Finite Group Databasebeta

← #357170

Subgroups — #357170

Order 512.

1174 conjugacy classes of subgroups.

The subgroup lattice has 682 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21083C211yesmaximal, characteristic
2C2×#1929C211yesmaximal, characteristic
3C2×#1866C211yesmaximal, characteristic
4C2×C4×#108C211yesmaximal, characteristic, complemented
5#20439C211yesmaximal, characteristic, complemented
6C2×#1764C211yesmaximal, characteristic
7#20387C211yesmaximal, characteristic
8#10933C211yesmaximal, characteristic
9#10929C211yesmaximal, characteristic
10#10910C211yesmaximal, characteristic, complemented
11#10910C211yesmaximal, characteristic, complemented
12#10886C211yesmaximal, characteristic, complemented
13#10882C211yesmaximal, characteristic, complemented
14#10706C211yesmaximal, characteristic, complemented
15C8×#95C211yesmaximal, characteristic, complemented
16C22×#131C2211yescharacteristic
17C22×#108C2211yescharacteristic
18C22×#108C2211yescharacteristic
19C2×#424C2211yescharacteristic
20C22×#104C2211yescharacteristic
21C22×#104C2211yescharacteristic
22C2×#416C2211yescharacteristic
23C2×#416C2211yescharacteristic
24C2×#408C2211yescharacteristic
25C2×#405C2211yescharacteristic
26C2×#405C2211yescharacteristic
27C2×#405C2211yescharacteristic
28C2×C4×M16C2211yescharacteristic
29C2×C4×M16C2211yescharacteristic
30C22×C4×C8C2211yescharacteristic
31C22×C4×C8C2211yescharacteristic
32C22×C4×C8C2211yescharacteristic
33#2297C2211yescharacteristic
34#2296C2211yescharacteristic
35#1934C412yescomplemented
36#1934C412yescomplemented
37#1933C412yescomplemented
38#1933C412yescomplemented
39#1929C418yescomplemented
40#1866C418yescomplemented
41C4×#108—24no—
42#1771—21no—
43#1771—21no—
44#1770—21no—
45C8×C4⋊C4—21no—
46#1764—24no—
47#1753—21no—
48#1753—21no—
49#1752—21no—
50C8×C22⋊C4—21no—
51C2×#341C2211yescharacteristic
52C2×#341C2211yescharacteristic
53#1735C2211yescharacteristic
54#1735C2211yescharacteristic
55C2×#339C2211yescharacteristic
56C2×#339C2211yescharacteristic
57C2×#339C2211yescharacteristic
58C2×#339C2211yescharacteristic
59C2×#339C2211yescharacteristic
60C2×#339C2211yescharacteristic
61#1729C2211yescharacteristic
62#1729C2211yescharacteristic
63C2×#336C2211yescharacteristic
64C2×#336C2211yescharacteristic
65C2×#336C2211yescharacteristic
66C4×#95C2211yescharacteristic
67C22×#41C2311yescharacteristic
68C2×#131—22no—
69C2×#131—22no—
70C2×#131—22no—
71C2×#131—41no—
72C2×#131—41no—
73C2×#131C2×C414yes—
74C22×M16—21no—
75C22×M16C2311yescharacteristic
76C22×M16C2311yescharacteristic
77C22×M16C2311yescharacteristic
78C22×M16C2311yescharacteristic
79C23×C8—21no—
80C23×C8C2311yescharacteristic
81C23×C8C2311yescharacteristic
82C23×C8C2311yescharacteristic
