Finite Group Databasebeta

← #7505063

Subgroups — #7505063

Order 512.

1824 conjugacy classes of subgroups.

The subgroup lattice has 708 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35620C211yesmaximal, characteristic, complemented
2#35515C211yesmaximal, characteristic, complemented
3#31547C214yesmaximal, complemented
4#31546C214yesmaximal, complemented
5#31531C214yesmaximal, complemented
6#31530C214yesmaximal, complemented
7C2×#2092C211yesmaximal, characteristic, complemented
8#23660C211yesmaximal, characteristic, complemented
9#23610C211yesmaximal, characteristic, complemented
10#23556C211yesmaximal, characteristic, complemented
11#23456C211yesmaximal, characteristic, complemented
12#23451C211yesmaximal, characteristic, complemented
13#21323C211yesmaximal, characteristic, complemented
14#21236C211yesmaximal, characteristic
15#21022C211yesmaximal, characteristic, complemented
16#20932C211yesmaximal, characteristic
17#20929C211yesmaximal, characteristic
18#20881C211yesmaximal, characteristic, complemented
19#20650C211yesmaximal, characteristic, complemented
20C22×AGL(1,ℤ/8)C2211yescharacteristic
21C2×#544C2211yescharacteristic
22C2×#533C2211yescharacteristic
23C2×#520C2211yescharacteristic
24C2×#518C2211yescharacteristic
25C2×#428C2211yescharacteristic
26C22×#107C2211yescharacteristic
27C22×#106C2211yescharacteristic
28C2×#424C2211yescharacteristic
29C2×#417C2211yescharacteristic
30C2×#418C2211yescharacteristic
31C2×#416C2211yescharacteristic
32C2×#416C2211yescharacteristic
33C22×#102C2211yescharacteristic
34C22×#102C2211yescharacteristic
35C2×C4×M16C2211yescharacteristic
36#2580—22no—
37#2568C2212yescomplemented
38#2566C2212yescomplemented
39#2565C2212yescomplemented
40#2563C2212yescomplemented
41#2538—22no—
42#2454C2214yes—
43#2452C2214yes—
44#2450—24no—
45#2442—22no—
46#2439—22no—
47#2437—24no—
48#2317C2211yescharacteristic
49#2317C2211yescharacteristic
50#2316C2211yescharacteristic
51#2311C2211yescharacteristic
52#2309C2211yescharacteristic
53#2297C2211yescharacteristic
54#2296C2211yescharacteristic
55C2×#390C2211yescharacteristic
56C2×#382C2211yescharacteristic
57#2092C2214yes—
58#2092C2214yes—
59#2085C2214yescomplemented
60#2083C2214yescomplemented
61#2066C2214yescomplemented
62#2065C2214yescomplemented
63#2059C2214yescomplemented
64#2058C2214yescomplemented
65#2026C2212yescomplemented
66#2025C2214yes—
67#2025C2212yescomplemented
68#2024C2212yescomplemented
69#2023C2214yes—
70#2023C2212yescomplemented
71#1993C2214yescomplemented
72#1991C2214yescomplemented
73#1973C2212yes—
74#1972C2212yes—
75#1970C2212yes—
76#1968C2212yes—
77#1917C2214yescomplemented
78#1916C2214yescomplemented
79#1880C2214yes—
80#1879C2214yes—
81#1879C2214yes—
82#1877C2214yes—
83#1863C2214yescomplemented
84#1862C2214yescomplemented
85#1804C2214yes—
86#1802C2214yes—
87#1736C2211yescharacteristic
88#1736C2211yescharacteristic
89#1736C2211yescharacteristic
90#1736C2211yescharacteristic
