Finite Group Databasebeta

← #6810395

Subgroups — #6810395

Order 512.

2153 conjugacy classes of subgroups.

The subgroup lattice has 750 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3215C212yesmaximal
2#36133C211yesmaximal, characteristic, complemented
3C2×#2547C211yesmaximal, characteristic
4#32905C211yesmaximal, characteristic
5#30901C214yesmaximal, complemented
6#30754C214yesmaximal, complemented
7#29626C212yesmaximal
8#28471C212yesmaximal, complemented
9#27776C212yesmaximal, complemented
10#25722C211yesmaximal, characteristic, complemented
11#25365C212yesmaximal, complemented
12#24729C212yesmaximal, complemented
13#24508C212yesmaximal, complemented
14#24212C211yesmaximal, characteristic, complemented
15#23934C212yesmaximal, complemented
16#22603C211yesmaximal, characteristic, complemented
17C4×#450C211yesmaximal, characteristic
18C2×#533C2212yes—
19C22×C4⋊Q8C2211yescharacteristic
20C22×#124C2212yes—
21C22×C4×D4C2211yescharacteristic
22#3215—216no—
23C2×#497C2214yes—
24C2×#494C2212yescomplemented
25C2×#489C2214yes—
26C2×#464C2214yes—
27C2×#464C2214yes—
28C2×#462C2212yescomplemented
29C2×#461C2212yes—
30C2×#451C2212yes—
31C2×#451C2212yes—
32C2×#451C2212yes—
33C2×#450C2212yes—
34C2×#450C2211yescharacteristic
35C2×C4×SD16C2212yes—
36C2×#432C2212yes—
37#2857C2211yescharacteristic
38#2855C2212yes—
39#2551C2211yescharacteristic, complemented
40#2547—21no—
41#2547—22no—
42#2547—21no—
43#2547C2212yes—
44#2547C2214yes—
45#2547C2212yes—
46#2546C2211yescharacteristic, complemented
47#2506—21no—
48#2462—22no—
49#2451C2212yescomplemented
50#2446C2212yescomplemented
51#2440—21no—
52#2438—22no—
53C2×#397—22no—
54#2365—21no—
55#2365C2211yescharacteristic, complemented
56Q8×C4⋊C4C2211yescharacteristic, complemented
57#2344—21no—
58C4×C22⋊Q8C2211yescharacteristic
59C2×#391C2212yes—
60C2×#380—21no—
61C2×#380—21no—
62C2×C4×C4⋊C4C2211yescharacteristic
63#2066C2214yescomplemented
64#2056C2214yescomplemented
65#2043C2212yescomplemented
66#2042C2214yes—
67#2042C2214yes—
68#2024C2214yescomplemented
69#2016C2212yescomplemented
70#2016C2214yescomplemented
71#2016C2214yescomplemented
72#2015C2214yes—
73#2015C2214yes—
74#1981C2212yescomplemented
75#1971C2214yescomplemented
76#1956C2212yescomplemented
77#1939C2212yescomplemented
78#1935C2214yescomplemented
79#1917C2214yescomplemented
80#1878C2214yescomplemented
81#1878C2214yescomplemented
82#1877C2212yes—
83#1877C2214yes—
84#1872C2211yescharacteristic
85#1872C2212yes—
86#1864C2212yes—
87#1853C2211yescharacteristic
88#1817C2212yes—
89#1802C2212yes—
