Finite Group Databasebeta

← #4458105

Subgroups — #4458105

Order 512.

2268 conjugacy classes of subgroups.

The subgroup lattice has 681 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2552C211yesmaximal, characteristic, complemented
2C2×#2314C211yesmaximal, characteristic
3#16984C212yesmaximal
4#16981C212yesmaximal, complemented
5#16605C212yesmaximal, complemented
6#16604C212yesmaximal
7#16332C214yesmaximal, complemented
8#16330C214yesmaximal
9#15225C212yesmaximal, complemented
10#15145C212yesmaximal, complemented
11#15131C212yesmaximal
12#15131C212yesmaximal
13#14340C211yesmaximal, characteristic
14#14150C211yesmaximal, characteristic, complemented
15#13954C211yesmaximal, characteristic, complemented
16#13910C211yesmaximal, characteristic, complemented
17#13852C211yesmaximal, characteristic
18C22×#125C2212yes—
19C22×#117C2211yescharacteristic
20#2552—28no—
21C2×#398C2214yes—
22C2×#398C2214yes—
23C2×#398C2214yes—
24C2×#396C2214yes—
25C2×#396C2214yes—
26#2314—28no—
27C2×#389C2212yes—
28C2×#389C2212yes—
29C2×#389C2212yes—
30C2×#389C2212yes—
31C2×#389C2212yes—
32C2×#389C2214yes—
33C2×#388C2212yes—
34C2×#388C2212yes—
35C2×#387C2212yes—
36C2×#387C2212yes—
37C2×#384C2211yescharacteristic
38C2×#384C2212yes—
39C2×#384C2211yescharacteristic
40C2×#384C2212yes—
41C2×#382C2211yescharacteristic
42C2×#382C2211yescharacteristic
43C2×#380C2211yescharacteristic
44C2×#380C2211yescharacteristic
45C2×#380C2212yes—
46C2×C4×C4⋊C4C2211yescharacteristic
47C2×C4×C4⋊C4C2211yescharacteristic
48C2×C4×C4⋊C4C2211yescharacteristic
49C2×C4×C4⋊C4C2211yescharacteristic
50C2×C4×C22⋊C4C2212yes—
51#1445C2214yes—
52#1444C2214yes—
53#1444C2214yes—
54#1444C2212yes—
55#1444C2212yes—
56#1444C2212yes—
57#1444C2214yes—
58#1442C2212yes—
59#1442C2214yes—
60#1441C2212yes—
61#1441C2212yes—
62#1441C2214yes—
63#1441C2214yes—
64#1441C2214yes—
65#1441C2214yes—
66#1441C2211yescharacteristic
67#1441C2212yes—
68#1441C2211yescharacteristic
69#1441C2214yes—
70#1441C2214yes—
71#1439C2212yes—
72#1439C2212yes—
73#1439C2214yes—
74#1439C2214yes—
75#1439C2211yescharacteristic
76#1439C2212yes—
77#1439C2211yescharacteristic
78#1436C2214yes—
79C4⋊#95C2212yes—
80C4⋊#95C2212yes—
81C4⋊#95C2212yes—
82C4⋊#95C2212yes—
83C4⋊#95C2212yes—
84C4×#95C2212yes—
85C24×C4C2311yescharacteristic
86C2×#125—28no—
87C2×#125—48no—
88C2×#117—24no—
89C2×#117—24no—
90C2×#117Q818yes—
91C22×C4⋊C4C2311yescharacteristic
92C22×C4⋊C4C2311yescharacteristic
93C22×C4⋊C4C2311yescharacteristic
94C22×C4⋊C4C2312yes—
95C22×C4⋊C4C2312yes—
96C22×C4⋊C4C2311yescharacteristic
97C22×C4⋊C4C2312yes—
98C22×C4⋊C4C2314yes—
99C22×C4⋊C4C2314yes—
100C22×C4⋊C4C2311yescharacteristic
101C22×C4⋊C4C2311yescharacteristic
102C22×C4⋊C4C2311yescharacteristic
103C22×C4⋊C4C2311yescharacteristic
104C22×C4⋊C4C2314yes—
105C22×C4⋊C4C2314yes—
106C22×C4⋊C4C2314yes—
107C22×C4⋊C4C2314yes—
108C22×C4⋊C4C2314yes—
109C22×C4⋊C4C2312yes—
110C22×C4⋊C4C2312yes—
111C22×C4⋊C4C2312yes—
112C22×C4⋊C4C2312yes—
113C22×C4⋊C4C2314yes—
114C22×C22⋊C4C2312yes—
115C22×C22⋊C4C2312yes—
116C22×C22⋊C4C2311yescharacteristic
117C22×C22⋊C4C2311yescharacteristic
118C22×C22⋊C4C2312yes—
119C22×C22⋊C4C2312yes—
120C22×C22⋊C4C2314yes—
121C22×C42C2312yes—
122C22×C42C2312yes—
