Finite Group Databasebeta

← #183116

Subgroups — #183116

Order 512.

1306 conjugacy classes of subgroups.

The subgroup lattice has 649 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34141C211yesmaximal, characteristic
2#20472C211yesmaximal, characteristic, complemented
3#20470C211yesmaximal, characteristic, complemented
4C2×#1496C211yesmaximal, characteristic
5C2×#1484C211yesmaximal, characteristic
6#8183C211yesmaximal, characteristic, complemented
7#8181C211yesmaximal, characteristic, complemented
8#8179C211yesmaximal, characteristic, complemented
9#8177C211yesmaximal, characteristic, complemented
10#8174C211yesmaximal, characteristic, complemented
11#8172C211yesmaximal, characteristic, complemented
12#8170C211yesmaximal, characteristic, complemented
13C4×D4⋊C8C211yesmaximal, characteristic, complemented
14#7904C211yesmaximal, characteristic, complemented
15#7892C211yesmaximal, characteristic, complemented
16C2×#532C2211yescharacteristic
17C2×C4×#41—21no—
18C2×#432C2211yescharacteristic
19C2×#431C2211yescharacteristic
20C2×#431C2211yescharacteristic
21C2×C4×M16C2211yescharacteristic
22#2367—22no—
23#2366C412yes—
24#2365C412yes—
25#2364—22no—
26#2361C412yes—
27#2361C412yes—
28#2297—21no—
29#2293C2211yescharacteristic
30#2293C2211yescharacteristic
31#2290—21no—
32#2290—21no—
33C42×D4C2211yescharacteristic
34C42×D4C2211yescharacteristic
35C4×#117C2211yescharacteristic
36C2×C4×C4⋊C4C2211yescharacteristic
37#1771—22no—
38#1767C2211yescharacteristic
39#1767C2211yescharacteristic
40C4×C4⋊C8C2211yescharacteristic
41C4×C4⋊C8C2211yescharacteristic
42#1753—22no—
43#1750—21no—
44#1750—21no—
45#1746—21no—
46#1744C2211yescharacteristic
47#1744C2211yescharacteristic
48#1726—21no—
49C4×#97C2211yescharacteristic
50C42×C8C2211yescharacteristic
51#1496C418yescomplemented
52#1484C418yescomplemented
53C2×D4⋊C8C2211yescharacteristic
54C2×D4⋊C8C2211yescharacteristic
55C2×D4⋊C8C2211yescharacteristic
56C2×D4⋊C8C2211yescharacteristic
57C2×D4⋊C8C2211yescharacteristic
58C2×D4⋊C8C2211yescharacteristic
59C2×D4⋊C8C2211yescharacteristic
60C2×D4⋊C8C2211yescharacteristic
61#1279C2211yescharacteristic
62#1279C2211yescharacteristic
63#1279C2211yescharacteristic
64#1279C2211yescharacteristic
65#1279C2211yescharacteristic
66#1279C2211yescharacteristic
67#1279C2211yescharacteristic
68#1279C2211yescharacteristic
69C22×#41—21no—
70C22×M16—21no—
71#532—22no—
72#532—22no—
73#532C2×C414yes—
74#532C2×C414yes—
75C2×C4⋊Q8C2×C411yescharacteristic
76C2×#127C2×C411yescharacteristic
77C2×#124—21no—
78C2×#121—21no—
79C2×#121C2×C411yescharacteristic
80C2×#121C2×C411yescharacteristic
81C4×#41—44no—
82C4×#41—44no—
83C2×C4×Q8—22no—
84C2×C4×D4—22no—
85C2×C4×D4—22no—
86C2×C4×D4—22no—
87C2×C4×D4—21no—
