Finite Group Databasebeta

← #220143

Subgroups — #220143

Order 512.

1264 conjugacy classes of subgroups.

The subgroup lattice has 622 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C4×SD16C211yesmaximal, characteristic
2#22510C211yesmaximal, characteristic, complemented
3#22401C211yesmaximal, characteristic
4C2×#1590C212yesmaximal, complemented
5#9655C212yesmaximal, complemented
6#8952C212yesmaximal, complemented
7#8949C212yesmaximal, complemented
8#8595C211yesmaximal, characteristic, complemented
9#8311C211yesmaximal, characteristic, complemented
10#8310C211yesmaximal, characteristic
11C8×C22⋊C8C211yesmaximal, characteristic, complemented
12C23×SD16C411yescharacteristic
13C22×C4×Q8C2211yescharacteristic
14C22×C4×D4C2211yescharacteristic
15C2×#446C2212yescomplemented
16C2×C4×SD16—28no—
17C2×C4×SD16C411yescharacteristic
18C2×C4×SD16C411yescharacteristic
19C2×C4×SD16C411yescharacteristic
20C2×C4×SD16C411yescharacteristic
21C2×C4×SD16C2211yescharacteristic
22C2×C4×SD16C2211yescharacteristic
23C2×C4×SD16C2211yescharacteristic
24C2×C4×SD16C2211yescharacteristic
25C2×#436C2212yes—
26C22×#106C411yescharacteristic
27C22×#103C411yescharacteristic
28C22×#102C411yescharacteristic
29C2×#407—21no—
30C22×C4×C8C2211yescharacteristic
31#1985—22no—
32#1984C2212yescomplemented
33#1954C2212yes—
34#1870—22no—
35#1797—21no—
36#1797C2211yescharacteristic
37#1797C2211yescharacteristic
38#1779—21no—
39#1749—21no—
40C4×C22⋊C8C2211yescharacteristic
41C4×C22⋊C8C2211yescharacteristic
42#1590—28no—
43C2×#334C2212yescomplemented
44C2×Q8⋊C8C2212yescomplemented
45C2×Q8⋊C8C2212yescomplemented
46C2×D4⋊C8C2212yes—
47C2×D4⋊C8C2212yes—
48C2×C82C2212yescomplemented
49#1277C2212yescomplemented
50#1273C2212yescomplemented
51C23×Q8C2×C411yescharacteristic
52C23×D4C2×C411yescharacteristic
53C24×C4—21no—
54C22×SD16—24no—
55C22×SD16—24no—
56C22×SD16—24no—
57C22×SD16C2×C411yescharacteristic
58C22×SD16C2×C411yescharacteristic
59C22×SD16C2×C411yescharacteristic
60C22×SD16C2×C411yescharacteristic
61C22×M16—22no—
62C23×C8C2×C411yescharacteristic
63C23×C8C2×C411yescharacteristic
64C2×C4×Q8—21no—
65C2×C4×Q8—21no—
66C2×C4×Q8—24no—
67C2×C4×Q8—42no—
68C2×C4×Q8C2×C411yescharacteristic
69C2×C4×Q8C2×C411yescharacteristic
70C2×C4×Q8C2311yescharacteristic
71C2×C4×Q8C2311yescharacteristic
72C2×C4×D4—24no—
73C2×C4×D4—21no—
74C2×C4×D4—21no—
75C2×C4×D4—42no—
76C2×C4×D4C2×C411yescharacteristic
77C2×C4×D4C2×C411yescharacteristic
78C2×C4×D4C2311yescharacteristic
79C2×C4×D4C2311yescharacteristic
80C22×C4⋊C4—21no—
81C22×C4⋊C4C2×C411yescharacteristic
82C22×C4⋊C4C2×C411yescharacteristic
83C22×C22⋊C4—21no—
