Finite Group Databasebeta

← #7505056

Subgroups — #7505056

Order 512.

1866 conjugacy classes of subgroups.

The subgroup lattice has 712 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35532C211yesmaximal, characteristic, complemented
2#35515C211yesmaximal, characteristic, complemented
3#31535C214yesmaximal, complemented
4#31534C214yesmaximal, complemented
5#31519C214yesmaximal, complemented
6#31518C214yesmaximal, complemented
7C2×#2093C211yesmaximal, characteristic, complemented
8#23660C211yesmaximal, characteristic, complemented
9#23611C211yesmaximal, characteristic, complemented
10#23555C211yesmaximal, characteristic, complemented
11#23456C211yesmaximal, characteristic, complemented
12#23451C211yesmaximal, characteristic, complemented
13#21322C211yesmaximal, 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×#107C2211yescharacteristic
28C2×#424C2211yescharacteristic
29C2×#417C2211yescharacteristic
30C2×#418C2211yescharacteristic
31C2×#416C2211yescharacteristic
32C2×#416C2211yescharacteristic
33C22×#102C2211yescharacteristic
34C22×#102C2211yescharacteristic
35C2×C4×M16C2211yescharacteristic
36#2580—22no—
37#2567C2212yescomplemented
38#2566C2212yescomplemented
39#2565C2212yescomplemented
40#2542C2212yescomplemented
41#2538—22no—
42#2454C2214yes—
43#2452C2214yes—
44#2449—24no—
45#2441—22no—
46#2440—22no—
47#2437—24no—
48#2317C2211yescharacteristic
49#2317C2211yescharacteristic
50#2316C2211yescharacteristic
51#2311C2211yescharacteristic
52#2309C2211yescharacteristic
53#2297C2211yescharacteristic
54#2296C2211yescharacteristic
55C2×#390C2211yescharacteristic
56C2×#384C2211yescharacteristic
57#2093C2214yes—
58#2093C2214yes—
59#2085C2214yescomplemented
60#2083C2214yescomplemented
61#2068C2214yescomplemented
62#2067C2214yescomplemented
63#2059C2214yescomplemented
64#2057C2214yescomplemented
65#2026C2212yescomplemented
66#2025C2214yes—
67#2025C2212yescomplemented
68#2024C2212yescomplemented
69#2023C2214yes—
70#2023C2212yescomplemented
71#1992C2214yescomplemented
72#1990C2214yescomplemented
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—24no—
121C2×C22⋊Q8—22no—
122#520—42no—
123#520—42no—
124#518—24no—
125#518—42no—
126C2×C4×Q8—22no—
127C2×C4×Q8C2312yes—
128C2×C4×D4—22no—
129C2×C4×D4—22no—
130C2×C4×D4C2314yes—
131C2×C4×D4C2312yes—
132C2×#117—21no—
133C2×#117C2311yescharacteristic
134C2×#117C2311yescharacteristic
135C2×#117C2311yescharacteristic
136C2×#117C2311yescharacteristic
137C2×#117C2311yescharacteristic
138C22×C4⋊C4—22no—
139C22×C4⋊C4—21no—
140C22×C4⋊C4C2311yescharacteristic
141C22×C4⋊C4C2311yescharacteristic
142C22×C4⋊C4C2311yescharacteristic
143C22×C22⋊C4—22no—
144C22×C22⋊C4—22no—
145C22×C42C2311yescharacteristic
146#428—42no—
147C2×#107—42no—
148C2×#107—42no—
149C2×#107C2312yes—
150C2×#107C2314yes—
151C2×#107C2314yes—
152C2×#106C2312yes—
153#424—42no—
154C2×C4⋊C8C2314yes—
155#418—42no—
156#417—42no—
157#416—24no—
158#416—42no—
159C2×#103C2312yes—
160C2×#103C2312yes—
161C2×#103C2312yes—
