Finite Group Databasebeta

← #800064

Subgroups — #800064

Order 512.

2342 conjugacy classes of subgroups.

The subgroup lattice has 674 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2355C211yesmaximal, characteristic, complemented
2C2×#2344C211yesmaximal, characteristic
3#16663C212yesmaximal, complemented
4#16031C214yesmaximal, complemented
5#15989C214yesmaximal
6#15242C211yesmaximal, characteristic, complemented
7#15195C211yesmaximal, characteristic
8#15158C211yesmaximal, characteristic, complemented
9#15088C212yesmaximal, complemented
10#14887C212yesmaximal
11#14030C212yesmaximal, complemented
12#13982C212yesmaximal, complemented
13#13973C212yesmaximal
14#13938C212yesmaximal
15#13921C211yesmaximal, characteristic, complemented
16#13881C212yesmaximal
17C4×#393C211yesmaximal, characteristic
18C22×C4×Q8C2211yescharacteristic
19C22×#117C2212yes—
20C23×C4⋊C4C2211yescharacteristic
21C2×#395C2212yes—
22C2×#393C2211yescharacteristic
23#2355—28no—
24#2344—28no—
25C2×#391C2212yes—
26C2×#389C2211yescharacteristic
27C2×#389C2212yes—
28C2×#389C2211yescharacteristic
29C2×#387C2212yes—
30C2×#387C2212yes—
31C2×#387C2214yes—
32C2×#384C2211yescharacteristic
33C2×#383C2212yes—
34C2×#382C2212yes—
35C2×#382C2212yes—
36C2×#381C2212yes—
37C2×#380C2212yes—
38C2×#380C2211yescharacteristic
39C2×C4×C4⋊C4C2211yescharacteristic
40C2×C4×C4⋊C4C2212yes—
41C2×C4×C4⋊C4C2212yes—
42C2×C4×C4⋊C4C2211yescharacteristic
43C2×C4×C22⋊C4C2212yes—
44C2×C4×C22⋊C4C2211yescharacteristic
45#1446C2211yescharacteristic
46#1445C2214yes—
47#1445C2214yes—
48#1443C2214yes—
49#1443C2214yes—
50#1443C2214yes—
51#1442C2214yes—
52#1442C2212yes—
53#1442C2214yes—
54#1442C2212yes—
55#1442C2211yescharacteristic
56#1442C2214yes—
57#1442C2214yes—
58#1442C2212yes—
59#1441C2212yes—
60#1441C2212yes—
61#1441C2212yes—
62#1441C2214yes—
63#1439C2211yescharacteristic
64#1439C2212yes—
65#1439C2212yes—
66#1439C2214yes—
67#1439C2212yes—
68#1437C2214yes—
69#1436C2214yes—
70#1436C2214yes—
71C4⋊#95C2212yes—
72C4⋊#95C2211yescharacteristic
73C4⋊#95C2212yes—
74C4⋊#95C2212yes—
75C4⋊#95C2212yes—
76C4⋊#95C2212yes—
77#1434C2212yes—
78#1434C2214yes—
79#1434C2214yes—
80#1434C2214yes—
81C22⋊#95C2214yes—
82C22⋊#95C2214yes—
83C4×#95C2211yescharacteristic
84C4×#95C2212yes—
85C23×Q8C2311yescharacteristic
86C24×C4C2311yescharacteristic
87C24×C4C2312yes—
88C2×C4×Q8—24no—
89C2×C4×Q8—28no—
90C2×#117—28no—
91C2×#117—28no—
92C2×#117—44no—
93C22×C4⋊C4—24no—
94C22×C4⋊C4—24no—
95C22×C4⋊C4—24no—
96C22×C4⋊C4C2312yes—
97C22×C4⋊C4C2312yes—
98C22×C4⋊C4C2311yescharacteristic
99C22×C4⋊C4C2312yes—
100C22×C4⋊C4C2312yes—
101C22×C4⋊C4C2312yes—
102C22×C4⋊C4C2311yescharacteristic
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2311yescharacteristic
105C22×C4⋊C4C2311yescharacteristic
106C22×C4⋊C4C2311yescharacteristic
107C22×C4⋊C4C2314yes—
108C22×C4⋊C4C2312yes—
109C22×C4⋊C4C2312yes—
110C22×C22⋊C4C2312yes—
111C22×C22⋊C4C2312yes—
112C22×C22⋊C4C2312yes—
113C22×C22⋊C4C2311yescharacteristic
114C22×C22⋊C4C2311yescharacteristic
115C22×C22⋊C4C2314yes—
116C22×C42C2312yes—
117C22×C42C2311yescharacteristic
118C22×C42C2312yes—
119C22×C42C2312yes—
120C22×C42C2311yescharacteristic
