Finite Group Databasebeta

← #776138

Subgroups — #776138

Order 512.

3578 conjugacy classes of subgroups.

The subgroup lattice has 675 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×Q82C211yesmaximal, characteristic
2C2×#2550C212yesmaximal
3C2×#2453C211yesmaximal, characteristic
4#16603C212yesmaximal
5#15783C214yesmaximal
6#15318C214yesmaximal
7#15230C211yesmaximal, characteristic
8#15188C212yesmaximal
9#15158C211yesmaximal, characteristic
10#15128C211yesmaximal, characteristic
11#14880C212yesmaximal
12#14186C212yesmaximal
13#14053C212yesmaximal
14#14017C212yesmaximal
15#13978C212yesmaximal
16C4×#395C211yesmaximal, characteristic
17#13817C211yesmaximal, characteristic
18C2×Q82—216no—
19C2×Q82—28no—
20C22×C4⋊Q8C2211yescharacteristic
21C22×C4⋊Q8C2212yes—
22C22×C4⋊Q8C2214yes—
23C22×C4⋊Q8C2212yes—
24C22×#125C2211yescharacteristic
25C22×C4×Q8C2212yes—
26C22×C4×Q8C2211yescharacteristic
27C22×C4×Q8C2212yes—
28C22×C4×Q8C2211yescharacteristic
29#2550—216no—
30#2453—28no—
31C2×#400C2212yes—
32C2×#398C2214yes—
33C2×#398C2212yes—
34C2×#395C2211yescharacteristic
35C2×#395C2212yes—
36C2×#395C2214yes—
37C2×#395C2212yes—
38C2×#391C2214yes—
39C2×#391C2211yescharacteristic
40C2×#391C2211yescharacteristic
41C2×#391C2214yes—
42C2×#391C2212yes—
43C2×#389C2212yes—
44C2×#389C2212yes—
45C2×#389C2212yes—
46C2×#389C2211yescharacteristic
47C2×#389C2214yes—
48C2×#389C2211yescharacteristic
49C2×#387C2212yes—
50C2×#382C2212yes—
51C2×C4×C4⋊C4C2211yescharacteristic
52C2×C4×C4⋊C4C2212yes—
53C2×C4×C4⋊C4C2212yes—
54#1446C2212yes—
55#1446C2211yescharacteristic
56#1446C2214yes—
57#1446C2214yes—
58#1444C2212yes—
59#1444C2212yes—
60#1444C2211yescharacteristic
61#1444C2214yes—
62#1442C2214yes—
63#1442C2214yes—
64#1442C2214yes—
65#1441C2212yes—
66#1441C2212yes—
67#1441C2214yes—
68#1441C2211yescharacteristic
69#1441C2212yes—
70#1441C2212yes—
71#1441C2214yes—
72#1441C2212yes—
73#1436C2214yes—
74#1436C2214yes—
75#1436C2214yes—
76#1436C2214yes—
77#1436C2214yes—
78#1436C2214yes—
79C4⋊#95C2211yescharacteristic
80C4⋊#95C2212yes—
81C4⋊#95C2212yes—
82C4⋊#95C2212yes—
83#1434C2212yes—
84C4×#95C2211yescharacteristic
85C4×#95C2212yes—
86C4×#95C2212yes—
87C23×Q8C2311yescharacteristic
88C23×Q8C2311yescharacteristic
89C23×Q8C2312yes—
90C23×Q8C2314yes—
91Q82—2128no—
92C2×C4⋊Q8—216no—
93C2×C4⋊Q8—216no—
94C2×C4⋊Q8—28no—
95C2×C4⋊Q8—216no—
96C2×C4⋊Q8—216no—
97C2×C4⋊Q8—28no—
98C2×C4⋊Q8—28no—
99C2×C4⋊Q8—24no—
100C2×C4⋊Q8—28no—
101C2×C4⋊Q8D4116yes—
102C2×#125—24no—
103C2×#125—28no—
104C2×C4×Q8—28no—
105C2×C4×Q8—28no—
106C2×C4×Q8—216no—
107C2×C4×Q8—28no—
108C2×C4×Q8—28no—
109C2×C4×Q8—28no—
110C2×C4×Q8—24no—
