Finite Group Databasebeta

← #825730

Subgroups — #825730

Order 512.

2450 conjugacy classes of subgroups.

The subgroup lattice has 659 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2567C211yesmaximal, characteristic
2C2×#2365C211yesmaximal, characteristic
3#17067C212yesmaximal
4#15592C214yesmaximal
5#15510C214yesmaximal
6#15230C211yesmaximal, characteristic
7#15158C211yesmaximal, characteristic
8#15128C211yesmaximal, characteristic
9#14881C212yesmaximal
10#14872C212yesmaximal
11#14053C212yesmaximal
12#14045C212yesmaximal
13#14006C212yesmaximal
14#13954C212yesmaximal
15#13926C212yesmaximal
16C4×#395C211yesmaximal, characteristic
17#13821C211yesmaximal, characteristic
18C22×C4⋊Q8C2211yescharacteristic
19C22×C4⋊Q8C2211yescharacteristic
20C22×C4×Q8C2212yes—
21C22×C4×Q8C2211yescharacteristic
22#2567—28no—
23C2×#395C2211yescharacteristic
24C2×#395C2214yes—
25C2×#395C2212yes—
26C2×#395C2212yes—
27#2365—28no—
28C2×#391C2212yes—
29C2×#391C2212yes—
30C2×#391C2214yes—
31C2×#389C2211yescharacteristic
32C2×#389C2211yescharacteristic
33C2×#389C2212yes—
34C2×#389C2212yes—
35C2×#389C2212yes—
36C2×#389C2212yes—
37C2×#389C2212yes—
38C2×#387C2212yes—
39C2×#387C2212yes—
40C2×#384C2212yes—
41C2×#382C2212yes—
42C2×C4×C4⋊C4C2211yescharacteristic
43C2×C4×C4⋊C4C2211yescharacteristic
44C2×C4×C4⋊C4C2211yescharacteristic
45C2×C4×C4⋊C4C2212yes—
46C2×C4×C4⋊C4C2212yes—
47C2×C4×C4⋊C4C2212yes—
48C2×C4×C4⋊C4C2211yescharacteristic
49#1446C2211yescharacteristic
50#1446C2212yes—
51#1446C2214yes—
52#1446C2214yes—
53#1444C2212yes—
54#1444C2214yes—
55#1444C2212yes—
56#1444C2211yescharacteristic
57#1444C2214yes—
58#1444C2214yes—
59#1442C2214yes—
60#1441C2212yes—
61#1441C2214yes—
62#1441C2214yes—
63#1441C2212yes—
64#1441C2212yes—
65#1441C2211yescharacteristic
66#1441C2212yes—
67#1441C2212yes—
68#1441C2214yes—
69#1441C2214yes—
70#1436C2214yes—
71#1436C2214yes—
72#1436C2214yes—
73#1436C2214yes—
74C4⋊#95C2214yes—
75C4⋊#95C2211yescharacteristic
76C4⋊#95C2212yes—
77C4⋊#95C2212yes—
78C4⋊#95C2214yes—
79C4⋊#95C2214yes—
80C4⋊#95C2214yes—
81C4⋊#95C2212yes—
82#1434C2212yes—
83C4×#95C2211yescharacteristic
84C4×#95C2212yes—
85C23×Q8C2311yescharacteristic
86C23×Q8C2312yes—
87C23×Q8C2312yes—
88C2×C4⋊Q8—24no—
89C2×C4⋊Q8—24no—
90C2×C4⋊Q8—24no—
91C2×C4⋊Q8—44no—
92C2×C4⋊Q8Q818yescomplemented
93C2×C4×Q8—28no—
94C2×C4×Q8—24no—
95C2×C4×Q8—28no—
96C2×C4×Q8—44no—
97C2×C4×Q8—44no—
98C22×C4⋊C4C2312yes—
99C22×C4⋊C4C2312yes—
100C22×C4⋊C4C2311yescharacteristic
101C22×C4⋊C4C2311yescharacteristic
102C22×C4⋊C4C2312yes—
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2311yescharacteristic
105C22×C4⋊C4C2312yes—
106C22×C4⋊C4C2312yes—
107C22×C4⋊C4C2311yescharacteristic
108C22×C4⋊C4C2312yes—
109C22×C4⋊C4C2314yes—
110C22×C4⋊C4C2314yes—
111C22×C4⋊C4C2314yes—
112C22×C4⋊C4C2311yescharacteristic
113C22×C4⋊C4C2312yes—
114C22×C4⋊C4C2314yes—
115C22×C4⋊C4C2312yes—
116C22×C4⋊C4C2314yes—
117C22×C4⋊C4C2311yescharacteristic
118C22×C4⋊C4C2311yescharacteristic
119C22×C42C2312yes—
120C22×C42C2311yescharacteristic
121C22×C42C2312yes—
122C22×C42C2312yes—
123C22×C42C2311yescharacteristic
124C22×C42C2312yes—
125C22×C42C2311yescharacteristic
