Finite Group Databasebeta

← #161054

Subgroups — #161054

Order 512.

1348 conjugacy classes of subgroups.

The subgroup lattice has 730 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#525C211yesmaximal, characteristic
2#20715C211yesmaximal, characteristic, complemented
3C2×#1797C211yesmaximal, characteristic, complemented
4#7648C212yesmaximal, complemented
5#7647C212yesmaximal, complemented
6#7613C212yesmaximal, complemented
7#7612C212yesmaximal, complemented
8#7589C212yesmaximal, complemented
9#7586C212yesmaximal, complemented
10C2×#525C411yescharacteristic
11C2×C4×#41C411yescharacteristic
12C23×C42C2211yescharacteristic
13C2×#432—21no—
14C22×C4⋊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#1817C2212yes—
37#1797—21no—
38#1797—21no—
39#1797—21no—
40#1797C2212yes—
41#1797C2212yes—
42#1797C2212yes—
43#1779—21no—
44#1749—21no—
45#1749C2212yes—
46C2×#336C2212yes—
47#1272C2214yescomplemented
48#1271C2214yescomplemented
49#1271C2214yescomplemented
50#1270C2214yescomplemented
51C22×#41C2×C411yescharacteristic
52C24×C4C2×C411yescharacteristic
53C24×C4D411yescharacteristic
54C24×C4D411yescharacteristic
55C23×C8—22no—
56#525—24no—
57#525—24no—
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×Q8C2×C411yescharacteristic
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×D4C2×C411yescharacteristic
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#432—44no—
96C2×C4⋊C8—22no—
97C2×C4⋊C8—22no—
98C2×C4⋊C8—22no—
99C2×C4⋊C8—44no—
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×#97—22no—
107#391C2×C411yescharacteristic
108#391C2311yescharacteristic
109#390C2×C411yescharacteristic
110#390C2311yescharacteristic
111#389C2×C411yescharacteristic
112#389C2×C411yescharacteristic
113#389C2311yescharacteristic
114#389C2311yescharacteristic
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⋊C4C2×C411yescharacteristic
150C4×C4⋊C4C2×C411yescharacteristic
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#336C2314yes—
170#336C2314yes—
171C25C22⋊C411yescharacteristic
172C2×#41—22no—
173C2×#41—22no—
174C2×#41—42no—
175C2×#41—42no—
176C22×Q8C22×C411yescharacteristic
177C22×D4—21no—
178C22×D4—21no—
179C22×D4—21no—
180C22×D4C22×C411yescharacteristic
181C23×C4—21no—
182C23×C4—22no—
183C23×C4—22no—
184C23×C4—24no—
185C23×C4—22no—
186C23×C4—22no—
187C23×C4—21no—
188C23×C4—21no—
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×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—
218#123—28no—
219C22⋊Q8—22no—
220C22⋊Q8—22no—
221C22⋊Q8—22no—
222C22⋊Q8M1614yes—
223#121—22no—
224#121—22no—
225#121—22no—
226#121M1614yes—
227#120—28no—
228C4×Q8—42no—
229C4×Q8—42no—
230C4×D4—28no—
231C4×D4—28no—
232C4×D4—42no—
233C4×D4—42no—
234C4×D4—44no—
235C4×D4—44no—
236C4×D4—44no—
237C4×D4—44no—
238C4×D4—44no—
239C4×D4—42no—
240C4×D4—42no—
241#117—22no—
242#117—22no—
243#117—42no—
244#117—44no—
245#117—42no—
246C2×C4⋊C4—21no—
247C2×C4⋊C4—21no—
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⋊C4C22×C411yescharacteristic
255C2×C4⋊C4C22×C411yescharacteristic
256C2×C4⋊C4C22×C411yescharacteristic
257C2×C4⋊C4C22×C411yescharacteristic
258C2×C4⋊C4C22×C411yescharacteristic
