Finite Group Databasebeta

← #6496360

Subgroups — #6496360

Order 512.

1792 conjugacy classes of subgroups.

The subgroup lattice has 707 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3104C211yesmaximal, characteristic
2C2×#3078C211yesmaximal, characteristic
3#35322C211yesmaximal, characteristic
4#34128C211yesmaximal, characteristic
5#33202C211yesmaximal, characteristic, complemented
6#33198C211yesmaximal, characteristic, complemented
7#33033C211yesmaximal, characteristic, complemented
8#33027C211yesmaximal, characteristic, complemented
9#29869C211yesmaximal, characteristic, complemented
10#29867C211yesmaximal, characteristic, complemented
11#29831C211yesmaximal, characteristic, complemented
12#29830C211yesmaximal, characteristic, complemented
13#24821C214yesmaximal, complemented
14#24687C214yesmaximal, complemented
15#24588C214yesmaximal
16#24540C214yesmaximal
17#21974C211yesmaximal, characteristic
18#21892C211yesmaximal, characteristic
19#21837C211yesmaximal, characteristic
20C2×#533C2211yescharacteristic
21C2×#533C2211yescharacteristic
22C22×C4×Q8C2211yescharacteristic
23#3104—28no—
24C2×#484C2212yes—
25#3078—28no—
26C2×#479C2212yes—
27C2×#479C2212yes—
28C2×#477C2211yescharacteristic
29C2×#477C2211yescharacteristic
30C2×#477C2211yescharacteristic
31C2×#477C2211yescharacteristic
32C2×#475C2211yescharacteristic
33C2×#475C2211yescharacteristic
34C2×#462C2211yescharacteristic
35C2×#462C2211yescharacteristic
36C2×#451C2212yes—
37C2×C4×Q16C2212yes—
38C2×C4×Q16C2212yes—
39C2×#431C2211yescharacteristic
40C2×#428C2212yes—
41C2×#423C2211yescharacteristic
42C2×#419C2212yes—
43#2868C2211yescharacteristic
44#2862C2211yescharacteristic
45#2862C2211yescharacteristic
46#2862C2211yescharacteristic
47#2621—22no—
48#2565—21no—
49#2565—21no—
50#2565C2214yes—
51#2499—21no—
52#2499—21no—
53#2499C2214yes—
54#2356—22no—
55#2356C2212yes—
56#2351C2212yes—
57#2344—22no—
58#2344C2212yes—
59Q8×C22⋊C4C2212yes—
60#2327—22no—
61#2295—21no—
62#2295—21no—
63#2295C2211yescharacteristic
64#2295C2211yescharacteristic
65#2283—21no—
66#2283C2211yescharacteristic
67C42×Q8C2211yescharacteristic
68C42×Q8C2211yescharacteristic
69C4×#117—21no—
70C4×#117C2211yescharacteristic
71#2085C2214yes—
72#2083C2214yes—
73#2074C2214yes—
74#2072C2214yes—
75#2068C2214yes—
76#2066C2214yes—
77#2056C2214yes—
78#2056C2214yes—
79#1991C2214yes—
80#1988C2211yescharacteristic
81#1988C2211yescharacteristic
82#1987C2211yescharacteristic
83#1987C2211yescharacteristic
84#1987C2211yescharacteristic
85#1987C2211yescharacteristic
86#1981C2211yescharacteristic
87#1981C2211yescharacteristic
88#1949C2214yes—
89#1947C2214yes—
90#1939C2214yes—
91#1928C2214yes—
92#1927C2214yes—
93#1894C2214yes—
94#1878C2214yes—
95#1878C2214yes—
96#1851C2214yes—
97#1842C2211yescharacteristic
98#1842C2211yescharacteristic
99#1831C2211yescharacteristic