126C22×C42C2312yes—
127C22×C42C2314yes—
128#395—24no—
129#395—216no—
130#395—44no—
131#395—44no—
132#391—28no—
133#391—28no—
134#391—48no—
135#389—28no—
136#389—28no—
137#389—24no—
138#389—24no—
139#389—44no—
140#389—44no—
141#389Q8116yescomplemented
142#387—28no—
143#387—28no—
144#384—28no—
145#382—44no—
146C4×C4⋊C4—24no—
147C4×C4⋊C4—24no—
148C4×C4⋊C4—28no—
149C4×C4⋊C4—24no—
150C4×C4⋊C4—44no—
151C4×C4⋊C4—44no—
152C4×C4⋊C4D418yes—
153C2×#95C2311yescharacteristic
154C2×#95C2312yes—
155C2×#95C2312yes—
156C2×#95C2314yes—
157C2×#95C2314yes—
158C2×#95C2314yes—
159C2×#95C2312yes—
160C2×#95C2312yes—
161C2×#95C2311yescharacteristic
162C2×#95C2314yes—
163C2×#95C2312yes—
164C2×#95C2312yes—
165C2×#95C2312yes—
166C2×#95C2311yescharacteristic
167C2×#95C2312yes—
168C2×#95C2311yescharacteristic
169C2×#95C2312yes—
170C2×#95C2314yes—
171C2×#95C2312yes—
172C2×#95C2312yes—
173C2×#95C2312yes—
174C2×#95C2314yes—
175C2×#95C2314yes—
176C2×#95C2314yes—
177C2×#95C2314yes—
178C2×#95C2314yes—
179C2×#95C2312yes—
180C2×#95C2314yes—
181C2×#95C2314yes—
182C2×#95C2314yes—
183C2×#95C2312yes—
184C2×#95C2314yes—
185C22×Q8—24no—
186C22×Q8—22no—
187C22×Q8—24no—
188C22×Q8—24no—
189C22×Q8—24no—
190C22×Q8—22no—
191C22×Q8—24no—
192C22×Q8—22no—
193C22×Q8—24no—
194C22×Q8—42no—
195C22×Q8—42no—
196C22×Q8—42no—
197C22×Q8—44no—
198C22×Q8—44no—
199C22×Q8—42no—
200C22×Q8—42no—
201C22×Q8C2×Q818yes—
202C23×C4C2411yescharacteristic
203C23×C4C2412yes—
204C23×C4C2414yes—
205C23×C4C2412yes—
206C23×C4C2411yescharacteristic
207C23×C4C2414yes—
208C23×C4C2412yes—
209C23×C4C2411yescharacteristic
210C23×C4C2412yes—
211C23×C4C2412yes—
212C23×C4C2411yescharacteristic
213C23×C4C2412yes—
214C23×C4C2412yes—
215C23×C4C2412yes—
216C23×C4C2411yescharacteristic
217C23×C4C2411yescharacteristic
218C23×C4C2411yescharacteristic
219C4⋊Q8—232no—
220C4⋊Q8—416no—
221C4×Q8—432no—
222C4×Q8—416no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—24no—
226C2×C4⋊C4—24no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—24no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—24no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—24no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—24no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—28no—
249C2×C4⋊C4—28no—
250C2×C4⋊C4—28no—
251C2×C4⋊C4—44no—
252C2×C4⋊C4—41no—
253C2×C4⋊C4—42no—
254C2×C4⋊C4—42no—
255C2×C4⋊C4—42no—
256C2×C4⋊C4—42no—
257C2×C4⋊C4—42no—
258C2×C4⋊C4—42no—
259C2×C4⋊C4—44no—
260C2×C4⋊C4—44no—
261C2×C4⋊C4—42no—
262C2×C4⋊C4—42no—
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⋊C4—44no—
270C2×C4⋊C4—44no—
271C2×C4⋊C4—44no—
272C2×C4⋊C4C2×D414yes—
273C2×C4⋊C4C2×D414yes—
274C2×C4⋊C4C2×Q818yes—
275C2×C4⋊C4C2×Q814yes—
276C2×C4⋊C4C2×Q818yes—
277C2×C4⋊C4C2×Q8116yes—
278C2×C4⋊C4C2×Q814yes—
279C2×C4⋊C4C2×Q818yes—
280C2×C4⋊C4#4114yes—
281C2×C4⋊C4#4114yes—
282C2×C42—22no—
283C2×C42—24no—
284C2×C42—22no—
285C2×C42—24no—
286C2×C42—24no—
287C2×C42—24no—
288C2×C42—24no—
289C2×C42—24no—
290C2×C42—24no—
291C2×C42—24no—
292C2×C42—22no—
293C2×C42—24no—
294C2×C42—24no—
295C2×C42—24no—
296C2×C42—24no—
297C2×C42—24no—
298C2×C42—28no—
299C2×C42—42no—
