Finite Group Databasebeta

← C3×#10591183 #13158362

Subgroups — C3×#10591183 #13158362

Order 1728.

1166 conjugacy classes of subgroups.

The subgroup lattice has 588 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×C4⋊(C3⋊D3)C211yesmaximal, characteristic
2C32×#924C211yesmaximal, characteristic, complemented
3C2×C3×#3717C212yesmaximal, complemented
4C32×#830C211yesmaximal, characteristic, complemented
5C3×#63564C212yesmaximal, complemented
6C3×#5046—31nomaximal
7C3×#4731—31nomaximal
8#10591183C311yesmaximal, characteristic, direct product factor
9C23×C3×C3⋊D3—21no—
10C22×C33×C4C2211yescharacteristic
11C2×C3×C4⋊(C3⋊D3)—21no—
12C2×C3×C4⋊(C3⋊D3)—24no—
13C2×C3×C4⋊(C3⋊D3)C2212yes—
14C2×C32×Dic6C2211yescharacteristic, complemented
15C32×#217—21no—
16C32×C4⋊Dic3C2211yescharacteristic, complemented
17C32×Dic3⋊C4—21no—
18C2×C32×C3⋊C8C2212yescomplemented
19C3×#3717—24no—
20C22×C4⋊(C3⋊D3)C2×C311yescharacteristic
21C22×C3×D12—31no—
22C22×C3×D12—31no—
23C22×C3×D12—61no—
24C32×C22⋊Q8—31no—
25C2×C3×#215—32no—
26C3×#924—21no—
27C3×#924D311yescharacteristic, complemented
28C3×#924C2×C311yescharacteristic, complemented
29C2×C3×#204—32no—
30C2×#3717C2×C312yescomplemented
31C32×C22⋊C8—31no—
32C3×#830—21no—
33C3×#830D311yescharacteristic, complemented
34C3×#830C2×C311yescharacteristic, complemented
35C3×#821—32no—
36C3×#809—32no—
37#63564C2×C312yescomplemented
38C23×C33D411yescharacteristic
39C22×C3×C3⋊D3—24no—
40C22×C3×C3⋊D3—22no—
41C22×C3×C3⋊D3—42no—
42C22×C3×C3⋊D3—42no—
43C2×C33×C4—22no—
44C2×C33×C4D411yescharacteristic
45C2×C33×C4C2311yescharacteristic
46C3×C4⋊(C3⋊D3)—24no—
47C3×C4⋊(C3⋊D3)—22no—
48C3×C4⋊(C3⋊D3)D414yes—
49C2×C32×Dic3—21no—
50C2×C32×Dic3—21no—
51C32×Dic6—22no—
52C32×C3⋊C8—22no—
53C3×#450—91no—
54#5046—31no—
55#4731—31no—
56C23×C3⋊D3—21no—
57C23×C3×D3—62no—
58C23×C3×D3—61no—
59C23×C3×D3—61no—
60C2×C32×Q8—31no—
61C22×C32×C4—21no—
62C22×C32×C4—21no—
63C22×C32×C4—21no—
64C22×C32×C4—41no—
65C22×C32×C4C2×D311yescharacteristic
66C22×C32×C4C2×D311yescharacteristic
67C22×C32×C4C22×C311yescharacteristic
68C2×C4⋊(C3⋊D3)—21no—
69C2×C4⋊(C3⋊D3)—24no—
70C2×C4⋊(C3⋊D3)C22×C312yes—
71C2×C3×D12—32no—
72C2×C3×D12—32no—
73C2×C3×D12—64no—
74C2×C3×D12—64no—
75C2×C3×D12—62no—
76C2×C3×D12—68no—
77C2×C3×D12—62no—
78C2×C3×D12—61no—
79C2×C3×D12—61no—
80C2×C3×Dic6—21no—
81C2×C3×Dic6C2×D311yescharacteristic, complemented
