Finite Group Databasebeta

← D3×#64160 #13178006

Subgroups — D3×#64160 #13178006

Order 1728.

1788 conjugacy classes of subgroups.

The subgroup lattice has 650 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C32×D3×#41C211yesmaximal, characteristic
2C2×D3×#634C211yesmaximal, characteristic, complemented
3#11723992C211yesmaximal, characteristic, complemented
4C3×#64160C211yesmaximal, characteristic, complemented
5#11723574C212yesmaximal, complemented
6D3×#3756C212yesmaximal, complemented
7#11723550C212yesmaximal, complemented
8D3×#3754C212yesmaximal, complemented
9#11723379C211yesmaximal, characteristic, complemented
10#11723373C211yesmaximal, characteristic, complemented
11D3×#3748C211yesmaximal, characteristic, complemented
12C2×#64160—31nomaximal
13D3×#951—34nomaximal
14C33×#41C2211yescharacteristic
15C2×#6308C2211yescharacteristic, complemented
16C2×C3×#634C2211yescharacteristic, complemented
17C32×#239C2211yescharacteristic
18C32×D3×Q8C2211yescharacteristic
19C32×#237—21no—
20C32×#237C2211yescharacteristic
21C32×D3×D4—21no—
22C32×D3×D4C2211yescharacteristic
23C32×#235—21no—
24C32×#235C2211yescharacteristic
25C2×C32×C4×D3—21no—
26C2×C32×C4×D3C2211yescharacteristic
27C2×#6293—21no—
28C2×D3×C3⋊Dic3—21no—
29D3×#634—21no—
30D3×#634—21no—
31D3×#634C2212yes—
32D3×#634C2212yescomplemented
33#88863C2212yescomplemented
34#88861C2212yescomplemented
35#88855C2211yescharacteristic, complemented
36C3×#3756C2212yescomplemented
37C3×#3754C2212yescomplemented
38C3×#3748C2211yescharacteristic, complemented
39#88799C2212yescomplemented
40#88798C2212yescomplemented
41#88794C2212yescomplemented
42#88792C2212yescomplemented
43#88786C2212yescomplemented
44D3×C3⋊(C3⋊C8)C2212yes—
45C2×C32×#41—31no—
46C22×#634—31no—
47C3×D3×#41D314yes—
48C2×D3×Dic6—34no—
49#64160—31no—
50#64160—31no—
51#64160—34no—
52#64160—32no—
53#64160—32no—
54#64160—31no—
55#64160—64no—
56#64160D311yescharacteristic, direct product factor
57C2×#3756—32no—
58C2×#3754—32no—
59C2×#3748—31no—
60C3×#951—34no—
61#63943—38no—
62D3×#216—38no—
63#63929—38no—
64D3×#214—38no—
65#63831—34no—
66#63827—34no—
67D3×#208—34no—
68C22×C32×D3—21no—
69C22×C32×D3D411yescharacteristic
70C33×Q8C2311yescharacteristic
71C33×D4—21no—
72C33×D4C2311yescharacteristic
73C2×C33×C4—21no—
74C2×C33×C4C2311yescharacteristic
75C2×C3⋊(C3⋊Dic3)—21no—
76#6308—21no—
77#6308C2312yes—
78C2×C3×C3⋊Dic3—21no—
79C3×#634—21no—
80C3×#634C2312yes—
81C32×#67—21no—
82C32×#67—21no—
83C32×#67—21no—
84C2×C32×Dic3—21no—
85C2×C32×Dic3D411yescharacteristic
86C32×D12—21no—
87C32×D12C2311yescharacteristic
88C32×C4×D3—21no—
89C32×C4×D3—21no—
90C32×C4×D3—21no—
91C32×C4×D3D411yescharacteristic
92C32×C4×D3D411yescharacteristic
