Finite Group Databasebeta

← C3×#10591438 #13158445

Subgroups — C3×#10591438 #13158445

Order 1728.

792 conjugacy classes of subgroups.

The subgroup lattice has 650 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C32×#940C211yesmaximal, characteristic
2C2×C32×#214C211yesmaximal, characteristic, complemented
3C3×Dic3×Dic6C211yesmaximal, characteristic, complemented
4C2×C3×#3715C211yesmaximal, characteristic, complemented
5C3×#63659C211yesmaximal, characteristic, complemented
6C3×Dic3⋊(C3⋊C8)C211yesmaximal, characteristic, complemented
7C3×#63563C211yesmaximal, characteristic, complemented
8C3×#5163—31nomaximal
9C3×#5075—31nomaximal
10#10591438C311yesmaximal, characteristic, direct product factor
11C2×C33×D4C2211yescharacteristic
12C2×C32×#67—21no—
13C2×C32×#67—21no—
14C2×C32×D12C2211yescharacteristic
15C2×C32×Dic6C2211yescharacteristic
16C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
17C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
18C32×#214—22no—
19C32×C4×Dic3C2211yescharacteristic
20C2×C32×C3⋊C8C2211yescharacteristic
21C3×Dic3⋊Dic3—21no—
22C3×Dic32—21no—
23C3×#3715—22no—
24C2×C32×SD16—31no—
25C32×#127—31no—
26C3×Q8×Dic3—31no—
27C2×C3×#215—31no—
28C3×#940—21no—
29C3×#940D311yescharacteristic
30C3×#940C2×C311yescharacteristic
31C2×C3×#214—21no—
32C2×C3×#214—31no—
33C2×C3×#214D311yescharacteristic, complemented
34C2×C3×#214C2×C311yescharacteristic, complemented
35C3×C4×Dic6—31no—
36Dic3×Dic6C2×C311yescharacteristic, complemented
37C2×#3715C2×C311yescharacteristic, complemented
38#63659C2×C311yescharacteristic, complemented
39C3×#832—31no—
40C3×#832—31no—
41C3×#832—31no—
42C3×#832—61no—
43C3×#821—31no—
44C3×Dic3⋊C8—31no—
45C3×C4⋊(C3⋊C8)—31no—
46Dic3⋊(C3⋊C8)C2×C311yescharacteristic, complemented
47#63563C2×C311yescharacteristic, complemented
48C23×C33—21no—
49C22×C32×D3—21no—
50C33×D4—21no—
51C33×D4—21no—
52C2×C33×C4C2311yescharacteristic
53C2×C3×C3⋊Dic3—21no—
54C32×#67—42no—
55C32×#67—42no—
56C2×C32×Dic3—21no—
57C2×C32×Dic3D411yescharacteristic
58C2×C32×Dic3D411yescharacteristic
59C32×D12—21no—
60C32×D12—21no—
61C32×Dic6—21no—
62C32×Dic6D412yescomplemented
63C3×C3⋊(C3⋊C8)—21no—
64C32×C3⋊C8—21no—
65C3×#460—91no—
66#5163—31no—
67#5075—31no—
68C2×C32×Q8—31no—
69C2×C32×D4—21no—
70C2×C32×D4—21no—
71C2×C32×D4—21no—
72C2×C32×D4—31no—
73C2×C32×D4—41no—
74C2×C32×D4—61no—
75C2×C32×D4—61no—
76C2×C32×D4C2×D311yescharacteristic
77C2×C32×D4C2×D311yescharacteristic
78C2×C32×D4C22×C311yescharacteristic
79C2×C3×#67—22no—
80C2×C3×#67—22no—
81C2×C3×#67—21no—
82C2×C3×#67—21no—
83C2×C3×#67—21no—
84C2×C3×#67—21no—
85C2×C3×D12—21no—
86C2×C3×D12C2×D311yescharacteristic
87C2×C3×D12C22×C311yescharacteristic
88C2×C3×Dic6—21no—
89C2×C3×Dic6C2×D311yescharacteristic
90C2×C3×Dic6C22×C311yescharacteristic
91C32×SD16—62no—
92C2×C32×C8—31no—
93C32×C42—31no—
