Finite Group Databasebeta

← C3×#10590366 #13157383

Subgroups — C3×#10590366 #13157383

Order 1728.

734 conjugacy classes of subgroups.

The subgroup lattice has 624 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#64084C211yesmaximal, characteristic, complemented
2C32×C4×D12C211yesmaximal, characteristic, complemented
3C2×C3×#3716C211yesmaximal, characteristic, complemented
4C2×C3×#3714C211yesmaximal, characteristic
5C32×C4⋊(C3⋊C8)C211yesmaximal, characteristic, complemented
6C3×#63567C211yesmaximal, characteristic, complemented
7C3×#63564C211yesmaximal, characteristic, complemented
8C3×#5016—31nomaximal
9C3×#4844—31nomaximal
10#10590366C311yesmaximal, characteristic, direct product factor
11C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
12C2×C3×#634C2211yescharacteristic
13C2×C32×D12C2211yescharacteristic
14C2×C32×C4×D3—21no—
15C33×C42C2211yescharacteristic
16C3×#3752—21no—
17C3×#3752—21no—
18C32×(C2×D3)⋊C4—21no—
19C32×C4⋊Dic3C2211yescharacteristic
20C2×C32×C3⋊C8C2211yescharacteristic
21C2×C32×C3⋊C8C2211yescharacteristic
22C3×#3716—22no—
23C3×#3714—22no—
24C32×C4×D4—31no—
25#64084C2×C311yescharacteristic, complemented
26C2×C3×#214—31no—
27C2×C3×#213—31no—
28C2×C3×D24—31no—
29C2×C3×#204—31no—
30C3×#875—31no—
31C3×#875—31no—
32C3×#875—61no—
33C3×C4×D12—21no—
34C3×C4×D12D311yescharacteristic, complemented
35C3×C4×D12C2×C311yescharacteristic, complemented
36C2×#3716C2×C311yescharacteristic, complemented
37C2×#3714C2×C311yescharacteristic
38C32×C4⋊C8—31no—
39C3×#821—31no—
40C3×#816—31no—
41C3×#810—31no—
42C3×#809—31no—
43C3×C4⋊(C3⋊C8)—21no—
44C3×C4⋊(C3⋊C8)D311yescharacteristic, complemented
45C3×C4⋊(C3⋊C8)C2×C311yescharacteristic, complemented
46#63567C2×C311yescharacteristic, complemented
47#63564C2×C311yescharacteristic, complemented
48C22×C3×C3⋊D3—21no—
49C22×C32×D3—21no—
50C2×C33×C4D411yescharacteristic
51C2×C33×C4D411yescharacteristic
52C2×C33×C4C2311yescharacteristic
53C2×C3×C3⋊Dic3—21no—
54C3×C4⋊(C3⋊D3)—21no—
55C3×C4⋊(C3⋊D3)—21no—
56C3×#634—21no—
57C3×#634—21no—
58C2×C32×Dic3—21no—
59C32×D12—21no—
60C32×D12D412yes—
61C32×C4×D3—42no—
62C32×C3⋊C8—21no—
63C32×C3⋊C8—21no—
64C3×#463—91no—
65#5016—31no—
66#4844—31no—
67C2×C32×D4—31no—
68C22×C32×C4—61no—
69C2×C4⋊(C3⋊D3)C22×C311yescharacteristic
70C2×#634C22×C311yescharacteristic
71C2×C3×D12—21no—
72C2×C3×D12—31no—
73C2×C3×D12—31no—
74C2×C3×D12—61no—
75C2×C3×D12C2×D311yescharacteristic
76C2×C3×D12C22×C311yescharacteristic
77C2×C3×C4×D3—22no—
78C2×C3×C4×D3—21no—
79C2×C3×C4×D3—21no—
80C2×C3×Dic6—31no—
81C2×C3×Dic6—31no—
82C2×C3×Dic6—61no—
83C2×C32×C8—31no—
84C2×C32×C8—31no—
85C32×C4⋊C4—31no—
86C32×C22⋊C4—61no—
87C32×C42—21no—
88C32×C42—21no—
89C32×C42—21no—
90C32×C42—41no—
91C32×C42C2×D311yescharacteristic
