Finite Group Databasebeta

← #204592

Subgroups — #204592

Order 512.

1159 conjugacy classes of subgroups.

The subgroup lattice has 660 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35241C211yesmaximal, characteristic
2#20535C211yesmaximal, characteristic, complemented
3C2×#1506C211yesmaximal, characteristic, complemented
4C2×#1509C211yesmaximal, characteristic
5C2×#1503C211yesmaximal, characteristic, complemented
6C2×#1500C211yesmaximal, characteristic
7#13783C211yesmaximal, characteristic, complemented
8#9683C212yesmaximal, complemented
9#9682C212yesmaximal
10#9681C212yesmaximal
11#9679C212yesmaximal, complemented
12C2×#533C2211yescharacteristic
13C2×#529C2211yescharacteristic
14C2×#432C2211yescharacteristic
15C2×#431C2211yescharacteristic
16#2857C2212yes—
17#2850C2212yes—
18#2664—22no—
19#2568C411yescharacteristic, complemented
20#2565C412yescomplemented
21#2547C411yescharacteristic, complemented
22#2546C411yescharacteristic, complemented
23#2545C412yescomplemented
24#2542C411yescharacteristic, complemented
25#2498—22no—
26#2417—21no—
27#2417—21no—
28#2323—21no—
29#2323—21no—
30C2×#389—21no—
31C2×C4×C4⋊C4C2211yescharacteristic
32C2×C4×C4⋊C4C2211yescharacteristic
33C2×#376C2211yescharacteristic
34#1771—22no—
35#1770—21no—
36C8×C4⋊C4—21no—
37#1749C2212yes—
38C4×C22⋊C8C2212yes—
39#1509—24no—
40#1506—24no—
41#1503—24no—
42#1500—24no—
43C2×#328C2211yescharacteristic
44C2×#328C2211yescharacteristic
45C2×#328C2212yes—
46C2×#328C2212yes—
47C2×#328C2212yes—
48C2×#328C2212yes—
49C2×#328C2212yes—
50C2×#327C2211yescharacteristic
51C2×#327C2211yescharacteristic
52C2×#327C2212yes—
53#1436—22no—
54C4⋊#95—21no—
55C4⋊#95C2212yes—
56#1434C2212yes—
57#533—22no—
58#533—22no—
59#533—22no—
60#533—22no—
61C2×C4⋊Q8C2×C411yescharacteristic
62C2×C4⋊Q8C2×C412yes—
63C2×C4⋊Q8C2×C411yescharacteristic
64#529—22no—
65#529—22no—
66#529—22no—
67#529—22no—
68C2×#126—21no—
69C2×#125—21no—
70C2×#125C2×C411yescharacteristic
71C2×#124C2×C411yescharacteristic
72C2×#124C2×C412yes—
73C2×#123—21no—
74C2×C22⋊Q8—21no—
75C2×C22⋊Q8—21no—
76C2×C22⋊Q8—21no—
77C2×C22⋊Q8C2312yes—
78C2×C22⋊Q8C2311yescharacteristic
79C2×#121C2311yescharacteristic
80C2×C4×Q8—21no—
81C2×C4×Q8C2×C412yes—
82C2×C4×Q8C2×C411yescharacteristic
83C2×C4×D4—21no—
84C2×C4×D4C2×C411yescharacteristic
85C2×#117—21no—
86C2×#117—21no—
87C22×C4⋊C4—21no—
88C22×C4⋊C4—22no—
89C22×C4⋊C4D411yescharacteristic
90C22×C4⋊C4D411yescharacteristic
91C22×C42—21no—
92C22×C42—21no—
93C22×C42C2312yes—
94C22×C42C2311yescharacteristic
95#432—22no—
96#432D414yes—
97#431—22no—
98#431D414yes—
