Finite Group Databasebeta

← C3×#10591011 #13158028

Subgroups — C3×#10591011 #13158028

Order 1728.

778 conjugacy classes of subgroups.

The subgroup lattice has 668 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#64308C211yesmaximal, characteristic
2C32×#909C211yesmaximal, characteristic, complemented
3C3×#63864C211yesmaximal, characteristic, complemented
4C3×#63827C211yesmaximal, characteristic, complemented
5C3×#63650C211yesmaximal, characteristic, complemented
6C32×#825C211yesmaximal, characteristic, complemented
7C3×#63569C211yesmaximal, characteristic, complemented
8C3×#5202—31nomaximal
9C3×#4825—31nomaximal
10#10591011C311yesmaximal, characteristic, direct product factor
11C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
12C32×#235C2211yescharacteristic
13C32×#235C2211yescharacteristic
14C2×C32×Dic6C2211yescharacteristic, complemented
15C3×#3811—22no—
16C3×D3×D12—21no—
17C3×D3×D12—21no—
18C3×#3799—21no—
19C3×#3799—21no—
20C33×M16C2211yescharacteristic, complemented
21C3×#3748C2211yescharacteristic, complemented
22C32×C4×Dic3C2211yescharacteristic, complemented
23C32×Dic12—21no—
24C32×#204—21no—
25C3×#3722—22no—
26C3×#3718—21no—
27C3×#3715—21no—
28C3×#1009—31no—
29C3×#1009—31no—
30#64308C2×C311yescharacteristic
31C32×#137—31no—
32C3×#951—31no—
33C3×#947—31no—
34C3×#942—31no—
35C3×#909—21no—
36C3×#909D311yescharacteristic, complemented
37C3×#909C2×C311yescharacteristic, complemented
38C3×#875—31no—
39#63864C2×C311yescharacteristic, complemented
40#63827C2×C311yescharacteristic, complemented
41C32×#104—31no—
42#63650C2×C311yescharacteristic, complemented
43C3×#837—31no—
44C3×#825—21no—
45C3×#825D311yescharacteristic, complemented
46C3×#825C2×C311yescharacteristic, complemented
47C3×#823—31no—
48C3×#823—31no—
49C3×#823—61no—
50C3×#805—31no—
51#63569C2×C311yescharacteristic, complemented
52C22×C3×C3⋊D3D412yes—
53C2×C3×D32—22no—
54C2×C33×C4C2311yescharacteristic
55C3×C4⋊(C3⋊D3)—21no—
56C3×C4⋊(C3⋊D3)—21no—
57C32×#67—21no—
58C32×#67—21no—
59C2×C32×Dic3—21no—
60C2×C32×Dic3—21no—
61C32×D12D411yescharacteristic
62C32×D12D411yescharacteristic
63C32×C4×D3—21no—
64C32×C4×D3—21no—
65C32×Dic6—21no—
66C32×Dic6—21no—
67C32×Dic6D411yescharacteristic
68C32×Dic6D411yescharacteristic
69C3×#626—22no—
70C3×#626—22no—
71C3×#624—22no—
72C33×C8—21no—
73C3×C3⋊(C3⋊C8)—21no—
74C3×#455—91no—
75#5202—31no—
76#4825—31no—
77C32×#41—31no—
78C32×#41—31no—
79C2×C32×Q8—31no—
80C2×C4⋊(C3⋊D3)C22×C311yescharacteristic
81C3×#239—61no—
82C3×#239—61no—
83C3×D3×D4—61no—
84C3×D3×D4—61no—
85C3×D3×D4—62no—
86C3×D3×D4—62no—
87C3×#235—21no—
88C3×#235—21no—
89C3×#235—61no—
90C3×#235—61no—
91C3×#235C2×D311yescharacteristic
92C3×#235C2×D311yescharacteristic
93C3×#235C22×C311yescharacteristic
94C3×#235C22×C311yescharacteristic
95C2×C3×D12—31no—
96C2×C3×D12—31no—
97C2×C3×D12—61no—
