Finite Group Databasebeta

← #13149857

Subgroups — #13149857

Order 1728.

842 conjugacy classes of subgroups.

The subgroup lattice has 710 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#88640C211yesmaximal, characteristic
2M16×C3⋊D9C211yesmaximal, characteristic, complemented
3D3×C9×M16C211yesmaximal, characteristic, complemented
4C3×M16×D9C211yesmaximal, characteristic, complemented
5D3×#3667C211yesmaximal, characteristic, complemented
6#11720863C211yesmaximal, characteristic, complemented
7D9×#208C211yesmaximal, characteristic, complemented
8#11720809C212yesmaximal, complemented
9#11720808C212yesmaximal, complemented
10#11720807C212yesmaximal, complemented
11#11720806C212yesmaximal, complemented
12#10594722—31nomaximal
13#10589847—31nomaximal
14C2×C4×C3⋊D9C2211yescharacteristic
15C2×C4×D3×C9C2211yescharacteristic
16C2×C3×C4×D9C2211yescharacteristic
17C2×#6195C411yescharacteristic
18C2×#6194C411yescharacteristic
19C2×#6192C411yescharacteristic
20C2×#6189C411yescharacteristic
21#88640C411yescharacteristic
22#88640C411yescharacteristic
23#88640C411yescharacteristic
24#88640C411yescharacteristic
25#88640C2211yescharacteristic
26#88640C2211yescharacteristic
27#88640C2211yescharacteristic
28#88640C2211yescharacteristic
29C3×C9×M16C2211yescharacteristic
30#88532C2211yescharacteristic
31#88523C2212yescomplemented
32C8×C3⋊D9C2212yescomplemented
33C9×#208C2211yescharacteristic
34C3×#3667C2211yescharacteristic
35C9×ΓL(1,𝔽25)C2212yescomplemented
36D3×C8×C9C2212yescomplemented
37C3×#3663C2212yescomplemented
38C3×C8×D9C2212yescomplemented
39#88420C2212yescomplemented
40D3×C9⋊C8C2212yescomplemented
41#88414C2212yescomplemented
42#88413C2212yescomplemented
43(C3⋊D9)⋊C8C2212yescomplemented
44D9×C3⋊C8C2212yescomplemented
45C2×#3798—31no—
46M16×C3⋊D3—31no—
47C3×D3×M16—31no—
48C3×D3×M16D311yescharacteristic
49#63818—31no—
50D3×#208—31no—
51D3×#208—31no—
52#63805—32no—
53#63804—32no—
54#63804—32no—
55#63803—32no—
56C2×#3697—31no—
57C2×C9×M16—31no—
58C2×#3667—31no—
59#63469—32no—
60#63469—32no—
61M16×D9—31no—
62M16×D9—31no—
63M16×D9—31no—
64M16×D9—61no—
65M16×D9D311yescharacteristic, complemented
66#63464—32no—
67#63464—32no—
68C22×C3⋊D9C2×C411yescharacteristic
69C22×D3×C9C2×C411yescharacteristic
70C22×C3×D9C2×C411yescharacteristic
71C2×C3×C4×C9C2311yescharacteristic
72C2×C3⋊Dic9C2×C411yescharacteristic
73C4×C3⋊D9C2×C411yescharacteristic
74C4×C3⋊D9C2×C411yescharacteristic
75C4×C3⋊D9C2311yescharacteristic
76C4×C3⋊D9C2311yescharacteristic
77C2×C9×Dic3C2×C411yescharacteristic
78C2×C3×Dic9C2×C411yescharacteristic
79C4×D3×C9C2×C411yescharacteristic
80C4×D3×C9C2×C411yescharacteristic
81C4×D3×C9C2311yescharacteristic
82C4×D3×C9C2311yescharacteristic
83C3×C4×D9C2×C411yescharacteristic
84C3×C4×D9C2×C411yescharacteristic
85C3×C4×D9C2311yescharacteristic
86C3×C4×D9C2311yescharacteristic
87#6195C2×C411yescharacteristic
88#6195C2×C411yescharacteristic
89#6195C2×C411yescharacteristic
90#6195C2×C411yescharacteristic
91#6194C2×C411yescharacteristic