100#1831C2211yescharacteristic
101#1825C2214yes—
102#1805C2214yes—
103#1805C2214yes—
104#1803C2214yes—
105C23×Q8C2311yescharacteristic
106C22×Q16C2312yes—
107C22×Q16C2312yes—
108C22×M16C2312yes—
109#533—24no—
110#533—24no—
111#533—24no—
112#533—24no—
113C2×C4⋊Q8—22no—
114C2×C4⋊Q8C2311yescharacteristic
115C2×C4⋊Q8C2311yescharacteristic
116C2×C4⋊Q8C2311yescharacteristic
117C2×C4⋊Q8C2311yescharacteristic
118C2×#125—21no—
119C2×#125C2312yes—
120C2×C22⋊Q8—21no—
121C2×C22⋊Q8—21no—
122C2×C22⋊Q8—21no—
123C2×C22⋊Q8C2312yes—
124C2×C4×Q8—21no—
125C2×C4×Q8—21no—
126C2×C4×Q8—21no—
127C2×C4×Q8—21no—
128C2×C4×Q8—21no—
129C2×C4×Q8—24no—
130C2×C4×Q8D418yes—
131C2×C4×Q8C2312yes—
132C2×C4×Q8C2312yes—
133C2×C4×Q8C2312yes—
134C2×C4×Q8C2311yescharacteristic
135C2×C4×Q8C2311yescharacteristic
136C2×C4×Q8C2311yescharacteristic
137C2×C4×Q8C2311yescharacteristic
138C2×#117—21no—
139C2×#117—22no—
140C2×#117D412yes—
141C2×#117C2312yes—
142C22×C4⋊C4—21no—
143C22×C4⋊C4C2311yescharacteristic
144C22×C42—21no—
145C22×C42C2311yescharacteristic
146#484—28no—
147#479—28no—
148#479—28no—
149#477—24no—
150#477—24no—
151#477—24no—
152#477—24no—
153#475—24no—
154#475—24no—
155#462—24no—
156#462—24no—
157#451—28no—
158C4×Q16—28no—
159C4×Q16—28no—
160#431—24no—
161#428—28no—
162C2×#107C2312yes—
163C2×#107C2312yes—
164C2×#107C2312yes—
165C2×#107C2312yes—
166C2×#106C2312yes—
167C2×#106C2312yes—
168#423—24no—
169C2×C4⋊C8C2311yescharacteristic
170C2×C4⋊C8C2311yescharacteristic
171C2×C4⋊C8C2311yescharacteristic
172C2×C4⋊C8C2311yescharacteristic
173C2×C4⋊C8C2312yes—
174C2×C4⋊C8C2314yes—
175C2×C4⋊C8C2314yes—
176#419—28no—
177C2×#103C2312yes—
178C2×#103C2312yes—
179C2×#103C2312yes—
180C2×#103C2312yes—
181C2×#103C2312yes—
182C2×#103C2312yes—
183C2×#103C2312yes—
184C2×#103C2312yes—
185C2×#103C2314yes—
186C2×#103C2314yes—
187C2×C22⋊C8C2312yes—
188C2×C4×C8C2312yes—
189#398—24no—
190#398—24no—
191#397—24no—
192#397—24no—
193#391—21no—
194#391—21no—
195#391—21no—
196#391—21no—
197#391—22no—
198#391—22no—
199#391—22no—
200#391—22no—
201#391C2314yes—
202#391C2314yes—
203#391C2312yes—
204#391C2312yes—
205#389—22no—
206#389—22no—
207#389C2312yes—
208#389C2312yes—
209#387—22no—
210#387—22no—
211#387—22no—
212#387—22no—
213#387—24no—
214#387—24no—
215#387—24no—
216#387—24no—
217#387—22no—
218#387—22no—
219#384—21no—
220#384—21no—
221#384C2314yes—
222#383—21no—
223#383—22no—
224#382—21no—
225#382—21no—
226#382—22no—
227#382—22no—
228#382C2314yes—
229#382C2312yes—
230#381—21no—
231#381—22no—
232#380—22no—
233#380C2312yes—
234C4×C4⋊C4—22no—
235C4×C4⋊C4—22no—
236C4×C4⋊C4—21no—
237C4×C4⋊C4—21no—
238C4×C4⋊C4—21no—
239C4×C4⋊C4—21no—
240C4×C4⋊C4—22no—
241C4×C4⋊C4D411yescharacteristic
242C4×C4⋊C4D411yescharacteristic
243C4×C4⋊C4C2312yes—
244C4×C4⋊C4C2312yes—
245C4×C4⋊C4C2311yescharacteristic
246C4×C4⋊C4C2311yescharacteristic
