Finite Group Databasebeta

← #536694

Subgroups — #536694

Order 512.

3218 conjugacy classes of subgroups.

The subgroup lattice has 670 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2580C211yesmaximal, characteristic
2C2×Q8×C4⋊C4C211yesmaximal, characteristic, complemented
3C2×D4×C4⋊C4C211yesmaximal, characteristic
4#14551C214yesmaximal, complemented
5#14392C214yesmaximal, complemented
6#14172C214yesmaximal, complemented
7#14150C211yesmaximal, characteristic, complemented
8#14132C212yesmaximal, complemented
9#14043C214yesmaximal, complemented
10#14005C212yesmaximal, complemented
11#13977C212yesmaximal
12#13850C211yesmaximal, characteristic, complemented
13#13820C212yesmaximal
14#13785C211yesmaximal, characteristic
15#13779C211yesmaximal, characteristic, complemented
16C22×C4×Q8C2211yescharacteristic
17C22×C4×Q8C2212yes—
18C22×C4×D4C2212yes—
19C22×C4×D4C2211yescharacteristic
20C23×C4⋊C4C2212yes—
21#2580C4116yescomplemented
22C2×#399C2212yes—
23C2×#393C2212yes—
24Q8×C4⋊C4—28no—
25D4×C4⋊C4—28no—
26C2×#390C2211yescharacteristic
27C2×#390C2212yes—
28C2×#390C2211yescharacteristic
29C2×#389C2212yes—
30C2×#389C2214yes—
31C2×#389C2212yes—
32C2×#389C2212yes—
33C2×#389C2214yes—
34C2×#389C2212yes—
35C2×#385C2214yes—
36C2×#385C2214yes—
37C2×#385C2214yes—
38C2×#385C2214yes—
39C2×#384C2212yes—
40C2×#384C2211yescharacteristic
41C2×#382C2211yescharacteristic
42C2×#380C2212yes—
43C2×C4×C4⋊C4C2211yescharacteristic
44C2×C4×C4⋊C4C2212yes—
45C2×C4×C4⋊C4C2211yescharacteristic
46C2×C4×C4⋊C4C2212yes—
47C2×C4×C22⋊C4C2214yes—
48C2×#376C2211yescharacteristic
49#1446C2212yes—
50#1445C2212yes—
51#1445C2214yes—
52#1444C2212yes—
53#1444C2214yes—
54#1442C2212yes—
55#1441C2212yes—
56#1441C2212yes—
57#1441C2214yes—
58#1441C2214yes—
59#1439C2214yes—
60#1439C2214yes—
61#1437C2214yes—
62#1437C2214yes—
63#1437C2214yes—
64#1436C2214yes—
65#1436C2214yes—
66#1436C2214yes—
67#1436C2214yes—
68#1436C2214yes—
69C4⋊#95C2211yescharacteristic
70C4⋊#95C2212yes—
71C4⋊#95C2212yes—
72C4⋊#95C2211yescharacteristic
73#1434C2214yes—
74C22⋊#95C2214yes—
75C22⋊#95C2214yes—
76C4×#95C2212yes—
77C4×#95C2212yes—
78C23×Q8C2311yescharacteristic
79C23×D4C2311yescharacteristic
80C24×C4C2314yes—
81C24×C4C2312yes—
82C2×C4×Q8—28no—
83C2×C4×Q8—24no—
84C2×C4×Q8—24no—
85C2×C4×Q8—48no—
86C2×C4×Q8C2×C418yescomplemented
87C2×C4×D4—28no—
88C2×C4×D4—28no—
89C2×C4×D4—24no—
90C2×C4×D4—24no—
91C2×C4×D4—44no—
92C2×C4×D4C2×C418yes—
93C22×C4⋊C4—28no—
94C22×C4⋊C4—48no—
95C22×C4⋊C4C2311yescharacteristic
96C22×C4⋊C4C2311yescharacteristic
97C22×C4⋊C4C2314yes—
98C22×C4⋊C4C2314yes—
