Finite Group Databasebeta

← #800430

Subgroups — #800430

Order 512.

2388 conjugacy classes of subgroups.

The subgroup lattice has 671 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2513C211yesmaximal, characteristic
2C2×#2365C211yesmaximal, characteristic, complemented
3#16982C212yesmaximal, complemented
4#15905C214yesmaximal, complemented
5#15898C214yesmaximal, complemented
6#15243C211yesmaximal, characteristic
7#15180C211yesmaximal, characteristic, complemented
8#15174C212yesmaximal, complemented
9#15147C211yesmaximal, characteristic
10#15047C212yesmaximal, complemented
11#14249C212yesmaximal, complemented
12#14153C212yesmaximal
13#14026C212yesmaximal
14#14017C212yesmaximal, complemented
15#14003C212yesmaximal, complemented
16C4×#398C211yesmaximal, characteristic, complemented
17#13812C211yesmaximal, characteristic, complemented
18C22×C4⋊Q8C2211yescharacteristic
19C22×#124C2212yes—
20C22×C4×Q8C2212yes—
21#2513—28no—
22C2×#398C2211yescharacteristic
23C2×#397C2212yes—
24C2×#397C2212yes—
25C2×#397C2212yes—
26#2365—28no—
27C2×#391C2212yes—
28C2×#391C2212yes—
29C2×#391C2212yes—
30C2×#391C2212yes—
31C2×#390C2211yescharacteristic
32C2×#390C2211yescharacteristic
33C2×#390C2211yescharacteristic
34C2×#390C2212yes—
35C2×#389C2212yes—
36C2×#389C2211yescharacteristic
37C2×#388C2212yes—
38C2×#388C2212yes—
39C2×#387C2214yes—
40C2×#382C2211yescharacteristic
41C2×C4×C4⋊C4C2211yescharacteristic
42C2×C4×C4⋊C4C2211yescharacteristic
43C2×C4×C4⋊C4C2212yes—
44C2×C4×C4⋊C4C2212yes—
45C2×C4×C4⋊C4C2212yes—
46C2×C4×C4⋊C4C2211yescharacteristic
47#1445C2211yescharacteristic
48#1444C2214yes—
49#1444C2211yescharacteristic
50#1443C2214yes—
51#1443C2214yes—
52#1442C2214yes—
53#1442C2212yes—
54#1442C2212yes—
55#1442C2212yes—
56#1441C2212yes—
57#1441C2212yes—
58#1441C2211yescharacteristic
59#1441C2212yes—
60#1440C2214yes—
61#1439C2214yes—
62#1439C2214yes—
63#1439C2214yes—
64#1439C2214yes—
65#1439C2214yes—
66#1439C2212yes—
67#1439C2212yes—
68#1439C2212yes—
69#1437C2214yes—
70#1437C2214yes—
71#1437C2214yes—
72#1437C2214yes—
73#1436C2214yes—
74#1436C2214yes—
75C4⋊#95C2211yescharacteristic
76C4⋊#95C2212yes—
77C4⋊#95C2212yes—
78#1434C2214yes—
79#1434C2214yes—
80#1434C2214yes—
81#1434C2212yes—
82C4×#95C2211yescharacteristic
83C4×#95C2212yes—
84C4×#95C2212yes—
85C23×Q8C2312yes—
86C23×D4C2311yescharacteristic
87C2×C4⋊Q8—28no—
88C2×C4⋊Q8—24no—
89C2×#124—28no—
90C2×#124—28no—
91C2×#124—44no—
92C2×C4×Q8—28no—
93C2×C4×Q8—44no—
94C2×C4×Q8—44no—
95C22×C4⋊C4C2311yescharacteristic
96C22×C4⋊C4C2312yes—
97C22×C4⋊C4C2312yes—
98C22×C4⋊C4C2312yes—
99C22×C4⋊C4C2311yescharacteristic
100C22×C4⋊C4C2311yescharacteristic
101C22×C4⋊C4C2312yes—
102C22×C4⋊C4C2311yescharacteristic
103C22×C4⋊C4C2311yescharacteristic
104C22×C4⋊C4C2311yescharacteristic
105C22×C4⋊C4C2312yes—
106C22×C4⋊C4C2314yes—
107C22×C4⋊C4C2311yescharacteristic
108C22×C4⋊C4C2312yes—
109C22×C22⋊C4C2312yes—
110C22×C22⋊C4C2312yes—
111C22×C22⋊C4C2314yes—
112C22×C22⋊C4C2314yes—
113C22×C22⋊C4C2312yes—
114C22×C22⋊C4C2314yes—
115C22×C22⋊C4C2312yes—
116C22×C22⋊C4C2312yes—
117C22×C22⋊C4C2312yes—
118C22×C22⋊C4C2314yes—
119C22×C42C2312yes—
120C22×C42C2311yescharacteristic
121C22×C42C2311yescharacteristic
122C22×C42C2311yescharacteristic
123C22×C42C2312yes—
124C22×C42C2314yes—
125C22×C42C2314yes—