92C32×C42C2×D311yescharacteristic
93C32×C42C22×C311yescharacteristic
94#3752—21no—
95#3752—21no—
96C3×#214—62no—
97C3×#213—62no—
98C3×(C2×D3)⋊C4—22no—
99C3×(C2×D3)⋊C4—21no—
100C3×(C2×D3)⋊C4—21no—
101C3×(C2×D3)⋊C4—62no—
102C3×(C2×D3)⋊C4—62no—
103C3×(C2×D3)⋊C4—61no—
104C3×(C2×D3)⋊C4—61no—
105C3×(C2×D3)⋊C4—61no—
106C3×(C2×D3)⋊C4—61no—
107C3×C4⋊Dic3—21no—
108C3×C4⋊Dic3C2×D311yescharacteristic
109C3×C4⋊Dic3C22×C311yescharacteristic
110C2×C3×C3⋊C8—21no—
111C2×C3×C3⋊C8—21no—
112C2×C3×C3⋊C8C2×D311yescharacteristic
113C2×C3×C3⋊C8C2×D311yescharacteristic
114C2×C3×C3⋊C8C22×C311yescharacteristic
115C2×C3×C3⋊C8C22×C311yescharacteristic
116C3×D24—62no—
117C3×#204—62no—
118#3716—22no—
119#3714—22no—
120C22×C33C2×D411yescharacteristic
121C2×C3×C3⋊D3—41no—
122C2×C3×C3⋊D3—41no—
123C2×C3×C3⋊D3—41no—
124C2×C32×D3—22no—
125C2×C32×D3—41no—
126C2×C32×D3—41no—
127C33×C4—21no—
128C33×C4—21no—
129C33×C4C2×D411yescharacteristic
130C33×C4#4111yescharacteristic
131C3×C3⋊Dic3—41no—
132C32×Dic3—41no—
133C2×C3×SD16—91no—
134C2×C3×D8—91no—
135C3×#124—91no—
136C3×C4×D4—31no—
137C3×C4×D4—31no—
138C3×C4×D4—61no—
139C2×#214—31no—
140C2×#213—31no—
141C2×D24—31no—
142C2×#204—31no—
143#875—31no—
144#875—31no—
145#875—61no—
146C4×D12C3×D311yescharacteristic, complemented
147C3×C4⋊C8—31no—
148C3×C4⋊C8—31no—
149C3×C4⋊C8—61no—
150C3×#103—91no—
151C3×#102—91no—
152#821—31no—
153#816—31no—
154#810—31no—
155#809—31no—
156C4⋊(C3⋊C8)C3×D311yescharacteristic, complemented
157C23×C32—61no—
158C22×C3⋊D3—21no—
159C22×C3×D3—22no—
160C22×C3×D3—21no—
161C22×C3×D3—21no—
162C22×C3×D3—62no—
163C22×C3×D3—61no—
164C22×C3×D3—61no—
165C32×D4—32no—
166C32×D4—61no—
167C2×C32×C4—21no—
168C2×C32×C4—21no—
169C2×C32×C4—21no—
170C2×C32×C4—21no—
171C2×C32×C4—21no—
172C2×C32×C4—21no—
173C2×C32×C4—21no—
174C2×C32×C4—21no—
175C2×C32×C4—21no—
176C2×C32×C4—41no—
177C2×C32×C4—41no—
178C2×C32×C4—41no—
179C2×C32×C4—61no—
180C2×C32×C4—122no—
181C2×C32×C4D1211yescharacteristic
182C2×C32×C4D1211yescharacteristic
183C2×C32×C4#6711yescharacteristic
184C2×C32×C4#6711yescharacteristic
185C2×C32×C4C3×D411yescharacteristic
186C2×C32×C4C3×D411yescharacteristic
187C2×C32×C4C22×D311yescharacteristic
188C2×C32×C4C22×D311yescharacteristic
189C2×C32×C4C23×C311yescharacteristic
190C2×C3⋊Dic3—21no—
191C4⋊(C3⋊D3)—21no—
192C4⋊(C3⋊D3)—21no—
193#634—21no—
194#634—21no—
195C2×C3×Dic3—22no—
196C2×C3×Dic3—21no—
197C2×C3×Dic3—21no—
198C2×C3×Dic3—62no—
199C2×C3×Dic3—61no—
200C2×C3×Dic3—61no—
201C3×D12—22no—
202C3×D12—21no—
203C3×D12—21no—
204C3×D12—22no—
205C3×D12—61no—
206C3×D12—61no—
207C3×D12—61no—
208C3×D12—61no—
209C3×D12—121no—
210C3×D12—121no—
211C3×D12D1212yes—
212C3×D12C3×D412yes—
213C3×C4×D3—44no—
214C3×C4×D3—42no—
215C3×C4×D3—42no—
216C3×Dic6—61no—
217C3×Dic6—61no—
