Finite Group Databasebeta

← C3×#10591087 #13158115

Subgroups — C3×#10591087 #13158115

Order 1728.

1026 conjugacy classes of subgroups.

The subgroup lattice has 622 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C32×#235C211yesmaximal, characteristic
2C2×C3×#3748C211yesmaximal, characteristic, complemented
3C2×C32×C4⋊Dic3C211yesmaximal, characteristic, complemented
4C3×#63565C212yesmaximal, complemented
5C3×#63562C212yesmaximal, complemented
6C3×#5232—31nomaximal
7C3×#4966—31nomaximal
8#10591087C311yesmaximal, characteristic, direct product factor
9C22×C33×C4C2211yescharacteristic
10C2×C32×#67—21no—
11C22×C32×Dic3—21no—
12C32×#235—22no—
13C32×#235C414yescomplemented
14C2×C32×D12C2211yescharacteristic
15C2×C32×C4×D3—21no—
16C2×C32×Dic6C2211yescharacteristic
17C3×#3748—22no—
18C2×C3×C3⋊(C3⋊C8)C2212yes—
19C32×C4⋊Dic3—21no—
20C32×C4⋊Dic3C2212yes—
21C2×C32×#41—31no—
22C2×C3×#235—21no—
23C2×C3×#235D311yescharacteristic
24C2×C3×#235C2×C311yescharacteristic
25C2×C32×C4⋊C4—31no—
26C2×#3748C2×C311yescharacteristic, complemented
27C2×C3×C4⋊Dic3—21no—
28C2×C3×C4⋊Dic3D311yescharacteristic, complemented
29C2×C3×C4⋊Dic3C2×C311yescharacteristic, complemented
30C2×C3×#208—31no—
31C2×C3×#208—31no—
32C2×C3×#208—61no—
33C3×#835—32no—
34C3×#832—32no—
35C3×#810—32no—
36C3×#809—32no—
37#63565C2×C312yescomplemented
38#63562C2×C312yescomplemented
39C23×C33D411yescharacteristic
40C22×C32×D3—21no—
41C2×C33×C4C2×C412yes—
42C2×C33×C4D411yescharacteristic
43C2×C33×C4D412yes—
44C2×C33×C4C2311yescharacteristic
45C32×#67—24no—
46C2×C32×Dic3—21no—
47C2×C32×Dic3—22no—
48C2×C32×Dic3—42no—
49C32×D12—21no—
50C32×D12C2×C412yescomplemented
51C32×C4×D3—24no—
52C32×Dic6—21no—
53C32×Dic6C2×C412yescomplemented
54C3×C3⋊(C3⋊C8)—22no—
55C3×#417—91no—
56#5232—31no—
57#4966—31no—
58C32×#41—34no—
59C32×#41—62no—
60C2×C32×Q8—31no—
61C2×C32×D4—31no—
62C2×C32×D4—61no—
63C22×C32×C4—21no—
64C22×C32×C4—21no—
65C22×C32×C4—21no—
66C22×C32×C4—41no—
67C22×C32×C4—61no—
68C22×C32×C4—61no—
69C22×C32×C4C2×D311yescharacteristic
70C22×C32×C4C2×D311yescharacteristic
71C22×C32×C4C22×C311yescharacteristic
72C2×C3×#67—22no—
73C2×C3×#67—21no—
74C2×C3×#67—21no—
75C22×C3×Dic3—22no—
76C22×C3×Dic3—21no—
77C22×C3×Dic3—21no—
78C3×#235—24no—
79C3×#235—24no—
80C3×#235—22no—
81C3×#235—22no—
82C3×#235Dic314yescomplemented
83C3×#235C3×C414yescomplemented
84C2×C3×D12—21no—
85C2×C3×D12C2×D311yescharacteristic
86C2×C3×D12C22×C311yescharacteristic
87C2×C3×C4×D3—22no—
88C2×C3×C4×D3—21no—
89C2×C3×C4×D3—21no—
90C2×C3×Dic6—21no—
91C2×C3×Dic6C2×D311yescharacteristic
92C2×C3×Dic6C22×C311yescharacteristic
93C32×C4⋊C4—32no—
94C32×C4⋊C4—61no—
95#3748—22no—
96C2×C3⋊(C3⋊C8)C22×C312yes—
97C3×C4⋊Dic3—22no—
98C3×C4⋊Dic3—22no—
99C3×C4⋊Dic3—21no—
100C3×C4⋊Dic3—21no—
101C3×C4⋊Dic3C2×D312yes—
