Finite Group Databasebeta

← #185024

Subgroups — #185024

Order 512.

3792 conjugacy classes of subgroups.

The subgroup lattice has 662 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#532C211yesmaximal, characteristic
2#20715C211yesmaximal, characteristic, complemented
3C2×#1505C214yesmaximal, complemented
4#13753C211yesmaximal, characteristic, complemented
5#8332C214yesmaximal, complemented
6#8328C214yesmaximal, complemented
7C23×#41—21no—
8C2×#532—21no—
9C2×#532—21no—
10C2×#532—21no—
11C2×#532—21no—
12C2×#532—216no—
13C2×#532C412yescomplemented
14C2×#532C412yescomplemented
15C2×#532C2212yes—
16C2×#532C2212yes—
17C22×C4⋊Q8C411yescharacteristic, complemented
18C22×#127C411yescharacteristic, complemented
19C22×#124—21no—
20C22×#121—21no—
21C22×#121C2212yes—
22C22×C4×D4—21no—
23C22×C4×D4C412yescomplemented
24C23×C42C2211yescharacteristic
25C2×#432C2212yescomplemented
26C2×C4×C22⋊C4—21no—
27C2×#376C2212yescomplemented
28#1817C2212yescomplemented
29#1797—21no—
30#1797—21no—
31#1779C2212yes—
32#1749—21no—
33#1749—21no—
34#1505—216no—
35C2×#327C22116yescomplemented
36#1434—21no—
37#1434C2212yescomplemented
38C22⋊#95—21no—
39C22⋊#95C2212yes—
40C4×#95—21no—
41C22×#41—216no—
42C22×#41—22no—
43C22×#41—22no—
44C22×#41—22no—
45C22×#41—22no—
46C22×#41—48no—
47C22×#41—48no—
48C23×Q8—21no—
49C23×D4—22no—
50C23×D4—21no—
51C23×D4—21no—
52C23×D4C2×C412yes—
53C24×C4—22no—
54C24×C4D411yescharacteristic
55C24×C4D411yescharacteristic
56C24×C4C2311yescharacteristic
57#532—216no—
58#532—216no—
59#532—216no—
60#532—216no—
61#532—28no—
62#532—28no—
63#532—216no—
64#532—216no—
65#532—28no—
66#532D4116yes—
67C2×C4⋊Q8—28no—
68C2×C4⋊Q8—21no—
69C2×C4⋊Q8—21no—
70C2×C4⋊Q8C2×C412yes—
71C2×C4⋊Q8C2×C412yes—
72C2×#127—28no—
73C2×#127—21no—
74C2×#127—21no—
75C2×#127C2×C412yes—
76C2×#127C2×C412yes—
77C2×#124—216no—
78C2×#124—22no—
79C2×#124—22no—
80C2×#124—22no—
81C2×#124—22no—
82C2×#121—216no—
83C2×#121—22no—
84C2×#121—22no—
85C2×#121—22no—
86C2×#121—22no—
87C2×#121—22no—
88C2×#121—22no—
89C2×#121—216no—
90C2×#121C2×C414yes—
91C2×#121C2314yes—
92C2×C4×D4—216no—
93C2×C4×D4—216no—
94C2×C4×D4—22no—
95C2×C4×D4—22no—
96C2×C4×D4—22no—
97C2×C4×D4—22no—
98C2×C4×D4—22no—
99C2×C4×D4—22no—
100C2×C4×D4C2×C414yes—
101C2×C4×D4C2×C414yes—
102C22×C4⋊C4—21no—
103C22×C4⋊C4C2×C412yes—
104C22×C22⋊C4—22no—
105C22×C22⋊C4—22no—
106C22×C22⋊C4—22no—
107C22×C42—28no—
108C22×C42—22no—
109C22×C42C2×C411yescharacteristic
110C22×C42C2×C411yescharacteristic
111C22×C42C2×C411yescharacteristic
112C22×C42C2×C411yescharacteristic
113C22×C42D411yescharacteristic
114C22×C42D411yescharacteristic
115C22×C42D411yescharacteristic
116C22×C42D411yescharacteristic
117C22×C42D411yescharacteristic
118C22×C42D411yescharacteristic
119C22×C42C2311yescharacteristic
120C22×C42C2311yescharacteristic
121#432—28no—
122C2×C4⋊C8—22no—
123C2×C22⋊C8—22no—
124C2×C22⋊C8—22no—
125C2×C22⋊C8C2314yes—
126C2×#97—22no—
127C4×C22⋊C4—28no—
128#376—28no—
129C2×#95—22no—
130C2×#95—22no—
131C2×#95—22no—
132C2×#95—22no—
133C2×#95C2314yes—
134#336—22no—
135#336—22no—
136#327—232no—
