Finite Group Databasebeta

← C3×#10590399 #13157416

Subgroups — C3×#10590399 #13157416

Order 1728.

754 conjugacy classes of subgroups.

The subgroup lattice has 644 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#64297C211yesmaximal, characteristic
2C32×#869C211yesmaximal, characteristic, complemented
3C3×#63833C211yesmaximal, characteristic, complemented
4C3×#63827C211yesmaximal, characteristic, complemented
5C3×#63632C211yesmaximal, characteristic, complemented
6C32×#805C211yesmaximal, characteristic, complemented
7C3×#63560C211yesmaximal, characteristic, complemented
8C3×#5095—31nomaximal
9C3×#4826—31nomaximal
10#10590399C311yesmaximal, characteristic, direct product factor
11C3×#3828C2211yescharacteristic
12C32×#235C2211yescharacteristic
13C2×C32×Dic6C2211yescharacteristic
14C2×C32×Dic6C2211yescharacteristic
15C3×#3804—22no—
16C3×#3799—21no—
17C3×#3797—21no—
18C3×#3796—21no—
19C3×D3×Dic6—21no—
20C33×C42C2211yescharacteristic
21C3×#3748C2211yescharacteristic
22C32×C4⋊Dic3—22no—
23C32×#208C2211yescharacteristic
24C3×#3719—21no—
25C3×#3718—21no—
26C3×#3717—21no—
27C3×#3715—21no—
28C3×#1010—31no—
29C3×#1007—31no—
30#64297C2×C311yescharacteristic
31C32×C4⋊Q8—31no—
32C3×#951—31no—
33C3×#942—31no—
34C3×#942—31no—
35C3×#909—31no—
36C3×#869—21no—
37C3×#869D311yescharacteristic, complemented
38C3×#869C2×C311yescharacteristic, complemented
39#63833C2×C311yescharacteristic, complemented
40#63827C2×C311yescharacteristic, complemented
41C32×#104—31no—
42#63632C2×C311yescharacteristic, complemented
43C3×#836—31no—
44C3×#824—31no—
45C3×#805—21no—
46C3×#805—31no—
47C3×#805—31no—
48C3×#805—61no—
49C3×#805D311yescharacteristic, complemented
50C3×#805C2×C311yescharacteristic, complemented
51#63560C2×C311yescharacteristic, complemented
52C2×C33×C4—21no—
53C2×C33×C4C2311yescharacteristic
54C3×#638—21no—
55C3×C4⋊(C3⋊D3)D411yescharacteristic
56C3×C4×C3⋊D3—21no—
57C3×#634D411yescharacteristic
58C32×#67—21no—
59C2×C32×Dic3—21no—
60C2×C32×Dic3D412yes—
61C32×D12D411yescharacteristic
62C32×C4×D3—21no—
63C32×Dic6—21no—
64C32×Dic6—21no—
65C32×Dic6—21no—
66C32×Dic6—21no—
67C32×Dic6D411yescharacteristic
68C3×#627—22no—
69C3×#626—22no—
70C3×#624—22no—
71C3×D3×Dic3—22no—
72C3×C3⋊(C3⋊C8)—21no—
73C32×C3⋊C8—21no—
74C3×#456—91no—
75#5095—31no—
76#4826—31no—
77C32×#41—31no—
78C2×C32×Q8—31no—
79C2×C32×Q8—31no—
80#3828C22×C311yescharacteristic
81C3×#239—61no—
82C3×#239—61no—
83C3×D3×Q8—61no—
84C3×D3×Q8—61no—
85C3×D3×Q8—61no—
86C3×#237—61no—
87C3×#237—62no—
88C3×#235—21no—
89C3×#235—31no—
90C3×#235—31no—
91C3×#235—61no—
92C3×#235—62no—
93C3×#235—61no—
94C3×#235C2×D311yescharacteristic
95C3×#235C22×C311yescharacteristic
96C2×C3×Dic6—21no—
97C2×C3×Dic6—21no—
98C2×C3×Dic6—61no—
99C2×C3×Dic6C2×D311yescharacteristic
100C2×C3×Dic6C2×D311yescharacteristic
101C2×C3×Dic6C22×C311yescharacteristic
