Finite Group Databasebeta

← #6477638

Subgroups — #6477638

Order 512.

1947 conjugacy classes of subgroups.

The subgroup lattice has 652 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3092C211yesmaximal, characteristic
2C22×#483C211yesmaximal, characteristic, complemented
3#37449C211yesmaximal, characteristic
4#34178C211yesmaximal, characteristic, complemented
5#33121C212yesmaximal, complemented
6#33119C212yesmaximal, complemented
7#29768C212yesmaximal, complemented
8#29764C212yesmaximal, complemented
9#24590C214yesmaximal, complemented
10#24581C212yesmaximal, complemented
11#24580C212yesmaximal, complemented
12#24491C214yesmaximal, complemented
13#24481C212yesmaximal, complemented
14#24480C212yesmaximal, complemented
15#21933C211yesmaximal, characteristic, complemented
16#21768C211yesmaximal, characteristic, complemented
17#21760C211yesmaximal, characteristic
18C2×#525C2211yescharacteristic
19C22×C22⋊Q8C2211yescharacteristic
20C23×C4⋊C4C2211yescharacteristic
21#3092—28no—
22C2×#484C2211yescharacteristic
23C2×#484C2211yescharacteristic
24C2×#481C2211yescharacteristic
25C2×#481C2211yescharacteristic
26C2×#483—28no—
27C2×#483C2212yes—
28C2×#483C2214yes—
29C2×#483C2212yes—
30C2×#483C2212yes—
31C2×#480C2212yes—
32C2×#428C2212yes—
33C22×#106C2212yes—
34C2×#417C2212yes—
35C22×#103C2212yes—
36C2×#407C2211yescharacteristic
37C22×C22⋊C8C2211yescharacteristic
38#2814—22no—
39#2638—22no—
40#2622—22no—
41#2621C2212yes—
42#2620C2212yes—
43#2579—22no—
44#2504C2211yescharacteristic
45#2502C2211yescharacteristic
46#2499C2212yes—
47#2436C2211yescharacteristic
48#2434C2211yescharacteristic
49#2432C2212yes—
50#2345—22no—
51#2345C2214yes—
52#2344—22no—
53#2344C2212yes—
54#2344C2212yes—
55#2316C2212yes—
56#2313—22no—
57#2280—22no—
58#2280C2211yescharacteristic
59#2280C2211yescharacteristic
60#2274C2212yes—
61#2085C2214yescomplemented
62#2084C2214yescomplemented
63#2068C2214yescomplemented
64#2066C2214yescomplemented
65#2066C2214yescomplemented
66#2065C2214yescomplemented
67#2043C2214yescomplemented
68#2042C2214yescomplemented
69#1983C2214yescomplemented
70#1981C2214yescomplemented
71#1949C2214yescomplemented
72#1948C2214yescomplemented
73#1928C2214yescomplemented
74#1927C2214yescomplemented
75#1927C2214yescomplemented
76#1926C2214yescomplemented
77#1894C2214yes—
78#1893C2212yes—
79#1892C2212yes—
80#1875C2212yes—
81#1873C2212yes—
82#1873C2212yes—
83#1873C2212yes—
84#1851C2214yes—
85#1850C2212yes—
86#1850C2212yes—
87#1825C2212yes—
88#1824C2212yes—
89#1798C2212yes—
90#1798C2212yes—
91#1788C2212yes—
92#1787C2212yes—
93C22×#41C2311yescharacteristic
