Finite Group Databasebeta

← #759342

Subgroups — #759342

Order 512.

3850 conjugacy classes of subgroups.

The subgroup lattice has 703 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2441C212yesmaximal, complemented
2C2×#2280C212yesmaximal
3#16298C214yesmaximal
4#15223C211yesmaximal, characteristic, complemented
5#15216C211yesmaximal, characteristic, complemented
6#15131C212yesmaximal, complemented
7#14772C214yesmaximal, complemented
8#14769C214yesmaximal, complemented
9#14210C214yesmaximal, complemented
10#14129C212yesmaximal, complemented
11#13951C212yesmaximal, complemented
12#13879C212yesmaximal, complemented
13C4×#379C211yesmaximal, characteristic
14C25×C4C2211yescharacteristic
15C22×#125C2211yescharacteristic, complemented
16C22×#117C2212yes—
17C22×#117C2212yes—
18#2441—216no—
19C2×#399C2214yes—
20C2×#396C2214yes—
21C2×#396C2214yes—
22C2×#389C2212yescomplemented
23C2×#389C2212yescomplemented
24C2×#389C2214yescomplemented
25C2×#389C2212yescomplemented
26C2×#387C2214yescomplemented
27C2×#384C2212yescomplemented
28C2×#382C2211yescharacteristic, complemented
29#2280—216no—
30C2×#381C2214yes—
31C2×#380C2212yes—
32C2×#380C2212yes—
33C2×#380C2212yes—
34C2×#380C2212yes—
35C2×#379C2212yes—
36C2×#379C2214yes—
37C2×C4×C22⋊C4C2212yes—
38C2×C4×C22⋊C4C2212yes—
39C2×C4×C22⋊C4C2214yes—
40C2×C4×C22⋊C4C2214yes—
41C2×C4×C22⋊C4C2211yescharacteristic
42C2×C4×C22⋊C4C2211yescharacteristic
43#1445C2214yescomplemented
44#1442C2214yescomplemented
45#1441C2214yescomplemented
46#1441C2214yescomplemented
47#1439C2214yes—
48#1439C2214yes—
49#1439C2214yes—
50#1439C2214yes—
51#1439C2214yes—
52#1439C2214yes—
53#1439C2214yes—
54#1439C2214yes—
55#1439C2214yes—
56#1436C2214yescomplemented
57#1436C2214yescomplemented
58#1436C2214yescomplemented
59#1436C2214yescomplemented
60C4⋊#95C2212yescomplemented
61C4⋊#95C2211yescharacteristic, complemented
62C4⋊#95C2212yescomplemented
63C4⋊#95C2211yescharacteristic, complemented
64C4⋊#95C2214yescomplemented
65#1434C2214yescomplemented
66C22⋊#95C2212yes—
67C22⋊#95C2212yes—
68C22⋊#95C2212yes—
69C26C2311yescharacteristic
70C24×C4—28no—
71C24×C4—28no—
72C24×C4—42no—
73C24×C4—42no—
74C24×C4Q818yes—
75C24×C4C2311yescharacteristic
76C24×C4C2311yescharacteristic
77C24×C4C2312yes—
78C24×C4C2312yes—
79C2×#125—28no—
80C2×#125—42no—
81C2×#117—28no—
82C2×#117—28no—
83C2×#117—28no—
84C2×#117—28no—
85C2×#117—44no—
86C2×#117—44no—
87C22×C4⋊C4C2312yes—
88C22×C4⋊C4C2312yes—
89C22×C4⋊C4C2314yes—
90C22×C4⋊C4C2314yes—
91C22×C4⋊C4C2314yes—
92C22×C4⋊C4C2312yes—
93C22×C4⋊C4C2312yes—
94C22×C4⋊C4C2314yes—
95C22×C4⋊C4C2314yes—
96C22×C4⋊C4C2314yes—
97C22×C22⋊C4C2312yes—
98C22×C22⋊C4C2314yes—
99C22×C22⋊C4C2314yes—
100C22×C22⋊C4C2314yes—
101C22×C22⋊C4C2312yes—
102C22×C22⋊C4C2314yes—
