Finite Group Databasebeta

← #6434314

Subgroups — #6434314

Order 512.

1710 conjugacy classes of subgroups.

The subgroup lattice has 690 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35338C211yesmaximal, characteristic
2#35313C211yesmaximal, characteristic, complemented
3#30352C214yesmaximal, complemented
4#30350C214yesmaximal, complemented
5#30347C214yesmaximal, complemented
6#30341C214yesmaximal, complemented
7#23613C211yesmaximal, characteristic, complemented
8#23612C211yesmaximal, characteristic
9#23551C211yesmaximal, characteristic, complemented
10#23542C211yesmaximal, characteristic, complemented
11#23541C211yesmaximal, characteristic
12#23506C211yesmaximal, characteristic
13#23455C211yesmaximal, characteristic, complemented
14C2×#2015C211yesmaximal, characteristic
15#21232C211yesmaximal, characteristic, complemented
16#21231C211yesmaximal, characteristic
17#21089C211yesmaximal, characteristic
18#21088C211yesmaximal, characteristic, complemented
19#20430C211yesmaximal, characteristic, complemented
20C23×Q16C2211yescharacteristic
21C23×SD16C2211yescharacteristic
22C2×#533C2211yescharacteristic
23C22×#124C2211yescharacteristic
24C2×#427C2211yescharacteristic
25C2×#427C2211yescharacteristic
26C2×#424C2211yescharacteristic
27C2×#424C2211yescharacteristic
28C2×#419C2211yescharacteristic
29C2×#419C2211yescharacteristic
30C2×#419C2211yescharacteristic
31C2×#419C2211yescharacteristic
32C2×#418C2211yescharacteristic
33C2×#418C2211yescharacteristic
34C22×#103C2211yescharacteristic
35C22×#102C2211yescharacteristic
36C2×#405C2211yescharacteristic
37C2×#405C2211yescharacteristic
38#2863C2214yescomplemented
39#2861C2214yes—
40#2645C2214yescomplemented
41#2640C2214yes—
42#2592—22no—
43#2592—22no—
44#2561—24no—
45#2514—22no—
46#2511—22no—
47#2493—24no—
48#2317C2211yescharacteristic
49#2316C2211yescharacteristic
50#2310C2211yescharacteristic
51#2310C2211yescharacteristic
52#2310C2211yescharacteristic
53#2310C2211yescharacteristic
54#2309C2211yescharacteristic
55#2309C2211yescharacteristic
56#2296C2211yescharacteristic
57C2×#391C2211yescharacteristic
58C4×#117C2211yescharacteristic
59#2068C2214yescomplemented
60#2067C2214yes—
61#2066C2214yescomplemented
62#2065C2214yes—
63#2059C2212yescomplemented
64#2059C2212yescomplemented
65#2058C2212yescomplemented
66#2057C2212yescomplemented
67#2056C2214yescomplemented
68#2056C2214yescomplemented
69#2055C2214yes—
70#2055C2214yes—
71#2042C2214yes—
72#2042C2214yes—
73#2026C2214yescomplemented
74#2024C2214yescomplemented
75#2015C2212yes—
76#2015C2212yes—
77#2015C2212yes—
78#2015C2212yes—
79#1973C2214yes—
80#1971C2214yescomplemented
81#1969C2214yescomplemented
82#1968C2214yes—
83#1936C2214yes—
84#1935C2214yescomplemented
85#1934C2214yes—
86#1933C2214yescomplemented
87#1768C2212yescomplemented
