Finite Group Databasebeta

← #214215

Subgroups — #214215

Order 512.

1422 conjugacy classes of subgroups.

The subgroup lattice has 739 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#525C211yesmaximal, characteristic
2#20626C211yesmaximal, characteristic, complemented
3C4×#407C211yesmaximal, characteristic, complemented
4#9078C212yesmaximal, complemented
5#9077C212yesmaximal, complemented
6#8944C212yesmaximal, complemented
7#8942C212yesmaximal, complemented
8#8708C212yesmaximal, complemented
9#8312C212yesmaximal, complemented
10C2×#525C411yescharacteristic
11C2×C4×#41C2211yescharacteristic
12C23×C42C2211yescharacteristic
13C2×#423C2212yes—
14C2×#407—21no—
15C2×#407—21no—
16C2×C4×M16C2212yes—
17C4×C22⋊Q8C411yescharacteristic
18C4×C22⋊Q8C411yescharacteristic
19C4×C22⋊Q8C2211yescharacteristic
20C4×C22⋊Q8C2211yescharacteristic
21C4×#123—21no—
22C4×#123—21no—
23C4×#121C411yescharacteristic
24C4×#121C411yescharacteristic
25C4×#121C2211yescharacteristic
26C4×#121C2211yescharacteristic
27C4×#120—21no—
28C4×#120—21no—
29#2297C411yescharacteristic
30#2296C411yescharacteristic
31#2290—21no—
32#2290—21no—
33#2280C411yescharacteristic
34C42×D4—21no—
35C42×D4—21no—
36C4×#117C2211yescharacteristic
37#1797—21no—
38#1797—21no—
39#1797C2212yes—
40#1797C2212yes—
41#1749—21no—
42#1749—21no—
43C4×C22⋊C8C2212yes—
44C4×C22⋊C8C2212yes—
45#1728—21no—
46#1728—21no—
47C2×Q8⋊C8C2214yescomplemented
48C2×D4⋊C8C2214yescomplemented
49#1279C2214yescomplemented
50#1279C2214yescomplemented
51C22×#41C2×C411yescharacteristic
52C24×C4C2×C411yescharacteristic
53C24×C4D411yescharacteristic
54C24×C4D411yescharacteristic
55C22×M16—22no—
56C22×M16—22no—
57#525—24no—
58#525—24no—
59C2×#123—21no—
60C2×C22⋊Q8C2×C411yescharacteristic
61C2×C22⋊Q8C2×C411yescharacteristic
62C2×#121C2×C411yescharacteristic
63C2×#121C2×C411yescharacteristic
64C2×#120—21no—
65C4×#41—44no—
66C2×C4×Q8C2×C411yescharacteristic
67C2×C4×Q8C2311yescharacteristic
68C2×C4×D4—21no—
69C2×C4×D4—21no—
70C2×C4×D4—21no—
71C2×C4×D4—21no—
72C2×C4×D4—21no—
73C2×C4×D4—21no—
74C2×C4×D4—21no—
75C2×C4×D4—21no—
76C2×C4×D4C2×C411yescharacteristic
77C2×C4×D4C2311yescharacteristic
78C2×#117—21no—
79C2×#117C2×C411yescharacteristic
80C2×#117C2×C411yescharacteristic
81C2×#117C2×C411yescharacteristic
82C22×C42—21no—
83C22×C42—44no—
84C22×C42C2×C411yescharacteristic
85C22×C42C2×C411yescharacteristic
86C22×C42C2×C411yescharacteristic
87C22×C42D411yescharacteristic
88C22×C42D411yescharacteristic
89C22×C42D411yescharacteristic
90C22×C42D411yescharacteristic
91C22×C42D411yescharacteristic
92C22×C42D411yescharacteristic
93C22×C42C2311yescharacteristic
94C22×C42C2311yescharacteristic
95C22×C42C2311yescharacteristic
96#423—44no—
97C2×C4⋊C8—22no—
98C2×C4⋊C8C2314yes—
99#407—28no—
100#407—44no—
101C2×C22⋊C8—22no—
102C2×C22⋊C8—22no—
103C2×C22⋊C8—22no—
104C2×C22⋊C8—22no—
105C4×M16—44no—
106C2×#97—22no—
107C2×C4×C8C2314yes—
108#391C2×C411yescharacteristic