103C22×C22⋊C4C2312yes—
104C22×C22⋊C4C2312yes—
105C22×C22⋊C4C2312yes—
106C22×C22⋊C4C2314yes—
107C22×C22⋊C4C2314yes—
108C22×C22⋊C4C2312yes—
109C22×C42C2311yescharacteristic
110C22×C42C2311yescharacteristic
111C22×C42C2312yes—
112C22×C42C2314yes—
113C22×C42C2314yes—
114#399—216no—
115#396—216no—
116#396—216no—
117#389—28no—
118#389—28no—
119#389—216no—
120#389—44no—
121#387—216no—
122#384—28no—
123#382—42no—
124#381—216no—
125#380—28no—
126#380—28no—
127#380—28no—
128#380D4116yes—
129#379—216no—
130#379—28no—
131C4×C22⋊C4—28no—
132C4×C22⋊C4—28no—
133C4×C22⋊C4—216no—
134C4×C22⋊C4—216no—
135C4×C22⋊C4—42no—
136C4×C22⋊C4—42no—
137C2×#95C2314yes—
138C2×#95C2314yes—
139C2×#95C2314yes—
140C2×#95C2312yes—
141C2×#95C2312yes—
142C2×#95C2312yes—
143C2×#95C2312yes—
144C2×#95C2314yes—
145C2×#95C2314yes—
146C2×#95C2314yes—
147C2×#95C2311yescharacteristic
148C2×#95C2311yescharacteristic
149C2×#95C2312yes—
150C2×#95C2314yes—
151C2×#95C2314yes—
152C2×#95C2314yes—
153C2×#95C2314yes—
154C2×#95C2314yes—
155C2×#95C2314yes—
156C2×#95C2314yes—
157C2×#95C2314yes—
158C25—24no—
159C25—24no—
160C25—24no—
161C25—41no—
162C25—41no—
163C25—42no—
164C25—42no—
165C25—42no—
166C25C2×Q814yes—
167C25C2412yes—
168C25C2411yescharacteristic
169C23×C4—24no—
170C23×C4—24no—
171C23×C4—24no—
172C23×C4—24no—
173C23×C4—24no—
174C23×C4—24no—
175C23×C4—264no—
176C23×C4—24no—
177C23×C4—24no—
178C23×C4—24no—
179C23×C4—24no—
180C23×C4—24no—
181C23×C4—24no—
182C23×C4—22no—
183C23×C4—22no—
184C23×C4—416no—
185C23×C4—41no—
186C23×C4—41no—
187C23×C4—42no—
188C23×C4—42no—
189C23×C4—416no—
190C23×C4—432no—
191C23×C4—416no—
192C23×C4C2×D418yes—
193C23×C4C2×Q814yes—
194C23×C4C2×Q818yes—
195C23×C4C2×Q814yes—
196C23×C4C2×Q818yes—
197C23×C4#4118yes—
198C23×C4C2412yes—
199C23×C4C2414yes—
200C23×C4C2414yes—
201C23×C4C2412yes—
202C23×C4C2414yes—
203C23×C4C2414yes—
204C23×C4C2411yescharacteristic
205C23×C4C2412yes—
206C23×C4C2412yes—
207C23×C4C2412yes—
208C23×C4C2411yescharacteristic
209#125—416no—
210#117—432no—
211#117—432no—
212C2×C4⋊C4—24no—
213C2×C4⋊C4—24no—
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—28no—
221C2×C4⋊C4—28no—
222C2×C4⋊C4—28no—
223C2×C4⋊C4—24no—
224C2×C4⋊C4—28no—
225C2×C4⋊C4—28no—
226C2×C4⋊C4—24no—
227C2×C4⋊C4—24no—
228C2×C4⋊C4—28no—
229C2×C4⋊C4—28no—
230C2×C4⋊C4—28no—
231C2×C4⋊C4—44no—
232C2×C4⋊C4—44no—
233C2×C4⋊C4—44no—
234C2×C4⋊C4—44no—
235C2×C4⋊C4—44no—
236C2×C4⋊C4—44no—
237C2×C4⋊C4—44no—
238C2×C4⋊C4—44no—
239C2×C4⋊C4—44no—
240C2×C4⋊C4C2×D418yes—
241C2×C4⋊C4C2×D418yes—
242C2×C22⋊C4—28no—
243C2×C22⋊C4—28no—
244C2×C22⋊C4—24no—
245C2×C22⋊C4—24no—
246C2×C22⋊C4—28no—
247C2×C22⋊C4—24no—
248C2×C22⋊C4—24no—
249C2×C22⋊C4—24no—
250C2×C22⋊C4—28no—
251C2×C22⋊C4—28no—
252C2×C22⋊C4—28no—
253C2×C22⋊C4—28no—
254C2×C22⋊C4—28no—