98C2×C3×D12—61no—
99C2×C3×D12—61no—
100C2×C3×Dic6—21no—
101C2×C3×Dic6C2×D311yescharacteristic, complemented
102C2×C3×Dic6C22×C311yescharacteristic, complemented
103#3811—22no—
104D3×D12—21no—
105D3×D12—21no—
106#3799—21no—
107#3799—21no—
108C32×Q16—61no—
109C32×SD16—61no—
110C32×M16—21no—
111C32×M16—21no—
112C32×M16—21no—
113C32×M16—41no—
114C32×M16C2×D311yescharacteristic, complemented
115C32×M16C2×D311yescharacteristic, complemented
116C32×M16C22×C311yescharacteristic, complemented
117C32×C42—31no—
118#3748C22×C311yescharacteristic, complemented
119C3×#216—61no—
120C3×#216—61no—
121C3×#215—61no—
122C3×#214—61no—
123C3×(C2×D3)⋊C4—62no—
124C3×(C2×D3)⋊C4—62no—
125C3×C4×Dic3—21no—
126C3×C4×Dic3C2×D311yescharacteristic, complemented
127C3×C4×Dic3C22×C311yescharacteristic, complemented
128C3×#208—31no—
129C3×#208—31no—
130C3×#208—61no—
131C3×Dic12—22no—
132C3×Dic12—21no—
133C3×Dic12—21no—
134C3×#204—22no—
135C3×#204—21no—
136C3×#204—21no—
137#3722—22no—
138#3718—21no—
139#3715—21no—
140C22×C33C2×D411yescharacteristic
141C2×C3×C3⋊D3—22no—
142C2×C3×C3⋊D3—42no—
143C2×C32×D3—21no—
144C2×C32×D3—21no—
145C3×D32—82no—
146C33×C4C2×D411yescharacteristic
147C33×C4C2×D411yescharacteristic
148C32×Dic3—21no—
149C32×Dic3—21no—
150C32×Dic3—41no—
151C32×Dic3—41no—
152C3×H(5,𝔽2)—91no—
153#1009—31no—
154#1009—31no—
155C3×#137—31no—
156C3×#137—31no—
157C3×#137—61no—
158C3×#137—91no—
159C3×#124—91no—
160#951—31no—
161#947—31no—
162#942—31no—
163#909C3×D311yescharacteristic, complemented
164#875—31no—
165C3×#104—31no—
166C3×#104—31no—
167C3×#104—61no—
168C3×#104—91no—
169C3×#100—91no—
170#837—31no—
171#825C3×D311yescharacteristic, complemented
172#823—31no—
173#823—31no—
174#823—61no—
175#805—31no—
176C22×C3⋊D3C3×D412yes—
177C22×C3×D3—32no—
178C22×C3×D3—32no—
179C22×C3×D3—62no—
180C22×C3×D3—62no—
181C22×C3×D3—62no—
182C2×D32—22no—
183C32×Q8—31no—
184C32×Q8—31no—
185C32×Q8—61no—
186C32×Q8—61no—
187C32×D4—31no—
188C32×D4—31no—
189C32×D4—61no—
190C32×D4—61no—
191C2×C32×C4—21no—
192C2×C32×C4—21no—
193C2×C32×C4—21no—
194C2×C32×C4—41no—
195C2×C32×C4—61no—
196C2×C32×C4—61no—
197C2×C32×C4—61no—
198C2×C32×C4—61no—
199C2×C32×C4C22×D311yescharacteristic
200C2×C32×C4C22×D311yescharacteristic
201C2×C32×C4C23×C311yescharacteristic
202C4⋊(C3⋊D3)—21no—
203C4⋊(C3⋊D3)—21no—
204C3×#67—21no—
205C3×#67—21no—
206C3×#67—21no—
207C3×#67—21no—
208C3×#67—41no—
209C3×#67—41no—
210C3×#67—62no—
211C3×#67—62no—
212C2×C3×Dic3—21no—
213C2×C3×Dic3—21no—
214C2×C3×Dic3—21no—
215C2×C3×Dic3—21no—
216C2×C3×Dic3—41no—
217C2×C3×Dic3—41no—
218C3×D12—21no—
219C3×D12—21no—
220C3×D12—62no—
221C3×D12—62no—
222C3×D12—61no—
223C3×D12—61no—
224C3×D12—61no—
225C3×D12—61no—
226C3×D12—121no—
227C3×D12—121no—
228C3×D12D1211yescharacteristic
229C3×D12#6711yescharacteristic
230C3×D12C3×D411yescharacteristic
