Finite Group Databasebeta

← #356752

Subgroups — #356752

Order 512.

1326 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×#1861C211yesmaximal, characteristic
2#20863C211yesmaximal, characteristic
3#20445C211yesmaximal, characteristic, complemented
4#20421C211yesmaximal, characteristic
5#20405C211yesmaximal, characteristic
6C2×C4×#104C211yesmaximal, characteristic
7#20343C211yesmaximal, characteristic
8#10897C211yesmaximal, characteristic, complemented
9#10895C211yesmaximal, characteristic, complemented
10#10894C211yesmaximal, characteristic, complemented
11#10893C211yesmaximal, characteristic, complemented
12#10839C211yesmaximal, characteristic, complemented
13#10826C211yesmaximal, characteristic, complemented
14#10772C211yesmaximal, characteristic
15C4×#339C211yesmaximal, characteristic, complemented
16C2×#568C2211yescharacteristic
17#3428C2211yescharacteristic
18C2×C4×#41C2211yescharacteristic
19C2×#430C2211yescharacteristic
20#2911C2211yescharacteristic
21C2×#423C2211yescharacteristic
22C2×#421C2211yescharacteristic
23C22×#104C2211yescharacteristic
24C22×#104C2211yescharacteristic
25#2896C2211yescharacteristic
26#2873C2211yescharacteristic
27#2873C2211yescharacteristic
28C2×C4×M16C2211yescharacteristic
29C2×C4×M16C2211yescharacteristic
30#2285C2211yescharacteristic
31C4×#117C2211yescharacteristic
32C2×C43C2211yescharacteristic
33#1861C418yescomplemented
34#1860C414yescomplemented
35#1859C414yescomplemented
36#1773—22no—
37#1772—22no—
38#1765—22no—
39#1764—22no—
40#1759—22no—
41C4×#104—22no—
42C4×#104C418yescomplemented
43C2×#344C2211yescharacteristic
44C2×#344C2211yescharacteristic
45#1742C2211yescharacteristic
46#1740C2211yescharacteristic
47#1738C2211yescharacteristic
48#1733—22no—
49#1733C2211yescharacteristic
50#1733C2211yescharacteristic
51#1733C2211yescharacteristic
52C2×#339—22no—
53C2×#339C2211yescharacteristic
54C2×#339C2211yescharacteristic
55C2×#339C2211yescharacteristic
56C2×#339C2211yescharacteristic
57C2×#339C2211yescharacteristic
58C2×#339C2211yescharacteristic
59C2×#339C2211yescharacteristic
60C2×#339C2211yescharacteristic
61C2×#339C2211yescharacteristic
62C2×#339C2211yescharacteristic
63C22×#41C2311yescharacteristic
64#568—22no—
65#568—22no—
66#568—41no—
67#568C2×C414yes—
68C2×#131—22no—
69C2×#131—41no—
70C22×M16—22no—
71C22×M16C2311yescharacteristic
72C22×M16C2311yescharacteristic
73C22×M16C2311yescharacteristic
74C22×M16C2311yescharacteristic
75C22×M16C2311yescharacteristic
76C22×M16C2311yescharacteristic
77#520—22no—
78#520C2×C414yes—
79#519—21no—
80#518—21no—
81#518—41no—
82C4×#41—22no—
83C4×#41—22no—
84C4×#41—22no—
85C4×#41—41no—
86C4×#41—41no—
87C4×#41C2×C414yes—
88C2×C4×Q8—21no—
89C2×C4×D4—21no—
90C2×C4×D4—41no—
91C2×#117—21no—
92C2×#117—22no—
93C2×#117C2311yescharacteristic
94C2×#117C2311yescharacteristic