102C3×C4⋊Dic3C22×C312yes—
103C3×#208—64no—
104C3×#208—62no—
105C3×#208—62no—
106C2×C3×C3⋊C8—32no—
107C2×C3×C3⋊C8—32no—
108C2×C3×C3⋊C8—62no—
109C22×C33—21no—
110C22×C33—21no—
111C22×C33C22⋊C412yes—
112C22×C33C2×D411yescharacteristic
113C2×C32×D3—22no—
114C2×C32×D3—41no—
115C2×C32×D3—41no—
116C33×C4C22⋊C412yes—
117C33×C4C22×C411yescharacteristic
118C33×C4C2×D411yescharacteristic
119C32×Dic3—22no—
120C32×Dic3—42no—
121C2×C3×#41—31no—
122C2×C3×#41—31no—
123C2×C3×#41—61no—
124C2×#235C3×D311yescharacteristic
125C2×C3×M16—91no—
126C2×C3×C4⋊C4—31no—
127C2×C3×C4⋊C4—31no—
128C2×C3×C4⋊C4—61no—
129C2×C4⋊Dic3C3×D311yescharacteristic, complemented
130C2×#208—31no—
131C2×#208—31no—
132C2×#208—61no—
133C3×#103—92no—
134C3×#102—92no—
135#835—32no—
136#832—32no—
137#810—32no—
138#809—32no—
139C23×C32—21no—
140C23×C32—21no—
141C23×C32—21no—
142C23×C32—41no—
143C23×C32—61no—
144C23×C32#6711yescharacteristic
145C23×C32#6711yescharacteristic
146C23×C32C3×D411yescharacteristic
147C22×C3×D3—22no—
148C22×C3×D3—21no—
149C22×C3×D3—21no—
150C32×Q8—32no—
151C32×Q8—61no—
152C32×D4—32no—
153C32×D4—64no—
154C32×D4—61no—
155C2×C32×C4—22no—
156C2×C32×C4—21no—
157C2×C32×C4—22no—
158C2×C32×C4—22no—
159C2×C32×C4—22no—
160C2×C32×C4—22no—
161C2×C32×C4—21no—
162C2×C32×C4—21no—
163C2×C32×C4—22no—
164C2×C32×C4—21no—
165C2×C32×C4—21no—
166C2×C32×C4—21no—
167C2×C32×C4—42no—
168C2×C32×C4—41no—
169C2×C32×C4—42no—
170C2×C32×C4—41no—
171C2×C32×C4—61no—
172C2×C32×C4—64no—
173C2×C32×C4—62no—
174C2×C32×C4—122no—
175C2×C32×C4C4×D312yes—
176C2×C32×C4D1212yes—
177C2×C32×C4C2×Dic312yes—
178C2×C32×C4#6711yescharacteristic
179C2×C32×C4#6711yescharacteristic
180C2×C32×C4#6712yes—
181C2×C32×C4C2×C3×C412yes—
182C2×C32×C4C3×D412yes—
183C2×C32×C4C3×D411yescharacteristic
184C2×C32×C4C22×D311yescharacteristic
185C2×C32×C4C22×D311yescharacteristic
186C2×C32×C4C23×C311yescharacteristic
187C3×#67—28no—
188C3×#67—24no—
189C3×#67—24no—
190C2×C3×Dic3—22no—
191C2×C3×Dic3—21no—
192C2×C3×Dic3—21no—
193C2×C3×Dic3—24no—
194C2×C3×Dic3—22no—
195C2×C3×Dic3—22no—
196C2×C3×Dic3—44no—
197C2×C3×Dic3—42no—
198C2×C3×Dic3—42no—
199C3×D12—22no—
200C3×D12—21no—
201C3×D12—21no—
202C3×D12—22no—
203C3×D12C2×Dic312yescomplemented
204C3×D12C2×C3×C412yescomplemented
205C3×C4×D3—28no—
206C3×C4×D3—24no—
207C3×C4×D3—24no—
208C3×Dic6—22no—
209C3×Dic6—21no—
210C3×Dic6—21no—
211C3×Dic6—22no—
212C3×Dic6C2×Dic312yescomplemented
213C3×Dic6C2×C3×C412yescomplemented
214C3⋊(C3⋊C8)—22no—
215C3×C3⋊C8—64no—
216C3×C3⋊C8—62no—
217C3×C3⋊C8—62no—
218#417—91noSylow
219C2×C33—22no—
220C2×C33C2×C22⋊C411yescharacteristic
221C2×C33AGL(1,ℤ/8)11yescharacteristic
222C2×C33#13711yescharacteristic
223C32×D3—42no—
224C3×#41—34no—
225C3×#41—34no—
226C3×#41—64no—
227C3×#41—64no—
228C3×#41—62no—
229C3×#41—62no—
230C2×C3×Q8—31no—
