Finite Group Databasebeta

← #6433830

Subgroups — #6433830

Order 512.

2112 conjugacy classes of subgroups.

The subgroup lattice has 721 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36625C211yesmaximal, characteristic
2C2×#2857C211yesmaximal, characteristic
3#30298C212yesmaximal, complemented
4#30296C212yesmaximal, complemented
5#30295C214yesmaximal, complemented
6#30294C214yesmaximal, complemented
7#30293C212yesmaximal, complemented
8#30291C212yesmaximal, complemented
9#23658C212yesmaximal, complemented
10#23657C212yesmaximal, complemented
11#23554C212yesmaximal, complemented
12#23552C212yesmaximal, complemented
13#20661C214yesmaximal
14C2×#1728C211yesmaximal, characteristic
15C23×M16C2212yes—
16C2×#533C2211yescharacteristic
17C22×C4⋊Q8C2211yescharacteristic
18C22×C22⋊Q8—21no—
19C23×C42C2211yescharacteristic
20C2×#428C2214yes—
21C2×#428C2214yes—
22#2857—21no—
23#2857—21no—
24#2857—21no—
25#2857—21no—
26#2857C2212yes—
27#2857C2212yes—
28#2857C2212yes—
29#2857C2212yes—
30#2857C2211yescharacteristic
31#2857C2211yescharacteristic
32#2857C2211yescharacteristic
33#2857C2211yescharacteristic
34#2855—21no—
35#2855—21no—
36#2851C2212yes—
37#2851C2212yes—
38#2848—21no—
39#2848—21no—
40C4×C22⋊Q8—21no—
41C4×C22⋊Q8—21no—
42C4×C22⋊Q8—21no—
43C4×C22⋊Q8—21no—
44#2296C2212yes—
45C2×#391—21no—
46C2×#389—21no—
47C2×#389C2212yes—
48#2285C2211yescharacteristic
49C2×#384C2211yescharacteristic
50#2280C2212yes—
51C2×#380—21no—
52C2×#380C2212yes—
53#2085C2218yescomplemented
54#2084C2214yescomplemented
55#2084C2214yescomplemented
56#2084C2214yescomplemented
57#2083C2214yescomplemented
58#2083C2214yescomplemented
59#2083C2214yescomplemented
60#2059C2218yescomplemented
61#2058C2214yescomplemented
62#2058C2214yescomplemented
63#2058C2214yescomplemented
64#2057C2214yescomplemented
65#2057C2214yescomplemented
66#2057C2214yescomplemented
67#1809C2218yes—
68#1809C2218yes—
69#1808C2218yes—
70#1807C2218yes—
71#1736C2214yes—
72#1736C2214yes—
73#1728C2212yes—
74#1728C2212yes—
75#1728C2212yes—
76#1728C2212yes—
77C2×#336C2212yes—
78C2×#336C2212yes—
79C23×Q8—21no—
80C24×C4D411yescharacteristic
81C24×C4D411yescharacteristic
82C24×C4C2311yescharacteristic
83C22×M16—44no—
84C22×M16—44no—
85C22×M16Q8116yescomplemented
86C22×M16C2314yes—
87C22×M16C2314yes—
88C23×C8C2314yes—
89#533—44no—
90C2×C4⋊Q8—21no—
91C2×C4⋊Q8—21no—
92C2×C4⋊Q8—44no—
93C2×C4⋊Q8C2311yescharacteristic
94C2×C4⋊Q8C2312yes—
95C2×C4⋊Q8C2312yes—
96C2×C4⋊Q8C2311yescharacteristic
97C2×#125C2312yes—
98C2×C22⋊Q8—21no—
99C2×C22⋊Q8—21no—
100C2×C22⋊Q8—21no—
101C2×C22⋊Q8—21no—
102C2×C22⋊Q8—21no—
103C2×C22⋊Q8—21no—
104C2×C22⋊Q8—22no—
105C2×C22⋊Q8—22no—
106C2×C22⋊Q8—22no—
107C2×C22⋊Q8—22no—
108C2×C22⋊Q8—48no—
109C2×C4×Q8—21no—
110C2×C4×Q8—21no—
111C2×#117C2312yes—