99C2×C4⋊C8—21no—
100C2×C4⋊C8—21no—
101C2×C22⋊C8C2311yescharacteristic
102C2×C22⋊C8C2311yescharacteristic
103C2×C22⋊C8C2312yes—
104C2×#97—21no—
105C2×#97—22no—
106C2×C4×C8—21no—
107C2×C4×C8—22no—
108#400—22no—
109#399—22no—
110#398—22no—
111#398—22no—
112#398—22no—
113#397—22no—
114#396—22no—
115#396—22no—
116#391—21no—
117#391—21no—
118#391C2×C412yes—
119#391C2×C412yes—
120#391C2×C412yes—
121#390C2×C412yes—
122#389—28no—
123#389—21no—
124#389—21no—
125#389—22no—
126#389—22no—
127#388—22no—
128#388—22no—
129#388—22no—
130#388—22no—
131#387—22no—
132#387—22no—
133#386—22no—
134#386—22no—
135#384C2×C412yes—
136#382C2×C412yes—
137#380—22no—
138#380—22no—
139#380—21no—
140#380—21no—
141C4×C4⋊C4—22no—
142C4×C4⋊C4—22no—
143C4×C4⋊C4C2×C412yes—
144C4×C4⋊C4C2×C411yescharacteristic
145C4×C4⋊C4C2×C411yescharacteristic
146C4×C4⋊C4D411yescharacteristic
147C4×C4⋊C4D411yescharacteristic
148C4×C4⋊C4D412yes—
149#376—22no—
150#376—22no—
151C2×#95—22no—
152C2×#95—21no—
153C2×#95—22no—
154C2×#95—22no—
155C2×#95D412yes—
156C2×#95C2311yescharacteristic
157C2×#95C2312yes—
158C2×#95C2311yescharacteristic
159#336—22no—
160#336—22no—
161#328—22no—
162#328—22no—
163#328—24no—
164#328—24no—
165#328—24no—
166#328—24no—
167#328—24no—
168#327—22no—
169#327—22no—
170#327—24no—
171C22×Q8—21no—
172C22×Q8C22×C411yescharacteristic
173C22×Q8C22×C412yes—
174C22×D4—21no—
175C22×D4C22×C411yescharacteristic
176C23×C4—22no—
177C23×C4—21no—
178C23×C4—21no—
179C23×C4—21no—
180C23×C4C2×D412yes—
181C23×C4C2×D411yescharacteristic
182C23×C4C2×D411yescharacteristic
183C23×C4C2×D412yes—
184C23×C4C2411yescharacteristic
185C22×C8—21no—
186C22×C8—21no—
187C22×C8—22no—
188C4⋊Q8—22no—
189C4⋊Q8—22no—
190C4⋊Q8—22no—
191C4⋊Q8—22no—
192C4⋊Q8—82no—
193#126—42no—
194#126—42no—
195#125—22no—
196#125—22no—
197#125—42no—
198#125—42no—
199#124—22no—
200#124—22no—
201#124—82no—
202#123—42no—
203#123—42no—
204C22⋊Q8—21no—
205C22⋊Q8—21no—
206C22⋊Q8—22no—
207C22⋊Q8—22no—
208C22⋊Q8—21no—
209C22⋊Q8—21no—
210C22⋊Q8—22no—
211C22⋊Q8—22no—
212C22⋊Q8—44no—
213C22⋊Q8—42no—
214C22⋊Q8—42no—
215C22⋊Q8—44no—
216#121—21no—
217#121—21no—
218#121—21no—
219#121—21no—
220C4×Q8—22no—
221C4×Q8—22no—
222C4×Q8—24no—
223C4×Q8—24no—
224C4×Q8—42no—
225C4×Q8—42no—
226C4×D4—22no—
227C4×D4—22no—
228C4×D4—42no—
229C4×D4—42no—
230#117—42no—
231#117—42no—
232#117—42no—
233#117—42no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—21no—
237C2×C4⋊C4—21no—
238C2×C4⋊C4—21no—
239C2×C4⋊C4—21no—