102C2×C3×Dic6C22×C311yescharacteristic
103#3804—22no—
104#3799—21no—
105#3797—21no—
106#3796—21no—
107D3×Dic6—21no—
108C32×M16—31no—
109C32×C4⋊C4—62no—
110C32×C42—21no—
111C32×C42—21no—
112C32×C42—21no—
113C32×C42—41no—
114C32×C42C2×D311yescharacteristic
115C32×C42C2×D311yescharacteristic
116C32×C42C22×C311yescharacteristic
117#3748C22×C311yescharacteristic
118C3×#216—61no—
119C3×#216—61no—
120C3×#216—61no—
121C3×#215—61no—
122C3×#215—61no—
123C3×#214—61no—
124C3×C4⋊Dic3—24no—
125C3×C4⋊Dic3—22no—
126C3×C4⋊Dic3—22no—
127C3×#208—21no—
128C3×#208—31no—
129C3×#208—31no—
130C3×#208—61no—
131C3×#208C2×D311yescharacteristic
132C3×#208C22×C311yescharacteristic
133C3×Dic12—61no—
134C3×#204—61no—
135#3719—21no—
136#3718—21no—
137#3717—21no—
138#3715—21no—
139C22×C33C2×D411yescharacteristic
140C2×C3×C3⋊D3—21no—
141C2×C32×D3—21no—
142C33×C4—22no—
143C33×C4C2×D411yescharacteristic
144C33×C4C2×D411yescharacteristic
145C3×C3⋊Dic3—21no—
146C32×Dic3—22no—
147C32×Dic3—21no—
148C32×Dic3—41no—
149C3×2−1+4—91no—
150#1010—31no—
151#1007—31no—
152C3×#137—91no—
153C3×#137—91no—
154C3×C4⋊Q8—31no—
155C3×C4⋊Q8—31no—
156C3×C4⋊Q8—61no—
157#951—31no—
158#942—31no—
159#942—31no—
160#909—31no—
161#869C3×D311yescharacteristic, complemented
162C3×#104—31no—
163C3×#104—31no—
164C3×#104—61no—
165C3×#104—91no—
166C3×#101—91no—
167#836—31no—
168#824—31no—
169#805—31no—
170#805—31no—
171#805—61no—
172#805C3×D311yescharacteristic, complemented
173C32×Q8—31no—
174C32×Q8—61no—
175C32×Q8—61no—
176C32×Q8—61no—
177C32×Q8—61no—
178C32×D4—31no—
179C32×D4—61no—
180C2×C32×C4—21no—
181C2×C32×C4—22no—
182C2×C32×C4—21no—
183C2×C32×C4—21no—
184C2×C32×C4—21no—
185C2×C32×C4—21no—
186C2×C32×C4—21no—
187C2×C32×C4—32no—
188C2×C32×C4—41no—
189C2×C32×C4—41no—
190C2×C32×C4—41no—
191C2×C32×C4—42no—
192C2×C32×C4—61no—
193C2×C32×C4—61no—
194C2×C32×C4C22×D311yescharacteristic
195C2×C32×C4C22×D311yescharacteristic
196C2×C32×C4C23×C311yescharacteristic
197#638—21no—
198C4⋊(C3⋊D3)C3×D411yescharacteristic
199C4×C3⋊D3—21no—
200#634C3×D411yescharacteristic
201C3×#67—21no—
202C3×#67—21no—
203C3×#67—41no—
204C3×#67—62no—
205C3×#67—61no—
206C3×#67—61no—
207C3×#67—121no—
208C2×C3×Dic3—22no—
209C2×C3×Dic3—21no—
210C2×C3×Dic3—21no—
211C2×C3×Dic3—22no—
212C2×C3×Dic3—62no—
213C2×C3×Dic3#6712yes—
214C2×C3×Dic3C3×D412yes—
215C3×D12—21no—
216C3×D12—31no—
217C3×D12—31no—
218C3×D12—61no—
219C3×D12—62no—
220C3×D12D1211yescharacteristic
221C3×D12C3×D411yescharacteristic
222C3×C4×D3—21no—
223C3×C4×D3—21no—
224C3×C4×D3—41no—
225C3×C4×D3—62no—
226C3×C4×D3—62no—
227C3×C4×D3—62no—
228C3×C4×D3—61no—
229C3×C4×D3—61no—
230C3×C4×D3—121no—
231C3×Dic6—21no—
232C3×Dic6—21no—
233C3×Dic6—21no—
234C3×Dic6—21no—
235C3×Dic6—21no—
236C3×Dic6—21no—
237C3×Dic6—21no—
238C3×Dic6—21no—
239C3×Dic6—21no—
240C3×Dic6—31no—
241C3×Dic6—31no—
242C3×Dic6—41no—
243C3×Dic6—41no—
244C3×Dic6—41no—
