Finite Group Databasebeta

← #7501919

Subgroups — #7501919

Order 512.

1847 conjugacy classes of subgroups.

The subgroup lattice has 717 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#499C211yesmaximal, characteristic, complemented
2C2×#3098C211yesmaximal, characteristic
3#35649C211yesmaximal, characteristic
4#35639C211yesmaximal, characteristic, complemented
5#30787C214yesmaximal, complemented
6#30786C214yesmaximal, complemented
7#30784C214yesmaximal, complemented
8#30781C214yesmaximal, complemented
9#23737C211yesmaximal, characteristic, complemented
10#23639C211yesmaximal, characteristic, complemented
11#23610C211yesmaximal, characteristic, complemented
12#23553C211yesmaximal, characteristic, complemented
13#23489C211yesmaximal, characteristic
14#23453C211yesmaximal, characteristic, complemented
15#21174C211yesmaximal, characteristic, complemented
16#21083C211yesmaximal, characteristic, complemented
17#21046C211yesmaximal, characteristic, complemented
18#21037C211yesmaximal, characteristic, complemented
19#20848C211yesmaximal, characteristic, complemented
20C2×#537C2211yescharacteristic
21C22×#125C2211yescharacteristic
22C22×#125C2211yescharacteristic
23C2×#520C2211yescharacteristic
24C2×#499—28no—
25C2×#499C2214yes—
26C2×#499C2214yes—
27#3098—28no—
28C2×#484C2212yes—
29C2×#485C2212yes—
30C2×#482C2212yes—
31C2×#480C2212yes—
32C2×#428C2211yescharacteristic
33C2×#428C2211yescharacteristic
34C22×#107C2211yescharacteristic
35C22×#107C2211yescharacteristic
36C22×#106C2211yescharacteristic
37C22×#106C2211yescharacteristic
38C2×#424C2211yescharacteristic
39C2×#417C2211yescharacteristic
40C2×#417C2211yescharacteristic
41C2×#416C2211yescharacteristic
42C2×#416C2211yescharacteristic
43C2×#408C2211yescharacteristic
44C2×#408C2211yescharacteristic
45C22×C4×C8C2211yescharacteristic
46#2591—22no—
47#2579—22no—
48#2504—22no—
49#2490—22no—
50#2490C2214yes—
51#2489—22no—
52#2489C2214yes—
53#2487—22no—
54#2454—22no—
55#2453—22no—
56#2442C2214yes—
57#2441C2214yes—
58#2317C2211yescharacteristic
59#2316C2211yescharacteristic
60#2311C2211yescharacteristic
61#2311C2211yescharacteristic
62#2311C2211yescharacteristic
63#2297C2211yescharacteristic
64#2297C2211yescharacteristic
65#2296C2211yescharacteristic
66#2274C2211yescharacteristic
67#2097C2212yes—
68#2095C2212yes—
69#2093C2212yes—
70#2092C2212yes—
71#2076C2214yes—
72#2075C2214yes—
73#2066C2214yes—
74#2065C2214yes—
75#2059C2214yes—
76#2059C2214yes—
77#2035C2214yes—
78#2034C2214yes—
79#2026C2214yes—
80#2025C2214yes—
81#1949C2214yes—
82#1948C2214yes—
83#1934C2212yes—
84#1934C2212yes—
85#1933C2212yes—
86#1933C2212yes—
87#1923C2214yes—
88#1923C2214yes—
89#1921C2214yes—
90#1918C2214yes—
91#1854C2214yes—
92#1853C2214yes—
93#1736C2211yescharacteristic
94#1736C2211yescharacteristic
95#1736C2211yescharacteristic
96#1736C2211yescharacteristic
