Finite Group Databasebeta

← #759369

Subgroups — #759369

Order 512.

3834 conjugacy classes of subgroups.

The subgroup lattice has 698 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2445C212yesmaximal, complemented
2C2×C4×#121C212yesmaximal, complemented
3#16334C214yesmaximal, complemented
4#15223C211yesmaximal, characteristic, complemented
5#15126C212yesmaximal, complemented
6#15124C211yesmaximal, characteristic
7#14806C214yesmaximal, complemented
8#14721C214yesmaximal, complemented
9#14141C212yesmaximal, complemented
10#14096C214yesmaximal, complemented
11#14045C212yesmaximal, complemented
12#13979C212yesmaximal, complemented
13C4×#385C211yesmaximal, characteristic, complemented
14C22×#127C2211yescharacteristic
15C22×#123C2214yes—
16C22×#121C2212yes—
17C22×C4×D4C2212yes—
18C22×C4×D4C2214yes—
19C23×C42C2211yescharacteristic
20#2445—216no—
21C2×#394C2214yes—
22C2×#394C2214yes—
23C2×#394C2214yes—
24C2×#392C2214yes—
25C4×#121—216no—
26C2×#390C2212yes—
27C2×#390C2212yes—
28C2×#390C2212yes—
29C2×#390C2212yes—
30C2×#389C2212yes—
31C2×#389C2212yes—
32C2×#389C2212yes—
33C2×#388C2214yes—
34C2×#388C2214yes—
35C2×#386C2214yes—
36C2×#386C2214yes—
37C2×#385C2212yes—
38C2×#382C2211yescharacteristic
39C2×#380C2212yes—
40C2×C4×C4⋊C4C2212yes—
41C2×C4×C4⋊C4C2212yes—
42C2×C4×C22⋊C4C2214yes—
43C2×C4×C22⋊C4C2211yescharacteristic
44C2×C4×C22⋊C4C2211yescharacteristic
45#1445C2214yes—
46#1444C2214yes—
47#1441C2214yes—
48#1440C2214yes—
49#1440C2214yes—
50#1439C2214yes—
51#1439C2214yes—
52#1439C2214yes—
53#1439C2214yes—
54#1439C2214yes—
55#1438C2214yes—
56#1438C2214yes—
57#1437C2214yes—
58#1436C2212yes—
59#1436C2214yes—
60#1436C2212yes—
61C4⋊#95C2214yes—
62C4⋊#95C2211yescharacteristic
63C4⋊#95C2212yes—
64C4⋊#95C2211yescharacteristic
65C22⋊#95C2214yes—
66C22⋊#95C2214yes—
67C22⋊#95C2212yes—
68C4×#95C2212yes—
69C23×D4C2312yes—
70C23×D4C2314yes—
71C23×D4C2314yes—
72C24×C4C2311yescharacteristic
73C24×C4C2312yes—
74C24×C4C2314yes—
75C2×#127—28no—
76C2×#127—42no—
77C2×#123—216no—
78C2×#123—216no—
79C2×#123—48no—
80C2×#121—28no—
81C2×#121—28no—
82C2×#121—28no—
83C2×C4×D4—28no—
84C2×C4×D4—216no—
85C2×C4×D4—28no—
86C2×C4×D4—44no—
87C2×C4×D4—48no—
88C2×C4×D4—48no—
89C22×C4⋊C4C2312yes—
90C22×C4⋊C4C2312yes—
91C22×C4⋊C4C2314yes—
92C22×C4⋊C4C2312yes—
93C22×C4⋊C4C2312yes—
94C22×C4⋊C4C2312yes—
95C22×C4⋊C4C2314yes—
96C22×C4⋊C4C2314yes—
97C22×C22⋊C4C2311yescharacteristic
98C22×C22⋊C4C2314yes—
99C22×C22⋊C4C2311yescharacteristic
100C22×C22⋊C4C2312yes—
101C22×C22⋊C4C2314yes—
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×C42—28no—
112C22×C42—42no—
113C22×C42C2311yescharacteristic
114C22×C42C2311yescharacteristic
115C22×C42C2312yes—
116C22×C42C2311yescharacteristic
117C22×C42C2311yescharacteristic
118C22×C42C2314yes—
119C22×C42C2312yes—
120#394—216no—
121#394—216no—
122#394—48no—
123#392—48no—
124#390—28no—
125#390—28no—
126#390—28no—
127#390D4116yescomplemented
128#389—28no—
129#389—28no—
130#389—28no—
131#388—216no—
132#388—48no—
133#386—216no—
134#386—216no—
135#385—28no—
