Finite Group Databasebeta

← #2736660

Subgroups — #2736660

Order 512.

2758 conjugacy classes of subgroups.

The subgroup lattice has 682 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2823C212yesmaximal, complemented
2C2×#2512C212yesmaximal, complemented
3#17135C212yesmaximal, complemented
4#17121C212yesmaximal, complemented
5#17096C214yesmaximal, complemented
6#16658C214yesmaximal, complemented
7#16514C212yesmaximal, complemented
8#15141C212yesmaximal, complemented
9#15140C212yesmaximal
10#15137C212yesmaximal, complemented
11#15134C212yesmaximal, complemented
12#15134C212yesmaximal, complemented
13#15127C211yesmaximal, characteristic
14#13910C211yesmaximal, characteristic, complemented
15C4×#398C211yesmaximal, characteristic, complemented
16C22×#127C2211yescharacteristic
17C22×#126C2214yes—
18#2823—216no—
19#2512—216no—
20C2×#398C2214yes—
21C2×#398C2211yescharacteristic
22C2×#397C2214yes—
23C2×#396C2214yes—
24C2×#396C2214yes—
25C2×#396C2214yes—
26C2×#396C2214yes—
27C2×#396C2214yes—
28C2×#396C2214yes—
29C2×#394C2212yes—
30C2×#394C2214yes—
31C2×#394C2212yes—
32C2×#394C2214yes—
33C2×#390C2212yes—
34C2×#390C2212yes—
35C2×#390C2212yes—
36C2×#390C2212yes—
37C2×#390C2212yes—
38C2×#390C2212yes—
39C2×#389C2212yes—
40C2×#389C2212yes—
41C2×#389C2212yes—
42C2×#389C2212yes—
43C2×#388C2214yes—
44C2×#384C2212yes—
45C2×#384C2212yes—
46C2×#384C2211yescharacteristic
47C2×C4×C4⋊C4C2212yes—
48C2×C4×C4⋊C4C2211yescharacteristic
49C2×C4×C4⋊C4C2211yescharacteristic
50C2×C4×C4⋊C4C2211yescharacteristic
51#1445C2211yescharacteristic
52#1444C2214yes—
53#1444C2211yescharacteristic
54#1444C2212yes—
55#1444C2212yes—
56#1442C2214yes—
57#1442C2212yes—
58#1441C2212yes—
59#1441C2214yes—
60#1441C2211yescharacteristic
61#1439C2214yes—
62#1439C2214yes—
63#1439C2214yes—
64#1439C2214yes—
65#1439C2214yes—
66#1439C2214yes—
67#1439C2214yes—
68#1438C2214yes—
69#1438C2214yes—
70#1438C2212yes—
71#1438C2212yes—
72C4⋊#95C2212yes—
73C4⋊#95C2211yescharacteristic
74C4⋊#95C2212yes—
75C4×#95C2212yes—
76C4×#95C2211yescharacteristic
77C23×D4C2312yes—
78C23×D4C2312yes—
79C23×D4C2312yes—
80C2×#127—42no—
81C2×#127—44no—
82C2×#126—216no—
83C2×#126—216no—
84C2×#126—48no—
85C22×C4⋊C4C2312yes—
86C22×C4⋊C4C2312yes—
87C22×C4⋊C4C2311yescharacteristic
88C22×C4⋊C4C2311yescharacteristic
89C22×C4⋊C4C2312yes—
90C22×C4⋊C4C2314yes—
91C22×C4⋊C4C2314yes—
92C22×C4⋊C4C2312yes—
93C22×C4⋊C4C2312yes—
94C22×C4⋊C4C2314yes—
95C22×C4⋊C4C2314yes—
96C22×C4⋊C4C2314yes—
97C22×C4⋊C4C2312yes—
98C22×C4⋊C4C2314yes—
99C22×C4⋊C4C2312yes—
100C22×C4⋊C4C2311yescharacteristic
101C22×C4⋊C4C2311yescharacteristic
102C22×C22⋊C4C2314yes—
103C22×C22⋊C4C2314yes—
