Finite Group Databasebeta

← #800887

Subgroups — #800887

Order 512.

2274 conjugacy classes of subgroups.

The subgroup lattice has 666 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2571C211yesmaximal, characteristic, complemented
2C2×#2357C211yesmaximal, characteristic
3#17040C212yesmaximal, complemented
4#15719C214yesmaximal, complemented
5#15481C214yesmaximal, complemented
6#15151C211yesmaximal, characteristic, complemented
7#15148C211yesmaximal, characteristic
8#15136C211yesmaximal, characteristic
9#15020C212yesmaximal, complemented
10#15016C212yesmaximal, complemented
11#14229C212yesmaximal, complemented
12#14141C212yesmaximal
13#14006C212yesmaximal, complemented
14#13982C212yesmaximal, complemented
15#13979C212yesmaximal
16C4×#400C211yesmaximal, characteristic, complemented
17#13812C211yesmaximal, characteristic, complemented
18C22×C4×D4C2211yescharacteristic
19#2571—28no—
20C2×#400C2211yescharacteristic
21C2×#400C2214yes—
22C2×#399C2212yes—
23C2×#398C2212yes—
24C2×#398C2212yes—
25C2×#397C2212yes—
26C2×#397C2212yes—
27#2357—28no—
28C2×#390C2211yescharacteristic
29C2×#390C2212yes—
30C2×#390C2212yes—
31C2×#390C2211yescharacteristic
32C2×#389C2212yes—
33C2×#389C2212yes—
34C2×#389C2212yes—
35C2×#389C2212yes—
36C2×#388C2212yes—
37C2×#388C2212yes—
38C2×#388C2214yes—
39C2×#384C2211yescharacteristic
40C2×#382C2211yescharacteristic
41C2×#382C2212yes—
42C2×#382C2212yes—
43C2×#381C2212yes—
44C2×C4×C4⋊C4C2211yescharacteristic
45C2×C4×C4⋊C4C2211yescharacteristic
46C2×C4×C4⋊C4C2212yes—
47C2×C4×C4⋊C4C2211yescharacteristic
48C2×C4×C4⋊C4C2211yescharacteristic
49C2×C4×C22⋊C4C2212yes—
50#1445C2211yescharacteristic
51#1445C2212yes—
52#1445C2212yes—
53#1445C2214yes—
54#1443C2212yes—
55#1443C2214yes—
56#1442C2212yes—
57#1442C2211yescharacteristic
58#1442C2214yes—
59#1441C2211yescharacteristic
60#1441C2214yes—
61#1440C2214yes—
62#1439C2214yes—
63#1439C2214yes—
64#1439C2212yes—
65#1439C2214yes—
66#1439C2214yes—
67#1439C2214yes—
68#1437C2214yes—
69#1436C2214yes—
70#1436C2214yes—
71C4⋊#95C2212yes—
72C4⋊#95C2211yescharacteristic
73C4⋊#95C2212yes—
74C4⋊#95C2214yes—
75C4⋊#95C2212yes—
76#1434C2214yes—
77#1434C2214yes—
78#1434C2212yes—
79C22⋊#95C2214yes—
80C22⋊#95C2214yes—
81C4×#95C2211yescharacteristic
82C4×#95C2212yes—
83C4×#95C2212yes—
84C4×#95C2212yes—
85C23×D4C2311yescharacteristic
86C24×C4C2312yes—
87C2×C4×D4—24no—
88C2×C4×D4—24no—
89C2×C4×D4—24no—
90C22×C4⋊C4C2312yes—
91C22×C4⋊C4C2311yescharacteristic
92C22×C4⋊C4C2311yescharacteristic
93C22×C4⋊C4C2311yescharacteristic
94C22×C4⋊C4C2312yes—
95C22×C4⋊C4C2311yescharacteristic
96C22×C4⋊C4C2311yescharacteristic
97C22×C4⋊C4C2311yescharacteristic
98C22×C4⋊C4C2312yes—
99C22×C4⋊C4C2314yes—
100C22×C4⋊C4C2312yes—
101C22×C4⋊C4C2314yes—
102C22×C4⋊C4C2314yes—
103C22×C4⋊C4C2311yescharacteristic
104C22×C22⋊C4C2312yes—
105C22×C22⋊C4C2312yes—
106C22×C22⋊C4C2314yes—
107C22×C22⋊C4C2312yes—
108C22×C22⋊C4C2314yes—
109C22×C22⋊C4C2314yes—
110C22×C22⋊C4C2314yes—
111C22×C22⋊C4C2312yes—
112C22×C22⋊C4C2312yes—
113C22×C42C2312yes—
114C22×C42C2311yescharacteristic
115C22×C42C2312yes—
116C22×C42C2311yescharacteristic
117C22×C42C2311yescharacteristic
118C22×C42C2314yes—
119C22×C42C2312yes—