109#391C2×C411yescharacteristic
110#390C2×C411yescharacteristic
111#390C2×C411yescharacteristic
112#389C2×C411yescharacteristic
113#389C2×C411yescharacteristic
114#389C2×C411yescharacteristic
115#389C2×C411yescharacteristic
116#388—21no—
117#388—21no—
118#388—21no—
119#388—21no—
120#387—21no—
121#387—21no—
122#387—21no—
123#387—21no—
124#386—21no—
125#386—21no—
126#386—21no—
127#386—21no—
128#386—21no—
129#386—21no—
130#385—21no—
131#385—21no—
132#385—21no—
133#385—21no—
134#385—21no—
135#385—21no—
136#381—21no—
137#380C2×C411yescharacteristic
138#380C2×C411yescharacteristic
139#380C2×C411yescharacteristic
140#380C2×C411yescharacteristic
141#379—21no—
142C4×C4⋊C4—21no—
143C4×C4⋊C4—21no—
144C4×C4⋊C4—21no—
145C4×C4⋊C4—21no—
146C4×C4⋊C4—21no—
147C4×C4⋊C4—21no—
148C4×C4⋊C4C2×C411yescharacteristic
149C4×C4⋊C4C2×C411yescharacteristic
150C4×C4⋊C4C2311yescharacteristic
151C4×C4⋊C4C2311yescharacteristic
152C4×C22⋊C4—21no—
153C4×C22⋊C4—21no—
154C4×C22⋊C4—21no—
155C4×C22⋊C4—21no—
156C4×C22⋊C4—21no—
157C4×C22⋊C4—21no—
158C4×C22⋊C4—21no—
159C4×C22⋊C4—21no—
160C4×C22⋊C4—21no—
161C4×C22⋊C4—21no—
162C4×C22⋊C4—21no—
163C4×C22⋊C4—21no—
164C4×C22⋊C4—21no—
165C4×C22⋊C4—21no—
166#376—21no—
167C43—21no—
168#336—22no—
169#336—22no—
170#336—22no—
171#336—22no—
172Q8⋊C8—44no—
173D4⋊C8—44no—
174C25C22⋊C411yescharacteristic
175C2×#41—22no—
176C2×#41—42no—
177C2×#41—42no—
178C2×#41M1614yescomplemented
179C22×Q8C22×C411yescharacteristic
180C22×D4—21no—
181C22×D4—21no—
182C22×D4—21no—
183C22×D4C22×C411yescharacteristic
184C23×C4—24no—
185C23×C4—21no—
186C23×C4—22no—
187C23×C4—22no—
188C23×C4—22no—
189C23×C4—22no—
190C23×C4—21no—
191C23×C4—21no—
192C23×C4—42no—
193C23×C4—42no—
194C23×C4—42no—
195C23×C4—42no—
196C23×C4—42no—
197C23×C4—42no—
198C23×C4C22⋊C411yescharacteristic
199C23×C4C22⋊C411yescharacteristic
200C23×C4C22⋊C411yescharacteristic
201C23×C4C22⋊C411yescharacteristic
202C23×C4C22⋊C411yescharacteristic
203C23×C4C22⋊C411yescharacteristic
204C23×C4C22⋊C411yescharacteristic
205C23×C4C22⋊C411yescharacteristic
206C23×C4C22⋊C411yescharacteristic
207C23×C4C22×C411yescharacteristic
208C23×C4C22×C411yescharacteristic
209C23×C4C22×C411yescharacteristic
210C23×C4C2×D411yescharacteristic
211C23×C4C2×D411yescharacteristic
212C23×C4C2×D411yescharacteristic
213C23×C4C2×D411yescharacteristic
214C23×C4C2×D411yescharacteristic
215C23×C4C2×D411yescharacteristic
216C2×M16—28no—
217C2×M16—44no—
218C2×M16—44no—
219C2×M16—44no—
220C2×M16—44no—
221C2×M16—44no—
222C22×C8—24no—
223C22×C8—24no—
224#123—28no—
225C22⋊Q8—22no—
226C22⋊Q8—22no—
227C22⋊Q8—22no—
228C22⋊Q8—22no—
229#121—22no—
230#121—22no—
231#121—22no—
232#121—22no—
233#120—28no—
234C4×Q8—24no—
235C4×Q8—42no—
236C4×D4—28no—
237C4×D4—28no—
238C4×D4—24no—
239C4×D4—42no—
240C4×D4—44no—
241C4×D4—44no—
242C4×D4—44no—
243C4×D4—44no—
244C4×D4—44no—
245C4×D4—42no—
246C4×D4—42no—
247#117—22no—
248#117—44no—
249#117—42no—
250#117—42no—
251#117M1614yescomplemented
