Finite Group Databasebeta

← #7505024

Subgroups — #7505024

Order 512.

1914 conjugacy classes of subgroups.

The subgroup lattice has 706 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35529C211yesmaximal, characteristic, complemented
2#35515C211yesmaximal, characteristic, complemented
3#31530C214yesmaximal, complemented
4#31527C214yesmaximal, complemented
5#31523C214yesmaximal, complemented
6#31518C214yesmaximal, complemented
7#23731C211yesmaximal, characteristic, complemented
8#23660C211yesmaximal, characteristic, complemented
9#23614C211yesmaximal, characteristic, complemented
10#23555C211yesmaximal, characteristic, complemented
11#23456C211yesmaximal, characteristic, complemented
12#23454C211yesmaximal, characteristic, complemented
13#21318C211yesmaximal, characteristic, complemented
14C2×#1970C211yesmaximal, characteristic
15#21020C211yesmaximal, characteristic, complemented
16#20934C211yesmaximal, characteristic
17#20934C211yesmaximal, characteristic
18#20876C211yesmaximal, characteristic, complemented
19#20650C211yesmaximal, characteristic, complemented
20C23×D8C2211yescharacteristic
21C2×#544C2211yescharacteristic
22C2×#533C2211yescharacteristic
23C2×#518C2211yescharacteristic
24C2×#518C2211yescharacteristic
25C2×#428C2211yescharacteristic
26C2×#428C2211yescharacteristic
27C22×#107C2211yescharacteristic
28C2×#424C2211yescharacteristic
29C2×#418C2211yescharacteristic
30C2×#418C2211yescharacteristic
31C2×#418C2211yescharacteristic
32C2×#418C2211yescharacteristic
33C22×#102C2211yescharacteristic
34C22×#102C2211yescharacteristic
35C22×C4×C8C2211yescharacteristic
36#2580—22no—
37#2566C2212yescomplemented
38#2566C2212yescomplemented
39#2563C2212yescomplemented
40#2542C2212yescomplemented
41#2538—22no—
42#2454—22no—
43#2454C2214yescomplemented
44#2452—22no—
45#2452C2214yescomplemented
46#2437—24no—
47#2437—24no—
48#2317C2211yescharacteristic
49#2317C2211yescharacteristic
50#2316C2211yescharacteristic
51#2309C2211yescharacteristic
52#2309C2211yescharacteristic
53#2296C2211yescharacteristic
54C2×#390C2211yescharacteristic
55C2×#390C2211yescharacteristic
56#2285C2211yescharacteristic
57#2094C2212yescomplemented
58#2093C2212yescomplemented
59#2092C2212yescomplemented
60#2092C2212yescomplemented
61#2085C2214yescomplemented
62#2083C2214yescomplemented
63#2067C2214yescomplemented
64#2065C2214yescomplemented
65#2059C2214yescomplemented
66#2057C2214yescomplemented
67#2025C2214yescomplemented
68#2025C2214yescomplemented
69#2023C2214yescomplemented
70#2023C2214yescomplemented
71#1991C2212yescomplemented
72#1991C2212yescomplemented
73#1990C2212yescomplemented
74#1990C2212yescomplemented
75#1970C2214yes—
76#1970C2214yes—
77#1916C2214yescomplemented
78#1916C2214yescomplemented
79#1879C2214yes—
80#1879C2214yes—
81#1879C2214yes—
82#1879C2214yes—
83#1864C2214yescomplemented
84#1862C2214yescomplemented
85#1804C2214yescomplemented
86#1802C2214yescomplemented
87#1736C2211yescharacteristic
88#1736C2211yescharacteristic
89#1735C2211yescharacteristic
