Finite Group Databasebeta

← #876770

Subgroups — #876770

Order 512.

3486 conjugacy classes of subgroups.

The subgroup lattice has 752 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2683C212yesmaximal
2C2×#2857C212yesmaximal, complemented
3C2×#2440C212yesmaximal, complemented
4#16305C212yesmaximal, complemented
5#16254C212yesmaximal, complemented
6#15623C214yesmaximal, complemented
7#15276C214yesmaximal, complemented
8#15143C212yesmaximal, complemented
9#15131C212yesmaximal, complemented
10#15131C212yesmaximal, complemented
11#14401C212yesmaximal
12#14005C212yesmaximal, complemented
13#13906C211yesmaximal, characteristic
14#13872C211yesmaximal, characteristic
15C4×#386C211yesmaximal, characteristic
16C22×C4⋊Q8C2212yes—
17C22×C22⋊Q8C2214yes—
18C22×C22⋊Q8C2214yes—
19C22×C4×D4C2211yescharacteristic
20C23×C4⋊C4C2211yescharacteristic
21C23×C42C2211yescharacteristic
22#2857—216no—
23#2683—216no—
24#2440—216no—
25C2×#398C2214yes—
26C2×#396C2214yes—
27C2×#396C2214yes—
28C2×#395C2214yes—
29C2×#395C2214yes—
30C2×#395C2214yes—
31C2×#391C2214yes—
32C2×#390C2212yes—
33C2×#389C2214yes—
34C2×#389C2214yes—
35C2×#389C2212yes—
36C2×#389C2212yes—
37C2×#389C2212yes—
38C2×#389C2212yes—
39C2×#386C2212yes—
40C2×#386C2211yescharacteristic
41C2×#386C2211yescharacteristic
42C2×#385C2214yes—
43C2×#384C2212yes—
44C2×#384C2212yes—
45C2×#384C2211yescharacteristic
46C2×#380C2212yes—
47C2×#380C2212yes—
48C2×#380C2214yes—
49C2×#380C2212yes—
50C2×#380C2211yescharacteristic
51C2×C4×C4⋊C4C2211yescharacteristic
52C2×C4×C22⋊C4C2212yes—
53C2×C4×C22⋊C4C2211yescharacteristic
54#1444C2214yes—
55#1444C2214yes—
56#1444C2214yes—
57#1441C2212yes—
58#1441C2214yes—
59#1441C2214yes—
60#1441C2214yes—
61#1441C2214yes—
62#1441C2214yes—
63#1439C2214yes—
64#1439C2214yes—
65#1439C2212yes—
66#1439C2212yes—
67#1439C2212yes—
68#1437C2212yes—
69#1436C2212yes—
70#1436C2214yes—
71C4⋊#95C2212yes—
72C4⋊#95C2212yes—
73C4⋊#95C2212yes—
74C22⋊#95C2212yes—
75C22⋊#95C2211yescharacteristic
76C22⋊#95C2211yescharacteristic
77C4×#95C2212yes—
78C23×Q8C2314yes—
79C23×D4C2311yescharacteristic
80C24×C4C2311yescharacteristic
81C24×C4C2311yescharacteristic
82C24×C4C2311yescharacteristic
83C24×C4C2311yescharacteristic
84C24×C4C2312yes—
85C2×C4⋊Q8—28no—
86C2×C4⋊Q8—28no—
87C2×C4⋊Q8—44no—
88C2×C22⋊Q8—216no—
89C2×C22⋊Q8—216no—
90C2×C22⋊Q8—216no—
91C2×C22⋊Q8—216no—
92C2×C22⋊Q8—48no—
93C2×C22⋊Q8—48no—
94C2×C4×D4—42no—
95C2×C4×D4—42no—
96C2×C4×D4Q818yescomplemented
97C22×C4⋊C4—28no—
98C22×C4⋊C4—42no—
99C22×C4⋊C4C2314yes—
100C22×C4⋊C4C2312yes—
101C22×C4⋊C4C2312yes—
102C22×C4⋊C4C2312yes—
103C22×C4⋊C4C2314yes—
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⋊C4C2314yes—
113C22×C4⋊C4C2314yes—
114C22×C4⋊C4C2312yes—
115C22×C4⋊C4C2311yescharacteristic
116C22×C22⋊C4C2314yes—
117C22×C22⋊C4C2314yes—
118C22×C22⋊C4C2314yes—
119C22×C22⋊C4C2312yes—
120C22×C22⋊C4C2312yes—
121C22×C22⋊C4C2312yes—
