Finite Group Databasebeta

← #6477637

Subgroups — #6477637

Order 512.

2077 conjugacy classes of subgroups.

The subgroup lattice has 678 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3092C211yesmaximal, characteristic, complemented
2C22×#480C211yesmaximal, characteristic, complemented
3#37448C211yesmaximal, characteristic
4#34178C211yesmaximal, characteristic, complemented
5#33120C212yesmaximal, complemented
6#33118C212yesmaximal, complemented
7#29767C212yesmaximal, complemented
8#29763C212yesmaximal, complemented
9#24591C214yesmaximal, complemented
10#24577C212yesmaximal, complemented
11#24576C212yesmaximal, complemented
12#24490C214yesmaximal, complemented
13#24477C212yesmaximal, complemented
14#24476C212yesmaximal, complemented
15#21933C211yesmaximal, characteristic, complemented
16#21767C211yesmaximal, characteristic, complemented
17#21760C211yesmaximal, characteristic, complemented
18C2×#525C2211yescharacteristic
19C22×#121C2211yescharacteristic
20C23×C4⋊C4C2211yescharacteristic
21#3092—28no—
22C2×#484C2211yescharacteristic, complemented
23C2×#484C2211yescharacteristic, complemented
24C2×#481C2211yescharacteristic, complemented
25C2×#481C2211yescharacteristic, complemented
26C2×#483C2212yes—
27C2×#480—28no—
28C2×#480C2212yes—
29C2×#480C2214yes—
30C2×#480C2212yes—
31C2×#480C2212yes—
32C2×#428C2212yescomplemented
33C22×#107C2212yes—
34C2×#417C2212yescomplemented
35C22×#102C2212yes—
36C2×#407C2211yescharacteristic
37C22×C22⋊C8C2211yescharacteristic
38#2813—22no—
39#2635—22no—
40#2620C2212yes—
41#2619C2212yes—
42#2613—22no—
43#2534—22no—
44#2503C2211yescharacteristic
45#2501C2211yescharacteristic
46#2500C2212yes—
47#2435C2211yescharacteristic
48#2433C2211yescharacteristic
49#2431C2212yes—
50#2345—22no—
51#2345C2214yescomplemented
52#2344—22no—
53#2344C2212yes—
54#2344C2212yes—
55#2316C2212yescomplemented
56#2313—22no—
57#2280—22no—
58#2280C2211yescharacteristic
59#2280C2211yescharacteristic
60#2274C2212yescomplemented
61#2085C2214yescomplemented
62#2083C2214yescomplemented
63#2068C2214yescomplemented
64#2067C2214yescomplemented
65#2067C2214yescomplemented
66#2065C2214yescomplemented
67#2042C2214yescomplemented
68#2041C2214yescomplemented
69#1983C2214yescomplemented
70#1982C2214yescomplemented
71#1949C2214yescomplemented
72#1947C2214yescomplemented
73#1928C2214yescomplemented
74#1926C2214yescomplemented
75#1925C2214yescomplemented
76#1925C2214yescomplemented
77#1895C2214yes—
78#1893C2212yes—
79#1892C2212yes—
80#1875C2212yes—
81#1875C2212yes—
82#1875C2212yescomplemented
83#1873C2212yescomplemented
84#1851C2214yescomplemented
85#1850C2212yes—
86#1850C2212yes—
87#1825C2212yescomplemented
88#1823C2212yes—
89#1798C2212yescomplemented
90#1798C2212yescomplemented
91#1788C2212yescomplemented
92#1786C2212yes—
93C22×#41C2311yescharacteristic
94C23×D4—21no—
95C23×D4C2311yescharacteristic
96C24×C4—21no—
