Finite Group Databasebeta

← C3×#10590267 #13157381

Subgroups — C3×#10590267 #13157381

Order 1728.

662 conjugacy classes of subgroups.

The subgroup lattice has 584 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C32×#875C211yesmaximal, characteristic
2C32×C4×Dic6C211yesmaximal, characteristic, complemented
3C2×C3×#3718C211yesmaximal, characteristic, complemented
4C2×C3×#3715C211yesmaximal, characteristic
5C3×C4⋊(C3⋊(C3⋊C8))C211yesmaximal, characteristic, complemented
6C3×#63565C211yesmaximal, characteristic, complemented
7C3×#63562C211yesmaximal, characteristic
8C3×#5054—31nomaximal
9C3×#5028—31nomaximal
10#10590267C311yesmaximal, characteristic, direct product factor
11C2×C32×D12C2211yescharacteristic
12C2×C32×Dic6C2211yescharacteristic
13C2×C32×Dic6C2211yescharacteristic
14C33×C42C2211yescharacteristic
15C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
16C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
17C32×(C2×D3)⋊C4—21no—
18C32×(C2×D3)⋊C4—21no—
19C32×C4⋊Dic3C2211yescharacteristic
20C32×Dic3⋊C4—21no—
21C32×C4×Dic3—21no—
22C3×#3718—22no—
23C3×#3715—22no—
24C32×#124—31no—
25C32×C4×Q8—31no—
26C2×C3×#216—31no—
27C2×C3×#216—31no—
28C2×C3×#215—31no—
29C2×C3×#214—31no—
30C3×#875—21no—
31C3×#875D311yescharacteristic
32C3×#875C2×C311yescharacteristic
33C3×C4×Dic6—21no—
34C3×C4×Dic6D311yescharacteristic, complemented
35C3×C4×Dic6C2×C311yescharacteristic, complemented
36C2×#3718C2×C311yescharacteristic, complemented
37C2×#3715C2×C311yescharacteristic
38C4⋊(C3⋊(C3⋊C8))C2×C311yescharacteristic, complemented
39C3×#835—31no—
40C3×#832—31no—
41C3×#810—31no—
42C3×#809—31no—
43C3×C4⋊(C3⋊C8)—31no—
44C3×C4⋊(C3⋊C8)—31no—
45C3×C4⋊(C3⋊C8)—61no—
46#63565C2×C311yescharacteristic, complemented
47#63562C2×C311yescharacteristic
48C22×C32×D3—21no—
49C2×C33×C4D411yescharacteristic
50C2×C33×C4D411yescharacteristic
51C2×C33×C4C2311yescharacteristic
52C2×C32×Dic3—21no—
53C2×C32×Dic3—21no—
54C2×C32×Dic3—21no—
55C32×D12—21no—
56C32×D12—21no—
57C32×Dic6—21no—
58C32×Dic6—21no—
59C32×Dic6—21no—
60C32×Dic6D412yes—
61C3×C3⋊(C3⋊C8)—21no—
62C3×C3⋊(C3⋊C8)—21no—
63C3×#464—91no—
64#5054—31no—
65#5028—31no—
66C2×C32×Q8—31no—
67C2×C32×Q8—31no—
68C2×C32×D4—31no—
69C2×C3×D12—21no—
70C2×C3×D12C2×D311yescharacteristic
71C2×C3×D12C22×C311yescharacteristic
72C2×C3×Dic6—21no—
73C2×C3×Dic6—21no—
74C2×C3×Dic6C2×D311yescharacteristic
75C2×C3×Dic6C2×D311yescharacteristic
76C2×C3×Dic6C22×C311yescharacteristic
77C2×C3×Dic6C22×C311yescharacteristic
78C32×C4⋊C4—31no—
79C32×C4⋊C4—61no—
80C32×C22⋊C4—61no—
81C32×C22⋊C4—61no—
82C32×C42—21no—
83C32×C42—21no—
84C32×C42—21no—
85C32×C42—41no—
86C32×C42—61no—
87C32×C42C2×D311yescharacteristic
88C32×C42C2×D311yescharacteristic
89C32×C42C22×C311yescharacteristic