136#382—42no—
137#380—28no—
138C4×C4⋊C4—28no—
139C4×C4⋊C4—28no—
140C4×C22⋊C4—24no—
141C4×C22⋊C4—216no—
142C4×C22⋊C4—42no—
143C2×#95C2312yes—
144C2×#95C2314yes—
145C2×#95C2312yes—
146C2×#95C2312yes—
147C2×#95C2312yes—
148C2×#95C2314yes—
149C2×#95C2312yes—
150C2×#95C2312yes—
151C2×#95C2314yes—
152C2×#95C2314yes—
153C2×#95C2314yes—
154C2×#95C2314yes—
155C2×#95C2314yes—
156C2×#95C2312yes—
157C2×#95C2314yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2312yes—
161C25C2414yes—
162C25C2411yescharacteristic
163C22×D4—28no—
164C22×D4—28no—
165C22×D4—28no—
166C22×D4—28no—
167C22×D4—28no—
168C22×D4—28no—
169C22×D4—24no—
170C22×D4—28no—
171C22×D4—24no—
172C22×D4—24no—
173C22×D4—24no—
174C22×D4—28no—
175C22×D4—24no—
176C22×D4—24no—
177C22×D4—44no—
178C22×D4—44no—
179C22×D4—44no—
180C22×D4—44no—
181C22×D4—44no—
182C22×D4—44no—
183C22×D4C2×D418yes—
184C23×C4—24no—
185C23×C4—24no—
186C23×C4—24no—
187C23×C4—24no—
188C23×C4—24no—
189C23×C4—22no—
190C23×C4—22no—
191C23×C4—28no—
192C23×C4—24no—
193C23×C4—24no—
194C23×C4—28no—
195C23×C4—28no—
196C23×C4—28no—
197C23×C4—42no—
198C23×C4—42no—
199C23×C4—44no—
200C23×C4—44no—
201C23×C4—44no—
202C23×C4#4114yes—
203C23×C4C2412yes—
204C23×C4C2412yes—
205C23×C4C2414yes—
206C23×C4C2414yes—
207C23×C4C2412yes—
208C23×C4C2412yes—
209C23×C4C2411yescharacteristic
210C23×C4C2414yes—
211C23×C4C2412yes—
212C23×C4C2412yes—
213C23×C4C2411yescharacteristic
214#127—416no—
215#123—464no—
216#121—264no—
217C4×D4—432no—
218C4×D4—464no—
219C2×C4⋊C4—24no—
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—28no—
227C2×C4⋊C4—24no—
228C2×C4⋊C4—28no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4—28no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4—24no—
233C2×C4⋊C4—28no—
234C2×C4⋊C4—42no—
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⋊C4#4118yes—
242C2×C4⋊C4#4118yes—
243C2×C22⋊C4—28no—
244C2×C22⋊C4—22no—
245C2×C22⋊C4—28no—
246C2×C22⋊C4—28no—
247C2×C22⋊C4—28no—
248C2×C22⋊C4—24no—
249C2×C22⋊C4—28no—
250C2×C22⋊C4—28no—
251C2×C22⋊C4—28no—
252C2×C22⋊C4—22no—
253C2×C22⋊C4—28no—
254C2×C22⋊C4—24no—
255C2×C22⋊C4—24no—
256C2×C22⋊C4—28no—
257C2×C22⋊C4—28no—
258C2×C22⋊C4—28no—
259C2×C22⋊C4—28no—
260C2×C22⋊C4—28no—
261C2×C22⋊C4—24no—
262C2×C22⋊C4—28no—
263C2×C22⋊C4—28no—
264C2×C22⋊C4—28no—
265C2×C22⋊C4—28no—
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—42no—
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⋊C4C2×D4116yes—
282C2×C22⋊C4C2×D4116yes—
283C2×C42—24no—
284C2×C42—24no—
285C2×C42—28no—
286C2×C42—24no—
287C2×C42—24no—
288C2×C42—24no—
289C2×C42—24no—
290C2×C42—24no—
291C2×C42—24no—
292C2×C42—24no—
293C2×C42—28no—
294C2×C42—22no—
295C2×C42—22no—
296C2×C42—416no—
297C2×C42—44no—
298C2×C42—41no—
299C2×C42—41no—
300C2×C42C2×D418yes—
301#95—28no—
302#95—28no—
303#95—24no—
304#95—28no—
305#95—24no—
306#95—28no—
307#95—24no—
308#95—24no—
309#95—24no—
310#95—28no—
311#95—24no—
312#95—44no—
313#95—42no—
314#95—42no—
315#95—44no—
316#95—44no—
317#95—44no—
318#95—44no—
319C24—22no—
320C24—22no—
321C24—22no—
322C24—21no—
