Finite Group Databasebeta

← #373183

Subgroups — #373183

Order 512.

982 conjugacy classes of subgroups.

The subgroup lattice has 678 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1953C211yesmaximal, characteristic, complemented
2#21072C211yesmaximal, characteristic
3C2×#1929C211yesmaximal, characteristic
4C2×#1861C211yesmaximal, characteristic
5#20852C211yesmaximal, characteristic
6#20459C211yesmaximal, characteristic, complemented
7#20422C211yesmaximal, characteristic
8#11326C211yesmaximal, characteristic, complemented
9#11250C211yesmaximal, characteristic, complemented
10#11114C211yesmaximal, characteristic
11#11104C211yesmaximal, characteristic
12#10909C211yesmaximal, characteristic, complemented
13#10901C211yesmaximal, characteristic, complemented
14#10894C211yesmaximal, characteristic, complemented
15#10872C211yesmaximal, characteristic, complemented
16C2×#568C2211yescharacteristic
17C2×#430C2211yescharacteristic
18C2×#430C2211yescharacteristic
19C2×#430C2211yescharacteristic
20C22×#108C2211yescharacteristic
21C22×#108C2211yescharacteristic
22C22×#108C2211yescharacteristic
23C2×#427C2211yescharacteristic
24C2×#423C2211yescharacteristic
25C2×#421C2211yescharacteristic
26C2×#421C2211yescharacteristic
27C22×#104C2211yescharacteristic
28C2×#416C2211yescharacteristic
29#2879C2211yescharacteristic
30C2×#408C2211yescharacteristic
31#2873C2211yescharacteristic
32#2873C2211yescharacteristic
33C2×#405C2211yescharacteristic
34C2×#405C2211yescharacteristic
35#1953—24no—
36#1929—22no—
37#1929—22no—
38#1929C412yescomplemented
39#1929C412yescomplemented
40#1929C412yescomplemented
41#1929C412yescomplemented
42#1861—24no—
43#1855—22no—
44#1855—22no—
45#1778—22no—
46C4×#108—22no—
47#1765—22no—
48#1764—22no—
49C2×#344C2211yescharacteristic
50C2×#344C2211yescharacteristic
51C2×#344C2211yescharacteristic
52C2×#344C2211yescharacteristic
53#1738C2211yescharacteristic
54C2×#341C2211yescharacteristic
55C2×#341C2211yescharacteristic
56C2×#341C2211yescharacteristic
57#1733C2211yescharacteristic
58#1733C2211yescharacteristic
59C2×#339C2211yescharacteristic
60C2×#339C2211yescharacteristic
61C2×#339C2211yescharacteristic
62#1729C2211yescharacteristic
63#1728C2211yescharacteristic
64C2×#336C2211yescharacteristic
65C22×#41C2311yescharacteristic
66#568—22no—
67#568—22no—
68#568—22no—
69#568—41no—
70C2×#131—22no—
71C2×#131—41no—
72C22×M16—22no—
73C22×M16C2311yescharacteristic
74C22×M16C2311yescharacteristic
75C22×M16C2311yescharacteristic
76C22×M16C2311yescharacteristic
77C22×M16C2311yescharacteristic
78C22×M16C2311yescharacteristic
79C22×M16C2311yescharacteristic
80C22×M16C2311yescharacteristic
81C23×C8C2311yescharacteristic
82C23×C8C2311yescharacteristic
83C2×#117C2311yescharacteristic
84C2×#117C2311yescharacteristic
85C2×#117C2311yescharacteristic
86C22×C42C2311yescharacteristic
