Finite Group Databasebeta

← #805103

Subgroups — #805103

Order 512.

2866 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×#2454C211yesmaximal, characteristic
2C2×D4×C4⋊C4C211yesmaximal, characteristic
3#17057C212yesmaximal, complemented
4#15545C214yesmaximal, complemented
5#15392C214yesmaximal, complemented
6#15196C211yesmaximal, characteristic, complemented
7#15194C211yesmaximal, characteristic, complemented
8#15145C211yesmaximal, characteristic, complemented
9#15092C212yesmaximal, complemented
10#14840C212yesmaximal, complemented
11#14054C212yesmaximal, complemented
12#14018C212yesmaximal
13#13977C212yesmaximal
14#13910C212yesmaximal, complemented
15#13880C212yesmaximal, complemented
16C4×#398C211yesmaximal, characteristic, complemented
17#13820C211yesmaximal, characteristic
18C22×#125C2211yescharacteristic
19C22×C4×D4C2211yescharacteristic
20C22×C4×D4C2212yes—
21C23×C4⋊C4C2212yes—
22#2454—28no—
23C2×#399C2212yes—
24C2×#398C2211yescharacteristic
25C2×#398C2214yes—
26C2×#396C2212yes—
27D4×C4⋊C4—28no—
28C2×#390C2211yescharacteristic
29C2×#390C2211yescharacteristic
30C2×#390C2212yes—
31C2×#389C2211yescharacteristic
32C2×#389C2212yes—
33C2×#389C2211yescharacteristic
34C2×#389C2212yes—
35C2×#389C2212yes—
36C2×#385C2214yes—
37C2×#385C2212yes—
38C2×#385C2214yes—
39C2×#385C2214yes—
40C2×#385C2214yes—
41C2×#384C2211yescharacteristic
42C2×#384C2214yes—
43C2×#384C2212yes—
44C2×#380C2212yes—
45C2×#380C2212yes—
46C2×C4×C4⋊C4C2211yescharacteristic
47C2×C4×C4⋊C4C2211yescharacteristic
48C2×C4×C4⋊C4C2212yes—
49C2×C4×C4⋊C4C2212yes—
50C2×C4×C22⋊C4C2212yes—
51#1445C2211yescharacteristic
52#1445C2212yes—
53#1445C2214yes—
54#1444C2212yes—
55#1444C2214yes—
56#1444C2211yescharacteristic
57#1444C2212yes—
58#1444C2212yes—
59#1444C2212yes—
60#1444C2212yes—
61#1444C2214yes—
62#1442C2212yes—
63#1441C2214yes—
64#1441C2214yes—
65#1441C2212yes—
66#1441C2211yescharacteristic
67#1441C2212yes—
68#1441C2212yes—
69#1439C2214yes—
70#1439C2214yes—
71#1439C2214yes—
72#1439C2214yes—
73#1437C2214yes—
74#1436C2214yes—
75C4⋊#95C2212yes—
76C4⋊#95C2214yes—
77C4⋊#95C2214yes—
78C4⋊#95C2211yescharacteristic
79C4⋊#95C2212yes—
80C4⋊#95C2212yes—
81C22⋊#95C2212yes—
82C22⋊#95C2214yes—
83C4×#95C2211yescharacteristic
84C4×#95C2212yes—
85C23×D4C2311yescharacteristic
86C24×C4C2312yes—
87C24×C4C2314yes—
88C2×#125—24no—
89C2×#125—28no—
90C2×C4×D4—28no—
91C2×C4×D4—28no—
92C2×C4×D4—44no—
93C2×C4×D4—44no—
94C2×C4×D4D418yes—
95C22×C4⋊C4—28no—
96C22×C4⋊C4—48no—
97C22×C4⋊C4C2312yes—
98C22×C4⋊C4C2311yescharacteristic
99C22×C4⋊C4C2311yescharacteristic
100C22×C4⋊C4C2312yes—
101C22×C4⋊C4C2311yescharacteristic
102C22×C4⋊C4C2312yes—
103C22×C4⋊C4C2311yescharacteristic