104C22×C22⋊C4C2314yes—
105C22×C22⋊C4C2314yes—
106C22×C22⋊C4C2314yes—
107C22×C22⋊C4C2314yes—
108C22×C22⋊C4C2314yes—
109C22×C22⋊C4C2314yes—
110C22×C22⋊C4C2314yes—
111C22×C22⋊C4C2314yes—
112C22×C22⋊C4C2314yes—
113C22×C22⋊C4C2314yes—
114C22×C42C2311yescharacteristic
115C22×C42C2312yes—
116C22×C42C2312yes—
117C22×C42C2311yescharacteristic
118C22×C42C2312yes—
119C22×C42C2311yescharacteristic
120#398—216no—
121#398D418yescomplemented
122#397—216no—
123#396—216no—
124#396—216no—
125#396—216no—
126#396—216no—
127#396—48no—
128#396—48no—
129#394—28no—
130#394—216no—
131#394—48no—
132#394—44no—
133#390—28no—
134#390—28no—
135#390—28no—
136#390—44no—
137#390—44no—
138#390—44no—
139#389—28no—
140#389—44no—
141#389—44no—
142#389—44no—
143#388—216no—
144#384—28no—
145#384—28no—
146#384—42no—
147C4×C4⋊C4—28no—
148C4×C4⋊C4—42no—
149C4×C4⋊C4—42no—
150C4×C4⋊C4D418yescomplemented
151C2×#95C2312yes—
152C2×#95C2312yes—
153C2×#95C2314yes—
154C2×#95C2314yes—
155C2×#95C2312yes—
156C2×#95C2314yes—
157C2×#95C2312yes—
158C2×#95C2312yes—
159C2×#95C2312yes—
160C2×#95C2312yes—
161C2×#95C2312yes—
162C2×#95C2311yescharacteristic
163C2×#95C2314yes—
164C2×#95C2314yes—
165C2×#95C2311yescharacteristic
166C2×#95C2311yescharacteristic
167C2×#95C2312yes—
168C2×#95C2314yes—
169C2×#95C2312yes—
170C2×#95C2311yescharacteristic
171C2×#95C2312yes—
172C25C2414yes—
173C22×D4—24no—
174C22×D4—24no—
175C22×D4—24no—
176C22×D4—24no—
177C22×D4—24no—
178C22×D4—24no—
179C22×D4—24no—
180C22×D4—24no—
181C22×D4—24no—
182C22×D4—44no—
183C22×D4—42no—
184C22×D4—42no—
185C22×D4—44no—
186C22×D4—44no—
187C22×D4—42no—
188C22×D4—44no—
189C22×D4—42no—
190C23×C4C2414yes—
191C23×C4C2414yes—
192C23×C4C2411yescharacteristic
193C23×C4C2412yes—
194C23×C4C2411yescharacteristic
195C23×C4C2414yes—
196C23×C4C2412yes—
197C23×C4C2412yes—
198C23×C4C2411yescharacteristic
199C23×C4C2411yescharacteristic
200C23×C4C2412yes—
201C23×C4C2411yescharacteristic
202C23×C4C2411yescharacteristic
203C23×C4C2411yescharacteristic
204#127—416no—
205#126—464no—
206C2×C4⋊C4—24no—
207C2×C4⋊C4—24no—
208C2×C4⋊C4—24no—
209C2×C4⋊C4—24no—
210C2×C4⋊C4—28no—
211C2×C4⋊C4—28no—
212C2×C4⋊C4—28no—
213C2×C4⋊C4—28no—
214C2×C4⋊C4—24no—
215C2×C4⋊C4—24no—
216C2×C4⋊C4—24no—
217C2×C4⋊C4—24no—
218C2×C4⋊C4—24no—
219C2×C4⋊C4—24no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—24no—
222C2×C4⋊C4—24no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—24no—
225C2×C4⋊C4—28no—
226C2×C4⋊C4—28no—
227C2×C4⋊C4—28no—
228C2×C4⋊C4—24no—
229C2×C4⋊C4—28no—
230C2×C4⋊C4—41no—
231C2×C4⋊C4—41no—
232C2×C4⋊C4—42no—
233C2×C4⋊C4—44no—
234C2×C4⋊C4—44no—
235C2×C4⋊C4—41no—
236C2×C4⋊C4—41no—
237C2×C4⋊C4—41no—
238C2×C4⋊C4—41no—