87#430—22no—
88#430—22no—
89#430—22no—
90#430—22no—
91#430—22no—
92#430D414yes—
93C2×#108—22no—
94C2×#108—22no—
95C2×#108—22no—
96C2×#108—22no—
97C2×#108—22no—
98C2×#108—22no—
99C2×#108—22no—
100C2×#108—21no—
101C2×#108—21no—
102C2×#108—21no—
103C2×#108—21no—
104C2×#108—41no—
105C2×#108—41no—
106C2×#108C2×C412yes—
107C2×#108C2×C412yes—
108#427—22no—
109#427—41no—
110C2×#107—21no—
111C2×#106—21no—
112#423—22no—
113#423—22no—
114C2×C4⋊C8—41no—
115#421—22no—
116#421—22no—
117#421—22no—
118#421—41no—
119C2×#104—22no—
120C2×#104—22no—
121C2×#104—22no—
122C2×#104—22no—
123C2×#104C2×C412yes—
124C2×#104C2×C412yes—
125#416C2×C412yes—
126#416C2×C412yes—
127#416D412yes—
128#416D412yes—
129C2×#103Q811yescharacteristic
130C2×#103Q811yescharacteristic
131C2×#102Q811yescharacteristic
132C2×#102Q811yescharacteristic
133#408—21no—
134#408—21no—
135#408—22no—
136#408—41no—
137#408C2×C412yes—
138#408C2×C412yes—
139#407—22no—
140C2×C22⋊C8—21no—
141C2×C22⋊C8—21no—
142#405—21no—
143#405—21no—
144#405—22no—
145#405—22no—
146#405—22no—
147#405—41no—
148#405—41no—
149#405—41no—
150#405C2×C412yes—
151#405C2×C412yes—
152C4×M16—22no—
153C4×M16—22no—
154C2×#97—21no—
155C2×#97—21no—
156C2×C4×C8—21no—
157C2×C4×C8—21no—
158#344—22no—
159#344—22no—
160#344—22no—
161#344—22no—
162#341—22no—
163#341—21no—
164#341—21no—
165#341—21no—
166#341—21no—
167#341D412yes—
168#341D412yes—
169#339—22no—
170#339—22no—
171#339—22no—
172#339—22no—
173#339C2×C412yes—
174#339C2×C412yes—
175#336—21no—
176#336—21no—
177#336—41no—
178#336C2×C411yescharacteristic
179#336C2×C411yescharacteristic
180#336C2×C411yescharacteristic
181#336C2×C411yescharacteristic
182C2×#41—21no—
183C2×#41—21no—
184C2×#41—21no—
185C2×#41—21no—
186C2×#41—21no—
187C2×#41—21no—
188C2×#41—21no—
189C2×#41—21no—
190C2×#41—21no—
191C2×#41—21no—
192C2×#41C22×C412yes—
193C2×#41C2×D412yes—
194C22×Q8C2×Q811yescharacteristic
195C22×D4—21no—
196C22×D4C2×Q811yescharacteristic
197C23×C4—21no—
198C23×C4—21no—
199C23×C4C2411yescharacteristic
200#131—28no—
201#131—44no—
202#131—44no—
203#131—44no—
204C2×M16—21no—
205C2×M16—21no—
206C2×M16—21no—
207C2×M16—21no—
208C2×M16—21no—
209C2×M16—21no—
210C2×M16—21no—
211C2×M16—21no—
212C2×M16—21no—
213C2×M16—21no—
214C2×M16—21no—
215C2×M16—21no—
216C2×M16—21no—
217C2×M16—21no—
218C2×M16—21no—
219C2×M16—21no—
220C2×M16—21no—
221C2×M16—21no—
222C2×M16—21no—
223C2×M16—21no—
224C2×M16—21no—
225C2×M16—21no—
226C2×M16—21no—
227C2×M16—21no—
228C2×M16—21no—
229C2×M16—21no—
230C2×M16—21no—
231C2×M16—21no—
232C2×M16—21no—
233C2×M16—21no—
234C2×M16—21no—
235C2×M16—21no—
236C2×M16—21no—
237C2×M16—21no—
238C2×M16—21no—
239C2×M16—21no—
240C2×M16—21no—
241C2×M16—42no—
242C2×M16—44no—
243C2×M16—44no—
244C2×M16C22×C412yes—
245C2×M16C22×C412yes—
246C2×M16C2×D412yes—
247C2×M16C2×D412yes—
248C2×M16C2×D412yes—