94C4⋊(C3⋊Dic3)C22×C311yescharacteristic
95C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
96C3×#215—62no—
97C3×#214—24no—
98C3×#214—22no—
99C3×#214—22no—
100C3×#214—62no—
101C3×C4⋊Dic3—31no—
102C3×C4⋊Dic3—31no—
103C3×C4⋊Dic3—61no—
104C3×C4⋊Dic3—61no—
105C3×Dic3⋊C4—61no—
106C3×C4×Dic3—21no—
107C3×C4×Dic3—61no—
108C3×C4×Dic3—61no—
109C3×C4×Dic3C2×D311yescharacteristic
110C3×C4×Dic3C22×C311yescharacteristic
111C2×C3×C3⋊C8—21no—
112C2×C3×C3⋊C8—31no—
113C2×C3×C3⋊C8—31no—
114C2×C3×C3⋊C8—61no—
115C2×C3×C3⋊C8C2×D311yescharacteristic
116C2×C3×C3⋊C8C22×C311yescharacteristic
117Dic3⋊Dic3—21no—
118Dic32—21no—
119#3715—22no—
120C22×C33—41no—
121C22×C33—41no—
122C22×C33—41no—
123C22×C33C2×D411yescharacteristic
124C2×C32×D3—41no—
125C2×C32×D3—41no—
126C2×C32×D3—41no—
127C33×C4C2×D411yescharacteristic
128C33×C4#4111yescharacteristic
129C3×C3⋊Dic3—41no—
130C32×Dic3—22no—
131C32×Dic3—21no—
132C32×Dic3SD1612yescomplemented
133C2×C3×SD16—31no—
134C2×C3×SD16—31no—
135C2×C3×SD16—61no—
136C2×C3×SD16—91no—
137C3×#127—31no—
138C3×#127—31no—
139C3×#127—61no—
140C3×C4×Q8—91no—
141Q8×Dic3—31no—
142C2×#215—31no—
143#940C3×D311yescharacteristic
144C2×#214—31no—
145C2×#214C3×D311yescharacteristic, complemented
146C4×Dic6—31no—
147C3×C4⋊C8—91no—
148C3×#102—91no—
149C3×#102—91no—
150#832—31no—
151#832—31no—
152#832—31no—
153#832—61no—
154#821—31no—
155Dic3⋊C8—31no—
156C4⋊(C3⋊C8)—31no—
157C23×C32—21no—
158C23×C32—22no—
159C23×C32—21no—
160C23×C32—21no—
161C23×C32—41no—
162C23×C32—41no—
163C23×C32—42no—
164C23×C32—61no—
165C22×C3×D3—22no—
166C22×C3×D3—21no—
167C22×C3×D3—21no—
168C32×Q8—32no—
169C32×Q8—61no—
170C32×D4—21no—
171C32×D4—21no—
172C32×D4—22no—
173C32×D4—22no—
174C32×D4—21no—
175C32×D4—21no—
176C32×D4—21no—
177C32×D4—21no—
178C32×D4—41no—
179C32×D4—41no—
180C32×D4—41no—
181C32×D4—41no—
182C32×D4—42no—
183C32×D4—42no—
184C32×D4—61no—
185C32×D4—61no—
186C32×D4—122no—
187C32×D4—122no—
188C2×C32×C4—21no—
189C2×C32×C4—21no—
190C2×C32×C4—21no—
191C2×C32×C4—31no—
192C2×C32×C4—31no—
193C2×C32×C4—41no—
194C2×C32×C4—61no—
195C2×C32×C4C22×D311yescharacteristic
196C2×C32×C4C22×D311yescharacteristic
197C2×C32×C4C23×C311yescharacteristic
198C2×C3⋊Dic3—21no—
199C3×#67—44no—
200C3×#67—42no—
201C3×#67—42no—
202C3×#67—44no—
203C3×#67—42no—
204C3×#67—42no—
205C2×C3×Dic3—21no—
206C2×C3×Dic3—21no—
207C2×C3×Dic3—21no—
208C2×C3×Dic3—21no—
209C2×C3×Dic3—41no—
210C2×C3×Dic3—61no—
211C2×C3×Dic3—61no—
212C2×C3×Dic3—121no—
213C2×C3×Dic3#6711yescharacteristic
214C2×C3×Dic3#6711yescharacteristic
215C2×C3×Dic3C3×D411yescharacteristic
216C2×C3×Dic3C3×D411yescharacteristic
217C3×D12—21no—
218C3×D12—21no—
219C3×D12—21no—
220C3×D12—21no—
221C3×D12—41no—
222C3×D12—41no—
223C3×Dic6—21no—
224C3×Dic6—21no—
225C3×Dic6—22no—
226C3×Dic6—41no—
227C3×Dic6#6712yescomplemented
228C3×Dic6C3×D412yescomplemented
229C32×C8—61no—
