Finite Group Databasebeta

← #6470509

Subgroups — #6470509

Order 512.

2276 conjugacy classes of subgroups.

The subgroup lattice has 678 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#37306C211yesmaximal, characteristic, complemented
2#34259C211yesmaximal, characteristic
3#32965C212yesmaximal, complemented
4#32958C212yesmaximal, complemented
5#29960C212yesmaximal, complemented
6#29951C212yesmaximal, complemented
7#25979C214yesmaximal, complemented
8#25978C214yesmaximal, complemented
9#25962C214yesmaximal, complemented
10#25955C214yesmaximal, complemented
11#21946C212yesmaximal, complemented
12#21936C211yesmaximal, characteristic, complemented
13#20603C212yesmaximal
14C22×2−1+4C2211yescharacteristic
15C2×#520C2211yescharacteristic
16C2×C4×#41C2211yescharacteristic
17C2×#477C2212yes—
18C2×#475C2212yes—
19C2×#476C2212yes—
20C2×#476C2212yes—
21C2×#417C2212yes—
22C2×#419C2212yes—
23C2×#416C2212yes—
24C22×#103C2212yes—
25C2×#408C2212yes—
26#2857C2212yes—
27#2811—21no—
28#2809—21no—
29#2804—21no—
30#2692—24no—
31#2683—21no—
32#2628C2212yes—
33#2440C2214yes—
34#2439C2214yes—
35Q8×C22⋊C4C2214yes—
36#2339—24no—
37#2339C2214yes—
38#2311—21no—
39#2311C2212yes—
40#2311C2211yescharacteristic
41#2310—21no—
42#2310C2211yescharacteristic
43#2297—21no—
44#2297C2211yescharacteristic
45C2×#391—21no—
46C2×#391C2211yescharacteristic
47#2097C2214yes—
48#2096C2214yes—
49#2068C2214yes—
50#2066C2214yes—
51#2066C2214yes—
52#2066C2214yes—
53#2024C2214yes—
54#2024C2214yes—
55#1865C2214yes—
56#1865C2214yes—
57#1864C2214yes—
58#1862C2214yes—
59#1854C2214yes—
60#1854C2214yes—
61#1853C2214yes—
62#1853C2214yes—
63#1851C2214yes—
64#1850C2214yes—
65#1809C2214yes—
66#1807C2214yes—
67#1805C2212yes—
68#1804C2212yes—
69#1803C2212yes—
70#1802C2212yes—
71#1798C2212yes—
72#1797C2212yes—
73#1788C2214yes—
74#1788C2214yes—
75#1787C2214yes—
76#1787C2214yes—
77C2×2−1+4—24no—
78C2×2−1+4—42no—
79C2×2−1+4—42no—
80C2×2−1+4—42no—
81C2×2−1+4—42no—
82C2×2−1+4Q818yescomplemented
83C22×#41—21no—
84C22×#41—22no—
85C22×#41C2311yescharacteristic
86C22×#41C2312yes—
87C22×#41C2311yescharacteristic
88C23×Q8—21no—
89C23×Q8C2312yes—
90C23×Q8C2311yescharacteristic
91C22×M16C2312yes—
92C23×C8C2312yes—
93C2×C4⋊Q8C2312yes—
94C2×C4⋊Q8C2312yes—
95C2×C22⋊Q8—22no—
96C2×C22⋊Q8—24no—
97#520—42no—
98#520—42no—
99C4×#41—42no—
100C4×#41—42no—
101C2×C4×Q8C2312yes—
102C2×C4×Q8C2312yes—
103C2×C4×D4—22no—
104C2×C4×D4—22no—
105C2×C4×D4C2312yes—
106C2×C4×D4C2312yes—
107C2×#117—21no—
108C2×#117—21no—
109C2×#117C2311yescharacteristic
110C2×#117C2312yes—
