Finite Group Databasebeta

← C3×#10591489 #13158502

Subgroups — C3×#10591489 #13158502

Order 1728.

764 conjugacy classes of subgroups.

The subgroup lattice has 612 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#3754C211yesmaximal, characteristic
2C32×D4×Dic3C211yesmaximal, characteristic
3C2×C32×#214C211yesmaximal, characteristic
4C3×#63866C211yesmaximal, characteristic, complemented
5C3×Dic3⋊(C3⋊C8)C211yesmaximal, characteristic, complemented
6C3×#63567C211yesmaximal, characteristic, complemented
7C3×#63565C211yesmaximal, characteristic, complemented
8C3×#5162—31nomaximal
9C3×#5019—31nomaximal
10#10591489C311yesmaximal, characteristic, direct product factor
11C2×C33×D4C2211yescharacteristic
12C2×C3×#634C2211yescharacteristic
13C22×C32×Dic3—21no—
14C2×C32×Dic6C2211yescharacteristic
15C3×#3754—22no—
16C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
17C32×#217—21no—
18C32×#214—22no—
19C32×C4⋊Dic3C2211yescharacteristic
20C32×C4×Dic3C2211yescharacteristic
21C2×C32×C3⋊C8C2211yescharacteristic
22C3×Dic3⋊Dic3—21no—
23C3×Dic3⋊Dic3—21no—
24C2×C32×SD16—31no—
25C32×C4×D4—31no—
26C2×#3754C2×C311yescharacteristic
27C3×#944—31no—
28C3×D4×Dic3—21no—
29C3×D4×Dic3D311yescharacteristic
30C3×D4×Dic3C2×C311yescharacteristic
31C2×C3×#214—21no—
32C2×C3×#214—31no—
33C2×C3×#214—31no—
34C2×C3×#214—61no—
35C2×C3×#214D311yescharacteristic
36C2×C3×#214C2×C311yescharacteristic
37C3×#869—31no—
38#63866C2×C311yescharacteristic, complemented
39C3×#835—31no—
40C3×#816—31no—
41C3×Dic3⋊C8—31no—
42C3×#810—31no—
43C3×#810—31no—
44C3×C4⋊(C3⋊C8)—31no—
45Dic3⋊(C3⋊C8)C2×C311yescharacteristic, complemented
46#63567C2×C311yescharacteristic, complemented
47#63565C2×C311yescharacteristic, complemented
48C23×C33—21no—
49C33×D4—21no—
50C33×D4D412yes—
51C2×C33×C4C2311yescharacteristic
52C2×C3×C3⋊Dic3—21no—
53C3×#634—21no—
54C3×#634—21no—
55C2×C32×Dic3—21no—
56C2×C32×Dic3—21no—
57C2×C32×Dic3—42no—
58C2×C32×Dic3D411yescharacteristic
59C2×C32×Dic3D411yescharacteristic
60C32×Dic6—21no—
61C32×Dic6—21no—
62C3×C3⋊(C3⋊C8)—21no—
63C32×C3⋊C8—21no—
64C3×#461—91no—
65#5162—31no—
66#5019—31no—
67C2×C32×Q8—31no—
68C2×C32×D4—21no—
69C2×C32×D4—21no—
70C2×C32×D4—21no—
71C2×C32×D4—41no—
72C2×C32×D4C2×D311yescharacteristic
73C2×C32×D4C2×D311yescharacteristic
74C2×C32×D4C22×C311yescharacteristic
75C22×C32×C4—61no—
76C2×#634C22×C311yescharacteristic
77C22×C3×Dic3—22no—
78C22×C3×Dic3—21no—
79C22×C3×Dic3—21no—
80C2×C3×Dic6—21no—
81C2×C3×Dic6—31no—
82C2×C3×Dic6—31no—
83C2×C3×Dic6—61no—
84C2×C3×Dic6C2×D311yescharacteristic
85C2×C3×Dic6C22×C311yescharacteristic
86C32×SD16—62no—
87C2×C32×C8—31no—
88C32×C4⋊C4—31no—
89C32×C22⋊C4—61no—
90C32×C42—31no—
91#3754—22no—
92C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
93C3×#217—22no—
94C3×#217—21no—
95C3×#217—21no—
96C3×#214—24no—
97C3×#214—22no—
