Finite Group Databasebeta

← #513666

Subgroups — #513666

Order 512.

3462 conjugacy classes of subgroups.

The subgroup lattice has 696 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2362C212yesmaximal
2C2×C4×C22⋊Q8C211yesmaximal, characteristic
3#13822C212yesmaximal
4#13821C211yesmaximal, characteristic, complemented
5#13819C214yesmaximal, complemented
6#13817C211yesmaximal, characteristic, complemented
7#13816C212yesmaximal
8Q8×#95C211yesmaximal, characteristic, complemented
9#13810C211yesmaximal, characteristic, complemented
10#13801C212yesmaximal
11#13789C214yesmaximal
12#13787C212yesmaximal, complemented
13C4×#391C212yesmaximal, complemented
14C4×#389C212yesmaximal, complemented
15#13756C211yesmaximal, characteristic
16#13750C212yesmaximal, complemented
17C42×C22⋊C4C211yesmaximal, characteristic
18C22×C22⋊Q8C2211yescharacteristic
19C22×C4×Q8C2211yescharacteristic
20C22×C4×Q8C2212yes—
21C22×#117C2212yes—
22C23×C42C2211yescharacteristic
23#2362C4132yescomplemented
24C4×C22⋊Q8C4116yescomplemented
25C2×#391C2211yescharacteristic
26C2×#391C2212yes—
27C2×#389C2211yescharacteristic
28C2×#389C2212yes—
29C2×#388C2214yes—
30C2×#388C2214yes—
31C2×#387C2212yes—
32C2×#387C2214yes—
33C2×#385C2212yes—
34C2×#380C2211yescharacteristic
35C2×C4×C4⋊C4C2214yes—
36C2×C4×C4⋊C4C2212yes—
37C2×C4×C4⋊C4C2212yes—
38C2×C4×C4⋊C4C2212yes—
39C2×C4×C4⋊C4C2212yes—
40C2×C4×C4⋊C4C2211yescharacteristic
41C2×C4×C4⋊C4C2214yes—
42C2×C4×C4⋊C4C2212yes—
43C2×C4×C22⋊C4C2212yes—
44C2×C4×C22⋊C4C2211yescharacteristic
45C2×C4×C22⋊C4C2211yescharacteristic
46C2×C4×C22⋊C4C2212yes—
47C2×C4×C22⋊C4C2212yes—
48C2×#376C2212yes—
49C2×C43C2212yes—
50#1437C2214yes—
51#1437C2214yes—
52#1436C2214yes—
53#1436C2214yes—
54#1436C2214yes—
55#1436C2214yes—
56#1436C2214yes—
57#1436C2214yes—
58C4⋊#95C2212yes—
59C4⋊#95C2211yescharacteristic
60C4⋊#95C2211yescharacteristic
61C4⋊#95C2212yes—
62C4⋊#95C2212yes—
63C4⋊#95C2212yes—
64C4⋊#95C2214yes—
65#1434C2214yes—
66#1434C2212yes—
67#1434C2212yes—
68#1434C2212yes—
69#1434C2214yes—
70C22⋊#95C2212yes—
71C22⋊#95C2212yes—
72C4×#95C2211yescharacteristic
73C4×#95C2211yescharacteristic
74C4×#95C2211yescharacteristic
75C4×#95C2211yescharacteristic
76C4×#95C2212yes—
77C4×#95C2212yes—
78C4×#95C2214yes—
79C4×#95C2214yes—
80C4×#95C2212yes—
81C4×#95C2212yes—
82C4×#95C2212yes—
83C4×#95C2214yes—
84C4×#95C2214yes—
85C23×Q8C2311yescharacteristic
86C24×C4C2311yescharacteristic
87C24×C4C2311yescharacteristic
88C24×C4C2311yescharacteristic
89C2×C22⋊Q8C2×C4116yes—
90C2×C22⋊Q8C2×C418yes—
91C2×C4×Q8—44no—
92C2×C4×Q8—44no—
93C2×C4×Q8—44no—
94C2×C4×Q8C2×C418yes—
95C2×C4×Q8C2×C4116yescomplemented
96C2×#117—44no—
97C2×#117—44no—
98C2×#117C2×C4116yes—
99C22×C4⋊C4C2312yes—
100C22×C4⋊C4C2312yes—
101C22×C4⋊C4C2311yescharacteristic
102C22×C4⋊C4C2311yescharacteristic
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2314yes—