90C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
91C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
92C3×#216—62no—
93C3×#216—62no—
94C3×#215—62no—
95C3×#214—62no—
96C3×(C2×D3)⋊C4—22no—
97C3×(C2×D3)⋊C4—22no—
98C3×(C2×D3)⋊C4—21no—
99C3×(C2×D3)⋊C4—21no—
100C3×(C2×D3)⋊C4—21no—
101C3×(C2×D3)⋊C4—21no—
102C3×C4⋊Dic3—21no—
103C3×C4⋊Dic3C2×D311yescharacteristic
104C3×C4⋊Dic3C22×C311yescharacteristic
105C3×Dic3⋊C4—22no—
106C3×Dic3⋊C4—21no—
107C3×Dic3⋊C4—21no—
108C3×C4×Dic3—22no—
109C3×C4×Dic3—21no—
110C3×C4×Dic3—21no—
111C2×C3×C3⋊C8—31no—
112C2×C3×C3⋊C8—31no—
113C2×C3×C3⋊C8—31no—
114C2×C3×C3⋊C8—31no—
115C2×C3×C3⋊C8—61no—
116C2×C3×C3⋊C8—61no—
117#3718—22no—
118#3715—22no—
119C22×C33C2×D411yescharacteristic
120C2×C32×D3—41no—
121C2×C32×D3—41no—
122C2×C32×D3—41no—
123C33×C4—21no—
124C33×C4—21no—
125C33×C4C2×D411yescharacteristic
126C33×C4#4111yescharacteristic
127C32×Dic3—22no—
128C32×Dic3—41no—
129C32×Dic3—41no—
130C2×C3×Q16—91no—
131C2×C3×SD16—91no—
132C3×#124—31no—
133C3×#124—31no—
134C3×#124—61no—
135C3×C4×Q8—31no—
136C3×C4×Q8—31no—
137C3×C4×Q8—61no—
138C2×#216—31no—
139C2×#216—31no—
140C2×#215—31no—
141C2×#214—31no—
142#875C3×D311yescharacteristic
143C4×Dic6C3×D311yescharacteristic, complemented
144C3×C4⋊C8—91no—
145C3×#103—91no—
146C3×#102—91no—
147#835—31no—
148#832—31no—
149#810—31no—
150#809—31no—
151C4⋊(C3⋊C8)—31no—
152C4⋊(C3⋊C8)—31no—
153C4⋊(C3⋊C8)—61no—
154C23×C32—61no—
155C22×C3×D3—22no—
156C22×C3×D3—21no—
157C22×C3×D3—21no—
158C32×Q8—32no—
159C32×Q8—61no—
160C32×Q8—61no—
161C32×Q8—61no—
162C32×D4—61no—
163C32×D4—61no—
164C2×C32×C4—21no—
165C2×C32×C4—21no—
166C2×C32×C4—21no—
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—41no—
174C2×C32×C4—41no—
175C2×C32×C4—41no—
176C2×C32×C4—61no—
177C2×C32×C4—61no—
178C2×C32×C4—61no—
179C2×C32×C4#6711yescharacteristic
180C2×C32×C4#6711yescharacteristic
181C2×C32×C4#6711yescharacteristic
182C2×C32×C4#6711yescharacteristic
183C2×C32×C4C3×D411yescharacteristic
184C2×C32×C4C3×D411yescharacteristic
185C2×C32×C4C22×D311yescharacteristic
186C2×C32×C4C22×D311yescharacteristic
187C2×C32×C4C23×C311yescharacteristic
188C2×C3×Dic3—22no—
189C2×C3×Dic3—21no—
190C2×C3×Dic3—21no—
191C2×C3×Dic3—22no—
192C2×C3×Dic3—22no—
193C2×C3×Dic3—21no—
194C2×C3×Dic3—21no—
195C2×C3×Dic3—21no—
196C2×C3×Dic3—21no—
197C3×D12—21no—
198C3×D12—21no—
199C3×D12—21no—
200C3×D12—21no—
201C3×D12—41no—
202C3×D12—41no—
203C3×Dic6—22no—
204C3×Dic6—21no—
205C3×Dic6—21no—
206C3×Dic6—21no—
207C3×Dic6—21no—
208C3×Dic6—21no—
209C3×Dic6—21no—
210C3×Dic6—22no—
211C3×Dic6—41no—
212C3×Dic6—41no—
213C3×Dic6D1212yes—
214C3×Dic6C3×D412yes—
215C3⋊(C3⋊C8)—21no—
216C3⋊(C3⋊C8)—21no—
217C3×C3⋊C8—61no—
218C3×C3⋊C8—61no—
219C3×C3⋊C8—61no—
220C3×C3⋊C8—61no—