231C2×C3×Q8—31no—
232C2×C3×Q8—61no—
233C2×C3×D4—31no—
234C2×C3×D4—31no—
235C2×C3×D4—61no—
236C2×C3×D4—62no—
237C2×C3×D4—61no—
238C2×C3×D4—61no—
239C22×C3×C4—21no—
240C22×C3×C4—21no—
241C22×C3×C4—21no—
242C22×C3×C4—41no—
243C22×C3×C4—62no—
244C22×C3×C4—61no—
245C22×C3×C4—61no—
246C22×C3×C4—62no—
247C22×C3×C4—61no—
248C22×C3×C4—61no—
249C22×C3×C4D3211yescharacteristic
250C22×C3×C4C2×C3×D311yescharacteristic
251C22×C3×C4C2×C3×D311yescharacteristic
252C2×#67—21no—
253C22×Dic3—21no—
254#235—22no—
255#235C3×Dic314yescomplemented
256C2×D12C2×C3×D311yescharacteristic
257C2×C4×D3—21no—
258C2×Dic6C2×C3×D311yescharacteristic
259C3×M16—182no—
260C2×C3×C8—92no—
261C3×C4⋊C4—32no—
262C3×C4⋊C4—32no—
263C3×C4⋊C4—62no—
264C3×C4⋊C4—62no—
265C3×C4⋊C4—61no—
266C3×C4⋊C4—61no—
267C4⋊Dic3—21no—
268C4⋊Dic3C2×C3×D312yes—
269#208—64no—
270#208—62no—
271#208—62no—
272C2×C3⋊C8—32no—
273C2×C3⋊C8—32no—
274C2×C3⋊C8—62no—
275C22×C32—21no—
276C22×C32—21no—
277C22×C32—22no—
278C22×C32—22no—
279C22×C32—21no—
280C22×C32—21no—
281C22×C32—21no—
282C22×C32—21no—
283C22×C32—22no—
284C22×C32—22no—
285C22×C32—22no—
286C22×C32—21no—
287C22×C32—21no—
288C22×C32—21no—
289C22×C32—42no—
290C22×C32—41no—
291C22×C32—41no—
292C22×C32—41no—
293C22×C32—41no—
294C22×C32—41no—
295C22×C32—42no—
296C22×C32—42no—
297C22×C32—62no—
298C22×C32—121no—
299C22×C32—121no—
300C22×C32(C2×D3)⋊C412yes—
301C22×C32#21712yes—
302C22×C32C3×C22⋊C412yes—
303C22×C32C2×#6711yescharacteristic
304C22×C32C2×#6711yescharacteristic
305C22×C32C2×C3×D411yescharacteristic
306C2×C3×D3—24no—
307C2×C3×D3—22no—
308C2×C3×D3—22no—
309C2×C3×D3—42no—
310C2×C3×D3—42no—
311C2×C3×D3—41no—
312C2×C3×D3—41no—
313C2×C3×D3—41no—
314C2×C3×D3—41no—
315C32×C4—22no—
316C32×C4—22no—
317C32×C4—22no—
318C32×C4—21no—
319C32×C4—21no—
320C32×C4—21no—
321C32×C4—21no—
322C32×C4—21no—
323C32×C4—21no—
324C32×C4—42no—
325C32×C4—41no—
326C32×C4—41no—
327C32×C4—62no—
328C32×C4—122no—
329C32×C4(C2×D3)⋊C412yes—
330C32×C4#21712yes—
331C32×C4C3×C22⋊C412yes—
332C32×C4C2×C4×D311yescharacteristic
333C32×C4C2×D1211yescharacteristic
334C32×C4C22×Dic311yescharacteristic
335C32×C4C2×#6711yescharacteristic
336C32×C4C22×C3×C411yescharacteristic
337C32×C4C2×C3×D411yescharacteristic
338C3×Dic3—24no—
339C3×Dic3—22no—
340C3×Dic3—22no—
341C3×Dic3—44no—
342C3×Dic3—42no—
343C3×Dic3—42no—
344C2×#41—31no—
345C2×M16—91no—
346C2×C4⋊C4—31no—
347#103—92no—
348#102—92no—
349C33#41711yescharacteristic, complemented, Sylow
350C23×C3—21no—
351C23×C3—21no—
352C23×C3—21no—
353C23×C3—41no—
354C23×C3—62no—
355C23×C3—61no—
356C23×C3—61no—
357C23×C3#62511yescharacteristic
358C23×C3C3×#6711yescharacteristic
359C23×C3C3×#6711yescharacteristic
360C22×D3—21no—
361C3×Q8—32no—
362C3×Q8—32no—
363C3×Q8—62no—
364C3×Q8—62no—
365C3×Q8—61no—
366C3×Q8—61no—
367C3×D4—32no—