240C2×C4⋊C4—21no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—21no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—21no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—28no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—42no—
251C2×C4⋊C4—42no—
252C2×C4⋊C4—42no—
253C2×C4⋊C4—82no—
254C2×C4⋊C4—82no—
255C2×C4⋊C4C22⋊C411yescharacteristic
256C2×C4⋊C4C22⋊C411yescharacteristic
257C2×C4⋊C4C22⋊C411yescharacteristic
258C2×C4⋊C4C22⋊C412yes—
259C2×C4⋊C4C22⋊C412yes—
260C2×C4⋊C4C22⋊C411yescharacteristic
261C2×C4⋊C4C22×C411yescharacteristic
262C2×C4⋊C4C22×C411yescharacteristic
263C2×C4⋊C4C22×C412yes—
264C2×C22⋊C4—22no—
265C2×C22⋊C4—21no—
266C2×C22⋊C4—21no—
267C2×C22⋊C4—21no—
268C2×C22⋊C4—21no—
269C2×C22⋊C4—22no—
270C2×C22⋊C4—22no—
271C2×C22⋊C4—21no—
272C2×C22⋊C4—22no—
273C2×C22⋊C4—22no—
274C2×C42—24no—
275C2×C42—21no—
276C2×C42—22no—
277C2×C42—42no—
278C2×C42—42no—
279C2×C42—42no—
280C2×C42—82no—
281C2×C42—81no—
282C2×C42—81no—
283C2×C42C22×C412yes—
284C2×C42C22×C412yes—
285C2×C42C22×C411yescharacteristic
286C2×C42C22×C411yescharacteristic
287C2×C42C2×D412yes—
288C2×C42C2×D412yes—
289C2×C42C2×D412yes—
290C2×C42C2×D412yes—
291C2×C42C2×D411yescharacteristic
292C2×C42C2×D411yescharacteristic
293C2×C42#4112yes—
294C2×C42#4112yes—
295C4⋊C8—24no—
296C4⋊C8—24no—
297C22⋊C8—21no—
298C22⋊C8—21no—
299C22⋊C8—22no—
300C22⋊C8C2×D412yes—
301C22⋊C8C2×D412yes—
302C22⋊C8C2×D414yes—
303#97—24no—
304#97—82no—
305C4×C8—24no—
306C4×C8—82no—
307#95—28no—
308#95—22no—
309#95—22no—
310#95—22no—
311#95—22no—
312#95—22no—
313#95—22no—
314#95—22no—
315#95—22no—
316#95—21no—
317#95—21no—
318#95—22no—
319#95—21no—
320#95—21no—
321#95—22no—
322#95—42no—
323#95—82no—
324#95—82no—
325#95C22⋊C412yes—
326#95C22⋊C412yes—
327#95C22⋊C412yes—
328#95C22⋊C412yes—
329C24—21no—
330C24C22×D411yescharacteristic
331C2×Q8—21no—
332C2×Q8—21no—
333C2×Q8—22no—
334C2×Q8—22no—
335C2×Q8—42no—
336C2×Q8—41no—
337C2×Q8—41no—
338C2×Q8—42no—
339C2×Q8—42no—
340C2×Q8—42no—
341C2×Q8—42no—
342C2×D4—21no—
343C2×D4—21no—
344C2×D4—41no—
345C2×D4—41no—
346C2×D4—41no—
347C2×D4—41no—
348C2×D4—41no—
349C2×D4—41no—
350C2×D4—41no—
351C2×D4—41no—
352C22×C4—21no—
353C22×C4—21no—
354C22×C4—21no—
355C22×C4—21no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—24no—
363C22×C4—22no—
364C22×C4—24no—
365C22×C4—22no—
366C22×C4—21no—
367C22×C4—22no—
368C22×C4—21no—
369C22×C4—22no—
370C22×C4—21no—
371C22×C4—41no—
372C22×C4—42no—
373C22×C4—41no—
374C22×C4—42no—
375C22×C4—42no—
376C22×C4—41no—
377C22×C4—42no—
378C22×C4—41no—
379C22×C4—41no—
380C22×C4—42no—
381C22×C4—41no—
382C22×C4—42no—
383C22×C4—41no—
384C22×C4—41no—
385C22×C4—41no—
386C22×C4—41no—
387C22×C4—42no—
388C22×C4—41no—