91#1735C2211yescharacteristic
92#1735C2211yescharacteristic
93#1729C2211yescharacteristic
94#1729C2211yescharacteristic
95#1729C2211yescharacteristic
96C2×#336C2211yescharacteristic
97C22×#41C2311yescharacteristic
98C23×D4C2311yescharacteristic
99C2×AGL(1,ℤ/8)—42no—
100C2×AGL(1,ℤ/8)—42no—
101C2×AGL(1,ℤ/8)Q818yescomplemented
102C22×SD16C2312yes—
103C22×D8C2312yes—
104C22×M16C2311yescharacteristic
105C22×M16C2311yescharacteristic
106C23×C8C2311yescharacteristic
107C23×C8C2311yescharacteristic
108#544—42no—
109#544—42no—
110#533—24no—
111#533—42no—
112C2×C4⋊Q8—21no—
113C2×C4⋊Q8C2312yes—
114C2×C4⋊Q8C2312yes—
115C2×#125—21no—
116C2×#125C2312yes—
117C2×#125C2312yes—
118C2×C22⋊Q8—22no—
119C2×C22⋊Q8—24no—
120C2×C22⋊Q8—22no—
121#520—42no—
122#520—42no—
123#518—24no—
124#518—42no—
125C2×C4×Q8—22no—
126C2×C4×Q8C2312yes—
127C2×C4×D4—22no—
128C2×C4×D4—22no—
129C2×C4×D4C2314yes—
130C2×C4×D4C2312yes—
131C2×#117—21no—
132C2×#117C2311yescharacteristic
133C2×#117C2311yescharacteristic
134C2×#117C2311yescharacteristic
135C2×#117C2311yescharacteristic
136C2×#117C2311yescharacteristic
137C22×C4⋊C4—21no—
138C22×C4⋊C4C2311yescharacteristic
139C22×C4⋊C4C2311yescharacteristic
140C22×C4⋊C4C2311yescharacteristic
141C22×C22⋊C4—22no—
142C22×C22⋊C4—22no—
143C22×C42C2311yescharacteristic
144#428—42no—
145C2×#107—42no—
146C2×#107C2312yes—
147C2×#107C2314yes—
148C2×#106—42no—
149C2×#106C2312yes—
150C2×#106C2314yes—
151#424—42no—
152C2×C4⋊C8C2314yes—
153#418—42no—
154#417—42no—
155#416—24no—
156#416—42no—
157C2×#103C2312yes—
158C2×#103C2312yes—
159C2×#103C2312yes—
160C2×#102—42no—
161C2×#102—42no—
162C2×#102C2312yes—
163C2×#102C2312yes—
164C2×#102C2314yes—
165C2×#102C2312yes—
166C2×#102C2312yes—
167C2×#102C2312yes—
168C2×#102C2314yes—
169C4×M16—42no—
170C2×#97C2312yes—
171C2×C4×C8C2312yes—
172#400—24no—
173#399—24no—
174#398—24no—
175#396—24no—
176#396—24no—
177#396—24no—
178#393—24no—
179#393—24no—
180#391—24no—
181#391C2311yescharacteristic
182#391C2312yes—
183#391C2311yescharacteristic
184#390—21no—
185#390—21no—
186#390—22no—
187#390C2311yescharacteristic
188#390C2312yes—
189#390C2314yes—
190#390C2311yescharacteristic
191#390C2312yes—
192#390C2312yes—
193#389—22no—
194#389—22no—
195#389D412yes—
196#389D412yes—
197#389C2314yes—
198#389C2314yes—
199#387—24no—
200#386—22no—
201#386—22no—
202#385—24no—
203#385—22no—
204#385—22no—
205#385—24no—
206#385—24no—
207#385D414yes—
208#384—21no—
209#384C2312yes—
210#382—21no—
211#382—22no—
212#382C2312yes—
213#382C2312yes—
214#382C2312yes—
215#380—24no—
216#380—22no—
217#380—22no—
218#380—24no—
219#379—22no—
220C4×C4⋊C4C2312yes—
221C4×C4⋊C4C2312yes—
222C4×C22⋊C4—22no—
223C2×#95—22no—
224#340C2312yes—
225#340C2314yes—
226#340C2312yes—
227#340C2314yes—
228#340C2314yes—
229#340C2312yes—
230#340C2314yes—
231#340C2312yes—
232#336C2314yes—
233#336C2314yes—
234#336C2312yes—
235#336C2312yes—
236#336C2314yes—
237C25—21no—
238C2×#41—22no—
239C2×#41—22no—
240C2×#41—22no—
241C2×#41—22no—
242C2×#41—22no—