300C2×C42—42no—
301C2×C42—44no—
302C2×C42—44no—
303C2×C42C2×D414yes—
304C2×C42C2×D418yes—
305C2×C42C2×Q814yes—
306C2×C42C2×Q818yes—
307#95—22no—
308#95—22no—
309#95—24no—
310#95—24no—
311#95—24no—
312#95—24no—
313#95—28no—
314#95—28no—
315#95—28no—
316#95—22no—
317#95—24no—
318#95—22no—
319#95—24no—
320#95—24no—
321#95—28no—
322#95—24no—
323#95—42no—
324#95—44no—
325#95—44no—
326#95—42no—
327#95—42no—
328#95—42no—
329#95—44no—
330#95—44no—
331#95—44no—
332#95—44no—
333#95—44no—
334#95—44no—
335#95—42no—
336#95—44no—
337#95C2×D418yes—
338#95C2×Q8116yes—
339C24C2511yescharacteristic
340C2×Q8—28no—
341C2×Q8—216no—
342C2×Q8—216no—
343C2×Q8—48no—
344C2×Q8—48no—
345C2×Q8—48no—
346C2×Q8—48no—
347C2×Q8—48no—
348C2×Q8—48no—
349C2×Q8—48no—
350C2×Q8—44no—
351C2×Q8—48no—
352C2×Q8—44no—
353C2×Q8—84no—
354C2×Q8—84no—
355C2×Q8—88no—
356C2×Q8—88no—
357C22×C4—21no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—21no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—21no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—24no—
368C22×C4—22no—
369C22×C4—21no—
370C22×C4—21no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—21no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—24no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—21no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—24no—
384C22×C4—24no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—24no—
392C22×C4—21no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—21no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—24no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—24no—
410C22×C4—22no—
411C22×C4—21no—
412C22×C4—21no—
413C22×C4—22no—
414C22×C4—24no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—22no—
421C22×C4—24no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—24no—
425C22×C4—22no—
426C22×C4—21no—
427C22×C4—24no—
428C22×C4—22no—
429C22×C4—24no—
430C22×C4—24no—
431C22×C4—21no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—24no—
436C22×C4C22×D412yes—
437C22×C4C22×D414yes—
438C22×C4C22×D414yes—
439C22×C4C22×D412yes—
440C22×C4C22×D412yes—
441C22×C4C22×Q814yes—
442C22×C4C22×Q818yes—
443C22×C4C22×Q812yes—
444C22×C4C22×Q814yes—
445C22×C4C22×Q814yes—
446C22×C4C22×Q814yes—
447C22×C4C22×Q818yes—
448C22×C4C22×Q812yes—
449C22×C4C22×Q812yes—
450C22×C4C22×Q814yes—
451C22×C4C2×#4112yes—
452C22×C4C2×#4114yes—
453C22×C4C2×#4112yes—
454C22×C4C2×#4112yes—
455C22×C4C2×#4112yes—
456C22×C4H(5,𝔽2)12yes—
457C22×C4H(5,𝔽2)12yes—
458C22×C4H(5,𝔽2)14yes—
459C22×C42−1+414yes—
460C4⋊C4—216no—
461C4⋊C4—28no—
462C4⋊C4—28no—
463C4⋊C4—216no—
464C4⋊C4—416no—
465C4⋊C4—44no—
466C4⋊C4—48no—
467C4⋊C4—48no—
468C4⋊C4—44no—
469C4⋊C4—48no—
470C4⋊C4—48no—
471C4⋊C4—48no—
472C4⋊C4—44no—
473C4⋊C4—48no—
474C4⋊C4—48no—
475C4⋊C4—88no—
476C4⋊C4—88no—
477C4⋊C4—88no—
478C4⋊C4—88no—
479C4⋊C4C22⋊Q8116yes—
480C4⋊C4C4⋊Q8116yes—
481C42—28no—
482C42—28no—
483C42—216no—
484C42—216no—
485C42—48no—
486C42—48no—
487C42—44no—
488C42—48no—
489C42—88no—
490C23C23×D411yescharacteristic
491C23C23×Q811yescharacteristic
492C23C23×Q812yes—
493C23C22×#4111yescharacteristic
494C23C22×#4111yescharacteristic
495C23C2×H(5,𝔽2)12yes—