249C2×M16C2×D412yes—
250C2×M16C2×D412yes—
251C2×M16#4112yes—
252C2×M16#4112yes—
253C2×M16#4112yes—
254C2×M16#4112yes—
255C22×C8—21no—
256C22×C8—21no—
257C22×C8—21no—
258C22×C8—21no—
259C22×C8—21no—
260C22×C8—21no—
261C22×C8—21no—
262C22×C8—21no—
263C22×C8—21no—
264C22×C8—21no—
265C22×C8—21no—
266C22×C8—21no—
267C22×C8—21no—
268C22×C8—21no—
269C22×C8—21no—
270C22×C8—21no—
271C22×C8—42no—
272C22×C8C22×C411yescharacteristic
273C22×C8C22×C411yescharacteristic
274C22×C8C22×C411yescharacteristic
275C22×C8C22×C411yescharacteristic
276C22×C8C2×D411yescharacteristic
277C22×C8C2×D411yescharacteristic
278C22×C8C2×D411yescharacteristic
279C22×C8C2×D411yescharacteristic
280C22×C8C2×Q811yescharacteristic
281C22×C8C2×Q811yescharacteristic
282C22×C8#4111yescharacteristic
283C22×C8#4111yescharacteristic
284#117—21no—
285#117—21no—
286#117—21no—
287#117—21no—
288#117—21no—
289#117—21no—
290#117—21no—
291#117—21no—
292#117—21no—
293#117C22×C412yes—
294#117C2×D412yes—
295#117#4112yes—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4C2×Q811yescharacteristic
299C2×C4⋊C4C2×Q811yescharacteristic
300C2×C22⋊C4—21no—
301C2×C22⋊C4—21no—
302C2×C22⋊C4—21no—
303C2×C42—21no—
304C2×C42—21no—
305C2×C42—21no—
306C2×C42—21no—
307C2×C42—21no—
308C2×C42—21no—
309C2×C42—21no—
310C2×C42—21no—
311C2×C42C22×C411yescharacteristic
312C2×C42C22×C411yescharacteristic
313#108—28no—
314#108—24no—
315#108—24no—
316#108—28no—
317#108—42no—
318#108—42no—
319#108—42no—
320#108—42no—
321#108—44no—
322#107—42no—
323#106—42no—
324C4⋊C8—44no—
325#104—28no—
326#104—24no—
327#104—24no—
328#104—44no—
329#103C4⋊C412yes—
330#103C4⋊C412yes—
331#103C4⋊C412yes—
332#103C4⋊C412yes—
333#102C4⋊C412yes—
334#102C4⋊C412yes—
335#102C4⋊C412yes—
336#102C4⋊C412yes—
337C22⋊C8—22no—
338C22⋊C8—22no—
339C22⋊C8—22no—
340C22⋊C8—22no—
341C22⋊C8—44no—
342#97—22no—
343#97—22no—
344#97—42no—
345#97—42no—
346#97—42no—
347C4×C8—22no—
348C4×C8—22no—
349C4×C8—42no—
350C4×C8—42no—
351C4×C8—42no—
352C24—21no—
353C242−1+411yescharacteristic
354#41—22no—
355#41—22no—
356#41—22no—
357#41—22no—
358#41—22no—
359#41—22no—
360#41—22no—
361#41—22no—
362#41—41no—
363#41—41no—
364#41—41no—
365#41—41no—
366#41—41no—
367#41—41no—
368#41—41no—
369#41—41no—
370C2×Q8—21no—
371C2×Q8—21no—
372C2×Q8—21no—
373C2×Q8—21no—
374C2×Q8—21no—
375C2×Q8C2×C4⋊C412yes—
376C2×D4—22no—
377C2×D4—22no—
378C2×D4—21no—
379C2×D4—21no—
380C2×D4—21no—
381C2×D4—21no—
382C2×D4—21no—
383C2×D4—41no—
384C2×D4—41no—
385C2×D4—41no—
386C2×D4—41no—
387C2×D4C2×C4⋊C412yes—
388C22×C4—21no—
389C22×C4—21no—
390C22×C4—21no—
391C22×C4—21no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—21no—
395C22×C4—42no—
396C22×C4—41no—
397C22×C4—41no—
398C22×C4—41no—
399C22×C4—41no—
400C22×C4—41no—
401C22×C4—41no—
402C22×C4—41no—
403C22×C4—41no—
404C22×C4—81no—
405C22×C4C23×C411yescharacteristic
