Finite Group Databasebeta

← #6508746

Subgroups — #6508746

Order 512.

2110 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×#3186C211yesmaximal, characteristic
2C22×#482C211yesmaximal, characteristic, complemented
3#35262C211yesmaximal, characteristic, complemented
4#34363C211yesmaximal, characteristic, complemented
5#28126C214yesmaximal, complemented
6#27655C214yesmaximal, complemented
7#25209C214yesmaximal, complemented
8#25207C214yesmaximal, complemented
9#24581C212yesmaximal, complemented
10#24576C212yesmaximal, complemented
11#24531C212yesmaximal, complemented
12#24526C212yesmaximal, complemented
13#21931C211yesmaximal, characteristic, complemented
14#21761C211yesmaximal, characteristic, complemented
15#20699C211yesmaximal, characteristic, complemented
16C22×#121C2211yescharacteristic
17C22×#121C2211yescharacteristic
18C22×#117C2211yescharacteristic
19#3186—28no—
20C2×#482—28no—
21C2×#482C2214yes—
22C2×#482C2214yescomplemented
23C2×#480C2212yes—
24C2×#480C2212yes—
25C2×#466C2212yes—
26C2×#466C2212yes—
27C2×#447C2212yes—
28C2×#447C2212yes—
29C22×#107C2211yescharacteristic, complemented
30C22×#106C2211yescharacteristic, complemented
31C22×#102C2211yescharacteristic, complemented
32C22×#102C2211yescharacteristic, complemented
33C22×C22⋊C8C2211yescharacteristic
34C22×C22⋊C8C2211yescharacteristic
35#2854—22no—
36#2624—22no—
37#2504C2212yes—
38#2501C2212yes—
39#2500C2214yescomplemented
40#2482—22no—
41#2405—21no—
42#2405—21no—
43#2377—22no—
44#2363C2214yescomplemented
45#2344C2212yescomplemented
46#2342—22no—
47D4×C22⋊C4C2212yescomplemented
48#2333—22no—
49#2324C2214yes—
50#2317C2212yescomplemented
51#2309—21no—
52#2309C2212yescomplemented
53#2308—21no—
54#2280C2211yescharacteristic
55#2280C2211yescharacteristic
56#2085C2214yescomplemented
57#2083C2214yescomplemented
58#2073C2214yescomplemented
59#2071C2214yescomplemented
60#2067C2214yescomplemented
61#2065C2214yescomplemented
62#2054C2214yescomplemented
63#2054C2214yescomplemented
64#1949C2214yescomplemented
65#1947C2214yescomplemented
66#1926C2214yescomplemented
67#1925C2214yescomplemented
68#1893C2212yes—
69#1893C2212yes—
70#1879C2214yescomplemented
71#1879C2214yescomplemented
72#1875C2214yescomplemented
73#1850C2212yescomplemented
74#1850C2212yescomplemented
75#1850C2212yescomplemented
76#1850C2212yescomplemented
77#1825C2214yescomplemented
78#1824C2212yescomplemented
79#1824C2212yescomplemented
80#1823C2212yescomplemented
81#1804C2214yescomplemented
82#1802C2214yescomplemented
83#1798C2214yescomplemented
84#1786C2212yescomplemented
85#1735C2212yescomplemented
86#1735C2212yescomplemented
87C23×D4—21no—
88C23×D4—21no—
89C23×D4C2311yescharacteristic
90C23×D4C2311yescharacteristic
91C24×C4C2311yescharacteristic
92C22×D8C2312yes—
93C22×D8C2312yes—
