Finite Group Databasebeta

← C3×#10594769 #13174366

Subgroups — C3×#10594769 #13174366

Order 1728.

1190 conjugacy classes of subgroups.

The subgroup lattice has 656 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#64315C212yesmaximal, complemented
2C3×Q16×C3⋊D3C211yesmaximal, characteristic, complemented
3C32×#918C212yesmaximal, complemented
4C3×#63951C212yesmaximal, complemented
5C3×#63947C212yesmaximal, complemented
6C3×D3×#216C212yesmaximal, complemented
7C3×#63809C211yesmaximal, characteristic, complemented
8C3×#63805C211yesmaximal, characteristic, complemented
9C3×#63800C212yesmaximal, complemented
10C3×#5779—32nomaximal
11#10594769C311yesmaximal, characteristic, direct product factor
12C3×Q8×C3⋊D3C2212yes—
13C32×#239C2212yes—
14C32×D3×Q8C2212yes—
15C3×#3798—22no—
16C3×#3798C2211yescharacteristic
17C3×#3796—22no—
18C3×#3796C2212yes—
19C3×#3795—22no—
20C3×#3795C2212yes—
21C3×D3×Dic6—22no—
22C3×D3×Dic6C2212yes—
23C33×Q16C2211yescharacteristic
24C3×#3756C2212yes—
25C3×#3746C2211yescharacteristic
26C3×C8×C3⋊D3C2211yescharacteristic
27C32×#216C2212yes—
28C32×#215C2212yes—
29C32×#204C2212yes—
30C32×ΓL(1,𝔽25)C2212yes—
31C3×#3719C2212yes—
32C3×#3718C2211yescharacteristic
33C3×#3716C2212yes—
34C3×#3713C2211yescharacteristic
35C3×#3712C2211yescharacteristic
36C3×D3×C3⋊C8C2212yes—
37C3×#1010—32no—
38C3×#1007—32no—
39#64315C2×C312yescomplemented
40C32×#137—32no—
41Q16×C3⋊D3C2×C311yescharacteristic, complemented
42C3×#950—32no—
43C2×C3×#216—32no—
44C3×#918—22no—
45C3×#918—32no—
46C3×#918D312yescomplemented
47C3×#918C2×C312yescomplemented
48C3×D3×Q16—32no—
49C3×D3×Q16—32no—
50C3×D3×Q16—61no—
51C3×#916—32no—
52C3×#916—32no—
53C3×#915—32no—
54C3×#915—32no—
55C3×#909—32no—
56C3×#907—32no—
57#63951C2×C312yescomplemented
58#63947C2×C312yescomplemented
59D3×#216C2×C312yescomplemented
60#63809C2×C311yescharacteristic, complemented
61#63805C2×C311yescharacteristic, complemented
62#63800C2×C312yescomplemented
63C33×Q8C2312yes—
64C3×C4×C3⋊D3—22no—
65C3×C4×C3⋊D3C2311yescharacteristic
66C3×#634—22no—
67C3×#634C2312yes—
68C32×D12—22no—
69C32×D12C2312yes—
70C32×C4×D3—22no—
71C32×C4×D3—22no—
72C32×C4×D3C2312yes—
73C32×Dic6—22no—
74C32×Dic6C2312yes—
75C3×#627—22no—
76C3×#627D411yescharacteristic
77C3×#626—22no—
78C3×#626D412yes—
79C3×#625—22no—
80C3×#625D411yescharacteristic
81C3×D3×Dic3—22no—
82C3×D3×Dic3—22no—
83C3×D3×Dic3—22no—
84C33×C8C2311yescharacteristic
85C3×C3⋊(C3⋊C8)C2311yescharacteristic
86C32×C3⋊C8C2312yes—
87C3×#578—91no—
88#5779—32no—
89C32×#41—32no—
90C2×C32×Q8—32no—
91Q8×C3⋊D3C22×C312yes—
92C3×#239—22no—
93C3×#239—32no—
94C3×#239—62no—
95C3×#239C2×D312yes—
96C3×#239C22×C312yes—
97C3×D3×Q8—22no—
98C3×D3×Q8—32no—
99C3×D3×Q8—32no—
100C3×D3×Q8—32no—
101C3×D3×Q8—62no—
102C3×D3×Q8—62no—
103C3×D3×Q8C2×D312yes—
104C3×D3×Q8C22×C312yes—
105C3×#237—32no—
106C3×#237—32no—
107C3×#237—62no—
108C3×#237—62no—
109C3×#235—32no—
110C3×#235—32no—
111C3×#235—62no—
112C3×#235—62no—
113C3×#235—62no—
114C2×C3×Dic6—32no—
115C2×C3×Dic6—62no—
116#3798—22no—
117#3798C22×C311yescharacteristic
118#3796—22no—
119#3796C22×C312yes—
120#3795—22no—
121#3795C22×C312yes—
122D3×Dic6—22no—
123D3×Dic6C22×C312yes—
124C32×Q16—21no—
125C32×Q16—22no—
126C32×Q16—32no—
127C32×Q16—41no—
