Finite Group Databasebeta

← #556945

Subgroups — #556945

Order 512.

2594 conjugacy classes of subgroups.

The subgroup lattice has 672 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×C22⋊Q8C211yesmaximal, characteristic
2#14672C214yesmaximal, complemented
3#14356C212yesmaximal, complemented
4#14352C212yesmaximal, complemented
5#14266C212yesmaximal, complemented
6#14265C214yesmaximal
7#14261C212yesmaximal, complemented
8#14139C212yesmaximal
9#14130C212yesmaximal
10#13935C211yesmaximal, characteristic, complemented
11#13931C211yesmaximal, characteristic, complemented
12#13921C211yesmaximal, characteristic, complemented
13#13918C211yesmaximal, characteristic, complemented
14#13901C212yesmaximal
15#13896C212yesmaximal
16C2×C4⋊#95C211yesmaximal, characteristic
17#13753C211yesmaximal, characteristic
18C22×C22⋊Q8C2211yescharacteristic
19C22×C4×Q8C2211yescharacteristic
20C23×C4⋊C4C2212yes—
21C23×C42C2211yescharacteristic
22C4×C22⋊Q8C4116yescomplemented
23C2×#391C2212yes—
24C2×#391C2212yes—
25C2×#391C2211yescharacteristic
26C2×#389C2212yes—
27C2×#389C2212yes—
28C2×#389C2211yescharacteristic
29C2×#387C2212yes—
30C2×#385C2214yes—
31C2×#385C2212yes—
32C2×#383C2212yes—
33C2×#382C2212yes—
34C2×#380C2211yescharacteristic
35C2×C4×C4⋊C4C2212yes—
36C2×C4×C4⋊C4C2211yescharacteristic
37C2×C4×C22⋊C4C2212yes—
38C2×C4×C22⋊C4C2212yes—
39C2×#376C2212yes—
40C22×#95C2212yes—
41C22×#95C2212yes—
42#1445C2214yes—
43#1445C2214yes—
44#1444C2214yes—
45#1443C2214yes—
46#1443C2214yes—
47#1443C2214yes—
48#1443C2214yes—
49#1442C2214yes—
50#1442C2214yes—
51#1442C2214yes—
52#1442C2214yes—
53#1442C2214yes—
54#1441C2214yes—
55#1441C2214yes—
56#1441C2214yes—
57#1441C2214yes—
58C4⋊#95C2212yes—
59C4⋊#95C2212yes—
60C4⋊#95C2212yes—
61C4⋊#95C2212yes—
62C4⋊#95C2212yes—
63C4⋊#95C2212yes—
64C4⋊#95C2212yes—
65C4⋊#95C2211yescharacteristic
66C4⋊#95C2211yescharacteristic
67C4⋊#95C2212yes—
68#1434C2211yescharacteristic
69#1434C2212yes—
70#1434C2212yes—
71#1434C2212yes—
72#1434C2212yes—
73#1434C2211yescharacteristic
74#1434C2212yes—
75#1434C2212yes—
76C22⋊#95C2214yes—
77C22⋊#95C2214yes—
78C22⋊#95C2214yes—
79C22⋊#95C2211yescharacteristic
80C22⋊#95C2211yescharacteristic
81C22⋊#95C2212yes—
82C4×#95C2211yescharacteristic
83C4×#95C2211yescharacteristic
84C23×Q8C2311yescharacteristic
85C24×C4C2311yescharacteristic
86C24×C4C2311yescharacteristic
87C24×C4C2314yes—
88C24×C4C2311yescharacteristic
89C2×C22⋊Q8—28no—
90C2×C22⋊Q8C2×C418yes—
91C2×C4×Q8—44no—
92C2×C4×Q8C2×C418yescomplemented
93C22×C4⋊C4—28no—
94C22×C4⋊C4—44no—
95C22×C4⋊C4—44no—
96C22×C4⋊C4C2312yes—
97C22×C4⋊C4C2312yes—
98C22×C4⋊C4C2311yescharacteristic
99C22×C4⋊C4C2311yescharacteristic
100C22×C4⋊C4C2312yes—
101C22×C4⋊C4C2311yescharacteristic
102C22×C4⋊C4C2314yes—
103C22×C4⋊C4C2314yes—
104C22×C4⋊C4C2314yes—
105C22×C4⋊C4C2312yes—
106C22×C4⋊C4C2312yes—
107C22×C22⋊C4C2312yes—
108C22×C22⋊C4C2312yes—
109C22×C22⋊C4C2312yes—
110C22×C22⋊C4C2312yes—
111C22×C42—44no—
112C22×C42C2×C418yes—
113C22×C42C2312yes—
114C22×C42C2312yes—
115C22×C42C2311yescharacteristic
116C22×C42C2311yescharacteristic
117C22×C42C2312yes—
118C22×C42C2311yescharacteristic
119C22×C42C2311yescharacteristic
120C22×C42C2312yes—
121#391—44no—
122#391—44no—
123#391C2×C418yescomplemented
124#389—44no—
125#389—44no—
126#389C2×C418yescomplemented
127#387C2×C4116yescomplemented