111C2×#117C2311yescharacteristic
112C22×C4⋊C4—21no—
113C22×C4⋊C4C2311yescharacteristic
114C22×C4⋊C4C2312yes—
115C22×C42—21no—
116C22×C42C2311yescharacteristic
117#477—44no—
118#476—44no—
119#476—44no—
120#475—44no—
121C2×#107C2314yes—
122C2×#106C2314yes—
123C2×#106C2314yes—
124C2×#106C2314yes—
125C2×C4⋊C8C2312yes—
126C2×C4⋊C8C2312yes—
127C2×C4⋊C8C2312yes—
128C2×C4⋊C8C2312yes—
129#419—44no—
130#417—44no—
131#416—44no—
132C2×#103—44no—
133C2×#103C2314yes—
134C2×#103C2314yes—
135C2×#103C2314yes—
136C2×#103C2314yes—
137C2×#103C2314yes—
138C2×#103C2314yes—
139C2×#102C2314yes—
140C2×#102C2314yes—
141#408—28no—
142C2×C22⋊C8C2314yes—
143C2×C22⋊C8C2314yes—
144#400—22no—
145#399—22no—
146#398—24no—
147#396—24no—
148#395—22no—
149#395—24no—
150#393—22no—
151#393—24no—
152#391—21no—
153#391—22no—
154#391—21no—
155#391—21no—
156#391—21no—
157#391—22no—
158#391—21no—
159#391—24no—
160#391—24no—
161#391—24no—
162#391D411yescharacteristic
163#391D411yescharacteristic
164#391C2314yes—
165#391C2314yes—
166#390—21no—
167#390—22no—
168#390D411yescharacteristic
169#390D411yescharacteristic
170#389—22no—
171#389C2314yes—
172#389C2314yes—
173#387—24no—
174#386—22no—
175#386—22no—
176#386—22no—
177#386—24no—
178#386—24no—
179#386D414yes—
180#385—24no—
181#384C2312yes—
182#382C2312yes—
183#381—22no—
184#381—24no—
185#380—22no—
186#380—24no—
187#380C2314yes—
188#380C2314yes—
189#380C2314yes—
190C4×C22⋊C4—24no—
191C4×C22⋊C4C2314yes—
192C2×#95—22no—
193C2×#95—24no—
194#340C2314yes—
195#340C2314yes—
196#340C2314yes—
197#340C2314yes—
198#336C2314yes—
199#336C2314yes—
2002−1+4—416no—
2012−1+4—416no—
2022−1+4—416no—
2032−1+4—416no—
204C2×#41—22no—
205C2×#41—22no—
206C2×#41—28no—
207C2×#41—216no—
208C2×#41—22no—
209C2×#41—22no—
210C2×#41—22no—
211C2×#41—24no—
212C2×#41—24no—
213C2×#41—24no—
214C2×#41—41no—
215C2×#41—41no—
216C2×#41—41no—
217C2×#41—41no—
218C2×#41—42no—
219C2×#41—42no—
220C2×#41—44no—
221C2×#41—44no—
222C2×#41—48no—
223C2×#41—48no—
224C2×#41C2×Q814yes—
225C2×#41C2×Q818yes—
226C2×#41C2×Q814yes—
227C2×#41#4114yes—
228C22×Q8—28no—
229C22×Q8—22no—
230C22×Q8—21no—
231C22×Q8—22no—
232C22×Q8—22no—
233C22×Q8—22no—
234C22×Q8—24no—
235C22×Q8—24no—
236C22×Q8—24no—
237C22×Q8—41no—
238C22×Q8—41no—
239C22×Q8—41no—
240C22×Q8—41no—
241C22×Q8—42no—
242C22×Q8—42no—
243C22×Q8—44no—
244C22×Q8—44no—
245C22×Q8C2×D411yescharacteristic
246C22×Q8C2×Q818yes—
247C22×Q8C2×Q814yes—
248C22×Q8C2412yes—
249C22×Q8C2411yescharacteristic
250C22×D4—21no—
251C22×D4—22no—
252C22×D4C2×D412yes—
253C22×D4C2×D411yescharacteristic