88C2×C4×D4C2×C412yes—
89C2×C4×D4C2×C412yes—
90C2×C4×D4C2311yescharacteristic
91C2×C4×D4C2311yescharacteristic
92C2×#117—21no—
93C2×#117—21no—
94C2×#117D411yescharacteristic
95C2×#117D411yescharacteristic
96C22×C4⋊C4C2×C412yes—
97C22×C42—21no—
98C22×C42—21no—
99C22×C42—21no—
100C22×C42C2311yescharacteristic
101#432—22no—
102#432C2×C414yescomplemented
103#431—22no—
104#431—22no—
105#431C2×C414yescomplemented
106#431C2×C414yescomplemented
107C2×C4⋊C8—21no—
108C2×C4⋊C8C2311yescharacteristic
109C2×C4⋊C8C2311yescharacteristic
110C2×C4⋊C8C2311yescharacteristic
111C2×C4⋊C8C2311yescharacteristic
112C2×C22⋊C8—21no—
113C2×C22⋊C8—21no—
114C2×C22⋊C8—21no—
115C4×M16—22no—
116C4×M16C2×C414yescomplemented
117C2×#97—21no—
118C2×#97—21no—
119C2×#97—22no—
120C2×#97—21no—
121C2×#97—21no—
122C2×C4×C8—21no—
123C2×C4×C8—22no—
124C2×C4×C8—21no—
125C2×C4×C8—21no—
126C2×C4×C8C2311yescharacteristic
127C2×C4×C8C2311yescharacteristic
128C2×C4×C8C2311yescharacteristic
129C2×C4×C8C2311yescharacteristic
130#391—22no—
131#390—22no—
132#390C2×C412yes—
133#390C2×C412yes—
134#389C2×C412yes—
135#389C2×C412yes—
136#388—22no—
137#388—22no—
138#388—22no—
139#388—22no—
140#387—22no—
141#387—22no—
142#386—22no—
143#386—22no—
144#386—22no—
145#386—22no—
146#385—22no—
147#385—22no—
148C4×C4⋊C4—21no—
149C4×C4⋊C4—21no—
150C4×C4⋊C4C2×C412yes—
151C4×C4⋊C4C2×C412yes—
152C4×C4⋊C4C2×C412yes—
153C4×C4⋊C4C2×C412yes—
154C4×C4⋊C4D412yes—
155C4×C4⋊C4C2311yescharacteristic
156C4×C4⋊C4C2311yescharacteristic
157C4×C22⋊C4—22no—
158C4×C22⋊C4—22no—
159C4×C22⋊C4—21no—
160C4×C22⋊C4—21no—
161C4×C22⋊C4—21no—
162C4×C22⋊C4—21no—
163C4×C22⋊C4C2×C412yes—
164C4×C22⋊C4D412yes—
165#376D411yescharacteristic
166#376D411yescharacteristic
167C2×#95—21no—
168C43C2311yescharacteristic
169C43C2311yescharacteristic
170#336—22no—
171#336—22no—
172#336—21no—
173#336—21no—
174#336—21no—
175#336—21no—
176D4⋊C8C2×C414yescomplemented
177D4⋊C8C2×C414yescomplemented
178D4⋊C8C2×C414yescomplemented
179D4⋊C8C2×C414yescomplemented
180D4⋊C8C2×C414yescomplemented
181D4⋊C8C2×C414yescomplemented
182D4⋊C8C2×C414yescomplemented
183D4⋊C8C2×C414yescomplemented
184C2×#41—24no—
185C2×#41—24no—
186C2×#41—42no—
187C2×#41—42no—
188C2×#41—42no—
189C2×#41—42no—
190C22×Q8—21no—
191C22×D4—21no—
192C22×D4—21no—
193C22×D4—21no—
194C22×D4—21no—
195C22×D4C22×C411yescharacteristic
196C22×D4C22×C411yescharacteristic
197C23×C4—21no—
198C23×C4—21no—
199C23×C4—22no—
200C23×C4—22no—
201C23×C4—22no—
202C23×C4C22×C411yescharacteristic
203C23×C4C2×D411yescharacteristic
204C23×C4C2×D411yescharacteristic
205C2×M16—24no—
206C2×M16—44no—
207C22×C8—21no—
208C22×C8—21no—
209C22×C8—21no—
210C22×C8—21no—
211C22×C8—21no—
212C22×C8—21no—
213C22×C8—21no—
214C22×C8—21no—
215C4⋊Q8—22no—
216C4⋊Q8C4214yes—
