Finite Group Databasebeta

← #674815

Subgroups — #674815

Order 512.

3138 conjugacy classes of subgroups.

The subgroup lattice has 750 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2859C211yesmaximal, characteristic, complemented
2C2×#2535C211yesmaximal, characteristic
3C2×#2490C212yesmaximal, complemented
4#15617C214yesmaximal, complemented
5#15532C214yesmaximal, complemented
6#15216C212yesmaximal, complemented
7#15130C212yesmaximal, complemented
8#14802C214yesmaximal, complemented
9#14205C214yesmaximal, complemented
10#13941C211yesmaximal, characteristic, complemented
11#13905C211yesmaximal, characteristic, complemented
12#13876C212yesmaximal, complemented
13#13804C211yesmaximal, characteristic, complemented
14C4×#390C211yesmaximal, characteristic, complemented
15C4×#390C211yesmaximal, characteristic, complemented
16C22×#125C2212yes—
17C22×#121C2212yes—
18C22×C4×D4C2211yescharacteristic
19C22×C4×D4C2211yescharacteristic
20#2859—28no—
21#2535—28no—
22#2490—216no—
23C2×#397C2212yes—
24C2×#397C2214yes—
25C2×#394C2214yes—
26C2×#394C2214yes—
27C2×#394C2214yes—
28C2×#392C2212yes—
29C2×#390C2212yes—
30C2×#390C2214yes—
31C2×#390C2211yescharacteristic
32C2×#390C2212yes—
33C2×#390C2211yescharacteristic
34C2×#389C2214yes—
35C2×#389C2214yes—
36C2×#388C2214yes—
37C2×#388C2214yes—
38C2×#386C2214yes—
39C2×#384C2212yes—
40C2×#382C2211yescharacteristic
41C2×#382C2211yescharacteristic
42C2×#380C2212yes—
43C2×#380C2212yes—
44C2×#380C2212yes—
45C2×C4×C4⋊C4C2212yes—
46C2×C4×C4⋊C4C2211yescharacteristic
47C2×C4×C4⋊C4C2211yescharacteristic
48C2×C4×C22⋊C4C2212yes—
49C2×C4×C22⋊C4C2212yes—
50C2×C4×C22⋊C4C2214yes—
51C2×C43C2211yescharacteristic
52#1445C2212yes—
53#1442C2214yes—
54#1442C2214yes—
55#1442C2214yes—
56#1439C2214yes—
57#1439C2214yes—
58#1439C2214yes—
59#1439C2214yes—
60#1439C2214yes—
61#1439C2214yes—
62#1438C2212yes—
63#1437C2214yes—
64#1437C2214yes—
65#1437C2212yes—
66#1437C2212yes—
67#1436C2214yes—
68#1436C2214yes—
69C4⋊#95C2212yes—
70C4⋊#95C2211yescharacteristic
71C4⋊#95C2212yes—
72C4⋊#95C2211yescharacteristic
73C4⋊#95C2212yes—
74#1434C2212yes—
75C22⋊#95C2212yes—
76C22⋊#95C2212yes—
77C4×#95C2212yes—
78C23×D4C2311yescharacteristic
79C23×D4C2311yescharacteristic
80C23×D4C2314yes—
81C24×C4C2312yes—
82C24×C4C2312yes—
83C2×#125—28no—
84C2×#125—28no—
85C2×#125—44no—
86C2×#121—28no—
87C2×#121—28no—
88C2×#121—44no—
89C2×C4×D4—24no—
90C2×C4×D4—24no—
91C2×C4×D4—24no—
92C2×C4×D4—24no—
93C2×C4×D4—24no—
94C2×C4×D4—24no—
95C22×C4⋊C4C2311yescharacteristic
96C22×C4⋊C4C2311yescharacteristic
97C22×C4⋊C4C2312yes—
98C22×C4⋊C4C2311yescharacteristic
99C22×C4⋊C4C2314yes—
100C22×C4⋊C4C2314yes—
101C22×C4⋊C4C2314yes—
102C22×C4⋊C4C2314yes—
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2311yescharacteristic