83C23×C8C2311yescharacteristic
84C2×#117C2311yescharacteristic
85C2×#117C2311yescharacteristic
86C2×#117C2311yescharacteristic
87C22×C42C2311yescharacteristic
88C22×C42C2311yescharacteristic
89C22×C42C2311yescharacteristic
90C2×#108—22no—
91C2×#108—22no—
92C2×#108—22no—
93C2×#108—22no—
94C2×#108C2×C414yes—
95C2×#108C2×C414yes—
96#424—22no—
97#424C2×C414yes—
98C2×C4⋊C8—21no—
99C2×C4⋊C8—21no—
100C2×#104—22no—
101C2×#104—22no—
102C2×#104—41no—
103C2×#104—41no—
104C2×#104C2×C414yes—
105C2×#104C2×C414yes—
106#416—22no—
107#416—22no—
108#416C2×C414yes—
109#416C2×C414yes—
110C2×#103C2×C412yes—
111C2×#103C2×C411yescharacteristic
112C2×#103C2×C411yescharacteristic
113C2×#102C2×C412yes—
114C2×#102C2×C411yescharacteristic
115C2×#102C2×C411yescharacteristic
116#408—22no—
117#408C2×C414yes—
118C2×C22⋊C8—21no—
119C2×C22⋊C8—21no—
120#405—22no—
121#405—22no—
122#405—22no—
123#405—41no—
124#405—41no—
125#405C2×C414yes—
126C4×M16—22no—
127C4×M16—22no—
128C4×M16—41no—
129C4×M16—41no—
130C2×#97—21no—
131C2×#97—21no—
132C2×#97—21no—
133C2×#97—21no—
134C2×#97—21no—
135C2×C4×C8—22no—
136C2×C4×C8—21no—
137C2×C4×C8—22no—
138C2×C4×C8—22no—
139C2×C4×C8—21no—
140C2×C4×C8—21no—
141C2×C4×C8—22no—
142C2×C4×C8—22no—
143C2×C4×C8—22no—
144C2×C4×C8—21no—
145C2×C4×C8—21no—
146C2×C4×C8—41no—
147C2×C4×C8—41no—
148C2×C4×C8C2×C412yes—
149C2×C4×C8C2×C412yes—
150#391C2×C412yes—
151#390C2×C412yes—
152#389C2×C412yes—
153#389C2×C412yes—
154#386—22no—
155#385—22no—
156C4×C4⋊C4—21no—
157C4×C4⋊C4—21no—
158C4×C22⋊C4—21no—
159C4×C22⋊C4—21no—
160C2×#95—22no—
161#341—22no—
162#341D414yes—
163#340C2×C412yes—
164#340C2×C412yes—
165#340C2×C412yes—
166#340C2×C412yes—
167#339—22no—
168#339—22no—
169#339—41no—
170#339—41no—
171#339C2×C414yes—
172#339C2×C414yes—
173#336—21no—
174#336—21no—
175#336—21no—
176#336—21no—
177#336C2×C414yes—
178#336C2×C414yes—
179#336D411yescharacteristic
180#336D411yescharacteristic
181#336D411yescharacteristic
182#336D411yescharacteristic
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×#41—21no—
193C2×#41C22×C412yes—
194C2×#41C22×C412yes—
195C22×Q8C22×C411yescharacteristic
196C22×D4—21no—
197C22×D4C22×C411yescharacteristic
198C23×C4—21no—
199C23×C4—21no—
200C23×C4—21no—
201C23×C4#4112yes—
202C23×C4C2411yescharacteristic
203#131—28no—
204#131—44no—
205#131—44no—
206#131—44no—
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—24no—
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—42no—
225C2×M16—42no—
226C2×M16C22×C412yes—
227C2×M16C22×C412yes—
228C2×M16C22×C412yes—
229C2×M16C22×C412yes—
230C2×M16C2×D412yes—
231C2×M16C2×D412yes—
232C2×M16#4112yes—
233C2×M16#4112yes—
234C22×C8—21no—
235C22×C8—21no—
236C22×C8—21no—
237C22×C8—21no—
238C22×C8—21no—
