Finite Group Databasebeta

← #6477640

Subgroups — #6477640

Order 512.

1947 conjugacy classes of subgroups.

The subgroup lattice has 647 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3092C211yesmaximal, characteristic
2C22×#484C211yesmaximal, characteristic, complemented
3#37449C211yesmaximal, characteristic
4#34178C211yesmaximal, characteristic, complemented
5#33120C212yesmaximal, complemented
6#33119C212yesmaximal, complemented
7#29767C212yesmaximal, complemented
8#29764C212yesmaximal, complemented
9#24588C214yesmaximal, complemented
10#24583C212yesmaximal, complemented
11#24582C212yesmaximal, complemented
12#24489C214yesmaximal, complemented
13#24483C212yesmaximal, complemented
14#24482C212yesmaximal, complemented
15#21933C211yesmaximal, characteristic, complemented
16#21768C211yesmaximal, characteristic, complemented
17#21760C211yesmaximal, characteristic
18C2×#525C2211yescharacteristic
19C22×C22⋊Q8C2211yescharacteristic
20C23×C4⋊C4C2211yescharacteristic
21#3092—28no—
22C2×#484—28no—
23C2×#484C2212yes—
24C2×#484C2214yes—
25C2×#484C2212yes—
26C2×#484C2212yes—
27C2×#481C2212yes—
28C2×#483C2211yescharacteristic
29C2×#483C2211yescharacteristic
30C2×#480C2211yescharacteristic
31C2×#480C2211yescharacteristic
32C2×#428C2212yes—
33C22×#107C2212yes—
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#2345C2212yes—
52#2345C2212yes—
53#2344—22no—
54#2344C2214yes—
55#2316C2212yes—
56#2313—22no—
57#2280—22no—
58#2280C2211yescharacteristic
59#2280C2211yescharacteristic
60#2274C2212yes—
61#2085C2214yescomplemented
62#2083C2214yescomplemented
63#2068C2214yescomplemented
64#2068C2214yescomplemented
65#2067C2214yescomplemented
66#2066C2214yescomplemented
67#2043C2214yescomplemented
68#2042C2214yescomplemented
69#1983C2214yescomplemented
70#1981C2214yescomplemented
71#1949C2214yescomplemented
72#1947C2214yescomplemented
73#1928C2214yescomplemented
74#1928C2214yescomplemented
75#1927C2214yescomplemented
76#1925C2214yescomplemented
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⋊Q8C2311yescharacteristic
109C2×C22⋊Q8C2312yes—
110C2×C22⋊Q8C2311yescharacteristic
111C2×C22⋊Q8C2312yes—
112C2×#121C2311yescharacteristic
113C2×#121C2311yescharacteristic
114C2×#120—21no—
115C2×C4×D4—21no—
116C2×C4×D4—21no—
117C2×#117—24no—
118C2×#117C2311yescharacteristic
119C2×#117C2312yes—
120C2×#117C2312yes—
121C22×C4⋊C4—21no—
122C22×C4⋊C4—21no—
123C22×C4⋊C4—21no—
124C22×C4⋊C4—24no—
125C22×C4⋊C4—21no—
126C22×C4⋊C4D418yes—
127C22×C4⋊C4C2311yescharacteristic
128C22×C4⋊C4C2311yescharacteristic
129C22×C4⋊C4C2311yescharacteristic
130C22×C4⋊C4C2312yes—
131C22×C4⋊C4C2312yes—
132C22×C22⋊C4—21no—
133#484—216no—
134#484—28no—
135#484—28no—
136#484—28no—
137#483—24no—
138#483—24no—
