Finite Group Databasebeta

← #11904607

Subgroups — #11904607

Order 1152.

2487 conjugacy classes of subgroups.

The subgroup lattice has 648 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#63715C211yesmaximal, characteristic, complemented
2C2×#41×D9C211yesmaximal, characteristic, complemented
3C9×#550C211yesmaximal, characteristic, complemented
4#10589936C211yesmaximal, characteristic, complemented
5#10589933C211yesmaximal, characteristic, complemented
6#10589819C213yesmaximal, complemented
7#10589817C213yesmaximal, complemented
8#10589714C213yesmaximal, complemented
9#10589711C213yesmaximal, complemented
10#10589710C213yesmaximal, complemented
11D9×#121C213yesmaximal, complemented
12#10589678C211yesmaximal, characteristic, complemented
13#10589676C211yesmaximal, characteristic, complemented
14#10589652C213yesmaximal, complemented
15#10589649C213yesmaximal, complemented
16#86769—31nomaximal
17C2×C9×#41C2211yescharacteristic, complemented
18C2×C9×#41C2211yescharacteristic, complemented
19#63715—28no—
20#41×D9—28no—
21C2×#3701C2211yescharacteristic
22C2×#3701C2211yescharacteristic, complemented
23C2×Q8×D9C2211yescharacteristic, complemented
24C2×#3699C2213yescomplemented
25C2×D4×D9C2213yescomplemented
26C2×D4×D9C2213yescomplemented
27C2×D4×D9C2213yescomplemented
28C2×D4×D9C2216yescomplemented
29C2×#3697C2213yescomplemented
30C2×#3697C2213yescomplemented
31C22×D36C2213yescomplemented
32C22×C4×D9C2213yescomplemented
33C9×#127C2213yescomplemented
34C9×#121C2213yescomplemented
35C9×#121C2213yescomplemented
36C4×Q8×C9C2211yescharacteristic, complemented
37C4×D4×C9C2213yescomplemented
38Q8×Dic9C2211yescharacteristic, complemented
39#63502C2213yescomplemented
40#63501C2216yescomplemented
41#63501C2213yescomplemented
42#63500C2213yescomplemented
43#63500C2216yescomplemented
44#63500C2213yescomplemented
45#63499C2216yescomplemented
46#63499C2216yescomplemented
47#63499C2213yescomplemented
48#63498C2213yescomplemented
49#63492C2213yescomplemented
50C4×#611C2213yescomplemented
51C4×#611C2213yescomplemented
52#63486C2213yescomplemented
53#63459C2213yescomplemented
54#63458C2211yescharacteristic, complemented
55#63457C2213yescomplemented
56#63456C2213yescomplemented
57#63455C2213yescomplemented
58C4⋊C4×D9C2213yescomplemented
59#63446C2213yescomplemented
60#63446C2216yescomplemented
61#63445C2213yescomplemented
62#63444C2213yescomplemented
63C22⋊C4×D9C2216yescomplemented
64#63440C2213yescomplemented
65#63437C2213yescomplemented
66C4×D36C2213yescomplemented
67C4×D36C2213yescomplemented
68#63434C2213yescomplemented
69C2×#1009—31no—
70C2×D3×#41—31no—
71C3×#550D311yescharacteristic
72#5834—31no—