243C2×#41C2×Q814yes—
244C22×Q8—22no—
245C22×Q8—22no—
246C22×Q8C2411yescharacteristic
247C22×D4—21no—
248C22×D4—21no—
249C22×D4—22no—
250C22×D4—22no—
251C22×D4—22no—
252C22×D4—22no—
253C22×D4—22no—
254C22×D4C2×Q814yes—
255C22×D4C2411yescharacteristic
256C22×D4C2412yes—
257C23×C4—21no—
258C23×C4—21no—
259C23×C4—21no—
260C23×C4—21no—
261C23×C4—24no—
262C23×C4—22no—
263C23×C4C2×D412yes—
264C23×C4C2411yescharacteristic
265AGL(1,ℤ/8)—416no—
266C2×SD16—42no—
267C2×SD16—42no—
268C2×SD16C2×Q818yes—
269C2×D8—42no—
270C2×D8—42no—
271C2×D8C2×Q818yes—
272C2×M16—22no—
273C2×M16—22no—
274C2×M16—22no—
275C2×M16—22no—
276C2×M16—22no—
277C2×M16C2×Q814yes—
278C22×C8—22no—
279C22×C8—22no—
280C22×C8—22no—
281C22×C8—22no—
282C22×C8—22no—
283C22×C8—22no—
284C22×C8C2412yes—
285C22×C8C2412yes—
286C22×C8C2412yes—
287C22×C8C2412yes—
288C4⋊Q8—24no—
289C4⋊Q8—42no—
290C4⋊Q8—42no—
291C4⋊Q8—42no—
292C4⋊Q8—44no—
293#125—24no—
294#125—42no—
295#125—42no—
296#125—42no—
297#125—44no—
298C22⋊Q8—48no—
299C22⋊Q8—48no—
300C22⋊Q8—88no—
301C4×Q8—48no—
302C4×Q8—42no—
303C4×Q8—42no—
304C4×D4—28no—
305C4×D4—44no—
306C4×D4—42no—
307C4×D4—42no—
308C4×D4—48no—
309C4×D4—48no—
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—44no—
320#117#4114yes—
321C2×C4⋊C4—22no—
322C2×C4⋊C4—22no—
323C2×C4⋊C4—22no—
324C2×C4⋊C4—22no—
325C2×C4⋊C4—22no—
326C2×C4⋊C4—22no—
327C2×C4⋊C4—21no—
328C2×C4⋊C4—22no—
329C2×C4⋊C4—21no—
330C2×C4⋊C4—21no—
331C2×C4⋊C4—22no—
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—24no—
340C2×C4⋊C4—44no—
341C2×C4⋊C4C2×D411yescharacteristic
342C2×C4⋊C4C2×D411yescharacteristic
343C2×C4⋊C4C2×D412yes—
344C2×C4⋊C4C2×D414yes—
345C2×C4⋊C4C2×D414yes—
346C2×C4⋊C4C2412yes—
347C2×C4⋊C4C2412yes—
348C2×C4⋊C4C2412yes—
349C2×C4⋊C4C2411yescharacteristic
350C2×C4⋊C4C2412yes—
351C2×C4⋊C4C2411yescharacteristic
352C2×C4⋊C4C2412yes—
353C2×C22⋊C4—21no—
354C2×C22⋊C4—21no—
355C2×C22⋊C4—21no—
356C2×C22⋊C4—24no—
357C2×C22⋊C4—24no—
358C2×C22⋊C4—22no—
359C2×C22⋊C4—24no—
360C2×C22⋊C4—24no—
361C2×C22⋊C4—22no—
362C2×C22⋊C4—22no—
363C2×C22⋊C4—24no—
364C2×C22⋊C4—24no—
365C2×C22⋊C4—48no—
366C2×C22⋊C4—84no—
367C2×C22⋊C4C2×D412yes—
368C2×C22⋊C4C2×D412yes—
369C2×C42—21no—
370C2×C42—22no—
371C2×C42—22no—
372C2×C42—22no—
373C2×C42—21no—
374C2×C42—21no—
375C2×C42C2×D412yes—
376C2×C42C2412yes—
377C2×C42C2412yes—
378C2×C42C2411yescharacteristic
379C2×C42C2411yescharacteristic
380#107—44no—
381#107—42no—
382#106—44no—
383#106—42no—
384C4⋊C8—44no—
385#103—24no—
386#103—42no—
387#103—42no—
388#102—24no—
389#102—44no—
390#102—44no—
391#102—42no—
392#102—42no—
393#102—42no—
394#102—42no—
395#97—42no—
396C4×C8—42no—
397#95—24no—
398#95—24no—
399#95—22no—
400#95—22no—
401#95—24no—
402#95—24no—
403#95—24no—
404#95—24no—
405#95C2×D414yes—
406#95C2×D414yes—
407C24—21no—
408C24—22no—
409C24—24no—
410C24—42no—
411C24—42no—
412C24—82no—
413C24—82no—
414C24C22×D411yescharacteristic
415#41—28no—
416#41—28no—
417#41—44no—
418#41—44no—
419C2×Q8—21no—
420C2×Q8—21no—