247C4×C4⋊C4C2312yes—
248C4×C22⋊C4—21no—
249C4×C22⋊C4—22no—
250C4×C22⋊C4D411yescharacteristic
251C4×C22⋊C4D411yescharacteristic
252C4×C22⋊C4C2312yes—
253#376—21no—
254#376—21no—
255#376—21no—
256#376—22no—
257#376D412yes—
258C43—21no—
259C43—21no—
260C43C2311yescharacteristic
261C43C2311yescharacteristic
262#340C2314yes—
263#340C2314yes—
264#340C2314yes—
265#340C2314yes—
266#340C2314yes—
267#340C2314yes—
268#336C2314yes—
269#336C2314yes—
270C22×Q8—21no—
271C22×Q8—21no—
272C22×Q8—21no—
273C22×Q8—21no—
274C22×Q8—22no—
275C22×Q8—24no—
276C22×Q8—42no—
277C22×Q8C2×D414yes—
278C22×Q8C2412yes—
279C22×Q8C2411yescharacteristic
280C22×Q8C2411yescharacteristic
281C23×C4—21no—
282C23×C4—21no—
283C23×C4C2×D411yescharacteristic
284C23×C4C2×D411yescharacteristic
285C23×C4C2411yescharacteristic
286C2×Q16—24no—
287C2×Q16—24no—
288C2×Q16—28no—
289C2×Q16—44no—
290C2×M16—24no—
291C2×M16—44no—
292C22×C8C2412yes—
293C22×C8C2412yes—
294C22×C8C2412yes—
295C22×C8C2412yes—
296C4⋊Q8—22no—
297C4⋊Q8—22no—
298C4⋊Q8—22no—
299C4⋊Q8—22no—
300C4⋊Q8—22no—
301C4⋊Q8—22no—
302C4⋊Q8—22no—
303C4⋊Q8—22no—
304C4⋊Q8—48no—
305#125—24no—
306#125—24no—
307#125—44no—
308C22⋊Q8—24no—
309C22⋊Q8—24no—
310C22⋊Q8—44no—
311C22⋊Q8—44no—
312C22⋊Q8—44no—
313C4×Q8—28no—
314C4×Q8—28no—
315C4×Q8—22no—
316C4×Q8—22no—
317C4×Q8—22no—
318C4×Q8—22no—
319C4×Q8—24no—
320C4×Q8—24no—
321C4×Q8—24no—
322C4×Q8—24no—
323C4×Q8—42no—
324C4×Q8—42no—
325C4×Q8—44no—
326C4×Q8—44no—
327C4×Q8—44no—
328C4×Q8C2×D414yes—
329C4×Q8C2×D414yes—
330C4×Q8C2×D414yes—
331C4×Q8C2×D414yes—
332#117—24no—
333#117—24no—
334#117—44no—
335#117—48no—
336#117—44no—
337C2×C4⋊C4—28no—
338C2×C4⋊C4—21no—
339C2×C4⋊C4—21no—
340C2×C4⋊C4—21no—
341C2×C4⋊C4—21no—
342C2×C4⋊C4—21no—
343C2×C4⋊C4—22no—
344C2×C4⋊C4—21no—
345C2×C4⋊C4—21no—
346C2×C4⋊C4—22no—
347C2×C4⋊C4—22no—
348C2×C4⋊C4—22no—
349C2×C4⋊C4—22no—
350C2×C4⋊C4—21no—
351C2×C4⋊C4—21no—
352C2×C4⋊C4—21no—
353C2×C4⋊C4—22no—
354C2×C4⋊C4—22no—
355C2×C4⋊C4—22no—
356C2×C4⋊C4—22no—
357C2×C4⋊C4—24no—
358C2×C4⋊C4—24no—
359C2×C4⋊C4C2×D412yes—
360C2×C4⋊C4C2×D414yes—
361C2×C4⋊C4C2×D412yes—
362C2×C4⋊C4C2412yes—
363C2×C4⋊C4C2412yes—
364C2×C4⋊C4C2412yes—
365C2×C4⋊C4C2412yes—
366C2×C4⋊C4C2412yes—
367C2×C4⋊C4C2411yescharacteristic
368C2×C4⋊C4C2411yescharacteristic
369C2×C22⋊C4—22no—
370C2×C22⋊C4—22no—
371C2×C22⋊C4—22no—
372C2×C22⋊C4C2×D412yes—
373C2×C22⋊C4C2×D412yes—
374C2×C42—28no—
375C2×C42—21no—
376C2×C42—21no—
377C2×C42—21no—
378C2×C42—21no—
379C2×C42—22no—
380C2×C42—22no—
381C2×C42—22no—
382C2×C42—22no—
383C2×C42—22no—
384C2×C42—22no—
385C2×C42—21no—
386C2×C42—21no—
387C2×C42C2×D412yes—
388C2×C42C2×D411yescharacteristic
389C2×C42C2×D411yescharacteristic
390C2×C42C2×D411yescharacteristic
391C2×C42C2×D411yescharacteristic
392C2×C42C2×D412yes—
393C2×C42C2×D411yescharacteristic
394C2×C42C2×D411yescharacteristic