389C22×C4—42no—
390C22×C4—42no—
391C22×C4—41no—
392C22×C4—41no—
393C22×C4—41no—
394C22×C4—42no—
395C22×C4—42no—
396C22×C4—42no—
397C22×C4C2×C22⋊C412yes—
398C22×C4C2×C22⋊C412yes—
399C22×C4C2×C22⋊C412yes—
400C22×C4C2×C22⋊C411yescharacteristic
401C22×C4C2×C22⋊C411yescharacteristic
402C22×C4C2×C22⋊C411yescharacteristic
403C22×C4C2×C22⋊C412yes—
404C22×C4C2×C22⋊C411yescharacteristic
405C22×C4#12412yes—
406C22×C4#12412yes—
407C22×C4#12712yes—
408C22×C4#12712yes—
409C22×C4C23×C411yescharacteristic
410C22×C4C22×D411yescharacteristic
411C22×C4C22×D411yescharacteristic
412C22×C4C22×D411yescharacteristic
413C22×C4C2×#4111yescharacteristic
414C22×C4C2×#4111yescharacteristic
415C2×C8—22no—
416C2×C8—22no—
417C2×C8—24no—
418C2×C8—41no—
419C2×C8—42no—
420C2×C8—41no—
421C2×C8—41no—
422C2×C8—41no—
423C2×C8—42no—
424C4⋊C4—24no—
425C4⋊C4—22no—
426C4⋊C4—22no—
427C4⋊C4—22no—
428C4⋊C4—21no—
429C4⋊C4—21no—
430C4⋊C4—21no—
431C4⋊C4—21no—
432C4⋊C4—22no—
433C4⋊C4—22no—
434C4⋊C4—42no—
435C4⋊C4—48no—
436C4⋊C4—44no—
437C4⋊C4—41no—
438C4⋊C4—41no—
439C4⋊C4—42no—
440C4⋊C4—42no—
441C4⋊C4—42no—
442C4⋊C4—42no—
443C4⋊C4—42no—
444C4⋊C4—42no—
445C4⋊C4—44no—
446C4⋊C4—42no—
447C4⋊C4—42no—
448C4⋊C4—44no—
449C4⋊C4—82no—
450C4⋊C4—82no—
451C4⋊C4—88no—
452C4⋊C4#10414yes—
453C22⋊C4—44no—
454C22⋊C4—42no—
455C22⋊C4—41no—
456C22⋊C4—42no—
457C22⋊C4—41no—
458C22⋊C4—41no—
459C22⋊C4—41no—
460C22⋊C4—42no—
461C22⋊C4—42no—
462C22⋊C4—42no—
463C22⋊C4—44no—
464C22⋊C4—82no—
465C22⋊C4—82no—
466C42—42no—
467C42—42no—
468C42—42no—
469C42—42no—
470C42—48no—
471C42—42no—
472C42—44no—
473C42—84no—
474C42C4×D412yes—
475C42C4×D412yes—
476C42C4×D412yes—
477C42C4×D412yes—
478C42#12112yes—
479C42#12112yes—
480C42#12112yes—
481C42#12112yes—
482C23—21no—
483C23—21no—
484C23—21no—
485C23—21no—
486C23—41no—
487C23—41no—
488C23—41no—
489C23—41no—
490C23—41no—
491C23—41no—
492C23—41no—
493C23C22×C22⋊C411yescharacteristic
494C23C2×#12411yescharacteristic
495C23C2×#12711yescharacteristic
496C23#57711yescharacteristic
497C23#57711yescharacteristic
498C23#57811yescharacteristic
499C23#57811yescharacteristic
500Q8—81no—
501Q8—81no—
502Q8—82no—
503Q8—82no—
504Q8—84no—
505D4—81no—
506D4—81no—
507D4—81no—
508D4—81no—
509D4—81no—
510D4—81no—
511C2×C4—22no—
512C2×C4—22no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—22no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—22no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—22no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—22no—
527C2×C4—24no—
528C2×C4—24no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—44no—
