Finite Group Databasebeta

← #7502010

Subgroups — #7502010

Order 512.

1914 conjugacy classes of subgroups.

The subgroup lattice has 705 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#489C211yesmaximal, characteristic
2C2×#3082C211yesmaximal, characteristic, complemented
3#35679C211yesmaximal, characteristic
4#35576C211yesmaximal, characteristic, complemented
5#30832C214yesmaximal, complemented
6#30830C214yesmaximal, complemented
7#30824C214yesmaximal, complemented
8#30822C214yesmaximal, complemented
9#23735C211yesmaximal, characteristic
10#23633C211yesmaximal, characteristic, complemented
11#23612C211yesmaximal, characteristic
12#23552C211yesmaximal, characteristic, complemented
13#23506C211yesmaximal, characteristic
14#23456C211yesmaximal, characteristic
15#21154C211yesmaximal, characteristic, complemented
16#21066C211yesmaximal, characteristic, complemented
17#21063C211yesmaximal, characteristic
18#21021C211yesmaximal, characteristic, complemented
19#20877C211yesmaximal, characteristic, complemented
20C2×#544C2211yescharacteristic
21C22×C4⋊Q8C2211yescharacteristic
22C22×#124C2211yescharacteristic
23C2×#519C2211yescharacteristic
24C2×#489—28no—
25C2×#489C2214yes—
26C2×#489C2214yes—
27#3082—28no—
28C2×#479C2212yescomplemented
29C2×#479C2212yescomplemented
30C2×#477C2212yescomplemented
31C2×#475C2212yescomplemented
32C2×#428C2211yescharacteristic
33C2×#428C2211yescharacteristic
34C2×#427C2211yescharacteristic
35C2×#427C2211yescharacteristic
36C2×#424C2211yescharacteristic
37C2×#419C2211yescharacteristic
38C2×#419C2211yescharacteristic
39C2×#418C2211yescharacteristic
40C2×#418C2211yescharacteristic
41C22×#103C2211yescharacteristic
42C22×#103C2211yescharacteristic
43C22×#102C2211yescharacteristic
44C22×#102C2211yescharacteristic
45C22×#97C2211yescharacteristic
46#2583—22no—
47#2551—22no—
48#2551C2214yescomplemented
49#2550—22no—
50#2550C2214yescomplemented
51#2482—22no—
52#2482—22no—
53#2448—22no—
54#2448—22no—
55#2448C2214yes—
56#2448C2214yes—
57#2434—22no—
58#2317C2211yescharacteristic
59#2317C2211yescharacteristic
60#2310C2211yescharacteristic
61#2309C2211yescharacteristic
62#2309C2211yescharacteristic
63#2296C2211yescharacteristic
64#2296C2211yescharacteristic
65#2296C2211yescharacteristic
66#2274C2211yescharacteristic
67#2094C2212yes—
68#2093C2212yes—
69#2092C2212yes—
70#2092C2212yes—
71#2074C2214yescomplemented
72#2072C2214yescomplemented
73#2067C2214yes—
74#2065C2214yes—
75#2058C2214yescomplemented
76#2057C2214yescomplemented
77#2042C2214yes—
78#2042C2214yes—
79#2025C2214yes—
80#2023C2214yes—
81#1944C2212yescomplemented
82#1944C2212yescomplemented
83#1943C2212yescomplemented
84#1943C2212yescomplemented
85#1928C2214yescomplemented
86#1927C2214yescomplemented
87#1926C2214yes—
88#1925C2214yes—
89#1917C2214yescomplemented
90#1917C2214yescomplemented
91#1865C2214yescomplemented
92#1863C2214yescomplemented
93#1736C2211yescharacteristic
94#1736C2211yescharacteristic
