Finite Group Databasebeta

← #6508853

Subgroups — #6508853

Order 512.

1910 conjugacy classes of subgroups.

The subgroup lattice has 680 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#485C211yesmaximal, characteristic, complemented
2C2×#2976C211yesmaximal, characteristic
3#37442C211yesmaximal, characteristic
4#35263C211yesmaximal, characteristic
5#28108C212yesmaximal, complemented
6#28098C212yesmaximal, complemented
7#27667C212yesmaximal, complemented
8#27658C212yesmaximal, complemented
9#25259C214yesmaximal, complemented
10#25247C214yesmaximal, complemented
11#24619C214yesmaximal, complemented
12#24569C214yesmaximal, complemented
13#21938C211yesmaximal, characteristic, complemented
14#21765C211yesmaximal, characteristic, complemented
15#20702C211yesmaximal, characteristic, complemented
16C2×#525C2211yescharacteristic
17C22×C22⋊Q8C2211yescharacteristic
18C22×#117C2211yescharacteristic
19C2×#485—28no—
20C2×#485C2214yes—
21C2×#485C2214yes—
22#2976—28no—
23C2×#436C2212yes—
24C2×#435C2212yes—
25C2×#435C2212yes—
26C2×C8×D4C2212yes—
27C2×#433C2212yes—
28C22×#107C2211yescharacteristic
29C22×#106C2211yescharacteristic
30C22×#103C2211yescharacteristic
31C22×#103C2211yescharacteristic
32C2×#408C2212yes—
33C2×#407C2211yescharacteristic
34C22×C22⋊C8C2211yescharacteristic
35#2855—21no—
36#2854—21no—
37#2814—22no—
38#2659—21no—
39#2639—21no—
40#2627—21no—
41#2625—21no—
42#2625C2212yes—
43#2624—21no—
44#2618—21no—
45#2616C2212yes—
46#2498—22no—
47#2487C2212yes—
48#2486C2212yes—
49#2486C2212yes—
50#2485C2214yes—
51#2417C2214yes—
52#2412—22no—
53#2412—22no—
54#2409C2212yes—
55#2316C2212yes—
56#2311C2212yes—
57C2×#380C2211yescharacteristic
58C2×#380C2211yescharacteristic
59#2088C2214yescomplemented
60#2087C2214yescomplemented
61#2077C2214yescomplemented
62#2076C2214yescomplemented
63#2066C2214yescomplemented
64#2066C2214yescomplemented
65#2035C2214yescomplemented
66#2035C2214yescomplemented
67#1944C2214yescomplemented
68#1941C2214yescomplemented
69#1935C2214yescomplemented
70#1933C2214yescomplemented
71#1894C2212yes—
72#1894C2212yes—
73#1894C2214yescomplemented
74#1892C2214yescomplemented
75#1872C2214yes—
76#1853C2214yes—
77#1853C2214yes—
78#1825C2214yes—
79#1824C2212yes—
80#1823C2212yes—
81#1823C2212yes—
82#1809C2214yescomplemented
83#1807C2212yes—
84#1807C2212yes—
85#1807C2214yescomplemented
86#1787C2212yes—
87#1736C2212yes—
88#1736C2212yes—
89C22×#41C2311yescharacteristic
90C23×Q8C2311yescharacteristic
91C24×C4C2311yescharacteristic
92C22×M16C2312yes—
93C22×M16C2311yescharacteristic
94C22×M16C2311yescharacteristic
95C23×C8C2312yes—
96C23×C8C2311yescharacteristic
97C23×C8C2311yescharacteristic
98#525—24no—
99#525—24no—
100C2×#123—22no—
