Finite Group Databasebeta

← #6496327

Subgroups — #6496327

Order 512.

1782 conjugacy classes of subgroups.

The subgroup lattice has 713 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3113C211yesmaximal, characteristic
2C2×#3078C211yesmaximal, characteristic
3#35322C211yesmaximal, characteristic
4Q8×#117C211yesmaximal, characteristic
5#33202C211yesmaximal, characteristic, complemented
6#33196C211yesmaximal, characteristic, complemented
7#33033C211yesmaximal, characteristic, complemented
8#33027C211yesmaximal, characteristic, complemented
9#29867C211yesmaximal, characteristic, complemented
10#29866C211yesmaximal, characteristic, complemented
11#29831C211yesmaximal, characteristic, complemented
12#29830C211yesmaximal, characteristic, complemented
13#24826C214yesmaximal, complemented
14#24691C214yesmaximal, complemented
15#24590C214yesmaximal
16#24540C214yesmaximal
17#21978C211yesmaximal, characteristic
18#21894C211yesmaximal, characteristic
19#21839C211yesmaximal, characteristic
20C2×#533C2211yescharacteristic
21C2×#533C2211yescharacteristic
22C22×C4×Q8C2211yescharacteristic
23#3113—28no—
24C2×#483C2212yes—
25#3078—28no—
26C2×#479C2212yes—
27C2×#479C2212yes—
28C2×#477C2211yescharacteristic
29C2×#477C2211yescharacteristic
30C2×#475C2211yescharacteristic
31C2×#475C2211yescharacteristic
32C2×#475C2211yescharacteristic
33C2×#475C2211yescharacteristic
34C2×#462C2211yescharacteristic
35C2×#462C2211yescharacteristic
36C2×#451C2212yes—
37C2×#441C2212yes—
38C2×#441C2212yes—
39C2×#432C2211yescharacteristic
40C2×#428C2212yes—
41C2×#423C2211yescharacteristic
42C2×#419C2212yes—
43#2868C2211yescharacteristic
44#2862C2211yescharacteristic
45#2862C2211yescharacteristic
46#2862C2211yescharacteristic
47#2621—21no—
48#2621—21no—
49#2565—22no—
50#2565C2214yes—
51#2499—22no—
52#2499C2214yes—
53#2355—22no—
54#2355C2212yes—
55Q8×C4⋊C4C2212yes—
56#2345—22no—
57#2345C2212yes—
58Q8×C22⋊C4C2212yes—
59#2327—21no—
60#2327—21no—
61#2295—21no—
62#2295—21no—
63#2295C2211yescharacteristic
64#2295C2211yescharacteristic
65#2285—21no—
66#2285C2211yescharacteristic
67C42×Q8C2211yescharacteristic
68C42×Q8C2211yescharacteristic
69C4×#117—21no—
70C4×#117C2211yescharacteristic
71#2085C2214yes—
72#2084C2214yes—
73#2074C2214yes—
74#2072C2214yes—
75#2068C2214yes—
76#2066C2214yes—
77#2056C2214yes—
78#2056C2214yes—
79#1992C2214yes—
80#1988C2211yescharacteristic
81#1988C2211yescharacteristic
82#1988C2211yescharacteristic
83#1988C2211yescharacteristic
84#1987C2211yescharacteristic
85#1987C2211yescharacteristic
86#1981C2211yescharacteristic