106C22×C22⋊C4C2312yes—
107C22×C22⋊C4C2312yes—
108C22×C22⋊C4C2314yes—
109C22×C22⋊C4C2312yes—
110C22×C22⋊C4C2312yes—
111C22×C22⋊C4C2314yes—
112C22×C22⋊C4C2314yes—
113C22×C22⋊C4C2312yes—
114C22×C22⋊C4C2312yes—
115C22×C22⋊C4C2314yes—
116C22×C22⋊C4C2314yes—
117C22×C22⋊C4C2314yes—
118C22×C22⋊C4C2314yes—
119C22×C22⋊C4C2314yes—
120C22×C42C2312yes—
121C22×C42C2312yes—
122C22×C42C2312yes—
123C22×C42C2311yescharacteristic
124C22×C42C2311yescharacteristic
125C22×C42C2312yes—
126C22×C42C2314yes—
127C22×C42C2311yescharacteristic
128#397—216no—
129#397—28no—
130#394—216no—
131#394—216no—
132#394—48no—
133#392—28no—
134#390—28no—
135#390—216no—
136#390—28no—
137#390—24no—
138#390D418yescomplemented
139#389—216no—
140#389—216no—
141#388—216no—
142#388—48no—
143#386—216no—
144#384—28no—
145#382—24no—
146#382D418yes—
147#380—28no—
148#380—28no—
149#380—44no—
150C4×C4⋊C4—28no—
151C4×C4⋊C4—24no—
152C4×C4⋊C4—24no—
153C4×C22⋊C4—28no—
154C4×C22⋊C4—28no—
155C4×C22⋊C4—48no—
156C2×#95C2314yes—
157C2×#95C2314yes—
158C2×#95C2312yes—
159C2×#95C2314yes—
160C2×#95C2314yes—
161C2×#95C2311yescharacteristic
162C2×#95C2311yescharacteristic
163C2×#95C2312yes—
164C2×#95C2312yes—
165C2×#95C2314yes—
166C2×#95C2312yes—
167C2×#95C2312yes—
168C2×#95C2312yes—
169C2×#95C2312yes—
170C2×#95C2314yes—
171C2×#95C2314yes—
172C2×#95C2314yes—
173C2×#95C2314yes—
174C2×#95C2314yes—
175C2×#95C2312yes—
176C2×#95C2312yes—
177C43—24no—
178C25C2412yes—
179C25C2412yes—
180C22×D4—24no—
181C22×D4—28no—
182C22×D4—28no—
183C22×D4—22no—
184C22×D4—22no—
185C22×D4—28no—
186C22×D4—22no—
187C22×D4—22no—
188C22×D4—22no—
189C22×D4—22no—
190C22×D4—22no—
191C22×D4—22no—
192C22×D4—22no—
193C22×D4—42no—
194C22×D4—44no—
195C22×D4—44no—
196C22×D4—44no—
197C22×D4—44no—
198C22×D4C2×D414yes—
199C23×C4—24no—
200C23×C4—24no—
201C23×C4—24no—
202C23×C4—24no—
203C23×C4—24no—
204C23×C4—24no—
205C23×C4—24no—
206C23×C4—24no—
207C23×C4—24no—
208C23×C4—24no—
209C23×C4—44no—
210C23×C4—44no—
211C23×C4C2414yes—
212C23×C4C2412yes—
213C23×C4C2412yes—
214C23×C4C2412yes—
215C23×C4C2412yes—
216C23×C4C2412yes—
217C23×C4C2412yes—
218C23×C4C2414yes—
219C23×C4C2412yes—
220C23×C4C2411yescharacteristic
221C23×C4C2411yescharacteristic
222C23×C4C2412yes—
223C23×C4C2411yescharacteristic
224#125—432no—
225#121—432no—
226C4×D4—416no—
227C4×D4—416no—
228C2×C4⋊C4—24no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—24no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—28no—
237C2×C4⋊C4—28no—
238C2×C4⋊C4—28no—
239C2×C4⋊C4—28no—
240C2×C4⋊C4—28no—
241C2×C4⋊C4—28no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—28no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—28no—
254C2×C4⋊C4—44no—
255C2×C4⋊C4—44no—
256C2×C4⋊C4—42no—
257C2×C4⋊C4—44no—
258C2×C4⋊C4—44no—
259C2×C4⋊C4C2×D418yes—
260C2×C4⋊C4C2×D414yes—