99C22×C4⋊C4C2311yescharacteristic
100C22×C4⋊C4C2311yescharacteristic
101C22×C4⋊C4C2312yes—
102C22×C4⋊C4C2312yes—
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2311yescharacteristic
105C22×C4⋊C4C2314yes—
106C22×C4⋊C4C2314yes—
107C22×C4⋊C4C2311yescharacteristic
108C22×C4⋊C4C2314yes—
109C22×C4⋊C4C2312yes—
110C22×C4⋊C4C2312yes—
111C22×C22⋊C4C2314yes—
112C22×C22⋊C4C2312yes—
113C22×C22⋊C4C2314yes—
114C22×C22⋊C4C2314yes—
115C22×C22⋊C4C2314yes—
116C22×C22⋊C4C2314yes—
117C22×C42C2312yes—
118C22×C42C2311yescharacteristic
119C22×C42C2314yes—
120C22×C42C2312yes—
121C22×C42C2311yescharacteristic
122C22×C42C2312yes—
123C22×C42C2314yes—
124C22×C42C2311yescharacteristic
125#399C2×C4116yescomplemented
126#393C2×C4116yescomplemented
127#390—24no—
128#390C2×C4116yes—
129#390Q818yescomplemented
130#389—28no—
131#389—216no—
132#389—28no—
133#389—48no—
134#389—44no—
135#389C2×C4116yescomplemented
136#385—216no—
137#385—48no—
138#385—48no—
139#385C2×C4132yescomplemented
140#384—28no—
141#384D418yes—
142#382C2×C418yescomplemented
143#380—28no—
144C4×C4⋊C4—28no—
145C4×C4⋊C4—24no—
146C4×C4⋊C4—24no—
147C4×C4⋊C4—44no—
148C4×C22⋊C4—48no—
149#376—24no—
150C2×#95C2312yes—
151C2×#95C2312yes—
152C2×#95C2312yes—
153C2×#95C2312yes—
154C2×#95C2312yes—
155C2×#95C2312yes—
156C2×#95C2312yes—
157C2×#95C2312yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2312yes—
161C2×#95C2312yes—
162C2×#95C2312yes—
163C2×#95C2312yes—
164C2×#95C2314yes—
165C2×#95C2314yes—
166C2×#95C2314yes—
167C2×#95C2314yes—
168C2×#95C2314yes—
169C2×#95C2314yes—
170C2×#95C2314yes—
171C2×#95C2314yes—
172C2×#95C2312yes—
173C2×#95C2314yes—
174C2×#95C2312yes—
175C25C2412yes—
176C22×Q8—22no—
177C22×Q8—22no—
178C22×Q8—24no—
179C22×Q8—24no—
180C22×Q8C22×C414yes—
181C22×D4—22no—
182C22×D4—22no—
183C22×D4—22no—
184C22×D4—22no—
185C22×D4C22×C414yes—
186C22×D4C2×Q814yes—
187C22×D4#4114yes—
188C23×C4—24no—
189C23×C4—24no—
190C23×C4—28no—
191C23×C4—28no—
192C23×C4—28no—
193C23×C4—44no—
194C23×C4—44no—
195C23×C4—44no—
196C23×C4—44no—
197C23×C4—44no—
198C23×C4—44no—
199C23×C4C22×C418yes—
200C23×C4C2412yes—
201C23×C4C2412yes—
202C23×C4C2412yes—
203C23×C4C2412yes—
204C23×C4C2414yes—
205C23×C4C2412yes—
206C23×C4C2414yes—
207C23×C4C2412yes—
208C23×C4C2412yes—
209C23×C4C2412yes—
210C23×C4C2411yescharacteristic
211C23×C4C2411yescharacteristic
212C23×C4C2412yes—
213C23×C4C2411yescharacteristic
214C4×Q8—232no—
215C4×Q8—432no—
216C4×D4—232no—
217C4×D4—432no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—24no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—24no—