87#1981C2211yescharacteristic
88#1949C2214yes—
89#1948C2214yes—
90#1940C2214yes—
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#1803C2214yes—
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—21no—
114C2×C4⋊Q8—21no—
115C2×C4⋊Q8C2311yescharacteristic
116C2×C4⋊Q8C2311yescharacteristic
117C2×C4⋊Q8C2311yescharacteristic
118C2×C4⋊Q8C2311yescharacteristic
119C2×#125—21no—
120C2×#125C2312yes—
121C2×C22⋊Q8—21no—
122C2×C22⋊Q8—21no—
123C2×C22⋊Q8—21no—
124C2×C22⋊Q8C2312yes—
125C2×C4×Q8—21no—
126C2×C4×Q8—21no—
127C2×C4×Q8—21no—
128C2×C4×Q8—21no—
129C2×C4×Q8—21no—
130C2×C4×Q8—24no—
131C2×C4×Q8D418yes—
132C2×C4×Q8C2312yes—
133C2×C4×Q8C2312yes—
134C2×C4×Q8C2312yes—
135C2×C4×Q8C2311yescharacteristic
136C2×C4×Q8C2311yescharacteristic
137C2×C4×Q8C2311yescharacteristic
138C2×C4×Q8C2311yescharacteristic
139C2×#117—21no—
140C2×#117—21no—
141C2×#117D411yescharacteristic
142C2×#117D411yescharacteristic
143C2×#117C2312yes—
144C22×C4⋊C4—21no—
145C22×C4⋊C4C2311yescharacteristic
146C22×C42—21no—
147C22×C42C2311yescharacteristic
148#483—28no—
149#479—28no—
150#479—28no—
151#477—24no—
152#477—24no—
153#475—24no—
154#475—24no—
155#475—24no—
156#475—24no—
157#462—24no—
158#462—24no—
159#451—28no—
160#441—28no—
161#441—28no—
162#432—24no—
163#428—28no—
164C2×#107C2312yes—
165C2×#107C2312yes—
166C2×#106C2312yes—
167C2×#106C2312yes—
168C2×#106C2312yes—
169C2×#106C2312yes—
170#423—24no—
171C2×C4⋊C8C2311yescharacteristic
172C2×C4⋊C8C2311yescharacteristic
173C2×C4⋊C8C2311yescharacteristic
174C2×C4⋊C8C2311yescharacteristic
175C2×C4⋊C8C2312yes—
176C2×C4⋊C8C2314yes—
177C2×C4⋊C8C2314yes—
178#419—28no—
179C2×#103C2312yes—
180C2×#103C2312yes—
181C2×#103C2312yes—
182C2×#103C2312yes—
183C2×#103C2312yes—
184C2×#103C2312yes—
185C2×#103C2312yes—
186C2×#103C2312yes—
187C2×#103C2314yes—
188C2×#103C2314yes—
189C2×C22⋊C8C2312yes—
190C2×#97C2312yes—
191#398—24no—
192#398—24no—
193#397—24no—
194#397—24no—
195#391—22no—
196#391—22no—
197#391—21no—
198#391—21no—
199#391—21no—
200#391—21no—
201#391—22no—
202#391—22no—
203#391C2314yes—
204#391C2314yes—
205#391C2312yes—
206#391C2312yes—
207#389—22no—
208#389—22no—
209#389C2312yes—
210#389C2312yes—
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#387—22no—
220#387—22no—
221#384—22no—
222#384—22no—
223#384C2314yes—
224#384C2312yes—
225#382—22no—
226#382—22no—
227#382—21no—
228#382—21no—
229#382—21no—
230#382C2314yes—
231#380—21no—
232#380—22no—
233#380—22no—
234#380C2312yes—
235C4×C4⋊C4—22no—
236C4×C4⋊C4—22no—
237C4×C4⋊C4—22no—
238C4×C4⋊C4—22no—
239C4×C4⋊C4—21no—
240C4×C4⋊C4—21no—
241C4×C4⋊C4—21no—
242C4×C4⋊C4—21no—
