Finite Group Databasebeta

← C3×#10590501 #13157484

Subgroups — C3×#10590501 #13157484

Order 1728.

682 conjugacy classes of subgroups.

The subgroup lattice has 606 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C32×#897C211yesmaximal, characteristic, complemented
2C3×#63908C211yesmaximal, characteristic
3C2×C3×#3715C211yesmaximal, characteristic, complemented
4C2×C3×D3×C3⋊C8C211yesmaximal, characteristic, complemented
5C3×#63635C211yesmaximal, characteristic, complemented
6C32×#809C211yesmaximal, characteristic, complemented
7C3×#63566C211yesmaximal, characteristic, complemented
8C3×#5048—31nomaximal
9C3×#4778—31nomaximal
10#10590501C311yesmaximal, characteristic, direct product factor
11C2×C32×D12C2211yescharacteristic
12C2×C32×C4×D3C2211yescharacteristic
13C2×C32×Dic6C2211yescharacteristic
14C2×C3×#625—21no—
15C33×C4⋊C4C2211yescharacteristic
16C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
17C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
18C32×(C2×D3)⋊C4—21no—
19C32×C4⋊Dic3—21no—
20C2×C32×C3⋊C8C2211yescharacteristic
21C3×(C2×D3)⋊Dic3—21no—
22C3×#3715—22no—
23C3×D3×C3⋊C8—22no—
24C32×C22⋊Q8—31no—
25C2×C3×#215—31no—
26C3×#938—31no—
27C2×C3×#214—31no—
28C3×#930—31no—
29C22×C3×C3⋊C8—31no—
30C2×C3×D3×C8—31no—
31C3×#897—21no—
32C3×#897D311yescharacteristic, complemented
33C3×#897C2×C311yescharacteristic, complemented
34#63908C2×C311yescharacteristic
35C2×#3715C2×C311yescharacteristic, complemented
36C2×D3×C3⋊C8C2×C311yescharacteristic, complemented
37C32×#102—31no—
38#63635C2×C311yescharacteristic, complemented
39C3×#835—31no—
40C3×#816—31no—
41C3×#809—21no—
42C3×#809D311yescharacteristic, complemented
43C3×#809C2×C311yescharacteristic, complemented
44C3×#808—31no—
45C3×#808—31no—
46C3×#808—61no—
47#63566C2×C311yescharacteristic, complemented
48C22×C32×D3—21no—
49C22×C32×D3D411yescharacteristic
50C2×C33×C4—21no—
51C2×C33×C4C2311yescharacteristic
52C2×C3×C3⋊Dic3—21no—
53C2×C32×Dic3—21no—
54C2×C32×Dic3D411yescharacteristic
55C32×D12—21no—
56C32×D12—21no—
57C32×C4×D3—21no—
58C32×C4×D3—21no—
59C32×Dic6—21no—
60C32×Dic6—21no—
61C3×#625—42no—
62C3×C3⋊(C3⋊C8)D412yescomplemented
63C32×C3⋊C8—21no—
64C3×#465—91no—
65#5048—31no—
66#4778—31no—
67C2×C32×Q8—31no—
68C2×C32×D4—31no—
69C22×C32×C4—31no—
70C2×C3×#67—61no—
71C2×C3×#67—61no—
72C2×C3×D12—21no—
73C2×C3×D12C2×D311yescharacteristic
74C2×C3×D12C22×C311yescharacteristic
75C2×C3×C4×D3—21no—
76C2×C3×C4×D3C2×D311yescharacteristic
77C2×C3×C4×D3C22×C311yescharacteristic
78C2×C3×Dic6—21no—
79C2×C3×Dic6C2×D311yescharacteristic
80C2×C3×Dic6C22×C311yescharacteristic
81C2×#625—21no—
82C2×C32×C8—31no—
83C32×C4⋊C4—21no—
84C32×C4⋊C4—21no—
85C32×C4⋊C4—21no—
86C32×C4⋊C4—41no—