101C2×#123—22no—
102C2×#123D412yes—
103C2×C22⋊Q8—21no—
104C2×C22⋊Q8—21no—
105C2×C22⋊Q8—24no—
106C2×C22⋊Q8—24no—
107C2×C22⋊Q8D411yescharacteristic
108C2×C22⋊Q8D411yescharacteristic
109C2×C22⋊Q8C2312yes—
110C2×C22⋊Q8C2312yes—
111C2×C22⋊Q8C2314yes—
112C2×#121D411yescharacteristic
113C2×#121D411yescharacteristic
114C2×#120D412yes—
115C2×C4×D4C2312yes—
116C2×C4×D4C2312yes—
117C2×#117—21no—
118C2×#117—21no—
119C2×#117—24no—
120C2×#117—24no—
121C2×#117C2312yes—
122C2×#117C2312yes—
123C2×#117C2311yescharacteristic
124C22×C4⋊C4—21no—
125C22×C4⋊C4C2311yescharacteristic
126C22×C4⋊C4C2311yescharacteristic
127C22×C4⋊C4C2311yescharacteristic
128C22×C4⋊C4C2312yes—
129C22×C4⋊C4C2312yes—
130C22×C22⋊C4—21no—
131C22×C22⋊C4—21no—
132C22×C22⋊C4—21no—
133C22×C22⋊C4—21no—
134C22×C42—21no—
135#485—216no—
136#485—216no—
137#436—28no—
138#435—28no—
139#435—28no—
140C8×D4—28no—
141#433—28no—
142C2×#107—24no—
143C2×#107C2314yes—
144C2×#106—24no—
145C2×#106C2314yes—
146C2×#106C2314yes—
147C2×#106C2314yes—
148C2×C4⋊C8C2312yes—
149C2×C4⋊C8C2312yes—
150C2×#103—24no—
151C2×#103—24no—
152C2×#103C2314yes—
153C2×#103C2314yes—
154C2×#103C2314yes—
155C2×#103C2314yes—
156#408—28no—
157#407—24no—
158C2×C22⋊C8D418yes—
159C2×C22⋊C8C2312yes—
160C2×C22⋊C8C2312yes—
161C2×C22⋊C8C2312yes—
162C2×C22⋊C8C2312yes—
163C2×C22⋊C8C2312yes—
164C2×C22⋊C8C2312yes—
165C2×#97C2312yes—
166C2×C4×C8C2312yes—
167#400—22no—
168#400—22no—
169#399—22no—
170#398—22no—
171#398—22no—
172#398—24no—
173#398—22no—
174#397—22no—
175#397—24no—
176#397—24no—
177#397—22no—
178#396—24no—
179#395—22no—
180#395—22no—
181#394—22no—
182#394—24no—
183#394—24no—
184#393—22no—
185#391—22no—
186#391C2314yes—
187#390—22no—
188#389—22no—
189#389—22no—
190#389C2314yes—
191#389C2314yes—
192#389C2314yes—
193#388—24no—
194#387—24no—
195#386—24no—
196#386—22no—
197#386—22no—
198#386—24no—
199#385—22no—
200#385—22no—
201#381—22no—
202#381—22no—
203#380—21no—
204#380—21no—
205#380—21no—
206#380—21no—
207#380—24no—
208#380—24no—
209#380C2314yes—
210#380C2314yes—
211#380C2312yes—
212#380C2312yes—
213#380C2314yes—
214#380C2312yes—
215#380C2312yes—
216C2×#95—21no—
217C2×#95—21no—
218#340C2314yes—
219#340C2314yes—
220#340C2314yes—
221#340C2314yes—
222#340C2314yes—
223#340C2314yes—
224#340C2314yes—
225#340C2314yes—
226C25C2×D411yescharacteristic
227C2×#41—22no—
228C2×#41—22no—
229C2×#41—42no—
230C2×#41—42no—
231C22×Q8—21no—
232C22×Q8—22no—
233C22×Q8—22no—
234C22×Q8—42no—
235C22×Q8—42no—
236C22×Q8C2×D411yescharacteristic
237C22×Q8C2412yes—
238C22×D4C2×D412yes—
239C22×D4C2×D412yes—
240C22×D4C2×D412yes—
241C22×D4C2×D411yescharacteristic
242C23×C4—21no—
243C23×C4—21no—
244C23×C4—21no—
245C23×C4—22no—
246C23×C4—22no—
