Finite Group Databasebeta

← #10575048

Subgroups — #10575048

Order 512.

7950 conjugacy classes of subgroups.

The subgroup lattice has 662 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3489C211yesmaximal, characteristic
2C2×#3428C211yesmaximal, characteristic, complemented
3#60526C218yesmaximal, complemented
4#60524C2116yesmaximal, complemented
5#60522C218yesmaximal, complemented
6#60520C216yesmaximal, complemented
7#60519C216yesmaximal, complemented
8#60511C216yesmaximal, complemented
9C2×#3095C216yesmaximal, complemented
10#60109C212yesmaximal, complemented
11#60088C212yesmaximal, complemented
12#60064C211yesmaximal, characteristic, complemented
13C22×2−1+4C2211yescharacteristic
14C23×#41C2211yescharacteristic
15C2×#568C2212yes—
16#3489—24no—
17#3489—24no—
18#3489C2218yes—
19#3489C2218yes—
20C2×D4×Q8—21no—
21C2×#548—21no—
22C2×#547—23no—
23C2×#546—23no—
24C2×#537C2213yes—
25C2×#536C2213yes—
26C2×#525C2213yes—
27C22×C22⋊Q8C2213yes—
28#3428—24no—
29#3428—23no—
30#3428—21no—
31#3428C2216yes—
32#3428C2212yes—
33#3428C2218yescomplemented
34C2×#520C2211yescharacteristic
35C2×#520C2214yescomplemented
36C2×#520C2214yescomplemented
37C2×#519—21no—
38C2×#518—23no—
39C2×C4×#41—24no—
40C22×#117C2216yes—
41#3099C22148yescomplemented
42#3098C22148yescomplemented
43#3097C22148yescomplemented
44#3097C22148yescomplemented
45#3095C22112yes—
46#3095C22124yescomplemented
47#3095C22112yescomplemented
48#3095C22124yescomplemented
49#3095C22124yescomplemented
50#3094C2214yes—
51#3094C22112yescomplemented
52#3093C22124yescomplemented
53#3092C22112yescomplemented
54#3092C22112yescomplemented
55#3091C22124yescomplemented
56#3091C22124yescomplemented
57C2×#484C22112yescomplemented
58C2×#485C22124yescomplemented
59C2×#483C22112yescomplemented
60#2913C22116yescomplemented
61#2912C22116yescomplemented
62#2911C22112yescomplemented
63C2×#428C22112yescomplemented
64#2898C22116yescomplemented
65#2898C22116yescomplemented
66#2896C22112yescomplemented
67C2×#419C22112yescomplemented
68#2880C2218yescomplemented
69#2880C2218yescomplemented
70#2879C2216yes—
71C2×#407C2216yes—
72C2×2−1+4—24no—
73C2×2−1+4—216no—
74C2×2−1+4C2318yes—
75C22×#41—216no—
76C22×#41—23no—
77C22×#41D414yes—
78C22×#41D413yes—
79C22×#41D411yescharacteristic
80C22×#41D414yes—
81C22×#41C2313yes—
82C22×#41C2313yes—
83C22×#41C2314yes—
84C22×#41C2314yes—
85C22×#41C2311yescharacteristic
86C23×Q8—21no—
87C23×Q8D411yescharacteristic
88C23×Q8C2313yes—
89C23×D4D413yes—
90C24×C4C2313yes—
91#568—216no—
92C2×#131C23116yes—
93C22×M16C23112yes—
94D4×Q8—28no—
95D4×Q8—28no—
96#548—28no—
97#548—28no—
98#547—224no—
99#547—224no—
100#546—224no—
101#546—224no—
102#537—212no—
103#537C23124yes—
104#536—212no—
105#536C23124yes—
106C2×C4⋊Q8—23no—
107C2×#124—23no—
108#525—26no—
109#525—26no—
110#525C23112yes—
111#525C23112yes—
112C2×#123—26no—
113C2×#123—23no—
114C2×C22⋊Q8—23no—
115C2×C22⋊Q8—23no—
116C2×C22⋊Q8—23no—
117C2×C22⋊Q8—26no—
118C2×C22⋊Q8—26no—
