Finite Group Databasebeta

← #2036930

Subgroups — #2036930

Order 512.

3438 conjugacy classes of subgroups.

The subgroup lattice has 704 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2679C211yesmaximal, characteristic
2C2×#2861C211yesmaximal, characteristic, complemented
3#17077C212yesmaximal, complemented
4#16310C214yesmaximal, complemented
5#15882C212yesmaximal, complemented
6#15833C212yesmaximal, complemented
7#15831C214yesmaximal, complemented
8#15281C214yesmaximal, complemented
9#15281C212yesmaximal, complemented
10#15259C212yesmaximal, complemented
11#14700C212yesmaximal, complemented
12#14445C211yesmaximal, characteristic, complemented
13#14358C214yesmaximal, complemented
14C22×#125C2211yescharacteristic, complemented
15C22×#124C2211yescharacteristic
16C22×#120C2214yes—
17C22×#120C2212yes—
18#2861—28no—
19#2679—28no—
20C2×#394C2212yes—
21C2×#394C2214yes—
22C2×#394C2212yes—
23C2×#394C2214yes—
24C2×#393C2214yescomplemented
25C2×#391C2211yescharacteristic, complemented
26C2×#391C2211yescharacteristic, complemented
27C2×#390C2211yescharacteristic
28C2×#390C2212yes—
29C2×#390C2212yes—
30C2×#390C2212yes—
31C2×#390C2212yes—
32C2×#390C2211yescharacteristic
33C2×#389C2212yescomplemented
34C2×#388C2214yescomplemented
35C2×#388C2214yescomplemented
36C2×#388C2214yescomplemented
37C2×#388C2214yescomplemented
38C2×#388C2214yescomplemented
39C2×#379C2212yescomplemented
40C2×#379C2214yescomplemented
41C2×#379C2212yescomplemented
42C2×C43C2211yescharacteristic, complemented
43#1445C2212yescomplemented
44#1445C2214yescomplemented
45#1442C2214yescomplemented
46#1440C2212yescomplemented
47#1440C2212yescomplemented
48#1439C2214yescomplemented
49#1439C2214yescomplemented
50#1439C2214yescomplemented
51#1439C2214yescomplemented
52#1439C2214yescomplemented
53#1439C2214yescomplemented
54#1439C2214yescomplemented
55#1439C2214yescomplemented
56#1438C2214yes—
57#1438C2214yes—
58#1438C2214yes—
59#1438C2214yes—
60#1438C2212yes—
61#1438C2214yes—
62#1437C2214yescomplemented
63#1437C2214yescomplemented
64#1437C2214yescomplemented
65#1436C2212yescomplemented
66#1436C2212yescomplemented
67C4⋊#95C2212yescomplemented
68C4⋊#95C2212yescomplemented
69C26C2312yes—
70C23×Q8C2311yescharacteristic
71C23×D4C2312yes—
72C23×D4C2314yes—
73C23×D4C2311yescharacteristic
74C23×D4C2312yes—
75C2×#125—24no—
76C2×#125—42no—
77C2×#125—42no—
78C2×#124—24no—
79C2×#124—42no—
80C2×#124—42no—
81C2×#120—28no—
82C2×#120—216no—
83C2×#120—216no—
84C2×#120—48no—
85C2×#120—48no—
86C22×C4⋊C4C2312yes—
87C22×C4⋊C4C2312yes—
88C22×C4⋊C4C2312yes—
89C22×C4⋊C4C2314yes—
90C22×C4⋊C4C2311yescharacteristic
91C22×C4⋊C4C2312yes—
92C22×C4⋊C4C2312yes—
93C22×C4⋊C4C2311yescharacteristic
94C22×C22⋊C4C2314yes—
95C22×C22⋊C4C2314yes—
