Finite Group Databasebeta

← #6396500

Subgroups — #6396500

Order 512.

2124 conjugacy classes of subgroups.

The subgroup lattice has 658 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3213C211yesmaximal, characteristic
2#34240C211yesmaximal, characteristic, complemented
3#34226C211yesmaximal, characteristic, complemented
4#23098C214yesmaximal, complemented
5#23005C214yesmaximal, complemented
6#22868C214yesmaximal, complemented
7#22865C214yesmaximal, complemented
8#22272C212yesmaximal, complemented
9#22223C212yesmaximal, complemented
10#21989C211yesmaximal, characteristic, complemented
11#21701C212yesmaximal, complemented
12C4×#489C212yesmaximal, complemented
13#21405C211yesmaximal, characteristic, complemented
14#21204C211yesmaximal, characteristic, complemented
15#20480C211yesmaximal, characteristic, complemented
16C2×#532C2211yescharacteristic
17C22×#124C2211yescharacteristic
18#3213C4116yescomplemented
19C2×#496C2212yes—
20C2×#489C2212yes—
21C2×#463C2214yes—
22C2×#448C2212yes—
23C2×#447C2212yes—
24C2×#442C2214yes—
25C2×#440C2214yescomplemented
26C2×#432C2211yescharacteristic
27C2×#419C2212yes—
28C2×#418C2212yes—
29C2×C4×M16C2211yescharacteristic
30#2372—21no—
31#2372—21no—
32#2367—22no—
33#2367C2212yes—
34#2367C2212yes—
35#2363C2214yes—
36#2357C2212yescomplemented
37#2349C2212yescomplemented
38#2342—22no—
39#2341—21no—
40Q8×C22⋊C4C2212yes—
41#2339—21no—
42D4×C22⋊C4C2212yes—
43#2337—22no—
44#2324—22no—
45#2319—21no—
46#2312—21no—
47C4×C4⋊Q8C2212yescomplemented
48C4×#127C2212yescomplemented
49C4×#124C2212yes—
50C4×#124C2212yes—
51#2285C2211yescharacteristic
52#2283C2211yescharacteristic
53#2275—21no—
54#2273—21no—
55C4×#117C2211yescharacteristic
56#1993C2211yescharacteristic, complemented
57#1993C2211yescharacteristic, complemented
58#1970C2214yes—
59#1968C2214yescomplemented
60#1961C2211yescharacteristic, complemented
61#1961C2211yescharacteristic, complemented
62#1959C2212yes—
63#1959C2212yes—
64#1958C2214yes—
65#1949C2214yescomplemented
66#1936C2214yes—
67#1935C2214yescomplemented
68#1928C2214yescomplemented
69#1926C2214yes—
70#1895C2214yes—
71#1880C2214yescomplemented
72#1879C2214yes—
73#1879C2214yes—
74#1877C2214yescomplemented
75#1865C2214yescomplemented
76#1862C2214yes—
77#1854C2214yes—
78#1847C2212yescomplemented
79#1836C2212yescomplemented
80#1770C2212yescomplemented
81C8×C4⋊C4C2212yescomplemented
82C4×#103C2214yescomplemented
83C4×#102C2214yes—
84C4×#102C2214yescomplemented
85#1753C2214yes—
86#1745C2212yescomplemented
87C4×#97C2212yescomplemented
88C22×#41—21no—
89C23×Q8C2311yescharacteristic
90C23×D4C2311yescharacteristic
91C22×SD16C2314yes—
92C22×D8C2314yes—
93C22×M16C2312yes—
94#532—24no—
95#532C2×C418yescomplemented
96C2×C4⋊Q8C2311yescharacteristic
97C2×#127C2311yescharacteristic
98C2×#124—21no—
