Finite Group Databasebeta

← #360448

Subgroups — #360448

Order 512.

1088 conjugacy classes of subgroups.

The subgroup lattice has 685 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1938C211yesmaximal, characteristic
2C2×#1937C211yesmaximal, characteristic
3#20793C211yesmaximal, characteristic, complemented
4C2×#1806C211yesmaximal, characteristic
5#20629C211yesmaximal, characteristic
6C2×#1771C211yesmaximal, characteristic, complemented
7#20411C211yesmaximal, characteristic
8#11291C211yesmaximal, characteristic, complemented
9#11291C211yesmaximal, characteristic, complemented
10#11068C211yesmaximal, characteristic
11#11068C211yesmaximal, characteristic
12#10883C211yesmaximal, characteristic, complemented
13#10882C211yesmaximal, characteristic, complemented
14#10852C211yesmaximal, characteristic, complemented
15#10845C211yesmaximal, characteristic, complemented
16C2×C4×#41C2211yescharacteristic
17C2×#424C2211yescharacteristic
18C2×#424C2211yescharacteristic
19C2×#424C2211yescharacteristic
20C2×#424C2211yescharacteristic
21C2×#423C2211yescharacteristic
22C2×#423C2211yescharacteristic
23C22×C4⋊C8C2211yescharacteristic
24C22×#104C2211yescharacteristic
25C22×#104C2211yescharacteristic
26C2×#416C2211yescharacteristic
27C2×#416C2211yescharacteristic
28C2×#408C2211yescharacteristic
29C2×#408C2211yescharacteristic
30C22×#97C2211yescharacteristic
31C22×#97C2211yescharacteristic
32C22×C4×C8C2211yescharacteristic
33C4×#117C2211yescharacteristic
34C2×C4×C4⋊C4C2211yescharacteristic
35#1938—24no—
36#1937—24no—
37#1843—22no—
38#1842—22no—
39#1806—24no—
40#1798—22no—
41#1797—22no—
42#1771—24no—
43#1763C412yescomplemented
44#1763C412yescomplemented
45#1762C412yescomplemented
46#1762C412yescomplemented
47C2×#341C2211yescharacteristic
48C2×#341C2211yescharacteristic
49C2×#341C2211yescharacteristic
50C2×#341C2211yescharacteristic
51#1735C2211yescharacteristic
52#1735C2211yescharacteristic
53C2×#339C2211yescharacteristic
54C2×#339C2211yescharacteristic
55C2×#339C2211yescharacteristic
56C2×#339C2211yescharacteristic
57C2×#339C2211yescharacteristic
58C2×#339C2211yescharacteristic
59#1729C2211yescharacteristic
60#1729C2211yescharacteristic
61C2×#336C2211yescharacteristic
62C2×#336C2211yescharacteristic
63C22×#41C2311yescharacteristic
64C22×M16C2311yescharacteristic
65C22×M16C2311yescharacteristic
66C22×M16C2311yescharacteristic
67C22×M16C2311yescharacteristic
68C23×C8C2311yescharacteristic
69C23×C8C2311yescharacteristic
70C23×C8C2311yescharacteristic
71C23×C8C2311yescharacteristic
72C4×#41—22no—
73C4×#41—22no—
74C4×#41—22no—
75C4×#41—22no—
76C2×C4×Q8C2×C411yescharacteristic
77C2×C4×Q8C2×C411yescharacteristic
78C2×C4×D4—21no—
79C2×C4×D4—21no—
80C2×C4×D4C2×C411yescharacteristic