95#1735C2211yescharacteristic
96#1735C2211yescharacteristic
97#1729C2211yescharacteristic
98#1729C2211yescharacteristic
99#1729C2211yescharacteristic
100#1728C2211yescharacteristic
101C23×Q8—21no—
102C23×Q8C2311yescharacteristic
103C23×D4C2311yescharacteristic
104C22×M16C2311yescharacteristic
105C22×M16C2311yescharacteristic
106C23×C8C2311yescharacteristic
107C23×C8C2311yescharacteristic
108#544—24no—
109#544—42no—
110C2×C4⋊Q8—21no—
111C2×C4⋊Q8—24no—
112C2×C4⋊Q8—24no—
113C2×C4⋊Q8—22no—
114C2×C4⋊Q8—42no—
115C2×C4⋊Q8C2312yes—
116C2×C4⋊Q8C2314yes—
117C2×#125—21no—
118C2×#125C2312yes—
119C2×#124—24no—
120C2×#124—22no—
121C2×#124—42no—
122C2×#124C2314yes—
123C2×C22⋊Q8—22no—
124#519—24no—
125#519—42no—
126C2×C4×Q8—21no—
127C2×C4×Q8—21no—
128C2×C4×Q8C2312yes—
129C2×C4×Q8C2312yes—
130C2×#117—22no—
131C2×#117C2311yescharacteristic
132C2×#117C2311yescharacteristic
133C2×#117C2311yescharacteristic
134C2×#117C2311yescharacteristic
135C2×#117C2311yescharacteristic
136C2×#117C2311yescharacteristic
137C22×C4⋊C4—21no—
138C22×C4⋊C4C2311yescharacteristic
139C22×C4⋊C4C2311yescharacteristic
140C22×C22⋊C4—21no—
141C22×C22⋊C4—21no—
142C22×C42C2311yescharacteristic
143#489—216no—
144#489—216no—
145#479—28no—
146#479—28no—
147#477—28no—
148#475—28no—
149#428—24no—
150#428—42no—
151#427—24no—
152#427—24no—
153C2×#107C2311yescharacteristic
154C2×#107C2312yes—
155C2×#107C2311yescharacteristic
156C2×#107C2312yes—
157C2×#107C2312yes—
158C2×#106C2311yescharacteristic
159C2×#106C2312yes—
160C2×#106C2311yescharacteristic
161C2×#106C2312yes—
162C2×#106C2312yes—
163#424—24no—
164C2×C4⋊C8C2312yes—
165C2×C4⋊C8C2312yes—
166#419—24no—
167#419—24no—
168#418—24no—
169#418—42no—
170C2×#103—24no—
171C2×#103D418yes—
172C2×#103C2312yes—
173C2×#103C2312yes—
174C2×#103C2314yes—
175C2×#103C2314yes—
176C2×#103C2312yes—
177C2×#103C2312yes—
178C2×#102—24no—
179C2×#102—24no—
180C2×#102C2312yes—
181C2×#102C2314yes—
182C2×#102C2312yes—
183C2×#102C2314yes—
184C2×#102C2312yes—
185C2×#102C2312yes—
186C2×#97—24no—
187C2×#97C2312yes—
188C2×#97C2312yes—
189#399—24no—
190#399—24no—
191#398—24no—
192#397—24no—
193#397—24no—
194#395—24no—
195#395—24no—
196#395—24no—
197#391—22no—
198#391—24no—
199#391C2314yes—
200#390—22no—
201#390—21no—
202#390—21no—
203#390C2314yes—
204#390C2312yes—
205#390C2312yes—
206#389—22no—
207#389—24no—
208#389—24no—
209#389D412yes—
210#389D412yes—
211#389C2312yes—
212#389C2312yes—
213#389C2314yes—
214#389C2312yes—
215#389C2312yes—
216#388—24no—
217#388—24no—
218#388—24no—
219#388—24no—
220#385—21no—
221#385—21no—
222#385—21no—
223#385—21no—
224#385—22no—
225#385—22no—
226#385D412yes—
227#385D412yes—
228#384—21no—
229#384C2312yes—
230#382—21no—
231#382C2312yes—
232#381—22no—
233#379—22no—
234#379—22no—
235C4×C4⋊C4—22no—
236C4×C4⋊C4—22no—
237C4×C4⋊C4—22no—
238C4×C4⋊C4C2314yes—
239C4×C4⋊C4C2314yes—
240C4×C22⋊C4—22no—
241#340C2314yes—
242#340C2314yes—
243#340C2312yes—
244#340C2312yes—
245#340C2312yes—
