Finite Group Databasebeta

← C3×#10591012 #13158029

Subgroups — C3×#10591012 #13158029

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×#64307C211yesmaximal, characteristic
2C32×#909C211yesmaximal, characteristic, complemented
3C3×#63863C211yesmaximal, characteristic, complemented
4C3×#63833C211yesmaximal, characteristic, complemented
5C3×#63652C211yesmaximal, characteristic, complemented
6C32×#823C211yesmaximal, characteristic, complemented
7C3×#63568C211yesmaximal, characteristic, complemented
8C3×#5202—31nomaximal
9C3×#4746—31nomaximal
10#10591012C311yesmaximal, characteristic, direct product factor
11C3×#3828C2211yescharacteristic
12C32×#235C2211yescharacteristic
13C2×C32×D12C2211yescharacteristic
14C2×C32×Dic6C2211yescharacteristic, complemented
15C3×D3×#67—22no—
16C3×#3802—21no—
17C3×D3×D12—21no—
18C3×#3796—21no—
19C3×#3795—21no—
20C33×M16C2211yescharacteristic, complemented
21C3×C4×C3⋊Dic3C2211yescharacteristic, complemented
22C32×#208C2211yescharacteristic, complemented
23C32×Dic12—21no—
24C32×#204—21no—
25C3×(C2×D3)⋊Dic3—22no—
26C3×#3719—21no—
27C3×#3717—21no—
28C3×#1009—31no—
29C3×#1004—31no—
30#64307C2×C311yescharacteristic
31C32×#137—31no—
32C3×#947—31no—
33C3×#942—31no—
34C3×#936—31no—
35C3×#909—21no—
36C3×#909—31no—
37C3×#909D311yescharacteristic, complemented
38C3×#909C2×C311yescharacteristic, complemented
39#63863C2×C311yescharacteristic, complemented
40#63833C2×C311yescharacteristic, complemented
41C32×#100—31no—
42#63652C2×C311yescharacteristic, complemented
43C3×#837—31no—
44C3×#825—31no—
45C3×#825—31no—
46C3×#825—31no—
47C3×#825—61no—
48C3×#823—21no—
49C3×#823D311yescharacteristic, complemented
50C3×#823C2×C311yescharacteristic, complemented
51#63568C2×C311yescharacteristic, complemented
52C22×C32×D3D412yes—
53C2×C3×D32—22no—
54C2×C33×C4C2311yescharacteristic
55C3×#638—21no—
56C2×C3×C3⋊Dic3—21no—
57C3×C4⋊(C3⋊D3)D411yescharacteristic
58C3×C4×C3⋊D3—21no—
59C3×#634D411yescharacteristic
60C32×#67—21no—
61C2×C32×Dic3—21no—
62C32×D12—21no—
63C32×D12—21no—
64C32×D12D411yescharacteristic
65C32×C4×D3—21no—
66C32×Dic6—21no—
67C32×Dic6—21no—
68C32×Dic6D411yescharacteristic
69C3×#626—22no—
70C3×#625—22no—
71C3×D3×Dic3—22no—
72C33×C8—21no—
73C32×C3⋊C8—21no—
74C3×#455—91no—
75#5202—31no—
76#4746—31no—
77C32×#41—31no—
78C2×C32×Q8—31no—
79C2×C32×D4—31no—
80#3828C22×C311yescharacteristic
81C2×C3×#67—62no—
82C3×#239—61no—
83C3×#237—61no—
84C3×#237—61no—
85C3×D3×D4—61no—
86C3×D3×D4—61no—
87C3×D3×D4—62no—
88C3×D3×D4—61no—
89C3×#235—21no—
90C3×#235—31no—
91C3×#235—31no—
92C3×#235—61no—
93C3×#235—61no—
94C3×#235C2×D311yescharacteristic
95C3×#235C22×C311yescharacteristic
96C2×C3×D12—21no—
97C2×C3×D12—61no—
98C2×C3×D12C2×D311yescharacteristic
99C2×C3×D12C22×C311yescharacteristic
100C2×C3×Dic6—21no—
101C2×C3×Dic6C2×D311yescharacteristic, complemented
102C2×C3×Dic6C22×C311yescharacteristic, complemented
