Finite Group Databasebeta

← #366888

Subgroups — #366888

Order 512.

1386 conjugacy classes of subgroups.

The subgroup lattice has 677 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
7#13755C211yesmaximal, characteristic, complemented
8#11370C212yesmaximal, complemented
9#11053C212yesmaximal
10#10924C212yesmaximal
11#10840C212yesmaximal, 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⋊C4C2211yescharacteristic
33C2×C4×C4⋊C4C2211yescharacteristic
34C2×#376C2212yes—
35C2×C43C2211yescharacteristic
36#1994—24no—
37#1986—24no—
38#1940—22no—
39#1939—22no—
40#1806—24no—
41C4×#104—24no—
42C2×#339C2212yes—
43C2×#339C2212yes—
44C2×#339C2212yes—
45C2×#339C2212yes—
46C2×#339C2211yescharacteristic
47C2×#339C2212yes—
48C2×#339C2211yescharacteristic
49#1728C2212yes—
50C4⋊#95—22no—
51#1434—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—21no—
87C22×C42C2311yescharacteristic
88C22×C42C2312yes—
89C22×C42C2312yes—
90C22×C42C2311yescharacteristic
91C22×C42C2311yescharacteristic
92#423—24no—
93#423—22no—
94#423—22no—
95#423—42no—
96C2×C4⋊C8—21no—
97C2×C4⋊C8—24no—
98C2×C4⋊C8—21no—
99C2×#104—22no—
100C2×#104—22no—
101C2×#104—22no—
102C2×#104—22no—
103C2×#104—22no—
104C2×#104—24no—
105C2×#104—24no—
106C2×#104—42no—
107C2×#104Q814yescomplemented
108C4×M16—22no—
109C4×M16—22no—
110C2×#97—21no—
111C2×C4×C8—21no—
112#400—22no—
113#399—22no—
114#398—24no—
115#396—24no—
116#395—22no—
117#393—22no—
118#391—22no—
119#391C2×C412yes—
120#391C2×C412yes—
121#390C2×C412yes—
122#390C2×C412yes—
123#389—22no—
124#387—24no—
125#386—24no—
126#385—24no—
127#384C2×C412yes—
128#384C2×C412yes—
129#382C2×C412yes—
130#382C2×C412yes—
131#380—24no—
132#380—22no—
133C4×C4⋊C4—24no—
134C4×C4⋊C4—22no—
135C4×C4⋊C4C2×C411yescharacteristic
136C4×C4⋊C4C2×C412yes—
137C4×C4⋊C4C2×C411yescharacteristic
138C4×C4⋊C4D414yes—
139C4×C4⋊C4D412yes—
140C4×C4⋊C4D412yes—
141C4×C22⋊C4—24no—
142#376—24no—
143#376—42no—
144C2×#95—21no—
145C2×#95—24no—
146C2×#95—24no—
147C2×#95D412yes—
148C43—22no—
149C43D414yes—
150#339—24no—
151#339—24no—
152#339—24no—
153#339—24no—
154#339—24no—
155#339—22no—
156#339—22no—
157#336—24no—
158C2×#41—21no—
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×#41C2×Q812yes—
169C2×#41#4112yes—
170C22×Q8—21no—
171C22×Q8—21no—
172C22×Q8C22×C411yescharacteristic
173C22×D4—21no—
174C22×D4C22×C411yescharacteristic
175C23×C4—21no—
176C23×C4—21no—
177C23×C4—21no—
178C23×C4—22no—
179C23×C4—21no—
180C23×C4—21no—
181C23×C4C2×D411yescharacteristic
182C23×C4C2×D414yes—
183C23×C4C2×D411yescharacteristic
184C23×C4C2411yescharacteristic
185C2×M16—21no—
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—22no—
196C2×M16—22no—
197C2×M16—22no—
198C2×M16—22no—
199C2×M16—22no—
200C2×M16—22no—
201C2×M16C2×Q812yes—
202C2×M16#4112yes—
203C22×C8—21no—
204C22×C8—21no—
205C22×C8—22no—
206C4⋊Q8—22no—
207C4⋊Q8—22no—
208C4⋊Q8—22no—
209C4⋊Q8—22no—
210#125—24no—
211#125—24no—
212C22⋊Q8—48no—
213C22⋊Q8—42no—
