Finite Group Databasebeta

← #7501984

Subgroups — #7501984

Order 512.

1967 conjugacy classes of subgroups.

The subgroup lattice has 728 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3136C211yesmaximal, characteristic
2C22×#478C211yesmaximal, characteristic, complemented
3#35640C211yesmaximal, characteristic, complemented
4#35152C211yesmaximal, characteristic
5#30827C214yesmaximal, complemented
6#30823C214yesmaximal, complemented
7#30811C214yesmaximal, complemented
8#30807C214yesmaximal, complemented
9#23737C211yesmaximal, characteristic, complemented
10#23632C211yesmaximal, characteristic, complemented
11#23609C211yesmaximal, characteristic, complemented
12#23556C211yesmaximal, characteristic, complemented
13#23509C211yesmaximal, characteristic
14#23453C211yesmaximal, characteristic, complemented
15#21156C211yesmaximal, characteristic, complemented
16#21068C211yesmaximal, characteristic, complemented
17#21065C211yesmaximal, characteristic, complemented
18#21024C211yesmaximal, characteristic, complemented
19#20873C211yesmaximal, characteristic, complemented
20C2×#544C2211yescharacteristic
21C2×#535C2211yescharacteristic
22C22×#125C2211yescharacteristic
23C22×C4×D4C2211yescharacteristic
24#3136—28no—
25C2×#490C2212yes—
26C2×#489C2212yes—
27C2×#488C2212yes—
28C2×#486C2212yes—
29C2×#478—28no—
30C2×#478C2214yes—
31C2×#478C2214yes—
32C2×#428C2211yescharacteristic
33C2×#427C2211yescharacteristic
34C22×#107C2211yescharacteristic
35C22×#106C2211yescharacteristic
36C22×C4⋊C8C2211yescharacteristic
37C2×#417C2211yescharacteristic
38C2×#417C2211yescharacteristic
39C2×#418C2211yescharacteristic
40C2×#418C2211yescharacteristic
41C2×#416C2211yescharacteristic
42C2×#416C2211yescharacteristic
43C22×#102C2211yescharacteristic
44C22×#102C2211yescharacteristic
45C2×#405C2211yescharacteristic
46#2849—22no—
47#2622—22no—
48#2583—22no—
49#2552—22no—
50#2552C2214yescomplemented
51#2550—22no—
52#2550C2214yescomplemented
53#2502—22no—
54#2447C2214yes—
55#2445C2214yes—
56#2422—22no—
57#2420—22no—
58#2317C2211yescharacteristic
59#2317C2211yescharacteristic
60#2317C2211yescharacteristic
61#2311C2211yescharacteristic
62#2309C2211yescharacteristic
63#2297C2211yescharacteristic
64#2296C2211yescharacteristic
65#2296C2211yescharacteristic
66#2274C2211yescharacteristic
67#2097C2212yescomplemented
68#2095C2212yescomplemented
69#2093C2212yescomplemented
70#2092C2212yescomplemented
71#2073C2214yes—
72#2071C2214yes—
73#2068C2212yescomplemented
74#2067C2212yescomplemented
75#2066C2212yescomplemented