246#340C2312yes—
247#336C2314yes—
248#336C2314yes—
249#336C2312yes—
250#336C2312yes—
251#336C2314yes—
252C25—21no—
253C22×Q8—21no—
254C22×Q8—24no—
255C22×Q8—22no—
256C22×Q8—22no—
257C22×Q8—22no—
258C22×Q8—22no—
259C22×Q8—22no—
260C22×Q8—22no—
261C22×Q8—44no—
262C22×Q8—84no—
263C22×Q8C2×D414yes—
264C22×Q8C2412yes—
265C22×D4—22no—
266C22×D4—21no—
267C22×D4—22no—
268C22×D4—22no—
269C22×D4—22no—
270C22×D4—22no—
271C22×D4—22no—
272C22×D4C2412yes—
273C23×C4—21no—
274C23×C4—21no—
275C23×C4—21no—
276C23×C4C2×D411yescharacteristic
277C23×C4C2×D411yescharacteristic
278C23×C4C2411yescharacteristic
279C2×M16—22no—
280C2×M16—22no—
281C2×M16—22no—
282C2×M16—22no—
283C2×M16—22no—
284C2×M16—22no—
285C22×C8—22no—
286C22×C8—22no—
287C22×C8—22no—
288C22×C8—22no—
289C22×C8—22no—
290C22×C8C2×D414yes—
291C22×C8C2412yes—
292C22×C8C2412yes—
293C22×C8C2412yes—
294C22×C8C2411yescharacteristic
295C22×C8C2411yescharacteristic
296C4⋊Q8—28no—
297C4⋊Q8—24no—
298C4⋊Q8—44no—
299C4⋊Q8—42no—
300C4⋊Q8—48no—
301C4⋊Q8—44no—
302C4⋊Q8—88no—
303#125—24no—
304#125—42no—
305#125—44no—
306#124—28no—
307#124—44no—
308#124—48no—
309C22⋊Q8—48no—
310C4×Q8—24no—
311C4×Q8—24no—
312C4×Q8—42no—
313C4×Q8—42no—
314C4×Q8—44no—
315C4×Q8—44no—
316#117—22no—
317#117—22no—
318#117—22no—
319#117—22no—
320#117—22no—
321#117—22no—
322#117—22no—
323#117—22no—
324#117—22no—
325#117—22no—
326#117—22no—
327#117—48no—
328#117#4114yes—
329C2×C4⋊C4—21no—
330C2×C4⋊C4—22no—
331C2×C4⋊C4—22no—
332C2×C4⋊C4—22no—
333C2×C4⋊C4—24no—
334C2×C4⋊C4—21no—
335C2×C4⋊C4—22no—
336C2×C4⋊C4—22no—
337C2×C4⋊C4—22no—
338C2×C4⋊C4—22no—
339C2×C4⋊C4—22no—
340C2×C4⋊C4—22no—
341C2×C4⋊C4—22no—
342C2×C4⋊C4—24no—
343C2×C4⋊C4—24no—
344C2×C4⋊C4—22no—
345C2×C4⋊C4—22no—
346C2×C4⋊C4—44no—
347C2×C4⋊C4C2×D411yescharacteristic
348C2×C4⋊C4C2×D411yescharacteristic
349C2×C4⋊C4C2×D412yes—
350C2×C4⋊C4C2×D414yes—
351C2×C4⋊C4C2×D414yes—
352C2×C4⋊C4C2×D414yes—
353C2×C4⋊C4C2412yes—
354C2×C4⋊C4C2411yescharacteristic
355C2×C4⋊C4C2411yescharacteristic
356C2×C4⋊C4C2411yescharacteristic
357C2×C4⋊C4C2411yescharacteristic
358C2×C4⋊C4C2412yes—
359C2×C4⋊C4C2412yes—
360C2×C4⋊C4C2412yes—
361C2×C22⋊C4—21no—
362C2×C22⋊C4—21no—
363C2×C22⋊C4—21no—
364C2×C22⋊C4—21no—
365C2×C22⋊C4—24no—
366C2×C22⋊C4—24no—
367C2×C22⋊C4—24no—
368C2×C22⋊C4—24no—
369C2×C22⋊C4—24no—
370C2×C22⋊C4—24no—
371C2×C22⋊C4—44no—
372C2×C22⋊C4—44no—
373C2×C22⋊C4C2×D412yes—
374C2×C22⋊C4C2×D412yes—
375C2×C42—21no—
376C2×C42—22no—
377C2×C42—22no—
378C2×C42—22no—
379C2×C42—22no—
380C2×C42—21no—
381C2×C42—21no—
382C2×C42—21no—
383C2×C42—24no—
384C2×C42C2×D412yes—
385C2×C42C2412yes—
386C2×C42C2412yes—
387C2×C42C2412yes—
388#107—22no—
389#107—22no—
390#107—24no—
391#107—24no—
392#107—42no—
393#106—22no—
394#106—22no—
395#106—24no—
396#106—24no—
397#106—42no—
398C4⋊C8—24no—
399C4⋊C8—24no—
400#103—28no—
401#103—24no—
402#103—24no—
403#103—28no—
404#103C2×D418yes—
405#103C2×D418yes—
406#102—24no—
407#102—24no—
408#102—28no—
409#102—28no—