84C22×C42—21no—
85C22×C42C2311yescharacteristic
86#446—24no—
87#446—42no—
88C4×SD16—24no—
89C4×SD16—24no—
90C4×SD16—24no—
91C4×SD16—24no—
92C4×SD16—24no—
93C4×SD16—24no—
94C4×SD16—24no—
95C4×SD16D418yes—
96#436—24no—
97#436—42no—
98C2×#106—24no—
99C2×#106C2×C411yescharacteristic
100C2×#106C2×C411yescharacteristic
101C2×#106C2×C411yescharacteristic
102C2×#106C2×C411yescharacteristic
103C2×C4⋊C8—22no—
104C2×C4⋊C8C2312yes—
105C2×C4⋊C8C2312yes—
106C2×#103—24no—
107C2×#103C2×C411yescharacteristic
108C2×#103C2×C411yescharacteristic
109C2×#103C2×C411yescharacteristic
110C2×#103C2×C411yescharacteristic
111C2×#102—24no—
112C2×#102C2×C411yescharacteristic
113C2×#102C2×C411yescharacteristic
114C2×#102C2×C411yescharacteristic
115C2×#102C2×C411yescharacteristic
116#407—44no—
117C2×C22⋊C8—21no—
118C2×C22⋊C8—21no—
119C2×C22⋊C8—21no—
120C2×C22⋊C8—22no—
121C2×C22⋊C8—22no—
122C2×C22⋊C8—22no—
123C2×C22⋊C8D411yescharacteristic
124C2×C22⋊C8D411yescharacteristic
125C2×#97—22no—
126C2×C4×C8—24no—
127C2×C4×C8C2×C411yescharacteristic
128C2×C4×C8C2×C411yescharacteristic
129C2×C4×C8C2312yes—
130C2×C4×C8C2311yescharacteristic
131C2×C4×C8C2312yes—
132C2×C4×C8C2311yescharacteristic
133#336—22no—
134#336—22no—
135#336—21no—
136#336—21no—
137#336—21no—
138#336D412yes—
139#334—24no—
140Q8⋊C8—24no—
141Q8⋊C8—42no—
142D4⋊C8—24no—
143D4⋊C8—42no—
144C82—24no—
145C25—21no—
146C22×Q8—21no—
147C22×Q8—22no—
148C22×Q8—22no—
149C22×Q8—22no—
150C22×Q8—22no—
151C22×Q8—41no—
152C22×Q8—41no—
153C22×Q8C22×C411yescharacteristic
154C22×Q8C22×C411yescharacteristic
155C22×D4—22no—
156C22×D4—22no—
157C22×D4—21no—
158C22×D4—22no—
159C22×D4—22no—
160C22×D4—41no—
161C22×D4—41no—
162C22×D4C22×C411yescharacteristic
163C22×D4C22×C411yescharacteristic
164C23×C4—21no—
165C23×C4—21no—
166C23×C4—21no—
167C23×C4—21no—
168C23×C4—21no—
169C23×C4—21no—
170C23×C4—21no—
171C23×C4—44no—
172C23×C4—42no—
173C23×C4—42no—
174C23×C4—82no—
175C23×C4C22×C411yescharacteristic
176C23×C4C2×D411yescharacteristic
177C23×C4#4111yescharacteristic
178C2×SD16—232no—
179C2×SD16—22no—
180C2×SD16—22no—
181C2×SD16—22no—
182C2×SD16—22no—
183C2×SD16—22no—
184C2×SD16—22no—
185C2×SD16—22no—
186C2×SD16—22no—
187C2×SD16—22no—
188C2×SD16—22no—
189C2×SD16C22⋊C414yes—
190C2×SD16M1614yescomplemented
191C2×M16—44no—
192C2×M16—44no—
193C2×M16—82no—
194C22×C8—22no—
195C22×C8—22no—
196C22×C8—22no—
197C22×C8—22no—
198C22×C8—22no—
199C22×C8—22no—
200C22×C8—22no—
201C22×C8—41no—
202C22×C8M1614yescomplemented
203C22×C8C22×C411yescharacteristic
204C22×C8C22×C411yescharacteristic
205C22×C8C22×C411yescharacteristic
206C22×C8C22×C411yescharacteristic