76#2065C2212yescomplemented
77#2059C2214yescomplemented
78#2058C2214yescomplemented
79#2042C2214yes—
80#2041C2214yes—
81#2026C2214yescomplemented
82#2025C2214yescomplemented
83#1944C2214yescomplemented
84#1942C2214yescomplemented
85#1927C2214yescomplemented
86#1926C2214yes—
87#1926C2214yescomplemented
88#1925C2214yes—
89#1917C2214yescomplemented
90#1916C2214yescomplemented
91#1864C2214yes—
92#1862C2214yes—
93#1736C2211yescharacteristic
94#1736C2211yescharacteristic
95#1736C2211yescharacteristic
96#1735C2211yescharacteristic
97#1729C2211yescharacteristic
98#1729C2211yescharacteristic
99#1729C2211yescharacteristic
100#1728C2211yescharacteristic
101C22×#41C2311yescharacteristic
102C23×D4C2311yescharacteristic
103C24×C4—21no—
104C22×M16C2311yescharacteristic
105C22×M16C2311yescharacteristic
106C23×C8C2311yescharacteristic
107C23×C8C2311yescharacteristic
108#544—24no—
109#544—42no—
110#535—24no—
111#535—42no—
112C2×C4⋊Q8—21no—
113C2×C4⋊Q8C2312yes—
114C2×#127C2312yes—
115C2×#125—21no—
116C2×#125—24no—
117C2×#125—22no—
118C2×#125—42no—
119C2×#125C2314yes—
120C2×#125C2312yes—
121C2×#124C2312yes—
122C2×#123—24no—
123C2×C22⋊Q8—22no—
124C2×#121—21no—
125C2×#121—21no—
126C2×#120—22no—
127C2×C4×D4—24no—
128C2×C4×D4—22no—
129C2×C4×D4—42no—
130C2×C4×D4C2314yes—
131C2×#117C2311yescharacteristic
132C2×#117C2311yescharacteristic
133C2×#117C2311yescharacteristic
134C2×#117C2311yescharacteristic
135C2×#117C2311yescharacteristic
136C22×C4⋊C4—21no—
137C22×C4⋊C4—21no—
138C22×C4⋊C4C2311yescharacteristic
139C22×C4⋊C4C2311yescharacteristic
140C22×C4⋊C4C2311yescharacteristic
141C22×C22⋊C4—21no—
142C22×C42C2311yescharacteristic
143#490—28no—
144#489—28no—
145#488—28no—
146#486—28no—
147#478—216no—
148#478—216no—
149#428—42no—
150#427—24no—
151C2×#107—24no—
152C2×#107C2312yes—
153C2×#107C2312yes—
154C2×#107C2312yes—
155C2×#107C2312yes—
156C2×#106—24no—
157C2×#106C2314yes—
158C2×#106C2312yes—
159C2×#106C2312yes—
160C2×C4⋊C8—24no—
161C2×C4⋊C8C2312yes—
162C2×C4⋊C8C2312yes—
163#418—24no—
164#418—24no—
165#417—24no—
166#417—42no—
167#416—24no—
168#416—24no—
169C2×#103C2311yescharacteristic
170C2×#103C2312yes—
171C2×#103C2311yescharacteristic
172C2×#103C2312yes—
173C2×#103C2312yes—
174C2×#102—24no—
175C2×#102D418yes—
176C2×#102C2314yes—
177C2×#102C2311yescharacteristic
178C2×#102C2312yes—
179C2×#102C2311yescharacteristic
180C2×#102C2312yes—
181C2×#102C2312yes—
182C2×#102C2312yes—
183C2×#102C2314yes—
184C2×#102C2312yes—
185C2×#102C2312yes—
186C2×#102C2312yes—
187#405—24no—
188C2×#97C2312yes—
189C2×C4×C8C2312yes—