90#1735C2211yescharacteristic
91#1735C2211yescharacteristic
92#1735C2211yescharacteristic
93#1729C2211yescharacteristic
94#1729C2211yescharacteristic
95#1729C2211yescharacteristic
96#1728C2211yescharacteristic
97C23×D4C2311yescharacteristic
98C23×D4C2311yescharacteristic
99C22×D8—42no—
100C22×D8—42no—
101C22×D8Q818yescomplemented
102C22×D8C2314yes—
103C22×M16C2311yescharacteristic
104C22×M16C2311yescharacteristic
105C23×C8C2311yescharacteristic
106C23×C8C2311yescharacteristic
107#544—42no—
108#544—42no—
109#533—24no—
110#533—42no—
111C2×C4⋊Q8—21no—
112C2×C4⋊Q8C2312yes—
113C2×C4⋊Q8C2312yes—
114C2×#125—21no—
115C2×#125C2312yes—
116C2×#125C2312yes—
117C2×C22⋊Q8—22no—
118C2×C22⋊Q8—24no—
119C2×C22⋊Q8—24no—
120C2×C22⋊Q8—22no—
121#518—24no—
122#518—42no—
123#518—42no—
124#518—42no—
125C2×C4×Q8—22no—
126C2×C4×D4—22no—
127C2×C4×D4—21no—
128C2×C4×D4—21no—
129C2×C4×D4C2314yes—
130C2×C4×D4C2312yes—
131C2×C4×D4C2312yes—
132C2×#117C2311yescharacteristic
133C2×#117C2311yescharacteristic
134C2×#117C2311yescharacteristic
135C2×#117C2311yescharacteristic
136C2×#117C2311yescharacteristic
137C2×#117C2311yescharacteristic
138C22×C4⋊C4C2311yescharacteristic
139C22×C4⋊C4C2311yescharacteristic
140C22×C22⋊C4—22no—
141C22×C22⋊C4—22no—
142C22×C22⋊C4—22no—
143C22×C22⋊C4—22no—
144C22×C42C2311yescharacteristic
145#428—42no—
146#428—42no—
147C2×#107—42no—
148C2×#107C2312yes—
149C2×#107C2314yes—
150C2×#107C2312yes—
151C2×#106C2312yes—
152C2×#106C2312yes—
153#424—42no—
154C2×C4⋊C8C2314yes—
155#418—24no—
156#418—42no—
157#418—42no—
158#418—42no—
159C2×#102—42no—
160C2×#102—42no—
161C2×#102C2314yes—
162C2×#102C2312yes—
163C2×#102C2314yes—
164C2×#102C2312yes—
165C2×#102C2314yes—
166C2×#102C2314yes—
167C2×#102C2312yes—
168C2×#102C2312yes—
169C2×C4×C8—42no—
170C2×C4×C8C2312yes—
171C2×C4×C8C2312yes—
172#399—24no—
173#399—24no—
174#396—24no—
175#396—24no—
176#396—24no—
177#396—24no—
178#393—24no—
179#393—24no—
180#390—21no—
181#390—22no—
182#390—21no—
183#390—22no—
184#390—22no—
185#390C2312yes—
186#390C2312yes—
187#390C2312yes—
188#390C2314yes—
189#390C2312yes—
190#390C2312yes—
191#390C2312yes—
192#389—22no—
193#389—22no—
194#389D412yes—
195#389D412yes—
196#389C2314yes—
197#389C2314yes—
198#385—24no—
199#385—22no—
200#385—22no—
201#385—24no—
202#385—24no—
203#385—24no—
204#385D414yes—
205#384—21no—
206#384C2311yescharacteristic
207#384C2312yes—
208#384C2311yescharacteristic
209#382—21no—
210#382C2311yescharacteristic
211#382C2312yes—
212#382C2311yescharacteristic
213#380—22no—
214#380—24no—
215#380—22no—
216#380—24no—
217#379—22no—
218#379—22no—
219C4×C4⋊C4—22no—
220C4×C4⋊C4C2312yes—
221C4×C4⋊C4C2312yes—
222#340C2312yes—
223#340C2314yes—
224#340C2312yes—
225#340C2312yes—
226#340C2312yes—
227#340C2312yes—
228#340C2312yes—
229#340C2312yes—
230#340C2314yes—
231#340C2312yes—
232#336C2314yes—
233#336C2314yes—
234#336C2312yes—
235#336C2312yes—
236#336C2314yes—
237C25—21no—
238C25—21no—
239C22×Q8—22no—
240C22×Q8—22no—
241C22×D4—21no—
242C22×D4—21no—
243C22×D4—22no—
244C22×D4—22no—