111C2×C4×Q8—28no—
112C2×C4×Q8D418yes—
113C22×C4⋊C4C2312yes—
114C22×C4⋊C4C2312yes—
115C22×C4⋊C4C2314yes—
116C22×C4⋊C4C2311yescharacteristic
117C22×C4⋊C4C2312yes—
118C22×C4⋊C4C2312yes—
119C22×C4⋊C4C2312yes—
120C22×C4⋊C4C2311yescharacteristic
121C22×C4⋊C4C2312yes—
122C22×C4⋊C4C2314yes—
123C22×C4⋊C4C2311yescharacteristic
124C22×C4⋊C4C2311yescharacteristic
125C22×C4⋊C4C2311yescharacteristic
126C22×C4⋊C4C2312yes—
127C22×C4⋊C4C2312yes—
128C22×C4⋊C4C2314yes—
129C22×C4⋊C4C2312yes—
130C22×C4⋊C4C2312yes—
131C22×C4⋊C4C2314yes—
132C22×C4⋊C4C2312yes—
133C22×C4⋊C4C2311yescharacteristic
134C22×C42C2314yes—
135C22×C42C2311yescharacteristic
136C22×C42C2311yescharacteristic
137C22×C42C2312yes—
138C22×C42C2312yes—
139C22×C42C2311yescharacteristic
140C22×C42C2312yes—
141C22×C42C2314yes—
142#400—28no—
143#398—28no—
144#398—216no—
145#395—216no—
146#395—24no—
147#395—28no—
148#395—44no—
149#391—24no—
150#391—24no—
151#391—216no—
152#391—48no—
153#391Q8116yescomplemented
154#389—24no—
155#389—216no—
156#389—28no—
157#389—28no—
158#389—24no—
159#389—28no—
160#387—28no—
161#382—28no—
162C4×C4⋊C4—28no—
163C4×C4⋊C4—24no—
164C4×C4⋊C4—28no—
165C2×#95C2314yes—
166C2×#95C2314yes—
167C2×#95C2311yescharacteristic
168C2×#95C2312yes—
169C2×#95C2312yes—
170C2×#95C2314yes—
171C2×#95C2314yes—
172C2×#95C2314yes—
173C2×#95C2312yes—
174C2×#95C2312yes—
175C2×#95C2312yes—
176C2×#95C2314yes—
177C2×#95C2311yescharacteristic
178C2×#95C2312yes—
179C2×#95C2312yes—
180C2×#95C2312yes—
181C2×#95C2311yescharacteristic
182C2×#95C2312yes—
183C2×#95C2312yes—
184C2×#95C2312yes—
185C2×#95C2314yes—
186C2×#95C2312yes—
187C2×#95C2311yescharacteristic
188C2×#95C2312yes—
189C2×#95C2312yes—
190C2×#95C2312yes—
191C2×#95C2314yes—
192C2×#95C2314yes—
193C2×#95C2314yes—
194C2×#95C2314yes—
195C2×#95C2314yes—
196C2×#95C2314yes—
197C22×Q8—24no—
198C22×Q8—22no—
199C22×Q8—24no—
200C22×Q8—28no—
201C22×Q8—24no—
202C22×Q8—28no—
203C22×Q8—28no—
204C22×Q8—22no—
205C22×Q8—24no—
206C22×Q8—22no—
207C22×Q8—22no—
208C22×Q8—44no—
209C22×Q8—44no—
210C22×Q8—44no—
211C22×Q8—44no—
212C22×Q8—42no—
213C22×Q8—44no—
214C22×Q8—42no—
215C22×Q8C2×D414yes—
216C22×Q8C2×D4116yes—
217C22×Q8C2×Q818yes—
218C22×Q8#4114yes—
219C23×C4C2411yescharacteristic
220C23×C4C2412yes—
221C23×C4C2411yescharacteristic
222C23×C4C2412yes—
223C23×C4C2411yescharacteristic
224C23×C4C2412yes—
225C23×C4C2414yes—
226C23×C4C2411yescharacteristic
227C23×C4C2414yes—
228C23×C4C2412yes—
229C23×C4C2412yes—
230C23×C4C2411yescharacteristic
231C23×C4C2412yes—
232C23×C4C2412yes—
233C23×C4C2412yes—
234C23×C4C2411yescharacteristic
235C23×C4C2411yescharacteristic
236C4⋊Q8—2128no—