97#1735C2211yescharacteristic
98#1735C2211yescharacteristic
99#1729C2211yescharacteristic
100#1728C2211yescharacteristic
101C22×#41C2311yescharacteristic
102C22×#41C2311yescharacteristic
103C22×M16C2311yescharacteristic
104C22×M16C2311yescharacteristic
105C23×C8C2311yescharacteristic
106C23×C8C2311yescharacteristic
107#537—24no—
108#537—42no—
109C2×#125—24no—
110C2×#125—24no—
111C2×#125—22no—
112C2×#125—22no—
113C2×#125—42no—
114C2×#125—42no—
115C2×#125C2314yes—
116C2×#125C2314yes—
117C2×C22⋊Q8—21no—
118C2×C22⋊Q8C2312yes—
119C2×#121—22no—
120C2×#121—21no—
121C2×#121C2312yes—
122#520—24no—
123#520—42no—
124C2×C4×Q8—21no—
125C2×C4×D4—22no—
126C2×C4×D4—21no—
127C2×#117C2311yescharacteristic
128C2×#117C2311yescharacteristic
129C2×#117C2311yescharacteristic
130C2×#117C2312yes—
131C2×#117C2311yescharacteristic
132C22×C4⋊C4—21no—
133C22×C4⋊C4—21no—
134C22×C4⋊C4—21no—
135C22×C4⋊C4—21no—
136C22×C4⋊C4C2311yescharacteristic
137C22×C4⋊C4C2311yescharacteristic
138C22×C4⋊C4C2312yes—
139C22×C4⋊C4C2311yescharacteristic
140C22×C4⋊C4C2311yescharacteristic
141C22×C42C2311yescharacteristic
142#499—216no—
143#499—216no—
144#485—28no—
145#484—28no—
146#482—28no—
147#480—28no—
148#428—24no—
149#428—42no—
150C2×#107—24no—
151C2×#107D418yes—
152C2×#107C2312yes—
153C2×#107C2312yes—
154C2×#107C2312yes—
155C2×#107C2312yes—
156C2×#107C2312yes—
157C2×#107C2312yes—
158C2×#106—24no—
159C2×#106—24no—
160C2×#106C2312yes—
161C2×#106C2314yes—
162C2×#106C2314yes—
163C2×#106C2312yes—
164#424—24no—
165C2×C4⋊C8C2314yes—
166#417—24no—
167#417—42no—
168#416—24no—
169#416—24no—
170C2×#103C2311yescharacteristic
171C2×#103C2312yes—
172C2×#103C2312yes—
173C2×#103C2311yescharacteristic
174C2×#103C2312yes—
175C2×#102C2311yescharacteristic
176C2×#102C2312yes—
177C2×#102C2311yescharacteristic
178C2×#102C2312yes—
179C2×#102C2312yes—
180#408—24no—
181#408—24no—
182C2×C22⋊C8C2312yes—
183C2×C22⋊C8C2312yes—
184C2×C22⋊C8C2314yes—
185C2×C4×C8—24no—
186C2×C4×C8C2312yes—
187C2×C4×C8C2312yes—
188#400—24no—
189#400—24no—
190#399—24no—
191#398—24no—
192#397—24no—
193#397—24no—
194#396—24no—
195#394—24no—
196#391—21no—
197#391—21no—
198#391D412yes—
199#391C2312yes—
200#391C2312yes—
201#390—21no—
202#390—21no—
203#390—24no—
204#390D412yes—
205#390C2312yes—
206#390C2312yes—
207#389—22no—
208#389—22no—
209#389—24no—
210#389C2312yes—
211#389C2312yes—
212#389C2314yes—
213#388—24no—
214#388—24no—
215#387—24no—
216#386—22no—
217#386—22no—
218#386—21no—
219#386—21no—
220#386—22no—
221#386D412yes—
222#386D412yes—
223#385—21no—
224#385—21no—
225#385—22no—
226#385—24no—
227#384C2312yes—
228#382—24no—
229#382C2312yes—
230#380—22no—
231#380—22no—
232#380—24no—
233#380C2314yes—
234#380C2314yes—
235C4×C4⋊C4—22no—
236C4×C4⋊C4—22no—
237C4×C4⋊C4C2314yes—
238C4×C22⋊C4—22no—
239C4×C22⋊C4—22no—
240C4×C22⋊C4C2314yes—
241#340C2314yes—
242#340C2314yes—
243#340C2312yes—