207C22×C8C2×D412yes—
208C22×C8C2×D412yes—
209C4×Q8—22no—
210C4×Q8—22no—
211C4×Q8—22no—
212C4×Q8—22no—
213C4×Q8—22no—
214C4×Q8—44no—
215C4×Q8—41no—
216C4×Q8—41no—
217C4×Q8—44no—
218C4×Q8C2×D414yes—
219C4×D4—22no—
220C4×D4—22no—
221C4×D4—22no—
222C4×D4—22no—
223C4×D4—22no—
224C4×D4—44no—
225C4×D4—41no—
226C4×D4—41no—
227C4×D4—44no—
228C4×D4C2×D414yes—
229C2×C4⋊C4—21no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—21no—
233C2×C4⋊C4—21no—
234C2×C4⋊C4—21no—
235C2×C4⋊C4—21no—
236C2×C4⋊C4—21no—
237C2×C4⋊C4—41no—
238C2×C4⋊C4—41no—
239C2×C4⋊C4—44no—
240C2×C4⋊C4C22×C411yescharacteristic
241C2×C4⋊C4C22×C411yescharacteristic
242C2×C4⋊C4C22×C411yescharacteristic
243C2×C4⋊C4C22×C411yescharacteristic
244C2×C22⋊C4—21no—
245C2×C22⋊C4—21no—
246C2×C22⋊C4—21no—
247C2×C22⋊C4—21no—
248C2×C22⋊C4—44no—
249C2×C42—21no—
250C2×C42—22no—
251C2×C42—22no—
252C2×C42—21no—
253C2×C42—21no—
254C2×C42—21no—
255C2×C42—44no—
256C2×C42C22×C411yescharacteristic
257C2×C42C2×D411yescharacteristic
258C2×C42#4111yescharacteristic
259C2×C42C2411yescharacteristic
260#106—22no—
261#106—22no—
262#106—22no—
263#106C22⋊C414yes—
264C4⋊C8—22no—
265C4⋊C8—22no—
266C4⋊C8—22no—
267C4⋊C8—22no—
268C4⋊C8—44no—
269#103—22no—
270#103—22no—
271#103—22no—
272#103C22⋊C414yes—
273#102—22no—
274#102—22no—
275#102—22no—
276#102C22⋊C414yes—
277C22⋊C8—24no—
278C22⋊C8—22no—
279C22⋊C8—44no—
280C22⋊C8—42no—
281C22⋊C8—42no—
282C22⋊C8—44no—
283C22⋊C8—82no—
284C22⋊C8SD1614yescomplemented
285#97—44no—
286C4×C8—22no—
287C4×C8—22no—
288C4×C8—22no—
289C4×C8—22no—
290C4×C8—22no—
291C4×C8—22no—
292C4×C8—22no—
293C4×C8C2×D414yes—
294C24—21no—
295C24—21no—
296C24—42no—
297C24—42no—
298C24—42no—
299C24—42no—
300C24—42no—
301C24—81no—
302C24—81no—
303C24C4×D411yescharacteristic
304C2×Q8—21no—
305C2×Q8—21no—
306C2×Q8—28no—
307C2×Q8—21no—
308C2×Q8—21no—
309C2×Q8—21no—
310C2×Q8—21no—
311C2×Q8—21no—
312C2×Q8—21no—
313C2×Q8—21no—
314C2×Q8—21no—
315C2×Q8—42no—
316C2×Q8—42no—
317C2×Q8—44no—
318C2×Q8—44no—
319C2×Q8—44no—
320C2×Q8—42no—
321C2×Q8C2×C22⋊C412yes—
322C2×Q8C2×M1612yes—
323C2×D4—28no—
324C2×D4—21no—
325C2×D4—21no—
326C2×D4—21no—
327C2×D4—21no—
328C2×D4—21no—
329C2×D4—21no—
330C2×D4—21no—
331C2×D4—21no—
332C2×D4—21no—
333C2×D4—21no—
334C2×D4—44no—
335C2×D4—44no—
336C2×D4—42no—
337C2×D4—42no—
338C2×D4—44no—
339C2×D4—42no—
340C2×D4C2×C22⋊C412yes—
341C2×D4C2×M1612yes—
342C22×C4—21no—
343C22×C4—21no—
344C22×C4—22no—
345C22×C4—21no—
346C22×C4—21no—
347C22×C4—22no—
348C22×C4—21no—
349C22×C4—21no—
350C22×C4—21no—
351C22×C4—21no—
352C22×C4—22no—
353C22×C4—24no—
354C22×C4—22no—
355C22×C4—21no—
356C22×C4—21no—
357C22×C4—41no—
358C22×C4—42no—