99C2×#124—21no—
100C2×#124—21no—
101C2×#124—24no—
102C2×#124C2×C418yescomplemented
103C2×#124C2312yes—
104C2×#124C2312yes—
105C2×#121—21no—
106C2×#121C2312yes—
107C2×C4×Q8—22no—
108C2×C4×Q8C2312yes—
109C2×C4×Q8C2312yes—
110C2×C4×D4—21no—
111C2×C4×D4—22no—
112C2×C4×D4—22no—
113C2×C4×D4C2312yes—
114C2×C4×D4C2314yes—
115C2×C4×D4C2312yes—
116C2×C4×D4C2312yes—
117C2×#117D412yes—
118C2×#117C2312yes—
119C22×C22⋊C4—21no—
120C22×C22⋊C4—21no—
121C22×C22⋊C4—22no—
122C22×C42C2311yescharacteristic
123#496C2×C4116yescomplemented
124#489C2×C4116yescomplemented
125#463C2×C4132yescomplemented
126#448C2×C4116yescomplemented
127#447C2×C4116yescomplemented
128#442—48no—
129#440—48no—
130#432C2×C418yescomplemented
131C2×#107C2314yes—
132C2×#106C2314yes—
133C2×C4⋊C8C2312yes—
134#419—44no—
135#418—44no—
136C2×#103C2314yes—
137C2×#103C2314yes—
138C2×#103C2314yes—
139C2×#102C2314yes—
140C2×#102C2314yes—
141C2×#102C2314yes—
142C2×#102C2314yes—
143C2×#102C2314yes—
144C2×C22⋊C8C2312yes—
145C4×M16—24no—
146C2×#97C2311yescharacteristic
147C2×#97C2311yescharacteristic
148C2×#97C2314yes—
149C2×#97C2312yes—
150C2×C4×C8C2312yes—
151C2×C4×C8C2314yes—
152#391—22no—
153#391—22no—
154#391—22no—
155#391C2312yes—
156#391C2312yes—
157#390—24no—
158#390—22no—
159#390—22no—
160#390C2314yes—
161#390C2312yes—
162#390C2312yes—
163#389—22no—
164#389C2314yes—
165#388—24no—
166#388—22no—
167#388—22no—
168#388—22no—
169#388—22no—
170#388—22no—
171#388—22no—
172#388—22no—
173#388—22no—
174#388—24no—
175#386—24no—
176#386—22no—
177#386—22no—
178#385—24no—
179#385—22no—
180#385—22no—
181#384C2312yes—
182#383—21no—
183#382C2311yescharacteristic
184#382C2312yes—
185#382C2311yescharacteristic
186#381—21no—
187#381—22no—
188#380—21no—
189#380—21no—
190#380—22no—
191#380—22no—
192#380C2312yes—
193#379—22no—
194C4×C4⋊C4—22no—
195C4×C4⋊C4—21no—
196C4×C4⋊C4—21no—
197C4×C4⋊C4D412yes—
198C4×C4⋊C4C2314yes—
199C4×C4⋊C4C2312yes—
200C4×C4⋊C4C2311yescharacteristic
201C4×C4⋊C4C2311yescharacteristic
202C4×C22⋊C4—21no—
203C4×C22⋊C4—22no—
204C4×C22⋊C4—22no—
205C4×C22⋊C4—22no—
206C4×C22⋊C4—22no—
207C4×C22⋊C4—22no—
208C4×C22⋊C4—22no—
209C4×C22⋊C4—24no—
210C4×C22⋊C4—22no—
211C4×C22⋊C4—22no—
212C4×C22⋊C4D411yescharacteristic
213C4×C22⋊C4D411yescharacteristic
214C4×C22⋊C4C2312yes—
215#376—21no—
216#376D411yescharacteristic
217#376D411yescharacteristic
218C2×#95—22no—
219C43C2312yes—
220#340C2314yes—
221#340C2314yes—
222#340C2314yes—
223#340C2314yes—
224#336C2314yes—
225#336C2314yes—
226C25—21no—
227C2×#41—28no—
228C2×#41—84no—
229C22×Q8—21no—
230C22×Q8—21no—
231C22×Q8—22no—
232C22×Q8—42no—
233C22×Q8—42no—
234C22×Q8C22×C414yes—
235C22×Q8C2412yes—
236C22×D4—22no—
237C22×D4—21no—
238C22×D4—21no—
239C22×D4—21no—
240C22×D4—22no—
241C22×D4—42no—
242C22×D4—42no—
243C22×D4C22×C414yes—
244C22×D4C2412yes—