119C2×C22⋊Q8C23112yes—
120C2×C22⋊Q8C23112yes—
121C2×C22⋊Q8C2313yes—
122C2×C22⋊Q8C2316yes—
123C2×C22⋊Q8C2313yes—
124C2×#121—26no—
125C2×#121—23no—
126C2×#121C2311yescharacteristic
127C2×#121C2313yes—
128C2×#120—23no—
129#520—24no—
130#520—212no—
131#520—212no—
132#520—24no—
133#520—24no—
134#520C23124yes—
135#520C23124yes—
136#520C2318yes—
137#520C2318yes—
138#520C2318yes—
139#519—26no—
140#519—22no—
141#519—28no—
142#518—212no—
143#518—26no—
144#518—26no—
145#518—224no—
146C4×#41—224no—
147C4×#41—28no—
148C4×#41—224no—
149C4×#41—28no—
150C2×C4×Q8—24no—
151C2×C4×Q8—24no—
152C2×C4×Q8—21no—
153C2×C4×D4—212no—
154C2×C4×D4—212no—
155C2×C4×D4—23no—
156C2×C4×D4—23no—
157C2×C4×D4—23no—
158C2×#117—26no—
159C2×#117—26no—
160C2×#117—26no—
161C2×#117—26no—
162C2×#117—26no—
163C2×#117—26no—
164C2×#117C23112yes—
165C2×#117C23112yes—
166C2×#117C23112yes—
167C2×#117C2314yes—
168C2×#117C23112yes—
169C2×#117C23112yes—
170C2×#117C2313yes—
171C22×C4⋊C4—23no—
172C22×C4⋊C4C2316yes—
173C22×C4⋊C4C2316yes—
174C22×C4⋊C4C2313yes—
175C22×C22⋊C4—26no—
176C22×C22⋊C4—23no—
177C22×C42—26no—
178#485C23148yes—
179#485C23196yescomplemented
180#485C23148yescomplemented
181#484C23124yes—
182#484C23148yescomplemented
183#484C23124yescomplemented
184#483C23124yes—
185#483C23148yescomplemented
186#483C23124yescomplemented
187#482C23116yes—
188#482C23148yescomplemented
189#481C2318yes—
190#481C23124yescomplemented
191#480C2318yes—
192#480C23124yescomplemented
193#428C2318yes—
194#428C23124yes—
195#428C23124yescomplemented
196#428C23124yescomplemented
197#428C23148yescomplemented
198#427C23148yescomplemented
199C2×#107C23124yescomplemented
200C2×#106C23124yescomplemented
201#419C23124yes—
202#419C23148yescomplemented
203#418C2318yes—
204#417C23124yescomplemented
205#417C23124yescomplemented
206#416C23148yescomplemented
207C2×#103C23124yescomplemented
208C2×#103C23124yescomplemented
209#408C23148yes—
210#407C23112yes—
211#407C23112yes—
212#407C2314yes—
213#407C23112yes—
214C2×C22⋊C8C23112yes—
215C2×C22⋊C8C23112yes—
216C25C2×D413yes—
2172−1+4—264no—
2182−1+4—432no—
219C2×#41—232no—
220C2×#41—232no—
221C2×#41—232no—
222C2×#41—232no—
223C2×#41—28no—
224C2×#41—224no—
225C2×#41—224no—
226C2×#41—26no—
227C2×#41—224no—
228C2×#41—224no—
229C2×#41—424no—
230C2×#41C2×D4112yes—
231C2×#41C2×D4112yes—
232C2×#41C2×D414yes—
233C2×#41C2×D4112yes—
234C2×#41C2×D414yes—
235C2×#41C2×D414yes—
236C2×#41#41116yes—
237C2×#41C24112yes—
238C2×#41C24112yes—
239C2×#41C2414yes—
240C22×Q8—28no—
241C22×Q8—28no—
242C22×Q8—23no—
243C22×Q8—26no—
244C22×Q8—224no—
245C22×Q8—48no—
246C22×Q8C2×D414yes—
247C22×Q8C2×D413yes—
248C22×Q8C2×D414yes—
249C22×Q8C2×D411yescharacteristic
250C22×Q8C2413yes—
251C22×Q8C24112yes—
252C22×D4—224no—
253C22×D4—21no—
254C22×D4—23no—
255C22×D4—23no—
256C22×D4C2×D4112yes—
257C22×D4C2×D416yes—
258C22×D4C2×D413yes—
259C22×D4C2×D4112yes—
260C22×D4C2×D413yes—
261C22×D4C2411yescharacteristic
262C23×C4—224no—
