Finite Group Databasebeta

← #6512629

Subgroups — #6512629

Order 512.

3376 conjugacy classes of subgroups.

The subgroup lattice has 667 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3506C211yesmaximal, characteristic
2C2×#3113C211yesmaximal, characteristic, complemented
3C2×#3112C211yesmaximal, characteristic, complemented
4C2×#2536C211yesmaximal, characteristic, complemented
5#24254C214yesmaximal, complemented
6#24253C214yesmaximal, complemented
7#24135C214yesmaximal, complemented
8#24133C214yesmaximal, complemented
9#24004C214yesmaximal, complemented
10#23922C214yesmaximal, complemented
11#21866C211yesmaximal, characteristic, complemented
12#21865C211yesmaximal, characteristic, complemented
13#21790C211yesmaximal, characteristic, complemented
14#3506—28no—
15#3506—28no—
16C2×#559—22no—
17C2×#556C2212yes—
18C2×#555C2212yes—
19C2×#553—22no—
20C2×#550C2212yes—
21C2×#547C2212yes—
22C2×#538—22no—
23C2×#535—21no—
24C2×#533C2211yescharacteristic
25C2×#532C2211yescharacteristic
26C2×#525C2214yes—
27C22×C22⋊Q8C2212yes—
28C2×#520—21no—
29C22×C4×Q8C2211yescharacteristic
30C22×C4×D4C2211yescharacteristic
31#3113—28no—
32#3112—28no—
33C2×#483C2212yes—
34C2×#480C2212yes—
35C2×#477C2212yescomplemented
36C2×#475C2212yescomplemented
37C2×#462C2212yescomplemented
38C2×#460C2212yescomplemented
39C2×#452C2218yescomplemented
40C2×#451C2212yes—
41C2×#448C2212yes—
42C2×#441C2214yescomplemented
43C2×#440C2214yescomplemented
44C2×#432C2211yescharacteristic
45C2×#432C2211yescharacteristic
46#2536—22no—
47#2536—22no—
48#2536C2214yes—
49#2536C2214yes—
50C2×#398—21no—
51C2×#394—21no—
52C2×#388—22no—
53C2×#382C2211yescharacteristic
54C2×#380C2212yes—
55#2036C2218yescomplemented
56#2034C2218yescomplemented
57#2001C2218yescomplemented
58#2000C2218yescomplemented
59#1993C2212yescomplemented
60#1993C2212yescomplemented
61#1961C2212yescomplemented
62#1960C2212yescomplemented
63#1955C2214yes—
64#1955C2214yes—
65#1874C2214yes—
66#1873C2218yescomplemented
67#1872C2214yes—
68#1854C2218yescomplemented
69#1805C2214yescomplemented
70#1804C2214yescomplemented
71#1788C2218yescomplemented
72#1779C2214yes—
73C22×#41—21no—
74C22×#41C2314yes—
75C23×Q8C2311yescharacteristic
76C23×D4D411yescharacteristic
77C24×C4C2312yes—
78C22×Q16C2314yes—
79C22×SD16C2314yes—
80#559—232no—
81#556—28no—
82#556—28no—
83#555—28no—
84#555—28no—
85#553—232no—
86#550—28no—
87#550—28no—
88#547—28no—
89#547—28no—
90#538—232no—
91#535—216no—
92#533—24no—
93#533D418yes—
94#532—24no—
95#532D418yes—
96C2×C4⋊Q8C2311yescharacteristic
97C2×C4⋊Q8C2311yescharacteristic
98C2×C4⋊Q8C2311yescharacteristic
99C2×#127—22no—
100C2×#127C2311yescharacteristic
101C2×#126—24no—