214C22⋊Q8—42no—
215C22⋊Q8—42no—
216C22⋊Q8—42no—
217C4×Q8—22no—
218C4×Q8—22no—
219C4×Q8—22no—
220C4×Q8—22no—
221C4×Q8—42no—
222C4×Q8—42no—
223C4×Q8—42no—
224C4×Q8—42no—
225C4×D4—22no—
226C4×D4—22no—
227C4×D4—22no—
228C4×D4—22no—
229C4×D4—42no—
230C4×D4—42no—
231C4×D4—42no—
232C4×D4—42no—
233#117—21no—
234#117—21no—
235#117—22no—
236#117—22no—
237#117—21no—
238#117—21no—
239#117—22no—
240#117—22no—
241#117—42no—
242#117—42no—
243C2×C4⋊C4—21no—
244C2×C4⋊C4—21no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—21no—
248C2×C4⋊C4—21no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—44no—
253C2×C4⋊C4—44no—
254C2×C4⋊C4C22⋊C412yes—
255C2×C4⋊C4C22⋊C412yes—
256C2×C4⋊C4C22⋊C414yes—
257C2×C4⋊C4C22×C411yescharacteristic
258C2×C4⋊C4C22×C411yescharacteristic
259C2×C4⋊C4C22×C412yes—
260C2×C4⋊C4C22×C412yes—
261C2×C22⋊C4—21no—
262C2×C22⋊C4—22no—
263C2×C22⋊C4—21no—
264C2×C22⋊C4—21no—
265C2×C22⋊C4—24no—
266C2×C22⋊C4—24no—
267C2×C42—21no—
268C2×C42—22no—
269C2×C42—22no—
270C2×C42—21no—
271C2×C42—22no—
272C2×C42—22no—
273C2×C42—21no—
274C2×C42—22no—
275C2×C42—21no—
276C2×C42—21no—
277C2×C42—22no—
278C2×C42—22no—
279C2×C42—21no—
280C2×C42—21no—
281C2×C42—22no—
282C2×C42—21no—
283C2×C42—21no—
284C2×C42—24no—
285C2×C42—24no—
286C2×C42—21no—
287C2×C42—22no—
288C2×C42—21no—
289C2×C42—22no—
290C2×C42—44no—
291C2×C42—44no—
292C2×C42—42no—
293C2×C42—42no—
294C2×C42C22×C412yes—
295C2×C42C22×C412yes—
296C2×C42C22×C411yescharacteristic
297C2×C42C22×C411yescharacteristic
298C2×C42C2×D412yes—
299C2×C42C2×D414yes—
300C2×C42C2×D414yes—
301C2×C42C2×D412yes—
302C2×C42C2×D412yes—
303C2×C42C2×D412yes—
304C2×C42C2×D411yescharacteristic
305C2×C42C2×D411yescharacteristic
306C2×C42C2×Q812yes—
307C2×C42#4112yes—
308C2×C42#4112yes—
309C2×C42#4112yes—
310C4⋊C8—48no—
311C4⋊C8—42no—
312C4⋊C8—42no—
313#104—28no—
314#104—28no—
315#104—48no—
316#97—42no—
317C4×C8—42no—
318#95—24no—
319#95—24no—
320#95—24no—
321#95—24no—
322#95—24no—
323#95—48no—
324#95—48no—
325#95C22⋊C414yes—
326#95C22⋊C412yes—
327#95C22⋊C412yes—
328C24—21no—
329C24C22×D411yescharacteristic
330#41—22no—
331#41—22no—
332#41—22no—
333#41—22no—
334#41—22no—
335#41—22no—
336#41—22no—
337#41—22no—
338#41—41no—
339#41—41no—
340#41—41no—
341#41—41no—
342#41—41no—
343#41—41no—
344#41—41no—
345#41—41no—
346C2×Q8—21no—
347C2×Q8—21no—
348C2×Q8—21no—
349C2×Q8—21no—
350C2×Q8—21no—
351C2×Q8—42no—
352C2×Q8—42no—
353C2×Q8—44no—
354C2×Q8—82no—
355C2×Q8C4×Q812yes—
356C2×D4—22no—
357C2×D4—22no—
358C2×D4—21no—
359C2×D4—21no—
360C2×D4—21no—
361C2×D4—21no—
362C2×D4—21no—
363C2×D4—41no—
364C2×D4—41no—
365C2×D4—41no—
366C2×D4—41no—
367C2×D4C4×Q812yes—
368C22×C4—24no—
369C22×C4—24no—
370C22×C4—21no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—24no—
376C22×C4—24no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—21no—
380C22×C4—21no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—24no—
384C22×C4—42no—
385C22×C4—44no—
386C22×C4—44no—
387C22×C4—42no—