93C32×C4×D3C2311yescharacteristic
94C32×C4×D3C2311yescharacteristic
95C32×Dic6—21no—
96C32×Dic6C2311yescharacteristic
97#6293—22no—
98#6293—22no—
99D3×C3⋊Dic3—22no—
100D3×C3⋊Dic3—22no—
101C3⋊(C3⋊(C3⋊C8))C2312yes—
102C3×C3⋊(C3⋊C8)C2312yes—
103C2×#951—94no—
104D3×#137—91no—
105C32×#41—24no—
106C32×#41—31no—
107C32×#41—31no—
108C32×#41—31no—
109C32×#41—61no—
110C32×#41—61no—
111C32×#41C2×D314yes—
112C32×#41C2×D311yescharacteristic
113C2×C32×Q8—31no—
114C2×C32×D4—31no—
115C2×C32×D4—61no—
116C22×C32×C4—31no—
117C22×C32×C4—61no—
118C22×C3⋊Dic3—61no—
119C2×#634—31no—
120C2×#634—34no—
121C2×#634—32no—
122C2×#634—32no—
123C2×#634—61no—
124C2×#634—61no—
125C2×#634—61no—
126C2×#634—64no—
127C2×#634C2×D311yescharacteristic, complemented
128C3×#239C2×D314yes—
129C3×D3×Q8C2×D314yes—
130C3×#237—24no—
131C3×#237C2×D314yes—
132C3×D3×D4—24no—
133C3×D3×D4C2×D314yes—
134C3×#235—24no—
135C3×#235C2×D314yes—
136C2×C3×C4×D3—24no—
137C2×C3×C4×D3C2×D314yes—
138C2×C3×Dic6—34no—
139C2×#627—64no—
140C2×D3×Dic3—64no—
141D3×Dic6—38no—
142D3×Dic6—38no—
143D3×Dic6—64no—
144D3×Dic6—64no—
145#3756—32no—
146#3756—38no—
147#3756—32no—
148#3756—32no—
149#3756—68no—
150#3756C2×D312yescomplemented
151#3754—32no—
152#3754—38no—
153#3754—32no—
154#3754—32no—
155#3754—68no—
156#3754C2×D312yescomplemented
157#3748—31no—
158#3748—34no—
159#3748—32no—
160#3748—64no—
161#3748C2×D311yescharacteristic, complemented
162C2×C3⋊(C3⋊C8)—32no—
163C3×#216—38no—
164C3×#214—38no—
165C3×#208—34no—
166#3719—38no—
167#3718—38no—
168#3716—38no—
169#3715—38no—
170#3711—38no—
171D3×C3⋊C8—38no—
172C22×C33—21no—
173C22×C33C2×D411yescharacteristic
174C2×C32×D3—22no—
175C2×C32×D3—21no—
176C2×C32×D3—42no—
177C2×C32×D3C2×D411yescharacteristic
178C2×C32×D3C2×D411yescharacteristic
179C33×C4—21no—
180C33×C4C2×D411yescharacteristic
181C33×C4C2411yescharacteristic
182C3⋊(C3⋊Dic3)—22no—
183C3×C3⋊Dic3—22no—
184C32×Dic3—21no—
185C32×Dic3C2×D411yescharacteristic
186C32×Dic3C2×D411yescharacteristic
187C2×C3×#41—34no—
188D3×#41C3⋊D311yescharacteristic
189C2×D3×Q8—91no—
190C22×Dic6—94no—
191C3×#137—91no—
192#951—34no—
193#951—94no—
194#951—98no—
195#951—98no—
196#951—94no—
197#951—94no—
198#951—91no—
199#951—184no—
200C2×#216—98no—
201#942—91no—
202C2×#214—98no—
203C2×#208—94no—
204#918—92no—
205D3×Q16—92no—
206#915—92no—
207D3×SD16—92no—
208#909—91no—
209D3×M16—91no—
210C23×C32—31no—
211C23×C32—61no—
212C22×C3×D3—24no—
213C22×C3×D3#6714yes—
214C32×Q8—24no—
215C32×Q8—31no—
216C32×Q8—61no—
217C32×Q8C22×D314yes—
218C32×Q8C22×D311yescharacteristic
219C32×D4—21no—
220C32×D4—24no—
221C32×D4—24no—
222C32×D4—31no—
223C32×D4—44no—
224C32×D4—61no—
225C32×D4—61no—
226C32×D4—61no—
227C32×D4—61no—
228C32×D4C22×D314yes—