162C2×#102—42no—
163C2×#102—42no—
164C2×#102C2312yes—
165C2×#102C2312yes—
166C2×#102C2314yes—
167C2×#102C2312yes—
168C2×#102C2312yes—
169C2×#102C2312yes—
170C2×#102C2314yes—
171C4×M16—42no—
172C2×#97C2312yes—
173C2×C4×C8C2312yes—
174#399—24no—
175#399—24no—
176#398—24no—
177#396—24no—
178#396—24no—
179#396—24no—
180#395—24no—
181#393—24no—
182#391C2311yescharacteristic
183#391C2312yes—
184#391C2311yescharacteristic
185#390—21no—
186#390—21no—
187#390—22no—
188#390C2314yes—
189#390C2311yescharacteristic
190#390C2312yes—
191#390C2312yes—
192#390C2312yes—
193#390C2311yescharacteristic
194#389—24no—
195#389—22no—
196#389—22no—
197#389D412yes—
198#389D412yes—
199#389C2314yes—
200#389C2314yes—
201#387—24no—
202#386—22no—
203#386—22no—
204#385—24no—
205#385—22no—
206#385—22no—
207#385—24no—
208#385—24no—
209#385D414yes—
210#384—21no—
211#384—22no—
212#384C2312yes—
213#384C2312yes—
214#384C2312yes—
215#382—21no—
216#382C2312yes—
217#380—22no—
218#380—22no—
219#380—24no—
220#379—22no—
221C4×C4⋊C4C2312yes—
222C4×C4⋊C4C2312yes—
223C4×C22⋊C4—22no—
224#340C2312yes—
225#340C2314yes—
226#340C2314yes—
227#340C2312yes—
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—
301C22⋊Q8—88no—
302C4×Q8—48no—
303C4×Q8—42no—
304C4×Q8—42no—
305C4×D4—28no—
306C4×D4—44no—
307C4×D4—42no—
308C4×D4—42no—
309C4×D4—48no—
310C4×D4—48no—
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—22no—
320#117—44no—
321#117#4114yes—
322C2×C4⋊C4—22no—
323C2×C4⋊C4—22no—
324C2×C4⋊C4—22no—
325C2×C4⋊C4—22no—
326C2×C4⋊C4—22no—
327C2×C4⋊C4—22no—
328C2×C4⋊C4—21no—
329C2×C4⋊C4—22no—
330C2×C4⋊C4—24no—
331C2×C4⋊C4—21no—
332C2×C4⋊C4—21no—
333C2×C4⋊C4—22no—
334C2×C4⋊C4—22no—
335C2×C4⋊C4—22no—
336C2×C4⋊C4—22no—
337C2×C4⋊C4—22no—
338C2×C4⋊C4—22no—
339C2×C4⋊C4—24no—
340C2×C4⋊C4—24no—
341C2×C4⋊C4—24no—
342C2×C4⋊C4—24no—
343C2×C4⋊C4—44no—
344C2×C4⋊C4—84no—
345C2×C4⋊C4C2×D411yescharacteristic
346C2×C4⋊C4C2×D411yescharacteristic
347C2×C4⋊C4C2×D412yes—
348C2×C4⋊C4C2×D414yes—
349C2×C4⋊C4C2×D414yes—
350C2×C4⋊C4C2412yes—
351C2×C4⋊C4C2412yes—
352C2×C4⋊C4C2412yes—
353C2×C4⋊C4C2411yescharacteristic
354C2×C4⋊C4C2412yes—
355C2×C4⋊C4C2411yescharacteristic
356C2×C4⋊C4C2412yes—
357C2×C22⋊C4—21no—
358C2×C22⋊C4—21no—
359C2×C22⋊C4—21no—
360C2×C22⋊C4—24no—
361C2×C22⋊C4—24no—
362C2×C22⋊C4—22no—
363C2×C22⋊C4—22no—
364C2×C22⋊C4—22no—
365C2×C22⋊C4—24no—
366C2×C22⋊C4—24no—
367C2×C22⋊C4—24no—
368C2×C22⋊C4—24no—
369C2×C22⋊C4—48no—
370C2×C22⋊C4—84no—
371C2×C22⋊C4C2×D412yes—
372C2×C22⋊C4C2×D412yes—
373C2×C42—21no—
374C2×C42—22no—
375C2×C42—22no—
376C2×C42—22no—
377C2×C42—21no—
378C2×C42—21no—
379C2×C42C2×D412yes—
380C2×C42C2412yes—
381C2×C42C2412yes—
382C2×C42C2411yescharacteristic
383C2×C42C2411yescharacteristic
384#107—44no—
385#107—44no—
386#107—42no—
387#106—42no—
388C4⋊C8—44no—
389#103—24no—
390#103—42no—
391#103—42no—
392#102—24no—
393#102—44no—
394#102—44no—
395#102—42no—
396#102—42no—
397#102—42no—
398#102—42no—
399#97—42no—
400C4×C8—42no—
401#95—22no—
402#95—22no—
