Finite Group Databasebeta

← #4465866

Subgroups — #4465866

Order 512.

3194 conjugacy classes of subgroups.

The subgroup lattice has 707 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2733C211yesmaximal, characteristic, complemented
2C2×#2620C211yesmaximal, characteristic, complemented
3#17091C214yesmaximal, complemented
4#16810C212yesmaximal, complemented
5#16535C214yesmaximal, complemented
6#16502C214yesmaximal, complemented
7#15763C212yesmaximal, complemented
8#15733C214yesmaximal, complemented
9#15643C212yesmaximal, complemented
10#15310C212yesmaximal, complemented
11#15264C212yesmaximal, complemented
12#14886C212yesmaximal, complemented
13#14489C211yesmaximal, characteristic, complemented
14C22×#127C2212yes—
15C22×#124C2211yescharacteristic
16C22×#124C2212yes—
17C22×C22⋊Q8C2211yescharacteristic
18C22×#121C2211yescharacteristic
19C22×#117C2211yescharacteristic
20#2733—28no—
21#2620—28no—
22C2×#397C2214yes—
23C2×#396C2214yes—
24C2×#394C2212yes—
25C2×#394C2214yes—
26C2×#394C2214yes—
27C2×#394C2212yes—
28C2×#394C2212yes—
29C2×#392C2214yes—
30C2×#392C2212yes—
31C2×#392C2212yes—
32C2×#392C2214yes—
33C2×#392C2214yes—
34C2×#392C2214yes—
35C2×#392C2214yes—
36C2×#392C2212yes—
37C2×#391C2211yescharacteristic
38C2×#390C2212yes—
39C2×#390C2211yescharacteristic
40C2×#388C2214yes—
41C2×#388C2214yes—
42C2×#386C2214yes—
43C2×#384C2211yescharacteristic
44#1446C2212yes—
45#1443C2214yes—
46#1442C2214yes—
47#1441C2212yes—
48#1439C2214yes—
49#1439C2214yes—
50#1439C2214yes—
51#1439C2212yes—
52#1439C2214yes—
53#1438C2214yes—
54#1438C2214yes—
55#1438C2214yes—
56#1438C2214yes—
57#1438C2214yes—
58#1438C2214yes—
59#1438C2212yes—
60#1438C2214yes—
61#1437C2214yes—
62#1437C2212yes—
63#1437C2214yes—
64#1437C2212yes—
65#1436C2212yes—
66#1436C2212yes—
67#1434C2212yes—
68#1434C2214yes—
69C23×Q8C2311yescharacteristic
70C23×D4C2311yescharacteristic
71C23×D4C2314yes—
72C23×D4C2312yes—
73C23×D4C2312yes—
74C23×D4C2312yes—
75C23×D4C2312yes—
76C23×D4C2314yes—
77C23×D4C2314yes—
78C24×C4C2311yescharacteristic
79C2×#127—48no—
80C2×#127—44no—
81C2×#124—24no—
82C2×#124—28no—
83C2×#124—44no—
84C2×#124—48no—
85C2×C22⋊Q8—24no—
86C2×C22⋊Q8—42no—
87C2×C22⋊Q8—42no—
88C2×#121—24no—
89C2×#121—42no—
90C2×#121—42no—
91C2×#117—24no—
92C2×#117—44no—
93C22×C4⋊C4C2312yes—
94C22×C4⋊C4C2312yes—
95C22×C4⋊C4C2314yes—
96C22×C4⋊C4C2312yes—
97C22×C4⋊C4C2311yescharacteristic
98C22×C4⋊C4C2311yescharacteristic
99C22×C22⋊C4C2314yes—
100C22×C22⋊C4C2314yes—
101C22×C22⋊C4C2312yes—
102C22×C22⋊C4C2312yes—
103C22×C22⋊C4C2314yes—
104C22×C22⋊C4C2312yes—
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×C22⋊C4C2314yes—
115C22×C22⋊C4C2314yes—
116C22×C42C2311yescharacteristic
117C22×C42C2312yes—
118C22×C42C2312yes—
119C22×C42C2311yescharacteristic
120#397—48no—
121#396—216no—
122#394—28no—
123#394—216no—
124#394—48no—
125#394—44no—
126#394—44no—
127#392—216no—
128#392—48no—
129#392—48no—
130#392—48no—
131#392—44no—
132#392—44no—
133#392—48no—
134#392—44no—
135#391D418yescomplemented
136#390—24no—
137#390—28no—