98C3×#214—22no—
99C3×#214—64no—
100C3×#214—62no—
101C3×#214—62no—
102C3×C4⋊Dic3—21no—
103C3×C4⋊Dic3—61no—
104C3×C4⋊Dic3—61no—
105C3×C4⋊Dic3C2×D311yescharacteristic
106C3×C4⋊Dic3C22×C311yescharacteristic
107C3×Dic3⋊C4—61no—
108C3×Dic3⋊C4—61no—
109C3×C4×Dic3—21no—
110C3×C4×Dic3C2×D311yescharacteristic
111C3×C4×Dic3C22×C311yescharacteristic
112C2×C3×C3⋊C8—21no—
113C2×C3×C3⋊C8—31no—
114C2×C3×C3⋊C8—31no—
115C2×C3×C3⋊C8—61no—
116C2×C3×C3⋊C8C2×D311yescharacteristic
117C2×C3×C3⋊C8C22×C311yescharacteristic
118Dic3⋊Dic3—21no—
119Dic3⋊Dic3—21no—
120C22×C33—22no—
121C22×C33—41no—
122C22×C33—41no—
123C22×C33C2×D411yescharacteristic
124C33×C4C2×D411yescharacteristic
125C33×C4#4111yescharacteristic
126C3×C3⋊Dic3—41no—
127C32×Dic3—21no—
128C32×Dic3—41no—
129C32×Dic3—41no—
130C32×Dic3SD1612yescomplemented
131C2×C3×SD16—31no—
132C2×C3×SD16—31no—
133C2×C3×SD16—61no—
134C2×C3×SD16—91no—
135C3×C4⋊Q8—91no—
136C3×C4×D4—31no—
137C3×C4×D4—31no—
138C3×C4×D4—61no—
139#944—31no—
140D4×Dic3C3×D311yescharacteristic
141C2×#214—31no—
142C2×#214—31no—
143C2×#214—61no—
144C2×#214C3×D311yescharacteristic
145#869—31no—
146C3×C4⋊C8—91no—
147C3×#103—91no—
148C3×#103—91no—
149#835—31no—
150#816—31no—
151Dic3⋊C8—31no—
152#810—31no—
153#810—31no—
154C4⋊(C3⋊C8)—31no—
155C23×C32—21no—
156C23×C32—22no—
157C23×C32—21no—
158C23×C32—21no—
159C23×C32—41no—
160C23×C32—41no—
161C23×C32—42no—
162C32×Q8—61no—
163C32×Q8—61no—
164C32×D4—21no—
165C32×D4—22no—
166C32×D4—21no—
167C32×D4—21no—
168C32×D4—22no—
169C32×D4—22no—
170C32×D4—22no—
171C32×D4—42no—
172C32×D4—41no—
173C32×D4—41no—
174C32×D4—42no—
175C32×D4D1212yes—
176C32×D4#6712yes—
177C32×D4C3×D412yes—
178C2×C32×C4—21no—
179C2×C32×C4—21no—
180C2×C32×C4—21no—
181C2×C32×C4—31no—
182C2×C32×C4—31no—
183C2×C32×C4—41no—
184C2×C32×C4—61no—
185C2×C32×C4—61no—
186C2×C32×C4—122no—
187C2×C32×C4C22×D311yescharacteristic
188C2×C32×C4C22×D311yescharacteristic
189C2×C32×C4C23×C311yescharacteristic
190C2×C3⋊Dic3—21no—
191#634—21no—
192#634—21no—
193C2×C3×Dic3—22no—
194C2×C3×Dic3—21no—
195C2×C3×Dic3—21no—
196C2×C3×Dic3—21no—
197C2×C3×Dic3—21no—
198C2×C3×Dic3—21no—
199C2×C3×Dic3—21no—
200C2×C3×Dic3—41no—
201C2×C3×Dic3—44no—
202C2×C3×Dic3—42no—
203C2×C3×Dic3—42no—
204C2×C3×Dic3—61no—
205C2×C3×Dic3—61no—
206C2×C3×Dic3—121no—
207C2×C3×Dic3#6711yescharacteristic
208C2×C3×Dic3#6711yescharacteristic
209C2×C3×Dic3C3×D411yescharacteristic
210C2×C3×Dic3C3×D411yescharacteristic
211C3×Dic6—22no—
212C3×Dic6—21no—
213C3×Dic6—21no—
214C3×Dic6—21no—
215C3×Dic6—21no—
216C3×Dic6—41no—
217C3×Dic6—62no—
218C3×Dic6—61no—
219C3×Dic6—61no—
220C3×Dic6—61no—
221C3×Dic6—61no—
222C3×Dic6—121no—
223C32×C8—61no—
224C3⋊(C3⋊C8)—21no—
225C3×C3⋊C8—22no—
226C3×C3⋊C8—21no—
227C3×C3⋊C8—21no—
228C3×C3⋊C8—62no—
229C3×C3⋊C8—61no—
230C3×C3⋊C8—61no—
231#461—91noSylow
232C2×C33—42no—
233C2×C33#12111yescharacteristic