97C24×C4C2311yescharacteristic
98C22×M16C2312yes—
99C23×C8C2312yes—
100C2×#127C2312yes—
101C2×#124C2312yes—
102#525—24no—
103#525—24no—
104C2×#123—21no—
105C2×#123—22no—
106C2×C22⋊Q8C2311yescharacteristic
107C2×C22⋊Q8C2311yescharacteristic
108C2×#121—21no—
109C2×#121—21no—
110C2×#121—24no—
111C2×#121—24no—
112C2×#121C2312yes—
113C2×#121C2311yescharacteristic
114C2×#121C2312yes—
115C2×#121C2311yescharacteristic
116C2×#120—21no—
117C2×#120—22no—
118C2×C4×D4—21no—
119C2×C4×D4—21no—
120C2×#117—22no—
121C2×#117—22no—
122C2×#117C2311yescharacteristic
123C2×#117C2312yes—
124C2×#117C2312yes—
125C22×C4⋊C4—21no—
126C22×C4⋊C4—21no—
127C22×C4⋊C4—21no—
128C22×C4⋊C4—24no—
129C22×C4⋊C4D418yescomplemented
130C22×C4⋊C4C2311yescharacteristic
131C22×C4⋊C4C2311yescharacteristic
132C22×C4⋊C4C2311yescharacteristic
133C22×C4⋊C4C2312yes—
134C22×C4⋊C4C2312yes—
135C22×C22⋊C4—21no—
136#484—24no—
137#484—24no—
138#483—28no—
139#481—24no—
140#481—24no—
141#480—216no—
142#480—28no—
143#480—28no—
144#480—28no—
145#428—28no—
146C2×#107—28no—
147C2×#107C2314yes—
148C2×#107C2314yes—
149C2×#107C2312yes—
150C2×#107C2312yes—
151C2×#106C2312yes—
152C2×#106C2312yes—
153C2×C4⋊C8C2314yes—
154C2×C4⋊C8C2314yes—
155#417—28no—
156C2×#103C2312yes—
157C2×#103C2312yes—
158C2×#103C2314yes—
159C2×#102—28no—
160C2×#102C2314yes—
161C2×#102C2314yes—
162C2×#102C2312yes—
163C2×#102C2312yes—
164C2×#102C2314yes—
165C2×#102C2314yes—
166C2×#102C2314yes—
167#407—24no—
168C2×C22⋊C8—24no—
169C2×C22⋊C8C2312yes—
170C2×C22⋊C8C2311yescharacteristic
171C2×C22⋊C8C2311yescharacteristic
172C2×C22⋊C8C2312yes—
173C2×C22⋊C8C2312yes—
174#400—22no—
175#399—22no—
176#396—24no—
177#394—24no—
178#394—24no—
179#394—24no—
180#394—22no—
181#392—24no—
182#392—24no—
183#392—22no—
184#391C2312yes—
185#390C2312yes—
186#390C2312yes—
187#390C2312yes—
188#389—24no—
189#389C2314yes—
190#389C2314yes—
191#388—24no—
192#387—24no—
193#387—24no—
194#387—24no—
195#387—24no—
196#386—24no—
197#384C2312yes—
198#382C2312yes—
199#381—21no—
200#381—22no—
201#381—22no—
202#381—22no—
203#381D412yes—
204#380—24no—
205#380—24no—
206#380D412yes—
207#380D412yes—
208#380C2312yes—
209#380C2312yes—
210#379—21no—
211#379—22no—
212#379D412yes—
213C4×C4⋊C4—24no—
214C4×C4⋊C4C2314yes—
215C4×C4⋊C4C2314yes—
216C4×C22⋊C4—21no—
217C4×C22⋊C4—21no—
218C4×C22⋊C4—24no—
219C4×C22⋊C4—24no—
220C4×C22⋊C4C2311yescharacteristic
221C4×C22⋊C4C2311yescharacteristic
222C4×C22⋊C4C2312yes—
223#340C2314yes—
224#340C2314yes—
225#340C2314yes—
226#340C2314yes—
227#336C2314yes—
228#336C2314yes—
229C25—21no—
230C25C2×D411yescharacteristic
231C2×#41—22no—
232C2×#41—22no—
233C2×#41—44no—
234C22×Q8C2411yescharacteristic
235C22×D4—21no—
236C22×D4—21no—
237C22×D4—21no—
238C22×D4—22no—
239C22×D4—22no—
240C22×D4—22no—
241C22×D4—21no—
242C22×D4—24no—
243C22×D4—22no—
244C22×D4—44no—
245C22×D4—44no—
246C22×D4—84no—
247C22×D4C2412yes—