323C24—22no—
324C24—21no—
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—
340C24C22×D418yes—
341C24C2×#4112yes—
342C24H(5,𝔽2)14yes—
343C24C2511yescharacteristic
344C2×D4—232no—
345C2×D4—216no—
346C2×D4—216no—
347C2×D4—216no—
348C2×D4—416no—
349C2×D4—416no—
350C2×D4—416no—
351C2×D4—416no—
352C2×D4—416no—
353C2×D4—416no—
354C2×D4—416no—
355C2×D4—416no—
356C2×D4—416no—
357C2×D4—48no—
358C2×D4—48no—
359C2×D4—48no—
360C2×D4—48no—
361C2×D4—88no—
362C2×D4—88no—
363C2×D4—88no—
364C2×D4—88no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—216no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—24no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—24no—
380C22×C4—22no—
381C22×C4—24no—
382C22×C4—22no—
383C22×C4—24no—
384C22×C4—21no—
385C22×C4—22no—
386C22×C4—24no—
387C22×C4—24no—
388C22×C4—22no—
389C22×C4—24no—
390C22×C4—24no—
391C22×C4—22no—
392C22×C4—24no—
393C22×C4—24no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—24no—
399C22×C4—21no—
400C22×C4—22no—
401C22×C4—24no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—216no—
405C22×C4—28no—
406C22×C4—22no—
407C22×C4—24no—
408C22×C4—24no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—24no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—24no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—24no—
427C22×C4—48no—
428C22×C4—48no—
429C22×C4—48no—
430C22×C4—48no—
431C22×C4—48no—
432C22×C4—416no—
433C22×C4—416no—
434C22×C4—416no—
435C22×C4—44no—
436C22×C4—48no—
437C22×C4—48no—
438C22×C4—44no—
439C22×C4—416no—
440C22×C4—88no—
441C22×C4—88no—
442C22×C4—88no—
443C22×C4C22×D418yes—
444C22×C4C22×D414yes—
445C22×C4C22×D414yes—
446C22×C4C22×D414yes—
447C22×C4C2×#4112yes—
448C22×C4C2×#4114yes—
449C22×C4C2×#4118yes—
450C22×C4C2×#4114yes—
451C22×C4C2×#4114yes—
452C22×C4C2×#4112yes—
453C22×C4C2×#4114yes—
454C22×C4H(5,𝔽2)14yes—
455C22×C4H(5,𝔽2)12yes—
456C22×C42−1+412yes—
457C4⋊C4—216no—
458C4⋊C4—216no—
459C4⋊C4—216no—
460C4⋊C4—48no—
461C4⋊C4—416no—
462C4⋊C4—416no—
463C4⋊C4—48no—
464C4⋊C4—88no—
465C22⋊C4—28no—
466C22⋊C4—232no—
467C22⋊C4—232no—
468C22⋊C4—416no—
469C22⋊C4—416no—
470C22⋊C4—48no—
471C22⋊C4—416no—
472C22⋊C4—416no—
473C22⋊C4—416no—
474C22⋊C4—44no—
475C22⋊C4—88no—
476C22⋊C4—88no—
477C22⋊C4—88no—
478C22⋊C4—88no—
479C42—216no—
480C42—44no—
481C42—44no—
482C42—416no—
483C42—48no—
484C42—44no—
485C42—44no—
486C23—24no—
487C23—22no—
488C23—22no—
489C23—24no—
490C23—24no—
491C23—24no—
492C23—24no—
493C23—24no—
494C23—24no—
495C23—24no—
496C23—28no—
497C23—28no—
498C23—28no—
499C23—28no—
500C23—28no—
501C23—28no—
502C23—28no—
503C23—41no—
504C23—42no—
505C23—42no—
506C23—42no—
507C23—41no—
508C23—42no—
509C23—42no—
510C23—42no—
511C23—41no—
512C23—41no—
513C23—44no—
514C23—44no—
515C23—44no—
516C23—44no—
517C23—44no—
518C23—44no—
519C23—44no—
520C23—44no—
521C23—44no—
522C23—44no—
523C23—44no—
524C23—44no—
525C23—44no—
526C23—44no—
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#54014yes—
542C23C23×D412yes—
543C23C22×#4111yescharacteristic
544C23C22×#4112yes—
545C23C22×#4112yes—