102C2×#125—21no—
103C2×#124—22no—
104C2×#124—24no—
105C2×#124—21no—
106#525—216no—
107#525—216no—
108C2×#123—24no—
109C2×#123—24no—
110C2×C22⋊Q8—21no—
111C2×C22⋊Q8—24no—
112C2×C22⋊Q8—48no—
113C2×C22⋊Q8C2314yes—
114C2×C22⋊Q8C2318yes—
115C2×C22⋊Q8C2312yes—
116C2×#121—22no—
117C2×#121—22no—
118C2×#121—21no—
119C2×#121—24no—
120C2×#121C2314yes—
121C2×#121C2312yes—
122C2×#120—24no—
123#520—216no—
124C2×C4×Q8—22no—
125C2×C4×Q8—21no—
126C2×C4×Q8—21no—
127C2×C4×Q8—21no—
128C2×C4×Q8—24no—
129C2×C4×Q8—24no—
130C2×C4×Q8C2312yes—
131C2×C4×Q8C2312yes—
132C2×C4×Q8C2312yes—
133C2×C4×D4—24no—
134C2×C4×D4—24no—
135C2×C4×D4—22no—
136C2×C4×D4—24no—
137C2×C4×D4—24no—
138C2×C4×D4—21no—
139C2×C4×D4D412yes—
140C2×C4×D4D412yes—
141C2×C4×D4C2312yes—
142C2×C4×D4C2312yes—
143C2×C4×D4C2312yes—
144C2×#117—21no—
145C2×#117—22no—
146C2×#117C2314yes—
147C22×C4⋊C4—22no—
148C22×C4⋊C4—21no—
149C22×C4⋊C4—22no—
150C22×C4⋊C4D411yescharacteristic
151C22×C4⋊C4C2311yescharacteristic
152C22×C4⋊C4C2312yes—
153C22×C22⋊C4—22no—
154C22×C22⋊C4—22no—
155C22×C22⋊C4D412yes—
156C22×C42—21no—
157C22×C42C2311yescharacteristic
158#483—28no—
159#480—28no—
160#477—28no—
161#475—28no—
162#462—28no—
163#460—28no—
164#452—416no—
165#451—28no—
166#448—28no—
167#441—216no—
168#440—216no—
169#432—24no—
170#432—24no—
171C2×#107C2314yes—
172C2×#106C2314yes—
173C2×C4⋊C8C2312yes—
174C2×C4⋊C8C2312yes—
175C2×#103C2314yes—
176C2×#103C2314yes—
177C2×#103C2314yes—
178C2×#103C2318yes—
179C2×#102C2314yes—
180C2×#102C2318yes—
181C2×C22⋊C8C2312yes—
182C2×C22⋊C8C2312yes—
183C2×C22⋊C8C2318yes—
184C2×C22⋊C8C2318yes—
185C2×#97C2312yes—
186C2×#97C2312yes—
187#398—24no—
188#398—24no—
189#394—24no—
190#394—24no—
191#388—28no—
192#388—28no—
193#382—21no—
194#382—21no—
195#382C2312yes—
196#382C2312yes—
197#380—24no—
198#380C2318yes—
199C2×#95—21no—
200C2×#95—22no—
201C2×#95—21no—
202#340C2318yes—
203#340C2318yes—
204C25C2×D412yes—
205C2×#41—28no—
206C2×#41—28no—
207C2×#41—28no—
208C2×#41—48no—
209C2×#41—48no—
210C2×#41—48no—
211C22×Q8—24no—
212C22×Q8—21no—
213C22×Q8—21no—
214C22×Q8—21no—
215C22×Q8—22no—
216C22×Q8—22no—
217C22×Q8—42no—
218C22×Q8C2412yes—
219C22×Q8C2412yes—
220C22×D4—22no—
221C22×D4—24no—
222C22×D4—24no—
223C22×D4—24no—
224C22×D4—21no—
225C22×D4—21no—
226C22×D4—44no—
227C22×D4C2×D412yes—
228C22×D4C2×D412yes—
229C22×D4C2412yes—
230C23×C4—21no—
231C23×C4—22no—
232C23×C4—24no—
233C23×C4—24no—
234C23×C4—24no—
235C23×C4—22no—
236C23×C4—21no—
237C23×C4—44no—
238C23×C4—44no—
239C23×C4C2×D411yescharacteristic
240C23×C4C2×D412yes—
241C23×C4C2×D414yes—
242C23×C4C2×D411yescharacteristic
243C23×C4C2414yes—
244C23×C4C2411yescharacteristic
245C2×Q16—28no—
246C2×Q16—28no—
247C2×Q16—44no—
