Finite Group Databasebeta

← #13164785

Subgroups — #13164785

Order 1728.

2596 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
1#11722672C211yesmaximal, characteristic, complemented
2D3×C9×#41C211yesmaximal, characteristic, complemented
3C3×#63716C211yesmaximal, characteristic, complemented
4D9×#239C212yesmaximal, complemented
5#11722509C212yesmaximal, complemented
6#11722500C216yesmaximal, complemented
7#11722498C216yesmaximal, complemented
8#11722478C213yesmaximal, complemented
9D3×#3697C213yesmaximal, complemented
10C2×#88642C213yesmaximal, complemented
11#11722464C213yesmaximal, complemented
12#10596697—31nomaximal
13#10593139—31nomaximal
14C3×C9×#41C2211yescharacteristic, complemented
15#88746C2212yescomplemented
16D4×C3⋊D9C2216yescomplemented
17#88736C2213yescomplemented
18C2×C4⋊(C3⋊D9)C2213yescomplemented
19C9×#239C2211yescharacteristic, complemented
20D3×Q8×C9C2211yescharacteristic, complemented
21C3×Q8×D9C2212yescomplemented
22C9×#237C2213yescomplemented
23D3×D4×C9C2213yescomplemented
24C3×#3699C2216yescomplemented
25C9×#235C2213yescomplemented
26C2×C4×D3×C9C2213yescomplemented
27C3×#3697C2213yescomplemented
28C2×C3×Dic18C2213yescomplemented
29#88668C22112yescomplemented
30D3×#611C2216yescomplemented
31C2×#6195C2216yescomplemented
32#88662C22112yescomplemented
33C2×(C3⋊D9)⋊C4C2216yescomplemented
34#88658C2216yescomplemented
35D9×D12C2216yescomplemented
36D3×D36C2213yescomplemented
37C4×D3×D9C2216yescomplemented
38#88642C2216yescomplemented
39#88642C2216yescomplemented
40#88642C2216yescomplemented
41#88642C2213yescomplemented
42#88641C2211yescharacteristic, complemented
43#88640C2216yescomplemented
44#88639C2216yescomplemented
45D3×Dic18C2213yescomplemented
46(C3⋊D9)⋊Q8C2216yescomplemented
47#88635C2216yescomplemented
48#64366—31no—
49C3×#1010—31no—
50C3×D3×#41D311yescharacteristic
51D3×#239—32no—
52#64316—32no—
53#64314—36no—
54#64312—36no—
55#64308—33no—
56D3×#235—33no—
57C2×#3799—33no—
58#64297—33no—
59C2×C9×#41—31no—
60#63716—33no—
61#63716D311yescharacteristic, complemented
62#63715—31no—
63#63715—33no—
64#63715—61no—
65#41×D9—32no—
66#41×D9—36no—
67#63713—32no—
68#63710—36no—
69C2×#3697—33no—
70C2×#3697—33no—
71C22×C3⋊D9C2316yes—
72C22×D3×C9C2313yes—
73C2×D3×D9C2316yes—
74C3×Q8×C9C2311yescharacteristic
75C3×D4×C9C2313yes—
76C2×C3×C4×C9C2313yes—
77#6233C2316yes—
78C4⋊(C3⋊D9)C2316yes—
79C4⋊(C3⋊D9)C2313yes—
80C4×C3⋊D9C2316yes—
81#6226C2311yescharacteristic
82C9×#67C2316yes—
83C3×#611C2316yes—
84C2×C9×Dic3C2313yes—