221C3×C3⋊C8—121no—
222C3×C3⋊C8—121no—
223#464—91noSylow
224C2×C33#12111yescharacteristic
225C2×C33#13511yescharacteristic
226C2×C33#13711yescharacteristic
227C32×D3—81no—
228C2×C3×Q8—31no—
229C2×C3×Q8—31no—
230C2×C3×Q8—31no—
231C2×C3×Q8—31no—
232C2×C3×Q8—61no—
233C2×C3×Q8—61no—
234C2×C3×D4—31no—
235C2×C3×D4—31no—
236C2×C3×D4—61no—
237C2×D12C2×C3×D311yescharacteristic
238C2×Dic6C2×C3×D311yescharacteristic
239C2×Dic6C2×C3×D311yescharacteristic
240C3×Q16—182no—
241C3×SD16—182no—
242C2×C3×C8—91no—
243C2×C3×C8—91no—
244C3×C4⋊C4—31no—
245C3×C4⋊C4—31no—
246C3×C4⋊C4—61no—
247C3×C4⋊C4—62no—
248C3×C4⋊C4—61no—
249C3×C4⋊C4—61no—
250C3×C22⋊C4—62no—
251C3×C22⋊C4—62no—
252C3×C22⋊C4—61no—
253C3×C22⋊C4—61no—
254C3×C22⋊C4—61no—
255C3×C22⋊C4—61no—
256C3×C42—21no—
257C3×C42—21no—
258C3×C42—21no—
259C3×C42—41no—
260C3×C42—62no—
261C3×C42—61no—
262C3×C42—61no—
263C3×C42D3211yescharacteristic
264C3×C42C2×C3×D311yescharacteristic
265C3×C42C2×C3×D311yescharacteristic
266#216—62no—
267#216—62no—
268#215—62no—
269#214—62no—
270(C2×D3)⋊C4—21no—
271(C2×D3)⋊C4—21no—
272C4⋊Dic3C2×C3×D311yescharacteristic
273Dic3⋊C4—21no—
274C4×Dic3—21no—
275C2×C3⋊C8—31no—
276C2×C3⋊C8—31no—
277C2×C3⋊C8—31no—
278C2×C3⋊C8—31no—
279C2×C3⋊C8—61no—
280C2×C3⋊C8—61no—
281C22×C32—21no—
282C22×C32—21no—
283C22×C32—21no—
284C22×C32—41no—
285C22×C32—121no—
286C22×C32—121no—
287C22×C32—121no—
288C22×C32C2×#6711yescharacteristic
289C22×C32C2×#6711yescharacteristic
290C22×C32C2×C3×D411yescharacteristic
291C2×C3×D3—42no—
292C2×C3×D3—42no—
293C2×C3×D3—42no—
294C2×C3×D3—41no—
295C2×C3×D3—41no—
296C2×C3×D3—41no—
297C2×C3×D3—41no—
298C2×C3×D3—41no—
299C2×C3×D3—41no—
300C32×C4—21no—
301C32×C4—21no—
302C32×C4—22no—
303C32×C4—22no—
304C32×C4—21no—
305C32×C4—21no—
306C32×C4—21no—
307C32×C4—21no—
308C32×C4—21no—
309C32×C4—21no—
310C32×C4—21no—
311C32×C4—21no—
312C32×C4—21no—
313C32×C4—21no—
314C32×C4—41no—
315C32×C4—41no—
316C32×C4—41no—
317C32×C4—41no—
318C32×C4—41no—
319C32×C4—41no—
320C32×C4—42no—
321C32×C4—42no—
322C32×C4—62no—
323C32×C4—121no—
324C32×C4—121no—
325C32×C4C2×D1211yescharacteristic
326C32×C4#23511yescharacteristic
327C32×C4D3×D411yescharacteristic
328C32×C4#23711yescharacteristic
329C32×C4C2×C3×D411yescharacteristic
330C32×C4C3×#4111yescharacteristic
331C3×Dic3—24no—
332C3×Dic3—22no—
333C3×Dic3—22no—
334C3×Dic3—42no—
335C3×Dic3—42no—
336C3×Dic3—41no—
337C3×Dic3—41no—
338C3×Dic3—41no—
339C3×Dic3—41no—
340C2×Q16—91no—
341C2×SD16—91no—
342#124—31no—
343C4×Q8—31no—
344C4⋊C8—91no—
345#103—91no—
346#102—91no—
347C33#46411yescharacteristic, complemented, Sylow
348C23×C3—62no—
349C23×C3—61no—
350C23×C3—61no—
351C22×D3—21no—
352C3×Q8—32no—
353C3×Q8—32no—
354C3×Q8—62no—
355C3×Q8—62no—
356C3×Q8—61no—
357C3×Q8—61no—
358C3×Q8—61no—