120C22×C42C2312yes—
121#400—216no—
122#400—24no—
123#399—44no—
124#398—44no—
125#398—44no—
126#397—28no—
127#397—44no—
128#390—28no—
129#390—24no—
130#390—28no—
131#390—24no—
132#389—28no—
133#389—28no—
134#389—28no—
135#389—28no—
136#388—28no—
137#388—28no—
138#388—48no—
139#384—24no—
140#382—28no—
141#382—24no—
142#382—28no—
143#381—28no—
144C4×C4⋊C4—24no—
145C4×C4⋊C4—28no—
146C4×C4⋊C4—24no—
147C4×C4⋊C4—24no—
148C4×C4⋊C4D418yes—
149C4×C22⋊C4—28no—
150C2×#95C2311yescharacteristic
151C2×#95C2314yes—
152C2×#95C2312yes—
153C2×#95C2314yes—
154C2×#95C2312yes—
155C2×#95C2312yes—
156C2×#95C2312yes—
157C2×#95C2314yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2312yes—
161C2×#95C2314yes—
162C2×#95C2314yes—
163C2×#95C2314yes—
164C2×#95C2312yes—
165C2×#95C2312yes—
166C2×#95C2311yescharacteristic
167C2×#95C2311yescharacteristic
168C2×#95C2312yes—
169C2×#95C2314yes—
170C2×#95C2312yes—
171C2×#95C2314yes—
172C2×#95C2312yes—
173C2×#95C2312yes—
174C2×#95C2314yes—
175C2×#95C2312yes—
176C2×#95C2312yes—
177C2×#95C2312yes—
178C2×#95C2312yes—
179C2×#95C2314yes—
180C2×#95C2312yes—
181C2×#95C2311yescharacteristic
182C25C2412yes—
183C22×D4—22no—
184C22×D4—22no—
185C22×D4—24no—
186C22×D4—24no—
187C22×D4#4114yes—
188C23×C4—24no—
189C23×C4—24no—
190C23×C4—24no—
191C23×C4—44no—
192C23×C4—44no—
193C23×C4C2414yes—
194C23×C4C2412yes—
195C23×C4C2412yes—
196C23×C4C2412yes—
197C23×C4C2411yescharacteristic
198C23×C4C2412yes—
199C23×C4C2411yescharacteristic
200C23×C4C2412yes—
201C23×C4C2412yes—
202C23×C4C2411yescharacteristic
203C23×C4C2414yes—
204C23×C4C2412yes—
205C23×C4C2411yescharacteristic
206C23×C4C2411yescharacteristic
207C23×C4C2411yescharacteristic
208C23×C4C2411yescharacteristic
209C4×D4—232no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—24no—
213C2×C4⋊C4—24no—
214C2×C4⋊C4—24no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—24no—
221C2×C4⋊C4—24no—
222C2×C4⋊C4—24no—
223C2×C4⋊C4—24no—
224C2×C4⋊C4—24no—
225C2×C4⋊C4—24no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—24no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—24no—
233C2×C4⋊C4—28no—
234C2×C4⋊C4—28no—
235C2×C4⋊C4—28no—
236C2×C4⋊C4—41no—
237C2×C4⋊C4—42no—
238C2×C4⋊C4—44no—
239C2×C4⋊C4—44no—
240C2×C4⋊C4—42no—
241C2×C4⋊C4—44no—
242C2×C4⋊C4—44no—
243C2×C4⋊C4—44no—
244C2×C4⋊C4—42no—
245C2×C4⋊C4—44no—
246C2×C4⋊C4C2×D414yes—
247C2×C4⋊C4C2×D414yes—
248C2×C4⋊C4#4118yes—
249C2×C22⋊C4—24no—
250C2×C22⋊C4—24no—
251C2×C22⋊C4—24no—
252C2×C22⋊C4—28no—
253C2×C22⋊C4—24no—
254C2×C22⋊C4—24no—
255C2×C22⋊C4—28no—
256C2×C22⋊C4—24no—
257C2×C22⋊C4—24no—
258C2×C22⋊C4—24no—
259C2×C22⋊C4—28no—
260C2×C22⋊C4—24no—
261C2×C22⋊C4—24no—
262C2×C22⋊C4—28no—
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×C42—22no—
275C2×C42—22no—
276C2×C42—24no—
277C2×C42—22no—
278C2×C42—24no—
279C2×C42—24no—
280C2×C42—24no—
281C2×C42—24no—
282C2×C42—24no—
283C2×C42—24no—
284C2×C42—24no—
285C2×C42—24no—
286C2×C42—28no—
287C2×C42—44no—
288C2×C42—44no—
289C2×C42—44no—
290C2×C42—44no—
291C2×C42C2×D414yes—
292C2×C42C2×D418yes—
293C2×C42#4114yes—
294#95—22no—
295#95—24no—
296#95—22no—
297#95—24no—
298#95—24no—
299#95—28no—
300#95—24no—
301#95—28no—
302#95—28no—
303#95—28no—
304#95—28no—