94C23×C8C2311yescharacteristic
95C23×C8C2311yescharacteristic
96C23×C8C2311yescharacteristic
97C23×C8C2311yescharacteristic
98C2×#123—22no—
99C2×#123—22no—
100C2×#121—21no—
101C2×#121—21no—
102C2×#121—24no—
103C2×#121—21no—
104C2×#121—21no—
105C2×#121—24no—
106C2×#121—24no—
107C2×#121D418yes—
108C2×#121C2312yes—
109C2×#121C2312yes—
110C2×#121C2312yes—
111C2×#121C2312yes—
112C2×C4×D4C2314yes—
113C2×#117—21no—
114C2×#117—21no—
115C2×#117—22no—
116C2×#117—24no—
117C2×#117—24no—
118C2×#117C2312yes—
119C2×#117C2312yes—
120C2×#117C2312yes—
121C2×#117C2312yes—
122C22×C4⋊C4C2311yescharacteristic
123C22×C4⋊C4C2311yescharacteristic
124C22×C4⋊C4C2311yescharacteristic
125C22×C4⋊C4C2311yescharacteristic
126C22×C22⋊C4—21no—
127C22×C22⋊C4—21no—
128C22×C22⋊C4—21no—
129C22×C22⋊C4—22no—
130C22×C42—21no—
131#482—216no—
132#482—216no—
133#480—28no—
134#480—28no—
135#466—28no—
136#466—28no—
137#447—28no—
138#447—28no—
139C2×#107—24no—
140C2×#107C2312yes—
141C2×#107C2312yes—
142C2×#107C2314yes—
143C2×#106—24no—
144C2×#106C2314yes—
145C2×C4⋊C8C2314yes—
146C2×C4⋊C8C2314yes—
147C2×#102—24no—
148C2×#102—24no—
149C2×#102C2314yes—
150C2×#102C2314yes—
151C2×#102C2312yes—
152C2×#102C2312yes—
153C2×#102C2312yes—
154C2×#102C2312yes—
155C2×#102C2314yes—
156C2×#102C2314yes—
157C2×C22⋊C8—24no—
158C2×C22⋊C8—24no—
159C2×C22⋊C8C2312yes—
160C2×C22⋊C8C2312yes—
161C2×C22⋊C8C2312yes—
162C2×C22⋊C8C2312yes—
163#399—24no—
164#397—24no—
165#394—24no—
166#394—24no—
167#390—24no—
168#390—24no—
169#390—24no—
170#390C2314yes—
171#390C2314yes—
172#390C2314yes—
173#389—24no—
174#389C2314yes—
175#388—24no—
176#388—24no—
177#388—24no—
178#388—24no—
179#387—24no—
180#386—24no—
181#386—24no—
182#386—22no—
183#386—22no—
184#386—24no—
185#385—24no—
186#385—22no—
187#385—22no—
188#384C2314yes—
189#382C2314yes—
190#381—21no—
191#381—22no—
192#381D412yes—
193#380D412yes—
194#380D412yes—
195#380C2312yes—
196#380C2312yes—
197#379—21no—
198#379—22no—
199#379—22no—
200#379—22no—
201#379D411yescharacteristic
202#379D411yescharacteristic
203C4×C4⋊C4C2314yes—
204C4×C22⋊C4—21no—
205C4×C22⋊C4—21no—
206C4×C22⋊C4—24no—
207C4×C22⋊C4C2312yes—
208C4×C22⋊C4C2312yes—
209#340C2314yes—
210#340C2314yes—
211#340C2314yes—
212#340C2314yes—
213#340C2314yes—
214#340C2314yes—
215#336C2314yes—
216#336C2314yes—
217C25—21no—
218C25—21no—
219C25C2×D411yescharacteristic
220C22×D4—28no—
221C22×D4—22no—
222C22×D4—22no—
223C22×D4—22no—
224C22×D4—22no—
225C22×D4—21no—
226C22×D4—22no—
227C22×D4—21no—
228C22×D4—22no—
229C22×D4—22no—
230C22×D4—42no—
231C22×D4—42no—
232C22×D4—42no—
233C22×D4—42no—
234C22×D4—44no—
235C22×D4—84no—
236C22×D4—84no—
237C22×D4C2×D414yes—
238C22×D4C2412yes—
239C22×D4C2412yes—
240C23×C4—21no—
241C23×C4—21no—
242C23×C4—21no—
243C23×C4—21no—
244C23×C4—24no—
245C23×C4—42no—
246C23×C4—42no—