123C22×C42C2311yescharacteristic
124C22×C42C2312yes—
125C22×C42C2312yes—
126C22×C42C2311yescharacteristic
127#398—216no—
128#398—48no—
129#398—48no—
130#396—48no—
131#396—48no—
132#389—28no—
133#389—28no—
134#389—48no—
135#389—44no—
136#389—44no—
137#389—44no—
138#388—28no—
139#388—28no—
140#387—28no—
141#387—28no—
142#384—24no—
143#384—28no—
144#384—24no—
145#384—44no—
146#382—24no—
147#382—24no—
148#380—24no—
149#380—24no—
150#380—44no—
151C4×C4⋊C4—24no—
152C4×C4⋊C4—24no—
153C4×C4⋊C4—24no—
154C4×C4⋊C4D418yes—
155C4×C22⋊C4—44no—
156C2×#95C2314yes—
157C2×#95C2314yes—
158C2×#95C2312yes—
159C2×#95C2312yes—
160C2×#95C2312yes—
161C2×#95C2314yes—
162C2×#95C2312yes—
163C2×#95C2312yes—
164C2×#95C2312yes—
165C2×#95C2311yescharacteristic
166C2×#95C2311yescharacteristic
167C2×#95C2312yes—
168C2×#95C2312yes—
169C2×#95C2312yes—
170C2×#95C2312yes—
171C2×#95C2312yes—
172C2×#95C2312yes—
173C2×#95C2314yes—
174C2×#95C2314yes—
175C2×#95C2314yes—
176C2×#95C2314yes—
177C2×#95C2314yes—
178C2×#95C2314yes—
179C2×#95C2314yes—
180C2×#95C2312yes—
181C2×#95C2312yes—
182C2×#95C2314yes—
183C2×#95C2312yes—
184C25C2411yescharacteristic
185C23×C4—22no—
186C23×C4—22no—
187C23×C4—22no—
188C23×C4—22no—
189C23×C4—22no—
190C23×C4—22no—
191C23×C4C2×Q814yes—
192C23×C4C2414yes—
193C23×C4C2412yes—
194C23×C4C2414yes—
195C23×C4C2412yes—
196C23×C4C2412yes—
197C23×C4C2412yes—
198C23×C4C2411yescharacteristic
199C23×C4C2411yescharacteristic
200C23×C4C2411yescharacteristic
201C23×C4C2412yes—
202C23×C4C2412yes—
203C23×C4C2412yes—
204C23×C4C2412yes—
205C23×C4C2411yescharacteristic
206C23×C4C2411yescharacteristic
207C23×C4C2411yescharacteristic
208#125—432no—
209#117—232no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—24no—
217C2×C4⋊C4—24no—
218C2×C4⋊C4—24no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—24no—
222C2×C4⋊C4—24no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—24no—
226C2×C4⋊C4—28no—
227C2×C4⋊C4—28no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—24no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4—28no—
233C2×C4⋊C4—28no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—28no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—28no—
247C2×C4⋊C4—28no—
248C2×C4⋊C4—28no—
249C2×C4⋊C4—28no—
250C2×C4⋊C4—44no—
251C2×C4⋊C4—44no—
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⋊C4—44no—
261C2×C4⋊C4—44no—
262C2×C4⋊C4—44no—
263C2×C4⋊C4—44no—
264C2×C4⋊C4—44no—
265C2×C4⋊C4—44no—
266C2×C4⋊C4—44no—
267C2×C4⋊C4—44no—
268C2×C4⋊C4—44no—
269C2×C4⋊C4C2×D414yes—
270C2×C4⋊C4C2×D414yes—
271C2×C4⋊C4C2×Q814yes—
272C2×C4⋊C4C2×Q814yes—
273C2×C22⋊C4—22no—
274C2×C22⋊C4—22no—
275C2×C22⋊C4—22no—
276C2×C22⋊C4—22no—
277C2×C22⋊C4—24no—
278C2×C22⋊C4—24no—
279C2×C22⋊C4—24no—
280C2×C22⋊C4—24no—
281C2×C22⋊C4—24no—
282C2×C22⋊C4—24no—
283C2×C22⋊C4—42no—
284C2×C22⋊C4—42no—
285C2×C22⋊C4—44no—
286C2×C22⋊C4—44no—
287C2×C22⋊C4—44no—
288C2×C22⋊C4—44no—
289C2×C22⋊C4—44no—
290C2×C22⋊C4C2×Q814yes—
291C2×C22⋊C4C2×Q814yes—
292C2×C42—22no—
293C2×C42—22no—
294C2×C42—24no—
295C2×C42—24no—
296C2×C42—24no—
297C2×C42—24no—
298C2×C42—24no—
299C2×C42—24no—
300C2×C42—24no—
301C2×C42—44no—
302C2×C42—44no—
303C2×C42C2×D418yes—
304C2×C42C2×D414yes—
305C2×C42C2×Q814yes—