104C22×C4⋊C4C2311yescharacteristic
105C22×C4⋊C4C2312yes—
106C22×C4⋊C4C2311yescharacteristic
107C22×C4⋊C4C2314yes—
108C22×C4⋊C4C2312yes—
109C22×C4⋊C4C2312yes—
110C22×C4⋊C4C2312yes—
111C22×C4⋊C4C2312yes—
112C22×C4⋊C4C2312yes—
113C22×C4⋊C4C2314yes—
114C22×C4⋊C4C2312yes—
115C22×C4⋊C4C2314yes—
116C22×C4⋊C4C2314yes—
117C22×C4⋊C4C2314yes—
118C22×C4⋊C4C2311yescharacteristic
119C22×C22⋊C4C2312yes—
120C22×C22⋊C4C2312yes—
121C22×C22⋊C4C2314yes—
122C22×C22⋊C4C2312yes—
123C22×C22⋊C4C2312yes—
124C22×C22⋊C4C2312yes—
125C22×C22⋊C4C2314yes—
126C22×C22⋊C4C2314yes—
127C22×C42C2312yes—
128C22×C42C2311yescharacteristic
129C22×C42C2311yescharacteristic
130C22×C42C2314yes—
131C22×C42C2311yescharacteristic
132C22×C42C2312yes—
133#399—28no—
134#398—24no—
135#398—48no—
136#396—28no—
137#390—24no—
138#390—24no—
139#390Q8116yes—
140#389—28no—
141#389—24no—
142#389—24no—
143#389—44no—
144#389—44no—
145#385—28no—
146#385—216no—
147#385—48no—
148#385—48no—
149#385—48no—
150#384—24no—
151#384—44no—
152#384—48no—
153#380—28no—
154#380—28no—
155C4×C4⋊C4—28no—
156C4×C4⋊C4—24no—
157C4×C4⋊C4—24no—
158C4×C4⋊C4—44no—
159C4×C22⋊C4—28no—
160C2×#95C2311yescharacteristic
161C2×#95C2312yes—
162C2×#95C2312yes—
163C2×#95C2314yes—
164C2×#95C2312yes—
165C2×#95C2311yescharacteristic
166C2×#95C2312yes—
167C2×#95C2312yes—
168C2×#95C2312yes—
169C2×#95C2314yes—
170C2×#95C2314yes—
171C2×#95C2312yes—
172C2×#95C2311yescharacteristic
173C2×#95C2312yes—
174C2×#95C2314yes—
175C2×#95C2312yes—
176C2×#95C2312yes—
177C2×#95C2314yes—
178C2×#95C2314yes—
179C2×#95C2314yes—
180C2×#95C2314yes—
181C2×#95C2314yes—
182C2×#95C2312yes—
183C2×#95C2314yes—
184C2×#95C2312yes—
185C2×#95C2311yescharacteristic
186C25C2412yes—
187C22×D4—24no—
188C22×D4—22no—
189C22×D4—22no—
190C22×D4C2×D414yes—
191C22×D4C2×Q818yes—
192C23×C4—28no—
193C23×C4—24no—
194C23×C4—24no—
195C23×C4—28no—
196C23×C4—28no—
197C23×C4—44no—
198C23×C4—44no—
199C23×C4—44no—
200C23×C4—44no—
201C23×C4—44no—
202C23×C4—44no—
203C23×C4C2×D418yes—
204C23×C4C2412yes—
205C23×C4C2411yescharacteristic
206C23×C4C2412yes—
207C23×C4C2411yescharacteristic
208C23×C4C2411yescharacteristic
209C23×C4C2412yes—
210C23×C4C2414yes—
211C23×C4C2411yescharacteristic
212C23×C4C2414yes—
213C23×C4C2412yes—
214C23×C4C2412yes—
215C23×C4C2411yescharacteristic
216C23×C4C2412yes—
217C23×C4C2411yescharacteristic
218C23×C4C2412yes—
219C23×C4C2411yescharacteristic
220#125—416no—
221C4×D4—232no—
222C4×D4—432no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—24no—
225C2×C4⋊C4—24no—
226C2×C4⋊C4—24no—
227C2×C4⋊C4—24no—
228C2×C4⋊C4—28no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4—24no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—28no—