139#481—28no—
140#480—24no—
141#480—24no—
142#428—28no—
143C2×#107—28no—
144C2×#107C2312yes—
145C2×#107C2312yes—
146C2×#107C2314yes—
147C2×#107C2314yes—
148C2×#106C2312yes—
149C2×#106C2312yes—
150C2×C4⋊C8C2314yes—
151C2×C4⋊C8C2314yes—
152#417—28no—
153C2×#103—28no—
154C2×#103C2312yes—
155C2×#103C2312yes—
156C2×#103C2314yes—
157C2×#103C2314yes—
158C2×#103C2314yes—
159C2×#103C2314yes—
160C2×#103C2314yes—
161C2×#102C2312yes—
162C2×#102C2312yes—
163C2×#102C2314yes—
164#407—24no—
165C2×C22⋊C8—24no—
166C2×C22⋊C8C2311yescharacteristic
167C2×C22⋊C8C2311yescharacteristic
168C2×C22⋊C8C2312yes—
169C2×C22⋊C8C2312yes—
170C2×C22⋊C8C2312yes—
171#398—24no—
172#398—22no—
173#397—24no—
174#397—24no—
175#397—22no—
176#396—24no—
177#395—22no—
178#394—24no—
179#394—24no—
180#393—22no—
181#391C2312yes—
182#391C2312yes—
183#391C2312yes—
184#390C2312yes—
185#389—24no—
186#389C2314yes—
187#389C2314yes—
188#387—24no—
189#387—24no—
190#387—24no—
191#387—24no—
192#387—24no—
193#386—22no—
194#386—22no—
195#385—22no—
196#385—22no—
197#385—24no—
198#384C2312yes—
199#382C2312yes—
200#381—21no—
201#381—24no—
202#381D412yes—
203#380—24no—
204#380—22no—
205#380—22no—
206#380D412yes—
207#380D412yes—
208#380C2312yes—
209#380C2312yes—
210#379—21no—
211#379D412yes—
212C4×C4⋊C4—24no—
213C4×C4⋊C4C2314yes—
214C4×C4⋊C4C2314yes—
215C4×C22⋊C4—21no—
216C4×C22⋊C4—21no—
217C4×C22⋊C4—24no—
218C4×C22⋊C4—22no—
219C4×C22⋊C4—22no—
220C4×C22⋊C4C2312yes—
221C4×C22⋊C4C2311yescharacteristic
222C4×C22⋊C4C2311yescharacteristic
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—88no—
293C2×C4⋊C4—28no—
294C2×C4⋊C4—28no—
295C2×C4⋊C4—28no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—24no—
301C2×C4⋊C4—24no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—21no—
311C2×C4⋊C4—24no—
312C2×C4⋊C4—24no—
313C2×C4⋊C4—24no—
314C2×C4⋊C4—24no—
315C2×C4⋊C4—44no—
316C2×C4⋊C4C2×D418yes—
317C2×C4⋊C4C2×D418yes—
318C2×C4⋊C4C2×D414yes—
319C2×C4⋊C4C2×D414yes—
320C2×C4⋊C4C2×D414yes—
321C2×C4⋊C4C2412yes—
322C2×C4⋊C4C2411yescharacteristic
323C2×C4⋊C4C2411yescharacteristic
324C2×C4⋊C4C2414yes—
325C2×C4⋊C4C2412yes—
326C2×C4⋊C4C2412yes—
327C2×C22⋊C4—22no—
328C2×C22⋊C4—22no—
329C2×C22⋊C4—22no—
330C2×C22⋊C4—21no—
331C2×C22⋊C4—22no—
332C2×C22⋊C4—21no—
333C2×C22⋊C4—21no—
334C2×C22⋊C4—22no—
335C2×C22⋊C4—21no—
336C2×C22⋊C4—21no—
337C2×C22⋊C4—24no—
338C2×C22⋊C4—24no—
339C2×C22⋊C4—44no—
340C2×C22⋊C4C2×D412yes—
341C2×C22⋊C4C2×D412yes—
342C2×C22⋊C4C2×D414yes—
343C2×C22⋊C4C2×D414yes—
344C2×C42—21no—
345C2×C42—22no—
346C2×C42—24no—
347C2×C42—24no—
348C2×C42C2412yes—
349C2×C42C2412yes—
350#107—28no—
351#107—28no—
352#107—24no—
353#107—24no—
354#106—24no—
355#106—24no—
356C4⋊C8—44no—
357C4⋊C8—44no—
358#103—28no—
359#103—28no—
360#103—24no—
361#103—24no—
