Finite Group Databasebeta

← D3×#63470 #13149850

Subgroups — D3×#63470 #13149850

Order 1728.

962 conjugacy classes of subgroups.

The subgroup lattice has 696 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1D3×#3697C211yesmaximal, characteristic, complemented
2C2×D3×D36C211yesmaximal, characteristic, complemented
3#11721783C211yesmaximal, characteristic, complemented
4D3×C9×M16C211yesmaximal, characteristic, complemented
5C3×#63470C211yesmaximal, characteristic, complemented
6#11720873C211yesmaximal, characteristic, complemented
7#11720868C211yesmaximal, characteristic, complemented
8#11720805C212yesmaximal, complemented
9#11720804C212yesmaximal, complemented
10D3×D72C212yesmaximal, complemented
11D3×#3664C212yesmaximal, complemented
12D3×#908—31nomaximal
13C2×#63470—31nomaximal
14C22×D3×D9—21no—
15#88736C2211yescharacteristic, complemented
16C2×C4⋊(C3⋊D9)C2211yescharacteristic, complemented
17C2×C4×D3×C9C2211yescharacteristic
18C3×#3697C2211yescharacteristic, complemented
19C2×C3×D36C2211yescharacteristic, complemented
20D3×#611—21no—
21C2×#6192—21no—
22#88658—21no—
23D3×D36—21no—
24D3×D36—21no—
25D3×D36C2212yes—
26D3×D36C2212yescomplemented
27D3×D36C2211yescharacteristic, complemented
28C4×D3×D9—21no—
29#88642C2211yescharacteristic, complemented
30#88641C2211yescharacteristic, complemented
31#88640—21no—
32D3×Dic18C2211yescharacteristic, complemented
33C3×C9×M16C2211yescharacteristic, complemented
34#88525C2212yescomplemented
35#88524C2212yescomplemented
36C9×#208C2211yescharacteristic, complemented
37C9×ΓL(1,𝔽25)C2212yescomplemented
38D3×C8×C9C2212yescomplemented
39C3×D72C2212yescomplemented
40C3×#3664C2212yescomplemented
41#88417C2212yescomplemented
42#88417C2212yescomplemented
43#88416C2212yescomplemented
44#88415C2212yescomplemented
45D3×#235—31no—
46C2×D3×D12—31no—
47#64117—31no—
48C3×#908—31no—
49C3×D3×M16D311yescharacteristic
50#63830—31no—
51#63825—31no—
52#63795—32no—
53#63794—32no—
54D3×D24—32no—
55D3×#204—32no—
56C2×#3697—31no—
57C22×D36—31no—
58C2×C9×M16—31no—
59#63470—31no—
60#63470—31no—
61#63470—31no—
62#63470—32no—
63#63470—32no—
64#63470—61no—
65#63470D311yescharacteristic, direct product factor
66C2×D72—32no—
67C2×#3664—32no—
68C22×C3⋊D9—21no—
69C22×D3×C9D411yescharacteristic
70C22×C3×D9—21no—
71C2×D3×D9—22no—
72C2×D3×D9—22no—
73C2×D3×D9—21no—
74C2×D3×D9—44no—
75C2×C3×C4×C9C2311yescharacteristic
76#6233—21no—
77C4⋊(C3⋊D9)—21no—
78C4⋊(C3⋊D9)C2312yes—
79C4⋊(C3⋊D9)C2311yescharacteristic, complemented
80C4×C3⋊D9—21no—
81#6226C2311yescharacteristic, complemented
82C3×#611—21no—
83C2×C9×Dic3D411yescharacteristic
84C4×D3×C9D411yescharacteristic
85C4×D3×C9D411yescharacteristic
86C4×D3×C9C2311yescharacteristic
87C4×D3×C9C2311yescharacteristic
88C3×D36—21no—
89C3×D36C2312yes—
90C3×D36C2311yescharacteristic, complemented
91C3×C4×D9—21no—
92C3×Dic18C2311yescharacteristic, complemented