225C2×C4⋊C4—24no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—24no—
228C2×C4⋊C4—24no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4—24no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—28no—
233C2×C4⋊C4—28no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—28no—
238C2×C4⋊C4—28no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—28no—
241C2×C4⋊C4—28no—
242C2×C4⋊C4—432no—
243C2×C4⋊C4—44no—
244C2×C4⋊C4—44no—
245C2×C4⋊C4—44no—
246C2×C4⋊C4—42no—
247C2×C4⋊C4—44no—
248C2×C4⋊C4—44no—
249C2×C4⋊C4—44no—
250C2×C4⋊C4—44no—
251C2×C4⋊C4—44no—
252C2×C4⋊C4—44no—
253C2×C4⋊C4—44no—
254C2×C4⋊C4C22×C414yes—
255C2×C4⋊C4C22×C418yes—
256C2×C4⋊C4C22×C414yes—
257C2×C4⋊C4C22×C418yes—
258C2×C4⋊C4C22×C4116yes—
259C2×C4⋊C4C2×D418yes—
260C2×C4⋊C4C2×D418yes—
261C2×C4⋊C4C2×D418yes—
262C2×C4⋊C4C2×Q814yes—
263C2×C22⋊C4—24no—
264C2×C22⋊C4—24no—
265C2×C22⋊C4—28no—
266C2×C22⋊C4—28no—
267C2×C22⋊C4—28no—
268C2×C22⋊C4—28no—
269C2×C22⋊C4—28no—
270C2×C22⋊C4—44no—
271C2×C22⋊C4—44no—
272C2×C22⋊C4—44no—
273C2×C22⋊C4—44no—
274C2×C22⋊C4—44no—
275C2×C22⋊C4—44no—
276C2×C22⋊C4—44no—
277C2×C22⋊C4C22×C418yes—
278C2×C22⋊C4C22×C4116yes—
279C2×C22⋊C4C22×C4116yes—
280C2×C22⋊C4C2×Q8116yes—
281C2×C42—24no—
282C2×C42—24no—
283C2×C42—22no—
284C2×C42—22no—
285C2×C42—24no—
286C2×C42—28no—
287C2×C42—24no—
288C2×C42—24no—
289C2×C42—22no—
290C2×C42—22no—
291C2×C42—24no—
292C2×C42—28no—
293C2×C42—22no—
294C2×C42—44no—
295C2×C42—44no—
296C2×C42—44no—
297C2×C42—44no—
298C2×C42—44no—
299C2×C42C22×C418yes—
300C2×C42C22×C414yes—
301C2×C42C2×D414yes—
302C2×C42C2×Q814yes—
303C2×C42#4114yes—
304#95—24no—
305#95—28no—
306#95—24no—
307#95—28no—
308#95—24no—
309#95—24no—
310#95—24no—
311#95—24no—
312#95—24no—
313#95—24no—
314#95—24no—
315#95—24no—
316#95—44no—
317#95—44no—
318#95—42no—
319#95—42no—
320#95—44no—
321#95—44no—
322#95—44no—
323#95—44no—
324#95—44no—
325#95—44no—
326#95C22×C418yes—
327#95C22×C418yes—
328#95C22×C4116yes—
329C24—22no—
330C24—22no—
331C24—24no—
332C24—24no—
333C24—22no—
334C24—24no—
335C24—22no—
336C24—24no—
337C24C23×C414yes—
338C24C22×Q814yes—
339C24C2×#4114yes—
340C24C2511yescharacteristic
341C2×Q8—28no—
342C2×Q8—28no—
343C2×Q8—216no—
344C2×Q8—28no—
345C2×Q8—216no—
346C2×D4—28no—
347C2×D4—28no—
348C2×D4—28no—
349C2×D4—28no—
350C2×D4—28no—
351C2×D4—28no—
352C2×D4C4×Q8116yescomplemented
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—24no—
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—24no—
380C22×C4—24no—
381C22×C4—24no—
382C22×C4—24no—
383C22×C4—24no—
384C22×C4—24no—
385C22×C4—24no—
386C22×C4—22no—
387C22×C4—24no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—24no—