245C22×D4—22no—
246C22×D4—22no—
247C22×D4—22no—
248C22×D4—22no—
249C22×D4—22no—
250C22×D4—22no—
251C22×D4—22no—
252C22×D4—22no—
253C22×D4C2×Q814yes—
254C22×D4C2×Q814yes—
255C22×D4C2412yes—
256C22×D4C2412yes—
257C23×C4—21no—
258C23×C4—21no—
259C23×C4—22no—
260C23×C4—24no—
261C23×C4—22no—
262C23×C4C2×D412yes—
263C23×C4C2411yescharacteristic
264C2×D8—416no—
265C2×D8—44no—
266C2×D8—44no—
267C2×D8C2×Q8116yes—
268C2×M16—22no—
269C2×M16—22no—
270C2×M16—22no—
271C2×M16—22no—
272C2×M16—22no—
273C2×M16—22no—
274C22×C8—22no—
275C22×C8—22no—
276C22×C8—22no—
277C22×C8—22no—
278C22×C8—22no—
279C22×C8C2×Q814yes—
280C22×C8C2412yes—
281C22×C8C2412yes—
282C22×C8C2412yes—
283C22×C8C2412yes—
284C4⋊Q8—24no—
285C4⋊Q8—42no—
286C4⋊Q8—42no—
287C4⋊Q8—42no—
288C4⋊Q8—44no—
289#125—24no—
290#125—42no—
291#125—42no—
292#125—42no—
293#125—44no—
294C22⋊Q8—48no—
295C22⋊Q8—48no—
296C22⋊Q8—88no—
297C22⋊Q8—88no—
298C4×Q8—48no—
299C4×D4—28no—
300C4×D4—44no—
301C4×D4—42no—
302C4×D4—42no—
303C4×D4—42no—
304C4×D4—42no—
305C4×D4—48no—
306C4×D4—44no—
307C4×D4—44no—
308#117—22no—
309#117—22no—
310#117—22no—
311#117—22no—
312#117—22no—
313#117—22no—
314#117—22no—
315#117—22no—
316#117—22no—
317#117—22no—
318#117—22no—
319#117#4114yes—
320C2×C4⋊C4—22no—
321C2×C4⋊C4—22no—
322C2×C4⋊C4—22no—
323C2×C4⋊C4—22no—
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—22no—
333C2×C4⋊C4—24no—
334C2×C4⋊C4—24no—
335C2×C4⋊C4—24no—
336C2×C4⋊C4—22no—
337C2×C4⋊C4C2×D411yescharacteristic
338C2×C4⋊C4C2×D411yescharacteristic
339C2×C4⋊C4C2×D412yes—
340C2×C4⋊C4C2×D414yes—
341C2×C4⋊C4C2×D414yes—
342C2×C4⋊C4C2412yes—
343C2×C4⋊C4C2412yes—
344C2×C4⋊C4C2411yescharacteristic
345C2×C4⋊C4C2411yescharacteristic
346C2×C4⋊C4C2411yescharacteristic
347C2×C4⋊C4C2412yes—
348C2×C4⋊C4C2411yescharacteristic
349C2×C4⋊C4C2412yes—
350C2×C22⋊C4—21no—
351C2×C22⋊C4—21no—
352C2×C22⋊C4—21no—
353C2×C22⋊C4—21no—
354C2×C22⋊C4—22no—
355C2×C22⋊C4—24no—
356C2×C22⋊C4—22no—
357C2×C22⋊C4—24no—
358C2×C22⋊C4—24no—
359C2×C22⋊C4—24no—
360C2×C22⋊C4—24no—
361C2×C22⋊C4—24no—
362C2×C22⋊C4—24no—
363C2×C22⋊C4—24no—
364C2×C22⋊C4—24no—
365C2×C22⋊C4—48no—
366C2×C22⋊C4—48no—
367C2×C22⋊C4—84no—
368C2×C22⋊C4—84no—
369C2×C22⋊C4C2×D412yes—
370C2×C22⋊C4C2×D412yes—
371C2×C42—21no—
372C2×C42—22no—
373C2×C42—22no—
374C2×C42—21no—
375C2×C42—21no—
376C2×C42—21no—
377C2×C42C2×D412yes—
378C2×C42C2412yes—
379C2×C42C2412yes—
380C2×C42C2411yescharacteristic
381C2×C42C2411yescharacteristic
382#107—44no—
383#107—42no—
384#107—42no—
385#106—42no—
386#106—42no—
387C4⋊C8—44no—
388#102—28no—
389#102—44no—
390#102—44no—
391#102—42no—
392#102—42no—
393#102—42no—
394#102—44no—
395#102—42no—
396C4×C8—42no—
397C4×C8—42no—
398#95—22no—
399#95—22no—
400#95—24no—
401#95—24no—
402#95—24no—
403#95C2×D414yes—
404#95C2×D414yes—
405C24—22no—
406C24—22no—
407C24—24no—
408C24—24no—
409C24—42no—
410C24—42no—
411C24—42no—
412C24—42no—
413C24—82no—
414C24—82no—
415C24—82no—
416C24—82no—
417C24C22×D411yescharacteristic
418C2×Q8—44no—
419C2×Q8—44no—
420C2×Q8—44no—
421C2×Q8—82no—
422C2×Q8—84no—