305#95—28no—
306#95—28no—
307#95—24no—
308#95—24no—
309#95—24no—
310#95—28no—
311#95—22no—
312#95—24no—
313#95—44no—
314#95—42no—
315#95—42no—
316#95—42no—
317#95—42no—
318#95—44no—
319#95—42no—
320#95—42no—
321#95—44no—
322#95—42no—
323#95—44no—
324#95C2×D418yes—
325#95#4114yes—
326C24—22no—
327C24—22no—
328C24—24no—
329C24—22no—
330C24—22no—
331C24—22no—
332C24—24no—
333C24—22no—
334C24—24no—
335C24—24no—
336C24C2×#4114yes—
337C24C2511yescharacteristic
338C2×D4—28no—
339C2×D4—216no—
340C2×D4—44no—
341C2×D4—44no—
342C2×D4—48no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—21no—
346C22×C4—21no—
347C22×C4—21no—
348C22×C4—21no—
349C22×C4—24no—
350C22×C4—24no—
351C22×C4—22no—
352C22×C4—21no—
353C22×C4—22no—
354C22×C4—21no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—24no—
358C22×C4—24no—
359C22×C4—22no—
360C22×C4—21no—
361C22×C4—22no—
362C22×C4—21no—
363C22×C4—21no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—24no—
368C22×C4—22no—
369C22×C4—21no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—24no—
373C22×C4—24no—
374C22×C4—24no—
375C22×C4—22no—
376C22×C4—24no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—24no—
380C22×C4—21no—
381C22×C4—22no—
382C22×C4—21no—
383C22×C4—216no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—21no—
387C22×C4—22no—
388C22×C4—24no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—21no—
394C22×C4—24no—
395C22×C4—24no—
396C22×C4—24no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—24no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—24no—
405C22×C4—21no—
406C22×C4—22no—
407C22×C4—24no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—24no—
420C22×C4—22no—
421C22×C4—24no—
422C22×C4—84no—
423C22×C4—84no—
424C22×C4—88no—
425C22×C4—88no—
426C22×C4C22×D414yes—
427C22×C4C22×D414yes—
428C22×C4C22×D412yes—
429C22×C4C22×D412yes—
430C22×C4C22×D412yes—
431C22×C4C2×#4114yes—
432C22×C4C2×#4112yes—
433C22×C4C2×#4118yes—
434C22×C4C2×#4112yes—
435C22×C4C2×#4112yes—
436C22×C4C2×#4112yes—
437C22×C4C2×#4112yes—
438C22×C4C2×#4114yes—
439C22×C4H(5,𝔽2)12yes—
440C22×C4H(5,𝔽2)12yes—
441C22×C42−1+414yes—
442C4⋊C4—28no—
443C4⋊C4—28no—
444C4⋊C4—216no—
445C4⋊C4—44no—
446C4⋊C4—44no—
447C4⋊C4—48no—
448C4⋊C4—44no—
449C4⋊C4—44no—
450C4⋊C4—44no—
451C4⋊C4—48no—
452C4⋊C4—88no—
453C4⋊C4—84no—
454C4⋊C4—88no—
455C4⋊C4—88no—
456C22⋊C4—216no—
457C22⋊C4—48no—
458C22⋊C4—416no—
459C22⋊C4—48no—
460C22⋊C4—84no—
461C22⋊C4—84no—
462C22⋊C4—88no—
463C22⋊C4—88no—
464C22⋊C4—88no—
465C42—28no—
466C42—28no—
467C42—48no—
468C42—48no—
469C42—44no—
470C42—48no—
471C42—84no—
472C42—88no—
473C23—24no—
474C23—28no—
475C23—24no—
476C23—24no—
477C23—28no—
478C23—42no—
479C23—42no—
480C23—42no—
481C23—44no—
482C23—44no—
483C23—42no—
484C23—42no—
485C23—42no—
486C23—42no—
487C23—42no—
488C23—44no—
489C23—44no—
490C23—44no—
491C23—44no—
492C23—44no—
493C23—44no—
494C23—44no—
495C23—44no—
496C23C23×D411yescharacteristic
497C23C22×#4111yescharacteristic
498C23C22×#4111yescharacteristic
499C23C22×#4112yes—
500C23C22×#4111yescharacteristic
501C23C2×H(5,𝔽2)11yescharacteristic
502C23C2×H(5,𝔽2)11yescharacteristic
503C23C2×2−1+412yes—
504C23#58511yescharacteristic
505C23#58512yes—
506C23#58512yes—
