Finite Group Databasebeta

← #160607

Subgroups — #160607

Order 512.

1394 conjugacy classes of subgroups.

The subgroup lattice has 731 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#525C211yesmaximal, characteristic
2C2×#1817C211yesmaximal, characteristic, complemented
3C2×#1797C211yesmaximal, characteristic, complemented
4#7600C212yesmaximal, complemented
5#7599C212yesmaximal, complemented
6#7589C212yesmaximal, complemented
7#7586C212yesmaximal, complemented
8#7579C212yesmaximal, complemented
9#7578C212yesmaximal, complemented
10C2×#525C411yescharacteristic
11C2×C4×#41C411yescharacteristic
12C23×C42C2211yescharacteristic
13C22×C4⋊C8—21no—
14C22×C22⋊C8—21no—
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#2297C2211yescharacteristic
29#2296C2211yescharacteristic
30#2290—21no—
31#2290—21no—
32#2280C411yescharacteristic
33C42×D4—21no—
34C42×D4—21no—
35C4×#117C411yescharacteristic
36#1817—21no—
37#1817—21no—
38#1817C2212yes—
39#1817C2212yes—
40#1797—21no—
41#1797—21no—
42#1797C2212yes—
43#1797C2212yes—
44C2×#336—21no—
45C2×#336C2212yes—
46C2×#336C2212yes—
47#1272C2214yescomplemented
48#1271C2214yescomplemented
49#1271C2214yescomplemented
50#1270C2214yescomplemented
51C22×#41C2×C411yescharacteristic
52C24×C4C2×C411yescharacteristic
53C24×C4D411yescharacteristic
54C24×C4D411yescharacteristic
55C23×C8—22no—
56C23×C8—22no—
57#525—24no—
58#525C818yescomplemented
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×Q8C2×C411yescharacteristic
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×D4C2×C411yescharacteristic
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
96C2×C4⋊C8—22no—
97C2×C4⋊C8—22no—
98C2×C4⋊C8—44no—
99C2×C22⋊C8—22no—
100C2×C22⋊C8—22no—
101C2×C22⋊C8—22no—
102C2×C22⋊C8—22no—
103C2×C22⋊C8—44no—
104C2×C22⋊C8—44no—
105#391C2×C411yescharacteristic
106#391C2311yescharacteristic
107#390C2×C411yescharacteristic
108#390C2311yescharacteristic
109#389C2×C411yescharacteristic
110#389C2×C411yescharacteristic
111#389C2311yescharacteristic
112#389C2311yescharacteristic
113#388—21no—
114#388—21no—
115#388—21no—
116#388—21no—
117#387—21no—
118#387—21no—
119#387—21no—
120#387—21no—
121#386—21no—
122#386—21no—
123#386—21no—
124#386—21no—
125#386—21no—
126#386—21no—
127#385—21no—
128#385—21no—
129#385—21no—
130#385—21no—
131#385—21no—
132#385—21no—
133#381—21no—
134#380C2×C411yescharacteristic
135#380C2×C411yescharacteristic
136#380C2×C411yescharacteristic
137#380C2×C411yescharacteristic
138#379—21no—
139C4×C4⋊C4—21no—
140C4×C4⋊C4—21no—
141C4×C4⋊C4—21no—
142C4×C4⋊C4—21no—
143C4×C4⋊C4—21no—
144C4×C4⋊C4—21no—
145C4×C4⋊C4C2×C411yescharacteristic
146C4×C4⋊C4C2×C411yescharacteristic
147C4×C4⋊C4C2×C411yescharacteristic
148C4×C4⋊C4C2×C411yescharacteristic
149C4×C22⋊C4—21no—
150C4×C22⋊C4—21no—
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—
163#376—21no—
164C43—21no—
165#336—22no—
166#336—22no—
167#336—22no—
168#336—22no—
169#336C2314yes—
170#336C2314yes—
171C25C22⋊C411yescharacteristic
172C2×#41—22no—
173C2×#41—42no—
174C2×#41—42no—
175C2×#41C2×C814yescomplemented
176C22×Q8C22×C411yescharacteristic
177C22×D4—21no—
178C22×D4—21no—
179C22×D4—21no—