73#5831—31no—
74#5717—33no—
75#5715—33no—
76#5612—33no—
77#5609—33no—
78#5608—33no—
79D3×#121—33no—
80#5576—31no—
81#5574—31no—
82#5550—33no—
83#5547—33no—
84C23×D9C2313yes—
85C23×D9C2316yes—
86C9×#41—24no—
87C9×#41—24no—
88C2×Q8×C9C2311yescharacteristic
89C2×D4×C9C2313yes—
90C2×D4×C9C2313yes—
91C2×D4×C9C2313yes—
92C2×D4×C9C2316yes—
93C22×C4×C9C2313yes—
94C22×C4×C9C2313yes—
95C2×#611C2313yes—
96C2×#611C2316yes—
97C2×#611C2313yes—
98C2×#611C2313yes—
99C2×#611C2313yes—
100C2×#611C2316yes—
101C22×Dic9C2313yes—
102#3701—24no—
103#3701D418yescomplemented
104Q8×D9—24no—
105#3699—212no—
106D4×D9—212no—
107D4×D9—212no—
108D4×D9—212no—
109D4×D9—412no—
110#3697—212no—
111#3697—212no—
112C2×D36—212no—
113C2×D36C2313yes—
114C2×D36C2313yes—
115C2×D36C2313yes—
116C2×D36C2313yes—
117C2×C4×D9—212no—
118C2×C4×D9C2313yes—
119C2×C4×D9C2313yes—
120C2×C4×D9C2313yes—
121C2×C4×D9C2311yescharacteristic
122C2×C4×D9C2316yes—
123C2×C4×D9C2313yes—
124C2×Dic18C2313yes—
125C2×Dic18C2311yescharacteristic
126C9×C4⋊C4C2311yescharacteristic
127C9×C4⋊C4C2313yes—
128C9×C22⋊C4C2313yes—
129C9×C22⋊C4C2313yes—
130C42×C9C2313yes—
131#3676C2313yes—
132#3676C2313yes—
133#3676C2316yes—
134(C2×D9)⋊C4C2311yescharacteristic
135(C2×D9)⋊C4C2313yes—
136(C2×D9)⋊C4C2313yes—
137(C2×D9)⋊C4C2311yescharacteristic
138(C2×D9)⋊C4C2313yes—
139(C2×D9)⋊C4C2313yes—
140(C2×D9)⋊C4C2316yes—
141C4⋊Dic9C2313yes—
142C4⋊Dic9C2313yes—
143Dic9⋊C4C2311yescharacteristic
144Dic9⋊C4C2313yes—
145Dic9⋊C4C2311yescharacteristic
146Dic9⋊C4C2313yes—
147C4×Dic9C2313yes—
148#3488—91noSylow
149C2×C3×#41C2×D311yescharacteristic
150C2×C3×#41C2×D311yescharacteristic
151#1009—68no—
152D3×#41—68no—
153C2×#239—31no—
154C2×#239—31no—
155C2×D3×Q8—31no—
156C2×#237—33no—
157C2×D3×D4—33no—
158C2×D3×D4—33no—
159C2×D3×D4—33no—
160C2×D3×D4—36no—
161C2×#235—33no—
162C2×#235—33no—
163C22×D12—33no—
164C22×C4×D3—33no—
165C3×#127C2×D313yes—
166C3×#121C2×D313yes—
167C3×#121C2×D313yes—
168C3×C4×Q8C2×D311yescharacteristic
169C3×C4×D4C2×D313yes—
170Q8×Dic3—31no—
171#940—33no—
172#939—33no—
173#939—36no—
174#938—33no—
175#938—33no—
176#938—36no—
177#937—36no—
178#937—36no—
179#937—33no—
180#936—33no—
181#930—33no—
182C4×#67—33no—
183C4×#67—33no—
184#924—33no—
185#897—33no—
186#896—31no—
187#895—33no—
188#894—33no—
189#893—33no—
190D3×C4⋊C4—33no—
191#884—33no—
192#884—36no—
193#883—33no—
194#882—33no—
195D3×C22⋊C4—36no—
196#878—33no—
197#875—33no—
198C4×D12—33no—
199C4×D12—33no—
200#872—33no—
201C23×C9C2413yes—
202C23×C9C2413yes—
203C22×D9—26no—
204C22×D9—26no—
205C22×D9—212no—
206C22×D9—43no—
207C22×D9—46no—
208C22×D9—46no—
209C22×D9C2413yes—
210C22×D9C2413yes—
211C22×D9C2411yescharacteristic