368C3×D4—32no—
369C3×D4—62no—
370C3×D4—62no—
371C3×D4—68no—
372C3×D4—64no—
373C3×D4—64no—
374C3×D4—61no—
375C3×D4—61no—
376C2×C3×C4—22no—
377C2×C3×C4—22no—
378C2×C3×C4—22no—
379C2×C3×C4—21no—
380C2×C3×C4—21no—
381C2×C3×C4—21no—
382C2×C3×C4—22no—
383C2×C3×C4—22no—
384C2×C3×C4—22no—
385C2×C3×C4—21no—
386C2×C3×C4—21no—
387C2×C3×C4—21no—
388C2×C3×C4—42no—
389C2×C3×C4—41no—
390C2×C3×C4—42no—
391C2×C3×C4—41no—
392C2×C3×C4—68no—
393C2×C3×C4—62no—
394C2×C3×C4—61no—
395C2×C3×C4—61no—
396C2×C3×C4—64no—
397C2×C3×C4—64no—
398C2×C3×C4—64no—
399C2×C3×C4—62no—
400C2×C3×C4—62no—
401C2×C3×C4—124no—
402C2×C3×C4—122no—
403C2×C3×C4—122no—
404C2×C3×C4D3×Dic312yes—
405C2×C3×C4#62511yescharacteristic
406C2×C3×C4#62612yes—
407C2×C3×C4C3×C4×D312yes—
408C2×C3×C4C3×D1212yes—
409C2×C3×C4C2×C3×Dic312yes—
410C2×C3×C4C3×#6711yescharacteristic
411C2×C3×C4C3×#6711yescharacteristic
412C2×C3×C4C3×#6712yes—
413C2×C3×C4C2×D3211yescharacteristic
414C2×C3×C4C22×C3×D311yescharacteristic
415C2×C3×C4C22×C3×D311yescharacteristic
416#67—24no—
417C2×Dic3—21no—
418C2×Dic3—22no—
419C2×Dic3—42no—
420D12—21no—
421D12C2×C3×Dic312yescomplemented
422C4×D3—24no—
423Dic6—21no—
424Dic6C2×C3×Dic312yescomplemented
425C3×C8—182no—
426C3⋊C8—64no—
427C3⋊C8—62no—
428C3⋊C8—62no—
429C2×C32—22no—
430C2×C32—24no—
431C2×C32—22no—
432C2×C32—22no—
433C2×C32—21no—
434C2×C32—21no—
435C2×C32—21no—
436C2×C32—21no—
437C2×C32—21no—
438C2×C32—21no—
439C2×C32—21no—
440C2×C32—21no—
441C2×C32—21no—
442C2×C32—41no—
443C2×C32—41no—
444C2×C32—41no—
445C2×C32—42no—
446C2×C32—42no—
447C2×C32—44no—
448C2×C32—122no—
449C2×C32C2×(C2×D3)⋊C411yescharacteristic
450C2×C32#93211yescharacteristic
451C2×C32#94211yescharacteristic
452C2×C32#94911yescharacteristic
453C2×C32#95111yescharacteristic
454C2×C32C2×#21711yescharacteristic
455C2×C32C2×C3×C22⋊C411yescharacteristic
456C2×C32C3×AGL(1,ℤ/8)11yescharacteristic
457C2×C32C3×#13711yescharacteristic
458C3×D3—44no—
459C3×D3—42no—
460C3×D3—42no—
461#41—34no—
462#41—62no—
463C2×Q8—31no—
464C2×D4—31no—
465C2×D4—61no—
466C22×C4—61no—
467C22×C4—61no—
468C22×C4C3×D3211yescharacteristic
469M16—182no—
470C2×C8—92no—
471C4⋊C4—32no—
472C4⋊C4—61no—
473C22×C3—22no—
474C22×C3—22no—
475C22×C3—21no—
476C22×C3—21no—
477C22×C3—21no—
478C22×C3—21no—
479C22×C3—21no—
480C22×C3—21no—
481C22×C3—21no—
482C22×C3—21no—
483C22×C3—21no—
484C22×C3—22no—
485C22×C3—22no—
486C22×C3—22no—
487C22×C3—41no—
488C22×C3—41no—
489C22×C3—41no—
490C22×C3—41no—
491C22×C3—42no—
492C22×C3—42no—
493C22×C3—41no—
494C22×C3—42no—
495C22×C3—64no—
496C22×C3—62no—
497C22×C3—62no—
498C22×C3—122no—
499C22×C3—122no—
500C22×C3—121no—
501C22×C3—121no—
502C22×C3—121no—
503C22×C3—121no—
504C22×C3(C2×D3)⋊Dic312yes—
505C22×C3C3×(C2×D3)⋊C412yes—
506C22×C3C3×#21712yes—
507C22×C3C2×#62511yescharacteristic