138#388—216no—
139#388—48no—
140#386—48no—
141#384—24no—
142C2×#95C2314yes—
143C2×#95C2314yes—
144C2×#95C2314yes—
145C2×#95C2312yes—
146C2×#95C2314yes—
147C2×#95C2312yes—
148C2×#95C2312yes—
149C2×#95C2314yes—
150C2×#95C2312yes—
151C2×#95C2312yes—
152C2×#95C2312yes—
153C2×#95C2314yes—
154C2×#95C2314yes—
155C2×#95C2314yes—
156C2×#95C2314yes—
157C2×#95C2314yes—
158C25C2412yes—
159C25C2414yes—
160C25C2411yescharacteristic
161C22×Q8—24no—
162C22×Q8—41no—
163C22×Q8—41no—
164C22×Q8—42no—
165C22×Q8C2×D414yes—
166C22×D4—24no—
167C22×D4—28no—
168C22×D4—22no—
169C22×D4—24no—
170C22×D4—24no—
171C22×D4—24no—
172C22×D4—24no—
173C22×D4—28no—
174C22×D4—28no—
175C22×D4—24no—
176C22×D4—41no—
177C22×D4—41no—
178C22×D4—42no—
179C22×D4—42no—
180C22×D4—44no—
181C22×D4—44no—
182C22×D4—44no—
183C22×D4—44no—
184C22×D4—44no—
185C22×D4—44no—
186C22×D4—42no—
187C22×D4—44no—
188C22×D4—44no—
189C22×D4—44no—
190C22×D4—44no—
191C22×D4—44no—
192C22×D4—44no—
193C22×D4—44no—
194C22×D4—44no—
195C22×D4—44no—
196C22×D4—44no—
197C22×D4—44no—
198C22×D4—44no—
199C22×D4—44no—
200C22×D4—42no—
201C22×D4—44no—
202C22×D4—42no—
203C22×D4—42no—
204C22×D4—44no—
205C22×D4—44no—
206C22×D4—44no—
207C22×D4—44no—
208C22×D4—44no—
209C22×D4—44no—
210C22×D4—42no—
211C22×D4—42no—
212C23×C4—22no—
213C23×C4—22no—
214C23×C4—22no—
215C23×C4—24no—
216C23×C4—42no—
217C23×C4C2412yes—
218C23×C4C2414yes—
219C23×C4C2414yes—
220C23×C4C2412yes—
221C23×C4C2412yes—
222C23×C4C2412yes—
223C23×C4C2412yes—
224C23×C4C2411yescharacteristic
225C23×C4C2414yes—
226C23×C4C2411yescharacteristic
227#127—432no—
228#124—416no—
229#124—432no—
230C22⋊Q8—416no—
231#121—416no—
232#117—416no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—24no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—28no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—42no—
241C2×C4⋊C4—41no—
242C2×C4⋊C4—41no—
243C2×C4⋊C4—42no—
244C2×C4⋊C4—41no—
245C2×C4⋊C4—41no—
246C2×C4⋊C4—44no—
247C2×C4⋊C4—44no—
248C2×C4⋊C4—44no—
249C2×C4⋊C4C2×D414yes—
250C2×C22⋊C4—24no—
251C2×C22⋊C4—24no—
252C2×C22⋊C4—24no—
253C2×C22⋊C4—24no—
254C2×C22⋊C4—28no—
255C2×C22⋊C4—28no—
256C2×C22⋊C4—28no—
257C2×C22⋊C4—28no—
258C2×C22⋊C4—24no—
259C2×C22⋊C4—24no—
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—28no—
271C2×C22⋊C4—44no—
272C2×C22⋊C4—44no—
273C2×C22⋊C4—44no—
274C2×C22⋊C4—42no—
275C2×C22⋊C4—42no—
276C2×C22⋊C4—42no—
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×C22⋊C4—44no—
292C2×C22⋊C4—44no—
293C2×C22⋊C4—44no—
294C2×C22⋊C4—44no—
295C2×C22⋊C4—44no—
296C2×C22⋊C4—44no—
297C2×C22⋊C4—44no—
298C2×C22⋊C4—44no—
299C2×C22⋊C4—44no—
300C2×C22⋊C4—44no—
301C2×C42—22no—
302C2×C42—24no—
303C2×C42—24no—
304C2×C42—28no—
305C2×C42—42no—
306C2×C42—44no—
307C2×C42—42no—
308C2×C42C2×D414yes—
309#95—24no—
310#95—28no—
311#95—28no—
312#95—28no—
313#95—28no—
314#95—44no—
315#95—44no—
316#95—44no—
317#95—42no—
318#95—44no—
319#95—42no—
320#95—42no—
321#95—42no—
322#95—42no—
323#95—44no—
324#95C2×D4116yes—
325C24—21no—
326C24—22no—
327C24—21no—
328C24—22no—
329C24—24no—
330C24—24no—
331C24—22no—