238C2×C4⋊C4—28no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—28no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—28no—
258C2×C4⋊C4—28no—
259C2×C4⋊C4—432no—
260C2×C4⋊C4—44no—
261C2×C4⋊C4—44no—
262C2×C4⋊C4—41no—
263C2×C4⋊C4—42no—
264C2×C4⋊C4—42no—
265C2×C4⋊C4—44no—
266C2×C4⋊C4—44no—
267C2×C4⋊C4—44no—
268C2×C4⋊C4—44no—
269C2×C4⋊C4—44no—
270C2×C4⋊C4—44no—
271C2×C4⋊C4—44no—
272C2×C4⋊C4—44no—
273C2×C4⋊C4—44no—
274C2×C4⋊C4—44no—
275C2×C4⋊C4—44no—
276C2×C4⋊C4—44no—
277C2×C4⋊C4—44no—
278C2×C4⋊C4C2×D414yes—
279C2×C4⋊C4C2×Q818yes—
280C2×C4⋊C4#4118yes—
281C2×C22⋊C4—28no—
282C2×C22⋊C4—28no—
283C2×C22⋊C4—28no—
284C2×C22⋊C4—28no—
285C2×C22⋊C4—24no—
286C2×C22⋊C4—24no—
287C2×C22⋊C4—24no—
288C2×C22⋊C4—24no—
289C2×C22⋊C4—24no—
290C2×C22⋊C4—44no—
291C2×C22⋊C4—42no—
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⋊C4C2×D418yes—
299C2×C22⋊C4C2×Q8116yes—
300C2×C22⋊C4C2×Q8116yes—
301C2×C42—24no—
302C2×C42—24no—
303C2×C42—24no—
304C2×C42—24no—
305C2×C42—22no—
306C2×C42—22no—
307C2×C42—24no—
308C2×C42—28no—
309C2×C42—28no—
310C2×C42—24no—
311C2×C42—24no—
312C2×C42—22no—
313C2×C42—22no—
314C2×C42—44no—
315C2×C42C2×D414yes—
316C2×C42C2×Q818yes—
317#95—22no—
318#95—24no—
319#95—22no—
320#95—28no—
321#95—24no—
322#95—24no—
323#95—24no—
324#95—28no—
325#95—24no—
326#95—24no—
327#95—24no—
328#95—28no—
329#95—24no—
330#95—22no—
331#95—42no—
332#95—44no—
333#95—42no—
334#95—42no—
335#95—42no—
336#95—44no—
337#95—44no—
338#95—44no—
339#95—44no—
340#95—44no—
341#95—44no—
342#95#4114yes—
343C24—22no—
344C24—24no—
345C24—24no—
346C24—24no—
347C24—22no—
348C24—22no—
349C24—22no—
350C24—22no—
351C24—22no—
352C24C22×D414yes—
353C24C22×Q818yes—
354C24C2511yescharacteristic
355C2×D4—28no—
356C2×D4—28no—
357C2×D4—216no—
358C2×D4—216no—
359C2×D4C4⋊Q8116yes—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—24no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—21no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—24no—
373C22×C4—24no—
374C22×C4—22no—
375C22×C4—21no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—21no—
381C22×C4—22no—
382C22×C4—21no—
383C22×C4—24no—
384C22×C4—24no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—24no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—21no—
392C22×C4—21no—
393C22×C4—24no—
394C22×C4—21no—
395C22×C4—22no—
396C22×C4—21no—
397C22×C4—216no—
398C22×C4—216no—
399C22×C4—22no—
400C22×C4—21no—
401C22×C4—22no—
402C22×C4—21no—
403C22×C4—21no—
404C22×C4—22no—
405C22×C4—24no—
406C22×C4—21no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—24no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—24no—
414C22×C4—24no—
415C22×C4—22no—
416C22×C4—21no—
417C22×C4—22no—
418C22×C4—24no—