229C32×D4C22×D311yescharacteristic
230C2×C32×C4—21no—
231C2×C32×C4—24no—
232C2×C32×C4—24no—
233C2×C32×C4—31no—
234C2×C32×C4—31no—
235C2×C32×C4—31no—
236C2×C32×C4—31no—
237C2×C32×C4—31no—
238C2×C32×C4—44no—
239C2×C32×C4—61no—
240C2×C32×C4—61no—
241C2×C32×C4—61no—
242C2×C32×C4—61no—
243C2×C32×C4C22×D314yes—
244C2×C32×C4C22×D311yescharacteristic
245C2×C3⋊Dic3—21no—
246C2×C3⋊Dic3—62no—
247C2×C3⋊Dic3—62no—
248C2×C3⋊Dic3—61no—
249C2×C3⋊Dic3—64no—
250C2×C3⋊Dic3—124no—
251#634—21no—
252#634—32no—
253#634—38no—
254#634—68no—
255#634—62no—
256#634—62no—
257#634—61no—
258#634—64no—
259#634—124no—
260#634C22×D312yes—
261C3×#67—24no—
262C3×#67—24no—
263C3×#67—24no—
264C2×C3×Dic3—24no—
265C2×C3×Dic3—64no—
266C2×C3×Dic3#6714yes—
267C3×D12—24no—
268C3×D12C22×D314yes—
269C3×C4×D3—24no—
270C3×C4×D3—24no—
271C3×C4×D3—24no—
272C3×C4×D3#6714yes—
273C3×C4×D3#6714yes—
274C3×C4×D3C22×D314yes—
275C3×C4×D3C22×D314yes—
276C3×Dic6—24no—
277C3×Dic6—38no—
278C3×Dic6—64no—
279C3×Dic6C22×D314yes—
280#627—68no—
281#627—68no—
282D3×Dic3—68no—
283D3×Dic3—68no—
284C3⋊(C3⋊C8)—32no—
285C3⋊(C3⋊C8)—38no—
286C3⋊(C3⋊C8)—68no—
287C3⋊(C3⋊C8)C22×D312yes—
288C3×C3⋊C8—38no—
289C2×#137—271noSylow
290C2×C33—21no—
291C2×C33—41no—
292C2×C33C22×D411yescharacteristic
293C32×D3—21no—
294C32×D3—41no—
295C32×D3#13712yescomplemented
296C3×#41—24no—
297C3×#41—34no—
298C3×#41—34no—
299C3×#41—34no—
300C3×#41—64no—
301C3×#41—64no—
302C3×#41D3214yes—
303C3×#41C2×C3⋊D311yescharacteristic
304C2×C3×Q8—34no—
305C2×C3×Q8—91no—
306C2×C3×D4—34no—
307C2×C3×D4—64no—
308C22×C3×C4—34no—
309C22×C3×C4—64no—
310C22×Dic3—184no—
311#239C2×C3⋊D311yescharacteristic
312D3×Q8—92no—
313D3×Q8—92no—
314D3×Q8—181no—
315D3×Q8—181no—
316D3×Q8C2×C3⋊D311yescharacteristic
317#237—21no—
318#237C2×C3⋊D311yescharacteristic
319D3×D4—21no—
320D3×D4C2×C3⋊D311yescharacteristic
321#235—21no—
322#235C2×C3⋊D311yescharacteristic
323C2×C4×D3—21no—
324C2×C4×D3—181no—
325C2×C4×D3C2×C3⋊D311yescharacteristic
326C2×Dic6—34no—
327C2×Dic6—98no—
328C2×Dic6—98no—
329C2×Dic6—94no—
330C2×Dic6—91no—
331C2×Dic6—184no—
332C2×Dic6—184no—
333C2×Dic6—184no—
334C2×Dic6—184no—
335C2×Dic6—181no—
336C3×Q16—92no—
337C3×SD16—92no—
338C3×M16—91no—
339#216—38no—
340#216—98no—
341#216—98no—
342#216—98no—
343#216—92no—
344#216—92no—
345#216—188no—
346#215—92no—
347#214—38no—
348#214—98no—
349#214—98no—
350#214—98no—
351#214—92no—
352#214—188no—
353#208—34no—
354#208—98no—
355#208—94no—
356#208—91no—
357#208—184no—
358C2×C3⋊C8—98no—
359Dic12—92no—
360#204—92no—
361ΓL(1,𝔽25)—92no—
362D3×C8—92no—
363C22×C32—21no—
364C22×C32—24no—
365C22×C32—24no—
366C22×C32—31no—
367C22×C32—31no—
368C22×C32—44no—
369C22×C32—62no—
370C22×C32—61no—
371C22×C32—122no—
