Finite Group Databasebeta

← C3×#10591475 #13158527

Subgroups — C3×#10591475 #13158527

Order 1728.

882 conjugacy classes of subgroups.

The subgroup lattice has 618 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#64149C211yesmaximal, characteristic, complemented
2C2×C32×#214C211yesmaximal, characteristic, complemented
3C2×C32×#213C211yesmaximal, characteristic, complemented
4C3×#63863C211yesmaximal, characteristic, complemented
5C2×C3×#3717C211yesmaximal, characteristic, complemented
6C2×C3×#3714C211yesmaximal, characteristic, complemented
7C3×#63555C211yesmaximal, characteristic, complemented
8C3×#5176—31nomaximal
9C3×#5154—31nomaximal
10#10591475C311yesmaximal, characteristic, direct product factor
11C2×C33×D4C2211yescharacteristic
12C2×C3×#638—22no—
13C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
14C2×C32×D12C2211yescharacteristic
15C2×C32×Dic6C2211yescharacteristic
16C3×C4×C3⋊Dic3C2211yescharacteristic
17C32×#214—22no—
18C32×#213—22no—
19C2×C32×C3⋊C8C2211yescharacteristic
20C2×C32×C3⋊C8C2211yescharacteristic
21C3×(C2×D3)⋊Dic3—22no—
22C3×#3717—22no—
23C3×#3714—22no—
24C2×C32×SD16—31no—
25C2×C32×D8—31no—
26#64149C2×C311yescharacteristic, complemented
27C3×#947—31no—
28C2×C3×#215—31no—
29C3×#940—31no—
30C3×#940—31no—
31C3×#940—61no—
32C3×#936—31no—
33C2×C3×#214—21no—
34C2×C3×#214D311yescharacteristic, complemented
35C2×C3×#214C2×C311yescharacteristic, complemented
36C2×C3×#213—21no—
37C2×C3×#213—31no—
38C2×C3×#213D311yescharacteristic, complemented
39C2×C3×#213C2×C311yescharacteristic, complemented
40C2×C3×D24—31no—
41C2×C3×#204—31no—
42#63863C2×C311yescharacteristic, complemented
43C2×#3717C2×C311yescharacteristic, complemented
44C2×#3714C2×C311yescharacteristic, complemented
45C3×#815—31no—
46C3×#815—31no—
47#63555C2×C311yescharacteristic, complemented
48C23×C33—21no—
49C22×C3×C3⋊D3—21no—
50C22×C32×D3—21no—
51C33×D4—21no—
52C33×D4—21no—
53C2×C33×C4C2311yescharacteristic
54C3×#638—44no—
55C2×C3×C3⋊Dic3D412yes—
56C3×C4⋊(C3⋊D3)—21no—
57C3×C4⋊(C3⋊D3)—21no—
58C2×C32×Dic3—21no—
59C32×D12—21no—
60C32×D12—21no—
61C32×Dic6—21no—
62C32×Dic6—21no—
63C32×C3⋊C8D412yescomplemented
64C32×C3⋊C8D412yescomplemented
65C3×#496—91no—
66#5176—31no—
67#5154—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×D4C2×D311yescharacteristic
75C2×C32×D4C2×D311yescharacteristic
76C2×C32×D4C22×C311yescharacteristic
77C2×#638—22no—
78C2×C4⋊(C3⋊D3)C22×C311yescharacteristic
79C2×C3×#67—62no—
80C2×C3×#67—62no—
81C2×C3×#67—64no—
82C2×C3×D12—21no—
83C2×C3×D12—31no—
84C2×C3×D12—31no—
85C2×C3×D12—61no—
86C2×C3×D12C2×D311yescharacteristic
87C2×C3×D12C22×C311yescharacteristic
88C2×C3×Dic6—21no—
89C2×C3×Dic6C2×D311yescharacteristic
90C2×C3×Dic6C22×C311yescharacteristic
91C32×SD16—62no—
92C32×D8—62no—
93C2×C32×C8—31no—
94C2×C32×C8—31no—
95C4×C3⋊Dic3C22×C311yescharacteristic