546C23C2×H(5,𝔽2)11yescharacteristic
547C23C2×H(5,𝔽2)12yes—
548C23C2×H(5,𝔽2)12yes—
549C23C2×2−1+411yescharacteristic
550C23#58512yes—
551D4—464no—
552D4—432no—
553D4—832no—
554C2×C4—24no—
555C2×C4—24no—
556C2×C4—24no—
557C2×C4—24no—
558C2×C4—24no—
559C2×C4—24no—
560C2×C4—24no—
561C2×C4—28no—
562C2×C4—24no—
563C2×C4—22no—
564C2×C4—24no—
565C2×C4—28no—
566C2×C4—24no—
567C2×C4—24no—
568C2×C4—28no—
569C2×C4—24no—
570C2×C4—24no—
571C2×C4—24no—
572C2×C4—28no—
573C2×C4—24no—
574C2×C4—28no—
575C2×C4—24no—
576C2×C4—28no—
577C2×C4—432no—
578C2×C4—42no—
579C2×C4—44no—
580C2×C4—42no—
581C2×C4—44no—
582C2×C4—41no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—41no—
588C2×C4—42no—
589C2×C4—44no—
590C2×C4—44no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—44no—
594C2×C4—42no—
595C2×C4—44no—
596C2×C4—42no—
597C2×C4—44no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—44no—
601C2×C4—416no—
602C2×C4—42no—
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—42no—
613C2×C4—42no—
614C2×C4—44no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—44no—
621C2×C4—42no—
622C2×C4—44no—
623C2×C4—832no—
624C2×C4#52518yes—
625C2×C4#53914yes—
626C2×C4D4214yes—
627C2×C4#54618yes—
628C2×C4#54718yes—
629C2×C4#55914yes—
630C22—24no—
631C22—28no—
632C22—42no—
633C22—44no—
634C22—42no—
635C22—44no—
636C22—44no—
637C22—44no—
638C22—44no—
639C22—48no—
640C22—48no—
641C22—48no—
642C22—48no—
643C22—48no—
644C22—48no—
645C22—48no—
646C22—84no—
647C22—84no—
648C22—84no—
649C22—84no—
650C22—84no—
651C22—84no—
652C22—84no—
653C22—84no—
654C22C2×#52512yes—
655C22#344411yescharacteristic
656C22C2×#53911yescharacteristic
657C22C2×#54011yescharacteristic
658C22#346012yes—
659C22C2×D4211yescharacteristic
660C22C2×#54612yes—
661C22C2×#54712yes—
662C22D4×#4112yes—
663C22C2×#55911yescharacteristic
664C22#348012yes—
665C22#348312yes—
666C22#348412yes—
667C22#348812yes—
668C22#351312yes—
669C22#351912yes—
670C22#351912yes—
671C22#352212yes—
672C22#354212yes—
673C22#354811yescharacteristic
674C22#355711yescharacteristic
675C4—28no—
676C4—42no—
677C4—44no—
678C4—44no—
679C4—48no—
680C4—44no—
681C4—48no—
682C4—84no—
683C4—82no—
684C4—82no—
685C4#245518yes—
686C2—44no—
687C2—88no—
688C2C2×#245511yescharacteristic
689C2#3471512yes—
690C2#3508211yescharacteristic
691C2#3522612yes—
692C2#3752011yescharacteristic
693C2#5900412yes—
694C2#5903912yes—
695C2#5907212yes—
696C2#5963312yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radical#15124C2
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×#2445C2semidirect
2. C2×C4×#121C2semidirect
3. #16334C2semidirect
4. #15223C2semidirect
5. #15126C2semidirect
7. #14806C2semidirect
8. #14721C2semidirect
9. #14141C2semidirect
10. #14096C2semidirect
11. #14045C2semidirect
12. #13979C2semidirect
13. C4×#385C2semidirect
127. #390D4semidirect

Maximal subgroups

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

subgroupindex
C2×#24452
C2×C4×#1212
#163342
#152232
#151262
#151242
#148062
#147212
#141412
#140962
#140452
#139792
C4×#3852