259C2×C4⋊C4C22×C411yescharacteristic
260C2×C22⋊C4—21no—
261C2×C22⋊C4—21no—
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×C42—21no—
278C2×C42—24no—
279C2×C42—24no—
280C2×C42—24no—
281C2×C42—21no—
282C2×C42—21no—
283C2×C42—21no—
284C2×C42—21no—
285C2×C42—21no—
286C2×C42—21no—
287C2×C42—21no—
288C2×C42—21no—
289C2×C42—42no—
290C2×C42—42no—
291C2×C42—42no—
292C2×C42—42no—
293C2×C42—42no—
294C2×C42—42no—
295C2×C42—42no—
296C2×C42—42no—
297C2×C42—42no—
298C2×C42—44no—
299C2×C42C22⋊C411yescharacteristic
300C2×C42C22⋊C411yescharacteristic
301C2×C42C22⋊C411yescharacteristic
302C2×C42C22⋊C411yescharacteristic
303C2×C42C22⋊C411yescharacteristic
304C2×C42C22⋊C411yescharacteristic
305C2×C42C22×C411yescharacteristic
306C2×C42C22×C411yescharacteristic
307C2×C42C22×C411yescharacteristic
308C2×C42C2×D411yescharacteristic
309C2×C42C2×D411yescharacteristic
310C2×C42C2×D411yescharacteristic
311C2×C42C2×D411yescharacteristic
312C2×C42C2×D411yescharacteristic
313C2×C42C2×D411yescharacteristic
314C2×C42C2411yescharacteristic
315C4⋊C8—44no—
316C4⋊C8—44no—
317C4⋊C8—44no—
318C22⋊C8—44no—
319C22⋊C8—44no—
320C22⋊C8—44no—
321C22⋊C8—44no—
322C22⋊C8—44no—
323#97—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—21no—
344C24—22no—
345C24—22no—
346C24—41no—
347C24—41no—
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×Q8C2×M1612yes—
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×D4C2×M1612yes—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—21no—
374C22×C4—21no—
375C22×C4—21no—
376C22×C4—21no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—24no—
382C22×C4—21no—
383C22×C4—21no—
384C22×C4—21no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—21no—
390C22×C4—21no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—21no—
394C22×C4—21no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—21no—
410C22×C4—21no—
411C22×C4—21no—
412C22×C4—21no—
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—48no—
434C22×C4—416no—
435C22×C4—48no—
436C22×C4—48no—
437C22×C4—48no—
438C22×C4—42no—
439C22×C4—42no—
440C22×C4—42no—
441C22×C4—42no—
442C22×C4—42no—
443C22×C4—42no—
444C22×C4—42no—
445C22×C4—42no—
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×C4C2×C22⋊C411yescharacteristic
453C22×C4C2×C22⋊C411yescharacteristic
454C22×C4C2×C22⋊C411yescharacteristic
455C22×C4C2×M1612yes—
456C22×C4C2×M1612yes—
457C22×C4#13514yes—
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—
470C4⋊C4—24no—
471C4⋊C4—21no—
472C4⋊C4—21no—
473C4⋊C4—24no—
474C4⋊C4—41no—
475C4⋊C4—41no—
476C4⋊C4—41no—
477C4⋊C4—41no—
478C4⋊C4—42no—
479C4⋊C4—42no—
480C4⋊C4—41no—
481C4⋊C4—41no—
482C4⋊C4—42no—
483C4⋊C4—42no—
484C4⋊C4—42no—
485C4⋊C4C2×M1612yes—
486C4⋊C4C2×M1612yes—
487C22⋊C4—24no—
488C22⋊C4—24no—
489C22⋊C4—24no—
490C22⋊C4—24no—
491C22⋊C4—24no—
492C22⋊C4—42no—
493C22⋊C4—42no—
494C22⋊C4—42no—
495C22⋊C4—42no—
496C22⋊C4—42no—
497C22⋊C4—42no—
498C22⋊C4—42no—
499C22⋊C4—42no—
500C22⋊C4—41no—
501C22⋊C4—41no—
502C22⋊C4—42no—
503C22⋊C4—42no—
504C22⋊C4—42no—
505C42—24no—
506C42—22no—
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—42no—
522C42—42no—
523C42—42no—
524C42—42no—
525C42—42no—
526C42—42no—