190#398—24no—
191#398—24no—
192#398—24no—
193#397—24no—
194#397—24no—
195#395—24no—
196#394—24no—
197#394—24no—
198#391—24no—
199#391C2312yes—
200#391C2312yes—
201#390—22no—
202#390C2314yes—
203#390C2312yes—
204#390C2312yes—
205#389—22no—
206#389—24no—
207#389—24no—
208#389D412yes—
209#389D412yes—
210#389C2314yes—
211#389C2314yes—
212#389C2312yes—
213#389C2312yes—
214#386—21no—
215#386—21no—
216#386—22no—
217#386—24no—
218#385—21no—
219#385—21no—
220#385—24no—
221#385—22no—
222#385—22no—
223#385—22no—
224#385D412yes—
225#385D412yes—
226#384—21no—
227#384—21no—
228#384—24no—
229#384C2312yes—
230#382—21no—
231#382—21no—
232#382C2312yes—
233#381—24no—
234#380—22no—
235#379—22no—
236C4×C4⋊C4—22no—
237C4×C4⋊C4—22no—
238C4×C4⋊C4C2314yes—
239C4×C4⋊C4C2314yes—
240C4×C22⋊C4—22no—
241C4×C22⋊C4—22no—
242#340C2312yes—
243#340C2314yes—
244#340C2312yes—
245#340C2314yes—
246#340C2314yes—
247#336C2314yes—
248#336C2314yes—
249#336C2312yes—
250#336C2312yes—
251#336C2314yes—
252C25—21no—
253C2×#41—22no—
254C2×#41—22no—
255C2×#41—22no—
256C2×#41—22no—
257C2×#41—22no—
258C2×#41—22no—
259C22×Q8—22no—
260C22×Q8C2411yescharacteristic
261C22×D4—21no—
262C22×D4—21no—
263C22×D4—22no—
264C22×D4—22no—
265C22×D4—22no—
266C22×D4—22no—
267C22×D4—22no—
268C22×D4—22no—
269C22×D4—22no—
270C22×D4C2×D414yes—
271C22×D4C2411yescharacteristic
272C22×D4C2412yes—
273C23×C4—21no—
274C23×C4—21no—
275C23×C4—21no—
276C23×C4—21no—
277C23×C4—24no—
278C23×C4—44no—
279C23×C4—84no—
280C23×C4C2×D411yescharacteristic
281C23×C4C2×D411yescharacteristic
282C23×C4C2411yescharacteristic
283C2×M16—22no—
284C2×M16—22no—
285C2×M16—22no—
286C2×M16—22no—
287C2×M16—22no—
288C2×M16—22no—
289C22×C8—22no—
290C22×C8—22no—
291C22×C8—22no—
292C22×C8—22no—
293C22×C8—22no—
294C22×C8C2×D414yes—
295C22×C8C2412yes—
296C22×C8C2412yes—
297C22×C8C2412yes—
298C22×C8C2411yescharacteristic
299C22×C8C2411yescharacteristic
300C4⋊Q8—24no—
301C4⋊Q8—42no—
302C4⋊Q8—44no—
303#127—24no—
304#127—42no—
305#125—28no—
306#125—24no—
307#125—44no—
308#125—42no—
309#125—48no—
310#125—44no—
311#124—24no—
312#124—42no—
313#123—88no—
314C22⋊Q8—48no—
315#121—44no—
316#121—44no—
317#120—48no—
318C4×D4—28no—
319C4×D4—44no—
320C4×D4—48no—
321#117—22no—
322#117—22no—
323#117—22no—
324#117—22no—
325#117—22no—
326#117—22no—
327#117—22no—
328#117—22no—
329#117—22no—
330#117#4114yes—
331C2×C4⋊C4—21no—
332C2×C4⋊C4—22no—
333C2×C4⋊C4—22no—
334C2×C4⋊C4—22no—
335C2×C4⋊C4—22no—
336C2×C4⋊C4—22no—
337C2×C4⋊C4—24no—
338C2×C4⋊C4—24no—
339C2×C4⋊C4—21no—
340C2×C4⋊C4—21no—
341C2×C4⋊C4—22no—
342C2×C4⋊C4—22no—