248C22×D4C2411yescharacteristic
249C23×C4—22no—
250C23×C4—28no—
251C23×C4—21no—
252C23×C4—22no—
253C23×C4—21no—
254C23×C4—21no—
255C23×C4—21no—
256C23×C4—44no—
257C23×C4—84no—
258C23×C4C2×D411yescharacteristic
259C23×C4C2×D411yescharacteristic
260C23×C4C2×D411yescharacteristic
261C23×C4C2×D414yes—
262C23×C4#4114yes—
263C23×C4C2411yescharacteristic
264C23×C4C2411yescharacteristic
265C23×C4C2411yescharacteristic
266C2×M16—24no—
267C2×M16—44no—
268C22×C8—24no—
269C22×C8—44no—
270C22×C8C2414yes—
271C22×C8C2412yes—
272C22×C8C2412yes—
273#127—44no—
274#124—44no—
275#123—44no—
276#123—84no—
277C22⋊Q8—22no—
278C22⋊Q8—22no—
279C22⋊Q8—22no—
280C22⋊Q8—22no—
281#121—24no—
282#121—24no—
283#121—22no—
284#121—22no—
285#121—24no—
286#121—24no—
287#121—22no—
288#121—22no—
289#121—44no—
290#121—44no—
291#120—44no—
292#120—84no—
293C4×D4—44no—
294C4×D4—44no—
295#117—22no—
296#117—22no—
297#117—44no—
298#117—44no—
299#117—84no—
300#117—84no—
301C2×C4⋊C4—28no—
302C2×C4⋊C4—28no—
303C2×C4⋊C4—28no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—24no—
309C2×C4⋊C4—24no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4—22no—
314C2×C4⋊C4—21no—
315C2×C4⋊C4—21no—
316C2×C4⋊C4—21no—
317C2×C4⋊C4—24no—
318C2×C4⋊C4—24no—
319C2×C4⋊C4—24no—
320C2×C4⋊C4C2×D418yes—
321C2×C4⋊C4C2×D418yes—
322C2×C4⋊C4C2×D414yes—
323C2×C4⋊C4C2×D414yes—
324C2×C4⋊C4C2×D414yes—
325C2×C4⋊C4C2412yes—
326C2×C4⋊C4C2411yescharacteristic
327C2×C4⋊C4C2411yescharacteristic
328C2×C4⋊C4C2414yes—
329C2×C4⋊C4C2412yes—
330C2×C4⋊C4C2412yes—
331C2×C22⋊C4—22no—
332C2×C22⋊C4—21no—
333C2×C22⋊C4—22no—
334C2×C22⋊C4—21no—
335C2×C22⋊C4—21no—
336C2×C22⋊C4—22no—
337C2×C22⋊C4—22no—
338C2×C22⋊C4—22no—
339C2×C22⋊C4—21no—
340C2×C22⋊C4—21no—
341C2×C22⋊C4—24no—
342C2×C22⋊C4—24no—
343C2×C22⋊C4—24no—
344C2×C22⋊C4—24no—
345C2×C22⋊C4—44no—
346C2×C22⋊C4C2×D412yes—
347C2×C22⋊C4C2×D412yes—
348C2×C22⋊C4C2×D414yes—
349C2×C22⋊C4C2×D414yes—
350C2×C42—21no—
351C2×C42—22no—
352C2×C42—24no—
353C2×C42—24no—
354C2×C42C2412yes—
355C2×C42C2412yes—
356#107—28no—
357#107—28no—
358#107—24no—
359#107—24no—
360#106—24no—
361#106—24no—
362C4⋊C8—44no—
363C4⋊C8—44no—
364#103—24no—
365#103—24no—
366#103—44no—
367#102—28no—
368#102—28no—
369#102—24no—
370#102—24no—
371#102—44no—
372#102—44no—
373#102—44no—
374C22⋊C8—24no—
375C22⋊C8—24no—
376C22⋊C8—24no—
377C22⋊C8—22no—
378C22⋊C8—22no—
379#95—24no—
380#95—24no—
381#95—24no—
382#95—24no—
383#95—22no—
384#95—22no—
385#95—24no—
386#95—24no—
387#95C2×D414yes—
388#95C2×D412yes—
389#95C2×D412yes—
390C24—24no—
391C24—22no—
392C24—21no—
393C24—42no—
394C24—42no—
395C24—42no—
396C24—44no—
397C24—82no—
398C24—82no—
399C24#12114yes—
400C24C22×D411yescharacteristic
401C24C22×D411yescharacteristic
402C24C22×D411yescharacteristic
403#41—48no—
404#41—84no—
405C2×Q8—21no—
406C2×Q8—21no—
407C2×Q8—24no—
408C2×D4—22no—
409C2×D4—21no—
410C2×D4—22no—
411C2×D4—21no—
412C2×D4—24no—