332C24—24no—
333C24—24no—
334C24—24no—
335C24—24no—
336C24—24no—
337C24—24no—
338C24—24no—
339C24—24no—
340C24—24no—
341C24—22no—
342C24—24no—
343C24—24no—
344C24—22no—
345C24—24no—
346C24—24no—
347C24—24no—
348C24—24no—
349C24—24no—
350C24—24no—
351C24—24no—
352C24—24no—
353C24—24no—
354C24—24no—
355C24H(5,𝔽2)12yes—
356C24C2511yescharacteristic
357C2×Q8—28no—
358C2×Q8—44no—
359C2×Q8—44no—
360C2×Q8—84no—
361C2×Q8—84no—
362C2×D4—48no—
363C2×D4—416no—
364C2×D4—44no—
365C2×D4—44no—
366C2×D4—48no—
367C2×D4—416no—
368C2×D4—48no—
369C2×D4—416no—
370C2×D4—48no—
371C2×D4—48no—
372C2×D4—44no—
373C2×D4—84no—
374C2×D4—84no—
375C2×D4—84no—
376C2×D4—84no—
377C2×D4—84no—
378C2×D4—84no—
379C2×D4—84no—
380C2×D4—88no—
381C2×D4—88no—
382C2×D4—88no—
383C2×D4—88no—
384C2×D4—88no—
385C2×D4—84no—
386C2×D4—88no—
387C2×D4—88no—
388C2×D4—84no—
389C2×D4—88no—
390C2×D4—88no—
391C2×D4—88no—
392C2×D4—88no—
393C2×D4—88no—
394C2×D4—88no—
395C2×D4—88no—
396C2×D4—88no—
397C2×D4—88no—
398C2×D4—88no—
399C2×D4—88no—
400C2×D4—88no—
401C2×D4—88no—
402C2×D4—88no—
403C2×D4—88no—
404C2×D4—88no—
405C2×D4—88no—
406C2×D4—88no—
407C2×D4—88no—
408C22×C4—22no—
409C22×C4—21no—
410C22×C4—21no—
411C22×C4—28no—
412C22×C4—22no—
413C22×C4—21no—
414C22×C4—24no—
415C22×C4—24no—
416C22×C4—24no—
417C22×C4—24no—
418C22×C4—22no—
419C22×C4—24no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—24no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—24no—
430C22×C4—24no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—21no—
434C22×C4—24no—
435C22×C4—24no—
436C22×C4—24no—
437C22×C4—24no—
438C22×C4—24no—
439C22×C4—22no—
440C22×C4—24no—
441C22×C4—24no—
442C22×C4—24no—
443C22×C4—24no—
444C22×C4—24no—
445C22×C4—24no—
446C22×C4—24no—
447C22×C4—24no—
448C22×C4—24no—
449C22×C4—24no—
450C22×C4—24no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—24no—
454C22×C4—24no—
455C22×C4—24no—
456C22×C4—24no—
457C22×C4—48no—
458C22×C4—48no—
459C22×C4—44no—
460C22×C4—44no—
461C22×C4C22×D414yes—
462C22×C4C22×D414yes—
463C22×C4C22×D414yes—
464C22×C4C22×D412yes—
465C22×C4H(5,𝔽2)18yes—
466C22×C4H(5,𝔽2)14yes—
467C22×C4H(5,𝔽2)12yes—
468C22×C4H(5,𝔽2)14yes—
469C4⋊C4—44no—
470C4⋊C4—48no—
471C4⋊C4—48no—
472C4⋊C4—44no—
473C4⋊C4—84no—
474C4⋊C4—88no—
475C22⋊C4—416no—
476C22⋊C4—48no—
477C22⋊C4—48no—
478C22⋊C4—416no—
479C22⋊C4—48no—
480C22⋊C4—416no—
481C22⋊C4—416no—
482C22⋊C4—416no—
483C22⋊C4—88no—
484C22⋊C4—88no—
485C22⋊C4—88no—
486C22⋊C4—88no—
487C22⋊C4—88no—
488C22⋊C4—88no—
489C22⋊C4—88no—
490C22⋊C4—88no—
491C22⋊C4—88no—
492C42—28no—
493C42—44no—
494C42—48no—
495C42—48no—
496C23—22no—
497C23—24no—
498C23—24no—
499C23—24no—
500C23—41no—
501C23—41no—
502C23—42no—
503C23—42no—
504C23—44no—
505C23—42no—
506C23—44no—
507C23—44no—
508C23—44no—
509C23—42no—
510C23—42no—
511C23—44no—
512C23—42no—
513C23—42no—
514C23—42no—
515C23—44no—
516C23—44no—
517C23—44no—
518C23—44no—
519C23—44no—
520C23—44no—
521C23—44no—
522C23—44no—
523C23—44no—
524C23—44no—
525C23—42no—
526C23—42no—
527C23—44no—
528C23—44no—
529C23—44no—
530C23—44no—