391C22×C4—24no—
392C22×C4—22no—
393C22×C4—24no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—24no—
397C22×C4—22no—
398C22×C4—24no—
399C22×C4—216no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—416no—
420C22×C4—416no—
421C22×C4—48no—
422C22×C4—48no—
423C22×C4—416no—
424C22×C4—416no—
425C22×C4—416no—
426C22×C4—88no—
427C22×C4—88no—
428C22×C4—88no—
429C22×C4—88no—
430C22×C4C23×C418yes—
431C22×C4C23×C414yes—
432C22×C4C23×C414yes—
433C22×C4C23×C412yes—
434C22×C4C23×C412yes—
435C22×C4C23×C414yes—
436C22×C4C23×C412yes—
437C22×C4C22×D418yes—
438C22×C4C22×D412yes—
439C22×C4C22×D412yes—
440C22×C4C22×D412yes—
441C22×C4C22×Q814yes—
442C22×C4C22×Q814yes—
443C22×C4C22×Q812yes—
444C22×C4C2×#4112yes—
445C22×C4C2×#4112yes—
446C22×C4C2×#4112yes—
447C22×C4C2×#4112yes—
448C22×C4H(5,𝔽2)14yes—
449C22×C4H(5,𝔽2)14yes—
450C22×C4H(5,𝔽2)12yes—
451C22×C42−1+414yes—
452C22×C42−1+412yes—
453C4⋊C4—28no—
454C4⋊C4—28no—
455C4⋊C4—216no—
456C4⋊C4—28no—
457C4⋊C4—216no—
458C4⋊C4—216no—
459C4⋊C4—28no—
460C4⋊C4—48no—
461C4⋊C4—416no—
462C4⋊C4—48no—
463C4⋊C4—416no—
464C4⋊C4—416no—
465C4⋊C4—44no—
466C4⋊C4—44no—
467C4⋊C4—88no—
468C4⋊C4—88no—
469C22⋊C4—216no—
470C22⋊C4—232no—
471C22⋊C4—416no—
472C22⋊C4—416no—
473C22⋊C4—416no—
474C22⋊C4—88no—
475C42—216no—
476C42—28no—
477C42—416no—
478C42—48no—
479C42—48no—
480C42—44no—
481C42—88no—
482C42C4×D4116yes—
483C23—24no—
484C23—28no—
485C23—24no—
486C23—28no—
487C23—24no—
488C23—24no—
489C23—24no—
490C23—24no—
491C23—28no—
492C23—28no—
493C23—28no—
494C23—28no—
495C23—42no—
496C23—42no—
497C23—42no—
498C23—42no—
499C23—44no—
500C23—44no—
501C23—44no—
502C23—44no—
503C23—44no—
504C23—44no—
505C23C2×C4×Q818yes—
506C23C24×C411yescharacteristic
507C23C23×D411yescharacteristic
508C23C23×Q811yescharacteristic
509C23C22×#4111yescharacteristic
510C23C22×#4111yescharacteristic
511C23C2×H(5,𝔽2)11yescharacteristic
512C23C2×H(5,𝔽2)12yes—
513C23C2×H(5,𝔽2)12yes—
514C23C2×2−1+411yescharacteristic
515C23C2×2−1+412yes—
516C23#58512yes—
517Q8—232no—
518D4—232no—
519C2×C4—24no—
520C2×C4—24no—
521C2×C4—24no—
522C2×C4—24no—
523C2×C4—22no—
524C2×C4—24no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—24no—
528C2×C4—22no—
529C2×C4—24no—
530C2×C4—24no—
531C2×C4—24no—
532C2×C4—24no—
533C2×C4—24no—
534C2×C4—24no—
535C2×C4—28no—
536C2×C4—28no—
537C2×C4—24no—
538C2×C4—28no—
539C2×C4—24no—
540C2×C4—28no—
541C2×C4—28no—
542C2×C4—24no—
543C2×C4—24no—
544C2×C4—28no—
545C2×C4—24no—
546C2×C4—28no—
547C2×C4—24no—
548C2×C4—28no—
549C2×C4—24no—
550C2×C4—24no—
551C2×C4—24no—