128#385—48no—
129#385C2×C4116yes—
130#383—44no—
131#382—44no—
132#380C2×C418yes—
133C4×C4⋊C4C2×C4116yescomplemented
134C4×C4⋊C4C2×C418yescomplemented
135C4×C22⋊C4—28no—
136C4×C22⋊C4C2×C4116yes—
137#376—44no—
138C2×#95—44no—
139C2×#95—44no—
140C2×#95C2311yescharacteristic
141C2×#95C2311yescharacteristic
142C2×#95C2311yescharacteristic
143C2×#95C2311yescharacteristic
144C2×#95C2312yes—
145C2×#95C2312yes—
146C2×#95C2312yes—
147C2×#95C2312yes—
148C2×#95C2312yes—
149C2×#95C2312yes—
150C2×#95C2314yes—
151C2×#95C2314yes—
152C2×#95C2314yes—
153C2×#95C2312yes—
154C2×#95C2314yes—
155C2×#95C2314yes—
156C2×#95C2314yes—
157C2×#95C2314yes—
158C2×#95C2312yes—
159C2×#95C2314yes—
160C2×#95C2312yes—
161C2×#95C2312yes—
162C2×#95C2314yes—
163C2×#95C2314yes—
164C2×#95C2314yes—
165C2×#95C2314yes—
166C2×#95C2314yes—
167C2×#95C2314yes—
168C2×#95C2312yes—
169C2×#95C2314yes—
170C2×#95C2314yes—
171C2×#95C2312yes—
172C2×#95C2312yes—
173C2×#95C2312yes—
174C2×#95C2312yes—
175C2×#95C2312yes—
176C2×#95C2312yes—
177C25C2411yescharacteristic
178C22×Q8—24no—
179C22×Q8—24no—
180C22×Q8—22no—
181C22×Q8—22no—
182C22×Q8C22×C414yes—
183C23×C4—22no—
184C23×C4—24no—
185C23×C4—22no—
186C23×C4—22no—
187C23×C4—24no—
188C23×C4—22no—
189C23×C4—22no—
190C23×C4—22no—
191C23×C4—28no—
192C23×C4—28no—
193C23×C4—28no—
194C23×C4—42no—
195C23×C4—42no—
196C23×C4—42no—
197C23×C4—44no—
198C23×C4—44no—
199C23×C4—44no—
200C23×C4—44no—
201C23×C4C22×C414yes—
202C23×C4C22×C414yes—
203C23×C4C22×C414yes—
204C23×C4#4118yes—
205C23×C4C2412yes—
206C23×C4C2412yes—
207C23×C4C2412yes—
208C23×C4C2412yes—
209C23×C4C2414yes—
210C23×C4C2414yes—
211C23×C4C2412yes—
212C23×C4C2412yes—
213C23×C4C2412yes—
214C23×C4C2412yes—
215C23×C4C2411yescharacteristic
216C23×C4C2411yescharacteristic
217C23×C4C2411yescharacteristic
218C23×C4C2411yescharacteristic
219C23×C4C2411yescharacteristic
220C23×C4C2411yescharacteristic
221C22⋊Q8—232no—
222C4×Q8—416no—
223C2×C4⋊C4—24no—
224C2×C4⋊C4—28no—
225C2×C4⋊C4—24no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—28no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4—24no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4—24no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—44no—
236C2×C4⋊C4—44no—
237C2×C4⋊C4—44no—
238C2×C4⋊C4—44no—
239C2×C4⋊C4—432no—
240C2×C4⋊C4—44no—
241C2×C4⋊C4—44no—
242C2×C4⋊C4—44no—
243C2×C4⋊C4—44no—
244C2×C4⋊C4—44no—
245C2×C4⋊C4—44no—
246C2×C4⋊C4C22×C414yes—
247C2×C4⋊C4C22×C418yes—
248C2×C4⋊C4C22×C414yes—
249C2×C4⋊C4C22×C414yes—
250C2×C4⋊C4C22×C418yes—
251C2×C4⋊C4C22×C418yes—
252C2×C22⋊C4—28no—
253C2×C22⋊C4—24no—
254C2×C22⋊C4—24no—
255C2×C22⋊C4—24no—
256C2×C22⋊C4—24no—
257C2×C22⋊C4—24no—
258C2×C22⋊C4—24no—
259C2×C22⋊C4—24no—
260C2×C22⋊C4—24no—
261C2×C22⋊C4C22×C418yes—
262C2×C22⋊C4C22×C418yes—
263C2×C42—24no—
264C2×C42—24no—
265C2×C42—24no—
266C2×C42—24no—
267C2×C42—24no—
268C2×C42—24no—
269C2×C42—24no—
270C2×C42—24no—
271C2×C42—24no—
272C2×C42—24no—
273C2×C42—416no—
274C2×C42—44no—
275C2×C42—42no—
276C2×C42—42no—
277C2×C42C22×C414yes—
278C2×C42C22×C418yes—
279C2×C42C22×C414yes—
280C2×C42C22×C418yes—
281C2×C42C22×C414yes—
282C2×C42C22×C414yes—
283#95—24no—
284#95—24no—
285#95—24no—
286#95—24no—
287#95—22no—
288#95—22no—
289#95—22no—
290#95—22no—
291#95—44no—
292#95—42no—
293#95—44no—
294#95—42no—
295#95—44no—
296#95—44no—
297#95—42no—