180C22×D4C22×C411yescharacteristic
181C23×C4—24no—
182C23×C4—21no—
183C23×C4—22no—
184C23×C4—22no—
185C23×C4—21no—
186C23×C4—21no—
187C23×C4—42no—
188C23×C4—42no—
189C23×C4—42no—
190C23×C4—42no—
191C23×C4—42no—
192C23×C4—42no—
193C23×C4C22⋊C411yescharacteristic
194C23×C4C22⋊C411yescharacteristic
195C23×C4C22⋊C411yescharacteristic
196C23×C4C22⋊C411yescharacteristic
197C23×C4C22⋊C411yescharacteristic
198C23×C4C22⋊C411yescharacteristic
199C23×C4C22⋊C411yescharacteristic
200C23×C4C22⋊C411yescharacteristic
201C23×C4C22⋊C411yescharacteristic
202C23×C4C2×C814yes—
203C23×C4M1614yes—
204C23×C4C22×C411yescharacteristic
205C23×C4C22×C411yescharacteristic
206C23×C4C22×C411yescharacteristic
207C23×C4C2×D411yescharacteristic
208C23×C4C2×D411yescharacteristic
209C23×C4C2×D411yescharacteristic
210C23×C4C2×D411yescharacteristic
211C23×C4C2×D411yescharacteristic
212C23×C4C2×D411yescharacteristic
213C22×C8—24no—
214C22×C8—24no—
215C22×C8—44no—
216C22×C8—44no—
217C22×C8—44no—
218C22×C8—44no—
219C22×C8—44no—
220C22×C8—44no—
221#123—28no—
222C22⋊Q8—22no—
223C22⋊Q8—22no—
224C22⋊Q8C2×C814yescomplemented
225C22⋊Q8C2×C814yescomplemented
226#121—22no—
227#121—22no—
228#121C2×C814yescomplemented
229#121C2×C814yescomplemented
230#120—28no—
231C4×Q8—42no—
232C4×Q8—42no—
233C4×D4—28no—
234C4×D4—28no—
235C4×D4—42no—
236C4×D4—42no—
237C4×D4—44no—
238C4×D4—44no—
239C4×D4—44no—
240C4×D4—44no—
241C4×D4—44no—
242C4×D4—42no—
243C4×D4—42no—
244#117—22no—
245#117—44no—
246#117—42no—
247#117—42no—
248#117C2×C814yescomplemented
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⋊C4—21no—
257C2×C4⋊C4C22×C411yescharacteristic
258C2×C4⋊C4C22×C411yescharacteristic
259C2×C4⋊C4C22×C411yescharacteristic
260C2×C4⋊C4C22×C411yescharacteristic
261C2×C4⋊C4C22×C411yescharacteristic
262C2×C4⋊C4C22×C411yescharacteristic
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×C22⋊C4—21no—
280C2×C42—24no—
281C2×C42—24no—
282C2×C42—21no—
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—44no—
300C2×C42—42no—
301C2×C42—42no—
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×C42C22×C411yescharacteristic
311C2×C42C2×D411yescharacteristic
312C2×C42C2×D411yescharacteristic
313C2×C42C2×D411yescharacteristic
314C2×C42C2×D411yescharacteristic
315C2×C42C2×D411yescharacteristic
316C2×C42C2×D411yescharacteristic
317C2×C42C2411yescharacteristic
318C4⋊C8—44no—
319C4⋊C8—44no—
320C22⋊C8—44no—
321C22⋊C8—44no—
322C22⋊C8—44no—
323C22⋊C8—44no—
324#95—21no—
325#95—21no—
326#95—21no—
327#95—21no—
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—
340C24—22no—
341C24—22no—
342C24—22no—
343C24—22no—
344C24—21no—
345C24—41no—
346C24—41no—
347C24C22⋊C814yes—
348C24C2×C22⋊C411yescharacteristic
349C24C2×C22⋊C411yescharacteristic
350C24C2×C22⋊C411yescharacteristic
351#41—48no—
352#41—48no—
353C2×Q8—21no—
354C2×Q8—41no—
355C2×Q8—41no—
356C2×Q8C22×C812yes—
357C2×D4—24no—
358C2×D4—24no—
359C2×D4—24no—
360C2×D4—21no—
361C2×D4—41no—
362C2×D4—41no—
363C2×D4—42no—
364C2×D4—42no—
365C2×D4—42no—
366C2×D4—42no—
367C2×D4—42no—
368C2×D4—42no—
369C2×D4C22×C812yes—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—21no—
377C22×C4—21no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—21no—