247C23×C4C2×D411yescharacteristic
248C23×C4C2×D411yescharacteristic
249C23×C4C2×D411yescharacteristic
250C23×C4C2×D414yes—
251C23×C4#4114yes—
252C23×C4C2411yescharacteristic
253C23×C4C2411yescharacteristic
254C23×C4C2411yescharacteristic
255C2×D8—24no—
256C2×D8—24no—
257C2×D8—44no—
258C2×D8—44no—
259C22×C8—22no—
260C22×C8—22no—
261C22×C8—22no—
262C22×C8—22no—
263C22×C8—42no—
264C22×C8—42no—
265C22×C8—42no—
266C22×C8—42no—
267C22×C8C2412yes—
268C22×C8C2412yes—
269C22×C8C2412yes—
270C22×C8C2412yes—
271#123—84no—
272#123—84no—
273#121—28no—
274#121—28no—
275#121—24no—
276#121—24no—
277#121—24no—
278#121—24no—
279#121—24no—
280#121—24no—
281#121—44no—
282#121—44no—
283#121C2×D418yes—
284#121C2×D418yes—
285C4×D4—44no—
286C4×D4—44no—
287#117—24no—
288#117—24no—
289#117—24no—
290#117—24no—
291#117—42no—
292#117—42no—
293#117—44no—
294#117—44no—
295#117—42no—
296#117—42no—
297#117—84no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—24no—
310C2×C4⋊C4—24no—
311C2×C4⋊C4—24no—
312C2×C4⋊C4C2×D414yes—
313C2×C4⋊C4C2×D414yes—
314C2×C4⋊C4C2412yes—
315C2×C4⋊C4C2412yes—
316C2×C4⋊C4C2412yes—
317C2×C4⋊C4C2414yes—
318C2×C4⋊C4C2412yes—
319C2×C22⋊C4—28no—
320C2×C22⋊C4—22no—
321C2×C22⋊C4—22no—
322C2×C22⋊C4—22no—
323C2×C22⋊C4—22no—
324C2×C22⋊C4—22no—
325C2×C22⋊C4—22no—
326C2×C22⋊C4—22no—
327C2×C22⋊C4—22no—
328C2×C22⋊C4—22no—
329C2×C22⋊C4—24no—
330C2×C22⋊C4—24no—
331C2×C22⋊C4—24no—
332C2×C22⋊C4—24no—
333C2×C22⋊C4—24no—
334C2×C22⋊C4—24no—
335C2×C22⋊C4—24no—
336C2×C22⋊C4—44no—
337C2×C22⋊C4—44no—
338C2×C22⋊C4—84no—
339C2×C22⋊C4C2×D412yes—
340C2×C22⋊C4C2×D412yes—
341C2×C22⋊C4C2×D412yes—
342C2×C22⋊C4C2×D412yes—
343C2×C22⋊C4C2×D412yes—
344C2×C22⋊C4C2×D412yes—
345C2×C42—22no—
346C2×C42—22no—
347C2×C42—24no—
348C2×C42—22no—
349C2×C42—44no—
350C2×C42C2414yes—
351#107—28no—
352#107—24no—
353#107—24no—
354#106—28no—
355C4⋊C8—44no—
356C4⋊C8—44no—
357#102—28no—
358#102—28no—
359#102—24no—
360#102—24no—
361#102—24no—
362#102—24no—
363#102—44no—
364#102—44no—
365C22⋊C8—24no—
366C22⋊C8—24no—
367C22⋊C8—24no—
368C22⋊C8—24no—
369#95—22no—
370#95—22no—
371#95—24no—
372#95—24no—
373#95—24no—
374#95—24no—
375#95C2×D414yes—
376#95C2×D414yes—
377C24—24no—
378C24—22no—
379C24—24no—
380C24—22no—
381C24—42no—
382C24—42no—
383C24—44no—
384C24—44no—
385C24—42no—
386C24—82no—
387C24—82no—
388C24—82no—
389C24—82no—
390C24#12114yes—
391C24C22×D411yescharacteristic
392C24C22×D411yescharacteristic
393C24C22×D411yescharacteristic
394C2×D4—28no—
395C2×D4—28no—
396C2×D4—24no—
397C2×D4—22no—
398C2×D4—22no—
399C2×D4—22no—
400C2×D4—24no—
401C2×D4—24no—
402C2×D4—24no—
403C2×D4—24no—
404C2×D4—44no—
405C2×D4—44no—
406C2×D4—44no—
407C2×D4—44no—
408C2×D4—44no—
409C2×D4—44no—
410C2×D4—44no—
411C2×D4—42no—
412C2×D4—82no—
413C2×D4—82no—
414C2×D4—84no—
415C2×D4—84no—
416C2×D4—82no—