87C32×C4⋊C4—61no—
88C32×C4⋊C4C2×D311yescharacteristic
89C32×C4⋊C4C2×D311yescharacteristic
90C32×C4⋊C4C22×C311yescharacteristic
91C32×C22⋊C4—61no—
92C4⋊(C3⋊Dic3)C22×C311yescharacteristic
93C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
94C3×#217—61no—
95C3×#215—62no—
96C3×#214—62no—
97C3×(C2×D3)⋊C4—22no—
98C3×(C2×D3)⋊C4—21no—
99C3×(C2×D3)⋊C4—21no—
100C3×(C2×D3)⋊C4—61no—
101C3×C4⋊Dic3—22no—
102C3×C4⋊Dic3—21no—
103C3×C4⋊Dic3—21no—
104C3×C4⋊Dic3—31no—
105C3×C4⋊Dic3—31no—
106C3×C4⋊Dic3—61no—
107C2×C3×C3⋊C8—21no—
108C2×C3×C3⋊C8—31no—
109C2×C3×C3⋊C8—31no—
110C2×C3×C3⋊C8—62no—
111C2×C3×C3⋊C8—61no—
112C2×C3×C3⋊C8C2×D311yescharacteristic
113C2×C3×C3⋊C8C22×C311yescharacteristic
114C3×D3×C8—62no—
115(C2×D3)⋊Dic3—21no—
116#3715—22no—
117D3×C3⋊C8—22no—
118C22×C33C2×D411yescharacteristic
119C2×C32×D3—21no—
120C2×C32×D3—21no—
121C2×C32×D3—41no—
122C2×C32×D3—41no—
123C2×C32×D3—41no—
124C2×C32×D3SD1612yescomplemented
125C33×C4—41no—
126C33×C4C2×D411yescharacteristic
127C33×C4#4111yescharacteristic
128C3×C3⋊Dic3—41no—
129C32×Dic3—21no—
130C32×Dic3—41no—
131C2×C3×SD16—91no—
132C22×C3×C8—91no—
133C3×C22⋊Q8—31no—
134C3×C22⋊Q8—31no—
135C3×C22⋊Q8—61no—
136C3×#121—91no—
137C2×#215—31no—
138#938—31no—
139C2×#214—31no—
140#930—31no—
141C22×C3⋊C8—31no—
142C2×D3×C8—31no—
143#897C3×D311yescharacteristic, complemented
144C3×#106—91no—
145C3×#103—91no—
146C3×#102—31no—
147C3×#102—31no—
148C3×#102—61no—
149#835—31no—
150#816—31no—
151#809C3×D311yescharacteristic, complemented
152#808—31no—
153#808—31no—
154#808—61no—
155C23×C32—31no—
156C23×C32—61no—
157C22×C3×D3—21no—
158C22×C3×D3—21no—
159C22×C3×D3—21no—
160C22×C3×D3—41no—
161C22×C3×D3#6711yescharacteristic
162C22×C3×D3C3×D411yescharacteristic
163C32×Q8—61no—
164C32×Q8—61no—
165C32×D4—61no—
166C32×D4—61no—
167C2×C32×C4—21no—
168C2×C32×C4—22no—
169C2×C32×C4—21no—
170C2×C32×C4—21no—
171C2×C32×C4—21no—
172C2×C32×C4—21no—
173C2×C32×C4—21no—
174C2×C32×C4—31no—
175C2×C32×C4—41no—
176C2×C32×C4—41no—
177C2×C32×C4—41no—
178C2×C32×C4—42no—
179C2×C32×C4—61no—
180C2×C32×C4—61no—
181C2×C32×C4—61no—
182C2×C32×C4C22×D311yescharacteristic
183C2×C32×C4C22×D311yescharacteristic
184C2×C32×C4C23×C311yescharacteristic
185C2×C3⋊Dic3—21no—
186C3×#67—122no—
187C3×#67—122no—
188C2×C3×Dic3—22no—
189C2×C3×Dic3—21no—
190C2×C3×Dic3—21no—
191C2×C3×Dic3—21no—
192C2×C3×Dic3—61no—
193C2×C3×Dic3—61no—
194C2×C3×Dic3—121no—
195C2×C3×Dic3#6711yescharacteristic
196C2×C3×Dic3C3×D411yescharacteristic
197C3×D12—22no—
198C3×D12—21no—
199C3×D12—21no—
200C3×D12—21no—
201C3×D12—21no—
202C3×D12—41no—
203C3×C4×D3—21no—
204C3×C4×D3—21no—
205C3×C4×D3—21no—
206C3×C4×D3—21no—
207C3×C4×D3—41no—
208C3×C4×D3—41no—
209C3×Dic6—21no—
210C3×Dic6—21no—
211C3×Dic6—21no—
