Finite Group Databasebeta

← #368518

Subgroups — #368518

Order 512.

1504 conjugacy classes of subgroups.

The subgroup lattice has 670 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1870C211yesmaximal, characteristic
2C2×#1869C211yesmaximal, characteristic
3C2×#1752C211yesmaximal, characteristic, complemented
4C2×#1729C211yesmaximal, characteristic, complemented
5#13839C211yesmaximal, characteristic
6#11298C212yesmaximal, complemented
7#11147C212yesmaximal, complemented
8#10922C212yesmaximal, complemented
9#10798C212yesmaximal, complemented
10#10715C211yesmaximal, characteristic, complemented
11#10713C211yesmaximal, characteristic, complemented
12C23×M16C2211yescharacteristic
13C24×C8C2211yescharacteristic
14C22×C4×D4C2211yescharacteristic
15C22×#117C2211yescharacteristic
16C22×C22⋊C8C2211yescharacteristic
17C22×C22⋊C8C2211yescharacteristic
18C22×C22⋊C8C2212yes—
19C22×C22⋊C8C2211yescharacteristic
20C22×C22⋊C8C2211yescharacteristic
21C22×#97C2212yes—
22C2×C4×C22⋊C4C2211yescharacteristic
23C2×C4×C22⋊C4C2212yes—
24#1870—24no—
25#1869—24no—
26#1752—24no—
27#1729—24no—
28C2×#336C2211yescharacteristic
29C2×#336C2212yes—
30C2×#336C2212yes—
31C2×#336C2212yes—
32C2×#336C2211yescharacteristic
33C2×#336C2212yes—
34C2×#336C2212yes—
35C2×#336C2212yes—
36C2×#336C2212yes—
37C2×#336C2212yes—
38#1441C411yescharacteristic
39#1441C411yescharacteristic
40#1440C412yes—
41#1439C411yescharacteristic
42#1439C411yescharacteristic
43#1438C412yes—
44C4×#95C2212yes—
45C23×D4C2×C411yescharacteristic
46C24×C4C2311yescharacteristic
47C24×C4C2311yescharacteristic
48C22×M16—22no—
49C22×M16—22no—
50C22×M16—22no—
51C22×M16—24no—
52C22×M16—42no—
53C23×C8—22no—
54C23×C8—22no—
55C23×C8—24no—
56C23×C8—42no—
57C23×C8D414yes—
58C23×C8C2311yescharacteristic
59C23×C8C2311yescharacteristic
60C23×C8C2312yes—
61C23×C8C2311yescharacteristic
62C23×C8C2312yes—
63C23×C8C2311yescharacteristic
64C2×C4×D4—22no—
65C2×C4×D4—22no—
66C2×C4×D4—22no—
67C2×C4×D4—22no—
68C2×C4×D4—22no—
69C2×C4×D4—22no—
70C2×#117—22no—
71C2×#117—22no—
72C2×#117—42no—
73C2×#117—42no—
74C22×C4⋊C4C2×C411yescharacteristic
75C22×C4⋊C4C2×C411yescharacteristic
76C22×C4⋊C4C2×C411yescharacteristic
77C22×C22⋊C4C2×C411yescharacteristic
78C22×C22⋊C4C2×C411yescharacteristic
79C22×C22⋊C4C2×C412yes—
80C22×C22⋊C4C2×C412yes—
81C22×C22⋊C4C2×C412yes—
82C22×C22⋊C4C2×C412yes—
83C22×C42C2312yes—
84C22×C42C2312yes—
85C22×C42C2311yescharacteristic
86C2×C22⋊C8—22no—
87C2×C22⋊C8—22no—
88C2×C22⋊C8—22no—
89C2×C22⋊C8—22no—
90C2×C22⋊C8—22no—
91C2×C22⋊C8—22no—
92C2×C22⋊C8—22no—
93C2×C22⋊C8—24no—
94C2×C22⋊C8—24no—
95C2×C22⋊C8—24no—
96C2×C22⋊C8—42no—
97C2×C22⋊C8—42no—
98C2×C22⋊C8D414yes—
99C2×#97—24no—
100C2×#97—24no—
101C2×#97—42no—
102C4×C22⋊C4—22no—
103C4×C22⋊C4—22no—
104C4×C22⋊C4—42no—
105C4×C22⋊C4—42no—
106C2×#95C2×C412yes—
107C2×#95C2×C411yescharacteristic
108C2×#95C2×C411yescharacteristic
109C2×#95C2×C412yes—
110C2×#95C2×C412yes—
111C2×#95C2×C412yes—
112C2×#95C2×C412yes—
113C2×#95C2×C412yes—
114#336—22no—
115#336—22no—
116#336—24no—