92#6194C2×C411yescharacteristic
93#6194C2×C411yescharacteristic
94#6194C2×C411yescharacteristic
95#6192C2×C411yescharacteristic
96#6192C2×C411yescharacteristic
97#6192C2×C411yescharacteristic
98#6192C2×C411yescharacteristic
99#6189C2×C411yescharacteristic
100#6189C2×C411yescharacteristic
101#6189C2×C411yescharacteristic
102#6189C2×C411yescharacteristic
103C3×C8×C9C2312yes—
104C3⋊(C9⋊C8)C2312yes—
105C9×C3⋊C8C2312yes—
106C3×C9⋊C8C2312yes—
107#5745—91no—
108#5745—91no—
109C2×C4×C3⋊D3—31no—
110C2×C3×C4×D3—31no—
111C2×C3×C4×D3C2×D311yescharacteristic
112C2×#627—31no—
113C2×#626—31no—
114C2×#626—31no—
115C2×#625—31no—
116#3798—31no—
117#3798—31no—
118#3798—31no—
119#3798—31no—
120#3798—31no—
121#3798—31no—
122#3798—31no—
123#3798—31no—
124C32×M16C2×D311yescharacteristic
125#3748—31no—
126#3743—32no—
127C8×C3⋊D3—32no—
128C3×#208—31no—
129C3×#208C2×D311yescharacteristic
130C3×ΓL(1,𝔽25)—32no—
131C3×ΓL(1,𝔽25)C2×D312yes—
132C3×D3×C8—32no—
133C3×D3×C8C2×D312yes—
134#3712—32no—
135#3711—32no—
136#3711—32no—
137#3710—32no—
138D3×C3⋊C8—32no—
139D3×C3⋊C8—32no—
140C22×C4×C9—31no—
141C2×#611—31no—
142C2×#611—31no—
143#3697—31no—
144#3697—31no—
145#3697—31no—
146#3697—31no—
147#3697—31no—
148#3697—31no—
149#3697—31no—
150#3697—31no—
151C2×D36—31no—
152C2×C4×D9—31no—
153C2×C4×D9—61no—
154C2×C4×D9C2×D311yescharacteristic
155C2×Dic18—31no—
156C9×M16—21no—
157C9×M16—31no—
158C9×M16—32no—
159C9×M16C2×D311yescharacteristic
160C2×C8×C9—32no—
161#3667—31no—
162#3667—32no—
163#3667—61no—
164#3667C2×D311yescharacteristic
165C2×C9⋊C8—32no—
166#3663—32no—
167#3663—32no—
168#3663—32no—
169#3663—62no—
170#3663C2×D312yescomplemented
171C8×D9—32no—
172C8×D9—32no—
173C8×D9—32no—
174C8×D9—62no—
175C8×D9C2×D312yescomplemented
176C22×C3×C9C22×C411yescharacteristic
177C2×C3⋊D9—41no—
178C2×C3⋊D9C22×C411yescharacteristic
179C2×C3⋊D9C22×C411yescharacteristic
180C2×D3×C9—41no—
181C2×D3×C9C22×C411yescharacteristic
182C2×D3×C9C22×C411yescharacteristic
183C2×C3×D9—41no—
184C2×C3×D9C22×C411yescharacteristic
185C2×C3×D9C22×C411yescharacteristic
186C3×C4×C9C22×C411yescharacteristic
187C3×C4×C9C2411yescharacteristic
188C3⋊Dic9C22×C411yescharacteristic
189C3⋊Dic9C22×C411yescharacteristic
190C9×Dic3C22×C411yescharacteristic
191C9×Dic3C22×C411yescharacteristic
192C3×Dic9C22×C411yescharacteristic
193C3×Dic9C22×C411yescharacteristic
194C2×#235—91no—
195C2×#235—91no—
196C2×C3×M16—31no—
197C2×C3×M16—91no—
198C2×#208—91no—
199C2×#208—91no—
200#907—92no—
201#907—92no—
202#907—92no—
203#907—92no—
204D3×M16—31no—
205D3×M16—91no—
206D3×M16—91no—
207D3×M16—91no—
208D3×M16—91no—
209D3×M16—91no—
210D3×M16—91no—
211D3×M16—181no—
212D3×M16D911yescharacteristic, complemented
213#901—92no—
214#901—92no—
215#901—92no—
216#901—92no—
217C22×C3⋊D3—31no—
218C22×C3×D3—31no—
219C22×C3×D3C4×D311yescharacteristic
220C2×C32×C4C22×D311yescharacteristic
221C2×C3⋊Dic3—31no—
222C4×C3⋊D3—31no—
223C4×C3⋊D3—31no—
224C4×C3⋊D3—31no—
225C4×C3⋊D3—31no—
226C2×C3×Dic3—31no—
