Finite Group Databasebeta

← #554200

Subgroups — #554200

Order 512.

3166 conjugacy classes of subgroups.

The subgroup lattice has 696 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2548C211yesmaximal, characteristic
2C2×#2328C211yesmaximal, characteristic
3C2×#2295C211yesmaximal, characteristic, complemented
4#14700C212yesmaximal, complemented
5#14573C214yesmaximal, complemented
6#14518C214yesmaximal, complemented
7#14429C214yesmaximal, complemented
8#14421C214yesmaximal, complemented
9#14260C212yesmaximal, complemented
10#14154C211yesmaximal, characteristic
11#14018C212yesmaximal, complemented
12#14018C212yesmaximal, complemented
13#13954C211yesmaximal, characteristic, complemented
14#13925C211yesmaximal, characteristic, complemented
15#13797C211yesmaximal, characteristic, complemented
16C22×#127C2211yescharacteristic
17C22×#124C2212yes—
18C22×C4×Q8C2211yescharacteristic
19C22×C4×D4C2212yes—
20#2548C4116yescomplemented
21C2×#394C2212yes—
22C2×#394C2212yes—
23#2328—28no—
24#2295—28no—
25C2×#391C2212yes—
26C2×#390C2212yes—
27C2×#390C2212yes—
28C2×#390C2212yes—
29C2×#390C2214yes—
30C2×#390C2212yes—
31C2×#390C2212yes—
32C2×#390C2212yes—
33C2×#389C2212yes—
34C2×#389C2212yes—
35C2×#389C2212yes—
36C2×#388C2214yes—
37C2×#388C2214yes—
38C2×#387C2212yes—
39C2×#387C2212yes—
40C2×#386C2214yes—
41C2×#386C2214yes—
42C2×#385C2214yes—
43C2×#385C2214yes—
44C2×#382C2211yescharacteristic
45C2×C4×C4⋊C4C2211yescharacteristic
46C2×C4×C4⋊C4C2211yescharacteristic
47C2×C4×C4⋊C4C2211yescharacteristic
48C2×C4×C4⋊C4C2211yescharacteristic
49C2×C4×C4⋊C4C2211yescharacteristic
50C2×#376C2211yescharacteristic
51C2×#376C2211yescharacteristic
52C2×C43C2211yescharacteristic
53#1446C2212yes—
54#1445C2212yes—
55#1444C2214yes—
56#1440C2214yes—
57#1440C2214yes—
58#1439C2214yes—
59#1439C2214yes—
60#1439C2214yes—
61#1439C2214yes—
62#1439C2214yes—
63#1439C2214yes—
64#1438C2214yes—
65#1437C2214yes—
66#1437C2214yes—
67#1436C2212yes—
68#1436C2212yes—
69C4⋊#95C2214yes—
70C4⋊#95C2212yes—
71C4⋊#95C2212yes—
72C4⋊#95C2212yes—
73C4⋊#95C2212yes—
74#1434C2214yes—
75#1434C2214yes—
76C22⋊#95C2214yes—
77C22⋊#95C2214yes—
78C23×Q8C2311yescharacteristic
79C23×D4C2312yes—
80C23×D4C2312yes—
81C23×D4C2312yes—
82C24×C4C2314yes—
83C2×#127—24no—
84C2×#127—24no—
85C2×#127C2×C418yes—
86C2×#124—48no—
87C2×#124C2×C4116yescomplemented
88C2×C4×Q8—24no—
89C2×C4×Q8—24no—
90C2×C4×Q8C2×C418yescomplemented
91C2×C4×D4—28no—
92C2×C4×D4—44no—
93C2×C4×D4—44no—
94C22×C4⋊C4C2312yes—
95C22×C4⋊C4C2312yes—
96C22×C4⋊C4C2312yes—
97C22×C4⋊C4C2312yes—
98C22×C4⋊C4C2312yes—
99C22×C4⋊C4C2311yescharacteristic
100C22×C4⋊C4C2311yescharacteristic