395C2×C42C2×D414yes—
396C2×C42#4114yes—
397C2×C42C2412yes—
398C2×C42C2412yes—
399C2×C42C2411yescharacteristic
400C2×C42C2411yescharacteristic
401#107—24no—
402#107—24no—
403#107—24no—
404#107—24no—
405#106—24no—
406#106—24no—
407C4⋊C8—22no—
408C4⋊C8—22no—
409C4⋊C8—24no—
410C4⋊C8—22no—
411C4⋊C8—22no—
412C4⋊C8—44no—
413C4⋊C8—44no—
414#103—24no—
415#103—24no—
416#103—24no—
417#103—24no—
418#103—24no—
419#103—24no—
420#103—24no—
421#103—24no—
422#103—44no—
423#103—44no—
424C22⋊C8—24no—
425C4×C8—24no—
426#95—22no—
427#95—22no—
428#95—22no—
429#95—22no—
430#95—22no—
431#95—22no—
432#95—22no—
433#95—22no—
434#95—22no—
435#95—22no—
436#95—22no—
437#95—21no—
438#95—21no—
439#95—24no—
440#95—24no—
441#95—24no—
442#95—24no—
443#95—24no—
444#95—24no—
445#95C2×D412yes—
446#95C2×D412yes—
447#95C2×D412yes—
448#95C2×D412yes—
449C24C22×D411yescharacteristic
450C2×Q8—24no—
451C2×Q8—21no—
452C2×Q8—21no—
453C2×Q8—22no—
454C2×Q8—28no—
455C2×Q8—22no—
456C2×Q8—22no—
457C2×Q8—22no—
458C2×Q8—24no—
459C2×Q8—22no—
460C2×Q8—24no—
461C2×Q8—44no—
462C2×Q8—44no—
463C2×Q8—48no—
464C2×Q8—44no—
465C2×Q8—44no—
466C2×Q8—42no—
467C2×Q8—42no—
468C2×Q8—42no—
469C2×Q8—42no—
470C2×Q8—42no—
471C2×Q8C22×D412yes—
472C2×Q8C22×D412yes—
473C2×Q82−1+414yes—
474C22×C4—21no—
475C22×C4—22no—
476C22×C4—22no—
477C22×C4—22no—
478C22×C4—22no—
479C22×C4—24no—
480C22×C4—24no—
481C22×C4—21no—
482C22×C4—21no—
483C22×C4—21no—
484C22×C4—22no—
485C22×C4—22no—
486C22×C4—22no—
487C22×C4—22no—
488C22×C4—22no—
489C22×C4—22no—
490C22×C4—42no—
491C22×C4—44no—
492C22×C4—44no—
493C22×C4#12114yes—
494C22×C4#12114yes—
495C22×C4C22×D412yes—
496C22×C4C22×D412yes—
497C22×C4C22×D412yes—
498C22×C4C22×D411yescharacteristic
499C22×C4C22×D411yescharacteristic
500C22×C4C22×D411yescharacteristic
501C22×C4C22×D411yescharacteristic
502C22×C4C22×D412yes—
503C22×C4C22×D412yes—
504C22×C4C22×D411yescharacteristic
505C22×C4C2×#4112yes—
506C22×C4C2511yescharacteristic
507Q16—48no—
508Q16—48no—
509M16—48no—
510C2×C8—22no—
511C2×C8—22no—
512C2×C8—22no—
513C2×C8—22no—
514C2×C8—24no—
515C2×C8—24no—
516C2×C8—24no—
517C2×C8—24no—
518C4⋊C4—24no—
519C4⋊C4—24no—
520C4⋊C4—24no—
521C4⋊C4—24no—
522C4⋊C4—24no—
523C4⋊C4—24no—
524C4⋊C4—21no—
525C4⋊C4—21no—
526C4⋊C4—22no—
527C4⋊C4—22no—
528C4⋊C4—22no—
529C4⋊C4—22no—
530C4⋊C4—22no—
531C4⋊C4—22no—
532C4⋊C4—22no—
533C4⋊C4—22no—
534C4⋊C4—22no—
535C4⋊C4—22no—
536C4⋊C4—42no—
537C4⋊C4—44no—
538C4⋊C4—44no—
539C4⋊C4—44no—
540C4⋊C4—48no—
541C4⋊C4—42no—
542C4⋊C4—42no—
543C4⋊C4—44no—
544C4⋊C4—44no—
545C4⋊C4—44no—
546C4⋊C4—44no—
547C4⋊C4—42no—
548C4⋊C4—42no—
549C4⋊C4—42no—
550C4⋊C4—44no—
551C4⋊C4—48no—
552C4⋊C4C22×D412yes—
553C4⋊C4C22×D412yes—
554C22⋊C4—24no—
555C22⋊C4—42no—
556C22⋊C4—44no—
557C22⋊C4—44no—
558C22⋊C4—44no—
559C42—24no—
560C42—24no—
561C42—24no—
562C42—24no—
563C42—24no—
564C42—22no—
565C42—22no—
566C42—22no—