227C2×C3×Dic3C4×D311yescharacteristic
228C3×C4×D3—31no—
229C3×C4×D3—31no—
230C3×C4×D3—31no—
231C3×C4×D3—31no—
232C3×C4×D3C4×D311yescharacteristic
233C3×C4×D3C4×D311yescharacteristic
234C3×C4×D3C22×D311yescharacteristic
235C3×C4×D3C22×D311yescharacteristic
236#627—31no—
237#627—31no—
238#627—31no—
239#627—31no—
240#626—31no—
241#626—31no—
242#626—31no—
243#626—31no—
244#626—31no—
245#626—31no—
246#626—31no—
247#626—31no—
248#625—31no—
249#625—31no—
250#625—31no—
251#625—31no—
252C23×C9—31no—
253C22×D9—31no—
254C22×D9—61no—
255C22×D9C4×D311yescharacteristic
256C32×C8C22×D312yes—
257C3⋊(C3⋊C8)—32no—
258C3×C3⋊C8—32no—
259C3×C3⋊C8C22×D312yes—
260C2×C4×C9—21no—
261C2×C4×C9—31no—
262C2×C4×C9—31no—
263C2×C4×C9—31no—
264C2×C4×C9—31no—
265C2×C4×C9—31no—
266C2×C4×C9C22×D311yescharacteristic
267#611—31no—
268#611—31no—
269#611—31no—
270#611—31no—
271#611—31no—
272#611—31no—
273#611—31no—
274#611—31no—
275C2×Dic9—31no—
276C2×Dic9—61no—
277C2×Dic9C4×D311yescharacteristic
278D36—31no—
279D36—31no—
280D36—31no—
281D36—31no—
282C4×D9—31no—
283C4×D9—31no—
284C4×D9—31no—
285C4×D9—31no—
286C4×D9—61no—
287C4×D9—61no—
288C4×D9—61no—
289C4×D9—61no—
290C4×D9C4×D311yescharacteristic
291C4×D9C4×D311yescharacteristic
292C4×D9C22×D311yescharacteristic
293C4×D9C22×D311yescharacteristic
294Dic18—31no—
295Dic18—31no—
296Dic18—31no—
297Dic18—31no—
298C8×C9—22no—
299C8×C9—32no—
300C8×C9C22×D312yes—
301C9⋊C8—32no—
302C9⋊C8—62no—
303C9⋊C8C22×D312yes—
304#568—271noSylow
305C2×C3×C9—21no—
306C2×C3×C9C23×C411yescharacteristic
307C3⋊D9—21no—
308C3⋊D9—21no—
309D3×C9—21no—
310D3×C9—21no—
311C3×D9—21no—
312C3×D9—21no—
313C22×C3×C4—31no—
314C22×C3×C4—91no—
315C2×#67—91no—
316C2×#67—91no—
317C2×#67—91no—
318C2×#67—91no—
319#235—91no—
320#235—91no—
321#235—91no—
322#235—91no—
323#235—91no—
324#235—91no—
325#235—91no—
326#235—91no—
327#235—91no—
328#235—91no—
329#235—91no—
330#235—91no—
331#235—91no—
332#235—91no—
333#235—91no—
334#235—91no—
335C2×D12—91no—
336C2×D12—91no—
337C2×C4×D3—31no—
338C2×C4×D3—91no—
339C2×C4×D3—91no—
340C2×C4×D3—181no—
341C2×C4×D3C2×D911yescharacteristic
342C2×Dic6—91no—
343C2×Dic6—91no—
344C3×M16—21no—
345C3×M16—31no—
346C3×M16—32no—
347C3×M16—91no—
348C3×M16—92no—
349C3×M16C2×D911yescharacteristic
350C3×M16D3211yescharacteristic
351C2×C3×C8—32no—
352C2×C3×C8—92no—
353#208—31no—
354#208—92no—
355#208—91no—
356#208—91no—
357#208—92no—
358#208—181no—
359#208C2×D911yescharacteristic
360C2×C3⋊C8—92no—
361C2×C3⋊C8—92no—
362ΓL(1,𝔽25)—32no—
363ΓL(1,𝔽25)—92no—
364ΓL(1,𝔽25)—92no—
365ΓL(1,𝔽25)—92no—
366ΓL(1,𝔽25)—92no—
367ΓL(1,𝔽25)—92no—
368ΓL(1,𝔽25)—92no—
369ΓL(1,𝔽25)—182no—
370ΓL(1,𝔽25)C2×D912yescomplemented
371D3×C8—32no—
372D3×C8—92no—
373D3×C8—92no—
374D3×C8—92no—
375D3×C8—92no—
376D3×C8—92no—
377D3×C8—92no—
378D3×C8—182no—
379D3×C8C2×D912yescomplemented
380C22×C32C2×C4×D311yescharacteristic
381C2×C3⋊D3—31no—
382C2×C3⋊D3—31no—
383C2×C3⋊D3—121no—
384C2×C3×D3—31no—
385C2×C3×D3—31no—
386C2×C3×D3—41no—
387C2×C3×D3—121no—
388C2×C3×D3C2×C4×D311yescharacteristic