90#1797C2211yescharacteristic
91#1787C2212yes—
92#1786C2211yescharacteristic
93#1770C2212yescomplemented
94C4×#102C2212yes—
95C4×C22⋊C8C2211yescharacteristic
96C23×Q8C2312yes—
97C23×D4C2311yescharacteristic
98C24×C4—21no—
99C22×Q16C2314yes—
100C22×Q16C2314yes—
101C22×Q16C2314yes—
102C22×SD16C2314yes—
103C22×SD16C2312yes—
104#533—28no—
105#533—44no—
106C2×C4⋊Q8—21no—
107C2×C4⋊Q8—21no—
108C2×C4⋊Q8—21no—
109C2×C4⋊Q8—44no—
110C2×C4⋊Q8C2311yescharacteristic
111C2×C4⋊Q8C2312yes—
112C2×C4⋊Q8C2312yes—
113C2×C4⋊Q8C2312yes—
114C2×C4⋊Q8C2312yes—
115C2×#125—22no—
116C2×#124—28no—
117C2×#124—22no—
118C2×#124—22no—
119C2×#124—44no—
120C2×#124C2314yes—
121C2×#124C2314yes—
122C2×C22⋊Q8—22no—
123C2×C22⋊Q8C2312yes—
124C2×C22⋊Q8C2311yescharacteristic
125C2×C4×Q8—22no—
126C2×C4×Q8C2311yescharacteristic
127C2×C4×Q8C2312yes—
128C2×C4×Q8C2312yes—
129C2×C4×D4—21no—
130C2×C4×D4—21no—
131C2×C4×D4—44no—
132C2×C4×D4C2312yes—
133C2×C4×D4C2312yes—
134C2×#117—22no—
135C22×C4⋊C4—21no—
136C22×C4⋊C4—21no—
137C22×C4⋊C4D411yescharacteristic
138C22×C4⋊C4D411yescharacteristic
139C22×C4⋊C4C2311yescharacteristic
140C22×C22⋊C4—21no—
141C22×C22⋊C4—22no—
142C22×C22⋊C4—22no—
143C22×C42C2311yescharacteristic
144C22×C42C2312yes—
145#497—216no—
146#494—44no—
147#489—216no—
148#464—216no—
149#464—216no—
150#462—44no—
151#461—44no—
152#451—28no—
153#451—28no—
154#451—44no—
155#450—28no—
156#450D418yes—
157C4×SD16—44no—
158#432—28no—
159C2×#106C2312yes—
160C2×C4⋊C8C2312yes—
161C2×C4⋊C8C2314yes—
162C2×#103C2314yes—
163C2×#103C2314yes—
164C2×#103C2314yes—
165C2×#103C2314yes—
166C2×#103C2314yes—
167C2×#103C2312yes—
168C2×#103C2314yes—
169C2×#102C2314yes—
170C2×#102C2312yes—
171C2×#102C2312yes—
172C2×#102C2312yes—
173C2×C22⋊C8C2312yes—
174C2×C22⋊C8C2311yescharacteristic
175C2×C22⋊C8C2311yescharacteristic
176C2×#97C2314yes—
177C2×C4×C8C2312yes—
178#400—22no—
179#398—22no—
180#398—22no—
181#398—22no—
182#397—24no—
183#397—24no—
184#397—24no—
185#397—24no—
186#395—22no—
187#395—22no—
188#391—22no—
189#391—22no—
190#391—22no—
191#391C2312yes—
192#391C2314yes—
193#391C2312yes—
194#391C2312yes—
195#391C2311yescharacteristic
196#391C2314yes—
197#389—21no—
198#389—21no—
199#389—21no—
200#389—21no—
201#389—21no—
202#389—21no—
203#389—22no—
204#389C2312yes—
205#389C2312yes—
206#389C2311yescharacteristic
207#387—22no—
208#387—22no—
209#387—22no—
210#387—22no—
211#387—21no—
212#387—22no—
213#387—22no—
214#385—22no—
215#385—21no—
216#384—21no—
217#384C2311yescharacteristic
218#381—22no—
219#380—21no—
220#380—21no—
221#380—22no—
222#380—22no—