128C32×Q16C2×D312yes—
129C32×Q16C22×C311yescharacteristic
130C32×SD16—32no—
131C32×SD16—32no—
132C32×M16—32no—
133#3756C22×C312yes—
134#3746C22×C311yescharacteristic
135C8×C3⋊D3C22×C311yescharacteristic
136C3×#216—22no—
137C3×#216—32no—
138C3×#216—32no—
139C3×#216—32no—
140C3×#216—32no—
141C3×#216—62no—
142C3×#216C2×D312yes—
143C3×#216C22×C312yes—
144C3×#215—22no—
145C3×#215C2×D312yes—
146C3×#215C22×C312yes—
147C3×#214—32no—
148C3×#213—32no—
149C2×C3×C3⋊C8—32no—
150C3×Dic12—32no—
151C3×Dic12—32no—
152C3×Dic12—61no—
153C3×#204—22no—
154C3×#204—32no—
155C3×#204C2×D312yes—
156C3×#204C22×C312yes—
157C3×ΓL(1,𝔽25)—22no—
158C3×ΓL(1,𝔽25)—32no—
159C3×ΓL(1,𝔽25)C2×D312yes—
160C3×ΓL(1,𝔽25)C22×C312yes—
161C3×D3×C8—32no—
162C3×D3×C8—32no—
163C3×D3×C8—61no—
164#3719C22×C312yes—
165#3718C22×C311yescharacteristic
166#3716C22×C312yes—
167#3713C22×C311yescharacteristic
168#3712C22×C311yescharacteristic
169D3×C3⋊C8C22×C312yes—
170C2×C3×C3⋊D3C2×D411yescharacteristic
171C2×C32×D3—22no—
172C2×C32×D3C2×D412yes—
173C33×C4—22no—
174C33×C4C2411yescharacteristic
175C3×C3⋊Dic3—22no—
176C3×C3⋊Dic3C2×D411yescharacteristic
177C32×Dic3—22no—
178C32×Dic3C2×D412yes—
179C3×2−1+4—92no—
180#1010—32no—
181#1007—32no—
182C3×#137—32no—
183C3×#137—32no—
184C3×#137—62no—
185C3×#137—92no—
186C3×#137—92no—
187C3×#135—91no—
188C3×#135—92no—
189C2×C3×Q16—92no—
190C2×C3×Q16—91no—
191C3×#131—91no—
192#950—32no—
193C2×#216—32no—
194#918—32no—
195#918C3×D312yescomplemented
196D3×Q16—32no—
197D3×Q16—32no—
198D3×Q16—61no—
199#916—32no—
200#916—32no—
201#915—32no—
202#915—32no—
203#909—32no—
204#907—32no—
205C32×Q8—22no—
206C32×Q8—22no—
207C32×Q8—22no—
208C32×Q8—32no—
209C32×Q8—42no—
210C32×Q8—62no—
211C32×Q8C22×D312yes—
212C32×Q8C22×D312yes—
213C32×Q8C23×C312yes—
214C32×D4—32no—
215C32×D4—62no—
216C2×C32×C4—32no—
217C2×C32×C4—62no—
218C2×C32×C4—62no—
219C4×C3⋊D3—22no—
220C4×C3⋊D3C23×C311yescharacteristic
221#634—22no—
222#634C23×C312yes—
223C3×#67—32no—
224C3×#67—32no—
225C3×#67—62no—
226C3×#67—62no—
227C3×#67—62no—
228C2×C3×Dic3—62no—
229C2×C3×Dic3—62no—
230C2×C3×Dic3—62no—
231C3×D12—22no—
232C3×D12—22no—
233C3×D12—22no—
234C3×D12—32no—
235C3×D12—42no—
236C3×D12—62no—
237C3×D12C22×D312yes—
238C3×D12C23×C312yes—
239C3×C4×D3—22no—
240C3×C4×D3—22no—
241C3×C4×D3—22no—
242C3×C4×D3—22no—
243C3×C4×D3—22no—
244C3×C4×D3—32no—
245C3×C4×D3—42no—
246C3×C4×D3—42no—
247C3×C4×D3—61no—
248C3×C4×D3—62no—
249C3×C4×D3—62no—
250C3×C4×D3—62no—
251C3×C4×D3—62no—
252C3×C4×D3—62no—
253C3×C4×D3—122no—
254C3×C4×D3C22×D312yes—
255C3×C4×D3C23×C312yes—
256C3×Dic6—22no—
257C3×Dic6—22no—
258C3×Dic6—22no—
259C3×Dic6—32no—
260C3×Dic6—32no—
261C3×Dic6—32no—
262C3×Dic6—42no—
263C3×Dic6—62no—
264C3×Dic6—62no—
265C3×Dic6—62no—
266C3×Dic6—62no—
267C3×Dic6—62no—
268C3×Dic6—122no—
269C3×Dic6C22×D312yes—
270C3×Dic6C23×C312yes—
271#627—22no—
272#627C3×D411yescharacteristic
273#626—22no—
274#626C3×D412yes—
275#625—22no—
276#625C3×D411yescharacteristic
277D3×Dic3—22no—
278D3×Dic3—22no—
279D3×Dic3—22no—
280C32×C8—21no—
281C32×C8—22no—
282C32×C8—32no—
283C32×C8—41no—
284C32×C8C22×D312yes—
285C32×C8C23×C311yescharacteristic
286C3⋊(C3⋊C8)C23×C311yescharacteristic
287C3×C3⋊C8—22no—
288C3×C3⋊C8—32no—
289C3×C3⋊C8—61no—
290C3×C3⋊C8C22×D312yes—