389C2×C3×D3C2×C4×D311yescharacteristic
390C32×C4C2×C4×D311yescharacteristic
391C32×C4C23×D311yescharacteristic
392C3⋊Dic3—31no—
393C3⋊Dic3—31no—
394C3×Dic3—31no—
395C3×Dic3—31no—
396C3×Dic3C2×C4×D311yescharacteristic
397C3×Dic3C2×C4×D311yescharacteristic
398C22×C9—21no—
399C22×C9—31no—
400C22×C9—31no—
401C22×C9—121no—
402C22×C9C2×C4×D311yescharacteristic
403C2×D9—31no—
404C2×D9—31no—
405C2×D9—41no—
406C2×D9—61no—
407C2×D9—61no—
408C2×D9—121no—
409C2×D9—122no—
410C2×D9C2×C4×D311yescharacteristic
411C2×D9C2×C4×D311yescharacteristic
412C4×C9—21no—
413C4×C9—21no—
414C4×C9—31no—
415C4×C9—31no—
416C4×C9C2×C4×D311yescharacteristic
417C4×C9C23×D311yescharacteristic
418Dic9—31no—
419Dic9—31no—
420Dic9—61no—
421Dic9—61no—
422Dic9C2×C4×D311yescharacteristic
423Dic9C2×C4×D311yescharacteristic
424C2×#41—271no—
425#131—272no—
426#131—272no—
427#131—272no—
428#131—272no—
429C2×M16—31no—
430C2×M16—91no—
431C2×M16—271no—
432C2×M16—271no—
433C2×M16—271no—
434C2×M16—271no—
435C3×C9#56811yescharacteristic, complemented, Sylow
436C23×C3—31no—
437C23×C3—91no—
438C22×D3—31no—
439C22×D3—91no—
440C22×D3—91no—
441C22×D3—181no—
442C22×D3C4×D911yescharacteristic
443C2×C3×C4—21no—
444C2×C3×C4—31no—
445C2×C3×C4—31no—
446C2×C3×C4—31no—
447C2×C3×C4—31no—
448C2×C3×C4—31no—
449C2×C3×C4—91no—
450C2×C3×C4—91no—
451C2×C3×C4—91no—
452C2×C3×C4—91no—
453C2×C3×C4—91no—
454C2×C3×C4C22×D911yescharacteristic
455C2×C3×C4C2×D3211yescharacteristic
456#67—91no—
457#67—91no—
458#67—91no—
459#67—91no—
460#67—91no—
461#67—91no—
462#67—91no—
463#67—91no—
464#67—91no—
465#67—91no—
466#67—91no—
467#67—91no—
468#67—91no—
469#67—91no—
470#67—91no—
471#67—91no—
472C2×Dic3—31no—
473C2×Dic3—91no—
474C2×Dic3—91no—
475C2×Dic3—181no—
476C2×Dic3C4×D911yescharacteristic
477D12—91no—
478D12—91no—
479D12—91no—
480D12—91no—
481D12—91no—
482D12—91no—
483D12—91no—
484D12—91no—
485C4×D3—31no—
486C4×D3—31no—
487C4×D3—31no—
488C4×D3—31no—
489C4×D3—91no—
490C4×D3—91no—
491C4×D3—91no—
492C4×D3—91no—
493C4×D3—91no—
494C4×D3—91no—
495C4×D3—91no—
496C4×D3—91no—
497C4×D3—181no—
498C4×D3—181no—
499C4×D3—181no—
500C4×D3—181no—
501C4×D3C4×D911yescharacteristic
502C4×D3C4×D911yescharacteristic
503C4×D3C22×D911yescharacteristic
504C4×D3C22×D911yescharacteristic
505Dic6—91no—
506Dic6—91no—
507Dic6—91no—
508Dic6—91no—
509Dic6—91no—
510Dic6—91no—
511Dic6—91no—
512Dic6—91no—
513C3×C8—22no—
514C3×C8—32no—
515C3×C8—92no—
516C3×C8C22×D912yes—
517C3×C8C2×D3212yes—
518C3⋊C8—32no—
519C3⋊C8—92no—
520C3⋊C8—92no—
521C3⋊C8—182no—
522C3⋊C8C22×D912yes—
523C2×C32—21no—
524C2×C32C22×C4×D311yescharacteristic
525C3⋊D3—61no—
526C3⋊D3—61no—
527C3×D3—21no—
528C3×D3—21no—
529C3×D3—61no—
530C3×D3—61no—
531C2×C9—21no—
532C2×C9—21no—
533C2×C9—41no—
534C2×C9—61no—
535C2×C9—61no—
536C2×C9C22×C4×D311yescharacteristic
537D9—21no—
538D9—21no—
539D9—62no—
540D9—61no—
541D9—61no—
542D9—121no—
543#41—271no—
544#41—271no—
545#41—271no—
546#41—271no—
547#41—271no—
548#41—271no—
549#41—271no—
550#41—271no—
551C2×Q8—271no—