218C3×Dic6—61no—
219C3×Dic6—61no—
220C3×Dic6—121no—
221C3×Dic6—121no—
222C32×C8—61no—
223C32×C8—61no—
224C3×C3⋊C8—21no—
225C3×C3⋊C8—21no—
226C3×C3⋊C8—21no—
227C3×C3⋊C8—21no—
228C3×C3⋊C8—41no—
229C3×C3⋊C8—41no—
230#463—91noSylow
231C2×C33#12111yescharacteristic
232C2×C33#13511yescharacteristic
233C2×C33AGL(1,ℤ/8)11yescharacteristic
234C3×C3⋊D3—81no—
235C32×D3—42no—
236C2×C3×Q8—91no—
237C2×C3×D4—31no—
238C2×C3×D4—31no—
239C2×C3×D4—61no—
240C2×C3×D4—91no—
241C22×C3×C4—62no—
242C22×C3×C4—61no—
243C22×C3×C4—61no—
244C2×D12—31no—
245C2×D12—31no—
246C2×D12—61no—
247C2×D12C2×C3×D311yescharacteristic
248C2×C4×D3—21no—
249C2×Dic6—31no—
250C2×Dic6—31no—
251C2×Dic6—61no—
252C3×SD16—182no—
253C3×D8—182no—
254C2×C3×C8—31no—
255C2×C3×C8—31no—
256C2×C3×C8—31no—
257C2×C3×C8—31no—
258C2×C3×C8—61no—
259C2×C3×C8—61no—
260C3×C4⋊C4—31no—
261C3×C4⋊C4—31no—
262C3×C4⋊C4—61no—
263C3×C22⋊C4—62no—
264C3×C22⋊C4—61no—
265C3×C22⋊C4—61no—
266C3×C22⋊C4—181no—
267C3×C22⋊C4—181no—
268C3×C42—21no—
269C3×C42—21no—
270C3×C42—21no—
271C3×C42—41no—
272C3×C42D3211yescharacteristic
273C3×C42C2×C3×D311yescharacteristic
274C3×C42C2×C3×D311yescharacteristic
275#214—62no—
276#213—62no—
277(C2×D3)⋊C4—21no—
278(C2×D3)⋊C4—62no—
279(C2×D3)⋊C4—62no—
280(C2×D3)⋊C4—61no—
281(C2×D3)⋊C4—61no—
282(C2×D3)⋊C4—61no—
283(C2×D3)⋊C4—61no—
284C4⋊Dic3C2×C3×D311yescharacteristic
285C2×C3⋊C8C2×C3×D311yescharacteristic
286C2×C3⋊C8C2×C3×D311yescharacteristic
287D24—62no—
288#204—62no—
289C22×C32—21no—
290C22×C32—21no—
291C22×C32—21no—
292C22×C32—41no—
293C22×C32—62no—
294C22×C32—121no—
295C22×C32—121no—
296C22×C32C2×D1211yescharacteristic
297C22×C32C2×#6711yescharacteristic
298C22×C32C2×C3×D411yescharacteristic
299C2×C3⋊D3—41no—
300C2×C3⋊D3—41no—
301C2×C3⋊D3—41no—
302C2×C3×D3—24no—
303C2×C3×D3—22no—
304C2×C3×D3—22no—
305C2×C3×D3—42no—
306C2×C3×D3—42no—
307C2×C3×D3—41no—
308C2×C3×D3—41no—
309C2×C3×D3—41no—
310C2×C3×D3—41no—
311C2×C3×D3—122no—
312C2×C3×D3—122no—
313C2×C3×D3—122no—
314C2×C3×D3—121no—
315C2×C3×D3—121no—
316C2×C3×D3—121no—
317C2×C3×D3—121no—
318C2×C3×D3—121no—
319C2×C3×D3—121no—
320C32×C4—21no—
321C32×C4—21no—
322C32×C4—22no—
323C32×C4—22no—
324C32×C4—21no—
325C32×C4—21no—
326C32×C4—21no—
327C32×C4—21no—
328C32×C4—21no—
329C32×C4—21no—
330C32×C4—21no—
331C32×C4—21no—
332C32×C4—21no—
333C32×C4—21no—
334C32×C4—41no—
335C32×C4—41no—
336C32×C4—41no—
337C32×C4—41no—
338C32×C4—41no—
339C32×C4—41no—
340C32×C4—42no—
341C32×C4—42no—
342C32×C4—121no—
343C32×C4C2×D1211yescharacteristic
344C32×C4#23511yescharacteristic
345C32×C4D3×D411yescharacteristic
346C32×C4#23911yescharacteristic
347C32×C4C2×C3×D411yescharacteristic
348C32×C4C3×#4111yescharacteristic
349C3⋊Dic3—41no—
350C3×Dic3—42no—
351C3×Dic3—41no—
352C3×Dic3—41no—
353C3×Dic3—122no—