417C2×D4—816no—
418C2×D4—82no—
419C2×D4—82no—
420C2×D4—82no—
421C2×D4—82no—
422C2×D4—82no—
423C2×D4—82no—
424C2×D4—816no—
425C2×D4C22×D414yes—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—24no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—24no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—44no—
439C22×C4—44no—
440C22×C4—48no—
441C22×C4—42no—
442C22×C4—42no—
443C22×C4—42no—
444C22×C4—44no—
445C22×C4—42no—
446C22×C4—42no—
447C22×C4—42no—
448C22×C4—44no—
449C22×C4—84no—
450C22×C4—84no—
451C22×C4—84no—
452C22×C4—82no—
453C22×C4—82no—
454C22×C4#12114yes—
455C22×C4#12114yes—
456C22×C4#12114yes—
457C22×C4C22×D414yes—
458C22×C4C22×D414yes—
459C22×C4C22×D411yescharacteristic
460C22×C4C22×D411yescharacteristic
461C22×C4C22×D411yescharacteristic
462C22×C4C22×D412yes—
463C22×C4C22×D412yes—
464C22×C4C22×D412yes—
465C22×C4C2×#4112yes—
466C22×C4C2×#4112yes—
467C22×C4C2×#4112yes—
468C22×C4C2511yescharacteristic
469D8—84no—
470D8—84no—
471C2×C8—22no—
472C2×C8—22no—
473C2×C8—22no—
474C2×C8—22no—
475C2×C8—24no—
476C2×C8—24no—
477C2×C8—24no—
478C2×C8—24no—
479C2×C8—44no—
480C2×C8—44no—
481C2×C8—82no—
482C2×C8—82no—
483C4⋊C4—24no—
484C4⋊C4—22no—
485C4⋊C4—22no—
486C4⋊C4—22no—
487C4⋊C4—22no—
488C4⋊C4—22no—
489C4⋊C4—22no—
490C4⋊C4—22no—
491C4⋊C4—24no—
492C4⋊C4—24no—
493C4⋊C4—44no—
494C4⋊C4—42no—
495C4⋊C4—42no—
496C4⋊C4—42no—
497C4⋊C4—84no—
498C4⋊C4—84no—
499C4⋊C4—84no—
500C4⋊C4C22×D414yes—
501C22⋊C4—24no—
502C22⋊C4—24no—
503C22⋊C4—28no—
504C22⋊C4—28no—
505C22⋊C4—42no—
506C22⋊C4—42no—
507C22⋊C4—42no—
508C22⋊C4—42no—
509C22⋊C4—44no—
510C22⋊C4—44no—
511C22⋊C4—44no—
512C22⋊C4—44no—
513C22⋊C4—44no—
514C22⋊C4—84no—
515C22⋊C4—84no—
516C22⋊C4—84no—
517C22⋊C4—84no—
518C22⋊C4—84no—
519C22⋊C4—84no—
520C22⋊C4—84no—
521C22⋊C4—82no—
522C22⋊C4—82no—
523C42—24no—
524C42—24no—
525C42—44no—
526C42—44no—
527C42—82no—
528C42—84no—
529C23—22no—
530C23—22no—
531C23—22no—
532C23—22no—
533C23—22no—
534C23—22no—
535C23—22no—
536C23—22no—
537C23—28no—
538C23—24no—
539C23—44no—
540C23—44no—
541C23—44no—
542C23—44no—
543C23—42no—
544C23—42no—
545C23—48no—
546C23—44no—
547C23—42no—
548C23—44no—
549C23—42no—
550C23—42no—
551C23—88no—
552C23—84no—
553C23—84no—
554C23—84no—
555C23—84no—
556C23—84no—
557C23—168no—
558C23—168no—
559C23C2×#12112yes—
560C23C2×#12112yes—
561C23C2×#12112yes—
562C23#57614yes—
563C23C23×D411yescharacteristic
564D4—24no—
565D4—24no—
566D4—42no—
567D4—42no—
568D4—44no—
569D4—44no—
570D4—84no—
571D4—84no—
572D4—88no—
573D4—88no—
574D4—88no—
575D4—88no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—22no—
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—24no—
595C2×C4—44no—
596C2×C4—44no—
597C2×C4—44no—
598C2×C4—48no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—44no—