261C2×C22⋊C4—24no—
262C2×C22⋊C4—24no—
263C2×C22⋊C4—28no—
264C2×C22⋊C4—28no—
265C2×C22⋊C4—24no—
266C2×C22⋊C4—24no—
267C2×C22⋊C4—24no—
268C2×C22⋊C4—28no—
269C2×C22⋊C4—24no—
270C2×C22⋊C4—24no—
271C2×C22⋊C4—24no—
272C2×C22⋊C4—24no—
273C2×C22⋊C4—24no—
274C2×C22⋊C4—24no—
275C2×C22⋊C4—24no—
276C2×C22⋊C4—24no—
277C2×C22⋊C4—28no—
278C2×C22⋊C4—24no—
279C2×C22⋊C4—28no—
280C2×C22⋊C4—28no—
281C2×C22⋊C4—28no—
282C2×C22⋊C4—24no—
283C2×C22⋊C4—28no—
284C2×C22⋊C4—28no—
285C2×C22⋊C4—28no—
286C2×C22⋊C4—28no—
287C2×C22⋊C4—44no—
288C2×C22⋊C4—42no—
289C2×C22⋊C4—42no—
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×C22⋊C4C2×D4116yes—
302C2×C22⋊C4#4118yes—
303C2×C42—22no—
304C2×C42—22no—
305C2×C42—22no—
306C2×C42—22no—
307C2×C42—24no—
308C2×C42—24no—
309C2×C42—24no—
310C2×C42—24no—
311C2×C42—24no—
312C2×C42—24no—
313C2×C42—24no—
314C2×C42—28no—
315C2×C42—24no—
316C2×C42—42no—
317C2×C42—42no—
318C2×C42—44no—
319C2×C42—44no—
320C2×C42—44no—
321C2×C42C2×D414yes—
322C2×C42C2×D414yes—
323C2×C42#4114yes—
324C2×C42#4114yes—
325C2×C42#4114yes—
326#95—24no—
327#95—24no—
328#95—28no—
329#95—28no—
330#95—28no—
331#95—24no—
332#95—24no—
333#95—24no—
334#95—24no—
335#95—22no—
336#95—24no—
337#95—22no—
338#95—28no—
339#95—28no—
340#95—44no—
341#95—44no—
342#95—44no—
343#95—44no—
344#95—44no—
345#95C2×D418yes—
346#95C2×D418yes—
347C24—22no—
348C24—22no—
349C24—22no—
350C24—22no—
351C24—22no—
352C24—22no—
353C24—24no—
354C24—22no—
355C24—22no—
356C24—22no—
357C24—22no—
358C24—22no—
359C24—24no—
360C24—22no—
361C24—22no—
362C24—22no—
363C24—22no—
364C24—22no—
365C24—22no—
366C24—22no—
367C24—24no—
368C24—24no—
369C24—22no—
370C24—22no—
371C24C22×D414yes—
372C24C2×#4114yes—
373C24C2511yescharacteristic
374C2×D4—28no—
375C2×D4—28no—
376C2×D4—28no—
377C2×D4—28no—
378C2×D4—48no—
379C2×D4—48no—
380C2×D4—44no—
381C2×D4—416no—
382C2×D4—416no—
383C2×D4—416no—
384C2×D4—44no—
385C2×D4—44no—
386C2×D4—44no—
387C2×D4—44no—
388C2×D4—44no—
389C2×D4—88no—
390C2×D4—88no—
391C2×D4—88no—
392C2×D4—88no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—24no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—24no—
408C22×C4—22no—
409C22×C4—24no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—24no—
413C22×C4—22no—
414C22×C4—24no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—24no—
419C22×C4—22no—
420C22×C4—24no—
421C22×C4—22no—
422C22×C4—24no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—24no—
428C22×C4—24no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—24no—
434C22×C4—24no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—24no—
442C22×C4—24no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—24no—
447C22×C4—24no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—24no—
451C22×C4—24no—
452C22×C4—24no—
453C22×C4—22no—