343C2×C4⋊C4—22no—
344C2×C4⋊C4—24no—
345C2×C4⋊C4—22no—
346C2×C4⋊C4—22no—
347C2×C4⋊C4—24no—
348C2×C4⋊C4—24no—
349C2×C4⋊C4—44no—
350C2×C4⋊C4—44no—
351C2×C4⋊C4C2×D411yescharacteristic
352C2×C4⋊C4C2×D411yescharacteristic
353C2×C4⋊C4C2×D412yes—
354C2×C4⋊C4C2×D414yes—
355C2×C4⋊C4C2×D414yes—
356C2×C4⋊C4C2×D414yes—
357C2×C4⋊C4C2412yes—
358C2×C4⋊C4C2412yes—
359C2×C4⋊C4C2411yescharacteristic
360C2×C4⋊C4C2411yescharacteristic
361C2×C4⋊C4C2412yes—
362C2×C4⋊C4C2412yes—
363C2×C4⋊C4C2412yes—
364C2×C22⋊C4—21no—
365C2×C22⋊C4—21no—
366C2×C22⋊C4—21no—
367C2×C22⋊C4—24no—
368C2×C22⋊C4—22no—
369C2×C22⋊C4—22no—
370C2×C22⋊C4—22no—
371C2×C22⋊C4—24no—
372C2×C22⋊C4—24no—
373C2×C22⋊C4—22no—
374C2×C22⋊C4—24no—
375C2×C22⋊C4—44no—
376C2×C22⋊C4C2×D412yes—
377C2×C22⋊C4C2×D412yes—
378C2×C42—21no—
379C2×C42—22no—
380C2×C42—22no—
381C2×C42—21no—
382C2×C42—21no—
383C2×C42—24no—
384C2×C42C2×D412yes—
385C2×C42C2412yes—
386C2×C42C2412yes—
387C2×C42C2412yes—
388#107—24no—
389#107—24no—
390#107—24no—
391#107—42no—
392#106—28no—
393#106—24no—
394#106—42no—
395C4⋊C8—24no—
396C4⋊C8—24no—
397#103—22no—
398#103—22no—
399#103—24no—
400#103—24no—
401#103—42no—
402#102—28no—
403#102—24no—
404#102—24no—
405#102—22no—
406#102—22no—
407#102—24no—
408#102—24no—
409#102—28no—
410#102—42no—
411#102C2×D418yes—
412#102C2×D418yes—
413#97—24no—
414C4×C8—24no—
415#95—24no—
416#95—22no—
417#95—22no—
418#95—22no—
419#95—22no—
420#95—24no—
421#95—24no—
422#95—24no—
423#95—24no—
424#95C2×D414yes—
425#95C2×D414yes—
426C24—22no—
427C24—24no—
428C24—21no—
429C24—42no—
430C24—42no—
431C24—82no—
432C24—82no—
433C24C22×D411yescharacteristic
434#41—44no—
435#41—44no—
436#41—44no—
437#41—44no—
438C2×Q8—21no—
439C2×Q8—21no—
440C2×Q8—21no—
441C2×Q8—21no—
442C2×Q8—21no—
443C2×Q8—21no—
444C2×Q8—44no—
445C2×Q8—84no—
446C2×D4—28no—
447C2×D4—24no—
448C2×D4—28no—
449C2×D4—21no—
450C2×D4—22no—
451C2×D4—21no—
452C2×D4—22no—
453C2×D4—21no—
454C2×D4—22no—
455C2×D4—22no—
456C2×D4—22no—
457C2×D4—21no—
458C2×D4—21no—
459C2×D4—21no—
460C2×D4—44no—
461C2×D4—44no—
462C2×D4—42no—
463C2×D4—42no—
464C2×D4—42no—
465C2×D4—42no—
466C2×D4—44no—
467C2×D4—42no—
468C2×D4—44no—
469C2×D4—84no—
470C2×D4—84no—
471C2×D4C22×D414yes—
472C22×C4—22no—
473C22×C4—22no—
474C22×C4—22no—
475C22×C4—22no—
476C22×C4—24no—
477C22×C4—21no—
478C22×C4—21no—
479C22×C4—22no—
480C22×C4—22no—
481C22×C4—22no—
482C22×C4—21no—
483C22×C4—42no—
484C22×C4—42no—
485C22×C4—48no—
486C22×C4—48no—
487C22×C4—42no—
488C22×C4—42no—
489C22×C4—42no—
490C22×C4—42no—
491C22×C4—42no—
492C22×C4—84no—
493C22×C4—82no—