245C22×D4C2412yes—
246C23×C4—21no—
247C23×C4—22no—
248C23×C4—22no—
249C23×C4—22no—
250C23×C4—24no—
251C23×C4C2×D411yescharacteristic
252C23×C4C2×D411yescharacteristic
253C23×C4C2411yescharacteristic
254C2×SD16—44no—
255C2×SD16—44no—
256C2×SD16C22×C4116yes—
257C2×D8—44no—
258C2×D8—44no—
259C2×D8C22×C4116yes—
260C2×M16—24no—
261C2×M16—24no—
262C2×M16—42no—
263C22×C8C2412yes—
264C22×C8C2414yes—
265C22×C8C2412yes—
266C4⋊Q8—22no—
267C4⋊Q8C22×C414yes—
268#127—22no—
269#127C22×C414yes—
270#124—28no—
271#124—28no—
272#124—28no—
273#124—24no—
274#124—24no—
275#124C22×C418yes—
276#124C22×C418yes—
277#121—28no—
278#121—24no—
279#121C22×C418yes—
280C4×Q8—44no—
281C4×Q8—44no—
282C4×Q8—84no—
283C4×D4—28no—
284C4×D4—24no—
285C4×D4—48no—
286C4×D4—44no—
287C4×D4—44no—
288C4×D4—44no—
289C4×D4—44no—
290C4×D4—84no—
291C4×D4C22×C418yes—
292#117—44no—
293#117—42no—
294#117—42no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—24no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4C2×D412yes—
302C2×C4⋊C4C2×D412yes—
303C2×C4⋊C4C2414yes—
304C2×C4⋊C4C2412yes—
305C2×C4⋊C4C2412yes—
306C2×C4⋊C4C2412yes—
307C2×C22⋊C4—28no—
308C2×C22⋊C4—28no—
309C2×C22⋊C4—22no—
310C2×C22⋊C4—22no—
311C2×C22⋊C4—22no—
312C2×C22⋊C4—22no—
313C2×C22⋊C4—22no—
314C2×C22⋊C4—22no—
315C2×C22⋊C4—22no—
316C2×C22⋊C4—22no—
317C2×C22⋊C4—22no—
318C2×C22⋊C4—24no—
319C2×C22⋊C4—22no—
320C2×C22⋊C4—24no—
321C2×C22⋊C4—22no—
322C2×C22⋊C4—22no—
323C2×C22⋊C4—24no—
324C2×C22⋊C4—22no—
325C2×C22⋊C4—22no—
326C2×C22⋊C4—48no—
327C2×C22⋊C4C2×D412yes—
328C2×C22⋊C4C2×D412yes—
329C2×C42—22no—
330C2×C42—24no—
331C2×C42—22no—
332C2×C42—22no—
333C2×C42—22no—
334C2×C42C22×C414yes—
335C2×C42C2×D412yes—
336C2×C42C2×D412yes—
337C2×C42C2×D412yes—
338C2×C42C2×D412yes—
339C2×C42C2×D411yescharacteristic
340C2×C42C2×D411yescharacteristic
341C2×C42#4114yes—
342C2×C42C2412yes—
343C2×C42C2412yes—
344C2×C42C2411yescharacteristic
345C2×C42C2411yescharacteristic
346#107—44no—
347#106—44no—
348C4⋊C8C22×C418yes—
349#103—28no—
350#103—44no—
351#103C22×C4116yes—
352#102—28no—
353#102—28no—
354#102—44no—
355#102—44no—
356#102C22×C4116yes—
357C22⋊C8C22×C418yes—
358#97—28no—
359#97—22no—
360#97—22no—
361#97C22×C418yes—
362C4×C8—24no—
363C4×C8—44no—
364#95—22no—
365#95—24no—
366#95—21no—
367#95—21no—
368#95—22no—
369#95—21no—
370#95—21no—
371#95—22no—
372#95—22no—
373#95—22no—
374#95—22no—
375#95—22no—
376#95—22no—
377#95—24no—
378#95C2×D411yescharacteristic
379#95C2×D411yescharacteristic
380#95C2×D411yescharacteristic
381#95C2×D412yes—
382#95C2×D411yescharacteristic
383#95C2×D412yes—
384C24—22no—
385C24—22no—
386C24—24no—
387C24—44no—
388C24—81no—
389C24—81no—
390C24—82no—
391C24C22×D411yescharacteristic
392#41—816no—
393C2×Q8—24no—
394C2×Q8—24no—
395C2×Q8—22no—
396C2×Q8—24no—
397C2×Q8—24no—
398C2×Q8—44no—
399C2×Q8—48no—
400C2×Q8—81no—
