Finite Group Databasebeta

← #4306634

Subgroups — #4306634

Order 512.

2574 conjugacy classes of subgroups.

The subgroup lattice has 688 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2369C211yesmaximal, characteristic, complemented
2C2×#2344C211yesmaximal, characteristic
3#15230C211yesmaximal, characteristic, complemented
4#15220C212yesmaximal
5#15194C212yesmaximal, complemented
6#15156C211yesmaximal, characteristic, complemented
7#15125C211yesmaximal, characteristic
8#15102C212yesmaximal, complemented
9#14958C214yesmaximal, complemented
10#14893C212yesmaximal
11#14869C212yesmaximal, complemented
12#14768C214yesmaximal
13C4×#399C211yesmaximal, characteristic
14#13852C212yesmaximal
15#13821C211yesmaximal, characteristic, complemented
16#13804C212yesmaximal, complemented
17C22⋊C4×C4⋊C4C212yesmaximal
18C22×C4⋊Q8C2211yescharacteristic
19C22×#117C2212yes—
20C23×C4⋊C4C2211yescharacteristic
21C2×#400C2212yes—
22C2×#399C2211yescharacteristic
23C2×#393C2212yes—
24#2369—28no—
25#2344—28no—
26C2×#391C2214yes—
27C2×#389C2212yes—
28C2×#389C2211yescharacteristic
29C2×#389C2211yescharacteristic
30C2×#389C2212yes—
31C2×#389C2212yes—
32C2×#387C2212yes—
33C2×#387C2212yes—
34C2×#382C2212yes—
35C2×#381C2212yes—
36C2×#380C2212yes—
37C2×#380C2211yescharacteristic
38C2×C4×C4⋊C4C2212yes—
39C2×C4×C4⋊C4C2211yescharacteristic
40C2×C4×C4⋊C4C2212yes—
41C2×C4×C4⋊C4C2211yescharacteristic
42C2×C4×C4⋊C4C2212yes—
43C2×C4×C4⋊C4C2212yes—
44C2×C4×C4⋊C4C2212yes—
45C2×C4×C4⋊C4C2211yescharacteristic
46C2×C4×C22⋊C4C2211yescharacteristic
47C2×C4×C22⋊C4C2212yes—
48C2×C4×C22⋊C4C2214yes—
49#1446C2212yes—
50#1446C2214yes—
51#1445C2211yescharacteristic
52#1445C2212yes—
53#1445C2214yes—
54#1444C2211yescharacteristic
55#1444C2214yes—
56#1443C2214yes—
57#1443C2214yes—
58#1442C2212yes—
59#1441C2212yes—
60#1441C2214yes—
61#1441C2214yes—
62#1441C2212yes—
63#1441C2214yes—
64#1441C2212yes—
65#1439C2212yes—
66#1439C2211yescharacteristic
67#1439C2214yes—
68#1439C2212yes—
69#1439C2214yes—
70#1436C2214yes—
71#1436C2214yes—
72#1436C2214yes—
73#1436C2214yes—
74#1436C2214yes—
75C4⋊#95C2211yescharacteristic
76C4⋊#95C2212yes—
77C4⋊#95C2212yes—
78C4⋊#95C2212yes—
79C4⋊#95C2212yes—
80#1434C2214yes—
81#1434C2212yes—
82C22⋊#95C2214yes—
83C4×#95C2214yes—
84C4×#95C2211yescharacteristic
85C23×Q8C2312yes—
86C24×C4C2311yescharacteristic
87C24×C4C2312yes—
88C2×C4⋊Q8—28no—
89C2×C4⋊Q8—24no—
90C2×#117—28no—
91C2×#117—28no—
92C2×#117—44no—
93C22×C4⋊C4—24no—
94C22×C4⋊C4—24no—
95C22×C4⋊C4Q818yescomplemented
96C22×C4⋊C4C2311yescharacteristic
97C22×C4⋊C4C2312yes—
98C22×C4⋊C4C2312yes—
99C22×C4⋊C4C2312yes—
100C22×C4⋊C4C2311yescharacteristic
101C22×C4⋊C4C2311yescharacteristic
102C22×C4⋊C4C2312yes—
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2311yescharacteristic
106C22×C4⋊C4C2311yescharacteristic
107C22×C4⋊C4C2312yes—
108C22×C4⋊C4C2312yes—
109C22×C4⋊C4C2314yes—
110C22×C4⋊C4C2314yes—
111C22×C4⋊C4C2311yescharacteristic
112C22×C22⋊C4C2312yes—
113C22×C22⋊C4C2311yescharacteristic
114C22×C22⋊C4C2314yes—
115C22×C22⋊C4C2312yes—
116C22×C22⋊C4C2312yes—
117C22×C22⋊C4C2311yescharacteristic
118C22×C42C2311yescharacteristic
119C22×C42C2312yes—
120C22×C42C2312yes—
121C22×C42C2312yes—
122C22×C42C2311yescharacteristic
123C22×C42C2312yes—
124C22×C42C2314yes—
125C22×C42C2314yes—
126#400—44no—
127#399—24no—
128#393—44no—
