Finite Group Databasebeta

← #6496393

Subgroups — #6496393

Order 512.

1910 conjugacy classes of subgroups.

The subgroup lattice has 679 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3101C211yesmaximal, characteristic
2C2×#3077C211yesmaximal, characteristic
3#37635C211yesmaximal, characteristic, complemented
4#34194C211yesmaximal, characteristic
5#33205C212yesmaximal, complemented
6#33040C212yesmaximal, complemented
7#29874C212yesmaximal, complemented
8#29836C212yesmaximal, complemented
9#24816C212yesmaximal, complemented
10#24815C212yesmaximal, complemented
11#24681C212yesmaximal, complemented
12#24680C212yesmaximal, complemented
13#24590C214yesmaximal, complemented
14#24538C214yesmaximal
15#21975C211yesmaximal, characteristic, complemented
16#21894C211yesmaximal, characteristic, complemented
17#21835C211yesmaximal, characteristic, complemented
18C2×#533C2211yescharacteristic
19C2×#533C2211yescharacteristic
20C22×C4×D4C2211yescharacteristic
21#3101—28no—
22C2×#483C2212yes—
23#3077—28no—
24C2×#478C2212yes—
25C2×#478C2212yes—
26C2×#476C2211yescharacteristic
27C2×#476C2211yescharacteristic
28C2×#476C2211yescharacteristic
29C2×#476C2211yescharacteristic
30C2×#474C2211yescharacteristic
31C2×#474C2211yescharacteristic
32C2×#461C2211yescharacteristic
33C2×#461C2211yescharacteristic
34C2×#450C2212yes—
35C2×C4×SD16C2212yes—
36C2×C4×SD16C2212yes—
37C2×#431C2211yescharacteristic
38C2×#428C2212yes—
39C2×#423C2211yescharacteristic
40C2×#418C2212yes—
41#2817—22no—
42#2660C2212yescomplemented
43#2642C2212yescomplemented
44#2634—22no—
45#2625—21no—
46#2625—21no—
47#2581—22no—
48#2568C2212yescomplemented
49#2567C2212yescomplemented
50#2499C2214yes—
51#2354C2212yes—
52#2352—22no—
53D4×C4⋊C4C2212yes—
54#2345C2212yes—
55#2342—22no—
56D4×C22⋊C4C2212yes—
57#2319C2212yes—
58#2318—22no—
59#2285C2211yescharacteristic
60#2283C2211yescharacteristic
61#2280—21no—
62C2×#380—21no—
63#2275C2212yes—
64#2085C2214yescomplemented
65#2084C2214yescomplemented
66#2073C2214yes—
67#2071C2214yes—
68#2068C2214yescomplemented
69#2066C2214yescomplemented
70#2055C2214yes—
71#2055C2214yes—
72#1990C2212yescomplemented
73#1990C2212yescomplemented
74#1989C2212yescomplemented
75#1989C2212yescomplemented
76#1988C2212yescomplemented
77#1981C2212yescomplemented
78#1949C2214yescomplemented
79#1948C2214yescomplemented
80#1939C2212yescomplemented
81#1939C2212yescomplemented
82#1928C2214yescomplemented
83#1927C2214yescomplemented
84#1894C2214yes—
85#1877C2214yes—
86#1877C2214yes—
87#1851C2214yes—
88#1843C2212yescomplemented
89#1832C2212yescomplemented
90#1825C2214yes—
91#1804C2214yes—
92#1803C2212yescomplemented
93#1803C2212yescomplemented
94#1802C2214yes—
95C23×D4C2311yescharacteristic
96C24×C4—21no—
97C22×SD16C2312yes—
98C22×SD16C2312yes—
99C22×M16C2312yes—