85C2×C3×Dic9C2316yes—
86C9×D12C2313yes—
87C4×D3×C9C2313yes—
88C4×D3×C9C2313yes—
89C4×D3×C9C2311yescharacteristic
90C4×D3×C9C2313yes—
91C3×D36C2311yescharacteristic
92C3×C4×D9C2316yes—
93C9×Dic6C2313yes—
94C3×Dic18C2316yes—
95C3×Dic18C2313yes—
96#6195C23112yes—
97#6195C2316yes—
98#6194C2312yes—
99D3×Dic9C2316yes—
100#6192C2316yes—
101(C3⋊D9)⋊C4C23112yes—
102(C3⋊D9)⋊C4C2316yes—
103Dic3×D9C2312yes—
104#6189C2316yes—
105#5968—91no—
106#5964—91no—
107C32×#41C2×D311yescharacteristic
108#3832—32no—
109D4×C3⋊D3—36no—
110#3828—33no—
111C2×C4⋊(C3⋊D3)—33no—
112C3×#239C2×D311yescharacteristic
113C3×D3×Q8—32no—
114C3×D3×Q8C2×D311yescharacteristic
115C3×#237—36no—
116C3×#237C2×D313yes—
117C3×D3×D4C2×D313yes—
118C3×#235—33no—
119C3×#235C2×D313yes—
120C2×C3×C4×D3C2×D313yes—
121C2×C3×Dic6—33no—
122#3811—312no—
123D3×#67—36no—
124C2×#626—36no—
125C2×#624—36no—
126#3804—312no—
127#3804—36no—
128D3×D12—36no—
129D3×D12—33no—
130C4×D32—36no—
131#3799—36no—
132#3799—33no—
133#3799—36no—
134#3799—36no—
135#3799—36no—
136#3798—36no—
137#3797—36no—
138#3796—36no—
139#3795—31no—
140D3×Dic6—33no—
141C9×#41—21no—
142C9×#41—31no—
143C9×#41—33no—
144C9×#41—33no—
145C9×#41C2×D311yescharacteristic, complemented
146C2×Q8×C9—31no—
147C2×D4×C9—33no—
148C22×C4×C9—33no—
149C2×#611—36no—
150C2×#611—36no—
151#3701—32no—
152#3701—36no—
153#3701—62no—
154Q8×D9—36no—
155Q8×D9C2×D312yescomplemented
156#3699—312no—
157#3699—36no—
158#3699C2×D316yescomplemented
159D4×D9—36no—
160D4×D9—312no—
161D4×D9—36no—
162D4×D9—66no—
163#3697—33no—
164#3697—36no—
165#3697—31no—
166#3697—36no—
167#3697—33no—
168#3697—36no—
169#3697—36no—
170#3697—36no—
171#3697—63no—
172#3697C2×D313yescomplemented
173C2×D36—33no—
174C2×D36—33no—
175C2×D36—63no—
176C2×C4×D9—36no—
177C2×C4×D9—36no—
178C2×Dic18—33no—
179C2×Dic18C2×D313yescomplemented
180C22×C3×C9C2413yes—
181C2×C3⋊D9—46no—
182C2×C3⋊D9C2416yes—
183C2×D3×C9—43no—
184C2×D3×C9C2413yes—
185C2×D3×C9C2411yescharacteristic
186C2×C3×D9C2412yes—
187D3×D9—46no—
188C3×C4×C9C2411yescharacteristic
189C3×C4×C9C2413yes—
190C3⋊Dic9C2412yes—
191C9×Dic3C2413yes—
192C9×Dic3C2411yescharacteristic
193C3×Dic9C2416yes—
194C3×2−1+4—91no—
195C2×C3×#41—31no—
196#1010—31no—
197#1010—93no—
198#1009—93no—
199#1009—91no—
200#1009—91no—
201#1009—96no—
202#1009—93no—
203#1009—181no—
204D3×#41—96no—
205D3×#41—92no—
206D3×#41—96no—
207D3×#41—93no—
208D3×#41D911yescharacteristic, complemented
209#1007—92no—
210#1007—92no—
211#1007—93no—
212C2×#239—92no—
213C2×#239—93no—