372C22×C32D3×D411yescharacteristic
373C22×C32C2×#6714yes—
374C2×C3×D3—28no—
375C2×C3×D3—24no—
376C2×C3×D3—48no—
377C2×C3×D3C2×#6714yes—
378C2×C3×D3C2×#6714yes—
379C32×C4—21no—
380C32×C4—24no—
381C32×C4—24no—
382C32×C4—24no—
383C32×C4—31no—
384C32×C4—31no—
385C32×C4—44no—
386C32×C4—61no—
387C32×C4D3×D411yescharacteristic
388C32×C4C2×#6714yes—
389C32×C4C23×D314yes—
390C32×C4C23×D311yescharacteristic
391C3⋊Dic3—22no—
392C3⋊Dic3—62no—
393C3⋊Dic3—68no—
394C3⋊Dic3—128no—
395C3×Dic3—24no—
396C3×Dic3—68no—
397C3×Dic3C2×#6714yes—
398C3×Dic3C2×#6714yes—
399C2×#41—31no—
400C22×Q8—271no—
401#137—91no—
402#137—271no—
403#137—272no—
404#137—272no—
405#137—271no—
406#137—271no—
407C2×Q16—272no—
408C2×SD16—272no—
409C2×M16—271no—
410C33C2×#13711yescharacteristic, complemented, Sylow
411C23×C3—34no—
412C23×C3—64no—
413C22×D3—21no—
414C22×D3#63811yescharacteristic
415C3×Q8—24no—
416C3×Q8—34no—
417C3×Q8—64no—
418C3×Q8—92no—
419C3×Q8—181no—
420C3×Q8C2×D3214yes—
421C3×Q8C22×C3⋊D311yescharacteristic
422C3×D4—24no—
423C3×D4—21no—
424C3×D4—24no—
425C3×D4—34no—
426C3×D4—44no—
427C3×D4—64no—
428C3×D4—64no—
429C3×D4—64no—
430C3×D4—64no—
431C3×D4C2×D3214yes—
432C3×D4C22×C3⋊D311yescharacteristic
433C2×C3×C4—24no—
434C2×C3×C4—21no—
435C2×C3×C4—24no—
436C2×C3×C4—34no—
437C2×C3×C4—34no—
438C2×C3×C4—34no—
439C2×C3×C4—34no—
440C2×C3×C4—34no—
441C2×C3×C4—44no—
442C2×C3×C4—64no—
443C2×C3×C4—64no—
444C2×C3×C4—64no—
445C2×C3×C4—64no—
446C2×C3×C4—181no—
447C2×C3×C4C2×D3214yes—
448C2×C3×C4C22×C3⋊D311yescharacteristic
449#67—21no—
450#67—21no—
451#67—21no—
452C2×Dic3—21no—
453C2×Dic3—64no—
454C2×Dic3—188no—
455C2×Dic3—188no—
456C2×Dic3—184no—
457C2×Dic3—181no—
458C2×Dic3—364no—
459C2×Dic3#63811yescharacteristic
460D12—21no—
461D12C22×C3⋊D311yescharacteristic
462C4×D3—21no—
463C4×D3—21no—
464C4×D3—21no—
465C4×D3—182no—
466C4×D3—182no—
467C4×D3#63811yescharacteristic
468C4×D3#63811yescharacteristic
469C4×D3C22×C3⋊D311yescharacteristic
470C4×D3C22×C3⋊D311yescharacteristic
471Dic6—21no—
472Dic6—38no—
473Dic6—64no—
474Dic6—98no—
475Dic6—92no—
476Dic6—188no—
477Dic6—188no—
478Dic6—188no—
479Dic6—184no—
480Dic6—181no—
481Dic6—182no—
482Dic6—182no—
483Dic6—364no—
484Dic6C22×C3⋊D311yescharacteristic
485C3×C8—92no—
486C3⋊C8—38no—
487C3⋊C8—98no—
488C3⋊C8—92no—
489C3⋊C8—188no—
490C2×C32—21no—
491C2×C32—24no—
492C2×C32—24no—
493C2×C32—32no—
494C2×C32—44no—
495C2×C32—41no—
496C2×C32—44no—
497C2×C32—61no—
498C2×C32—84no—
499C2×C32—121no—
500C2×C32C2×D3×D411yescharacteristic
501C2×C32C22×#6714yes—
502C3×D3—24no—
503C3×D3—44no—
504C3×D3#95118yescomplemented
505#41—31no—
506#41—31no—
507#41—31no—
508#41—61no—
509#41—61no—
510#41D3×C3⋊D311yescharacteristic
511C2×Q8—31no—
512C2×Q8—91no—