101C22×C4⋊C4C2311yescharacteristic
102C22×C4⋊C4C2312yes—
103C22×C4⋊C4C2314yes—
104C22×C4⋊C4C2314yes—
105C22×C22⋊C4C2314yes—
106C22×C22⋊C4C2314yes—
107C22×C22⋊C4C2314yes—
108C22×C22⋊C4C2314yes—
109C22×C22⋊C4C2314yes—
110C22×C22⋊C4C2314yes—
111C22×C22⋊C4C2314yes—
112C22×C22⋊C4C2314yes—
113C22×C22⋊C4C2314yes—
114C22×C22⋊C4C2314yes—
115C22×C22⋊C4C2314yes—
116C22×C42C2311yescharacteristic
117C22×C42C2311yescharacteristic
118C22×C42C2312yes—
119C22×C42C2312yes—
120C22×C42C2311yescharacteristic
121C22×C42C2311yescharacteristic
122C22×C42C2311yescharacteristic
123C22×C42C2311yescharacteristic
124C22×C42C2312yes—
125C22×C42C2312yes—
126C22×C42C2311yescharacteristic
127#394C2×C4116yescomplemented
128#394C2×C4116yescomplemented
129#391—28no—
130#390—28no—
131#390—44no—
132#390—44no—
133#390—48no—
134#390C2×C4116yescomplemented
135#390C2×C4116yescomplemented
136#390C2×C4116yescomplemented
137#389—28no—
138#389—44no—
139#389—44no—
140#388—48no—
141#388—48no—
142#387—28no—
143#387—28no—
144#386—216no—
145#386—216no—
146#385—48no—
147#385—48no—
148#382—24no—
149C4×C4⋊C4—24no—
150C4×C4⋊C4—24no—
151C4×C4⋊C4—24no—
152C4×C4⋊C4—24no—
153C4×C4⋊C4C2×C418yescomplemented
154#376—24no—
155#376D418yes—
156C2×#95C2312yes—
157C2×#95C2312yes—
158C2×#95C2312yes—
159C2×#95C2312yes—
160C2×#95C2312yes—
161C2×#95C2314yes—
162C2×#95C2312yes—
163C2×#95C2312yes—
164C2×#95C2312yes—
165C2×#95C2314yes—
166C2×#95C2314yes—
167C2×#95C2314yes—
168C2×#95C2314yes—
169C2×#95C2314yes—
170C2×#95C2314yes—
171C2×#95C2314yes—
172C2×#95C2314yes—
173C2×#95C2314yes—
174C2×#95C2312yes—
175C2×#95C2312yes—
176C2×#95C2312yes—
177C43—24no—
178C25C2414yes—
179C22×Q8—22no—
180C22×Q8—22no—
181C22×Q8—24no—
182C22×Q8—42no—
183C22×Q8C22×C414yes—
184C22×D4—24no—
185C22×D4—24no—
186C22×D4—24no—
187C22×D4—24no—
188C22×D4—24no—
189C22×D4—24no—
190C22×D4—24no—
191C22×D4—24no—
192C22×D4—24no—
193C22×D4—24no—
194C22×D4—44no—
195C22×D4—44no—
196C22×D4—44no—
197C22×D4—44no—
198C22×D4C22×C418yes—
199C22×D4C22×C418yes—
200C22×D4C22×C418yes—
201C23×C4—28no—
202C23×C4—44no—
203C23×C4—44no—
204C23×C4—44no—
205C23×C4—44no—
206C23×C4—44no—
207C23×C4—44no—
208C23×C4C2412yes—
209C23×C4C2414yes—
210C23×C4C2414yes—
211C23×C4C2411yescharacteristic
212C23×C4C2414yes—
213C23×C4C2411yescharacteristic
214C23×C4C2411yescharacteristic
215C23×C4C2412yes—
216C23×C4C2412yes—
217C23×C4C2412yes—
218C23×C4C2411yescharacteristic
219C23×C4C2411yescharacteristic
220C23×C4C2411yescharacteristic
221C23×C4C2411yescharacteristic
222#127—232no—
223#124—432no—
224C4×Q8—232no—
225C4×D4—432no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—22no—
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—24no—