214#1004—96no—
215C2×#235—93no—
216C2×#235—93no—
217C22×C3⋊D3—36no—
218C22×C3×D3C22×D313yes—
219C2×D32—36no—
220C32×Q8C22×D311yescharacteristic
221C32×D4C22×D313yes—
222C2×C32×C4C22×D313yes—
223#638—36no—
224C4⋊(C3⋊D3)—33no—
225C4⋊(C3⋊D3)—36no—
226C4×C3⋊D3—36no—
227#634—31no—
228C3×#67—36no—
229C3×#67C22×D316yes—
230C2×C3×Dic3—36no—
231C2×C3×Dic3C22×D313yes—
232C3×D12—31no—
233C3×D12C22×D313yes—
234C3×C4×D3—36no—
235C3×C4×D3C22×D313yes—
236C3×C4×D3C22×D311yescharacteristic
237C3×C4×D3C22×D313yes—
238C3×C4×D3C22×D313yes—
239C3×Dic6—33no—
240C3×Dic6—36no—
241C3×Dic6C22×D313yes—
242#627—36no—
243#626—312no—
244#626—36no—
245#626—36no—
246#625—32no—
247#624—312no—
248#624—36no—
249D3×Dic3—36no—
250D3×Dic3—32no—
251C23×C9—33no—
252C22×D9—36no—
253C22×D9—36no—
254C22×D9—66no—
255Q8×C9—21no—
256Q8×C9—33no—
257Q8×C9C22×D311yescharacteristic
258D4×C9—23no—
259D4×C9—36no—
260D4×C9—33no—
261D4×C9C22×D313yes—
262C2×C4×C9—23no—
263C2×C4×C9—33no—
264C2×C4×C9—33no—
265C2×C4×C9—33no—
266C2×C4×C9—31no—
267C2×C4×C9—33no—
268C2×C4×C9C22×D313yes—
269#611—36no—
270#611—32no—
271#611—36no—
272#611—312no—
273#611—66no—
274#611C22×D316yes—
275C2×Dic9—36no—
276C2×Dic9C22×D316yes—
277D36—36no—
278D36—33no—
279D36—36no—
280D36—63no—
281D36—66no—
282D36C22×D311yescharacteristic
283C4×D9—32no—
284C4×D9—36no—
285C4×D9—36no—
286C4×D9—312no—
287C4×D9—66no—
288C4×D9C22×D316yes—
289Dic18—31no—
290Dic18—36no—
291Dic18—61no—
292Dic18C22×D313yes—
293Dic18C22×D316yes—
294#585—271noSylow
295C2×C3×C9—23no—
296C2×C3×C9C2511yescharacteristic
297C3⋊D9—26no—
298D3×C9—23no—
299D3×C9—21no—
300C3×D9—22no—
301C3×#41—21no—
302C3×#41—31no—
303C3×#41—33no—
304C3×#41—33no—
305C3×#41—96no—
306C3×#41—93no—
307C3×#41C2×D911yescharacteristic, complemented
308C3×#41D3211yescharacteristic
309C2×C3×Q8—31no—
310C2×C3×Q8—92no—
311C2×C3×Q8—93no—
312C2×C3×D4—33no—
313C22×C3×C4—33no—
314C2×#67—96no—
315C2×#67—96no—
316#239—96no—
317#239—92no—
318#239—92no—
319#239—96no—
320#239—96no—
321#239—93no—
322#239—96no—
323#239—96no—
324#239—182no—
325#239C2×D911yescharacteristic, complemented
326D3×Q8—32no—
327D3×Q8—96no—
328D3×Q8—93no—
329D3×Q8—96no—
330D3×Q8C2×D911yescharacteristic, complemented
331#237—36no—
332#237—96no—
333#237—912no—
334#237—96no—
335#237—91no—
336#237C2×D913yescomplemented
337D3×D4—96no—
338D3×D4—912no—
339D3×D4—96no—
340D3×D4—96no—
341D3×D4—96no—
342D3×D4—912no—
343D3×D4—93no—
344D3×D4—186no—
345D3×D4C2×D913yescomplemented
346#235—33no—
347#235—93no—
348#235—96no—