96C3×#217—62no—
97C3×#215—62no—
98C3×#214—24no—
99C3×#214—22no—
100C3×#214—22no—
101C3×#213—24no—
102C3×#213—22no—
103C3×#213—22no—
104C3×#213—62no—
105C3×(C2×D3)⋊C4—62no—
106C3×C4×Dic3—31no—
107C3×C4×Dic3—31no—
108C3×C4×Dic3—61no—
109C2×C3×C3⋊C8—21no—
110C2×C3×C3⋊C8—21no—
111C2×C3×C3⋊C8C2×D311yescharacteristic
112C2×C3×C3⋊C8C2×D311yescharacteristic
113C2×C3×C3⋊C8C22×C311yescharacteristic
114C2×C3×C3⋊C8C22×C311yescharacteristic
115C3×D24—62no—
116C3×#204—62no—
117(C2×D3)⋊Dic3—22no—
118#3717—22no—
119#3714—22no—
120C22×C33—41no—
121C22×C33—42no—
122C22×C33C2×D411yescharacteristic
123C2×C3×C3⋊D3—41no—
124C2×C3×C3⋊D3—42no—
125C2×C32×D3—41no—
126C2×C32×D3—42no—
127C33×C4C2×D411yescharacteristic
128C33×C4C2×D411yescharacteristic
129C3×C3⋊Dic3—22no—
130C32×Dic3—41no—
131C2×C3×SD16—31no—
132C2×C3×SD16—31no—
133C2×C3×SD16—61no—
134C2×C3×SD16—91no—
135C2×C3×D8—31no—
136C2×C3×D8—31no—
137C2×C3×D8—61no—
138C2×C3×D8—91no—
139C3×#127—91no—
140C3×#124—91no—
141#947—31no—
142C2×#215—31no—
143#940—31no—
144#940—31no—
145#940—61no—
146#936—31no—
147C2×#214C3×D311yescharacteristic, complemented
148C2×#213—31no—
149C2×#213C3×D311yescharacteristic, complemented
150C2×D24—31no—
151C2×#204—31no—
152C3×#97—91no—
153#815—31no—
154#815—31no—
155C23×C32—21no—
156C23×C32—22no—
157C23×C32—21no—
158C23×C32—21no—
159C23×C32—41no—
160C23×C32—41no—
161C23×C32—42no—
162C23×C32—61no—
163C22×C3⋊D3—21no—
164C22×C3×D3—21no—
165C22×C3×D3—21no—
166C22×C3×D3—41no—
167C22×C3×D3—62no—
168C22×C3×D3—61no—
169C22×C3×D3—61no—
170C32×Q8—61no—
171C32×Q8—61no—
172C32×D4—21no—
173C32×D4—21no—
174C32×D4—22no—
175C32×D4—22no—
176C32×D4—21no—
177C32×D4—21no—
178C32×D4—21no—
179C32×D4—21no—
180C32×D4—41no—
181C32×D4—41no—
182C32×D4—41no—
183C32×D4—41no—
184C32×D4—42no—
185C32×D4—42no—
186C32×D4—61no—
187C32×D4—61no—
188C2×C32×C4—21no—
189C2×C32×C4—21no—
190C2×C32×C4—21no—
191C2×C32×C4—41no—
192C2×C32×C4—61no—
193C2×C32×C4C22×D311yescharacteristic
194C2×C32×C4C22×D311yescharacteristic
195C2×C32×C4C23×C311yescharacteristic
196#638—44no—
197C2×C3⋊Dic3C3×D412yes—
198C4⋊(C3⋊D3)—21no—
199C4⋊(C3⋊D3)—21no—
200C3×#67—128no—
201C3×#67—124no—
202C3×#67—124no—
203C2×C3×Dic3—21no—
204C2×C3×Dic3—21no—
205C2×C3×Dic3—32no—
206C2×C3×Dic3—32no—
207C2×C3×Dic3—41no—
208C2×C3×Dic3—62no—
209C3×D12—22no—
210C3×D12—21no—
211C3×D12—21no—
212C3×D12—21no—
213C3×D12—21no—
214C3×D12—41no—
215C3×D12—61no—
216C3×D12—61no—
217C3×D12—61no—
218C3×D12—61no—
219C3×D12—121no—
220C3×D12—121no—
221C3×Dic6—22no—
222C3×Dic6—21no—
223C3×Dic6—21no—
224C3×Dic6—21no—
225C3×Dic6—21no—
226C3×Dic6—41no—
227C32×C8—32no—
228C32×C8—32no—
229C3×C3⋊C8—22no—
230C3×C3⋊C8—22no—
231C3×C3⋊C8#6712yescomplemented
232C3×C3⋊C8#6712yescomplemented
233C3×C3⋊C8C3×D412yescomplemented
234C3×C3⋊C8C3×D412yescomplemented
235#496—91noSylow
236C2×C33—81no—
237C2×C33#12711yescharacteristic