298#95—42no—
299#95—44no—
300#95—42no—
301#95—42no—
302#95—42no—
303#95—42no—
304#95—44no—
305#95—44no—
306#95—44no—
307#95—44no—
308#95—44no—
309#95—44no—
310#95—44no—
311#95—44no—
312#95—44no—
313#95—44no—
314#95—44no—
315#95C22×C418yes—
316#95C22×C418yes—
317#95C22×C418yes—
318#95C22×C418yes—
319#95C22×C418yes—
320C24—22no—
321C24—21no—
322C24—21no—
323C24—21no—
324C24—21no—
325C24—22no—
326C24C23×C412yes—
327C24C2×#4114yes—
328C24H(5,𝔽2)12yes—
329C242−1+414yes—
330C242−1+412yes—
331C24C2511yescharacteristic
332C2×Q8—28no—
333C2×Q8—28no—
334C2×Q8—28no—
335C2×Q8—48no—
336C2×Q8—48no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—21no—
340C22×C4—22no—
341C22×C4—21no—
342C22×C4—28no—
343C22×C4—28no—
344C22×C4—216no—
345C22×C4—216no—
346C22×C4—28no—
347C22×C4—28no—
348C22×C4—24no—
349C22×C4—24no—
350C22×C4—22no—
351C22×C4—21no—
352C22×C4—21no—
353C22×C4—21no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—24no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—24no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—21no—
371C22×C4—21no—
372C22×C4—21no—
373C22×C4—22no—
374C22×C4—24no—
375C22×C4—22no—
376C22×C4—24no—
377C22×C4—22no—
378C22×C4—28no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—24no—
384C22×C4—24no—
385C22×C4—22no—
386C22×C4—24no—
387C22×C4—22no—
388C22×C4—24no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—24no—
395C22×C4—24no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—24no—
399C22×C4—24no—
400C22×C4—24no—
401C22×C4—24no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—24no—
414C22×C4—24no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—24no—
419C22×C4—22no—
420C22×C4—24no—
421C22×C4—44no—
422C22×C4—44no—
423C22×C4—44no—
424C22×C4—48no—
425C22×C4—48no—
426C22×C4—48no—
427C22×C4—48no—
428C22×C4—416no—
429C22×C4—416no—
430C22×C4—416no—
431C22×C4—88no—
432C22×C4—88no—
433C22×C4—88no—
434C22×C4—88no—
435C22×C4#117116yes—
436C22×C4C23×C412yes—
437C22×C4C23×C414yes—
438C22×C4C23×C412yes—
439C22×C4C23×C412yes—
440C22×C4C23×C414yes—
441C22×C4C23×C414yes—
442C22×C4C23×C414yes—
443C22×C4C23×C412yes—
444C22×C4C23×C412yes—
445C22×C4C23×C412yes—
446C22×C4C2×#4114yes—
447C22×C4C2×#4114yes—
448C22×C4H(5,𝔽2)12yes—
449C22×C4H(5,𝔽2)14yes—
450C22×C4H(5,𝔽2)12yes—
451C22×C42−1+412yes—
452C22×C42−1+412yes—
453C4⋊C4—28no—
454C4⋊C4—216no—
455C4⋊C4—28no—
456C4⋊C4—28no—
457C4⋊C4—48no—
458C4⋊C4—48no—
459C4⋊C4—416no—
460C4⋊C4—48no—
461C4⋊C4—48no—
462C4⋊C4—88no—
463C4⋊C4—88no—
464C22⋊C4—216no—
465C22⋊C4—216no—
466C22⋊C4—48no—
467C22⋊C4—48no—
468C42—28no—
469C42—28no—
470C42—216no—
471C42—44no—
472C42—44no—
473C42—48no—
474C42—48no—
475C42—48no—
476C23—22no—
477C23—22no—
478C23—22no—
479C23—22no—
480C23—24no—
481C23—24no—
482C23—24no—
483C23—24no—
484C23—24no—
485C23—24no—
486C23—24no—
487C23—24no—
488C23—22no—
489C23—22no—
490C23—22no—
491C23—22no—
492C23—24no—
493C23—24no—
494C23C2×#11714yes—
495C23#51914yes—
496C23#52014yes—
497C23#54118yes—
498C23#54218yes—
499C23C24×C411yescharacteristic
500C23C22×#4112yes—
501C23C2×H(5,𝔽2)12yes—
502C23C2×H(5,𝔽2)11yescharacteristic
503C23C2×H(5,𝔽2)11yescharacteristic
504C23C2×H(5,𝔽2)11yescharacteristic
505C23C2×2−1+412yes—
506C23C2×2−1+411yescharacteristic
507C23C2×2−1+411yescharacteristic
508C23C2×2−1+411yescharacteristic
509C23#58512yes—
510Q8—416no—