567C42—22no—
568C42—22no—
569C42—22no—
570C42—22no—
571C42—22no—
572C42—42no—
573C42—44no—
574C42—44no—
575C42—44no—
576C42—44no—
577C42—44no—
578C42—44no—
579C42—42no—
580C42—42no—
581C42#12114yes—
582C42#12114yes—
583C42C22×D412yes—
584C42C22×D412yes—
585C42C2×#4112yes—
586C42C2×#4112yes—
587C23—22no—
588C23—22no—
589C23—21no—
590C23—21no—
591C23C2×#12112yes—
592C23C23×D411yescharacteristic
593Q8—22no—
594Q8—24no—
595Q8—22no—
596Q8—22no—
597Q8—48no—
598Q8—42no—
599Q8—42no—
600Q8—42no—
601Q8—44no—
602Q8—84no—
603Q8—84no—
604Q8—84no—
605Q8#53614yes—
606C2×C4—22no—
607C2×C4—22no—
608C2×C4—24no—
609C2×C4—22no—
610C2×C4—22no—
611C2×C4—22no—
612C2×C4—22no—
613C2×C4—21no—
614C2×C4—21no—
615C2×C4—22no—
616C2×C4—22no—
617C2×C4—24no—
618C2×C4—28no—
619C2×C4—22no—
620C2×C4—22no—
621C2×C4—22no—
622C2×C4—22no—
623C2×C4—22no—
624C2×C4—22no—
625C2×C4—22no—
626C2×C4—22no—
627C2×C4—22no—
628C2×C4—22no—
629C2×C4—22no—
630C2×C4—22no—
631C2×C4—44no—
632C2×C4—48no—
633C2×C4—48no—
634C2×C4—42no—
635C2×C4—42no—
636C2×C4—42no—
637C2×C4—42no—
638C2×C4—42no—
639C2×C4—42no—
640C2×C4—42no—
641C2×C4—42no—
642C2×C4—42no—
643C2×C4—44no—
644C2×C4—42no—
645C2×C4—42no—
646C2×C4—44no—
647C2×C4—44no—
648C2×C4—42no—
649C2×C4—42no—
650C2×C4—42no—
651C2×C4—42no—
652C2×C4C2×#12112yes—
653C2×C4C2×#12112yes—
654C2×C4C2×#12112yes—
655C2×C4C2×#12112yes—
656C2×C4C2×#12112yes—
657C2×C4#57514yes—
658C2×C4#57812yes—
659C2×C4#57812yes—
660C2×C4C23×D411yescharacteristic
661C2×C4C22×#4111yescharacteristic
662C2×C4C2×2−1+412yes—
663C2×C4#58512yes—
664C8—42no—
665C8—42no—
666C8—42no—
667C8—42no—
668C22—22no—
669C22—22no—
670C22—22no—
671C22—42no—
672C22—42no—
673C22C22×#12111yescharacteristic
674C22#348912yes—
675C22C2×#57512yes—
676C22C2×#57811yescharacteristic
677C22C2×#57811yescharacteristic
678C4—22no—
679C4—24no—
680C4—22no—
681C4—24no—
682C4—22no—
683C4—22no—
684C4—42no—
685C4—42no—
686C4—42no—
687C4—42no—
688C4—42no—
689C4—42no—
690C4—82no—
691C4—82no—
692C4—82no—
693C4—82no—
694C4—82no—
695C4—82no—
696C4#304414yes—
697C4C2×#53611yescharacteristic
698C4#344911yescharacteristic
699C4#349512yes—
700C2—42no—
701C2#3530611yescharacteristic
702C2#6041511yescharacteristic
703C2C2×#304411yescharacteristic
704C2#6213012yes—
705C2#6214612yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC24C22×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
5. #33202C2semidirect
6. #33198C2semidirect
7. #33033C2semidirect
8. #33027C2semidirect
9. #29869C2semidirect
10. #29867C2semidirect
11. #29831C2semidirect
12. #29830C2semidirect
13. #24821C2semidirect
14. #24687C2semidirect

Maximal subgroups

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

subgroupindex
C2×#31042
C2×#30782
#353222
#341282
#332022
#331982
#330332
#330272
#298692
#298672
#298312
#298302
#248212
#246872
#245882
#245402
#219742
#218922
#218372