103D3×#67—22no—
104#3802—21no—
105D3×D12—21no—
106#3796—21no—
107#3795—21no—
108C32×Q16—61no—
109C32×SD16—61no—
110C32×M16—21no—
111C32×M16—21no—
112C32×M16—21no—
113C32×M16—31no—
114C32×M16—41no—
115C32×M16C2×D311yescharacteristic, complemented
116C32×M16C2×D311yescharacteristic, complemented
117C32×M16C22×C311yescharacteristic, complemented
118C4×C3⋊Dic3C22×C311yescharacteristic, complemented
119C3×#217—62no—
120C3×#216—61no—
121C3×#215—61no—
122C3×(C2×D3)⋊C4—62no—
123C3×C4×Dic3—31no—
124C3×C4×Dic3—31no—
125C3×C4×Dic3—61no—
126C3×#208—21no—
127C3×#208C2×D311yescharacteristic, complemented
128C3×#208C22×C311yescharacteristic, complemented
129C3×Dic12—22no—
130C3×Dic12—21no—
131C3×Dic12—21no—
132C3×Dic12—61no—
133C3×#204—22no—
134C3×#204—21no—
135C3×#204—21no—
136C3×#204—61no—
137(C2×D3)⋊Dic3—22no—
138#3719—21no—
139#3717—21no—
140C22×C33C2×D411yescharacteristic
141C2×C3×C3⋊D3—21no—
142C2×C32×D3—22no—
143C2×C32×D3—21no—
144C2×C32×D3—42no—
145C3×D32—82no—
146C33×C4C2×D411yescharacteristic
147C33×C4C2×D411yescharacteristic
148C3×C3⋊Dic3—21no—
149C3×C3⋊Dic3—41no—
150C32×Dic3—21no—
151C32×Dic3—41no—
152C3×H(5,𝔽2)—91no—
153#1009—31no—
154#1004—31no—
155C3×#137—31no—
156C3×#137—31no—
157C3×#137—61no—
158C3×#137—91no—
159C3×#124—91no—
160#947—31no—
161#942—31no—
162#936—31no—
163#909—31no—
164#909C3×D311yescharacteristic, complemented
165C3×#104—91no—
166C3×#104—91no—
167C3×#100—31no—
168C3×#100—31no—
169C3×#100—61no—
170#837—31no—
171#825—31no—
172#825—31no—
173#825—31no—
174#825—61no—
175#823C3×D311yescharacteristic, complemented
176C23×C32—32no—
177C22×C3×D3—22no—
178C22×C3×D3—62no—
179C22×C3×D3—62no—
180C22×C3×D3D1212yes—
181C22×C3×D3C3×D412yes—
182C2×D32—22no—
183C32×Q8—31no—
184C32×Q8—61no—
185C32×Q8—61no—
186C32×D4—31no—
187C32×D4—61no—
188C32×D4—61no—
189C32×D4—61no—
190C2×C32×C4—21no—
191C2×C32×C4—21no—
192C2×C32×C4—21no—
193C2×C32×C4—41no—
194C2×C32×C4—61no—
195C2×C32×C4—61no—
196C2×C32×C4C22×D311yescharacteristic
197C2×C32×C4C22×D311yescharacteristic
198C2×C32×C4C23×C311yescharacteristic
199#638—21no—
200C2×C3⋊Dic3—21no—
201C4⋊(C3⋊D3)C3×D411yescharacteristic
202C4×C3⋊D3—21no—
203#634C3×D411yescharacteristic
204C3×#67—21no—
205C3×#67—21no—
206C3×#67—41no—
207C3×#67—62no—
208C3×#67—62no—
209C3×#67—62no—
210C3×#67—61no—
211C3×#67—61no—
212C3×#67—121no—
213C2×C3×Dic3—21no—
214C2×C3×Dic3—21no—
215C2×C3×Dic3—41no—
216C2×C3×Dic3—62no—
217C2×C3×Dic3—61no—
218C2×C3×Dic3—61no—
219C2×C3×Dic3—121no—
220C3×D12—21no—
221C3×D12—21no—
222C3×D12—21no—
223C3×D12—21no—
224C3×D12—21no—
225C3×D12—31no—
226C3×D12—31no—
227C3×D12—41no—
228C3×D12—41no—
229C3×D12—61no—
230C3×D12—62no—
231C3×D12#6711yescharacteristic
232C3×D12C3×D411yescharacteristic
233C3×C4×D3—21no—
234C3×C4×D3—21no—
235C3×C4×D3—41no—
236C3×C4×D3—62no—
237C3×C4×D3—61no—
238C3×C4×D3—61no—
239C3×C4×D3—121no—
240C3×Dic6—22no—