81C2×C4×D4C2×C411yescharacteristic
82C2×#117—21no—
83C2×#117C2311yescharacteristic
84C2×#117C2311yescharacteristic
85C2×#117C2311yescharacteristic
86C22×C4⋊C4—21no—
87C22×C42—21no—
88C22×C42C2311yescharacteristic
89C22×C42C2311yescharacteristic
90C22×C42C2311yescharacteristic
91#424—22no—
92#424—22no—
93#424—22no—
94#424—22no—
95#424—41no—
96#424—41no—
97#424—41no—
98#424—41no—
99#423—22no—
100#423—22no—
101#423—22no—
102#423—22no—
103C2×C4⋊C8—22no—
104C2×C4⋊C8—22no—
105C2×C4⋊C8—22no—
106C2×C4⋊C8—22no—
107C2×C4⋊C8—22no—
108C2×C4⋊C8—22no—
109C2×C4⋊C8—21no—
110C2×C4⋊C8—21no—
111C2×C4⋊C8—21no—
112C2×C4⋊C8—21no—
113C2×C4⋊C8D412yes—
114C2×C4⋊C8D412yes—
115C2×#104—22no—
116C2×#104—22no—
117C2×#104—22no—
118C2×#104—22no—
119C2×#104—41no—
120C2×#104—41no—
121#416—22no—
122#416—22no—
123#416D414yes—
124#416Q814yes—
125C2×#103C2×C412yes—
126C2×#103C2×C412yes—
127C2×#102C2×C412yes—
128C2×#102C2×C412yes—
129#408—22no—
130#408—22no—
131#408—41no—
132#408—41no—
133C2×C22⋊C8—22no—
134C2×C22⋊C8—22no—
135C2×#97—22no—
136C2×#97—22no—
137C2×#97—22no—
138C2×#97—22no—
139C2×#97—41no—
140C2×#97C2×C412yes—
141C2×#97C2×C412yes—
142C2×C4×C8—22no—
143C2×C4×C8—22no—
144C2×C4×C8—41no—
145C4×C4⋊C4—22no—
146C4×C4⋊C4C2×C412yes—
147C4×C4⋊C4C2×C412yes—
148C4×C4⋊C4D414yes—
149C4×C22⋊C4—21no—
150C4×C22⋊C4—21no—
151#376—21no—
152C2×#95—21no—
153C43—21no—
154#341—22no—
155#341—22no—
156#341—41no—
157#341—41no—
158#340C2×C412yes—
159#340C2×C412yes—
160#340C2×C412yes—
161#340C2×C412yes—
162#339—22no—
163#339—22no—
164#339—22no—
165#339—22no—
166#339—22no—
167#339—22no—
168#336—22no—
169#336—22no—
170#336—22no—
171#336—22no—
172C2×#41—21no—
173C2×#41—21no—
174C2×#41—21no—
175C2×#41—21no—
176C2×#41—21no—
177C2×#41—21no—
178C2×#41—21no—
179C2×#41—21no—
180C2×#41—21no—
181C2×#41—21no—
182C2×#41C2×D412yes—
183C2×#41C2×Q812yes—
184C22×Q8C22×C411yescharacteristic
185C22×D4—21no—
186C22×D4C22×C411yescharacteristic
187C23×C4—21no—
188C23×C4—21no—
189C23×C4—21no—
190C23×C4—21no—
191C23×C4—21no—
192C23×C4#4111yescharacteristic
193C23×C4#4111yescharacteristic
194C23×C4C2411yescharacteristic
195C2×M16—21no—
196C2×M16—21no—
197C2×M16—21no—
198C2×M16—21no—
199C2×M16—21no—
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—
219C22×C8—21no—
220C22×C8—21no—
221C22×C8—21no—
222C22×C8—21no—
223C22×C8—21no—
224C22×C8—21no—
225C22×C8—21no—
226C22×C8—21no—
227C22×C8—21no—
228C22×C8—21no—
229C22×C8—21no—
230C22×C8—21no—
231C22×C8—21no—
232C22×C8—21no—
233C22×C8—21no—
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×C8C22×C412yes—
246C22×C8C22×C412yes—
247C22×C8C2×D412yes—
248C22×C8C2×D412yes—
249C22×C8C2×D412yes—
250C22×C8C2×Q812yes—
251C4×Q8—22no—
252C4×Q8—22no—
253C4×Q8—22no—
254C4×Q8—22no—
255C4×D4—22no—