362#103—44no—
363#103—44no—
364#103—44no—
365#102—24no—
366#102—24no—
367#102—44no—
368C22⋊C8—24no—
369C22⋊C8—24no—
370C22⋊C8—22no—
371C22⋊C8—22no—
372C22⋊C8—24no—
373#95—24no—
374#95—24no—
375#95—24no—
376#95—24no—
377#95—22no—
378#95—22no—
379#95—24no—
380#95—24no—
381#95—24no—
382#95—24no—
383#95C2×D414yes—
384#95C2×D412yes—
385#95C2×D412yes—
386C24—24no—
387C24—21no—
388C24—42no—
389C24—42no—
390C24#12114yes—
391C24C22×D411yescharacteristic
392C24C22×D411yescharacteristic
393C24C22×D411yescharacteristic
394#41—48no—
395#41—84no—
396C2×Q8—22no—
397C2×Q8—21no—
398C2×Q8—22no—
399C2×Q8—21no—
400C2×Q8—24no—
401C2×Q8—24no—
402C2×Q8—24no—
403C2×Q8—48no—
404C2×Q8—42no—
405C2×Q8—82no—
406C2×Q8—84no—
407C2×D4—21no—
408C2×D4—21no—
409C2×D4—24no—
410C2×D4—42no—
411C2×D4—42no—
412C2×D4—42no—
413C2×D4—44no—
414C2×D4—82no—
415C2×D4—82no—
416C22×C4—24no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—28no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—24no—
423C22×C4—24no—
424C22×C4—28no—
425C22×C4—21no—
426C22×C4—24no—
427C22×C4—21no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—21no—
431C22×C4—48no—
432C22×C4—48no—
433C22×C4—42no—
434C22×C4—42no—
435C22×C4—42no—
436C22×C4—48no—
437C22×C4—42no—
438C22×C4—42no—
439C22×C4—42no—
440C22×C4—82no—
441C22×C4—82no—
442C22×C4—82no—
443C22×C4—84no—
444C22×C4—82no—
445C22×C4—82no—
446C22×C4—816no—
447C22×C4#12114yes—
448C22×C4#12114yes—
449C22×C4#12114yes—
450C22×C4C22×D414yes—
451C22×C4C22×D414yes—
452C22×C4C22×D411yescharacteristic
453C22×C4C22×D411yescharacteristic
454C22×C4C22×D411yescharacteristic
455C22×C4C22×D412yes—
456C22×C4C22×D412yes—
457C22×C4C22×D412yes—
458C22×C4C2×#4112yes—
459C22×C4C2×#4112yes—
460C22×C4C2×#4112yes—
461C22×C4C2511yescharacteristic
462M16—48no—
463C2×C8—22no—
464C2×C8—22no—
465C2×C8—24no—
466C2×C8—24no—
467C2×C8—24no—
468C2×C8—24no—
469C2×C8—24no—
470C2×C8—48no—
471C4⋊C4—28no—
472C4⋊C4—24no—
473C4⋊C4—24no—
474C4⋊C4—28no—
475C4⋊C4—22no—
476C4⋊C4—22no—
477C4⋊C4—21no—
478C4⋊C4—22no—
479C4⋊C4—21no—
480C4⋊C4—22no—
481C4⋊C4—21no—
482C4⋊C4—21no—
483C4⋊C4—24no—
484C4⋊C4—24no—
485C4⋊C4—44no—
486C4⋊C4—42no—
487C4⋊C4—44no—
488C4⋊C4—44no—
489C4⋊C4—42no—
490C4⋊C4—42no—
491C4⋊C4—84no—
492C4⋊C4—84no—
493C4⋊C4—84no—
494C4⋊C4—84no—
495C4⋊C4C22×D414yes—
496C4⋊C4C22×D414yes—
497C22⋊C4—24no—
498C22⋊C4—24no—
499C22⋊C4—44no—
500C22⋊C4—44no—
501C22⋊C4—44no—
502C22⋊C4—44no—
503C22⋊C4—44no—
504C22⋊C4—44no—
505C22⋊C4—42no—
506C22⋊C4—42no—
507C22⋊C4—42no—
508C22⋊C4—44no—
509C22⋊C4—42no—
510C22⋊C4—42no—
511C22⋊C4—84no—
512C22⋊C4—84no—
513C42—24no—
514C42—24no—
515C42—44no—
516C42—42no—
517C42—84no—
518C42—84no—
519C23—22no—
520C23—22no—
521C23—22no—
522C23—22no—
523C23—22no—
524C23—22no—
525C23—28no—