96C22×C22⋊C4C2314yes—
97C22×C22⋊C4C2314yes—
98C22×C22⋊C4C2314yes—
99C22×C22⋊C4C2314yes—
100C22×C22⋊C4C2312yes—
101C22×C22⋊C4C2312yes—
102C22×C22⋊C4C2314yes—
103C22×C22⋊C4C2312yes—
104C22×C22⋊C4C2314yes—
105C22×C22⋊C4C2314yes—
106C22×C22⋊C4C2314yes—
107C22×C22⋊C4C2314yes—
108C22×C22⋊C4C2314yes—
109C22×C22⋊C4C2312yes—
110C22×C22⋊C4C2314yes—
111C22×C22⋊C4C2312yes—
112C22×C22⋊C4C2314yes—
113C22×C42C2311yescharacteristic
114C22×C42C2312yes—
115C22×C42C2311yescharacteristic
116C22×C42C2312yes—
117C22×C42C2312yes—
118C22×C42C2311yescharacteristic
119#394—28no—
120#394—216no—
121#394—44no—
122#394—48no—
123#393—48no—
124#391—24no—
125#391—24no—
126#390—24no—
127#390—28no—
128#390—28no—
129#390—44no—
130#390—44no—
131#390D418yes—
132#389—44no—
133#388—216no—
134#388—216no—
135#388—48no—
136#388—48no—
137#388—48no—
138#379—44no—
139#379—44no—
140#379—48no—
141C2×#95C2314yes—
142C2×#95C2312yes—
143C2×#95C2314yes—
144C2×#95C2314yes—
145C2×#95C2314yes—
146C2×#95C2314yes—
147C2×#95C2314yes—
148C2×#95C2312yes—
149C2×#95C2314yes—
150C2×#95C2314yes—
151C2×#95C2314yes—
152C2×#95C2314yes—
153C2×#95C2314yes—
154C2×#95C2314yes—
155C43—24no—
156C25—24no—
157C25—28no—
158C25—44no—
159C25—44no—
160C25—44no—
161C25—44no—
162C25—42no—
163C25—44no—
164C25—42no—
165C25C2412yes—
166C25C2414yes—
167C22×Q8—22no—
168C22×Q8—24no—
169C22×Q8—41no—
170C22×Q8—41no—
171C22×Q8—42no—
172C22×D4—28no—
173C22×D4—28no—
174C22×D4—28no—
175C22×D4—24no—
176C22×D4—24no—
177C22×D4—24no—
178C22×D4—24no—
179C22×D4—44no—
180C22×D4—44no—
181C22×D4—44no—
182C22×D4—44no—
183C22×D4—41no—
184C22×D4—41no—
185C22×D4—42no—
186C22×D4—42no—
187C22×D4—42no—
188C22×D4—42no—
189C22×D4—42no—
190C22×D4—42no—
191C22×D4—44no—
192C22×D4—42no—
193C22×D4—44no—
194C22×D4C2×D414yes—
195C22×D4#4118yes—
196C23×C4C2412yes—
197C23×C4C2414yes—
198C23×C4C2414yes—
199C23×C4C2412yes—
200C23×C4C2414yes—
201C23×C4C2412yes—
202C23×C4C2412yes—
203C23×C4C2411yescharacteristic
204C23×C4C2411yescharacteristic
205C23×C4C2412yes—
206C23×C4C2411yescharacteristic
207#125—416no—
208#124—416no—
209#120—464no—
210#120—432no—
211C2×C4⋊C4—24no—
212C2×C4⋊C4—24no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—24no—
215C2×C4⋊C4—24no—
216C2×C4⋊C4—24no—
217C2×C4⋊C4—24no—
218C2×C4⋊C4—24no—
219C2×C4⋊C4—28no—
220C2×C4⋊C4—42no—
221C2×C4⋊C4—41no—
222C2×C4⋊C4—41no—
223C2×C4⋊C4—42no—
224C2×C4⋊C4—42no—
225C2×C4⋊C4—41no—
226C2×C4⋊C4—41no—
227C2×C4⋊C4—42no—
228C2×C4⋊C4—42no—
229C2×C4⋊C4—42no—
230C2×C4⋊C4—44no—
231C2×C4⋊C4—44no—
232C2×C4⋊C4—42no—
233C2×C4⋊C4—42no—
234C2×C4⋊C4C2×D414yes—
235C2×C22⋊C4—28no—
236C2×C22⋊C4—28no—
237C2×C22⋊C4—24no—
238C2×C22⋊C4—24no—
239C2×C22⋊C4—24no—
240C2×C22⋊C4—28no—
241C2×C22⋊C4—28no—
242C2×C22⋊C4—28no—
243C2×C22⋊C4—28no—
244C2×C22⋊C4—28no—