494C22×C4—82no—
495C22×C4—82no—
496C22×C4—816no—
497C22×C4C22×D412yes—
498C22×C4C22×D414yes—
499C22×C4C22×D412yes—
500C22×C4C22×D412yes—
501C22×C4C22×D412yes—
502C22×C4C22×D411yescharacteristic
503C22×C4C22×D412yes—
504C22×C4C2×#4112yes—
505C22×C4H(5,𝔽2)12yes—
506C22×C42−1+412yes—
507C22×C42−1+412yes—
508C22×C42−1+412yes—
509C22×C4C2511yescharacteristic
510M16—44no—
511M16—44no—
512C2×C8—28no—
513C2×C8—21no—
514C2×C8—21no—
515C2×C8—21no—
516C2×C8—21no—
517C2×C8—22no—
518C2×C8—22no—
519C2×C8—22no—
520C2×C8—22no—
521C2×C8—22no—
522C2×C8—22no—
523C2×C8—22no—
524C2×C8—22no—
525C2×C8—22no—
526C2×C8—44no—
527C2×C8C22×D412yes—
528C2×C8C22×D412yes—
529C4⋊C4—24no—
530C4⋊C4—24no—
531C4⋊C4—22no—
532C4⋊C4—22no—
533C4⋊C4—22no—
534C4⋊C4—22no—
535C4⋊C4—21no—
536C4⋊C4—21no—
537C4⋊C4—22no—
538C4⋊C4—22no—
539C4⋊C4—22no—
540C4⋊C4—22no—
541C4⋊C4—21no—
542C4⋊C4—22no—
543C4⋊C4—21no—
544C4⋊C4—22no—
545C4⋊C4—44no—
546C4⋊C4—44no—
547C4⋊C4—48no—
548C4⋊C4—48no—
549C4⋊C4—44no—
550C4⋊C4—44no—
551C4⋊C4—44no—
552C4⋊C4—42no—
553C4⋊C4—42no—
554C4⋊C4—82no—
555C4⋊C4—84no—
556C4⋊C4—84no—
557C4⋊C4—82no—
558C4⋊C4—84no—
559C4⋊C4C22×D414yes—
560C4⋊C4C2×#4114yes—
561C22⋊C4—24no—
562C22⋊C4—24no—
563C22⋊C4—24no—
564C22⋊C4—48no—
565C22⋊C4—44no—
566C22⋊C4—44no—
567C22⋊C4—44no—
568C22⋊C4—42no—
569C22⋊C4—42no—
570C22⋊C4—84no—
571C22⋊C4—82no—
572C22⋊C4—84no—
573C22⋊C4—84no—
574C42—24no—
575C42—24no—
576C42—22no—
577C42—22no—
578C42—22no—
579C42—22no—
580C42—22no—
581C42—22no—
582C42—42no—
583C42—42no—
584C42—84no—
585C23—22no—
586C23—22no—
587C23—24no—
588C23—44no—
589C23—44no—
590C23—48no—
591C23—42no—
592C23—42no—
593C23—42no—
594C23—42no—
595C23—41no—
596C23—41no—
597C23—41no—
598C23—84no—
599C23—84no—
600C23—88no—
601C23—88no—
602C23#53612yes—
603C23D4212yes—
604C23#54812yes—
605C23C23×D411yescharacteristic
606Q8—41no—
607Q8—41no—
608Q8—41no—
609Q8—41no—
610Q8—164no—
611D4—24no—
612D4—24no—
613D4—48no—
614D4—41no—
615D4—42no—
616D4—41no—
617D4—41no—
618D4—42no—
619D4—41no—
620D4—84no—
621D4—164no—
622D4—164no—
623C2×C4—24no—
624C2×C4—22no—
625C2×C4—22no—
626C2×C4—24no—
627C2×C4—22no—
628C2×C4—24no—
629C2×C4—24no—
630C2×C4—22no—
631C2×C4—24no—
632C2×C4—22no—
633C2×C4—22no—
634C2×C4—22no—
635C2×C4—22no—
636C2×C4—24no—
637C2×C4—24no—
638C2×C4—44no—
639C2×C4—44no—
640C2×C4—42no—
641C2×C4—44no—
642C2×C4—42no—
643C2×C4—42no—
644C2×C4—42no—
645C2×C4—42no—
646C2×C4—44no—
647C2×C4—42no—
648C2×C4—44no—
649C2×C4—41no—
650C2×C4—41no—
651C2×C4—41no—
652C2×C4—41no—
653C2×C4—42no—
654C2×C4—42no—
655C2×C4—42no—
656C2×C4—41no—
657C2×C4—84no—