543C2×C4—44no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—44no—
547C2×C4—42no—
548C2×C4—44no—
549C2×C4—42no—
550C2×C4—44no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—41no—
555C2×C4—42no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—42no—
560C2×C4—41no—
561C2×C4—42no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—41no—
568C2×C4—82no—
569C2×C4—82no—
570C2×C4—82no—
571C2×C4—81no—
572C2×C4—81no—
573C2×C4—81no—
574C2×C4—82no—
575C2×C4—81no—
576C2×C4—81no—
577C2×C4—82no—
578C2×C4—81no—
579C2×C4—82no—
580C2×C4—81no—
581C2×C4—81no—
582C2×C4—84no—
583C2×C4#39012yes—
584C2×C4#39012yes—
585C2×C4#39012yes—
586C2×C4#39012yes—
587C2×C4C2×#10414yes—
588C2×C4C2×#10412yes—
589C2×C4#42112yes—
590C2×C4C2×C4×D411yescharacteristic
591C2×C4C2×C4×D411yescharacteristic
592C2×C4C2×#12111yescharacteristic
593C2×C4C2×#12111yescharacteristic
594C8—42no—
595C8—42no—
596C8—44no—
597C22—21no—
598C22—21no—
599C22—21no—
600C22—21no—
601C22—21no—
602C22—21no—
603C22—21no—
604C22—21no—
605C22—21no—
606C22—21no—
607C22—21no—
608C22—21no—
609C22—81no—
610C22—81no—
611C22—81no—
612C22—81no—
613C22—81no—
614C22—81no—
615C22—81no—
616C22C2×#39011yescharacteristic
617C22#288311yescharacteristic
618C22#288411yescharacteristic
619C22#289811yescharacteristic
620C22#289811yescharacteristic
621C22C22×#10411yescharacteristic
622C22C2×#42111yescharacteristic
623C22#313711yescharacteristic
624C22#313711yescharacteristic
625C22#315411yescharacteristic
626C22#315611yescharacteristic
627C4—22no—
628C4—24no—
629C4—22no—
630C4—22no—
631C4—21no—
632C4—21no—
633C4—21no—
634C4—21no—
635C4—42no—
636C4—42no—
637C4—42no—
638C4—82no—
639C4—84no—
640C4—82no—
641C4—81no—
642C4—82no—
643C4—81no—
644C4—82no—
645C4—81no—
646C4#198014yes—
647C2—21no—
648C2—21no—
649C2—21no—
650C2—21no—
651C2—161no—
652C2#1754811yescharacteristic
653C2#1754911yescharacteristic
654C2#1760011yescharacteristic
655C2#1760111yescharacteristic
656C2#2120511yescharacteristic
657C2#2121411yescharacteristic
658C2C2×#198011yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC2×#121C23
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #20535C2semidirect
3. C2×#1506C2semidirect
5. C2×#1503C2semidirect
7. #13783C2semidirect
8. #9683C2semidirect
11. #9679C2semidirect
19. #2568C4semidirect
20. #2565C4semidirect
21. #2547C4semidirect
22. #2546C4semidirect
23. #2545C4semidirect
24. #2542C4semidirect

Maximal subgroups

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

subgroupindex
#352412
#205352
C2×#15062
C2×#15092
C2×#15032
C2×#15002
#137832
#96832
#96822
#96812
#96792