354C3×Dic3—121no—
355C3×Dic3—121no—
356C2×SD16—91no—
357C2×D8—91no—
358#124—91no—
359C4×D4—31no—
360C4⋊C8—31no—
361#103—91no—
362#102—91no—
363C33#46311yescharacteristic, complemented, Sylow
364C23×C3—62no—
365C23×C3—61no—
366C23×C3—61no—
367C23×C3—181no—
368C22×D3—21no—
369C22×D3—62no—
370C22×D3—61no—
371C22×D3—61no—
372C3×Q8—181no—
373C3×Q8—181no—
374C3×D4—32no—
375C3×D4—32no—
376C3×D4—62no—
377C3×D4—62no—
378C3×D4—61no—
379C3×D4—61no—
380C3×D4—181no—
381C3×D4—181no—
382C2×C3×C4—21no—
383C2×C3×C4—21no—
384C2×C3×C4—21no—
385C2×C3×C4—21no—
386C2×C3×C4—21no—
387C2×C3×C4—21no—
388C2×C3×C4—21no—
389C2×C3×C4—21no—
390C2×C3×C4—21no—
391C2×C3×C4—41no—
392C2×C3×C4—41no—
393C2×C3×C4—41no—
394C2×C3×C4—62no—
395C2×C3×C4—61no—
396C2×C3×C4—61no—
397C2×C3×C4—124no—
398C2×C3×C4—122no—
399C2×C3×C4—122no—
400C2×C3×C4—181no—
401C2×C3×C4#62611yescharacteristic
402C2×C3×C4#62611yescharacteristic
403C2×C3×C4C3×D1211yescharacteristic
404C2×C3×C4C3×D1211yescharacteristic
405C2×C3×C4C3×#6711yescharacteristic
406C2×C3×C4C3×#6711yescharacteristic
407C2×C3×C4C2×D3211yescharacteristic
408C2×C3×C4C22×C3×D311yescharacteristic
409C2×C3×C4C22×C3×D311yescharacteristic
410C2×Dic3—21no—
411C2×Dic3—62no—
412C2×Dic3—61no—
413C2×Dic3—61no—
414D12—21no—
415D12—61no—
416D12—61no—
417D12—61no—
418D12—61no—
419D12—121no—
420D12—121no—
421D12C3×D1212yes—
422C4×D3—42no—
423Dic6—61no—
424Dic6—61no—
425Dic6—61no—
426Dic6—61no—
427Dic6—121no—
428Dic6—121no—
429C3×C8—61no—
430C3×C8—61no—
431C3×C8—61no—
432C3×C8—61no—
433C3×C8—121no—
434C3×C8—121no—
435C3⋊C8—21no—
436C3⋊C8—21no—
437C2×C32—21no—
438C2×C32—21no—
439C2×C32—21no—
440C2×C32—21no—
441C2×C32—21no—
442C2×C32—21no—
443C2×C32—21no—
444C2×C32—21no—
445C2×C32—21no—
446C2×C32—41no—
447C2×C32—41no—
448C2×C32—41no—
449C2×C32—122no—
450C2×C32#89511yescharacteristic
451C2×C32#90411yescharacteristic
452C2×C32#90811yescharacteristic
453C2×C32#93011yescharacteristic
454C2×C32#93211yescharacteristic
455C2×C32#95011yescharacteristic
456C2×C32C3×#12111yescharacteristic
457C2×C32C3×#13511yescharacteristic
458C2×C32C3×AGL(1,ℤ/8)11yescharacteristic
459C3⋊D3—81no—
460C3×D3—44no—
461C3×D3—42no—
462C3×D3—42no—
463C3×D3—242no—
464C3×D3—241no—
465C3×D3—241no—
466C2×Q8—91no—
467C2×D4—31no—
468C2×D4—91no—
469C22×C4—61no—
470SD16—182no—
471D8—182no—
472C2×C8—31no—
473C2×C8—31no—
474C4⋊C4—31no—
475C22⋊C4—61no—
476C22⋊C4—181no—
477C22⋊C4—181no—
478C42C3×D3211yescharacteristic
479C22×C3—21no—
480C22×C3—21no—
481C22×C3—21no—
482C22×C3—41no—
483C22×C3—64no—
484C22×C3—62no—
485C22×C3—62no—
486C22×C3—122no—
487C22×C3—122no—
488C22×C3—121no—
489C22×C3—121no—
490C22×C3—121no—
491C22×C3—121no—
492C22×C3—361no—
493C22×C3—361no—
494C22×C3—361no—
495C22×C3C2×#62611yescharacteristic
496C22×C3C2×C3×D1211yescharacteristic