212C3×Dic6—21no—
213C3×Dic6—41no—
214C3×Dic6—41no—
215#625—42no—
216C32×C8—61no—
217C3⋊(C3⋊C8)C3×D412yescomplemented
218C3×C3⋊C8—21no—
219C3×C3⋊C8—21no—
220C3×C3⋊C8—32no—
221C3×C3⋊C8—32no—
222C3×C3⋊C8—41no—
223C3×C3⋊C8—62no—
224#465—91noSylow
225C2×C33#12111yescharacteristic
226C2×C33C2×SD1611yescharacteristic
227C2×C33#13511yescharacteristic
228C32×D3—22no—
229C32×D3—81no—
230C2×C3×Q8—31no—
231C2×C3×Q8—31no—
232C2×C3×Q8—61no—
233C2×C3×D4—31no—
234C2×C3×D4—31no—
235C2×C3×D4—61no—
236C2×C3×D4—181no—
237C22×C3×C4—31no—
238C22×C3×C4—31no—
239C22×C3×C4—61no—
240C2×#67—61no—
241C2×#67—61no—
242C2×D12C2×C3×D311yescharacteristic
243C2×C4×D3C2×C3×D311yescharacteristic
244C2×Dic6C2×C3×D311yescharacteristic
245C3×SD16—182no—
246C2×C3×C8—31no—
247C2×C3×C8—31no—
248C2×C3×C8—61no—
249C2×C3×C8—91no—
250C2×C3×C8—182no—
251C3×C4⋊C4—21no—
252C3×C4⋊C4—21no—
253C3×C4⋊C4—21no—
254C3×C4⋊C4—41no—
255C3×C4⋊C4—61no—
256C3×C4⋊C4—61no—
257C3×C4⋊C4—62no—
258C3×C4⋊C4—91no—
259C3×C4⋊C4D3211yescharacteristic
260C3×C4⋊C4C2×C3×D311yescharacteristic
261C3×C4⋊C4C2×C3×D311yescharacteristic
262C3×C22⋊C4—61no—
263C3×C22⋊C4—61no—
264C3×C22⋊C4—62no—
265C3×C22⋊C4—181no—
266#217—61no—
267#215—62no—
268#214—62no—
269(C2×D3)⋊C4—21no—
270(C2×D3)⋊C4—61no—
271C4⋊Dic3—21no—
272C4⋊Dic3—31no—
273C4⋊Dic3—31no—
274C4⋊Dic3—61no—
275C2×C3⋊C8—31no—
276C2×C3⋊C8—31no—
277C2×C3⋊C8—61no—
278C2×C3⋊C8—62no—
279C2×C3⋊C8C2×C3×D311yescharacteristic
280D3×C8—62no—
281C22×C32—21no—
282C22×C32—21no—
283C22×C32—21no—
284C22×C32—32no—
285C22×C32—41no—
286C22×C32—61no—
287C22×C32—61no—
288C22×C32—121no—
289C22×C32—121no—
290C22×C32—121no—
291C22×C32D3×D411yescharacteristic
292C22×C32C2×#6711yescharacteristic
293C22×C32C2×C3×D411yescharacteristic
294C2×C3×D3—21no—
295C2×C3×D3—21no—
296C2×C3×D3—21no—
297C2×C3×D3—21no—
298C2×C3×D3—22no—
299C2×C3×D3—41no—
300C2×C3×D3—41no—
301C2×C3×D3—42no—
302C2×C3×D3—41no—
303C2×C3×D3—41no—
304C2×C3×D3—41no—
305C2×C3×D3—41no—
306C2×C3×D3—41no—
307C2×C3×D3—41no—
308C2×C3×D3—81no—
309C2×C3×D3—81no—
310C2×C3×D3#21512yescomplemented
311C2×C3×D3C3×SD1612yescomplemented
312C32×C4—21no—
313C32×C4—21no—
314C32×C4—21no—
315C32×C4—21no—
316C32×C4—21no—
317C32×C4—21no—
318C32×C4—41no—
319C32×C4—41no—
320C32×C4—41no—
321C32×C4—42no—
322C32×C4—41no—
323C32×C4—41no—
324C32×C4—61no—
325C32×C4—81no—
326C32×C4—81no—
327C32×C4—82no—
328C32×C4—121no—
329C32×C4#23511yescharacteristic
330C32×C4D3×D411yescharacteristic
331C32×C4D3×D411yescharacteristic
332C32×C4#23711yescharacteristic
333C32×C4C2×C3×D411yescharacteristic
334C32×C4C3×#4111yescharacteristic
335C3⋊Dic3—41no—
336C3×Dic3—21no—
337C3×Dic3—21no—
338C3×Dic3—41no—
339C3×Dic3—42no—
340C3×Dic3—41no—
341C3×Dic3—41no—
342C3×Dic3—121no—
