Finite Group Databasebeta

← #553058

Subgroups — #553058

Order 512.

2590 conjugacy classes of subgroups.

The subgroup lattice has 677 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2346C211yesmaximal, characteristic
2C2×#2346C211yesmaximal, characteristic
3C2×#2317C211yesmaximal, characteristic
4#14677C212yesmaximal, complemented
5#14488C214yesmaximal
6#14442C214yesmaximal
7#14348C212yesmaximal, complemented
8#14277C212yesmaximal, complemented
9#14264C211yesmaximal, characteristic
10#14130C211yesmaximal, characteristic
11#14046C212yesmaximal, complemented
12#14006C212yesmaximal, complemented
13#13963C212yesmaximal, complemented
14#13954C212yesmaximal, complemented
15#13933C212yesmaximal, complemented
16#13881C211yesmaximal, characteristic
17#13844C211yesmaximal, characteristic
18C22×#117C2211yescharacteristic
19C22×#117C2211yescharacteristic
20C23×C4⋊C4C2211yescharacteristic
21#2346—28no—
22#2346C4116yescomplemented
23#2317—28no—
24C2×#389C2212yes—
25C2×#389C2212yes—
26C2×#389C2211yescharacteristic
27C2×#389C2211yescharacteristic
28C2×#389C2212yes—
29C2×#389C2214yes—
30C2×#389C2212yes—
31C2×#389C2212yes—
32C2×#388C2212yes—
33C2×#388C2212yes—
34C2×#388C2214yes—
35C2×#385C2214yes—
36C2×#385C2212yes—
37C2×#384C2212yes—
38C2×#382C2212yes—
39C2×#382C2212yes—
40C2×#382C2212yes—
41C2×#382C2212yes—
42C2×#380C2211yescharacteristic
43C2×#380C2211yescharacteristic
44C2×C4×C4⋊C4C2211yescharacteristic
45C2×C4×C4⋊C4C2211yescharacteristic
46C2×C4×C4⋊C4C2212yes—
47C2×C4×C4⋊C4C2212yes—
48C2×C4×C22⋊C4C2211yescharacteristic
49C2×C4×C22⋊C4C2211yescharacteristic
50C2×#376C2212yes—
51C2×#376C2212yes—
52#1444C2211yescharacteristic
53#1444C2211yescharacteristic
54#1444C2212yes—
55#1444C2214yes—
56#1444C2212yes—
57#1444C2214yes—
58#1442C2211yescharacteristic
59#1442C2212yes—
60#1442C2212yes—
61#1442C2214yes—
62#1442C2214yes—
63#1442C2212yes—
64#1442C2211yescharacteristic
65#1442C2212yes—
66#1442C2214yes—
67#1442C2214yes—
68#1441C2214yes—
69#1441C2214yes—
70#1439C2212yes—
71#1439C2212yes—
72#1439C2214yes—
73#1436C2214yes—
74#1436C2214yes—
75C4⋊#95C2214yes—
76C4⋊#95C2214yes—
77C4⋊#95C2212yes—
78C4⋊#95C2212yes—
79C4⋊#95C2214yes—
80C4⋊#95C2212yes—
81C4⋊#95C2214yes—
82#1434C2214yes—
83C22⋊#95C2212yes—
84C22⋊#95C2214yes—
85C4×#95C2212yes—
86C24×C4C2311yescharacteristic
87C24×C4C2312yes—
88C2×#117—24no—
89C2×#117—44no—
90C2×#117C2×C418yes—
91C2×#117D418yes—
92C2×#117Q818yes—
93C22×C4⋊C4—24no—
94C22×C4⋊C4—44no—
95C22×C4⋊C4C2312yes—
96C22×C4⋊C4C2311yescharacteristic
97C22×C4⋊C4C2311yescharacteristic
98C22×C4⋊C4C2312yes—
99C22×C4⋊C4C2312yes—
100C22×C4⋊C4C2311yescharacteristic
101C22×C4⋊C4C2311yescharacteristic
102C22×C4⋊C4C2312yes—
103C22×C4⋊C4C2314yes—