93#6195—21no—
94#6194—21no—
95D3×Dic9—21no—
96#6192—21no—
97#6192—22no—
98#6192—22no—
99(C3⋊D9)⋊C4—21no—
100Dic3×D9—21no—
101#6189—21no—
102C3×C8×C9C2312yescomplemented
103C9×C3⋊C8C2312yescomplemented
104D3×AGL(1,ℤ/8)—91no—
105C2×#908—91no—
106C22×D32—61no—
107#3828—31no—
108C2×C4⋊(C3⋊D3)—31no—
109C3×#235—31no—
110C2×C3×D12—31no—
111C2×C3×C4×D3C2×D311yescharacteristic
112D3×#67—61no—
113C2×#626—61no—
114#3804—61no—
115D3×D12—32no—
116D3×D12—32no—
117D3×D12—31no—
118D3×D12—61no—
119D3×D12—61no—
120C4×D32—61no—
121#3799—31no—
122#3798—61no—
123#3795—31no—
124D3×Dic6—31no—
125C23×D9—61no—
126C32×M16C2×D311yescharacteristic
127#3745—32no—
128#3744—32no—
129C3×#208C2×D311yescharacteristic
130C3×D24—32no—
131C3×#204—32no—
132C3×ΓL(1,𝔽25)C2×D312yes—
133C3×D3×C8C2×D312yes—
134#3717—32no—
135#3716—32no—
136#3714—32no—
137#3714—32no—
138C22×C4×C9—31no—
139C2×#611—61no—
140#3697—31no—
141#3697—31no—
142#3697—31no—
143#3697—61no—
144#3697—61no—
145#3697—61no—
146#3697C2×D311yescharacteristic, complemented
147C2×D36—31no—
148C2×D36—32no—
149C2×D36—32no—
150C2×D36—31no—
151C2×D36—61no—
152C2×D36—61no—
153C2×D36—61no—
154C2×D36—61no—
155C2×D36C2×D311yescharacteristic, complemented
156C2×C4×D9—61no—
157C2×Dic18—31no—
158C9×M16—21no—
159C9×M16—31no—
160C9×M16—32no—
161C9×M16C2×D311yescharacteristic, complemented
162C2×C8×C9—32no—
163D72—32no—
164D72—32no—
165D72—32no—
166D72—62no—
167D72C2×D312yescomplemented
168#3664—32no—
169#3664—32no—
170#3664—32no—
171#3664—62no—
172#3664C2×D312yescomplemented
173C22×C3×C9C2×D411yescharacteristic
174C2×C3⋊D9—21no—
175C2×C3⋊D9—22no—
176C2×C3⋊D9—42no—
177C2×D3×C9—22no—
178C2×D3×C9C2×D411yescharacteristic
179C2×D3×C9C2×D411yescharacteristic
180C2×C3×D9—21no—
181C2×C3×D9—22no—
182C2×C3×D9—42no—
183D3×D9—44no—
184D3×D9—42no—
185D3×D9—44no—
186C3×C4×C9C2×D411yescharacteristic
187C3×C4×C9C2411yescharacteristic
188C3⋊Dic9—21no—
189C9×Dic3C2×D411yescharacteristic
190C9×Dic3C2×D411yescharacteristic
191C3×Dic9—21no—
192D3×#41—91no—
193C2×D3×D4—91no—
194C2×#235—91no—
195C22×D12—91no—
196C3×AGL(1,ℤ/8)—91no—
197C2×C3×M16—31no—
198#949—91no—
199#932—91no—
200#914—92no—
201D3×SD16—92no—
202#911—92no—
203D3×D8—92no—
204#908—31no—
205#908—92no—
206#908—92no—
207#908—91no—
208#908—91no—
209#908—91no—
210#908—91no—
211#908—181no—
212D3×M16D911yescharacteristic, complemented
213C2×D24—92no—
214C2×#204—92no—
215C22×C3⋊D3—61no—
216C22×C3×D3—61no—
217C22×C3×D3D1211yescharacteristic
218C2×D32—62no—
219C2×D32—62no—
220C2×D32—61no—
221C2×D32—124no—
222C2×C32×C4C22×D311yescharacteristic
223#638—61no—
224C4⋊(C3⋊D3)—32no—
225C4⋊(C3⋊D3)—31no—
226C4⋊(C3⋊D3)—61no—
227C4×C3⋊D3—61no—
228#634—31no—
229C3×#67—61no—
230C2×C3×Dic3D1211yescharacteristic
231C3×D12—32no—
232C3×D12—31no—
233C3×D12—61no—
234C3×C4×D3—61no—
235C3×C4×D3D1211yescharacteristic
236C3×C4×D3D1211yescharacteristic