410#102—42no—
411#102—42no—
412#97—24no—
413#97—24no—
414#95—24no—
415#95—22no—
416#95—22no—
417#95—22no—
418#95—22no—
419#95—24no—
420#95—24no—
421#95—24no—
422#95—24no—
423#95C2×D414yes—
424#95C2×D414yes—
425C24—22no—
426C24—42no—
427C24—42no—
428C24—42no—
429C24—82no—
430C24—82no—
431C24C22×D411yescharacteristic
432C2×Q8—24no—
433C2×Q8—22no—
434C2×Q8—28no—
435C2×Q8—28no—
436C2×Q8—22no—
437C2×Q8—22no—
438C2×Q8—22no—
439C2×Q8—22no—
440C2×Q8—48no—
441C2×Q8—44no—
442C2×Q8—48no—
443C2×Q8—42no—
444C2×Q8—44no—
445C2×Q8—42no—
446C2×Q8—44no—
447C2×Q8—84no—
448C2×Q8—84no—
449C2×Q8—816no—
450C2×Q8C22×D414yes—
451C2×D4—22no—
452C2×D4—22no—
453C2×D4—22no—
454C2×D4—22no—
455C2×D4—22no—
456C2×D4—22no—
457C2×D4—44no—
458C2×D4—42no—
459C2×D4—44no—
460C2×D4—42no—
461C2×D4—44no—
462C2×D4—42no—
463C2×D4—44no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—24no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—24no—
471C22×C4—22no—
472C22×C4—21no—
473C22×C4—22no—
474C22×C4—21no—
475C22×C4—21no—
476C22×C4—21no—
477C22×C4—42no—
478C22×C4—42no—
479C22×C4—42no—
480C22×C4—42no—
481C22×C4—42no—
482C22×C4—82no—
483C22×C4—82no—
484C22×C4C22×D414yes—
485C22×C4C22×D412yes—
486C22×C4C22×D412yes—
487C22×C4C22×D412yes—
488C22×C4C22×D412yes—
489C22×C4C22×D411yescharacteristic
490C22×C4C22×D412yes—
491C22×C4C2×#4112yes—
492C22×C4H(5,𝔽2)12yes—
493C22×C42−1+412yes—
494C22×C42−1+412yes—
495C22×C42−1+412yes—
496C22×C4C2511yescharacteristic
497M16—44no—
498M16—44no—
499C2×C8—28no—
500C2×C8—21no—
501C2×C8—21no—
502C2×C8—21no—
503C2×C8—21no—
504C2×C8—22no—
505C2×C8—22no—
506C2×C8—22no—
507C2×C8—22no—
508C2×C8—22no—
509C2×C8—22no—
510C2×C8—22no—
511C2×C8—22no—
512C2×C8—22no—
513C2×C8—44no—
514C2×C8C22×D412yes—
515C2×C8C22×D412yes—
516C4⋊C4—24no—
517C4⋊C4—24no—
518C4⋊C4—22no—
519C4⋊C4—22no—
520C4⋊C4—22no—
521C4⋊C4—22no—
522C4⋊C4—22no—
523C4⋊C4—21no—
524C4⋊C4—21no—
525C4⋊C4—21no—
526C4⋊C4—21no—
527C4⋊C4—22no—
528C4⋊C4—21no—
529C4⋊C4—21no—
530C4⋊C4—21no—
531C4⋊C4—22no—
532C4⋊C4—21no—
533C4⋊C4—22no—
534C4⋊C4—44no—
535C4⋊C4—44no—
536C4⋊C4—48no—
537C4⋊C4—44no—
538C4⋊C4—44no—
539C4⋊C4—44no—
540C4⋊C4—44no—
541C4⋊C4—44no—
542C4⋊C4—44no—
543C4⋊C4—44no—
544C4⋊C4—42no—
545C4⋊C4—82no—
546C4⋊C4—82no—
547C4⋊C4—84no—
548C4⋊C4—84no—
549C4⋊C4C22×D414yes—
550C4⋊C4C2×#4114yes—
551C22⋊C4—24no—
552C22⋊C4—24no—
553C22⋊C4—24no—
554C22⋊C4—48no—
555C22⋊C4—48no—
556C22⋊C4—42no—
557C22⋊C4—42no—
558C22⋊C4—42no—
559C22⋊C4—84no—
560C22⋊C4—84no—
561C22⋊C4—84no—
562C22⋊C4—84no—
563C42—24no—
564C42—24no—
565C42—22no—
566C42—22no—
567C42—22no—
568C42—22no—
569C42—22no—
570C42—22no—
571C42—44no—
572C42—44no—
573C42—42no—
574C42—42no—
575C42—42no—
576C42—84no—
577C23—22no—
578C23—22no—
579C23—44no—
580C23—44no—
581C23—42no—
582C23—42no—
583C23—42no—
584C23—84no—
585C23—84no—
586C23—84no—
587C23—164no—
588C23—164no—
589C23#53612yes—
590C23D4212yes—
591C23#54812yes—
592C23C23×D411yescharacteristic
593Q8—24no—
594Q8—24no—
595Q8—48no—