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2314yes—
106C22×C4⋊C4C2314yes—
107C22×C4⋊C4C2312yes—
108C22×C4⋊C4C2314yes—
109C22×C4⋊C4C2312yes—
110C22×C4⋊C4C2314yes—
111C22×C4⋊C4C2311yescharacteristic
112C22×C4⋊C4C2311yescharacteristic
113C22×C22⋊C4C2311yescharacteristic
114C22×C22⋊C4C2311yescharacteristic
115C22×C22⋊C4C2312yes—
116C22×C22⋊C4C2312yes—
117C22×C22⋊C4C2312yes—
118C22×C22⋊C4C2312yes—
119C22×C22⋊C4C2312yes—
120C22×C42C2312yes—
121C22×C42C2311yescharacteristic
122C22×C42C2311yescharacteristic
123C22×C42C2312yes—
124C22×C42C2312yes—
125C22×C42C2312yes—
126C22×C42C2314yes—
127C22×C42C2312yes—
128#389—28no—
129#389—28no—
130#389—24no—
131#389—24no—
132#389—44no—
133#389—44no—
134#389—48no—
135#389C2×C4116yescomplemented
136#388—28no—
137#388—28no—
138#388C2×C4132yes—
139#385—216no—
140#385—44no—
141#384—28no—
142#382—28no—
143#382—28no—
144#382—44no—
145#382C2×C4116yescomplemented
146#380—24no—
147#380C2×C418yes—
148C4×C4⋊C4—24no—
149C4×C4⋊C4—24no—
150C4×C4⋊C4—44no—
151C4×C4⋊C4C2×C4116yescomplemented
152C4×C22⋊C4—24no—
153C4×C22⋊C4C2×C418yes—
154#376—28no—
155#376C2×C4116yescomplemented
156C2×#95C2312yes—
157C2×#95C2312yes—
158C2×#95C2314yes—
159C2×#95C2312yes—
160C2×#95C2314yes—
161C2×#95C2312yes—
162C2×#95C2312yes—
163C2×#95C2312yes—
164C2×#95C2311yescharacteristic
165C2×#95C2311yescharacteristic
166C2×#95C2312yes—
167C2×#95C2312yes—
168C2×#95C2312yes—
169C2×#95C2314yes—
170C2×#95C2314yes—
171C2×#95C2314yes—
172C2×#95C2314yes—
173C2×#95C2314yes—
174C2×#95C2314yes—
175C2×#95C2314yes—
176C2×#95C2312yes—
177C2×#95C2312yes—
178C2×#95C2314yes—
179C2×#95C2314yes—
180C2×#95C2314yes—
181C2×#95C2314yes—
182C2×#95C2311yescharacteristic
183C2×#95C2312yes—
184C2×#95C2311yescharacteristic
185C2×#95C2314yes—
186C25C2411yescharacteristic
187C23×C4—24no—
188C23×C4—24no—
189C23×C4—24no—
190C23×C4—44no—
191C23×C4—42no—
192C23×C4—44no—
193C23×C4—42no—
194C23×C4C22×C414yes—
195C23×C4C2×D414yes—
196C23×C4C2×Q814yes—
197C23×C4C2412yes—
198C23×C4C2412yes—
199C23×C4C2411yescharacteristic
200C23×C4C2412yes—
201C23×C4C2412yes—
202C23×C4C2411yescharacteristic
203C23×C4C2414yes—
204C23×C4C2412yes—
205C23×C4C2414yes—
206C23×C4C2412yes—
207C23×C4C2411yescharacteristic
208C23×C4C2411yescharacteristic
209C23×C4C2412yes—
210C23×C4C2412yes—
211C23×C4C2411yescharacteristic
212C23×C4C2411yescharacteristic
213#117—416no—
214#117C4⋊C4164yescomplemented
215C2×C4⋊C4—24no—
216C2×C4⋊C4—24no—
217C2×C4⋊C4—24no—
218C2×C4⋊C4—24no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—24no—
224C2×C4⋊C4—24no—
225C2×C4⋊C4—24no—
226C2×C4⋊C4—24no—
227C2×C4⋊C4—24no—
228C2×C4⋊C4—24no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4—28no—
231C2×C4⋊C4—28no—