359C3×Q8—61no—
360C3×Q8—61no—
361C3×Q8—61no—
362C3×Q8—121no—
363C3×Q8—121no—
364C3×D4—61no—
365C3×D4—61no—
366C3×D4—61no—
367C3×D4—61no—
368C3×D4—121no—
369C3×D4—121no—
370C2×C3×C4—21no—
371C2×C3×C4—21no—
372C2×C3×C4—21no—
373C2×C3×C4—21no—
374C2×C3×C4—21no—
375C2×C3×C4—21no—
376C2×C3×C4—21no—
377C2×C3×C4—21no—
378C2×C3×C4—21no—
379C2×C3×C4—41no—
380C2×C3×C4—41no—
381C2×C3×C4—41no—
382C2×C3×C4—62no—
383C2×C3×C4—61no—
384C2×C3×C4—61no—
385C2×C3×C4—62no—
386C2×C3×C4—62no—
387C2×C3×C4—61no—
388C2×C3×C4—61no—
389C2×C3×C4—61no—
390C2×C3×C4—61no—
391C2×C3×C4#62511yescharacteristic
392C2×C3×C4#62511yescharacteristic
393C2×C3×C4C3×#6711yescharacteristic
394C2×C3×C4C3×#6711yescharacteristic
395C2×C3×C4C3×#6711yescharacteristic
396C2×C3×C4C3×#6711yescharacteristic
397C2×C3×C4C2×D3211yescharacteristic
398C2×C3×C4C22×C3×D311yescharacteristic
399C2×C3×C4C22×C3×D311yescharacteristic
400C2×Dic3—21no—
401C2×Dic3—21no—
402C2×Dic3—21no—
403D12—21no—
404D12—21no—
405Dic6—21no—
406Dic6—21no—
407Dic6—21no—
408Dic6C3×D1212yes—
409C3×C8—181no—
410C3×C8—181no—
411C3⋊C8—61no—
412C3⋊C8—61no—
413C3⋊C8—61no—
414C3⋊C8—61no—
415C3⋊C8—121no—
416C3⋊C8—121no—
417C2×C32—21no—
418C2×C32—21no—
419C2×C32—21no—
420C2×C32—21no—
421C2×C32—21no—
422C2×C32—21no—
423C2×C32—21no—
424C2×C32—21no—
425C2×C32—21no—
426C2×C32—41no—
427C2×C32—41no—
428C2×C32—41no—
429C2×C32—241no—
430C2×C32#93011yescharacteristic
431C2×C32#93811yescharacteristic
432C2×C32#94211yescharacteristic
433C2×C32#95011yescharacteristic
434C2×C32#95011yescharacteristic
435C2×C32#95111yescharacteristic
436C2×C32C3×#12111yescharacteristic
437C2×C32C3×#13511yescharacteristic
438C2×C32C3×#13711yescharacteristic
439C3×D3—82no—
440C3×D3—81no—
441C3×D3—81no—
442C2×Q8—31no—
443C2×Q8—31no—
444C2×D4—31no—
445Q16—182no—
446SD16—182no—
447C2×C8—91no—
448C2×C8—91no—
449C4⋊C4—31no—
450C4⋊C4—61no—
451C22⋊C4—61no—
452C22⋊C4—61no—
453C42—61no—
454C42C3×D3211yescharacteristic
455C22×C3—21no—
456C22×C3—21no—
457C22×C3—21no—
458C22×C3—41no—
459C22×C3—122no—
460C22×C3—122no—
461C22×C3—122no—
462C22×C3—121no—
463C22×C3—121no—
464C22×C3—121no—
465C22×C3—121no—
466C22×C3—121no—
467C22×C3—121no—
468C22×C3C2×#62511yescharacteristic
469C22×C3C2×C3×#6711yescharacteristic
470C22×C3C2×C3×#6711yescharacteristic
471C2×D3—41no—
472C2×D3—41no—
473C2×D3—41no—
474C3×C4—22no—
475C3×C4—22no—
476C3×C4—21no—
477C3×C4—21no—
478C3×C4—21no—
479C3×C4—21no—
480C3×C4—21no—
481C3×C4—21no—
482C3×C4—21no—
483C3×C4—21no—
484C3×C4—21no—
485C3×C4—21no—
486C3×C4—21no—
487C3×C4—21no—
488C3×C4—41no—
489C3×C4—41no—
490C3×C4—41no—
491C3×C4—41no—
492C3×C4—42no—
493C3×C4—42no—
494C3×C4—41no—
495C3×C4—41no—
496C3×C4—64no—
497C3×C4—62no—
498C3×C4—62no—
499C3×C4—122no—
500C3×C4—122no—
501C3×C4—121no—
