Finite Group Databasebeta

← #366887

Subgroups — #366887

Order 512.

1562 conjugacy classes of subgroups.

The subgroup lattice has 681 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34947C211yesmaximal, characteristic
2C2×#1994C211yesmaximal, characteristic, complemented
3C2×#1986C211yesmaximal, characteristic, complemented
4#21146C211yesmaximal, characteristic, complemented
5C2×#1806C211yesmaximal, characteristic
6C2×C4×#104C211yesmaximal, characteristic
7C42×C4⋊C4C211yesmaximal, characteristic, complemented
8#11370C212yesmaximal, complemented
9#11053C212yesmaximal
10#10923C212yesmaximal
11#10839C212yesmaximal, complemented
12C2×#533C2211yescharacteristic
13C2×C4×#41C2211yescharacteristic
14C2×#423C2211yescharacteristic
15C2×#423C2212yes—
16C22×#104C2211yescharacteristic
17C22×#104C2212yes—
18C22×#104C2211yescharacteristic
19C2×C4×M16C2211yescharacteristic
20#2568C411yescharacteristic, complemented
21#2567C411yescharacteristic, complemented
22#2566C412yescomplemented
23#2565C412yescomplemented
24#2564—22no—
25#2563C411yescharacteristic, complemented
26#2542C411yescharacteristic, complemented
27#2417—22no—
28#2396—22no—
29C4×C22⋊Q8—22no—
30#2297C2212yes—
31#2285C2212yes—
32C2×C4×C4⋊C4—22no—
33C2×C4×C4⋊C4C2211yescharacteristic
34C2×C4×C4⋊C4C2211yescharacteristic
35C2×C43C2212yes—
36C2×C43C2211yescharacteristic
37#1994—24no—
38#1986—24no—
39#1940—22no—
40#1939—22no—
41#1806—24no—
42C4×#104—24no—
43C2×#339C2212yes—
44C2×#339C2212yes—
45C2×#339C2212yes—
46C2×#339C2212yes—
47C2×#339C2211yescharacteristic
48C2×#339C2212yes—
49C2×#339C2211yescharacteristic
50#1728C2212yes—
51C4×#95—22no—
52C4×#95C2212yes—
53C22×#41C2311yescharacteristic
54C22×M16C2312yes—
55C22×M16C2311yescharacteristic
56C22×M16C2311yescharacteristic
57#533—22no—
58#533—22no—
59#533—22no—
60#533—22no—
61C2×C4⋊Q8C2×C411yescharacteristic
62C2×C4⋊Q8C2×C411yescharacteristic
63C2×#125C2×C412yes—
64C2×C22⋊Q8—21no—
65C2×C22⋊Q8—21no—
66C2×C22⋊Q8—22no—
67C4×#41—22no—
68C4×#41—22no—
69C4×#41—22no—
70C4×#41—22no—
71C2×C4×Q8—21no—
72C2×C4×Q8—21no—
73C2×C4×Q8C2×C411yescharacteristic
74C2×C4×Q8C2×C411yescharacteristic
75C2×C4×D4—21no—
76C2×C4×D4—21no—
77C2×C4×D4C2×C411yescharacteristic
78C2×C4×D4C2×C411yescharacteristic
79C2×#117—21no—
80C2×#117C2311yescharacteristic
81C2×#117C2312yes—
82C22×C4⋊C4—21no—
83C22×C4⋊C4D412yes—
84C22×C42—21no—
85C22×C42—21no—
86C22×C42—24no—
87C22×C42—21no—
88C22×C42C2311yescharacteristic
89C22×C42C2312yes—
90C22×C42C2312yes—
91C22×C42C2311yescharacteristic
92C22×C42C2311yescharacteristic
93#423—24no—
94#423—22no—
95#423—22no—
96#423—42no—
97C2×C4⋊C8—21no—
98C2×C4⋊C8—24no—
99C2×C4⋊C8—21no—
100C2×#104—22no—
101C2×#104—22no—
102C2×#104—22no—
103C2×#104—22no—
104C2×#104—22no—
105C2×#104—24no—
106C2×#104—24no—
107C2×#104—42no—
108C2×#104Q814yescomplemented