255C2×C22⋊C4—28no—
256C2×C22⋊C4—24no—
257C2×C22⋊C4—24no—
258C2×C22⋊C4—24no—
259C2×C22⋊C4—24no—
260C2×C22⋊C4—44no—
261C2×C22⋊C4—44no—
262C2×C22⋊C4—42no—
263C2×C22⋊C4—42no—
264C2×C22⋊C4—44no—
265C2×C22⋊C4—44no—
266C2×C22⋊C4—42no—
267C2×C22⋊C4—42no—
268C2×C22⋊C4—42no—
269C2×C22⋊C4—44no—
270C2×C22⋊C4—42no—
271C2×C22⋊C4—44no—
272C2×C22⋊C4—44no—
273C2×C22⋊C4—44no—
274C2×C22⋊C4—42no—
275C2×C22⋊C4—42no—
276C2×C22⋊C4C2×D4116yes—
277C2×C22⋊C4#4118yes—
278C2×C42—28no—
279C2×C42—28no—
280C2×C42—28no—
281C2×C42—24no—
282C2×C42—24no—
283C2×C42—28no—
284C2×C42—24no—
285C2×C42—24no—
286C2×C42—22no—
287C2×C42—22no—
288C2×C42—24no—
289C2×C42—44no—
290C2×C42—44no—
291#95—28no—
292#95—28no—
293#95—28no—
294#95—24no—
295#95—24no—
296#95—28no—
297#95—28no—
298#95—24no—
299#95—24no—
300#95—28no—
301#95—44no—
302#95—44no—
303#95—44no—
304#95—44no—
305#95—44no—
306#95—41no—
307#95—41no—
308#95—42no—
309#95—44no—
310#95—44no—
311#95C2×D4116yes—
312C24—22no—
313C24—22no—
314C24—22no—
315C24—22no—
316C24—22no—
317C24—22no—
318C24—22no—
319C24—22no—
320C24—21no—
321C24—21no—
322C24—22no—
323C24—22no—
324C24—22no—
325C24—22no—
326C24—22no—
327C24—22no—
328C24—22no—
329C24—22no—
330C24—22no—
331C24—216no—
332C24—28no—
333C24—44no—
334C24—48no—
335C24—44no—
336C24—44no—
337C24—48no—
338C24—44no—
339C24—48no—
340C24—48no—
341C24—48no—
342C24—48no—
343C24—48no—
344C24—48no—
345C24—48no—
346C24—48no—
347C24—48no—
348C24—48no—
349C24—48no—
350C24—44no—
351C24—44no—
352C24C22×D414yes—
353C24C22×Q814yes—
354C24C22×Q812yes—
355C24C2×#4114yes—
356C24C2×#4114yes—
357C24C2511yescharacteristic
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—216no—
363C22×C4—216no—
364C22×C4—216no—
365C22×C4—216no—
366C22×C4—216no—
367C22×C4—216no—
368C22×C4—216no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—24no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—24no—
376C22×C4—24no—
377C22×C4—24no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—21no—
381C22×C4—21no—
382C22×C4—22no—
383C22×C4—24no—
384C22×C4—24no—
385C22×C4—22no—
386C22×C4—24no—
387C22×C4—24no—
388C22×C4—24no—
389C22×C4—24no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—24no—
394C22×C4—24no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—216no—
398C22×C4—216no—
399C22×C4—216no—
400C22×C4—28no—
401C22×C4—22no—
402C22×C4—24no—
403C22×C4—24no—
404C22×C4—24no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—24no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—24no—
415C22×C4—22no—
416C22×C4—24no—
417C22×C4—22no—
418C22×C4—24no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—24no—
422C22×C4—22no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—22no—
426C22×C4—24no—
427C22×C4—24no—
428C22×C4—22no—
429C22×C4—44no—
430C22×C4—48no—