122C22×C22⋊C4C2311yescharacteristic
123C22×C22⋊C4C2311yescharacteristic
124C22×C22⋊C4C2311yescharacteristic
125C22×C22⋊C4C2311yescharacteristic
126C22×C42—28no—
127C22×C42—42no—
128C22×C42C2312yes—
129C22×C42C2312yes—
130C22×C42C2312yes—
131C22×C42C2311yescharacteristic
132C22×C42C2312yes—
133#398—216no—
134#396—216no—
135#396—216no—
136#395—216no—
137#395—216no—
138#395—48no—
139#391—216no—
140#390—28no—
141#389—216no—
142#389—28no—
143#389—28no—
144#389—216no—
145#389—44no—
146#389D4116yescomplemented
147#386—28no—
148#386D418yes—
149#386Q818yescomplemented
150#385—216no—
151#384—28no—
152#384—28no—
153#384—42no—
154#380—28no—
155#380—28no—
156#380—28no—
157#380—216no—
158#380—42no—
159C4×C4⋊C4—42no—
160C4×C22⋊C4—28no—
161C4×C22⋊C4—42no—
162C2×#95C2312yes—
163C2×#95C2312yes—
164C2×#95C2314yes—
165C2×#95C2314yes—
166C2×#95C2314yes—
167C2×#95C2314yes—
168C2×#95C2312yes—
169C2×#95C2312yes—
170C2×#95C2314yes—
171C2×#95C2314yes—
172C2×#95C2314yes—
173C2×#95C2314yes—
174C2×#95C2314yes—
175C2×#95C2312yes—
176C2×#95C2314yes—
177C2×#95C2312yes—
178C2×#95C2312yes—
179C2×#95C2312yes—
180C2×#95C2314yes—
181C2×#95C2312yes—
182C25C2411yescharacteristic
183C25C2411yescharacteristic
184C22×Q8—28no—
185C22×Q8—28no—
186C22×Q8—28no—
187C22×Q8—28no—
188C22×Q8—44no—
189C22×Q8—44no—
190C22×Q8—44no—
191C22×D4—41no—
192C22×D4—41no—
193C22×D4—41no—
194C22×D4—41no—
195C22×D4C2×D414yes—
196C22×D4C2×Q814yes—
197C22×D4C2×Q814yes—
198C23×C4—24no—
199C23×C4—24no—
200C23×C4—24no—
201C23×C4—24no—
202C23×C4—22no—
203C23×C4—22no—
204C23×C4—22no—
205C23×C4—22no—
206C23×C4—24no—
207C23×C4—24no—
208C23×C4—24no—
209C23×C4—24no—
210C23×C4—24no—
211C23×C4—28no—
212C23×C4—41no—
213C23×C4—41no—
214C23×C4—41no—
215C23×C4—41no—
216C23×C4—42no—
217C23×C4—44no—
218C23×C4—42no—
219C23×C4C2×D414yes—
220C23×C4C2×Q814yes—
221C23×C4C2×Q814yes—
222C23×C4C2×Q814yes—
223C23×C4C2414yes—
224C23×C4C2414yes—
225C23×C4C2412yes—
226C23×C4C2414yes—
227C23×C4C2414yes—
228C23×C4C2412yes—
229C23×C4C2411yescharacteristic
230C23×C4C2411yescharacteristic
231C23×C4C2411yescharacteristic
232C23×C4C2412yes—
233C23×C4C2411yescharacteristic
234C23×C4C2412yes—
235C23×C4C2411yescharacteristic
236C4⋊Q8—432no—
237C22⋊Q8—464no—
238C22⋊Q8—464no—
239C4×D4—416no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—28no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—28no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—28no—
247C2×C4⋊C4—28no—
248C2×C4⋊C4—28no—
249C2×C4⋊C4—28no—
250C2×C4⋊C4—28no—
251C2×C4⋊C4—28no—
252C2×C4⋊C4—28no—
253C2×C4⋊C4—28no—
254C2×C4⋊C4—28no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—28no—
257C2×C4⋊C4—24no—
258C2×C4⋊C4—28no—
259C2×C4⋊C4—28no—
260C2×C4⋊C4—24no—
261C2×C4⋊C4—28no—
262C2×C4⋊C4—24no—
263C2×C4⋊C4—24no—
264C2×C4⋊C4—28no—
265C2×C4⋊C4—24no—
266C2×C4⋊C4—28no—
267C2×C4⋊C4—28no—
268C2×C4⋊C4—24no—
269C2×C4⋊C4—416no—
270C2×C4⋊C4—42no—
271C2×C4⋊C4—44no—