237C4⋊Q8—264no—
238C4⋊Q8—232no—
239C4⋊Q8—264no—
240#125—416no—
241C4×Q8—264no—
242C4×Q8—264no—
243C4×Q8—232no—
244C4×Q8—232no—
245C2×C4⋊C4—28no—
246C2×C4⋊C4—28no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—28no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—28no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—28no—
256C2×C4⋊C4—28no—
257C2×C4⋊C4—28no—
258C2×C4⋊C4—24no—
259C2×C4⋊C4—24no—
260C2×C4⋊C4—24no—
261C2×C4⋊C4—24no—
262C2×C4⋊C4—24no—
263C2×C4⋊C4—24no—
264C2×C4⋊C4—22no—
265C2×C4⋊C4—28no—
266C2×C4⋊C4—24no—
267C2×C4⋊C4—28no—
268C2×C4⋊C4—22no—
269C2×C4⋊C4—24no—
270C2×C4⋊C4—22no—
271C2×C4⋊C4—24no—
272C2×C4⋊C4—24no—
273C2×C4⋊C4—22no—
274C2×C4⋊C4—24no—
275C2×C4⋊C4—28no—
276C2×C4⋊C4—24no—
277C2×C4⋊C4—28no—
278C2×C4⋊C4—22no—
279C2×C4⋊C4—24no—
280C2×C4⋊C4—24no—
281C2×C4⋊C4—22no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—24no—
285C2×C4⋊C4—28no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—24no—
288C2×C4⋊C4—28no—
289C2×C4⋊C4—42no—
290C2×C4⋊C4—41no—
291C2×C4⋊C4—42no—
292C2×C4⋊C4—42no—
293C2×C4⋊C4—44no—
294C2×C4⋊C4—44no—
295C2×C4⋊C4C2×D418yes—
296C2×C4⋊C4C2×D4116yes—
297C2×C4⋊C4C2×D418yes—
298C2×C4⋊C4C2×D418yes—
299C2×C4⋊C4C2×D414yes—
300C2×C4⋊C4C2×Q818yes—
301C2×C42—28no—
302C2×C42—28no—
303C2×C42—28no—
304C2×C42—28no—
305C2×C42—24no—
306C2×C42—24no—
307C2×C42—24no—
308C2×C42—24no—
309C2×C42—24no—
310C2×C42—24no—
311C2×C42—22no—
312C2×C42—22no—
313C2×C42—22no—
314C2×C42—22no—
315C2×C42—28no—
316C2×C42—44no—
317C2×C42—44no—
318C2×C42C2×D418yes—
319C2×C42C2×D414yes—
320C2×C42C2×D418yes—
321C2×C42C2×Q818yes—
322#95—22no—
323#95—24no—
324#95—24no—
325#95—28no—
326#95—22no—
327#95—28no—
328#95—28no—
329#95—24no—
330#95—24no—
331#95—24no—
332#95—28no—
333#95—28no—
334#95—24no—
335#95—24no—
336#95—24no—
337#95—22no—
338#95—24no—
339#95—24no—
340#95—28no—
341#95—24no—
342#95—28no—
343#95—22no—
344#95—24no—
345#95—42no—
346#95—42no—
347#95—44no—
348#95—44no—
349#95—44no—
350#95—44no—
351#95—42no—
352#95C2×Q8116yes—
353#95C2×Q8116yes—
354C24C2511yescharacteristic
355C2×Q8—28no—
356C2×Q8—28no—
357C2×Q8—216no—
358C2×Q8—216no—
359C2×Q8—28no—
360C2×Q8—232no—
361C2×Q8—28no—
362C2×Q8—216no—
363C2×Q8—28no—
364C2×Q8—216no—
365C2×Q8—216no—
366C2×Q8—416no—
367C2×Q8—416no—
368C2×Q8—88no—
369C2×Q8—88no—
370C2×Q8—88no—
371C2×Q8—88no—
372C2×Q8—88no—
373C2×Q8—84no—
374C2×Q8—88no—
375C2×Q8—84no—
376C2×Q8C4⋊Q8116yes—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—24no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—24no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—24no—
392C22×C4—21no—