247C23×C4—21no—
248C23×C4—42no—
249C23×C4—42no—
250C23×C4C2×D411yescharacteristic
251C23×C4C2×D411yescharacteristic
252C23×C4C2×D411yescharacteristic
253C23×C4C2×D414yes—
254C23×C4#4114yes—
255C23×C4C2412yes—
256C23×C4C2411yescharacteristic
257C23×C4C2411yescharacteristic
258C23×C4C2411yescharacteristic
259C2×M16—22no—
260C2×M16—22no—
261C2×M16—24no—
262C2×M16—44no—
263C2×M16—42no—
264C2×M16—42no—
265C22×C8—24no—
266C22×C8—42no—
267C22×C8—42no—
268C22×C8—44no—
269C22×C8C2×D414yes—
270C22×C8C2×D414yes—
271C22×C8C2412yes—
272C22×C8C2412yes—
273C22×C8C2412yes—
274C22×C8C2412yes—
275#123—28no—
276#123—84no—
277#123—84no—
278C22⋊Q8—24no—
279C22⋊Q8—24no—
280C22⋊Q8—24no—
281C22⋊Q8—24no—
282C22⋊Q8—24no—
283C22⋊Q8—24no—
284C22⋊Q8—44no—
285C22⋊Q8—44no—
286C22⋊Q8—44no—
287C22⋊Q8—44no—
288#121—24no—
289#121—24no—
290#120—28no—
291C4×D4—24no—
292C4×D4—24no—
293C4×D4—24no—
294C4×D4—24no—
295#117—22no—
296#117—22no—
297#117—24no—
298#117—24no—
299#117—24no—
300#117—24no—
301#117—44no—
302#117—44no—
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—22no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—21no—
312C2×C4⋊C4—21no—
313C2×C4⋊C4—21no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—22no—
316C2×C4⋊C4—24no—
317C2×C4⋊C4—24no—
318C2×C4⋊C4—24no—
319C2×C4⋊C4—24no—
320C2×C4⋊C4—42no—
321C2×C4⋊C4—42no—
322C2×C4⋊C4—42no—
323C2×C4⋊C4—42no—
324C2×C4⋊C4—44no—
325C2×C4⋊C4C2×D411yescharacteristic
326C2×C4⋊C4C2×D411yescharacteristic
327C2×C4⋊C4C2×D412yes—
328C2×C4⋊C4C2×D412yes—
329C2×C4⋊C4C2412yes—
330C2×C4⋊C4C2412yes—
331C2×C4⋊C4C2412yes—
332C2×C4⋊C4C2414yes—
333C2×C4⋊C4C2414yes—
334C2×C22⋊C4—22no—
335C2×C22⋊C4—22no—
336C2×C22⋊C4—22no—
337C2×C22⋊C4—22no—
338C2×C22⋊C4—22no—
339C2×C22⋊C4—22no—
340C2×C22⋊C4—22no—
341C2×C22⋊C4—22no—
342C2×C22⋊C4—24no—
343C2×C22⋊C4—24no—
344C2×C22⋊C4—24no—
345C2×C22⋊C4—24no—
346C2×C22⋊C4—44no—
347C2×C22⋊C4—44no—
348C2×C22⋊C4—82no—
349C2×C22⋊C4—82no—
350C2×C22⋊C4C2×D412yes—
351C2×C22⋊C4C2×D412yes—
352C2×C22⋊C4C2×D412yes—
353C2×C22⋊C4C2×D412yes—
354C2×C22⋊C4C2×D412yes—
355C2×C42—22no—
356C2×C42—22no—
357C2×C42—44no—
358C2×C42C2412yes—
359#107—28no—
360#106—28no—
361#106—44no—
362#106—44no—
363C4⋊C8—24no—
364C4⋊C8—24no—
365#103—28no—
366#103—28no—
367#103—44no—
368#103—44no—
369C22⋊C8—24no—
370C22⋊C8—24no—
371C22⋊C8—24no—
372C22⋊C8—24no—
373C22⋊C8C2×D418yes—
374C22⋊C8C2×D418yes—
375#97—24no—
376C4×C8—24no—
377#95—22no—
378#95—22no—
379#95—22no—
380#95—22no—
381#95—24no—
382#95—24no—
383#95—24no—
384#95—24no—
385#95—24no—
386#95—24no—
387#95—24no—
388#95—24no—
389C24—22no—
390C24—22no—
391C24—42no—
392C24—42no—
393C24#12114yes—
394C24C22×D412yes—
395C24C22×D411yescharacteristic
396C24C22×D411yescharacteristic