272C2×C4⋊C4—41no—
273C2×C4⋊C4—41no—
274C2×C4⋊C4—44no—
275C2×C4⋊C4—44no—
276C2×C4⋊C4—44no—
277C2×C4⋊C4—44no—
278C2×C4⋊C4—42no—
279C2×C4⋊C4—44no—
280C2×C4⋊C4—44no—
281C2×C4⋊C4—42no—
282C2×C4⋊C4—42no—
283C2×C4⋊C4—44no—
284C2×C4⋊C4—44no—
285C2×C4⋊C4—44no—
286C2×C4⋊C4C2×D418yes—
287C2×C4⋊C4C2×D4116yes—
288C2×C4⋊C4C2×D418yes—
289C2×C4⋊C4C2×Q814yes—
290C2×C22⋊C4—24no—
291C2×C22⋊C4—24no—
292C2×C22⋊C4—24no—
293C2×C22⋊C4—24no—
294C2×C22⋊C4—28no—
295C2×C22⋊C4—28no—
296C2×C22⋊C4—28no—
297C2×C22⋊C4—28no—
298C2×C22⋊C4—28no—
299C2×C22⋊C4—28no—
300C2×C22⋊C4—24no—
301C2×C22⋊C4—41no—
302C2×C22⋊C4—41no—
303C2×C22⋊C4—41no—
304C2×C22⋊C4—41no—
305C2×C22⋊C4—41no—
306C2×C22⋊C4—41no—
307C2×C22⋊C4—44no—
308C2×C22⋊C4—44no—
309C2×C22⋊C4—44no—
310C2×C22⋊C4—41no—
311C2×C22⋊C4—41no—
312C2×C22⋊C4—42no—
313C2×C22⋊C4—42no—
314C2×C22⋊C4C2×D418yes—
315C2×C22⋊C4C2×D414yes—
316C2×C22⋊C4C2×Q818yes—
317C2×C22⋊C4C2×Q814yes—
318C2×C22⋊C4C2×Q814yes—
319C2×C22⋊C4C2×Q814yes—
320C2×C42—24no—
321C2×C42—22no—
322C2×C42—22no—
323C2×C42—24no—
324C2×C42—24no—
325C2×C42—24no—
326C2×C42—24no—
327C2×C42—24no—
328C2×C42—24no—
329C2×C42—24no—
330C2×C42—416no—
331C2×C42—42no—
332C2×C42—42no—
333C2×C42—42no—
334C2×C42C2×D418yes—
335C2×C42C2×Q814yes—
336#95—24no—
337#95—24no—
338#95—28no—
339#95—28no—
340#95—28no—
341#95—28no—
342#95—28no—
343#95—28no—
344#95—28no—
345#95—24no—
346#95—24no—
347#95—42no—
348#95—42no—
349#95—42no—
350#95—44no—
351#95—44no—
352#95—44no—
353#95C2×D4116yes—
354#95C2×D418yes—
355#95C2×Q818yes—
356C24—22no—
357C24—22no—
358C24—22no—
359C24—22no—
360C24—21no—
361C24—21no—
362C24—21no—
363C24—21no—
364C24—22no—
365C24—22no—
366C24—22no—
367C24—22no—
368C24—21no—
369C24—21no—
370C24—21no—
371C24—21no—
372C24C22×D412yes—
373C24C22×D412yes—
374C24C22×Q812yes—
375C24C22×Q812yes—
376C24C22×Q812yes—
377C24C22×Q812yes—
378C24C2511yescharacteristic
379C2×Q8—416no—
380C2×Q8—416no—
381C2×Q8—416no—
382C2×Q8—416no—
383C2×Q8—88no—
384C2×Q8—88no—
385C2×Q8—88no—
386C2×D4—44no—
387C2×D4—44no—
388C2×D4—44no—
389C2×D4—44no—
390C2×D4—44no—
391C2×D4—44no—
392C2×D4C4⋊Q8116yescomplemented
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—21no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—21no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—21no—
402C22×C4—22no—
403C22×C4—21no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—21no—
416C22×C4—24no—
417C22×C4—21no—
418C22×C4—21no—
419C22×C4—22no—
420C22×C4—24no—
421C22×C4—21no—
422C22×C4—21no—
423C22×C4—22no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—21no—
427C22×C4—24no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—24no—
434C22×C4—24no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—24no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—24no—
441C22×C4—28no—
442C22×C4—28no—
443C22×C4—216no—
444C22×C4—216no—
445C22×C4—216no—
446C22×C4—216no—
447C22×C4—24no—
448C22×C4—24no—