109C4×M16—22no—
110C4×M16—22no—
111C2×#97—21no—
112C2×C4×C8—21no—
113#400—22no—
114#399—22no—
115#398—24no—
116#396—24no—
117#395—22no—
118#393—22no—
119#391—22no—
120#391C2×C412yes—
121#391C2×C412yes—
122#390C2×C412yes—
123#390C2×C412yes—
124#389—22no—
125#387—24no—
126#386—24no—
127#385—24no—
128#384C2×C412yes—
129#384C2×C412yes—
130#382C2×C412yes—
131#382C2×C412yes—
132#380—24no—
133#380—22no—
134C4×C4⋊C4—216no—
135C4×C4⋊C4—24no—
136C4×C4⋊C4—22no—
137C4×C4⋊C4C2×C411yescharacteristic
138C4×C4⋊C4C2×C412yes—
139C4×C4⋊C4C2×C411yescharacteristic
140C4×C4⋊C4D414yes—
141C4×C4⋊C4D412yes—
142C4×C4⋊C4D412yes—
143C4×C22⋊C4—24no—
144C2×#95—21no—
145C2×#95—24no—
146C2×#95D412yes—
147C43—22no—
148C43—24no—
149C43—42no—
150C43D414yes—
151#339—24no—
152#339—24no—
153#339—24no—
154#339—24no—
155#339—24no—
156#339—22no—
157#339—22no—
158#336—24no—
159C2×#41—21no—
160C2×#41—21no—
161C2×#41—21no—
162C2×#41—21no—
163C2×#41—21no—
164C2×#41—21no—
165C2×#41—21no—
166C2×#41—21no—
167C2×#41—21no—
168C2×#41—21no—
169C2×#41C2×Q812yes—
170C2×#41#4112yes—
171C22×Q8—21no—
172C22×Q8—21no—
173C22×Q8C22×C411yescharacteristic
174C22×D4—21no—
175C22×D4C22×C411yescharacteristic
176C23×C4—21no—
177C23×C4—22no—
178C23×C4—21no—
179C23×C4—21no—
180C23×C4—21no—
181C23×C4—21no—
182C23×C4C2×D414yes—
183C23×C4C2×D411yescharacteristic
184C23×C4C2×D411yescharacteristic
185C23×C4C2411yescharacteristic
186C2×M16—21no—
187C2×M16—21no—
188C2×M16—21no—
189C2×M16—21no—
190C2×M16—21no—
191C2×M16—21no—
192C2×M16—21no—
193C2×M16—21no—
194C2×M16—21no—
195C2×M16—21no—
196C2×M16—22no—
197C2×M16—22no—
198C2×M16—22no—
199C2×M16—22no—
200C2×M16—22no—
201C2×M16—22no—
202C2×M16C2×Q812yes—
203C2×M16#4112yes—
204C22×C8—21no—
205C22×C8—21no—
206C22×C8—22no—
207C4⋊Q8—22no—
208C4⋊Q8—22no—
209C4⋊Q8—22no—
210C4⋊Q8—22no—
211#125—24no—
212#125—24no—
213C22⋊Q8—42no—
214C22⋊Q8—42no—
215C22⋊Q8—42no—
216C22⋊Q8—42no—
217C22⋊Q8—48no—
218C4×Q8—22no—
219C4×Q8—22no—
220C4×Q8—22no—
221C4×Q8—22no—
222C4×Q8—42no—
223C4×Q8—42no—
224C4×Q8—42no—
225C4×Q8—42no—
226C4×D4—22no—
227C4×D4—22no—
228C4×D4—22no—
229C4×D4—22no—
230C4×D4—42no—
231C4×D4—42no—
232C4×D4—42no—
233C4×D4—42no—
234#117—21no—
235#117—21no—
236#117—22no—
237#117—22no—
238#117—21no—
239#117—21no—
240#117—22no—
241#117—22no—
242#117—42no—
243#117—42no—
244C2×C4⋊C4—28no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—28no—
247C2×C4⋊C4—21no—
248C2×C4⋊C4—21no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—21no—
252C2×C4⋊C4—21no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4C22⋊C412yes—
256C2×C4⋊C4C22⋊C414yes—
257C2×C4⋊C4C22⋊C412yes—
258C2×C4⋊C4C22×C411yescharacteristic
259C2×C4⋊C4C22×C411yescharacteristic
260C2×C4⋊C4C22×C412yes—
261C2×C4⋊C4C22×C412yes—
262C2×C22⋊C4—21no—
263C2×C22⋊C4—22no—
264C2×C22⋊C4—21no—
265C2×C22⋊C4—21no—