306C2×C42C2×Q814yes—
307C2×C42#4114yes—
308#95—24no—
309#95—24no—
310#95—24no—
311#95—24no—
312#95—24no—
313#95—24no—
314#95—28no—
315#95—22no—
316#95—22no—
317#95—24no—
318#95—24no—
319#95—28no—
320#95—28no—
321#95—24no—
322#95—24no—
323#95—24no—
324#95—24no—
325#95—42no—
326#95—42no—
327#95—44no—
328#95—44no—
329#95—44no—
330#95—44no—
331#95—44no—
332#95—44no—
333#95—44no—
334#95—44no—
335#95C2×D418yes—
336C24—21no—
337C24—21no—
338C24—22no—
339C24—22no—
340C24—21no—
341C24—21no—
342C24—22no—
343C24—22no—
344C24—21no—
345C24—21no—
346C24C22×Q812yes—
347C24C2511yescharacteristic
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—21no—
353C22×C4—22no—
354C22×C4—21no—
355C22×C4—21no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—21no—
360C22×C4—21no—
361C22×C4—21no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—21no—
367C22×C4—21no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—24no—
378C22×C4—24no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—24no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—24no—
396C22×C4—21no—
397C22×C4—21no—
398C22×C4—28no—
399C22×C4—28no—
400C22×C4—28no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—24no—
404C22×C4—24no—
405C22×C4—24no—
406C22×C4—24no—
407C22×C4—24no—
408C22×C4—24no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—22no—
412C22×C4—24no—
413C22×C4—24no—
414C22×C4—24no—
415C22×C4—24no—
416C22×C4—24no—
417C22×C4—24no—
418C22×C4—22no—
419C22×C4—24no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—24no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—44no—
426C22×C4—44no—
427C22×C4—44no—
428C22×C4—44no—
429C22×C4C22×D414yes—
430C22×C4C22×D414yes—
431C22×C4C22×D412yes—
432C22×C4C22×D412yes—
433C22×C4C22×D412yes—
434C22×C4C22×Q812yes—
435C22×C4C22×Q812yes—
436C22×C4C22×Q812yes—
437C22×C4C22×Q812yes—
438C22×C4C22×Q812yes—
439C22×C4C22×Q812yes—
440C22×C4C2×#4112yes—
441C22×C4C2×#4112yes—
442C22×C4C2×#4112yes—
443C22×C4H(5,𝔽2)14yes—
444C22×C4H(5,𝔽2)14yes—
445C22×C4H(5,𝔽2)12yes—
446C22×C4H(5,𝔽2)12yes—
447C22×C42−1+414yes—
448C22×C42−1+414yes—
449C22×C42−1+412yes—
450C22×C42−1+412yes—
451C4⋊C4—28no—
452C4⋊C4—216no—
453C4⋊C4—28no—
454C4⋊C4—28no—
455C4⋊C4—28no—
456C4⋊C4—28no—
457C4⋊C4—28no—
458C4⋊C4—416no—
459C4⋊C4—416no—
460C4⋊C4—48no—
461C4⋊C4—44no—
462C4⋊C4—44no—
463C4⋊C4—48no—
464C4⋊C4—48no—
465C4⋊C4—84no—
466C4⋊C4—84no—
467C4⋊C4—84no—
468C4⋊C4—88no—
469C4⋊C4—88no—
470C4⋊C4—88no—
471C4⋊C4—88no—
472C4⋊C4—88no—
473C4⋊C4—88no—
474C22⋊C4—28no—
475C22⋊C4—28no—
476C22⋊C4—48no—
477C22⋊C4—48no—
478C22⋊C4—84no—
479C22⋊C4—84no—
480C22⋊C4—88no—
481C42—28no—
482C42—48no—
483C42—48no—
484C42—48no—
485C42—84no—
486C42C22⋊Q8116yes—
487C23—22no—
488C23—22no—
489C23—22no—
490C23—24no—
491C23—24no—
492C23—41no—
493C23—41no—
494C23—41no—
495C23—41no—
496C23—41no—
497C23—41no—
498C23—42no—
499C23—41no—
500C23—41no—
501C23—42no—
502C23—42no—
503C23—42no—
504C23—42no—
505C23—42no—
506C23—42no—
507C23—42no—
508C23—42no—
509C23—42no—
510C23C23×D411yescharacteristic
511C23C23×Q811yescharacteristic
512C23C22×#4111yescharacteristic
513C23C2×H(5,𝔽2)11yescharacteristic
514C23C2×H(5,𝔽2)11yescharacteristic
515C23C2×H(5,𝔽2)12yes—
516C23C2×H(5,𝔽2)12yes—
517C23C2×2−1+411yescharacteristic