223#380—22no—
224#380—24no—
225#380—22no—
226#380—24no—
227#380C2311yescharacteristic
228C4×C4⋊C4—21no—
229C4×C4⋊C4—21no—
230C4×C4⋊C4—21no—
231C4×C4⋊C4D411yescharacteristic
232C4×C4⋊C4D412yes—
233C4×C4⋊C4D411yescharacteristic
234C4×C4⋊C4C2311yescharacteristic
235C4×C4⋊C4C2311yescharacteristic
236C4×C4⋊C4C2312yes—
237C4×C4⋊C4C2311yescharacteristic
238C4×C22⋊C4—21no—
239C2×#95—22no—
240C2×#95—22no—
241C2×#95D412yes—
242#340C2312yes—
243#340C2314yes—
244#340C2312yes—
245#336C2312yes—
246#336C2314yes—
247#336C2314yes—
248#336C2312yes—
249C25—21no—
250C22×Q8—22no—
251C22×Q8—22no—
252C22×Q8—24no—
253C22×Q8—24no—
254C22×Q8—44no—
255C22×Q8—44no—
256C22×Q8C2412yes—
257C22×Q8C2414yes—
258C22×Q8C2411yescharacteristic
259C22×D4—21no—
260C22×D4—22no—
261C22×D4—42no—
262C22×D4—42no—
263C22×D4C2×D414yes—
264C22×D4C2412yes—
265C23×C4—22no—
266C23×C4—21no—
267C23×C4—22no—
268C23×C4—22no—
269C23×C4—44no—
270C23×C4—44no—
271C23×C4—44no—
272C23×C4C2×D412yes—
273C23×C4C2×D411yescharacteristic
274C23×C4C2×D411yescharacteristic
275C23×C4C2×D412yes—
276C23×C4C2411yescharacteristic
277C2×Q16—28no—
278C2×Q16—28no—
279C2×Q16—44no—
280C2×Q16—44no—
281C2×Q16—44no—
282C2×Q16—44no—
283C2×Q16—44no—
284C2×Q16—44no—
285C2×Q16—44no—
286C2×SD16—28no—
287C2×SD16—42no—
288C2×SD16—42no—
289C2×SD16—44no—
290C2×SD16—44no—
291C2×SD16C2×D418yes—
292C22×C8C2414yes—
293C22×C8C2412yes—
294C22×C8C2412yes—
295C4⋊Q8—24no—
296C4⋊Q8—24no—
297C4⋊Q8—24no—
298C4⋊Q8—24no—
299C4⋊Q8—24no—
300C4⋊Q8—42no—
301C4⋊Q8—44no—
302C4⋊Q8—42no—
303C4⋊Q8—44no—
304C4⋊Q8—44no—
305C4⋊Q8—44no—
306#125—48no—
307#124—28no—
308#124—28no—
309#124—44no—
310#124—44no—
311#124—48no—
312#124—48no—
313C22⋊Q8—22no—
314C22⋊Q8—24no—
315C22⋊Q8—24no—
316C22⋊Q8—48no—
317C22⋊Q8C2×D414yes—
318C4×Q8—24no—
319C4×Q8—24no—
320C4×Q8—24no—
321C4×Q8—42no—
322C4×Q8—42no—
323C4×Q8—48no—
324C4×D4—24no—
325C4×D4—24no—
326C4×D4—42no—
327C4×D4—42no—
328C4×D4—44no—
329C4×D4—44no—
330#117—48no—
331C2×C4⋊C4—24no—
332C2×C4⋊C4—21no—
333C2×C4⋊C4—22no—
334C2×C4⋊C4—22no—
335C2×C4⋊C4—22no—
336C2×C4⋊C4—22no—
337C2×C4⋊C4—21no—
338C2×C4⋊C4—22no—
339C2×C4⋊C4—22no—
340C2×C4⋊C4—22no—
341C2×C4⋊C4—22no—
342C2×C4⋊C4—22no—
343C2×C4⋊C4—22no—
344C2×C4⋊C4—21no—
345C2×C4⋊C4—21no—
346C2×C4⋊C4—22no—
347C2×C4⋊C4—21no—
348C2×C4⋊C4—21no—
349C2×C4⋊C4—21no—
350C2×C4⋊C4—21no—
351C2×C4⋊C4—22no—
352C2×C4⋊C4—42no—
353C2×C4⋊C4—42no—
354C2×C4⋊C4—44no—
355C2×C4⋊C4—44no—
356C2×C4⋊C4C2×D411yescharacteristic
357C2×C4⋊C4C2×D411yescharacteristic
358C2×C4⋊C4C2×D411yescharacteristic
359C2×C4⋊C4C2×D412yes—