513C2×Q8—272no—
514C2×Q8—272no—
515C2×Q8—271no—
516C2×Q8—541no—
517C2×Q8—541no—
518C2×Q8—541no—
519C2×D4—31no—
520C2×D4—61no—
521C22×C4—31no—
522C22×C4—61no—
523C22×C4—541no—
524Q16—92no—
525Q16—272no—
526Q16—272no—
527Q16—272no—
528SD16—92no—
529SD16—272no—
530SD16—272no—
531SD16—272no—
532M16—91no—
533M16—272no—
534M16—271no—
535C2×C8—272no—
536C22×C3—24no—
537C22×C3—21no—
538C22×C3—24no—
539C22×C3—34no—
540C22×C3—34no—
541C22×C3—44no—
542C22×C3—68no—
543C22×C3—64no—
544C22×C3—128no—
545C22×C3D3×#6714yes—
546C22×C3C2×#63811yescharacteristic
547C2×D3—22no—
548C2×D3—21no—
549C2×D3—42no—
550C2×D3C2×#63811yescharacteristic
551C2×D3C2×#63811yescharacteristic
552C3×C4—24no—
553C3×C4—21no—
554C3×C4—24no—
555C3×C4—24no—
556C3×C4—34no—
557C3×C4—34no—
558C3×C4—44no—
559C3×C4—64no—
560C3×C4—182no—
561C3×C4D3×#6714yes—
562C3×C4C2×#63811yescharacteristic
563C3×C4C22×D3214yes—
564C3×C4C23×C3⋊D311yescharacteristic
565Dic3—21no—
566Dic3—68no—
567Dic3—188no—
568Dic3—182no—
569Dic3—368no—
570Dic3C2×#63811yescharacteristic
571Dic3C2×#63811yescharacteristic
572C32—24no—
573C32D3×#13711yescharacteristic, complemented
574C32C2×#95114yescomplemented
575C23—31no—
576C23—61no—
577Q8—31no—
578Q8—61no—
579Q8—92no—
580Q8—181no—
581Q8—272no—
582Q8—542no—
583Q8—542no—
584Q8—541no—
585Q8C2×D3×C3⋊D311yescharacteristic
586D4—21no—
587D4—31no—
588D4—61no—
589D4—61no—
590D4—61no—
591D4—61no—
592D4C2×D3×C3⋊D311yescharacteristic
593C2×C4—21no—
594C2×C4—31no—
595C2×C4—31no—
596C2×C4—31no—
597C2×C4—31no—
598C2×C4—31no—
599C2×C4—61no—
600C2×C4—61no—
601C2×C4—61no—
602C2×C4—61no—
603C2×C4—181no—
604C2×C4—542no—
605C2×C4—542no—
606C2×C4—541no—
607C2×C4C2×D3×C3⋊D311yescharacteristic
608C8—92no—
609C8—272no—
610C2×C3—24no—
611C2×C3—21no—
612C2×C3—24no—
613C2×C3—38no—
614C2×C3—44no—
615C2×C3—44no—
616C2×C3—41no—
617C2×C3—64no—
618C2×C3—84no—
619C2×C3—124no—
620C2×C3C2×D3×#6714yes—
621C2×C3C22×#63811yescharacteristic
622D3—21no—
623D3—41no—
624D3#6416012yesdirect product factor
625C22—21no—
626C22—31no—
627C22—31no—
628C22—62no—
629C22—61no—
630C22—122no—
631C22D3×#63811yescharacteristic
632C4—21no—
633C4—31no—
634C4—31no—
635C4—61no—
636C4—182no—
637C4—542no—
638C4D3×#63811yescharacteristic
639C4C22×D3×C3⋊D311yescharacteristic
640C3—24no—
641C3D3×#95114yescomplemented
642C3C2×#6416011yescharacteristic, complemented
643C2—21no—
644C2—32no—
645C2—41no—
646C2—61no—
647C2—121no—
648C2C2×D3×#63811yescharacteristic
subgroupG/H
centerC2C2×D3×#638
commutator, cosocleC33×C4C24
FrattiniC4C22×D3×C3⋊D3
FittingC33×#41C22
socleC2×C33C22×D4
prime radicalC32×D3×#41C2
hypercenter#41D3×C3⋊D3
hypocenterC33C2×#137
LCSG ▷ C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C2 ◁ C2×C4 ◁ #41