244#340C2314yes—
245#340C2312yes—
246#340C2312yes—
247#340C2314yes—
248#340C2312yes—
249#336C2314yes—
250#336C2312yes—
251#336C2312yes—
252C2×#41—22no—
253C2×#41—22no—
254C2×#41—22no—
255C2×#41—22no—
256C2×#41—22no—
257C2×#41—22no—
258C2×#41—22no—
259C2×#41—22no—
260C2×#41—22no—
261C2×#41—22no—
262C2×#41—22no—
263C2×#41—22no—
264C22×Q8C2×D411yescharacteristic
265C22×Q8C2411yescharacteristic
266C22×D4—21no—
267C22×D4—22no—
268C22×D4C2×D412yes—
269C22×D4C2×D411yescharacteristic
270C22×D4C2411yescharacteristic
271C23×C4—21no—
272C23×C4—21no—
273C23×C4—21no—
274C23×C4—21no—
275C23×C4—22no—
276C23×C4—21no—
277C23×C4C2×D411yescharacteristic
278C23×C4C2×D411yescharacteristic
279C23×C4C2412yes—
280C23×C4C2411yescharacteristic
281C2×M16—22no—
282C2×M16—22no—
283C2×M16—22no—
284C2×M16—22no—
285C2×M16—22no—
286C2×M16—22no—
287C22×C8—22no—
288C22×C8—22no—
289C22×C8—22no—
290C22×C8—22no—
291C22×C8C2×D414yes—
292C22×C8#4114yes—
293C22×C8C2412yes—
294C22×C8C2412yes—
295C22×C8C2411yescharacteristic
296C22×C8C2411yescharacteristic
297C22×C8C2412yes—
298#125—28no—
299#125—28no—
300#125—44no—
301#125—44no—
302#125—48no—
303#125—48no—
304C22⋊Q8—24no—
305C22⋊Q8—42no—
306C22⋊Q8—44no—
307#121—24no—
308#121—42no—
309#121—44no—
310#121—48no—
311C4×Q8—44no—
312C4×D4—44no—
313C4×D4—48no—
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—24no—
323#117—42no—
324C2×C4⋊C4—21no—
325C2×C4⋊C4—21no—
326C2×C4⋊C4—22no—
327C2×C4⋊C4—22no—
328C2×C4⋊C4—22no—
329C2×C4⋊C4—22no—
330C2×C4⋊C4—22no—
331C2×C4⋊C4—22no—
332C2×C4⋊C4—24no—
333C2×C4⋊C4—24no—
334C2×C4⋊C4—24no—
335C2×C4⋊C4—24no—
336C2×C4⋊C4—24no—
337C2×C4⋊C4—22no—
338C2×C4⋊C4—21no—
339C2×C4⋊C4—21no—
340C2×C4⋊C4—22no—
341C2×C4⋊C4—22no—
342C2×C4⋊C4—22no—
343C2×C4⋊C4—24no—
344C2×C4⋊C4—24no—
345C2×C4⋊C4—42no—
346C2×C4⋊C4—44no—
347C2×C4⋊C4—44no—
348C2×C4⋊C4—44no—
349C2×C4⋊C4—44no—
350C2×C4⋊C4C2×D412yes—
351C2×C4⋊C4C2×D414yes—
352C2×C4⋊C4C2×D414yes—
353C2×C4⋊C4C2411yescharacteristic
354C2×C4⋊C4C2412yes—
355C2×C4⋊C4C2412yes—
356C2×C4⋊C4C2411yescharacteristic
357C2×C4⋊C4C2412yes—
358C2×C4⋊C4C2412yes—
359C2×C4⋊C4C2412yes—
360C2×C4⋊C4C2412yes—
361C2×C22⋊C4—21no—
362C2×C22⋊C4—22no—
363C2×C22⋊C4—21no—
364C2×C22⋊C4—21no—
365C2×C22⋊C4—22no—
366C2×C22⋊C4—22no—
367C2×C22⋊C4—24no—
368C2×C22⋊C4—24no—
369C2×C22⋊C4—22no—
370C2×C22⋊C4—22no—
371C2×C22⋊C4—24no—
372C2×C22⋊C4C2×D412yes—
373C2×C22⋊C4C2×D414yes—
374C2×C22⋊C4C2×D414yes—
375C2×C42—22no—
376C2×C42—22no—
377C2×C42—21no—
378C2×C42—22no—
379C2×C42—21no—
380C2×C42—21no—
381C2×C42—24no—
382C2×C42C2×D412yes—
383C2×C42C2412yes—
384C2×C42C2412yes—
385#107—24no—
386#107—24no—
387#107—24no—
388#107—42no—
389#107C2×D418yes—
390#107C2×D418yes—
391#106—28no—
392#106—28no—
393#106—24no—
394#106—42no—
395C4⋊C8—28no—
396#103—22no—
397#103—22no—
398#103—24no—
399#103—24no—
400#103—42no—
401#102—22no—
402#102—22no—
403#102—24no—
404#102—24no—
405#102—42no—