393C22×C4—24no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—24no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—24no—
400C22×C4—21no—
401C22×C4—22no—
402C22×C4—24no—
403C22×C4—22no—
404C22×C4—24no—
405C22×C4—21no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—21no—
411C22×C4—21no—
412C22×C4—24no—
413C22×C4—22no—
414C22×C4—24no—
415C22×C4—21no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—24no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—21no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—21no—
428C22×C4—24no—
429C22×C4—21no—
430C22×C4—22no—
431C22×C4—21no—
432C22×C4—22no—
433C22×C4—21no—
434C22×C4—22no—
435C22×C4—24no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—21no—
440C22×C4—22no—
441C22×C4—21no—
442C22×C4—22no—
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—21no—
449C22×C4—24no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—21no—
455C22×C4—22no—
456C22×C4C22×D418yes—
457C22×C4C22×D414yes—
458C22×C4C22×D412yes—
459C22×C4C22×D418yes—
460C22×C4C22×D414yes—
461C22×C4C22×D414yes—
462C22×C4C22×D414yes—
463C22×C4C22×D412yes—
464C22×C4C22×D412yes—
465C22×C4C22×D414yes—
466C22×C4C22×Q814yes—
467C22×C4C22×Q814yes—
468C22×C4C22×Q818yes—
469C22×C4C22×Q818yes—
470C22×C4C22×Q814yes—
471C22×C4C2×#4112yes—
472C22×C4C2×#4114yes—
473C22×C4H(5,𝔽2)12yes—
474C22×C42−1+412yes—
475C22×C42−1+414yes—
476C22×C42−1+412yes—
477C22×C42−1+412yes—
478C4⋊C4—232no—
479C4⋊C4—216no—
480C4⋊C4—216no—
481C4⋊C4—28no—
482C4⋊C4—232no—
483C4⋊C4—216no—
484C4⋊C4—216no—
485C4⋊C4—216no—
486C4⋊C4—48no—
487C4⋊C4—44no—
488C4⋊C4—48no—
489C4⋊C4—48no—
490C4⋊C4—48no—
491C4⋊C4—44no—
492C4⋊C4—44no—
493C4⋊C4—44no—
494C4⋊C4—48no—
495C4⋊C4—416no—
496C4⋊C4—48no—
497C4⋊C4—88no—
498C4⋊C4#120116yes—
499C42—232no—
500C42—216no—
501C42—216no—
502C42—28no—
503C42—216no—
504C42—44no—
505C42—44no—
506C42—88no—
507C23C23×D411yescharacteristic
508C23C23×D412yes—
509C23C23×Q812yes—
510C23C22×#4111yescharacteristic
511C23C2×H(5,𝔽2)11yescharacteristic
512C23C2×2−1+411yescharacteristic
513C23C2×2−1+411yescharacteristic
514C23C2×2−1+411yescharacteristic
515C23C2×2−1+412yes—
516C23#58512yes—
517C23#58511yescharacteristic
518Q8—232no—
519Q8—232no—
520Q8—832no—
521Q8—816no—
522C2×C4—24no—
523C2×C4—24no—
524C2×C4—22no—
525C2×C4—24no—
526C2×C4—24no—
527C2×C4—24no—
528C2×C4—24no—
529C2×C4—28no—
530C2×C4—28no—
531C2×C4—28no—
532C2×C4—22no—
533C2×C4—24no—
534C2×C4—24no—
535C2×C4—24no—
536C2×C4—24no—
537C2×C4—24no—
538C2×C4—28no—
539C2×C4—24no—
540C2×C4—28no—
541C2×C4—24no—
542C2×C4—22no—
543C2×C4—24no—
544C2×C4—24no—
545C2×C4—28no—
546C2×C4—28no—
547C2×C4—24no—
548C2×C4—24no—
