Finite Group Databasebeta

← #949706

Subgroups — #949706

Order 512.

2276 conjugacy classes of subgroups.

The subgroup lattice has 686 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#17176C214yesmaximal, complemented
2#17019C212yesmaximal, complemented
3#17003C212yesmaximal, complemented
4#16707C212yesmaximal, complemented
5#16305C212yesmaximal, complemented
6#15335C214yesmaximal
7#15137C212yesmaximal, complemented
8#15135C212yesmaximal, complemented
9#15131C212yesmaximal, complemented
10#14264C212yesmaximal, complemented
11#13981C212yesmaximal, complemented
12#13907C212yesmaximal, complemented
13#13906C211yesmaximal, characteristic
14#13872C211yesmaximal, characteristic
15#13839C211yesmaximal, characteristic
16C22×C4×D4C2211yescharacteristic
17C22×#117C2211yescharacteristic
18C23×C4⋊C4C2211yescharacteristic
19C2×#399C2214yes—
20C2×#398C2214yes—
21C2×#396C2214yes—
22C2×#390C2212yes—
23C2×#389C2212yes—
24C2×#389C2212yes—
25C2×#389C2212yes—
26C2×#389C2212yes—
27C2×#387C2214yes—
28C2×#386C2212yes—
29C2×#384C2212yes—
30C2×#384C2211yescharacteristic
31C2×#380C2212yes—
32C2×#380C2211yescharacteristic
33C2×#380C2212yes—
34C2×#380C2212yes—
35C2×#380C2214yes—
36C2×C4×C4⋊C4C2211yescharacteristic
37C2×C4×C22⋊C4C2212yes—
38C2×C4×C22⋊C4C2211yescharacteristic
39#1445C2214yes—
40#1444C2214yes—
41#1444C2214yes—
42#1444C2214yes—
43#1442C2214yes—
44#1442C2214yes—
45#1442C2214yes—
46#1441C2211yescharacteristic
47#1441C2211yescharacteristic
48#1441C2214yes—
49#1441C2214yes—
50#1441C2214yes—
51#1441C2214yes—
52#1441C2214yes—
53#1441C2214yes—
54#1441C2212yes—
55#1440C2212yes—
56#1439C2211yescharacteristic
57#1439C2211yescharacteristic
58#1439C2212yes—
59#1439C2212yes—
60#1439C2214yes—
61#1439C2214yes—
62#1439C2214yes—
63#1439C2214yes—
64#1439C2212yes—
65#1438C2212yes—
66#1436C2214yes—
67C4⋊#95C2212yes—
68C4⋊#95C2212yes—
69C4⋊#95C2212yes—
70C4⋊#95C2212yes—
71C4⋊#95C2212yes—
72C22⋊#95C2212yes—
73C22⋊#95C2214yes—
74C4×#95C2212yes—
75C23×D4C2311yescharacteristic
76C24×C4C2311yescharacteristic
77C24×C4C2311yescharacteristic
78C24×C4C2312yes—
79C2×C4×D4—42no—
80C2×C4×D4—42no—
81C2×C4×D4Q818yes—
82C2×#117—44no—
83C2×#117—42no—
84C22×C4⋊C4—28no—
85C22×C4⋊C4—42no—
86C22×C4⋊C4C2312yes—
87C22×C4⋊C4C2312yes—
88C22×C4⋊C4C2312yes—
89C22×C4⋊C4C2314yes—
90C22×C4⋊C4C2312yes—
91C22×C4⋊C4C2312yes—
92C22×C4⋊C4C2314yes—
93C22×C4⋊C4C2314yes—
94C22×C4⋊C4C2312yes—
95C22×C4⋊C4C2314yes—
96C22×C4⋊C4C2314yes—
97C22×C4⋊C4C2312yes—
98C22×C4⋊C4C2314yes—
99C22×C4⋊C4C2311yescharacteristic
100C22×C4⋊C4C2311yescharacteristic
101C22×C4⋊C4C2311yescharacteristic
102C22×C22⋊C4C2312yes—
