Finite Group Databasebeta

← #385391

Subgroups — #385391

Order 512.

934 conjugacy classes of subgroups.

The subgroup lattice has 676 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1953C211yesmaximal, characteristic, complemented
2#21098C211yesmaximal, characteristic
3C2×#1929C211yesmaximal, characteristic
4#20886C211yesmaximal, characteristic
5C2×#1861C211yesmaximal, characteristic
6#20852C211yesmaximal, characteristic
7#20807C211yesmaximal, characteristic, complemented
8#11341C211yesmaximal, characteristic, complemented
9#11326C211yesmaximal, characteristic, complemented
10#11275C211yesmaximal, characteristic, complemented
11#11250C211yesmaximal, characteristic, complemented
12#11114C211yesmaximal, characteristic
13#11108C211yesmaximal, characteristic
14#10910C211yesmaximal, characteristic, complemented
15#10828C211yesmaximal, characteristic, complemented
16C2×#568C2211yescharacteristic
17C2×#430C2211yescharacteristic
18C2×#430C2211yescharacteristic
19C2×#430C2211yescharacteristic
20C22×#108C2211yescharacteristic
21C22×#108C2211yescharacteristic
22C22×#108C2211yescharacteristic
23C22×#108C2211yescharacteristic
24#2905C2211yescharacteristic
25#2905C2211yescharacteristic
26C2×#423C2211yescharacteristic
27C2×#421C2211yescharacteristic
28C2×#421C2211yescharacteristic
29C22×#104C2211yescharacteristic
30C2×#416C2211yescharacteristic
31#2879C2211yescharacteristic
32C2×#408C2211yescharacteristic
33C2×#405C2211yescharacteristic
34C2×#405C2211yescharacteristic
35#1953—24no—
36#1938—22no—
37#1937—22no—
38#1929C412yescomplemented
39#1929C412yescomplemented
40#1929C412yescomplemented
41#1929C412yescomplemented
42#1866—22no—
43#1866—22no—
44#1861—24no—
45#1855—22no—
46#1855—22no—
47#1849—22no—
48#1848—22no—
49C2×#344C2211yescharacteristic
50C2×#344C2211yescharacteristic
51C2×#344C2211yescharacteristic
52C2×#344C2211yescharacteristic
53#1738C2211yescharacteristic
54C2×#341C2211yescharacteristic
55C2×#341C2211yescharacteristic
56#1735C2211yescharacteristic
57#1733C2211yescharacteristic
58#1733C2211yescharacteristic
59C2×#339C2211yescharacteristic
60C2×#339C2211yescharacteristic
61C2×#339C2211yescharacteristic
62#1729C2211yescharacteristic
63#1728C2211yescharacteristic
64C2×#336C2211yescharacteristic
65C22×#41C2311yescharacteristic
66#568—22no—
67#568—41no—
68#568—41no—
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—41no—
91#430—41no—
92#430—41no—
93C2×#108—22no—
94C2×#108—22no—
95C2×#108—22no—
96C2×#108—22no—
97C2×#108—22no—
98C2×#108—22no—
99C2×#108—21no—
100C2×#108—21no—
101C2×#108—21no—
102C2×#108—21no—
103C2×#108—41no—
104C2×#108—41no—
105C2×#108—41no—
106C2×#108—41no—
107C2×#108—41no—
108C2×#108C2×C412yes—
109C2×#108C2×C412yes—
110C2×#108D412yes—
111C2×#108D412yes—
112#424—22no—
113#424—22no—
114#424—41no—
115#424—41no—
116#423—22no—
117#423—22no—
118#423—22no—
119#423—41no—
120C2×C4⋊C8—41no—
121#421—22no—
122#421—41no—
123#421—41no—
124#421—41no—
125C2×#104—22no—
126C2×#104—22no—
127C2×#104—22no—
128C2×#104—22no—
129C2×#104C2×C412yes—
130C2×#104C2×C412yes—
131#416C2×C412yes—
132#416C2×C412yes—
133#416D412yes—
134#416D412yes—
135C2×#103Q811yescharacteristic
136C2×#103Q811yescharacteristic
137C2×#102Q811yescharacteristic
138C2×#102Q811yescharacteristic
139#408—21no—
140#408—21no—
141#408—22no—
142#408—41no—
143#408C2×C412yes—
144#408C2×C412yes—
145#407—22no—
146C2×C22⋊C8—21no—
147C2×C22⋊C8—21no—
148#405—21no—
149#405—21no—
150#405—22no—
151#405—41no—
152#405C2×C412yes—
153#405C2×C412yes—
154C2×#97—21no—
155C2×#97—21no—
156C2×C4×C8—21no—
157C2×C4×C8—21no—
158#344—22no—