263C23×C4—23no—
264C23×C4—23no—
265C23×C4—26no—
266C23×C4—26no—
267C23×C4—23no—
268C23×C4C2×D413yes—
269C23×C4C2×D413yes—
270C23×C4C2×D413yes—
271C23×C4C2×D416yes—
272C23×C4C2×D416yes—
273C23×C4C2416yes—
274C23×C4C2411yescharacteristic
275C23×C4C2413yes—
276C23×C4C2416yes—
277C23×C4C2413yes—
278#131—264no—
279C2×M16—248no—
280C2×M16C2418yes—
281C2×M16C24124yes—
282C22×C8C24124yes—
283C4⋊Q8—224no—
284#124—224no—
285#123—248no—
286#123—212no—
287#123—212no—
288C22⋊Q8—224no—
289C22⋊Q8—23no—
290C22⋊Q8—23no—
291C22⋊Q8—212no—
292C22⋊Q8—212no—
293C22⋊Q8—212no—
294C22⋊Q8—212no—
295C22⋊Q8—212no—
296C22⋊Q8—26no—
297C22⋊Q8C2416yes—
298C22⋊Q8C2416yes—
299C22⋊Q8C24124yes—
300C22⋊Q8C24112yes—
301#121—248no—
302#121—224no—
303#121—21no—
304#121—21no—
305#121—26no—
306#121C2412yes—
307#121C2412yes—
308#121C24112yes—
309#120—212no—
310#120—212no—
311C4×Q8—224no—
312C4×Q8—224no—
313C4×Q8—28no—
314C4×Q8—28no—
315C4×Q8—28no—
316C4×D4—248no—
317C4×D4—248no—
318C4×D4—224no—
319C4×D4—224no—
320C4×D4—224no—
321C4×D4—224no—
322C4×D4—224no—
323C4×D4—212no—
324C4×D4—212no—
325C4×D4—212no—
326C4×D4—212no—
327#117—24no—
328#117—24no—
329#117—212no—
330#117—212no—
331#117—212no—
332#117—212no—
333#117—212no—
334#117—212no—
335#117—212no—
336#117—212no—
337#117—212no—
338#117—212no—
339#117—26no—
340#117C2418yes—
341#117C2418yes—
342#117C24124yes—
343#117C24124yes—
344#117C24112yes—
345C2×C4⋊C4—23no—
346C2×C4⋊C4—26no—
347C2×C4⋊C4—212no—
348C2×C4⋊C4—212no—
349C2×C4⋊C4—26no—
350C2×C4⋊C4—26no—
351C2×C4⋊C4—26no—
352C2×C4⋊C4—26no—
353C2×C4⋊C4—26no—
354C2×C4⋊C4—26no—
355C2×C4⋊C4—23no—
356C2×C4⋊C4—26no—
357C2×C4⋊C4—23no—
358C2×C4⋊C4C2414yes—
359C2×C4⋊C4C2414yes—
360C2×C4⋊C4C2413yes—
361C2×C4⋊C4C24112yes—
362C2×C4⋊C4C24112yes—
363C2×C4⋊C4C24112yes—
364C2×C4⋊C4C24112yes—
365C2×C4⋊C4C24112yes—
366C2×C4⋊C4C2411yescharacteristic
367C2×C22⋊C4—24no—
368C2×C22⋊C4—23no—
369C2×C22⋊C4—212no—
370C2×C22⋊C4—212no—
371C2×C22⋊C4—212no—
372C2×C22⋊C4—212no—
373C2×C22⋊C4—212no—
374C2×C22⋊C4—26no—
375C2×C22⋊C4—212no—
376C2×C22⋊C4—212no—
377C2×C22⋊C4—23no—
378C2×C22⋊C4—23no—
379C2×C22⋊C4—21no—
380C2×C22⋊C4—23no—
381C2×C42—24no—
382C2×C42—212no—
383C2×C42—212no—
384C2×C42—212no—
385C2×C42—212no—
386C2×C42—212no—
387C2×C42—23no—
388#107C24148yes—
389#107C24116yes—
390#106C24148yes—
391#106C24116yes—
392#103C24148yes—
393#103C24148yes—
394#102C24116yes—
395#102C24116yes—
396C22⋊C8C2418yes—
397C22⋊C8C2418yes—
398C22⋊C8C24124yes—
399C22⋊C8C24124yes—
400C24—212no—
401C24—21no—
402C24—46no—
403C24—83no—
404C24C22×D411yescharacteristic
405C24C22×D416yes—
406C24C22×D413yes—
407C24C22×D416yes—
408C24C22×D413yes—
409#41—216no—
410#41—216no—
411#41—248no—
412#41—248no—
413#41—248no—
414#41—248no—
415#41—248no—
416#41—496no—
417#41—424no—
418#41#123132yes—
419#41C2×#41132yes—
420C2×Q8—216no—
421C2×Q8—216no—
422C2×Q8—212no—
423C2×Q8—23no—
424C2×Q8—212no—
425C2×Q8—212no—
426C2×Q8—248no—
427C2×Q8—432no—
428C2×Q8—412no—
429C2×Q8—424no—