245C2×C22⋊C4—28no—
246C2×C22⋊C4—28no—
247C2×C22⋊C4—28no—
248C2×C22⋊C4—28no—
249C2×C22⋊C4—28no—
250C2×C22⋊C4—28no—
251C2×C22⋊C4—24no—
252C2×C22⋊C4—24no—
253C2×C22⋊C4—28no—
254C2×C22⋊C4—28no—
255C2×C22⋊C4—28no—
256C2×C22⋊C4—28no—
257C2×C22⋊C4—28no—
258C2×C22⋊C4—44no—
259C2×C22⋊C4—44no—
260C2×C22⋊C4—44no—
261C2×C22⋊C4—42no—
262C2×C22⋊C4—42no—
263C2×C22⋊C4—44no—
264C2×C22⋊C4—44no—
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—42no—
272C2×C22⋊C4—44no—
273C2×C22⋊C4—44no—
274C2×C22⋊C4—44no—
275C2×C22⋊C4—42no—
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⋊C4C2×D4116yes—
289C2×C42—22no—
290C2×C42—22no—
291C2×C42—22no—
292C2×C42—24no—
293C2×C42—24no—
294C2×C42—22no—
295C2×C42—24no—
296C2×C42—24no—
297C2×C42—24no—
298C2×C42—24no—
299C2×C42—42no—
300C2×C42—42no—
301C2×C42—44no—
302C2×C42—42no—
303C2×C42C2×D414yes—
304#95—28no—
305#95—28no—
306#95—24no—
307#95—28no—
308#95—28no—
309#95—44no—
310#95—44no—
311#95—44no—
312#95—44no—
313#95—44no—
314#95—44no—
315#95—44no—
316#95—42no—
317#95—44no—
318C24—24no—
319C24—24no—
320C24—24no—
321C24—24no—
322C24—24no—
323C24—24no—
324C24—22no—
325C24—22no—
326C24—24no—
327C24—24no—
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—416no—
341C24—48no—
342C24—48no—
343C24—416no—
344C24—48no—
345C24—84no—
346C24—84no—
347C24—88no—
348C24—88no—
349C24—88no—
350C24—88no—
351C24—88no—
352C24—88no—
353C24—88no—
354C24—84no—
355C24—84no—
356C24—88no—
357C24—88no—
358C24—88no—
359C24—88no—
360C24—88no—
361C24C22×D414yes—
362C24C2×#4118yes—
363C24C2511yescharacteristic
364C2×Q8—44no—
365C2×Q8—44no—
366C2×Q8—44no—
367C2×Q8—84no—
368C2×Q8—84no—
369C2×D4—28no—
370C2×D4—216no—
371C2×D4—44no—
372C2×D4—44no—
373C2×D4—48no—
374C2×D4—48no—
375C2×D4—416no—
376C2×D4—416no—
377C2×D4—48no—
378C2×D4—416no—
379C2×D4—84no—
380C2×D4—84no—
381C2×D4—84no—
382C2×D4—84no—
383C2×D4—84no—
384C2×D4—84no—
385C2×D4—84no—
386C2×D4—88no—
387C2×D4—88no—
388C2×D4—88no—
389C2×D4—88no—
390C2×D4—84no—
391C2×D4—88no—
392C2×D4—88no—
393C22×C4—22no—
394C22×C4—21no—
395C22×C4—21no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—21no—
402C22×C4—21no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—24no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—24no—
410C22×C4—22no—
411C22×C4—21no—
412C22×C4—21no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—24no—
421C22×C4—24no—
422C22×C4—24no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—24no—
427C22×C4—24no—
428C22×C4—24no—
429C22×C4—24no—
430C22×C4—24no—
431C22×C4—24no—
432C22×C4—24no—
433C22×C4—24no—
434C22×C4—24no—
435C22×C4—24no—
436C22×C4—24no—
437C22×C4—24no—
438C22×C4—24no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—24no—
444C22×C4—24no—
445C22×C4—24no—
446C22×C4—24no—