423C2×D4—28no—
424C2×D4—28no—
425C2×D4—24no—
426C2×D4—24no—
427C2×D4—28no—
428C2×D4—28no—
429C2×D4—22no—
430C2×D4—22no—
431C2×D4—22no—
432C2×D4—22no—
433C2×D4—22no—
434C2×D4—22no—
435C2×D4—22no—
436C2×D4—22no—
437C2×D4—22no—
438C2×D4—22no—
439C2×D4—42no—
440C2×D4—42no—
441C2×D4—44no—
442C2×D4—44no—
443C2×D4—42no—
444C2×D4—42no—
445C2×D4—44no—
446C2×D4—44no—
447C2×D4C22×Q814yes—
448C2×D4C22×Q814yes—
449C22×C4—24no—
450C22×C4—22no—
451C22×C4—24no—
452C22×C4—21no—
453C22×C4—22no—
454C22×C4—21no—
455C22×C4—22no—
456C22×C4—21no—
457C22×C4—21no—
458C22×C4—22no—
459C22×C4—42no—
460C22×C4—48no—
461C22×C4—44no—
462C22×C4—44no—
463C22×C4—42no—
464C22×C4—42no—
465C22×C4—84no—
466C22×C4—84no—
467C22×C4—84no—
468C22×C4—84no—
469C22×C4—82no—
470C22×C4—82no—
471C22×C4C22×D412yes—
472C22×C4C22×D412yes—
473C22×C4C22×D414yes—
474C22×C4C22×D412yes—
475C22×C4C22×D412yes—
476C22×C4C22×D411yescharacteristic
477C22×C4C22×Q812yes—
478C22×C4C2×#4112yes—
479C22×C4H(5,𝔽2)12yes—
480C22×C4H(5,𝔽2)12yes—
481C22×C4H(5,𝔽2)12yes—
482C22×C42−1+412yes—
483C22×C4C2511yescharacteristic
484D8—416no—
485M16—44no—
486M16—44no—
487C2×C8—28no—
488C2×C8—22no—
489C2×C8—22no—
490C2×C8—22no—
491C2×C8—22no—
492C2×C8—22no—
493C2×C8—22no—
494C2×C8—22no—
495C2×C8—22no—
496C2×C8—22no—
497C2×C8—22no—
498C2×C8—22no—
499C2×C8—44no—
500C2×C8C22×Q814yes—
501C4⋊C4—22no—
502C4⋊C4—22no—
503C4⋊C4—24no—
504C4⋊C4—28no—
505C4⋊C4—22no—
506C4⋊C4—22no—
507C4⋊C4—22no—
508C4⋊C4—22no—
509C4⋊C4—21no—
510C4⋊C4—21no—
511C4⋊C4—21no—
512C4⋊C4—22no—
513C4⋊C4—21no—
514C4⋊C4—21no—
515C4⋊C4—21no—
516C4⋊C4—22no—
517C4⋊C4—22no—
518C4⋊C4—21no—
519C4⋊C4—21no—
520C4⋊C4—44no—
521C4⋊C4—44no—
522C4⋊C4—44no—
523C4⋊C4—44no—
524C4⋊C4—44no—
525C4⋊C4—44no—
526C4⋊C4—44no—
527C4⋊C4—42no—
528C4⋊C4—42no—
529C4⋊C4—44no—
530C4⋊C4—82no—
531C4⋊C4—84no—
532C4⋊C4—84no—
533C4⋊C4—84no—
534C4⋊C4C2×#4114yes—
535C22⋊C4—24no—
536C22⋊C4—24no—
537C22⋊C4—24no—
538C22⋊C4—48no—
539C22⋊C4—44no—
540C22⋊C4—44no—
541C22⋊C4—42no—
542C22⋊C4—42no—
543C22⋊C4—42no—
544C22⋊C4—48no—
545C22⋊C4—48no—
546C22⋊C4—84no—
547C22⋊C4—84no—
548C22⋊C4—84no—
549C22⋊C4—84no—
550C22⋊C4—84no—
551C22⋊C4—84no—
552C42—24no—
553C42—24no—
554C42—21no—
555C42—21no—
556C42—22no—
557C42—21no—
558C42—21no—
559C42—22no—
560C42—22no—
561C42—22no—
562C42—42no—
563C42—42no—
564C42—42no—
565C23—22no—
566C23—22no—
567C23—24no—
568C23—24no—
569C23—44no—
570C23—44no—
571C23—42no—
572C23—42no—
573C23—42no—
574C23—42no—
575C23—44no—
576C23—44no—
577C23—48no—
578C23—48no—
579C23—88no—
580C23—88no—
581C23—162no—
582C23—162no—
583C23—162no—
584C23—162no—
585C23—164no—
586C23—164no—
587C23#53512yes—
588C23#54612yes—
589C23D4×Q812yes—
590C23C23×D411yescharacteristic
591Q8—88no—
592Q8—164no—
593D4—24no—
594D4—24no—
595D4—24no—
596D4—24no—
597D4—42no—
598D4—42no—
599D4—48no—
600D4—48no—
601D4—42no—
602D4—42no—
603C2×C4—22no—
604C2×C4—22no—
605C2×C4—24no—
606C2×C4—24no—
607C2×C4—22no—
608C2×C4—22no—
609C2×C4—22no—
610C2×C4—22no—
611C2×C4—24no—
612C2×C4—24no—
613C2×C4—22no—
614C2×C4—24no—
615C2×C4—24no—