230C3⋊(C3⋊C8)—21no—
231C3×C3⋊C8—22no—
232C3×C3⋊C8—21no—
233C3×C3⋊C8—21no—
234C3×C3⋊C8—61no—
235C3×C3⋊C8—61no—
236C3×C3⋊C8—121no—
237#460—91noSylow
238C2×C33—81no—
239C2×C33#12111yescharacteristic
240C2×C33C2×SD1611yescharacteristic
241C2×C33AGL(1,ℤ/8)11yescharacteristic
242C32×D3—81no—
243C2×C3×Q8—31no—
244C2×C3×Q8—31no—
245C2×C3×Q8—61no—
246C2×C3×D4—21no—
247C2×C3×D4—21no—
248C2×C3×D4—21no—
249C2×C3×D4—31no—
250C2×C3×D4—31no—
251C2×C3×D4—41no—
252C2×C3×D4—61no—
253C2×C3×D4—61no—
254C2×C3×D4—61no—
255C2×C3×D4—61no—
256C2×C3×D4—61no—
257C2×C3×D4—62no—
258C2×C3×D4—62no—
259C2×C3×D4D3211yescharacteristic
260C2×C3×D4C2×C3×D311yescharacteristic
261C2×C3×D4C2×C3×D311yescharacteristic
262C2×#67—21no—
263C2×#67—21no—
264C2×D12C2×C3×D311yescharacteristic
265C2×Dic6C2×C3×D311yescharacteristic
266C3×SD16—62no—
267C3×SD16—62no—
268C3×SD16—64no—
269C3×SD16—182no—
270C2×C3×C8—31no—
271C2×C3×C8—31no—
272C2×C3×C8—61no—
273C2×C3×C8—91no—
274C3×C4⋊C4—91no—
275C3×C4⋊C4—181no—
276C3×C42—31no—
277C3×C42—31no—
278C3×C42—61no—
279C3×C42—181no—
280#215—62no—
281#214—22no—
282#214—62no—
283C4⋊Dic3—31no—
284C4⋊Dic3—31no—
285C4⋊Dic3—61no—
286C4⋊Dic3—61no—
287Dic3⋊C4—61no—
288C4×Dic3—61no—
289C4×Dic3—61no—
290C4×Dic3C2×C3×D311yescharacteristic
291C2×C3⋊C8—31no—
292C2×C3⋊C8—31no—
293C2×C3⋊C8—61no—
294C2×C3⋊C8C2×C3×D311yescharacteristic
295C22×C32—21no—
296C22×C32—21no—
297C22×C32—21no—
298C22×C32—41no—
299C22×C32—42no—
300C22×C32—42no—
301C22×C32—42no—
302C22×C32—44no—
303C22×C32—41no—
304C22×C32—41no—
305C22×C32—41no—
306C22×C32—41no—
307C22×C32—41no—
308C22×C32—42no—
309C22×C32—42no—
310C22×C32—41no—
311C22×C32—41no—
312C22×C32—41no—
313C22×C32—41no—
314C22×C32—81no—
315C22×C32—81no—
316C22×C32—81no—
317C22×C32—81no—
318C22×C32—82no—
319C22×C32—82no—
320C22×C32—121no—
321C22×C32—121no—
322C22×C32—121no—
323C22×C32D3×D411yescharacteristic
324C22×C32C2×#6711yescharacteristic
325C22×C32C2×C3×D411yescharacteristic
326C2×C3×D3—42no—
327C2×C3×D3—42no—
328C2×C3×D3—42no—
329C2×C3×D3—41no—
330C2×C3×D3—41no—
331C2×C3×D3—41no—
332C2×C3×D3—41no—
333C2×C3×D3—41no—
334C2×C3×D3—41no—
335C32×C4—21no—
336C32×C4—21no—
337C32×C4—21no—
338C32×C4—21no—
339C32×C4—21no—
340C32×C4—21no—
341C32×C4—32no—
342C32×C4—41no—
343C32×C4—41no—
344C32×C4—61no—
345C32×C4—62no—
346C32×C4D3×D411yescharacteristic
347C32×C4#23711yescharacteristic
348C32×C4#23711yescharacteristic
349C32×C4C2×#6711yescharacteristic
350C32×C4C2×C3×D411yescharacteristic
351C32×C4C3×#4111yescharacteristic
352C3⋊Dic3—41no—
353C3×Dic3—22no—
354C3×Dic3—21no—
355C3×Dic3—22no—
356C3×Dic3—22no—
357C3×Dic3—21no—
358C3×Dic3—22no—
359C3×Dic3—42no—
360C3×Dic3—121no—
361C3×Dic3—121no—
362C3×Dic3—241no—
363C3×Dic3#21412yescomplemented
364C3×Dic3C3×SD1612yescomplemented
365C2×SD16—31no—
366C2×SD16—91no—
367#127—31no—
368C4×Q8—91no—
369C4⋊C8—91no—
