Finite Group Databasebeta

← #6500307

Subgroups — #6500307

Order 512.

2683 conjugacy classes of subgroups.

The subgroup lattice has 688 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3196C211yesmaximal, characteristic
2C22×#451C211yesmaximal, characteristic, complemented
3#35242C211yesmaximal, characteristic
4#34370C211yesmaximal, characteristic, complemented
5#28132C214yesmaximal, complemented
6#25107C214yesmaximal, complemented
7#25095C214yesmaximal, complemented
8#24483C212yesmaximal, complemented
9#24482C212yesmaximal, complemented
10#24361C212yesmaximal, complemented
11#24360C212yesmaximal, complemented
12#24221C214yesmaximal, complemented
13#21933C211yesmaximal, characteristic, complemented
14#21760C211yesmaximal, characteristic
15#20599C211yesmaximal, characteristic, complemented
16C24×Q8C2211yescharacteristic
17C23×Q16C2212yes—
18C2×#536C2212yes—
19C2×#525C2211yescharacteristic
20C22×C22⋊Q8C2211yescharacteristic
21#3196—28no—
22C2×#484C2211yescharacteristic
23C2×#481C2211yescharacteristic
24C2×#485C2212yes—
25C2×#470C2212yes—
26C2×#468C2212yes—
27C2×#464C2214yescomplemented
28C2×#462C2214yescomplemented
29C2×#452C2212yes—
30C2×#451—28no—
31C2×#451C2214yes—
32C2×#451C2211yescharacteristic
33C2×#451C2214yes—
34C2×#448C2211yescharacteristic
35C2×#419C2214yes—
36C22×#103C2212yes—
37C2×#407C2211yescharacteristic
38C22×C22⋊C8C2211yescharacteristic
39#2849—21no—
40#2849—21no—
41#2614—22no—
42#2436C2212yes—
43#2436C2212yes—
44#2434C2212yes—
45#2434C2212yes—
46#2409—22no—
47#2409—22no—
48#2378—22no—
49#2362C2214yes—
50#2345—22no—
51#2345C2212yes—
52Q8×C22⋊C4C2212yes—
53#2332—22no—
54#2323C2214yes—
55#2312—21no—
56#2312—21no—
57#2310C2212yes—
58#2280C2211yescharacteristic
59#2280C2211yescharacteristic
60#2068C2214yescomplemented
61#2066C2214yescomplemented
62#2056C2214yescomplemented
63#2056C2214yescomplemented
64#2043C2214yescomplemented
65#2042C2214yescomplemented
66#2016C2214yescomplemented
67#2015C2214yescomplemented
68#1983C2214yescomplemented
69#1981C2214yescomplemented
70#1928C2214yescomplemented
71#1927C2214yescomplemented
72#1893C2212yes—
73#1892C2212yes—
74#1878C2214yescomplemented
75#1878C2214yescomplemented
76#1875C2212yes—
77#1873C2212yes—
78#1851C2214yes—
79#1850C2212yes—
80#1850C2212yes—
81#1825C2212yes—
82#1798C2214yes—
83#1788C2212yes—
84#1787C2212yes—
85#1787C2212yes—
86#1787C2214yes—
87C22×#41C2311yescharacteristic
88C23×Q8—21no—
89C23×Q8—21no—
90C23×Q8—21no—
91C23×Q8—24no—
92C23×Q8—24no—
93C23×Q8—48no—
94C23×Q8C2312yes—
95C23×Q8C2312yes—
96C23×Q8C2312yes—
97C23×Q8C2311yescharacteristic
98C24×C4—21no—
99C24×C4C2311yescharacteristic
100C22×Q16—28no—
101C22×Q16—48no—
102C22×Q16C2314yes—
103C22×Q16C2314yes—
104C22×Q16C2312yes—