245C3×Dic6—41no—
246C3×Dic6—61no—
247C3×Dic6—62no—
248C3×Dic6—62no—
249C3×Dic6D1211yescharacteristic
250C3×Dic6C3×D411yescharacteristic
251#627—22no—
252#626—22no—
253#624—22no—
254D3×Dic3—22no—
255C32×C8—61no—
256C3⋊(C3⋊C8)—21no—
257C3×C3⋊C8—21no—
258C3×C3⋊C8—21no—
259C3×C3⋊C8—41no—
260C3×C3⋊C8—61no—
261C3×C3⋊C8—61no—
262C3×C3⋊C8—121no—
263#456—91noSylow
264C2×C33—21no—
265C2×C33#12011yescharacteristic
266C3×C3⋊D3—41no—
267C32×D3—41no—
268C3×#41—31no—
269C3×#41—31no—
270C3×#41—61no—
271C3×#41—91no—
272C3×#41—182no—
273C3×#41—181no—
274C3×#41—181no—
275C2×C3×Q8—31no—
276C2×C3×Q8—31no—
277C2×C3×Q8—31no—
278C2×C3×Q8—31no—
279C2×C3×Q8—61no—
280C2×C3×Q8—61no—
281C2×C3×Q8—181no—
282C2×C3×Q8—181no—
283#239—61no—
284#239—61no—
285D3×Q8—61no—
286D3×Q8—61no—
287D3×Q8—61no—
288#237—61no—
289#237—62no—
290#235—31no—
291#235—31no—
292#235—61no—
293#235—61no—
294#235—62no—
295#235C2×C3×D311yescharacteristic
296C2×Dic6—61no—
297C2×Dic6C2×C3×D311yescharacteristic
298C2×Dic6C2×C3×D311yescharacteristic
299C3×Q16—181no—
300C3×Q16—181no—
301C3×SD16—181no—
302C3×SD16—181no—
303C3×M16—31no—
304C3×M16—31no—
305C3×M16—61no—
306C3×M16—91no—
307C3×C4⋊C4—62no—
308C3×C4⋊C4—62no—
309C3×C4⋊C4—64no—
310C3×C42—21no—
311C3×C42—21no—
312C3×C42—21no—
313C3×C42—41no—
314C3×C42D3211yescharacteristic
315C3×C42C2×C3×D311yescharacteristic
316C3×C42C2×C3×D311yescharacteristic
317#216—61no—
318#216—61no—
319#216—61no—
320#215—61no—
321#215—61no—
322#214—61no—
323C4⋊Dic3—22no—
324#208—31no—
325#208—31no—
326#208—61no—
327#208C2×C3×D311yescharacteristic
328Dic12—61no—
329#204—61no—
330C22×C32—21no—
331C22×C32—21no—
332C22×C32—21no—
333C22×C32—41no—
334C22×C32—61no—
335C22×C32D3×D411yescharacteristic
336C22×C32C2×#6711yescharacteristic
337C22×C32C2×C3×D411yescharacteristic
338C2×C3⋊D3—21no—
339C2×C3×D3—21no—
340C2×C3×D3—21no—
341C2×C3×D3—41no—
342C2×C3×D3—61no—
343C2×C3×D3—61no—
344C2×C3×D3—121no—
345C32×C4—22no—
346C32×C4—24no—
347C32×C4—22no—
348C32×C4—22no—
349C32×C4—21no—
350C32×C4—21no—
351C32×C4—21no—
352C32×C4—21no—
353C32×C4—21no—
354C32×C4—21no—
355C32×C4—41no—
356C32×C4—41no—
357C32×C4—42no—
358C32×C4—42no—
359C32×C4—44no—
360C32×C4—62no—
361C32×C4—61no—
362C32×C4—121no—
363C32×C4C2×D1211yescharacteristic
364C32×C4D3×D411yescharacteristic
365C32×C4D3×D411yescharacteristic
366C32×C4D3×D411yescharacteristic
367C32×C4C2×C3×D411yescharacteristic
368C32×C4C2×C3×D411yescharacteristic
369C3⋊Dic3—21no—
370C3×Dic3—21no—
371C3×Dic3—22no—
372C3×Dic3—22no—
373C3×Dic3—21no—
374C3×Dic3—41no—
375C3×Dic3—42no—
376C3×Dic3—42no—
377C3×Dic3—41no—
378C3×Dic3—41no—
379C3×Dic3—61no—
380C3×Dic3—61no—
381C3×Dic3—121no—
3822−1+4—91no—
383#137—91no—
384#137—91no—
385C4⋊Q8—31no—
386#104—31no—
387#104—91no—
388#101—91no—
389C33#45611yescharacteristic, complemented, Sylow
390C3×Q8—31no—
391C3×Q8—31no—
392C3×Q8—61no—
393C3×Q8—61no—
