Finite Group Databasebeta

← #645281

Subgroups — #645281

Order 512.

2288 conjugacy classes of subgroups.

The subgroup lattice has 660 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×C4⋊Q8C211yesmaximal, characteristic, complemented
2C2×C4×#126C211yesmaximal, characteristic
3#17067C211yesmaximal, characteristic, complemented
4#17066C211yesmaximal, characteristic
5#16901C212yesmaximal
6#16898C212yesmaximal, complemented
7#16603C211yesmaximal, characteristic, complemented
8#16602C211yesmaximal, characteristic
9#15206C212yesmaximal, complemented
10#15204C212yesmaximal
11#15108C212yesmaximal, complemented
12#15102C212yesmaximal, complemented
13#15100C212yesmaximal
14#14969C212yesmaximal
15#14689C211yesmaximal, characteristic, complemented
16#14296C212yesmaximal, complemented
17#14291C212yesmaximal
18#14174C212yesmaximal, complemented
19#14172C212yesmaximal
20C22×C4⋊Q8C2211yescharacteristic
21C22×#126C2211yescharacteristic
22C22×C4×Q8C2212yes—
23C2×#395C2212yes—
24C2×#393C2212yes—
25C4×C4⋊Q8—28no—
26C4×#126—28no—
27C2×#391C2212yes—
28C2×#389C2212yes—
29C2×#389C2212yes—
30C2×#388C2211yescharacteristic
31C2×#388C2211yescharacteristic
32C2×#388C2211yescharacteristic
33C2×#387C2211yescharacteristic
34C2×#387C2211yescharacteristic
35C2×#387C2211yescharacteristic
36C2×#387C2214yes—
37C2×#385C2214yes—
38C2×#384C2211yescharacteristic
39C2×#383C2211yescharacteristic
40C2×#382C2212yes—
41C2×C4×C4⋊C4C2211yescharacteristic
42C2×C4×C4⋊C4C2211yescharacteristic
43C2×C4×C4⋊C4C2211yescharacteristic
44C2×C4×C4⋊C4C2212yes—
45C2×C4×C4⋊C4C2212yes—
46C2×C4×C22⋊C4C2211yescharacteristic
47C2×C4×C22⋊C4C2211yescharacteristic
48C2×C4×C22⋊C4C2211yescharacteristic
49C2×C43C2211yescharacteristic
50#1446C2212yes—
51#1446C2212yes—
52#1445C2214yes—
53#1444C2212yes—
54#1444C2212yes—
55#1444C2212yes—
56#1444C2212yes—
57#1444C2212yes—
58#1443C2214yes—
59#1442C2214yes—
60#1442C2212yes—
61#1442C2212yes—
62#1442C2212yes—
63#1442C2212yes—
64#1442C2212yes—
65#1441C2214yes—
66#1441C2214yes—
67#1441C2212yes—
68#1441C2212yes—
69#1441C2212yes—
70#1441C2214yes—
71#1441C2212yes—
72#1441C2212yes—
73#1439C2212yes—
74#1439C2214yes—
75#1439C2214yes—
76#1439C2212yes—
77#1439C2212yes—
78#1439C2212yes—
79#1439C2212yes—
80#1437C2214yes—
81#1436C2214yes—
82#1436C2214yes—
83#1436C2214yes—
84#1436C2214yes—
85#1436C2214yes—
86C4⋊#95C2212yes—
87C4⋊#95C2212yes—
88#1434C2212yes—
89#1434C2212yes—
90#1434C2211yescharacteristic
91#1434C2211yescharacteristic
92#1434C2212yes—
93C23×Q8C2312yes—
94C24×C4C2311yescharacteristic
95C2×C4⋊Q8—28no—
96C2×C4⋊Q8—24no—
97C2×#126—24no—
98C2×#126—24no—
99C2×#126Q818yescomplemented
100C2×C4×Q8—28no—
101C2×C4×Q8—48no—
102C22×C4⋊C4C2311yescharacteristic
103C22×C4⋊C4C2314yes—
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2312yes—
106C22×C4⋊C4C2311yescharacteristic
107C22×C4⋊C4C2311yescharacteristic
108C22×C4⋊C4C2311yescharacteristic
109C22×C4⋊C4C2311yescharacteristic
110C22×C4⋊C4C2314yes—
111C22×C4⋊C4C2312yes—
112C22×C4⋊C4C2314yes—
113C22×C4⋊C4C2312yes—
114C22×C4⋊C4C2312yes—
115C22×C4⋊C4C2311yescharacteristic
116C22×C4⋊C4C2314yes—
117C22×C22⋊C4C2311yescharacteristic
118C22×C22⋊C4C2311yescharacteristic
119C22×C22⋊C4C2311yescharacteristic
120C22×C22⋊C4C2311yescharacteristic