159#344—22no—
160#344—41no—
161#344D414yes—
162#341—22no—
163#341—21no—
164#341—21no—
165#341—21no—
166#341—21no—
167#340—21no—
168#340—21no—
169#339—22no—
170#339—22no—
171#339—22no—
172#339—41no—
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—44no—
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—21no—
242C2×M16—21no—
243C2×M16—21no—
244C2×M16—21no—
245C2×M16—42no—
246C2×M16—44no—
247C2×M16—44no—
248C2×M16C22×C412yes—
249C2×M16C22×C412yes—
250C2×M16C2×D412yes—
251C2×M16C2×D412yes—
252C2×M16C2×D412yes—
253C2×M16C2×D412yes—
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#117C2×D412yes—
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—24no—
314#108—24no—
315#108—42no—
316#108—42no—
317#108—42no—
318#108—42no—
319#108—44no—
320#108—44no—
321#108—42no—
322#108—42no—
323#108—44no—
324C4⋊C8—44no—
325C4⋊C8—44no—
326C4⋊C8—44no—
327#104—24no—
328#104—24no—
329#104—44no—
330#104—44no—
331#103C4⋊C412yes—
332#103C4⋊C412yes—
333#103C4⋊C412yes—
334#103C4⋊C412yes—
335#102C4⋊C412yes—
336#102C4⋊C412yes—
337#102C4⋊C412yes—
338#102C4⋊C412yes—
339C22⋊C8—22no—
340C22⋊C8—22no—
341C22⋊C8—22no—
342C22⋊C8—22no—
343C22⋊C8—44no—
344#97—22no—
345#97—22no—
346#97—42no—
347C4×C8—22no—
348C4×C8—22no—
349C4×C8—42no—
350C24—21no—
351C242−1+411yescharacteristic
352#41—22no—
353#41—22no—
354#41—22no—
355#41—22no—
356#41—22no—
357#41—22no—
358#41—22no—
359#41—22no—
360#41—41no—
361#41—41no—
362#41—41no—
363#41—41no—
364#41—41no—
365#41—41no—
366#41—41no—
367#41—41no—
368C2×Q8—21no—
369C2×Q8—21no—
370C2×Q8—21no—
371C2×Q8—21no—
372C2×Q8—21no—
373C2×Q8C2×C4⋊C412yes—
374C2×D4—22no—
375C2×D4—22no—
376C2×D4—21no—
377C2×D4—21no—
378C2×D4—21no—
379C2×D4—21no—
380C2×D4—21no—
381C2×D4—41no—
382C2×D4—41no—
383C2×D4—41no—
384C2×D4—41no—
385C2×D4C2×C4⋊C412yes—
386C22×C4—21no—
387C22×C4—21no—
388C22×C4—21no—
389C22×C4—21no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—21no—
393C22×C4—42no—
394C22×C4—41no—
395C22×C4—41no—
396C22×C4—41no—
397C22×C4—41no—
398C22×C4—41no—
399C22×C4—41no—
400C22×C4—41no—
401C22×C4—41no—
402C22×C4—81no—
403C22×C4C23×C411yescharacteristic
404C22×C4C22×D411yescharacteristic
405C22×C4C22×D411yescharacteristic
406C22×C4C22×D411yescharacteristic
407C22×C4C22×Q811yescharacteristic
408C22×C4C2×#4111yescharacteristic
409C22×C4C2×#4111yescharacteristic
410C22×C4H(5,𝔽2)11yescharacteristic
411C22×C4H(5,𝔽2)11yescharacteristic
412C22×C4H(5,𝔽2)11yescharacteristic
413C22×C4H(5,𝔽2)11yescharacteristic
414C22×C42−1+411yescharacteristic
415C22×C42−1+411yescharacteristic
416C22×C42−1+411yescharacteristic
417M16—22no—
418M16—22no—
419M16—22no—
420M16—22no—
421M16—22no—
422M16—22no—
423M16—22no—
424M16—22no—
425M16—22no—
426M16—22no—
427M16—22no—
428M16—48no—
429M16—41no—
430M16—41no—
431M16—41no—
432M16—41no—
433M16—41no—
434M16—41no—
435M16—41no—
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—
449M16C4×D414yes—
450C2×C8—22no—
451C2×C8—21no—
452C2×C8—21no—
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—21no—
462C2×C8—21no—
463C2×C8—21no—
464C2×C8—21no—
465C2×C8—21no—
466C2×C8—21no—
467C2×C8—21no—
468C2×C8—21no—
469C2×C8—22no—
470C2×C8—21no—
471C2×C8—21no—
472C2×C8—22no—
473C2×C8—21no—
474C2×C8—21no—
475C2×C8—21no—
476C2×C8—21no—
477C2×C8—22no—
478C2×C8—22no—
479C2×C8—22no—
480C2×C8—22no—
481C2×C8—48no—
482C2×C8—41no—
483C2×C8—41no—
484C2×C8—41no—
485C2×C8—41no—
486C2×C8—41no—
487C2×C8—41no—
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×C8C2×C4⋊C411yescharacteristic
500C2×C8C2×C4⋊C411yescharacteristic
501C2×C8C2×C4⋊C411yescharacteristic
502C2×C8C2×C4⋊C411yescharacteristic