137C25—22no—
138C25C2×D411yescharacteristic
139C2×#41—216no—
140C2×#41—216no—
141C2×#41—216no—
142C2×#41—216no—
143C2×#41—4128no—
144C2×#41—48no—
145C2×#41—48no—
146C2×#41—48no—
147C2×#41—48no—
148C2×#41—48no—
149C2×#41—48no—
150C2×#41—48no—
151C2×#41—48no—
152C22×Q8—28no—
153C22×Q8—22no—
154C22×Q8—22no—
155C22×Q8—44no—
156C22×Q8—44no—
157C22×D4—28no—
158C22×D4—216no—
159C22×D4—28no—
160C22×D4—24no—
161C22×D4—24no—
162C22×D4—28no—
163C22×D4—22no—
164C22×D4—22no—
165C22×D4—22no—
166C22×D4—22no—
167C22×D4—22no—
168C22×D4—44no—
169C22×D4—44no—
170C22×D4—44no—
171C22×D4—44no—
172C22×D4—44no—
173C22×D4—48no—
174C22×D4—48no—
175C22×D4—44no—
176C22×D4C22×C414yes—
177C23×C4—24no—
178C23×C4—24no—
179C23×C4—24no—
180C23×C4—216no—
181C23×C4—22no—
182C23×C4—24no—
183C23×C4—24no—
184C23×C4—24no—
185C23×C4—24no—
186C23×C4—24no—
187C23×C4—24no—
188C23×C4—42no—
189C23×C4—42no—
190C23×C4—42no—
191C23×C4—48no—
192C23×C4—48no—
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×C4C22×C411yescharacteristic
203C23×C4C2×D411yescharacteristic
204C23×C4C2×D411yescharacteristic
205C23×C4C2×D411yescharacteristic
206C23×C4C2×D411yescharacteristic
207C23×C4C2×D411yescharacteristic
208C23×C4C2×D411yescharacteristic
209C23×C4C2×D411yescharacteristic
210C23×C4C2411yescharacteristic
211C22×C8—24no—
212C4⋊Q8—28no—
213C4⋊Q8—28no—
214C4⋊Q8—28no—
215C4⋊Q8—24no—
216C4⋊Q8C22⋊C418yescomplemented
217#127—28no—
218#127—28no—
219#127—28no—
220#127—24no—
221#127C22⋊C418yescomplemented
222#124—216no—
223#124—216no—
224#124—216no—
225#124—216no—
226#121—216no—
227#121—216no—
228#121—216no—
229#121—216no—
230#121—216no—
231#121—28no—
232#121—216no—
233#121—28no—
234#121—28no—
235#121C2×D4116yes—
236C4×D4—216no—
237C4×D4—216no—
238C4×D4—216no—
239C4×D4—216no—
240C4×D4—216no—
241C4×D4—28no—
242C4×D4—28no—
243C4×D4—28no—
244C4×D4—216no—
245C4×D4C22⋊C4116yescomplemented
246C2×C4⋊C4—28no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—28no—
251C2×C4⋊C4C22×C414yes—
252C2×C22⋊C4—216no—
253C2×C22⋊C4—216no—
254C2×C22⋊C4—24no—
255C2×C22⋊C4—24no—
256C2×C22⋊C4—24no—
257C2×C22⋊C4—24no—
258C2×C22⋊C4—28no—
259C2×C22⋊C4—44no—
260C2×C22⋊C4—44no—
261C2×C42—24no—
262C2×C42—24no—
263C2×C42—24no—
264C2×C42—24no—
265C2×C42—24no—
266C2×C42—24no—
267C2×C42—24no—
268C2×C42—24no—
269C2×C42—24no—
270C2×C42—24no—
271C2×C42—24no—
272C2×C42—28no—
273C2×C42—44no—
274C2×C42—44no—
275C2×C42C22⋊C411yescharacteristic
276C2×C42C22⋊C411yescharacteristic
277C2×C42C22⋊C411yescharacteristic
278C2×C42C22⋊C411yescharacteristic
279C2×C42C22⋊C411yescharacteristic
280C2×C42C22⋊C411yescharacteristic
281C2×C42C22⋊C411yescharacteristic
282C2×C42C22⋊C411yescharacteristic
283C2×C42C22×C411yescharacteristic
284C2×C42C22×C411yescharacteristic
285C2×C42C22×C411yescharacteristic
286C2×C42C22×C411yescharacteristic
287C2×C42C2×D411yescharacteristic
288C2×C42C2×D411yescharacteristic
289C2×C42C2×D418yes—
290C2×C42C2×D411yescharacteristic
291C2×C42C2×D411yescharacteristic
292C4⋊C8—44no—
293C22⋊C8—28no—
294C22⋊C8—44no—
295C22⋊C8—44no—
296#97—44no—
297#95—28no—
298#95—28no—
299#95—44no—