121C22×C22⋊C4C2311yescharacteristic
122C22×C22⋊C4C2314yes—
123C22×C22⋊C4C2311yescharacteristic
124C22×C22⋊C4C2314yes—
125C22×C42C2311yescharacteristic
126C22×C42C2311yescharacteristic
127C22×C42C2311yescharacteristic
128C22×C42C2311yescharacteristic
129C22×C42C2314yes—
130C22×C42C2311yescharacteristic
131C22×C42C2311yescharacteristic
132C22×C42C2311yescharacteristic
133C22×C42C2311yescharacteristic
134C22×C42C2311yescharacteristic
135C22×C42C2311yescharacteristic
136#395—28no—
137#393—28no—
138#391—28no—
139#389—28no—
140#389—28no—
141#388—24no—
142#388—24no—
143#388—24no—
144#387—24no—
145#387—24no—
146#387—24no—
147#387—48no—
148#385—48no—
149#384—24no—
150#383—24no—
151#382—28no—
152C4×C4⋊C4—24no—
153C4×C4⋊C4—24no—
154C4×C4⋊C4—24no—
155C4×C4⋊C4—28no—
156C4×C4⋊C4—28no—
157C4×C22⋊C4—24no—
158C4×C22⋊C4—24no—
159C4×C22⋊C4—24no—
160C2×#95C2312yes—
161C2×#95C2314yes—
162C2×#95C2312yes—
163C2×#95C2314yes—
164C2×#95C2314yes—
165C2×#95C2312yes—
166C2×#95C2312yes—
167C2×#95C2314yes—
168C2×#95C2312yes—
169C2×#95C2312yes—
170C2×#95C2311yescharacteristic
171C2×#95C2311yescharacteristic
172C2×#95C2314yes—
173C2×#95C2314yes—
174C2×#95C2311yescharacteristic
175C2×#95C2311yescharacteristic
176C2×#95C2311yescharacteristic
177C2×#95C2311yescharacteristic
178C2×#95C2311yescharacteristic
179C2×#95C2311yescharacteristic
180C2×#95C2311yescharacteristic
181C2×#95C2311yescharacteristic
182C2×#95C2311yescharacteristic
183C2×#95C2311yescharacteristic
184C2×#95C2314yes—
185C2×#95C2314yes—
186C2×#95C2314yes—
187C2×#95C2314yes—
188C2×#95C2314yes—
189C2×#95C2314yes—
190C2×#95C2314yes—
191C2×#95C2314yes—
192C2×#95C2314yes—
193C2×#95C2314yes—
194C2×#95C2314yes—
195C43D418yes—
196C25C2411yescharacteristic
197C22×Q8—24no—
198C22×Q8—24no—
199C22×Q8—44no—
200C22×Q8—44no—
201C22×Q8#4118yes—
202C23×C4—22no—
203C23×C4—24no—
204C23×C4—42no—
205C23×C4—42no—
206C23×C4C2411yescharacteristic
207C23×C4C2414yes—
208C23×C4C2414yes—
209C23×C4C2414yes—
210C23×C4C2411yescharacteristic
211C23×C4C2411yescharacteristic
212C23×C4C2411yescharacteristic
213C23×C4C2411yescharacteristic
214C23×C4C2411yescharacteristic
215C23×C4C2411yescharacteristic
216C23×C4C2411yescharacteristic
217C23×C4C2414yes—
218C23×C4C2411yescharacteristic
219C23×C4C2411yescharacteristic
220C23×C4C2411yescharacteristic
221C23×C4C2411yescharacteristic
222C23×C4C2411yescharacteristic
223C23×C4C2411yescharacteristic
224C4⋊Q8—232no—
225#126—232no—
226C4×Q8—432no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—24no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—28no—
236C2×C4⋊C4—28no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—28no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—28no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—28no—
247C2×C4⋊C4—28no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—28no—
250C2×C4⋊C4—44no—
251C2×C4⋊C4—44no—
252C2×C4⋊C4—42no—
253C2×C4⋊C4—44no—
254C2×C4⋊C4—44no—
255C2×C4⋊C4—44no—
256C2×C4⋊C4—44no—
257C2×C4⋊C4—44no—
258C2×C4⋊C4—44no—
259C2×C4⋊C4—44no—
260C2×C4⋊C4—44no—
261C2×C4⋊C4C2×Q814yes—
262C2×C4⋊C4C2×Q814yes—
263C2×C4⋊C4C2×Q814yes—
264C2×C22⋊C4—22no—
265C2×C22⋊C4—22no—
266C2×C22⋊C4—22no—
267C2×C22⋊C4—22no—
268C2×C22⋊C4—22no—
269C2×C22⋊C4—22no—
270C2×C22⋊C4—22no—
