Finite Group Databasebeta

← #4306638

Subgroups — #4306638

Order 512.

2722 conjugacy classes of subgroups.

The subgroup lattice has 700 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2369C211yesmaximal, characteristic, complemented
2C2×#2354C211yesmaximal, characteristic
3#15148C211yesmaximal, characteristic
4#15144C211yesmaximal, characteristic
5#15141C212yesmaximal
6#15137C212yesmaximal, complemented
7#15128C211yesmaximal, characteristic, complemented
8#15100C212yesmaximal, complemented
9#14956C214yesmaximal, complemented
10#14893C212yesmaximal, complemented
11#14870C212yesmaximal, complemented
12#14770C214yesmaximal, complemented
13C4×#400C211yesmaximal, characteristic, complemented
14#13852C212yesmaximal, complemented
15#13821C211yesmaximal, characteristic, complemented
16#13820C212yesmaximal
17#13818C212yesmaximal, complemented
18C22×C4⋊Q8C2211yescharacteristic
19C22×C4×D4C2211yescharacteristic
20C22×#117C2212yes—
21C2×#400C2211yescharacteristic
22C2×#399C2212yes—
23C2×#398C2214yes—
24C2×#396C2214yes—
25C2×#395C2212yes—
26C2×#393C2214yes—
27#2369—28no—
28#2354—28no—
29C2×#391C2214yes—
30C2×#390C2211yescharacteristic
31C2×#390C2212yes—
32C2×#390C2212yes—
33C2×#390C2211yescharacteristic
34C2×#389C2212yes—
35C2×#389C2212yes—
36C2×#389C2212yes—
37C2×#389C2212yes—
38C2×#389C2212yes—
39C2×#388C2212yes—
40C2×#388C2212yes—
41C2×#384C2211yescharacteristic
42C2×#380C2212yes—
43C2×C4×C4⋊C4C2211yescharacteristic
44C2×C4×C4⋊C4C2212yes—
45C2×C4×C4⋊C4C2211yescharacteristic
46C2×C4×C4⋊C4C2211yescharacteristic
47C2×C4×C4⋊C4C2212yes—
48C2×C4×C4⋊C4C2212yes—
49C2×C4×C4⋊C4C2212yes—
50C2×C4×C4⋊C4C2211yescharacteristic
51C2×C4×C22⋊C4C2214yes—
52#1446C2212yes—
53#1445C2211yescharacteristic
54#1444C2212yes—
55#1444C2214yes—
56#1443C2212yes—
57#1442C2214yes—
58#1442C2212yes—
59#1442C2211yescharacteristic
60#1441C2212yes—
61#1441C2211yescharacteristic
62#1441C2214yes—
63#1439C2212yes—
64#1439C2214yes—
65#1439C2212yes—
66#1439C2214yes—
67#1439C2214yes—
68#1439C2212yes—
69#1439C2214yes—
70#1437C2214yes—
71#1437C2214yes—
72#1436C2214yes—
73#1436C2214yes—
74#1436C2214yes—
75C4⋊#95C2214yes—
76C4⋊#95C2212yes—
77C4⋊#95C2211yescharacteristic
78C4⋊#95C2212yes—
79#1434C2212yes—
80C22⋊#95C2214yes—
81C4×#95C2214yes—
82C4×#95C2212yes—
83C4×#95C2212yes—
84C4×#95C2211yescharacteristic
85C23×Q8C2312yes—
86C23×D4C2311yescharacteristic
87C24×C4C2312yes—
88C2×C4⋊Q8—28no—
89C2×C4⋊Q8—24no—
90C2×C4×D4—24no—
91C2×C4×D4—24no—
92C2×C4×D4Q818yescomplemented
93C2×#117—28no—
94C2×#117—28no—
95C2×#117—44no—
96C22×C4⋊C4C2311yescharacteristic
97C22×C4⋊C4C2312yes—
98C22×C4⋊C4C2311yescharacteristic
99C22×C4⋊C4C2311yescharacteristic
100C22×C4⋊C4C2311yescharacteristic