100#533—24no—
101#533—24no—
102#533—24no—
103#533—24no—
104C2×C4⋊Q8C2311yescharacteristic
105C2×C4⋊Q8C2311yescharacteristic
106C2×C4⋊Q8C2311yescharacteristic
107C2×C4⋊Q8C2311yescharacteristic
108C2×#125—21no—
109C2×#125C2312yes—
110C2×C22⋊Q8—21no—
111C2×C22⋊Q8—21no—
112C2×C22⋊Q8—21no—
113C2×C22⋊Q8C2312yes—
114C2×C4×Q8—21no—
115C2×C4×Q8—21no—
116C2×C4×Q8—21no—
117C2×C4×Q8C2312yes—
118C2×C4×D4—21no—
119C2×C4×D4—21no—
120C2×C4×D4—24no—
121C2×C4×D4D418yes—
122C2×C4×D4C2314yes—
123C2×C4×D4C2311yescharacteristic
124C2×C4×D4C2311yescharacteristic
125C2×C4×D4C2311yescharacteristic
126C2×C4×D4C2311yescharacteristic
127C2×#117—21no—
128C2×#117—21no—
129C2×#117—22no—
130C2×#117C2312yes—
131C22×C4⋊C4—22no—
132C22×C4⋊C4C2311yescharacteristic
133C22×C22⋊C4—21no—
134C22×C22⋊C4—22no—
135C22×C42C2311yescharacteristic
136#483—28no—
137#478—28no—
138#478—28no—
139#476—24no—
140#476—24no—
141#476—24no—
142#476—24no—
143#474—24no—
144#474—24no—
145#461—24no—
146#461—24no—
147#450—28no—
148C4×SD16—28no—
149C4×SD16—28no—
150#431—24no—
151#428—28no—
152C2×#107C2312yes—
153C2×#107C2312yes—
154C2×#106C2312yes—
155C2×#106C2312yes—
156C2×#106C2312yes—
157C2×#106C2312yes—
158#423—24no—
159C2×C4⋊C8C2312yes—
160C2×C4⋊C8C2311yescharacteristic
161C2×C4⋊C8C2311yescharacteristic
162C2×C4⋊C8C2311yescharacteristic
163C2×C4⋊C8C2311yescharacteristic
164C2×C4⋊C8C2314yes—
165C2×C4⋊C8C2314yes—
166#418—28no—
167C2×#103C2312yes—
168C2×#103C2312yes—
169C2×#103C2314yes—
170C2×#103C2314yes—
171C2×#102C2312yes—
172C2×#102C2312yes—
173C2×#102C2312yes—
174C2×#102C2312yes—
175C2×#102C2312yes—
176C2×#102C2312yes—
177C2×C22⋊C8C2312yes—
178C2×C4×C8C2312yes—
179#400—24no—
180#400—24no—
181#400—22no—
182#400—22no—
183#398—22no—
184#398—22no—
185#397—24no—
186#397—24no—
187#395—24no—
188#395—24no—
189#391—22no—
190#391—22no—
191#391C2314yes—
192#391C2314yes—
193#390C2314yes—
194#389—22no—
195#389—22no—
196#389C2314yes—
197#388—24no—
198#388—24no—
199#387—24no—
200#387—24no—
201#387—24no—
202#386—24no—
203#385—24no—
204#384C2314yes—
205#384C2312yes—
206#383D412yes—
207#382—21no—
208#382D412yes—
209#382C2314yes—
210#381—22no—
211#381D412yes—
212#380—21no—
213#380—22no—
214#380—22no—
215#380—22no—
216#380—22no—
217#380D411yescharacteristic
218#380D411yescharacteristic
219#380C2312yes—
220#379—22no—
221C4×C4⋊C4—21no—
222C4×C4⋊C4C2312yes—
223C4×C4⋊C4C2314yes—
224C4×C4⋊C4C2312yes—
225C4×C22⋊C4—22no—
226C4×C22⋊C4—22no—
227C4×C22⋊C4—24no—
228C4×C22⋊C4C2312yes—
229#376C2312yes—
230C2×#95—22no—
231#340C2314yes—
232#340C2314yes—
233#340C2314yes—
234#340C2314yes—
235#340C2314yes—
236#340C2314yes—
237#336C2314yes—
238#336C2314yes—
239C25—21no—
240C22×Q8—21no—
241C22×Q8—21no—
242C22×Q8—21no—
243C22×Q8C2412yes—
244C22×D4—24no—
245C22×D4—21no—
246C22×D4—22no—
247C22×D4—42no—
248C22×D4C2×D414yes—