421C2×Q8—21no—
422C2×Q8—21no—
423C2×Q8—21no—
424C2×Q8—44no—
425C2×Q8—44no—
426C2×Q8—44no—
427C2×Q8—82no—
428C2×Q8—84no—
429C2×Q8C22×Q812yes—
430C2×D4—24no—
431C2×D4—28no—
432C2×D4—24no—
433C2×D4—28no—
434C2×D4—21no—
435C2×D4—22no—
436C2×D4—22no—
437C2×D4—21no—
438C2×D4—21no—
439C2×D4—22no—
440C2×D4—21no—
441C2×D4—22no—
442C2×D4—22no—
443C2×D4—21no—
444C2×D4—42no—
445C2×D4—42no—
446C2×D4—42no—
447C2×D4—44no—
448C2×D4—42no—
449C2×D4—44no—
450C2×D4C22×Q812yes—
451C2×D4C22×Q814yes—
452C22×C4—21no—
453C22×C4—22no—
454C22×C4—24no—
455C22×C4—24no—
456C22×C4—21no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—21no—
462C22×C4—42no—
463C22×C4—42no—
464C22×C4—48no—
465C22×C4—44no—
466C22×C4—42no—
467C22×C4—42no—
468C22×C4—42no—
469C22×C4—42no—
470C22×C4—42no—
471C22×C4—84no—
472C22×C4—84no—
473C22×C4—84no—
474C22×C4—82no—
475C22×C4—82no—
476C22×C4—82no—
477C22×C4C22×D412yes—
478C22×C4C22×D412yes—
479C22×C4C22×D414yes—
480C22×C4C22×D412yes—
481C22×C4C22×D412yes—
482C22×C4C22×D411yescharacteristic
483C22×C4C22×Q812yes—
484C22×C4C2×#4112yes—
485C22×C4H(5,𝔽2)12yes—
486C22×C4H(5,𝔽2)12yes—
487C22×C4H(5,𝔽2)12yes—
488C22×C42−1+412yes—
489C22×C4C2511yescharacteristic
490SD16—48no—
491D8—48no—
492M16—28no—
493M16—44no—
494C2×C8—22no—
495C2×C8—22no—
496C2×C8—22no—
497C2×C8—22no—
498C2×C8—22no—
499C2×C8—22no—
500C2×C8—22no—
501C2×C8—22no—
502C2×C8—22no—
503C2×C8—22no—
504C2×C8—22no—
505C2×C8—44no—
506C2×C8—44no—
507C2×C8C22×Q814yes—
508C4⋊C4—22no—
509C4⋊C4—22no—
510C4⋊C4—24no—
511C4⋊C4—22no—
512C4⋊C4—22no—
513C4⋊C4—22no—
514C4⋊C4—22no—
515C4⋊C4—21no—
516C4⋊C4—21no—
517C4⋊C4—22no—
518C4⋊C4—22no—
519C4⋊C4—22no—
520C4⋊C4—21no—
521C4⋊C4—22no—
522C4⋊C4—22no—
523C4⋊C4—21no—
524C4⋊C4—44no—
525C4⋊C4—44no—
526C4⋊C4—44no—
527C4⋊C4—44no—
528C4⋊C4—44no—
529C4⋊C4—44no—
530C4⋊C4—44no—
531C4⋊C4—44no—
532C4⋊C4—44no—
533C4⋊C4—42no—
534C4⋊C4—42no—
535C4⋊C4—42no—
536C4⋊C4—84no—
537C4⋊C4—84no—
538C4⋊C4—84no—
539C4⋊C4C2×#4114yes—
540C22⋊C4—24no—
541C22⋊C4—24no—
542C22⋊C4—24no—
543C22⋊C4—48no—
544C22⋊C4—48no—
545C22⋊C4—44no—
546C22⋊C4—42no—
547C22⋊C4—42no—
548C22⋊C4—84no—
549C22⋊C4—84no—
550C22⋊C4—82no—
551C22⋊C4—82no—
552C22⋊C4—84no—
553C22⋊C4—84no—
554C42—24no—
555C42—24no—
556C42—21no—
557C42—21no—
558C42—22no—
559C42—21no—
560C42—21no—
561C42—22no—
562C42—22no—
563C42—22no—
564C42—44no—
565C42—42no—
566C42—42no—
567C23—22no—
568C23—22no—
569C23—22no—
570C23—24no—
571C23—42no—
572C23—44no—
573C23—44no—
574C23—42no—
575C23—41no—
576C23—42no—
577C23—42no—
578C23—41no—
579C23—48no—
580C23—88no—
581C23—164no—
582C23—162no—
583C23—162no—
584C23#53512yes—
585C23#54612yes—
586C23D4×Q812yes—
587C23C23×D411yescharacteristic
588Q8—22no—
589Q8—22no—
590Q8—41no—
591Q8—41no—
592Q8—88no—
593Q8—164no—
594D4—22no—
595D4—24no—
596D4—24no—
597D4—22no—
598D4—48no—
599D4—42no—
600D4—41no—
601D4—48no—
602D4—42no—
603D4—41no—
604C2×C4—24no—
605C2×C4—24no—
606C2×C4—22no—
607C2×C4—22no—
608C2×C4—22no—
609C2×C4—22no—
610C2×C4—24no—