234C2×C33C2×SD1611yescharacteristic
235C2×C33#13711yescharacteristic
236C2×C3×Q8—31no—
237C2×C3×Q8—31no—
238C2×C3×Q8—61no—
239C2×C3×Q8—91no—
240C2×C3×D4—21no—
241C2×C3×D4—21no—
242C2×C3×D4—21no—
243C2×C3×D4—41no—
244C2×C3×D4D3211yescharacteristic
245C2×C3×D4C2×C3×D311yescharacteristic
246C2×C3×D4C2×C3×D311yescharacteristic
247C22×C3×C4—61no—
248C22×C3×C4—61no—
249C22×C3×C4—62no—
250C22×Dic3—21no—
251C2×Dic6—31no—
252C2×Dic6—31no—
253C2×Dic6—61no—
254C2×Dic6C2×C3×D311yescharacteristic
255C3×SD16—64no—
256C3×SD16—62no—
257C3×SD16—62no—
258C3×SD16—182no—
259C2×C3×C8—31no—
260C2×C3×C8—31no—
261C2×C3×C8—61no—
262C2×C3×C8—91no—
263C3×C4⋊C4—31no—
264C3×C4⋊C4—31no—
265C3×C4⋊C4—61no—
266C3×C4⋊C4—181no—
267C3×C4⋊C4—181no—
268C3×C22⋊C4—61no—
269C3×C22⋊C4—61no—
270C3×C22⋊C4—62no—
271C3×C42—31no—
272C3×C42—31no—
273C3×C42—61no—
274#217—21no—
275#214—22no—
276#214—64no—
277#214—62no—
278#214—62no—
279C4⋊Dic3—61no—
280C4⋊Dic3—61no—
281C4⋊Dic3C2×C3×D311yescharacteristic
282Dic3⋊C4—61no—
283Dic3⋊C4—61no—
284C4×Dic3C2×C3×D311yescharacteristic
285C2×C3⋊C8—31no—
286C2×C3⋊C8—31no—
287C2×C3⋊C8—61no—
288C2×C3⋊C8C2×C3×D311yescharacteristic
289C22×C32—22no—
290C22×C32—24no—
291C22×C32—22no—
292C22×C32—22no—
293C22×C32—21no—
294C22×C32—21no—
295C22×C32—21no—
296C22×C32—41no—
297C22×C32—42no—
298C22×C32—42no—
299C22×C32—44no—
300C22×C32—41no—
301C22×C32—41no—
302C22×C32—42no—
303C22×C32—42no—
304C22×C32—41no—
305C22×C32—41no—
306C22×C32—41no—
307C22×C32—41no—
308C22×C32—81no—
309C22×C32—81no—
310C22×C32—81no—
311C22×C32—81no—
312C22×C32—82no—
313C22×C32—82no—
314C22×C32D3×D411yescharacteristic
315C22×C32C2×#6711yescharacteristic
316C22×C32C2×C3×D411yescharacteristic
317C32×C4—21no—
318C32×C4—21no—
319C32×C4—21no—
320C32×C4—21no—
321C32×C4—21no—
322C32×C4—21no—
323C32×C4—32no—
324C32×C4—41no—
325C32×C4—41no—
326C32×C4—61no—
327C32×C4—121no—
328C32×C4—121no—
329C32×C4C2×D1211yescharacteristic
330C32×C4#23511yescharacteristic
331C32×C4#23711yescharacteristic
332C32×C4C2×#6711yescharacteristic
333C32×C4C2×C3×D411yescharacteristic
334C32×C4C3×#4111yescharacteristic
335C3⋊Dic3—41no—
336C3×Dic3—21no—
337C3×Dic3—21no—
338C3×Dic3—22no—
339C3×Dic3—41no—
340C3×Dic3—42no—
341C3×Dic3—42no—
342C3×Dic3—41no—
343C3×Dic3—41no—
344C3×Dic3—41no—
345C3×Dic3—41no—
346C3×Dic3—122no—
347C3×Dic3—121no—
348C3×Dic3—121no—
349C3×Dic3#21412yescomplemented
350C3×Dic3C3×SD1612yescomplemented
351C2×SD16—31no—
352C2×SD16—91no—
353C4⋊Q8—91no—
354C4×D4—31no—
355C4⋊C8—91no—
356#103—91no—
357#103—91no—
358C33#46111yescharacteristic, complemented, Sylow
359C23×C3—22no—
360C23×C3—21no—
361C23×C3—21no—
362C23×C3—21no—
363C23×C3—41no—
364C23×C3—41no—
365C23×C3—42no—
366C3×Q8—62no—
367C3×Q8—61no—
368C3×Q8—61no—
369C3×Q8—61no—
370C3×Q8—61no—
371C3×Q8—121no—
372C3×Q8—181no—
373C3×Q8—181no—
374C3×D4—22no—