403#95—24no—
404#95—24no—
405#95—24no—
406#95—24no—
407#95C2×D414yes—
408#95C2×D414yes—
409C24—21no—
410C24—22no—
411C24—24no—
412C24—42no—
413C24—42no—
414C24—82no—
415C24—82no—
416C24C22×D411yescharacteristic
417#41—28no—
418#41—28no—
419#41—44no—
420#41—44no—
421C2×Q8—21no—
422C2×Q8—21no—
423C2×Q8—21no—
424C2×Q8—21no—
425C2×Q8—21no—
426C2×Q8—44no—
427C2×Q8—44no—
428C2×Q8—44no—
429C2×Q8—82no—
430C2×Q8—84no—
431C2×Q8C22×Q812yes—
432C2×D4—24no—
433C2×D4—28no—
434C2×D4—24no—
435C2×D4—28no—
436C2×D4—21no—
437C2×D4—22no—
438C2×D4—22no—
439C2×D4—21no—
440C2×D4—21no—
441C2×D4—22no—
442C2×D4—21no—
443C2×D4—22no—
444C2×D4—22no—
445C2×D4—21no—
446C2×D4—42no—
447C2×D4—42no—
448C2×D4—42no—
449C2×D4—44no—
450C2×D4—42no—
451C2×D4—44no—
452C2×D4C22×Q812yes—
453C2×D4C22×Q814yes—
454C22×C4—24no—
455C22×C4—21no—
456C22×C4—22no—
457C22×C4—24no—
458C22×C4—24no—
459C22×C4—21no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—21no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—42no—
466C22×C4—48no—
467C22×C4—44no—
468C22×C4—42no—
469C22×C4—42no—
470C22×C4—42no—
471C22×C4—42no—
472C22×C4—42no—
473C22×C4—84no—
474C22×C4—84no—
475C22×C4—84no—
476C22×C4—82no—
477C22×C4—82no—
478C22×C4—82no—
479C22×C4C22×D414yes—
480C22×C4C22×D412yes—
481C22×C4C22×D412yes—
482C22×C4C22×D412yes—
483C22×C4C22×D412yes—
484C22×C4C22×D411yescharacteristic
485C22×C4C22×Q812yes—
486C22×C4C2×#4112yes—
487C22×C4H(5,𝔽2)12yes—
488C22×C4H(5,𝔽2)12yes—
489C22×C4H(5,𝔽2)12yes—
490C22×C42−1+412yes—
491C22×C4C2511yescharacteristic
492SD16—48no—
493D8—48no—
494M16—28no—
495M16—44no—
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—22no—
506C2×C8—22no—
507C2×C8—44no—
508C2×C8—44no—
509C2×C8C22×Q814yes—
510C4⋊C4—22no—
511C4⋊C4—22no—
512C4⋊C4—24no—
513C4⋊C4—28no—
514C4⋊C4—22no—
515C4⋊C4—22no—
516C4⋊C4—22no—
517C4⋊C4—22no—
518C4⋊C4—21no—
519C4⋊C4—21no—
520C4⋊C4—22no—
521C4⋊C4—22no—
522C4⋊C4—21no—
523C4⋊C4—22no—
524C4⋊C4—22no—
525C4⋊C4—22no—
526C4⋊C4—21no—
527C4⋊C4—44no—
528C4⋊C4—44no—
529C4⋊C4—44no—
530C4⋊C4—44no—
531C4⋊C4—44no—
532C4⋊C4—44no—
533C4⋊C4—44no—
534C4⋊C4—44no—
535C4⋊C4—42no—
536C4⋊C4—42no—
537C4⋊C4—42no—
538C4⋊C4—48no—
539C4⋊C4—84no—
540C4⋊C4—84no—
541C4⋊C4—84no—
542C4⋊C4—84no—
543C4⋊C4C2×#4114yes—
544C22⋊C4—24no—
545C22⋊C4—24no—
546C22⋊C4—24no—
547C22⋊C4—48no—
548C22⋊C4—44no—
549C22⋊C4—44no—
550C22⋊C4—42no—
551C22⋊C4—42no—
552C22⋊C4—48no—
553C22⋊C4—44no—
554C22⋊C4—84no—
555C22⋊C4—84no—
556C22⋊C4—84no—
557C22⋊C4—84no—
558C42—24no—
559C42—24no—
560C42—21no—
561C42—21no—
562C42—22no—
563C42—21no—
564C42—21no—
565C42—22no—
566C42—22no—
567C42—22no—
568C42—44no—
569C42—42no—
570C42—42no—
571C23—22no—
572C23—22no—
573C23—22no—
574C23—24no—
575C23—42no—
576C23—44no—
577C23—41no—
578C23—42no—
579C23—42no—
580C23—41no—
581C23—42no—
582C23—44no—
583C23—48no—
584C23—88no—
585C23—164no—
586C23—162no—
587C23—162no—
588C23#53512yes—
589C23#54612yes—
590C23D4×Q812yes—
591C23C23×D411yescharacteristic
592Q8—22no—
593Q8—22no—
594Q8—41no—
595Q8—41no—
596Q8—88no—