508C22×C3C2×C3×#6711yescharacteristic
509C22×C3C2×C3×#6711yescharacteristic
510C2×D3—22no—
511C2×D3—41no—
512C2×D3—41no—
513C3×C4—21no—
514C3×C4—21no—
515C3×C4—21no—
516C3×C4—21no—
517C3×C4—21no—
518C3×C4—21no—
519C3×C4—22no—
520C3×C4—22no—
521C3×C4—22no—
522C3×C4—41no—
523C3×C4—41no—
524C3×C4—42no—
525C3×C4—64no—
526C3×C4—62no—
527C3×C4—62no—
528C3×C4—124no—
529C3×C4—122no—
530C3×C4—122no—
531C3×C4(C2×D3)⋊Dic312yes—
532C3×C4C3×(C2×D3)⋊C412yes—
533C3×C4C3×#21712yes—
534C3×C4C2×D3×Dic311yescharacteristic
535C3×C4C2×#62611yescharacteristic
536C3×C4C2×C3×C4×D311yescharacteristic
537C3×C4C2×C3×D1211yescharacteristic
538C3×C4C22×C3×Dic311yescharacteristic
539C3×C4C2×C3×#6711yescharacteristic
540Dic3—22no—
541Dic3—42no—
542C32—21no—
543C32—21no—
544C32—21no—
545C32—41no—
546C32#496611yescharacteristic, complemented
547C32#523211yescharacteristic, complemented
548C32C3×#41711yescharacteristic, complemented
549C23—61no—
550C23C3×#62511yescharacteristic
551Q8—32no—
552Q8—61no—
553D4—32no—
554D4—64no—
555D4—61no—
556C2×C4—61no—
557C2×C4—64no—
558C2×C4—62no—
559C2×C4—122no—
560C2×C4C3×D3×Dic312yes—
561C2×C4C3×#62511yescharacteristic
562C2×C4C3×#62612yes—
563C2×C4C2×C3×D3211yescharacteristic
564C8—182no—
565C2×C3—24no—
566C2×C3—22no—
567C2×C3—22no—
568C2×C3—22no—
569C2×C3—21no—
570C2×C3—21no—
571C2×C3—21no—
572C2×C3—21no—
573C2×C3—21no—
574C2×C3—21no—
575C2×C3—21no—
576C2×C3—21no—
577C2×C3—21no—
578C2×C3—42no—
579C2×C3—42no—
580C2×C3—44no—
581C2×C3—41no—
582C2×C3—41no—
583C2×C3—41no—
584C2×C3—124no—
585C2×C3—122no—
586C2×C3—122no—
587C2×C3#6382311yescharacteristic
588C2×C3#6382711yescharacteristic
589C2×C3C2×(C2×D3)⋊Dic311yescharacteristic
590C2×C3C2×C3×(C2×D3)⋊C411yescharacteristic
591C2×C3C3×#93211yescharacteristic
592C2×C3C3×#94211yescharacteristic
593C2×C3C3×#94911yescharacteristic
594C2×C3C3×#95111yescharacteristic
595C2×C3C2×C3×#21711yescharacteristic
596D3—42no—
597C22—21no—
598C22—21no—
599C22—62no—
600C22—121no—
601C22—121no—
602C22C3×(C2×D3)⋊Dic312yes—
603C22C2×C3×#62511yescharacteristic
604C4—62no—
605C4—122no—
606C4C3×(C2×D3)⋊Dic312yes—
607C4C2×C3×D3×Dic311yescharacteristic
608C4C2×C3×#62611yescharacteristic
609C3—21no—
610C3—21no—
611C3—21no—
612C3—41no—
613C3#1059108711yescharacteristic, direct product factor
614C3C3×#496611yescharacteristic, complemented
615C3C3×#523211yescharacteristic, complemented
616C2—22no—
617C2—122no—
618C2C3×#6382311yescharacteristic
619C2C3×#6382711yescharacteristic
620C2C2×C3×(C2×D3)⋊Dic311yescharacteristic
subgroupG/H
centerC22×C3C2×#625
commutatorC32×C4C22×C3×C4
FrattiniC2×C4C2×C3×D32
FittingC22×C33×C4C22
socleC22×C33C2×D4
prime radicalC2×C32×#235C2
cosocleC2×C32×C4C23×C3
hypercenterC22×C3×C4D32
hypocenterC32C3×#417
LCSG ▷ C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C22×C3×C4