94C23×Q8C2311yescharacteristic
95C24×C4—21no—
96C24×C4C2311yescharacteristic
97C22×M16C2312yes—
98C23×C8C2312yes—
99C2×C4⋊Q8C2312yes—
100C2×#124C2312yes—
101#525—24no—
102#525—24no—
103C2×#123—21no—
104C2×C22⋊Q8—21no—
105C2×C22⋊Q8—21no—
106C2×C22⋊Q8—24no—
107C2×C22⋊Q8—24no—
108C2×C22⋊Q8C2312yes—
109C2×C22⋊Q8C2311yescharacteristic
110C2×C22⋊Q8C2312yes—
111C2×C22⋊Q8C2311yescharacteristic
112C2×#121C2311yescharacteristic
113C2×#121C2311yescharacteristic
114C2×#120—21no—
115C2×C4×D4—21no—
116C2×C4×D4—21no—
117C2×#117—22no—
118C2×#117—22no—
119C2×#117C2311yescharacteristic
120C2×#117C2312yes—
121C2×#117C2312yes—
122C22×C4⋊C4—21no—
123C22×C4⋊C4—21no—
124C22×C4⋊C4—21no—
125C22×C4⋊C4—24no—
126C22×C4⋊C4—21no—
127C22×C4⋊C4D418yes—
128C22×C4⋊C4C2311yescharacteristic
129C22×C4⋊C4C2311yescharacteristic
130C22×C4⋊C4C2312yes—
131C22×C4⋊C4C2311yescharacteristic
132C22×C4⋊C4C2312yes—
133C22×C22⋊C4—21no—
134#484—24no—
135#484—24no—
136#483—216no—
137#483—28no—
138#483—28no—
139#483—28no—
140#481—24no—
141#481—24no—
142#480—28no—
143#428—28no—
144C2×#107C2312yes—
145C2×#107C2312yes—
146C2×#106—28no—
147C2×#106C2314yes—
148C2×#106C2314yes—
149C2×#106C2312yes—
150C2×#106C2312yes—
151C2×C4⋊C8C2314yes—
152C2×C4⋊C8C2314yes—
153#417—28no—
154C2×#103—28no—
155C2×#103C2314yes—
156C2×#103C2314yes—
157C2×#103C2312yes—
158C2×#103C2312yes—
159C2×#103C2314yes—
160C2×#103C2314yes—
161C2×#103C2314yes—
162C2×#102C2312yes—
163C2×#102C2312yes—
164C2×#102C2314yes—
165#407—24no—
166C2×C22⋊C8—24no—
167C2×C22⋊C8C2312yes—
168C2×C22⋊C8C2311yescharacteristic
169C2×C22⋊C8C2311yescharacteristic
170C2×C22⋊C8C2312yes—
171C2×C22⋊C8C2312yes—
172#398—24no—
173#398—22no—
174#397—24no—
175#397—24no—
176#397—22no—
177#396—24no—
178#395—22no—
179#394—24no—
180#394—24no—
181#393—22no—
182#391C2312yes—
183#391C2312yes—
184#391C2312yes—
185#390C2312yes—
186#389—24no—
187#389C2314yes—
188#389C2314yes—
189#387—24no—
190#387—24no—
191#387—24no—
192#387—24no—
193#387—24no—
194#386—22no—
195#386—22no—
196#385—22no—
197#385—22no—
198#385—24no—
199#384C2312yes—
200#382C2312yes—
201#381—21no—
202#381—22no—
203#381—22no—
204#381D412yes—
205#380—24no—
206#380—24no—
207#380D412yes—
208#380D412yes—
209#380C2312yes—
210#380C2312yes—
211#379—21no—
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—
229C25C2×D411yescharacteristic
230C2×#41—22no—
231C2×#41—22no—
232C2×#41—44no—
233C22×Q8—21no—
234C22×Q8—22no—
235C22×Q8—22no—
236C22×Q8—44no—
237C22×Q8C2412yes—
238C22×Q8C2411yescharacteristic
239C22×D4—21no—
240C22×D4—21no—
241C22×D4—21no—
242C22×D4C2411yescharacteristic
243C23×C4—22no—