243C4×C4⋊C4—21no—
244C4×C4⋊C4D412yes—
245C4×C4⋊C4C2312yes—
246C4×C4⋊C4C2312yes—
247C4×C4⋊C4C2311yescharacteristic
248C4×C4⋊C4C2311yescharacteristic
249C4×C4⋊C4C2312yes—
250C4×C22⋊C4—22no—
251C4×C22⋊C4—22no—
252C4×C22⋊C4D412yes—
253C4×C22⋊C4C2312yes—
254#376—21no—
255#376—21no—
256#376D411yescharacteristic
257#376D411yescharacteristic
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—44no—
305C4⋊Q8—44no—
306#125—24no—
307#125—24no—
308#125—44no—
309C22⋊Q8—24no—
310C22⋊Q8—24no—
311C22⋊Q8—44no—
312C22⋊Q8—44no—
313C22⋊Q8—44no—
314C4×Q8—28no—
315C4×Q8—28no—
316C4×Q8—22no—
317C4×Q8—22no—
318C4×Q8—22no—
319C4×Q8—22no—
320C4×Q8—24no—
321C4×Q8—24no—
322C4×Q8—24no—
323C4×Q8—24no—
324C4×Q8—42no—
325C4×Q8—42no—
326C4×Q8—44no—
327C4×Q8—44no—
328C4×Q8—44no—
329C4×Q8C2×D414yes—
330C4×Q8C2×D414yes—
331C4×Q8C2×D414yes—
332C4×Q8C2×D414yes—
333#117—24no—
334#117—24no—
335#117—24no—
336#117—42no—
337#117—44no—
338#117—44no—
339C2×C4⋊C4—28no—
340C2×C4⋊C4—21no—
341C2×C4⋊C4—21no—
342C2×C4⋊C4—21no—
343C2×C4⋊C4—21no—
344C2×C4⋊C4—21no—
345C2×C4⋊C4—22no—
346C2×C4⋊C4—21no—
347C2×C4⋊C4—21no—
348C2×C4⋊C4—22no—
349C2×C4⋊C4—22no—
350C2×C4⋊C4—22no—
351C2×C4⋊C4—21no—
352C2×C4⋊C4—21no—
353C2×C4⋊C4—21no—
354C2×C4⋊C4—22no—
355C2×C4⋊C4—22no—
356C2×C4⋊C4—22no—
357C2×C4⋊C4—22no—
358C2×C4⋊C4—24no—
359C2×C4⋊C4—24no—
360C2×C4⋊C4—22no—
361C2×C4⋊C4—22no—
362C2×C4⋊C4—22no—
363C2×C4⋊C4C2×D412yes—
364C2×C4⋊C4C2×D414yes—
365C2×C4⋊C4C2×D412yes—
366C2×C4⋊C4C2412yes—
367C2×C4⋊C4C2412yes—
368C2×C4⋊C4C2412yes—
369C2×C4⋊C4C2412yes—
370C2×C4⋊C4C2412yes—
371C2×C4⋊C4C2411yescharacteristic
372C2×C4⋊C4C2411yescharacteristic
373C2×C22⋊C4—22no—
374C2×C22⋊C4—22no—
375C2×C22⋊C4—22no—
376C2×C22⋊C4C2×D412yes—
377C2×C22⋊C4C2×D412yes—
378C2×C42—28no—
379C2×C42—21no—
380C2×C42—21no—
381C2×C42—21no—
382C2×C42—21no—
383C2×C42—22no—
384C2×C42—21no—
385C2×C42—21no—
386C2×C42—22no—
387C2×C42—22no—
388C2×C42—22no—
389C2×C42—22no—
390C2×C42—22no—
391C2×C42C2×D411yescharacteristic
392C2×C42C2×D411yescharacteristic
393C2×C42C2×D411yescharacteristic
394C2×C42C2×D412yes—
395C2×C42C2×D412yes—
396C2×C42C2×D411yescharacteristic
397C2×C42C2×D411yescharacteristic
398C2×C42C2×D414yes—
399C2×C42C2×D411yescharacteristic
400C2×C42#4114yes—
401C2×C42C2412yes—
402C2×C42C2412yes—
403C2×C42C2411yescharacteristic
404C2×C42C2411yescharacteristic
405#107—24no—
406#107—24no—
407#106—24no—
408#106—24no—
409#106—24no—
410#106—24no—
411C4⋊C8—22no—
412C4⋊C8—22no—
413C4⋊C8—24no—
414C4⋊C8—22no—
415C4⋊C8—22no—
416C4⋊C8—44no—
417C4⋊C8—44no—
418#103—24no—
419#103—24no—
420#103—24no—
421#103—24no—
422#103—24no—