552C2×C4—432no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—44no—
566C2×C4—44no—
567C2×C4—44no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—44no—
571C2×C4—44no—
572C2×C4—44no—
573C2×C4—42no—
574C2×C4—44no—
575C2×C4—44no—
576C2×C4—42no—
577C2×C4—44no—
578C2×C4—44no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—44no—
593C2×C4—44no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—832no—
599C2×C4C2×C4×D414yes—
600C2×C4C2×C4×D414yes—
601C2×C4C2×C4×D414yes—
602C2×C4C2×C4×Q814yes—
603C2×C4#52018yes—
604C2×C4#52014yes—
605C2×C4#54614yes—
606C2×C4D4×Q814yes—
607C2×C4#55214yes—
608C2×C4#55414yes—
609C22—216no—
610C22—28no—
611C22—44no—
612C22—44no—
613C22—48no—
614C22—48no—
615C22—44no—
616C22—44no—
617C22—44no—
618C22—48no—
619C22—44no—
620C22C22×C4×D411yescharacteristic
621C22C22×C4×Q811yescharacteristic
622C22C2×#52012yes—
623C22C2×#52011yescharacteristic
624C22C4×H(5,𝔽2)12yes—
625C22C2×#54611yescharacteristic
626C22C2×D4×Q811yescharacteristic
627C22C2×#55211yescharacteristic
628C22C2×#55411yescharacteristic
629C22#348411yescharacteristic
630C22#348411yescharacteristic
631C22#348611yescharacteristic
632C22#348711yescharacteristic
633C22#349212yes—
634C22#350112yes—
635C22#350312yes—
636C22#350612yes—
637C22#351512yes—
638C22#351912yes—
639C22#354312yes—
640C22#355012yes—
641C22#355412yes—
642C22#355412yes—
643C4—24no—
644C4—28no—
645C4—24no—
646C4—44no—
647C4—44no—
648C4—48no—
649C4—44no—
650C4—48no—
651C4—44no—
652C4—44no—
653C4—82no—
654C4—82no—
655C4—82no—
656C4#236018yes—
657C2—48no—
658C2C2×#236011yescharacteristic
659C2#3420011yescharacteristic
660C2#3426511yescharacteristic
661C2#3433212yes—
662C2#3449812yes—
663C2#4490812yes—
664C2#4491012yes—
665C2#4493411yescharacteristic
666C2#4493611yescharacteristic
667C2#4533511yescharacteristic
668C2#4535711yescharacteristic
subgroupG/H
center, Frattini, socle, cosocleC24C25
commutatorC23C24×C4
prime radicalC23×D4C23
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×Q8×C4⋊C4C2semidirect
4. #14551C2semidirect
5. #14392C2semidirect
6. #14172C2semidirect
7. #14150C2semidirect
8. #14132C2semidirect
9. #14043C2semidirect
10. #14005C2semidirect
12. #13850C2semidirect
15. #13779C2semidirect
21. #2580C4semidirect
86. C2×C4×Q8C2×C4semidirect
125. #399C2×C4semidirect
126. #393C2×C4semidirect
129. #390Q8semidirect
135. #389C2×C4semidirect
139. #385C2×C4semidirect
142. #382C2×C4semidirect
352. C2×D4C4×Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#25802
C2×Q8×C4⋊C42
C2×D4×C4⋊C42
#145512
#143922
#141722
#141502
#141322
#140432
#140052
#139772
#138502
#138202
#137852
#137792