82C2×C3×Dic6C22×C311yescharacteristic, complemented
83C2×C32×C8—32no—
84C32×C4⋊C4—31no—
85C32×C4⋊C4—61no—
86C32×C22⋊C4—61no—
87C3×#217—22no—
88C3×#217—21no—
89C3×#217—21no—
90C3×#215—64no—
91C3×C4⋊Dic3—21no—
92C3×C4⋊Dic3C2×D311yescharacteristic, complemented
93C3×C4⋊Dic3C22×C311yescharacteristic, complemented
94C3×Dic3⋊C4—22no—
95C3×Dic3⋊C4—21no—
96C3×Dic3⋊C4—21no—
97C2×C3×C3⋊C8—22no—
98C2×C3×C3⋊C8C2×D312yescomplemented
99C2×C3×C3⋊C8C22×C312yescomplemented
100C3×#204—64no—
101#3717—24no—
102C22×C33—21no—
103C22×C33—21no—
104C22×C33SD1612yescomplemented
105C22×C33C2×D411yescharacteristic
106C2×C3×C3⋊D3—24no—
107C2×C3×C3⋊D3—48no—
108C2×C3×C3⋊D3—42no—
109C2×C3×C3⋊D3—42no—
110C33×C4—21no—
111C33×C4C2×D412yes—
112C32×Dic3—41no—
113C32×Dic3—41no—
114C22×C3×D4—91no—
115C22×D12—31no—
116C22×D12—31no—
117C22×D12—61no—
118C2×C3×SD16—92no—
119C3×C22⋊Q8—31no—
120C3×C22⋊Q8—31no—
121C3×C22⋊Q8—61no—
122C2×#215—32no—
123#924C3×D311yescharacteristic, complemented
124C2×#204—32no—
125C3×#102—92no—
126C3×C22⋊C8—31no—
127C3×C22⋊C8—31no—
128C3×C22⋊C8—61no—
129#830C3×D311yescharacteristic, complemented
130#821—32no—
131#809—32no—
132C23×C32—21no—
133C23×C32—21no—
134C23×C32—21no—
135C23×C32—41no—
136C23×C32D1211yescharacteristic
137C23×C32#6711yescharacteristic
138C23×C32C3×D411yescharacteristic
139C22×C3⋊D3—24no—
140C22×C3⋊D3—22no—
141C22×C3⋊D3—42no—
142C22×C3⋊D3—42no—
143C22×C3×D3—68no—
144C22×C3×D3—64no—
145C22×C3×D3—64no—
146C22×C3×D3—64no—
147C22×C3×D3—62no—
148C22×C3×D3—62no—
149C22×C3×D3—124no—
150C22×C3×D3—122no—
151C22×C3×D3—122no—
152C22×C3×D3—124no—
153C22×C3×D3—122no—
154C22×C3×D3—122no—
155C32×Q8—62no—
156C2×C32×C4—22no—
157C2×C32×C4—24no—
158C2×C32×C4—22no—
159C2×C32×C4—22no—
160C2×C32×C4—21no—
161C2×C32×C4—21no—
162C2×C32×C4—21no—
163C2×C32×C4—21no—
164C2×C32×C4—21no—
165C2×C32×C4—21no—
166C2×C32×C4—41no—
167C2×C32×C4—41no—
168C2×C32×C4—42no—
169C2×C32×C4—42no—
170C2×C32×C4—44no—
171C2×C32×C4—61no—
172C2×C32×C4—61no—
173C2×C32×C4D1211yescharacteristic
174C2×C32×C4#6711yescharacteristic
175C2×C32×C4C3×D411yescharacteristic
176C2×C32×C4C22×D311yescharacteristic
177C2×C32×C4C22×D311yescharacteristic
178C2×C32×C4C23×C311yescharacteristic
179C4⋊(C3⋊D3)—24no—
180C4⋊(C3⋊D3)—22no—
181C4⋊(C3⋊D3)C3×D414yes—
182C2×C3×Dic3—22no—
183C2×C3×Dic3—22no—
184C2×C3×Dic3—21no—
185C2×C3×Dic3—21no—
186C2×C3×Dic3—21no—
187C2×C3×Dic3—21no—