497C22×C3C2×C3×#6711yescharacteristic
498C2×D3—22no—
499C2×D3—41no—
500C2×D3—41no—
501C2×D3—122no—
502C2×D3—122no—
503C2×D3—122no—
504C2×D3—121no—
505C2×D3—121no—
506C2×D3—121no—
507C2×D3—121no—
508C2×D3—121no—
509C2×D3—121no—
510C3×C4—22no—
511C3×C4—22no—
512C3×C4—21no—
513C3×C4—21no—
514C3×C4—21no—
515C3×C4—21no—
516C3×C4—21no—
517C3×C4—21no—
518C3×C4—21no—
519C3×C4—21no—
520C3×C4—21no—
521C3×C4—21no—
522C3×C4—21no—
523C3×C4—21no—
524C3×C4—41no—
525C3×C4—41no—
526C3×C4—41no—
527C3×C4—41no—
528C3×C4—42no—
529C3×C4—42no—
530C3×C4—41no—
531C3×C4—41no—
532C3×C4—122no—
533C3×C4—121no—
534C3×C4—121no—
535C3×C4—361no—
536C3×C4#379911yescharacteristic
537C3×C4D3×D1211yescharacteristic
538C3×C4C2×C3×D1211yescharacteristic
539C3×C4C3×#23511yescharacteristic
540C3×C4C3×D3×D411yescharacteristic
541C3×C4C3×#23911yescharacteristic
542Dic3—41no—
543Dic3—122no—
544Dic3—121no—
545Dic3—121no—
546C32—21no—
547C32—21no—
548C32—21no—
549C32—41no—
550C32#484411yescharacteristic, complemented
551C32#501611yescharacteristic, complemented
552C32C3×#46311yescharacteristic, complemented
553C23—61no—
554C23—181no—
555Q8—181no—
556Q8—181no—
557D4—32no—
558D4—61no—
559D4—181no—
560D4—181no—
561C2×C4—61no—
562C2×C4—122no—
563C2×C4—181no—
564C2×C4C3×#62611yescharacteristic
565C2×C4C3×#62611yescharacteristic
566C2×C4C2×C3×D3211yescharacteristic
567C8—61no—
568C8—61no—
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—41no—
579C2×C3—41no—
580C2×C3—41no—
581C2×C3—124no—
582C2×C3—122no—
583C2×C3—122no—
584C2×C3—721no—
585C2×C3#6382911yescharacteristic
586C2×C3#6383011yescharacteristic
587C2×C3#6390911yescharacteristic
588C2×C3C3×#89511yescharacteristic
589C2×C3C3×#90411yescharacteristic
590C2×C3C3×#90811yescharacteristic
591C2×C3C3×#93011yescharacteristic
592C2×C3C3×#93211yescharacteristic
593C2×C3C3×#95011yescharacteristic
594D3—42no—
595D3—242no—
596D3—241no—
597D3—241no—
598C22—62no—
599C22—121no—
600C22—121no—
601C22—361no—
602C22—361no—
603C22—361no—
604C22C2×C3×#62611yescharacteristic
605C4—21no—
606C4—21no—
607C4—121no—
608C4—361no—
609C4C3×#379911yescharacteristic
610C4C3×D3×D1211yescharacteristic
611C3—21no—
612C3—21no—
613C3—21no—
614C3—41no—
615C3#1059036611yescharacteristic, direct product factor
616C3C3×#484411yescharacteristic, complemented
617C3C3×#501611yescharacteristic, complemented
618C2—122no—
619C2—721no—
620C2C3×#6382911yescharacteristic
621C2C3×#6383011yescharacteristic
622C2C3×#6390911yescharacteristic
subgroupG/H
centerC22×C3C2×#626
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C4C2×C3×D32
FittingC33×C42C22
socleC22×C33C2×D4
prime radicalC2×C3×#3714C2
hypercenterC3×C42D32
hypocenterC32C3×#463
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C3×C42