105C22×SD16C2312yes—
106C22×M16C2312yes—
107C23×C8C2312yes—
108#536—48no—
109C2×C4⋊Q8C2314yes—
110C2×#124C2314yes—
111#525—24no—
112#525—24no—
113C2×#123—21no—
114C2×#123—22no—
115C2×#123—22no—
116C2×C22⋊Q8—21no—
117C2×C22⋊Q8—21no—
118C2×C22⋊Q8—24no—
119C2×C22⋊Q8—24no—
120C2×C22⋊Q8D418yes—
121C2×C22⋊Q8C2311yescharacteristic
122C2×C22⋊Q8C2311yescharacteristic
123C2×C22⋊Q8C2312yes—
124C2×C22⋊Q8C2312yes—
125C2×#121C2311yescharacteristic
126C2×#121C2311yescharacteristic
127C2×#120—21no—
128C2×C4×Q8C2314yes—
129C2×C4×D4—21no—
130C2×C4×D4—21no—
131C2×#117C2311yescharacteristic
132C2×#117C2312yes—
133C2×#117C2312yes—
134C22×C4⋊C4—21no—
135C22×C4⋊C4C2311yescharacteristic
136C22×C22⋊C4—21no—
137#485—28no—
138#484—24no—
139#481—24no—
140#470—28no—
141#468—28no—
142#464—48no—
143#462—48no—
144#452—28no—
145#451—24no—
146#451—216no—
147#451—216no—
148#448—24no—
149C2×#107C2312yes—
150C2×#106C2312yes—
151C2×C4⋊C8C2314yes—
152C2×C4⋊C8C2314yes—
153#419—48no—
154C2×#103—28no—
155C2×#103C2312yes—
156C2×#103C2312yes—
157C2×#103C2314yes—
158C2×#103C2314yes—
159C2×#103C2314yes—
160C2×#103C2312yes—
161C2×#103C2314yes—
162C2×#103C2314yes—
163C2×#103C2314yes—
164C2×#103C2314yes—
165C2×#102C2312yes—
166C2×#102C2314yes—
167#407—24no—
168C2×C22⋊C8—24no—
169C2×C22⋊C8C2312yes—
170C2×C22⋊C8C2312yes—
171C2×C22⋊C8C2311yescharacteristic
172C2×C22⋊C8C2311yescharacteristic
173C2×C22⋊C8C2312yes—
174#395—24no—
175#395—24no—
176#393—24no—
177#393—24no—
178#391—24no—
179#391—24no—
180#391C2314yes—
181#391C2314yes—
182#391C2314yes—
183#389—24no—
184#389C2314yes—
185#388—24no—
186#388—24no—
187#388—24no—
188#388—24no—
189#387—24no—
190#387—24no—
191#387—24no—
192#386—22no—
193#386—22no—
194#386—22no—
195#386—22no—
196#385—22no—
197#385—22no—
198#385—22no—
199#385—22no—
200#381—21no—
201#381—22no—
202#381—22no—
203#381D412yes—
204#380—22no—
205#380—22no—
206#380D412yes—
207#380D412yes—
208#380C2312yes—
209#380C2312yes—
210#379—21no—
211#379D411yescharacteristic
212#379D411yescharacteristic
213C4×C4⋊C4—24no—
214C4×C4⋊C4C2314yes—
215C4×C22⋊C4—21no—
216C4×C22⋊C4—21no—
217C4×C22⋊C4—22no—
218C4×C22⋊C4—22no—
219C4×C22⋊C4C2312yes—
220C4×C22⋊C4C2312yes—
221#340C2314yes—
222#340C2314yes—
223#336C2314yes—
224#336C2314yes—
225C25C2×D411yescharacteristic
226C2×#41—22no—
227C2×#41—22no—
228C2×#41—44no—
229C22×Q8—28no—
230C22×Q8—22no—
231C22×Q8—21no—
232C22×Q8—22no—
233C22×Q8—22no—
234C22×Q8—22no—
235C22×Q8—28no—
236C22×Q8—28no—
237C22×Q8—24no—
238C22×Q8—24no—
239C22×Q8—24no—
240C22×Q8—24no—
241C22×Q8—24no—
242C22×Q8—24no—
243C22×Q8—44no—
244C22×Q8—44no—
245C22×Q8—44no—
246C22×Q8—48no—
247C22×Q8—44no—
248C22×Q8—432no—
249C22×Q8—432no—
250C22×Q8—48no—
251C22×Q8—48no—
252C22×Q8—44no—
253C22×Q8—44no—