239C2×C4⋊C4—42no—
240C2×C4⋊C4—44no—
241C2×C4⋊C4—42no—
242C2×C4⋊C4—42no—
243C2×C4⋊C4—44no—
244C2×C4⋊C4—44no—
245C2×C4⋊C4—44no—
246C2×C4⋊C4—44no—
247C2×C4⋊C4—44no—
248C2×C4⋊C4—44no—
249C2×C4⋊C4—44no—
250C2×C4⋊C4—44no—
251C2×C4⋊C4C2×D418yes—
252C2×C4⋊C4C2×D418yes—
253C2×C4⋊C4C2×D414yes—
254C2×C4⋊C4C2×D414yes—
255C2×C22⋊C4—28no—
256C2×C22⋊C4—28no—
257C2×C22⋊C4—28no—
258C2×C22⋊C4—28no—
259C2×C22⋊C4—28no—
260C2×C22⋊C4—28no—
261C2×C22⋊C4—28no—
262C2×C22⋊C4—28no—
263C2×C22⋊C4—28no—
264C2×C22⋊C4—28no—
265C2×C22⋊C4—28no—
266C2×C22⋊C4—28no—
267C2×C22⋊C4—28no—
268C2×C22⋊C4—28no—
269C2×C22⋊C4—28no—
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×C22⋊C4—44no—
277C2×C22⋊C4—44no—
278C2×C22⋊C4—44no—
279C2×C22⋊C4—44no—
280C2×C22⋊C4—44no—
281C2×C22⋊C4—44no—
282C2×C22⋊C4—44no—
283C2×C22⋊C4—44no—
284C2×C22⋊C4—44no—
285C2×C22⋊C4—44no—
286C2×C22⋊C4—44no—
287C2×C22⋊C4—44no—
288C2×C22⋊C4—44no—
289C2×C22⋊C4—44no—
290C2×C22⋊C4—44no—
291C2×C42—24no—
292C2×C42—24no—
293C2×C42—22no—
294C2×C42—22no—
295C2×C42—24no—
296C2×C42—24no—
297C2×C42—22no—
298C2×C42—24no—
299C2×C42—22no—
300C2×C42—24no—
301C2×C42—24no—
302C2×C42—24no—
303C2×C42—24no—
304C2×C42—24no—
305C2×C42C2×D418yes—
306C2×C42C2×D414yes—
307#95—24no—
308#95—24no—
309#95—28no—
310#95—24no—
311#95—28no—
312#95—24no—
313#95—28no—
314#95—24no—
315#95—42no—
316#95—42no—
317#95—41no—
318#95—41no—
319#95—41no—
320#95—42no—
321#95—44no—
322#95—44no—
323#95—44no—
324#95—42no—
325#95C2×D418yes—
326#95C2×D418yes—
327#95C2×D414yes—
328C24—24no—
329C24—24no—
330C24—24no—
331C24—24no—
332C24—24no—
333C24—24no—
334C24—24no—
335C24—24no—
336C24—24no—
337C24—24no—
338C24—24no—
339C24—24no—
340C24—24no—
341C24—24no—
342C24—24no—
343C24C2511yescharacteristic
344C2×D4—48no—
345C2×D4—48no—
346C2×D4—48no—
347C2×D4—48no—
348C2×D4—48no—
349C2×D4—48no—
350C2×D4—48no—
351C2×D4—48no—
352C2×D4—48no—
353C2×D4—84no—
354C2×D4—88no—
355C2×D4—84no—
356C2×D4—88no—
357C2×D4—84no—
358C2×D4—88no—
359C2×D4—88no—
360C2×D4—84no—
361C22×C4—21no—
362C22×C4—21no—
363C22×C4—21no—
364C22×C4—21no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—24no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—21no—
371C22×C4—22no—
372C22×C4—24no—
373C22×C4—21no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—21no—
377C22×C4—21no—
378C22×C4—21no—
379C22×C4—24no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—24no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—21no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—24no—
396C22×C4—22no—
397C22×C4—21no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—24no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—21no—
405C22×C4—22no—