503C2×C8C4×D411yescharacteristic
504C2×C8C4×D411yescharacteristic
505C2×C8C4×D411yescharacteristic
506C2×C8C4×D411yescharacteristic
507C4⋊C4—21no—
508C4⋊C4—21no—
509C4⋊C4—22no—
510C4⋊C4—41no—
511C4⋊C4—41no—
512C4⋊C4—41no—
513C4⋊C4C2×C4⋊C412yes—
514C4⋊C4C2×C4⋊C412yes—
515C22⋊C4—22no—
516C22⋊C4—22no—
517C22⋊C4—22no—
518C22⋊C4—41no—
519C22⋊C4—41no—
520C22⋊C4—41no—
521C42—22no—
522C42—22no—
523C42—21no—
524C42—21no—
525C42—21no—
526C42—21no—
527C42—22no—
528C42—41no—
529C42—41no—
530C42—41no—
531C42—41no—
532C42—41no—
533C23—22no—
534C23—41no—
535C23—41no—
536C23—41no—
537C23—41no—
538C23—81no—
539C23—81no—
540C23#52011yescharacteristic
541C23#53611yescharacteristic
542C23#54211yescharacteristic
543C23#54711yescharacteristic
544C23D4×Q811yescharacteristic
545C23#56311yescharacteristic
546C23#56311yescharacteristic
547Q8—22no—
548Q8—22no—
549Q8—41no—
550Q8—41no—
551D4—22no—
552D4—22no—
553D4—42no—
554D4—42no—
555D4—42no—
556D4—42no—
557D4—41no—
558D4—41no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—42no—
575C2×C4—81no—
576C2×C4—81no—
577C2×C4—82no—
578C2×C4—82no—
579C2×C4—81no—
580C2×C4—81no—
581C2×C4—81no—
582C2×C4—81no—
583C2×C4—81no—
584C2×C4C22×C4⋊C411yescharacteristic
585C2×C4C2×C4×D411yescharacteristic
586C2×C4C2×C4×D411yescharacteristic
587C2×C4#52011yescharacteristic
588C2×C4#52011yescharacteristic
589C2×C4#52011yescharacteristic
590C2×C4#53411yescharacteristic
591C2×C4#53511yescharacteristic
592C2×C4#53511yescharacteristic
593C2×C4#53611yescharacteristic
594C2×C4#53611yescharacteristic
595C2×C4#53911yescharacteristic
596C2×C4#53911yescharacteristic
597C2×C4#54211yescharacteristic
598C2×C4#54411yescharacteristic
599C2×C4#54411yescharacteristic
600C2×C4D4211yescharacteristic
601C2×C4D4211yescharacteristic
602C2×C4D4211yescharacteristic
603C2×C4#54711yescharacteristic
604C2×C4#54711yescharacteristic
605C2×C4#54811yescharacteristic
606C2×C4D4×Q811yescharacteristic
607C2×C4#55311yescharacteristic
608C2×C4#55411yescharacteristic
609C2×C4#55411yescharacteristic
610C2×C4#56011yescharacteristic
611C2×C4#56311yescharacteristic
612C8—21no—
613C8—21no—
614C8—21no—
615C8—21no—
616C8—21no—
617C8—21no—
618C8—21no—
619C8—21no—
620C8—24no—
621C8—42no—
622C8—42no—
623C8—42no—
624C8—42no—
625C8—81no—
626C8—81no—
627C8—81no—
628C22—21no—
629C22—21no—
630C22—21no—
631C22—21no—
632C22—21no—
633C22—21no—
634C22—41no—
635C22—41no—
636C22—41no—
637C22—41no—
638C22—42no—
639C22—42no—
640C22—42no—
641C22—161no—
642C22—161no—
643C22#232611yescharacteristic
644C22D4×C4⋊C411yescharacteristic
645C22#238311yescharacteristic
646C22#267711yescharacteristic
647C22#268011yescharacteristic
648C22#271811yescharacteristic
649C22#272911yescharacteristic
650C4—42no—
651C4—42no—
652C4—42no—
653C4—42no—
654C4—81no—
655C4—82no—
656C4—161no—
657C4#231711yescharacteristic
658C4#232211yescharacteristic
659C4D4×C4⋊C411yescharacteristic
660C4#236111yescharacteristic
661C4#266911yescharacteristic
662C4#267411yescharacteristic
663C4#272011yescharacteristic
664C4#279011yescharacteristic
665C2—21no—
666C2—21no—
667C2—21no—
668C2—21no—
669C2—21no—
670C2—21no—
671C2—82no—
672C2#1440511yescharacteristic
673C2#2269411yescharacteristic
674C2#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
7. #20807C2semidirect
8. #11341C2semidirect
9. #11326C2semidirect
10. #11275C2semidirect
11. #11250C2semidirect
14. #10910C2semidirect
15. #10828C2semidirect
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
#210982
C2×#19292
#208862
C2×#18612
#208522
#208072
#113412
#113262
#112752
#112502
#111142
#111082
#109102
#108282