239C22×C8—21no—
240C22×C8—21no—
241C22×C8—21no—
242C22×C8—21no—
243C22×C8—21no—
244C22×C8—21no—
245C22×C8—21no—
246C22×C8—21no—
247C22×C8—21no—
248C22×C8—21no—
249C22×C8—21no—
250C22×C8—21no—
251C22×C8—21no—
252C22×C8—21no—
253C22×C8—21no—
254C22×C8—24no—
255C22×C8—42no—
256C22×C8—42no—
257C22×C8C22×C412yes—
258C22×C8C22×C412yes—
259C22×C8C22×C411yescharacteristic
260C22×C8C22×C411yescharacteristic
261C22×C8C22×C412yes—
262C22×C8C22×C412yes—
263C22×C8C22×C412yes—
264C22×C8C2×D412yes—
265C22×C8C2×D411yescharacteristic
266C22×C8C2×D411yescharacteristic
267C22×C8C2×D411yescharacteristic
268C22×C8C2×D411yescharacteristic
269C22×C8#4112yes—
270C22×C8#4112yes—
271C22×C8#4112yes—
272#117—21no—
273#117—21no—
274#117—21no—
275#117—21no—
276#117—21no—
277#117—21no—
278#117—21no—
279#117—21no—
280#117—21no—
281#117—21no—
282#117C22×C412yes—
283#117C22×C412yes—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4C22×C412yes—
287C2×C4⋊C4C22×C412yes—
288C2×C4⋊C4C22×C411yescharacteristic
289C2×C4⋊C4C22×C411yescharacteristic
290C2×C22⋊C4—21no—
291C2×C22⋊C4—21no—
292C2×C22⋊C4—21no—
293C2×C22⋊C4—22no—
294C2×C22⋊C4—22no—
295C2×C42—21no—
296C2×C42—21no—
297C2×C42—21no—
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—21no—
311C2×C42—21no—
312C2×C42C22×C412yes—
313C2×C42C22×C412yes—
314C2×C42C22×C412yes—
315C2×C42C2×D411yescharacteristic
316C2×C42C2×D411yescharacteristic
317#108—28no—
318#108—28no—
319C4⋊C8—24no—
320C4⋊C8—24no—
321#104—28no—
322#104—44no—
323#103—24no—
324#103C4214yes—
325#103C4214yes—
326#102—24no—
327#102C4214yes—
328#102C4214yes—
329C22⋊C8—24no—
330C22⋊C8—24no—
331#97—24no—
332#97—42no—
333#97—42no—
334#97—42no—
335#97—42no—
336C4×C8—24no—
337C4×C8—24no—
338C4×C8—24no—
339C4×C8—42no—
340C4×C8—42no—
341C4×C8—44no—
342C4×C8—44no—
343C4×C8—42no—
344C4×C8—42no—
345#95—28no—
346#95—22no—
347#95—22no—
348#95—22no—
349#95—22no—
350C24—21no—
351C24C2×#4111yescharacteristic
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×C4212yes—
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×C4212yes—
386C22×C4—24no—
387C22×C4—24no—
388C22×C4—21no—
389C22×C4—21no—
390C22×C4—21no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—24no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—42no—
401C22×C4—42no—
402C22×C4—42no—
403C22×C4—41no—
404C22×C4—41no—
405C22×C4—41no—
406C22×C4—41no—
407C22×C4—41no—
408C22×C4—41no—
409C22×C4—41no—
410C22×C4—41no—
411C22×C4C4×D412yes—
412C22×C4C4×D412yes—
413C22×C4C23×C411yescharacteristic
414C22×C4C23×C411yescharacteristic
415C22×C4C23×C411yescharacteristic
416C22×C4C22×D411yescharacteristic
417C22×C4C22×D411yescharacteristic
418C22×C4C2×#4111yescharacteristic
419C22×C4C2×#4111yescharacteristic
420C22×C4C2×#4111yescharacteristic