502C3×C4—121no—
503C3×C4—121no—
504C3×C4—121no—
505C3×C4#379511yescharacteristic
506C3×C4D3×D1211yescharacteristic
507C3×C4C2×C3×D1211yescharacteristic
508C3×C4C3×#23511yescharacteristic
509C3×C4C3×D3×D411yescharacteristic
510C3×C4C3×#23711yescharacteristic
511Dic3—22no—
512Dic3—41no—
513Dic3—41no—
514C32—21no—
515C32—21no—
516C32—21no—
517C32—41no—
518C32#502811yescharacteristic, complemented
519C32#505411yescharacteristic, complemented
520C32C3×#46411yescharacteristic, complemented
521C23—61no—
522Q8—32no—
523Q8—61no—
524Q8—61no—
525Q8—61no—
526D4—61no—
527D4—61no—
528C2×C4—61no—
529C2×C4—61no—
530C2×C4—61no—
531C2×C4C3×#62511yescharacteristic
532C2×C4C3×#62511yescharacteristic
533C2×C4C2×C3×D3211yescharacteristic
534C8—181no—
535C8—181no—
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—21no—
545C2×C3—41no—
546C2×C3—41no—
547C2×C3—41no—
548C2×C3—242no—
549C2×C3—241no—
550C2×C3—241no—
551C2×C3#6382211yescharacteristic
552C2×C3#6382711yescharacteristic
553C2×C3#6390811yescharacteristic
554C2×C3C3×#93011yescharacteristic
555C2×C3C3×#93811yescharacteristic
556C2×C3C3×#94211yescharacteristic
557C2×C3C3×#95011yescharacteristic
558C2×C3C3×#95011yescharacteristic
559C2×C3C3×#95111yescharacteristic
560D3—81no—
561C22—121no—
562C22—121no—
563C22—121no—
564C22C2×C3×#62511yescharacteristic
565C4—21no—
566C4—21no—
567C4—62no—
568C4—121no—
569C4—121no—
570C4C3×#379511yescharacteristic
571C4C3×D3×D1211yescharacteristic
572C3—21no—
573C3—21no—
574C3—21no—
575C3—41no—
576C3#1059026711yescharacteristic, direct product factor
577C3C3×#502811yescharacteristic, complemented
578C3C3×#505411yescharacteristic, complemented
579C2—241no—
580C2C3×#6382211yescharacteristic
581C2C3×#6382711yescharacteristic
582C2C3×#6390811yescharacteristic
subgroupG/H
centerC22×C3C2×#625
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C4C2×C3×D32
FittingC33×C42C22
socleC22×C33C2×D4
prime radicalC2×C32×D12C22
hypercenterC3×C42D32
hypocenterC32C3×#464
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
2. C32×C4×Dic6C2semidirect
3. C2×C3×#3718C2semidirect
5. C3×C4⋊(C3⋊(C3⋊C8))C2semidirect
6. C3×#63565C2semidirect
10. #10590267C3direct (N × H)
34. C3×C4×Dic6D3semidirect
35. C3×C4×Dic6C2×C3semidirect
36. C2×#3718C2×C3semidirect
38. C4⋊(C3⋊(C3⋊C8))C2×C3semidirect
46. #63565C2×C3semidirect
143. C4×Dic6C3×D3semidirect
347. C33#464semidirect
518. C32#5028semidirect
519. C32#5054semidirect
520. C32C3×#464semidirect
576. C3#10590267direct (N × H)
577. C3C3×#5028semidirect
578. C3C3×#5054semidirect

Maximal subgroups

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

subgroupindex
C32×#8752
C32×C4×Dic62
C2×C3×#37182
C2×C3×#37152
C3×C4⋊(C3⋊(C3⋊C8))2
C3×#635652
C3×#635622
C3×#50543
C3×#50283
#105902673