254C22×D4C2411yescharacteristic
255C23×C4—21no—
256C23×C4—21no—
257C23×C4—22no—
258C23×C4—22no—
259C23×C4—21no—
260C23×C4—22no—
261C23×C4—22no—
262C23×C4—22no—
263C23×C4C2×D412yes—
264C23×C4C2411yescharacteristic
265C23×C4C2412yes—
266C2×M16—24no—
267C2×M16—42no—
268C2×M16—42no—
269C22×C8—24no—
270C22×C8—42no—
271C22×C8—42no—
272C22×C8C2414yes—
273C22×C8C2414yes—
274C4⋊Q8—42no—
275C4⋊Q8—42no—
276C4⋊Q8—42no—
277C4⋊Q8—42no—
278C22⋊Q8—88no—
279C22⋊Q8—84no—
280C4×Q8—24no—
281C4×Q8—24no—
282C4×Q8—42no—
283C4×Q8—42no—
284C4×D4—24no—
285C4×D4—24no—
286C4×D4—42no—
287C4×D4—42no—
288C4×D4—48no—
289C4×D4—48no—
290#117—41no—
291#117—41no—
292#117—41no—
293#117—41no—
294#117—44no—
295#117—44no—
296#117—42no—
297#117—42no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—24no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—24no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—41no—
309C2×C4⋊C4—41no—
310C2×C4⋊C4—42no—
311C2×C4⋊C4—42no—
312C2×C4⋊C4—44no—
313C2×C4⋊C4C2×D411yescharacteristic
314C2×C4⋊C4C2×D411yescharacteristic
315C2×C4⋊C4C2412yes—
316C2×C4⋊C4C2414yes—
317C2×C4⋊C4C2412yes—
318C2×C4⋊C4C2414yes—
319C2×C22⋊C4—21no—
320C2×C22⋊C4—22no—
321C2×C22⋊C4—22no—
322C2×C22⋊C4—22no—
323C2×C22⋊C4—24no—
324C2×C22⋊C4—24no—
325C2×C22⋊C4—24no—
326C2×C22⋊C4C2×D414yes—
327C2×C22⋊C4C2×D412yes—
328C2×C22⋊C4C2×D414yes—
329C2×C42—21no—
330C2×C42—22no—
331C2×C42—22no—
332C2×C42—22no—
333C2×C42—41no—
334C2×C42—41no—
335C2×C42—44no—
336C2×C42C2×D411yescharacteristic
337C2×C42C2×D411yescharacteristic
338C2×C42C2412yes—
339C2×C42C2412yes—
340#107—44no—
341#106—44no—
342#106—44no—
343#106—44no—
344C4⋊C8—42no—
345C4⋊C8—42no—
346C4⋊C8—42no—
347C4⋊C8—42no—
348#103—28no—
349#103—28no—
350#103—44no—
351#103—44no—
352#103—44no—
353#103—44no—
354#102—28no—
355#102—28no—
356C22⋊C8—28no—
357C22⋊C8—28no—
358#95—24no—
359#95—24no—
360#95—24no—
361#95—24no—
362#95—24no—
363#95—24no—
364#95—24no—
365#95—24no—
366#95—22no—
367#95—22no—
368#95—24no—
369#95C2×D414yes—
370#95C2×D414yes—
371C24—21no—
372C24C22×D412yes—
373C24C22×D411yescharacteristic
374#41—28no—
375#41—28no—
376#41—28no—
377#41—28no—
378#41—216no—
379#41—44no—
380#41—44no—
381#41—44no—
382#41—44no—
383#41—48no—
384#41—48no—
385#41—48no—
386#41—432no—
387#41—464no—
388C2×Q8—28no—
389C2×Q8—24no—
390C2×Q8—21no—
391C2×Q8—22no—
392C2×Q8—21no—
393C2×Q8—21no—
394C2×Q8—216no—
395C2×Q8—22no—
396C2×Q8—22no—
397C2×Q8—22no—
398C2×Q8—22no—
399C2×Q8—41no—
400C2×Q8—41no—
401C2×Q8—44no—
402C2×Q8—44no—
403C2×Q8—48no—
404C2×Q8—44no—
405C2×Q8—44no—
406C2×Q8—48no—
407C2×Q8—48no—