343C3×Dic3—121no—
344C3×Dic3—241no—
345C2×SD16—91no—
346C22×C8—91no—
347C22⋊Q8—31no—
348#121—91no—
349#106—91no—
350#103—91no—
351#102—31no—
352C33#46511yescharacteristic, complemented, Sylow
353C23×C3—31no—
354C23×C3—31no—
355C23×C3—61no—
356C23×C3—61no—
357C23×C3—61no—
358C23×C3—121no—
359C22×D3—21no—
360C22×D3C3×#6711yescharacteristic
361C3×Q8—61no—
362C3×Q8—61no—
363C3×Q8—61no—
364C3×Q8—61no—
365C3×Q8—121no—
366C3×Q8—121no—
367C3×D4—62no—
368C3×D4—61no—
369C3×D4—61no—
370C3×D4—61no—
371C3×D4—61no—
372C3×D4—121no—
373C3×D4—362no—
374C2×C3×C4—22no—
375C2×C3×C4—21no—
376C2×C3×C4—21no—
377C2×C3×C4—21no—
378C2×C3×C4—21no—
379C2×C3×C4—21no—
380C2×C3×C4—21no—
381C2×C3×C4—31no—
382C2×C3×C4—31no—
383C2×C3×C4—41no—
384C2×C3×C4—41no—
385C2×C3×C4—42no—
386C2×C3×C4—41no—
387C2×C3×C4—61no—
388C2×C3×C4—61no—
389C2×C3×C4—61no—
390C2×C3×C4—61no—
391C2×C3×C4—61no—
392C2×C3×C4—62no—
393C2×C3×C4—61no—
394C2×C3×C4—61no—
395C2×C3×C4—121no—
396C2×C3×C4—121no—
397C2×C3×C4—181no—
398C2×C3×C4C2×D3211yescharacteristic
399C2×C3×C4C22×C3×D311yescharacteristic
400C2×C3×C4C22×C3×D311yescharacteristic
401#67—122no—
402#67—122no—
403C2×Dic3—21no—
404C2×Dic3—61no—
405C2×Dic3—61no—
406C2×Dic3—121no—
407C2×Dic3C3×#6711yescharacteristic
408D12—21no—
409D12—21no—
410C4×D3—21no—
411C4×D3—21no—
412Dic6—21no—
413Dic6—21no—
414C3×C8—61no—
415C3×C8—61no—
416C3×C8—92no—
417C3×C8—121no—
418C3⋊C8—21no—
419C3⋊C8—32no—
420C3⋊C8—32no—
421C3⋊C8—62no—
422C2×C32—21no—
423C2×C32—21no—
424C2×C32—21no—
425C2×C32—21no—
426C2×C32—21no—
427C2×C32—21no—
428C2×C32—21no—
429C2×C32—21no—
430C2×C32—21no—
431C2×C32—41no—
432C2×C32—41no—
433C2×C32—41no—
434C2×C32—62no—
435C2×C32—241no—
436C2×C32#88411yescharacteristic
437C2×C32#91211yescharacteristic
438C2×C32D3×SD1611yescharacteristic
439C2×C32#93911yescharacteristic
440C2×C32C2×#21511yescharacteristic
441C2×C32#95011yescharacteristic
442C2×C32C3×#12111yescharacteristic
443C2×C32C2×C3×SD1611yescharacteristic
444C2×C32C3×#13511yescharacteristic
445C3×D3—22no—
446C3×D3—22no—
447C3×D3—42no—
448C3×D3—82no—
449C3×D3—81no—
450C3×D3—81no—
451C2×Q8—31no—
452C2×D4—31no—
453C2×D4—181no—
454C22×C4—31no—
455SD16—182no—
456C2×C8—31no—
457C2×C8—91no—
458C2×C8—182no—
459C4⋊C4—61no—
460C4⋊C4—91no—
461C4⋊C4C3×D3211yescharacteristic
462C22⋊C4—61no—
463C22⋊C4—181no—
464C22×C3—21no—
465C22×C3—21no—
466C22×C3—21no—
467C22×C3—32no—
468C22×C3—32no—
469C22×C3—41no—
470C22×C3—62no—
471C22×C3—61no—
472C22×C3—61no—
473C22×C3—61no—
474C22×C3—61no—
475C22×C3—121no—
476C22×C3—121no—
477C22×C3—122no—
478C22×C3—121no—
479C22×C3—121no—
480C22×C3—121no—
481C22×C3—121no—
482C22×C3—121no—
483C22×C3—121no—
484C22×C3—241no—
485C22×C3—241no—
486C22×C3D3×#6711yescharacteristic