238C2×C33AGL(1,ℤ/8)12yes—
239C3×C3⋊D3—81no—
240C32×D3—81no—
241C2×C3×Q8—31no—
242C2×C3×Q8—31no—
243C2×C3×Q8—61no—
244C2×C3×D4—21no—
245C2×C3×D4—21no—
246C2×C3×D4—21no—
247C2×C3×D4—31no—
248C2×C3×D4—31no—
249C2×C3×D4—41no—
250C2×C3×D4—61no—
251C2×C3×D4—91no—
252C2×C3×D4—182no—
253C2×C3×D4D3211yescharacteristic
254C2×C3×D4C2×C3×D311yescharacteristic
255C2×C3×D4C2×C3×D311yescharacteristic
256C2×#67—62no—
257C2×#67—62no—
258C2×#67—64no—
259C2×D12—31no—
260C2×D12—31no—
261C2×D12—61no—
262C2×D12C2×C3×D311yescharacteristic
263C2×Dic6C2×C3×D311yescharacteristic
264C3×SD16—62no—
265C3×SD16—62no—
266C3×SD16—64no—
267C3×SD16—182no—
268C3×D8—62no—
269C3×D8—62no—
270C3×D8—64no—
271C3×D8—182no—
272C2×C3×C8—31no—
273C2×C3×C8—31no—
274C2×C3×C8—31no—
275C2×C3×C8—31no—
276C2×C3×C8—61no—
277C2×C3×C8—61no—
278C3×C22⋊C4—182no—
279C3×C42—91no—
280#217—62no—
281#215—62no—
282#214—22no—
283#213—22no—
284#213—62no—
285(C2×D3)⋊C4—62no—
286C4×Dic3—31no—
287C4×Dic3—31no—
288C4×Dic3—61no—
289C2×C3⋊C8C2×C3×D311yescharacteristic
290C2×C3⋊C8C2×C3×D311yescharacteristic
291D24—62no—
292#204—62no—
293C22×C32—21no—
294C22×C32—21no—
295C22×C32—21no—
296C22×C32—41no—
297C22×C32—42no—
298C22×C32—42no—
299C22×C32—42no—
300C22×C32—44no—
301C22×C32—41no—
302C22×C32—41no—
303C22×C32—41no—
304C22×C32—42no—
305C22×C32—44no—
306C22×C32—42no—
307C22×C32—42no—
308C22×C32—82no—
309C22×C32—82no—
310C22×C32—84no—
311C22×C32—121no—
312C22×C32—122no—
313C22×C32D3×D411yescharacteristic
314C22×C32D3×D411yescharacteristic
315C22×C32C2×C3×D411yescharacteristic
316C2×C3⋊D3—41no—
317C2×C3⋊D3—42no—
318C2×C3×D3—42no—
319C2×C3×D3—42no—
320C2×C3×D3—41no—
321C2×C3×D3—41no—
322C2×C3×D3—42no—
323C2×C3×D3—82no—
324C2×C3×D3—122no—
325C2×C3×D3—124no—
326C2×C3×D3—122no—
327C2×C3×D3—122no—
328C2×C3×D3—121no—
329C2×C3×D3—121no—
330C32×C4—21no—
331C32×C4—21no—
332C32×C4—21no—
333C32×C4—21no—
334C32×C4—21no—
335C32×C4—21no—
336C32×C4—41no—
337C32×C4—41no—
338C32×C4—121no—
339C32×C4D3×D411yescharacteristic
340C32×C4D3×D411yescharacteristic
341C32×C4C2×#6711yescharacteristic
342C32×C4C2×#6711yescharacteristic
343C32×C4C2×C3×D411yescharacteristic
344C32×C4C2×C3×D411yescharacteristic
345C3⋊Dic3—22no—
346C3×Dic3—42no—
347C3×Dic3—41no—
348C3×Dic3—41no—
349C3×Dic3—64no—
350C3×Dic3—62no—
351C3×Dic3—62no—
352C2×SD16—31no—
353C2×SD16—91no—
354C2×D8—31no—
355C2×D8—91no—
356#127—91no—
357#124—91no—
358#97—91no—
359C33#49611yescharacteristic, complemented, Sylow
360C23×C3—22no—
361C23×C3—21no—
362C23×C3—21no—
363C23×C3—21no—
364C23×C3—41no—
365C23×C3—41no—
366C23×C3—42no—
367C23×C3—61no—
368C23×C3—61no—
369C23×C3—121no—
370C23×C3—181no—
371C22×D3—21no—
372C22×D3—62no—
373C22×D3—61no—
374C22×D3—61no—
375C3×Q8—62no—
376C3×Q8—61no—
377C3×Q8—61no—
378C3×Q8—61no—
379C3×Q8—61no—
380C3×Q8—121no—
381C3×D4—22no—
382C3×D4—22no—
383C3×D4—21no—