254C22×Q8—44no—
255C22×Q8C2×D414yes—
256C22×Q8C2411yescharacteristic
257C22×Q8C2412yes—
258C22×Q8C2414yes—
259C22×D4—21no—
260C22×D4—21no—
261C22×D4—21no—
262C22×D4C2411yescharacteristic
263C23×C4—22no—
264C23×C4—21no—
265C23×C4—21no—
266C23×C4—22no—
267C23×C4—22no—
268C23×C4—21no—
269C23×C4—44no—
270C23×C4—48no—
271C23×C4—44no—
272C23×C4C2×D411yescharacteristic
273C23×C4C2×D411yescharacteristic
274C23×C4C2×D411yescharacteristic
275C23×C4C2×D414yes—
276C23×C4#4114yes—
277C23×C4C2411yescharacteristic
278C23×C4C2411yescharacteristic
279C23×C4C2411yescharacteristic
280C2×Q16—28no—
281C2×Q16—28no—
282C2×Q16—28no—
283C2×Q16—28no—
284C2×Q16—24no—
285C2×Q16—44no—
286C2×Q16—44no—
287C2×Q16—44no—
288C2×Q16—432no—
289C2×SD16—24no—
290C2×SD16—44no—
291C2×M16—24no—
292C2×M16—44no—
293C22×C8—24no—
294C22×C8—44no—
295C22×C8C2414yes—
296C22×C8C2412yes—
297C22×C8C2412yes—
298C4⋊Q8—28no—
299C4⋊Q8—44no—
300#124—28no—
301#124—44no—
302#123—48no—
303#123—48no—
304#123—44no—
305C22⋊Q8—28no—
306C22⋊Q8—28no—
307C22⋊Q8—24no—
308C22⋊Q8—24no—
309C22⋊Q8—22no—
310C22⋊Q8—22no—
311C22⋊Q8—22no—
312C22⋊Q8—22no—
313C22⋊Q8—416no—
314C22⋊Q8C2×D418yes—
315C22⋊Q8C2×D418yes—
316#121—22no—
317#121—22no—
318#121—22no—
319#121—22no—
320#120—44no—
321C4×Q8—44no—
322C4×Q8—44no—
323C4×D4—44no—
324C4×D4—44no—
325#117—22no—
326#117—22no—
327#117—44no—
328#117—44no—
329C2×C4⋊C4—28no—
330C2×C4⋊C4—22no—
331C2×C4⋊C4—22no—
332C2×C4⋊C4—21no—
333C2×C4⋊C4—22no—
334C2×C4⋊C4—21no—
335C2×C4⋊C4—21no—
336C2×C4⋊C4—24no—
337C2×C4⋊C4—24no—
338C2×C4⋊C4—24no—
339C2×C4⋊C4—24no—
340C2×C4⋊C4C2×D414yes—
341C2×C4⋊C4C2×D414yes—
342C2×C4⋊C4C2414yes—
343C2×C4⋊C4C2411yescharacteristic
344C2×C4⋊C4C2411yescharacteristic
345C2×C4⋊C4C2412yes—
346C2×C22⋊C4—28no—
347C2×C22⋊C4—22no—
348C2×C22⋊C4—22no—
349C2×C22⋊C4—21no—
350C2×C22⋊C4—21no—
351C2×C22⋊C4—21no—
352C2×C22⋊C4—22no—
353C2×C22⋊C4—21no—
354C2×C22⋊C4—21no—
355C2×C22⋊C4—24no—
356C2×C22⋊C4—24no—
357C2×C22⋊C4—22no—
358C2×C22⋊C4—22no—
359C2×C22⋊C4C2×D412yes—
360C2×C22⋊C4C2×D412yes—
361C2×C22⋊C4C2×D412yes—
362C2×C22⋊C4C2×D412yes—
363C2×C22⋊C4C2×D412yes—
364C2×C22⋊C4C2×D412yes—
365C2×C42—21no—
366C2×C42—22no—
367C2×C42—24no—
368C2×C42—24no—
369C2×C42C2414yes—
370#107—24no—
371#106—24no—
372C4⋊C8—44no—
373C4⋊C8—44no—
374#103—24no—
375#103—24no—
376#103—28no—
377#103—28no—
378#103—28no—
379#103—28no—
380#103—24no—
381#103—44no—
382#103—44no—
383#103—44no—
384#102—24no—
385#102—44no—
386C22⋊C8—24no—
387C22⋊C8—24no—
388C22⋊C8—22no—
389C22⋊C8—22no—
390C22⋊C8—24no—
391#95—22no—
392#95—22no—
393#95—24no—
394#95—24no—
395#95—24no—
396#95—24no—
397#95—24no—
398#95—24no—
399#95—24no—
400#95—24no—
401#95C2×D414yes—
402#95C2×D414yes—
403C24—24no—
404C24—21no—
405C24—42no—
406C24—42no—