112C2×#117C2312yes—
113C22×C4⋊C4—21no—
114C22×C4⋊C4—22no—
115C22×C4⋊C4—21no—
116C22×C4⋊C4—21no—
117C22×C4⋊C4C2312yes—
118C22×C4⋊C4C2312yes—
119C22×C22⋊C4—22no—
120C22×C42—44no—
121C22×C42D411yescharacteristic
122C22×C42D411yescharacteristic
123C22×C42D411yescharacteristic
124C22×C42D411yescharacteristic
125C22×C42D411yescharacteristic
126C22×C42D411yescharacteristic
127C22×C42C2311yescharacteristic
128C22×C42C2311yescharacteristic
129C22×C42C2311yescharacteristic
130C22×C42C2311yescharacteristic
131C22×C42C2311yescharacteristic
132C22×C42C2311yescharacteristic
133#428—48no—
134#428—48no—
135C2×#107C2318yes—
136C2×#107C2318yes—
137C2×#106C2318yes—
138C2×#106C2318yes—
139#391—21no—
140#391—21no—
141#391—21no—
142#391—21no—
143#391—22no—
144#391—22no—
145#391—22no—
146#391—22no—
147#389—21no—
148#389—21no—
149#389—21no—
150#389—21no—
151#389—22no—
152#389—22no—
153#389—22no—
154#389—22no—
155#389—22no—
156#389—22no—
157#389C2312yes—
158#389C2312yes—
159#389C2314yes—
160#389C2312yes—
161#389C2312yes—
162#389C2314yes—
163#387—22no—
164#387—22no—
165#387—22no—
166#387—22no—
167#385—22no—
168#385—22no—
169#385—22no—
170#385—22no—
171#385—22no—
172#385—22no—
173#384—21no—
174#384—21no—
175#384C2311yescharacteristic
176#384C2312yes—
177#384C2312yes—
178#384C2311yescharacteristic
179#382C2312yes—
180#381—22no—
181#380—21no—
182#380—21no—
183#380—21no—
184#380—21no—
185#380—21no—
186#380—21no—
187#380—22no—
188#380—22no—
189#380—22no—
190#380—22no—
191#380—22no—
192#380—22no—
193#380C2312yes—
194#380C2314yes—
195#380C2312yes—
196#380C2312yes—
197#380C2312yes—
198#380C2314yes—
199#379—22no—
200C4×C4⋊C4—21no—
201C4×C4⋊C4—21no—
202C4×C4⋊C4—22no—
203C4×C4⋊C4—22no—
204C4×C22⋊C4—22no—
205C4×C22⋊C4—22no—
206C2×#95—22no—
207C2×#95—22no—
208#340C2318yes—
209#340C2318yes—
210#340C2318yes—
211#340C2318yes—
212#336C2314yes—
213#336C2314yes—
214#336C2314yes—
215#336C2314yes—
216C25C2×D411yescharacteristic
217C22×Q8—21no—
218C22×Q8—21no—
219C22×Q8—22no—
220C22×Q8—22no—
221C22×Q8—44no—
222C22×Q8—44no—
223C22×Q8—44no—
224C23×C4—22no—
225C23×C4—24no—
226C23×C4—22no—
227C23×C4—24no—
228C23×C4—24no—
229C23×C4—22no—
230C23×C4—22no—
231C23×C4—42no—
232C23×C4—42no—
233C23×C4—42no—
234C23×C4—42no—
235C23×C4C2×D411yescharacteristic
236C23×C4C2×D411yescharacteristic
237C23×C4C2×D411yescharacteristic
238C23×C4C2×D411yescharacteristic
239C23×C4C2×D411yescharacteristic
240C23×C4C2×D411yescharacteristic
241C23×C4C2×D411yescharacteristic
242C23×C4C2×D411yescharacteristic
243C23×C4C2×D411yescharacteristic
244C23×C4C2×D411yescharacteristic
245C23×C4C2×D411yescharacteristic
246C23×C4C2×D411yescharacteristic
247C23×C4C2×D411yescharacteristic
248C23×C4C2×D411yescharacteristic
249C23×C4C2×D411yescharacteristic