423#103—24no—
424#103—24no—
425#103—24no—
426#103—44no—
427#103—44no—
428C22⋊C8—24no—
429#97—24no—
430#95—22no—
431#95—22no—
432#95—22no—
433#95—22no—
434#95—22no—
435#95—22no—
436#95—21no—
437#95—21no—
438#95—22no—
439#95—22no—
440#95—22no—
441#95—24no—
442#95—24no—
443#95—24no—
444#95—24no—
445#95—24no—
446#95—24no—
447#95C2×D412yes—
448#95C2×D412yes—
449#95C2×D412yes—
450#95C2×D412yes—
451C24C22×D411yescharacteristic
452C2×Q8—24no—
453C2×Q8—21no—
454C2×Q8—21no—
455C2×Q8—22no—
456C2×Q8—28no—
457C2×Q8—22no—
458C2×Q8—22no—
459C2×Q8—22no—
460C2×Q8—24no—
461C2×Q8—22no—
462C2×Q8—24no—
463C2×Q8—44no—
464C2×Q8—44no—
465C2×Q8—48no—
466C2×Q8—44no—
467C2×Q8—42no—
468C2×Q8—42no—
469C2×Q8—42no—
470C2×Q8—42no—
471C2×Q8—42no—
472C2×Q8—42no—
473C2×Q8—42no—
474C2×Q8C22×D412yes—
475C2×Q8C22×D412yes—
476C2×Q82−1+414yes—
477C22×C4—21no—
478C22×C4—22no—
479C22×C4—22no—
480C22×C4—22no—
481C22×C4—22no—
482C22×C4—24no—
483C22×C4—21no—
484C22×C4—21no—
485C22×C4—21no—
486C22×C4—24no—
487C22×C4—22no—
488C22×C4—22no—
489C22×C4—22no—
490C22×C4—22no—
491C22×C4—22no—
492C22×C4—22no—
493C22×C4—42no—
494C22×C4—44no—
495C22×C4—82no—
496C22×C4#12114yes—
497C22×C4#12114yes—
498C22×C4C22×D412yes—
499C22×C4C22×D412yes—
500C22×C4C22×D412yes—
501C22×C4C22×D411yescharacteristic
502C22×C4C22×D411yescharacteristic
503C22×C4C22×D411yescharacteristic
504C22×C4C22×D411yescharacteristic
505C22×C4C22×D412yes—
506C22×C4C22×D412yes—
507C22×C4C22×D411yescharacteristic
508C22×C4C2×#4112yes—
509C22×C4C2511yescharacteristic
510Q16—48no—
511Q16—48no—
512M16—48no—
513C2×C8—22no—
514C2×C8—22no—
515C2×C8—22no—
516C2×C8—22no—
517C2×C8—24no—
518C2×C8—24no—
519C2×C8—24no—
520C2×C8—24no—
521C4⋊C4—24no—
522C4⋊C4—24no—
523C4⋊C4—24no—
524C4⋊C4—24no—
525C4⋊C4—24no—
526C4⋊C4—24no—
527C4⋊C4—21no—
528C4⋊C4—21no—
529C4⋊C4—22no—
530C4⋊C4—22no—
531C4⋊C4—22no—
532C4⋊C4—22no—
533C4⋊C4—22no—
534C4⋊C4—22no—
535C4⋊C4—22no—
536C4⋊C4—22no—
537C4⋊C4—22no—
538C4⋊C4—22no—
539C4⋊C4—42no—
540C4⋊C4—44no—
541C4⋊C4—44no—
542C4⋊C4—44no—
543C4⋊C4—48no—
544C4⋊C4—42no—
545C4⋊C4—42no—
546C4⋊C4—44no—
547C4⋊C4—44no—
548C4⋊C4—44no—
549C4⋊C4—44no—
550C4⋊C4—42no—
551C4⋊C4—42no—
552C4⋊C4—42no—
553C4⋊C4—48no—
554C4⋊C4—82no—
555C4⋊C4—82no—
556C4⋊C4—82no—
557C4⋊C4C22×D412yes—
558C4⋊C4C22×D412yes—
559C22⋊C4—24no—
560C22⋊C4—42no—
561C22⋊C4—44no—
562C22⋊C4—44no—
563C22⋊C4—44no—
564C42—24no—
565C42—24no—
566C42—24no—
567C42—24no—
568C42—22no—
569C42—22no—
570C42—22no—
571C42—22no—
572C42—22no—
573C42—22no—
574C42—22no—
575C42—42no—
576C42—42no—
577C42—44no—
578C42—44no—
579C42—44no—
580C42—44no—
581C42—42no—