237C3×C4×D3C22×D311yescharacteristic
238C3×C4×D3C22×D311yescharacteristic
239C3×Dic6—31no—
240#627—61no—
241#626—61no—
242#626—61no—
243#626—62no—
244#626—62no—
245#625—61no—
246#624—61no—
247D3×Dic3—61no—
248D3×Dic3—61no—
249C23×C9—31no—
250C22×D9—21no—
251C22×D9—61no—
252C22×D9—62no—
253C22×D9—62no—
254C22×D9—61no—
255C22×D9—121no—
256C22×D9—124no—
257C32×C8C22×D312yes—
258C3×C3⋊C8C22×D312yes—
259C2×C4×C9—21no—
260C2×C4×C9—31no—
261C2×C4×C9—31no—
262C2×C4×C9—31no—
263C2×C4×C9—31no—
264C2×C4×C9—31no—
265C2×C4×C9C22×D311yescharacteristic
266#611—21no—
267#611—61no—
268#611—61no—
269#611—61no—
270#611—121no—
271C2×Dic9—61no—
272D36—21no—
273D36—32no—
274D36—31no—
275D36—62no—
276D36—61no—
277D36—62no—
278D36—62no—
279D36—61no—
280D36—61no—
281D36—121no—
282D36C22×D312yes—
283D36C22×D311yescharacteristic, complemented
284C4×D9—21no—
285C4×D9—61no—
286C4×D9—61no—
287C4×D9—61no—
288C4×D9—121no—
289Dic18—31no—
290Dic18—61no—
291Dic18—61no—
292Dic18C22×D311yescharacteristic, complemented
293C8×C9—22no—
294C8×C9—32no—
295C8×C9C22×D312yescomplemented
296C2×AGL(1,ℤ/8)—271noSylow
297C2×C3×C9—21no—
298C2×C3×C9C22×D411yescharacteristic
299C3⋊D9—42no—
300C3⋊D9—41no—
301D3×C9—21no—
302D3×C9AGL(1,ℤ/8)12yescomplemented
303C3×D9—42no—
304C3×D9—41no—
305C23×D3—181no—
306C23×D3—181no—
307C3×#41—91no—
308C2×C3×D4—91no—
309C22×C3×C4—31no—
310C2×#67—181no—
311C2×#67—181no—
312#239—91no—
313D3×Q8—91no—
314#237—91no—
315#237—181no—
316D3×D4—91no—
317D3×D4—92no—
318D3×D4—92no—
319D3×D4—181no—
320D3×D4—181no—
321D3×D4—181no—
322#235—31no—
323#235—91no—
324#235—91no—
325#235—91no—
326#235—91no—
327#235—181no—
328#235—181no—
329#235—181no—
330#235—181no—
331C2×D12—31no—
332C2×D12—92no—
333C2×D12—91no—
334C2×D12—92no—
335C2×D12—91no—
336C2×D12—91no—
337C2×D12—181no—
338C2×D12—181no—
339C2×D12—181no—
340C2×D12—181no—
341C2×C4×D3—181no—
342C2×C4×D3—181no—
343C2×C4×D3C2×D911yescharacteristic
344C2×Dic6—91no—
345C3×SD16—92no—
346C3×D8—92no—
347C3×M16—21no—
348C3×M16—31no—
349C3×M16—32no—
350C3×M16C2×D911yescharacteristic, complemented
351C3×M16D3211yescharacteristic
352C2×C3×C8—32no—
353#215—92no—
354#214—92no—
355#213—92no—
356#213—92no—
357#208C2×D911yescharacteristic, complemented
358D24—32no—
359D24—92no—
360D24—92no—
361D24—92no—
362D24—92no—
363D24—182no—
364#204—32no—
365#204—92no—
366#204—92no—
367#204—92no—
368#204—92no—
369#204—182no—
370ΓL(1,𝔽25)C2×D912yescomplemented
371D3×C8C2×D912yescomplemented
372C22×C32C2×D1211yescharacteristic
373C2×C3⋊D3—61no—
374C2×C3⋊D3—62no—
375C2×C3⋊D3—122no—
376C2×C3×D3—22no—
377C2×C3×D3—61no—
378C2×C3×D3—62no—
379C2×C3×D3—122no—
380C2×C3×D3C2×D1211yescharacteristic
381C2×C3×D3C2×D1211yescharacteristic
382D32—124no—
383D32—122no—
384D32—124no—
385C32×C4C2×D1211yescharacteristic
386C32×C4C23×D311yescharacteristic
387C3⋊Dic3—61no—
388C3×Dic3—61no—
389C3×Dic3C2×D1211yescharacteristic