212Q8×C9C2×D414yescomplemented
213D4×C9—26no—
214D4×C9—26no—
215D4×C9—46no—
216D4×C9—43no—
217C2×C4×C9—26no—
218C2×C4×C9—26no—
219C2×C4×C9C2411yescharacteristic
220C2×C4×C9C2413yes—
221C2×C4×C9C2411yescharacteristic
222C2×C4×C9C2411yescharacteristic
223C2×C4×C9C2413yes—
224#611—26no—
225#611—212no—
226#611—26no—
227#611—26no—
228#611—46no—
229#611—43no—
230C2×Dic9—26no—
231C2×Dic9C2411yescharacteristic
232C2×Dic9C2413yes—
233C2×Dic9C2411yescharacteristic
234C2×Dic9C2413yes—
235C2×Dic9C2411yescharacteristic
236D36—26no—
237D36—26no—
238D36—26no—
239D36C2×D4112yescomplemented
240C4×D9—26no—
241C4×D9—26no—
242C4×D9—26no—
243C4×D9—22no—
244C4×D9—46no—
245C4×D9C2×D4112yescomplemented
246Dic18—26no—
247Dic18—22no—
248C2×H(5,𝔽2)—91no—
249C22×#41—91no—
250#550—91no—
251#550D911yescharacteristic, complemented
252#548—91no—
253#548—91no—
254#546—93no—
255#546—93no—
256D42—93no—
257D42—93no—
258#537—93no—
259#535—93no—
260#525—93no—
261C2×#121—93no—
262#520—91no—
263C23×D3—33no—
264C23×D3—36no—
265C3×#41—24no—
266C3×#41—24no—
267C2×C3×Q8C22×D311yescharacteristic
268C2×C3×D4C22×D313yes—
269C2×C3×D4C22×D313yes—
270C2×C3×D4C22×D313yes—
271C2×C3×D4C22×D316yes—
272C22×C3×C4C22×D313yes—
273C22×C3×C4C22×D313yes—
274C2×#67—33no—
275C2×#67—33no—
276C2×#67—36no—
277C2×#67—33no—
278C2×#67—33no—
279C2×#67—36no—
280C22×Dic3—33no—
281#239—38no—
282#239—64no—
283D3×Q8—64no—
284#237—612no—
285D3×D4—612no—
286D3×D4—612no—
287D3×D4—612no—
288D3×D4—1212no—
289#235—612no—
290#235—612no—
291C2×D12—33no—
292C2×D12—33no—
293C2×D12—33no—
294C2×D12—33no—
295C2×D12—612no—
296C2×C4×D3—33no—
297C2×C4×D3—33no—
298C2×C4×D3—33no—
299C2×C4×D3—33no—
300C2×C4×D3—31no—
301C2×C4×D3—36no—
302C2×C4×D3—612no—
303C2×Dic6—33no—
304C2×Dic6—31no—
305C3×C4⋊C4C22×D313yes—
306C3×C4⋊C4C22×D311yescharacteristic
307C3×C22⋊C4C22×D313yes—
308C3×C22⋊C4C22×D313yes—
309C3×C42C22×D313yes—
310#217—33no—
311#217—36no—
312#217—33no—
313(C2×D3)⋊C4—33no—
314(C2×D3)⋊C4—33no—
315(C2×D3)⋊C4—31no—
316(C2×D3)⋊C4—33no—
317(C2×D3)⋊C4—36no—
318(C2×D3)⋊C4—33no—
319(C2×D3)⋊C4—31no—
320C4⋊Dic3—33no—
321C4⋊Dic3—33no—
322Dic3⋊C4—31no—
323Dic3⋊C4—33no—
324Dic3⋊C4—33no—
325Dic3⋊C4—31no—
326C4×Dic3—33no—
327C22×C9—23no—
328C22×C9—23no—
329C22×C9—23no—
330C22×C9—23no—
331C22×C9—23no—
332C22×C9—23no—
333C22×C9C2511yescharacteristic
334C2×D9—21no—
335C2×D9—23no—
336C2×D9—21no—
337C2×D9—23no—
338C2×D9—23no—
339C2×D9—23no—
340C2×D9—23no—
341C2×D9—412no—
342C2×D9—812no—
343C2×D9C22×D416yes—
344C2×D9H(5,𝔽2)12yescomplemented
345C4×C9—21no—
346C4×C9—21no—
347C4×C9—23no—
348C4×C9—21no—
349C4×C9C22×D416yes—
350Dic9—23no—
351Dic9—21no—
352Dic9—23no—
353Dic9—21no—