382C22×C4—21no—
383C22×C4—21no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—21no—
389C22×C4—21no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—24no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—21no—
401C22×C4—21no—
402C22×C4—21no—
403C22×C4—21no—
404C22×C4—41no—
405C22×C4—41no—
406C22×C4—41no—
407C22×C4—41no—
408C22×C4—41no—
409C22×C4—41no—
410C22×C4—41no—
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—48no—
421C22×C4—48no—
422C22×C4—41no—
423C22×C4—41no—
424C22×C4—41no—
425C22×C4—41no—
426C22×C4—41no—
427C22×C4—41no—
428C22×C4—42no—
429C22×C4—48no—
430C22×C4—416no—
431C22×C4—48no—
432C22×C4—42no—
433C22×C4—42no—
434C22×C4—42no—
435C22×C4—42no—
436C22×C4—42no—
437C22×C4—42no—
438C22×C4—42no—
439C22×C4C22⋊C814yes—
440C22×C4C22⋊C814yes—
441C22×C4C22⋊C814yes—
442C22×C4#9914yes—
443C22×C4C2×C22⋊C411yescharacteristic
444C22×C4C2×C22⋊C411yescharacteristic
445C22×C4C2×C22⋊C411yescharacteristic
446C22×C4C2×C22⋊C411yescharacteristic
447C22×C4C2×C22⋊C411yescharacteristic
448C22×C4C2×C22⋊C411yescharacteristic
449C22×C4C2×C22⋊C411yescharacteristic
450C22×C4C2×C22⋊C411yescharacteristic
451C22×C4C2×C22⋊C411yescharacteristic
452C22×C4C22×C812yes—
453C22×C4C22×C812yes—
454C22×C4C22×C812yes—
455C22×C4C2×M1612yes—
456C22×C4C2×M1612yes—
457C22×C4C2×M1612yes—
458C22×C4AGL(1,ℤ/8)12yes—
459C22×C4#13712yes—
460C22×C4C23×C411yescharacteristic
461C22×C4C22×D411yescharacteristic
462C22×C4C22×D411yescharacteristic
463C2×C8—44no—
464C2×C8—44no—
465C2×C8—44no—
466C2×C8—44no—
467C2×C8—44no—
468C2×C8—44no—
469C2×C8—88no—
470C2×C8—88no—
471C4⋊C4—21no—
472C4⋊C4—21no—
473C4⋊C4—24no—
474C4⋊C4—24no—
475C4⋊C4—41no—
476C4⋊C4—41no—
477C4⋊C4—41no—
478C4⋊C4—41no—
479C4⋊C4—42no—
480C4⋊C4—42no—
481C4⋊C4—41no—
482C4⋊C4—41no—
483C4⋊C4—42no—
484C4⋊C4—42no—
485C4⋊C4—42no—
486C4⋊C4C22×C812yes—
487C4⋊C4C22×C812yes—
488C22⋊C4—24no—
489C22⋊C4—24no—
490C22⋊C4—24no—
491C22⋊C4—24no—
492C22⋊C4—24no—
493C22⋊C4—42no—
494C22⋊C4—42no—
495C22⋊C4—42no—
496C22⋊C4—42no—
497C22⋊C4—42no—
498C22⋊C4—42no—
499C22⋊C4—42no—
500C22⋊C4—42no—
501C22⋊C4—41no—
502C22⋊C4—41no—
503C22⋊C4—42no—
504C22⋊C4—42no—
505C22⋊C4—42no—
506C42—24no—
507C42—22no—
508C42—22no—
509C42—22no—
510C42—22no—
511C42—22no—
512C42—22no—
513C42—22no—
514C42—22no—
515C42—41no—
516C42—41no—
517C42—41no—
518C42—41no—
519C42—41no—
520C42—41no—
521C42—41no—
522C42—41no—
523C42—42no—
524C42—42no—
525C42—42no—
526C42—42no—
527C42—42no—
528C42—42no—
529C42—42no—
530C42—42no—
531C23—21no—
532C23—21no—
533C23—21no—
534C23—21no—
535C23—21no—
536C23—21no—
537C23—21no—
538C23—21no—
539C23—21no—
540C23—21no—
541C23—21no—
542C23—21no—
543C23—21no—
544C23—21no—
545C23—28no—
546C23—28no—
547C23—22no—
548C23—44no—
549C23—44no—
550C23—44no—
551C23—44no—
552C23—44no—
553C23—41no—
554C23—41no—
555C23—41no—
556C23—41no—
557C23—41no—
558C23—41no—
559C23C2×C22⋊C812yes—
560C23C2×C22⋊C812yes—
561C23C2×C22⋊C812yes—
562C23C2×#9912yes—
563C23#41312yes—
564C23#41712yes—
565C23#42112yes—
566C23C22×C22⋊C411yescharacteristic
567Q8—42no—
568Q8—42no—
569D4—42no—
570D4—42no—
571D4—48no—
572D4—48no—
573D4—48no—
574C2×C4—24no—
575C2×C4—28no—