117#336—24no—
118#336—24no—
119#336—24no—
120#336—24no—
121#336—42no—
122#336—42no—
123#336—42no—
124C25C22×C411yescharacteristic
125C25C22×C411yescharacteristic
126C22×D4—22no—
127C22×D4—22no—
128C22×D4—22no—
129C22×D4—22no—
130C22×D4—22no—
131C22×D4—41no—
132C22×D4—41no—
133C23×C4—22no—
134C23×C4—22no—
135C23×C4—22no—
136C23×C4—22no—
137C23×C4—21no—
138C23×C4—21no—
139C23×C4—21no—
140C23×C4—21no—
141C23×C4—21no—
142C23×C4—21no—
143C23×C4—22no—
144C23×C4—22no—
145C23×C4—22no—
146C23×C4—22no—
147C23×C4C22×C412yes—
148C23×C4C22×C412yes—
149C23×C4C22×C411yescharacteristic
150C23×C4C22×C412yes—
151C23×C4C22×C412yes—
152C23×C4C22×C411yescharacteristic
153C23×C4C22×C411yescharacteristic
154C23×C4C22×C411yescharacteristic
155C23×C4C2×D412yes—
156C23×C4C2×D412yes—
157C23×C4#4112yes—
158C23×C4#4112yes—
159C23×C4#4112yes—
160C23×C4#4112yes—
161C23×C4C2411yescharacteristic
162C2×M16—28no—
163C2×M16—48no—
164C2×M16—48no—
165C2×M16—44no—
166C2×M16—44no—
167C22×C8—21no—
168C22×C8—21no—
169C22×C8—22no—
170C22×C8—28no—
171C22×C8—21no—
172C22×C8—22no—
173C22×C8—22no—
174C22×C8—21no—
175C22×C8—22no—
176C22×C8—22no—
177C22×C8—22no—
178C22×C8—22no—
179C22×C8—22no—
180C22×C8—22no—
181C22×C8—22no—
182C22×C8—22no—
183C22×C8—22no—
184C22×C8—22no—
185C22×C8—22no—
186C22×C8—22no—
187C22×C8—28no—
188C22×C8—21no—
189C22×C8—21no—
190C22×C8—22no—
191C22×C8—22no—
192C22×C8—21no—
193C22×C8—21no—
194C22×C8—21no—
195C22×C8—21no—
196C22×C8—22no—
197C22×C8—22no—
198C22×C8—44no—
199C22×C8—48no—
200C22×C8—48no—
201C22×C8C2×D412yes—
202C22×C8C2×D414yes—
203C22×C8C2×D412yes—
204C22×C8#4114yes—
205C22×C8#4114yes—
206C4×D4—44no—
207C4×D4—44no—
208C4×D4—44no—
209C4×D4—44no—
210#117—44no—
211#117—44no—
212#117—48no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—42no—
219C2×C4⋊C4—42no—
220C2×C22⋊C4—22no—
221C2×C22⋊C4—24no—
222C2×C22⋊C4—22no—
223C2×C22⋊C4—22no—
224C2×C22⋊C4—24no—
225C2×C22⋊C4—24no—
226C2×C22⋊C4—22no—
227C2×C22⋊C4—22no—
228C2×C22⋊C4—22no—
229C2×C22⋊C4—42no—
230C2×C22⋊C4—42no—
231C2×C22⋊C4—42no—
232C2×C22⋊C4—42no—
233C2×C22⋊C4—42no—
234C2×C22⋊C4—42no—
235C2×C22⋊C4—42no—
236C2×C22⋊C4—42no—
237C2×C22⋊C4—42no—
238C2×C42—22no—
239C2×C42—21no—
240C2×C42—21no—
241C2×C42—22no—
242C2×C42—22no—
243C2×C42—21no—
244C2×C42—21no—
245C2×C42—21no—
246C2×C42—21no—
247C2×C42—22no—
248C2×C42—22no—
249C2×C42—22no—
250C2×C42—22no—
251C2×C42—22no—
252C2×C42—22no—
253C2×C42—22no—
254C2×C42—22no—
255C2×C42—22no—
256C22⋊C8—28no—
257C22⋊C8—48no—
258C22⋊C8—44no—
259C22⋊C8—44no—
260C22⋊C8—44no—
261#97—48no—
262#95—22no—
263#95—22no—
264#95—24no—
265#95—42no—
266#95—42no—
267#95—42no—
268#95—42no—
269#95—42no—
270C24—22no—
271C24—22no—
272C24—21no—
273C24—21no—
274C24—22no—
275C24—22no—
276C24—21no—
277C24—21no—
278C24—21no—
279C24—21no—
280C24—21no—
281C24—22no—
282C24—22no—
283C24—21no—
284C24—21no—
285C24—22no—
286C24—22no—
287C24—21no—
288C24#11712yes—
289C24C4×D412yes—
290C24C4×D412yes—
291C24#13112yes—
292C24C23×C411yescharacteristic
293C2×D4—28no—
294C2×D4—44no—
295C2×D4—44no—
296C2×D4—44no—