511C2×C4—22no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—24no—
516C2×C4—24no—
517C2×C4—24no—
518C2×C4—24no—
519C2×C4—232no—
520C2×C4—24no—
521C2×C4—24no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—22no—
525C2×C4—24no—
526C2×C4—24no—
527C2×C4—22no—
528C2×C4—24no—
529C2×C4—24no—
530C2×C4—24no—
531C2×C4—24no—
532C2×C4—24no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—24no—
537C2×C4—22no—
538C2×C4—24no—
539C2×C4—24no—
540C2×C4—22no—
541C2×C4—416no—
542C2×C4—416no—
543C2×C4—44no—
544C2×C4—44no—
545C2×C4—44no—
546C2×C4—44no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—44no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—44no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—44no—
562C2×C4—44no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—44no—
570C2×C4—44no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—44no—
574C2×C4—44no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—44no—
580C2×C4—44no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—44no—
587C2×C4—42no—
588C2×C4—44no—
589C2×C4—42no—
590C2×C4—44no—
591C2×C4—44no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—44no—
598C2×C4—44no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—832no—
602C2×C4C2×#11714yes—
603C2×C4C2×#11714yes—
604C2×C4#51814yes—
605C2×C4#52014yes—
606C2×C4#52014yes—
607C2×C4#53818yes—
608C2×C4#53918yes—
609C22—24no—
610C22—24no—
611C22—24no—
612C22—24no—
613C22—24no—
614C22—24no—
615C22—28no—
616C22—28no—
617C22—28no—
618C22—28no—
619C22#232018yes—
620C22C22×#11711yescharacteristic
621C22C2×#51811yescharacteristic
622C22C2×#51911yescharacteristic
623C22C2×#52011yescharacteristic
624C22C2×#52011yescharacteristic
625C22C2×#52011yescharacteristic
626C22#342811yescharacteristic
627C22C2×#53812yes—
628C22C2×#53912yes—
629C22C2×#54112yes—
630C22C2×#54212yes—
631C22#346012yes—
632C22#346112yes—
633C22#353612yes—
634C22#353812yes—
635C22#354112yes—
636C22#354212yes—
637C22#354311yescharacteristic
638C22#354411yescharacteristic
639C22#355211yescharacteristic
640C22#355211yescharacteristic
641C22#356112yes—
642C22#356312yes—
643C4—24no—
644C4—24no—
645C4—44no—
646C4—44no—
647C4—42no—
648C4—44no—
649C4—44no—
650C4—42no—
651C4—42no—
652C4—82no—
653C4—84no—
654C4—84no—
655C4—82no—
656C4—82no—
657C4—82no—
658C4#231818yes—
659C2—28no—
660C2C2×#231811yescharacteristic
661C2C2×#232011yescharacteristic
662C2#3406011yescharacteristic
663C2#3438011yescharacteristic
664C2#3438411yescharacteristic
665C2#3439911yescharacteristic
666C2#3440411yescharacteristic
667C2#3749212yes—
668C2#3749512yes—
669C2#3750312yes—
670C2#3750412yes—
subgroupG/H
center, Frattini, socle, cosocleC24C25
commutatorC23C24×C4
prime radicalC25C24
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #14672C2semidirect
3. #14356C2semidirect
4. #14352C2semidirect
5. #14266C2semidirect
7. #14261C2semidirect
10. #13935C2semidirect
11. #13931C2semidirect
12. #13921C2semidirect
13. #13918C2semidirect
22. C4×C22⋊Q8C4semidirect
92. C2×C4×Q8C2×C4semidirect
123. #391C2×C4semidirect
126. #389C2×C4semidirect
127. #387C2×C4semidirect
133. C4×C4⋊C4C2×C4semidirect
134. C4×C4⋊C4C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×C4×C22⋊Q82
#146722
#143562
#143522
#142662
#142652
#142612
#141392
#141302
#139352
#139312
#139212
#139182
#139012
#138962
C2×C4⋊#952
#137532