95C2×#117C2311yescharacteristic
96C2×#117C2311yescharacteristic
97C2×#117C2311yescharacteristic
98C22×C42—21no—
99C22×C42—41no—
100C22×C42C2311yescharacteristic
101C22×C42C2311yescharacteristic
102C22×C42C2311yescharacteristic
103#430—22no—
104#430C2×C414yescomplemented
105C2×#108—22no—
106#428—22no—
107#428C2×C414yescomplemented
108#427—22no—
109#423—41no—
110#423C2×C414yescomplemented
111C2×C4⋊C8—41no—
112#421—22no—
113#421C2×C414yes—
114C2×#104—22no—
115C2×#104C2×C412yes—
116C2×#104C2×C412yes—
117C2×#104C2×C414yes—
118C2×#104C2×C414yes—
119C2×#104C2×C414yes—
120C2×#104D412yes—
121C2×#104D412yes—
122C2×#104Q814yes—
123#419—21no—
124#418—21no—
125#417C2×C414yes—
126#416—22no—
127#405—22no—
128#405—22no—
129#405—22no—
130#405—41no—
131C4×M16—22no—
132C4×M16—22no—
133C4×M16—22no—
134C4×M16—41no—
135C4×M16—41no—
136C4×M16C2×C414yescomplemented
137C2×#97—21no—
138C2×#97—21no—
139C2×C4×C8—21no—
140C2×C4×C8—21no—
141#384C2×C412yescomplemented
142#382C2×C412yescomplemented
143#380—41no—
144C4×C4⋊C4—22no—
145C4×C4⋊C4—41no—
146C4×C22⋊C4—41no—
147#376—21no—
148C43—21no—
149C43—41no—
150C43C2×C414yescomplemented
151#344—41no—
152#344C2×C414yescomplemented
153#343C2×C414yescomplemented
154#342—41no—
155#341—22no—
156#339—22no—
157#339—22no—
158#339—22no—
159#339—21no—
160#339—21no—
161#339—21no—
162#339—21no—
163#339—21no—
164#339—21no—
165#339—21no—
166#339—21no—
167#339—22no—
168#339C2×C412yescomplemented
169#339C2×C412yescomplemented
170#339C2×C414yescomplemented
171#339C2×C414yescomplemented
172#339C2×C414yescomplemented
173#339D412yes—
174#339D412yes—
175C2×#41—21no—
176C2×#41—21no—
177C2×#41—21no—
178C2×#41—21no—
179C2×#41—21no—
180C2×#41—21no—
181C2×#41C22×C412yes—
182C2×#41C22×C412yes—
183C2×#41C22×C412yes—
184C2×#41C2×D412yes—
185C2×#41C2×Q812yes—
186C2×#41#4112yes—
187C22×Q8#4111yescharacteristic
188C22×D4—21no—
189C22×D4#4111yescharacteristic
190C23×C4—21no—
191C23×C4—21no—
192C23×C4—21no—
193C23×C4—21no—
194C23×C4—41no—
195C23×C4C2411yescharacteristic
196#131—28no—
197#131—44no—
198#131—44no—
199#131—44no—
200C2×M16—21no—
201C2×M16—21no—
202C2×M16—21no—
203C2×M16—21no—
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—28no—
225C2×M16—42no—
226C2×M16—44no—
227C2×M16C22×C412yes—
228C2×M16C22×C412yes—
229C2×M16C22×C412yes—
230C2×M16C22×C412yes—
231C2×M16C22×C412yes—
232C2×M16C22×C412yes—
233C2×M16C22×C412yes—
234C2×M16C22×C412yes—
235C2×M16C2×D412yes—
236C2×M16C2×D412yes—
237C2×M16C2×Q812yes—
238C2×M16#4112yes—
239C22×C8—21no—
240C22×C8—21no—
241C22×C8—21no—
242C22×C8—21no—
243C22×C8—21no—
244C22×C8—21no—
245C22×C8—42no—
246C4×Q8—24no—
247C4×Q8—24no—
248C4×Q8—42no—
249C4×Q8—42no—
250C4×D4—24no—
251C4×D4—24no—
252C4×D4—42no—
253C4×D4—44no—
254C4×D4—44no—
255C4×D4—42no—
256C4×D4—44no—
257C4×D4—44no—
258#117—21no—
259#117—21no—
260#117—21no—
261#117—21no—