271C2×C22⋊C4—22no—
272C2×C22⋊C4—22no—
273C2×C22⋊C4—28no—
274C2×C22⋊C4—24no—
275C2×C22⋊C4—24no—
276C2×C22⋊C4—44no—
277C2×C22⋊C4—42no—
278C2×C22⋊C4—44no—
279C2×C22⋊C4—44no—
280C2×C22⋊C4—44no—
281C2×C22⋊C4—44no—
282C2×C22⋊C4C2×Q814yes—
283C2×C22⋊C4C2×Q814yes—
284C2×C22⋊C4C2×Q814yes—
285C2×C42—24no—
286C2×C42—24no—
287C2×C42—24no—
288C2×C42—28no—
289C2×C42—22no—
290C2×C42—22no—
291C2×C42—22no—
292C2×C42—24no—
293C2×C42—44no—
294C2×C42—44no—
295C2×C42—42no—
296C2×C42—42no—
297C2×C42—42no—
298C2×C42—42no—
299C2×C42—42no—
300C2×C42C2×D414yes—
301C2×C42C2×D414yes—
302C2×C42C2×D414yes—
303C2×C42C2×D414yes—
304C2×C42C2×D414yes—
305C2×C42C2×D414yes—
306C2×C42C2×D414yes—
307C2×C42C2×Q814yes—
308C2×C42#4114yes—
309#95—22no—
310#95—22no—
311#95—22no—
312#95—22no—
313#95—22no—
314#95—22no—
315#95—22no—
316#95—22no—
317#95—22no—
318#95—22no—
319#95—22no—
320#95—22no—
321#95—28no—
322#95—28no—
323#95—28no—
324#95—24no—
325#95—24no—
326#95—28no—
327#95—24no—
328#95—28no—
329#95—24no—
330#95—28no—
331#95—24no—
332#95—24no—
333#95—44no—
334#95—44no—
335#95—44no—
336#95—44no—
337#95—44no—
338#95—44no—
339#95—44no—
340#95—44no—
341#95—44no—
342#95—44no—
343#95—44no—
344C24—21no—
345C24—21no—
346C24—22no—
347C24—22no—
348C24—22no—
349C24—22no—
350C24—22no—
351C24—22no—
352C24C22×Q812yes—
353C24C2511yescharacteristic
354C2×Q8—216no—
355C2×Q8—416no—
356C2×Q8—48no—
357C2×Q8—48no—
358C2×Q8—416no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—21no—
372C22×C4—21no—
373C22×C4—21no—
374C22×C4—24no—
375C22×C4—22no—
376C22×C4—21no—
377C22×C4—21no—
378C22×C4—21no—
379C22×C4—24no—
380C22×C4—22no—
381C22×C4—24no—
382C22×C4—24no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—24no—
390C22×C4—24no—
391C22×C4—22no—
392C22×C4—21no—
393C22×C4—21no—
394C22×C4—21no—
395C22×C4—21no—
396C22×C4—24no—
397C22×C4—22no—
398C22×C4—24no—
399C22×C4—24no—
400C22×C4—24no—
401C22×C4—24no—
402C22×C4—22no—
403C22×C4—24no—
404C22×C4—24no—
405C22×C4—22no—
406C22×C4—24no—
407C22×C4—22no—
408C22×C4—24no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—24no—
416C22×C4—22no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—24no—
422C22×C4—24no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—44no—
430C22×C4—44no—
431C22×C4—44no—
432C22×C4—88no—
433C22×C4C22×D412yes—
434C22×C4C22×D412yes—
435C22×C4C22×D412yes—
436C22×C4C22×D412yes—
437C22×C4C22×D412yes—
438C22×C4C22×D412yes—
439C22×C4C22×D412yes—
440C22×C4C22×Q812yes—
441C22×C4C22×Q812yes—
442C22×C4C22×Q812yes—
443C22×C4C22×Q812yes—
444C22×C4C22×Q812yes—
445C22×C4C22×Q812yes—
446C22×C4C2×#4118yes—
447C22×C4C2×#4112yes—
448C22×C4C2×#4112yes—
449C22×C4C2×#4112yes—
450C22×C4C2×#4114yes—
451C22×C4H(5,𝔽2)14yes—
452C22×C42−1+414yes—
453C4⋊C4—216no—
454C4⋊C4—216no—
455C4⋊C4—28no—
456C4⋊C4—28no—
457C4⋊C4—28no—
458C4⋊C4—48no—
459C4⋊C4—48no—
460C4⋊C4—44no—
461C4⋊C4—416no—
462C4⋊C4—48no—
463C4⋊C4—44no—
464C4⋊C4—416no—
465C4⋊C4—44no—
466C4⋊C4—88no—
467C4⋊C4—88no—
468C22⋊C4—28no—
469C22⋊C4—28no—
470C22⋊C4—28no—
471C22⋊C4—44no—
472C22⋊C4—416no—
473C22⋊C4—44no—
474C22⋊C4—44no—
475C22⋊C4—88no—
476C42—28no—
477C42—28no—
478C42—28no—
479C42—44no—
480C42—44no—
481C42—44no—
482C42—44no—
483C42—44no—
484C42—416no—
485C42—44no—
486C42C22⋊Q8116yes—
487C23—22no—
488C23—24no—