447C22×C4—24no—
448C22×C4—24no—
449C22×C4C22×D414yes—
450C22×C4C22×D412yes—
451C22×C4C22×D414yes—
452C22×C4C2×#4114yes—
453C22×C4H(5,𝔽2)12yes—
454C22×C4H(5,𝔽2)12yes—
455C22×C4H(5,𝔽2)14yes—
456C22×C4H(5,𝔽2)14yes—
457C22×C4H(5,𝔽2)14yes—
458C4⋊C4—44no—
459C4⋊C4—48no—
460C4⋊C4—48no—
461C4⋊C4—48no—
462C4⋊C4—44no—
463C4⋊C4—84no—
464C4⋊C4—84no—
465C4⋊C4—88no—
466C22⋊C4—232no—
467C22⋊C4—416no—
468C22⋊C4—416no—
469C22⋊C4—416no—
470C22⋊C4—416no—
471C22⋊C4—48no—
472C22⋊C4—48no—
473C22⋊C4—48no—
474C22⋊C4—416no—
475C22⋊C4—416no—
476C22⋊C4—88no—
477C22⋊C4—88no—
478C22⋊C4—84no—
479C22⋊C4—84no—
480C22⋊C4—88no—
481C22⋊C4—88no—
482C22⋊C4—88no—
483C22⋊C4—88no—
484C22⋊C4—88no—
485C42—28no—
486C42—44no—
487C42—48no—
488C42—44no—
489C42—48no—
490C42—84no—
491C23—24no—
492C23—28no—
493C23—28no—
494C23—28no—
495C23—28no—
496C23—24no—
497C23—24no—
498C23—28no—
499C23—28no—
500C23—28no—
501C23—28no—
502C23—28no—
503C23—44no—
504C23—44no—
505C23—44no—
506C23—44no—
507C23—42no—
508C23—44no—
509C23—44no—
510C23—44no—
511C23—44no—
512C23—44no—
513C23—44no—
514C23—44no—
515C23—44no—
516C23—44no—
517C23—44no—
518C23—44no—
519C23—44no—
520C23—44no—
521C23—42no—
522C23—44no—
523C23—44no—
524C23—44no—
525C23—44no—
526C23—44no—
527C23—44no—
528C23—44no—
529C23—42no—
530C23—42no—
531C23—44no—
532C23—44no—
533C23—44no—
534C23—44no—
535C23—44no—
536C23—44no—
537C23—44no—
538C23—44no—
539C23—44no—
540C23—44no—
541C23—44no—
542C23—44no—
543C23—44no—
544C23—44no—
545C23—44no—
546C23—44no—
547C23—816no—
548C23—832no—
549C23—832no—
550C23—832no—
551C23—816no—
552C23—832no—
553C23—832no—
554C23—816no—
555C23—832no—
556C23C23×D411yescharacteristic
557C23C22×#4112yes—
558C23C2×H(5,𝔽2)12yes—
559C23C2×H(5,𝔽2)12yes—
560C23C2×H(5,𝔽2)12yes—
561C23C2×H(5,𝔽2)11yescharacteristic
562C23C2×H(5,𝔽2)11yescharacteristic
563C23#58512yes—
564C23#58512yes—
565Q8—88no—
566D4—88no—
567D4—816no—
568D4—816no—
569D4—832no—
570C2×C4—24no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—24no—
574C2×C4—24no—
575C2×C4—24no—
576C2×C4—24no—
577C2×C4—24no—
578C2×C4—24no—
579C2×C4—24no—
580C2×C4—28no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—24no—
584C2×C4—24no—
585C2×C4—24no—
586C2×C4—24no—
587C2×C4—24no—
588C2×C4—24no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—41no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—44no—
597C2×C4—41no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—41no—
601C2×C4—41no—
602C2×C4—44no—
603C2×C4—42no—
604C2×C4—44no—
605C2×C4—44no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—44no—
610C2×C4—44no—
611C2×C4—44no—
612C2×C4—44no—
613C2×C4—44no—
614C2×C4—44no—
615C2×C4—42no—
616C2×C4—44no—
617C2×C4—44no—
618C2×C4—44no—
619C2×C4—44no—
620C2×C4—44no—
621C2×C4—44no—
622C2×C4—44no—
