Finite Group Databasebeta

← #214125

Subgroups — #214125

Order 512.

1408 conjugacy classes of subgroups.

The subgroup lattice has 735 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#525C211yesmaximal, characteristic
2#20714C211yesmaximal, characteristic, complemented
3C2×#1797C211yesmaximal, characteristic, complemented
4#9074C212yesmaximal, complemented
5#9073C212yesmaximal, complemented
6#8939C212yesmaximal, complemented
7#8937C212yesmaximal, complemented
8#8708C212yesmaximal, complemented
9#8312C212yesmaximal, complemented
10C2×#525C411yescharacteristic
11C2×C4×#41C2211yescharacteristic
12C23×C42C2211yescharacteristic
13C2×#431C2212yes—
14C22×C4⋊C8C2212yes—
15C22×C22⋊C8—21no—
16C4×C22⋊Q8C411yescharacteristic
17C4×C22⋊Q8C411yescharacteristic
18C4×C22⋊Q8C2211yescharacteristic
19C4×C22⋊Q8C2211yescharacteristic
20C4×#123—21no—
21C4×#123—21no—
22C4×#121C411yescharacteristic
23C4×#121C411yescharacteristic
24C4×#121C2211yescharacteristic
25C4×#121C2211yescharacteristic
26C4×#120—21no—
27C4×#120—21no—
28#2297C411yescharacteristic
29#2296C411yescharacteristic
30#2290—21no—
31#2290—21no—
32#2280C411yescharacteristic
33C42×D4—21no—
34C42×D4—21no—
35C4×#117C2211yescharacteristic
36#1817—21no—
37#1797—21no—
38#1797—21no—
39#1797—21no—
40#1797—21no—
41#1797C2212yes—
42#1797C2212yes—
43#1779—21no—
44C4×C22⋊C8C2212yes—
45C4×C22⋊C8C2212yes—
46C2×#336—21no—
47C2×Q8⋊C8C2214yescomplemented
48C2×D4⋊C8C2214yescomplemented
49#1279C2214yescomplemented
50#1279C2214yescomplemented
51C22×#41C2×C411yescharacteristic
52C24×C4C2×C411yescharacteristic
53C24×C4D411yescharacteristic
54C24×C4D411yescharacteristic
55C23×C8—22no—
56#525—24no—
57#525C818yescomplemented
58C2×#123—21no—
59C2×C22⋊Q8C2×C411yescharacteristic
60C2×C22⋊Q8C2×C411yescharacteristic
61C2×#121C2×C411yescharacteristic
62C2×#121C2×C411yescharacteristic
63C2×#120—21no—
64C4×#41—44no—
65C2×C4×Q8C2×C411yescharacteristic
66C2×C4×Q8C2311yescharacteristic
67C2×C4×D4—21no—
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×D4C2×C411yescharacteristic
76C2×C4×D4C2311yescharacteristic
77C2×#117—21no—
78C2×#117C2×C411yescharacteristic
79C2×#117C2×C411yescharacteristic
80C2×#117C2×C411yescharacteristic
81C22×C42—21no—
82C22×C42—44no—
83C22×C42C2×C411yescharacteristic
84C22×C42C2×C411yescharacteristic
85C22×C42C2×C411yescharacteristic
86C22×C42D411yescharacteristic
87C22×C42D411yescharacteristic
88C22×C42D411yescharacteristic
89C22×C42D411yescharacteristic
90C22×C42D411yescharacteristic
91C22×C42D411yescharacteristic
92C22×C42C2311yescharacteristic
93C22×C42C2311yescharacteristic
94C22×C42C2311yescharacteristic
95#431—44no—
96C2×C4⋊C8—22no—
97C2×C4⋊C8—22no—
98C2×C4⋊C8—44no—
99C2×C4⋊C8C2314yes—
100C2×C22⋊C8—22no—
101C2×C22⋊C8—22no—