Complements

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

NH ≅ G/Ntype
2. C2×D3×#634C2semidirect
3. #11723992C2semidirect
4. C3×#64160C2semidirect
5. #11723574C2semidirect
6. D3×#3756C2semidirect
7. #11723550C2semidirect
8. D3×#3754C2semidirect
9. #11723379C2semidirect
10. #11723373C2semidirect
11. D3×#3748C2semidirect
15. C2×#6308C22semidirect
16. C2×C3×#634C22semidirect
32. D3×#634C22semidirect
33. #88863C22semidirect
34. #88861C22semidirect
35. #88855C22semidirect
36. C3×#3756C22semidirect
37. C3×#3754C22semidirect
38. C3×#3748C22semidirect
39. #88799C22semidirect
40. #88798C22semidirect
41. #88794C22semidirect
42. #88792C22semidirect
43. #88786C22semidirect
56. #64160D3direct (N × H)
127. C2×#634C2×D3semidirect
150. #3756C2×D3semidirect
156. #3754C2×D3semidirect
161. #3748C2×D3semidirect
295. C32×D3#137semidirect
410. C33C2×#137semidirect
504. C3×D3#951semidirect
573. C32D3×#137semidirect
574. C32C2×#951semidirect
624. D3#64160direct (N × H)
641. C3D3×#951semidirect
642. C3C2×#64160semidirect

Maximal subgroups

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

subgroupindex
C32×D3×#412
C2×D3×#6342
#117239922
C3×#641602
#117235742
D3×#37562
#117235502
D3×#37542
#117233792
#117233732
D3×#37482
C2×#641603
D3×#9513