88#1768C2212yescomplemented
89#1767C2212yescomplemented
90C4×C4⋊C8C2212yescomplemented
91#1736C2211yescharacteristic
92#1736C2211yescharacteristic
93#1735C2211yescharacteristic
94#1735C2211yescharacteristic
95#1728C2211yescharacteristic
96C2×#336C2211yescharacteristic
97C23×Q8C2311yescharacteristic
98C23×Q8C2311yescharacteristic
99C23×D4C2311yescharacteristic
100C22×Q16—24no—
101C22×Q16—42no—
102C22×Q16—42no—
103C22×Q16C2312yes—
104C22×Q16C2312yes—
105C22×SD16—24no—
106C22×SD16—42no—
107C22×SD16—42no—
108C22×SD16C2312yes—
109C22×SD16C2312yes—
110C22×M16C2311yescharacteristic
111C22×M16C2311yescharacteristic
112C23×C8C2311yescharacteristic
113C23×C8C2311yescharacteristic
114#533—42no—
115#533—42no—
116C2×C4⋊Q8C2312yes—
117C2×#125C2312yes—
118C2×#124—22no—
119C2×#124—42no—
120C2×#124—42no—
121C2×#124C2312yes—
122C2×#124C2312yes—
123C2×C22⋊Q8—22no—
124C2×C4×Q8—22no—
125C2×#117C2311yescharacteristic
126C2×#117C2311yescharacteristic
127C2×#117C2311yescharacteristic
128C2×#117C2311yescharacteristic
129C2×#117C2311yescharacteristic
130C2×#117C2311yescharacteristic
131C22×C4⋊C4C2311yescharacteristic
132C22×C22⋊C4—22no—
133C22×C42C2311yescharacteristic
134#427—42no—
135#427—42no—
136C2×#107C2312yes—
137C2×#107C2312yes—
138C2×#106C2312yes—
139C2×#106C2312yes—
140#424—24no—
141#424—24no—
142C2×C4⋊C8C2312yes—
143C2×C4⋊C8C2312yes—
144C2×C4⋊C8C2312yes—
145C2×C4⋊C8C2312yes—
146#419—42no—
147#419—42no—
148#419—42no—
149#419—42no—
150#418—42no—
151#418—42no—
152C2×#103—24no—
153C2×#103C2312yes—
154C2×#103C2312yes—
155C2×#103C2314yes—
156C2×#103C2314yes—
157C2×#103C2314yes—
158C2×#103C2314yes—
159C2×#102—24no—
160C2×#102C2312yes—
161C2×#102C2312yes—
162C2×#102C2314yes—
163C2×#102C2314yes—
164#405—42no—
165#405—42no—
166C2×#97C2312yes—
167C2×#97C2312yes—
168C2×C4×C8C2312yes—
169C2×C4×C8C2312yes—
170#401—24no—
171#400—24no—
172#400—24no—
173#400—24no—
174#397—24no—
175#397—24no—
176#393—24no—
177#393—24no—
178#391—21no—
179#391—21no—
180#391—22no—
181#391—21no—
182#391—22no—
183#391—21no—
184#391—22no—
185#391C2314yes—
186#391C2314yes—
187#391C2314yes—
188#391C2314yes—
189#391C2312yes—
190#391C2312yes—
191#390—21no—
192#390—21no—
193#390—22no—
194#390C2314yes—
195#390C2314yes—
196#389—22no—
197#389C2312yes—
198#389C2312yes—
199#389C2314yes—
200#389C2314yes—
201#388—24no—
202#388—24no—
203#388—24no—
204#388—24no—
205#387—24no—
206#387—24no—
207#387—24no—
208#385—22no—
209#385—22no—
210#384—21no—
211#383—24no—
212#382—21no—
213#381—22no—
214#381—22no—
215#381—22no—
216#381—22no—
217#380—22no—
218#379—22no—
219#379—22no—
220C4×C4⋊C4—22no—
221C4×C4⋊C4D412yes—
222C4×C4⋊C4D412yes—
223C4×C22⋊C4D414yes—
224#376C2312yes—
225C2×#95—22no—
226C43C2312yes—
227#340C2312yes—
228#340C2312yes—
229#340C2312yes—
230#340C2312yes—
231#340C2314yes—
232#340C2314yes—
233#336C2311yescharacteristic