244C23×C4—28no—
245C23×C4—21no—
246C23×C4—22no—
247C23×C4—21no—
248C23×C4—21no—
249C23×C4—21no—
250C23×C4—21no—
251C23×C4—44no—
252C23×C4—84no—
253C23×C4C2×D411yescharacteristic
254C23×C4C2×D411yescharacteristic
255C23×C4C2×D411yescharacteristic
256C23×C4C2×D414yes—
257C23×C4#4114yes—
258C23×C4C2411yescharacteristic
259C23×C4C2411yescharacteristic
260C23×C4C2411yescharacteristic
261C2×M16—24no—
262C2×M16—44no—
263C22×C8—24no—
264C22×C8—44no—
265C22×C8C2414yes—
266C22×C8C2412yes—
267C22×C8C2412yes—
268C4⋊Q8—44no—
269#124—44no—
270#123—44no—
271C22⋊Q8—24no—
272C22⋊Q8—24no—
273C22⋊Q8—22no—
274C22⋊Q8—22no—
275C22⋊Q8—24no—
276C22⋊Q8—24no—
277C22⋊Q8—22no—
278C22⋊Q8—22no—
279C22⋊Q8—44no—
280C22⋊Q8—44no—
281#121—22no—
282#121—22no—
283#121—22no—
284#121—22no—
285#120—44no—
286C4×D4—44no—
287C4×D4—44no—
288#117—22no—
289#117—22no—
290#117—44no—
291#117—44no—
292#117—84no—
293#117—84no—
294C2×C4⋊C4—28no—
295C2×C4⋊C4—28no—
296C2×C4⋊C4—28no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—24no—
302C2×C4⋊C4—24no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—21no—
311C2×C4⋊C4—21no—
312C2×C4⋊C4—24no—
313C2×C4⋊C4—24no—
314C2×C4⋊C4—24no—
315C2×C4⋊C4—24no—
316C2×C4⋊C4—44no—
317C2×C4⋊C4C2×D418yes—
318C2×C4⋊C4C2×D418yes—
319C2×C4⋊C4C2×D414yes—
320C2×C4⋊C4C2×D414yes—
321C2×C4⋊C4C2×D414yes—
322C2×C4⋊C4C2412yes—
323C2×C4⋊C4C2411yescharacteristic
324C2×C4⋊C4C2411yescharacteristic
325C2×C4⋊C4C2414yes—
326C2×C4⋊C4C2412yes—
327C2×C4⋊C4C2412yes—
328C2×C22⋊C4—22no—
329C2×C22⋊C4—22no—
330C2×C22⋊C4—22no—
331C2×C22⋊C4—21no—
332C2×C22⋊C4—22no—
333C2×C22⋊C4—21no—
334C2×C22⋊C4—21no—
335C2×C22⋊C4—22no—
336C2×C22⋊C4—21no—
337C2×C22⋊C4—21no—
338C2×C22⋊C4—24no—
339C2×C22⋊C4—24no—
340C2×C22⋊C4—44no—
341C2×C22⋊C4C2×D412yes—
342C2×C22⋊C4C2×D412yes—
343C2×C22⋊C4C2×D414yes—
344C2×C22⋊C4C2×D414yes—
345C2×C42—21no—
346C2×C42—22no—
347C2×C42—24no—
348C2×C42—24no—
349C2×C42C2412yes—
350C2×C42C2412yes—
351#107—24no—
352#107—24no—
353#106—28no—
354#106—28no—
355#106—24no—
356#106—24no—
357C4⋊C8—44no—
358C4⋊C8—44no—
359#103—28no—
360#103—28no—
361#103—24no—
362#103—24no—
363#103—44no—
364#103—44no—
365#103—44no—
366#102—24no—
367#102—24no—
368#102—44no—
369C22⋊C8—24no—
370C22⋊C8—24no—
371C22⋊C8—24no—
372C22⋊C8—22no—
373C22⋊C8—22no—
374#95—24no—
375#95—24no—
376#95—24no—
377#95—24no—
378#95—22no—
379#95—22no—
380#95—24no—
381#95—24no—
382#95—24no—
383#95—24no—
384#95C2×D414yes—
385#95C2×D412yes—
386#95C2×D412yes—
387C24—24no—
388C24—21no—
389C24—42no—
390C24—42no—
391C24#12114yes—
392C24C22×D411yescharacteristic
393C24C22×D411yescharacteristic