449C22×C4—24no—
450C22×C4—24no—
451C22×C4—24no—
452C22×C4—24no—
453C22×C4—24no—
454C22×C4—24no—
455C22×C4—24no—
456C22×C4—24no—
457C22×C4—24no—
458C22×C4—24no—
459C22×C4—24no—
460C22×C4—24no—
461C22×C4—44no—
462C22×C4—44no—
463C22×C4—44no—
464C22×C4—44no—
465C22×C4—48no—
466C22×C4—48no—
467C22×C4—44no—
468C22×C4—44no—
469C22×C4—48no—
470C22×C4—416no—
471C22×C4—48no—
472C22×C4—48no—
473C22×C4—44no—
474C22×C4—44no—
475C22×C4—44no—
476C22×C4—44no—
477C22×C4—88no—
478C22×C4—84no—
479C22×C4—84no—
480C22×C4C22×D418yes—
481C22×C4C22×D414yes—
482C22×C4C22×D418yes—
483C22×C4C22×D412yes—
484C22×C4C22×D414yes—
485C22×C4C22×D414yes—
486C22×C4C22×D414yes—
487C22×C4C22×D414yes—
488C22×C4C22×Q814yes—
489C22×C4C22×Q812yes—
490C22×C4C22×Q812yes—
491C22×C4C22×Q812yes—
492C22×C4C22×Q812yes—
493C22×C4C22×Q812yes—
494C22×C4C22×Q814yes—
495C22×C4C22×Q812yes—
496C22×C4H(5,𝔽2)14yes—
497C22×C4H(5,𝔽2)12yes—
498C22×C4H(5,𝔽2)12yes—
499C22×C42−1+414yes—
500C22×C42−1+414yes—
501C4⋊C4—216no—
502C4⋊C4—216no—
503C4⋊C4—232no—
504C4⋊C4—44no—
505C4⋊C4—416no—
506C4⋊C4—48no—
507C4⋊C4—416no—
508C4⋊C4—416no—
509C4⋊C4—416no—
510C4⋊C4—416no—
511C4⋊C4—48no—
512C4⋊C4—416no—
513C4⋊C4—48no—
514C4⋊C4—84no—
515C4⋊C4—88no—
516C4⋊C4—88no—
517C4⋊C4—84no—
518C22⋊C4—216no—
519C22⋊C4—216no—
520C22⋊C4—44no—
521C22⋊C4—44no—
522C22⋊C4—44no—
523C22⋊C4—44no—
524C22⋊C4—416no—
525C22⋊C4—416no—
526C22⋊C4—416no—
527C22⋊C4—84no—
528C42—28no—
529C42—216no—
530C42—48no—
531C42—48no—
532C42—84no—
533C23—22no—
534C23—22no—
535C23—22no—
536C23—22no—
537C23—24no—
538C23—24no—
539C23—24no—
540C23—24no—
541C23—24no—
542C23—24no—
543C23—22no—
544C23—22no—
545C23—22no—
546C23—22no—
547C23—24no—
548C23—24no—
549C23—22no—
550C23—22no—
551C23—22no—
552C23—22no—
553C23—24no—
554C23—24no—
555C23—24no—
556C23—24no—
557C23—42no—
558C23—42no—
559C23—41no—
560C23—41no—
561C23—42no—
562C23—42no—
563C23—42no—
564C23—42no—
565C23—41no—
566C23—41no—
567C23—41no—
568C23—41no—
569C23—41no—
570C23—41no—
571C23—42no—
572C23—42no—
573C23—42no—
574C23—42no—
575C23—42no—
576C23—42no—
577C23C2×C4⋊Q814yes—
578C23C2×C4⋊Q814yes—
579C23C23×D411yescharacteristic
580C23C23×D412yes—
581C23C23×Q811yescharacteristic
582C23C23×Q811yescharacteristic
583C23C2×H(5,𝔽2)11yescharacteristic
584C23C2×H(5,𝔽2)11yescharacteristic
585C23C2×H(5,𝔽2)12yes—
586C23C2×2−1+412yes—
587C23C2×2−1+412yes—
588C23#58511yescharacteristic
589C23#58511yescharacteristic
590Q8—832no—
591D4—416no—
592C2×C4—22no—
593C2×C4—22no—
594C2×C4—24no—
595C2×C4—24no—
596C2×C4—24no—
597C2×C4—24no—
598C2×C4—24no—
599C2×C4—24no—
600C2×C4—24no—
601C2×C4—24no—
602C2×C4—22no—
603C2×C4—22no—
604C2×C4—24no—
605C2×C4—24no—
606C2×C4—22no—
607C2×C4—22no—
608C2×C4—24no—
609C2×C4—24no—
610C2×C4—28no—
611C2×C4—22no—
612C2×C4—22no—
613C2×C4—22no—
614C2×C4—24no—
615C2×C4—28no—
616C2×C4—28no—
617C2×C4—28no—
618C2×C4—24no—
619C2×C4—24no—
620C2×C4—24no—
621C2×C4—28no—