105C22×C4⋊C4C2312yes—
106C22×C4⋊C4C2314yes—
107C22×C4⋊C4C2312yes—
108C22×C4⋊C4C2311yescharacteristic
109C22×C22⋊C4C2312yes—
110C22×C22⋊C4C2312yes—
111C22×C22⋊C4C2314yes—
112C22×C22⋊C4C2312yes—
113C22×C22⋊C4C2312yes—
114C22×C42—44no—
115C22×C42C2×C418yes—
116C22×C42C2312yes—
117C22×C42C2312yes—
118C22×C42C2311yescharacteristic
119C22×C42C2311yescharacteristic
120C22×C42C2312yes—
121C22×C42C2314yes—
122C22×C42C2312yes—
123C22×C42C2312yes—
124C22×C42C2314yes—
125C22×C42C2314yes—
126C22×C42C2312yes—
127C22×C42C2311yescharacteristic
128C22×C42C2311yescharacteristic
129C22×C42C2314yes—
130C22×C42C2314yes—
131C22×C42C2312yes—
132C22×C42C2312yes—
133#391C2×C418yes—
134#391C2×C4116yescomplemented
135#389C2×C418yes—
136#389C2×C4116yescomplemented
137#388C2×C4132yes—
138#388C2×C4132yes—
139#387C2×C4116yes—
140#387C2×C4132yescomplemented
141#385C2×C4116yes—
142#380C2×C418yes—
143C4×C4⋊C4—216no—
144C4×C4⋊C4—28no—
145C4×C4⋊C4—44no—
146C4×C4⋊C4—44no—
147C4×C4⋊C4C2×C4116yes—
148C4×C4⋊C4C2×C418yes—
149C4×C4⋊C4C2×C4132yescomplemented
150C4×C4⋊C4C2×C4116yescomplemented
151C4×C22⋊C4—28no—
152C4×C22⋊C4—24no—
153C4×C22⋊C4—24no—
154C4×C22⋊C4C2×C4116yes—
155C4×C22⋊C4C2×C4116yes—
156#376—28no—
157C2×#95C2314yes—
158C2×#95C2312yes—
159C2×#95C2312yes—
160C2×#95C2314yes—
161C2×#95C2311yescharacteristic
162C2×#95C2311yescharacteristic
163C2×#95C2312yes—
164C2×#95C2314yes—
165C2×#95C2312yes—
166C2×#95C2314yes—
167C2×#95C2312yes—
168C2×#95C2312yes—
169C2×#95C2314yes—
170C2×#95C2314yes—
171C2×#95C2314yes—
172C2×#95C2314yes—
173C2×#95C2312yes—
174C2×#95C2312yes—
175C2×#95C2312yes—
176C2×#95C2312yes—
177C2×#95C2312yes—
178C2×#95C2312yes—
179C2×#95C2314yes—
180C2×#95C2314yes—
181C2×#95C2314yes—
182C2×#95C2312yes—
183C2×#95C2314yes—
184C2×#95C2311yescharacteristic
185C2×#95C2311yescharacteristic
186C43—44no—
187C25C2411yescharacteristic
188C22×Q8—24no—
189C22×Q8—22no—
190C22×Q8—22no—
191C22×Q8C22×C414yes—
192C22×Q8C22×C418yes—
193C23×C4—22no—
194C23×C4—24no—
195C23×C4—22no—
196C23×C4—22no—
197C23×C4—22no—
198C23×C4—24no—
199C23×C4—24no—
200C23×C4—41no—
201C23×C4—41no—
202C23×C4—42no—
203C23×C4—42no—
204C23×C4C22×C414yes—
205C23×C4C22×C418yes—
206C23×C4C22×C414yes—
207C23×C4C22×C414yes—
208C23×C4C2412yes—
209C23×C4C2414yes—
210C23×C4C2412yes—
211C23×C4C2414yes—
212C23×C4C2412yes—
213C23×C4C2412yes—
214C23×C4C2411yescharacteristic
215C23×C4C2412yes—
216C23×C4C2412yes—
217C23×C4C2411yescharacteristic
218C23×C4C2412yes—
219C23×C4C2412yes—
220C23×C4C2411yescharacteristic
221C23×C4C2411yescharacteristic
222C23×C4C2411yescharacteristic
223C23×C4C2411yescharacteristic
224C22⋊Q8C42164yescomplemented
225C4×Q8—416no—
226C4×Q8—432no—
227#117—432no—
228C2×C4⋊C4—28no—
229C2×C4⋊C4—28no—
230C2×C4⋊C4—28no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4—24no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—44no—
239C2×C4⋊C4—44no—
240C2×C4⋊C4—44no—