291C3×C3⋊C8C23×C312yes—
292#578—91noSylow
293C2×C33C22×D411yescharacteristic
294C3×C3⋊D3—21no—
295C32×D3—22no—
296C32×D3—42no—
297C3×#41—32no—
298C3×#41—32no—
299C3×#41—62no—
300C3×#41—91no—
301C3×#41—92no—
302C3×#41—92no—
303C3×#41—182no—
304C3×#41—182no—
305C3×#41—182no—
306C2×C3×Q8—32no—
307C2×C3×Q8—32no—
308C2×C3×Q8—62no—
309C2×C3×Q8—92no—
310C2×C3×Q8—92no—
311C2×C3×Q8—182no—
312#239—32no—
313#239—62no—
314#239C2×C3×D312yes—
315D3×Q8—32no—
316D3×Q8—32no—
317D3×Q8—32no—
318D3×Q8—62no—
319D3×Q8—62no—
320D3×Q8C2×C3×D312yes—
321#237—32no—
322#237—32no—
323#237—62no—
324#237—62no—
325#235—32no—
326#235—32no—
327#235—62no—
328#235—62no—
329#235—62no—
330C2×Dic6—32no—
331C2×Dic6—62no—
332C3×Q16—21no—
333C3×Q16—22no—
334C3×Q16—32no—
335C3×Q16—32no—
336C3×Q16—41no—
337C3×Q16—62no—
338C3×Q16—92no—
339C3×Q16—91no—
340C3×Q16—92no—
341C3×Q16—91no—
342C3×Q16D3211yescharacteristic
343C3×Q16C2×C3×D312yes—
344C3×SD16—32no—
345C3×SD16—32no—
346C3×SD16—32no—
347C3×SD16—32no—
348C3×SD16—62no—
349C3×SD16—62no—
350C3×SD16—92no—
351C3×D8—91no—
352C3×M16—32no—
353C3×M16—32no—
354C3×M16—62no—
355C3×M16—91no—
356C2×C3×C8—92no—
357C2×C3×C8—91no—
358#216—32no—
359#216—32no—
360#216—32no—
361#216—32no—
362#216—62no—
363#216C2×C3×D312yes—
364#215C2×C3×D312yes—
365#214—32no—
366#213—32no—
367C2×C3⋊C8—32no—
368Dic12—32no—
369Dic12—32no—
370Dic12—61no—
371#204—32no—
372#204C2×C3×D312yes—
373ΓL(1,𝔽25)—32no—
374ΓL(1,𝔽25)C2×C3×D312yes—
375D3×C8—32no—
376D3×C8—32no—
377D3×C8—61no—
378C22×C32—32no—
379C22×C32—62no—
380C2×C3⋊D3C2×C3×D411yescharacteristic
381C2×C3×D3—22no—
382C2×C3×D3—22no—
383C2×C3×D3—22no—
384C2×C3×D3—32no—
385C2×C3×D3—42no—
386C2×C3×D3—61no—
387C2×C3×D3D3×D412yes—
388C2×C3×D3C2×C3×D412yes—
389C32×C4—22no—
390C32×C4—22no—
391C32×C4—22no—
392C32×C4—21no—
393C32×C4—22no—
394C32×C4—32no—
395C32×C4—41no—
396C32×C4—42no—
397C32×C4—42no—
398C32×C4—42no—
399C32×C4—62no—
400C32×C4—82no—
401C32×C4C23×D312yes—
402C32×C4C24×C311yescharacteristic
403C3⋊Dic3—22no—
404C3⋊Dic3C2×C3×D411yescharacteristic
405C3×Dic3—22no—
406C3×Dic3—22no—
407C3×Dic3—22no—
408C3×Dic3—32no—
409C3×Dic3—42no—
410C3×Dic3—61no—
411C3×Dic3—62no—
412C3×Dic3—62no—
413C3×Dic3—122no—
414C3×Dic3D3×D412yes—
415C3×Dic3C2×C3×D412yes—
4162−1+4—92no—
417#137—32no—
418#137—92no—
419#137—92no—
420#135—91no—
421#135—92no—
422C2×Q16—92no—
423C2×Q16—91no—
424#131—91no—
425C33#57811yescharacteristic, complemented, Sylow
426C3×Q8—22no—
427C3×Q8—22no—
428C3×Q8—22no—
429C3×Q8—32no—
430C3×Q8—32no—
431C3×Q8—42no—
432C3×Q8—62no—
433C3×Q8—62no—
434C3×Q8—62no—
435C3×Q8—91no—
436C3×Q8—92no—
437C3×Q8—122no—
438C3×Q8—182no—
439C3×Q8—182no—
440C3×Q8C2×D3212yes—
441C3×Q8C22×C3×D312yes—
442C3×Q8C22×C3×D312yes—
443C3×D4—32no—
444C3×D4—32no—
445C3×D4—62no—
446C3×D4—62no—
447C3×D4—62no—
448C3×D4—92no—
449C3×D4—91no—
450C3×D4—122no—
451C3×D4—182no—
452C3×D4—182no—
453C2×C3×C4—32no—
454C2×C3×C4—32no—
455C2×C3×C4—62no—
456C2×C3×C4—62no—
457C2×C3×C4—62no—
458C2×C3×C4—62no—
459C2×C3×C4—62no—
460C2×C3×C4—91no—
461C2×C3×C4—122no—
462C2×C3×C4—122no—
463C2×C3×C4—182no—
464C2×C3×C4—182no—
465C2×C3×C4—182no—
466C2×C3×C4—182no—
467#67—32no—
468#67—32no—
469#67—62no—
470#67—62no—
471#67—62no—
472C2×Dic3—62no—
473C2×Dic3—62no—