232C2×C4⋊C4—28no—
233C2×C4⋊C4—44no—
234C2×C4⋊C4—44no—
235C2×C4⋊C4—42no—
236C2×C4⋊C4—42no—
237C2×C4⋊C4—42no—
238C2×C4⋊C4—42no—
239C2×C4⋊C4—42no—
240C2×C4⋊C4—42no—
241C2×C4⋊C4—42no—
242C2×C4⋊C4—44no—
243C2×C4⋊C4—416no—
244C2×C4⋊C4—44no—
245C2×C4⋊C4—44no—
246C2×C4⋊C4—44no—
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⋊C4—44no—
255C2×C4⋊C4C22×C418yes—
256C2×C4⋊C4C22×C4116yes—
257C2×C4⋊C4C22×C414yes—
258C2×C4⋊C4C22×C414yes—
259C2×C4⋊C4C2×D414yes—
260C2×C4⋊C4C2×D414yes—
261C2×C4⋊C4C2×Q814yes—
262C2×C4⋊C4C2×Q814yes—
263C2×C22⋊C4—24no—
264C2×C22⋊C4—22no—
265C2×C22⋊C4—22no—
266C2×C22⋊C4—22no—
267C2×C22⋊C4—22no—
268C2×C22⋊C4—24no—
269C2×C22⋊C4—42no—
270C2×C22⋊C4—42no—
271C2×C22⋊C4—42no—
272C2×C22⋊C4—41no—
273C2×C22⋊C4—41no—
274C2×C22⋊C4—42no—
275C2×C22⋊C4—42no—
276C2×C22⋊C4—42no—
277C2×C22⋊C4—44no—
278C2×C22⋊C4C22×C418yes—
279C2×C22⋊C4C22×C418yes—
280C2×C22⋊C4C22×C418yes—
281C2×C22⋊C4C2×D418yes—
282C2×C22⋊C4C2×Q818yes—
283C2×C42—24no—
284C2×C42—24no—
285C2×C42—24no—
286C2×C42—24no—
287C2×C42—24no—
288C2×C42—22no—
289C2×C42—22no—
290C2×C42—24no—
291C2×C42—24no—
292C2×C42—24no—
293C2×C42—24no—
294C2×C42—24no—
295C2×C42—24no—
296C2×C42—44no—
297C2×C42—44no—
298C2×C42—42no—
299C2×C42—42no—
300C2×C42C22×C418yes—
301C2×C42C22×C4116yes—
302C2×C42C22×C418yes—
303C2×C42C2×D418yes—
304C2×C42C2×Q818yes—
305#95—22no—
306#95—24no—
307#95—24no—
308#95—24no—
309#95—22no—
310#95—24no—
311#95—24no—
312#95—28no—
313#95—28no—
314#95—24no—
315#95—44no—
316#95—42no—
317#95—44no—
318#95—42no—
319#95—44no—
320#95—42no—
321#95—42no—
322#95—42no—
323#95—44no—
324#95—44no—
325#95—44no—
326#95—44no—
327#95—44no—
328#95—44no—
329#95—44no—
330#95C22×C414yes—
331#95C22×C418yes—
332#95C22×C4116yes—
333#95C22×C414yes—
334#95C22×C4116yes—
335C24—21no—
336C24—22no—
337C24—21no—
338C24—22no—
339C24—22no—
340C24—21no—
341C24—22no—
342C24—21no—
343C24C23×C412yes—
344C24C22×D412yes—
345C24C22×Q812yes—
346C24C2511yescharacteristic
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—24no—
352C22×C4—24no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—216no—
356C22×C4—28no—
357C22×C4—28no—
358C22×C4—216no—
359C22×C4—21no—
360C22×C4—21no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—24no—
364C22×C4—24no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—24no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—24no—
375C22×C4—24no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—21no—
381C22×C4—21no—
382C22×C4—24no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—21no—
387C22×C4—22no—
388C22×C4—21no—
389C22×C4—22no—
390C22×C4—24no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—21no—
394C22×C4—22no—
