Finite Group Databasebeta

← #385586

Subgroups — #385586

Order 512.

886 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
1#21178C211yesmaximal, characteristic, complemented
2#21098C211yesmaximal, characteristic
3#21072C211yesmaximal, characteristic
4C2×#1866C211yesmaximal, characteristic
5#20668C211yesmaximal, characteristic
6#11326C211yesmaximal, characteristic, complemented
7#11275C211yesmaximal, characteristic, complemented
8#11250C211yesmaximal, characteristic, complemented
9#11240C211yesmaximal, characteristic, complemented
10#11114C211yesmaximal, characteristic
11#11080C211yesmaximal, characteristic
12#10910C211yesmaximal, characteristic, complemented
13#10894C211yesmaximal, characteristic, complemented
14#10873C211yesmaximal, characteristic, complemented
15#10773C211yesmaximal, characteristic
16C22×#131C2211yescharacteristic
17C2×#430C2211yescharacteristic
18C22×#108C2211yescharacteristic
19C22×#108C2211yescharacteristic
20C22×#108C2211yescharacteristic
21C22×#108C2211yescharacteristic
22#2905C2211yescharacteristic
23C2×#424C2211yescharacteristic
24C2×#423C2211yescharacteristic
25C2×#421C2211yescharacteristic
26C2×#421C2211yescharacteristic
27C22×#104C2211yescharacteristic
28C2×#416C2211yescharacteristic
29#2879C2211yescharacteristic
30#2879C2211yescharacteristic
31#2873C2211yescharacteristic
32C2×#405C2211yescharacteristic
33#1950—22no—
34#1950—22no—
35#1938—22no—
36#1937—22no—
37#1929—22no—
38#1929—22no—
39#1866C412yescomplemented
40#1866C412yescomplemented
41#1866C412yescomplemented
42#1866C412yescomplemented
43#1811—22no—
44#1810—22no—
45C2×#344C2211yescharacteristic
46C2×#344C2211yescharacteristic
47C2×#344C2211yescharacteristic
48C2×#344C2211yescharacteristic
49#1738C2211yescharacteristic
50#1738C2211yescharacteristic
51#1738C2211yescharacteristic
52C2×#341C2211yescharacteristic
53C2×#341C2211yescharacteristic
54#1735C2211yescharacteristic
55#1733C2211yescharacteristic
56#1733C2211yescharacteristic
57C2×#339C2211yescharacteristic
58C2×#339C2211yescharacteristic
59C2×#339C2211yescharacteristic
60#1729C2211yescharacteristic
61#1728C2211yescharacteristic
62C2×#336C2211yescharacteristic
63C22×#41C2311yescharacteristic
64C2×#131—21no—
65C2×#131—21no—
66C2×#131—41no—
67C2×#131—41no—
68C2×#131—41no—
69C2×#131—41no—
70C2×#131C2×C412yes—
71C2×#131C2×C412yes—
72C22×M16—21no—
73C22×M16C2311yescharacteristic
74C22×M16C2311yescharacteristic
75C22×M16C2311yescharacteristic
76C22×M16C2311yescharacteristic
77C22×M16C2311yescharacteristic
78C22×M16C2311yescharacteristic
79C22×M16C2311yescharacteristic
80C22×M16C2311yescharacteristic
81C23×C8—21no—
82C23×C8C2311yescharacteristic
83C23×C8C2311yescharacteristic
84C2×#117C2311yescharacteristic
85C2×#117C2311yescharacteristic
86C2×#117C2311yescharacteristic
87C22×C42C2311yescharacteristic
88#430—41no—
89#430—41no—
90C2×#108—22no—
91C2×#108—22no—
92C2×#108—21no—
93C2×#108—21no—
94C2×#108—21no—
95C2×#108—21no—
96C2×#108—41no—
97C2×#108—41no—
98C2×#108—41no—
99C2×#108—41no—
100C2×#108—41no—
101C2×#108—41no—
102C2×#108—41no—
103C2×#108C2×C412yes—
104C2×#108C2×C412yes—
105C2×#108D412yes—
106C2×#108D412yes—
107#424—21no—
108#424—21no—
109#424—22no—
110#424—41no—
111#424C2×C412yes—
112#424C2×C412yes—
113#423—22no—
114#423—22no—
115#423—41no—
116C2×C4⋊C8—21no—
117C2×C4⋊C8—21no—
118C2×C4⋊C8—41no—
119#421—22no—
120#421—22no—
121#421—41no—
122#421—41no—
123C2×#104—22no—
124C2×#104—22no—
125C2×#104—41no—
126C2×#104—41no—
127C2×#104C2×C412yes—
128C2×#104C2×C412yes—
129#416C2×C412yes—
130#416C2×C412yes—
131#416D412yes—
132#416D412yes—
133C2×#103Q811yescharacteristic
134C2×#103Q811yescharacteristic
135C2×#102Q811yescharacteristic