217#127—22no—
218#127C4214yes—
219#124—24no—
220#124—24no—
221#121—22no—
222#121—22no—
223#121—24no—
224#121—24no—
225#121C4214yes—
226#121C4214yes—
227C4×Q8—48no—
228C4×D4—24no—
229C4×D4—24no—
230C4×D4—24no—
231C4×D4—24no—
232C4×D4—21no—
233C4×D4—21no—
234C4×D4—21no—
235C4×D4—21no—
236C4×D4—42no—
237C4×D4—42no—
238C4×D4—48no—
239C4×D4—48no—
240C4×D4—48no—
241C4×D4C22×C412yes—
242C4×D4C22×C412yes—
243C4×D4C22×C412yes—
244C4×D4C22×C412yes—
245#117—22no—
246#117—22no—
247#117—41no—
248#117—41no—
249#117—42no—
250#117—42no—
251#117—42no—
252#117—42no—
253C2×C4⋊C4—21no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—42no—
259C2×C4⋊C4C22⋊C412yes—
260C2×C4⋊C4C22⋊C412yes—
261C2×C4⋊C4C22×C412yes—
262C2×C4⋊C4C22×C411yescharacteristic
263C2×C4⋊C4C22×C411yescharacteristic
264C2×C4⋊C4C22×C412yes—
265C2×C22⋊C4—21no—
266C2×C22⋊C4—21no—
267C2×C22⋊C4—21no—
268C2×C22⋊C4—21no—
269C2×C22⋊C4—22no—
270C2×C22⋊C4—22no—
271C2×C22⋊C4—22no—
272C2×C22⋊C4—22no—
273C2×C22⋊C4—22no—
274C2×C22⋊C4—22no—
275C2×C22⋊C4C22⋊C412yes—
276C2×C22⋊C4C22⋊C412yes—
277C2×C42—22no—
278C2×C42—22no—
279C2×C42—22no—
280C2×C42—21no—
281C2×C42—21no—
282C2×C42—42no—
283C2×C42—42no—
284C2×C42—42no—
285C2×C42—42no—
286C2×C42—42no—
287C2×C42—42no—
288C2×C42C22×C412yes—
289C2×C42C22×C412yes—
290C2×C42C22×C411yescharacteristic
291C2×C42C22×C412yes—
292C2×C42C22×C412yes—
293C2×C42C2×D411yescharacteristic
294C2×C42C2×D411yescharacteristic
295C2×C42C2×D411yescharacteristic
296C2×C42C2×D412yes—
297C2×C42C2×D412yes—
298C2×C42C2×D411yescharacteristic
299C2×C42C2×D411yescharacteristic
300C2×C42C2×D411yescharacteristic
301C2×C42#4112yes—
302C2×C42#4112yes—
303C2×C42C2411yescharacteristic
304C4⋊C8—24no—
305C4⋊C8—21no—
306C4⋊C8—21no—
307C4⋊C8—21no—
308C4⋊C8—21no—
309C4⋊C8C22×C412yes—
310C4⋊C8C22×C412yes—
311C4⋊C8C22×C412yes—
312C4⋊C8C22×C412yes—
313C22⋊C8—24no—
314C22⋊C8—24no—
315C22⋊C8—24no—
316#97—24no—
317#97—24no—
318#97—42no—
319#97—44no—
320#97—42no—
321C4×C8—24no—
322C4×C8—21no—
323C4×C8—21no—
324C4×C8—21no—
325C4×C8—21no—
326C4×C8—42no—
327C4×C8—44no—
328C4×C8—42no—
329C4×C8C22×C412yes—
330C4×C8C22×C412yes—
331C4×C8C22×C412yes—
332C4×C8C22×C412yes—
333#95—22no—
334#95—22no—
335#95—22no—
336#95—22no—
337#95—22no—
338#95—21no—
339#95—21no—
340#95C22⋊C412yes—
341#95C22⋊C412yes—
342#95C22⋊C412yes—
343#95C22⋊C412yes—
344C24—21no—
345C24—21no—
346C24C2×C22⋊C411yescharacteristic
347#41—48no—
348#41—48no—
349#41—48no—
350#41—48no—
351C2×Q8—24no—
352C2×Q8—42no—
353C2×Q8—42no—
354C2×D4—24no—
355C2×D4—24no—
356C2×D4—21no—
357C2×D4—21no—
358C2×D4—22no—
359C2×D4—22no—
360C2×D4—22no—
361C2×D4—22no—
362C2×D4—24no—
363C2×D4—24no—
364C2×D4—42no—
365C2×D4—42no—
366C2×D4—42no—
367C2×D4—42no—
368C2×D4—42no—
369C2×D4—42no—