419C22×C4—22no—
420C22×C4—24no—
421C22×C4—24no—
422C22×C4—21no—
423C22×C4—22no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—24no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—24no—
430C22×C4—22no—
431C22×C4—21no—
432C22×C4—22no—
433C22×C4—21no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—21no—
438C22×C4—48no—
439C22×C4—416no—
440C22×C4—416no—
441C22×C4—416no—
442C22×C4—416no—
443C22×C4—88no—
444C22×C4—88no—
445C22×C4—88no—
446C22×C4—88no—
447C22×C4—88no—
448C22×C4C22×D412yes—
449C22×C4C22×D414yes—
450C22×C4C22×D412yes—
451C22×C4C22×D412yes—
452C22×C4C22×Q814yes—
453C22×C4C22×Q818yes—
454C22×C4C22×Q814yes—
455C22×C4C22×Q814yes—
456C22×C4C2×#4118yes—
457C22×C4C2×#4112yes—
458C22×C4C2×#4114yes—
459C22×C4C2×#4114yes—
460C22×C4H(5,𝔽2)14yes—
461C22×C4H(5,𝔽2)14yes—
462C22×C4H(5,𝔽2)12yes—
463C22×C4H(5,𝔽2)12yes—
464C22×C42−1+412yes—
465C22×C42−1+412yes—
466C4⋊C4—28no—
467C4⋊C4—216no—
468C4⋊C4—48no—
469C4⋊C4—48no—
470C4⋊C4—44no—
471C4⋊C4—416no—
472C4⋊C4—48no—
473C4⋊C4—48no—
474C4⋊C4—48no—
475C4⋊C4—44no—
476C4⋊C4—44no—
477C4⋊C4—48no—
478C4⋊C4—44no—
479C4⋊C4—44no—
480C4⋊C4—84no—
481C4⋊C4—88no—
482C4⋊C4—84no—
483C4⋊C4—84no—
484C4⋊C4—88no—
485C4⋊C4—88no—
486C4⋊C4—88no—
487C4⋊C4#123116yes—
488C22⋊C4—216no—
489C22⋊C4—232no—
490C22⋊C4—416no—
491C22⋊C4—48no—
492C22⋊C4—416no—
493C22⋊C4—84no—
494C22⋊C4—84no—
495C22⋊C4—84no—
496C42—28no—
497C42—216no—
498C42—416no—
499C42—44no—
500C42—44no—
501C42—48no—
502C23—24no—
503C23—24no—
504C23—24no—
505C23—28no—
506C23—28no—
507C23—28no—
508C23—24no—
509C23—24no—
510C23—24no—
511C23—28no—
512C23—28no—
513C23—28no—
514C23—44no—
515C23—42no—
516C23—44no—
517C23—44no—
518C23—42no—
519C23—44no—
520C23—44no—
521C23—44no—
522C23—42no—
523C23—42no—
524C23C2×C4⋊Q818yes—
525C23C23×D411yescharacteristic
526C23C23×Q812yes—
527C23C22×#4112yes—
528C23C22×#4111yescharacteristic
529C23C2×H(5,𝔽2)11yescharacteristic
530C23C2×H(5,𝔽2)11yescharacteristic
531C23C2×H(5,𝔽2)12yes—
532C23C2×H(5,𝔽2)12yes—
533C23C2×2−1+411yescharacteristic
534C23C2×2−1+411yescharacteristic
535C23#58511yescharacteristic
536D4—232no—
537C2×C4—22no—
538C2×C4—24no—
539C2×C4—24no—
540C2×C4—24no—
541C2×C4—28no—
542C2×C4—24no—
543C2×C4—22no—
544C2×C4—24no—
545C2×C4—24no—
546C2×C4—24no—
547C2×C4—24no—
548C2×C4—28no—
549C2×C4—24no—
550C2×C4—24no—
551C2×C4—22no—
552C2×C4—24no—
553C2×C4—28no—
554C2×C4—28no—
555C2×C4—24no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—24no—
560C2×C4—28no—
561C2×C4—24no—
562C2×C4—22no—
563C2×C4—24no—
564C2×C4—24no—
565C2×C4—22no—
566C2×C4—432no—
567C2×C4—41no—
568C2×C4—42no—
569C2×C4—41no—
570C2×C4—44no—
571C2×C4—42no—
572C2×C4—41no—
573C2×C4—42no—
574C2×C4—44no—