256C4×D4—22no—
257C4×D4—22no—
258C4×D4—22no—
259C4×D4—42no—
260C4×D4—42no—
261C4×D4—42no—
262C4×D4—42no—
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—42no—
272#117—42no—
273#117C2×D412yes—
274#117C2×Q812yes—
275#117#4112yes—
276#117#4112yes—
277C2×C4⋊C4—21no—
278C2×C4⋊C4—21no—
279C2×C4⋊C4—24no—
280C2×C4⋊C4—44no—
281C2×C4⋊C4C22×C411yescharacteristic
282C2×C4⋊C4C22×C411yescharacteristic
283C2×C4⋊C4C22×C412yes—
284C2×C4⋊C4C22×C412yes—
285C2×C22⋊C4—21no—
286C2×C22⋊C4—21no—
287C2×C22⋊C4—21no—
288C2×C22⋊C4—21no—
289C2×C22⋊C4—21no—
290C2×C42—21no—
291C2×C42—21no—
292C2×C42—21no—
293C2×C42—21no—
294C2×C42—21no—
295C2×C42—21no—
296C2×C42—21no—
297C2×C42—21no—
298C2×C42—21no—
299C2×C42—21no—
300C2×C42—21no—
301C2×C42—21no—
302C2×C42—22no—
303C2×C42—21no—
304C2×C42—21no—
305C2×C42—21no—
306C2×C42—22no—
307C2×C42—42no—
308C2×C42—42no—
309C2×C42C22×C411yescharacteristic
310C2×C42C22×C411yescharacteristic
311C2×C42C2×D412yes—
312C2×C42C2×D412yes—
313C2×C42C2×D412yes—
314C2×C42C2×D411yescharacteristic
315C2×C42C2×D411yescharacteristic
316C2×C42#4112yes—
317C2×C42#4112yes—
318C2×C42#4112yes—
319C4⋊C8—24no—
320C4⋊C8—24no—
321C4⋊C8—44no—
322C4⋊C8—44no—
323C4⋊C8—44no—
324C4⋊C8—44no—
325C4⋊C8—42no—
326C4⋊C8—42no—
327C4⋊C8—42no—
328C4⋊C8—42no—
329#104—44no—
330#104—44no—
331#103—24no—
332#103C4⋊C418yescomplemented
333#102—24no—
334#102C4⋊C418yescomplemented
335C22⋊C8—44no—
336C22⋊C8—44no—
337#97—24no—
338#97—24no—
339#97—44no—
340C4×C8—44no—
341#95—22no—
342#95—22no—
343#95—24no—
344C24—21no—
345C24C2×#4111yescharacteristic
346#41—22no—
347#41—22no—
348#41—22no—
349#41—22no—
350#41—22no—
351#41—22no—
352#41—22no—
353#41—22no—
354#41—41no—
355#41—41no—
356#41—41no—
357#41—41no—
358#41—41no—
359#41—41no—
360#41—41no—
361#41—41no—
362C2×Q8—21no—
363C2×Q8—21no—
364C2×Q8—21no—
365C2×Q8—21no—
366C2×Q8—21no—
367C2×Q8C2×C4⋊C412yes—
368C2×D4—22no—
369C2×D4—22no—
370C2×D4—21no—
371C2×D4—21no—
372C2×D4—21no—
373C2×D4—21no—
374C2×D4—21no—
375C2×D4—41no—
376C2×D4—41no—
377C2×D4—41no—
378C2×D4—41no—
379C2×D4C2×C4⋊C412yes—
380C22×C4—24no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—21no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—21no—
388C22×C4—21no—
389C22×C4—22no—
390C22×C4—42no—
391C22×C4—42no—
392C22×C4—44no—
393C22×C4—42no—
394C22×C4—41no—
395C22×C4—41no—
396C22×C4—41no—
397C22×C4—41no—
398C22×C4—41no—
399C22×C4—41no—
400C22×C4—42no—
401C22×C4—42no—
402C22×C4—41no—
403C22×C4—41no—
404C22×C4—42no—
405C22×C4—42no—
406C22×C4—82no—
407C22×C4C4×D411yescharacteristic
408C22×C4C4×D411yescharacteristic
409C22×C4C4×D411yescharacteristic
410C22×C4C4×D411yescharacteristic
411C22×C4#12312yes—
412C22×C4#12312yes—
413C22×C4#12612yes—
414C22×C4#12612yes—