421C22×C4C2×#4111yescharacteristic
422C22×C4C2×#4111yescharacteristic
423C22×C4H(5,𝔽2)11yescharacteristic
424C22×C4H(5,𝔽2)11yescharacteristic
425C22×C42−1+411yescharacteristic
426C22×C42−1+411yescharacteristic
427M16—22no—
428M16—22no—
429M16—22no—
430M16—22no—
431M16—22no—
432M16—22no—
433M16—22no—
434M16—22no—
435M16—22no—
436M16—22no—
437M16—48no—
438M16—41no—
439M16—41no—
440M16—41no—
441M16—41no—
442M16C4×D414yes—
443M16C4×D414yes—
444C2×C8—21no—
445C2×C8—21no—
446C2×C8—22no—
447C2×C8—22no—
448C2×C8—22no—
449C2×C8—22no—
450C2×C8—22no—
451C2×C8—22no—
452C2×C8—22no—
453C2×C8—22no—
454C2×C8—22no—
455C2×C8—21no—
456C2×C8—21no—
457C2×C8—22no—
458C2×C8—22no—
459C2×C8—22no—
460C2×C8—22no—
461C2×C8—21no—
462C2×C8—22no—
463C2×C8—21no—
464C2×C8—21no—
465C2×C8—22no—
466C2×C8—22no—
467C2×C8—22no—
468C2×C8—21no—
469C2×C8—21no—
470C2×C8—21no—
471C2×C8—22no—
472C2×C8—21no—
473C2×C8—21no—
474C2×C8—22no—
475C2×C8—22no—
476C2×C8—22no—
477C2×C8—22no—
478C2×C8—22no—
479C2×C8—22no—
480C2×C8—22no—
481C2×C8—22no—
482C2×C8—48no—
483C2×C8—41no—
484C2×C8—41no—
485C2×C8—41no—
486C2×C8—41no—
487C2×C8C2×C4212yes—
488C2×C8C2×C4212yes—
489C2×C8C4×D412yes—
490C2×C8C4×D414yes—
491C2×C8C4×D412yes—
492C2×C8C4×D412yes—
493C2×C8C4×D412yes—
494C4⋊C4—21no—
495C4⋊C4—21no—
496C4⋊C4—22no—
497C4⋊C4—22no—
498C4⋊C4—22no—
499C4⋊C4—22no—
500C4⋊C4—44no—
501C4⋊C4—82no—
502C4⋊C4C2×C4212yes—
503C4⋊C4C2×C4212yes—
504C22⋊C4—22no—
505C22⋊C4—22no—
506C22⋊C4—41no—
507C22⋊C4—41no—
508C22⋊C4—41no—
509C22⋊C4—41no—
510C22⋊C4—44no—
511C22⋊C4—82no—
512C42—22no—
513C42—22no—
514C42—22no—
515C42—22no—
516C42—21no—
517C42—21no—
518C42—21no—
519C42—21no—
520C42—22no—
521C42—22no—
522C42—22no—
523C42—22no—
524C42—41no—
525C42—41no—
526C42—41no—
527C42—41no—
528C42—41no—
529C42—41no—
530C42—41no—
531C42—41no—
532C23—22no—
533C23—22no—
534C23—22no—
535C23—41no—
536C23—41no—
537C23—42no—
538C23—42no—
539C23C2×C4×D411yescharacteristic
540C23C4×#4111yescharacteristic
541C23C4×#4111yescharacteristic
542C23#54811yescharacteristic
543C23#55311yescharacteristic
544C23#55611yescharacteristic
545C23#55911yescharacteristic
546Q8—22no—
547Q8—22no—
548Q8—41no—
549Q8—41no—
550D4—22no—
551D4—22no—
552D4—42no—
553D4—42no—
554D4—42no—
555D4—42no—
556D4—41no—
557D4—41no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—24no—
561C2×C4—24no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—44no—
569C2×C4—44no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—44no—
573C2×C4—44no—
574C2×C4—44no—
575C2×C4—44no—
576C2×C4—41no—
577C2×C4—42no—
578C2×C4—41no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4#44314yes—
593C2×C4C22×C4211yescharacteristic
594C2×C4C2×C4×D411yescharacteristic
595C2×C4C2×C4×D411yescharacteristic