102C2×C22⋊C8—22no—
103C2×C22⋊C8—22no—
104C2×C22⋊C8—22no—
105C2×C22⋊C8—44no—
106C2×C4×C8C2314yes—
107#391C2×C411yescharacteristic
108#391C2×C411yescharacteristic
109#390C2×C411yescharacteristic
110#390C2×C411yescharacteristic
111#389C2×C411yescharacteristic
112#389C2×C411yescharacteristic
113#389C2×C411yescharacteristic
114#389C2×C411yescharacteristic
115#388—21no—
116#388—21no—
117#388—21no—
118#388—21no—
119#387—21no—
120#387—21no—
121#387—21no—
122#387—21no—
123#386—21no—
124#386—21no—
125#386—21no—
126#386—21no—
127#386—21no—
128#386—21no—
129#385—21no—
130#385—21no—
131#385—21no—
132#385—21no—
133#385—21no—
134#385—21no—
135#381—21no—
136#380C2×C411yescharacteristic
137#380C2×C411yescharacteristic
138#380C2×C411yescharacteristic
139#380C2×C411yescharacteristic
140#379—21no—
141C4×C4⋊C4—21no—
142C4×C4⋊C4—21no—
143C4×C4⋊C4—21no—
144C4×C4⋊C4—21no—
145C4×C4⋊C4—21no—
146C4×C4⋊C4—21no—
147C4×C4⋊C4C2×C411yescharacteristic
148C4×C4⋊C4C2×C411yescharacteristic
149C4×C4⋊C4C2311yescharacteristic
150C4×C4⋊C4C2311yescharacteristic
151C4×C22⋊C4—21no—
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—
165#376—21no—
166C43—21no—
167#336—22no—
168#336—22no—
169#336—22no—
170#336—22no—
171Q8⋊C8—44no—
172D4⋊C8—44no—
173C25C22⋊C411yescharacteristic
174C2×#41—22no—
175C2×#41—42no—
176C2×#41—42no—
177C2×#41C2×C814yescomplemented
178C22×Q8C22×C411yescharacteristic
179C22×D4—21no—
180C22×D4—21no—
181C22×D4—21no—
182C22×D4C22×C411yescharacteristic
183C23×C4—21no—
184C23×C4—22no—
185C23×C4—22no—
186C23×C4—21no—
187C23×C4—21no—
188C23×C4—42no—
189C23×C4—42no—
190C23×C4—42no—
191C23×C4—42no—
192C23×C4—42no—
193C23×C4—42no—
194C23×C4—42no—
195C23×C4C22⋊C411yescharacteristic
196C23×C4C22⋊C411yescharacteristic
197C23×C4C22⋊C411yescharacteristic
198C23×C4C22⋊C411yescharacteristic
199C23×C4C22⋊C411yescharacteristic
200C23×C4C22⋊C411yescharacteristic
201C23×C4C22⋊C411yescharacteristic
202C23×C4C22⋊C411yescharacteristic
203C23×C4C22⋊C411yescharacteristic
204C23×C4C2×C814yes—
205C23×C4M1614yes—
206C23×C4C22×C411yescharacteristic
207C23×C4C22×C411yescharacteristic
208C23×C4C22×C411yescharacteristic
209C23×C4C2×D411yescharacteristic
210C23×C4C2×D411yescharacteristic
211C23×C4C2×D411yescharacteristic
212C23×C4C2×D411yescharacteristic
213C23×C4C2×D411yescharacteristic
214C23×C4C2×D411yescharacteristic
215C22×C8—24no—
216C22×C8—24no—
217C22×C8—44no—
218C22×C8—44no—
219C22×C8—44no—
220#123—28no—
221C22⋊Q8—22no—
222C22⋊Q8—22no—
223C22⋊Q8C2×C814yes—
224C22⋊Q8C2×C814yescomplemented
225#121—22no—
226#121—22no—
227#121C2×C814yes—
228#121C2×C814yescomplemented
229#120—28no—
230C4×Q8—24no—
231C4×Q8—42no—
232C4×D4—28no—
233C4×D4—28no—
234C4×D4—24no—
235C4×D4—42no—
236C4×D4—44no—
237C4×D4—44no—
238C4×D4—44no—
239C4×D4—44no—