415C22×C4C23×C411yescharacteristic
416C22×C4C22×D411yescharacteristic
417C22×C4C22×D411yescharacteristic
418C22×C4C22×D411yescharacteristic
419C22×C4C22×Q811yescharacteristic
420C22×C4C2×#4111yescharacteristic
421C22×C4C2×#4111yescharacteristic
422C22×C4C2×#4111yescharacteristic
423C22×C4C2×#4111yescharacteristic
424C22×C4C2×#4111yescharacteristic
425C22×C4H(5,𝔽2)11yescharacteristic
426C22×C4H(5,𝔽2)11yescharacteristic
427C22×C42−1+411yescharacteristic
428C22×C42−1+411yescharacteristic
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—
445C2×C8—22no—
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—22no—
456C2×C8—22no—
457C2×C8—22no—
458C2×C8—22no—
459C2×C8—22no—
460C2×C8—41no—
461C2×C8—41no—
462C2×C8—41no—
463C2×C8—41no—
464C2×C8—41no—
465C2×C8—41no—
466C2×C8—41no—
467C2×C8—41no—
468C2×C8—41no—
469C2×C8—41no—
470C2×C8—41no—
471C2×C8—41no—
472C2×C8—41no—
473C2×C8—41no—
474C2×C8—41no—
475C2×C8—41no—
476C2×C8—41no—
477C2×C8—41no—
478C2×C8—41no—
479C2×C8—41no—
480C2×C8—41no—
481C2×C8—41no—
482C2×C8—41no—
483C2×C8—41no—
484C2×C8C2×C4⋊C414yes—
485C4⋊C4—22no—
486C4⋊C4—22no—
487C4⋊C4—21no—
488C4⋊C4—21no—
489C4⋊C4—21no—
490C4⋊C4—21no—
491C4⋊C4—48no—
492C4⋊C4—42no—
493C4⋊C4—42no—
494C4⋊C4C2×C4⋊C414yes—
495C4⋊C4#11714yes—
496C22⋊C4—22no—
497C22⋊C4—22no—
498C22⋊C4—22no—
499C22⋊C4—22no—
500C22⋊C4—41no—
501C22⋊C4—41no—
502C22⋊C4—42no—
503C22⋊C4—42no—
504C42—22no—
505C42—22no—
506C42—22no—
507C42—22no—
508C42—22no—
509C42—22no—
510C42—22no—
511C42—22no—
512C42—44no—
513C42—41no—
514C42—41no—
515C42—41no—
516C42—41no—
517C42—41no—
518C42—41no—
519C42—42no—
520C42—42no—
521C42—82no—
522C42#11712yes—
523C42#11712yes—
524C42C4×D412yes—
525C42C4×D412yes—
526C42#12112yes—
527C42#12112yes—
528C42#12412yes—
529C42#12412yes—
530C23—22no—
531C23—22no—
532C23—22no—
533C23—41no—
534C23—42no—
535C23—42no—
536C23—41no—
537C23C2×C4×D411yescharacteristic
538C23C2×#12311yescharacteristic
539C23C2×#12611yescharacteristic
540C23#54711yescharacteristic
541C23#55211yescharacteristic
542C23#55311yescharacteristic
543C23#55411yescharacteristic
544Q8—22no—
545Q8—22no—
546Q8—41no—
547Q8—41no—
548D4—22no—
549D4—22no—
550D4—42no—
551D4—42no—
552D4—42no—
553D4—42no—
554D4—41no—
555D4—41no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—24no—
561C2×C4—24no—
562C2×C4—22no—
563C2×C4—24no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—44no—
569C2×C4—42no—
570C2×C4—44no—
571C2×C4—41no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—41no—
578C2×C4—42no—
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—41no—
588C2×C4—41no—
589C2×C4—84no—
590C2×C4—84no—
591C2×C4—82no—
592C2×C4—82no—
593C2×C4#38812yes—
594C2×C4#38812yes—
595C2×C4#38812yes—
596C2×C4#38812yes—
597C2×C4C22×C4⋊C411yescharacteristic