394C24C22×D411yescharacteristic
395#41—48no—
396#41—84no—
397C2×Q8—22no—
398C2×Q8—21no—
399C2×Q8—22no—
400C2×Q8—21no—
401C2×Q8—24no—
402C2×Q8—24no—
403C2×Q8—24no—
404C2×Q8—48no—
405C2×Q8—42no—
406C2×Q8—82no—
407C2×Q8—84no—
408C2×D4—21no—
409C2×D4—21no—
410C2×D4—24no—
411C2×D4—42no—
412C2×D4—42no—
413C2×D4—42no—
414C2×D4—44no—
415C2×D4—82no—
416C2×D4—82no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—28no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—28no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—21no—
427C22×C4—24no—
428C22×C4—21no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—21no—
432C22×C4—48no—
433C22×C4—48no—
434C22×C4—42no—
435C22×C4—42no—
436C22×C4—42no—
437C22×C4—44no—
438C22×C4—44no—
439C22×C4—42no—
440C22×C4—42no—
441C22×C4—42no—
442C22×C4—82no—
443C22×C4—82no—
444C22×C4—816no—
445C22×C4—82no—
446C22×C4—82no—
447C22×C4—82no—
448C22×C4—82no—
449C22×C4—82no—
450C22×C4#12114yes—
451C22×C4#12114yes—
452C22×C4#12114yes—
453C22×C4C22×D414yes—
454C22×C4C22×D414yes—
455C22×C4C22×D411yescharacteristic
456C22×C4C22×D411yescharacteristic
457C22×C4C22×D411yescharacteristic
458C22×C4C22×D412yes—
459C22×C4C22×D412yes—
460C22×C4C22×D412yes—
461C22×C4C2×#4112yes—
462C22×C4C2×#4112yes—
463C22×C4C2×#4112yes—
464C22×C4C2511yescharacteristic
465M16—48no—
466C2×C8—22no—
467C2×C8—22no—
468C2×C8—24no—
469C2×C8—24no—
470C2×C8—24no—
471C2×C8—24no—
472C2×C8—24no—
473C2×C8—48no—
474C4⋊C4—28no—
475C4⋊C4—24no—
476C4⋊C4—24no—
477C4⋊C4—28no—
478C4⋊C4—22no—
479C4⋊C4—22no—
480C4⋊C4—22no—
481C4⋊C4—21no—
482C4⋊C4—21no—
483C4⋊C4—22no—
484C4⋊C4—21no—
485C4⋊C4—21no—
486C4⋊C4—24no—
487C4⋊C4—24no—
488C4⋊C4—44no—
489C4⋊C4—42no—
490C4⋊C4—44no—
491C4⋊C4—44no—
492C4⋊C4—42no—
493C4⋊C4—42no—
494C4⋊C4—84no—
495C4⋊C4—84no—
496C4⋊C4—84no—
497C4⋊C4—84no—
498C4⋊C4C22×D414yes—
499C4⋊C4C22×D414yes—
500C22⋊C4—24no—
501C22⋊C4—24no—
502C22⋊C4—44no—
503C22⋊C4—44no—
504C22⋊C4—44no—
505C22⋊C4—44no—
506C22⋊C4—44no—
507C22⋊C4—44no—
508C22⋊C4—42no—
509C22⋊C4—42no—
510C22⋊C4—42no—
511C22⋊C4—44no—
512C22⋊C4—42no—
513C22⋊C4—42no—
514C22⋊C4—84no—
515C22⋊C4—84no—
516C42—24no—
517C42—24no—
518C42—44no—
519C42—42no—
520C42—84no—
521C42—84no—
522C23—22no—
523C23—22no—
524C23—22no—
525C23—22no—
526C23—22no—
527C23—22no—
528C23—28no—
529C23—22no—
530C23—22no—
531C23—44no—
532C23—44no—
533C23—416no—
534C23—42no—
535C23—41no—
536C23—42no—
537C23—42no—
538C23C2×#12112yes—
539C23C2×#12112yes—
540C23C2×#12112yes—
541C23#57714yes—
542C23C23×D411yescharacteristic
543Q8—44no—
544Q8—42no—
545Q8—44no—
546Q8—42no—
547Q8—84no—
548D4—42no—
549D4—42no—
550D4—84no—
551D4—84no—