188C3×D12—34no—
189C3×D12—34no—
190C3×D12—64no—
191C3×D12—68no—
192C3×D12—64no—
193C3×D12—64no—
194C3×D12—64no—
195C3×D12—62no—
196C3×D12—62no—
197C3×Dic6—22no—
198C3×Dic6—22no—
199C3×Dic6—42no—
200C32×C8—62no—
201C3×C3⋊C8—22no—
202C3×C3⋊C8—22no—
203C3×C3⋊C8—42no—
204#450—91noSylow
205C2×C33—22no—
206C2×C33#12011yescharacteristic
207C2×C33C2×SD1611yescharacteristic
208C2×C33AGL(1,ℤ/8)11yescharacteristic
209C3×C3⋊D3—44no—
210C24×C3—181no—
211C23×D3—62no—
212C23×D3—61no—
213C23×D3—61no—
214C2×C3×Q8—31no—
215C2×C3×Q8—31no—
216C2×C3×Q8—61no—
217C2×C3×D4—92no—
218C2×C3×D4—184no—
219C2×C3×D4—181no—
220C22×C3×C4—21no—
221C22×C3×C4—21no—
222C22×C3×C4—21no—
223C22×C3×C4—41no—
224C22×C3×C4D3211yescharacteristic
225C22×C3×C4C2×C3×D311yescharacteristic
226C22×C3×C4C2×C3×D311yescharacteristic
227C2×D12—32no—
228C2×D12—32no—
229C2×D12—62no—
230C2×D12—64no—
231C2×D12—64no—
232C2×D12—68no—
233C2×D12—62no—
234C2×D12—61no—
235C2×D12—61no—
236C2×Dic6C2×C3×D311yescharacteristic, complemented
237C3×SD16—184no—
238C2×C3×C8—32no—
239C2×C3×C8—32no—
240C2×C3×C8—62no—
241C3×C4⋊C4—31no—
242C3×C4⋊C4—31no—
243C3×C4⋊C4—61no—
244C3×C4⋊C4—62no—
245C3×C4⋊C4—61no—
246C3×C4⋊C4—61no—
247C3×C22⋊C4—62no—
248C3×C22⋊C4—61no—
249C3×C22⋊C4—61no—
250#217—21no—
251#215—64no—
252C4⋊Dic3C2×C3×D311yescharacteristic, complemented
253Dic3⋊C4—21no—
254C2×C3⋊C8C2×C3×D312yescomplemented
255#204—64no—
256C22×C32—21no—
257C22×C32—21no—
258C22×C32—22no—
259C22×C32—22no—
260C22×C32—21no—
261C22×C32—21no—
262C22×C32—21no—
263C22×C32—21no—
264C22×C32—22no—
265C22×C32—22no—
266C22×C32—22no—
267C22×C32—21no—
268C22×C32—21no—
269C22×C32—21no—
270C22×C32—42no—
271C22×C32—41no—
272C22×C32—41no—
273C22×C32—41no—
274C22×C32—41no—
275C22×C32—41no—
276C22×C32—42no—
277C22×C32—42no—
278C22×C32#20412yescomplemented
279C22×C32#21512yescomplemented
280C22×C32C3×SD1612yescomplemented
281C22×C32C2×D1211yescharacteristic
282C22×C32C2×#6711yescharacteristic
283C22×C32C2×C3×D411yescharacteristic
284C2×C3⋊D3—24no—
285C2×C3⋊D3—48no—
286C2×C3⋊D3—42no—
287C2×C3⋊D3—42no—
288C2×C3×D3—68no—
289C2×C3×D3—64no—
290C2×C3×D3—64no—
291C2×C3×D3—1216no—
292C2×C3×D3—128no—
293C2×C3×D3—128no—
294C2×C3×D3—124no—
295C2×C3×D3—124no—
296C2×C3×D3—122no—
297C2×C3×D3—122no—
298C2×C3×D3—122no—
299C2×C3×D3—122no—
300C32×C4—21no—
301C32×C4—22no—
302C32×C4—21no—
303C32×C4—21no—
304C32×C4—22no—
305C32×C4—22no—