300#95—44no—
301#95—44no—
302C24—22no—
303C24—22no—
304C24—22no—
305C24—22no—
306C24—28no—
307C24—24no—
308C24—41no—
309C24—41no—
310C24—44no—
311C24—44no—
312C24—44no—
313C24—44no—
314C24—44no—
315C24—44no—
316C24C2×C22⋊C411yescharacteristic
317C24C2×C22⋊C411yescharacteristic
318C24AGL(1,ℤ/8)14yes—
319C24C22×D411yescharacteristic
320#41—464no—
321#41—464no—
322#41—464no—
323#41—464no—
324C2×Q8—28no—
325C2×Q8—28no—
326C2×Q8—432no—
327C2×Q8—44no—
328C2×Q8—44no—
329C2×Q8—44no—
330C2×Q8—44no—
331C2×D4—28no—
332C2×D4—28no—
333C2×D4—216no—
334C2×D4—28no—
335C2×D4—216no—
336C2×D4—24no—
337C2×D4—28no—
338C2×D4—28no—
339C2×D4—44no—
340C2×D4—44no—
341C2×D4—432no—
342C2×D4—44no—
343C2×D4—44no—
344C2×D4—432no—
345C2×D4—432no—
346C2×D4—44no—
347C2×D4—44no—
348C2×D4—44no—
349C2×D4—464no—
350C2×D4—48no—
351C2×D4—48no—
352C2×D4—48no—
353C2×D4—48no—
354C2×D4—44no—
355C2×D4—44no—
356C2×D4—44no—
357C2×D4—44no—
358C2×D4—44no—
359C2×D4C2×C22⋊C418yes—
360C22×C4—232no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
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—22no—
383C22×C4—22no—
384C22×C4—28no—
385C22×C4—216no—
386C22×C4—24no—
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—22no—
396C22×C4—22no—
397C22×C4—216no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—28no—
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—416no—
416C22×C4—416no—
417C22×C4—464no—
418C22×C4—48no—
419C22×C4—48no—
420C22×C4—48no—
421C22×C4—44no—
422C22×C4—44no—
423C22×C4—48no—
424C22×C4—44no—
425C22×C4—44no—
426C22×C4—44no—
427C22×C4—44no—
428C22×C4—44no—
429C22×C4—44no—
430C22×C4#10414yes—
431C22×C4C2×C22⋊C411yescharacteristic
432C22×C4C2×C22⋊C411yescharacteristic
433C22×C4C2×C22⋊C411yescharacteristic
434C22×C4C2×C22⋊C411yescharacteristic
435C22×C4C2×C22⋊C411yescharacteristic
436C22×C4C2×C22⋊C411yescharacteristic
437C22×C4C2×C22⋊C411yescharacteristic
438C22×C4C2×C22⋊C411yescharacteristic
439C22×C4C2×C22⋊C411yescharacteristic
440C22×C4C2×C22⋊C411yescharacteristic
441C22×C4#12014yes—
442C22×C4#12014yes—
443C22×C4AGL(1,ℤ/8)14yes—
444C22×C4C23×C411yescharacteristic
445C22×C4C22×D411yescharacteristic
446C22×C4C22×D414yes—
447C2×C8—44no—
448C2×C8—44no—
449C2×C8—44no—
450C4⋊C4—28no—
451C4⋊C4—28no—
452C4⋊C4—24no—
453C4⋊C4—28no—
454C4⋊C4C2×C22⋊C418yes—
455C22⋊C4—216no—
456C22⋊C4—216no—
457C22⋊C4—216no—
458C22⋊C4—216no—
459C22⋊C4—416no—
460C42—22no—
461C42—22no—
462C42—22no—
463C42—22no—
464C42—22no—
465C42—22no—
466C42—22no—
467C42—22no—
468C42—22no—
469C42—22no—
470C42—22no—
471C42—22no—
472C42—416no—
473C42C2×C22⋊C414yes—
474C42C2×C22⋊C414yes—
475C42#12014yes—
476C42#12014yes—
477C23—21no—
478C23—21no—
479C23—21no—
480C23—21no—
481C23—21no—
482C23—21no—
483C23—21no—
484C23—21no—
485C23—21no—
486C23—21no—
487C23—21no—
488C23—21no—
489C23—21no—
490C23—28no—
491C23—28no—
492C23—21no—
493C23—28no—
494C23—44no—
495C23—44no—
496C23—44no—
497C23—44no—
498C23—44no—
499C23—416no—
500C23—416no—
501C23—416no—
502C23—44no—
503C23—44no—
504C23—44no—
505C23—44no—
506C23—44no—
507C23—44no—
508C23—88no—
509C23—88no—