262#117—24no—
263#117—21no—
264#117—21no—
265#117—21no—
266#117—21no—
267#117—21no—
268#117—21no—
269#117—21no—
270#117—21no—
271#117—21no—
272#117—21no—
273#117—21no—
274#117—21no—
275#117—24no—
276#117—42no—
277#117—42no—
278#117—42no—
279#117C22×C412yes—
280#117C22×C412yes—
281#117C22×C412yes—
282#117C2×D412yes—
283C2×C4⋊C4—21no—
284C2×C4⋊C4—21no—
285C2×C4⋊C4—24no—
286C2×C4⋊C4—41no—
287C2×C4⋊C4—42no—
288C2×C4⋊C4C22×C412yes—
289C2×C4⋊C4C22×C412yes—
290C2×C4⋊C4#4112yes—
291C2×C22⋊C4—21no—
292C2×C22⋊C4—21no—
293C2×C22⋊C4—21no—
294C2×C22⋊C4—21no—
295C2×C22⋊C4—21no—
296C2×C22⋊C4—42no—
297C2×C22⋊C4—41no—
298C2×C42—21no—
299C2×C42—21no—
300C2×C42—21no—
301C2×C42—21no—
302C2×C42—21no—
303C2×C42—21no—
304C2×C42—21no—
305C2×C42—21no—
306C2×C42—21no—
307C2×C42—21no—
308C2×C42—21no—
309C2×C42—21no—
310C2×C42—24no—
311C2×C42—21no—
312C2×C42—21no—
313C2×C42—42no—
314C2×C42—42no—
315C2×C42—44no—
316C2×C42—41no—
317C2×C42—48no—
318C2×C42—42no—
319C2×C42C22×C412yes—
320C2×C42C22×C411yescharacteristic
321C2×C42C22×C411yescharacteristic
322C2×C42C22×C412yes—
323C2×C42C22×C412yes—
324C2×C42C22×C412yes—
325C2×C42C2×D411yescharacteristic
326C2×C42C2×D411yescharacteristic
327C2×C42C2×D411yescharacteristic
328C2×C42C2×D411yescharacteristic
329C2×C42C2×Q812yes—
330C2×C42#4112yes—
331#108—28no—
332#107—24no—
333#106—24no—
334C4⋊C8—44no—
335#104—28no—
336#104C4218yescomplemented
337#104C4218yescomplemented
338#104C4⋊C418yescomplemented
339#104C4⋊C418yescomplemented
340#103—24no—
341#102—24no—
342#97—24no—
343#97—42no—
344#97—42no—
345#97—42no—
346C4×C8—24no—
347C4×C8—42no—
348C4×C8—42no—
349C4×C8—42no—
350#95—42no—
351#95—42no—
352C24—21no—
353C24H(5,𝔽2)11yescharacteristic
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—22no—
363#41—22no—
364#41—22no—
365#41—22no—
366#41C2×C4214yes—
367#41C2×C4⋊C414yes—
368#41C4×D414yes—
369#41C4×Q814yes—
370C2×Q8—21no—
371C2×Q8—21no—
372C2×Q8—21no—
373C2×Q8#11712yes—
374C2×Q8C4×D412yes—
375C2×Q8C4×Q812yes—
376C2×D4—22no—
377C2×D4—22no—
378C2×D4—22no—
379C2×D4—22no—
380C2×D4—22no—
381C2×D4—22no—
382C2×D4—21no—
383C2×D4—21no—
384C2×D4—21no—
385C2×D4#11712yes—
386C2×D4C4×D412yes—
387C2×D4C4×Q812yes—
388C22×C4—21no—
389C22×C4—21no—
390C22×C4—21no—
391C22×C4—21no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—21no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—21no—
404C22×C4—21no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—44no—
411C22×C4—44no—
412C22×C4—44no—
413C22×C4—44no—
414C22×C4—44no—
415C22×C4—41no—
416C22×C4—41no—
417C22×C4—84no—
418C22×C4C23×C411yescharacteristic
419C22×C4C23×C411yescharacteristic
420C22×C4C23×C411yescharacteristic
421C22×C4C22×D411yescharacteristic
422C22×C4C22×D411yescharacteristic
423C22×C4C22×Q811yescharacteristic
424C22×C4C2×#4111yescharacteristic
425C22×C4C2×#4111yescharacteristic