241C2×C4⋊C4C22×C418yes—
242C2×C4⋊C4C22×C418yes—
243C2×C4⋊C4C22×C414yes—
244C2×C4⋊C4C22×C414yes—
245C2×C4⋊C4C22×C418yes—
246C2×C4⋊C4C22×C4116yes—
247C2×C4⋊C4C22×C418yes—
248C2×C4⋊C4C22×C4116yes—
249C2×C4⋊C4C22×C418yes—
250C2×C4⋊C4C22×C414yes—
251C2×C4⋊C4C22×C4116yes—
252C2×C4⋊C4C22×C418yes—
253C2×C22⋊C4—24no—
254C2×C22⋊C4—24no—
255C2×C22⋊C4—24no—
256C2×C22⋊C4—24no—
257C2×C22⋊C4—42no—
258C2×C22⋊C4—42no—
259C2×C22⋊C4—44no—
260C2×C22⋊C4—44no—
261C2×C22⋊C4C22×C418yes—
262C2×C22⋊C4C22×C418yes—
263C2×C22⋊C4C22×C4116yes—
264C2×C22⋊C4C22×C418yes—
265C2×C22⋊C4C22×C418yes—
266C2×C22⋊C4C22×C4116yes—
267C2×C42—24no—
268C2×C42—28no—
269C2×C42—28no—
270C2×C42—24no—
271C2×C42—24no—
272C2×C42—24no—
273C2×C42—24no—
274C2×C42—24no—
275C2×C42—24no—
276C2×C42—24no—
277C2×C42—24no—
278C2×C42—24no—
279C2×C42—24no—
280C2×C42—24no—
281C2×C42—24no—
282C2×C42—28no—
283C2×C42—28no—
284C2×C42—416no—
285C2×C42—44no—
286C2×C42—44no—
287C2×C42—44no—
288C2×C42—44no—
289C2×C42—44no—
290C2×C42—42no—
291C2×C42—42no—
292C2×C42—42no—
293C2×C42—42no—
294C2×C42—42no—
295C2×C42—42no—
296C2×C42—42no—
297C2×C42—42no—
298C2×C42—44no—
299C2×C42—44no—
300C2×C42—44no—
301C2×C42—44no—
302C2×C42C22×C414yes—
303C2×C42C22×C418yes—
304C2×C42C22×C418yes—
305C2×C42C22×C418yes—
306C2×C42C22×C4116yes—
307C2×C42C22×C414yes—
308C2×C42C22×C4116yes—
309C2×C42C22×C418yes—
310C2×C42C22×C4116yes—
311C2×C42C22×C414yes—
312C2×C42C22×C414yes—
313C2×C42#4118yes—
314#95—24no—
315#95—28no—
316#95—24no—
317#95—24no—
318#95—22no—
319#95—22no—
320#95—28no—
321#95—28no—
322#95—24no—
323#95—28no—
324#95—42no—
325#95—44no—
326#95—44no—
327#95—42no—
328#95—44no—
329#95—44no—
330#95C22×C418yes—
331#95C22×C418yes—
332#95C22×C418yes—
333#95C22×C418yes—
334#95C22×C418yes—
335#95C22×C418yes—
336#95C22×C418yes—
337#95C22×C4116yes—
338#95C22×C4116yes—
339#95C22×C4116yes—
340#95C22×C4116yes—
341#95#4114yes—
342#95#4114yes—
343C24—22no—
344C24—21no—
345C24—21no—
346C24—21no—
347C24—22no—
348C24—22no—
349C24—21no—
350C24—21no—
351C24—21no—
352C24C23×C414yes—
353C24C23×C412yes—
354C24C2511yescharacteristic
355C2×Q8—28no—
356C2×Q8—216no—
357C2×Q8—216no—
358C2×Q8—28no—
359C2×Q8C2×C42116yes—
360C22×C4—22no—
361C22×C4—21no—
362C22×C4—21no—
363C22×C4—24no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—21no—
373C22×C4—24no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—21no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—21no—
384C22×C4—21no—
385C22×C4—28no—
386C22×C4—216no—
387C22×C4—216no—
388C22×C4—28no—
389C22×C4—24no—
390C22×C4—22no—
391C22×C4—24no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—24no—
398C22×C4—24no—
399C22×C4—24no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—24no—
405C22×C4—24no—
406C22×C4—24no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—24no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—24no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—21no—