136C2×#102Q811yescharacteristic
137#408—22no—
138#408—22no—
139#408—41no—
140#408—41no—
141#407—22no—
142#407—22no—
143#405—22no—
144#405—22no—
145#405—41no—
146#405—41no—
147C4×M16—22no—
148C2×#97—21no—
149C2×C4×C8—21no—
150#344—41no—
151#344—41no—
152#344—41no—
153#344—41no—
154#341—22no—
155#341—22no—
156#341—22no—
157#341—21no—
158#341—21no—
159#341—21no—
160#341—21no—
161#340—21no—
162#340—21no—
163#339—22no—
164#339—22no—
165#339—22no—
166#339—41no—
167#339C2×C412yes—
168#339C2×C412yes—
169#336—21no—
170#336—21no—
171#336—41no—
172#336C2×C411yescharacteristic
173#336C2×C411yescharacteristic
174#336C2×C411yescharacteristic
175#336C2×C411yescharacteristic
176C2×#41—21no—
177C2×#41—21no—
178C2×#41—21no—
179C2×#41—21no—
180C2×#41—21no—
181C2×#41—21no—
182C2×#41—21no—
183C2×#41—21no—
184C2×#41—21no—
185C2×#41—21no—
186C2×#41C22×C412yes—
187C2×#41C2×D412yes—
188C22×Q8C2×Q811yescharacteristic
189C22×D4—21no—
190C22×D4C2×Q811yescharacteristic
191C23×C4—21no—
192C23×C4—21no—
193C23×C4C2411yescharacteristic
194#131—42no—
195#131—42no—
196#131—42no—
197#131—42no—
198#131—42no—
199#131—42no—
200#131—42no—
201#131—42no—
202C2×M16—21no—
203C2×M16—21no—
204C2×M16—21no—
205C2×M16—21no—
206C2×M16—21no—
207C2×M16—22no—
208C2×M16—22no—
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—21no—
246C2×M16—21no—
247C2×M16—42no—
248C2×M16—42no—
249C2×M16C22×C412yes—
250C2×M16C22×C412yes—
251C2×M16C2×D412yes—
252C2×M16C2×D412yes—
253C2×M16#4112yes—
254C22×C8—21no—
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—22no—
268C22×C8—22no—
269C22×C8—21no—
270C22×C8—21no—
271C22×C8—21no—
272C22×C8—42no—
273C22×C8—42no—
274C22×C8C22×C411yescharacteristic
275C22×C8C22×C411yescharacteristic
276C22×C8C22×C411yescharacteristic
277C22×C8C22×C411yescharacteristic
278C22×C8C2×D411yescharacteristic
279C22×C8C2×D411yescharacteristic
280C22×C8C2×D411yescharacteristic
281C22×C8C2×D411yescharacteristic
282C22×C8C2×Q811yescharacteristic
283C22×C8C2×Q811yescharacteristic
284C22×C8#4111yescharacteristic
285C22×C8#4111yescharacteristic
286#117—21no—
287#117—21no—
288#117—21no—
289#117—21no—
290#117—21no—
291#117—21no—
292#117—21no—
293#117—21no—
294#117—21no—
295#117C22×C412yes—
296#117C2×D412yes—
297#117#4112yes—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4C2×Q811yescharacteristic
301C2×C4⋊C4C2×Q811yescharacteristic
302C2×C22⋊C4—21no—
303C2×C22⋊C4—21no—
304C2×C22⋊C4—21no—
305C2×C42—21no—
306C2×C42—21no—
307C2×C42—21no—
308C2×C42—21no—
309C2×C42—21no—
310C2×C42—21no—
311C2×C42—21no—
312C2×C42—21no—
313C2×C42C22×C411yescharacteristic
314C2×C42C22×C411yescharacteristic
315#108—42no—
316#108—42no—
317#108—42no—
318#108—42no—
319#108—42no—
320#108—42no—
321#108—42no—
322#108—42no—
323#108—44no—
324C4⋊C8—22no—
325C4⋊C8—22no—
326C4⋊C8—22no—
327C4⋊C8—22no—
328C4⋊C8—44no—
329C4⋊C8—44no—
330#104—42no—
331#104—42no—
332#104—44no—
333#104—44no—
334#103C4⋊C412yes—
335#103C4⋊C412yes—
336#103C4⋊C412yes—
337#103C4⋊C412yes—
338#102C4⋊C412yes—
339#102C4⋊C412yes—
340#102C4⋊C412yes—
341#102C4⋊C412yes—
342C22⋊C8—44no—
343C22⋊C8—44no—
344#97—42no—
345#97—42no—
346C4×C8—42no—
347C4×C8—42no—
348C24—21no—
349C242−1+411yescharacteristic
350#41—22no—
351#41—22no—
352#41—22no—
353#41—22no—
354#41—22no—
355#41—22no—
356#41—22no—
357#41—22no—
358#41—41no—
359#41—41no—
360#41—41no—
361#41—41no—
362#41—41no—
363#41—41no—
364#41—41no—
365#41—41no—
366C2×Q8—21no—
367C2×Q8—21no—
368C2×Q8—21no—
369C2×Q8—21no—
370C2×Q8—21no—
371C2×Q8C2×C4⋊C412yes—
372C2×D4—22no—
373C2×D4—22no—
374C2×D4—21no—
375C2×D4—21no—
376C2×D4—21no—
377C2×D4—21no—
378C2×D4—21no—
379C2×D4—41no—
380C2×D4—41no—
381C2×D4—41no—
382C2×D4—41no—