658C2×C4—816no—
659C2×C4—84no—
660C2×C4—84no—
661C2×C4—84no—
662C2×C4#53612yes—
663C2×C4#53614yes—
664C2×C4#54214yes—
665C2×C4D4212yes—
666C2×C4#54714yes—
667C2×C4#54812yes—
668C2×C4#56314yes—
669C2×C4C23×D411yescharacteristic
670C2×C4C22×#4111yescharacteristic
671C2×C4C2×H(5,𝔽2)11yescharacteristic
672C2×C4C2×2−1+411yescharacteristic
673C2×C4C2×2−1+411yescharacteristic
674C2×C4C2×2−1+411yescharacteristic
675C8—22no—
676C8—22no—
677C8—42no—
678C8—42no—
679C8—42no—
680C22—22no—
681C22—22no—
682C22—24no—
683C22—24no—
684C22—44no—
685C22—44no—
686C22—44no—
687C22—816no—
688C22—84no—
689C22—84no—
690C22—81no—
691C22—81no—
692C22—81no—
693C22—82no—
694C22—82no—
695C22#267414yes—
696C22C2×#53611yescharacteristic
697C22C2×D4211yescharacteristic
698C22C2×#54811yescharacteristic
699C22#358412yes—
700C22#358512yes—
701C4—42no—
702C4—42no—
703C4—44no—
704C4—42no—
705C4—42no—
706C4—44no—
707C4—44no—
708C4—81no—
709C4—81no—
710C4—81no—
711C4—82no—
712C4—82no—
713C4—82no—
714C4#267414yes—
715C4C2×#53611yescharacteristic
716C4C2×#54211yescharacteristic
717C4C2×#54711yescharacteristic
718C4C2×#56311yescharacteristic
719C2—24no—
720C2—84no—
721C2—161no—
722C2C2×#267411yescharacteristic
723C2#6089012yes—
724C2#6105111yescharacteristic
725C2#6105911yescharacteristic
726C2#6134912yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#535C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ #2296 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C22×#478C2semidirect
3. #35640C2semidirect
5. #30827C2semidirect
6. #30823C2semidirect
7. #30811C2semidirect
8. #30807C2semidirect
9. #23737C2semidirect
10. #23632C2semidirect
11. #23609C2semidirect
12. #23556C2semidirect
14. #23453C2semidirect
15. #21156C2semidirect
16. #21068C2semidirect
17. #21065C2semidirect
18. #21024C2semidirect
19. #20873C2semidirect
50. #2552C22semidirect
52. #2550C22semidirect
67. #2097C22semidirect
68. #2095C22semidirect
69. #2093C22semidirect
70. #2092C22semidirect
73. #2068C22semidirect
74. #2067C22semidirect
75. #2066C22semidirect
76. #2065C22semidirect
77. #2059C22semidirect
78. #2058C22semidirect
81. #2026C22semidirect
82. #2025C22semidirect
83. #1944C22semidirect
84. #1942C22semidirect
85. #1927C22semidirect
87. #1926C22semidirect
89. #1917C22semidirect
90. #1916C22semidirect

Maximal subgroups

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

subgroupindex
C2×#31362
C22×#4782
#356402
#351522
#308272
#308232
#308112
#308072
#237372
#236322
#236092
#235562
#235092
#234532
#211562
#210682
#210652
#210242
#208732