370#102—91no—
371#102—91no—
372C33#46011yescharacteristic, complemented, Sylow
373C23×C3—22no—
374C23×C3—21no—
375C23×C3—21no—
376C23×C3—21no—
377C23×C3—41no—
378C23×C3—41no—
379C23×C3—42no—
380C23×C3—62no—
381C23×C3—61no—
382C23×C3—61no—
383C22×D3—21no—
384C3×Q8—32no—
385C3×Q8—32no—
386C3×Q8—62no—
387C3×Q8—61no—
388C3×Q8—61no—
389C3×Q8—121no—
390C3×D4—22no—
391C3×D4—22no—
392C3×D4—21no—
393C3×D4—21no—
394C3×D4—21no—
395C3×D4—21no—
396C3×D4—21no—
397C3×D4—21no—
398C3×D4—41no—
399C3×D4—41no—
400C3×D4—41no—
401C3×D4—41no—
402C3×D4—42no—
403C3×D4—42no—
404C3×D4—61no—
405C3×D4—61no—
406C3×D4—61no—
407C3×D4—61no—
408C3×D4—121no—
409C3×D4—121no—
410C3×D4—122no—
411C3×D4—122no—
412C3×D4—124no—
413C3×D4—124no—
414C3×D4—122no—
415C3×D4—122no—
416C2×C3×C4—21no—
417C2×C3×C4—21no—
418C2×C3×C4—21no—
419C2×C3×C4—31no—
420C2×C3×C4—31no—
421C2×C3×C4—31no—
422C2×C3×C4—31no—
423C2×C3×C4—41no—
424C2×C3×C4—61no—
425C2×C3×C4—61no—
426C2×C3×C4—61no—
427C2×C3×C4—61no—
428C2×C3×C4—121no—
429C2×C3×C4—181no—
430C2×C3×C4C2×D3211yescharacteristic
431C2×C3×C4C22×C3×D311yescharacteristic
432C2×C3×C4C22×C3×D311yescharacteristic
433#67—42no—
434#67—42no—
435C2×Dic3—21no—
436C2×Dic3—61no—
437C2×Dic3—61no—
438C2×Dic3—121no—
439C2×Dic3C3×#6711yescharacteristic
440C2×Dic3C3×#6711yescharacteristic
441D12—21no—
442D12—21no—
443Dic6—21no—
444Dic6C3×#6712yescomplemented
445C3×C8—62no—
446C3×C8—61no—
447C3×C8—61no—
448C3×C8—181no—
449C3⋊C8—21no—
450C3⋊C8—61no—
451C3⋊C8—61no—
452C3⋊C8—121no—
453C2×C32—21no—
454C2×C32—21no—
455C2×C32—21no—
456C2×C32—21no—
457C2×C32—21no—
458C2×C32—21no—
459C2×C32—21no—
460C2×C32—21no—
461C2×C32—21no—
462C2×C32—41no—
463C2×C32—41no—
464C2×C32—41no—
465C2×C32—82no—
466C2×C32—82no—
467C2×C32—82no—
468C2×C32—84no—
469C2×C32—81no—
470C2×C32—81no—
471C2×C32—81no—
472C2×C32—241no—
473C2×C32D3×SD1611yescharacteristic
474C2×C32#91411yescharacteristic
475C2×C32#93211yescharacteristic
476C2×C32C2×#21411yescharacteristic
477C2×C32#93811yescharacteristic
478C2×C32#93911yescharacteristic
479C2×C32C3×#12111yescharacteristic
480C2×C32C2×C3×SD1611yescharacteristic
481C2×C32C3×AGL(1,ℤ/8)11yescharacteristic
482C3×D3—82no—
483C3×D3—81no—
484C3×D3—81no—
485C2×Q8—31no—
486C2×D4—31no—
487C2×D4—61no—
488C2×D4—61no—
489C2×D4C3×D3211yescharacteristic
490SD16—62no—
491SD16—182no—
492C2×C8—31no—
493C2×C8—91no—
494C4⋊C4—91no—
495C4⋊C4—181no—
496C42—31no—
497C42—181no—
498C22×C3—21no—
499C22×C3—21no—
500C22×C3—21no—
501C22×C3—42no—
502C22×C3—42no—
503C22×C3—42no—
504C22×C3—42no—
505C22×C3—42no—
506C22×C3—44no—
507C22×C3—41no—
508C22×C3—41no—
509C22×C3—41no—
510C22×C3—41no—
511C22×C3—41no—
512C22×C3—41no—
513C22×C3—41no—
514C22×C3—41no—
515C22×C3—41no—
516C22×C3—41no—
517C22×C3—81no—
518C22×C3—81no—
519C22×C3—81no—
520C22×C3—81no—
521C22×C3—82no—
522C22×C3—82no—
523C22×C3—122no—
524C22×C3—122no—