426C22×C4C2×#4111yescharacteristic
427C22×C4C2×#4111yescharacteristic
428C22×C4C2×#4111yescharacteristic
429C22×C4H(5,𝔽2)11yescharacteristic
430C22×C4H(5,𝔽2)11yescharacteristic
431C22×C42−1+411yescharacteristic
432M16—22no—
433M16—22no—
434M16—22no—
435M16—22no—
436M16—22no—
437M16—22no—
438M16—22no—
439M16—22no—
440M16—22no—
441M16—22no—
442M16—22no—
443M16—22no—
444M16—22no—
445M16—22no—
446M16—22no—
447M16—48no—
448M16—41no—
449M16—41no—
450M16—41no—
451M16—41no—
452M16—41no—
453M16—41no—
454M16C2×C4214yes—
455M16C2×C4⋊C414yes—
456M16C4×D414yes—
457C2×C8—22no—
458C2×C8—22no—
459C2×C8—22no—
460C2×C8—22no—
461C2×C8—22no—
462C2×C8—22no—
463C2×C8—22no—
464C2×C8—22no—
465C2×C8—22no—
466C2×C8—22no—
467C2×C8—22no—
468C2×C8—22no—
469C2×C8—48no—
470C2×C8—41no—
471C2×C8—41no—
472C2×C8—41no—
473C2×C8—41no—
474C2×C8—41no—
475C2×C8—41no—
476C4⋊C4—22no—
477C4⋊C4—22no—
478C4⋊C4—22no—
479C4⋊C4—22no—
480C4⋊C4—22no—
481C4⋊C4—22no—
482C4⋊C4—22no—
483C4⋊C4—44no—
484C4⋊C4—44no—
485C4⋊C4—41no—
486C4⋊C4—41no—
487C4⋊C4—84no—
488C4⋊C4C4×D414yes—
489C22⋊C4—22no—
490C22⋊C4—22no—
491C22⋊C4—22no—
492C22⋊C4—22no—
493C22⋊C4—44no—
494C22⋊C4—44no—
495C22⋊C4—41no—
496C22⋊C4—41no—
497C22⋊C4—41no—
498C22⋊C4—41no—
499C22⋊C4—41no—
500C22⋊C4—41no—
501C42—22no—
502C42—22no—
503C42—22no—
504C42—22no—
505C42—21no—
506C42—22no—
507C42—21no—
508C42—22no—
509C42—21no—
510C42—21no—
511C42—21no—
512C42—21no—
513C42—22no—
514C42—22no—
515C42—22no—
516C42—22no—
517C42—44no—
518C42—44no—
519C42—416no—
520C42—41no—
521C42—41no—
522C42—41no—
523C42—41no—
524C42—41no—
525C42—41no—
526C42—84no—
527C42C2×C4212yes—
528C42C2×C4212yes—
529C42C2×C4⋊C412yes—
530C42C2×C4⋊C412yes—
531C42C4×D412yes—
532C42C4×D412yes—
533C23—22no—
534C23—22no—
535C23—22no—
536C23—42no—
537C23—82no—
538C23#51811yescharacteristic
539C23#51811yescharacteristic
540C23#52011yescharacteristic
541C23#54611yescharacteristic
542C23#54611yescharacteristic
543C23#55111yescharacteristic
544C23#55411yescharacteristic
545Q8—22no—
546Q8—22no—
547Q8—22no—
548Q8C4×C4⋊C414yes—
549D4—24no—
550D4—24no—
551D4—24no—
552D4—24no—
553D4—22no—
554D4—22no—
555D4—22no—
556D4C4×C4⋊C414yes—
557C2×C4—24no—
558C2×C4—24no—
559C2×C4—24no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—24no—
567C2×C4—24no—
568C2×C4—24no—
569C2×C4—24no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—24no—
573C2×C4—22no—
574C2×C4—44no—
575C2×C4—44no—
576C2×C4—44no—
577C2×C4—44no—
578C2×C4—44no—
579C2×C4—44no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—88no—
588C2×C4—82no—
589C2×C4—82no—
590C2×C4—82no—
591C2×C4—82no—
592C2×C4—82no—
593C2×C4—164no—
594C2×C4C22×C4211yescharacteristic
595C2×C4C22×C4⋊C411yescharacteristic
596C2×C4C2×#11711yescharacteristic
597C2×C4C2×C4×D411yescharacteristic
598C2×C4C2×C4×D411yescharacteristic
599C2×C4C2×C4×D411yescharacteristic