354Dic9C22×D412yes—
355H(5,𝔽2)—188no—
356C2×#41—91no—
357C2×#41—91no—
358C2×#41—93no—
359C2×#41—93no—
360C2×#41—93no—
361C2×#41—188no—
362C2×#41C2×D911yescharacteristic, complemented
363C2×#41C2×D911yescharacteristic, complemented
364C22×Q8—91no—
365C22×D4—93no—
366C22×D4—93no—
367C22×D4—93no—
368C22×D4—96no—
369C22×D4—93no—
370C23×C4—93no—
371#127—93no—
372#127C2×D913yescomplemented
373#124—93no—
374#124—93no—
375#123—93no—
376#123—93no—
377C22⋊Q8—93no—
378C22⋊Q8—91no—
379C22⋊Q8—93no—
380C22⋊Q8—93no—
381#121—96no—
382#121—93no—
383#121—93no—
384#121—96no—
385#121—93no—
386#121—93no—
387#121—93no—
388#121—93no—
389#121—96no—
390#121C2×D913yescomplemented
391#121C2×D913yescomplemented
392#120—93no—
393#120—96no—
394#120—96no—
395#120—93no—
396C4×Q8—91no—
397C4×Q8C2×D911yescharacteristic, complemented
398C4×D4—93no—
399C4×D4—93no—
400C4×D4—93no—
401C4×D4—93no—
402C4×D4—93no—
403C4×D4C2×D913yescomplemented
404#117—93no—
405C2×C4⋊C4—93no—
406C2×C22⋊C4—96no—
407C23×C3C23×D313yes—
408C23×C3C23×D313yes—
409C22×D3—33no—
410C22×D3—31no—
411C22×D3—33no—
412C22×D3—66no—
413C22×D3—66no—
414C22×D3—612no—
415C22×D3—123no—
416C22×D3—126no—
417C22×D3—126no—
418C3×Q8D3×D414yes—
419C3×D4—26no—
420C3×D4—26no—
421C3×D4—46no—
422C3×D4—43no—
423C2×C3×C4—26no—
424C2×C3×C4—26no—
425C2×C3×C4C23×D311yescharacteristic
426C2×C3×C4C23×D313yes—
427C2×C3×C4C23×D311yescharacteristic
428C2×C3×C4C23×D311yescharacteristic
429C2×C3×C4C23×D313yes—
430#67—66no—
431#67—66no—
432#67—612no—
433#67—66no—
434#67—126no—
435#67—123no—
436C2×Dic3—31no—
437C2×Dic3—31no—
438C2×Dic3—33no—
439C2×Dic3—31no—
440C2×Dic3—33no—
441C2×Dic3—66no—
442D12—312no—
443D12—66no—
444D12—66no—
445D12—66no—
446C4×D3—312no—
447C4×D3—62no—
448C4×D3—66no—
449C4×D3—66no—
450C4×D3—66no—
451C4×D3—126no—
452Dic6—62no—
453Dic6—66no—
454C2×C9—43no—
455C2×C9—43no—
456C2×C9C23×D411yescharacteristic
457C2×C9C2×H(5,𝔽2)11yescharacteristic
458C2×C9#58511yescharacteristic
459D9—22no—
460D9—26no—
461D9—43no—
462C24—93no—
463C24—96no—
464#41—24no—
465#41—24no—
466#41—98no—
467#41—1812no—
468#41—1812no—
469#41—1812no—
470#41—184no—
471C2×Q8—91no—
472C2×Q8—93no—
473C2×Q8—184no—
474C2×Q8C22×D911yescharacteristic
475C2×D4—93no—
476C2×D4—96no—
477C2×D4—93no—
478C2×D4—93no—
479C2×D4—93no—
480C2×D4—96no—
481C2×D4—93no—
482C2×D4—93no—
483C2×D4—93no—
484C2×D4—93no—
485C2×D4—1812no—
486C2×D4—1812no—
487C2×D4—1812no—
488C2×D4—1812no—
489C2×D4—3612no—
490C2×D4C22×D913yes—
491C2×D4C22×D913yes—
492C2×D4C22×D913yes—
493C2×D4C22×D916yes—
494C22×C4—93no—
495C22×C4—93no—
496C22×C4—93no—
497C22×C4—93no—
498C22×C4—93no—
499C22×C4—91no—
500C22×C4—96no—
501C22×C4—1812no—