525C22×C3—122no—
526C22×C3—121no—
527C22×C3—121no—
528C22×C3—121no—
529C22×C3—121no—
530C22×C3—121no—
531C22×C3—121no—
532C22×C3D3×#6711yescharacteristic
533C22×C3C3×D3×D411yescharacteristic
534C22×C3C2×C3×#6711yescharacteristic
535C2×D3—41no—
536C2×D3—41no—
537C2×D3—41no—
538C3×C4—21no—
539C3×C4—21no—
540C3×C4—21no—
541C3×C4—21no—
542C3×C4—21no—
543C3×C4—21no—
544C3×C4—32no—
545C3×C4—32no—
546C3×C4—41no—
547C3×C4—41no—
548C3×C4—62no—
549C3×C4—62no—
550C3×C4—61no—
551C3×C4—62no—
552C3×C4—61no—
553C3×C4—62no—
554C3×C4—122no—
555C3×C4—361no—
556C3×C4#380511yescharacteristic
557C3×C4D3×#6711yescharacteristic
558C3×C4C3×D3×D411yescharacteristic
559C3×C4C3×#23711yescharacteristic
560C3×C4C3×#23711yescharacteristic
561C3×C4C2×C3×#6711yescharacteristic
562Dic3—21no—
563Dic3—22no—
564Dic3—121no—
565Dic3—121no—
566Dic3—241no—
567Dic3C3×#21412yescomplemented
568C32—21no—
569C32—21no—
570C32—21no—
571C32—41no—
572C32#507511yescharacteristic, complemented
573C32#516311yescharacteristic, complemented
574C32C3×#46011yescharacteristic, complemented
575C23—21no—
576C23—61no—
577Q8—32no—
578Q8—61no—
579D4—21no—
580D4—21no—
581D4—61no—
582D4—61no—
583D4—122no—
584D4—122no—
585C2×C4—31no—
586C2×C4—31no—
587C2×C4—61no—
588C2×C4—181no—
589C2×C4C2×C3×D3211yescharacteristic
590C8—61no—
591C8—181no—
592C2×C3—21no—
593C2×C3—21no—
594C2×C3—21no—
595C2×C3—21no—
596C2×C3—21no—
597C2×C3—21no—
598C2×C3—21no—
599C2×C3—21no—
600C2×C3—21no—
601C2×C3—41no—
602C2×C3—41no—
603C2×C3—41no—
604C2×C3—82no—
605C2×C3—82no—
606C2×C3—82no—
607C2×C3—84no—
608C2×C3—81no—
609C2×C3—81no—
610C2×C3—81no—
611C2×C3—242no—
612C2×C3—241no—
613C2×C3—241no—
614C2×C3D3×#21411yescharacteristic
615C2×C3#6393411yescharacteristic
616C2×C3#6397011yescharacteristic
617C2×C3C3×D3×SD1611yescharacteristic
618C2×C3C3×#91411yescharacteristic
619C2×C3C3×#93211yescharacteristic
620C2×C3C2×C3×#21411yescharacteristic
621C2×C3C3×#93811yescharacteristic
622C2×C3C3×#93911yescharacteristic
623D3—81no—
624C22—41no—
625C22—41no—
626C22—41no—
627C22—121no—
628C22—121no—
629C22—121no—
630C22C3×D3×#6711yescharacteristic
631C4—32no—
632C4—62no—
633C4—61no—
634C4—361no—
635C4C3×#380511yescharacteristic
636C4C3×D3×#6711yescharacteristic
637C3—21no—
638C3—21no—
639C3—21no—
640C3—41no—
641C3#1059143811yescharacteristic, direct product factor
642C3C3×#507511yescharacteristic, complemented
643C3C3×#516311yescharacteristic, complemented
644C2—81no—
645C2—241no—
646C2C3×D3×#21411yescharacteristic
647C2C3×#6393411yescharacteristic
648C2C3×#6397011yescharacteristic
subgroupG/H
centerC22×C3D3×#67
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C4C2×C3×D32
FittingC2×C33×D4C22
socleC22×C33C2×D4
prime radicalC32×#940C2
hypercenterC2×C3×D4D32
hypocenterC32C3×#460
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C2×C3×C4 ◁ C2×C3×D4