406C22×C4—24no—
407C22×C4—24no—
408C22×C4—24no—
409C22×C4—22no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—21no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—24no—
416C22×C4—24no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—21no—
421C22×C4—24no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—24no—
425C22×C4—21no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—24no—
430C22×C4—24no—
431C22×C4C22×D412yes—
432C22×C4C22×D412yes—
433C22×C4C22×D414yes—
434C22×C4C22×D418yes—
435C22×C4C22×D414yes—
436C22×C4C22×D412yes—
437C22×C4C22×D412yes—
438C22×C4C22×D412yes—
439C22×C4C22×D412yes—
440C22×C4H(5,𝔽2)14yes—
441C22×C4H(5,𝔽2)12yes—
442C22×C4H(5,𝔽2)12yes—
443C22×C4H(5,𝔽2)14yes—
444C22×C42−1+412yes—
445C22×C42−1+412yes—
446C22×C42−1+414yes—
447C4⋊C4—216no—
448C4⋊C4—216no—
449C4⋊C4—44no—
450C4⋊C4—44no—
451C4⋊C4—44no—
452C4⋊C4—48no—
453C4⋊C4—44no—
454C4⋊C4—416no—
455C4⋊C4—416no—
456C4⋊C4—48no—
457C4⋊C4—48no—
458C4⋊C4—84no—
459C4⋊C4—88no—
460C4⋊C4—88no—
461C4⋊C4—88no—
462C4⋊C4—88no—
463C4⋊C4—84no—
464C22⋊C4—416no—
465C22⋊C4—416no—
466C22⋊C4—416no—
467C22⋊C4—416no—
468C22⋊C4—88no—
469C22⋊C4—88no—
470C22⋊C4—88no—
471C22⋊C4—88no—
472C22⋊C4—88no—
473C22⋊C4—88no—
474C22⋊C4—88no—
475C22⋊C4—88no—
476C42—28no—
477C42—216no—
478C42—48no—
479C42—48no—
480C42—44no—
481C42—44no—
482C23—44no—
483C23—44no—
484C23—44no—
485C23—44no—
486C23—44no—
487C23—44no—
488C23—44no—
489C23—44no—
490C23—44no—
491C23—44no—
492C23—44no—
493C23—44no—
494C23—44no—
495C23—44no—
496C23—44no—
497C23—44no—
498C23—44no—
499C23—44no—
500C23—44no—
501C23—44no—
502C23—44no—
503C23—44no—
504C23—44no—
505C23—44no—
506C23—44no—
507C23—44no—
508C23—44no—
509C23—44no—
510C23—44no—
511C23—44no—
512C23—44no—
513C23—44no—
514C23—44no—
515C23—44no—
516C23—44no—
517C23C23×D411yescharacteristic
518C23C23×D411yescharacteristic
519C23C2×H(5,𝔽2)11yescharacteristic
520C23C2×H(5,𝔽2)12yes—
521C23C2×H(5,𝔽2)11yescharacteristic
522C23C2×H(5,𝔽2)12yes—
523C23C2×2−1+412yes—
524C23C2×2−1+411yescharacteristic
525C23C2×2−1+411yescharacteristic
526C23#58512yes—
527C23#58511yescharacteristic
528D4—816no—
529D4—816no—
530D4—816no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—28no—
539C2×C4—28no—
540C2×C4—24no—
541C2×C4—24no—
542C2×C4—24no—
543C2×C4—28no—
544C2×C4—24no—
545C2×C4—24no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—24no—
549C2×C4—24no—
550C2×C4—24no—
551C2×C4—24no—
552C2×C4—24no—
553C2×C4—22no—
554C2×C4—24no—
555C2×C4—24no—
556C2×C4—24no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—44no—
563C2×C4—42no—
564C2×C4—44no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—42no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—44no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—44no—