390C3×Dic3C2×D1211yescharacteristic
391C22×C9—21no—
392C22×C9—31no—
393C22×C9—31no—
394C22×C9—62no—
395C22×C9D3×D411yescharacteristic
396C2×D9—21no—
397C2×D9—22no—
398C2×D9—42no—
399C2×D9—61no—
400C2×D9—62no—
401C2×D9—121no—
402C2×D9—122no—
403C2×D9—122no—
404C2×D9—124no—
405C2×D9—124no—
406C2×D9—122no—
407C2×D9—242no—
408C4×C9—21no—
409C4×C9—21no—
410C4×C9—31no—
411C4×C9—31no—
412C4×C9D3×D411yescharacteristic
413C4×C9C23×D311yescharacteristic
414Dic9—21no—
415Dic9—61no—
416Dic9—121no—
417C2×#41—271no—
418C22×D4—271no—
419AGL(1,ℤ/8)—91no—
420AGL(1,ℤ/8)—272no—
421AGL(1,ℤ/8)—272no—
422AGL(1,ℤ/8)—271no—
423AGL(1,ℤ/8)—271no—
424AGL(1,ℤ/8)—271no—
425C2×SD16—272no—
426C2×D8—272no—
427C2×M16—31no—
428C3×C9C2×AGL(1,ℤ/8)11yescharacteristic, complemented, Sylow
429C23×C3—31no—
430C23×C3—181no—
431C22×D3—61no—
432C22×D3—181no—
433C22×D3—182no—
434C22×D3—182no—
435C22×D3—181no—
436C22×D3—181no—
437C22×D3—181no—
438C22×D3—182no—
439C22×D3—182no—
440C22×D3—361no—
441C22×D3—364no—
442C22×D3—364no—
443C22×D3D3611yescharacteristic
444C3×Q8—91no—
445C3×D4—91no—
446C3×D4—92no—
447C3×D4—181no—
448C3×D4—181no—
449C2×C3×C4—21no—
450C2×C3×C4—31no—
451C2×C3×C4—31no—
452C2×C3×C4—31no—
453C2×C3×C4—31no—
454C2×C3×C4—31no—
455C2×C3×C4—181no—
456C2×C3×C4C22×D911yescharacteristic
457C2×C3×C4C2×D3211yescharacteristic
458#67—61no—
459#67—181no—
460#67—181no—
461#67—181no—
462#67—181no—
463#67—181no—
464#67—181no—
465#67—182no—
466#67—182no—
467#67—361no—
468C2×Dic3—181no—
469C2×Dic3—181no—
470C2×Dic3D3611yescharacteristic
471D12—32no—
472D12—31no—
473D12—61no—
474D12—92no—
475D12—91no—
476D12—92no—
477D12—91no—
478D12—182no—
479D12—181no—
480D12—182no—
481D12—182no—
482D12—181no—
483D12—181no—
484D12—181no—
485D12—181no—
486D12—361no—
487C4×D3—61no—
488C4×D3—181no—
489C4×D3—181no—
490C4×D3—181no—
491C4×D3—181no—
492C4×D3—181no—
493C4×D3—181no—
494C4×D3—361no—
495C4×D3D3611yescharacteristic
496C4×D3D3611yescharacteristic
497C4×D3C22×D911yescharacteristic
498C4×D3C22×D911yescharacteristic
499Dic6—31no—
500Dic6—91no—
501Dic6—91no—
502Dic6—181no—
503Dic6—181no—
504Dic6—181no—
505C3×C8—22no—
506C3×C8—32no—
507C3×C8C22×D912yescomplemented
508C3×C8C2×D3212yes—
509C3⋊C8C22×D912yescomplemented
510C2×C32—21no—
511C2×C32C22×D1211yescharacteristic
512C3⋊D3—122no—
513C3⋊D3—121no—
514C3×D3—21no—
515C3×D3—122no—
516C3×D3—121no—
517C3×D3#90812yes—
518C2×C9—21no—
519C2×C9—21no—
520C2×C9—32no—
521C2×C9—41no—
522C2×C9—61no—
523C2×C9C2×D3×D411yescharacteristic
524D9—42no—
525D9—41no—
526D9—121no—
527D9—122no—
528D9—241no—
529D9—242no—
530C24—541no—
531#41—91no—
532#41—271no—
533#41—271no—
534#41—271no—
535#41—541no—
536#41—541no—
537C2×Q8—271no—
538C2×D4—91no—
539C2×D4—271no—
540C2×D4—272no—
541C2×D4—272no—
542C2×D4—271no—
543C2×D4—541no—
544C2×D4—541no—
545C2×D4—541no—
546C2×D4—541no—
547C22×C4—31no—
548C22×C4—541no—