406C22×C4C22×D411yescharacteristic
407C22×C4C22×D411yescharacteristic
408C22×C4C22×D411yescharacteristic
409C22×C4C22×Q811yescharacteristic
410C22×C4C2×#4111yescharacteristic
411C22×C4C2×#4111yescharacteristic
412C22×C4C2×#4111yescharacteristic
413C22×C4C2×#4111yescharacteristic
414C22×C4C2×#4111yescharacteristic
415C22×C4C2×#4111yescharacteristic
416C22×C4H(5,𝔽2)11yescharacteristic
417C22×C4H(5,𝔽2)11yescharacteristic
418C22×C42−1+411yescharacteristic
419M16—22no—
420M16—22no—
421M16—22no—
422M16—22no—
423M16—22no—
424M16—22no—
425M16—22no—
426M16—22no—
427M16—22no—
428M16—22no—
429M16—22no—
430M16—22no—
431M16—22no—
432M16—22no—
433M16—22no—
434M16—22no—
435M16—48no—
436M16—41no—
437M16—41no—
438M16—41no—
439M16—41no—
440M16—41no—
441M16—41no—
442M16—41no—
443M16—41no—
444M16—41no—
445M16—41no—
446M16—41no—
447M16—41no—
448M16—41no—
449M16—41no—
450M16C4×D414yes—
451M16#12114yes—
452C2×C8—22no—
453C2×C8—21no—
454C2×C8—21no—
455C2×C8—21no—
456C2×C8—21no—
457C2×C8—21no—
458C2×C8—21no—
459C2×C8—21no—
460C2×C8—21no—
461C2×C8—22no—
462C2×C8—22no—
463C2×C8—22no—
464C2×C8—21no—
465C2×C8—22no—
466C2×C8—21no—
467C2×C8—22no—
468C2×C8—21no—
469C2×C8—21no—
470C2×C8—21no—
471C2×C8—21no—
472C2×C8—21no—
473C2×C8—21no—
474C2×C8—21no—
475C2×C8—21no—
476C2×C8—21no—
477C2×C8—21no—
478C2×C8—21no—
479C2×C8—21no—
480C2×C8—21no—
481C2×C8—21no—
482C2×C8—22no—
483C2×C8—22no—
484C2×C8—22no—
485C2×C8—22no—
486C2×C8—22no—
487C2×C8—48no—
488C2×C8—41no—
489C2×C8—41no—
490C2×C8—41no—
491C2×C8—41no—
492C2×C8—41no—
493C2×C8—41no—
494C2×C8—41no—
495C2×C8—41no—
496C2×C8—41no—
497C2×C8—41no—
498C2×C8—41no—
499C2×C8—41no—
500C2×C8—41no—
501C2×C8C2×C4⋊C411yescharacteristic
502C2×C8C2×C4⋊C411yescharacteristic
503C2×C8C2×C4⋊C411yescharacteristic
504C2×C8C2×C4⋊C411yescharacteristic
505C2×C8C4×D411yescharacteristic
506C2×C8C4×D411yescharacteristic
507C2×C8C4×D411yescharacteristic
508C2×C8C4×D411yescharacteristic
509C4⋊C4—21no—
510C4⋊C4—21no—
511C4⋊C4—22no—
512C4⋊C4—41no—
513C4⋊C4—41no—
514C4⋊C4—41no—
515C4⋊C4C2×C4⋊C412yes—
516C4⋊C4C2×C4⋊C412yes—
517C22⋊C4—22no—
518C22⋊C4—22no—
519C22⋊C4—22no—
520C22⋊C4—41no—
521C22⋊C4—41no—
522C22⋊C4—41no—
523C42—22no—
524C42—22no—
525C42—21no—
526C42—21no—
527C42—21no—
528C42—21no—
529C42—22no—
530C42—41no—
531C42—41no—
532C42—41no—
533C42—41no—
534C42—41no—
535C23—22no—
536C23—41no—
537C23—41no—
538C23—41no—
539C23—41no—
540C23—81no—
541C23—81no—
542C23#52011yescharacteristic
543C23#54111yescharacteristic
544C23#54211yescharacteristic
545C23#54711yescharacteristic
546C23#54811yescharacteristic
547C23D4×Q811yescharacteristic
548C23#55611yescharacteristic
549Q8—22no—
550Q8—22no—
551Q8—41no—
552Q8—41no—
553D4—22no—
554D4—22no—
555D4—42no—
556D4—42no—
557D4—42no—
558D4—42no—
559D4—41no—
560D4—41no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—42no—
577C2×C4—81no—