597Q8—164no—
598D4—22no—
599D4—24no—
600D4—24no—
601D4—22no—
602D4—48no—
603D4—42no—
604D4—41no—
605D4—48no—
606D4—42no—
607D4—41no—
608C2×C4—22no—
609C2×C4—22no—
610C2×C4—22no—
611C2×C4—24no—
612C2×C4—24no—
613C2×C4—22no—
614C2×C4—22no—
615C2×C4—22no—
616C2×C4—24no—
617C2×C4—24no—
618C2×C4—22no—
619C2×C4—22no—
620C2×C4—24no—
621C2×C4—24no—
622C2×C4—48no—
623C2×C4—42no—
624C2×C4—41no—
625C2×C4—41no—
626C2×C4—42no—
627C2×C4—48no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—41no—
631C2×C4—42no—
632C2×C4—41no—
633C2×C4—42no—
634C2×C4—42no—
635C2×C4—42no—
636C2×C4—44no—
637C2×C4—42no—
638C2×C4—42no—
639C2×C4—44no—
640C2×C4—44no—
641C2×C4—168no—
642C2×C4—164no—
643C2×C4—162no—
644C2×C4—162no—
645C2×C4—162no—
646C2×C4#53512yes—
647C2×C4#54214yes—
648C2×C4#54314yes—
649C2×C4#54612yes—
650C2×C4D4×Q812yes—
651C2×C4#55414yes—
652C2×C4#56214yes—
653C2×C4#57614yes—
654C2×C4C23×Q811yescharacteristic
655C2×C4C22×#4111yescharacteristic
656C2×C4C2×H(5,𝔽2)11yescharacteristic
657C2×C4C2×H(5,𝔽2)11yescharacteristic
658C2×C4C2×H(5,𝔽2)11yescharacteristic
659C2×C4C2×2−1+411yescharacteristic
660C8—24no—
661C8—42no—
662C8—42no—
663C8—42no—
664C22—24no—
665C22—22no—
666C22—22no—
667C22—24no—
668C22—42no—
669C22—44no—
670C22—44no—
671C22—48no—
672C22—84no—
673C22—82no—
674C22—81no—
675C22—82no—
676C22—81no—
677C22—82no—
678C22—168no—
679C22#269214yes—
680C22C2×#53511yescharacteristic
681C22C2×#54611yescharacteristic
682C22C2×D4×Q811yescharacteristic
683C22C2×#57612yes—
684C22#358612yes—
685C4—24no—
686C4—42no—
687C4—44no—
688C4—44no—
689C4—42no—
690C4—42no—
691C4—42no—
692C4—82no—
693C4—82no—
694C4—82no—
695C4—82no—
696C4—81no—
697C4—81no—
698C4#269214yes—
699C4C2×#54211yescharacteristic
700C4C2×#54311yescharacteristic
701C4C2×#55411yescharacteristic
702C4C2×#56211yescharacteristic
703C2—24no—
704C2—82no—
705C2—84no—
706C2C2×#269211yescharacteristic
707C2#6081711yescharacteristic
708C2#6082811yescharacteristic
709C2#6119112yes—
710C2#6138112yes—
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. #35532C2semidirect
2. #35515C2semidirect
3. #31535C2semidirect
4. #31534C2semidirect
5. #31519C2semidirect
6. #31518C2semidirect
7. C2×#2093C2semidirect
8. #23660C2semidirect
9. #23611C2semidirect
10. #23555C2semidirect
11. #23456C2semidirect
12. #23451C2semidirect
13. #21322C2semidirect
15. #21022C2semidirect
18. #20881C2semidirect
19. #20650C2semidirect
37. #2567C22semidirect
38. #2566C22semidirect
39. #2565C22semidirect
40. #2542C22semidirect
59. #2085C22semidirect
60. #2083C22semidirect
61. #2068C22semidirect
62. #2067C22semidirect
63. #2059C22semidirect
64. #2057C22semidirect
65. #2026C22semidirect
67. #2025C22semidirect
68. #2024C22semidirect
70. #2023C22semidirect
71. #1992C22semidirect
72. #1990C22semidirect
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
#355322
#355152
#315352
#315342
#315192
#315182
C2×#20932
#236602
#236112
#235552
#234562
#234512
#213222
#212362
#210222
#209322
#209292
#208812
#206502