496C23C2×H(5,𝔽2)11yescharacteristic
497C23C2×H(5,𝔽2)11yescharacteristic
498C23C2×2−1+412yes—
499C23#58511yescharacteristic
500C23#58512yes—
501Q8—432no—
502Q8—416no—
503Q8—816no—
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—24no—
507C2×C4—24no—
508C2×C4—22no—
509C2×C4—24no—
510C2×C4—24no—
511C2×C4—22no—
512C2×C4—28no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—24no—
516C2×C4—24no—
517C2×C4—28no—
518C2×C4—24no—
519C2×C4—22no—
520C2×C4—24no—
521C2×C4—22no—
522C2×C4—24no—
523C2×C4—24no—
524C2×C4—22no—
525C2×C4—24no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—24no—
529C2×C4—24no—
530C2×C4—28no—
531C2×C4—24no—
532C2×C4—24no—
533C2×C4—24no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—24no—
537C2×C4—28no—
538C2×C4—24no—
539C2×C4—24no—
540C2×C4—24no—
541C2×C4—24no—
542C2×C4—28no—
543C2×C4—24no—
544C2×C4—24no—
545C2×C4—24no—
546C2×C4—42no—
547C2×C4—41no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—42no—
555C2×C4—41no—
556C2×C4—44no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—44no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—44no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—44no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—41no—
576C2×C4—44no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—41no—
581C2×C4—42no—
582C2×C4—44no—
583C2×C4—42no—
584C2×C4—44no—
585C2×C4—41no—
586C2×C4—44no—
587C2×C4—44no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—44no—
592C2×C4—44no—
593C2×C4—42no—
594C2×C4—44no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—44no—
598C2×C4C2×C22⋊Q818yes—
599C2×C4C2×C22⋊Q814yes—
600C2×C4C2×C4⋊Q818yes—
601C2×C4C2×C4⋊Q814yes—
602C2×C4#53514yes—
603C2×C4#54614yes—
604C2×C4#55414yes—
605C2×C4#55418yes—
606C2×C4Q8218yes—
607C22C22×C22⋊Q811yescharacteristic
608C22C22×C4⋊Q811yescharacteristic
609C22C2×#53511yescharacteristic
610C22C2×#54611yescharacteristic
611C22C2×#55412yes—
612C22C2×#55411yescharacteristic
613C22#347511yescharacteristic
614C22C2×Q8212yes—
615C22Q8×#4112yes—
616C22#348611yescharacteristic
617C22#348811yescharacteristic
618C22#350011yescharacteristic
619C22#350112yes—
620C22#350111yescharacteristic
621C22#350212yes—
622C22#350212yes—
623C22#350312yes—
624C22#351812yes—
625C22#351911yescharacteristic
626C22#352012yes—
627C22#353612yes—
628C22#354012yes—
629C22#354612yes—
630C4—28no—
631C4—24no—
632C4—28no—
633C4—24no—
634C4—44no—
635C4—48no—
636C4—44no—
637C4—42no—
638C4—44no—
639C4—42no—
640C4—42no—
641C4—44no—
642C4—82no—
643C4—81no—
644C4—84no—
645C4—82no—
646C4#256718yes—
647C2C2×#256711yescharacteristic
648C2#3571111yescharacteristic
649C2#3593711yescharacteristic
650C2#3615411yescharacteristic
651C2#3627611yescharacteristic
652C2#3890511yescharacteristic
653C2#4298711yescharacteristic
654C2#4536012yes—
655C2#4536412yes—
656C2#5945912yes—
657C2#5955112yes—
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
92. C2×C4⋊Q8Q8semidirect
141. #389Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#25672
C2×#23652
#170672
#155922
#155102
#152302
#151582
#151282
#148812
#148722
#140532
#140452
#140062
#139542
#139262
C4×#3952
#138212