596Q8—42no—
597Q8—42no—
598Q8—816no—
599Q8—164no—
600D4—42no—
601D4—42no—
602D4—42no—
603D4—42no—
604D4—84no—
605C2×C4—24no—
606C2×C4—24no—
607C2×C4—22no—
608C2×C4—22no—
609C2×C4—24no—
610C2×C4—24no—
611C2×C4—22no—
612C2×C4—22no—
613C2×C4—24no—
614C2×C4—22no—
615C2×C4—22no—
616C2×C4—22no—
617C2×C4—22no—
618C2×C4—24no—
619C2×C4—24no—
620C2×C4—24no—
621C2×C4—44no—
622C2×C4—44no—
623C2×C4—42no—
624C2×C4—44no—
625C2×C4—42no—
626C2×C4—48no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4—42no—
632C2×C4—44no—
633C2×C4—42no—
634C2×C4—42no—
635C2×C4—42no—
636C2×C4—41no—
637C2×C4—41no—
638C2×C4—41no—
639C2×C4—41no—
640C2×C4—42no—
641C2×C4—42no—
642C2×C4—84no—
643C2×C4—84no—
644C2×C4#53612yes—
645C2×C4#53614yes—
646C2×C4#54214yes—
647C2×C4D4212yes—
648C2×C4#54714yes—
649C2×C4#54812yes—
650C2×C4#56314yes—
651C2×C4C23×D411yescharacteristic
652C2×C4C22×#4111yescharacteristic
653C2×C4C2×H(5,𝔽2)11yescharacteristic
654C2×C4C2×2−1+411yescharacteristic
655C2×C4C2×2−1+411yescharacteristic
656C2×C4C2×2−1+411yescharacteristic
657C8—22no—
658C8—22no—
659C8—42no—
660C8—42no—
661C8—42no—
662C22—22no—
663C22—22no—
664C22—24no—
665C22—24no—
666C22—82no—
667C22—82no—
668C22—82no—
669C22—84no—
670C22—84no—
671C22—168no—
672C22#267414yes—
673C22C2×#53611yescharacteristic
674C22C2×D4211yescharacteristic
675C22C2×#54811yescharacteristic
676C22#358512yes—
677C22#358612yes—
678C4—44no—
679C4—42no—
680C4—42no—
681C4—44no—
682C4—42no—
683C4—42no—
684C4—44no—
685C4—44no—
686C4—81no—
687C4—82no—
688C4—81no—
689C4—81no—
690C4—81no—
691C4—82no—
692C4#267414yes—
693C4C2×#53611yescharacteristic
694C4C2×#54211yescharacteristic
695C4C2×#54711yescharacteristic
696C4C2×#56311yescharacteristic
697C2—24no—
698C2—162no—
699C2C2×#267411yescharacteristic
700C2#6088512yes—
701C2#6105911yescharacteristic
702C2#6106111yescharacteristic
703C2#6133612yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ #2296 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#3082C2semidirect
4. #35576C2semidirect
5. #30832C2semidirect
6. #30830C2semidirect
7. #30824C2semidirect
8. #30822C2semidirect
10. #23633C2semidirect
12. #23552C2semidirect
15. #21154C2semidirect
16. #21066C2semidirect
18. #21021C2semidirect
19. #20877C2semidirect
28. C2×#479C22semidirect
29. C2×#479C22semidirect
30. C2×#477C22semidirect
31. C2×#475C22semidirect
48. #2551C22semidirect
50. #2550C22semidirect
71. #2074C22semidirect
72. #2072C22semidirect
75. #2058C22semidirect
76. #2057C22semidirect
81. #1944C22semidirect
82. #1944C22semidirect
83. #1943C22semidirect
84. #1943C22semidirect
85. #1928C22semidirect
86. #1927C22semidirect
89. #1917C22semidirect
90. #1917C22semidirect
91. #1865C22semidirect
92. #1863C22semidirect

Maximal subgroups

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

subgroupindex
C22×#4892
C2×#30822
#356792
#355762
#308322
#308302
#308242
#308222
#237352
#236332
#236122
#235522
#235062
#234562
#211542
#210662
#210632
#210212
#208772