408C2×Q8—42no—
409C2×Q8—42no—
410C2×Q8—432no—
411C2×Q8—44no—
412C2×Q8—44no—
413C2×Q8—44no—
414C2×Q8—84no—
415C2×Q8—82no—
416C2×Q8—82no—
417C2×Q8—82no—
418C2×Q8C22⋊Q814yes—
419C2×Q8C22×Q814yes—
420C2×Q8C22×Q812yes—
421C2×Q8H(5,𝔽2)12yes—
422C2×Q8H(5,𝔽2)12yes—
423C2×D4—24no—
424C2×D4—24no—
425C2×D4—24no—
426C2×D4—28no—
427C2×D4—21no—
428C2×D4—21no—
429C2×D4—21no—
430C2×D4—41no—
431C2×D4—42no—
432C2×D4—41no—
433C2×D4—42no—
434C2×D4—44no—
435C2×D4—44no—
436C2×D4C22⋊Q818yes—
437C2×D4C22⋊Q814yes—
438C2×D4C22×Q812yes—
439C2×D4H(5,𝔽2)12yes—
440C2×D4H(5,𝔽2)12yes—
441C22×C4—24no—
442C22×C4—24no—
443C22×C4—21no—
444C22×C4—24no—
445C22×C4—21no—
446C22×C4—21no—
447C22×C4—21no—
448C22×C4—21no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—28no—
454C22×C4—24no—
455C22×C4—22no—
456C22×C4—28no—
457C22×C4—24no—
458C22×C4—22no—
459C22×C4—24no—
460C22×C4—22no—
461C22×C4—24no—
462C22×C4—42no—
463C22×C4—42no—
464C22×C4—44no—
465C22×C4—44no—
466C22×C4—44no—
467C22×C4—44no—
468C22×C4—44no—
469C22×C4—44no—
470C22×C4—42no—
471C22×C4—42no—
472C22×C4—42no—
473C22×C4—44no—
474C22×C4—82no—
475C22×C4—84no—
476C22×C4—82no—
477C22×C4—84no—
478C22×C4—82no—
479C22×C4C22×D412yes—
480C22×C4C22×D412yes—
481C22×C4C22×D412yes—
482C22×C4C22×D412yes—
483C22×C4C22×D411yescharacteristic
484C22×C4C22×D412yes—
485C22×C4C22×Q814yes—
486C22×C4C22×Q812yes—
487C22×C4C2×#4112yes—
488C22×C4C2×#4114yes—
489C22×C4C2511yescharacteristic
490M16—84no—
491C2×C8—24no—
492C2×C8—24no—
493C2×C8—24no—
494C2×C8—24no—
495C2×C8—24no—
496C2×C8—24no—
497C2×C8—84no—
498C4⋊C4—24no—
499C4⋊C4—24no—
500C4⋊C4—24no—
501C4⋊C4—24no—
502C4⋊C4—22no—
503C4⋊C4—22no—
504C4⋊C4—22no—
505C4⋊C4—22no—
506C4⋊C4—41no—
507C4⋊C4—41no—
508C4⋊C4—44no—
509C4⋊C4—44no—
510C4⋊C4—42no—
511C4⋊C4—48no—
512C4⋊C4—44no—
513C4⋊C4—82no—
514C4⋊C4—84no—
515C4⋊C4—82no—
516C4⋊C4—82no—
517C4⋊C4—82no—
518C22⋊C4—28no—
519C22⋊C4—44no—
520C22⋊C4—42no—
521C22⋊C4—44no—
522C22⋊C4—44no—
523C22⋊C4—42no—
524C22⋊C4—84no—
525C22⋊C4—82no—
526C22⋊C4—84no—
527C22⋊C4—84no—
528C42—22no—
529C42—22no—
530C42—22no—
531C42—22no—
532C42—41no—
533C42—41no—
534C42—44no—
535C42—44no—
536C42—42no—
537C42—82no—
538C23—24no—
539C23—22no—
540C23—22no—
541C23—21no—
542C23—21no—
543C23—22no—
544C23—22no—
545C23—22no—
546C23—22no—
547C23—24no—
548C23—22no—
549C23—42no—
550C23—42no—
551C23C2×C22⋊Q812yes—
552C23C2×C22⋊Q814yes—
553C23C23×D411yescharacteristic
554Q8—22no—
555Q8—22no—
556Q8—22no—
557Q8—24no—
558Q8—24no—
559Q8—44no—
560Q8—416no—
561Q8—48no—
562Q8—42no—
563Q8—42no—
564Q8—164no—
565Q8—164no—
566Q8#54314yes—