240C4×D4—44no—
241C4×D4—42no—
242C4×D4—42no—
243#117—22no—
244#117—44no—
245#117—42no—
246#117—42no—
247#117C2×C814yescomplemented
248C2×C4⋊C4—21no—
249C2×C4⋊C4—21no—
250C2×C4⋊C4—21no—
251C2×C4⋊C4—21no—
252C2×C4⋊C4—21no—
253C2×C4⋊C4—21no—
254C2×C4⋊C4—21no—
255C2×C4⋊C4—21no—
256C2×C4⋊C4C22×C411yescharacteristic
257C2×C4⋊C4C22×C411yescharacteristic
258C2×C4⋊C4C22×C411yescharacteristic
259C2×C4⋊C4C22×C411yescharacteristic
260C2×C4⋊C4C22×C411yescharacteristic
261C2×C4⋊C4C22×C411yescharacteristic
262C2×C22⋊C4—21no—
263C2×C22⋊C4—21no—
264C2×C22⋊C4—21no—
265C2×C22⋊C4—21no—
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×C42—21no—
280C2×C42—24no—
281C2×C42—24no—
282C2×C42—24no—
283C2×C42—21no—
284C2×C42—21no—
285C2×C42—21no—
286C2×C42—21no—
287C2×C42—21no—
288C2×C42—21no—
289C2×C42—21no—
290C2×C42—21no—
291C2×C42—42no—
292C2×C42—42no—
293C2×C42—42no—
294C2×C42—42no—
295C2×C42—42no—
296C2×C42—42no—
297C2×C42—42no—
298C2×C42—42no—
299C2×C42—42no—
300C2×C42—44no—
301C2×C42C22⋊C411yescharacteristic
302C2×C42C22⋊C411yescharacteristic
303C2×C42C22⋊C411yescharacteristic
304C2×C42C22⋊C411yescharacteristic
305C2×C42C22⋊C411yescharacteristic
306C2×C42C22⋊C411yescharacteristic
307C2×C42C22×C411yescharacteristic
308C2×C42C22×C411yescharacteristic
309C2×C42C22×C411yescharacteristic
310C2×C42C2×D411yescharacteristic
311C2×C42C2×D411yescharacteristic
312C2×C42C2×D411yescharacteristic
313C2×C42C2×D411yescharacteristic
314C2×C42C2×D411yescharacteristic
315C2×C42C2×D411yescharacteristic
316C2×C42C2411yescharacteristic
317C4⋊C8—24no—
318C4⋊C8—24no—
319C4⋊C8—44no—
320C4⋊C8—44no—
321C22⋊C8—44no—
322C22⋊C8—44no—
323C22⋊C8—44no—
324C22⋊C8—44no—
325C22⋊C8—44no—
326C4×C8—24no—
327C4×C8—24no—
328#95—21no—
329#95—21no—
330#95—21no—
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—
344C24—22no—
345C24—22no—
346C24—21no—
347C24—41no—
348C24—41no—
349C24—41no—
350C24—41no—
351C24C22⋊C814yes—
352C24C2×C22⋊C411yescharacteristic
353C24C2×C22⋊C411yescharacteristic
354C24C2×C22⋊C411yescharacteristic
355#41—48no—
356#41—48no—
357C2×Q8—21no—
358C2×Q8—22no—
359C2×Q8—22no—
360C2×Q8C22×C812yes—
361C2×D4—24no—
362C2×D4—24no—
363C2×D4—24no—
364C2×D4—22no—
365C2×D4—21no—
366C2×D4—22no—
367C2×D4—42no—
368C2×D4—42no—
369C2×D4—42no—
370C2×D4—42no—
371C2×D4—42no—
372C2×D4—42no—
373C2×D4C22×C812yes—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—21no—
383C22×C4—21no—
384C22×C4—21no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—21no—
394C22×C4—21no—
395C22×C4—21no—
396C22×C4—21no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—24no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—21no—
406C22×C4—21no—
407C22×C4—21no—
408C22×C4—21no—
409C22×C4—48no—
410C22×C4—48no—
411C22×C4—41no—
412C22×C4—41no—