241C3×Dic6—21no—
242C3×Dic6—21no—
243C3×Dic6—21no—
244C3×Dic6—21no—
245C3×Dic6—21no—
246C3×Dic6—31no—
247C3×Dic6—31no—
248C3×Dic6—41no—
249C3×Dic6—61no—
250C3×Dic6#6711yescharacteristic
251C3×Dic6C3×D411yescharacteristic
252#626—22no—
253#625—22no—
254D3×Dic3—22no—
255C32×C8—21no—
256C32×C8—22no—
257C32×C8—21no—
258C32×C8—21no—
259C32×C8—41no—
260C32×C8—41no—
261C32×C8—42no—
262C32×C8—61no—
263C3×C3⋊C8—21no—
264C3×C3⋊C8—21no—
265C3×C3⋊C8—41no—
266#455—91noSylow
267C2×C33—21no—
268C2×C33#12011yescharacteristic
269C3×C3⋊D3—41no—
270C32×D3—41no—
271C32×D3—42no—
272C3×#41—31no—
273C3×#41—31no—
274C3×#41—61no—
275C3×#41—91no—
276C3×#41—181no—
277C3×#41—181no—
278C2×C3×Q8—31no—
279C2×C3×Q8—31no—
280C2×C3×Q8—61no—
281C2×C3×D4—31no—
282C2×C3×D4—31no—
283C2×C3×D4—61no—
284C2×C3×D4—181no—
285C2×C3×D4—181no—
286C2×C3×D4—182no—
287C2×#67—62no—
288#239—61no—
289#237—61no—
290#237—61no—
291D3×D4—61no—
292D3×D4—61no—
293D3×D4—62no—
294D3×D4—61no—
295#235—31no—
296#235—31no—
297#235—61no—
298#235—61no—
299#235C2×C3×D311yescharacteristic
300C2×D12—61no—
301C2×D12C2×C3×D311yescharacteristic
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—31no—
315C3×M16—31no—
316C3×M16—41no—
317C3×M16—61no—
318C3×M16D3211yescharacteristic, complemented
319C3×M16C2×C3×D311yescharacteristic, complemented
320C3×M16C2×C3×D311yescharacteristic, complemented
321C3×C22⋊C4—182no—
322C3×C42—91no—
323#217—62no—
324#216—61no—
325#215—61no—
326(C2×D3)⋊C4—62no—
327C4×Dic3—31no—
328C4×Dic3—31no—
329C4×Dic3—61no—
330#208C2×C3×D311yescharacteristic, complemented
331Dic12—21no—
332Dic12—61no—
333#204—21no—
334#204—61no—
335C22×C32—21no—
336C22×C32—21no—
337C22×C32—21no—
338C22×C32—41no—
339C22×C32—61no—
340C22×C32—62no—
341C22×C32—122no—
342C22×C32C2×D1211yescharacteristic
343C22×C32D3×D411yescharacteristic
344C22×C32C2×C3×D411yescharacteristic
345C2×C3⋊D3—21no—
346C2×C3×D3—22no—
347C2×C3×D3—21no—
348C2×C3×D3—22no—
349C2×C3×D3—21no—
350C2×C3×D3—41no—
351C2×C3×D3—42no—
352C2×C3×D3—42no—
353C2×C3×D3—42no—
354C2×C3×D3—61no—
355C2×C3×D3—61no—
356C2×C3×D3—82no—
357C2×C3×D3—121no—
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—121no—
371C32×C4D3×D411yescharacteristic
372C32×C4D3×D411yescharacteristic
373C32×C4D3×D411yescharacteristic
374C32×C4C2×#6711yescharacteristic
375C32×C4C2×C3×D411yescharacteristic
376C32×C4C2×C3×D411yescharacteristic
377C3⋊Dic3—21no—
378C3⋊Dic3—41no—
379C3×Dic3—21no—
380C3×Dic3—21no—
381C3×Dic3—41no—
382C3×Dic3—42no—
383C3×Dic3—41no—
384C3×Dic3—41no—
385C3×Dic3—61no—
386C3×Dic3—61no—
387C3×Dic3—121no—
388C3×Dic3—121no—
389C3×Dic3—121no—
390C3×Dic3—122no—
391H(5,𝔽2)—91no—
392#137—31no—
393#137—91no—
394#124—91no—
395#104—91no—
396#104—91no—
397#100—31no—
398C33#45511yescharacteristic, complemented, Sylow
399C23×C3—32no—
400C23×C3—32no—
401C23×C3—62no—
402C23×C3—182no—
403C22×D3—62no—
404C22×D3—62no—