395C22×C4—24no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—24no—
399C22×C4—24no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—24no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—21no—
408C22×C4—24no—
409C22×C4—22no—
410C22×C4—21no—
411C22×C4—22no—
412C22×C4—24no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—22no—
420C22×C4—21no—
421C22×C4—24no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—24no—
426C22×C4—48no—
427C22×C4—84no—
428C22×C4—84no—
429C22×C4—88no—
430C22×C4—88no—
431C22×C4C2×C4⋊C4116yes—
432C22×C4C23×C414yes—
433C22×C4C23×C414yes—
434C22×C4C23×C418yes—
435C22×C4C23×C412yes—
436C22×C4C23×C414yes—
437C22×C4C23×C414yes—
438C22×C4C23×C412yes—
439C22×C4C22×D414yes—
440C22×C4C22×D412yes—
441C22×C4C22×D414yes—
442C22×C4C22×D412yes—
443C22×C4C22×Q812yes—
444C22×C4C22×Q814yes—
445C22×C4C22×Q814yes—
446C22×C4C22×Q812yes—
447C22×C4H(5,𝔽2)12yes—
448C22×C4H(5,𝔽2)14yes—
449C22×C42−1+412yes—
450C22×C42−1+414yes—
451C4⋊C4—216no—
452C4⋊C4—416no—
453C4⋊C4—48no—
454C4⋊C4—48no—
455C4⋊C4—44no—
456C4⋊C4—44no—
457C4⋊C4—48no—
458C4⋊C4—44no—
459C4⋊C4—48no—
460C4⋊C4—44no—
461C4⋊C4—88no—
462C4⋊C4—84no—
463C4⋊C4—84no—
464C4⋊C4—88no—
465C4⋊C4—88no—
466C4⋊C4—88no—
467C4⋊C4C2×C4⋊C4116yes—
468C4⋊C4C2×C4⋊C4116yes—
469C22⋊C4—48no—
470C22⋊C4—48no—
471C22⋊C4—44no—
472C22⋊C4—44no—
473C22⋊C4—48no—
474C22⋊C4—84no—
475C22⋊C4C2×C4⋊C4132yes—
476C42—216no—
477C42—416no—
478C42—48no—
479C42—48no—
480C42—44no—
481C42—48no—
482C42—44no—
483C42C2×C4⋊C4132yes—
484C23—24no—
485C23—24no—
486C23—22no—
487C23—22no—
488C23—22no—
489C23—22no—
490C23—24no—
491C23—22no—
492C23—22no—
493C23—22no—
494C23—22no—
495C23—22no—
496C23—24no—
497C23—22no—
498C23—42no—
499C23—42no—
500C23—42no—
501C23—42no—
502C23—42no—
503C23—42no—
504C23—42no—
505C23—42no—
506C23C22×C4⋊C414yes—
507C23C24×C411yescharacteristic
508C23C23×D411yescharacteristic
509C23C23×Q811yescharacteristic
510C23C2×H(5,𝔽2)12yes—
511C23C2×H(5,𝔽2)11yescharacteristic
512C23C2×2−1+412yes—
513C23C2×2−1+411yescharacteristic
514C23#58512yes—
515C23#58512yes—
516C23#58511yescharacteristic
517C23#58511yescharacteristic
518C2×C4—24no—
519C2×C4—28no—
520C2×C4—24no—
521C2×C4—24no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—24no—
525C2×C4—24no—
526C2×C4—24no—
527C2×C4—24no—
528C2×C4—24no—
529C2×C4—24no—
530C2×C4—28no—
531C2×C4—24no—
532C2×C4—232no—
533C2×C4—24no—
534C2×C4—24no—
535C2×C4—28no—
536C2×C4—24no—
537C2×C4—28no—
538C2×C4—24no—
539C2×C4—28no—
540C2×C4—24no—
541C2×C4—24no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—24no—
545C2×C4—24no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—42no—
551C2×C4—44no—
552C2×C4—42no—
553C2×C4—44no—
554C2×C4—44no—
555C2×C4—42no—
556C2×C4—42no—