231C3×D12C3×D411yescharacteristic
232C3×C4×D3—21no—
233C3×C4×D3—21no—
234C3×C4×D3—21no—
235C3×C4×D3—21no—
236C3×C4×D3—41no—
237C3×C4×D3—41no—
238C3×C4×D3—62no—
239C3×C4×D3—62no—
240C3×Dic6—22no—
241C3×Dic6—21no—
242C3×Dic6—21no—
243C3×Dic6—21no—
244C3×Dic6—21no—
245C3×Dic6—21no—
246C3×Dic6—21no—
247C3×Dic6—41no—
248C3×Dic6D1211yescharacteristic
249C3×Dic6#6711yescharacteristic
250C3×Dic6C3×D411yescharacteristic
251C3×Dic6C3×D411yescharacteristic
252#626—22no—
253#626—22no—
254#624—22no—
255C32×C8—21no—
256C32×C8—22no—
257C32×C8—21no—
258C32×C8—21no—
259C32×C8—41no—
260C32×C8—41no—
261C32×C8—42no—
262C3⋊(C3⋊C8)—21no—
263C3×C3⋊C8—61no—
264C3×C3⋊C8—61no—
265C3×C3⋊C8—121no—
266#455—91noSylow
267C2×C33—21no—
268C2×C33#12011yescharacteristic
269C3×C3⋊D3—42no—
270C32×D3—41no—
271C32×D3—41no—
272C3×#41—31no—
273C3×#41—31no—
274C3×#41—31no—
275C3×#41—31no—
276C3×#41—61no—
277C3×#41—61no—
278C3×#41—181no—
279C3×#41—181no—
280C2×C3×Q8—31no—
281C2×C3×Q8—31no—
282C2×C3×Q8—61no—
283C2×C3×D4—91no—
284C2×C3×D4—181no—
285C2×C3×D4—181no—
286C2×C3×D4—182no—
287#239—61no—
288#239—61no—
289D3×D4—61no—
290D3×D4—61no—
291D3×D4—62no—
292D3×D4—62no—
293#235—61no—
294#235—61no—
295#235C2×C3×D311yescharacteristic
296#235C2×C3×D311yescharacteristic
297C2×D12—31no—
298C2×D12—31no—
299C2×D12—61no—
300C2×D12—61no—
301C2×D12—61no—
302C2×Dic6C2×C3×D311yescharacteristic, complemented
303C3×Q16—61no—
304C3×Q16—61no—
305C3×Q16—62no—
306C3×Q16—181no—
307C3×SD16—61no—
308C3×SD16—61no—
309C3×SD16—62no—
310C3×SD16—181no—
311C3×M16—21no—
312C3×M16—21no—
313C3×M16—21no—
314C3×M16—41no—
315C3×M16—91no—
316C3×M16D3211yescharacteristic, complemented
317C3×M16C2×C3×D311yescharacteristic, complemented
318C3×M16C2×C3×D311yescharacteristic, complemented
319C3×C22⋊C4—182no—
320C3×C42—31no—
321C3×C42—31no—
322C3×C42—61no—
323#216—61no—
324#216—61no—
325#215—61no—
326#214—61no—
327(C2×D3)⋊C4—62no—
328(C2×D3)⋊C4—62no—
329C4×Dic3C2×C3×D311yescharacteristic, complemented
330#208—31no—
331#208—31no—
332#208—61no—
333Dic12—21no—
334#204—21no—
335C22×C32—21no—
336C22×C32—21no—
337C22×C32—21no—
338C22×C32—41no—
339C22×C32—61no—
340C22×C32—61no—
341C22×C32D3×D411yescharacteristic
342C22×C32D3×D411yescharacteristic
343C22×C32C2×C3×D411yescharacteristic
344C2×C3⋊D3—22no—
345C2×C3⋊D3—42no—
346C2×C3×D3—21no—
347C2×C3×D3—21no—
348C2×C3×D3—21no—
349C2×C3×D3—21no—
350C2×C3×D3—41no—
351C2×C3×D3—41no—
352C2×C3×D3—62no—
353C2×C3×D3—62no—
354C2×C3×D3—122no—
355C2×C3×D3—122no—
356C2×C3×D3—122no—
357C2×C3×D3—242no—
358C2×C3×D3—242no—
359C2×C3×D3—242no—
360D32—82no—
361C32×C4—21no—
362C32×C4—21no—
363C32×C4—21no—
364C32×C4—21no—
365C32×C4—21no—
366C32×C4—21no—
367C32×C4—41no—
368C32×C4—41no—
369C32×C4—61no—
370C32×C4—61no—
371C32×C4—121no—
372C32×C4—121no—
373C32×C4C2×D1211yescharacteristic
374C32×C4D3×D411yescharacteristic