598C2×C4C2×#11711yescharacteristic
599C2×C4C2×#11711yescharacteristic
600C2×C4C2×C4×D411yescharacteristic
601C2×C4#52011yescharacteristic
602C2×C4#52011yescharacteristic
603C2×C4C2×#12111yescharacteristic
604C2×C4C2×#12411yescharacteristic
605C2×C4#53211yescharacteristic
606C2×C4#53211yescharacteristic
607C2×C4#53311yescharacteristic
608C2×C4#53311yescharacteristic
609C2×C4#54011yescharacteristic
610C2×C4#54111yescharacteristic
611C2×C4#54211yescharacteristic
612C2×C4#54211yescharacteristic
613C2×C4D4211yescharacteristic
614C2×C4D4211yescharacteristic
615C2×C4#54611yescharacteristic
616C2×C4#54711yescharacteristic
617C2×C4#54711yescharacteristic
618C2×C4#54811yescharacteristic
619C2×C4D4×Q811yescharacteristic
620C2×C4D4×Q811yescharacteristic
621C2×C4#55011yescharacteristic
622C2×C4#55411yescharacteristic
623C2×C4#55511yescharacteristic
624C2×C4#55611yescharacteristic
625C8—24no—
626C8—42no—
627C8—42no—
628C8—42no—
629C8—81no—
630C8—81no—
631C8—81no—
632C8—81no—
633C22—22no—
634C22—22no—
635C22—21no—
636C22—21no—
637C22—21no—
638C22—21no—
639C22—22no—
640C22—22no—
641C22—42no—
642C22—44no—
643C22—82no—
644C22—81no—
645C22—81no—
646C22C2×#38811yescharacteristic
647C22D4×C4⋊C411yescharacteristic
648C22#235211yescharacteristic
649C22#242211yescharacteristic
650C22#242311yescharacteristic
651C22#249611yescharacteristic
652C22#249711yescharacteristic
653C22#295512yes—
654C22#295612yes—
655C4—22no—
656C4—22no—
657C4—22no—
658C4—22no—
659C4—44no—
660C4—44no—
661C4—42no—
662C4—44no—
663C4—82no—
664C4—81no—
665C4—81no—
666C4#228511yescharacteristic
667C4#231411yescharacteristic
668C4D4×C4⋊C411yescharacteristic
669C4#235411yescharacteristic
670C4#242811yescharacteristic
671C4#242911yescharacteristic
672C4#249111yescharacteristic
673C4#249211yescharacteristic
674C2—21no—
675C2—21no—
676C2—21no—
677C2—21no—
678C2—82no—
679C2#1387611yescharacteristic
680C2#2109911yescharacteristic
681C2#2110011yescharacteristic
682C2#2268712yes—
683C2#2278412yes—
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
socleC23C2×C4×D4
prime radicalC22×#41C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C22×C4⋊C8 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #20793C2semidirect
6. C2×#1771C2semidirect
8. #11291C2semidirect
9. #11291C2semidirect
12. #10883C2semidirect
13. #10882C2semidirect
14. #10852C2semidirect
15. #10845C2semidirect
43. #1763C4semidirect
44. #1763C4semidirect
45. #1762C4semidirect
46. #1762C4semidirect
332. #103C4⋊C4semidirect
334. #102C4⋊C4semidirect

Maximal subgroups

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

subgroupindex
C2×#19382
C2×#19372
#207932
C2×#18062
#206292
C2×#17712
#204112
#112912
#110682
#108832
#108822
#108522
#108452