407C24#12114yes—
408C24C22×D411yescharacteristic
409C24C22×D411yescharacteristic
410C24C22×D411yescharacteristic
411#41—84no—
412#41—84no—
413C2×Q8—28no—
414C2×Q8—24no—
415C2×Q8—22no—
416C2×Q8—24no—
417C2×Q8—216no—
418C2×Q8—216no—
419C2×Q8—24no—
420C2×Q8—21no—
421C2×Q8—24no—
422C2×Q8—21no—
423C2×Q8—24no—
424C2×Q8—24no—
425C2×Q8—24no—
426C2×Q8—24no—
427C2×Q8—24no—
428C2×Q8—48no—
429C2×Q8—48no—
430C2×Q8—416no—
431C2×Q8—416no—
432C2×Q8—416no—
433C2×Q8—416no—
434C2×Q8—44no—
435C2×Q8—48no—
436C2×Q8—48no—
437C2×Q8—48no—
438C2×Q8—432no—
439C2×Q8—432no—
440C2×Q8—432no—
441C2×Q8—44no—
442C2×Q8—44no—
443C2×Q8—44no—
444C2×Q8—82no—
445C2×Q8—84no—
446C2×Q8C22×D414yes—
447C2×Q8H(5,𝔽2)18yes—
448C2×D4—21no—
449C2×D4—21no—
450C2×D4—24no—
451C2×D4—44no—
452C2×D4—44no—
453C2×D4—42no—
454C2×D4—42no—
455C2×D4—42no—
456C2×D4—44no—
457C22×C4—24no—
458C22×C4—24no—
459C22×C4—24no—
460C22×C4—28no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—24no—
464C22×C4—24no—
465C22×C4—24no—
466C22×C4—21no—
467C22×C4—21no—
468C22×C4—24no—
469C22×C4—21no—
470C22×C4—48no—
471C22×C4—48no—
472C22×C4—42no—
473C22×C4—42no—
474C22×C4—42no—
475C22×C4—48no—
476C22×C4—48no—
477C22×C4—44no—
478C22×C4—44no—
479C22×C4—432no—
480C22×C4—44no—
481C22×C4—44no—
482C22×C4—44no—
483C22×C4—42no—
484C22×C4—82no—
485C22×C4—82no—
486C22×C4#12114yes—
487C22×C4#12114yes—
488C22×C4#12114yes—
489C22×C4C22×D414yes—
490C22×C4C22×D414yes—
491C22×C4C22×D411yescharacteristic
492C22×C4C22×D411yescharacteristic
493C22×C4C22×D411yescharacteristic
494C22×C4C22×D412yes—
495C22×C4C22×D412yes—
496C22×C4C22×D412yes—
497C22×C4C2×#4112yes—
498C22×C4C2×#4112yes—
499C22×C4C2×#4112yes—
500C22×C4C2511yescharacteristic
501Q16—416no—
502Q16—416no—
503Q16—84no—
504SD16—84no—
505M16—84no—
506C2×C8—22no—
507C2×C8—22no—
508C2×C8—24no—
509C2×C8—24no—
510C2×C8—24no—
511C2×C8—24no—
512C2×C8—24no—
513C2×C8—48no—
514C4⋊C4—28no—
515C4⋊C4—28no—
516C4⋊C4—24no—
517C4⋊C4—22no—
518C4⋊C4—21no—
519C4⋊C4—21no—
520C4⋊C4—21no—
521C4⋊C4—21no—
522C4⋊C4—24no—
523C4⋊C4—24no—
524C4⋊C4—44no—
525C4⋊C4—48no—
526C4⋊C4—48no—
527C4⋊C4—42no—
528C4⋊C4—42no—
529C4⋊C4—84no—
530C4⋊C4—84no—
531C4⋊C4C22×D414yes—
532C22⋊C4—28no—
533C22⋊C4—28no—
534C22⋊C4—24no—
535C22⋊C4—24no—
536C22⋊C4—42no—
537C22⋊C4—42no—
538C22⋊C4—42no—
539C22⋊C4—42no—
540C22⋊C4—48no—
541C22⋊C4—48no—
542C22⋊C4—42no—
543C22⋊C4—42no—
544C22⋊C4—42no—
545C22⋊C4—44no—
546C22⋊C4—42no—
547C22⋊C4—42no—
548C22⋊C4—82no—
549C22⋊C4—82no—
550C42—24no—
551C42—24no—
552C42—42no—
553C42—84no—
554C42—82no—
555C42—84no—
556C23—22no—
557C23—22no—
558C23—22no—
559C23—22no—
560C23—22no—
561C23—22no—
562C23—28no—
563C23—22no—
564C23—22no—
565C23—44no—
566C23—44no—
567C23—416no—
568C23—42no—
569C23—41no—
570C23—42no—