370C2×D4—82no—
371C2×D4C2×C4212yes—
372C2×D4C2×C4212yes—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—21no—
376C22×C4—21no—
377C22×C4—21no—
378C22×C4—21no—
379C22×C4—22no—
380C22×C4—21no—
381C22×C4—21no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—42no—
388C22×C4—44no—
389C22×C4—44no—
390C22×C4—44no—
391C22×C4—44no—
392C22×C4—41no—
393C22×C4—41no—
394C22×C4—41no—
395C22×C4—41no—
396C22×C4—41no—
397C22×C4—41no—
398C22×C4—41no—
399C22×C4—41no—
400C22×C4—44no—
401C22×C4—44no—
402C22×C4—82no—
403C22×C4—82no—
404C22×C4—82no—
405C22×C4C2×C4212yes—
406C22×C4C2×C22⋊C412yes—
407C22×C4C2×C22⋊C412yes—
408C22×C4C2×C22⋊C412yes—
409C22×C4C2×C22⋊C411yescharacteristic
410C22×C4C2×C22⋊C411yescharacteristic
411C22×C4C2×C22⋊C411yescharacteristic
412C22×C4C2×C22⋊C412yes—
413C22×C4#11712yes—
414C22×C4C4×D412yes—
415C22×C4C4×D412yes—
416C22×C4C4×D412yes—
417C22×C4C4×D412yes—
418C22×C4C23×C411yescharacteristic
419C22×C4C22×D411yescharacteristic
420C22×C4C22×D411yescharacteristic
421M16—48no—
422C2×C8—24no—
423C2×C8—22no—
424C2×C8—22no—
425C2×C8—22no—
426C2×C8—22no—
427C2×C8—22no—
428C2×C8—22no—
429C2×C8—22no—
430C2×C8—22no—
431C2×C8—42no—
432C2×C8—42no—
433C2×C8—42no—
434C2×C8—41no—
435C2×C8—41no—
436C2×C8—41no—
437C2×C8—41no—
438C2×C8—41no—
439C2×C8—41no—
440C2×C8—41no—
441C2×C8—41no—
442C4⋊C4—21no—
443C4⋊C4—21no—
444C4⋊C4—22no—
445C4⋊C4—24no—
446C4⋊C4—24no—
447C4⋊C4—22no—
448C4⋊C4—22no—
449C4⋊C4—42no—
450C4⋊C4—44no—
451C4⋊C4—44no—
452C4⋊C4—44no—
453C4⋊C4C2×C4212yes—
454C4⋊C4C2×C4212yes—
455C4⋊C4#13114yes—
456C22⋊C4—22no—
457C22⋊C4—22no—
458C22⋊C4—22no—
459C22⋊C4—22no—
460C22⋊C4—24no—
461C22⋊C4—24no—
462C22⋊C4—24no—
463C22⋊C4—42no—
464C22⋊C4—44no—
465C22⋊C4—44no—
466C22⋊C4—44no—
467C22⋊C4—44no—
468C22⋊C4—82no—
469C22⋊C4—82no—
470C42—22no—
471C42—22no—
472C42—22no—
473C42—22no—
474C42—21no—
475C42—21no—
476C42—21no—
477C42—21no—
478C42—21no—
479C42—21no—
480C42—42no—
481C42—42no—
482C42—44no—
483C42—44no—
484C42—41no—
485C42—41no—
486C42—41no—
487C42—41no—
488C42—44no—
489C42C2×C4212yes—
490C42#11712yes—
491C42C4×D412yes—
492C42C4×D412yes—
493C42C4×D412yes—
494C42C4×D412yes—
495C42AGL(1,ℤ/8)12yes—
496C42AGL(1,ℤ/8)12yes—
497C42C23×C411yescharacteristic
498C42C23×C411yescharacteristic
499C42C2×#4111yescharacteristic
500C42C2×#4111yescharacteristic
501C23—22no—
502C23—21no—
503C23—21no—
504C23—22no—
505C23—22no—
506C23—22no—
507C23—42no—
508C23—42no—
509C23—41no—
510C23—41no—
511C23—41no—
512C23—41no—
513C23—42no—
514C23—42no—
515C23C4×C22⋊C412yes—
516C23C22×C22⋊C411yescharacteristic
517Q8—48no—
518D4—22no—
519D4—22no—
520D4—22no—
521D4—22no—
522D4—48no—
523D4—48no—
524D4—48no—
525D4—42no—
526D4—42no—