252C2×C4⋊C4—21no—
253C2×C4⋊C4—21no—
254C2×C4⋊C4—21no—
255C2×C4⋊C4—21no—
256C2×C4⋊C4—21no—
257C2×C4⋊C4—21no—
258C2×C4⋊C4—21no—
259C2×C4⋊C4—21no—
260C2×C4⋊C4C22×C411yescharacteristic
261C2×C4⋊C4C22×C411yescharacteristic
262C2×C4⋊C4C22×C411yescharacteristic
263C2×C4⋊C4C22×C411yescharacteristic
264C2×C4⋊C4C22×C411yescharacteristic
265C2×C4⋊C4C22×C411yescharacteristic
266C2×C22⋊C4—21no—
267C2×C22⋊C4—21no—
268C2×C22⋊C4—21no—
269C2×C22⋊C4—21no—
270C2×C22⋊C4—21no—
271C2×C22⋊C4—21no—
272C2×C22⋊C4—21no—
273C2×C22⋊C4—21no—
274C2×C22⋊C4—21no—
275C2×C22⋊C4—21no—
276C2×C22⋊C4—21no—
277C2×C22⋊C4—21no—
278C2×C22⋊C4—21no—
279C2×C22⋊C4—21no—
280C2×C22⋊C4—21no—
281C2×C22⋊C4—21no—
282C2×C22⋊C4—21no—
283C2×C42—21no—
284C2×C42—24no—
285C2×C42—24no—
286C2×C42—24no—
287C2×C42—21no—
288C2×C42—21no—
289C2×C42—21no—
290C2×C42—21no—
291C2×C42—21no—
292C2×C42—21no—
293C2×C42—21no—
294C2×C42—21no—
295C2×C42—42no—
296C2×C42—42no—
297C2×C42—42no—
298C2×C42—42no—
299C2×C42—42no—
300C2×C42—42no—
301C2×C42—42no—
302C2×C42—42no—
303C2×C42—42no—
304C2×C42—44no—
305C2×C42C22⋊C411yescharacteristic
306C2×C42C22⋊C411yescharacteristic
307C2×C42C22⋊C411yescharacteristic
308C2×C42C22⋊C411yescharacteristic
309C2×C42C22⋊C411yescharacteristic
310C2×C42C22⋊C411yescharacteristic
311C2×C42C22×C411yescharacteristic
312C2×C42C22×C411yescharacteristic
313C2×C42C22×C411yescharacteristic
314C2×C42C2×D411yescharacteristic
315C2×C42C2×D411yescharacteristic
316C2×C42C2×D411yescharacteristic
317C2×C42C2×D411yescharacteristic
318C2×C42C2×D411yescharacteristic
319C2×C42C2×D411yescharacteristic
320C2×C42C2411yescharacteristic
321C4⋊C8—24no—
322C4⋊C8—24no—
323C4⋊C8—44no—
324C22⋊C8—28no—
325C22⋊C8—28no—
326C22⋊C8—44no—
327C22⋊C8—44no—
328#97—44no—
329C4×C8—24no—
330C4×C8—24no—
331#95—21no—
332#95—21no—
333#95—21no—
334#95—21no—
335#95—21no—
336#95—21no—
337#95—21no—
338#95—21no—
339#95—21no—
340#95—21no—
341#95—21no—
342#95—21no—
343#95—21no—
344#95—21no—
345#95—21no—
346#95—21no—
347C24—21no—
348C24—22no—
349C24—22no—
350C24—22no—
351C24—41no—
352C24—41no—
353C24—41no—
354C24—41no—
355C24C2×C22⋊C411yescharacteristic
356C24C2×C22⋊C411yescharacteristic
357C24C2×C22⋊C411yescharacteristic
358#41—48no—
359#41—48no—
360C2×Q8—21no—
361C2×Q8—22no—
362C2×Q8—22no—
363C2×Q8C2×M1612yes—
364C2×D4—24no—
365C2×D4—24no—
366C2×D4—24no—
367C2×D4—22no—
368C2×D4—21no—
369C2×D4—22no—
370C2×D4—42no—
371C2×D4—42no—
372C2×D4—42no—
373C2×D4—42no—
374C2×D4—42no—
375C2×D4—42no—
376C2×D4C2×M1612yes—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—21no—
385C22×C4—21no—
386C22×C4—21no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—21no—
396C22×C4—21no—
397C22×C4—21no—
398C22×C4—21no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—21no—
402C22×C4—21no—
403C22×C4—21no—
404C22×C4—21no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—24no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—21no—
419C22×C4—21no—
420C22×C4—21no—
421C22×C4—21no—