397C24C22×D411yescharacteristic
398#41—82no—
399#41—82no—
400#41—84no—
401C2×Q8—22no—
402C2×Q8—22no—
403C2×Q8—22no—
404C2×Q8—24no—
405C2×Q8—24no—
406C2×Q8—42no—
407C2×Q8—44no—
408C2×Q8—44no—
409C2×Q8—42no—
410C2×Q8—82no—
411C2×Q8—84no—
412C2×D4—24no—
413C2×D4—24no—
414C2×D4—24no—
415C2×D4—24no—
416C2×D4—24no—
417C2×D4—22no—
418C2×D4—44no—
419C2×D4—42no—
420C2×D4—44no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—24no—
430C22×C4—24no—
431C22×C4—24no—
432C22×C4—24no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—44no—
438C22×C4—44no—
439C22×C4—48no—
440C22×C4—42no—
441C22×C4—42no—
442C22×C4—42no—
443C22×C4—42no—
444C22×C4—42no—
445C22×C4—42no—
446C22×C4—42no—
447C22×C4—42no—
448C22×C4—42no—
449C22×C4—42no—
450C22×C4—42no—
451C22×C4—44no—
452C22×C4—44no—
453C22×C4—82no—
454C22×C4—82no—
455C22×C4—82no—
456C22×C4—82no—
457C22×C4—82no—
458C22×C4—82no—
459C22×C4#12114yes—
460C22×C4#12114yes—
461C22×C4#12114yes—
462C22×C4C22×D412yes—
463C22×C4C22×D412yes—
464C22×C4C22×D412yes—
465C22×C4C22×D411yescharacteristic
466C22×C4C22×D411yescharacteristic
467C22×C4C22×D411yescharacteristic
468C22×C4C22×D412yes—
469C22×C4C22×D412yes—
470C22×C4C22×D412yes—
471C22×C4C2×#4112yes—
472C22×C4C2×#4112yes—
473C22×C4C2×#4112yes—
474C22×C4C2511yescharacteristic
475M16—84no—
476M16—82no—
477M16—82no—
478C2×C8—24no—
479C2×C8—24no—
480C2×C8—22no—
481C2×C8—22no—
482C2×C8—24no—
483C2×C8—24no—
484C2×C8—44no—
485C2×C8—44no—
486C2×C8—84no—
487C2×C8C22×D414yes—
488C2×C8C22×D414yes—
489C4⋊C4—22no—
490C4⋊C4—22no—
491C4⋊C4—24no—
492C4⋊C4—24no—
493C4⋊C4—22no—
494C4⋊C4—22no—
495C4⋊C4—22no—
496C4⋊C4—22no—
497C4⋊C4—22no—
498C4⋊C4—22no—
499C4⋊C4—24no—
500C4⋊C4—24no—
501C4⋊C4—24no—
502C4⋊C4—24no—
503C4⋊C4—44no—
504C4⋊C4—42no—
505C4⋊C4—42no—
506C4⋊C4—42no—
507C4⋊C4—44no—
508C4⋊C4—44no—
509C4⋊C4—84no—
510C4⋊C4—82no—
511C4⋊C4—84no—
512C4⋊C4—84no—
513C4⋊C4—84no—
514C22⋊C4—24no—
515C22⋊C4—24no—
516C22⋊C4—24no—
517C22⋊C4—24no—
518C22⋊C4—24no—
519C22⋊C4—44no—
520C22⋊C4—44no—
521C22⋊C4—44no—
522C22⋊C4—44no—
523C22⋊C4—44no—
524C22⋊C4—44no—
525C22⋊C4—84no—
526C22⋊C4—82no—
527C22⋊C4—82no—
528C22⋊C4—84no—
529C22⋊C4—84no—
530C22⋊C4—84no—
531C42—22no—
532C42—22no—
533C42—44no—
534C42—44no—
535C23—22no—
536C23—22no—
537C23—22no—
538C23—22no—
539C23—22no—
540C23—22no—
541C23—21no—
542C23—21no—
543C23—21no—
544C23—21no—
545C23—28no—
546C23—24no—
547C23—22no—
548C23—24no—
549C23—22no—
550C23—22no—
551C23—44no—
552C23—44no—
553C23—84no—
554C23—84no—
555C23C2×#12112yes—
556C23C2×#12112yes—
557C23C2×#12112yes—
558C23#57712yes—
559C23#57712yes—
560C23C23×D411yescharacteristic
561Q8—42no—
562Q8—42no—
563Q8—44no—
564Q8—82no—
565Q8—84no—
566D4—48no—
567D4—82no—
568D4—84no—