578C2×C4—42no—
579C2×C4—41no—
580C2×C4—42no—
581C2×C4—41no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—44no—
587C2×C4—44no—
588C2×C4—42no—
589C2×C4—44no—
590C2×C4—44no—
591C2×C4—42no—
592C2×C4—44no—
593C2×C4—44no—
594C2×C4—44no—
595C2×C4—44no—
596C2×C4—41no—
597C2×C4—44no—
598C2×C4—41no—
599C2×C4—44no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—44no—
603C2×C4—44no—
604C2×C4—44no—
605C2×C4—44no—
606C2×C4—42no—
607C2×C4—41no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4#53714yes—
612C2×C4#53718yes—
613C2×C4D4214yes—
614C2×C4#56014yes—
615C2×C4#56314yes—
616C2×C4#56318yes—
617C22—84no—
618C22—84no—
619C22—84no—
620C22—84no—
621C22—84no—
622C22—84no—
623C22—84no—
624C22—84no—
625C22—84no—
626C22—84no—
627C22—84no—
628C22—84no—
629C22—84no—
630C22—84no—
631C22—84no—
632C22C2×#53712yes—
633C22C2×#53711yescharacteristic
634C22C2×D4211yescharacteristic
635C22#348812yes—
636C22#349611yescharacteristic
637C22#349711yescharacteristic
638C22#349712yes—
639C22#349812yes—
640C22#349911yescharacteristic
641C22C2×#56011yescharacteristic
642C22C2×#56311yescharacteristic
643C22C2×#56312yes—
644C22#353611yescharacteristic
645C22#353711yescharacteristic
646C22#354612yes—
647C22#354612yes—
648C22#354812yes—
649C22#355012yes—
650C22#355111yescharacteristic
651C22#355712yes—
652C22#355911yescharacteristic
653C22#355912yes—
654C22#356112yes—
655C4—24no—
656C4—42no—
657C4—48no—
658C4—44no—
659C4—42no—
660C4—42no—
661C4—44no—
662C4—42no—
663C4—44no—
664C4—81no—
665C4—82no—
666C4—84no—
667C4—84no—
668C4#282318yescomplemented
669C2—164no—
670C2C2×#282311yescharacteristic
671C2#3934112yes—
672C2#3935911yescharacteristic
673C2#4279912yes—
674C2#4295211yescharacteristic
675C2#4548711yescharacteristic
676C2#4599812yes—
677C2#4632911yescharacteristic
678C2#5301811yescharacteristic
679C2#5309911yescharacteristic
680C2#5313212yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC22×#127C22
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×#2823C2semidirect
2. C2×#2512C2semidirect
3. #17135C2semidirect
4. #17121C2semidirect
5. #17096C2semidirect
6. #16658C2semidirect
7. #16514C2semidirect
8. #15141C2semidirect
10. #15137C2semidirect
11. #15134C2semidirect
12. #15134C2semidirect
14. #13910C2semidirect
15. C4×#398C2semidirect
121. #398D4semidirect
150. C4×C4⋊C4D4semidirect
668. C4#2823semidirect

Maximal subgroups

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

subgroupindex
C2×#28232
C2×#25122
#171352
#171212
#170962
#166582
#165142
#151412
#151402
#151372
#151342
#151272
#139102
C4×#3982