250C23×C4C2×Q818yes—
251C23×C4C2411yescharacteristic
252C23×C4C2411yescharacteristic
253C23×C4C2411yescharacteristic
254C2×M16—28no—
255C2×M16—28no—
256C2×M16—44no—
257C2×M16—44no—
258C2×M16—432no—
259C2×M16C2×Q8116yes—
260C2×M16C2×Q8116yescomplemented
261C22×C8—44no—
262C22×C8—44no—
263C22×C8C2×Q8116yescomplemented
264C22×C8C2418yes—
265C4⋊Q8—44no—
266C4⋊Q8—44no—
267C4⋊Q8—42no—
268C4⋊Q8—42no—
269C4⋊Q8—44no—
270C4⋊Q8—44no—
271#125—44no—
272C22⋊Q8—44no—
273C22⋊Q8—44no—
274C22⋊Q8—48no—
275C22⋊Q8—48no—
276C22⋊Q8—48no—
277C22⋊Q8—44no—
278C22⋊Q8—44no—
279C22⋊Q8—44no—
280C22⋊Q8—48no—
281C22⋊Q8—44no—
282C4×Q8—44no—
283C4×Q8—44no—
284#117—28no—
285#117—24no—
286#117—24no—
287C2×C4⋊C4—28no—
288C2×C4⋊C4—24no—
289C2×C4⋊C4—24no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—24no—
293C2×C4⋊C4—24no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—21no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—44no—
313C2×C4⋊C4—44no—
314C2×C4⋊C4—48no—
315C2×C4⋊C4—44no—
316C2×C4⋊C4C2414yes—
317C2×C4⋊C4C2414yes—
318C2×C4⋊C4C2412yes—
319C2×C4⋊C4C2412yes—
320C2×C4⋊C4C2412yes—
321C2×C4⋊C4C2412yes—
322C2×C22⋊C4—24no—
323C2×C22⋊C4—24no—
324C2×C22⋊C4—22no—
325C2×C22⋊C4—22no—
326C2×C22⋊C4—22no—
327C2×C22⋊C4—22no—
328C2×C22⋊C4—22no—
329C2×C22⋊C4—22no—
330C2×C22⋊C4—48no—
331C2×C42—24no—
332C2×C42—24no—
333C2×C42—22no—
334C2×C42—22no—
335C2×C42—42no—
336C2×C42—42no—
337C2×C42—42no—
338C2×C42—42no—
339C2×C42—42no—
340C2×C42—42no—
341C2×C42—42no—
342C2×C42—42no—
343C2×C42—42no—
344C2×C42—42no—
345C2×C42C2×D411yescharacteristic
346C2×C42C2×D411yescharacteristic
347C2×C42C2×D411yescharacteristic
348C2×C42C2×D411yescharacteristic
349C2×C42C2×D411yescharacteristic
350C2×C42C2×D411yescharacteristic
351C2×C42C2×D411yescharacteristic
352C2×C42C2×D411yescharacteristic
353C2×C42C2×D411yescharacteristic
354C2×C42C2×D411yescharacteristic
355C2×C42C2×D411yescharacteristic
356C2×C42C2×D411yescharacteristic
357C2×C42C2411yescharacteristic
358C2×C42C2411yescharacteristic
359C2×C42C2411yescharacteristic
360C2×C42C2411yescharacteristic
361#107—48no—
362#107—48no—
363#106—48no—
364#106—48no—
365#95—24no—
366#95—24no—
367#95—24no—
368#95—24no—
369#95—22no—
370#95—22no—
371#95—22no—
372#95—22no—
373#95—22no—
374#95—22no—
375#95—22no—
376#95—22no—
377#95—22no—
378#95—22no—
379C24—22no—
380C24—22no—
381C24—22no—
382C24—22no—
383C24—41no—
384C24—41no—
385C24C4⋊Q814yes—
386C24C22×D411yescharacteristic
387C24C22×D411yescharacteristic
388C24C22×D411yescharacteristic
389C2×Q8—42no—
390C2×Q8—42no—
391C2×Q8—44no—
392C2×Q8—42no—
393C2×Q8—42no—
394C2×Q8—44no—
395C2×Q8—44no—
396C2×Q8—44no—
397C2×Q8—44no—
398C2×Q8—42no—
399C2×Q8—42no—
400C2×Q8—44no—
401C2×Q8—816no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—216no—
411C22×C4—216no—