360C2×C4⋊C4C2×D411yescharacteristic
361C2×C4⋊C4C2×D412yes—
362C2×C4⋊C4C2412yes—
363C2×C4⋊C4C2412yes—
364C2×C4⋊C4C2411yescharacteristic
365C2×C4⋊C4C2411yescharacteristic
366C2×C4⋊C4C2411yescharacteristic
367C2×C22⋊C4—21no—
368C2×C22⋊C4—22no—
369C2×C22⋊C4—21no—
370C2×C22⋊C4—22no—
371C2×C22⋊C4—22no—
372C2×C22⋊C4—24no—
373C2×C22⋊C4—24no—
374C2×C22⋊C4—24no—
375C2×C22⋊C4—24no—
376C2×C22⋊C4—48no—
377C2×C22⋊C4—48no—
378C2×C22⋊C4—44no—
379C2×C42—24no—
380C2×C42—24no—
381C2×C42—24no—
382C2×C42—22no—
383C2×C42—22no—
384C2×C42—21no—
385C2×C42—21no—
386C2×C42C2×D411yescharacteristic
387C2×C42C2×D411yescharacteristic
388C2×C42C2×D412yes—
389C2×C42C2×D412yes—
390C2×C42C2411yescharacteristic
391C2×C42C2412yes—
392C2×C42C2412yes—
393C2×C42C2411yescharacteristic
394#106—42no—
395C4⋊C8—28no—
396C4⋊C8—24no—
397#103—28no—
398#103—28no—
399#103—44no—
400#103—42no—
401#103—44no—
402#103—44no—
403#103—44no—
404#102—28no—
405#102—42no—
406#102—42no—
407#102C2×D418yes—
408C22⋊C8—24no—
409C22⋊C8—22no—
410C22⋊C8C2×D414yes—
411#97—28no—
412C4×C8—24no—
413#95—24no—
414#95—24no—
415#95—24no—
416#95—24no—
417#95—21no—
418#95—21no—
419#95—21no—
420#95—22no—
421#95—22no—
422#95—21no—
423#95—21no—
424#95—22no—
425#95—22no—
426#95—22no—
427#95—22no—
428#95—22no—
429#95—22no—
430#95C2×D412yes—
431#95C2×D412yes—
432#95C2×D412yes—
433#95C2×D412yes—
434C24—24no—
435C24—22no—
436C24—42no—
437C24—42no—
438C24—42no—
439C24—42no—
440C24—42no—
441C24—42no—
442C24C22×D411yescharacteristic
443C2×Q8—21no—
444C2×Q8—24no—
445C2×Q8—22no—
446C2×Q8—22no—
447C2×Q8—24no—
448C2×Q8—24no—
449C2×Q8—24no—
450C2×Q8—22no—
451C2×Q8—24no—
452C2×Q8—22no—
453C2×Q8—24no—
454C2×Q8—24no—
455C2×Q8—24no—
456C2×Q8—44no—
457C2×Q8—44no—
458C2×Q8—44no—
459C2×Q8—44no—
460C2×Q8—44no—
461C2×Q8—82no—
462C2×Q8—88no—
463C2×Q8—88no—
464C2×Q8C22×D412yes—
465C2×D4—24no—
466C2×D4—22no—
467C2×D4—22no—
468C2×D4—22no—
469C2×D4—22no—
470C2×D4—22no—
471C2×D4—48no—
472C2×D4—42no—
473C2×D4—42no—
474C2×D4—48no—
475C2×D4C22×D414yes—
476C22×C4—22no—
477C22×C4—22no—
478C22×C4—22no—
479C22×C4—22no—
480C22×C4—24no—
481C22×C4—24no—
482C22×C4—22no—
483C22×C4—24no—
484C22×C4—21no—
485C22×C4—21no—
486C22×C4—24no—
487C22×C4—22no—
488C22×C4—21no—
489C22×C4—21no—
490C22×C4—22no—
491C22×C4—22no—
492C22×C4—22no—
493C22×C4—24no—
494C22×C4—41no—
495C22×C4—42no—
496C22×C4—42no—
497C22×C4—44no—
498C22×C4—44no—
499C22×C4—42no—
500C22×C4—44no—
501C22×C4—42no—
502C22×C4—44no—
503C22×C4—44no—
504C22×C4—42no—
505C22×C4—44no—
506C22×C4—44no—
507C22×C4—44no—
508C22×C4—44no—
509C22×C4—816no—
510C22×C4#13714yes—