394C3×Q8—61no—
395C3×Q8—61no—
396C3×Q8—61no—
397C3×Q8—61no—
398C3×Q8—61no—
399C3×Q8—61no—
400C3×Q8—61no—
401C3×Q8—91no—
402C3×Q8—121no—
403C3×Q8—121no—
404C3×Q8—121no—
405C3×Q8—121no—
406C3×Q8—182no—
407C3×D4—31no—
408C3×D4—31no—
409C3×D4—61no—
410C3×D4—61no—
411C3×D4—61no—
412C3×D4—91no—
413C3×D4—121no—
414C3×D4—182no—
415C3×D4—181no—
416C2×C3×C4—22no—
417C2×C3×C4—21no—
418C2×C3×C4—21no—
419C2×C3×C4—21no—
420C2×C3×C4—21no—
421C2×C3×C4—21no—
422C2×C3×C4—21no—
423C2×C3×C4—32no—
424C2×C3×C4—32no—
425C2×C3×C4—41no—
426C2×C3×C4—41no—
427C2×C3×C4—42no—
428C2×C3×C4—41no—
429C2×C3×C4—62no—
430C2×C3×C4—61no—
431C2×C3×C4—61no—
432C2×C3×C4—62no—
433C2×C3×C4—61no—
434C2×C3×C4—61no—
435C2×C3×C4—121no—
436C2×C3×C4—182no—
437C2×C3×C4—182no—
438C2×C3×C4—181no—
439C2×C3×C4C2×D3211yescharacteristic
440C2×C3×C4C22×C3×D311yescharacteristic
441C2×C3×C4C22×C3×D311yescharacteristic
442#67—21no—
443#67—62no—
444#67—61no—
445#67—61no—
446#67—121no—
447C2×Dic3—21no—
448C2×Dic3—62no—
449C2×Dic3C3×#6712yes—
450D12—31no—
451D12—31no—
452D12—61no—
453D12—62no—
454D12C3×D1211yescharacteristic
455C4×D3—21no—
456C4×D3—62no—
457C4×D3—62no—
458C4×D3—62no—
459C4×D3—61no—
460C4×D3—61no—
461C4×D3—121no—
462Dic6—21no—
463Dic6—21no—
464Dic6—21no—
465Dic6—21no—
466Dic6—31no—
467Dic6—31no—
468Dic6—61no—
469Dic6—62no—
470Dic6—62no—
471Dic6C3×D1211yescharacteristic
472C3×C8—61no—
473C3×C8—61no—
474C3×C8—121no—
475C3×C8—181no—
476C3⋊C8—21no—
477C3⋊C8—61no—
478C3⋊C8—61no—
479C3⋊C8—121no—
480C2×C32—21no—
481C2×C32—22no—
482C2×C32—21no—
483C2×C32—21no—
484C2×C32—21no—
485C2×C32—21no—
486C2×C32—21no—
487C2×C32—41no—
488C2×C32—41no—
489C2×C32—41no—
490C2×C32—42no—
491C2×C32—121no—
492C2×C32#88211yescharacteristic
493C2×C32#93711yescharacteristic
494C2×C32C3×#12011yescharacteristic
495C3⋊D3—41no—
496C3×D3—41no—
497C3×D3—41no—
498C3×D3—81no—
499C3×D3—121no—
500C3×D3—121no—
501C3×D3—241no—
502#41—31no—
503#41—91no—
504#41—182no—
505#41—181no—
506#41—181no—
507C2×Q8—31no—
508C2×Q8—31no—
509C2×Q8—181no—
510C2×Q8—181no—
511Q16—181no—
512Q16—181no—
513SD16—181no—
514SD16—181no—
515M16—31no—
516M16—91no—
517C4⋊C4—62no—
518C42C3×D3211yescharacteristic
519C22×C3—21no—
520C22×C3—21no—
521C22×C3—21no—
522C22×C3—41no—
523C22×C3—61no—
524C22×C3—61no—
525C22×C3—121no—
526C22×C3—181no—
527C22×C3D3×#6711yescharacteristic
528C22×C3C3×D3×D411yescharacteristic
529C22×C3C2×C3×#6711yescharacteristic
530C2×D3—21no—
531C2×D3—61no—
532C2×D3—61no—
533C2×D3—121no—
534C3×C4—22no—
535C3×C4—24no—
536C3×C4—22no—
537C3×C4—22no—
538C3×C4—21no—
539C3×C4—21no—
540C3×C4—21no—
541C3×C4—21no—
542C3×C4—21no—
543C3×C4—21no—
544C3×C4—42no—
545C3×C4—42no—
546C3×C4—44no—
547C3×C4—41no—
548C3×C4—41no—
549C3×C4—62no—
550C3×C4—61no—
551C3×C4—62no—
552C3×C4—61no—
553C3×C4—122no—
554C3×C4—121no—
555C3×C4—122no—
556C3×C4—121no—
557C3×C4—121no—
558C3×C4—181no—