567D4—216no—
568D4—22no—
569D4—22no—
570D4—22no—
571D4—44no—
572D4—48no—
573D4—416no—
574D4#54314yes—
575C2×C4—216no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—21no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—21no—
584C2×C4—22no—
585C2×C4—24no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—24no—
589C2×C4—24no—
590C2×C4—24no—
591C2×C4—22no—
592C2×C4—22no—
593C2×C4—22no—
594C2×C4—24no—
595C2×C4—24no—
596C2×C4—22no—
597C2×C4—22no—
598C2×C4—22no—
599C2×C4—24no—
600C2×C4—22no—
601C2×C4—416no—
602C2×C4—416no—
603C2×C4—48no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—44no—
608C2×C4—44no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—44no—
612C2×C4—44no—
613C2×C4—44no—
614C2×C4—44no—
615C2×C4—44no—
616C2×C4—42no—
617C2×C4—84no—
618C2×C4—88no—
619C2×C4—164no—
620C2×C4—162no—
621C2×C4—164no—
622C2×C4C2×C22⋊Q812yes—
623C2×C4C2×C22⋊Q814yes—
624C2×C4#57714yes—
625C2×C4C23×Q811yescharacteristic
626C2×C4C22×#4111yescharacteristic
627C2×C4C2×H(5,𝔽2)11yescharacteristic
628C2×C4C2×H(5,𝔽2)11yescharacteristic
629C2×C4C2×H(5,𝔽2)11yescharacteristic
630C2×C4C2×H(5,𝔽2)11yescharacteristic
631C8—44no—
632C8—44no—
633C22—28no—
634C22—24no—
635C22—22no—
636C22—24no—
637C22—24no—
638C22—44no—
639C22—42no—
640C22—44no—
641C22—44no—
642C22—42no—
643C22—41no—
644C22—42no—
645C22—41no—
646C22—42no—
647C22—42no—
648C22C22×C22⋊Q811yescharacteristic
649C22#348411yescharacteristic
650C22#348411yescharacteristic
651C22C2×#57712yes—
652C22#358612yes—
653C4—22no—
654C4—24no—
655C4—24no—
656C4—24no—
657C4—44no—
658C4—42no—
659C4—42no—
660C4—42no—
661C4—82no—
662C4—84no—
663C4—84no—
664C4—82no—
665C4C2×#54311yescharacteristic
666C4C2×#54311yescharacteristic
667C4#349011yescharacteristic
668C4#349011yescharacteristic
669C2—42no—
670C2—44no—
671C2—44no—
672C2#3498211yescharacteristic
673C2#6049712yes—
674C2#6162012yes—
675C2#6166411yescharacteristic
676C2#6167411yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×2−1+4C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ #2297 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #37306C2semidirect
3. #32965C2semidirect
4. #32958C2semidirect
5. #29960C2semidirect
6. #29951C2semidirect
7. #25979C2semidirect
8. #25978C2semidirect
9. #25962C2semidirect
10. #25955C2semidirect
11. #21946C2semidirect
12. #21936C2semidirect
82. C2×2−1+4Q8semidirect

Maximal subgroups

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

subgroupindex
#373062
#342592
#329652
#329582
#299602
#299512
#259792
#259782
#259622
#259552
#219462
#219362
#206032