578C2×C4—81no—
579C2×C4—82no—
580C2×C4—82no—
581C2×C4—81no—
582C2×C4—81no—
583C2×C4—81no—
584C2×C4—81no—
585C2×C4—81no—
586C2×C4C22×C4⋊C411yescharacteristic
587C2×C4C2×C4×D411yescharacteristic
588C2×C4C2×C4×D411yescharacteristic
589C2×C4C4×#4111yescharacteristic
590C2×C4C4×#4111yescharacteristic
591C2×C4#52011yescharacteristic
592C2×C4C2×#12111yescharacteristic
593C2×C4#52511yescharacteristic
594C2×C4#52511yescharacteristic
595C2×C4#52511yescharacteristic
596C2×C4#52911yescharacteristic
597C2×C4#53211yescharacteristic
598C2×C4#53311yescharacteristic
599C2×C4#53311yescharacteristic
600C2×C4#53811yescharacteristic
601C2×C4#53911yescharacteristic
602C2×C4D4211yescharacteristic
603C2×C4D4211yescharacteristic
604C2×C4#54611yescharacteristic
605C2×C4#54611yescharacteristic
606C2×C4#54711yescharacteristic
607C2×C4#54711yescharacteristic
608C2×C4D4×Q811yescharacteristic
609C2×C4#55111yescharacteristic
610C2×C4#55211yescharacteristic
611C2×C4#55411yescharacteristic
612C2×C4#55411yescharacteristic
613C2×C4#55911yescharacteristic
614C8—21no—
615C8—21no—
616C8—21no—
617C8—21no—
618C8—21no—
619C8—21no—
620C8—21no—
621C8—21no—
622C8—24no—
623C8—24no—
624C8—42no—
625C8—42no—
626C8—42no—
627C8—42no—
628C8—42no—
629C8—81no—
630C22—21no—
631C22—21no—
632C22—21no—
633C22—21no—
634C22—21no—
635C22—21no—
636C22—41no—
637C22—41no—
638C22—41no—
639C22—41no—
640C22—42no—
641C22—42no—
642C22—42no—
643C22—161no—
644C22—161no—
645C22#233411yescharacteristic
646C22D4×C4⋊C411yescharacteristic
647C22#236711yescharacteristic
648C22#242911yescharacteristic
649C22#246111yescharacteristic
650C22#246511yescharacteristic
651C22#247811yescharacteristic
652C4—42no—
653C4—42no—
654C4—42no—
655C4—42no—
656C4—81no—
657C4—82no—
658C4—161no—
659C4#229611yescharacteristic
660C4C4×#12111yescharacteristic
661C4D4×C4⋊C411yescharacteristic
662C4#235911yescharacteristic
663C4#242411yescharacteristic
664C4#245511yescharacteristic
665C4#255611yescharacteristic
666C4#258911yescharacteristic
667C2—21no—
668C2—21no—
669C2—21no—
670C2—21no—
671C2—21no—
672C2—21no—
673C2—82no—
674C2#1411211yescharacteristic
675C2#2269411yescharacteristic
676C2#2269511yescharacteristic
subgroupG/H
centerC2×C4D42
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
socleC22D4×C4⋊C4
prime radicalC22×#41C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#1953C2semidirect
6. #20459C2semidirect
8. #11326C2semidirect
9. #11250C2semidirect
12. #10909C2semidirect
13. #10901C2semidirect
14. #10894C2semidirect
15. #10872C2semidirect
38. #1929C4semidirect
39. #1929C4semidirect
40. #1929C4semidirect
41. #1929C4semidirect

Maximal subgroups

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

subgroupindex
C2×#19532
#210722
C2×#19292
C2×#18612
#208522
#204592
#204222
#113262
#112502
#111142
#111042
#109092
#109012
#108942
#108722