423C22×C4—21no—
424C22×C4—24no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—28no—
428C22×C4—28no—
429C22×C4—28no—
430C22×C4—44no—
431C22×C4—44no—
432C22×C4—44no—
433C22×C4—48no—
434C22×C4—48no—
435C22×C4—48no—
436C22×C4—48no—
437C22×C4C2×C42116yes—
438C22×C4C23×C414yes—
439C22×C4C23×C414yes—
440C22×C4C23×C414yes—
441C22×C4C23×C418yes—
442C22×C4C23×C412yes—
443C22×C4C23×C414yes—
444C22×C4C23×C418yes—
445C22×C4C23×C414yes—
446C22×C4C23×C412yes—
447C22×C4C23×C414yes—
448C22×C4C23×C412yes—
449C22×C4C23×C412yes—
450C22×C4C23×C412yes—
451C22×C4C23×C414yes—
452C22×C4C23×C414yes—
453C22×C4C23×C418yes—
454C22×C4C23×C414yes—
455C22×C4C23×C418yes—
456C22×C4C23×C412yes—
457C22×C4C23×C414yes—
458C22×C4C2×#4112yes—
459C22×C4C2×#4112yes—
460C22×C4C2×#4114yes—
461C22×C4C2×#4114yes—
462C22×C4C2×#4114yes—
463C22×C4C2×#4114yes—
464C22×C4C2×#4114yes—
465C22×C4H(5,𝔽2)14yes—
466C22×C4H(5,𝔽2)12yes—
467C22×C42−1+414yes—
468C22×C42−1+412yes—
469C4⋊C4—232no—
470C4⋊C4—216no—
471C4⋊C4—216no—
472C4⋊C4—216no—
473C4⋊C4—28no—
474C4⋊C4—28no—
475C4⋊C4—48no—
476C4⋊C4—416no—
477C4⋊C4C2×C42116yes—
478C4⋊C4C2×C42132yes—
479C22⋊C4—216no—
480C22⋊C4—216no—
481C22⋊C4—48no—
482C22⋊C4—416no—
483C22⋊C4C2×C42132yes—
484C42—232no—
485C42—216no—
486C42—28no—
487C42—28no—
488C42—216no—
489C42—216no—
490C42—44no—
491C42—44no—
492C42—416no—
493C42—48no—
494C42—416no—
495C42—48no—
496C42—48no—
497C42—48no—
498C42—48no—
499C42—88no—
500C42—88no—
501C23—24no—
502C23—24no—
503C23—22no—
504C23—22no—
505C23—24no—
506C23—22no—
507C23—22no—
508C23—24no—
509C23—22no—
510C23—22no—
511C23—24no—
512C23—24no—
513C23—42no—
514C23—42no—
515C23—42no—
516C23—42no—
517C23—41no—
518C23—41no—
519C23—41no—
520C23—41no—
521C23—42no—
522C23—42no—
523C23C22×C4214yes—
524C23C24×C411yescharacteristic
525C23C24×C412yes—
526C23C22×#4112yes—
527C23C22×#4111yescharacteristic
528C23C22×#4111yescharacteristic
529C23C2×H(5,𝔽2)12yes—
530C23C2×H(5,𝔽2)11yescharacteristic
531C23C2×2−1+412yes—
532C23C2×2−1+411yescharacteristic
533C23#58511yescharacteristic
534C23#58511yescharacteristic
535Q8—232no—
536C2×C4—28no—
537C2×C4—24no—
538C2×C4—28no—
539C2×C4—24no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—24no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—24no—
546C2×C4—22no—
547C2×C4—28no—
548C2×C4—24no—
549C2×C4—22no—
550C2×C4—24no—
551C2×C4—24no—
552C2×C4—22no—
553C2×C4—28no—
554C2×C4—24no—
555C2×C4—22no—
556C2×C4—24no—
557C2×C4—24no—
558C2×C4—24no—
559C2×C4—24no—
560C2×C4—24no—
561C2×C4—24no—
562C2×C4—24no—
563C2×C4—24no—
564C2×C4—24no—
565C2×C4—24no—
566C2×C4—24no—
567C2×C4—24no—
568C2×C4—24no—
569C2×C4—24no—
570C2×C4—24no—
571C2×C4—24no—
572C2×C4—232no—
573C2×C4—28no—
574C2×C4—24no—
575C2×C4—24no—
576C2×C4—24no—
577C2×C4—28no—
578C2×C4—24no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—24no—
582C2×C4—28no—
583C2×C4—28no—