Complements

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

NH ≅ G/Ntype
2. C2×C32×#214C2semidirect
3. C3×Dic3×Dic6C2semidirect
4. C2×C3×#3715C2semidirect
5. C3×#63659C2semidirect
6. C3×Dic3⋊(C3⋊C8)C2semidirect
7. C3×#63563C2semidirect
10. #10591438C3direct (N × H)
33. C2×C3×#214D3semidirect
34. C2×C3×#214C2×C3semidirect
36. Dic3×Dic6C2×C3semidirect
37. C2×#3715C2×C3semidirect
38. #63659C2×C3semidirect
46. Dic3⋊(C3⋊C8)C2×C3semidirect
47. #63563C2×C3semidirect
62. C32×Dic6D4semidirect
132. C32×Dic3SD16semidirect
145. C2×#214C3×D3semidirect
227. C3×Dic6#67semidirect
228. C3×Dic6C3×D4semidirect
363. C3×Dic3#214semidirect
364. C3×Dic3C3×SD16semidirect
372. C33#460semidirect
444. Dic6C3×#67semidirect
567. Dic3C3×#214semidirect
572. C32#5075semidirect
573. C32#5163semidirect
574. C32C3×#460semidirect
641. C3#10591438direct (N × H)
642. C3C3×#5075semidirect
643. C3C3×#5163semidirect

Maximal subgroups

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

subgroupindex
C32×#9402
C2×C32×#2142
C3×Dic3×Dic62
C2×C3×#37152
C3×#636592
C3×Dic3⋊(C3⋊C8)2
C3×#635632
C3×#51633
C3×#50753
#105914383