431C22×C4—44no—
432C22×C4—48no—
433C22×C4—44no—
434C22×C4—4128no—
435C22×C4—48no—
436C22×C4—48no—
437C22×C4—48no—
438C22×C4—48no—
439C22×C4—44no—
440C22×C4—48no—
441C22×C4—44no—
442C22×C4C22⋊Q8132yes—
443C22×C4C22×D418yes—
444C22×C4C22×D418yes—
445C22×C4C22×D414yes—
446C22×C4C22×D414yes—
447C22×C4C22×Q812yes—
448C22×C4C22×Q814yes—
449C22×C4C22×Q812yes—
450C22×C4C2×#4114yes—
451C22×C4C2×#4114yes—
452C22×C4C2×#4114yes—
453C22×C4C2×#4114yes—
454C22×C4H(5,𝔽2)12yes—
455C22×C4H(5,𝔽2)12yes—
456C4⋊C4—216no—
457C4⋊C4—216no—
458C4⋊C4—416no—
459C4⋊C4—416no—
460C4⋊C4—416no—
461C4⋊C4—48no—
462C4⋊C4—48no—
463C4⋊C4—88no—
464C4⋊C4—88no—
465C4⋊C4—88no—
466C22⋊C4—416no—
467C22⋊C4—48no—
468C22⋊C4—48no—
469C22⋊C4—48no—
470C22⋊C4—416no—
471C22⋊C4—416no—
472C22⋊C4—48no—
473C22⋊C4—48no—
474C22⋊C4—416no—
475C22⋊C4—416no—
476C22⋊C4—416no—
477C22⋊C4—48no—
478C42—416no—
479C42—416no—
480C42—44no—
481C42—48no—
482C42—44no—
483C23—24no—
484C23—24no—
485C23—24no—
486C23—24no—
487C23—22no—
488C23—24no—
489C23—24no—
490C23—24no—
491C23—24no—
492C23—22no—
493C23—24no—
494C23—24no—
495C23—24no—
496C23—24no—
497C23—24no—
498C23—24no—
499C23—24no—
500C23—24no—
501C23—24no—
502C23—24no—
503C23—24no—
504C23—24no—
505C23—24no—
506C23—24no—
507C23—24no—
508C23—24no—
509C23—24no—
510C23—24no—
511C23—416no—
512C23—432no—
513C23—432no—
514C23—416no—
515C23—432no—
516C23—416no—
517C23—432no—
518C23—432no—
519C23—432no—
520C23—41no—
521C23—42no—
522C23—42no—
523C23—42no—
524C23—41no—
525C23—42no—
526C23—42no—
527C23—42no—
528C23—42no—
529C23—42no—
530C23—42no—
531C23—42no—
532C23—42no—
533C23—42no—
534C23—42no—
535C23—42no—
536C23—42no—
537C23—42no—
538C23—42no—
539C23—42no—
540C23—42no—
541C23—42no—
542C23—42no—
543C23—41no—
544C23—42no—
545C23—41no—
546C23C2×C22⋊Q818yes—
547C23#53314yes—
548C23C23×D412yes—
549C23C23×Q811yescharacteristic
550C23C22×#4112yes—
551C23C22×#4112yes—
552C23C2×H(5,𝔽2)11yescharacteristic
553C23C2×H(5,𝔽2)11yescharacteristic
554C23#58512yes—
555C23#58512yes—
556C23#58512yes—
557C2×C4—24no—
558C2×C4—24no—
559C2×C4—24no—
560C2×C4—24no—
561C2×C4—24no—
562C2×C4—24no—
563C2×C4—264no—
564C2×C4—24no—
565C2×C4—22no—
566C2×C4—28no—
567C2×C4—28no—
568C2×C4—24no—
569C2×C4—24no—
570C2×C4—28no—
571C2×C4—28no—
572C2×C4—24no—
573C2×C4—24no—
574C2×C4—28no—
575C2×C4—28no—
576C2×C4—24no—
577C2×C4—24no—
578C2×C4—416no—
579C2×C4—432no—
580C2×C4—41no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—44no—
585C2×C4—41no—
586C2×C4—44no—
587C2×C4—42no—
588C2×C4—44no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—44no—
593C2×C4—44no—
594C2×C4—44no—
595C2×C4—44no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—44no—
599C2×C4—416no—
600C2×C4—44no—
601C2×C4—42no—
602C2×C4—44no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—44no—