571C23—42no—
572C23C2×#12112yes—
573C23C2×#12112yes—
574C23C2×#12112yes—
575C23#57814yes—
576C23C23×D411yescharacteristic
577Q8—24no—
578Q8—24no—
579Q8—28no—
580Q8—28no—
581Q8—416no—
582Q8—416no—
583Q8—416no—
584Q8—44no—
585Q8—44no—
586Q8—44no—
587Q8—42no—
588Q8—42no—
589Q8—84no—
590D4—42no—
591D4—42no—
592D4—84no—
593D4—84no—
594D4—84no—
595C2×C4—22no—
596C2×C4—22no—
597C2×C4—22no—
598C2×C4—24no—
599C2×C4—24no—
600C2×C4—24no—
601C2×C4—24no—
602C2×C4—24no—
603C2×C4—24no—
604C2×C4—24no—
605C2×C4—24no—
606C2×C4—24no—
607C2×C4—24no—
608C2×C4—24no—
609C2×C4—24no—
610C2×C4—28no—
611C2×C4—24no—
612C2×C4—24no—
613C2×C4—44no—
614C2×C4—44no—
615C2×C4—44no—
616C2×C4—416no—
617C2×C4—416no—
618C2×C4—416no—
619C2×C4—44no—
620C2×C4—44no—
621C2×C4—44no—
622C2×C4—44no—
623C2×C4—44no—
624C2×C4—42no—
625C2×C4—41no—
626C2×C4—41no—
627C2×C4—41no—
628C2×C4—44no—
629C2×C4—42no—
630C2×C4—44no—
631C2×C4—42no—
632C2×C4—84no—
633C2×C4—84no—
634C2×C4—84no—
635C2×C4C2×#12112yes—
636C2×C4C2×#12112yes—
637C2×C4C2×#12112yes—
638C2×C4C23×D411yescharacteristic
639C2×C4C22×#4111yescharacteristic
640C2×C4C2×H(5,𝔽2)14yes—
641C8—44no—
642C8—42no—
643C8—42no—
644C22—22no—
645C22—22no—
646C22—28no—
647C22—22no—
648C22—22no—
649C22—22no—
650C22—22no—
651C22—22no—
652C22—22no—
653C22—416no—
654C22—44no—
655C22—44no—
656C22—81no—
657C22—81no—
658C22—81no—
659C22—84no—
660C22#304414yes—
661C22C22×#12111yescharacteristic
662C22#348412yes—
663C22C2×#57812yes—
664C22#358612yes—
665C4—22no—
666C4—22no—
667C4—22no—
668C4—44no—
669C4—48no—
670C4—44no—
671C4—44no—
672C4—44no—
673C4—81no—
674C4—81no—
675C4—81no—
676C4—84no—
677C4#348412yes—
678C4#349012yes—
679C2—24no—
680C2—42no—
681C2—42no—
682C2—161no—
683C2#3534811yescharacteristic
684C2C2×#304411yescharacteristic
685C2#6042711yescharacteristic
686C2#6161214yes—
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×#451C2semidirect
4. #34370C2semidirect
5. #28132C2semidirect
6. #25107C2semidirect
7. #25095C2semidirect
8. #24483C2semidirect
9. #24482C2semidirect
10. #24361C2semidirect
11. #24360C2semidirect
12. #24221C2semidirect
13. #21933C2semidirect
15. #20599C2semidirect
27. C2×#464C22semidirect
28. C2×#462C22semidirect
60. #2068C22semidirect
61. #2066C22semidirect
62. #2056C22semidirect
63. #2056C22semidirect
64. #2043C22semidirect
65. #2042C22semidirect
66. #2016C22semidirect
67. #2015C22semidirect
68. #1983C22semidirect
69. #1981C22semidirect
70. #1928C22semidirect
71. #1927C22semidirect
74. #1878C22semidirect
75. #1878C22semidirect

Maximal subgroups

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

subgroupindex
C2×#31962
C22×#4512
#352422
#343702
#281322
#251072
#250952
#244832
#244822
#243612
#243602
#242212
#219332
#217602
#205992