375C32×C4D3×D411yescharacteristic
376C32×C4C2×#6711yescharacteristic
377C32×C4C2×C3×D411yescharacteristic
378C32×C4C2×C3×D411yescharacteristic
379C3×Dic3—21no—
380C3×Dic3—21no—
381C3×Dic3—21no—
382C3×Dic3—21no—
383C3×Dic3—41no—
384C3×Dic3—41no—
385C3×Dic3—42no—
386C3×Dic3—41no—
387C3×Dic3—41no—
388C3×Dic3—42no—
389C3×Dic3—41no—
390C3×Dic3—41no—
391H(5,𝔽2)—91no—
392#137—31no—
393#137—91no—
394#124—91no—
395#104—31no—
396#104—91no—
397#100—91no—
398C33#45511yescharacteristic, complemented, Sylow
399C23×C3—92no—
400C23×C3—182no—
401C22×D3—32no—
402C22×D3—32no—
403C22×D3—62no—
404C22×D3—62no—
405C22×D3—62no—
406C3×Q8—31no—
407C3×Q8—31no—
408C3×Q8—31no—
409C3×Q8—31no—
410C3×Q8—61no—
411C3×Q8—61no—
412C3×Q8—62no—
413C3×Q8—61no—
414C3×Q8—61no—
415C3×Q8—61no—
416C3×Q8—61no—
417C3×Q8—121no—
418C3×D4—31no—
419C3×D4—31no—
420C3×D4—31no—
421C3×D4—31no—
422C3×D4—61no—
423C3×D4—61no—
424C3×D4—61no—
425C3×D4—61no—
426C3×D4—61no—
427C3×D4—61no—
428C3×D4—121no—
429C3×D4—121no—
430C3×D4—182no—
431C3×D4—182no—
432C3×D4—181no—
433C3×D4—181no—
434C2×C3×C4—21no—
435C2×C3×C4—21no—
436C2×C3×C4—21no—
437C2×C3×C4—41no—
438C2×C3×C4—61no—
439C2×C3×C4—61no—
440C2×C3×C4—61no—
441C2×C3×C4—61no—
442C2×C3×C4—61no—
443C2×C3×C4—61no—
444C2×C3×C4—61no—
445C2×C3×C4—61no—
446C2×C3×C4—121no—
447C2×C3×C4—121no—
448C2×C3×C4—121no—
449C2×C3×C4—121no—
450C2×C3×C4—182no—
451C2×C3×C4C2×D3211yescharacteristic
452C2×C3×C4C22×C3×D311yescharacteristic
453C2×C3×C4C22×C3×D311yescharacteristic
454#67—21no—
455#67—21no—
456#67—62no—
457#67—62no—
458C2×Dic3—21no—
459C2×Dic3—21no—
460D12—62no—
461D12—62no—
462D12—61no—
463D12—61no—
464D12—61no—
465D12—61no—
466D12—121no—
467D12—121no—
468D12C3×D1211yescharacteristic
469D12C3×#6711yescharacteristic
470C4×D3—21no—
471C4×D3—21no—
472C4×D3—62no—
473C4×D3—62no—
474Dic6—21no—
475Dic6—21no—
476Dic6C3×D1211yescharacteristic
477Dic6C3×#6711yescharacteristic
478C3×C8—22no—
479C3×C8—21no—
480C3×C8—21no—
481C3×C8—21no—
482C3×C8—41no—
483C3×C8—41no—
484C3×C8—42no—
485C3×C8—181no—
486C3⋊C8—61no—
487C3⋊C8—61no—
488C3⋊C8—121no—
489C2×C32—21no—
490C2×C32—21no—
491C2×C32—21no—
492C2×C32—21no—
493C2×C32—21no—
494C2×C32—21no—
495C2×C32—41no—
496C2×C32—41no—
497C2×C32—41no—
498C2×C32—41no—
499C2×C32—81no—
500C2×C32—121no—
501C2×C32—121no—
502C2×C32#88211yescharacteristic
503C2×C32#93711yescharacteristic
504C2×C32C3×#12011yescharacteristic
505C3⋊D3—42no—
506C3×D3—41no—
507C3×D3—41no—
508C3×D3—41no—
509C3×D3—41no—
510C3×D3—81no—
511C3×D3—81no—
512C3×D3—122no—
513C3×D3—122no—
514C3×D3—242no—
515#41—31no—
516#41—31no—
517#41—181no—
518#41—181no—
519C2×Q8—31no—
520C2×D4—91no—
521C2×D4—181no—
522C2×D4—181no—
523C2×D4—182no—
524Q16—61no—
525Q16—181no—
526SD16—61no—
527SD16—181no—
528M16—91no—
529M16C3×D3211yescharacteristic, complemented
530C22⋊C4—182no—