Complements

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

NH ≅ G/Ntype
2. C2×C3×#3748C2semidirect
3. C2×C32×C4⋊Dic3C2semidirect
4. C3×#63565C2semidirect
5. C3×#63562C2semidirect
8. #10591087C3direct (N × H)
13. C32×#235C4semidirect
26. C2×#3748C2×C3semidirect
28. C2×C3×C4⋊Dic3D3semidirect
29. C2×C3×C4⋊Dic3C2×C3semidirect
37. #63565C2×C3semidirect
38. #63562C2×C3semidirect
50. C32×D12C2×C4semidirect
53. C32×Dic6C2×C4semidirect
82. C3×#235Dic3semidirect
83. C3×#235C3×C4semidirect
129. C2×C4⋊Dic3C3×D3semidirect
203. C3×D12C2×Dic3semidirect
204. C3×D12C2×C3×C4semidirect
212. C3×Dic6C2×Dic3semidirect
213. C3×Dic6C2×C3×C4semidirect
255. #235C3×Dic3semidirect
349. C33#417semidirect
421. D12C2×C3×Dic3semidirect
424. Dic6C2×C3×Dic3semidirect
546. C32#4966semidirect
547. C32#5232semidirect
548. C32C3×#417semidirect
613. C3#10591087direct (N × H)
614. C3C3×#4966semidirect
615. C3C3×#5232semidirect

Maximal subgroups

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

subgroupindex
C2×C32×#2352
C2×C3×#37482
C2×C32×C4⋊Dic32
C3×#635652
C3×#635622
C3×#52323
C3×#49663
#105910873