103C22×C22⋊C4C2312yes—
104C22×C22⋊C4C2312yes—
105C22×C22⋊C4C2312yes—
106C22×C22⋊C4C2314yes—
107C22×C22⋊C4C2314yes—
108C22×C22⋊C4C2314yes—
109C22×C22⋊C4C2312yes—
110C22×C22⋊C4C2311yescharacteristic
111C22×C22⋊C4C2311yescharacteristic
112C22×C42C2312yes—
113C22×C42C2312yes—
114C22×C42C2311yescharacteristic
115C22×C42C2312yes—
116#399—48no—
117#398—48no—
118#396—48no—
119#390—44no—
120#389—28no—
121#389—44no—
122#389—44no—
123#389—44no—
124#387—48no—
125#386—44no—
126#384—42no—
127#384—44no—
128#380—28no—
129#380—48no—
130#380—44no—
131#380—44no—
132#380—42no—
133C4×C4⋊C4—42no—
134C4×C22⋊C4—44no—
135C4×C22⋊C4—42no—
136C2×#95C2312yes—
137C2×#95C2314yes—
138C2×#95C2312yes—
139C2×#95C2312yes—
140C2×#95C2312yes—
141C2×#95C2314yes—
142C2×#95C2314yes—
143C2×#95C2314yes—
144C2×#95C2314yes—
145C2×#95C2314yes—
146C2×#95C2314yes—
147C2×#95C2314yes—
148C2×#95C2314yes—
149C2×#95C2314yes—
150C2×#95C2314yes—
151C2×#95C2314yes—
152C2×#95C2314yes—
153C2×#95C2312yes—
154C2×#95C2314yes—
155C2×#95C2314yes—
156C2×#95C2312yes—
157C2×#95C2311yescharacteristic
158C2×#95C2311yescharacteristic
159C2×#95C2312yes—
160C2×#95C2312yes—
161C25C2411yescharacteristic
162C25C2411yescharacteristic
163C22×D4—22no—
164C22×D4—22no—
165C22×D4—22no—
166C22×D4—22no—
167C22×D4—41no—
168C22×D4—41no—
169C22×D4C2×Q814yes—
170C23×C4—24no—
171C23×C4—24no—
172C23×C4—22no—
173C23×C4—22no—
174C23×C4—24no—
175C23×C4—22no—
176C23×C4—22no—
177C23×C4—24no—
178C23×C4—24no—
179C23×C4—28no—
180C23×C4—42no—
181C23×C4—44no—
182C23×C4—42no—
183C23×C4C2×Q814yes—
184C23×C4C2×Q814yes—
185C23×C4C2414yes—
186C23×C4C2414yes—
187C23×C4C2412yes—
188C23×C4C2414yes—
189C23×C4C2412yes—
190C23×C4C2411yescharacteristic
191C23×C4C2411yescharacteristic
192C23×C4C2411yescharacteristic
193C23×C4C2414yes—
194C23×C4C2412yes—
195C23×C4C2412yes—
196C23×C4C2411yescharacteristic
197C23×C4C2411yescharacteristic
198C4×D4—416no—
199#117—416no—
200C2×C4⋊C4—24no—
201C2×C4⋊C4—24no—
202C2×C4⋊C4—28no—
203C2×C4⋊C4—24no—
204C2×C4⋊C4—28no—
205C2×C4⋊C4—24no—
206C2×C4⋊C4—24no—
207C2×C4⋊C4—24no—
208C2×C4⋊C4—24no—
209C2×C4⋊C4—24no—
210C2×C4⋊C4—24no—
211C2×C4⋊C4—28no—
212C2×C4⋊C4—24no—
213C2×C4⋊C4—24no—
214C2×C4⋊C4—28no—
215C2×C4⋊C4—28no—
216C2×C4⋊C4—24no—
217C2×C4⋊C4—416no—
218C2×C4⋊C4—42no—
219C2×C4⋊C4—42no—
220C2×C4⋊C4—42no—
221C2×C4⋊C4—41no—
222C2×C4⋊C4—41no—
223C2×C4⋊C4—41no—
224C2×C4⋊C4—41no—
225C2×C4⋊C4—44no—
226C2×C4⋊C4—44no—
227C2×C4⋊C4—42no—
228C2×C4⋊C4—44no—
229C2×C4⋊C4—44no—
230C2×C4⋊C4—42no—
231C2×C4⋊C4—44no—
232C2×C4⋊C4—44no—
233C2×C4⋊C4—44no—
234C2×C4⋊C4—44no—
235C2×C4⋊C4—42no—
236C2×C4⋊C4—44no—
237C2×C4⋊C4—44no—