349#235—96no—
350#235—96no—
351#235—96no—
352#235—91no—
353#235—96no—
354#235—93no—
355#235—93no—
356#235—912no—
357#235—96no—
358#235—96no—
359#235—183no—
360#235C2×D913yescomplemented
361C2×D12—93no—
362C2×D12—96no—
363C2×D12—93no—
364C2×D12—93no—
365C2×D12—96no—
366C2×D12—183no—
367C2×C4×D3—96no—
368C2×C4×D3—96no—
369C2×C4×D3—96no—
370C2×C4×D3—96no—
371C2×C4×D3C2×D913yescomplemented
372C2×Dic6—33no—
373C2×Dic6—93no—
374C22×C32C23×D313yes—
375C2×C3⋊D3—36no—
376C2×C3⋊D3—126no—
377C2×C3×D3—32no—
378C2×C3×D3—43no—
379C2×C3×D3C23×D313yes—
380C2×C3×D3C23×D311yescharacteristic
381D32—126no—
382C32×C4C23×D311yescharacteristic
383C32×C4C23×D313yes—
384C3⋊Dic3—32no—
385C3×Dic3—36no—
386C3×Dic3C23×D311yescharacteristic
387C3×Dic3C23×D313yes—
388C22×C9—23no—
389C22×C9—33no—
390C22×C9—31no—
391C22×C9—123no—
392C22×C9C23×D313yes—
393C2×D9—36no—
394C2×D9—66no—
395C2×D9—126no—
396C2×D9—126no—
397C2×D9—246no—
398C2×D9C23×D312yes—
399C4×C9—21no—
400C4×C9—23no—
401C4×C9—33no—
402C4×C9—31no—
403C4×C9C23×D311yescharacteristic
404C4×C9C23×D313yes—
405Dic9—32no—
406Dic9—62no—
407Dic9C23×D316yes—
4082−1+4—91no—
4092−1+4—272no—
4102−1+4—273no—
411H(5,𝔽2)—276no—
412H(5,𝔽2)—273no—
413H(5,𝔽2)—271no—
414C2×#41—31no—
415C2×#41—273no—
416C2×#41—273no—
417C2×#41—272no—
418C2×#41—276no—
419C3×C9#58511yescharacteristic, complemented, Sylow
420C23×C3—33no—
421C22×D3—96no—
422C22×D3—96no—
423C22×D3—96no—
424C22×D3—96no—
425C22×D3—186no—
426C22×D3C22×D913yes—
427C3×Q8—21no—
428C3×Q8—33no—
429C3×Q8—93no—
430C3×Q8—96no—
431C3×Q8C22×D911yescharacteristic
432C3×Q8C2×D3211yescharacteristic
433C3×D4—23no—
434C3×D4—36no—
435C3×D4—33no—
436C3×D4—96no—
437C3×D4—91no—
438C3×D4C22×D913yes—
439C3×D4C2×D3213yes—
440C2×C3×C4—23no—
441C2×C3×C4—33no—
442C2×C3×C4—33no—
443C2×C3×C4—33no—
444C2×C3×C4—31no—
445C2×C3×C4—33no—
446C2×C3×C4—96no—
447C2×C3×C4—96no—
448C2×C3×C4C22×D913yes—
449C2×C3×C4C2×D3213yes—
450#67—36no—
451#67—96no—
452#67—912no—
453#67—92no—
454#67—96no—
455#67—96no—
456#67—92no—
457#67—96no—
458#67—186no—
459#67C22×D916yes—
460C2×Dic3—36no—
461C2×Dic3—96no—
462C2×Dic3—92no—
463C2×Dic3C22×D913yes—
464D12—31no—
465D12—96no—
466D12—912no—
467D12—96no—
468D12—93no—
469D12—96no—
470D12—93no—
471D12—96no—
472D12—183no—
473D12—186no—
474D12C22×D913yes—
475C4×D3—36no—
476C4×D3—96no—
477C4×D3—912no—
478C4×D3—92no—
479C4×D3—96no—
480C4×D3—96no—
481C4×D3—96no—
482C4×D3—96no—
483C4×D3—912no—
484C4×D3—186no—
485C4×D3C22×D913yes—
486C4×D3C22×D911yescharacteristic
487C4×D3C22×D913yes—