297C2×D4—44no—
298C2×D4—44no—
299C2×D4—44no—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—24no—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—24no—
308C22×C4—24no—
309C22×C4—24no—
310C22×C4—24no—
311C22×C4—24no—
312C22×C4—24no—
313C22×C4—24no—
314C22×C4—24no—
315C22×C4—24no—
316C22×C4—21no—
317C22×C4—21no—
318C22×C4—21no—
319C22×C4—21no—
320C22×C4—21no—
321C22×C4—21no—
322C22×C4—21no—
323C22×C4—21no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—21no—
327C22×C4—21no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—21no—
331C22×C4—21no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—41no—
361C22×C4—41no—
362C22×C4—41no—
363C22×C4—41no—
364C22×C4—41no—
365C22×C4—41no—
366C22×C4—42no—
367C22×C4—41no—
368C22×C4—41no—
369C22×C4—42no—
370C22×C4—42no—
371C22×C4—42no—
372C22×C4—42no—
373C22×C4—42no—
374C22×C4—42no—
375C22×C4—42no—
376C22×C4—42no—
377C22×C4—42no—
378C22×C4#11712yes—
379C22×C4C4×D412yes—
380C22×C4C4×D412yes—
381C22×C4#12114yes—
382C22×C4#12314yes—
383C22×C4#12414yes—
384C22×C4#12614yes—
385C22×C4#13112yes—
386C22×C4C22×D411yescharacteristic
387C22×C4C22×D411yescharacteristic
388C22×C4C2×#4111yescharacteristic
389C22×C4C2×#4111yescharacteristic
390C22×C4C2×#4111yescharacteristic
391C22×C4C2×#4112yes—
392C22×C4C2×#4112yes—
393C22×C4C2×#4111yescharacteristic
394C22×C4H(5,𝔽2)12yes—
395C22×C4H(5,𝔽2)12yes—
396M16—416no—
397C2×C8—22no—
398C2×C8—22no—
399C2×C8—24no—
400C2×C8—24no—
401C2×C8—22no—
402C2×C8—24no—
403C2×C8—22no—
404C2×C8—24no—
405C2×C8—22no—
406C2×C8—24no—
407C2×C8—24no—
408C2×C8—24no—
409C2×C8—24no—
410C2×C8—416no—
411C2×C8—42no—
412C2×C8—42no—
413C2×C8—42no—
414C2×C8—42no—
415C2×C8—42no—
416C2×C8—42no—
417C2×C8—42no—
418C2×C8—42no—
419C2×C8—42no—
420C2×C8—42no—
421C2×C8—41no—
422C2×C8—41no—
423C2×C8—41no—
424C2×C8—41no—
425C2×C8—41no—
426C2×C8—41no—
427C2×C8—42no—
428C2×C8—42no—
429C2×C8—42no—
430C2×C8—42no—
431C2×C8#12314yes—
432C4⋊C4—44no—
433C4⋊C4—44no—
434C4⋊C4—44no—
435C22⋊C4—48no—
436C22⋊C4—48no—
437C22⋊C4—48no—
438C22⋊C4—48no—
439C22⋊C4—44no—
440C22⋊C4—44no—
441C42—41no—
442C42—41no—
443C42—41no—
444C42—41no—
445C42—42no—
446C42—42no—
447C42—42no—
448C42—42no—
449C42—42no—
450C42—42no—
451C42—42no—
452C42—42no—
453C23—22no—
454C23—24no—
455C23—24no—
456C23—24no—
457C23—22no—
458C23—22no—
459C23—22no—
460C23—24no—
461C23—22no—
462C23—22no—
463C23—22no—
464C23—22no—
465C23—22no—
466C23—22no—
467C23—22no—
468C23—22no—
469C23—22no—
470C23—24no—
471C23—24no—
472C23—42no—
473C23—42no—
474C23—42no—
475C23—42no—
476C23—42no—
477C23—42no—
478C23—42no—
479C23—42no—
480C23—41no—
481C23—42no—
482C23—42no—
483C23—42no—
484C23—41no—
485C23—41no—
486C23—42no—
487C23—41no—
488C23—41no—
489C23—41no—
490C23—42no—
491C23—42no—
492C23—42no—
493C23—42no—
494C23—42no—
495C23—42no—
496C23—41no—
497C23—41no—
498C23#38814yes—
499C23C2×#11711yescharacteristic
500C23C2×C4×D411yescharacteristic
501C23C2×C4×D411yescharacteristic
502C23C4×#4112yes—
503C23#51812yes—
504C23C2×#13111yescharacteristic
505C23C2×#13111yescharacteristic