552C2×D4—271no—
553C2×D4—271no—
554C2×D4—271no—
555C22×C4—31no—
556C22×C4—91no—
557C22×C4—271no—
558M16—31no—
559M16—32no—
560M16—91no—
561M16—92no—
562M16—272no—
563M16—272no—
564M16—272no—
565M16—271no—
566M16—272no—
567M16D3×D911yescharacteristic
568C2×C8—32no—
569C2×C8—92no—
570C2×C8—272no—
571C2×C8—272no—
572C2×C8—272no—
573C2×C8—272no—
574C22×C3—21no—
575C22×C3—31no—
576C22×C3—31no—
577C22×C3—91no—
578C22×C3—91no—
579C22×C3—121no—
580C22×C3—361no—
581C22×C3C2×C4×D911yescharacteristic
582C22×C3C4×D3211yescharacteristic
583C2×D3—31no—
584C2×D3—31no—
585C2×D3—41no—
586C2×D3—91no—
587C2×D3—91no—
588C2×D3—91no—
589C2×D3—91no—
590C2×D3—121no—
591C2×D3—181no—
592C2×D3—181no—
593C2×D3—361no—
594C2×D3—361no—
595C2×D3—362no—
596C2×D3C2×C4×D911yescharacteristic
597C2×D3C2×C4×D911yescharacteristic
598C3×C4—21no—
599C3×C4—21no—
600C3×C4—31no—
601C3×C4—31no—
602C3×C4—91no—
603C3×C4—91no—
604C3×C4C2×C4×D911yescharacteristic
605C3×C4C23×D911yescharacteristic
606C3×C4C4×D3211yescharacteristic
607C3×C4C22×D3211yescharacteristic
608Dic3—31no—
609Dic3—31no—
610Dic3—91no—
611Dic3—91no—
612Dic3—91no—
613Dic3—91no—
614Dic3—181no—
615Dic3—181no—
616Dic3C2×C4×D911yescharacteristic
617Dic3C2×C4×D911yescharacteristic
618C32#574511yescharacteristic
619C9—21no—
620C9#574511yescharacteristic, complemented
621C23—31no—
622C23—91no—
623C23—271no—
624Q8—271no—
625Q8—271no—
626Q8—271no—
627Q8—271no—
628D4—271no—
629D4—271no—
630D4—271no—
631D4—271no—
632D4—271no—
633D4—271no—
634D4—271no—
635D4—271no—
636D4—271no—
637D4—271no—
638D4—271no—
639D4—271no—
640C2×C4—31no—
641C2×C4—31no—
642C2×C4—31no—
643C2×C4—31no—
644C2×C4—31no—
645C2×C4—91no—
646C2×C4—91no—
647C2×C4—91no—
648C2×C4—91no—
649C2×C4—91no—
650C2×C4—271no—
651C2×C4—271no—
652C2×C4—271no—
653C2×C4—271no—
654C2×C4—271no—
655C2×C4C2×D3×D911yescharacteristic
656C8—32no—
657C8—92no—
658C8—272no—
659C8C2×D3×D912yes—
660C2×C3—21no—
661C2×C3—21no—
662C2×C3—21no—
663C2×C3—41no—
664C2×C3—61no—
665C2×C3—61no—
666C2×C3—181no—
667C2×C3—181no—
668C2×C3C22×C4×D911yescharacteristic
669C2×C3C2×C4×D3211yescharacteristic
670D3—21no—
671D3—21no—
672D3—61no—
673D3—61no—
674D3—182no—
675D3—181no—
676D3—181no—
677D3—181no—
678D3—181no—
679D3—361no—
680C22—31no—
681C22—31no—
682C22—91no—
683C22—91no—
684C22—121no—
685C22—271no—
686C22—271no—
687C22—361no—
688C22—1081no—
689C22C4×D3×D911yescharacteristic
690C4—31no—
691C4—31no—
692C4—91no—
693C4—91no—
694C4—271no—
695C4—271no—
696C4C4×D3×D911yescharacteristic
697C4C22×D3×D911yescharacteristic
698C3—21no—
699C3#1058984711yescharacteristic, complemented
700C3#1059472211yescharacteristic
701C2—21no—
702C2—61no—
703C2—61no—
704C2—181no—
705C2—181no—
706C2—541no—
707C2—541no—
708C2C2×C4×D3×D911yescharacteristic
subgroupG/H
centerC4C22×D3×D9
commutatorC2×C3×C9C23×C4
FrattiniC3×C4C22×D32
FittingC3×C9×M16C22
socleC2×C32C22×C4×D3
prime radicalC2×#88640C2
cosocleC3×C4×C9C24
hypercenterM16D3×D9
hypocenterC3×C9#568
LCSG ▷ C2×C3×C9 ▷ C3×C9
UCSC1 ◁ C4 ◁ M16