101C22×C4⋊C4C2314yes—
102C22×C4⋊C4C2312yes—
103C22×C4⋊C4C2311yescharacteristic
104C22×C4⋊C4C2311yescharacteristic
105C22×C4⋊C4C2312yes—
106C22×C4⋊C4C2312yes—
107C22×C4⋊C4C2314yes—
108C22×C4⋊C4C2314yes—
109C22×C4⋊C4C2312yes—
110C22×C4⋊C4C2314yes—
111C22×C4⋊C4C2311yescharacteristic
112C22×C22⋊C4C2312yes—
113C22×C22⋊C4C2314yes—
114C22×C22⋊C4C2312yes—
115C22×C22⋊C4C2314yes—
116C22×C22⋊C4C2312yes—
117C22×C22⋊C4C2312yes—
118C22×C22⋊C4C2314yes—
119C22×C22⋊C4C2312yes—
120C22×C22⋊C4C2314yes—
121C22×C42C2312yes—
122C22×C42C2312yes—
123C22×C42C2311yescharacteristic
124C22×C42C2312yes—
125C22×C42C2312yes—
126C22×C42C2311yescharacteristic
127C22×C42C2311yescharacteristic
128C22×C42C2314yes—
129C22×C42C2314yes—
130#400—24no—
131#399—44no—
132#398—48no—
133#396—48no—
134#395—44no—
135#393—48no—
136#391—216no—
137#390—24no—
138#390—28no—
139#390—28no—
140#390—24no—
141#389—28no—
142#389—28no—
143#389—28no—
144#389—28no—
145#389—28no—
146#388—28no—
147#388—28no—
148#384—24no—
149#380—28no—
150C4×C4⋊C4—24no—
151C4×C4⋊C4—28no—
152C4×C4⋊C4—28no—
153C4×C4⋊C4—24no—
154C4×C4⋊C4—28no—
155C4×C4⋊C4—24no—
156C4×C4⋊C4—44no—
157C4×C4⋊C4D418yes—
158C4×C22⋊C4—48no—
159C2×#95C2312yes—
160C2×#95C2314yes—
161C2×#95C2314yes—
162C2×#95C2312yes—
163C2×#95C2312yes—
164C2×#95C2312yes—
165C2×#95C2312yes—
166C2×#95C2311yescharacteristic
167C2×#95C2312yes—
168C2×#95C2312yes—
169C2×#95C2314yes—
170C2×#95C2314yes—
171C2×#95C2312yes—
172C2×#95C2311yescharacteristic
173C2×#95C2312yes—
174C2×#95C2314yes—
175C2×#95C2314yes—
176C2×#95C2314yes—
177C2×#95C2312yes—
178C2×#95C2314yes—
179C2×#95C2312yes—
180C2×#95C2311yescharacteristic
181C2×#95C2311yescharacteristic
182C2×#95C2314yes—
183C2×#95C2312yes—
184C2×#95C2314yes—
185C2×#95C2312yes—
186C2×#95C2312yes—
187C25C2412yes—
188C22×Q8—24no—
189C22×Q8—28no—
190C22×Q8—42no—
191C22×Q8—44no—
192C22×Q8—42no—
193C22×D4—22no—
194C22×D4—22no—
195C22×D4—24no—
196C22×D4—24no—
197C22×D4C2×Q814yes—
198C23×C4—24no—
199C23×C4—24no—
200C23×C4—24no—
201C23×C4—24no—
202C23×C4—42no—
203C23×C4—42no—
204C23×C4C2×Q818yes—
205C23×C4C2411yescharacteristic
206C23×C4C2412yes—
207C23×C4C2412yes—
208C23×C4C2412yes—
209C23×C4C2412yes—
210C23×C4C2411yescharacteristic
211C23×C4C2411yescharacteristic
212C23×C4C2414yes—
213C23×C4C2414yes—
214C23×C4C2411yescharacteristic
215C23×C4C2412yes—
216C23×C4C2412yes—
217C23×C4C2412yes—
218C23×C4C2411yescharacteristic
219C23×C4C2411yescharacteristic
220C23×C4C2411yescharacteristic
221C4⋊Q8—232no—
222C4×D4—232no—
223#117—432no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—24no—