359C22×C4—42no—
360C22×C4—42no—
361C22×C4—42no—
362C22×C4—42no—
363C22×C4—42no—
364C22×C4—42no—
365C22×C4—41no—
366C22×C4—41no—
367C22×C4—42no—
368C22×C4—42no—
369C22×C4—42no—
370C22×C4—42no—
371C22×C4—41no—
372C22×C4—41no—
373C22×C4—42no—
374C22×C4—42no—
375C22×C4—42no—
376C22×C4—81no—
377C22×C4—81no—
378C22×C4—88no—
379C22×C4—88no—
380C22×C4—81no—
381C22×C4—81no—
382C22×C4C4×D411yescharacteristic
383C22×C4C4×D411yescharacteristic
384C22×C4C4×D411yescharacteristic
385C22×C4C2×M1612yes—
386C22×C4#13112yes—
387C22×C4C2×SD1612yes—
388C22×C4#13512yes—
389C22×C4C23×C411yescharacteristic
390C22×C4C22×D411yescharacteristic
391C22×C4C2×#4111yescharacteristic
392SD16—28no—
393SD16—28no—
394SD16—28no—
395SD16—28no—
396M16—88no—
397C2×C8—21no—
398C2×C8—22no—
399C2×C8—21no—
400C2×C8—21no—
401C2×C8—21no—
402C2×C8—22no—
403C2×C8—28no—
404C2×C8—24no—
405C2×C8—21no—
406C2×C8—22no—
407C2×C8—21no—
408C2×C8—21no—
409C2×C8—22no—
410C2×C8—22no—
411C2×C8—44no—
412C2×C8—42no—
413C2×C8—42no—
414C2×C8—42no—
415C2×C8—42no—
416C2×C8—42no—
417C2×C8—42no—
418C2×C8—42no—
419C2×C8—42no—
420C2×C8C2×C22⋊C412yes—
421C2×C8C2×C22⋊C412yes—
422C2×C8C2×M1612yes—
423C2×C8C2×M1612yes—
424C2×C8C2×M1612yes—
425C2×C8C2×SD1614yescomplemented
426C4⋊C4—21no—
427C4⋊C4—21no—
428C4⋊C4—24no—
429C4⋊C4—21no—
430C4⋊C4—21no—
431C4⋊C4—21no—
432C4⋊C4—21no—
433C4⋊C4—42no—
434C4⋊C4—42no—
435C4⋊C4—42no—
436C4⋊C4—42no—
437C4⋊C4—42no—
438C4⋊C4C2×C22⋊C412yes—
439C4⋊C4C2×C22⋊C412yes—
440C22⋊C4—24no—
441C22⋊C4—42no—
442C22⋊C4—42no—
443C22⋊C4—42no—
444C42—21no—
445C42—21no—
446C42—24no—
447C42—22no—
448C42—22no—
449C42—42no—
450C42—42no—
451C42—42no—
452C42C22×D412yes—
453C42H(5,𝔽2)12yes—
454C23—21no—
455C23—21no—
456C23—21no—
457C23—21no—
458C23—22no—
459C23—22no—
460C23—48no—
461C23—41no—
462C23—41no—
463C23—41no—
464C23—41no—
465C23—41no—
466C23—41no—
467C23—41no—
468C23—41no—
469C23—41no—
470C23—41no—
471C23—41no—
472C23—41no—
473C23—84no—
474C23—84no—
475C23—84no—
476C23—84no—
477C23—84no—
478C23—84no—
479C23#43612yes—
480C23C4×SD1612yes—
481C23#44312yes—
482C23C2×C4×D411yescharacteristic
483Q8—22no—
484Q8—22no—
485Q8—22no—
486Q8—22no—
487Q8—48no—
488Q8—41no—
489Q8—41no—
490Q8—41no—
491Q8—41no—
492D4—22no—
493D4—22no—
494D4—22no—
495D4—22no—
496D4—48no—
497D4—41no—
498D4—41no—
499D4—41no—
500D4—41no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—22no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—22no—
511C2×C4—24no—
512C2×C4—24no—
513C2×C4—21no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—44no—
521C2×C4—48no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—44no—