126C22×C42C2312yes—
127#398—24no—
128#397—28no—
129#397—28no—
130#397—28no—
131#391—28no—
132#391—28no—
133#391—28no—
134#391—44no—
135#390—24no—
136#390—24no—
137#390—24no—
138#390—44no—
139#389—24no—
140#389—28no—
141#388—28no—
142#388—28no—
143#387—216no—
144#382—24no—
145C4×C4⋊C4—24no—
146C4×C4⋊C4—28no—
147C4×C4⋊C4—24no—
148C4×C4⋊C4—28no—
149C4×C4⋊C4—44no—
150C4×C4⋊C4D418yes—
151C2×#95C2311yescharacteristic
152C2×#95C2312yes—
153C2×#95C2312yes—
154C2×#95C2312yes—
155C2×#95C2314yes—
156C2×#95C2312yes—
157C2×#95C2311yescharacteristic
158C2×#95C2312yes—
159C2×#95C2311yescharacteristic
160C2×#95C2314yes—
161C2×#95C2314yes—
162C2×#95C2314yes—
163C2×#95C2314yes—
164C2×#95C2314yes—
165C2×#95C2314yes—
166C2×#95C2314yes—
167C2×#95C2312yes—
168C2×#95C2311yescharacteristic
169C2×#95C2312yes—
170C2×#95C2312yes—
171C2×#95C2312yes—
172C2×#95C2314yes—
173C2×#95C2312yes—
174C2×#95C2314yes—
175C2×#95C2312yes—
176C2×#95C2314yes—
177C2×#95C2312yes—
178C2×#95C2314yes—
179C2×#95C2314yes—
180C2×#95C2312yes—
181C2×#95C2312yes—
182C25C2412yes—
183C22×Q8—24no—
184C22×Q8—24no—
185C22×Q8—24no—
186C22×Q8—24no—
187C22×Q8—24no—
188C22×Q8—24no—
189C22×Q8#4118yes—
190C22×D4—22no—
191C22×D4—24no—
192C22×D4—24no—
193C22×D4—22no—
194C22×D4—22no—
195C23×C4C2412yes—
196C23×C4C2411yescharacteristic
197C23×C4C2412yes—
198C23×C4C2411yescharacteristic
199C23×C4C2411yescharacteristic
200C23×C4C2414yes—
201C23×C4C2412yes—
202C23×C4C2411yescharacteristic
203C23×C4C2412yes—
204C23×C4C2414yes—
205C23×C4C2412yes—
206C23×C4C2412yes—
207C23×C4C2412yes—
208C23×C4C2411yescharacteristic
209C23×C4C2411yescharacteristic
210C23×C4C2411yescharacteristic
211C4⋊Q8—232no—
212#124—432no—
213C4×Q8—432no—
214C2×C4⋊C4—24no—
215C2×C4⋊C4—24no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—28no—
218C2×C4⋊C4—28no—
219C2×C4⋊C4—24no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—28no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—22no—
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—22no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—41no—
239C2×C4⋊C4—42no—
240C2×C4⋊C4—42no—
241C2×C4⋊C4—42no—
242C2×C4⋊C4—44no—
243C2×C4⋊C4—44no—
244C2×C4⋊C4—42no—
245C2×C4⋊C4—42no—
246C2×C4⋊C4—44no—
247C2×C4⋊C4C2×D414yes—
248C2×C4⋊C4C2×D414yes—
249C2×C4⋊C4#4114yes—
250C2×C22⋊C4—24no—
251C2×C22⋊C4—24no—
252C2×C22⋊C4—28no—
253C2×C22⋊C4—28no—
254C2×C22⋊C4—28no—
255C2×C22⋊C4—28no—
256C2×C22⋊C4—24no—
257C2×C22⋊C4—24no—
258C2×C22⋊C4—24no—
259C2×C22⋊C4—24no—
260C2×C22⋊C4—24no—
261C2×C22⋊C4—24no—
262C2×C22⋊C4—24no—
263C2×C22⋊C4—28no—
264C2×C22⋊C4—42no—
265C2×C22⋊C4—44no—
266C2×C22⋊C4—44no—
267C2×C22⋊C4—44no—
268C2×C22⋊C4—44no—
269C2×C22⋊C4—44no—
270C2×C22⋊C4—44no—
271C2×C22⋊C4—44no—
272C2×C22⋊C4—44no—
273C2×C22⋊C4—44no—
274C2×C22⋊C4—44no—
275C2×C22⋊C4—44no—
276C2×C42—24no—
277C2×C42—24no—
278C2×C42—24no—
279C2×C42—28no—
280C2×C42—24no—
281C2×C42—22no—
282C2×C42—22no—
283C2×C42—24no—
284C2×C42—24no—
285C2×C42—24no—
286C2×C42—22no—
287C2×C42—22no—
288C2×C42—28no—
289C2×C42—44no—
290C2×C42—44no—
291C2×C42—44no—
292C2×C42—44no—
293C2×C42—44no—
294C2×C42C2×D414yes—
295C2×C42C2×D418yes—
296C2×C42#4118yes—
297#95—24no—
298#95—28no—
299#95—22no—
300#95—22no—
301#95—24no—
302#95—24no—
303#95—28no—
304#95—24no—
305#95—28no—
306#95—28no—
307#95—28no—