129#391—216no—
130#389—24no—
131#389—28no—
132#389—24no—
133#389—28no—
134#389—28no—
135#387—28no—
136#387—28no—
137#382—28no—
138#381—28no—
139#380—24no—
140#380—28no—
141C4×C4⋊C4—24no—
142C4×C4⋊C4—28no—
143C4×C4⋊C4—28no—
144C4×C4⋊C4—28no—
145C4×C4⋊C4—24no—
146C4×C4⋊C4—28no—
147C4×C4⋊C4—44no—
148C4×C4⋊C4D418yes—
149C4×C22⋊C4—24no—
150C4×C22⋊C4—28no—
151C4×C22⋊C4—48no—
152C2×#95C2311yescharacteristic
153C2×#95C2312yes—
154C2×#95C2312yes—
155C2×#95C2314yes—
156C2×#95C2314yes—
157C2×#95C2311yescharacteristic
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2312yes—
161C2×#95C2312yes—
162C2×#95C2314yes—
163C2×#95C2314yes—
164C2×#95C2312yes—
165C2×#95C2314yes—
166C2×#95C2312yes—
167C2×#95C2314yes—
168C2×#95C2312yes—
169C2×#95C2312yes—
170C2×#95C2312yes—
171C2×#95C2311yescharacteristic
172C2×#95C2312yes—
173C2×#95C2311yescharacteristic
174C2×#95C2314yes—
175C2×#95C2312yes—
176C2×#95C2314yes—
177C2×#95C2314yes—
178C2×#95C2314yes—
179C2×#95C2312yes—
180C2×#95C2314yes—
181C2×#95C2314yes—
182C2×#95C2314yes—
183C2×#95C2312yes—
184C25C2411yescharacteristic
185C22×Q8—24no—
186C22×Q8—28no—
187C22×Q8—42no—
188C22×Q8—44no—
189C22×Q8—42no—
190C23×C4—24no—
191C23×C4—24no—
192C23×C4—24no—
193C23×C4—22no—
194C23×C4—24no—
195C23×C4—22no—
196C23×C4—24no—
197C23×C4—24no—
198C23×C4—42no—
199C23×C4—42no—
200C23×C4C2×Q814yes—
201C23×C4C2×Q818yes—
202C23×C4C2412yes—
203C23×C4C2411yescharacteristic
204C23×C4C2412yes—
205C23×C4C2414yes—
206C23×C4C2412yes—
207C23×C4C2411yescharacteristic
208C23×C4C2412yes—
209C23×C4C2411yescharacteristic
210C23×C4C2411yescharacteristic
211C23×C4C2414yes—
212C23×C4C2412yes—
213C23×C4C2412yes—
214C23×C4C2412yes—
215C23×C4C2412yes—
216C23×C4C2411yescharacteristic
217C23×C4C2411yescharacteristic
218C4⋊Q8—232no—
219#117—432no—
220C2×C4⋊C4—24no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—24no—
223C2×C4⋊C4—24no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—24no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—28no—
228C2×C4⋊C4—24no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4—24no—
231C2×C4⋊C4—232no—
232C2×C4⋊C4—24no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—24no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—28no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—28no—
251C2×C4⋊C4—42no—
252C2×C4⋊C4—42no—
253C2×C4⋊C4—44no—
254C2×C4⋊C4—42no—
255C2×C4⋊C4—44no—
256C2×C4⋊C4—44no—
257C2×C4⋊C4—44no—
258C2×C4⋊C4C2×D414yes—
259C2×C4⋊C4C2×D414yes—
260C2×C4⋊C4C2×Q818yes—
261C2×C4⋊C4C2×Q814yes—
262C2×C4⋊C4C2×Q814yes—
263C2×C4⋊C4#4118yes—
264C2×C4⋊C4#4114yes—
265C2×C22⋊C4—24no—
266C2×C22⋊C4—22no—
267C2×C22⋊C4—24no—
268C2×C22⋊C4—22no—
269C2×C22⋊C4—24no—
270C2×C22⋊C4—24no—
271C2×C22⋊C4—24no—
272C2×C22⋊C4—28no—
273C2×C22⋊C4—22no—
274C2×C22⋊C4—24no—
275C2×C22⋊C4—22no—
276C2×C22⋊C4—24no—
277C2×C22⋊C4—28no—
278C2×C22⋊C4—42no—
279C2×C22⋊C4—44no—
280C2×C22⋊C4—41no—
281C2×C22⋊C4—42no—
282C2×C22⋊C4#4114yes—
283C2×C42—24no—
284C2×C42—24no—
285C2×C42—24no—
286C2×C42—24no—
287C2×C42—24no—
288C2×C42—24no—
289C2×C42—22no—
290C2×C42—24no—
291C2×C42—24no—
292C2×C42—24no—
293C2×C42—24no—
294C2×C42—24no—
295C2×C42—28no—
296C2×C42—28no—
297C2×C42—42no—
298C2×C42—44no—
299C2×C42—44no—
300C2×C42—44no—
301C2×C42—44no—
302C2×C42C2×D418yes—
303C2×C42C2×D414yes—
304C2×C42#4118yes—
305#95—22no—
306#95—22no—
307#95—24no—