383C2×D4C2×C4⋊C412yes—
384C22×C4—21no—
385C22×C4—21no—
386C22×C4—21no—
387C22×C4—21no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—21no—
391C22×C4—42no—
392C22×C4—41no—
393C22×C4—41no—
394C22×C4—41no—
395C22×C4—41no—
396C22×C4—41no—
397C22×C4—41no—
398C22×C4—41no—
399C22×C4—41no—
400C22×C4—81no—
401C22×C4C23×C411yescharacteristic
402C22×C4C22×D411yescharacteristic
403C22×C4C22×D411yescharacteristic
404C22×C4C22×Q811yescharacteristic
405C22×C4C2×#4111yescharacteristic
406C22×C4C2×#4111yescharacteristic
407C22×C4C2×#4111yescharacteristic
408C22×C4H(5,𝔽2)11yescharacteristic
409C22×C4H(5,𝔽2)11yescharacteristic
410C22×C4H(5,𝔽2)11yescharacteristic
411C22×C4H(5,𝔽2)11yescharacteristic
412C22×C4H(5,𝔽2)11yescharacteristic
413C22×C42−1+411yescharacteristic
414C22×C42−1+411yescharacteristic
415M16—22no—
416M16—22no—
417M16—22no—
418M16—22no—
419M16—22no—
420M16—22no—
421M16—22no—
422M16—22no—
423M16—22no—
424M16—22no—
425M16—44no—
426M16—44no—
427M16—41no—
428M16—41no—
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—
449C2×C8—21no—
450C2×C8—21no—
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—21no—
470C2×C8—21no—
471C2×C8—21no—
472C2×C8—21no—
473C2×C8—22no—
474C2×C8—22no—
475C2×C8—22no—
476C2×C8—22no—
477C2×C8—22no—
478C2×C8—44no—
479C2×C8—44no—
480C2×C8—41no—
481C2×C8—41no—
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#53911yescharacteristic
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×C4C4×#4111yescharacteristic
587C2×C4#51811yescharacteristic
588C2×C4#51811yescharacteristic
589C2×C4#51911yescharacteristic
590C2×C4#53411yescharacteristic
591C2×C4#53511yescharacteristic
592C2×C4#53711yescharacteristic
593C2×C4#53711yescharacteristic
594C2×C4#53811yescharacteristic
595C2×C4#53811yescharacteristic
596C2×C4#53811yescharacteristic
597C2×C4#54011yescharacteristic
598C2×C4#54411yescharacteristic
599C2×C4#54411yescharacteristic
600C2×C4D4211yescharacteristic
601C2×C4#54611yescharacteristic
602C2×C4#54611yescharacteristic
603C2×C4#54711yescharacteristic
604C2×C4#54811yescharacteristic
605C2×C4#55211yescharacteristic
606C2×C4#55411yescharacteristic
607C2×C4#55411yescharacteristic
608C2×C4#55411yescharacteristic
609C2×C4#55611yescharacteristic
610C2×C4#56211yescharacteristic
611C2×C4#56211yescharacteristic
612C8—21no—
613C8—21no—
614C8—21no—
615C8—21no—
616C8—21no—
617C8—21no—
618C8—21no—
619C8—21no—
620C8—42no—
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—82no—
642C22—82no—
643C22#233511yescharacteristic
644C22D4×C4⋊C411yescharacteristic
645C22#237811yescharacteristic
646C22#269511yescharacteristic
647C22#271511yescharacteristic
648C22#272011yescharacteristic
649C22#272911yescharacteristic
650C4—42no—
651C4—42no—
652C4—42no—
653C4—42no—
654C4—81no—
655C4—82no—
656C4—161no—
657C4#231611yescharacteristic
658C4#232111yescharacteristic
659C4#234911yescharacteristic
660C4#236211yescharacteristic
661C4#267111yescharacteristic
662C4#267611yescharacteristic
663C4#276811yescharacteristic
664C4#278811yescharacteristic
665C2—21no—
666C2—21no—
667C2—21no—
668C2—21no—
669C2—21no—
670C2—21no—
671C2—82no—
672C2#1440811yescharacteristic
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. #21178C2semidirect
6. #11326C2semidirect
7. #11275C2semidirect
8. #11250C2semidirect
9. #11240C2semidirect
12. #10910C2semidirect
13. #10894C2semidirect
14. #10873C2semidirect
39. #1866C4semidirect
40. #1866C4semidirect
41. #1866C4semidirect
42. #1866C4semidirect

Maximal subgroups

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

subgroupindex
#211782
#210982
#210722
C2×#18662
#206682
#113262
#112752
#112502
#112402
#111142
#110802
#109102
#108942
#108732
#107732