511C22×C4C22×D412yes—
512C22×C4C22×D412yes—
513C22×C4C22×D411yescharacteristic
514C22×C4C22×D412yes—
515C22×C4C22×D412yes—
516C22×C4C22×D411yescharacteristic
517C22×C4C22×D411yescharacteristic
518C22×C4C22×D412yes—
519C22×C4C22×D411yescharacteristic
520C22×C4C22×D411yescharacteristic
521C22×C4H(5,𝔽2)12yes—
522C22×C4C2511yescharacteristic
523Q16—88no—
524Q16—88no—
525Q16—88no—
526SD16—48no—
527SD16—88no—
528C2×C8—22no—
529C2×C8—22no—
530C2×C8—24no—
531C2×C8—22no—
532C2×C8—22no—
533C2×C8—24no—
534C2×C8—22no—
535C2×C8—24no—
536C2×C8C22×D414yes—
537C4⋊C4—24no—
538C4⋊C4—24no—
539C4⋊C4—22no—
540C4⋊C4—22no—
541C4⋊C4—22no—
542C4⋊C4—22no—
543C4⋊C4—21no—
544C4⋊C4—22no—
545C4⋊C4—22no—
546C4⋊C4—22no—
547C4⋊C4—22no—
548C4⋊C4—22no—
549C4⋊C4—22no—
550C4⋊C4—42no—
551C4⋊C4—44no—
552C4⋊C4—44no—
553C4⋊C4—44no—
554C4⋊C4—44no—
555C4⋊C4—44no—
556C4⋊C4—42no—
557C4⋊C4—44no—
558C4⋊C4—44no—
559C4⋊C4—44no—
560C4⋊C4—44no—
561C4⋊C4—44no—
562C4⋊C4—42no—
563C4⋊C4—42no—
564C4⋊C4—42no—
565C4⋊C4—44no—
566C4⋊C4—42no—
567C4⋊C4—81no—
568C4⋊C4—81no—
569C4⋊C4—82no—
570C4⋊C4C22×D412yes—
571C22⋊C4—24no—
572C22⋊C4—48no—
573C22⋊C4—48no—
574C22⋊C4—48no—
575C22⋊C4—48no—
576C22⋊C4—44no—
577C22⋊C4—44no—
578C22⋊C4—42no—
579C22⋊C4—44no—
580C42—22no—
581C42—22no—
582C42—24no—
583C42—24no—
584C42—22no—
585C42—22no—
586C42—22no—
587C42—22no—
588C42—22no—
589C42—44no—
590C42—44no—
591C42—42no—
592C42—42no—
593C42H(5,𝔽2)14yes—
594C23—21no—
595C23—21no—
596C23—21no—
597C23—21no—
598C23—24no—
599C23—48no—
600C23—48no—
601C23—42no—
602C23—42no—
603C23—42no—
604C23—48no—
605C23—42no—
606C23—42no—
607C23—42no—
608C23—84no—
609C23—84no—
610C23—84no—
611C23—84no—
612C23D4212yes—
613C23C2×#13712yes—
614C23#57812yes—
615C23C23×D411yescharacteristic
616Q8—24no—
617Q8—42no—
618Q8—44no—
619Q8—44no—
620Q8—44no—
621Q8—44no—
622Q8—44no—
623Q8—44no—
624Q8—88no—
625Q8—88no—
626D4—24no—
627D4—24no—
628D4—48no—
629D4—42no—
630D4—42no—
631C2×C4—21no—
632C2×C4—21no—
633C2×C4—21no—
634C2×C4—21no—
635C2×C4—21no—
636C2×C4—21no—
637C2×C4—28no—
638C2×C4—24no—
639C2×C4—24no—
640C2×C4—22no—
641C2×C4—21no—
642C2×C4—22no—
643C2×C4—21no—
644C2×C4—22no—
645C2×C4—22no—
646C2×C4—22no—
647C2×C4—22no—
648C2×C4—22no—
649C2×C4—22no—
650C2×C4—22no—
651C2×C4—22no—
652C2×C4—22no—
653C2×C4—22no—
654C2×C4—22no—
655C2×C4—22no—
656C2×C4—21no—
657C2×C4—44no—
658C2×C4—41no—
659C2×C4—41no—
660C2×C4—41no—
661C2×C4—48no—
662C2×C4—41no—
663C2×C4—41no—
664C2×C4—42no—
665C2×C4—41no—
666C2×C4—42no—
667C2×C4—44no—
668C2×C4—42no—
669C2×C4—42no—
670C2×C4—42no—
671C2×C4—44no—