405C22×D3C3×D1212yes—
406C3×Q8—31no—
407C3×Q8—31no—
408C3×Q8—61no—
409C3×Q8—62no—
410C3×Q8—61no—
411C3×Q8—61no—
412C3×Q8—61no—
413C3×Q8—61no—
414C3×Q8—91no—
415C3×Q8—121no—
416C3×D4—31no—
417C3×D4—31no—
418C3×D4—61no—
419C3×D4—61no—
420C3×D4—61no—
421C3×D4—61no—
422C3×D4—61no—
423C3×D4—61no—
424C3×D4—61no—
425C3×D4—91no—
426C3×D4—121no—
427C3×D4—121no—
428C3×D4—121no—
429C3×D4—182no—
430C3×D4—182no—
431C3×D4—181no—
432C2×C3×C4—21no—
433C2×C3×C4—21no—
434C2×C3×C4—21no—
435C2×C3×C4—41no—
436C2×C3×C4—61no—
437C2×C3×C4—61no—
438C2×C3×C4—61no—
439C2×C3×C4—61no—
440C2×C3×C4—121no—
441C2×C3×C4—121no—
442C2×C3×C4—182no—
443C2×C3×C4—181no—
444C2×C3×C4—181no—
445C2×C3×C4C2×D3211yescharacteristic
446C2×C3×C4C22×C3×D311yescharacteristic
447C2×C3×C4C22×C3×D311yescharacteristic
448#67—21no—
449#67—62no—
450#67—62no—
451#67—62no—
452#67—61no—
453#67—61no—
454#67—121no—
455C2×Dic3—21no—
456C2×Dic3—62no—
457C2×Dic3—61no—
458C2×Dic3—61no—
459C2×Dic3—121no—
460D12—21no—
461D12—21no—
462D12—31no—
463D12—31no—
464D12—61no—
465D12—62no—
466D12C3×#6711yescharacteristic
467C4×D3—21no—
468C4×D3—62no—
469C4×D3—61no—
470C4×D3—61no—
471C4×D3—121no—
472Dic6—21no—
473Dic6—21no—
474Dic6—31no—
475Dic6—31no—
476Dic6—61no—
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—61no—
486C3×C8—61no—
487C3×C8—121no—
488C3⋊C8—21no—
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—122no—
501C2×C32—121no—
502C2×C32#88211yescharacteristic
503C2×C32#93711yescharacteristic
504C2×C32C3×#12011yescharacteristic
505C3⋊D3—41no—
506C3×D3—41no—
507C3×D3—42no—
508C3×D3—41no—
509C3×D3—42no—
510C3×D3—81no—
511C3×D3—82no—
512C3×D3—121no—
513C3×D3—121no—
514C3×D3—241no—
515#41—31no—
516#41—91no—
517#41—181no—
518#41—181no—
519C2×Q8—31no—
520C2×D4—31no—
521C2×D4—181no—
522C2×D4—181no—
523C2×D4—182no—
524Q16—61no—
525Q16—181no—
526SD16—61no—
527SD16—181no—
528M16—31no—
529M16C3×D3211yescharacteristic, complemented
530C22⋊C4—182no—
531C42—91no—
532C22×C3—21no—
533C22×C3—21no—
534C22×C3—21no—
535C22×C3—41no—
536C22×C3—61no—
537C22×C3—62no—
538C22×C3—61no—
539C22×C3—62no—
540C22×C3—121no—
541C22×C3—122no—
542C22×C3—122no—
543C22×C3—122no—
544C22×C3—181no—
545C22×C3—242no—
546C22×C3—722no—
547C22×C3D3×D1211yescharacteristic
548C22×C3C2×C3×D1211yescharacteristic
549C22×C3C3×D3×D411yescharacteristic
550C2×D3—22no—
551C2×D3—21no—
552C2×D3—42no—
553C2×D3—61no—
554C2×D3—61no—
555C2×D3—121no—
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—121no—
569C3×C4—122no—
570C3×C4—121no—
571C3×C4—121no—
572C3×C4—181no—
573C3×C4—361no—
574C3×C4#380211yescharacteristic
575C3×C4D3×#6711yescharacteristic
576C3×C4C3×D3×D411yescharacteristic
577C3×C4C3×D3×D411yescharacteristic
578C3×C4C3×D3×D411yescharacteristic
579C3×C4C2×C3×#6711yescharacteristic
580Dic3—21no—
581Dic3—41no—
582Dic3—61no—
583Dic3—61no—
584Dic3—121no—
585Dic3—121no—
586Dic3—121no—
587Dic3—122no—