406C22⋊C8—24no—
407C22⋊C8—24no—
408C22⋊C8—28no—
409C4×C8—24no—
410C4×C8—24no—
411#95—24no—
412#95—22no—
413#95—22no—
414#95—22no—
415#95—22no—
416#95—24no—
417#95—24no—
418#95—24no—
419#95—24no—
420#95—24no—
421#95C2×D414yes—
422#95C2×D414yes—
423C24—21no—
424C24C22×D412yes—
425C24C22×D411yescharacteristic
426#41—44no—
427#41—44no—
428#41—44no—
429#41—44no—
430#41—44no—
431#41—44no—
432#41—44no—
433#41—44no—
434C2×Q8—22no—
435C2×Q8—22no—
436C2×Q8—22no—
437C2×Q8—21no—
438C2×Q8—21no—
439C2×Q8—21no—
440C2×Q8—21no—
441C2×Q8—21no—
442C2×Q8—21no—
443C2×D4—24no—
444C2×D4—24no—
445C2×D4—24no—
446C2×D4—22no—
447C2×D4—22no—
448C2×D4—22no—
449C2×D4—21no—
450C2×D4—21no—
451C2×D4—21no—
452C2×D4—21no—
453C2×D4—21no—
454C2×D4—21no—
455C2×D4—44no—
456C2×D4—42no—
457C2×D4—42no—
458C2×D4—42no—
459C2×D4—84no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—24no—
465C22×C4—24no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—21no—
473C22×C4—22no—
474C22×C4—22no—
475C22×C4—21no—
476C22×C4—22no—
477C22×C4—22no—
478C22×C4—21no—
479C22×C4—42no—
480C22×C4—42no—
481C22×C4—44no—
482C22×C4—42no—
483C22×C4—42no—
484C22×C4—42no—
485C22×C4—42no—
486C22×C4—42no—
487C22×C4—82no—
488C22×C4—82no—
489C22×C4—82no—
490C22×C4—82no—
491C22×C4—84no—
492C22×C4C22×D412yes—
493C22×C4C22×D414yes—
494C22×C4C22×D412yes—
495C22×C4C22×D412yes—
496C22×C4C22×D411yescharacteristic
497C22×C4C22×D412yes—
498C22×C4C2×#4112yes—
499C22×C4H(5,𝔽2)12yes—
500C22×C42−1+412yes—
501C22×C42−1+412yes—
502C22×C42−1+412yes—
503C22×C4C2511yescharacteristic
504M16—44no—
505M16—44no—
506C2×C8—28no—
507C2×C8—28no—
508C2×C8—22no—
509C2×C8—21no—
510C2×C8—21no—
511C2×C8—21no—
512C2×C8—21no—
513C2×C8—22no—
514C2×C8—22no—
515C2×C8—22no—
516C2×C8—22no—
517C2×C8—22no—
518C2×C8—22no—
519C2×C8—22no—
520C2×C8C22×D412yes—
521C2×C8C22×D412yes—
522C2×C8C2×#4114yes—
523C4⋊C4—24no—
524C4⋊C4—24no—
525C4⋊C4—24no—
526C4⋊C4—22no—
527C4⋊C4—22no—
528C4⋊C4—21no—
529C4⋊C4—21no—
530C4⋊C4—22no—
531C4⋊C4—22no—
532C4⋊C4—22no—
533C4⋊C4—22no—
534C4⋊C4—21no—
535C4⋊C4—22no—
536C4⋊C4—21no—
537C4⋊C4—22no—
538C4⋊C4—22no—
539C4⋊C4—22no—
540C4⋊C4—44no—
541C4⋊C4—48no—
542C4⋊C4—48no—
543C4⋊C4—48no—
544C4⋊C4—48no—
545C4⋊C4—44no—
546C4⋊C4—42no—
547C4⋊C4—42no—
548C4⋊C4—84no—
549C4⋊C4—84no—
550C4⋊C4—82no—
551C4⋊C4—82no—
552C4⋊C4C22×D414yes—
553C4⋊C4C22×D414yes—
554C22⋊C4—24no—
555C22⋊C4—44no—
556C22⋊C4—44no—
557C22⋊C4—44no—
558C22⋊C4—44no—
559C22⋊C4—44no—
560C22⋊C4—42no—
561C22⋊C4—42no—
562C22⋊C4—42no—
563C22⋊C4—84no—
564C22⋊C4—82no—
565C22⋊C4—82no—
566C22⋊C4—84no—
567C22⋊C4—84no—
568C42—24no—
569C42—22no—
570C42—22no—
571C42—22no—
572C42—22no—
573C42—44no—
574C42—42no—
575C42—42no—
576C42—42no—
577C42—84no—
578C23—22no—
579C23—22no—
580C23—24no—
581C23—24no—
582C23—22no—
583C23—22no—
584C23—22no—
585C23—42no—
586C23—42no—
587C23—41no—
588C23—41no—
589C23—41no—
590C23#53612yes—