606C2×C4—44no—
607C2×C4—44no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—44no—
612C2×C4—44no—
613C2×C4—42no—
614C2×C4—44no—
615C2×C4—44no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—44no—
620C2×C4—44no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—44no—
624C2×C4—42no—
625C2×C4—44no—
626C2×C4—42no—
627C2×C4—44no—
628C2×C4C2×C22⋊Q818yes—
629C2×C4C2×C22⋊Q818yes—
630C2×C4#54314yes—
631C2×C4D4214yes—
632C2×C4#54618yes—
633C2×C4#55114yes—
634C22—28no—
635C22—28no—
636C22—28no—
637C22—28no—
638C22—24no—
639C22—464no—
640C22—44no—
641C22—44no—
642C22—44no—
643C22—44no—
644C22—44no—
645C22—42no—
646C22—44no—
647C22—42no—
648C22—44no—
649C22—44no—
650C22—44no—
651C22—44no—
652C22—44no—
653C22—44no—
654C22—44no—
655C22—44no—
656C22—44no—
657C22—44no—
658C22—44no—
659C22C22×C22⋊Q812yes—
660C22C2×#53311yescharacteristic
661C22C2×#54311yescharacteristic
662C22#345811yescharacteristic
663C22#345812yes—
664C22C2×D4211yescharacteristic
665C22C2×#54612yes—
666C22D4×#4112yes—
667C22D4×#4112yes—
668C22C2×#55111yescharacteristic
669C22#347512yes—
670C22#347512yes—
671C22#348512yes—
672C22#348712yes—
673C22#349312yes—
674C22#349312yes—
675C22#350512yes—
676C22#350612yes—
677C22#353912yes—
678C22#353911yescharacteristic
679C22#353911yescharacteristic
680C4—28no—
681C4—42no—
682C4—48no—
683C4—44no—
684C4—48no—
685C4—44no—
686C4—82no—
687C4—84no—
688C4—82no—
689C4—84no—
690C4#243718yes—
691C2—48no—
692C2—44no—
693C2C2×#243711yescharacteristic
694C2#3473912yes—
695C2#3503311yescharacteristic
696C2#3515112yes—
697C2#3748111yescharacteristic
698C2#5899912yes—
699C2#5901912yes—
700C2#5902012yes—
701C2#5920012yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC26C23
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×#2441C2semidirect
4. #15223C2semidirect
5. #15216C2semidirect
6. #15131C2semidirect
7. #14772C2semidirect
8. #14769C2semidirect
9. #14210C2semidirect
10. #14129C2semidirect
11. #13951C2semidirect
12. #13879C2semidirect
15. C22×#125C22semidirect
22. C2×#389C22semidirect
23. C2×#389C22semidirect
24. C2×#389C22semidirect
25. C2×#389C22semidirect
26. C2×#387C22semidirect
27. C2×#384C22semidirect
28. C2×#382C22semidirect
43. #1445C22semidirect
44. #1442C22semidirect
45. #1441C22semidirect
46. #1441C22semidirect
56. #1436C22semidirect
57. #1436C22semidirect
58. #1436C22semidirect
59. #1436C22semidirect
60. C4⋊#95C22semidirect
61. C4⋊#95C22semidirect
62. C4⋊#95C22semidirect
63. C4⋊#95C22semidirect
64. C4⋊#95C22semidirect
65. #1434C22semidirect

Maximal subgroups

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

subgroupindex
C2×#24412
C2×#22802
#162982
#152232
#152162
#151312
#147722
#147692
#142102
#141292
#139512
#138792
C4×#3792