—
468C22×C3—124no—
469C22×C3—124no—
470C22×C3C2×D3×Dic311yescharacteristic
471C22×C3C2×C3×C4×D311yescharacteristic
472C22×C3C22×C3×Dic311yescharacteristic
473C2×D3—22no—
474C2×D3—22no—
475C2×D3—44no—
476C2×D3—44no—
477C2×D3—42no—
478C2×D3C3×#21711yescharacteristic
479C2×D3C3×#21711yescharacteristic
480C3×C4—22no—
481C3×C4—22no—
482C3×C4—22no—
483C3×C4—32no—
484C3×C4—32no—
485C3×C4—42no—
486C3×C4—62no—
487C3×C4D3×#6712yes—
488C3×C4C3×D3×D412yes—
489C3×C4C2×C3×#6712yes—
490Dic3C3×#21712yes—
491C32—21no—
492C32—21no—
493C32—21no—
494C32—41no—
495C32D3×#10011yescharacteristic, complemented
496C32C2×#83311yescharacteristic, complemented
497C32C2×C3×#10011yescharacteristic, complemented
498C23—31no—
499C23—32no—
500C23—62no—
501C23—62no—
502C23—124no—
503C23C3×D3×Dic312yes—
504D4—22no—
505D4—64no—
506D4—62no—
507C2×C4—31no—
508C2×C4—32no—
509C2×C4—32no—
510C2×C4C2×C3×D3211yescharacteristic
511C8—62no—
512C8—182no—
513C2×C3—22no—
514C2×C3—21no—
515C2×C3—21no—
516C2×C3—21no—
517C2×C3—21no—
518C2×C3—21no—
519C2×C3—21no—
520C2×C3—32no—
521C2×C3—32no—
522C2×C3—41no—
523C2×C3—41no—
524C2×C3—42no—
525C2×C3—44no—
526C2×C3—42no—
527C2×C3—42no—
528C2×C3—42no—
529C2×C3—41no—
530C2×C3—62no—
531C2×C3—62no—
532C2×C3—61no—
533C2×C3—61no—
534C2×C3—82no—
535C2×C3—82no—
536C2×C3—84no—
537C2×C3—124no—
538C2×C3—122no—
539C2×C3—122no—
540C2×C3D3×#21711yescharacteristic
541C2×C3C3×D3×C22⋊C411yescharacteristic
542C2×C3C2×C3×#21711yescharacteristic
543D3—21no—
544D3—42no—
545D3C3×#83312yesdirect product factor
546C22—22no—
547C22—31no—
548C22—31no—
549C22—42no—
550C22—62no—
551C22—62no—
552C22—122no—
553C22—124no—
554C22—124no—
555C22C2×C3×D3×Dic311yescharacteristic
556C4—32no—
557C4C3×D3×#6712yes—
558C3—21no—
559C3—21no—
560C3—21no—
561C3—41no—
562C3D3×#83311yescharacteristic, direct product factor
563C3C3×D3×#10011yescharacteristic, complemented
564C3C2×C3×#83311yescharacteristic, complemented
565C2—21no—
566C2—32no—
567C2—42no—
568C2—61no—
569C2—122no—
570C2C3×D3×#21711yescharacteristic
subgroupG/H
centerC2×C3D3×#217
commutatorC22×C32C22×C3×C4
FrattiniC2×C4C2×C3×D32
FittingC2×C33×D4C22
socleC2×C33C2×C22⋊C4
prime radicalC2×C32×D3×D4C2
cosocleC2×C32×C4C23×C3
hypercenterC2×C3×D4D32
hypocenterC32C2×C3×#100
LCSG ▷ C22×C32 ▷ 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
2. C3×D3×#208C2semidirect
3. C3×#63660C2semidirect
4. C32×#833C2semidirect
5. C3×#63558C2semidirect
8. D3×#833C3direct (N × H)
16. C3×#3748C22semidirect
17. C32×#208C22semidirect
26. D3×#208C2×C3semidirect
28. #63660C2×C3semidirect
34. C3×#833D3direct (N × H)
35. C3×#833C2×C3semidirect
37. #63558C2×C3semidirect
88. #3748C22×C3semidirect
94. C3×#208C2×D3semidirect
95. C3×#208C22×C3semidirect
129. #833C3×D3direct (N × H)
206. C32×D3#100semidirect
248. #208C2×C3×D3semidirect
317. C33C2×#100semidirect
418. C3×D3#833direct (N × H)
419. C3×D3C3×#100semidirect
495. C32D3×#100semidirect
496. C32C2×#833semidirect
497. C32C2×C3×#100semidirect
545. D3C3×#833direct (N × H)
562. C3D3×#833direct (N × H)
563. C3C3×D3×#100semidirect
564. C3C2×C3×#833semidirect

Maximal subgroups

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

subgroupindex
C2×C32×D3×D42
C3×D3×#2082
C3×#636602
C32×#8332
C3×#635582
C2×C3×#8333
C3×D3×#1003
D3×#8333