266C2×C22⋊C4—24no—
267C2×C22⋊C4—24no—
268C2×C42—28no—
269C2×C42—24no—
270C2×C42—22no—
271C2×C42—21no—
272C2×C42—22no—
273C2×C42—21no—
274C2×C42—22no—
275C2×C42—22no—
276C2×C42—21no—
277C2×C42—22no—
278C2×C42—21no—
279C2×C42—21no—
280C2×C42—22no—
281C2×C42—22no—
282C2×C42—21no—
283C2×C42—21no—
284C2×C42—22no—
285C2×C42—216no—
286C2×C42—21no—
287C2×C42—21no—
288C2×C42—24no—
289C2×C42—21no—
290C2×C42—22no—
291C2×C42—21no—
292C2×C42—22no—
293C2×C42—48no—
294C2×C42—44no—
295C2×C42—48no—
296C2×C42—42no—
297C2×C42—42no—
298C2×C42C22×C412yes—
299C2×C42C22×C412yes—
300C2×C42C22×C411yescharacteristic
301C2×C42C22×C411yescharacteristic
302C2×C42C2×D412yes—
303C2×C42C2×D414yes—
304C2×C42C2×D414yes—
305C2×C42C2×D412yes—
306C2×C42C2×D412yes—
307C2×C42C2×D412yes—
308C2×C42C2×D411yescharacteristic
309C2×C42C2×D411yescharacteristic
310C2×C42C2×Q812yes—
311C2×C42#4112yes—
312C2×C42#4112yes—
313C2×C42#4112yes—
314C4⋊C8—48no—
315C4⋊C8—42no—
316C4⋊C8—42no—
317#104—28no—
318#104—28no—
319#104—48no—
320#97—42no—
321C4×C8—42no—
322#95—216no—
323#95—24no—
324#95—24no—
325#95—24no—
326#95—24no—
327#95—24no—
328#95C22⋊C414yes—
329#95C22⋊C412yes—
330#95C22⋊C412yes—
331C24—21no—
332C24C22×D411yescharacteristic
333#41—22no—
334#41—22no—
335#41—22no—
336#41—22no—
337#41—22no—
338#41—22no—
339#41—22no—
340#41—22no—
341#41—41no—
342#41—41no—
343#41—41no—
344#41—41no—
345#41—41no—
346#41—41no—
347#41—41no—
348#41—41no—
349C2×Q8—21no—
350C2×Q8—21no—
351C2×Q8—21no—
352C2×Q8—21no—
353C2×Q8—21no—
354C2×Q8—42no—
355C2×Q8—42no—
356C2×Q8—44no—
357C2×Q8—82no—
358C2×Q8C4×Q812yes—
359C2×D4—22no—
360C2×D4—22no—
361C2×D4—21no—
362C2×D4—21no—
363C2×D4—21no—
364C2×D4—21no—
365C2×D4—21no—
366C2×D4—41no—
367C2×D4—41no—
368C2×D4—41no—
369C2×D4—41no—
370C2×D4C4×Q812yes—
371C22×C4—24no—
372C22×C4—24no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—24no—
377C22×C4—24no—
378C22×C4—28no—
379C22×C4—21no—
380C22×C4—24no—
381C22×C4—22no—
382C22×C4—24no—
383C22×C4—24no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—28no—
387C22×C4—21no—
388C22×C4—21no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—44no—
392C22×C4—42no—
393C22×C4—44no—
394C22×C4—42no—
395C22×C4—42no—
396C22×C4—41no—
397C22×C4—41no—
398C22×C4—42no—
399C22×C4—41no—
400C22×C4—41no—
401C22×C4—41no—
402C22×C4—41no—
403C22×C4—42no—
404C22×C4—42no—
405C22×C4—44no—
406C22×C4—82no—
407C22×C4C2×C22⋊C414yes—
408C22×C4C2×C22⋊C411yescharacteristic
409C22×C4C2×C22⋊C411yescharacteristic
410C22×C4C2×C22⋊C411yescharacteristic
411C22×C4C2×C22⋊C414yes—
412C22×C4C2×C22⋊C411yescharacteristic
413C22×C4#12112yes—
414C22×C4#12112yes—
415C22×C4#12112yes—
416C22×C4#12112yes—
417C22×C4C23×C411yescharacteristic
418C22×C4C22×D411yescharacteristic
419C22×C4C22×D411yescharacteristic
420C22×C4C22×D411yescharacteristic