249C22×D4C2411yescharacteristic
250C22×D4C2411yescharacteristic
251C23×C4—21no—
252C23×C4—21no—
253C23×C4—21no—
254C23×C4—28no—
255C23×C4—21no—
256C23×C4—21no—
257C23×C4—24no—
258C23×C4—84no—
259C23×C4C2×D411yescharacteristic
260C23×C4C2×D411yescharacteristic
261C23×C4C2411yescharacteristic
262C2×SD16—24no—
263C2×SD16—24no—
264C2×SD16—24no—
265C2×SD16—24no—
266C2×SD16—42no—
267C2×SD16—42no—
268C2×M16—24no—
269C2×M16—44no—
270C22×C8C2412yes—
271C22×C8C2412yes—
272C22×C8C2412yes—
273C22×C8C2412yes—
274C4⋊Q8—22no—
275C4⋊Q8—22no—
276C4⋊Q8—22no—
277C4⋊Q8—22no—
278C4⋊Q8—22no—
279C4⋊Q8—22no—
280C4⋊Q8—22no—
281C4⋊Q8—22no—
282#125—24no—
283#125—24no—
284#125—44no—
285C22⋊Q8—24no—
286C22⋊Q8—24no—
287C22⋊Q8—44no—
288C22⋊Q8—44no—
289C22⋊Q8—44no—
290C4×Q8—24no—
291C4×Q8—24no—
292C4×Q8—44no—
293C4×Q8—44no—
294C4×Q8—44no—
295C4×D4—28no—
296C4×D4—28no—
297C4×D4—28no—
298C4×D4—22no—
299C4×D4—22no—
300C4×D4—22no—
301C4×D4—22no—
302C4×D4—44no—
303C4×D4C2×D414yes—
304C4×D4C2×D414yes—
305C4×D4C2×D414yes—
306C4×D4C2×D414yes—
307#117—24no—
308#117—24no—
309#117—44no—
310#117—82no—
311#117—84no—
312C2×C4⋊C4—21no—
313C2×C4⋊C4—22no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—21no—
316C2×C4⋊C4—21no—
317C2×C4⋊C4—22no—
318C2×C4⋊C4—22no—
319C2×C4⋊C4—21no—
320C2×C4⋊C4—21no—
321C2×C4⋊C4—21no—
322C2×C4⋊C4—22no—
323C2×C4⋊C4—22no—
324C2×C4⋊C4—24no—
325C2×C4⋊C4—22no—
326C2×C4⋊C4—24no—
327C2×C4⋊C4—24no—
328C2×C4⋊C4—84no—
329C2×C4⋊C4C2×D412yes—
330C2×C4⋊C4C2×D412yes—
331C2×C4⋊C4C2×D414yes—
332C2×C4⋊C4C2412yes—
333C2×C4⋊C4C2412yes—
334C2×C4⋊C4C2414yes—
335C2×C4⋊C4C2412yes—
336C2×C4⋊C4C2411yescharacteristic
337C2×C4⋊C4C2411yescharacteristic
338C2×C22⋊C4—28no—
339C2×C22⋊C4—21no—
340C2×C22⋊C4—21no—
341C2×C22⋊C4—21no—
342C2×C22⋊C4—21no—
343C2×C22⋊C4—22no—
344C2×C22⋊C4—22no—
345C2×C22⋊C4—22no—
346C2×C22⋊C4—24no—
347C2×C22⋊C4—24no—
348C2×C22⋊C4—24no—
349C2×C22⋊C4—84no—
350C2×C22⋊C4C2×D412yes—
351C2×C22⋊C4C2×D412yes—
352C2×C42—22no—
353C2×C42—22no—
354C2×C42—24no—
355C2×C42—22no—
356C2×C42C2×D411yescharacteristic
357C2×C42C2×D411yescharacteristic
358C2×C42C2×D414yes—
359C2×C42#4114yes—
360C2×C42C2414yes—
361C2×C42C2411yescharacteristic
362C2×C42C2411yescharacteristic
363#107—24no—
364#107—24no—
365#106—24no—
366#106—24no—
367#106—24no—
368#106—24no—
369C4⋊C8—22no—
370C4⋊C8—22no—
371C4⋊C8—24no—
372C4⋊C8—22no—
373C4⋊C8—22no—
374C4⋊C8—44no—
375C4⋊C8—44no—
376#103—24no—
377#103—24no—
378#103—44no—
379#103—44no—
380#102—24no—
381#102—24no—
382#102—24no—
383#102—24no—
384#102—24no—
385#102—24no—
386C22⋊C8—24no—
387C4×C8—24no—
388#95—22no—
389#95—24no—
390#95—24no—
391#95—22no—
392#95—24no—
393#95—24no—
394#95—24no—
395#95—24no—
396#95—24no—
397#95—24no—
398#95C2×D412yes—
399#95C2×D412yes—
400#95C2×D414yes—
401#95C2×D414yes—
402#95C2×D414yes—
403C24—21no—