474C2×Dic3—62no—
475D12—22no—
476D12—32no—
477D12—62no—
478D12C22×C3×D312yes—
479C4×D3—22no—
480C4×D3—22no—
481C4×D3—32no—
482C4×D3—61no—
483C4×D3—62no—
484C4×D3—62no—
485C4×D3—62no—
486C4×D3—62no—
487C4×D3—62no—
488C4×D3—122no—
489C4×D3C22×C3×D312yes—
490Dic6—22no—
491Dic6—32no—
492Dic6—32no—
493Dic6—32no—
494Dic6—62no—
495Dic6—62no—
496Dic6—62no—
497Dic6—62no—
498Dic6—62no—
499Dic6—122no—
500Dic6C22×C3×D312yes—
501C3×C8—22no—
502C3×C8—21no—
503C3×C8—32no—
504C3×C8—32no—
505C3×C8—41no—
506C3×C8—62no—
507C3×C8—91no—
508C3×C8C2×D3211yescharacteristic
509C3×C8C22×C3×D312yes—
510C3⋊C8—32no—
511C3⋊C8—61no—
512C3⋊C8C22×C3×D312yes—
513C2×C32—21no—
514C2×C32—22no—
515C2×C32—41no—
516C2×C32—62no—
517C2×C32—122no—
518C2×C32C2×D3×D412yes—
519C2×C32C22×C3×D411yescharacteristic
520C3⋊D3—21no—
521C3×D3—22no—
522C3×D3—22no—
523C3×D3—42no—
524C3×D3—42no—
525C3×D3—42no—
526C3×D3—62no—
527C3×D3—62no—
528C3×D3—82no—
529#41—32no—
530#41—91no—
531#41—92no—
532#41—92no—
533#41—182no—
534#41—182no—
535#41—182no—
536C2×Q8—32no—
537C2×Q8—92no—
538C2×Q8—92no—
539C2×Q8—182no—
540Q16—32no—
541Q16—92no—
542Q16—91no—
543Q16—92no—
544Q16—91no—
545Q16C3×D3211yescharacteristic
546SD16—32no—
547SD16—32no—
548SD16—92no—
549D8—91no—
550M16—32no—
551M16—91no—
552C2×C8—92no—
553C2×C8—91no—
554C22×C3—32no—
555C22×C3—32no—
556C22×C3—62no—
557C22×C3—62no—
558C22×C3—62no—
559C22×C3—91no—
560C22×C3—122no—
561C2×D3—22no—
562C2×D3—32no—
563C2×D3—61no—
564C2×D3C3×D3×D412yes—
565C3×C4—22no—
566C3×C4—22no—
567C3×C4—22no—
568C3×C4—21no—
569C3×C4—22no—
570C3×C4—32no—
571C3×C4—32no—
572C3×C4—42no—
573C3×C4—42no—
574C3×C4—42no—
575C3×C4—41no—
576C3×C4—62no—
577C3×C4—62no—
578C3×C4—62no—
579C3×C4—82no—
580C3×C4—91no—
581C3×C4—122no—
582C3×C4—182no—
583C3×C4C22×D3211yescharacteristic
584C3×C4C23×C3×D312yes—
585Dic3—22no—
586Dic3—32no—
587Dic3—61no—
588Dic3—62no—
589Dic3—62no—
590Dic3—122no—
591Dic3C3×D3×D412yes—
592C32—22no—
593C32—21no—
594C32—41no—
595C32#577912yescomplemented
596C32C3×#57811yescharacteristic, complemented
597Q8—32no—
598Q8—62no—
599Q8—91no—
600Q8—92no—
601Q8—182no—
602Q8—182no—
603Q8C2×C3×D3212yes—
604D4—32no—
605D4—62no—
606D4—92no—
607D4—91no—
608D4—182no—
609D4—182no—
610C2×C4—32no—
611C2×C4—62no—
612C2×C4—62no—
613C2×C4—91no—
614C2×C4—182no—
615C2×C4—182no—
616C2×C4—182no—
617C2×C4—182no—
618C8—32no—
619C8—91no—
620C8C2×C3×D3211yescharacteristic
621C2×C3—21no—
622C2×C3—22no—
623C2×C3—41no—
624C2×C3—62no—
625C2×C3—62no—
626C2×C3—122no—
627C2×C3—122no—
628C2×C3—122no—
629C2×C3—181no—
630C2×C3—242no—
631C2×C3D32×D411yescharacteristic
632C2×C3C2×C3×D3×D412yes—
633D3—22no—
634D3—42no—
635D3—62no—
636D3—62no—
637C22—32no—
638C22—62no—
639C22—91no—
640C4—22no—
641C4—32no—
642C4—62no—
643C4—91no—
644C4—182no—
645C4C22×C3×D3211yescharacteristic
646C3—21no—
647C3—22no—
648C3—41no—
649C3#1059476911yescharacteristic, direct product factor
650C3C3×#577912yescomplemented
651C2—62no—
652C2—122no—
653C2—181no—
654C2C3×D32×D411yescharacteristic
subgroupG/H
centerC2×C3D32×D4
commutator, cosocleC32×C4C24×C3
FrattiniC4C22×C3×D32
FittingC33×Q16C22
socleC2×C33C22×D4
hypercenterC3×Q16D32
hypocenterC32C3×#578
LCSG ▷ C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C2×C3 ◁ C3×C4 ◁ C3×Q16