582C42—42no—
583C42—82no—
584C42—82no—
585C42#12114yes—
586C42#12114yes—
587C42#13714yes—
588C42C22×D412yes—
589C42C22×D412yes—
590C42C2×#4112yes—
591C42C2×#4112yes—
592C23—22no—
593C23—22no—
594C23—21no—
595C23—21no—
596C23C2×#12112yes—
597C23C23×D411yescharacteristic
598Q8—22no—
599Q8—24no—
600Q8—22no—
601Q8—22no—
602Q8—48no—
603Q8—42no—
604Q8—42no—
605Q8—42no—
606Q8—44no—
607Q8—84no—
608Q8—84no—
609Q8—84no—
610Q8#53614yes—
611C2×C4—22no—
612C2×C4—22no—
613C2×C4—22no—
614C2×C4—22no—
615C2×C4—22no—
616C2×C4—24no—
617C2×C4—22no—
618C2×C4—21no—
619C2×C4—21no—
620C2×C4—22no—
621C2×C4—22no—
622C2×C4—24no—
623C2×C4—28no—
624C2×C4—22no—
625C2×C4—22no—
626C2×C4—22no—
627C2×C4—22no—
628C2×C4—22no—
629C2×C4—22no—
630C2×C4—22no—
631C2×C4—22no—
632C2×C4—22no—
633C2×C4—22no—
634C2×C4—22no—
635C2×C4—22no—
636C2×C4—44no—
637C2×C4—48no—
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—42no—
647C2×C4—42no—
648C2×C4—42no—
649C2×C4—42no—
650C2×C4—44no—
651C2×C4—42no—
652C2×C4—42no—
653C2×C4—42no—
654C2×C4—42no—
655C2×C4—42no—
656C2×C4—42no—
657C2×C4—84no—
658C2×C4C2×#12112yes—
659C2×C4C2×#12112yes—
660C2×C4C2×#12112yes—
661C2×C4C2×#12112yes—
662C2×C4C2×#12112yes—
663C2×C4C2×#13714yes—
664C2×C4C2×#13712yes—
665C2×C4#57512yes—
666C2×C4C23×D411yescharacteristic
667C2×C4C22×#4111yescharacteristic
668C2×C4C2×2−1+412yes—
669C2×C4#58512yes—
670C8—42no—
671C8—42no—
672C8—42no—
673C8—42no—
674C22—22no—
675C22—22no—
676C22—22no—
677C22—42no—
678C22—42no—
679C22C22×#12111yescharacteristic
680C22#348912yes—
681C22C22×#13711yescharacteristic
682C22C2×#57511yescharacteristic
683C22#358612yes—
684C4—22no—
685C4—22no—
686C4—24no—
687C4—22no—
688C4—24no—
689C4—22no—
690C4—42no—
691C4—42no—
692C4—42no—
693C4—42no—
694C4—42no—
695C4—42no—
696C4—82no—
697C4—82no—
698C4—82no—
699C4—82no—
700C4—82no—
701C4—82no—
702C4#303714yes—
703C4C2×#53611yescharacteristic
704C4#344911yescharacteristic
705C4#349512yes—
706C2—42no—
707C2#3530611yescharacteristic
708C2C2×#303711yescharacteristic
709C2#6042711yescharacteristic
710C2#6211312yes—
711C2#6211812yes—
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. #33196C2semidirect
7. #33033C2semidirect
8. #33027C2semidirect
9. #29867C2semidirect
10. #29866C2semidirect
11. #29831C2semidirect
12. #29830C2semidirect
13. #24826C2semidirect
14. #24691C2semidirect

Maximal subgroups

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

subgroupindex
C2×#31132
C2×#30782
#353222
Q8×#1172
#332022
#331962
#330332
#330272
#298672
#298662
#298312
#298302
#248262
#246912
#245902
#245402
#219782
#218942
#218392