430C2×Q8#12318yes—
431C2×Q8C22×D4112yes—
432C2×Q8C22×D414yes—
433C2×Q8C2516yes—
434C2×D4—224no—
435C2×D4—248no—
436C2×D4—248no—
437C2×D4—22no—
438C2×D4—22no—
439C2×D4—212no—
440C2×D4—212no—
441C2×D4—212no—
442C2×D4—21no—
443C2×D4—24no—
444C2×D4—44no—
445C2×D4—412no—
446C2×D4—86no—
447C2×D4#123124yes—
448C2×D4C22×D4124yes—
449C2×D4C22×D4112yes—
450C2×D4C22×D4112yes—
451C2×D4C2512yes—
452C22×C4—212no—
453C22×C4—212no—
454C22×C4—212no—
455C22×C4—224no—
456C22×C4—224no—
457C22×C4—24no—
458C22×C4—212no—
459C22×C4—224no—
460C22×C4—224no—
461C22×C4—24no—
462C22×C4—24no—
463C22×C4—21no—
464C22×C4—23no—
465C22×C4—23no—
466C22×C4—212no—
467C22×C4—212no—
468C22×C4—212no—
469C22×C4—212no—
470C22×C4—212no—
471C22×C4—212no—
472C22×C4—212no—
473C22×C4—412no—
474C22×C4—412no—
475C22×C4—812no—
476C22×C4—812no—
477C22×C4—86no—
478C22×C4C22×D411yescharacteristic
479C22×C4C22×D416yes—
480C22×C4C22×D416yes—
481C22×C4C22×D416yes—
482C22×C4C22×D416yes—
483C22×C4C22×D411yescharacteristic
484C22×C4C22×D413yes—
485C22×C4C22×D416yes—
486C22×C4C22×D416yes—
487C22×C4C22×D416yes—
488C22×C4C22×D411yescharacteristic
489C22×C4C22×D416yes—
490C22×C4C22×D413yes—
491C22×C4C2×#41124yes—
492C22×C4C2511yescharacteristic
493C22×C4C2516yes—
494C22×C4C2511yescharacteristic
495C22×C4C2516yes—
496C22×C4C2511yescharacteristic
497M16—216no—
498M16—248no—
499C2×C8—248no—
500C2×C8C25116yes—
501C4⋊C4—24no—
502C4⋊C4—24no—
503C4⋊C4—26no—
504C4⋊C4—26no—
505C4⋊C4—23no—
506C4⋊C4—224no—
507C4⋊C4—212no—
508C4⋊C4—212no—
509C4⋊C4—212no—
510C4⋊C4—212no—
511C4⋊C4—212no—
512C4⋊C4—212no—
513C4⋊C4—21no—
514C4⋊C4C2518yes—
515C4⋊C4C2518yes—
516C4⋊C4C2516yes—
517C4⋊C4C2512yes—
518C22⋊C4—28no—
519C22⋊C4—28no—
520C22⋊C4—26no—
521C22⋊C4—26no—
522C22⋊C4—224no—
523C22⋊C4—224no—
524C22⋊C4—224no—
525C22⋊C4—212no—
526C22⋊C4—212no—
527C22⋊C4—22no—
528C22⋊C4—22no—
529C22⋊C4—212no—
530C42—28no—
531C42—28no—
532C42—224no—
533C42—224no—
534C42—212no—
535C23—212no—
536C23—22no—
537C23—22no—
538C23—212no—
539C23—212no—
540C23—26no—
541C23—22no—
542C23—46no—
543C23—43no—
544C23—43no—
545C23—41no—
546C23—43no—
547C23—43no—
548C23—46no—
549C23—46no—
550C23—42no—
551C23—42no—
552C23—42no—
553C23—824no—
554C23C2×#123112yes—
555C23C23×D411yescharacteristic
556C23C23×D416yes—
557C23C23×D411yescharacteristic
558C23C23×D411yescharacteristic
559C23C23×D416yes—
560Q8—224no—
561Q8—212no—
562Q8—46no—
563Q8—424no—
564Q8C2×#123116yes—
565D4—248no—
566D4—24no—
567D4—224no—
568D4—44no—
569D4—44no—
570D4—424no—
571D4—42no—
572D4—812no—
573D4C2×#123148yes—
574C2×C4—22no—
575C2×C4—26no—
576C2×C4—212no—
577C2×C4—212no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—212no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—212no—
584C2×C4—212no—
585C2×C4—212no—
586C2×C4—212no—
587C2×C4—212no—
588C2×C4—28no—
589C2×C4—28no—
590C2×C4—22no—
591C2×C4—26no—
592C2×C4—26no—
593C2×C4—48no—
594C2×C4—46no—
595C2×C4—42no—
596C2×C4—48no—
597C2×C4—46no—
598C2×C4—424no—