121C22×C42C2312yes—
122C22×C42C2314yes—
123#395—44no—
124#393—24no—
125#391—28no—
126#389—24no—
127#389—24no—
128#389—28no—
129#387—28no—
130#387—28no—
131#387—216no—
132#384—24no—
133#383—44no—
134#382—28no—
135#382—28no—
136#381—28no—
137#380—28no—
138#380—24no—
139C4×C4⋊C4—24no—
140C4×C4⋊C4—28no—
141C4×C4⋊C4—28no—
142C4×C4⋊C4D418yes—
143C4×C22⋊C4—24no—
144C4×C22⋊C4—28no—
145C2×#95C2311yescharacteristic
146C2×#95C2312yes—
147C2×#95C2312yes—
148C2×#95C2311yescharacteristic
149C2×#95C2314yes—
150C2×#95C2314yes—
151C2×#95C2314yes—
152C2×#95C2314yes—
153C2×#95C2314yes—
154C2×#95C2312yes—
155C2×#95C2312yes—
156C2×#95C2312yes—
157C2×#95C2314yes—
158C2×#95C2314yes—
159C2×#95C2312yes—
160C2×#95C2312yes—
161C2×#95C2311yescharacteristic
162C2×#95C2311yescharacteristic
163C2×#95C2312yes—
164C2×#95C2314yes—
165C2×#95C2312yes—
166C2×#95C2312yes—
167C2×#95C2312yes—
168C2×#95C2312yes—
169C2×#95C2314yes—
170C2×#95C2314yes—
171C2×#95C2314yes—
172C2×#95C2314yes—
173C2×#95C2314yes—
174C2×#95C2314yes—
175C2×#95C2314yes—
176C2×#95C2314yes—
177C2×#95C2314yes—
178C2×#95C2312yes—
179C2×#95C2312yes—
180C25C2411yescharacteristic
181C22×Q8—22no—
182C22×Q8—24no—
183C22×Q8—22no—
184C22×Q8—22no—
185C22×Q8—42no—
186C23×C4—24no—
187C23×C4—24no—
188C23×C4—22no—
189C23×C4—24no—
190C23×C4—22no—
191C23×C4—24no—
192C23×C4—24no—
193C23×C4—24no—
194C23×C4—24no—
195C23×C4—42no—
196C23×C4—42no—
197C23×C4#4114yes—
198C23×C4C2412yes—
199C23×C4C2414yes—
200C23×C4C2414yes—
201C23×C4C2411yescharacteristic
202C23×C4C2412yes—
203C23×C4C2412yes—
204C23×C4C2411yescharacteristic
205C23×C4C2412yes—
206C23×C4C2412yes—
207C23×C4C2412yes—
208C23×C4C2412yes—
209C23×C4C2411yescharacteristic
210C23×C4C2412yes—
211C23×C4C2411yescharacteristic
212C23×C4C2411yescharacteristic
213C23×C4C2411yescharacteristic
214C4×Q8—232no—
215#117—432no—
216C2×C4⋊C4—232no—
217C2×C4⋊C4—24no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—24no—
221C2×C4⋊C4—24no—
222C2×C4⋊C4—24no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—24no—
227C2×C4⋊C4—24no—
228C2×C4⋊C4—24no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4—24no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—24no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—28no—
246C2×C4⋊C4—44no—
247C2×C4⋊C4—42no—
248C2×C4⋊C4—44no—
249C2×C4⋊C4—44no—
250C2×C4⋊C4—44no—
251C2×C4⋊C4C2×D414yes—
252C2×C4⋊C4C2×D414yes—
253C2×C4⋊C4#4114yes—
254C2×C22⋊C4—22no—
255C2×C22⋊C4—22no—
256C2×C22⋊C4—24no—
257C2×C22⋊C4—24no—
258C2×C22⋊C4—24no—
259C2×C22⋊C4—22no—
260C2×C22⋊C4—24no—
261C2×C22⋊C4—22no—
262C2×C22⋊C4—24no—
263C2×C22⋊C4—24no—
264C2×C22⋊C4—28no—
265C2×C22⋊C4—41no—
266C2×C22⋊C4—42no—
267C2×C22⋊C4—42no—
268C2×C22⋊C4—42no—
269C2×C22⋊C4—44no—
270C2×C22⋊C4—44no—
271C2×C22⋊C4#4114yes—
272C2×C42—24no—
273C2×C42—24no—
274C2×C42—24no—
275C2×C42—24no—
276C2×C42—24no—
277C2×C42—24no—
278C2×C42—22no—
279C2×C42—24no—
280C2×C42—24no—
281C2×C42—24no—
282C2×C42—24no—
283C2×C42—24no—
284C2×C42—28no—
285C2×C42—42no—
286C2×C42—44no—
287C2×C42—44no—
288C2×C42C2×D418yes—
289C2×C42C2×D414yes—
290C2×C42#4118yes—
291#95—22no—
292#95—22no—
293#95—24no—
294#95—28no—
295#95—28no—
296#95—24no—