375C3×D4—21no—
376C3×D4—21no—
377C3×D4—21no—
378C3×D4—22no—
379C3×D4—22no—
380C3×D4—22no—
381C3×D4—41no—
382C3×D4—41no—
383C3×D4—42no—
384C3×D4—42no—
385C3×D4#62612yes—
386C3×D4C3×D1212yes—
387C3×D4C3×#6712yes—
388C2×C3×C4—21no—
389C2×C3×C4—21no—
390C2×C3×C4—21no—
391C2×C3×C4—31no—
392C2×C3×C4—31no—
393C2×C3×C4—31no—
394C2×C3×C4—31no—
395C2×C3×C4—41no—
396C2×C3×C4—61no—
397C2×C3×C4—61no—
398C2×C3×C4—62no—
399C2×C3×C4—61no—
400C2×C3×C4—61no—
401C2×C3×C4—61no—
402C2×C3×C4—61no—
403C2×C3×C4—121no—
404C2×C3×C4—122no—
405C2×C3×C4—122no—
406C2×C3×C4—124no—
407C2×C3×C4—181no—
408C2×C3×C4C2×D3211yescharacteristic
409C2×C3×C4C22×C3×D311yescharacteristic
410C2×C3×C4C22×C3×D311yescharacteristic
411C2×Dic3—21no—
412C2×Dic3—21no—
413C2×Dic3—42no—
414C2×Dic3—61no—
415C2×Dic3—61no—
416C2×Dic3—121no—
417C2×Dic3C3×#6711yescharacteristic
418C2×Dic3C3×#6711yescharacteristic
419Dic6—21no—
420Dic6—21no—
421Dic6—62no—
422Dic6—61no—
423Dic6—61no—
424Dic6—61no—
425Dic6—61no—
426Dic6—121no—
427C3×C8—62no—
428C3×C8—61no—
429C3×C8—61no—
430C3×C8—181no—
431C3⋊C8—21no—
432C3⋊C8—62no—
433C3⋊C8—61no—
434C3⋊C8—61no—
435C2×C32—21no—
436C2×C32—21no—
437C2×C32—21no—
438C2×C32—21no—
439C2×C32—21no—
440C2×C32—21no—
441C2×C32—21no—
442C2×C32—21no—
443C2×C32—21no—
444C2×C32—41no—
445C2×C32—41no—
446C2×C32—41no—
447C2×C32—42no—
448C2×C32—44no—
449C2×C32—42no—
450C2×C32—42no—
451C2×C32—82no—
452C2×C32—82no—
453C2×C32—84no—
454C2×C32D3×SD1611yescharacteristic
455C2×C32#91511yescharacteristic
456C2×C32#93011yescharacteristic
457C2×C32C2×#21411yescharacteristic
458C2×C32#93911yescharacteristic
459C2×C32#95111yescharacteristic
460C2×C32C3×#12111yescharacteristic
461C2×C32C2×C3×SD1611yescharacteristic
462C2×C32C3×#13711yescharacteristic
463C2×Q8—31no—
464C2×Q8—91no—
465C2×D4C3×D3211yescharacteristic
466C22×C4—61no—
467SD16—62no—
468SD16—182no—
469C2×C8—31no—
470C2×C8—91no—
471C4⋊C4—31no—
472C4⋊C4—181no—
473C4⋊C4—181no—
474C22⋊C4—61no—
475C42—31no—
476C22×C3—24no—
477C22×C3—22no—
478C22×C3—22no—
479C22×C3—22no—
480C22×C3—21no—
481C22×C3—21no—
482C22×C3—21no—
483C22×C3—42no—
484C22×C3—42no—
485C22×C3—44no—
486C22×C3—42no—
487C22×C3—42no—
488C22×C3—41no—
489C22×C3—41no—
490C22×C3—41no—
491C22×C3—41no—
492C22×C3—41no—
493C22×C3—41no—
494C22×C3—41no—
495C22×C3—81no—
496C22×C3—81no—
497C22×C3—81no—
498C22×C3—81no—
499C22×C3—82no—
500C22×C3—82no—
501C22×C3D3×#6711yescharacteristic
502C22×C3C3×D3×D411yescharacteristic
503C22×C3C2×C3×#6711yescharacteristic
504C3×C4—21no—
505C3×C4—21no—
506C3×C4—21no—
507C3×C4—21no—
508C3×C4—21no—
509C3×C4—21no—
510C3×C4—32no—
511C3×C4—32no—
512C3×C4—41no—
513C3×C4—41no—
514C3×C4—62no—
515C3×C4—61no—
516C3×C4—61no—
517C3×C4—121no—
518C3×C4—122no—
519C3×C4—122no—
520C3×C4—121no—
521C3×C4—121no—
522C3×C4—121no—
523C3×C4—121no—
524C3×C4—361no—
525C3×C4#380411yescharacteristic