404C24—21no—
405C24—24no—
406C24—44no—
407C24—42no—
408C24—42no—
409C24—82no—
410C24C22×D411yescharacteristic
411C2×Q8—22no—
412C2×Q8—22no—
413C2×Q8—22no—
414C2×Q8—22no—
415C2×Q8—22no—
416C2×Q8—22no—
417C2×Q8—42no—
418C2×Q8—42no—
419C2×Q8—42no—
420C2×Q8—44no—
421C2×Q8—81no—
422C2×Q8—81no—
423C2×Q8—81no—
424C2×Q8—81no—
425C2×D4—28no—
426C2×D4—24no—
427C2×D4—21no—
428C2×D4—21no—
429C2×D4—22no—
430C2×D4—22no—
431C2×D4—22no—
432C2×D4—44no—
433C2×D4—48no—
434C2×D4—44no—
435C2×D4C22×D412yes—
436C2×D4C22×D412yes—
437C2×D42−1+414yes—
438C22×C4—24no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—21no—
442C22×C4—21no—
443C22×C4—24no—
444C22×C4—24no—
445C22×C4—24no—
446C22×C4—24no—
447C22×C4—24no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—24no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—41no—
454C22×C4—41no—
455C22×C4—48no—
456C22×C4—44no—
457C22×C4—44no—
458C22×C4—816no—
459C22×C4—82no—
460C22×C4—82no—
461C22×C4—84no—
462C22×C4—84no—
463C22×C4—82no—
464C22×C4#12114yes—
465C22×C4#12114yes—
466C22×C4C22×D414yes—
467C22×C4C22×D414yes—
468C22×C4C22×D411yescharacteristic
469C22×C4C22×D411yescharacteristic
470C22×C4C22×D411yescharacteristic
471C22×C4C22×D411yescharacteristic
472C22×C4C22×D412yes—
473C22×C4C22×D411yescharacteristic
474C22×C4C2×#4112yes—
475C22×C4C2511yescharacteristic
476SD16—48no—
477SD16—48no—
478M16—48no—
479C2×C8—22no—
480C2×C8—22no—
481C2×C8—22no—
482C2×C8—22no—
483C2×C8—22no—
484C2×C8—22no—
485C2×C8—24no—
486C2×C8—24no—
487C2×C8—22no—
488C2×C8—22no—
489C4⋊C4—24no—
490C4⋊C4—24no—
491C4⋊C4—24no—
492C4⋊C4—21no—
493C4⋊C4—21no—
494C4⋊C4—22no—
495C4⋊C4—22no—
496C4⋊C4—22no—
497C4⋊C4—22no—
498C4⋊C4—22no—
499C4⋊C4—22no—
500C4⋊C4—24no—
501C4⋊C4—24no—
502C4⋊C4—42no—
503C4⋊C4—42no—
504C4⋊C4—44no—
505C4⋊C4—44no—
506C4⋊C4—44no—
507C4⋊C4—44no—
508C4⋊C4—42no—
509C4⋊C4—42no—
510C4⋊C4—42no—
511C4⋊C4—44no—
512C4⋊C4—84no—
513C4⋊C4—84no—
514C4⋊C4—84no—
515C4⋊C4—82no—
516C4⋊C4C22×D412yes—
517C4⋊C4C22×D412yes—
518C22⋊C4—24no—
519C22⋊C4—24no—
520C22⋊C4—24no—
521C22⋊C4—24no—
522C22⋊C4—42no—
523C22⋊C4—42no—
524C22⋊C4—48no—
525C22⋊C4—44no—
526C22⋊C4—44no—
527C22⋊C4—44no—
528C22⋊C4—84no—
529C22⋊C4—84no—
530C42—24no—
531C42—24no—
532C42—44no—
533C42—44no—
534C42—82no—
535C42—84no—
536C42#12114yes—
537C42#12114yes—
538C42C22×D412yes—
539C42C22×D412yes—
540C42C2×#4112yes—
541C42C2×#4112yes—
542C23—21no—
543C23—21no—
544C23—22no—
545C23—24no—
546C23—22no—
547C23—22no—
548C23—22no—
549C23—48no—
550C23—48no—
551C23—41no—
552C23—41no—
553C23—42no—
554C23—41no—
555C23—41no—
556C23—42no—
557C23—42no—
558C23—84no—
559C23—88no—
560C23—88no—
561C23C2×#12112yes—
562C23C23×D411yescharacteristic
563Q8—42no—
564Q8—42no—
565Q8—42no—
566Q8—42no—
567Q8—84no—
568Q8—84no—
569Q8—84no—
570D4—22no—
571D4—22no—
572D4—22no—
573D4—24no—