611C2×C4—24no—
612C2×C4—22no—
613C2×C4—22no—
614C2×C4—22no—
615C2×C4—22no—
616C2×C4—24no—
617C2×C4—24no—
618C2×C4—22no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—41no—
622C2×C4—41no—
623C2×C4—48no—
624C2×C4—42no—
625C2×C4—44no—
626C2×C4—42no—
627C2×C4—44no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—44no—
631C2×C4—41no—
632C2×C4—42no—
633C2×C4—41no—
634C2×C4—42no—
635C2×C4—42no—
636C2×C4—42no—
637C2×C4—84no—
638C2×C4—168no—
639C2×C4—164no—
640C2×C4—162no—
641C2×C4—162no—
642C2×C4—162no—
643C2×C4#53512yes—
644C2×C4#54214yes—
645C2×C4#54314yes—
646C2×C4#54612yes—
647C2×C4D4×Q812yes—
648C2×C4#55414yes—
649C2×C4#56214yes—
650C2×C4C23×Q811yescharacteristic
651C2×C4C22×#4111yescharacteristic
652C2×C4C2×H(5,𝔽2)11yescharacteristic
653C2×C4C2×H(5,𝔽2)11yescharacteristic
654C2×C4C2×H(5,𝔽2)11yescharacteristic
655C2×C4C2×2−1+411yescharacteristic
656C8—24no—
657C8—42no—
658C8—42no—
659C8—42no—
660C22—24no—
661C22—22no—
662C22—22no—
663C22—24no—
664C22—42no—
665C22—44no—
666C22—44no—
667C22—48no—
668C22—84no—
669C22—82no—
670C22—81no—
671C22—82no—
672C22—81no—
673C22—82no—
674C22—168no—
675C22#269214yes—
676C22C2×#53511yescharacteristic
677C22C2×#54611yescharacteristic
678C22C2×D4×Q811yescharacteristic
679C22#358412yes—
680C22#358512yes—
681C4—42no—
682C4—42no—
683C4—44no—
684C4—44no—
685C4—42no—
686C4—42no—
687C4—42no—
688C4—82no—
689C4—82no—
690C4—81no—
691C4—81no—
692C4—82no—
693C4—82no—
694C4#269214yes—
695C4C2×#54211yescharacteristic
696C4C2×#54311yescharacteristic
697C4C2×#55411yescharacteristic
698C4C2×#56211yescharacteristic
699C2—24no—
700C2—82no—
701C2—84no—
702C2C2×#269211yescharacteristic
703C2#6081911yescharacteristic
704C2#6082511yescharacteristic
705C2#6119512yes—
706C2#6138512yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×AGL(1,ℤ/8)C22
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. #35620C2semidirect
2. #35515C2semidirect
3. #31547C2semidirect
4. #31546C2semidirect
5. #31531C2semidirect
6. #31530C2semidirect
7. C2×#2092C2semidirect
8. #23660C2semidirect
9. #23610C2semidirect
10. #23556C2semidirect
11. #23456C2semidirect
12. #23451C2semidirect
13. #21323C2semidirect
15. #21022C2semidirect
18. #20881C2semidirect
19. #20650C2semidirect
37. #2568C22semidirect
38. #2566C22semidirect
39. #2565C22semidirect
40. #2563C22semidirect
59. #2085C22semidirect
60. #2083C22semidirect
61. #2066C22semidirect
62. #2065C22semidirect
63. #2059C22semidirect
64. #2058C22semidirect
65. #2026C22semidirect
67. #2025C22semidirect
68. #2024C22semidirect
70. #2023C22semidirect
71. #1993C22semidirect
72. #1991C22semidirect
77. #1917C22semidirect
78. #1916C22semidirect
83. #1863C22semidirect
84. #1862C22semidirect
101. C2×AGL(1,ℤ/8)Q8semidirect

Maximal subgroups

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

subgroupindex
#356202
#355152
#315472
#315462
#315312
#315302
C2×#20922
#236602
#236102
#235562
#234562
#234512
#213232
#212362
#210222
#209322
#209292
#208812
#206502