575C2×C4—44no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—44no—
582C2×C4—42no—
583C2×C4—41no—
584C2×C4—42no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—41no—
590C2×C4—44no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—41no—
594C2×C4—42no—
595C2×C4—41no—
596C2×C4—42no—
597C2×C4—44no—
598C2×C4—41no—
599C2×C4—42no—
600C2×C4—41no—
601C2×C4—42no—
602C2×C4—41no—
603C2×C4—41no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—44no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—44no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—44no—
614C2×C4—42no—
615C2×C4—44no—
616C2×C4—44no—
617C2×C4—44no—
618C2×C4—42no—
619C2×C4—44no—
620C2×C4—44no—
621C2×C4—44no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—832no—
625C2×C4C2×#12318yes—
626C2×C4C2×#12314yes—
627C2×C4C2×C4⋊Q814yes—
628C2×C4#53514yes—
629C2×C4#54714yes—
630C2×C4#55418yes—
631C2×C4#55418yes—
632C2×C4#55418yes—
633C22—28no—
634C22—28no—
635C22—28no—
636C22—44no—
637C22—48no—
638C22—48no—
639C22—48no—
640C22—48no—
641C22—44no—
642C22—44no—
643C22—44no—
644C22C22×#12311yescharacteristic
645C22C22×C4⋊Q811yescharacteristic
646C22C2×#53511yescharacteristic
647C22C2×#54711yescharacteristic
648C22C2×#55412yes—
649C22C2×#55412yes—
650C22C2×#55412yes—
651C22#348112yes—
652C22#348411yescharacteristic
653C22#348911yescharacteristic
654C22#349811yescharacteristic
655C22#350112yes—
656C22#350312yes—
657C22#350312yes—
658C22#351711yescharacteristic
659C22#351712yes—
660C22#352011yescharacteristic
661C22#352012yes—
662C22#352212yes—
663C22#353712yes—
664C22#354312yes—
665C22#354711yescharacteristic
666C22#355211yescharacteristic
667C4—28no—
668C4—24no—
669C4—28no—
670C4—42no—
671C4—44no—
672C4—48no—
673C4—42no—
674C4—44no—
675C4—82no—
676C4—81no—
677C4—81no—
678C4—82no—
679C4—81no—
680C4—82no—
681C4—84no—
682C4#250218yes—
683C2—48no—
684C2C2×#250211yescharacteristic
685C2#3592811yescharacteristic
686C2#3600511yescharacteristic
687C2#3618011yescharacteristic
688C2#3623711yescharacteristic
689C2#3892011yescharacteristic
690C2#4243011yescharacteristic
691C2#4508612yes—
692C2#4535912yes—
693C2#5945412yes—
694C2#5946612yes—
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
3. #17057C2semidirect
4. #15545C2semidirect
5. #15392C2semidirect
6. #15196C2semidirect
7. #15194C2semidirect
8. #15145C2semidirect
9. #15092C2semidirect
10. #14840C2semidirect
11. #14054C2semidirect
14. #13910C2semidirect
15. #13880C2semidirect
16. C4×#398C2semidirect

Maximal subgroups

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

subgroupindex
C2×#24542
C2×D4×C4⋊C42
#170572
#155452
#153922
#151962
#151942
#151452
#150922
#148402
#140542
#140182
#139772
#139102
#138802
C4×#3982
#138202