549C2×C4—28no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—28no—
553C2×C4—24no—
554C2×C4—22no—
555C2×C4—24no—
556C2×C4—24no—
557C2×C4—28no—
558C2×C4—24no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—41no—
562C2×C4—44no—
563C2×C4—42no—
564C2×C4—41no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—44no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—44no—
572C2×C4—42no—
573C2×C4—44no—
574C2×C4—42no—
575C2×C4—44no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—44no—
579C2×C4—42no—
580C2×C4—41no—
581C2×C4—44no—
582C2×C4—42no—
583C2×C4—44no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—42no—
590C2×C4—41no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—44no—
595C2×C4—41no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—41no—
599C2×C4—41no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—44no—
603C2×C4—41no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—41no—
607C2×C4—41no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4C2×#12018yes—
615C2×C4C2×#12014yes—
616C2×C4C2×C4⋊Q818yes—
617C2×C4C2×C4⋊Q814yes—
618C2×C4#53614yes—
619C2×C4#54818yes—
620C2×C4#54814yes—
621C2×C4D4×Q818yes—
622C2×C4D4×Q818yes—
623C22C22×#12011yescharacteristic
624C22C22×C4⋊Q811yescharacteristic
625C22C2×#53611yescharacteristic
626C22C2×#54812yes—
627C22C2×#54811yescharacteristic
628C22C2×D4×Q812yes—
629C22C2×D4×Q812yes—
630C22Q8×#4112yes—
631C22#348512yes—
632C22#348611yescharacteristic
633C22#348612yes—
634C22#348811yescharacteristic
635C22#349612yes—
636C22#350011yescharacteristic
637C22#350212yes—
638C22#350212yes—
639C22#350312yes—
640C22#351811yescharacteristic
641C22#352211yescharacteristic
642C22#354411yescharacteristic
643C22#355512yes—
644C22#355512yes—
645C22#355511yescharacteristic
646C4—28no—
647C4—28no—
648C4—28no—
649C4—24no—
650C4—48no—
651C4—44no—
652C4—44no—
653C4—48no—
654C4—42no—
655C4—44no—
656C4—42no—
657C4—81no—
658C4—81no—
659C4—82no—
660C4—81no—
661C4—82no—
662C4#243418yes—
663C2C2×#243411yescharacteristic
664C2#3575011yescharacteristic
665C2#3593711yescharacteristic
666C2#3647411yescharacteristic
667C2#3657511yescharacteristic
668C2#3912511yescharacteristic
669C2#4279311yescharacteristic
670C2#4489412yes—
671C2#4538912yes—
672C2#5908112yes—
673C2#5965512yes—
subgroupG/H
center, commutator, Frattini, socle, prime radical, cosocleC24C25
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
153. #391Q8semidirect

Maximal subgroups

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

subgroupindex
C22×Q822
C2×#25502
C2×#24532
#166032
#157832
#153182
#152302
#151882
#151582
#151282
#148802
#141862
#140532
#140172
#139782
C4×#3952
#138172