421C22×C4C22×Q811yescharacteristic
422C22×C4C2×#4111yescharacteristic
423C22×C4C2×#4111yescharacteristic
424C22×C4C2×#4111yescharacteristic
425C22×C4C2×#4111yescharacteristic
426C22×C4C2×#4111yescharacteristic
427C22×C4H(5,𝔽2)11yescharacteristic
428C22×C42−1+411yescharacteristic
429C22×C42−1+411yescharacteristic
430C22×C42−1+411yescharacteristic
431M16—22no—
432M16—22no—
433M16—22no—
434M16—22no—
435M16—41no—
436M16—41no—
437M16—41no—
438M16—41no—
439M16—42no—
440M16—42no—
441M16—42no—
442M16—42no—
443C2×C8—22no—
444C2×C8—22no—
445C2×C8—41no—
446C2×C8—41no—
447C2×C8—41no—
448C2×C8—41no—
449C2×C8—42no—
450C2×C8—42no—
451C2×C8—42no—
452C4⋊C4—216no—
453C4⋊C4—28no—
454C4⋊C4—24no—
455C4⋊C4—21no—
456C4⋊C4—21no—
457C4⋊C4—21no—
458C4⋊C4—21no—
459C4⋊C4—22no—
460C4⋊C4—22no—
461C4⋊C4—22no—
462C4⋊C4—22no—
463C4⋊C4—42no—
464C4⋊C4—42no—
465C4⋊C4—44no—
466C4⋊C4—44no—
467C4⋊C4—42no—
468C4⋊C4—42no—
469C4⋊C4—48no—
470C4⋊C4—84no—
471C4⋊C4#10418yescomplemented
472C22⋊C4—41no—
473C22⋊C4—41no—
474C22⋊C4—42no—
475C22⋊C4—42no—
476C22⋊C4—42no—
477C22⋊C4—42no—
478C22⋊C4—48no—
479C22⋊C4—84no—
480C42—216no—
481C42—22no—
482C42—22no—
483C42—22no—
484C42—22no—
485C42—22no—
486C42—24no—
487C42—24no—
488C42—22no—
489C42—22no—
490C42—22no—
491C42—22no—
492C42—24no—
493C42—24no—
494C42—432no—
495C42—48no—
496C42—41no—
497C42—42no—
498C42—42no—
499C42—41no—
500C42—42no—
501C42—42no—
502C42—41no—
503C42—41no—
504C42—42no—
505C42—42no—
506C42C4×D412yes—
507C42C4×D412yes—
508C42C4×D412yes—
509C42C4×D412yes—
510C42C22⋊Q814yescomplemented
511C42#12412yes—
512C42#12412yes—
513C42#12712yes—
514C42#12712yes—
515C23—22no—
516C23—22no—
517C23—22no—
518C23—41no—
519C23—42no—
520C23—42no—
521C23—41no—
522C23C22×C22⋊C411yescharacteristic
523C23C2×#12111yescharacteristic
524C23C2×#12111yescharacteristic
525C23#54611yescharacteristic
526C23#54711yescharacteristic
527C23#54811yescharacteristic
528C23D4×Q811yescharacteristic
529Q8—22no—
530Q8—22no—
531Q8—41no—
532Q8—41no—
533Q8—84no—
534Q8—84no—
535D4—22no—
536D4—22no—
537D4—42no—
538D4—42no—
539D4—42no—
540D4—42no—
541D4—41no—
542D4—41no—
543C2×C4—24no—
544C2×C4—28no—
545C2×C4—24no—
546C2×C4—28no—
547C2×C4—24no—
548C2×C4—22no—
549C2×C4—24no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—24no—
553C2×C4—24no—
554C2×C4—24no—
555C2×C4—24no—
556C2×C4—28no—
557C2×C4—24no—
558C2×C4—22no—
559C2×C4—24no—
560C2×C4—24no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—48no—
564C2×C4—48no—
565C2×C4—42no—
566C2×C4—48no—
567C2×C4—44no—
568C2×C4—48no—
569C2×C4—44no—
570C2×C4—44no—
571C2×C4—41no—
572C2×C4—42no—
573C2×C4—44no—
574C2×C4—41no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—44no—
580C2×C4—42no—
581C2×C4—44no—
582C2×C4—42no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—84no—
588C2×C4—84no—
589C2×C4#39012yes—