238C2×C4⋊C4—44no—
239C2×C4⋊C4—44no—
240C2×C4⋊C4—44no—
241C2×C4⋊C4—44no—
242C2×C4⋊C4—44no—
243C2×C4⋊C4C2×Q814yes—
244C2×C4⋊C4#4118yes—
245C2×C22⋊C4—24no—
246C2×C22⋊C4—24no—
247C2×C22⋊C4—24no—
248C2×C22⋊C4—24no—
249C2×C22⋊C4—24no—
250C2×C22⋊C4—24no—
251C2×C22⋊C4—28no—
252C2×C22⋊C4—24no—
253C2×C22⋊C4—28no—
254C2×C22⋊C4—42no—
255C2×C22⋊C4—42no—
256C2×C22⋊C4—42no—
257C2×C22⋊C4—41no—
258C2×C22⋊C4—41no—
259C2×C22⋊C4—41no—
260C2×C22⋊C4—41no—
261C2×C22⋊C4—42no—
262C2×C22⋊C4—42no—
263C2×C22⋊C4—42no—
264C2×C22⋊C4—44no—
265C2×C22⋊C4—44no—
266C2×C22⋊C4—42no—
267C2×C22⋊C4—44no—
268C2×C22⋊C4—44no—
269C2×C22⋊C4—42no—
270C2×C22⋊C4—44no—
271C2×C22⋊C4—44no—
272C2×C22⋊C4—44no—
273C2×C22⋊C4C2×Q814yes—
274C2×C22⋊C4C2×Q814yes—
275C2×C42—22no—
276C2×C42—22no—
277C2×C42—24no—
278C2×C42—24no—
279C2×C42—24no—
280C2×C42—24no—
281C2×C42—24no—
282C2×C42—24no—
283C2×C42—24no—
284C2×C42—42no—
285C2×C42—42no—
286C2×C42C2×Q814yes—
287#95—28no—
288#95—28no—
289#95—28no—
290#95—28no—
291#95—24no—
292#95—24no—
293#95—24no—
294#95—24no—
295#95—42no—
296#95—42no—
297#95—41no—
298#95—41no—
299#95—42no—
300#95—44no—
301#95—44no—
302#95—42no—
303#95—44no—
304#95—44no—
305#95—44no—
306#95—44no—
307#95—44no—
308#95—44no—
309#95—44no—
310#95—44no—
311#95—44no—
312C24—22no—
313C24—22no—
314C24—22no—
315C24—21no—
316C24—21no—
317C24—21no—
318C24—22no—
319C24—21no—
320C24—22no—
321C24—22no—
322C24—22no—
323C24—21no—
324C24—21no—
325C24—21no—
326C24—21no—
327C24—22no—
328C24—21no—
329C24—21no—
330C24C22×Q812yes—
331C24C22×Q812yes—
332C24H(5,𝔽2)12yes—
333C242−1+412yes—
334C24C2511yescharacteristic
335C2×D4—28no—
336C2×D4—28no—
337C2×D4—44no—
338C2×D4—44no—
339C2×D4—44no—
340C2×D4—44no—
341C2×D4—44no—
342C22×C4—21no—
343C22×C4—21no—
344C22×C4—21no—
345C22×C4—21no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—24no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—24no—
353C22×C4—24no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—21no—
358C22×C4—22no—
359C22×C4—21no—
360C22×C4—21no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—24no—
366C22×C4—24no—
367C22×C4—21no—
368C22×C4—22no—
369C22×C4—21no—
370C22×C4—21no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—24no—
376C22×C4—28no—
377C22×C4—28no—
378C22×C4—216no—
379C22×C4—216no—
380C22×C4—24no—
381C22×C4—24no—
382C22×C4—22no—
383C22×C4—24no—
384C22×C4—24no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—24no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—24no—
392C22×C4—24no—
393C22×C4—24no—
394C22×C4—24no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—24no—
398C22×C4—24no—
399C22×C4—24no—