297#95—24no—
298#95—24no—
299#95—28no—
300#95—24no—
301#95—24no—
302#95—28no—
303#95—28no—
304#95—28no—
305#95—28no—
306#95—24no—
307#95—22no—
308#95—22no—
309#95—24no—
310#95—44no—
311#95—42no—
312#95—42no—
313#95—44no—
314#95—42no—
315#95—44no—
316#95—44no—
317#95—42no—
318#95—42no—
319#95—44no—
320#95—44no—
321#95—44no—
322#95—44no—
323#95—44no—
324#95—44no—
325#95C2×D418yes—
326C24—21no—
327C24—22no—
328C24—21no—
329C24—21no—
330C24—22no—
331C24—22no—
332C24—22no—
333C24—21no—
334C24C2×#4112yes—
335C24C2×#4112yes—
336C242−1+412yes—
337C24C2511yescharacteristic
338C2×Q8—28no—
339C2×Q8—48no—
340C2×Q8—48no—
341C2×Q8—44no—
342C2×Q8—44no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—216no—
346C22×C4—22no—
347C22×C4—21no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—21no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—21no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—21no—
368C22×C4—22no—
369C22×C4—21no—
370C22×C4—22no—
371C22×C4—24no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—21no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—24no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—28no—
387C22×C4—216no—
388C22×C4—21no—
389C22×C4—22no—
390C22×C4—24no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—24no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—24no—
398C22×C4—24no—
399C22×C4—24no—
400C22×C4—24no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—24no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—21no—
409C22×C4—22no—
410C22×C4—24no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—24no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—21no—
418C22×C4—22no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—24no—
422C22×C4—24no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—48no—
427C22×C4—48no—
428C22×C4—48no—
429C22×C4—48no—
430C22×C4—48no—
431C22×C4—48no—
432C22×C4—44no—
433C22×C4—44no—
434C22×C4—48no—
435C22×C4C22×D414yes—
436C22×C4C22×D414yes—
437C22×C4C22×D412yes—
438C22×C4C22×D412yes—
439C22×C4C22×D412yes—
440C22×C4C2×#4112yes—
441C22×C4C2×#4114yes—
442C22×C4C2×#4114yes—
443C22×C4C2×#4112yes—
444C22×C4C2×#4114yes—
445C22×C4C2×#4112yes—
446C22×C4C2×#4112yes—
447C22×C4C2×#4114yes—
448C22×C4C2×#4112yes—
449C22×C4H(5,𝔽2)12yes—
450C22×C42−1+414yes—
451C4⋊C4—216no—
452C4⋊C4—28no—
453C4⋊C4—216no—
454C4⋊C4—28no—
455C4⋊C4—28no—
456C4⋊C4—28no—
457C4⋊C4—48no—
458C4⋊C4—48no—
459C4⋊C4—44no—
460C4⋊C4—48no—
461C4⋊C4—48no—
462C4⋊C4—48no—
463C4⋊C4—88no—
464C4⋊C4—84no—
465C22⋊C4—28no—
466C22⋊C4—48no—
467C22⋊C4—44no—
468C22⋊C4—48no—
469C22⋊C4—48no—
470C22⋊C4—88no—
471C42—28no—
472C42—216no—
473C42—216no—
474C42—48no—
475C42—44no—
476C42—48no—
477C42—88no—
478C23—22no—
479C23—22no—
480C23—24no—
481C23—24no—
482C23—22no—
483C23—22no—
484C23—24no—
485C23—22no—
486C23—24no—
487C23—24no—
488C23—22no—
489C23—24no—
490C23—41no—
491C23—41no—
492C23—42no—
493C23—41no—
494C23—42no—
495C23—41no—
496C23—42no—
497C23—42no—
498C23—42no—
499C23—42no—
500C23#55614yes—
501C23C23×D411yescharacteristic
502C23C22×#4111yescharacteristic
503C23C22×#4112yes—
504C23C22×#4111yescharacteristic
505C23C22×#4111yescharacteristic