Complements

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

NH ≅ G/Ntype
1. C3×#64084C2semidirect
2. C32×C4×D12C2semidirect
3. C2×C3×#3716C2semidirect
5. C32×C4⋊(C3⋊C8)C2semidirect
6. C3×#63567C2semidirect
7. C3×#63564C2semidirect
10. #10590366C3direct (N × H)
25. #64084C2×C3semidirect
34. C3×C4×D12D3semidirect
35. C3×C4×D12C2×C3semidirect
36. C2×#3716C2×C3semidirect
44. C3×C4⋊(C3⋊C8)D3semidirect
45. C3×C4⋊(C3⋊C8)C2×C3semidirect
46. #63567C2×C3semidirect
47. #63564C2×C3semidirect
146. C4×D12C3×D3semidirect
156. C4⋊(C3⋊C8)C3×D3semidirect
363. C33#463semidirect
550. C32#4844semidirect
551. C32#5016semidirect
552. C32C3×#463semidirect
615. C3#10590366direct (N × H)
616. C3C3×#4844semidirect
617. C3C3×#5016semidirect

Maximal subgroups

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

subgroupindex
C3×#640842
C32×C4×D122
C2×C3×#37162
C2×C3×#37142
C32×C4⋊(C3⋊C8)2
C3×#635672
C3×#635642
C3×#50163
C3×#48443
#105903663