401C2×Q8—81no—
402C2×Q8—84no—
403C2×Q8C23×C414yes—
404C2×D4—28no—
405C2×D4—24no—
406C2×D4—24no—
407C2×D4—24no—
408C2×D4—22no—
409C2×D4—22no—
410C2×D4—24no—
411C2×D4—24no—
412C2×D4—24no—
413C2×D4—24no—
414C2×D4—48no—
415C2×D4—44no—
416C2×D4—44no—
417C2×D4—84no—
418C2×D4—81no—
419C2×D4—81no—
420C2×D4—84no—
421C2×D4C23×C414yes—
422C2×D4C23×C414yes—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—24no—
428C22×C4—28no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—24no—
432C22×C4—42no—
433C22×C4—42no—
434C22×C4—44no—
435C22×C4—48no—
436C22×C4—44no—
437C22×C4—84no—
438C22×C4—82no—
439C22×C4—82no—
440C22×C4—84no—
441C22×C4—84no—
442C22×C4—82no—
443C22×C4—82no—
444C22×C4—81no—
445C22×C4—81no—
446C22×C4C4×D414yes—
447C22×C4C4×D414yes—
448C22×C4C23×C412yes—
449C22×C4C22×D412yes—
450C22×C4C22×D412yes—
451C22×C4C22×D412yes—
452C22×C4C22×D411yescharacteristic
453C22×C4C22×D411yescharacteristic
454C22×C4C22×D411yescharacteristic
455C22×C4C22×D411yescharacteristic
456C22×C4C22×D411yescharacteristic
457C22×C4C22×D412yes—
458C22×C4C2×#4112yes—
459C22×C4C2511yescharacteristic
460SD16—416no—
461D8—416no—
462M16—48no—
463C2×C8—22no—
464C2×C8—24no—
465C2×C8—22no—
466C2×C8—24no—
467C2×C8—22no—
468C2×C8—22no—
469C2×C8—22no—
470C2×C8C23×C418yes—
471C2×C8H(5,𝔽2)14yes—
472C4⋊C4—24no—
473C4⋊C4—24no—
474C4⋊C4—22no—
475C4⋊C4—24no—
476C4⋊C4—24no—
477C4⋊C4—24no—
478C4⋊C4—24no—
479C4⋊C4—42no—
480C4⋊C4—44no—
481C4⋊C4—44no—
482C4⋊C4—82no—
483C4⋊C4—82no—
484C4⋊C4—84no—
485C4⋊C4C23×C414yes—
486C22⋊C4—28no—
487C22⋊C4—28no—
488C22⋊C4—28no—
489C22⋊C4—28no—
490C22⋊C4—28no—
491C22⋊C4—28no—
492C22⋊C4—24no—
493C22⋊C4—42no—
494C22⋊C4—44no—
495C22⋊C4—44no—
496C22⋊C4—44no—
497C22⋊C4—48no—
498C22⋊C4—82no—
499C22⋊C4—82no—
500C22⋊C4—84no—
501C22⋊C4—82no—
502C22⋊C4—82no—
503C22⋊C4—84no—
504C22⋊C4—82no—
505C42—24no—
506C42—24no—
507C42—24no—
508C42—42no—
509C42—42no—
510C42—42no—
511C42—82no—
512C42—82no—
513C42—82no—
514C42—82no—
515C42—84no—
516C42C4×D414yes—
517C42C4×D414yes—
518C42C23×C412yes—
519C42C23×C412yes—
520C42C2×#4112yes—
521C42C2×#4112yes—
522C23—22no—
523C23—21no—
524C23—21no—
525C23—22no—
526C23—24no—
527C23—24no—
528C23—48no—
529C23—44no—
530C23—42no—
531C23—42no—
532C23—42no—
533C23—42no—
534C23—42no—
535C23—42no—
536C23—42no—
537C23—44no—
538C23—42no—
539C23—84no—
540C23—84no—
541C23—164no—
542C23—164no—
543C23C2×C4×D412yes—
544C23C23×D411yescharacteristic
545Q8—28no—
546Q8—48no—
547Q8—44no—
548Q8—84no—
549D4—28no—
550D4—28no—
551D4—48no—
552D4—48no—
553D4—44no—
554D4—44no—
555D4—84no—
556D4—88no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—21no—
561C2×C4—22no—
562C2×C4—21no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—24no—
566C2×C4—24no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—24no—