234#336C2311yescharacteristic
235#336C2311yescharacteristic
236#336C2311yescharacteristic
237#336C2312yes—
238#336C2312yes—
239C25—21no—
240C22×Q8—21no—
241C22×Q8—21no—
242C22×Q8—22no—
243C22×Q8—22no—
244C22×Q8—22no—
245C22×Q8—22no—
246C22×Q8—22no—
247C22×Q8—22no—
248C22×Q8—22no—
249C22×Q8—22no—
250C22×Q8—22no—
251C22×Q8—22no—
252C22×Q8—22no—
253C22×Q8—22no—
254C22×Q8—22no—
255C22×Q8C2412yes—
256C22×Q8C2412yes—
257C22×D4—21no—
258C22×D4—22no—
259C22×D4—22no—
260C22×D4—22no—
261C22×D4—22no—
262C22×D4—22no—
263C22×D4—22no—
264C22×D4C2412yes—
265C23×C4—21no—
266C23×C4—21no—
267C23×C4—21no—
268C23×C4C2×D412yes—
269C23×C4C2411yescharacteristic
270C2×Q16—24no—
271C2×Q16—24no—
272C2×Q16—42no—
273C2×Q16—42no—
274C2×Q16—42no—
275C2×Q16—42no—
276C2×Q16—416no—
277C2×SD16—24no—
278C2×SD16—24no—
279C2×SD16—42no—
280C2×SD16—42no—
281C2×SD16—42no—
282C2×SD16—42no—
283C2×SD16—416no—
284C2×M16—22no—
285C2×M16—22no—
286C2×M16—22no—
287C2×M16—22no—
288C2×M16—22no—
289C2×M16—22no—
290C22×C8—22no—
291C22×C8—22no—
292C22×C8—22no—
293C22×C8—22no—
294C22×C8#4114yes—
295C22×C8#4114yes—
296C22×C8C2412yes—
297C22×C8C2412yes—
298C22×C8C2411yescharacteristic
299C22×C8C2411yescharacteristic
300C22×C8C2411yescharacteristic
301C22×C8C2411yescharacteristic
302C4⋊Q8—42no—
303C4⋊Q8—42no—
304#125—42no—
305#125—42no—
306#124—42no—
307#124—42no—
308#124—42no—
309#124—42no—
310#124—48no—
311C22⋊Q8—48no—
312C4×Q8—48no—
313#117—22no—
314#117—22no—
315#117—22no—
316#117—22no—
317#117—22no—
318#117—22no—
319#117—22no—
320#117—22no—
321#117—22no—
322#117—22no—
323#117—22no—
324#117—22no—
325C2×C4⋊C4—22no—
326C2×C4⋊C4—22no—
327C2×C4⋊C4—21no—
328C2×C4⋊C4—22no—
329C2×C4⋊C4—22no—
330C2×C4⋊C4—22no—
331C2×C4⋊C4—22no—
332C2×C4⋊C4—22no—
333C2×C4⋊C4—24no—
334C2×C4⋊C4—24no—
335C2×C4⋊C4C2×D411yescharacteristic
336C2×C4⋊C4C2×D411yescharacteristic
337C2×C4⋊C4C2×D412yes—
338C2×C4⋊C4C2412yes—
339C2×C4⋊C4C2412yes—
340C2×C4⋊C4C2412yes—
341C2×C4⋊C4C2412yes—
342C2×C4⋊C4C2412yes—
343C2×C22⋊C4—21no—
344C2×C22⋊C4—21no—
345C2×C22⋊C4—21no—
346C2×C22⋊C4—21no—
347C2×C22⋊C4—24no—
348C2×C22⋊C4—24no—
349C2×C22⋊C4—24no—
350C2×C22⋊C4—24no—
351C2×C22⋊C4—24no—
352C2×C22⋊C4—24no—
353C2×C22⋊C4—48no—
354C2×C22⋊C4C2×D412yes—
355C2×C22⋊C4C2×D412yes—
356C2×C42—22no—
357C2×C42—22no—
358C2×C42—21no—
359C2×C42—21no—
360C2×C42—21no—
361C2×C42—21no—
362C2×C42C2×D412yes—
363C2×C42C2×D414yes—
364C2×C42C2×D414yes—
365C2×C42C2412yes—
366C2×C42C2412yes—
367C2×C42C2411yescharacteristic
368C2×C42C2411yescharacteristic
369#107—42no—
370#107—42no—
371#106—42no—
372#106—42no—
373C4⋊C8—24no—
374C4⋊C8—24no—
375C4⋊C8—24no—
376C4⋊C8—24no—
377#103—24no—
378#103—24no—
379#103—44no—
380#103—44no—
381#103—44no—
382#103—44no—
383#102—24no—
384#102—24no—
385#102—44no—
386#102—44no—
387#97—42no—
388#97—42no—
389C4×C8—42no—