576C2×C4—24no—
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—21no—
590C2×C4—21no—
591C2×C4—21no—
592C2×C4—21no—
593C2×C4—21no—
594C2×C4—21no—
595C2×C4—21no—
596C2×C4—21no—
597C2×C4—28no—
598C2×C4—24no—
599C2×C4—24no—
600C2×C4—22no—
601C2×C4—22no—
602C2×C4—22no—
603C2×C4—21no—
604C2×C4—21no—
605C2×C4—21no—
606C2×C4—21no—
607C2×C4—21no—
608C2×C4—21no—
609C2×C4—44no—
610C2×C4—44no—
611C2×C4—44no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—44no—
619C2×C4—44no—
620C2×C4—44no—
621C2×C4—44no—
622C2×C4—44no—
623C2×C4—44no—
624C2×C4—44no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—48no—
628C2×C4—41no—
629C2×C4—41no—
630C2×C4—41no—
631C2×C4—41no—
632C2×C4—41no—
633C2×C4—41no—
634C2×C4—41no—
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—84no—
647C2×C4—84no—
648C2×C4C2×C22⋊C812yes—
649C2×C4C2×C22⋊C812yes—
650C2×C4C2×C22⋊C812yes—
651C2×C4C2×#9912yes—
652C2×C4C2×#9912yes—
653C2×C4#41712yes—
654C2×C4#42112yes—
655C2×C4C23×C811yescharacteristic
656C2×C4C22×M1611yescharacteristic
657C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
658C2×C4C2×#13711yescharacteristic
659C2×C4#57512yes—
660C8—84no—
661C8—84no—
662C22—22no—
663C22—22no—
664C22—22no—
665C22—22no—
666C22—22no—
667C22—22no—
668C22—22no—
669C22—22no—
670C22—22no—
671C22—22no—
672C22—22no—
673C22—22no—
674C22—22no—
675C22—22no—
676C22—22no—
677C22—416no—
678C22—41no—
679C22—41no—
680C22—41no—
681C22—41no—
682C22—42no—
683C22—42no—
684C22—42no—
685C22—81no—
686C22—81no—
687C22—81no—
688C22—81no—
689C22#145914yes—
690C22#147714yes—
691C22C22×C22⋊C811yescharacteristic
692C22C22×#9911yescharacteristic
693C22C2×#41311yescharacteristic
694C22C2×#41711yescharacteristic
695C22#289611yescharacteristic
696C22C2×#42111yescharacteristic
697C22C2×#42111yescharacteristic
698C4—22no—
699C4—22no—
700C4—22no—
701C4—21no—
702C4—21no—
703C4—22no—
704C4—22no—
705C4—22no—
706C4—21no—
707C4—21no—
708C4—22no—
709C4—22no—
710C4—42no—
711C4—42no—
712C4—41no—
713C4—41no—
714C4—42no—
715C4—81no—
716C4—81no—
717C4—81no—
718C4—81no—
719C2—24no—
720C2—24no—
721C2—42no—
722C2—82no—
723C2C2×#145911yescharacteristic
724C2C2×#147711yescharacteristic
725C2#1743711yescharacteristic
726C2#1758711yescharacteristic
727C2#1759211yescharacteristic
728C2#1767711yescharacteristic
729C2#1775311yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC2×C4C23×C8
Frattini, cosocleC2×C42C24
prime radicalC2×#525C4
LCSG ▷ C2×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C23×C42 ◁ G
LECSG ▷ C2×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#1817C2semidirect
3. C2×#1797C2semidirect
4. #7600C2semidirect
5. #7599C2semidirect
6. #7589C2semidirect
7. #7586C2semidirect
8. #7579C2semidirect
9. #7578C2semidirect
47. #1272C22semidirect
48. #1271C22semidirect
49. #1271C22semidirect
50. #1270C22semidirect
58. #525C8semidirect
175. C2×#41C2×C8semidirect
224. C22⋊Q8C2×C8semidirect
225. C22⋊Q8C2×C8semidirect
228. #121C2×C8semidirect
229. #121C2×C8semidirect
248. #117C2×C8semidirect

Maximal subgroups

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

subgroupindex
C4×#5252
C2×#18172
C2×#17972
#76002
#75992
#75892
#75862
#75792
#75782