226C2×C4⋊C4—24no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—28no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4—24no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4—24no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—28no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—28no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—28no—
248C2×C4⋊C4—28no—
249C2×C4⋊C4—28no—
250C2×C4⋊C4—41no—
251C2×C4⋊C4—42no—
252C2×C4⋊C4—44no—
253C2×C4⋊C4—44no—
254C2×C4⋊C4—44no—
255C2×C4⋊C4—44no—
256C2×C4⋊C4—44no—
257C2×C4⋊C4—44no—
258C2×C4⋊C4—44no—
259C2×C4⋊C4—44no—
260C2×C4⋊C4C2×D414yes—
261C2×C4⋊C4C2×D414yes—
262C2×C4⋊C4C2×Q814yes—
263C2×C4⋊C4#4114yes—
264C2×C4⋊C4#4118yes—
265C2×C4⋊C4#4118yes—
266C2×C4⋊C4#4114yes—
267C2×C22⋊C4—24no—
268C2×C22⋊C4—24no—
269C2×C22⋊C4—24no—
270C2×C22⋊C4—24no—
271C2×C22⋊C4—24no—
272C2×C22⋊C4—24no—
273C2×C22⋊C4—28no—
274C2×C22⋊C4—28no—
275C2×C22⋊C4—24no—
276C2×C22⋊C4—28no—
277C2×C22⋊C4—28no—
278C2×C22⋊C4—24no—
279C2×C22⋊C4—24no—
280C2×C22⋊C4—28no—
281C2×C22⋊C4—28no—
282C2×C22⋊C4—42no—
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⋊C4C2×Q818yes—
292C2×C42—22no—
293C2×C42—24no—
294C2×C42—22no—
295C2×C42—24no—
296C2×C42—24no—
297C2×C42—24no—
298C2×C42—24no—
299C2×C42—22no—
300C2×C42—24no—
301C2×C42—24no—
302C2×C42—24no—
303C2×C42—24no—
304C2×C42—24no—
305C2×C42—24no—
306C2×C42—24no—
307C2×C42—28no—
308C2×C42—28no—
309C2×C42—42no—
310C2×C42—44no—
311C2×C42—44no—
312C2×C42—44no—
313C2×C42—44no—
314C2×C42C2×D418yes—
315C2×C42C2×D414yes—
316C2×C42C2×Q814yes—
317#95—24no—
318#95—22no—
319#95—22no—
320#95—28no—
321#95—28no—
322#95—24no—
323#95—28no—
324#95—28no—
325#95—28no—
326#95—28no—
327#95—22no—
328#95—24no—
329#95—24no—
330#95—24no—
331#95—28no—
332#95—22no—
333#95—24no—
334#95—24no—
335#95—28no—
336#95—24no—
337#95—24no—
338#95—24no—
339#95—24no—
340#95—42no—
341#95—44no—
342#95—42no—
343#95—44no—
344#95C2×D418yes—
345C24—22no—
346C24—22no—
347C24—24no—
348C24—24no—
349C24—22no—
350C24—24no—
351C24—22no—
352C24—22no—
353C24—24no—
354C24—22no—
355C24C22×Q814yes—
356C24C2511yescharacteristic
357C2×Q8—216no—
358C2×Q8—84no—
359C2×Q8—84no—
360C2×Q8—88no—
361C2×Q8—88no—
362C2×D4—28no—
363C2×D4—216no—
364C2×D4—44no—
365C2×D4—48no—
366C2×D4—44no—
367C22×C4—21no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—21no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—21no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—21no—
377C22×C4—22no—
378C22×C4—21no—
379C22×C4—24no—
380C22×C4—21no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—24no—
384C22×C4—24no—
385C22×C4—24no—
386C22×C4—22no—
387C22×C4—21no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—24no—
394C22×C4—22no—
395C22×C4—24no—
396C22×C4—24no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—24no—
401C22×C4—24no—
402C22×C4—22no—
403C22×C4—21no—
404C22×C4—24no—