502C22×C4C22×D913yes—
503C22×C4C22×D913yes—
504C4⋊C4—93no—
505C4⋊C4—93no—
506C4⋊C4—91no—
507C4⋊C4—91no—
508C4⋊C4—93no—
509C4⋊C4—93no—
510C4⋊C4C22×D913yes—
511C4⋊C4C22×D911yescharacteristic
512C22⋊C4—96no—
513C22⋊C4—93no—
514C22⋊C4—93no—
515C22⋊C4—93no—
516C22⋊C4—96no—
517C22⋊C4—93no—
518C22⋊C4—93no—
519C22⋊C4—91no—
520C22⋊C4—91no—
521C22⋊C4—93no—
522C22⋊C4C22×D913yes—
523C22⋊C4C22×D913yes—
524C42—93no—
525C42C22×D913yes—
526C22×C3—23no—
527C22×C3—23no—
528C22×C3—23no—
529C22×C3—23no—
530C22×C3—23no—
531C22×C3—23no—
532C22×C3C24×D311yescharacteristic
533C2×D3—36no—
534C2×D3—32no—
535C2×D3—61no—
536C2×D3—61no—
537C2×D3—63no—
538C2×D3—63no—
539C2×D3—63no—
540C2×D3—63no—
541C2×D3—63no—
542C2×D3—1212no—
543C2×D3—2412no—
544C3×C4—21no—
545C3×C4—21no—
546C3×C4—21no—
547C3×C4—23no—
548C3×C4C2×D3×D416yes—
549Dic3—32no—
550Dic3—63no—
551Dic3—61no—
552Dic3—61no—
553Dic3—63no—
554C9#348811yescharacteristic, complemented, Sylow
555C23—93no—
556C23—91no—
557C23—93no—
558C23—186no—
559C23—186no—
560C23—1812no—
561C23—363no—
562C23—366no—
563C23—366no—
564C23C23×D913yes—
565C23C23×D913yes—
566Q8—182no—
567Q8—186no—
568Q8D4×D914yescomplemented
569D4—26no—
570D4—26no—
571D4—46no—
572D4—43no—
573D4—912no—
574D4—186no—
575D4—186no—
576D4—186no—
577D4—186no—
578D4—1812no—
579D4—186no—
580D4—186no—
581D4—366no—
582D4—363no—
583C2×C4—26no—
584C2×C4—26no—
585C2×C4—91no—
586C2×C4—912no—
587C2×C4—91no—
588C2×C4—93no—
589C2×C4—93no—
590C2×C4—91no—
591C2×C4—182no—
592C2×C4—186no—
593C2×C4—186no—
594C2×C4—186no—
595C2×C4—186no—
596C2×C4—366no—
597C2×C4C23×D911yescharacteristic
598C2×C4C23×D913yes—
599C2×C4C23×D911yescharacteristic
600C2×C4C23×D911yescharacteristic
601C2×C4C23×D913yes—
602C2×C3—43no—
603C2×C3—43no—
604C2×C3C22×D3×D411yescharacteristic
605C2×C3#596411yescharacteristic
606C2×C3D3×H(5,𝔽2)11yescharacteristic
607D3—62no—
608D3—66no—
609D3—123no—
610C22—23no—
611C22—23no—
612C22—23no—
613C22—23no—
614C22—23no—
615C22—23no—
616C22—96no—
617C22—92no—
618C22—181no—
619C22—181no—
620C22—183no—
621C22—183no—
622C22—183no—
623C22—183no—
624C22—183no—
625C22—3612no—
626C22—7212no—
627C22C24×D911yescharacteristic
628C4—21no—
629C4—21no—
630C4—21no—
631C4—23no—
632C4—92no—
633C4—181no—
634C4—183no—
635C4—181no—
636C4—183no—
637C4C2×D4×D916yes—
638C3#8676911yescharacteristic
639C2—43no—
640C2—43no—
641C2—182no—
642C2—186no—
643C2—363no—
644C2C22×D4×D911yescharacteristic
645C2#1059313911yescharacteristic
646C2D9×H(5,𝔽2)11yescharacteristic
subgroupG/H
centerC22C24×D9
commutator, cosocleC22×C9C25
Frattini, socleC22×C3C24×D3
FittingC9×#550C2
hypercenter#550D9
hypocenterC9#3488
LCSG ▷ C22×C9 ▷ C9
UCSC1 ◁ C22 ◁ #550