Complements

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

NH ≅ G/Ntype
1. C3×#64315C2semidirect
2. C3×Q16×C3⋊D3C2semidirect
3. C32×#918C2semidirect
4. C3×#63951C2semidirect
5. C3×#63947C2semidirect
6. C3×D3×#216C2semidirect
7. C3×#63809C2semidirect
8. C3×#63805C2semidirect
9. C3×#63800C2semidirect
11. #10594769C3direct (N × H)
39. #64315C2×C3semidirect
41. Q16×C3⋊D3C2×C3semidirect
46. C3×#918D3semidirect
47. C3×#918C2×C3semidirect
57. #63951C2×C3semidirect
58. #63947C2×C3semidirect
59. D3×#216C2×C3semidirect
60. #63809C2×C3semidirect
61. #63805C2×C3semidirect
62. #63800C2×C3semidirect
195. #918C3×D3semidirect
425. C33#578semidirect
595. C32#5779semidirect
596. C32C3×#578semidirect
649. C3#10594769direct (N × H)
650. C3C3×#5779semidirect

Maximal subgroups

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

subgroupindex
C3×#643152
C3×Q16×C3⋊D32
C32×#9182
C3×#639512
C3×#639472
C3×D3×#2162
C3×#638092
C3×#638052
C3×#638002
C3×#57793
#105947693