596C2×C4C2×C4×D411yescharacteristic
597C2×C4C2×C4×D411yescharacteristic
598C2×C4C2×C4×D411yescharacteristic
599C2×C4C4×#4111yescharacteristic
600C2×C4C4×#4111yescharacteristic
601C2×C4C4×#4111yescharacteristic
602C2×C4C4×#4111yescharacteristic
603C2×C4#51811yescharacteristic
604C2×C4#51911yescharacteristic
605C2×C4#52011yescharacteristic
606C2×C4#52011yescharacteristic
607C2×C4#52011yescharacteristic
608C2×C4#52011yescharacteristic
609C2×C4D4211yescharacteristic
610C2×C4#54611yescharacteristic
611C2×C4#54611yescharacteristic
612C2×C4#54711yescharacteristic
613C2×C4#54711yescharacteristic
614C2×C4#54811yescharacteristic
615C2×C4#55111yescharacteristic
616C2×C4#55111yescharacteristic
617C2×C4#55211yescharacteristic
618C2×C4#55211yescharacteristic
619C2×C4#55311yescharacteristic
620C2×C4#55611yescharacteristic
621C8—22no—
622C8—22no—
623C8—24no—
624C8—22no—
625C8—22no—
626C8—24no—
627C8—24no—
628C8—22no—
629C8—22no—
630C8—42no—
631C8—42no—
632C22—21no—
633C22—21no—
634C22—21no—
635C22—21no—
636C22—22no—
637C22—22no—
638C22—22no—
639C22—22no—
640C22—42no—
641C22—42no—
642C22—42no—
643C22—82no—
644C22—82no—
645C22C42×D411yescharacteristic
646C22#236111yescharacteristic
647C22#236211yescharacteristic
648C22#236711yescharacteristic
649C22#236711yescharacteristic
650C22#237311yescharacteristic
651C22#237411yescharacteristic
652C22C2×#44312yes—
653C22#295712yes—
654C4—24no—
655C4—44no—
656C4—44no—
657C4—42no—
658C4—42no—
659C4—42no—
660C4—42no—
661C4—82no—
662C4—82no—
663C4C42×D411yescharacteristic
664C4#227411yescharacteristic
665C4#235811yescharacteristic
666C4#235911yescharacteristic
667C4#235911yescharacteristic
668C4#236111yescharacteristic
669C4#236211yescharacteristic
670C4#237311yescharacteristic
671C2—21no—
672C2—21no—
673C2—21no—
674C2—21no—
675C2—82no—
676C2#1382311yescharacteristic
677C2C4×#44311yescharacteristic
678C2#2051311yescharacteristic
679C2#2301812yes—
680C2#2323412yes—
subgroupG/H
centerC22×C4C22×D4
commutatorC2×C4C22×C42
Frattini, cosocleC23×C4C24
socleC23C2×C4×D4
prime radicalC22×#41C23
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C2×#336 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. C2×C4×#108C2semidirect
5. #20439C2semidirect
10. #10910C2semidirect
11. #10910C2semidirect
12. #10886C2semidirect
13. #10882C2semidirect
14. #10706C2semidirect
15. C8×#95C2semidirect
35. #1934C4semidirect
36. #1934C4semidirect
37. #1933C4semidirect
38. #1933C4semidirect
39. #1929C4semidirect
40. #1866C4semidirect

Maximal subgroups

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

subgroupindex
#210832
C2×#19292
C2×#18662
C2×C4×#1082
#204392
C2×#17642
#203872
#109332
#109292
#109102
#108862
#108822
#107062
C8×#952