623C2×C4—42no—
624C2×C4—44no—
625C2×C4—44no—
626C2×C4—44no—
627C2×C4—44no—
628C2×C4—44no—
629C2×C4—44no—
630C2×C4—44no—
631C2×C4—44no—
632C2×C4—44no—
633C2×C4#53514yes—
634C2×C4#54618yes—
635C22—44no—
636C22—48no—
637C22—48no—
638C22—48no—
639C22—48no—
640C22—48no—
641C22—48no—
642C22—44no—
643C22—48no—
644C22—44no—
645C22—48no—
646C22—48no—
647C22—864no—
648C22—84no—
649C22—84no—
650C22—84no—
651C22—84no—
652C22—84no—
653C22—84no—
654C22—84no—
655C22—84no—
656C22—84no—
657C22—84no—
658C22—84no—
659C22—84no—
660C22C2×#53511yescharacteristic
661C22#345912yes—
662C22C2×#54612yes—
663C22#348111yescharacteristic
664C22#348411yescharacteristic
665C22#348511yescharacteristic
666C22#348812yes—
667C22#349112yes—
668C22#351412yes—
669C22#351912yes—
670C22#351912yes—
671C22#352911yescharacteristic
672C22#354312yes—
673C22#354612yes—
674C22#354911yescharacteristic
675C22#354911yescharacteristic
676C22#355812yes—
677C22#355812yes—
678C22#355812yes—
679C22#355812yes—
680C22#356012yes—
681C4—24no—
682C4—28no—
683C4—42no—
684C4—42no—
685C4—44no—
686C4—44no—
687C4—82no—
688C4—84no—
689C4—84no—
690C4—82no—
691C4—84no—
692C2—84no—
693C2—88no—
694C2#3696711yescharacteristic
695C2#3724912yes—
696C2#3774612yes—
697C2#3876812yes—
698C2#3891012yes—
699C2#4594712yes—
700C2#4622511yescharacteristic
701C2#5337512yes—
702C2#5904111yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC2×#2679C2
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×#2861C2semidirect
3. #17077C2semidirect
4. #16310C2semidirect
5. #15882C2semidirect
6. #15833C2semidirect
7. #15831C2semidirect
8. #15281C2semidirect
9. #15281C2semidirect
10. #15259C2semidirect
11. #14700C2semidirect
12. #14445C2semidirect
13. #14358C2semidirect
14. C22×#125C22semidirect
24. C2×#393C22semidirect
25. C2×#391C22semidirect
26. C2×#391C22semidirect
33. C2×#389C22semidirect
34. C2×#388C22semidirect
35. C2×#388C22semidirect
36. C2×#388C22semidirect
37. C2×#388C22semidirect
38. C2×#388C22semidirect
39. C2×#379C22semidirect
40. C2×#379C22semidirect
41. C2×#379C22semidirect
42. C2×C43C22semidirect
43. #1445C22semidirect
44. #1445C22semidirect
45. #1442C22semidirect
46. #1440C22semidirect
47. #1440C22semidirect
48. #1439C22semidirect
49. #1439C22semidirect
50. #1439C22semidirect
51. #1439C22semidirect
52. #1439C22semidirect
53. #1439C22semidirect
54. #1439C22semidirect
55. #1439C22semidirect
62. #1437C22semidirect
63. #1437C22semidirect
64. #1437C22semidirect
65. #1436C22semidirect
66. #1436C22semidirect
67. C4⋊#95C22semidirect
68. C4⋊#95C22semidirect

Maximal subgroups

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

subgroupindex
C2×#26792
C2×#28612
#170772
#163102
#158822
#158332
#158312
#152812
#152592
#147002
#144452
#143582