584C2×C4—24no—
585C2×C4—24no—
586C2×C4—24no—
587C2×C4—24no—
588C2×C4—24no—
589C2×C4—22no—
590C2×C4—28no—
591C2×C4—24no—
592C2×C4—24no—
593C2×C4—28no—
594C2×C4—416no—
595C2×C4—416no—
596C2×C4—44no—
597C2×C4—44no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—44no—
602C2×C4—44no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—44no—
606C2×C4—42no—
607C2×C4—44no—
608C2×C4—42no—
609C2×C4—44no—
610C2×C4—42no—
611C2×C4—44no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—44no—
619C2×C4—44no—
620C2×C4C22×C4218yes—
621C2×C4C22×C4218yes—
622C2×C4C22×C4214yes—
623C2×C4C22×C4214yes—
624C2×C4C4×#4118yes—
625C2×C4C4×#4114yes—
626C2×C4C4×#4118yes—
627C2×C4C4×#4114yes—
628C2×C4#51814yes—
629C2×C4#51818yes—
630C2×C4#51914yes—
631C2×C4#52018yes—
632C2×C4#52014yes—
633C22—24no—
634C22—24no—
635C22—24no—
636C22—44no—
637C22—44no—
638C22—44no—
639C22—44no—
640C22—42no—
641C22—42no—
642C22—42no—
643C22—42no—
644C22C23×C4211yescharacteristic
645C22C2×C4×#4112yes—
646C22C2×C4×#4112yes—
647C22C2×C4×#4111yescharacteristic
648C22C2×C4×#4111yescharacteristic
649C22C2×#51812yes—
650C22C2×#51811yescharacteristic
651C22C2×#51911yescharacteristic
652C22C2×#52012yes—
653C22C2×#52011yescharacteristic
654C22#342811yescharacteristic
655C22C4×2−1+412yes—
656C22C4×2−1+412yes—
657C22#350512yes—
658C22#350612yes—
659C22#350912yes—
660C22#351112yes—
661C22#351411yescharacteristic
662C22#351511yescharacteristic
663C22#351711yescharacteristic
664C22#351811yescharacteristic
665C22#352012yes—
666C22#352012yes—
667C4—28no—
668C4—28no—
669C4—24no—
670C4—24no—
671C4—24no—
672C4—28no—
673C4—24no—
674C4—28no—
675C4—48no—
676C4—44no—
677C4—44no—
678C4—48no—
679C4—44no—
680C4—42no—
681C4—44no—
682C4#227518yes—
683C2—44no—
684C2C42×#4111yescharacteristic
685C2C2×#227511yescharacteristic
686C2C4×#51911yescharacteristic
687C2#3422311yescharacteristic
688C2#3422511yescharacteristic
689C2#3424211yescharacteristic
690C2#3424511yescharacteristic
691C2#3431412yes—
692C2#3431612yes—
693C2#3432412yes—
694C2#3432712yes—
subgroupG/H
center, Frattini, socle, cosocleC24C25
commutatorC22C23×C42
prime radicalC25C24
LCSG ▷ C22 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
4. #13821C2semidirect
5. #13819C2semidirect
6. #13817C2semidirect
8. Q8×#95C2semidirect
9. #13810C2semidirect
12. #13787C2semidirect
13. C4×#391C2semidirect
14. C4×#389C2semidirect
16. #13750C2semidirect
23. #2362C4semidirect
24. C4×C22⋊Q8C4semidirect
95. C2×C4×Q8C2×C4semidirect
134. #391C2×C4semidirect
136. #389C2×C4semidirect
140. #387C2×C4semidirect
149. C4×C4⋊C4C2×C4semidirect
150. C4×C4⋊C4C2×C4semidirect
224. C22⋊Q8C42semidirect

Maximal subgroups

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

subgroupindex
C2×#23622
C2×C4×C22⋊Q82
#138222
#138212
#138192
#138172
#138162
Q8×#952
#138102
#138012
#137892
#137872
C4×#3912
C4×#3892
#137562
#137502
C42×C22⋊C42