506C23#56812yes—
507C23#56812yes—
508C23#56811yescharacteristic
509C23#56811yescharacteristic
510D4—416no—
511C2×C4—22no—
512C2×C4—24no—
513C2×C4—24no—
514C2×C4—22no—
515C2×C4—24no—
516C2×C4—24no—
517C2×C4—24no—
518C2×C4—28no—
519C2×C4—24no—
520C2×C4—28no—
521C2×C4—22no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—41no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—81no—
567C2×C4—81no—
568C2×C4—81no—
569C2×C4—82no—
570C2×C4—81no—
571C2×C4—82no—
572C2×C4#38814yes—
573C2×C4C2×#12111yescharacteristic
574C2×C4C2×#12311yescharacteristic
575C2×C4C2×#12311yescharacteristic
576C2×C4#52511yescharacteristic
577C2×C4C2×#12411yescharacteristic
578C2×C4C2×#12611yescharacteristic
579C2×C4#52912yes—
580C2×C4#53411yescharacteristic
581C2×C4#53511yescharacteristic
582C2×C4#53812yes—
583C2×C4#54612yes—
584C2×C4#54612yes—
585C2×C4#55112yes—
586C2×C4#55112yes—
587C2×C4#55312yes—
588C2×C4#55312yes—
589C2×C4#55312yes—
590C2×C4#55312yes—
591C8—24no—
592C8—44no—
593C8—42no—
594C8—44no—
595C8—81no—
596C8—81no—
597C22—24no—
598C22—24no—
599C22—24no—
600C22—44no—
601C22—44no—
602C22—42no—
603C22—42no—
604C22—42no—
605C22—42no—
606C22—44no—
607C22—44no—
608C22—44no—
609C22—42no—
610C22—42no—
611C22—42no—
612C22—44no—
613C22—82no—
614C22—81no—
615C22—81no—
616C22—81no—
617C22—82no—
618C22—81no—
619C22C2×#38811yescharacteristic
620C22C4×#12311yescharacteristic
621C22#232111yescharacteristic
622C22#235312yes—
623C22#236412yes—
624C22#292312yes—
625C22#292411yescharacteristic
626C22#292411yescharacteristic
627C22#292911yescharacteristic
628C22#293011yescharacteristic
629C22#293012yes—
630C22#293211yescharacteristic
631C22#293211yescharacteristic
632C22#293311yescharacteristic
633C22#293311yescharacteristic
634C22#296712yes—
635C22#296712yes—
636C22#296812yes—
637C22#296912yes—
638C22#297612yes—
639C22#297812yes—
640C22#298012yes—
641C22#298112yes—
642C4—24no—
643C4—42no—
644C4—82no—
645C4—82no—
646C4—81no—
647C4—82no—
648C4—82no—
649C4—81no—
650C4#241111yescharacteristic
651C4#241211yescharacteristic
652C4#244412yes—
653C4#250411yescharacteristic
654C4#251012yes—
655C4#259111yescharacteristic
656C2—44no—
657C2—82no—
658C2#1402011yescharacteristic
659C2#2104711yescharacteristic
660C2#2105311yescharacteristic
661C2#2156711yescharacteristic
662C2#2156911yescharacteristic
663C2#2203111yescharacteristic
664C2#2203511yescharacteristic
665C2#2279012yes—
666C2#2279412yes—
667C2#2312412yes—
668C2#2312512yes—
subgroupG/H
center, Frattini, cosocleC23×C4C24
commutator, socleC24C23×C4
prime radicalC23×D4C2×C4
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C23×C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. C2×#1752C2semidirect
4. C2×#1729C2semidirect
6. #11298C2semidirect
7. #11147C2semidirect
8. #10922C2semidirect
9. #10798C2semidirect
10. #10715C2semidirect
11. #10713C2semidirect

Maximal subgroups

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

subgroupindex
C2×#18702
C2×#18692
C2×#17522
C2×#17292
#138392
#112982
#111472
#109222
#107982
#107152
#107132