413C22×C4—41no—
414C22×C4—41no—
415C22×C4—41no—
416C22×C4—41no—
417C22×C4—41no—
418C22×C4—41no—
419C22×C4—41no—
420C22×C4—41no—
421C22×C4—41no—
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—42no—
434C22×C4—48no—
435C22×C4—48no—
436C22×C4—42no—
437C22×C4—42no—
438C22×C4—42no—
439C22×C4—416no—
440C22×C4—42no—
441C22×C4—41no—
442C22×C4—41no—
443C22×C4—42no—
444C22×C4—42no—
445C22×C4C22⋊C814yes—
446C22×C4C22⋊C814yes—
447C22×C4C22⋊C814yes—
448C22×C4C2×C22⋊C411yescharacteristic
449C22×C4C2×C22⋊C411yescharacteristic
450C22×C4C2×C22⋊C411yescharacteristic
451C22×C4C2×C22⋊C411yescharacteristic
452C22×C4C2×C22⋊C411yescharacteristic
453C22×C4C2×C22⋊C411yescharacteristic
454C22×C4C2×C22⋊C411yescharacteristic
455C22×C4C2×C22⋊C411yescharacteristic
456C22×C4C2×C22⋊C411yescharacteristic
457C22×C4C22×C812yes—
458C22×C4C22×C812yes—
459C22×C4C22×C812yes—
460C22×C4C2×M1612yes—
461C22×C4C2×M1612yes—
462C22×C4C2×M1612yes—
463C22×C4AGL(1,ℤ/8)12yes—
464C22×C4#13712yes—
465C22×C4C23×C411yescharacteristic
466C22×C4C22×D411yescharacteristic
467C22×C4C22×D411yescharacteristic
468C2×C8—44no—
469C2×C8—44no—
470C2×C8—44no—
471C2×C8—44no—
472C2×C8—44no—
473C2×C8—44no—
474C2×C8—88no—
475C4⋊C4—21no—
476C4⋊C4—21no—
477C4⋊C4—24no—
478C4⋊C4—24no—
479C4⋊C4—22no—
480C4⋊C4—22no—
481C4⋊C4—22no—
482C4⋊C4—22no—
483C4⋊C4—42no—
484C4⋊C4—42no—
485C4⋊C4—42no—
486C4⋊C4—42no—
487C4⋊C4—42no—
488C4⋊C4—42no—
489C4⋊C4C22×C812yes—
490C4⋊C4C22×C812yes—
491C22⋊C4—24no—
492C22⋊C4—24no—
493C22⋊C4—24no—
494C22⋊C4—24no—
495C22⋊C4—24no—
496C22⋊C4—42no—
497C22⋊C4—42no—
498C22⋊C4—42no—
499C22⋊C4—42no—
500C22⋊C4—42no—
501C22⋊C4—42no—
502C22⋊C4—42no—
503C22⋊C4—42no—
504C22⋊C4—42no—
505C22⋊C4—42no—
506C22⋊C4—42no—
507C22⋊C4—42no—
508C42—22no—
509C42—22no—
510C42—22no—
511C42—22no—
512C42—22no—
513C42—22no—
514C42—22no—
515C42—22no—
516C42—22no—
517C42—24no—
518C42—41no—
519C42—41no—
520C42—41no—
521C42—41no—
522C42—41no—
523C42—41no—
524C42—42no—
525C42—42no—
526C42—42no—
527C42—42no—
528C42—42no—
529C42—41no—
530C42—41no—
531C42—42no—
532C42—42no—
533C42H(5,𝔽2)14yes—
534C23—21no—
535C23—21no—
536C23—21no—
537C23—21no—
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—28no—
551C23—22no—
552C23—44no—
553C23—44no—
554C23—44no—
555C23—44no—
556C23—44no—
557C23—44no—
558C23—41no—
559C23—41no—
560C23—41no—
561C23—41no—
562C23—41no—
563C23—41no—
564C23C2×C22⋊C812yes—
565C23C2×C22⋊C812yes—
566C23C2×C22⋊C812yes—
567C23#41712yes—
568C23#42112yes—
569C23C22×C22⋊C411yescharacteristic
570Q8—24no—
571Q8—42no—
572D4—24no—
573D4—48no—
574D4—48no—
575D4—48no—
576D4—42no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—21no—
589C2×C4—24no—
590C2×C4—24no—
591C2×C4—24no—
592C2×C4—24no—
593C2×C4—24no—
594C2×C4—24no—
595C2×C4—24no—