488C4×D3C22×D913yes—
489Dic6—33no—
490Dic6—36no—
491Dic6—96no—
492Dic6—91no—
493Dic6—91no—
494Dic6—96no—
495Dic6—181no—
496Dic6C22×D913yes—
497C2×C32—23no—
498C2×C32C24×D311yescharacteristic
499C3⋊D3—66no—
500C3×D3—23no—
501C3×D3—21no—
502C3×D3—62no—
503C2×C9—23no—
504C2×C9—21no—
505C2×C9—43no—
506C2×C9—63no—
507C2×C9—61no—
508C2×C9C24×D311yescharacteristic
509D9—22no—
510D9—66no—
511D9—126no—
512#41—31no—
513#41—33no—
514#41—33no—
515#41—96no—
516#41—93no—
517#41—276no—
518#41—276no—
519#41—271no—
520#41—273no—
521#41—276no—
522#41—276no—
523#41—276no—
524#41—2712no—
525#41—276no—
526#41—276no—
527#41—272no—
528#41—273no—
529#41D3×D911yescharacteristic, complemented
530C2×Q8—31no—
531C2×Q8—92no—
532C2×Q8—93no—
533C2×Q8—273no—
534C2×Q8—276no—
535C2×D4—33no—
536C2×D4—273no—
537C2×D4—276no—
538C2×D4—276no—
539C2×D4—2712no—
540C2×D4—276no—
541C2×D4—276no—
542C2×D4—273no—
543C22×C4—33no—
544C22×C4—276no—
545C22×C4—276no—
546C22×C3—23no—
547C22×C3—33no—
548C22×C3—31no—
549C22×C3—92no—
550C22×C3—123no—
551C22×C3C23×D913yes—
552C22×C3C22×D3213yes—
553C2×D3—32no—
554C2×D3—43no—
555C2×D3—96no—
556C2×D3—96no—
557C2×D3—186no—
558C2×D3—366no—
559C2×D3—366no—
560C2×D3—366no—
561C2×D3—366no—
562C2×D3—726no—
563C2×D3C23×D913yes—
564C2×D3C23×D911yescharacteristic
565C3×C4—21no—
566C3×C4—23no—
567C3×C4—33no—
568C3×C4—31no—
569C3×C4—96no—
570C3×C4C23×D911yescharacteristic
571C3×C4C23×D913yes—
572C3×C4C22×D3211yescharacteristic
573C3×C4C22×D3213yes—
574Dic3—36no—
575Dic3—92no—
576Dic3—92no—
577Dic3—182no—
578Dic3C23×D913yes—
579Dic3C23×D911yescharacteristic
580C32#596411yescharacteristic
581C9—21no—
582C9#596811yescharacteristic, complemented
583C23—33no—
584C23—276no—
585C23—276no—
586Q8—33no—
587Q8—93no—
588Q8—96no—
589Q8—276no—
590Q8—271no—
591Q8C2×D3×D911yescharacteristic
592D4—36no—
593D4—33no—
594D4—96no—
595D4—91no—
596D4—276no—
597D4—276no—
598D4—2712no—
599D4—272no—
600D4—276no—
601D4—273no—
602D4—276no—
603D4C2×D3×D913yes—
604C2×C4—33no—
605C2×C4—31no—
606C2×C4—33no—
607C2×C4—33no—
608C2×C4—33no—
609C2×C4—96no—
610C2×C4—96no—
611C2×C4—276no—
612C2×C4—2712no—
613C2×C4—272no—
614C2×C4—276no—
615C2×C4—276no—
616C2×C4C2×D3×D913yes—
617C2×C3—23no—
618C2×C3—23no—
619C2×C3—21no—
620C2×C3—43no—
621C2×C3—63no—
622C2×C3—61no—
623C2×C3—182no—
624C2×C3C24×D911yescharacteristic
625C2×C3C23×D3211yescharacteristic
626D3—21no—
627D3—23no—
628D3—62no—
629D3—186no—
630D3—186no—
631D3—366no—
632C22—31no—
633C22—33no—
634C22—92no—
635C22—123no—