306C32×C4—22no—
307C32×C4—42no—
308C32×C4—41no—
309C32×C4—41no—
310C32×C4—42no—
311C32×C4—121no—
312C32×C4—121no—
313C32×C4D3×D412yes—
314C32×C4D3×D412yes—
315C32×C4C2×C3×D412yes—
316C3×Dic3—42no—
317C3×Dic3—42no—
318C3×Dic3—41no—
319C3×Dic3—41no—
320C3×Dic3—41no—
321C3×Dic3—41no—
322C22×D4—91no—
323C2×SD16—92no—
324C22⋊Q8—31no—
325#102—92no—
326C22⋊C8—31no—
327C33#45011yescharacteristic, complemented, Sylow
328C23×C3—21no—
329C23×C3—21no—
330C23×C3—21no—
331C23×C3—41no—
332C23×C3—184no—
333C23×C3—182no—
334C23×C3—362no—
335C23×C3—362no—
336C23×C3#62611yescharacteristic
337C23×C3C3×D1211yescharacteristic
338C23×C3C3×#6711yescharacteristic
339C22×D3—68no—
340C22×D3—64no—
341C22×D3—64no—
342C22×D3—64no—
343C22×D3—62no—
344C22×D3—62no—
345C22×D3—124no—
346C22×D3—122no—
347C22×D3—122no—
348C22×D3—124no—
349C22×D3—122no—
350C22×D3—122no—
351C3×Q8—62no—
352C3×Q8—62no—
353C3×Q8—122no—
354C3×D4—94no—
355C3×D4—184no—
356C3×D4—182no—
357C2×C3×C4—24no—
358C2×C3×C4—22no—
359C2×C3×C4—22no—
360C2×C3×C4—22no—
361C2×C3×C4—21no—
362C2×C3×C4—21no—
363C2×C3×C4—21no—
364C2×C3×C4—21no—
365C2×C3×C4—21no—
366C2×C3×C4—21no—
367C2×C3×C4—42no—
368C2×C3×C4—42no—
369C2×C3×C4—44no—
370C2×C3×C4—41no—
371C2×C3×C4—41no—
372C2×C3×C4—62no—
373C2×C3×C4—62no—
374C2×C3×C4—61no—
375C2×C3×C4—61no—
376C2×C3×C4—61no—
377C2×C3×C4—61no—
378C2×C3×C4#62611yescharacteristic
379C2×C3×C4C3×D1211yescharacteristic
380C2×C3×C4C3×#6711yescharacteristic
381C2×C3×C4C2×D3211yescharacteristic
382C2×C3×C4C22×C3×D311yescharacteristic
383C2×C3×C4C22×C3×D311yescharacteristic
384C2×Dic3—21no—
385C2×Dic3—21no—
386D12—34no—
387D12—34no—
388D12—64no—
389D12—68no—
390D12—64no—
391D12—64no—
392D12—64no—
393D12—62no—
394D12—62no—
395Dic6—22no—
396C3×C8—62no—
397C3×C8—62no—
398C3×C8—122no—
399C3⋊C8—22no—
400C2×C32—22no—
401C2×C32—24no—
402C2×C32—22no—
403C2×C32—22no—
404C2×C32—21no—
405C2×C32—21no—
406C2×C32—21no—
407C2×C32—21no—
408C2×C32—21no—
409C2×C32—21no—
410C2×C32—21no—
411C2×C32—21no—
412C2×C32—21no—
413C2×C32—41no—
414C2×C32—41no—
415C2×C32—41no—
416C2×C32—42no—
417C2×C32—42no—
418C2×C32—44no—
419C2×C32#88211yescharacteristic
420C2×C32C2×#20411yescharacteristic
421C2×C32#90811yescharacteristic
422C2×C32#93711yescharacteristic
423C2×C32C2×#21511yescharacteristic
424C2×C32#94911yescharacteristic
425C2×C32C3×#12011yescharacteristic
426C2×C32C2×C3×SD1611yescharacteristic