384C3×D4—21no—
385C3×D4—21no—
386C3×D4—21no—
387C3×D4—21no—
388C3×D4—21no—
389C3×D4—41no—
390C3×D4—41no—
391C3×D4—41no—
392C3×D4—41no—
393C3×D4—42no—
394C3×D4—42no—
395C3×D4—62no—
396C3×D4—61no—
397C3×D4—61no—
398C3×D4—61no—
399C3×D4—61no—
400C3×D4—121no—
401C3×D4—181no—
402C3×D4—181no—
403C3×D4—364no—
404C2×C3×C4—21no—
405C2×C3×C4—21no—
406C2×C3×C4—21no—
407C2×C3×C4—41no—
408C2×C3×C4—61no—
409C2×C3×C4—61no—
410C2×C3×C4—92no—
411C2×C3×C4—121no—
412C2×C3×C4C2×D3211yescharacteristic
413C2×C3×C4C22×C3×D311yescharacteristic
414C2×C3×C4C22×C3×D311yescharacteristic
415#67—128no—
416#67—124no—
417#67—124no—
418C2×Dic3—21no—
419C2×Dic3—32no—
420C2×Dic3—32no—
421C2×Dic3—62no—
422D12—21no—
423D12—21no—
424D12—61no—
425D12—61no—
426D12—61no—
427D12—61no—
428D12—121no—
429D12—121no—
430Dic6—21no—
431Dic6—21no—
432C3×C8—32no—
433C3×C8—32no—
434C3×C8—32no—
435C3×C8—32no—
436C3×C8—62no—
437C3×C8—62no—
438C3⋊C8C3×#6712yescomplemented
439C3⋊C8C3×#6712yescomplemented
440C2×C32—22no—
441C2×C32—22no—
442C2×C32—21no—
443C2×C32—21no—
444C2×C32—22no—
445C2×C32—21no—
446C2×C32—41no—
447C2×C32—42no—
448C2×C32—82no—
449C2×C32—82no—
450C2×C32—82no—
451C2×C32—84no—
452C2×C32—81no—
453C2×C32—81no—
454C2×C32—81no—
455C2×C32—241no—
456C2×C32#91112yes—
457C2×C32#91412yes—
458C2×C32#94011yescharacteristic
459C2×C32#94011yescharacteristic
460C2×C32C3×#12711yescharacteristic
461C2×C32C3×AGL(1,ℤ/8)12yes—
462C3⋊D3—81no—
463C3×D3—82no—
464C3×D3—81no—
465C3×D3—81no—
466C3×D3—242no—
467C3×D3—241no—
468C3×D3—241no—
469C2×Q8—31no—
470C2×D4—31no—
471C2×D4—91no—
472C2×D4—182no—
473C2×D4C3×D3211yescharacteristic
474SD16—62no—
475SD16—182no—
476D8—62no—
477D8—182no—
478C2×C8—31no—
479C2×C8—31no—
480C22⋊C4—182no—
481C42—91no—
482C22×C3—21no—
483C22×C3—21no—
484C22×C3—21no—
485C22×C3—44no—
486C22×C3—42no—
487C22×C3—42no—
488C22×C3—42no—
489C22×C3—44no—
490C22×C3—42no—
491C22×C3—42no—
492C22×C3—41no—
493C22×C3—41no—
494C22×C3—42no—
495C22×C3—41no—
496C22×C3—41no—
497C22×C3—82no—
498C22×C3—82no—
499C22×C3—84no—
500C22×C3—122no—
501C22×C3—122no—
502C22×C3—121no—
503C22×C3—122no—
504C22×C3—121no—
505C22×C3—242no—
506C22×C3—362no—
507C22×C3—361no—
508C22×C3#381111yescharacteristic
509C22×C3C3×D3×D411yescharacteristic
510C22×C3C3×D3×D411yescharacteristic
511C2×D3—42no—
512C2×D3—41no—
513C2×D3—122no—
514C2×D3—124no—
515C2×D3—122no—
516C2×D3—122no—
517C2×D3—121no—
518C2×D3—121no—
519C3×C4—21no—
520C3×C4—21no—
521C3×C4—21no—
522C3×C4—21no—
523C3×C4—21no—
524C3×C4—21no—
525C3×C4—41no—
526C3×C4—41no—
527C3×C4—122no—
528C3×C4—121no—
529C3×C4—121no—
530C3×C4—182no—
531C3×C4D3×#6711yescharacteristic
532C3×C4D3×#6711yescharacteristic
533C3×C4C3×D3×D411yescharacteristic
534C3×C4C3×D3×D411yescharacteristic
535C3×C4C2×C3×#6711yescharacteristic
536C3×C4C2×C3×#6711yescharacteristic
537Dic3—41no—
538Dic3—64no—
539Dic3—62no—