588C32—21no—
589C32—21no—
590C32—21no—
591C32—41no—
592C32#474611yescharacteristic, complemented
593C32#520211yescharacteristic, complemented
594C32C3×#45511yescharacteristic, complemented
595C23—32no—
596C23—182no—
597Q8—31no—
598Q8—61no—
599Q8—61no—
600Q8—91no—
601D4—31no—
602D4—61no—
603D4—61no—
604D4—61no—
605D4—91no—
606D4—182no—
607D4—182no—
608D4—181no—
609C2×C4—61no—
610C2×C4—61no—
611C2×C4—182no—
612C2×C4—181no—
613C2×C4—181no—
614C2×C4C2×C3×D3211yescharacteristic
615C8—21no—
616C8—61no—
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—122no—
629C2×C3—121no—
630C2×C3—122no—
631C2×C3—121no—
632C2×C3—242no—
633C2×C3—241no—
634C2×C3—361no—
635C2×C3#6392211yescharacteristic
636C2×C3C3×#88211yescharacteristic
637C2×C3C3×#93711yescharacteristic
638D3—41no—
639D3—42no—
640D3—121no—
641D3—121no—
642D3—241no—
643C22—61no—
644C22—62no—
645C22—122no—
646C22—181no—
647C22—722no—
648C22C3×D3×D1211yescharacteristic
649C4—61no—
650C4—121no—
651C4—181no—
652C4—361no—
653C4C3×#380211yescharacteristic
654C4C3×D3×#6711yescharacteristic
655C3—21no—
656C3—21no—
657C3—21no—
658C3—41no—
659C3#1059101211yescharacteristic, direct product factor
660C3C3×#474611yescharacteristic, complemented
661C3C3×#520211yescharacteristic, complemented
662C2—21no—
663C2—122no—
664C2—121no—
665C2—361no—
666C2C3×#6392211yescharacteristic
subgroupG/H
centerC2×C3#63922
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C4C2×C3×D32
FittingC33×M16C22
socleC2×C33#120
prime radicalC3×#64307C2
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×#63863C2semidirect
4. C3×#63833C2semidirect
5. C3×#63652C2semidirect
6. C32×#823C2semidirect
7. C3×#63568C2semidirect
10. #10591012C3direct (N × H)
14. C2×C32×Dic6C22semidirect
20. C33×M16C22semidirect
21. C3×C4×C3⋊Dic3C22semidirect
22. C32×#208C22semidirect
37. C3×#909D3semidirect
38. C3×#909C2×C3semidirect
39. #63863C2×C3semidirect
40. #63833C2×C3semidirect
42. #63652C2×C3semidirect
49. C3×#823D3semidirect
50. C3×#823C2×C3semidirect
51. #63568C2×C3semidirect
101. C2×C3×Dic6C2×D3semidirect
102. C2×C3×Dic6C22×C3semidirect
115. C32×M16C2×D3semidirect
116. C32×M16C2×D3semidirect
117. C32×M16C22×C3semidirect
118. C4×C3⋊Dic3C22×C3semidirect
127. C3×#208C2×D3semidirect
128. C3×#208C22×C3semidirect
164. #909C3×D3semidirect
175. #823C3×D3semidirect
302. C2×Dic6C2×C3×D3semidirect
318. C3×M16D32semidirect
319. C3×M16C2×C3×D3semidirect
320. C3×M16C2×C3×D3semidirect
330. #208C2×C3×D3semidirect
398. C33#455semidirect
529. M16C3×D32semidirect
592. C32#4746semidirect
593. C32#5202semidirect
594. C32C3×#455semidirect
659. C3#10591012direct (N × H)
660. C3C3×#4746semidirect
661. C3C3×#5202semidirect

Maximal subgroups

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

subgroupindex
C3×#643072
C32×#9092
C3×#638632
C3×#638332
C3×#636522
C32×#8232
C3×#635682
C3×#52023
C3×#47463
#105910123