531C42—31no—
532C22×C3—21no—
533C22×C3—21no—
534C22×C3—21no—
535C22×C3—41no—
536C22×C3—61no—
537C22×C3—61no—
538C22×C3—61no—
539C22×C3—61no—
540C22×C3—121no—
541C22×C3—121no—
542C22×C3—182no—
543C22×C3—362no—
544C22×C3—722no—
545C22×C3#380211yescharacteristic
546C22×C3C3×D3×D411yescharacteristic
547C22×C3C3×D3×D411yescharacteristic
548C2×D3—21no—
549C2×D3—21no—
550C2×D3—62no—
551C2×D3—62no—
552C2×D3—122no—
553C2×D3—122no—
554C2×D3—122no—
555C2×D3—242no—
556C2×D3—242no—
557C2×D3—242no—
558C3×C4—21no—
559C3×C4—21no—
560C3×C4—21no—
561C3×C4—21no—
562C3×C4—21no—
563C3×C4—21no—
564C3×C4—41no—
565C3×C4—41no—
566C3×C4—61no—
567C3×C4—61no—
568C3×C4—61no—
569C3×C4—61no—
570C3×C4—121no—
571C3×C4—121no—
572C3×C4—121no—
573C3×C4—121no—
574C3×C4—122no—
575C3×C4—122no—
576C3×C4—121no—
577C3×C4—121no—
578C3×C4D3×D1211yescharacteristic
579C3×C4D3×#6711yescharacteristic
580C3×C4C2×C3×D1211yescharacteristic
581C3×C4C3×D3×D411yescharacteristic
582C3×C4C3×D3×D411yescharacteristic
583C3×C4C2×C3×#6711yescharacteristic
584Dic3—21no—
585Dic3—21no—
586Dic3—41no—
587Dic3—41no—
588C32—21no—
589C32—21no—
590C32—21no—
591C32—41no—
592C32#482511yescharacteristic, complemented
593C32#520211yescharacteristic, complemented
594C32C3×#45511yescharacteristic, complemented
595C23—92no—
596C23—182no—
597Q8—31no—
598Q8—31no—
599Q8—61no—
600Q8—61no—
601D4—31no—
602D4—31no—
603D4—61no—
604D4—61no—
605D4—182no—
606D4—182no—
607D4—181no—
608D4—181no—
609C2×C4—61no—
610C2×C4—61no—
611C2×C4—61no—
612C2×C4—61no—
613C2×C4—182no—
614C2×C4C2×C3×D3211yescharacteristic
615C8—21no—
616C8—181no—
617C2×C3—21no—
618C2×C3—21no—
619C2×C3—21no—
620C2×C3—21no—
621C2×C3—21no—
622C2×C3—21no—
623C2×C3—41no—
624C2×C3—41no—
625C2×C3—41no—
626C2×C3—41no—
627C2×C3—81no—
628C2×C3—121no—
629C2×C3—121no—
630C2×C3—121no—
631C2×C3—121no—
632C2×C3—241no—
633C2×C3—241no—
634C2×C3—362no—
635C2×C3#6392211yescharacteristic
636C2×C3C3×#88211yescharacteristic
637C2×C3C3×#93711yescharacteristic
638D3—41no—
639D3—41no—
640D3—122no—
641D3—122no—
642D3—242no—
643C22—61no—
644C22—61no—
645C22—182no—
646C22—362no—
647C22—722no—
648C22C3×#380211yescharacteristic
649C4—61no—
650C4—61no—
651C4—121no—
652C4—121no—
653C4C3×D3×D1211yescharacteristic
654C4C3×D3×#6711yescharacteristic
655C3—21no—
656C3—21no—
657C3—21no—
658C3—41no—
659C3#1059101111yescharacteristic, direct product factor
660C3C3×#482511yescharacteristic, complemented
661C3C3×#520211yescharacteristic, complemented
662C2—21no—
663C2—121no—
664C2—121no—
665C2—362no—
666C2C3×#6392211yescharacteristic
subgroupG/H
centerC2×C3#63922
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C4C2×C3×D32
FittingC33×M16C22
socleC2×C33#120
prime radicalC3×#64308C2
hypercenterC3×M16D32
hypocenterC32C3×#455
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C2×C3 ◁ C2×C3×C4 ◁ C3×M16