487C22×C3C3×D3×D411yescharacteristic
488C22×C3C2×C3×#6711yescharacteristic
489C2×D3—21no—
490C2×D3—21no—
491C2×D3—41no—
492C2×D3—41no—
493C2×D3—41no—
494C2×D3C3×#21512yescomplemented
495C3×C4—21no—
496C3×C4—21no—
497C3×C4—21no—
498C3×C4—21no—
499C3×C4—21no—
500C3×C4—21no—
501C3×C4—42no—
502C3×C4—41no—
503C3×C4—41no—
504C3×C4—41no—
505C3×C4—41no—
506C3×C4—41no—
507C3×C4—61no—
508C3×C4—61no—
509C3×C4—81no—
510C3×C4—81no—
511C3×C4—82no—
512C3×C4—121no—
513C3×C4—122no—
514C3×C4—121no—
515C3×C4—121no—
516C3×C4—361no—
517C3×C4#379511yescharacteristic
518C3×C4#380211yescharacteristic
519C3×C4C3×#23511yescharacteristic
520C3×C4C3×D3×D411yescharacteristic
521C3×C4C3×D3×D411yescharacteristic
522C3×C4C3×#23711yescharacteristic
523Dic3—21no—
524Dic3—41no—
525Dic3—121no—
526Dic3—121no—
527Dic3—241no—
528C32—21no—
529C32—21no—
530C32—21no—
531C32—41no—
532C32#477811yescharacteristic, complemented
533C32#504811yescharacteristic, complemented
534C32C3×#46511yescharacteristic, complemented
535C23—31no—
536C23—61no—
537Q8—61no—
538Q8—61no—
539D4—61no—
540D4—61no—
541D4—362no—
542C2×C4—21no—
543C2×C4—31no—
544C2×C4—61no—
545C2×C4—61no—
546C2×C4—61no—
547C2×C4—181no—
548C2×C4C2×C3×D3211yescharacteristic
549C8—61no—
550C8—92no—
551C2×C3—21no—
552C2×C3—21no—
553C2×C3—21no—
554C2×C3—21no—
555C2×C3—21no—
556C2×C3—21no—
557C2×C3—21no—
558C2×C3—21no—
559C2×C3—21no—
560C2×C3—41no—
561C2×C3—41no—
562C2×C3—41no—
563C2×C3—62no—
564C2×C3—62no—
565C2×C3—122no—
566C2×C3—242no—
567C2×C3—241no—
568C2×C3—241no—
569C2×C3#6388511yescharacteristic
570C2×C3#6393311yescharacteristic
571C2×C3D3×#21511yescharacteristic
572C2×C3C3×#88411yescharacteristic
573C2×C3C3×#91211yescharacteristic
574C2×C3C3×D3×SD1611yescharacteristic
575C2×C3C3×#93911yescharacteristic
576C2×C3C2×C3×#21511yescharacteristic
577C2×C3C3×#95011yescharacteristic
578D3—22no—
579D3—81no—
580C22—32no—
581C22—61no—
582C22—61no—
583C22—121no—
584C22—121no—
585C22—121no—
586C22C3×D3×#6711yescharacteristic
587C4—41no—
588C4—61no—
589C4—121no—
590C4—361no—
591C4C3×#379511yescharacteristic
592C4C3×#380211yescharacteristic
593C3—21no—
594C3—21no—
595C3—21no—
596C3—41no—
597C3#1059050111yescharacteristic, direct product factor
598C3C3×#477811yescharacteristic, complemented
599C3C3×#504811yescharacteristic, complemented
600C2—62no—
601C2—241no—
602C2C3×#6388511yescharacteristic
603C2C3×#6393311yescharacteristic
604C2C3×D3×#21511yescharacteristic
subgroupG/H
centerC22×C3D3×#67
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C4C2×C3×D32
FittingC33×C4⋊C4C22
socleC22×C33C2×D4
prime radicalC3×#63908C2
hypercenterC3×C4⋊C4D32
hypocenterC32C3×#465
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C2×C3×C4 ◁ C3×C4⋊C4