308#95—28no—
309#95—28no—
310#95—24no—
311#95—24no—
312#95—24no—
313#95—24no—
314#95—24no—
315#95—24no—
316#95—28no—
317#95—24no—
318#95—24no—
319#95—28no—
320#95—28no—
321#95—28no—
322#95—28no—
323#95—22no—
324#95—22no—
325#95—24no—
326#95—28no—
327#95—44no—
328#95—44no—
329#95—42no—
330#95—44no—
331#95—44no—
332#95—44no—
333#95—42no—
334#95—44no—
335#95—44no—
336#95C2×D418yes—
337C24—21no—
338C24—22no—
339C24—22no—
340C24—21no—
341C24—21no—
342C24—22no—
343C24—22no—
344C24—21no—
345C24C22×Q812yes—
346C24C2×#4112yes—
347C24H(5,𝔽2)12yes—
348C24C2511yescharacteristic
349C2×Q8—216no—
350C2×Q8—84no—
351C2×Q8—84no—
352C2×Q8—88no—
353C2×Q8—88no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—216no—
361C22×C4—21no—
362C22×C4—22no—
363C22×C4—24no—
364C22×C4—22no—
365C22×C4—21no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—24no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—21no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—24no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—24no—
388C22×C4—21no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—21no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—24no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—24no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—24no—
405C22×C4—21no—
406C22×C4—24no—
407C22×C4—216no—
408C22×C4—28no—
409C22×C4—216no—
410C22×C4—21no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—24no—
414C22×C4—22no—
415C22×C4—24no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—24no—
423C22×C4—21no—
424C22×C4—22no—
425C22×C4—24no—
426C22×C4—22no—
427C22×C4—24no—
428C22×C4—24no—
429C22×C4—22no—
430C22×C4—24no—
431C22×C4—22no—
432C22×C4—21no—
433C22×C4—22no—
434C22×C4—48no—
435C22×C4—48no—
436C22×C4—48no—
437C22×C4—48no—
438C22×C4—48no—
439C22×C4—44no—
440C22×C4—48no—
441C22×C4—44no—
442C22×C4C22×D412yes—
443C22×C4C22×D414yes—
444C22×C4C22×D414yes—
445C22×C4C22×D412yes—
446C22×C4C22×D412yes—
447C22×C4C22×Q812yes—
448C22×C4C22×Q814yes—
449C22×C4C22×Q814yes—
450C22×C4C22×Q812yes—
451C22×C4C2×#4114yes—
452C22×C4C2×#4114yes—
453C22×C4C2×#4118yes—
454C22×C4C2×#4112yes—
455C22×C4C2×#4114yes—
456C22×C4C2×#4112yes—
457C22×C4C2×#4112yes—
458C22×C4C2×#4114yes—
459C22×C4C2×#4114yes—
460C22×C4H(5,𝔽2)12yes—
461C22×C42−1+414yes—
462C4⋊C4—28no—
463C4⋊C4—216no—
464C4⋊C4—216no—
465C4⋊C4—28no—
466C4⋊C4—216no—
467C4⋊C4—28no—
468C4⋊C4—48no—
469C4⋊C4—48no—
470C4⋊C4—48no—
471C4⋊C4—416no—
472C4⋊C4—44no—
473C4⋊C4—48no—
474C4⋊C4—44no—
475C4⋊C4—48no—
476C4⋊C4—88no—
477C4⋊C4C22⋊Q8116yes—
478C22⋊C4—216no—
479C22⋊C4—28no—
480C22⋊C4—48no—
481C22⋊C4—44no—
482C22⋊C4—416no—
483C22⋊C4—48no—
484C42—28no—
485C42—216no—
486C42—216no—
487C42—48no—
488C42—44no—
489C42—48no—
490C42—88no—
491C42—88no—
492C23—22no—
493C23—24no—
494C23—22no—
495C23—24no—
496C23—22no—
497C23—24no—
498C23—22no—
499C23—24no—
500C23—22no—
501C23—24no—
502C23—24no—
503C23—22no—
504C23—41no—
505C23—42no—
506C23—41no—
507C23—41no—
508C23—42no—
509C23—41no—
510C23—42no—
511C23—42no—
512C23—42no—
513C23—42no—
514C23#55414yes—
515C23C23×D411yescharacteristic
516C23C23×Q811yescharacteristic
517C23C22×#4111yescharacteristic
518C23C22×#4112yes—
519C23C22×#4112yes—
520C23C22×#4111yescharacteristic
521C23C2×H(5,𝔽2)11yescharacteristic
522C23C2×H(5,𝔽2)11yescharacteristic
523C23C2×2−1+412yes—
524C23#58511yescharacteristic