Complements

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

NH ≅ G/Ntype
2. C32×#909C2semidirect
3. C3×#63864C2semidirect
4. C3×#63827C2semidirect
5. C3×#63650C2semidirect
6. C32×#825C2semidirect
7. C3×#63569C2semidirect
10. #10591011C3direct (N × H)
14. C2×C32×Dic6C22semidirect
20. C33×M16C22semidirect
21. C3×#3748C22semidirect
22. C32×C4×Dic3C22semidirect
36. C3×#909D3semidirect
37. C3×#909C2×C3semidirect
39. #63864C2×C3semidirect
40. #63827C2×C3semidirect
42. #63650C2×C3semidirect
45. C3×#825D3semidirect
46. C3×#825C2×C3semidirect
51. #63569C2×C3semidirect
101. C2×C3×Dic6C2×D3semidirect
102. C2×C3×Dic6C22×C3semidirect
114. C32×M16C2×D3semidirect
115. C32×M16C2×D3semidirect
116. C32×M16C22×C3semidirect
118. #3748C22×C3semidirect
126. C3×C4×Dic3C2×D3semidirect
127. C3×C4×Dic3C22×C3semidirect
163. #909C3×D3semidirect
171. #825C3×D3semidirect
302. C2×Dic6C2×C3×D3semidirect
316. C3×M16D32semidirect
317. C3×M16C2×C3×D3semidirect
318. C3×M16C2×C3×D3semidirect
329. C4×Dic3C2×C3×D3semidirect
398. C33#455semidirect
529. M16C3×D32semidirect
592. C32#4825semidirect
593. C32#5202semidirect
594. C32C3×#455semidirect
659. C3#10591011direct (N × H)
660. C3C3×#4825semidirect
661. C3C3×#5202semidirect

Maximal subgroups

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

subgroupindex
C3×#643082
C32×#9092
C3×#638642
C3×#638272
C3×#636502
C32×#8252
C3×#635692
C3×#52023
C3×#48253
#105910113