388C22×C4—42no—
389C22×C4—42no—
390C22×C4—41no—
391C22×C4—41no—
392C22×C4—41no—
393C22×C4—41no—
394C22×C4—41no—
395C22×C4—41no—
396C22×C4—44no—
397C22×C4—44no—
398C22×C4—42no—
399C22×C4—42no—
400C22×C4—42no—
401C22×C4—42no—
402C22×C4—44no—
403C22×C4—82no—
404C22×C4C2×C22⋊C414yes—
405C22×C4C2×C22⋊C414yes—
406C22×C4C2×C22⋊C411yescharacteristic
407C22×C4C2×C22⋊C411yescharacteristic
408C22×C4C2×C22⋊C411yescharacteristic
409C22×C4C2×C22⋊C411yescharacteristic
410C22×C4#12112yes—
411C22×C4#12112yes—
412C22×C4#12112yes—
413C22×C4#12112yes—
414C22×C4C23×C411yescharacteristic
415C22×C4C22×D411yescharacteristic
416C22×C4C22×D411yescharacteristic
417C22×C4C22×D411yescharacteristic
418C22×C4C22×Q811yescharacteristic
419C22×C4C2×#4111yescharacteristic
420C22×C4C2×#4111yescharacteristic
421C22×C4C2×#4111yescharacteristic
422C22×C4C2×#4111yescharacteristic
423C22×C4C2×#4111yescharacteristic
424C22×C4H(5,𝔽2)11yescharacteristic
425C22×C42−1+411yescharacteristic
426C22×C42−1+411yescharacteristic
427C22×C42−1+411yescharacteristic
428M16—22no—
429M16—22no—
430M16—22no—
431M16—22no—
432M16—41no—
433M16—41no—
434M16—41no—
435M16—41no—
436M16—42no—
437M16—42no—
438M16—42no—
439M16—42no—
440C2×C8—22no—
441C2×C8—22no—
442C2×C8—41no—
443C2×C8—41no—
444C2×C8—41no—
445C2×C8—41no—
446C2×C8—42no—
447C2×C8—42no—
448C2×C8—42no—
449C4⋊C4—24no—
450C4⋊C4—24no—
451C4⋊C4—21no—
452C4⋊C4—21no—
453C4⋊C4—21no—
454C4⋊C4—21no—
455C4⋊C4—22no—
456C4⋊C4—22no—
457C4⋊C4—22no—
458C4⋊C4—22no—
459C4⋊C4—44no—
460C4⋊C4—48no—
461C4⋊C4—48no—
462C4⋊C4—42no—
463C4⋊C4—42no—
464C4⋊C4—44no—
465C4⋊C4—44no—
466C4⋊C4—42no—
467C4⋊C4—42no—
468C4⋊C4—84no—
469C22⋊C4—48no—
470C22⋊C4—41no—
471C22⋊C4—41no—
472C22⋊C4—42no—
473C22⋊C4—42no—
474C22⋊C4—42no—
475C22⋊C4—42no—
476C22⋊C4—84no—
477C42—22no—
478C42—22no—
479C42—22no—
480C42—22no—
481C42—22no—
482C42—24no—
483C42—24no—
484C42—22no—
485C42—22no—
486C42—22no—
487C42—22no—
488C42—24no—
489C42—24no—
490C42—48no—
491C42—48no—
492C42—41no—
493C42—42no—
494C42—42no—
495C42—41no—
496C42—42no—
497C42—42no—
498C42—41no—
499C42—41no—
500C42—42no—
501C42—42no—
502C42C4×D412yes—
503C42C4×D412yes—
504C42C4×D412yes—
505C42C4×D412yes—
506C42C22⋊Q814yescomplemented
507C42#12412yes—
508C42#12412yes—
509C42#12712yes—
510C42#12712yes—
511C23—22no—
512C23—22no—
513C23—22no—
514C23—41no—
515C23—42no—
516C23—42no—
517C23—41no—
518C23C22×C22⋊C411yescharacteristic
519C23C2×#12111yescharacteristic
520C23C2×#12111yescharacteristic
521C23#54611yescharacteristic
522C23#54711yescharacteristic
523C23#54811yescharacteristic
524C23D4×Q811yescharacteristic
525Q8—22no—
526Q8—22no—
527Q8—41no—
528Q8—41no—
529Q8—84no—
530Q8—84no—
531D4—22no—
532D4—22no—
533D4—42no—
534D4—42no—
535D4—42no—
536D4—42no—
537D4—41no—
538D4—41no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—24no—
543C2×C4—24no—
544C2×C4—24no—
545C2×C4—24no—
546C2×C4—24no—
547C2×C4—22no—
548C2×C4—24no—
549C2×C4—24no—
550C2×C4—24no—
551C2×C4—24no—