Complements

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

NH ≅ G/Ntype
1. C2×#63715C2semidirect
2. C2×#41×D9C2semidirect
3. C9×#550C2semidirect
4. #10589936C2semidirect
5. #10589933C2semidirect
6. #10589819C2semidirect
7. #10589817C2semidirect
8. #10589714C2semidirect
9. #10589711C2semidirect
10. #10589710C2semidirect
11. D9×#121C2semidirect
12. #10589678C2semidirect
13. #10589676C2semidirect
14. #10589652C2semidirect
15. #10589649C2semidirect
17. C2×C9×#41C22semidirect
18. C2×C9×#41C22semidirect
22. C2×#3701C22semidirect
23. C2×Q8×D9C22semidirect
24. C2×#3699C22semidirect
25. C2×D4×D9C22semidirect
26. C2×D4×D9C22semidirect
27. C2×D4×D9C22semidirect
28. C2×D4×D9C22semidirect
29. C2×#3697C22semidirect
30. C2×#3697C22semidirect
31. C22×D36C22semidirect
32. C22×C4×D9C22semidirect
33. C9×#127C22semidirect
34. C9×#121C22semidirect
35. C9×#121C22semidirect
36. C4×Q8×C9C22semidirect
37. C4×D4×C9C22semidirect
38. Q8×Dic9C22semidirect
39. #63502C22semidirect
40. #63501C22semidirect
41. #63501C22semidirect
42. #63500C22semidirect
43. #63500C22semidirect
44. #63500C22semidirect
45. #63499C22semidirect
46. #63499C22semidirect
47. #63499C22semidirect
48. #63498C22semidirect
49. #63492C22semidirect
50. C4×#611C22semidirect
51. C4×#611C22semidirect
52. #63486C22semidirect
53. #63459C22semidirect
54. #63458C22semidirect
55. #63457C22semidirect
56. #63456C22semidirect
57. #63455C22semidirect
58. C4⋊C4×D9C22semidirect
59. #63446C22semidirect
60. #63446C22semidirect
61. #63445C22semidirect
62. #63444C22semidirect
63. C22⋊C4×D9C22semidirect
64. #63440C22semidirect
65. #63437C22semidirect
66. C4×D36C22semidirect
67. C4×D36C22semidirect
68. #63434C22semidirect
103. #3701D4semidirect
212. Q8×C9C2×D4semidirect
239. D36C2×D4semidirect
245. C4×D9C2×D4semidirect
251. #550D9semidirect
344. C2×D9H(5,𝔽2)semidirect
362. C2×#41C2×D9semidirect
363. C2×#41C2×D9semidirect
372. #127C2×D9semidirect
390. #121C2×D9semidirect
391. #121C2×D9semidirect
397. C4×Q8C2×D9semidirect
403. C4×D4C2×D9semidirect
554. C9#3488semidirect
568. Q8D4×D9semidirect

Maximal subgroups

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

subgroupindex
C2×#637152
C2×#41×D92
C9×#5502
#105899362
#105899332
#105898192
#105898172
#105897142
#105897112
#105897102
D9×#1212
#105896782
#105896762
#105896522
#105896492
#867693