552D4—84no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—24no—
557C2×C4—24no—
558C2×C4—24no—
559C2×C4—24no—
560C2×C4—24no—
561C2×C4—24no—
562C2×C4—24no—
563C2×C4—24no—
564C2×C4—24no—
565C2×C4—24no—
566C2×C4—24no—
567C2×C4—24no—
568C2×C4—24no—
569C2×C4—24no—
570C2×C4—44no—
571C2×C4—44no—
572C2×C4—44no—
573C2×C4—416no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—44no—
579C2×C4—44no—
580C2×C4—42no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—42no—
584C2×C4—41no—
585C2×C4—42no—
586C2×C4—44no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—44no—
591C2×C4—42no—
592C2×C4—84no—
593C2×C4—84no—
594C2×C4—88no—
595C2×C4—84no—
596C2×C4—84no—
597C2×C4—84no—
598C2×C4C2×#12112yes—
599C2×C4C2×#12112yes—
600C2×C4C2×#12112yes—
601C2×C4C23×D411yescharacteristic
602C2×C4C22×#4111yescharacteristic
603C2×C4#58514yes—
604C8—44no—
605C8—42no—
606C8—42no—
607C22—22no—
608C22—22no—
609C22—28no—
610C22—22no—
611C22—22no—
612C22—22no—
613C22—22no—
614C22—22no—
615C22—22no—
616C22—416no—
617C22—44no—
618C22—44no—
619C22—81no—
620C22—81no—
621C22—81no—
622C22—84no—
623C22#304214yes—
624C22C22×#12111yescharacteristic
625C22#348512yes—
626C22C2×#57712yes—
627C22#358612yes—
628C4—22no—
629C4—22no—
630C4—22no—
631C4—44no—
632C4—44no—
633C4—44no—
634C4—44no—
635C4—81no—
636C4—82no—
637C4—81no—
638C4—81no—
639C4—82no—
640C4—84no—
641C4#344912yes—
642C4#349112yes—
643C2—24no—
644C2—42no—
645C2—42no—
646C2—161no—
647C2#3508911yescharacteristic
648C2C2×#304211yescharacteristic
649C2#6042411yescharacteristic
650C2#6174514yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#525C22
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
2. C22×#483C2semidirect
4. #34178C2semidirect
5. #33121C2semidirect
6. #33119C2semidirect
7. #29768C2semidirect
8. #29764C2semidirect
9. #24590C2semidirect
10. #24581C2semidirect
11. #24580C2semidirect
12. #24491C2semidirect
13. #24481C2semidirect
14. #24480C2semidirect
15. #21933C2semidirect
16. #21768C2semidirect
61. #2085C22semidirect
62. #2084C22semidirect
63. #2068C22semidirect
64. #2066C22semidirect
65. #2066C22semidirect
66. #2065C22semidirect
67. #2043C22semidirect
68. #2042C22semidirect
69. #1983C22semidirect
70. #1981C22semidirect
71. #1949C22semidirect
72. #1948C22semidirect
73. #1928C22semidirect
74. #1927C22semidirect
75. #1927C22semidirect
76. #1926C22semidirect

Maximal subgroups

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

subgroupindex
C2×#30922
C22×#4832
#374492
#341782
#331212
#331192
#297682
#297642
#245902
#245812
#245802
#244912
#244812
#244802
#219332
#217682
#217602