413C2×D4—24no—
414C2×D4—24no—
415C2×D4—48no—
416C2×D4—42no—
417C2×D4—42no—
418C2×D4—42no—
419C2×D4—44no—
420C2×D4—44no—
421C2×D4—44no—
422C2×D4—42no—
423C2×D4—84no—
424C2×D4—82no—
425C2×D4—84no—
426C2×D4—84no—
427C2×D4—84no—
428C2×D4—82no—
429C2×D4—84no—
430C2×D4—84no—
431C2×D4—82no—
432C2×D4—816no—
433C22×C4—24no—
434C22×C4—24no—
435C22×C4—24no—
436C22×C4—28no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—28no—
440C22×C4—24no—
441C22×C4—24no—
442C22×C4—24no—
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—22no—
446C22×C4—21no—
447C22×C4—48no—
448C22×C4—48no—
449C22×C4—42no—
450C22×C4—42no—
451C22×C4—42no—
452C22×C4—44no—
453C22×C4—44no—
454C22×C4—42no—
455C22×C4—44no—
456C22×C4—42no—
457C22×C4—82no—
458C22×C4—816no—
459C22×C4—82no—
460C22×C4—82no—
461C22×C4—82no—
462C22×C4—82no—
463C22×C4#12114yes—
464C22×C4#12114yes—
465C22×C4#12114yes—
466C22×C4C22×D414yes—
467C22×C4C22×D414yes—
468C22×C4C22×D411yescharacteristic
469C22×C4C22×D411yescharacteristic
470C22×C4C22×D411yescharacteristic
471C22×C4C22×D412yes—
472C22×C4C22×D412yes—
473C22×C4C22×D412yes—
474C22×C4C2×#4112yes—
475C22×C4C2×#4112yes—
476C22×C4C2×#4112yes—
477C22×C4C2511yescharacteristic
478M16—48no—
479C2×C8—24no—
480C2×C8—22no—
481C2×C8—22no—
482C2×C8—24no—
483C2×C8—24no—
484C2×C8—24no—
485C2×C8—24no—
486C2×C8—48no—
487C4⋊C4—28no—
488C4⋊C4—24no—
489C4⋊C4—24no—
490C4⋊C4—28no—
491C4⋊C4—22no—
492C4⋊C4—22no—
493C4⋊C4—21no—
494C4⋊C4—22no—
495C4⋊C4—21no—
496C4⋊C4—22no—
497C4⋊C4—21no—
498C4⋊C4—21no—
499C4⋊C4—24no—
500C4⋊C4—24no—
501C4⋊C4—44no—
502C4⋊C4—42no—
503C4⋊C4—42no—
504C4⋊C4—42no—
505C4⋊C4—84no—
506C4⋊C4—84no—
507C4⋊C4—84no—
508C4⋊C4C22×D414yes—
509C4⋊C4C22×D414yes—
510C22⋊C4—24no—
511C22⋊C4—24no—
512C22⋊C4—44no—
513C22⋊C4—44no—
514C22⋊C4—44no—
515C22⋊C4—44no—
516C22⋊C4—42no—
517C22⋊C4—42no—
518C22⋊C4—42no—
519C22⋊C4—44no—
520C22⋊C4—44no—
521C22⋊C4—44no—
522C22⋊C4—42no—
523C22⋊C4—42no—
524C22⋊C4—84no—
525C22⋊C4—84no—
526C22⋊C4—84no—
527C22⋊C4—84no—
528C42—24no—
529C42—24no—
530C42—44no—
531C42—42no—
532C42—84no—
533C42—84no—
534C23—22no—
535C23—22no—
536C23—22no—
537C23—22no—
538C23—22no—
539C23—22no—
540C23—28no—
541C23—22no—
542C23—22no—
543C23—44no—
544C23—44no—
545C23—416no—
546C23—44no—
547C23—42no—
548C23—41no—
549C23—42no—
550C23—42no—
551C23—42no—
552C23—42no—
553C23—42no—
554C23—44no—
555C23—84no—
556C23—84no—
557C23—84no—
558C23—168no—
559C23C2×#12112yes—
560C23C2×#12112yes—
561C23C2×#12112yes—
562C23#57614yes—
563C23C23×D411yescharacteristic
564Q8—42no—
565Q8—42no—
566D4—44no—
567D4—42no—
568D4—44no—
569D4—42no—
570D4—84no—
571D4—84no—
572D4—88no—
573D4—88no—
574D4—84no—
575D4—84no—
576D4—168no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—24no—
581C2×C4—24no—
582C2×C4—24no—
583C2×C4—24no—
584C2×C4—24no—
585C2×C4—24no—
586C2×C4—24no—
587C2×C4—24no—
588C2×C4—24no—
589C2×C4—24no—
590C2×C4—24no—
591C2×C4—24no—