672C2×C4—41no—
673C2×C4—44no—
674C2×C4—42no—
675C2×C4—42no—
676C2×C4—41no—
677C2×C4—42no—
678C2×C4—42no—
679C2×C4—41no—
680C2×C4—41no—
681C2×C4—88no—
682C2×C4—84no—
683C2×C4—84no—
684C2×C4—88no—
685C2×C4—88no—
686C2×C4#53512yes—
687C2×C4#53512yes—
688C2×C4D4212yes—
689C2×C4C2×#13712yes—
690C2×C4C2×#13712yes—
691C2×C4#57512yes—
692C2×C4C23×D411yescharacteristic
693C2×C4C2×H(5,𝔽2)12yes—
694C2×C4C2×H(5,𝔽2)11yescharacteristic
695C2×C4#58512yes—
696C8—24no—
697C8—44no—
698C8—42no—
699C22—22no—
700C22—22no—
701C22—22no—
702C22—22no—
703C22—22no—
704C22—22no—
705C22—44no—
706C22—44no—
707C22—44no—
708C22—816no—
709C22—82no—
710C22—82no—
711C22—82no—
712C22—82no—
713C22#327614yes—
714C22C2×#53511yescharacteristic
715C22C2×D4211yescharacteristic
716C22#348711yescharacteristic
717C22C22×#13711yescharacteristic
718C22C2×#57511yescharacteristic
719C22C2×#57811yescharacteristic
720C22#358611yescharacteristic
721C4—22no—
722C4—22no—
723C4—22no—
724C4—42no—
725C4—42no—
726C4—42no—
727C4—42no—
728C4—42no—
729C4—41no—
730C4—42no—
731C4—82no—
732C4—82no—
733C4—81no—
734C4—84no—
735C4—82no—
736C4—81no—
737C4#321514yescomplemented
738C4#349612yes—
739C4#354212yes—
740C2—24no—
741C2—84no—
742C2#3933911yescharacteristic
743C2C2×#321511yescharacteristic
744C2#6081411yescharacteristic
745C2C2×#327611yescharacteristic
746C2#6106711yescharacteristic
747C2#6188911yescharacteristic
748C2#6190511yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #36133C2semidirect
5. #30901C2semidirect
6. #30754C2semidirect
8. #28471C2semidirect
9. #27776C2semidirect
10. #25722C2semidirect
11. #25365C2semidirect
12. #24729C2semidirect
13. #24508C2semidirect
14. #24212C2semidirect
15. #23934C2semidirect
16. #22603C2semidirect
24. C2×#494C22semidirect
28. C2×#462C22semidirect
39. #2551C22semidirect
46. #2546C22semidirect
49. #2451C22semidirect
50. #2446C22semidirect
55. #2365C22semidirect
56. Q8×C4⋊C4C22semidirect
63. #2066C22semidirect
64. #2056C22semidirect
65. #2043C22semidirect
68. #2024C22semidirect
69. #2016C22semidirect
70. #2016C22semidirect
71. #2016C22semidirect
74. #1981C22semidirect
75. #1971C22semidirect
76. #1956C22semidirect
77. #1939C22semidirect
78. #1935C22semidirect
79. #1917C22semidirect
80. #1878C22semidirect
81. #1878C22semidirect
93. #1770C22semidirect
737. C4#3215semidirect

Maximal subgroups

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

subgroupindex
C2×#32152
#361332
C2×#25472
#329052
#309012
#307542
#296262
#284712
#277762
#257222
#253652
#247292
#245082
#242122
#239342
#226032
C4×#4502