427C2×C32C3×AGL(1,ℤ/8)11yescharacteristic
428C3⋊D3—44no—
429C3×D3—128no—
430C3×D3—124no—
431C3×D3—124no—
432C24—181no—
433C2×Q8—31no—
434C2×D4—92no—
435C2×D4—184no—
436C2×D4—181no—
437C22×C4C3×D3211yescharacteristic
438SD16—184no—
439C2×C8—32no—
440C4⋊C4—31no—
441C4⋊C4—61no—
442C22⋊C4—61no—
443C22×C3—22no—
444C22×C3—22no—
445C22×C3—21no—
446C22×C3—21no—
447C22×C3—21no—
448C22×C3—21no—
449C22×C3—21no—
450C22×C3—21no—
451C22×C3—21no—
452C22×C3—21no—
453C22×C3—21no—
454C22×C3—22no—
455C22×C3—22no—
456C22×C3—22no—
457C22×C3—41no—
458C22×C3—41no—
459C22×C3—41no—
460C22×C3—41no—
461C22×C3—42no—
462C22×C3—42no—
463C22×C3—41no—
464C22×C3—42no—
465C22×C3—184no—
466C22×C3—368no—
467C22×C3—362no—
468C22×C3—362no—
469C22×C3#371712yescomplemented
470C22×C3C3×#20412yescomplemented
471C22×C3C3×#21512yescomplemented
472C22×C3C2×#62611yescharacteristic
473C22×C3C2×C3×D1211yescharacteristic
474C22×C3C2×C3×#6711yescharacteristic
475C2×D3—68no—
476C2×D3—64no—
477C2×D3—64no—
478C2×D3—1216no—
479C2×D3—128no—
480C2×D3—128no—
481C2×D3—124no—
482C2×D3—124no—
483C2×D3—122no—
484C2×D3—122no—
485C2×D3—122no—
486C2×D3—122no—
487C3×C4—22no—
488C3×C4—21no—
489C3×C4—21no—
490C3×C4—21no—
491C3×C4—22no—
492C3×C4—22no—
493C3×C4—22no—
494C3×C4—41no—
495C3×C4—41no—
496C3×C4—42no—
497C3×C4—42no—
498C3×C4—122no—
499C3×C4—122no—
500C3×C4—121no—
501C3×C4—121no—
502C3×C4—121no—
503C3×C4—121no—
504C3×C4#381112yes—
505C3×C4C3×D3×D412yes—
506C3×C4C3×D3×D412yes—
507Dic3—41no—
508Dic3—41no—
509C32—21no—
510C32—21no—
511C32—21no—
512C32—41no—
513C32#473111yescharacteristic, complemented
514C32#504611yescharacteristic, complemented
515C32C3×#45011yescharacteristic, complemented
516C23—184no—
517C23—182no—
518C23—362no—
519C23—362no—
520C23C3×#62611yescharacteristic
521Q8—62no—
522D4—94no—
523D4—184no—
524D4—182no—
525C2×C4—22no—
526C2×C4—61no—
527C2×C4—61no—
528C2×C4C3×#62611yescharacteristic
529C2×C4C2×C3×D3211yescharacteristic
530C8—62no—
531C2×C3—24no—
532C2×C3—22no—
533C2×C3—22no—
534C2×C3—22no—
535C2×C3—21no—
536C2×C3—21no—
537C2×C3—21no—
538C2×C3—21no—
539C2×C3—21no—
540C2×C3—21no—
541C2×C3—21no—
542C2×C3—21no—
543C2×C3—21no—
544C2×C3—42no—
545C2×C3—42no—
546C2×C3—44no—
547C2×C3—41no—
548C2×C3—41no—
549C2×C3—41no—
550C2×C3—364no—
551C2×C3#6382511yescharacteristic
552C2×C3C2×#371711yescharacteristic
553C2×C3#6398111yescharacteristic