602C2×C4—44no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—44no—
607C2×C4—44no—
608C2×C4—44no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—84no—
612C2×C4—84no—
613C2×C4—84no—
614C2×C4—168no—
615C2×C4C2×#12112yes—
616C2×C4C2×#12112yes—
617C2×C4C2×#12112yes—
618C2×C4C23×D411yescharacteristic
619C2×C4C22×#4111yescharacteristic
620C2×C4#58512yes—
621C2×C4#58512yes—
622C8—42no—
623C8—42no—
624C8—42no—
625C8—42no—
626C22—22no—
627C22—28no—
628C22—22no—
629C22—22no—
630C22—22no—
631C22—22no—
632C22—22no—
633C22—22no—
634C22—22no—
635C22—44no—
636C22—44no—
637C22—44no—
638C22—44no—
639C22—44no—
640C22—88no—
641C22—84no—
642C22—84no—
643C22—82no—
644C22—82no—
645C22—84no—
646C22—84no—
647C22—82no—
648C22—168no—
649C22—168no—
650C22#304114yes—
651C22C22×#12111yescharacteristic
652C22#348512yes—
653C22C2×#57612yes—
654C22#358412yes—
655C4—22no—
656C4—22no—
657C4—22no—
658C4—44no—
659C4—44no—
660C4—44no—
661C4—82no—
662C4—82no—
663C4—82no—
664C4—84no—
665C4#348511yescharacteristic
666C4#348511yescharacteristic
667C4#349112yes—
668C2—24no—
669C2—42no—
670C2—42no—
671C2—84no—
672C2—162no—
673C2#3544711yescharacteristic
674C2C2×#304111yescharacteristic
675C2#6042211yescharacteristic
676C2#6177214yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#3186C2
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×#482C2semidirect
3. #35262C2semidirect
4. #34363C2semidirect
5. #28126C2semidirect
6. #27655C2semidirect
7. #25209C2semidirect
8. #25207C2semidirect
9. #24581C2semidirect
10. #24576C2semidirect
11. #24531C2semidirect
12. #24526C2semidirect
13. #21931C2semidirect
14. #21761C2semidirect
15. #20699C2semidirect
22. C2×#482C22semidirect
29. C22×#107C22semidirect
30. C22×#106C22semidirect
31. C22×#102C22semidirect
32. C22×#102C22semidirect
39. #2500C22semidirect
44. #2363C22semidirect
45. #2344C22semidirect
47. D4×C22⋊C4C22semidirect
50. #2317C22semidirect
52. #2309C22semidirect
56. #2085C22semidirect
57. #2083C22semidirect
58. #2073C22semidirect
59. #2071C22semidirect
60. #2067C22semidirect
61. #2065C22semidirect
62. #2054C22semidirect
63. #2054C22semidirect
64. #1949C22semidirect
65. #1947C22semidirect
66. #1926C22semidirect
67. #1925C22semidirect
70. #1879C22semidirect
71. #1879C22semidirect
72. #1875C22semidirect
73. #1850C22semidirect
74. #1850C22semidirect
75. #1850C22semidirect
76. #1850C22semidirect
77. #1825C22semidirect
78. #1824C22semidirect
79. #1824C22semidirect
80. #1823C22semidirect
81. #1804C22semidirect
82. #1802C22semidirect
83. #1798C22semidirect
84. #1786C22semidirect
85. #1735C22semidirect
86. #1735C22semidirect

Maximal subgroups

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

subgroupindex
C2×#31862
C22×#4822
#352622
#343632
#281262
#276552
#252092
#252072
#245812
#245762
#245312
#245262
#219312
#217612
#206992