238C2×C4⋊C4—24no—
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—28no—
247C2×C4⋊C4—42no—
248C2×C4⋊C4—42no—
249C2×C4⋊C4—42no—
250C2×C4⋊C4—44no—
251C2×C4⋊C4—44no—
252C2×C4⋊C4—44no—
253C2×C4⋊C4—44no—
254C2×C4⋊C4—44no—
255C2×C4⋊C4C22×C414yes—
256C2×C4⋊C4C22×C414yes—
257C2×C4⋊C4C22×C414yes—
258C2×C4⋊C4C22×C418yes—
259C2×C22⋊C4—28no—
260C2×C22⋊C4—28no—
261C2×C22⋊C4—28no—
262C2×C22⋊C4—28no—
263C2×C22⋊C4—28no—
264C2×C22⋊C4—28no—
265C2×C22⋊C4—28no—
266C2×C22⋊C4—44no—
267C2×C22⋊C4—44no—
268C2×C22⋊C4—44no—
269C2×C22⋊C4—44no—
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⋊C4—44no—
278C2×C22⋊C4—44no—
279C2×C22⋊C4—44no—
280C2×C22⋊C4—44no—
281C2×C22⋊C4—44no—
282C2×C22⋊C4—44no—
283C2×C22⋊C4—44no—
284C2×C22⋊C4—44no—
285C2×C22⋊C4—44no—
286C2×C22⋊C4—44no—
287C2×C22⋊C4—44no—
288C2×C22⋊C4C22×C4116yes—
289C2×C22⋊C4C22×C4116yes—
290C2×C22⋊C4C22×C4116yes—
291C2×C22⋊C4C22×C4116yes—
292C2×C42—22no—
293C2×C42—22no—
294C2×C42—22no—
295C2×C42—22no—
296C2×C42—24no—
297C2×C42—24no—
298C2×C42—22no—
299C2×C42—24no—
300C2×C42—24no—
301C2×C42—22no—
302C2×C42—24no—
303C2×C42—24no—
304C2×C42—24no—
305C2×C42—24no—
306C2×C42—24no—
307C2×C42—44no—
308C2×C42—42no—
309C2×C42—42no—
310C2×C42—44no—
311C2×C42C22×C414yes—
312C2×C42C22×C414yes—
313C2×C42C22×C418yes—
314C2×C42C22×C414yes—
315C2×C42C2×D414yes—
316C2×C42C2×D414yes—
317C2×C42C2×D414yes—
318C2×C42#4114yes—
319#95—24no—
320#95—24no—
321#95—24no—
322#95—24no—
323#95—24no—
324#95—24no—
325#95—24no—
326#95—24no—
327#95—28no—
328#95—44no—
329#95—44no—
330#95—44no—
331#95—44no—
332#95—44no—
333#95—44no—
334#95—44no—
335#95—44no—
336#95—44no—
337#95C22×C418yes—
338#95C2×D418yes—
339#95C2×D418yes—
340C24—24no—
341C24—24no—
342C24—24no—
343C24—24no—
344C24—24no—
345C24—24no—
346C24—24no—
347C24—24no—
348C24—24no—
349C24—24no—
350C24—24no—
351C24—24no—
352C24—24no—
353C24—24no—
354C24C23×C418yes—
355C24C2511yescharacteristic
356C2×Q8—28no—
357C2×Q8—28no—
358C2×Q8—44no—
359C2×Q8—48no—
360C2×Q8—48no—
361C2×D4—216no—
362C2×D4—216no—
363C2×D4—216no—
364C2×D4—216no—
365C2×D4—216no—
366C2×D4—48no—
367C2×D4—48no—
368C2×D4—416no—
369C2×D4—48no—
370C2×D4—48no—
371C2×D4—48no—
372C2×D4—48no—
373C2×D4—88no—
374C2×D4—88no—
375C2×D4—88no—
376C2×D4—84no—
377C2×D4—84no—
378C22×C4—21no—
379C22×C4—21no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—21no—
386C22×C4—21no—
387C22×C4—22no—
388C22×C4—21no—
389C22×C4—21no—
390C22×C4—21no—
391C22×C4—21no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—24no—
398C22×C4—24no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—24no—
402C22×C4—24no—