554C2×C3C3×#88211yescharacteristic
555C2×C3C2×C3×#20411yescharacteristic
556C2×C3C3×#90811yescharacteristic
557C2×C3C3×#93711yescharacteristic
558C2×C3C2×C3×#21511yescharacteristic
559C2×C3C3×#94911yescharacteristic
560D3—128no—
561D3—124no—
562D3—124no—
563C22—21no—
564C22—21no—
565C22—184no—
566C22—368no—
567C22—362no—
568C22—362no—
569C22C3×#371712yescomplemented
570C22C2×C3×#62611yescharacteristic
571C4—21no—
572C4—121no—
573C4—121no—
574C4C3×#381112yes—
575C3—21no—
576C3—21no—
577C3—21no—
578C3—41no—
579C3#1059118311yescharacteristic, direct product factor
580C3C3×#473111yescharacteristic, complemented
581C3C3×#504611yescharacteristic, complemented
582C2—22no—
583C2—364no—
584C2C3×#6382511yescharacteristic
585C2C2×C3×#371711yescharacteristic
586C2C3×#6398111yescharacteristic
subgroupG/H
centerC22×C3C2×#626
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C4C2×C3×D32
FittingC22×C33×C4C22
socleC22×C33C2×D4
prime radicalC22×C3×C4⋊(C3⋊D3)C2
hypercenterC22×C3×C4D32
hypocenterC32C3×#450
LCSG ▷ C2×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. C32×#924C2semidirect
3. C2×C3×#3717C2semidirect
4. C32×#830C2semidirect
5. C3×#63564C2semidirect
8. #10591183C3direct (N × H)
14. C2×C32×Dic6C22semidirect
16. C32×C4⋊Dic3C22semidirect
18. C2×C32×C3⋊C8C22semidirect
27. C3×#924D3semidirect
28. C3×#924C2×C3semidirect
30. C2×#3717C2×C3semidirect
33. C3×#830D3semidirect
34. C3×#830C2×C3semidirect
37. #63564C2×C3semidirect
81. C2×C3×Dic6C2×D3semidirect
82. C2×C3×Dic6C22×C3semidirect
92. C3×C4⋊Dic3C2×D3semidirect
93. C3×C4⋊Dic3C22×C3semidirect
98. C2×C3×C3⋊C8C2×D3semidirect
99. C2×C3×C3⋊C8C22×C3semidirect
104. C22×C33SD16semidirect
123. #924C3×D3semidirect
129. #830C3×D3semidirect
236. C2×Dic6C2×C3×D3semidirect
252. C4⋊Dic3C2×C3×D3semidirect
254. C2×C3⋊C8C2×C3×D3semidirect
278. C22×C32#204semidirect
279. C22×C32#215semidirect
280. C22×C32C3×SD16semidirect
327. C33#450semidirect
469. C22×C3#3717semidirect
470. C22×C3C3×#204semidirect
471. C22×C3C3×#215semidirect
513. C32#4731semidirect
514. C32#5046semidirect
515. C32C3×#450semidirect
569. C22C3×#3717semidirect
579. C3#10591183direct (N × H)
580. C3C3×#4731semidirect
581. C3C3×#5046semidirect

Maximal subgroups

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

subgroupindex
C22×C3×C4⋊(C3⋊D3)2
C32×#9242
C2×C3×#37172
C32×#8302
C3×#635642
C3×#50463
C3×#47313
#105911833