489C23—24no—
490C23—24no—
491C23—41no—
492C23—41no—
493C23—41no—
494C23—41no—
495C23—41no—
496C23—41no—
497C23—44no—
498C23—42no—
499C23—42no—
500C23—42no—
501C23—44no—
502C23—44no—
503C23—44no—
504C23C23×D411yescharacteristic
505C23C23×Q811yescharacteristic
506C23C22×#4111yescharacteristic
507C23C22×#4112yes—
508C23C2×H(5,𝔽2)12yes—
509C23C2×2−1+412yes—
510C23#58512yes—
511C23#58512yes—
512C23#58512yes—
513Q8—432no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—28no—
527C2×C4—24no—
528C2×C4—24no—
529C2×C4—24no—
530C2×C4—24no—
531C2×C4—24no—
532C2×C4—24no—
533C2×C4—28no—
534C2×C4—28no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—28no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—28no—
543C2×C4—28no—
544C2×C4—24no—
545C2×C4—24no—
546C2×C4—24no—
547C2×C4—41no—
548C2×C4—44no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—44no—
555C2×C4—44no—
556C2×C4—44no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—44no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—44no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—44no—
573C2×C4—44no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—44no—
577C2×C4—44no—
578C2×C4—44no—
579C2×C4—44no—
580C2×C4—44no—
581C2×C4—44no—
582C2×C4—44no—
583C2×C4—44no—
584C2×C4—44no—
585C2×C4—44no—
586C2×C4—44no—
587C2×C4—44no—
588C2×C4—44no—
589C2×C4—44no—
590C2×C4—44no—
591C2×C4—44no—
592C2×C4—44no—
593C2×C4—44no—
594C2×C4—44no—
595C2×C4—44no—
596C2×C4—44no—
597C2×C4—44no—
598C2×C4—88no—
599C2×C4C2×C22⋊Q814yes—
600C2×C4C2×C22⋊Q814yes—
601C2×C4C2×C22⋊Q814yes—
602C2×C4#52514yes—
603C2×C4#53514yes—
604C2×C4#53614yes—
605C22—42no—
606C22—48no—
607C22—42no—
608C22—42no—
609C22—81no—
610C22—84no—
611C22—81no—
612C22—81no—
613C22—81no—
614C22C22×C22⋊Q811yescharacteristic
615C22C2×#52511yescharacteristic
616C22C2×#53511yescharacteristic
617C22C2×#53611yescharacteristic
618C22#344911yescharacteristic
619C22#344911yescharacteristic
620C22#344911yescharacteristic
621C22#345914yes—
622C22Q8×#4112yes—
623C22#348012yes—
624C22#350312yes—
625C22#350412yes—
626C22#351012yes—
627C22#351112yes—
628C22#351614yes—
629C22#356014yes—
630C22#356514yes—
631C4—24no—
632C4—24no—
633C4—24no—
634C4—24no—
635C4—24no—
636C4—24no—
637C4—42no—
638C4—48no—
639C4—42no—
640C4—42no—
641C4—42no—
642C4—81no—
643C4—84no—
644C4—84no—
645C4—81no—
646C4—84no—
647C4—81no—
648C4—81no—
649C2—82no—
650C2#3459311yescharacteristic
651C2#3495211yescharacteristic
652C2#3524611yescharacteristic
653C2#3527911yescharacteristic
654C2#3533411yescharacteristic
655C2#3754511yescharacteristic
656C2#3754711yescharacteristic
657C2#3970714yes—
658C2#3983514yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC25C24
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×C4×C4⋊Q8C2semidirect
3. #17067C2semidirect
6. #16898C2semidirect
7. #16603C2semidirect
9. #15206C2semidirect
11. #15108C2semidirect
12. #15102C2semidirect
15. #14689C2semidirect
16. #14296C2semidirect
18. #14174C2semidirect
99. C2×#126Q8semidirect

Maximal subgroups

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

subgroupindex
C2×C4×C4⋊Q82
C2×C4×#1262
#170672
#170662
#169012
#168982
#166032
#166022
#152062
#152042
#151082
#151022
#151002
#149692
#146892
#142962
#142912
#141742
#141722