527C42—42no—
528C42—42no—
529C42#9914yes—
530C23—21no—
531C23—21no—
532C23—21no—
533C23—21no—
534C23—28no—
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—22no—
548C23—44no—
549C23—44no—
550C23—44no—
551C23—44no—
552C23—44no—
553C23—44no—
554C23—41no—
555C23—41no—
556C23—41no—
557C23—41no—
558C23—41no—
559C23—41no—
560C23#40712yes—
561C23#41312yes—
562C23#41612yes—
563C23#41712yes—
564C23#42112yes—
565C23C22×C22⋊C411yescharacteristic
566Q8—42no—
567Q8—42no—
568D4—42no—
569D4—42no—
570D4—48no—
571D4—48no—
572D4—48no—
573C2×C4—24no—
574C2×C4—28no—
575C2×C4—24no—
576C2×C4—24no—
577C2×C4—24no—
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—21no—
598C2×C4—22no—
599C2×C4—22no—
600C2×C4—21no—
601C2×C4—22no—
602C2×C4—21no—
603C2×C4—21no—
604C2×C4—21no—
605C2×C4—21no—
606C2×C4—21no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—44no—
610C2×C4—42no—
611C2×C4—42no—
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—42no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—48no—
627C2×C4—41no—
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—84no—
646C2×C4—84no—
647C2×C4#40712yes—
648C2×C4C2×#9912yes—
649C2×C4C2×#9912yes—
650C2×C4C2×#9912yes—
651C2×C4#41612yes—
652C2×C4#41712yes—
653C2×C4#42112yes—
654C2×C4C22×M1611yescharacteristic
655C2×C4C22×M1611yescharacteristic
656C2×C4C2×#13512yes—
657C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
658C2×C4C2×#13711yescharacteristic
659C8—84no—
660C8—84no—
661C22—22no—
662C22—22no—
663C22—22no—
664C22—22no—
665C22—22no—
666C22—22no—
667C22—22no—
668C22—22no—
669C22—22no—
670C22—22no—
671C22—22no—
672C22—22no—
673C22—416no—
674C22—41no—
675C22—41no—
676C22—41no—
677C22—41no—
678C22—41no—
679C22—41no—
680C22—41no—
681C22—41no—
682C22—42no—
683C22—42no—
684C22—42no—
685C22—81no—
686C22—81no—
687C22—81no—
688C22—81no—
689C22#148814yes—
690C22C2×#40711yescharacteristic
691C22C22×#9911yescharacteristic
692C22C2×#41311yescharacteristic
693C22C2×#41611yescharacteristic
694C22C2×#41711yescharacteristic
695C22C2×#42111yescharacteristic
696C22C2×#42111yescharacteristic
697C4—22no—
698C4—22no—
699C4—22no—
700C4—22no—
701C4—21no—
702C4—21no—
703C4—22no—
704C4—22no—
705C4—22no—
706C4—21no—
707C4—21no—
708C4—22no—
709C4—42no—
710C4—42no—
711C4—41no—
712C4—41no—
713C4—42no—
714C4—81no—
715C4—81no—
716C4—81no—
717C4—81no—
718C2—24no—
719C2—42no—
720C2—42no—
721C2—82no—
722C2#1736211yescharacteristic
723C2C2×#148811yescharacteristic
724C2#1745111yescharacteristic
725C2#1757211yescharacteristic
726C2#1759211yescharacteristic
727C2#1767711yescharacteristic
728C2#1771911yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC2×#525C4
LCSG ▷ C22×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. #20715C2semidirect
3. C2×#1797C2semidirect
4. #7648C2semidirect
5. #7647C2semidirect
6. #7613C2semidirect
7. #7612C2semidirect
8. #7589C2semidirect
9. #7586C2semidirect
47. #1272C22semidirect
48. #1271C22semidirect
49. #1271C22semidirect
50. #1270C22semidirect

Maximal subgroups

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

subgroupindex
C4×#5252
#207152
C2×#17972
#76482
#76472
#76132
#76122
#75892
#75862