422C22×C4—41no—
423C22×C4—41no—
424C22×C4—41no—
425C22×C4—41no—
426C22×C4—41no—
427C22×C4—41no—
428C22×C4—41no—
429C22×C4—41no—
430C22×C4—41no—
431C22×C4—41no—
432C22×C4—41no—
433C22×C4—41no—
434C22×C4—41no—
435C22×C4—41no—
436C22×C4—41no—
437C22×C4—41no—
438C22×C4—41no—
439C22×C4—41no—
440C22×C4—42no—
441C22×C4—416no—
442C22×C4—48no—
443C22×C4—48no—
444C22×C4—48no—
445C22×C4—48no—
446C22×C4—42no—
447C22×C4—42no—
448C22×C4—42no—
449C22×C4—42no—
450C22×C4—41no—
451C22×C4—41no—
452C22×C4—42no—
453C22×C4—42no—
454C22×C4#10414yes—
455C22×C4C2×C22⋊C411yescharacteristic
456C22×C4C2×C22⋊C411yescharacteristic
457C22×C4C2×C22⋊C411yescharacteristic
458C22×C4C2×C22⋊C411yescharacteristic
459C22×C4C2×C22⋊C411yescharacteristic
460C22×C4C2×C22⋊C411yescharacteristic
461C22×C4C2×C22⋊C411yescharacteristic
462C22×C4C2×C22⋊C411yescharacteristic
463C22×C4C2×C22⋊C411yescharacteristic
464C22×C4C2×M1612yes—
465C22×C4C2×M1612yes—
466C22×C4AGL(1,ℤ/8)12yes—
467C22×C4#13712yes—
468C22×C4C23×C411yescharacteristic
469C22×C4C22×D411yescharacteristic
470C22×C4C22×D411yescharacteristic
471M16—416no—
472M16—88no—
473C2×C8—28no—
474C2×C8—44no—
475C2×C8—44no—
476C2×C8—44no—
477C2×C8—44no—
478C2×C8—44no—
479C4⋊C4—24no—
480C4⋊C4—24no—
481C4⋊C4—21no—
482C4⋊C4—21no—
483C4⋊C4—22no—
484C4⋊C4—22no—
485C4⋊C4—22no—
486C4⋊C4—22no—
487C4⋊C4—42no—
488C4⋊C4—42no—
489C4⋊C4—42no—
490C4⋊C4—42no—
491C4⋊C4—42no—
492C4⋊C4—42no—
493C4⋊C4C2×M1612yes—
494C4⋊C4C2×M1612yes—
495C22⋊C4—24no—
496C22⋊C4—24no—
497C22⋊C4—24no—
498C22⋊C4—24no—
499C22⋊C4—24no—
500C22⋊C4—42no—
501C22⋊C4—42no—
502C22⋊C4—42no—
503C22⋊C4—42no—
504C22⋊C4—42no—
505C22⋊C4—42no—
506C22⋊C4—42no—
507C22⋊C4—42no—
508C22⋊C4—42no—
509C22⋊C4—42no—
510C22⋊C4—42no—
511C22⋊C4—42no—
512C42—22no—
513C42—22no—
514C42—22no—
515C42—22no—
516C42—22no—
517C42—22no—
518C42—22no—
519C42—22no—
520C42—22no—
521C42—24no—
522C42—41no—
523C42—41no—
524C42—41no—
525C42—41no—
526C42—41no—
527C42—41no—
528C42—42no—
529C42—42no—
530C42—42no—
531C42—42no—
532C42—42no—
533C42—41no—
534C42—41no—
535C42—42no—
536C42—42no—
537C42H(5,𝔽2)14yes—
538C23—21no—
539C23—21no—
540C23—21no—
541C23—21no—
542C23—21no—
543C23—21no—
544C23—21no—
545C23—21no—
546C23—21no—
547C23—21no—
548C23—21no—
549C23—21no—
550C23—21no—
551C23—21no—
552C23—21no—
553C23—21no—
554C23—21no—
555C23—21no—
556C23—22no—
557C23—44no—
558C23—44no—
559C23—44no—
560C23—44no—
561C23—44no—
562C23—44no—
563C23—44no—
564C23—41no—
565C23—41no—
566C23—41no—
567C23—41no—
568C23—41no—
569C23—41no—
570C23#40712yes—
571C23#41712yes—
572C23C2×#10412yes—
573C23C22×C22⋊C411yescharacteristic
574Q8—24no—
575Q8—42no—
576D4—24no—
577D4—48no—
578D4—48no—
579D4—48no—
580D4—42no—
581C2×C4—28no—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—21no—
592C2×C4—21no—
593C2×C4—21no—
594C2×C4—24no—
595C2×C4—24no—
596C2×C4—24no—
597C2×C4—24no—
598C2×C4—21no—
599C2×C4—21no—
600C2×C4—21no—
601C2×C4—21no—
602C2×C4—22no—
603C2×C4—22no—
604C2×C4—21no—