526C23—22no—
527C23—22no—
528C23—44no—
529C23—44no—
530C23—416no—
531C23—42no—
532C23—41no—
533C23—42no—
534C23—42no—
535C23C2×#12112yes—
536C23C2×#12112yes—
537C23C2×#12112yes—
538C23#57814yes—
539C23C23×D411yescharacteristic
540Q8—44no—
541Q8—42no—
542Q8—44no—
543Q8—42no—
544Q8—84no—
545D4—42no—
546D4—42no—
547D4—84no—
548D4—84no—
549D4—84no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—24no—
554C2×C4—24no—
555C2×C4—24no—
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—44no—
568C2×C4—44no—
569C2×C4—44no—
570C2×C4—416no—
571C2×C4—44no—
572C2×C4—44no—
573C2×C4—44no—
574C2×C4—44no—
575C2×C4—42no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—42no—
579C2×C4—41no—
580C2×C4—42no—
581C2×C4—44no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—44no—
586C2×C4—42no—
587C2×C4—84no—
588C2×C4—88no—
589C2×C4—84no—
590C2×C4—84no—
591C2×C4—84no—
592C2×C4—84no—
593C2×C4C2×#12112yes—
594C2×C4C2×#12112yes—
595C2×C4C2×#12112yes—
596C2×C4C23×D411yescharacteristic
597C2×C4C22×#4111yescharacteristic
598C2×C4#58514yes—
599C8—44no—
600C8—42no—
601C8—42no—
602C22—22no—
603C22—22no—
604C22—28no—
605C22—22no—
606C22—22no—
607C22—22no—
608C22—22no—
609C22—22no—
610C22—22no—
611C22—416no—
612C22—44no—
613C22—44no—
614C22—81no—
615C22—81no—
616C22—81no—
617C22—84no—
618C22#304414yes—
619C22C22×#12111yescharacteristic
620C22#348512yes—
621C22C2×#57812yes—
622C22#358512yes—
623C4—22no—
624C4—22no—
625C4—22no—
626C4—44no—
627C4—44no—
628C4—44no—
629C4—44no—
630C4—81no—
631C4—82no—
632C4—81no—
633C4—81no—
634C4—82no—
635C4—84no—
636C4#344912yes—
637C4#349112yes—
638C2—24no—
639C2—42no—
640C2—42no—
641C2—161no—
642C2#3508911yescharacteristic
643C2C2×#304411yescharacteristic
644C2#6042311yescharacteristic
645C2#6174714yes—
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×#484C2semidirect
4. #34178C2semidirect
5. #33120C2semidirect
6. #33119C2semidirect
7. #29767C2semidirect
8. #29764C2semidirect
9. #24588C2semidirect
10. #24583C2semidirect
11. #24582C2semidirect
12. #24489C2semidirect
13. #24483C2semidirect
14. #24482C2semidirect
15. #21933C2semidirect
16. #21768C2semidirect
61. #2085C22semidirect
62. #2083C22semidirect
63. #2068C22semidirect
64. #2068C22semidirect
65. #2067C22semidirect
66. #2066C22semidirect
67. #2043C22semidirect
68. #2042C22semidirect
69. #1983C22semidirect
70. #1981C22semidirect
71. #1949C22semidirect
72. #1947C22semidirect
73. #1928C22semidirect
74. #1928C22semidirect
75. #1927C22semidirect
76. #1925C22semidirect

Maximal subgroups

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

subgroupindex
C2×#30922
C22×#4842
#374492
#341782
#331202
#331192
#297672
#297642
#245882
#245832
#245822
#244892
#244832
#244822
#219332
#217682
#217602