599C2×C4—424no—
600C2×C4—812no—
601C2×C4—812no—
602C2×C4—812no—
603C2×C4—812no—
604C2×C4—812no—
605C2×C4C2×#123112yes—
606C2×C4C2×#123124yes—
607C2×C4C23×D411yescharacteristic
608C2×C4C23×D416yes—
609C2×C4C23×D411yescharacteristic
610C2×C4C23×D411yescharacteristic
611C2×C4C23×D416yes—
612C2×C4C22×#41124yes—
613C2×C4C22×#4114yes—
614C2×C4C2611yescharacteristic
615C8—216no—
616C22—21no—
617C22—26no—
618C22—21no—
619C22—21no—
620C22—26no—
621C22—21no—
622C22—26no—
623C22—21no—
624C22—26no—
625C22—21no—
626C22—21no—
627C22—26no—
628C22—412no—
629C22—424no—
630C22—44no—
631C22—42no—
632C22—42no—
633C22—42no—
634C22—86no—
635C22—812no—
636C22—84no—
637C22C22×#12312yes—
638C22C22×#123112yes—
639C22#358612yes—
640C22C24×D411yescharacteristic
641C4—21no—
642C4—21no—
643C4—26no—
644C4—48no—
645C4—42no—
646C4—48no—
647C4—46no—
648C4—46no—
649C4C22×#12312yes—
650C4C22×#123112yes—
651C4C23×#4112yes—
652C2—22no—
653C2—22no—
654C2—212no—
655C2—46no—
656C2—41no—
657C2—82no—
658C2C23×#12311yescharacteristic
659C2C2×#358611yescharacteristic
660C2#6309011yescharacteristic
subgroupG/H
center, socleC22C24×D4
commutator, Frattini, cosocleC2×C4C26
prime radicalC2×#3489C2
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23×#41 ◁ G
LECSG ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#3428C2semidirect
3. #60526C2semidirect
4. #60524C2semidirect
5. #60522C2semidirect
6. #60520C2semidirect
7. #60519C2semidirect
8. #60511C2semidirect
9. C2×#3095C2semidirect
10. #60109C2semidirect
11. #60088C2semidirect
12. #60064C2semidirect
33. #3428C22semidirect
35. C2×#520C22semidirect
36. C2×#520C22semidirect
41. #3099C22semidirect
42. #3098C22semidirect
43. #3097C22semidirect
44. #3097C22semidirect
46. #3095C22semidirect
47. #3095C22semidirect
48. #3095C22semidirect
49. #3095C22semidirect
51. #3094C22semidirect
52. #3093C22semidirect
53. #3092C22semidirect
54. #3092C22semidirect
55. #3091C22semidirect
56. #3091C22semidirect
57. C2×#484C22semidirect
58. C2×#485C22semidirect
59. C2×#483C22semidirect
60. #2913C22semidirect
61. #2912C22semidirect
62. #2911C22semidirect
63. C2×#428C22semidirect
64. #2898C22semidirect
65. #2898C22semidirect
66. #2896C22semidirect
67. C2×#419C22semidirect
68. #2880C22semidirect
69. #2880C22semidirect
179. #485C23semidirect
180. #485C23semidirect
182. #484C23semidirect
183. #484C23semidirect
185. #483C23semidirect
186. #483C23semidirect
188. #482C23semidirect
190. #481C23semidirect
192. #480C23semidirect
195. #428C23semidirect
196. #428C23semidirect
197. #428C23semidirect
198. #427C23semidirect
199. C2×#107C23semidirect
200. C2×#106C23semidirect
202. #419C23semidirect
204. #417C23semidirect
205. #417C23semidirect
206. #416C23semidirect
207. C2×#103C23semidirect
208. C2×#103C23semidirect

Maximal subgroups

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

subgroupindex
C2×#34892
C2×#34282
#605262
#605242
#605222
#605202
#605192
#605112
C2×#30952
#601092
#600882
#600642