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—42no—
566C2×C4—41no—
567C2×C4—44no—
568C2×C4—42no—
569C2×C4—41no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—41no—
573C2×C4—42no—
574C2×C4—44no—
575C2×C4—42no—
576C2×C4—41no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—44no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—44no—
589C2×C4—42no—
590C2×C4—44no—
591C2×C4—42no—
592C2×C4—44no—
593C2×C4—44no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—44no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—44no—
602C2×C4—44no—
603C2×C4—42no—
604C2×C4—44no—
605C2×C4—816no—
606C2×C4C22×C4⋊C414yes—
607C2×C4C22×C4⋊C418yes—
608C2×C4C22×C4⋊C418yes—
609C2×C4C22×C4⋊C414yes—
610C2×C4#52018yes—
611C2×C4#53514yes—
612C2×C4#53614yes—
613C2×C4#54418yes—
614C22—24no—
615C22—24no—
616C22—24no—
617C22—44no—
618C22—44no—
619C22—42no—
620C22—42no—
621C22—44no—
622C22—42no—
623C22—44no—
624C22—42no—
625C22C23×C4⋊C411yescharacteristic
626C22C2×#52012yes—
627C22#342812yes—
628C22C4×H(5,𝔽2)11yescharacteristic
629C22C4×2−1+411yescharacteristic
630C22C2×#53511yescharacteristic
631C22C2×#53611yescharacteristic
632C22C2×#54412yes—
633C22#348811yescharacteristic
634C22#348911yescharacteristic
635C22#350011yescharacteristic
636C22#350011yescharacteristic
637C22#350311yescharacteristic
638C22#350412yes—
639C22#350411yescharacteristic
640C22#354012yes—
641C22#354212yes—
642C22#354712yes—
643C22#354712yes—
644C22#355112yes—
645C22#356312yes—
646C22#356512yes—
647C22#356512yes—
648C4—24no—
649C4—28no—
650C4—28no—
651C4—44no—
652C4—48no—
653C4—44no—
654C4—42no—
655C4—42no—
656C4—82no—
657C4—84no—
658C4—81no—
659C4—82no—
660C4—82no—
661C4—81no—
662C4—82no—
663C4#231718yes—
664C2—44no—
665C2C2×#231711yescharacteristic
666C2#3418411yescharacteristic
667C2#3425011yescharacteristic
668C2#3446712yes—
669C2#3448212yes—
670C2#4296211yescharacteristic
671C2#4296511yescharacteristic
672C2#4297212yes—
673C2#4297811yescharacteristic
674C2#4298112yes—
675C2#4299511yescharacteristic
subgroupG/H
center, Frattini, socle, cosocleC24C25
commutatorC23C24×C4
prime radicalC25C24
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #14677C2semidirect
7. #14348C2semidirect
8. #14277C2semidirect
11. #14046C2semidirect
12. #14006C2semidirect
13. #13963C2semidirect
14. #13954C2semidirect
15. #13933C2semidirect
22. #2346C4semidirect
135. #389C2×C4semidirect
145. #382C2×C4semidirect
151. C4×C4⋊C4C2×C4semidirect
155. #376C2×C4semidirect
214. #117C4⋊C4semidirect

Maximal subgroups

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

subgroupindex
C2×#23462
C2×#23172
#146772
#144882
#144422
#143482
#142772
#142642
#141302
#140462
#140062
#139632
#139542
#139332
#138812
#138442