Complements

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

NH ≅ G/Ntype
1. C32×#897C2semidirect
3. C2×C3×#3715C2semidirect
4. C2×C3×D3×C3⋊C8C2semidirect
5. C3×#63635C2semidirect
6. C32×#809C2semidirect
7. C3×#63566C2semidirect
10. #10590501C3direct (N × H)
32. C3×#897D3semidirect
33. C3×#897C2×C3semidirect
35. C2×#3715C2×C3semidirect
36. C2×D3×C3⋊C8C2×C3semidirect
38. #63635C2×C3semidirect
42. C3×#809D3semidirect
43. C3×#809C2×C3semidirect
47. #63566C2×C3semidirect
62. C3×C3⋊(C3⋊C8)D4semidirect
124. C2×C32×D3SD16semidirect
143. #897C3×D3semidirect
151. #809C3×D3semidirect
217. C3⋊(C3⋊C8)C3×D4semidirect
310. C2×C3×D3#215semidirect
311. C2×C3×D3C3×SD16semidirect
352. C33#465semidirect
494. C2×D3C3×#215semidirect
532. C32#4778semidirect
533. C32#5048semidirect
534. C32C3×#465semidirect
597. C3#10590501direct (N × H)
598. C3C3×#4778semidirect
599. C3C3×#5048semidirect

Maximal subgroups

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

subgroupindex
C32×#8972
C3×#639082
C2×C3×#37152
C2×C3×D3×C3⋊C82
C3×#636352
C32×#8092
C3×#635662
C3×#50483
C3×#47783
#105905013