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—44no—
548C23—44no—
549C23—44no—
550C23—44no—
551C23—44no—
552C23—44no—
553C23—44no—
554C23—44no—
555C23—44no—
556C23—44no—
557C23—44no—
558C23—44no—
559C23—44no—
560C23—44no—
561C23—44no—
562C23—44no—
563C23—42no—
564C23—44no—
565C23—44no—
566C23—42no—
567C23C23×D411yescharacteristic
568C23C2×H(5,𝔽2)12yes—
569C23C2×H(5,𝔽2)11yescharacteristic
570C23C2×H(5,𝔽2)12yes—
571C23C2×H(5,𝔽2)11yescharacteristic
572C23C2×H(5,𝔽2)14yes—
573C23#58514yes—
574Q8—88no—
575D4—88no—
576D4—832no—
577D4—816no—
578D4—816no—
579D4—816no—
580D4—832no—
581D4—816no—
582D4—832no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—24no—
586C2×C4—24no—
587C2×C4—24no—
588C2×C4—24no—
589C2×C4—24no—
590C2×C4—22no—
591C2×C4—24no—
592C2×C4—24no—
593C2×C4—24no—
594C2×C4—24no—
595C2×C4—24no—
596C2×C4—28no—
597C2×C4—28no—
598C2×C4—28no—
599C2×C4—28no—
600C2×C4—24no—
601C2×C4—28no—
602C2×C4—416no—
603C2×C4—41no—
604C2×C4—41no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—44no—
610C2×C4—42no—
611C2×C4—44no—
612C2×C4—44no—
613C2×C4—44no—
614C2×C4—44no—
615C2×C4—44no—
616C2×C4—44no—
617C2×C4—44no—
618C2×C4—44no—
619C2×C4—44no—
620C2×C4—44no—
621C2×C4—42no—
622C2×C4—44no—
623C2×C4—44no—
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—44no—
634C2×C4—44no—
635C2×C4—42no—
636C2×C4—44no—
637C2×C4—42no—
638C2×C4—44no—
639C2×C4#53514yes—
640C22—42no—
641C22—42no—
642C22—48no—
643C22—42no—
644C22—82no—
645C22—82no—
646C22—84no—
647C22—84no—
648C22—84no—
649C22—84no—
650C22—82no—
651C22—84no—
652C22—82no—
653C22—84no—
654C22—84no—
655C22—84no—
656C22—84no—
657C22—84no—
658C22—84no—
659C22—84no—
660C22—84no—
661C22—84no—
662C22—84no—
663C22—82no—
664C22—84no—
665C22—84no—
666C22—84no—
667C22—84no—
668C22—84no—
669C22—84no—
670C22—84no—
671C22C2×#53511yescharacteristic
672C22#348411yescharacteristic
673C22#348411yescharacteristic
674C22#348712yes—
675C22#349812yes—
676C22#353212yes—
677C22#353414yes—
678C22#353712yes—
679C22#353712yes—
680C22#354214yes—
681C22#354512yes—
682C22#354614yes—
683C22#354614yes—
684C22#354912yes—
685C22#355112yes—
686C4—24no—
687C4—44no—
688C4—44no—
689C4—42no—
690C4—44no—
691C4—48no—
692C4—44no—
693C4—84no—
694C4—82no—
695C4—84no—
696C2—82no—
697C2—164no—
698C2—162no—
699C2#3671711yescharacteristic
700C2#4287612yes—
701C2#4642512yes—
702C2#5190211yescharacteristic
703C2#5208311yescharacteristic
704C2#5394014yes—
705C2#5858314yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
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×#2733C2semidirect
2. C2×#2620C2semidirect
3. #17091C2semidirect
4. #16810C2semidirect
5. #16535C2semidirect
6. #16502C2semidirect
7. #15763C2semidirect
8. #15733C2semidirect
9. #15643C2semidirect
10. #15310C2semidirect
11. #15264C2semidirect
12. #14886C2semidirect
13. #14489C2semidirect
135. #391D4semidirect

Maximal subgroups

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

subgroupindex
C2×#27332
C2×#26202
#170912
#168102
#165352
#165022
#157632
#157332
#156432
#153102
#152642
#148862
#144892