605C2×C4—21no—
606C2×C4—22no—
607C2×C4—21no—
608C2×C4—21no—
609C2×C4—21no—
610C2×C4—21no—
611C2×C4—21no—
612C2×C4—21no—
613C2×C4—44no—
614C2×C4—44no—
615C2×C4—44no—
616C2×C4—44no—
617C2×C4—44no—
618C2×C4—44no—
619C2×C4—44no—
620C2×C4—44no—
621C2×C4—44no—
622C2×C4—48no—
623C2×C4—44no—
624C2×C4—44no—
625C2×C4—41no—
626C2×C4—41no—
627C2×C4—41no—
628C2×C4—41no—
629C2×C4—41no—
630C2×C4—41no—
631C2×C4—42no—
632C2×C4—42no—
633C2×C4—42no—
634C2×C4—42no—
635C2×C4—42no—
636C2×C4—42no—
637C2×C4—42no—
638C2×C4—42no—
639C2×C4—41no—
640C2×C4—41no—
641C2×C4—41no—
642C2×C4—41no—
643C2×C4—41no—
644C2×C4—41no—
645C2×C4—41no—
646C2×C4—41no—
647C2×C4—41no—
648C2×C4—41no—
649C2×C4—41no—
650C2×C4—41no—
651C2×C4—84no—
652C2×C4—82no—
653C2×C4—82no—
654C2×C4#40712yes—
655C2×C4#41712yes—
656C2×C4C2×#10412yes—
657C2×C4C2×#10412yes—
658C2×C4#42112yes—
659C2×C4#51812yes—
660C2×C4#53412yes—
661C2×C4#53412yes—
662C2×C4C22×M1611yescharacteristic
663C2×C4C22×M1611yescharacteristic
664C2×C4#56812yes—
665C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
666C2×C4C2×#13711yescharacteristic
667C2×C4#57512yes—
668C8—48no—
669C8—84no—
670C22—22no—
671C22—22no—
672C22—22no—
673C22—22no—
674C22—22no—
675C22—22no—
676C22—22no—
677C22—22no—
678C22—22no—
679C22—416no—
680C22—41no—
681C22—41no—
682C22—41no—
683C22—41no—
684C22—41no—
685C22—41no—
686C22—41no—
687C22—41no—
688C22—41no—
689C22—41no—
690C22—41no—
691C22—41no—
692C22—42no—
693C22—42no—
694C22—42no—
695C22—81no—
696C22—81no—
697C22—81no—
698C22—81no—
699C22#230811yescharacteristic
700C22C2×#40711yescharacteristic
701C22#287911yescharacteristic
702C22C2×#41711yescharacteristic
703C22#289611yescharacteristic
704C22C22×#10411yescharacteristic
705C22C2×#42111yescharacteristic
706C4—22no—
707C4—22no—
708C4—22no—
709C4—22no—
710C4—22no—
711C4—22no—
712C4—41no—
713C4—41no—
714C4—42no—
715C4—42no—
716C4—41no—
717C4—41no—
718C4—42no—
719C4—81no—
720C4—81no—
721C4—81no—
722C4—81no—
723C4#297312yes—
724C4#297512yes—
725C4#319412yes—
726C4#319612yes—
727C2—42no—
728C2—42no—
729C2—42no—
730C2—82no—
731C2#1744711yescharacteristic
732C2#1745111yescharacteristic
733C2#1749311yescharacteristic
734C2#1750411yescharacteristic
735C2#2174211yescharacteristic
736C2#2176911yescharacteristic
737C2#2177311yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC23C22×C22⋊C4
prime radicalC2×#525C4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C23×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #20626C2semidirect
3. C4×#407C2semidirect
4. #9078C2semidirect
5. #9077C2semidirect
6. #8944C2semidirect
7. #8942C2semidirect
8. #8708C2semidirect
9. #8312C2semidirect
47. C2×Q8⋊C8C22semidirect
48. C2×D4⋊C8C22semidirect
49. #1279C22semidirect
50. #1279C22semidirect
178. C2×#41M16semidirect
251. #117M16semidirect

Maximal subgroups

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

subgroupindex
C4×#5252
#206262
C4×#4072
#90782
#90772
#89442
#89422
#87082
#83122