510C23—88no—
511C23—88no—
512C23#41012yes—
513C23#41812yes—
514C23#41812yes—
515C23C2×#10412yes—
516C23C22×C22⋊C411yescharacteristic
517C23C2×#12012yes—
518C23C2×AGL(1,ℤ/8)12yes—
519C23C2×AGL(1,ℤ/8)12yes—
520Q8—416no—
521Q8—416no—
522D4—48no—
523D4—48no—
524D4—416no—
525D4—416no—
526D4—416no—
527D4—416no—
528D4—432no—
529D4—432no—
530D4—416no—
531C2×C4—28no—
532C2×C4—28no—
533C2×C4—28no—
534C2×C4—28no—
535C2×C4—28no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—28no—
551C2×C4—28no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—28no—
557C2×C4—28no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—28no—
565C2×C4—44no—
566C2×C4—44no—
567C2×C4—44no—
568C2×C4—44no—
569C2×C4—44no—
570C2×C4—44no—
571C2×C4—44no—
572C2×C4—44no—
573C2×C4—432no—
574C2×C4—432no—
575C2×C4—416no—
576C2×C4—44no—
577C2×C4—44no—
578C2×C4—44no—
579C2×C4—48no—
580C2×C4—48no—
581C2×C4—48no—
582C2×C4—84no—
583C2×C4—84no—
584C2×C4—84no—
585C2×C4—84no—
586C2×C4#37912yes—
587C2×C4#37912yes—
588C2×C4#37912yes—
589C2×C4#37912yes—
590C2×C4#41012yes—
591C2×C4#41812yes—
592C2×C4#41812yes—
593C2×C4C2×#10412yes—
594C2×C4C2×#10412yes—
595C2×C4#42112yes—
596C2×C4C22×C22⋊C412yes—
597C2×C4C2×#12012yes—
598C8—84no—
599C22—22no—
600C22—22no—
601C22—22no—
602C22—22no—
603C22—22no—
604C22—22no—
605C22—22no—
606C22—22no—
607C22—22no—
608C22—22no—
609C22—22no—
610C22—22no—
611C22—22no—
612C22—22no—
613C22—22no—
614C22—416no—
615C22—41no—
616C22—41no—
617C22—41no—
618C22—41no—
619C22—48no—
620C22—48no—
621C22—48no—
622C22—832no—
623C22—84no—
624C22—84no—
625C22—84no—
626C22—84no—
627C22#150514yescomplemented
628C22C2×#37911yescharacteristic
629C22C2×#41011yescharacteristic
630C22C2×#41011yescharacteristic
631C22C2×#41811yescharacteristic
632C22C2×#41811yescharacteristic
633C22C22×#10411yescharacteristic
634C22C2×#42111yescharacteristic
635C22#300314yes—
636C4—22no—
637C4—22no—
638C4—22no—
639C4—22no—
640C4—22no—
641C4—22no—
642C4—22no—
643C4—22no—
644C4—22no—
645C4—41no—
646C4—48no—
647C4—41no—
648C4—88no—
649C4#178914yes—
650C2—24no—
651C2—24no—
652C2—42no—
653C2—88no—
654C2C2×#150511yescharacteristic
655C2#1754411yescharacteristic
656C2#1758911yescharacteristic
657C2#1759511yescharacteristic
658C2#2058411yescharacteristic
659C2#2059811yescharacteristic
660C2C2×#178911yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC22×#532C2
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C23×C42 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #20715C2semidirect
3. C2×#1505C2semidirect
4. #13753C2semidirect
5. #8332C2semidirect
6. #8328C2semidirect
13. C2×#532C4semidirect
14. C2×#532C4semidirect
17. C22×C4⋊Q8C4semidirect
18. C22×#127C4semidirect
23. C22×C4×D4C4semidirect
25. C2×#432C22semidirect
27. C2×#376C22semidirect
28. #1817C22semidirect
35. C2×#327C22semidirect
37. #1434C22semidirect
216. C4⋊Q8C22⋊C4semidirect
221. #127C22⋊C4semidirect
245. C4×D4C22⋊C4semidirect
627. C22#1505semidirect

Maximal subgroups

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

subgroupindex
C22×#5322
#207152
C2×#15052
#137532
#83322
#83282