616C2×C4—48no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—41no—
623C2×C4—41no—
624C2×C4—41no—
625C2×C4—41no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—44no—
630C2×C4—44no—
631C2×C4—42no—
632C2×C4—42no—
633C2×C4—42no—
634C2×C4—168no—
635C2×C4—164no—
636C2×C4—164no—
637C2×C4—162no—
638C2×C4—162no—
639C2×C4#53512yes—
640C2×C4#54214yes—
641C2×C4#54314yes—
642C2×C4#54612yes—
643C2×C4D4×Q812yes—
644C2×C4#55414yes—
645C2×C4#56214yes—
646C2×C4#57614yes—
647C2×C4C23×Q811yescharacteristic
648C2×C4C22×#4111yescharacteristic
649C2×C4C2×H(5,𝔽2)11yescharacteristic
650C2×C4C2×H(5,𝔽2)11yescharacteristic
651C2×C4C2×H(5,𝔽2)11yescharacteristic
652C2×C4C2×2−1+411yescharacteristic
653C8—24no—
654C8—42no—
655C8—42no—
656C8—42no—
657C22—22no—
658C22—22no—
659C22—24no—
660C22—24no—
661C22—44no—
662C22—44no—
663C22—48no—
664C22—48no—
665C22—82no—
666C22—82no—
667C22—82no—
668C22—82no—
669C22—84no—
670C22—84no—
671C22—168no—
672C22—168no—
673C22#269214yes—
674C22C2×#53511yescharacteristic
675C22C2×#54611yescharacteristic
676C22C2×D4×Q811yescharacteristic
677C22C2×#57612yes—
678C22#358412yes—
679C4—24no—
680C4—42no—
681C4—42no—
682C4—42no—
683C4—44no—
684C4—44no—
685C4—82no—
686C4—81no—
687C4—82no—
688C4—81no—
689C4—81no—
690C4—81no—
691C4—82no—
692C4#269214yes—
693C4C2×#54211yescharacteristic
694C4C2×#54311yescharacteristic
695C4C2×#55411yescharacteristic
696C4C2×#56211yescharacteristic
697C2—24no—
698C2—84no—
699C2—84no—
700C2C2×#269211yescharacteristic
701C2#6081711yescharacteristic
702C2#6081911yescharacteristic
703C2#6117112yes—
704C2#6137012yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×D8C22
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
1. #35529C2semidirect
2. #35515C2semidirect
3. #31530C2semidirect
4. #31527C2semidirect
5. #31523C2semidirect
6. #31518C2semidirect
7. #23731C2semidirect
8. #23660C2semidirect
9. #23614C2semidirect
10. #23555C2semidirect
11. #23456C2semidirect
12. #23454C2semidirect
13. #21318C2semidirect
15. #21020C2semidirect
18. #20876C2semidirect
19. #20650C2semidirect
37. #2566C22semidirect
38. #2566C22semidirect
39. #2563C22semidirect
40. #2542C22semidirect
43. #2454C22semidirect
45. #2452C22semidirect
57. #2094C22semidirect
58. #2093C22semidirect
59. #2092C22semidirect
60. #2092C22semidirect
61. #2085C22semidirect
62. #2083C22semidirect
63. #2067C22semidirect
64. #2065C22semidirect
65. #2059C22semidirect
66. #2057C22semidirect
67. #2025C22semidirect
68. #2025C22semidirect
69. #2023C22semidirect
70. #2023C22semidirect
71. #1991C22semidirect
72. #1991C22semidirect
73. #1990C22semidirect
74. #1990C22semidirect
77. #1916C22semidirect
78. #1916C22semidirect
83. #1864C22semidirect
84. #1862C22semidirect
85. #1804C22semidirect
86. #1802C22semidirect
101. C22×D8Q8semidirect

Maximal subgroups

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

subgroupindex
#355292
#355152
#315302
#315272
#315232
#315182
#237312
#236602
#236142
#235552
#234562
#234542
#213182
C2×#19702
#210202
#209342
#208762
#206502