308#95—28no—
309#95—28no—
310#95—28no—
311#95—24no—
312#95—22no—
313#95—24no—
314#95—22no—
315#95—42no—
316#95—44no—
317#95—44no—
318#95—42no—
319#95—42no—
320#95—42no—
321#95—42no—
322#95—44no—
323#95—42no—
324#95—42no—
325#95—44no—
326#95—44no—
327#95C2×D418yes—
328C24—22no—
329C24—22no—
330C24—24no—
331C24—24no—
332C24—22no—
333C24—22no—
334C24—22no—
335C24—22no—
336C24—24no—
337C24—24no—
338C24—22no—
339C24C2511yescharacteristic
340C2×Q8—216no—
341C2×Q8—216no—
342C2×Q8—216no—
343C2×Q8—48no—
344C2×Q8—48no—
345C2×Q8—48no—
346C2×Q8—48no—
347C2×D4—44no—
348C2×D4—44no—
349C2×D4—44no—
350C2×D4—48no—
351C2×D4—48no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—21no—
356C22×C4—24no—
357C22×C4—22no—
358C22×C4—24no—
359C22×C4—24no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—24no—
364C22×C4—21no—
365C22×C4—24no—
366C22×C4—21no—
367C22×C4—22no—
368C22×C4—21no—
369C22×C4—24no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—24no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—21no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—21no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—21no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—24no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—24no—
393C22×C4—24no—
394C22×C4—22no—
395C22×C4—21no—
396C22×C4—21no—
397C22×C4—21no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—21no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—24no—
405C22×C4—24no—
406C22×C4—21no—
407C22×C4—24no—
408C22×C4—24no—
409C22×C4—21no—
410C22×C4—21no—
411C22×C4—24no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—21no—
416C22×C4—21no—
417C22×C4—22no—
418C22×C4—24no—
419C22×C4—21no—
420C22×C4—24no—
421C22×C4—24no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—24no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4C22×D414yes—
431C22×C4C22×D412yes—
432C22×C4C22×D414yes—
433C22×C4C22×D412yes—
434C22×C4C22×D412yes—
435C22×C4C2×#4114yes—
436C22×C4C2×#4114yes—
437C22×C4C2×#4114yes—
438C22×C4C2×#4118yes—
439C22×C4C2×#4112yes—
440C22×C4C2×#4114yes—
441C22×C4C2×#4114yes—
442C22×C4H(5,𝔽2)12yes—
443C22×C4H(5,𝔽2)12yes—
444C22×C42−1+412yes—
445C22×C42−1+412yes—
446C4⋊C4—216no—
447C4⋊C4—216no—
448C4⋊C4—28no—
449C4⋊C4—28no—
450C4⋊C4—28no—
451C4⋊C4—44no—
452C4⋊C4—48no—
453C4⋊C4—44no—
454C4⋊C4—416no—
455C4⋊C4—48no—
456C4⋊C4—44no—
457C4⋊C4—44no—
458C4⋊C4—48no—
459C4⋊C4—84no—
460C22⋊C4—48no—
461C22⋊C4—416no—
462C22⋊C4—416no—
463C22⋊C4—48no—
464C22⋊C4—84no—
465C22⋊C4—84no—
466C22⋊C4—84no—
467C22⋊C4—84no—
468C22⋊C4—88no—
469C22⋊C4—88no—
470C42—28no—
471C42—216no—
472C42—48no—
473C42—416no—
474C42—44no—
475C42—44no—
476C42—48no—
477C42—88no—
478C23—42no—
479C23—42no—
480C23—42no—
481C23—42no—
482C23—44no—
483C23—44no—
484C23—44no—
485C23—44no—
486C23—42no—
487C23—42no—
488C23—42no—
489C23—44no—
490C23—44no—
491C23—44no—
492C23—42no—
493C23—44no—
494C23—44no—
495C23—42no—
496C23—44no—
497C23—44no—
498C23—42no—
499C23—42no—
500C23—44no—
501C23C23×D411yescharacteristic
502C23C22×#4112yes—
503C23C22×#4112yes—
504C23C22×#4111yescharacteristic
505C23C2×H(5,𝔽2)11yescharacteristic
506C23C2×H(5,𝔽2)11yescharacteristic
507C23C2×2−1+411yescharacteristic
508C23C2×2−1+411yescharacteristic
509C23#58512yes—
510C23#58512yes—
511C23#58511yescharacteristic
512Q8—432no—