507D4—416no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—24no—
511C2×C4—24no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—24no—
515C2×C4—24no—
516C2×C4—22no—
517C2×C4—28no—
518C2×C4—24no—
519C2×C4—22no—
520C2×C4—24no—
521C2×C4—24no—
522C2×C4—24no—
523C2×C4—22no—
524C2×C4—24no—
525C2×C4—22no—
526C2×C4—24no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—24no—
530C2×C4—28no—
531C2×C4—28no—
532C2×C4—24no—
533C2×C4—24no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—42no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—44no—
543C2×C4—44no—
544C2×C4—44no—
545C2×C4—42no—
546C2×C4—44no—
547C2×C4—41no—
548C2×C4—42no—
549C2×C4—44no—
550C2×C4—42no—
551C2×C4—44no—
552C2×C4—42no—
553C2×C4—44no—
554C2×C4—44no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—44no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—41no—
563C2×C4—42no—
564C2×C4—44no—
565C2×C4—41no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—44no—
569C2×C4—44no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—44no—
573C2×C4—41no—
574C2×C4—42no—
575C2×C4—44no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—44no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—44no—
584C2×C4—42no—
585C2×C4—44no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—816no—
597C2×C4#52514yes—
598C2×C4#53214yes—
599C2×C4#53614yes—
600C2×C4#55014yes—
601C2×C4#55218yes—
602C2×C4#55218yes—
603C22—44no—
604C22—48no—
605C22—48no—
606C22—44no—
607C22—44no—
608C22—84no—
609C22—84no—
610C22—82no—
611C22—82no—
612C22—82no—
613C22—82no—
614C22C2×#52511yescharacteristic
615C22C2×#53211yescharacteristic
616C22C2×#53611yescharacteristic
617C22C2×#55011yescharacteristic
618C22C2×#55212yes—
619C22C2×#55212yes—
620C22#348011yescharacteristic
621C22#348012yes—
622C22#348111yescharacteristic
623C22#348511yescharacteristic
624C22#348811yescharacteristic
625C22#350011yescharacteristic
626C22#350811yescharacteristic
627C22#351012yes—
628C22#351412yes—
629C22#351411yescharacteristic
630C22#351412yes—
631C22#351512yes—
632C22#352112yes—
633C22#352412yes—
634C22#355912yes—
635C22#355912yes—
636C22#356512yes—
637C4—24no—
638C4—44no—
639C4—42no—
640C4—48no—
641C4—42no—
642C4—42no—
643C4—44no—
644C4—42no—
645C4—44no—
646C4—82no—
647C4—84no—
648C4—82no—
649C4—81no—
650C4—82no—
651C4—82no—
652C4#261118yes—
653C2—84no—
654C2C2×#261111yescharacteristic
655C2#3603311yescharacteristic
656C2#3605111yescharacteristic
657C2#3649311yescharacteristic
658C2#3659711yescharacteristic
659C2#3913411yescharacteristic
660C2#4294411yescharacteristic
661C2#4517212yes—
662C2#4521112yes—
663C2#5944912yes—
664C2#5950712yes—
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. C2×#2571C2semidirect
3. #17040C2semidirect
4. #15719C2semidirect
5. #15481C2semidirect
6. #15151C2semidirect
9. #15020C2semidirect
10. #15016C2semidirect
11. #14229C2semidirect
13. #14006C2semidirect
14. #13982C2semidirect
16. C4×#400C2semidirect
17. #13812C2semidirect

Maximal subgroups

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

subgroupindex
C2×#25712
C2×#23572
#170402
#157192
#154812
#151512
#151482
#151362
#150202
#150162
#142292
#141412
#140062
#139822
#139792
C4×#4002
#138122