549SD16—92no—
550SD16—272no—
551SD16—272no—
552SD16—272no—
553D8—92no—
554D8—272no—
555D8—272no—
556D8—272no—
557M16—31no—
558M16—32no—
559M16D3×D911yescharacteristic, complemented
560C2×C8—32no—
561C22×C3—21no—
562C22×C3—31no—
563C22×C3—31no—
564C22×C3—62no—
565C22×C3—181no—
566C22×C3—182no—
567C22×C3—362no—
568C22×C3C2×D3611yescharacteristic
569C22×C3D3×D1211yescharacteristic
570C2×D3—22no—
571C2×D3—61no—
572C2×D3—62no—
573C2×D3—122no—
574C2×D3—181no—
575C2×D3—182no—
576C2×D3—181no—
577C2×D3—182no—
578C2×D3—361no—
579C2×D3—362no—
580C2×D3—364no—
581C2×D3—364no—
582C2×D3—362no—
583C2×D3—364no—
584C2×D3—362no—
585C2×D3—362no—
586C2×D3—364no—
587C2×D3—362no—
588C2×D3—722no—
589C2×D3C2×D3611yescharacteristic
590C2×D3C2×D3611yescharacteristic
591C3×C4—21no—
592C3×C4—21no—
593C3×C4—31no—
594C3×C4—31no—
595C3×C4—181no—
596C3×C4C2×D3611yescharacteristic
597C3×C4C23×D911yescharacteristic
598C3×C4D3×D1211yescharacteristic
599C3×C4C22×D3211yescharacteristic
600Dic3—61no—
601Dic3—181no—
602Dic3—181no—
603Dic3—361no—
604Dic3C2×D3611yescharacteristic
605Dic3C2×D3611yescharacteristic
606C32C2×#90811yescharacteristic
607C9—21no—
608C9D3×AGL(1,ℤ/8)11yescharacteristic, complemented
609C23—31no—
610C23—181no—
611C23—541no—
612C23—542no—
613C23—542no—
614C23—541no—
615C23—1084no—
616Q8—91no—
617Q8—271no—
618Q8—541no—
619D4—91no—
620D4—92no—
621D4—181no—
622D4—181no—
623D4—271no—
624D4—272no—
625D4—542no—
626D4—542no—
627D4—541no—
628D4—541no—
629D4—541no—
630D4—541no—
631D4—541no—
632C2×C4—31no—
633C2×C4—31no—
634C2×C4—31no—
635C2×C4—31no—
636C2×C4—31no—
637C2×C4—181no—
638C2×C4—541no—
639C2×C4—541no—
640C2×C4—541no—
641C2×C4—541no—
642C2×C4C2×D3×D911yescharacteristic
643C8—32no—
644C8C2×D3×D912yescomplemented
645C2×C3—21no—
646C2×C3—21no—
647C2×C3—21no—
648C2×C3—32no—
649C2×C3—41no—
650C2×C3—61no—
651C2×C3—362no—
652C2×C3—361no—
653C2×C3C22×D3611yescharacteristic
654C2×C3C2×D3×D1211yescharacteristic
655D3—21no—
656D3—122no—
657D3—121no—
658D3—362no—
659D3—361no—
660D3—362no—
661D3—361no—
662D3—722no—
663D3—721no—
664D3#6347012yesdirect product factor
665C22—31no—
666C22—31no—
667C22—62no—
668C22—181no—
669C22—182no—
670C22—362no—
671C22—541no—
672C22—542no—
673C22—1082no—
674C22—1084no—
675C22—1082no—
676C22—1084no—
677C22D3×D3611yescharacteristic
678C4—31no—
679C4—31no—
680C4—181no—
681C4—541no—
682C4D3×D3611yescharacteristic
683C4C22×D3×D911yescharacteristic
684C3—21no—
685C3C2×#6347011yescharacteristic, complemented
686C3D3×#90811yescharacteristic
687C2—21no—
688C2—32no—
689C2—61no—
690C2—362no—
691C2—361no—
692C2—1082no—
693C2—1081no—
694C2C2×D3×D3611yescharacteristic
subgroupG/H
centerC2C2×D3×D36
commutator, cosocleC3×C4×C9C24
FrattiniC3×C4C22×D32
FittingC3×C9×M16C22
socleC2×C32C22×D12
hypercenterM16D3×D9
hypocenterC3×C9C2×AGL(1,ℤ/8)
LCSG ▷ C3×C4×C9 ▷ C2×C3×C9 ▷ C3×C9
UCSC1 ◁ C2 ◁ C2×C4 ◁ M16