513D4—88no—
514C2×C4—22no—
515C2×C4—24no—
516C2×C4—22no—
517C2×C4—24no—
518C2×C4—22no—
519C2×C4—28no—
520C2×C4—24no—
521C2×C4—24no—
522C2×C4—24no—
523C2×C4—24no—
524C2×C4—24no—
525C2×C4—28no—
526C2×C4—28no—
527C2×C4—24no—
528C2×C4—24no—
529C2×C4—24no—
530C2×C4—22no—
531C2×C4—24no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—24no—
535C2×C4—24no—
536C2×C4—24no—
537C2×C4—24no—
538C2×C4—24no—
539C2×C4—22no—
540C2×C4—24no—
541C2×C4—22no—
542C2×C4—44no—
543C2×C4—42no—
544C2×C4—44no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—44no—
548C2×C4—44no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—44no—
552C2×C4—44no—
553C2×C4—41no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—44no—
563C2×C4—42no—
564C2×C4—41no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—44no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—41no—
573C2×C4—42no—
574C2×C4—41no—
575C2×C4—44no—
576C2×C4—44no—
577C2×C4—44no—
578C2×C4—44no—
579C2×C4—42no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—42no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—42no—
586C2×C4—41no—
587C2×C4—42no—
588C2×C4—44no—
589C2×C4—44no—
590C2×C4—44no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—44no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—44no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4#52514yes—
603C2×C4#53214yes—
604C2×C4#53514yes—
605C2×C4#54814yes—
606C2×C4#55118yes—
607C2×C4#55318yes—
608C22—84no—
609C22—82no—
610C22—84no—
611C22—84no—
612C22—82no—
613C22—84no—
614C22—82no—
615C22—82no—
616C22—82no—
617C22—82no—
618C22—82no—
619C22C2×#52511yescharacteristic
620C22C2×#53211yescharacteristic
621C22C2×#53511yescharacteristic
622C22C2×#54811yescharacteristic
623C22C2×#55112yes—
624C22C2×#55312yes—
625C22#348012yes—
626C22#348012yes—
627C22#348511yescharacteristic
628C22#348911yescharacteristic
629C22#350011yescharacteristic
630C22#350712yes—
631C22#350712yes—
632C22#351012yes—
633C22#351411yescharacteristic
634C22#351612yes—
635C22#351812yes—
636C22#352011yescharacteristic
637C22#352412yes—
638C22#353912yes—
639C22#354711yescharacteristic
640C22#355211yescharacteristic
641C22#356212yes—
642C4—24no—
643C4—48no—
644C4—44no—
645C4—44no—
646C4—44no—
647C4—44no—
648C4—44no—
649C4—42no—
650C4—42no—
651C4—84no—
652C4—82no—
653C4—81no—
654C4—82no—
655C4—81no—
656C4—81no—
657C4#260318yes—
658C2—162no—
659C2C2×#260311yescharacteristic
660C2#3595911yescharacteristic
661C2#3607711yescharacteristic
662C2#3653811yescharacteristic
663C2#3658211yescharacteristic
664C2#3912411yescharacteristic
665C2#4265211yescharacteristic
666C2#4515612yes—
667C2#4539112yes—
668C2#5968412yes—
669C2#5976912yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC23×D4C23
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2365C2semidirect
3. #16982C2semidirect
4. #15905C2semidirect
5. #15898C2semidirect
7. #15180C2semidirect
8. #15174C2semidirect
10. #15047C2semidirect
11. #14249C2semidirect
14. #14017C2semidirect
15. #14003C2semidirect
16. C4×#398C2semidirect
17. #13812C2semidirect

Maximal subgroups

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

subgroupindex
C2×#25132
C2×#23652
#169822
#159052
#158982
#152432
#151802
#151742
#151472
#150472
#142492
#141532
#140262
#140172
#140032
C4×#3982
#138122