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—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—24no—
441C22×C4—24no—
442C22×C4—41no—
443C22×C4—41no—
444C22×C4—41no—
445C22×C4—41no—
446C22×C4—41no—
447C22×C4—41no—
448C22×C4—41no—
449C22×C4—41no—
450C22×C4—41no—
451C22×C4—41no—
452C22×C4—41no—
453C22×C4—41no—
454C22×C4—41no—
455C22×C4—41no—
456C22×C4—416no—
457C22×C4—416no—
458C22×C4—44no—
459C22×C4—44no—
460C22×C4—44no—
461C22×C4—44no—
462C22×C4—44no—
463C22×C4—44no—
464C22×C4C4⋊Q814yes—
465C22×C4C4⋊Q814yes—
466C22×C4C4⋊Q814yes—
467C22×C4AGL(1,ℤ/8)14yes—
468C22×C4#13714yes—
469C22×C4C22×D411yescharacteristic
470C22×C4C22×D411yescharacteristic
471C22×C4C22×D411yescharacteristic
472C22×C4C22×D411yescharacteristic
473C22×C4C22×D411yescharacteristic
474C22×C4C22×D411yescharacteristic
475C22×C4C22×D411yescharacteristic
476C22×C4C22×D411yescharacteristic
477C22×C4C22×D411yescharacteristic
478C22×C4C22×D411yescharacteristic
479C22×C4C22×D411yescharacteristic
480C22×C4C22×Q814yes—
481C22×C4C22×Q814yes—
482C22×C4C22×Q814yes—
483C22×C4H(5,𝔽2)14yes—
484C22×C4H(5,𝔽2)12yes—
485C22×C42−1+412yes—
486C22×C4C2511yescharacteristic
487M16—232no—
488M16—416no—
489C2×C8—28no—
490C2×C8—28no—
491C2×C8—416no—
492C2×C8C22×Q8116yes—
493C4⋊C4—22no—
494C4⋊C4—24no—
495C4⋊C4—24no—
496C4⋊C4—28no—
497C4⋊C4—22no—
498C4⋊C4—24no—
499C4⋊C4—24no—
500C4⋊C4—22no—
501C4⋊C4—22no—
502C4⋊C4—44no—
503C4⋊C4—44no—
504C4⋊C4—48no—
505C4⋊C4—44no—
506C4⋊C4—44no—
507C4⋊C4—44no—
508C4⋊C4—44no—
509C4⋊C4—42no—
510C4⋊C4—42no—
511C4⋊C4—42no—
512C4⋊C4—42no—
513C4⋊C4—44no—
514C4⋊C4—44no—
515C4⋊C4—44no—
516C4⋊C4—48no—
517C4⋊C4—42no—
518C4⋊C4—44no—
519C4⋊C4—42no—
520C4⋊C4—44no—
521C4⋊C4—44no—
522C4⋊C4—44no—
523C4⋊C4—44no—
524C22⋊C4—48no—
525C22⋊C4—44no—
526C22⋊C4—44no—
527C22⋊C4—44no—
528C22⋊C4—44no—
529C22⋊C4—48no—
530C22⋊C4—44no—
531C22⋊C4—44no—
532C42—22no—
533C42—22no—
534C42—22no—
535C42—22no—
536C42—22no—
537C42—22no—
538C42—22no—
539C42—22no—
540C42—41no—
541C42—41no—
542C42—41no—
543C42—41no—
544C42—41no—
545C42—41no—
546C42—41no—
547C42—41no—
548C42—44no—
549C42—44no—
550C23—21no—
551C23—21no—
552C23—21no—
553C23—21no—
554C23—21no—
555C23—21no—
556C23—21no—
557C23—21no—
558C23—28no—
559C23—28no—
560C23—21no—
561C23—21no—
562C23—21no—
563C23—21no—
564C23—21no—
565C23—21no—
566C23—44no—
567C23—44no—
568C23—44no—
569C23—44no—
570C23—44no—
571C23C2×C4⋊Q812yes—
572C23C2×C4⋊Q812yes—
573C23C2×C4⋊Q812yes—
574C23#53412yes—
575C23#53712yes—
576C23C2×AGL(1,ℤ/8)12yes—
577C23C2×#13712yes—
578C23C23×D411yescharacteristic
579Q8—84no—
580Q8—84no—
581Q8—88no—
582Q8—88no—
583C2×C4—28no—
584C2×C4—28no—
585C2×C4—28no—
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—21no—
599C2×C4—21no—
600C2×C4—21no—
601C2×C4—21no—
602C2×C4—28no—