400C22×C4—24no—
401C22×C4—24no—
402C22×C4—24no—
403C22×C4—24no—
404C22×C4—24no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—24no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—24no—
413C22×C4—24no—
414C22×C4—24no—
415C22×C4—48no—
416C22×C4—416no—
417C22×C4—48no—
418C22×C4—48no—
419C22×C4—44no—
420C22×C4—44no—
421C22×C4—44no—
422C22×C4—44no—
423C22×C4—88no—
424C22×C4—84no—
425C22×C4—84no—
426C22×C4C22×Q812yes—
427C22×C4C22×Q812yes—
428C22×C4C22×Q812yes—
429C22×C4C22×Q812yes—
430C22×C4C22×Q812yes—
431C22×C4C2×#4118yes—
432C22×C4C2×#4114yes—
433C22×C4H(5,𝔽2)14yes—
434C22×C4H(5,𝔽2)12yes—
435C22×C4H(5,𝔽2)12yes—
436C22×C4H(5,𝔽2)14yes—
437C22×C4H(5,𝔽2)12yes—
438C22×C42−1+414yes—
439C22×C42−1+412yes—
440C4⋊C4—216no—
441C4⋊C4—44no—
442C4⋊C4—416no—
443C4⋊C4—48no—
444C4⋊C4—44no—
445C4⋊C4—44no—
446C4⋊C4—48no—
447C4⋊C4—48no—
448C4⋊C4—48no—
449C4⋊C4—84no—
450C4⋊C4—88no—
451C4⋊C4—88no—
452C4⋊C4—88no—
453C4⋊C4—84no—
454C4⋊C4—88no—
455C4⋊C4—88no—
456C22⋊C4—44no—
457C22⋊C4—44no—
458C22⋊C4—48no—
459C22⋊C4—48no—
460C22⋊C4—48no—
461C22⋊C4—48no—
462C22⋊C4—88no—
463C22⋊C4—88no—
464C22⋊C4—84no—
465C22⋊C4—88no—
466C42—28no—
467C42—48no—
468C42—48no—
469C42—84no—
470C23—22no—
471C23—22no—
472C23—24no—
473C23—24no—
474C23—24no—
475C23—24no—
476C23—22no—
477C23—22no—
478C23—24no—
479C23—22no—
480C23—24no—
481C23—22no—
482C23—22no—
483C23—22no—
484C23—22no—
485C23—24no—
486C23—24no—
487C23—42no—
488C23—41no—
489C23—42no—
490C23—42no—
491C23—41no—
492C23—42no—
493C23—42no—
494C23—42no—
495C23—41no—
496C23—41no—
497C23—42no—
498C23—41no—
499C23—42no—
500C23—42no—
501C23—42no—
502C23—42no—
503C23—41no—
504C23—42no—
505C23—41no—
506C23—41no—
507C23—42no—
508C23—41no—
509C23—41no—
510C23—42no—
511C23—42no—
512C23—42no—
513C23—41no—
514C23—41no—
515C23#54414yes—
516C23C23×Q811yescharacteristic
517C23C22×#4112yes—
518C23C2×H(5,𝔽2)11yescharacteristic
519C23C2×H(5,𝔽2)12yes—
520C23C2×H(5,𝔽2)11yescharacteristic
521C23C2×H(5,𝔽2)11yescharacteristic
522C23C2×H(5,𝔽2)11yescharacteristic
523C23C2×H(5,𝔽2)12yes—
524C23C2×2−1+411yescharacteristic
525C23C2×2−1+411yescharacteristic
526C23C2×2−1+412yes—
527D4—416no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—28no—
538C2×C4—24no—
539C2×C4—24no—
540C2×C4—24no—
541C2×C4—28no—
542C2×C4—24no—
543C2×C4—28no—
544C2×C4—24no—
545C2×C4—24no—
546C2×C4—24no—
547C2×C4—24no—
548C2×C4—24no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—44no—
556C2×C4—42no—
557C2×C4—44no—
558C2×C4—42no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—42no—
562C2×C4—44no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—41no—