518C23C2×2−1+411yescharacteristic
519C23C2×2−1+412yes—
520C23C2×2−1+412yes—
521C2×C4—22no—
522C2×C4—22no—
523C2×C4—24no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—24no—
527C2×C4—24no—
528C2×C4—24no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—24no—
533C2×C4—24no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—24no—
537C2×C4—24no—
538C2×C4—24no—
539C2×C4—24no—
540C2×C4—24no—
541C2×C4—24no—
542C2×C4—24no—
543C2×C4—24no—
544C2×C4—24no—
545C2×C4—24no—
546C2×C4—24no—
547C2×C4—24no—
548C2×C4—24no—
549C2×C4—24no—
550C2×C4—24no—
551C2×C4—24no—
552C2×C4—24no—
553C2×C4—24no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—41no—
567C2×C4—42no—
568C2×C4—41no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—44no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—44no—
579C2×C4—44no—
580C2×C4—42no—
581C2×C4—44no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—416no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—44no—
588C2×C4—44no—
589C2×C4—42no—
590C2×C4—44no—
591C2×C4—42no—
592C2×C4—44no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—44no—
596C2×C4—44no—
597C2×C4—44no—
598C2×C4—44no—
599C2×C4—44no—
600C2×C4—44no—
601C2×C4—44no—
602C2×C4—44no—
603C2×C4—44no—
604C2×C4—44no—
605C2×C4—44no—
606C2×C4—44no—
607C2×C4—44no—
608C2×C4—44no—
609C2×C4C2×C22⋊Q814yes—
610C2×C4C2×C22⋊Q814yes—
611C2×C4C2×C22⋊Q814yes—
612C2×C4#53714yes—
613C2×C4#53714yes—
614C2×C4#54014yes—
615C2×C4#54114yes—
616C2×C4#54414yes—
617C2×C4#54414yes—
618C22—42no—
619C22—44no—
620C22—44no—
621C22—42no—
622C22—42no—
623C22—81no—
624C22—81no—
625C22—81no—
626C22—81no—
627C22—82no—
628C22—82no—
629C22C22×C22⋊Q811yescharacteristic
630C22C2×#53711yescharacteristic
631C22C2×#53711yescharacteristic
632C22C2×#54011yescharacteristic
633C22C2×#54111yescharacteristic
634C22C2×#54411yescharacteristic
635C22C2×#54411yescharacteristic
636C22#348411yescharacteristic
637C22#348611yescharacteristic
638C22#349212yes—
639C22#349811yescharacteristic
640C22#349911yescharacteristic
641C22#350312yes—
642C22#350312yes—
643C22#351112yes—
644C22#353612yes—
645C22#353612yes—
646C22#354312yes—
647C22#354412yes—
648C22#354812yes—
649C22#355212yes—
650C22#355212yes—
651C22#355612yes—
652C4—24no—
653C4—24no—
654C4—44no—
655C4—42no—
656C4—44no—
657C4—44no—
658C4—44no—
659C4—42no—
660C4—44no—
661C4—44no—
662C4—42no—
663C4—82no—
664C4—82no—
665C4—84no—
666C4—84no—
667C4#262918yes—
668C2—82no—
669C2C2×#262911yescharacteristic
670C2#3553211yescharacteristic
671C2#3557411yescharacteristic
672C2#3730912yes—
673C2#3731212yes—
674C2#4242511yescharacteristic
675C2#4242811yescharacteristic
676C2#4245512yes—
677C2#4246012yes—
678C2#4279111yescharacteristic
679C2#4279411yescharacteristic
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×#2552C2semidirect
4. #16981C2semidirect
5. #16605C2semidirect
7. #16332C2semidirect
9. #15225C2semidirect
10. #15145C2semidirect
14. #14150C2semidirect
15. #13954C2semidirect
16. #13910C2semidirect

Maximal subgroups

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

subgroupindex
C2×#25522
C2×#23142
#169842
#169812
#166052
#166042
#163322
#163302
#152252
#151452
#151312
#143402
#141502
#139542
#139102
#138522