248C2×SD16—28no—
249C2×SD16—28no—
250C2×SD16—44no—
251C22×C8C2414yes—
252C22×C8C2414yes—
253C4⋊Q8—22no—
254C4⋊Q8—22no—
255C4⋊Q8—22no—
256C4⋊Q8C2×D414yes—
257C4⋊Q8C2×D414yes—
258C4⋊Q8C2×D414yes—
259#127—216no—
260#127—22no—
261#127C2×D414yes—
262#126—232no—
263#125—28no—
264#124—216no—
265#124—232no—
266#124—28no—
267#123—232no—
268#123—232no—
269C22⋊Q8—28no—
270C22⋊Q8—28no—
271C22⋊Q8—24no—
272C22⋊Q8—28no—
273C22⋊Q8—216no—
274C22⋊Q8—48no—
275C22⋊Q8—416no—
276C22⋊Q8C2×D418yes—
277#121—216no—
278#121—216no—
279#121—232no—
280#121—28no—
281#121—28no—
282#121—24no—
283#121—28no—
284#121C2×D418yes—
285#120—232no—
286C4×Q8—216no—
287C4×Q8—28no—
288C4×Q8—24no—
289C4×Q8—24no—
290C4×Q8—24no—
291C4×Q8—24no—
292C4×Q8—24no—
293C4×Q8—44no—
294C4×Q8—44no—
295C4×Q8C2×D418yes—
296C4×D4—232no—
297C4×D4—216no—
298C4×D4—232no—
299C4×D4—28no—
300C4×D4—28no—
301C4×D4—28no—
302C4×D4—24no—
303C4×D4—24no—
304C4×D4—24no—
305C4×D4—24no—
306C4×D4—24no—
307C4×D4C2×D418yes—
308#117—216no—
309#117—28no—
310#117—28no—
311#117—28no—
312C2×C4⋊C4—216no—
313C2×C4⋊C4—24no—
314C2×C4⋊C4—21no—
315C2×C4⋊C4—24no—
316C2×C4⋊C4—21no—
317C2×C4⋊C4—22no—
318C2×C4⋊C4—21no—
319C2×C4⋊C4—21no—
320C2×C4⋊C4—22no—
321C2×C4⋊C4—22no—
322C2×C4⋊C4—22no—
323C2×C4⋊C4—22no—
324C2×C4⋊C4—22no—
325C2×C4⋊C4—28no—
326C2×C4⋊C4—21no—
327C2×C4⋊C4—22no—
328C2×C4⋊C4—44no—
329C2×C4⋊C4—48no—
330C2×C4⋊C4—44no—
331C2×C4⋊C4C2×D412yes—
332C2×C4⋊C4C2×D412yes—
333C2×C4⋊C4C2412yes—
334C2×C4⋊C4C2414yes—
335C2×C4⋊C4C2412yes—
336C2×C4⋊C4C2412yes—
337C2×C22⋊C4—22no—
338C2×C22⋊C4—24no—
339C2×C22⋊C4—24no—
340C2×C22⋊C4—28no—
341C2×C22⋊C4—24no—
342C2×C22⋊C4—22no—
343C2×C22⋊C4—22no—
344C2×C22⋊C4—28no—
345C2×C22⋊C4—28no—
346C2×C22⋊C4—48no—
347C2×C22⋊C4—48no—
348C2×C22⋊C4C2×D414yes—
349C2×C22⋊C4C2×D414yes—
350C2×C42—24no—
351C2×C42—22no—
352C2×C42—22no—
353C2×C42—22no—
354C2×C42—44no—
355C2×C42C2×D411yescharacteristic
356C2×C42C2×D411yescharacteristic
357C2×C42C2×D414yes—
358C2×C42C2×D414yes—
359C2×C42C2411yescharacteristic
360C2×C42C2411yescharacteristic
361#107—28no—
362#106—28no—
363C4⋊C8—24no—
364C4⋊C8—24no—
365#103—28no—
366#103—28no—
367#103—28no—
368#103—48no—
369#102—28no—
370#102—48no—
371C22⋊C8—24no—
372C22⋊C8—24no—
373C22⋊C8—48no—
374C22⋊C8—48no—
375#97—24no—
376#97—24no—
377#95—28no—
378#95—22no—
379#95—22no—
380#95—22no—
381#95—22no—
382C24—24no—
383C24—22no—
384C24—24no—
385C24—24no—
386C24—42no—
387C24—42no—
388C24—42no—
389C24—42no—
390C24C22×D414yes—
391C24C22×D411yescharacteristic
392#41—432no—
393#41—816no—
394#41—816no—
395C2×Q8—24no—
396C2×Q8—24no—
397C2×Q8—24no—
398C2×Q8—22no—
399C2×Q8—22no—
400C2×Q8—22no—
401C2×Q8—22no—
402C2×Q8—24no—
403C2×Q8—22no—
404C2×Q8—22no—