591C23D4212yes—
592C23#54712yes—
593C23C23×D411yescharacteristic
594Q8—44no—
595Q8—41no—
596Q8—41no—
597Q8—41no—
598Q8—41no—
599D4—48no—
600D4—44no—
601D4—41no—
602D4—41no—
603D4—41no—
604D4—41no—
605D4—84no—
606D4—164no—
607C2×C4—24no—
608C2×C4—22no—
609C2×C4—22no—
610C2×C4—24no—
611C2×C4—22no—
612C2×C4—22no—
613C2×C4—22no—
614C2×C4—22no—
615C2×C4—22no—
616C2×C4—22no—
617C2×C4—24no—
618C2×C4—24no—
619C2×C4—24no—
620C2×C4—22no—
621C2×C4—44no—
622C2×C4—44no—
623C2×C4—44no—
624C2×C4—44no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4—42no—
632C2×C4—44no—
633C2×C4—44no—
634C2×C4—42no—
635C2×C4—42no—
636C2×C4—42no—
637C2×C4—42no—
638C2×C4—41no—
639C2×C4—41no—
640C2×C4—41no—
641C2×C4—41no—
642C2×C4—42no—
643C2×C4—42no—
644C2×C4—42no—
645C2×C4—41no—
646C2×C4—84no—
647C2×C4—84no—
648C2×C4—84no—
649C2×C4—84no—
650C2×C4—84no—
651C2×C4—164no—
652C2×C4#53612yes—
653C2×C4#53614yes—
654C2×C4#54214yes—
655C2×C4D4212yes—
656C2×C4#54712yes—
657C2×C4#54814yes—
658C2×C4#56314yes—
659C2×C4C23×D411yescharacteristic
660C2×C4C22×#4111yescharacteristic
661C2×C4C2×H(5,𝔽2)11yescharacteristic
662C2×C4C2×2−1+411yescharacteristic
663C2×C4C2×2−1+411yescharacteristic
664C2×C4C2×2−1+411yescharacteristic
665C8—22no—
666C8—22no—
667C8—24no—
668C8—42no—
669C8—42no—
670C22—24no—
671C22—22no—
672C22—22no—
673C22—24no—
674C22—44no—
675C22—42no—
676C22—42no—
677C22—42no—
678C22—44no—
679C22—82no—
680C22—81no—
681C22—81no—
682C22—81no—
683C22—82no—
684C22#267414yes—
685C22C2×#53611yescharacteristic
686C22C2×D4211yescharacteristic
687C22C2×#54711yescharacteristic
688C22#358412yes—
689C22#358412yes—
690C4—44no—
691C4—42no—
692C4—44no—
693C4—44no—
694C4—42no—
695C4—42no—
696C4—82no—
697C4—81no—
698C4—81no—
699C4—81no—
700C4—82no—
701C4—82no—
702C4—82no—
703C4#267414yes—
704C4C2×#53611yescharacteristic
705C4C2×#54211yescharacteristic
706C4C2×#54811yescharacteristic
707C4C2×#56311yescharacteristic
708C2—24no—
709C2—82no—
710C2—161no—
711C2C2×#267411yescharacteristic
712C2#6088212yes—
713C2#6105111yescharacteristic
714C2#6105411yescharacteristic
715C2#6129012yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#537C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ #2297 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C22×#499C2semidirect
4. #35639C2semidirect
5. #30787C2semidirect
6. #30786C2semidirect
7. #30784C2semidirect
8. #30781C2semidirect
9. #23737C2semidirect
10. #23639C2semidirect
11. #23610C2semidirect
12. #23553C2semidirect
14. #23453C2semidirect
15. #21174C2semidirect
16. #21083C2semidirect
17. #21046C2semidirect
18. #21037C2semidirect
19. #20848C2semidirect

Maximal subgroups

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

subgroupindex
C22×#4992
C2×#30982
#356492
#356392
#307872
#307862
#307842
#307812
#237372
#236392
#236102
#235532
#234892
#234532
#211742
#210832
#210462
#210372
#208482