525C23#58512yes—
526Q8—816no—
527C2×C4—22no—
528C2×C4—24no—
529C2×C4—24no—
530C2×C4—24no—
531C2×C4—24no—
532C2×C4—22no—
533C2×C4—24no—
534C2×C4—24no—
535C2×C4—24no—
536C2×C4—24no—
537C2×C4—24no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—28no—
541C2×C4—24no—
542C2×C4—24no—
543C2×C4—24no—
544C2×C4—24no—
545C2×C4—24no—
546C2×C4—22no—
547C2×C4—24no—
548C2×C4—24no—
549C2×C4—24no—
550C2×C4—24no—
551C2×C4—22no—
552C2×C4—24no—
553C2×C4—24no—
554C2×C4—24no—
555C2×C4—24no—
556C2×C4—24no—
557C2×C4—22no—
558C2×C4—24no—
559C2×C4—24no—
560C2×C4—28no—
561C2×C4—28no—
562C2×C4—432no—
563C2×C4—41no—
564C2×C4—42no—
565C2×C4—41no—
566C2×C4—42no—
567C2×C4—41no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—44no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—44no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—44no—
579C2×C4—41no—
580C2×C4—44no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—44no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—44no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—416no—
594C2×C4—42no—
595C2×C4—44no—
596C2×C4—42no—
597C2×C4—41no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—44no—
603C2×C4—42no—
604C2×C4—41no—
605C2×C4—41no—
606C2×C4—42no—
607C2×C4—44no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—44no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—44no—
614C2×C4—44no—
615C2×C4—44no—
616C2×C4—44no—
617C2×C4—44no—
618C2×C4C2×C22⋊Q818yes—
619C2×C4C2×C22⋊Q814yes—
620C2×C4#52514yes—
621C2×C4#52918yes—
622C2×C4#53214yes—
623C2×C4#53318yes—
624C2×C4#54614yes—
625C22—24no—
626C22—28no—
627C22—44no—
628C22—42no—
629C22—42no—
630C22—42no—
631C22—42no—
632C22—44no—
633C22—42no—
634C22—44no—
635C22—42no—
636C22C22×C22⋊Q811yescharacteristic
637C22C2×#52511yescharacteristic
638C22C2×#52912yes—
639C22C2×#53211yescharacteristic
640C22C2×#53312yes—
641C22C2×#54611yescharacteristic
642C22C2×#55411yescharacteristic
643C22#347511yescharacteristic
644C22#347512yes—
645C22#348212yes—
646C22#348611yescharacteristic
647C22#348811yescharacteristic
648C22#349212yes—
649C22#349512yes—
650C22#350011yescharacteristic
651C22#350111yescharacteristic
652C22#350212yes—
653C22#350612yes—
654C22#351312yes—
655C22#351812yes—
656C22#351911yescharacteristic
657C22#352112yes—
658C22#352212yes—
659C4—28no—
660C4—24no—
661C4—44no—
662C4—44no—
663C4—44no—
664C4—42no—
665C4—44no—
666C4—44no—
667C4—48no—
668C4—42no—
669C4—42no—
670C4—82no—
671C4—82no—
672C4—81no—
673C4—84no—
674C4#239618yes—
675C2—44no—
676C2C2×#239611yescharacteristic
677C2#3571111yescharacteristic
678C2#3572212yes—
679C2#3578312yes—
680C2#3592611yescharacteristic
681C2#3598211yescharacteristic
682C2#3615411yescharacteristic
683C2#3640912yes—
684C2#3641511yescharacteristic
685C2#3642211yescharacteristic
686C2#3642512yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC25C24
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×#2369C2semidirect
3. #15230C2semidirect
5. #15194C2semidirect
6. #15156C2semidirect
8. #15102C2semidirect
9. #14958C2semidirect
11. #14869C2semidirect
15. #13821C2semidirect
16. #13804C2semidirect
95. C22×C4⋊C4Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#23692
C2×#23442
#152302
#152202
#151942
#151562
#151252
#151022
#149582
#148932
#148692
#147682
C4×#3992
#138522
#138212
#138042
C22⋊C4×C4⋊C42