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—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—24no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—24no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—22no—
427C22×C4—24no—
428C22×C4—24no—
429C22×C4—24no—
430C22×C4—24no—
431C22×C4—24no—
432C22×C4—22no—
433C22×C4—24no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—24no—
437C22×C4—24no—
438C22×C4—24no—
439C22×C4—22no—
440C22×C4—24no—
441C22×C4—22no—
442C22×C4—416no—
443C22×C4—88no—
444C22×C4—88no—
445C22×C4—88no—
446C22×C4—88no—
447C22×C4—88no—
448C22×C4—88no—
449C22×C4C23×C418yes—
450C22×C4C23×C412yes—
451C22×C4C23×C412yes—
452C22×C4C23×C412yes—
453C22×C4C23×C412yes—
454C22×C4C23×C412yes—
455C22×C4C23×C412yes—
456C22×C4C23×C412yes—
457C22×C4C22×D414yes—
458C22×C4C22×D414yes—
459C22×C4C22×D412yes—
460C22×C4C22×D412yes—
461C22×C4C22×D412yes—
462C22×C4C2×#4112yes—
463C22×C4C2×#4112yes—
464C22×C4C2×#4112yes—
465C22×C4H(5,𝔽2)14yes—
466C22×C4H(5,𝔽2)14yes—
467C22×C4H(5,𝔽2)14yes—
468C22×C42−1+414yes—
469C4⋊C4—28no—
470C4⋊C4—28no—
471C4⋊C4—28no—
472C4⋊C4—216no—
473C4⋊C4—48no—
474C4⋊C4—48no—
475C4⋊C4—48no—
476C4⋊C4—48no—
477C4⋊C4—48no—
478C4⋊C4—88no—
479C4⋊C4—88no—
480C22⋊C4—232no—
481C22⋊C4—416no—
482C22⋊C4—416no—
483C22⋊C4—416no—
484C22⋊C4—416no—
485C22⋊C4—88no—
486C22⋊C4—88no—
487C22⋊C4—88no—
488C22⋊C4—88no—
489C22⋊C4—88no—
490C22⋊C4—88no—
491C42—28no—
492C42—28no—
493C42—28no—
494C42—28no—
495C42—48no—
496C42—48no—
497C42—48no—
498C42—44no—
499C42—44no—
500C42—84no—
501C42C4×D4116yes—
502C23—28no—
503C23—28no—
504C23—28no—
505C23—28no—
506C23—28no—
507C23—28no—
508C23—28no—
509C23—44no—
510C23—44no—
511C23—44no—
512C23—44no—
513C23—44no—
514C23—44no—
515C23—44no—
516C23—44no—
517C23—44no—
518C23—44no—
519C23—44no—
520C23—44no—
521C23—44no—
522C23—44no—
523C23—44no—
524C23—44no—
525C23—44no—
526C23—44no—
527C23—44no—
528C23—44no—
529C23—44no—
530C23—44no—
531C23—44no—
532C23—44no—
533C23—44no—
534C23—44no—
535C23—44no—
536C23—44no—
537C23C24×C411yescharacteristic
538C23C23×D411yescharacteristic
539C23C22×#4111yescharacteristic
540C23C2×H(5,𝔽2)12yes—
541C23C2×H(5,𝔽2)12yes—
542C23C2×H(5,𝔽2)12yes—
543C23C2×2−1+412yes—
544C23#58512yes—
545C23#58512yes—
546Q8—416no—
547D4—432no—
548D4—816no—
549D4—816no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—24no—
555C2×C4—24no—
556C2×C4—24no—
557C2×C4—24no—
558C2×C4—24no—
559C2×C4—24no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—28no—
566C2×C4—22no—
567C2×C4—28no—
568C2×C4—22no—
569C2×C4—28no—
570C2×C4—24no—
571C2×C4—24no—
572C2×C4—24no—
573C2×C4—24no—
574C2×C4—24no—
575C2×C4—24no—
576C2×C4—24no—
577C2×C4—24no—
578C2×C4—24no—
579C2×C4—24no—