596C2×C4—24no—
597C2×C4—21no—
598C2×C4—21no—
599C2×C4—21no—
600C2×C4—21no—
601C2×C4—22no—
602C2×C4—22no—
603C2×C4—22no—
604C2×C4—21no—
605C2×C4—21no—
606C2×C4—21no—
607C2×C4—21no—
608C2×C4—21no—
609C2×C4—21no—
610C2×C4—21no—
611C2×C4—21no—
612C2×C4—44no—
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—41no—
636C2×C4—41no—
637C2×C4—41no—
638C2×C4—41no—
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—84no—
648C2×C4—82no—
649C2×C4—82no—
650C2×C4C2×C22⋊C812yes—
651C2×C4C2×C22⋊C812yes—
652C2×C4C2×C22⋊C812yes—
653C2×C4#41712yes—
654C2×C4#42112yes—
655C2×C4#51812yes—
656C2×C4#53412yes—
657C2×C4#53412yes—
658C2×C4C23×C811yescharacteristic
659C2×C4C22×M1611yescharacteristic
660C2×C4#56812yes—
661C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
662C2×C4C2×#13711yescharacteristic
663C2×C4#57512yes—
664C8—84no—
665C8—84no—
666C22—22no—
667C22—22no—
668C22—22no—
669C22—22no—
670C22—22no—
671C22—22no—
672C22—22no—
673C22—22no—
674C22—22no—
675C22—22no—
676C22—22no—
677C22—22no—
678C22—416no—
679C22—41no—
680C22—41no—
681C22—41no—
682C22—41no—
683C22—41no—
684C22—41no—
685C22—41no—
686C22—41no—
687C22—42no—
688C22—42no—
689C22—42no—
690C22—81no—
691C22—81no—
692C22—81no—
693C22—81no—
694C22#147714yes—
695C22#230811yescharacteristic
696C22C22×C22⋊C811yescharacteristic
697C22#287911yescharacteristic
698C22C2×#41711yescharacteristic
699C22#289611yescharacteristic
700C22C2×#42111yescharacteristic
701C22C2×#42111yescharacteristic
702C4—22no—
703C4—22no—
704C4—22no—
705C4—22no—
706C4—22no—
707C4—22no—
708C4—41no—
709C4—41no—
710C4—41no—
711C4—41no—
712C4—42no—
713C4—42no—
714C4—42no—
715C4—81no—
716C4—81no—
717C4—81no—
718C4—81no—
719C4#297212yes—
720C4#297312yes—
721C4#319412yes—
722C4#319612yes—
723C2—24no—
724C2—42no—
725C2—42no—
726C2—82no—
727C2C2×#147711yescharacteristic
728C2#1743711yescharacteristic
729C2#1749411yescharacteristic
730C2#1749811yescharacteristic
731C2#2173811yescharacteristic
732C2#2176911yescharacteristic
733C2#2177411yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC2×C4C23×C8
Frattini, cosocleC2×C42C24
socleC23C22×C22⋊C4
prime radicalC2×#525C4
LCSG ▷ C2×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. #20714C2semidirect
3. C2×#1797C2semidirect
4. #9074C2semidirect
5. #9073C2semidirect
6. #8939C2semidirect
7. #8937C2semidirect
8. #8708C2semidirect
9. #8312C2semidirect
47. C2×Q8⋊C8C22semidirect
48. C2×D4⋊C8C22semidirect
49. #1279C22semidirect
50. #1279C22semidirect
57. #525C8semidirect
177. C2×#41C2×C8semidirect
224. C22⋊Q8C2×C8semidirect
228. #121C2×C8semidirect
247. #117C2×C8semidirect

Maximal subgroups

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

subgroupindex
C4×#5252
#207142
C2×#17972
#90742
#90732
#89392
#89372
#87082
#83122