571C2×C4—24no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—44no—
579C2×C4—44no—
580C2×C4—44no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—44no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—44no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—44no—
592C2×C4—88no—
593C2×C4—84no—
594C2×C4—164no—
595C2×C4—164no—
596C2×C4—168no—
597C2×C4C2×C4×D412yes—
598C2×C4C2×C4×D412yes—
599C2×C4C2×C4×D412yes—
600C2×C4C2×C4×D412yes—
601C2×C4C2×C4×D412yes—
602C2×C4#57514yes—
603C2×C4C24×C411yescharacteristic
604C2×C4C22×#4111yescharacteristic
605C2×C4C2×H(5,𝔽2)12yes—
606C2×C4#58512yes—
607C8—28no—
608C8—24no—
609C8—42no—
610C22—22no—
611C22—22no—
612C22—22no—
613C22—42no—
614C22—42no—
615C22—44no—
616C22—44no—
617C22—44no—
618C22—44no—
619C22—44no—
620C22—44no—
621C22—82no—
622C22—84no—
623C22—82no—
624C22—84no—
625C22—82no—
626C22—82no—
627C22—168no—
628C22C22×C4×D411yescharacteristic
629C22#349811yescharacteristic
630C22#350011yescharacteristic
631C22C2×#57512yes—
632C22#358512yes—
633C4—22no—
634C4—22no—
635C4—24no—
636C4—22no—
637C4—22no—
638C4—22no—
639C4—44no—
640C4—42no—
641C4—44no—
642C4—42no—
643C4—42no—
644C4—82no—
645C4—84no—
646C4—82no—
647C4C4×H(5,𝔽2)12yes—
648C4#349412yes—
649C2—42no—
650C2—84no—
651C2—84no—
652C2#3427511yescharacteristic
653C2C4×#57511yescharacteristic
654C2#6016811yescharacteristic
655C2#6245512yes—
656C2#6258412yes—
subgroupG/H
center, socleC23C23×D4
commutatorC2×C4C24×C4
Frattini, cosocleC22×C4C25
prime radicalC2×#3213C2
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #34240C2semidirect
3. #34226C2semidirect
4. #23098C2semidirect
5. #23005C2semidirect
6. #22868C2semidirect
7. #22865C2semidirect
8. #22272C2semidirect
9. #22223C2semidirect
10. #21989C2semidirect
11. #21701C2semidirect
12. C4×#489C2semidirect
13. #21405C2semidirect
14. #21204C2semidirect
15. #20480C2semidirect
18. #3213C4semidirect
25. C2×#440C22semidirect
36. #2357C22semidirect
37. #2349C22semidirect
47. C4×C4⋊Q8C22semidirect
48. C4×#127C22semidirect
56. #1993C22semidirect
57. #1993C22semidirect
59. #1968C22semidirect
60. #1961C22semidirect
61. #1961C22semidirect
65. #1949C22semidirect
67. #1935C22semidirect
68. #1928C22semidirect
71. #1880C22semidirect
74. #1877C22semidirect
75. #1865C22semidirect
78. #1847C22semidirect
79. #1836C22semidirect
80. #1770C22semidirect
81. C8×C4⋊C4C22semidirect
82. C4×#103C22semidirect
84. C4×#102C22semidirect
86. #1745C22semidirect
87. C4×#97C22semidirect
95. #532C2×C4semidirect
102. C2×#124C2×C4semidirect
123. #496C2×C4semidirect
124. #489C2×C4semidirect
125. #463C2×C4semidirect
126. #448C2×C4semidirect
127. #447C2×C4semidirect
130. #432C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#32132
#342402
#342262
#230982
#230052
#228682
#228652
#222722
#222232
#219892
#217012
C4×#4892
#214052
#212042
#204802