592C2×C4—24no—
593C2×C4—24no—
594C2×C4—44no—
595C2×C4—44no—
596C2×C4—44no—
597C2×C4—416no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—44no—
603C2×C4—42no—
604C2×C4—41no—
605C2×C4—41no—
606C2×C4—42no—
607C2×C4—41no—
608C2×C4—44no—
609C2×C4—42no—
610C2×C4—44no—
611C2×C4—42no—
612C2×C4—44no—
613C2×C4—84no—
614C2×C4—84no—
615C2×C4—88no—
616C2×C4—84no—
617C2×C4—84no—
618C2×C4C2×#12112yes—
619C2×C4C2×#12112yes—
620C2×C4C2×#12112yes—
621C2×C4C23×D411yescharacteristic
622C2×C4C22×#4111yescharacteristic
623C2×C4#58514yes—
624C8—44no—
625C8—42no—
626C8—42no—
627C22—22no—
628C22—22no—
629C22—28no—
630C22—22no—
631C22—22no—
632C22—22no—
633C22—22no—
634C22—22no—
635C22—22no—
636C22—416no—
637C22—44no—
638C22—44no—
639C22—82no—
640C22—81no—
641C22—81no—
642C22—82no—
643C22—81no—
644C22—82no—
645C22—84no—
646C22—84no—
647C22—84no—
648C22—168no—
649C22#304114yes—
650C22C22×#12111yescharacteristic
651C22#348512yes—
652C22C2×#57612yes—
653C22#358612yes—
654C4—22no—
655C4—22no—
656C4—22no—
657C4—44no—
658C4—44no—
659C4—44no—
660C4—44no—
661C4—82no—
662C4—81no—
663C4—81no—
664C4—81no—
665C4—84no—
666C4#344912yes—
667C4#349112yes—
668C2—24no—
669C2—42no—
670C2—42no—
671C2—161no—
672C2—162no—
673C2#3508911yescharacteristic
674C2C2×#304111yescharacteristic
675C2#6042411yescharacteristic
676C2#6174414yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#37448C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ #2280 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#3092C2semidirect
2. C22×#480C2semidirect
4. #34178C2semidirect
5. #33120C2semidirect
6. #33118C2semidirect
7. #29767C2semidirect
8. #29763C2semidirect
9. #24591C2semidirect
10. #24577C2semidirect
11. #24576C2semidirect
12. #24490C2semidirect
13. #24477C2semidirect
14. #24476C2semidirect
15. #21933C2semidirect
16. #21767C2semidirect
17. #21760C2semidirect
22. C2×#484C22semidirect
23. C2×#484C22semidirect
24. C2×#481C22semidirect
25. C2×#481C22semidirect
32. C2×#428C22semidirect
34. C2×#417C22semidirect
51. #2345C22semidirect
55. #2316C22semidirect
60. #2274C22semidirect
61. #2085C22semidirect
62. #2083C22semidirect
63. #2068C22semidirect
64. #2067C22semidirect
65. #2067C22semidirect
66. #2065C22semidirect
67. #2042C22semidirect
68. #2041C22semidirect
69. #1983C22semidirect
70. #1982C22semidirect
71. #1949C22semidirect
72. #1947C22semidirect
73. #1928C22semidirect
74. #1926C22semidirect
75. #1925C22semidirect
76. #1925C22semidirect
82. #1875C22semidirect
83. #1873C22semidirect
84. #1851C22semidirect
87. #1825C22semidirect
89. #1798C22semidirect
90. #1798C22semidirect
91. #1788C22semidirect
129. C22×C4⋊C4D4semidirect

Maximal subgroups

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

subgroupindex
C2×#30922
C22×#4802
#374482
#341782
#331202
#331182
#297672
#297632
#245912
#245772
#245762
#244902
#244772
#244762
#219332
#217672
#217602