405C2×Q8—24no—
406C2×Q8—48no—
407C2×Q8—48no—
408C2×Q8—42no—
409C2×Q8—44no—
410C2×Q8—42no—
411C2×Q8—82no—
412C2×Q8C22×D414yes—
413C2×Q8H(5,𝔽2)14yes—
414C2×D4—28no—
415C2×D4—216no—
416C2×D4—216no—
417C2×D4—24no—
418C2×D4—24no—
419C2×D4—24no—
420C2×D4—24no—
421C2×D4—22no—
422C2×D4—24no—
423C2×D4—24no—
424C2×D4—44no—
425C2×D4—44no—
426C2×D4—416no—
427C2×D4—44no—
428C2×D4—48no—
429C2×D4—82no—
430C2×D4—84no—
431C2×D4—84no—
432C2×D4—84no—
433C2×D4—84no—
434C2×D4C22×D414yes—
435C22×C4—24no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—24no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—216no—
444C22×C4—24no—
445C22×C4—28no—
446C22×C4—24no—
447C22×C4—24no—
448C22×C4—28no—
449C22×C4—28no—
450C22×C4—28no—
451C22×C4—24no—
452C22×C4—22no—
453C22×C4—42no—
454C22×C4—42no—
455C22×C4—44no—
456C22×C4—44no—
457C22×C4—42no—
458C22×C4—48no—
459C22×C4—42no—
460C22×C4—42no—
461C22×C4—44no—
462C22×C4—44no—
463C22×C4—42no—
464C22×C4—44no—
465C22×C4—44no—
466C22×C4—88no—
467C22×C4—88no—
468C22×C4—84no—
469C22×C4—84no—
470C22×C4#12014yes—
471C22×C4#12014yes—
472C22×C4C22×D414yes—
473C22×C4C22×D411yescharacteristic
474C22×C4C22×D411yescharacteristic
475C22×C4C22×D411yescharacteristic
476C22×C4C22×D411yescharacteristic
477C22×C4C22×D411yescharacteristic
478C22×C4C22×D412yes—
479C22×C4C22×D412yes—
480C22×C4C2511yescharacteristic
481Q16—416no—
482SD16—416no—
483C2×C8—24no—
484C2×C8—24no—
485C2×C8—24no—
486C2×C8—24no—
487C2×C8—24no—
488C2×C8—24no—
489C4⋊C4—216no—
490C4⋊C4—24no—
491C4⋊C4—24no—
492C4⋊C4—24no—
493C4⋊C4—24no—
494C4⋊C4—28no—
495C4⋊C4—24no—
496C4⋊C4—24no—
497C4⋊C4—28no—
498C4⋊C4—22no—
499C4⋊C4—22no—
500C4⋊C4—22no—
501C4⋊C4—22no—
502C4⋊C4—28no—
503C4⋊C4—416no—
504C4⋊C4—44no—
505C4⋊C4—42no—
506C4⋊C4—44no—
507C4⋊C4—44no—
508C4⋊C4C22×D414yes—
509C4⋊C4C22×D414yes—
510C22⋊C4—216no—
511C22⋊C4—216no—
512C22⋊C4—216no—
513C22⋊C4—28no—
514C22⋊C4—28no—
515C22⋊C4—28no—
516C22⋊C4—28no—
517C22⋊C4—28no—
518C22⋊C4—416no—
519C22⋊C4—416no—
520C42—216no—
521C42—28no—
522C42—44no—
523C42—44no—
524C42#12014yes—
525C42#12014yes—
526C42C22×D412yes—
527C42C22×D412yes—
528C42C22×D412yes—
529C42C22×D412yes—
530C23—22no—
531C23—21no—
532C23—21no—
533C23—24no—
534C23—24no—
535C23—24no—
536C23—24no—
537C23—24no—
538C23—24no—
539C23—24no—
540C23—24no—
541C23—48no—
542C23—48no—
543C23—44no—
544C23—42no—
545C23—42no—
546C23—48no—
547C23—44no—
548C23—84no—
549C23—84no—
550C23—84no—
551C23—84no—
552C23C2×#12012yes—
553C23#57514yes—
554C23C23×D411yescharacteristic
555Q8—24no—
556Q8—24no—
557Q8—28no—
558Q8—48no—
559Q8—48no—
560Q8—42no—
561Q8—42no—
562Q8—44no—
563Q8—84no—
564D4—28no—
565D4—48no—
566D4—48no—
567D4—48no—
568D4—416no—
569D4—44no—