569D4—84no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—24no—
580C2×C4—24no—
581C2×C4—24no—
582C2×C4—24no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—24no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—44no—
590C2×C4—44no—
591C2×C4—44no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—44no—
595C2×C4—42no—
596C2×C4—44no—
597C2×C4—42no—
598C2×C4—44no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—44no—
604C2×C4—44no—
605C2×C4—44no—
606C2×C4—44no—
607C2×C4—44no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—84no—
613C2×C4—84no—
614C2×C4—84no—
615C2×C4—84no—
616C2×C4—164no—
617C2×C4—164no—
618C2×C4C2×#12112yes—
619C2×C4C2×#12112yes—
620C2×C4C2×#12112yes—
621C2×C4C23×D411yescharacteristic
622C2×C4C22×#4111yescharacteristic
623C2×C4#58512yes—
624C2×C4#58512yes—
625C8—24no—
626C8—24no—
627C8—42no—
628C8—42no—
629C22—22no—
630C22—24no—
631C22—24no—
632C22—22no—
633C22—22no—
634C22—22no—
635C22—22no—
636C22—22no—
637C22—22no—
638C22—22no—
639C22—42no—
640C22—42no—
641C22—42no—
642C22—42no—
643C22—42no—
644C22—42no—
645C22—44no—
646C22—44no—
647C22—42no—
648C22—88no—
649C22#306014yes—
650C22C22×#12111yescharacteristic
651C22#348512yes—
652C22C2×#57711yescharacteristic
653C22C2×#57711yescharacteristic
654C22#358612yes—
655C4—22no—
656C4—22no—
657C4—22no—
658C4—42no—
659C4—42no—
660C4—42no—
661C4—82no—
662C4—82no—
663C4—82no—
664C4—84no—
665C4—84no—
666C4—82no—
667C4#348511yescharacteristic
668C4#348511yescharacteristic
669C4#349112yes—
670C2—24no—
671C2—42no—
672C2—42no—
673C2—82no—
674C2#3544711yescharacteristic
675C2C2×#306011yescharacteristic
676C2#6045511yescharacteristic
677C2#6175312yes—
678C2#6175512yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#525C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×#525 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C22×#485C2semidirect
5. #28108C2semidirect
6. #28098C2semidirect
7. #27667C2semidirect
8. #27658C2semidirect
9. #25259C2semidirect
10. #25247C2semidirect
11. #24619C2semidirect
12. #24569C2semidirect
13. #21938C2semidirect
14. #21765C2semidirect
15. #20702C2semidirect
59. #2088C22semidirect
60. #2087C22semidirect
61. #2077C22semidirect
62. #2076C22semidirect
63. #2066C22semidirect
64. #2066C22semidirect
65. #2035C22semidirect
66. #2035C22semidirect
67. #1944C22semidirect
68. #1941C22semidirect
69. #1935C22semidirect
70. #1933C22semidirect
73. #1894C22semidirect
74. #1892C22semidirect
82. #1809C22semidirect
85. #1807C22semidirect

Maximal subgroups

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

subgroupindex
C22×#4852
C2×#29762
#374422
#352632
#281082
#280982
#276672
#276582
#252592
#252472
#246192
#245692
#219382
#217652
#207022