454C22×C4—416no—
455C22×C4—48no—
456C22×C4—48no—
457C22×C4—48no—
458C22×C4—48no—
459C22×C4—48no—
460C22×C4—48no—
461C22×C4—48no—
462C22×C4—48no—
463C22×C4—88no—
464C22×C4—84no—
465C22×C4—84no—
466C22×C4C22×D414yes—
467C22×C4C22×D414yes—
468C22×C4C22×D418yes—
469C22×C4C22×D412yes—
470C22×C4C22×D412yes—
471C22×C4C22×D412yes—
472C22×C4C22×D412yes—
473C22×C4C2×#4114yes—
474C22×C4C2×#4112yes—
475C22×C4C2×#4112yes—
476C22×C4C2×#4112yes—
477C22×C4C2×#4112yes—
478C22×C4C2×#4114yes—
479C22×C4C2×#4114yes—
480C22×C4C2×#4112yes—
481C22×C4C2×#4112yes—
482C22×C4C2×#4112yes—
483C22×C4H(5,𝔽2)14yes—
484C22×C4H(5,𝔽2)12yes—
485C22×C42−1+412yes—
486C4⋊C4—216no—
487C4⋊C4—28no—
488C4⋊C4—416no—
489C4⋊C4—48no—
490C4⋊C4—416no—
491C4⋊C4—416no—
492C4⋊C4—416no—
493C4⋊C4—44no—
494C4⋊C4—44no—
495C4⋊C4—48no—
496C4⋊C4—44no—
497C22⋊C4—216no—
498C22⋊C4—416no—
499C22⋊C4—48no—
500C22⋊C4—416no—
501C22⋊C4—48no—
502C22⋊C4—416no—
503C22⋊C4—48no—
504C22⋊C4—48no—
505C22⋊C4—48no—
506C22⋊C4—416no—
507C22⋊C4—88no—
508C22⋊C4—88no—
509C22⋊C4—88no—
510C22⋊C4—88no—
511C42—28no—
512C42—28no—
513C42—48no—
514C42—48no—
515C42—48no—
516C42—84no—
517C42—88no—
518C42#124116yes—
519C23—24no—
520C23—24no—
521C23—24no—
522C23—24no—
523C23—24no—
524C23—28no—
525C23—24no—
526C23—24no—
527C23—24no—
528C23—28no—
529C23—24no—
530C23—24no—
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—42no—
544C23—44no—
545C23—42no—
546C23—42no—
547C23—44no—
548C23—42no—
549C23—42no—
550C23—42no—
551C23—44no—
552C23—42no—
553C23—44no—
554C23—42no—
555C23—42no—
556C23—42no—
557C23—44no—
558C23—42no—
559C23—42no—
560C23—42no—
561C23—44no—
562C23—42no—
563C23—42no—
564C23—42no—
565C23—42no—
566C23—42no—
567C23—42no—
568C23—44no—
569C23—42no—
570C23—44no—
571C23—42no—
572C23—42no—
573C23—44no—
574C23—42no—
575C23—44no—
576C23—42no—
577C23C23×D411yescharacteristic
578C23C23×D411yescharacteristic
579C23C22×#4111yescharacteristic
580C23C22×#4111yescharacteristic
581C23C22×#4111yescharacteristic
582C23C22×#4112yes—
583C23C2×H(5,𝔽2)11yescharacteristic
584C23C2×H(5,𝔽2)12yes—
585C23C2×2−1+411yescharacteristic
586C23#58512yes—
587C23#58512yes—
588D4—416no—
589D4—416no—
590D4—832no—
591C2×C4—22no—
592C2×C4—22no—
593C2×C4—24no—
594C2×C4—24no—
595C2×C4—24no—
596C2×C4—24no—
597C2×C4—24no—
598C2×C4—24no—
599C2×C4—24no—
600C2×C4—22no—
601C2×C4—22no—
602C2×C4—24no—
603C2×C4—24no—
604C2×C4—24no—
605C2×C4—24no—
606C2×C4—28no—
607C2×C4—24no—
608C2×C4—24no—
609C2×C4—24no—
610C2×C4—28no—
611C2×C4—24no—
612C2×C4—24no—
613C2×C4—28no—
614C2×C4—28no—
615C2×C4—24no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—44no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—44no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—44no—
627C2×C4—42no—
628C2×C4—44no—
629C2×C4—42no—
630C2×C4—44no—