603C2×C4—28no—
604C2×C4—24no—
605C2×C4—24no—
606C2×C4—24no—
607C2×C4—24no—
608C2×C4—21no—
609C2×C4—21no—
610C2×C4—21no—
611C2×C4—21no—
612C2×C4—21no—
613C2×C4—21no—
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—44no—
623C2×C4—44no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—44no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—44no—
630C2×C4—44no—
631C2×C4—44no—
632C2×C4—42no—
633C2×C4—42no—
634C2×C4—42no—
635C2×C4—42no—
636C2×C4—42no—
637C2×C4—42no—
638C2×C4—42no—
639C2×C4—42no—
640C2×C4—44no—
641C2×C4—44no—
642C2×C4—88no—
643C2×C4—88no—
644C2×C4C2×C4⋊Q812yes—
645C2×C4C2×C4⋊Q812yes—
646C2×C4C2×C4⋊Q812yes—
647C2×C4#53412yes—
648C2×C4#53712yes—
649C2×C4#54318yes—
650C2×C4#54418yes—
651C2×C4C2×AGL(1,ℤ/8)12yes—
652C2×C4C2×AGL(1,ℤ/8)12yes—
653C2×C4C2×#13712yes—
654C2×C4C2×#13712yes—
655C2×C4#57512yes—
656C2×C4#57512yes—
657C2×C4C23×Q812yes—
658C2×C4C2×H(5,𝔽2)11yescharacteristic
659C2×C4C2×H(5,𝔽2)12yes—
660C2×C4C2×2−1+411yescharacteristic
661C8—216no—
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#265914yes—
683C22#316514yes—
684C22C22×C4⋊Q811yescharacteristic
685C22C2×#53411yescharacteristic
686C22C2×#53711yescharacteristic
687C22C22×AGL(1,ℤ/8)11yescharacteristic
688C22C22×#13711yescharacteristic
689C22C2×#57511yescharacteristic
690C22C2×#57511yescharacteristic
691C4—22no—
692C4—22no—
693C4—22no—
694C4—22no—
695C4—22no—
696C4—22no—
697C4—22no—
698C4—22no—
699C4—41no—
700C4—41no—
701C4—84no—
702C4—82no—
703C4—82no—
704C4—82no—
705C4—84no—
706C4—82no—
707C4#265914yes—
708C4C2×#54312yes—
709C4C2×#54412yes—
710C2—24no—
711C2—24no—
712C2—42no—
713C2C2×#265911yescharacteristic
714C2C2×#316511yescharacteristic
715C2#6066311yescharacteristic
716C2#6075011yescharacteristic
717C2#6075111yescharacteristic
718C2#6091611yescharacteristic
719C2#6091811yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C42 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #30298C2semidirect
4. #30296C2semidirect
5. #30295C2semidirect
6. #30294C2semidirect
7. #30293C2semidirect
8. #30291C2semidirect
9. #23658C2semidirect
10. #23657C2semidirect
11. #23554C2semidirect
12. #23552C2semidirect
53. #2085C22semidirect
54. #2084C22semidirect
55. #2084C22semidirect
56. #2084C22semidirect
57. #2083C22semidirect
58. #2083C22semidirect
59. #2083C22semidirect
60. #2059C22semidirect
61. #2058C22semidirect
62. #2058C22semidirect
63. #2058C22semidirect
64. #2057C22semidirect
65. #2057C22semidirect
66. #2057C22semidirect
85. C22×M16Q8semidirect
260. C2×M16C2×Q8semidirect
263. C22×C8C2×Q8semidirect

Maximal subgroups

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

subgroupindex
#366252
C2×#28572
#302982
#302962
#302952
#302942
#302932
#302912
#236582
#236572
#235542
#235522
#206612
C2×#17282