540Dic3—62no—
541C32—21no—
542C32—21no—
543C32—21no—
544C32—41no—
545C32#515411yescharacteristic, complemented
546C32#517611yescharacteristic, complemented
547C32C3×#49611yescharacteristic, complemented
548C23—21no—
549C23—61no—
550C23—181no—
551Q8—61no—
552Q8—61no—
553D4—21no—
554D4—21no—
555D4—61no—
556D4—61no—
557D4—181no—
558D4—181no—
559D4—364no—
560C2×C4—61no—
561C2×C4—92no—
562C2×C4C2×C3×D3211yescharacteristic
563C8—32no—
564C8—32no—
565C2×C3—21no—
566C2×C3—21no—
567C2×C3—21no—
568C2×C3—22no—
569C2×C3—22no—
570C2×C3—22no—
571C2×C3—41no—
572C2×C3—42no—
573C2×C3—82no—
574C2×C3—82no—
575C2×C3—82no—
576C2×C3—84no—
577C2×C3—81no—
578C2×C3—81no—
579C2×C3—81no—
580C2×C3—242no—
581C2×C3—241no—
582C2×C3—241no—
583C2×C3—721no—
584C2×C3#6393712yes—
585C2×C3#6397911yescharacteristic
586C2×C3C3×#91112yes—
587C2×C3C3×#91412yes—
588C2×C3C3×#94011yescharacteristic
589C2×C3C3×#94011yescharacteristic
590D3—81no—
591D3—242no—
592D3—241no—
593D3—241no—
594C22—41no—
595C22—42no—
596C22—121no—
597C22—122no—
598C22—361no—
599C22—362no—
600C22C3×#381111yescharacteristic
601C4—121no—
602C4—182no—
603C4C3×D3×#6711yescharacteristic
604C4C3×D3×#6711yescharacteristic
605C3—21no—
606C3—21no—
607C3—21no—
608C3—41no—
609C3#1059147511yescharacteristic, direct product factor
610C3C3×#515411yescharacteristic, complemented
611C3C3×#517611yescharacteristic, complemented
612C2—81no—
613C2—241no—
614C2—721no—
615C2C3×#6393712yes—
616C2C3×#6397911yescharacteristic
subgroupG/H
centerC22×C3#3811
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C4C2×C3×D32
FittingC2×C33×D4C22
socleC22×C33C2×D4
hypercenterC2×C3×D4D32
hypocenterC32C3×#496
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
1. C3×#64149C2semidirect
2. C2×C32×#214C2semidirect
3. C2×C32×#213C2semidirect
4. C3×#63863C2semidirect
5. C2×C3×#3717C2semidirect
6. C2×C3×#3714C2semidirect
7. C3×#63555C2semidirect
10. #10591475C3direct (N × H)
26. #64149C2×C3semidirect
34. C2×C3×#214D3semidirect
35. C2×C3×#214C2×C3semidirect
38. C2×C3×#213D3semidirect
39. C2×C3×#213C2×C3semidirect
42. #63863C2×C3semidirect
43. C2×#3717C2×C3semidirect
44. C2×#3714C2×C3semidirect
47. #63555C2×C3semidirect
63. C32×C3⋊C8D4semidirect
64. C32×C3⋊C8D4semidirect
147. C2×#214C3×D3semidirect
149. C2×#213C3×D3semidirect
231. C3×C3⋊C8#67semidirect
232. C3×C3⋊C8#67semidirect
233. C3×C3⋊C8C3×D4semidirect
234. C3×C3⋊C8C3×D4semidirect
359. C33#496semidirect
438. C3⋊C8C3×#67semidirect
439. C3⋊C8C3×#67semidirect
545. C32#5154semidirect
546. C32#5176semidirect
547. C32C3×#496semidirect
609. C3#10591475direct (N × H)
610. C3C3×#5154semidirect
611. C3C3×#5176semidirect

Maximal subgroups

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

subgroupindex
C3×#641492
C2×C32×#2142
C2×C32×#2132
C3×#638632
C2×C3×#37172
C2×C3×#37142
C3×#635552
C3×#51763
C3×#51543
#105914753