Complements

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

NH ≅ G/Ntype
2. M16×C3⋊D9C2semidirect
3. D3×C9×M16C2semidirect
4. C3×M16×D9C2semidirect
5. D3×#3667C2semidirect
6. #11720863C2semidirect
7. D9×#208C2semidirect
8. #11720809C2semidirect
9. #11720808C2semidirect
10. #11720807C2semidirect
11. #11720806C2semidirect
31. #88523C22semidirect
32. C8×C3⋊D9C22semidirect
35. C9×ΓL(1,𝔽25)C22semidirect
36. D3×C8×C9C22semidirect
37. C3×#3663C22semidirect
38. C3×C8×D9C22semidirect
39. #88420C22semidirect
40. D3×C9⋊C8C22semidirect
41. #88414C22semidirect
42. #88413C22semidirect
43. (C3⋊D9)⋊C8C22semidirect
44. D9×C3⋊C8C22semidirect
65. M16×D9D3semidirect
170. #3663C2×D3semidirect
175. C8×D9C2×D3semidirect
212. D3×M16D9semidirect
370. ΓL(1,𝔽25)C2×D9semidirect
379. D3×C8C2×D9semidirect
435. C3×C9#568semidirect
620. C9#5745semidirect
699. C3#10589847semidirect

Maximal subgroups

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

subgroupindex
C2×#886402
M16×C3⋊D92
D3×C9×M162
C3×M16×D92
D3×#36672
#117208632
D9×#2082
#117208092
#117208082
#117208072
#117208062
#105947223
#105898473