531C2×C4—44no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—84no—
539C2×C4—84no—
540C2×C4—84no—
541C2×C4—84no—
542C2×C4—82no—
543C2×C4—82no—
544C2×C4#43612yes—
545C2×C4C4×SD1612yes—
546C2×C4C22×C22⋊C411yescharacteristic
547C2×C4#51811yescharacteristic
548C2×C4D4212yes—
549C2×C4#54612yes—
550C2×C4C22×M1611yescharacteristic
551C2×C4C22×M1611yescharacteristic
552C2×C4C2×#13111yescharacteristic
553C2×C4#56811yescharacteristic
554C2×C4C22×SD1611yescharacteristic
555C2×C4C2×#13511yescharacteristic
556C2×C4#57712yes—
557C8—22no—
558C8—22no—
559C8—24no—
560C8—44no—
561C8—41no—
562C8—42no—
563C8—82no—
564C8—82no—
565C8#40714yescomplemented
566C22—22no—
567C22—22no—
568C22—22no—
569C22—22no—
570C22—22no—
571C22—22no—
572C22—42no—
573C22—42no—
574C22—42no—
575C22—42no—
576C22—42no—
577C22—42no—
578C22—816no—
579C22—81no—
580C22—81no—
581C22—81no—
582C22—81no—
583C22—81no—
584C22—81no—
585C22—81no—
586C22—81no—
587C22#159014yescomplemented
588C22D4×C22⋊C411yescharacteristic
589C22C2×#43611yescharacteristic
590C22D4×M1611yescharacteristic
591C22C2×C4×SD1611yescharacteristic
592C22#294911yescharacteristic
593C22C2×#44311yescharacteristic
594C22#295711yescharacteristic
595C4—22no—
596C4—22no—
597C4—22no—
598C4—42no—
599C4—42no—
600C4—42no—
601C4—41no—
602C4—42no—
603C4—81no—
604C4—81no—
605C4C2×#40711yescharacteristic
606C4#288011yescharacteristic
607C4#297411yescharacteristic
608C4#297511yescharacteristic
609C4D4×SD1612yes—
610C4#328212yes—
611C2—24no—
612C2—82no—
613C2—82no—
614C2C2×#159011yescharacteristic
615C2#1789611yescharacteristic
616C2M16×SD1611yescharacteristic
617C2#1805611yescharacteristic
618C2#2240111yescharacteristic
619C2C22⋊C4×SD1611yescharacteristic
620C2#2245411yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC23C2×C4×D4
prime radicalC23×D4C2×C4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C4×C22⋊C8 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #22510C2semidirect
4. C2×#1590C2semidirect
5. #9655C2semidirect
6. #8952C2semidirect
7. #8949C2semidirect
8. #8595C2semidirect
9. #8311C2semidirect
11. C8×C22⋊C8C2semidirect
15. C2×#446C22semidirect
32. #1984C22semidirect
43. C2×#334C22semidirect
44. C2×Q8⋊C8C22semidirect
45. C2×Q8⋊C8C22semidirect
48. C2×C82C22semidirect
49. #1277C22semidirect
50. #1273C22semidirect
190. C2×SD16M16semidirect
202. C22×C8M16semidirect
284. C22⋊C8SD16semidirect
425. C2×C8C2×SD16semidirect
565. C8#407semidirect
587. C22#1590semidirect

Maximal subgroups

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

subgroupindex
C22×C4×SD162
#225102
#224012
C2×#15902
#96552
#89522
#89492
#85952
#83112
#83102
C8×C22⋊C82