390C4×C8—42no—
391#95—24no—
392#95—24no—
393#95—22no—
394#95—22no—
395#95—24no—
396#95—24no—
397#95—24no—
398#95—24no—
399#95—24no—
400#95—24no—
401#95—24no—
402#95—24no—
403#95—24no—
404#95—24no—
405#95C2×D414yes—
406#95C2×D414yes—
407C24—22no—
408C24—42no—
409C24—42no—
410C24—42no—
411C24—82no—
412C24—82no—
413C24C22×D411yescharacteristic
414C2×Q8—22no—
415C2×Q8—22no—
416C2×Q8—22no—
417C2×Q8—22no—
418C2×Q8—22no—
419C2×Q8—22no—
420C2×Q8—22no—
421C2×Q8—22no—
422C2×Q8—22no—
423C2×Q8—22no—
424C2×Q8—22no—
425C2×Q8—22no—
426C2×Q8—44no—
427C2×Q8—42no—
428C2×Q8—42no—
429C2×Q8—42no—
430C2×Q8—42no—
431C2×Q8—42no—
432C2×Q8—42no—
433C2×Q8—44no—
434C2×Q8—44no—
435C2×Q8—44no—
436C2×Q8—44no—
437C2×Q8—44no—
438C2×Q8—44no—
439C2×Q8—44no—
440C2×Q8—44no—
441C2×Q8—82no—
442C2×Q8—82no—
443C2×D4—22no—
444C2×D4—22no—
445C2×D4—22no—
446C2×D4—22no—
447C2×D4—22no—
448C2×D4—22no—
449C2×D4—42no—
450C2×D4—42no—
451C2×D4—42no—
452C2×D4—44no—
453C2×D4—44no—
454C2×D4—44no—
455C2×D4—44no—
456C22×C4—24no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—42no—
465C22×C4—42no—
466C22×C4—42no—
467C22×C4—42no—
468C22×C4—42no—
469C22×C4—42no—
470C22×C4—42no—
471C22×C4—42no—
472C22×C4—42no—
473C22×C4—82no—
474C22×C4C22×D412yes—
475C22×C4C22×D414yes—
476C22×C4C22×D412yes—
477C22×C4C22×D412yes—
478C22×C4C22×D412yes—
479C22×C4C22×D411yescharacteristic
480C22×C4C2×#4112yes—
481C22×C4C2×#4112yes—
482C22×C4H(5,𝔽2)12yes—
483C22×C4H(5,𝔽2)12yes—
484C22×C4H(5,𝔽2)12yes—
485C22×C42−1+412yes—
486C22×C4C2511yescharacteristic
487Q16—48no—
488Q16—48no—
489SD16—48no—
490SD16—48no—
491M16—44no—
492M16—44no—
493C2×C8—28no—
494C2×C8—28no—
495C2×C8—21no—
496C2×C8—21no—
497C2×C8—21no—
498C2×C8—21no—
499C2×C8—21no—
500C2×C8—21no—
501C2×C8—21no—
502C2×C8—21no—
503C2×C8—22no—
504C2×C8—22no—
505C2×C8—22no—
506C2×C8—22no—
507C2×C8—22no—
508C2×C8—22no—
509C2×C8C2×#4112yes—
510C2×C8C2×#4112yes—
511C2×C8C2×#4112yes—
512C2×C8C2×#4112yes—
513C4⋊C4—22no—
514C4⋊C4—22no—
515C4⋊C4—24no—
516C4⋊C4—22no—
517C4⋊C4—22no—
518C4⋊C4—22no—
519C4⋊C4—22no—
520C4⋊C4—22no—
521C4⋊C4—22no—
522C4⋊C4—22no—
523C4⋊C4—22no—
524C4⋊C4—22no—
525C4⋊C4—22no—
526C4⋊C4—44no—
527C4⋊C4—44no—
528C4⋊C4—44no—
529C4⋊C4—44no—
530C4⋊C4—42no—
531C4⋊C4—84no—
532C4⋊C4—84no—
533C4⋊C4—82no—
534C22⋊C4—24no—
535C22⋊C4—24no—
536C22⋊C4—48no—
537C22⋊C4—48no—
538C22⋊C4—42no—
539C22⋊C4—42no—
540C22⋊C4—42no—
541C22⋊C4—42no—
542C22⋊C4—84no—
543C22⋊C4—84no—
544C22⋊C4—84no—
545C22⋊C4—84no—
546C42—24no—
547C42—22no—
548C42—22no—
549C42—22no—
550C42—21no—
551C42—21no—
552C42—21no—
553C42—21no—
554C42—22no—
555C42—44no—
556C42—44no—
557C42—42no—
558C42—42no—
559C42—42no—
560C42—42no—
561C23—22no—
562C23—22no—
563C23—44no—
564C23—44no—
565C23—42no—
566C23—42no—
567C23—42no—
568C23—88no—
569C23—84no—
570C23—162no—
571C23—162no—
572C23—164no—
573C23#53212yes—