631C2×C4—42no—
632C2×C4—42no—
633C2×C4—42no—
634C2×C4—42no—
635C2×C4—42no—
636C2×C4—42no—
637C2×C4—42no—
638C2×C4—44no—
639C2×C4—42no—
640C2×C4—42no—
641C2×C4—42no—
642C2×C4—44no—
643C2×C4—44no—
644C2×C4—44no—
645C2×C4—42no—
646C2×C4—42no—
647C2×C4—42no—
648C2×C4—42no—
649C2×C4—44no—
650C2×C4—44no—
651C2×C4—42no—
652C2×C4—44no—
653C2×C4—42no—
654C2×C4—44no—
655C2×C4—42no—
656C2×C4—44no—
657C2×C4—44no—
658C2×C4—42no—
659C2×C4—42no—
660C2×C4—42no—
661C2×C4—44no—
662C2×C4—816no—
663C2×C4—816no—
664C2×C4C2×#12414yes—
665C2×C4C2×#12414yes—
666C2×C4C2×#12414yes—
667C2×C4#53214yes—
668C2×C4#54214yes—
669C2×C4D4214yes—
670C2×C4#54714yes—
671C2×C4#54814yes—
672C2×C4#55318yes—
673C2×C4#55514yes—
674C22—44no—
675C22—44no—
676C22—44no—
677C22—44no—
678C22—44no—
679C22—48no—
680C22—44no—
681C22—44no—
682C22—44no—
683C22—44no—
684C22—48no—
685C22—44no—
686C22—82no—
687C22—84no—
688C22—82no—
689C22—84no—
690C22—82no—
691C22—82no—
692C22—82no—
693C22—82no—
694C22—82no—
695C22—82no—
696C22—82no—
697C22—82no—
698C22—82no—
699C22—82no—
700C22C22×#12411yescharacteristic
701C22C2×#53211yescharacteristic
702C22#344411yescharacteristic
703C22C2×#54211yescharacteristic
704C22#345511yescharacteristic
705C22C2×D4211yescharacteristic
706C22C2×#54711yescharacteristic
707C22C2×#54811yescharacteristic
708C22D4×#4112yes—
709C22D4×#4112yes—
710C22C2×#55312yes—
711C22C2×#55511yescharacteristic
712C22#347512yes—
713C22#348812yes—
714C22#349612yes—
715C22#350612yes—
716C22#350812yes—
717C22#350811yescharacteristic
718C22#351011yescharacteristic
719C22#351212yes—
720C22#351512yes—
721C22#355912yes—
722C22#356212yes—
723C4—28no—
724C4—24no—
725C4—24no—
726C4—44no—
727C4—44no—
728C4—44no—
729C4—48no—
730C4—82no—
731C4—82no—
732C4—82no—
733C4—82no—
734C4—84no—
735C4#249118yes—
736C2—84no—
737C2—84no—
738C2C2×#249111yescharacteristic
739C2#3481311yescharacteristic
740C2#3555611yescharacteristic
741C2#3613511yescharacteristic
742C2#3621111yescharacteristic
743C2#3646812yes—
744C2#3937211yescharacteristic
745C2#3950211yescharacteristic
746C2#4000312yes—
747C2#5912912yes—
748C2#5913912yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC2×#2535C2
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×#2859C2semidirect
3. C2×#2490C2semidirect
4. #15617C2semidirect
5. #15532C2semidirect
6. #15216C2semidirect
7. #15130C2semidirect
8. #14802C2semidirect
9. #14205C2semidirect
10. #13941C2semidirect
11. #13905C2semidirect
12. #13876C2semidirect
13. #13804C2semidirect
14. C4×#390C2semidirect
15. C4×#390C2semidirect
138. #390D4semidirect

Maximal subgroups

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

subgroupindex
C2×#28592
C2×#25352
C2×#24902
#156172
#155322
#152162
#151302
#148022
#142052
#139412
#139052
#138762
#138042
C4×#3902