559C3×C4D3×D1211yescharacteristic
560C3×C4#380211yescharacteristic
561C3×C4C2×C3×D1211yescharacteristic
562C3×C4C3×D3×D411yescharacteristic
563C3×C4C3×D3×D411yescharacteristic
564C3×C4C3×D3×D411yescharacteristic
565Dic3—21no—
566Dic3—22no—
567Dic3—41no—
568Dic3—61no—
569Dic3—61no—
570Dic3—121no—
571C32—21no—
572C32—21no—
573C32—21no—
574C32—41no—
575C32#482611yescharacteristic, complemented
576C32#509511yescharacteristic, complemented
577C32C3×#45611yescharacteristic, complemented
578Q8—31no—
579Q8—61no—
580Q8—61no—
581Q8—61no—
582Q8—61no—
583Q8—91no—
584Q8—182no—
585D4—31no—
586D4—61no—
587D4—91no—
588D4—182no—
589D4—181no—
590C2×C4—21no—
591C2×C4—32no—
592C2×C4—61no—
593C2×C4—61no—
594C2×C4—182no—
595C2×C4—182no—
596C2×C4—181no—
597C2×C4C2×C3×D3211yescharacteristic
598C8—61no—
599C8—181no—
600C2×C3—22no—
601C2×C3—21no—
602C2×C3—21no—
603C2×C3—21no—
604C2×C3—21no—
605C2×C3—21no—
606C2×C3—21no—
607C2×C3—41no—
608C2×C3—41no—
609C2×C3—42no—
610C2×C3—41no—
611C2×C3—121no—
612C2×C3—121no—
613C2×C3—241no—
614C2×C3—361no—
615C2×C3#6392211yescharacteristic
616C2×C3C3×#88211yescharacteristic
617C2×C3C3×#93711yescharacteristic
618D3—41no—
619D3—121no—
620D3—121no—
621D3—241no—
622C22—61no—
623C22—181no—
624C22C3×D3×#6711yescharacteristic
625C4—22no—
626C4—62no—
627C4—61no—
628C4—121no—
629C4—181no—
630C4C3×D3×D1211yescharacteristic
631C4C3×#380211yescharacteristic
632C3—21no—
633C3—21no—
634C3—21no—
635C3—41no—
636C3#1059039911yescharacteristic, direct product factor
637C3C3×#482611yescharacteristic, complemented
638C3C3×#509511yescharacteristic, complemented
639C2—21no—
640C2—121no—
641C2—361no—
642C2C3×#6392211yescharacteristic
subgroupG/H
centerC2×C3#63922
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C4C2×C3×D32
FittingC33×C42C22
socleC2×C33#120
prime radicalC3×#64297C2
hypercenterC3×C42D32
hypocenterC32C3×#456
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C2×C3 ◁ C2×C3×C4 ◁ C3×C42

Complements

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

NH ≅ G/Ntype
2. C32×#869C2semidirect
3. C3×#63833C2semidirect
4. C3×#63827C2semidirect
5. C3×#63632C2semidirect
6. C32×#805C2semidirect
7. C3×#63560C2semidirect
10. #10590399C3direct (N × H)
37. C3×#869D3semidirect
38. C3×#869C2×C3semidirect
39. #63833C2×C3semidirect
40. #63827C2×C3semidirect
42. #63632C2×C3semidirect
49. C3×#805D3semidirect
50. C3×#805C2×C3semidirect
51. #63560C2×C3semidirect
161. #869C3×D3semidirect
172. #805C3×D3semidirect
389. C33#456semidirect
575. C32#4826semidirect
576. C32#5095semidirect
577. C32C3×#456semidirect
636. C3#10590399direct (N × H)
637. C3C3×#4826semidirect
638. C3C3×#5095semidirect

Maximal subgroups

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

subgroupindex
C3×#642972
C32×#8692
C3×#638332
C3×#638272
C3×#636322
C32×#8052
C3×#635602
C3×#50953
C3×#48263
#105903993