600C2×C4C2×C4×D411yescharacteristic
601C2×C4C2×C4×D411yescharacteristic
602C2×C4C2×C4×Q811yescharacteristic
603C2×C4C2×C4×Q811yescharacteristic
604C2×C4C4×#4111yescharacteristic
605C2×C4C4×#4111yescharacteristic
606C2×C4C4×#4111yescharacteristic
607C2×C4#51811yescharacteristic
608C2×C4#52011yescharacteristic
609C2×C4#52011yescharacteristic
610C2×C4D4211yescharacteristic
611C2×C4#54611yescharacteristic
612C2×C4#54611yescharacteristic
613C2×C4#54711yescharacteristic
614C2×C4D4×Q811yescharacteristic
615C2×C4#55111yescharacteristic
616C2×C4#55111yescharacteristic
617C2×C4#55111yescharacteristic
618C2×C4#55211yescharacteristic
619C2×C4#55211yescharacteristic
620C2×C4#55411yescharacteristic
621C2×C4#55411yescharacteristic
622C8—24no—
623C8—24no—
624C8—24no—
625C8—42no—
626C8—42no—
627C8—42no—
628C22—22no—
629C22—22no—
630C22—22no—
631C22—24no—
632C22—41no—
633C22—41no—
634C22—41no—
635C22—41no—
636C22—44no—
637C22—82no—
638C22—82no—
639C22#227511yescharacteristic
640C22#234211yescharacteristic
641C22#234911yescharacteristic
642C22#235811yescharacteristic
643C22#235911yescharacteristic
644C22#236011yescharacteristic
645C22#236811yescharacteristic
646C4—24no—
647C4—24no—
648C4—24no—
649C4—24no—
650C4—42no—
651C4—42no—
652C4—42no—
653C4—44no—
654C4—82no—
655C4—82no—
656C4—82no—
657C4—82no—
658C4C2×C4×C4⋊C411yescharacteristic
659C4C42×D411yescharacteristic
660C4#234311yescharacteristic
661C4D4×C4⋊C411yescharacteristic
662C4#235811yescharacteristic
663C4#235911yescharacteristic
664C4#236011yescharacteristic
665C4#236811yescharacteristic
666C2—21no—
667C2—21no—
668C2—21no—
669C2—21no—
670C2—21no—
671C2—21no—
672C2—44no—
673C2#1381411yescharacteristic
674C2#2289912yes—
subgroupG/H
centerC2×C4D42
commutatorC2×C4C22×C42
Frattini, cosocleC23×C4C24
socleC22#2358
prime radicalC22×#41C23
LCSG ▷ C2×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
3. #20445C2semidirect
8. #10897C2semidirect
9. #10895C2semidirect
10. #10894C2semidirect
11. #10893C2semidirect
12. #10839C2semidirect
13. #10826C2semidirect
15. C4×#339C2semidirect
33. #1861C4semidirect
34. #1860C4semidirect
35. #1859C4semidirect
42. C4×#104C4semidirect
104. #430C2×C4semidirect
107. #428C2×C4semidirect
110. #423C2×C4semidirect
136. C4×M16C2×C4semidirect
141. #384C2×C4semidirect
142. #382C2×C4semidirect
150. C43C2×C4semidirect
152. #344C2×C4semidirect
153. #343C2×C4semidirect
168. #339C2×C4semidirect
169. #339C2×C4semidirect
170. #339C2×C4semidirect
171. #339C2×C4semidirect
172. #339C2×C4semidirect
336. #104C42semidirect
337. #104C42semidirect
338. #104C4⋊C4semidirect
339. #104C4⋊C4semidirect

Maximal subgroups

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

subgroupindex
C2×#18612
#208632
#204452
#204212
#204052
C2×C4×#1042
#203432
#108972
#108952
#108942
#108932
#108392
#108262
#107722
C4×#3392