622C2×C4—28no—
623C2×C4—24no—
624C2×C4—24no—
625C2×C4—24no—
626C2×C4—24no—
627C2×C4—24no—
628C2×C4—416no—
629C2×C4—416no—
630C2×C4—42no—
631C2×C4—42no—
632C2×C4—42no—
633C2×C4—42no—
634C2×C4—41no—
635C2×C4—41no—
636C2×C4—44no—
637C2×C4—42no—
638C2×C4—44no—
639C2×C4—42no—
640C2×C4—44no—
641C2×C4—44no—
642C2×C4—44no—
643C2×C4—41no—
644C2×C4—41no—
645C2×C4—41no—
646C2×C4—44no—
647C2×C4—41no—
648C2×C4—42no—
649C2×C4—44no—
650C2×C4—44no—
651C2×C4—42no—
652C2×C4—42no—
653C2×C4—42no—
654C2×C4—42no—
655C2×C4—416no—
656C2×C4—416no—
657C2×C4—44no—
658C2×C4—44no—
659C2×C4—44no—
660C2×C4—44no—
661C2×C4—44no—
662C2×C4—44no—
663C2×C4—44no—
664C2×C4—44no—
665C2×C4—44no—
666C2×C4—44no—
667C2×C4—44no—
668C2×C4—44no—
669C2×C4—44no—
670C2×C4—44no—
671C2×C4—816no—
672C2×C4C2×C4⋊Q814yes—
673C2×C4#53718yes—
674C2×C4#54314yes—
675C2×C4D4218yes—
676C2×C4D4214yes—
677C2×C4D4×Q818yes—
678C2×C4D4×Q818yes—
679C2×C4#56314yes—
680C22—28no—
681C22—28no—
682C22—24no—
683C22—24no—
684C22—44no—
685C22—44no—
686C22—42no—
687C22—42no—
688C22—42no—
689C22—42no—
690C22—42no—
691C22—42no—
692C22—44no—
693C22—44no—
694C22—44no—
695C22—44no—
696C22—42no—
697C22—42no—
698C22—42no—
699C22—42no—
700C22—42no—
701C22—42no—
702C22C22×C4⋊Q811yescharacteristic
703C22C2×#53712yes—
704C22C2×#54311yescharacteristic
705C22#345811yescharacteristic
706C22C2×D4212yes—
707C22C2×D4211yescharacteristic
708C22C2×D4×Q812yes—
709C22C2×D4×Q812yes—
710C22#348812yes—
711C22#348912yes—
712C22#349611yescharacteristic
713C22#349611yescharacteristic
714C22#349912yes—
715C22#350111yescharacteristic
716C22#350111yescharacteristic
717C22#350312yes—
718C22#350312yes—
719C22C2×#56311yescharacteristic
720C22#353711yescharacteristic
721C22#353811yescharacteristic
722C22#354712yes—
723C22#354812yes—
724C22#355512yes—
725C4—28no—
726C4—28no—
727C4—24no—
728C4—48no—
729C4—42no—
730C4—42no—
731C4—48no—
732C4—44no—
733C4—44no—
734C4—42no—
735C4—84no—
736C4—84no—
737C4#268318yescomplemented
738C2—44no—
739C2—44no—
740C2#3500211yescharacteristic
741C2D4×C4⋊Q812yes—
742C2#3730711yescharacteristic
743C2#3769012yes—
744C2#3934012yes—
745C2#3959812yes—
746C2#4548811yescharacteristic
747C2#4550211yescharacteristic
748C2C2×#268311yescharacteristic
749C2#5899711yescharacteristic
750C2#5899811yescharacteristic
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
2. C2×#2857C2semidirect
3. C2×#2440C2semidirect
4. #16305C2semidirect
5. #16254C2semidirect
6. #15623C2semidirect
7. #15276C2semidirect
8. #15143C2semidirect
9. #15131C2semidirect
10. #15131C2semidirect
12. #14005C2semidirect
96. C2×C4×D4Q8semidirect
146. #389D4semidirect
149. #386Q8semidirect
392. C2×D4C4⋊Q8semidirect
737. C4#2683semidirect

Maximal subgroups

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

subgroupindex
C2×#26832
C2×#28572
C2×#24402
#163052
#162542
#156232
#152762
#151432
#151312
#144012
#140052
#139062
#138722
C4×#3862