405C22×C4—24no—
406C22×C4—22no—
407C22×C4—21no—
408C22×C4—24no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—24no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—24no—
416C22×C4—21no—
417C22×C4—216no—
418C22×C4—216no—
419C22×C4—21no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—24no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—24no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—21no—
440C22×C4—22no—
441C22×C4—21no—
442C22×C4—22no—
443C22×C4—48no—
444C22×C4—48no—
445C22×C4—48no—
446C22×C4—48no—
447C22×C4—48no—
448C22×C4C22×D412yes—
449C22×C4C22×D414yes—
450C22×C4C22×D414yes—
451C22×C4C22×D412yes—
452C22×C4C22×D412yes—
453C22×C4C22×Q812yes—
454C22×C4C22×Q814yes—
455C22×C4C22×Q812yes—
456C22×C4C22×Q812yes—
457C22×C4C2×#4112yes—
458C22×C4C2×#4118yes—
459C22×C4C2×#4118yes—
460C22×C4C2×#4114yes—
461C22×C4C2×#4112yes—
462C22×C4C2×#4112yes—
463C22×C4C2×#4112yes—
464C22×C4C2×#4114yes—
465C22×C4C2×#4114yes—
466C22×C4H(5,𝔽2)14yes—
467C22×C4H(5,𝔽2)12yes—
468C22×C4H(5,𝔽2)12yes—
469C22×C42−1+414yes—
470C4⋊C4—28no—
471C4⋊C4—216no—
472C4⋊C4—216no—
473C4⋊C4—216no—
474C4⋊C4—28no—
475C4⋊C4—28no—
476C4⋊C4—44no—
477C4⋊C4—416no—
478C4⋊C4—48no—
479C4⋊C4—44no—
480C4⋊C4—44no—
481C4⋊C4—84no—
482C4⋊C4—88no—
483C4⋊C4—88no—
484C4⋊C4—88no—
485C4⋊C4C22⋊Q8116yes—
486C22⋊C4—216no—
487C22⋊C4—416no—
488C22⋊C4—416no—
489C22⋊C4—48no—
490C22⋊C4—48no—
491C22⋊C4—84no—
492C22⋊C4—84no—
493C22⋊C4—88no—
494C22⋊C4—88no—
495C42—28no—
496C42—28no—
497C42—216no—
498C42—48no—
499C42—44no—
500C42—48no—
501C42—48no—
502C42—88no—
503C42—88no—
504C23—24no—
505C23—28no—
506C23—24no—
507C23—24no—
508C23—28no—
509C23—42no—
510C23—44no—
511C23—42no—
512C23—42no—
513C23—42no—
514C23—44no—
515C23—44no—
516C23—42no—
517C23—42no—
518C23—42no—
519C23—44no—
520C23—44no—
521C23—44no—
522C23—44no—
523C23—42no—
524C23—44no—
525C23—44no—
526C23—44no—
527C23C23×D411yescharacteristic
528C23C23×Q811yescharacteristic
529C23C22×#4111yescharacteristic
530C23C22×#4112yes—
531C23C22×#4112yes—
532C23C22×#4111yescharacteristic
533C23C2×H(5,𝔽2)11yescharacteristic
534C23C2×H(5,𝔽2)11yescharacteristic
535C23C2×H(5,𝔽2)12yes—
536C23C2×2−1+412yes—
537C23#58511yescharacteristic
538Q8—816no—
539D4—416no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—24no—
543C2×C4—22no—
544C2×C4—24no—
545C2×C4—22no—
546C2×C4—24no—
547C2×C4—24no—
548C2×C4—22no—
549C2×C4—24no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—24no—
553C2×C4—24no—
554C2×C4—22no—
555C2×C4—24no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—28no—
559C2×C4—28no—
560C2×C4—24no—
561C2×C4—24no—
562C2×C4—22no—
563C2×C4—24no—
564C2×C4—24no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—24no—
568C2×C4—24no—