574D4—48no—
575D4—42no—
576D4#53614yes—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—24no—
586C2×C4—28no—
587C2×C4—24no—
588C2×C4—24no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—24no—
592C2×C4—24no—
593C2×C4—24no—
594C2×C4—24no—
595C2×C4—44no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—48no—
599C2×C4—48no—
600C2×C4—44no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—44no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—44no—
613C2×C4—44no—
614C2×C4—84no—
615C2×C4—84no—
616C2×C4—84no—
617C2×C4—168no—
618C2×C4C2×#12112yes—
619C2×C4C2×#12112yes—
620C2×C4C2×#12112yes—
621C2×C4C2×#12112yes—
622C2×C4C2×#12112yes—
623C2×C4#57714yes—
624C2×C4C23×D411yescharacteristic
625C2×C4C22×#4111yescharacteristic
626C2×C4C2×2−1+412yes—
627C2×C4#58512yes—
628C8—42no—
629C8—42no—
630C8—42no—
631C8—42no—
632C22—22no—
633C22—22no—
634C22—22no—
635C22—24no—
636C22—42no—
637C22—42no—
638C22—44no—
639C22—42no—
640C22—42no—
641C22—44no—
642C22—42no—
643C22—42no—
644C22—42no—
645C22—816no—
646C22—82no—
647C22—82no—
648C22C22×#12111yescharacteristic
649C22#348912yes—
650C22C2×#57712yes—
651C22#358511yescharacteristic
652C22#358611yescharacteristic
653C4—22no—
654C4—22no—
655C4—22no—
656C4—22no—
657C4—42no—
658C4—42no—
659C4—44no—
660C4—44no—
661C4—82no—
662C4—82no—
663C4—82no—
664C4—82no—
665C4—84no—
666C4#304214yes—
667C4C2×#53611yescharacteristic
668C4#344911yescharacteristic
669C4#349512yes—
670C2—42no—
671C2—44no—
672C2—82no—
673C2#3530611yescharacteristic
674C2C2×#304211yescharacteristic
675C2#6042511yescharacteristic
676C2#6218812yes—
677C2#6218912yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ #2283 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #37635C2semidirect
5. #33205C2semidirect
6. #33040C2semidirect
7. #29874C2semidirect
8. #29836C2semidirect
9. #24816C2semidirect
10. #24815C2semidirect
11. #24681C2semidirect
12. #24680C2semidirect
13. #24590C2semidirect
15. #21975C2semidirect
16. #21894C2semidirect
17. #21835C2semidirect
42. #2660C22semidirect
43. #2642C22semidirect
48. #2568C22semidirect
49. #2567C22semidirect
64. #2085C22semidirect
65. #2084C22semidirect
68. #2068C22semidirect
69. #2066C22semidirect
72. #1990C22semidirect
73. #1990C22semidirect
74. #1989C22semidirect
75. #1989C22semidirect
76. #1988C22semidirect
77. #1981C22semidirect
78. #1949C22semidirect
79. #1948C22semidirect
80. #1939C22semidirect
81. #1939C22semidirect
82. #1928C22semidirect
83. #1927C22semidirect
88. #1843C22semidirect
89. #1832C22semidirect
92. #1803C22semidirect
93. #1803C22semidirect

Maximal subgroups

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

subgroupindex
C2×#31012
C2×#30772
#376352
#341942
#332052
#330402
#298742
#298362
#248162
#248152
#246812
#246802
#245902
#245382
#219752
#218942
#218352