574C23#53512yes—
575C23#53712yes—
576C23C23×D411yescharacteristic
577Q8—42no—
578Q8—42no—
579Q8—42no—
580Q8—42no—
581Q8—42no—
582Q8—42no—
583Q8—42no—
584Q8—42no—
585Q8—88no—
586Q8—84no—
587Q8—84no—
588D4—42no—
589D4—42no—
590D4—42no—
591D4—42no—
592D4—84no—
593C2×C4—22no—
594C2×C4—24no—
595C2×C4—24no—
596C2×C4—22no—
597C2×C4—22no—
598C2×C4—24no—
599C2×C4—24no—
600C2×C4—22no—
601C2×C4—22no—
602C2×C4—22no—
603C2×C4—22no—
604C2×C4—22no—
605C2×C4—24no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—44no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—84no—
627C2×C4—84no—
628C2×C4—84no—
629C2×C4—162no—
630C2×C4#53212yes—
631C2×C4#53512yes—
632C2×C4#53712yes—
633C2×C4#53814yes—
634C2×C4#53814yes—
635C2×C4#54214yes—
636C2×C4#54214yes—
637C2×C4C22×#4111yescharacteristic
638C2×C4C22×#4111yescharacteristic
639C2×C4C2×H(5,𝔽2)11yescharacteristic
640C2×C4C2×H(5,𝔽2)11yescharacteristic
641C2×C4C2×H(5,𝔽2)11yescharacteristic
642C2×C4C2×2−1+411yescharacteristic
643C8—22no—
644C8—22no—
645C8—22no—
646C8—22no—
647C8—42no—
648C8—42no—
649C22—22no—
650C22—22no—
651C22—24no—
652C22—24no—
653C22—82no—
654C22—82no—
655C22—82no—
656C22—84no—
657C22—84no—
658C22—168no—
659C22#266214yes—
660C22C2×#53211yescharacteristic
661C22C2×#53511yescharacteristic
662C22C2×#53711yescharacteristic
663C22#358412yes—
664C22#358612yes—
665C4—42no—
666C4—42no—
667C4—42no—
668C4—42no—
669C4—44no—
670C4—82no—
671C4—82no—
672C4—82no—
673C4—82no—
674C4—82no—
675C4—82no—
676C4—82no—
677C4#266214yes—
678C4C2×#53811yescharacteristic
679C4C2×#53811yescharacteristic
680C4C2×#54211yescharacteristic
681C4C2×#54211yescharacteristic
682C2—24no—
683C2—162no—
684C2C2×#266211yescharacteristic
685C2#6071311yescharacteristic
686C2#6072011yescharacteristic
687C2#6081612yes—
688C2#6092712yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #35313C2semidirect
3. #30352C2semidirect
4. #30350C2semidirect
5. #30347C2semidirect
6. #30341C2semidirect
7. #23613C2semidirect
9. #23551C2semidirect
10. #23542C2semidirect
13. #23455C2semidirect
15. #21232C2semidirect
18. #21088C2semidirect
19. #20430C2semidirect
38. #2863C22semidirect
40. #2645C22semidirect
59. #2068C22semidirect
61. #2066C22semidirect
63. #2059C22semidirect
64. #2059C22semidirect
65. #2058C22semidirect
66. #2057C22semidirect
67. #2056C22semidirect
68. #2056C22semidirect
73. #2026C22semidirect
74. #2024C22semidirect
80. #1971C22semidirect
81. #1969C22semidirect
84. #1935C22semidirect
86. #1933C22semidirect
87. #1768C22semidirect
88. #1768C22semidirect
89. #1767C22semidirect
90. C4×C4⋊C8C22semidirect

Maximal subgroups

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

subgroupindex
#353382
#353132
#303522
#303502
#303472
#303412
#236132
#236122
#235512
#235422
#235412
#235062
#234552
C2×#20152
#212322
#212312
#210892
#210882
#204302