569C2×C4—28no—
570C2×C4—24no—
571C2×C4—24no—
572C2×C4—28no—
573C2×C4—28no—
574C2×C4—24no—
575C2×C4—24no—
576C2×C4—28no—
577C2×C4—24no—
578C2×C4—432no—
579C2×C4—41no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—44no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—42no—
586C2×C4—44no—
587C2×C4—41no—
588C2×C4—44no—
589C2×C4—44no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—44no—
594C2×C4—44no—
595C2×C4—44no—
596C2×C4—44no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—44no—
602C2×C4—42no—
603C2×C4—44no—
604C2×C4—42no—
605C2×C4—41no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—44no—
616C2×C4—42no—
617C2×C4—41no—
618C2×C4—41no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—44no—
628C2×C4—42no—
629C2×C4C2×C22⋊Q818yes—
630C2×C4C2×C22⋊Q814yes—
631C2×C4#52514yes—
632C2×C4#52918yes—
633C2×C4#53214yes—
634C2×C4#53318yes—
635C2×C4#54614yes—
636C2×C4#55414yes—
637C22—48no—
638C22—48no—
639C22—44no—
640C22—44no—
641C22—44no—
642C22—82no—
643C22—84no—
644C22—82no—
645C22—82no—
646C22—84no—
647C22—82no—
648C22C22×C22⋊Q811yescharacteristic
649C22C2×#52511yescharacteristic
650C22C2×#52912yes—
651C22C2×#53211yescharacteristic
652C22C2×#53312yes—
653C22C2×#54611yescharacteristic
654C22C2×#55411yescharacteristic
655C22#347511yescharacteristic
656C22#348212yes—
657C22#348312yes—
658C22#348611yescharacteristic
659C22#348811yescharacteristic
660C22#349811yescharacteristic
661C22#350111yescharacteristic
662C22#350312yes—
663C22#351412yes—
664C22#351612yes—
665C22#351912yes—
666C22#351912yes—
667C22#351911yescharacteristic
668C22#352012yes—
669C22#352212yes—
670C22#352212yes—
671C4—24no—
672C4—28no—
673C4—24no—
674C4—44no—
675C4—44no—
676C4—42no—
677C4—48no—
678C4—44no—
679C4—42no—
680C4—48no—
681C4—42no—
682C4—82no—
683C4—82no—
684C4—82no—
685C4—81no—
686C4#239618yes—
687C2—84no—
688C2C2×#239611yescharacteristic
689C2#3574811yescharacteristic
690C2#3587112yes—
691C2#3590911yescharacteristic
692C2#3591612yes—
693C2#3600411yescharacteristic
694C2#3615411yescharacteristic
695C2#3640712yes—
696C2#3641211yescharacteristic
697C2#3641312yes—
698C2#3642211yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC23×D4C23
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
6. #15137C2semidirect
7. #15128C2semidirect
8. #15100C2semidirect
9. #14956C2semidirect
10. #14893C2semidirect
11. #14870C2semidirect
12. #14770C2semidirect
13. C4×#400C2semidirect
14. #13852C2semidirect
15. #13821C2semidirect
17. #13818C2semidirect
92. C2×C4×D4Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#23692
C2×#23542
#151482
#151442
#151412
#151372
#151282
#151002
#149562
#148932
#148702
#147702
C4×#4002
#138522
#138212
#138202
#138182