636C22—276no—
637C22—1086no—
638C22—1086no—
639C22C22×D3×D913yes—
640C4—31no—
641C4—33no—
642C4—96no—
643C4—272no—
644C4C22×D3×D911yescharacteristic
645C4C22×D3×D913yes—
646C3—21no—
647C3#1059313911yescharacteristic, complemented
648C3#1059669711yescharacteristic
649C2—23no—
650C2—61no—
651C2—63no—
652C2—182no—
653C2—546no—
654C2C23×D3×D911yescharacteristic
subgroupG/H
centerC2C23×D3×D9
commutator, cosocleC2×C3×C9C25
FrattiniC2×C3C23×D32
FittingC3×C9×#41C22
socleC2×C32C24×D3
hypercenter#41D3×D9
hypocenterC3×C9#585
LCSG ▷ C2×C3×C9 ▷ C3×C9
UCSC1 ◁ C2 ◁ #41

Complements

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

NH ≅ G/Ntype
1. #11722672C2semidirect
2. D3×C9×#41C2semidirect
3. C3×#63716C2semidirect
4. D9×#239C2semidirect
5. #11722509C2semidirect
6. #11722500C2semidirect
7. #11722498C2semidirect
8. #11722478C2semidirect
9. D3×#3697C2semidirect
10. C2×#88642C2semidirect
11. #11722464C2semidirect
14. C3×C9×#41C22semidirect
15. #88746C22semidirect
16. D4×C3⋊D9C22semidirect
17. #88736C22semidirect
18. C2×C4⋊(C3⋊D9)C22semidirect
19. C9×#239C22semidirect
20. D3×Q8×C9C22semidirect
21. C3×Q8×D9C22semidirect
22. C9×#237C22semidirect
23. D3×D4×C9C22semidirect
24. C3×#3699C22semidirect
25. C9×#235C22semidirect
26. C2×C4×D3×C9C22semidirect
27. C3×#3697C22semidirect
28. C2×C3×Dic18C22semidirect
29. #88668C22semidirect
30. D3×#611C22semidirect
31. C2×#6195C22semidirect
32. #88662C22semidirect
33. C2×(C3⋊D9)⋊C4C22semidirect
34. #88658C22semidirect
35. D9×D12C22semidirect
36. D3×D36C22semidirect
37. C4×D3×D9C22semidirect
38. #88642C22semidirect
39. #88642C22semidirect
40. #88642C22semidirect
41. #88642C22semidirect
42. #88641C22semidirect
43. #88640C22semidirect
44. #88639C22semidirect
45. D3×Dic18C22semidirect
46. (C3⋊D9)⋊Q8C22semidirect
47. #88635C22semidirect
61. #63716D3semidirect
145. C9×#41C2×D3semidirect
155. Q8×D9C2×D3semidirect
158. #3699C2×D3semidirect
172. #3697C2×D3semidirect
179. C2×Dic18C2×D3semidirect
208. D3×#41D9semidirect
307. C3×#41C2×D9semidirect
325. #239C2×D9semidirect
330. D3×Q8C2×D9semidirect
336. #237C2×D9semidirect
345. D3×D4C2×D9semidirect
360. #235C2×D9semidirect
371. C2×C4×D3C2×D9semidirect
419. C3×C9#585semidirect
529. #41D3×D9semidirect
582. C9#5968semidirect
647. C3#10593139semidirect

Maximal subgroups

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

subgroupindex
#117226722
D3×C9×#412
C3×#637162
D9×#2392
#117225092
#117225002
#117224982
#117224782
D3×#36972
C2×#886422
#117224642
#105966973
#105931393