570D4—84no—
571D4—816no—
572D4—816no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—28no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—24no—
581C2×C4—24no—
582C2×C4—24no—
583C2×C4—24no—
584C2×C4—24no—
585C2×C4—24no—
586C2×C4—24no—
587C2×C4—24no—
588C2×C4—24no—
589C2×C4—44no—
590C2×C4—416no—
591C2×C4—48no—
592C2×C4—44no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—44no—
598C2×C4—48no—
599C2×C4—48no—
600C2×C4—44no—
601C2×C4—42no—
602C2×C4—44no—
603C2×C4—44no—
604C2×C4—44no—
605C2×C4—88no—
606C2×C4—84no—
607C2×C4—816no—
608C2×C4—88no—
609C2×C4—84no—
610C2×C4C2×#12012yes—
611C2×C4C2×#12012yes—
612C2×C4C2×#12012yes—
613C2×C4C2×#12012yes—
614C2×C4C2×#12012yes—
615C2×C4C23×D411yescharacteristic
616C2×C4C23×D411yescharacteristic
617C2×C4C2×H(5,𝔽2)12yes—
618C2×C4#58512yes—
619C8—44no—
620C8—44no—
621C22—24no—
622C22—24no—
623C22—22no—
624C22—22no—
625C22—44no—
626C22—44no—
627C22—44no—
628C22—44no—
629C22—44no—
630C22—44no—
631C22—44no—
632C22—44no—
633C22—44no—
634C22—44no—
635C22—816no—
636C22—84no—
637C22—84no—
638C22#300514yes—
639C22C22×#12011yescharacteristic
640C22#348411yescharacteristic
641C22#348511yescharacteristic
642C22C2×#57512yes—
643C22#358612yes—
644C4—22no—
645C4—22no—
646C4—22no—
647C4—22no—
648C4—22no—
649C4—44no—
650C4—44no—
651C4—44no—
652C4—44no—
653C4—44no—
654C4—84no—
655C4—82no—
656C4#348812yes—
657C4#348812yes—
658C2—24no—
659C2—84no—
660C2—84no—
661C2#3551211yescharacteristic
662C2C2×#300511yescharacteristic
663C2#6037011yescharacteristic
664C2#6157812yes—
665C2#6171112yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#3506C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C4×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#3113C2semidirect
3. C2×#3112C2semidirect
4. C2×#2536C2semidirect
5. #24254C2semidirect
6. #24253C2semidirect
7. #24135C2semidirect
8. #24133C2semidirect
9. #24004C2semidirect
10. #23922C2semidirect
11. #21866C2semidirect
12. #21865C2semidirect
13. #21790C2semidirect
35. C2×#477C22semidirect
36. C2×#475C22semidirect
37. C2×#462C22semidirect
38. C2×#460C22semidirect
39. C2×#452C22semidirect
42. C2×#441C22semidirect
43. C2×#440C22semidirect
55. #2036C22semidirect
56. #2034C22semidirect
57. #2001C22semidirect
58. #2000C22semidirect
59. #1993C22semidirect
60. #1993C22semidirect
61. #1961C22semidirect
62. #1960C22semidirect
66. #1873C22semidirect
68. #1854C22semidirect
69. #1805C22semidirect
70. #1804C22semidirect
71. #1788C22semidirect

Maximal subgroups

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

subgroupindex
C2×#35062
C2×#31132
C2×#31122
C2×#25362
#242542
#242532
#241352
#241332
#240042
#239222
#218662
#218652
#217902