526C3×C4C2×#62611yescharacteristic
527C3×C4C2×C3×D1211yescharacteristic
528C3×C4C3×#23511yescharacteristic
529C3×C4C3×#23711yescharacteristic
530C3×C4C2×C3×#6711yescharacteristic
531Dic3—21no—
532Dic3—41no—
533Dic3—41no—
534Dic3—122no—
535Dic3—121no—
536Dic3—121no—
537Dic3C3×#21412yescomplemented
538C32—21no—
539C32—21no—
540C32—21no—
541C32—41no—
542C32#501911yescharacteristic, complemented
543C32#516211yescharacteristic, complemented
544C32C3×#46111yescharacteristic, complemented
545C23—21no—
546Q8—61no—
547Q8—61no—
548Q8—181no—
549Q8—181no—
550D4—21no—
551D4C3×#62612yes—
552C2×C4—31no—
553C2×C4—31no—
554C2×C4—61no—
555C2×C4—61no—
556C2×C4—122no—
557C2×C4—181no—
558C2×C4C2×C3×D3211yescharacteristic
559C8—61no—
560C8—181no—
561C2×C3—21no—
562C2×C3—21no—
563C2×C3—21no—
564C2×C3—21no—
565C2×C3—21no—
566C2×C3—21no—
567C2×C3—21no—
568C2×C3—21no—
569C2×C3—21no—
570C2×C3—44no—
571C2×C3—42no—
572C2×C3—42no—
573C2×C3—42no—
574C2×C3—41no—
575C2×C3—41no—
576C2×C3—41no—
577C2×C3—82no—
578C2×C3—82no—
579C2×C3—84no—
580C2×C3D3×#21411yescharacteristic
581C2×C3#6392911yescharacteristic
582C2×C3#6397811yescharacteristic
583C2×C3C3×D3×SD1611yescharacteristic
584C2×C3C3×#91511yescharacteristic
585C2×C3C3×#93011yescharacteristic
586C2×C3C2×C3×#21411yescharacteristic
587C2×C3C3×#93911yescharacteristic
588C2×C3C3×#95111yescharacteristic
589C22—22no—
590C22—41no—
591C22—41no—
592C22C3×D3×#6711yescharacteristic
593C4—32no—
594C4—61no—
595C4—121no—
596C4—121no—
597C4—361no—
598C4C3×#380411yescharacteristic
599C4C2×C3×#62611yescharacteristic
600C3—21no—
601C3—21no—
602C3—21no—
603C3—41no—
604C3#1059148911yescharacteristic, direct product factor
605C3C3×#501911yescharacteristic, complemented
606C3C3×#516211yescharacteristic, complemented
607C2—42no—
608C2C3×D3×#21411yescharacteristic
609C2C3×#6392911yescharacteristic
610C2C3×#6397811yescharacteristic
subgroupG/H
centerC22×C3D3×#67
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C4C2×C3×D32
Fitting, prime radicalC2×C33×D4C22
socleC22×C33C2×D4
hypercenterC2×C3×D4D32
hypocenterC32C3×#461
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
4. C3×#63866C2semidirect
5. C3×Dic3⋊(C3⋊C8)C2semidirect
6. C3×#63567C2semidirect
7. C3×#63565C2semidirect
10. #10591489C3direct (N × H)
38. #63866C2×C3semidirect
45. Dic3⋊(C3⋊C8)C2×C3semidirect
46. #63567C2×C3semidirect
47. #63565C2×C3semidirect
130. C32×Dic3SD16semidirect
349. C3×Dic3#214semidirect
350. C3×Dic3C3×SD16semidirect
358. C33#461semidirect
537. Dic3C3×#214semidirect
542. C32#5019semidirect
543. C32#5162semidirect
544. C32C3×#461semidirect
604. C3#10591489direct (N × H)
605. C3C3×#5019semidirect
606. C3C3×#5162semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#37542
C32×D4×Dic32
C2×C32×#2142
C3×#638662
C3×Dic3⋊(C3⋊C8)2
C3×#635672
C3×#635652
C3×#51623
C3×#50193
#105914893