590C2×C4#39012yes—
591C2×C4#39012yes—
592C2×C4#39012yes—
593C2×C4C2×#10418yes—
594C2×C4C2×#10414yes—
595C2×C4C2×C4×D411yescharacteristic
596C2×C4C2×C4×D411yescharacteristic
597C2×C4C2×C4×Q811yescharacteristic
598C2×C4C4×#4111yescharacteristic
599C2×C4#51911yescharacteristic
600C2×C4#52011yescharacteristic
601C2×C4C2×C22⋊Q811yescharacteristic
602C2×C4#52511yescharacteristic
603C2×C4C2×#12411yescharacteristic
604C2×C4#52911yescharacteristic
605C2×C4C2×#12711yescharacteristic
606C2×C4#53311yescharacteristic
607C2×C4#53611yescharacteristic
608C2×C4#53711yescharacteristic
609C2×C4#54111yescharacteristic
610C2×C4#54211yescharacteristic
611C2×C4#54611yescharacteristic
612C2×C4#54611yescharacteristic
613C2×C4#54711yescharacteristic
614C2×C4#54711yescharacteristic
615C2×C4#54811yescharacteristic
616C2×C4#54811yescharacteristic
617C2×C4D4×Q811yescharacteristic
618C2×C4D4×Q811yescharacteristic
619C2×C4#55211yescharacteristic
620C2×C4#55211yescharacteristic
621C2×C4#55411yescharacteristic
622C2×C4#55611yescharacteristic
623C8—42no—
624C8—42no—
625C8—82no—
626C22—22no—
627C22—22no—
628C22—21no—
629C22—21no—
630C22—21no—
631C22—21no—
632C22—22no—
633C22—22no—
634C22—44no—
635C22—42no—
636C22—81no—
637C22—81no—
638C22—82no—
639C22C2×#39011yescharacteristic
640C22#233911yescharacteristic
641C22Q8×C22⋊C411yescharacteristic
642C22#241511yescharacteristic
643C22#241711yescharacteristic
644C22#242711yescharacteristic
645C22#242911yescharacteristic
646C22C22×#10412yes—
647C22#290112yes—
648C4—28no—
649C4—22no—
650C4—22no—
651C4—22no—
652C4—28no—
653C4—48no—
654C4—44no—
655C4—42no—
656C4—44no—
657C4—81no—
658C4—82no—
659C4—81no—
660C4—82no—
661C4#196214yes—
662C4C4×C22⋊Q811yescharacteristic
663C4#232311yescharacteristic
664C4#236011yescharacteristic
665C4#236211yescharacteristic
666C4#248911yescharacteristic
667C4#249211yescharacteristic
668C4#254011yescharacteristic
669C4#254211yescharacteristic
670C2—21no—
671C2—21no—
672C2—21no—
673C2—21no—
674C2—82no—
675C2#1399711yescharacteristic
676C2C2×#196211yescharacteristic
677C2#2121011yescharacteristic
678C2#2250512yes—
679C2Q8×#10412yes—
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
socleC23C22×C22⋊C4
prime radicalC22×#41C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#1994C2semidirect
3. C2×#1986C2semidirect
4. #21146C2semidirect
7. C42×C4⋊C4C2semidirect
8. #11370C2semidirect
11. #10839C2semidirect
20. #2568C4semidirect
21. #2567C4semidirect
22. #2566C4semidirect
23. #2565C4semidirect
25. #2563C4semidirect
26. #2542C4semidirect
108. C2×#104Q8semidirect
471. C4⋊C4#104semidirect
510. C42C22⋊Q8semidirect

Maximal subgroups

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

subgroupindex
#349472
C2×#19942
C2×#19862
#211462
C2×#18062
C2×C4×#1042
C42×C4⋊C42
#113702
#110532
#109232
#108392