527D4—84no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—24no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—24no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—88no—
574C2×C4—88no—
575C2×C4—88no—
576C2×C4C4×C22⋊C412yes—
577C2×C4C4×C22⋊C412yes—
578C2×C4C4×C22⋊C412yes—
579C2×C4#41814yes—
580C2×C4C22×C4211yescharacteristic
581C2×C4C2×#11711yescharacteristic
582C2×C4C2×C4×D411yescharacteristic
583C2×C4C2×C4×D411yescharacteristic
584C2×C4C2×C4×D411yescharacteristic
585C2×C4C2×C4×D411yescharacteristic
586C2×C4C2×#13112yes—
587C2×C4#56812yes—
588C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
589C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
590C2×C4#57511yescharacteristic
591C2×C4#57511yescharacteristic
592C8—24no—
593C8—42no—
594C8—42no—
595C8—42no—
596C8—42no—
597C8—42no—
598C8—42no—
599C8—42no—
600C22—22no—
601C22—22no—
602C22—22no—
603C22—42no—
604C22—42no—
605C22—42no—
606C22—42no—
607C22—42no—
608C22—42no—
609C22—42no—
610C22—42no—
611C22—82no—
612C22—82no—
613C22—82no—
614C22—82no—
615C22C2×C4×C22⋊C411yescharacteristic
616C22#288012yes—
617C22C2×#41811yescharacteristic
618C22#289611yescharacteristic
619C22#290112yes—
620C4—24no—
621C4—22no—
622C4—21no—
623C4—21no—
624C4—22no—
625C4—22no—
626C4—42no—
627C4—44no—
628C4—42no—
629C4—42no—
630C4—42no—
631C4—42no—
632C4—82no—
633C4C4×#13111yescharacteristic
634C4#287511yescharacteristic
635C4#292512yes—
636C4C4×AGL(1,ℤ/8)11yescharacteristic
637C4C4×AGL(1,ℤ/8)11yescharacteristic
638C4C4×AGL(1,ℤ/8)11yescharacteristic
639C4C4×AGL(1,ℤ/8)11yescharacteristic
640C2—42no—
641C2—82no—
642C2—82no—
643C2#1751112yes—
644C2#1751612yes—
645C2#2037811yescharacteristic
646C2#2040511yescharacteristic
647C2C4×#41811yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC2×C4C22×C42
Frattini, cosocleC2×C42C24
socleC23C22×C22⋊C4
prime radicalC2×#532C22
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C4×#117 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #20472C2semidirect
3. #20470C2semidirect
6. #8183C2semidirect
7. #8181C2semidirect
8. #8179C2semidirect
9. #8177C2semidirect
10. #8174C2semidirect
11. #8172C2semidirect
12. #8170C2semidirect
13. C4×D4⋊C8C2semidirect
14. #7904C2semidirect
15. #7892C2semidirect
51. #1496C4semidirect
52. #1484C4semidirect
102. #432C2×C4semidirect
105. #431C2×C4semidirect
106. #431C2×C4semidirect
116. C4×M16C2×C4semidirect
176. D4⋊C8C2×C4semidirect
177. D4⋊C8C2×C4semidirect
178. D4⋊C8C2×C4semidirect
179. D4⋊C8C2×C4semidirect
180. D4⋊C8C2×C4semidirect
181. D4⋊C8C2×C4semidirect
182. D4⋊C8C2×C4semidirect
183. D4⋊C8C2×C4semidirect

Maximal subgroups

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

subgroupindex
#341412
#204722
#204702
C2×#14962
C2×#14842
#81832
#81812
#81792
#81772
#81742
#81722
#81702
C4×D4⋊C82
#79042
#78922