568C2×C4—44no—
569C2×C4—41no—
570C2×C4—416no—
571C2×C4—416no—
572C2×C4—44no—
573C2×C4—44no—
574C2×C4—44no—
575C2×C4—44no—
576C2×C4—44no—
577C2×C4—44no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—44no—
581C2×C4—44no—
582C2×C4—44no—
583C2×C4—44no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—44no—
587C2×C4—42no—
588C2×C4—44no—
589C2×C4—44no—
590C2×C4—44no—
591C2×C4—44no—
592C2×C4—44no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—44no—
599C2×C4—44no—
600C2×C4—44no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—44no—
604C2×C4—44no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—816no—
608C2×C4#54018yes—
609C2×C4#54314yes—
610C2×C4#54414yes—
611C2×C4#55114yes—
612C2×C4#55418yes—
613C2×C4#56014yes—
614C22—24no—
615C22—28no—
616C22—42no—
617C22—42no—
618C22—42no—
619C22—44no—
620C22—44no—
621C22—44no—
622C22—42no—
623C22—42no—
624C22—42no—
625C22—44no—
626C22—42no—
627C22—42no—
628C22—44no—
629C22—42no—
630C22—81no—
631C22—81no—
632C22—81no—
633C22—82no—
634C22—82no—
635C22—81no—
636C22C2×#54012yes—
637C22C2×#54311yescharacteristic
638C22C2×#54411yescharacteristic
639C22C2×#54411yescharacteristic
640C22C2×#55111yescharacteristic
641C22C2×#55412yes—
642C22#350312yes—
643C22#351912yes—
644C22#352012yes—
645C22#352212yes—
646C22#352312yes—
647C22C2×#56011yescharacteristic
648C22#353611yescharacteristic
649C22#353611yescharacteristic
650C22#353612yes—
651C22#353711yescharacteristic
652C22#353712yes—
653C22#353812yes—
654C22#353811yescharacteristic
655C22#354812yes—
656C22#354812yes—
657C22#354911yescharacteristic
658C22#355011yescharacteristic
659C4—24no—
660C4—48no—
661C4—44no—
662C4—44no—
663C4—42no—
664C4—42no—
665C4—42no—
666C4—82no—
667C4—84no—
668C4—84no—
669C4—82no—
670C4—84no—
671C4#274818yes—
672C2—44no—
673C2—82no—
674C2#3730611yescharacteristic
675C2#3747711yescharacteristic
676C2#3809812yes—
677C2#3896512yes—
678C2#4260812yes—
679C2#4263312yes—
680C2#4508811yescharacteristic
681C2#4535111yescharacteristic
682C2C2×#274811yescharacteristic
683C2#5295711yescharacteristic
684C2#5300711yescharacteristic
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
1. #17176C2semidirect
2. #17019C2semidirect
3. #17003C2semidirect
4. #16707C2semidirect
5. #16305C2semidirect
7. #15137C2semidirect
8. #15135C2semidirect
9. #15131C2semidirect
10. #14264C2semidirect
11. #13981C2semidirect
12. #13907C2semidirect

Maximal subgroups

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

subgroupindex
#171762
#170192
#170032
#167072
#163052
#153352
#151372
#151352
#151312
#142642
#139812
#139072
#139062
#138722
#138392