506C23C2×H(5,𝔽2)11yescharacteristic
507C23C2×2−1+412yes—
508C23C2×2−1+411yescharacteristic
509C23#58512yes—
510C23#58511yescharacteristic
511C23#58512yes—
512Q8—416no—
513C2×C4—22no—
514C2×C4—24no—
515C2×C4—22no—
516C2×C4—24no—
517C2×C4—22no—
518C2×C4—24no—
519C2×C4—24no—
520C2×C4—24no—
521C2×C4—22no—
522C2×C4—24no—
523C2×C4—24no—
524C2×C4—24no—
525C2×C4—24no—
526C2×C4—22no—
527C2×C4—24no—
528C2×C4—24no—
529C2×C4—24no—
530C2×C4—22no—
531C2×C4—24no—
532C2×C4—24no—
533C2×C4—24no—
534C2×C4—24no—
535C2×C4—22no—
536C2×C4—24no—
537C2×C4—24no—
538C2×C4—24no—
539C2×C4—432no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—41no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—44no—
563C2×C4—42no—
564C2×C4—44no—
565C2×C4—42no—
566C2×C4—416no—
567C2×C4—42no—
568C2×C4—44no—
569C2×C4—41no—
570C2×C4—42no—
571C2×C4—44no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—44no—
577C2×C4—44no—
578C2×C4—42no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—44no—
582C2×C4—44no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—44no—
586C2×C4—44no—
587C2×C4—44no—
588C2×C4—44no—
589C2×C4—44no—
590C2×C4—42no—
591C2×C4—44no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—44no—
597C2×C4—44no—
598C2×C4—42no—
599C2×C4—44no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—44no—
605C2×C4#52514yes—
606C2×C4#52514yes—
607C2×C4#53614yes—
608C2×C4#54614yes—
609C2×C4#55218yes—
610C2×C4#55618yes—
611C22—24no—
612C22—28no—
613C22—44no—
614C22—44no—
615C22—42no—
616C22—42no—
617C22—42no—
618C22—42no—
619C22—44no—
620C22—42no—
621C22—42no—
622C22C2×#52511yescharacteristic
623C22C2×#52511yescharacteristic
624C22C2×#53611yescharacteristic
625C22C2×#54611yescharacteristic
626C22C2×#55212yes—
627C22C2×#55612yes—
628C22C2×#55611yescharacteristic
629C22#347511yescharacteristic
630C22#347512yes—
631C22#348711yescharacteristic
632C22#348911yescharacteristic
633C22#349412yes—
634C22#349512yes—
635C22#350011yescharacteristic
636C22#350611yescharacteristic
637C22#351012yes—
638C22#351212yes—
639C22#351512yes—
640C22#352011yescharacteristic
641C22#352412yes—
642C22#354712yes—
643C22#356312yes—
644C22#356512yes—
645C4—24no—
646C4—44no—
647C4—42no—
648C4—42no—
649C4—44no—
650C4—44no—
651C4—44no—
652C4—42no—
653C4—44no—
654C4—82no—
655C4—82no—
656C4—82no—
657C4—81no—
658C4—84no—
659C4—84no—
660C4#259918yes—
661C2—44no—
662C2C2×#259911yescharacteristic
663C2#3604111yescharacteristic
664C2#3608011yescharacteristic
665C2#3616111yescharacteristic
666C2#3642611yescharacteristic
667C2#3895511yescharacteristic
668C2#4288811yescharacteristic
669C2#4495812yes—
670C2#4508512yes—
671C2#5933312yes—
672C2#5943212yes—
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×#2355C2semidirect
3. #16663C2semidirect
4. #16031C2semidirect
6. #15242C2semidirect
8. #15158C2semidirect
9. #15088C2semidirect
11. #14030C2semidirect
12. #13982C2semidirect
15. #13921C2semidirect

Maximal subgroups

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

subgroupindex
C2×#23552
C2×#23442
#166632
#160312
#159892
#152422
#151952
#151582
#150882
#148872
#140302
#139822
#139732
#139382
#139212
#138812
C4×#3932