Complements

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

NH ≅ G/Ntype
1. D3×#3697C2semidirect
2. C2×D3×D36C2semidirect
3. #11721783C2semidirect
4. D3×C9×M16C2semidirect
5. C3×#63470C2semidirect
6. #11720873C2semidirect
7. #11720868C2semidirect
8. #11720805C2semidirect
9. #11720804C2semidirect
10. D3×D72C2semidirect
11. D3×#3664C2semidirect
15. #88736C22semidirect
16. C2×C4⋊(C3⋊D9)C22semidirect
18. C3×#3697C22semidirect
19. C2×C3×D36C22semidirect
26. D3×D36C22semidirect
27. D3×D36C22semidirect
29. #88642C22semidirect
30. #88641C22semidirect
32. D3×Dic18C22semidirect
33. C3×C9×M16C22semidirect
34. #88525C22semidirect
35. #88524C22semidirect
36. C9×#208C22semidirect
37. C9×ΓL(1,𝔽25)C22semidirect
38. D3×C8×C9C22semidirect
39. C3×D72C22semidirect
40. C3×#3664C22semidirect
41. #88417C22semidirect
42. #88417C22semidirect
43. #88416C22semidirect
44. #88415C22semidirect
65. #63470D3direct (N × H)
79. C4⋊(C3⋊D9)C23semidirect
81. #6226C23semidirect
90. C3×D36C23semidirect
92. C3×Dic18C23semidirect
102. C3×C8×C9C23semidirect
103. C9×C3⋊C8C23semidirect
146. #3697C2×D3semidirect
155. C2×D36C2×D3semidirect
161. C9×M16C2×D3semidirect
167. D72C2×D3semidirect
172. #3664C2×D3semidirect
212. D3×M16D9semidirect
283. D36C22×D3semidirect
292. Dic18C22×D3semidirect
295. C8×C9C22×D3semidirect
302. D3×C9AGL(1,ℤ/8)semidirect
350. C3×M16C2×D9semidirect
357. #208C2×D9semidirect
370. ΓL(1,𝔽25)C2×D9semidirect
371. D3×C8C2×D9semidirect
428. C3×C9C2×AGL(1,ℤ/8)semidirect
507. C3×C8C22×D9semidirect
509. C3⋊C8C22×D9semidirect
559. M16D3×D9semidirect
608. C9D3×AGL(1,ℤ/8)semidirect
644. C8C2×D3×D9semidirect
664. D3#63470direct (N × H)
685. C3C2×#63470semidirect

Maximal subgroups

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

subgroupindex
D3×#36972
C2×D3×D362
#117217832
D3×C9×M162
C3×#634702
#117208732
#117208682
#117208052
#117208042
D3×D722
D3×#36642
D3×#9083
C2×#634703