580C2×C4—24no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—44no—
592C2×C4—42no—
593C2×C4—44no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—44no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—44no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—44no—
607C2×C4—44no—
608C2×C4—44no—
609C2×C4—44no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—44no—
613C2×C4—44no—
614C2×C4—42no—
615C2×C4—44no—
616C2×C4—44no—
617C2×C4—42no—
618C2×C4—44no—
619C2×C4—44no—
620C2×C4—44no—
621C2×C4—42no—
622C2×C4—44no—
623C2×C4—44no—
624C2×C4—42no—
625C2×C4—44no—
626C2×C4—832no—
627C2×C4C2×C4×D414yes—
628C2×C4C2×C4×D414yes—
629C2×C4C2×C4×D414yes—
630C2×C4#51814yes—
631C2×C4#52018yes—
632C2×C4#53514yes—
633C2×C4#53614yes—
634C2×C4#54218yes—
635C22—48no—
636C22—48no—
637C22—48no—
638C22—48no—
639C22—48no—
640C22—48no—
641C22—48no—
642C22—84no—
643C22—84no—
644C22—84no—
645C22—84no—
646C22—84no—
647C22—84no—
648C22—84no—
649C22—84no—
650C22C22×C4×D411yescharacteristic
651C22C2×#51811yescharacteristic
652C22C2×#52012yes—
653C22#342811yescharacteristic
654C22C4×H(5,𝔽2)12yes—
655C22C2×#53511yescharacteristic
656C22C2×#53611yescharacteristic
657C22C2×#54212yes—
658C22#348812yes—
659C22#348812yes—
660C22#349412yes—
661C22#350811yescharacteristic
662C22#351011yescharacteristic
663C22#354312yes—
664C22#354612yes—
665C22#354812yes—
666C22#355412yes—
667C22#355412yes—
668C22#355712yes—
669C22#355812yes—
670C22#356212yes—
671C4—24no—
672C4—24no—
673C4—24no—
674C4—44no—
675C4—42no—
676C4—44no—
677C4—44no—
678C4—48no—
679C4—42no—
680C4—42no—
681C4—82no—
682C4—84no—
683C4—84no—
684C4#232618yes—
685C2—88no—
686C2C2×#232611yescharacteristic
687C2#3420411yescharacteristic
688C2#3426911yescharacteristic
689C2#3449412yes—
690C2#3450012yes—
691C2#3727412yes—
692C2#3727812yes—
693C2#3733612yes—
694C2#3735712yes—
subgroupG/H
center, Frattini, socle, cosocleC24C25
commutatorC23C24×C4
prime radicalC22×#127C22
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
3. C2×#2295C2semidirect
4. #14700C2semidirect
5. #14573C2semidirect
6. #14518C2semidirect
7. #14429C2semidirect
8. #14421C2semidirect
9. #14260C2semidirect
11. #14018C2semidirect
12. #14018C2semidirect
13. #13954C2semidirect
14. #13925C2semidirect
15. #13797C2semidirect
20. #2548C4semidirect
87. C2×#124C2×C4semidirect
90. C2×C4×Q8C2×C4semidirect
127. #394C2×C4semidirect
128. #394C2×C4semidirect
134. #390C2×C4semidirect
135. #390C2×C4semidirect
136. #390C2×C4semidirect
153. C4×C4⋊C4C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#25482
C2×#23282
C2×#22952
#147002
#145732
#145182
#144292
#144212
#142602
#141542
#140182
#139542
#139252
#137972