552C2×C4—24no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—44no—
559C2×C4—44no—
560C2×C4—44no—
561C2×C4—44no—
562C2×C4—41no—
563C2×C4—42no—
564C2×C4—41no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—44no—
570C2×C4—44no—
571C2×C4—44no—
572C2×C4—42no—
573C2×C4—44no—
574C2×C4—42no—
575C2×C4—44no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—84no—
581C2×C4—84no—
582C2×C4—84no—
583C2×C4—84no—
584C2×C4—84no—
585C2×C4—84no—
586C2×C4#39012yes—
587C2×C4#39012yes—
588C2×C4#39012yes—
589C2×C4#39012yes—
590C2×C4#42114yes—
591C2×C4C2×C4×D411yescharacteristic
592C2×C4C2×C4×D411yescharacteristic
593C2×C4C2×C4×Q811yescharacteristic
594C2×C4C4×#4111yescharacteristic
595C2×C4#51911yescharacteristic
596C2×C4#52011yescharacteristic
597C2×C4C2×C22⋊Q811yescharacteristic
598C2×C4#52511yescharacteristic
599C2×C4C2×#12411yescharacteristic
600C2×C4#52911yescharacteristic
601C2×C4C2×#12711yescharacteristic
602C2×C4#53311yescharacteristic
603C2×C4#53611yescharacteristic
604C2×C4#53711yescharacteristic
605C2×C4#54111yescharacteristic
606C2×C4#54211yescharacteristic
607C2×C4#54611yescharacteristic
608C2×C4#54611yescharacteristic
609C2×C4#54711yescharacteristic
610C2×C4#54711yescharacteristic
611C2×C4#54811yescharacteristic
612C2×C4#54811yescharacteristic
613C2×C4D4×Q811yescharacteristic
614C2×C4D4×Q811yescharacteristic
615C2×C4#55211yescharacteristic
616C2×C4#55211yescharacteristic
617C2×C4#55411yescharacteristic
618C2×C4#55611yescharacteristic
619C8—42no—
620C8—42no—
621C8—82no—
622C22—22no—
623C22—22no—
624C22—21no—
625C22—21no—
626C22—21no—
627C22—21no—
628C22—22no—
629C22—22no—
630C22—44no—
631C22—42no—
632C22—81no—
633C22—81no—
634C22—82no—
635C22C2×#39011yescharacteristic
636C22#233911yescharacteristic
637C22Q8×C22⋊C411yescharacteristic
638C22#241511yescharacteristic
639C22#241711yescharacteristic
640C22#242711yescharacteristic
641C22#242911yescharacteristic
642C22C2×#42112yes—
643C22#290112yes—
644C4—22no—
645C4—22no—
646C4—22no—
647C4—44no—
648C4—44no—
649C4—42no—
650C4—44no—
651C4—44no—
652C4—84no—
653C4—81no—
654C4—82no—
655C4—81no—
656C4—82no—
657C4#196314yes—
658C4C4×C22⋊Q811yescharacteristic
659C4#232311yescharacteristic
660C4#236011yescharacteristic
661C4#236211yescharacteristic
662C4#248911yescharacteristic
663C4#249211yescharacteristic
664C4#254011yescharacteristic
665C4#254211yescharacteristic
666C2—21no—
667C2—21no—
668C2—21no—
669C2—21no—
670C2—82no—
671C2#1399711yescharacteristic
672C2C2×#196311yescharacteristic
673C2#2121011yescharacteristic
674C2#2250712yes—
675C2#2252912yes—
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. #13755C2semidirect
8. #11370C2semidirect
11. #10840C2semidirect
20. #2568C4semidirect
21. #2567C4semidirect
22. #2566C4semidirect
23. #2565C4semidirect
25. #2563C4semidirect
26. #2542C4semidirect
107. C2×#104Q8semidirect
506. 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
#137552
#113702
#110532
#109242
#108402