Finite Group Databasebeta

← #388983

Subgroups — #388983

Order 512.

1678 conjugacy classes of subgroups.

The subgroup lattice has 664 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1870C212yesmaximal, complemented
2C2×#1870C212yesmaximal, complemented
3C4×#392C211yesmaximal, characteristic
4#11298C212yesmaximal, complemented
5#11147C211yesmaximal, characteristic, complemented
6#10964C212yesmaximal, complemented
7#10922C211yesmaximal, characteristic, complemented
8#10922C212yesmaximal, complemented
9#10715C212yesmaximal, complemented
10C23×M16C2212yes—
11C22×C4×D4C2211yescharacteristic
12C22×C4×D4C2212yes—
13C22×C22⋊C8C2211yescharacteristic
14C22×C22⋊C8C2212yes—
15C22×C22⋊C8C2212yes—
16C22×C22⋊C8C2212yes—
17C22×C22⋊C8C2211yescharacteristic
18C22×C22⋊C8C2212yes—
19C22×#97C2211yescharacteristic
20C2×#392C411yescharacteristic
21C2×C4×C22⋊C4C2211yescharacteristic
22C2×C4×C22⋊C4C2212yes—
23#1870—28no—
24#1870—28no—
25C2×#336C2212yes—
26C2×#336C2212yes—
27C2×#336C2212yes—
28C2×#336C2212yes—
29C2×#336C2212yes—
30C2×#336C2212yes—
31C2×#336C2212yes—
32C2×#336C2211yescharacteristic
33#1444C411yescharacteristic
34#1439C412yes—
35#1439C411yescharacteristic
36#1438C412yes—
37#1438C411yescharacteristic
38C4×#95C2211yescharacteristic
39C23×D4C2×C411yescharacteristic
40C23×D4C2×C412yes—
41C24×C4C2311yescharacteristic
42C24×C4C2312yes—
43C22×M16—24no—
44C22×M16—24no—
45C22×M16—24no—
46C22×M16—42no—
47C22×M16—42no—
48C22×M16—42no—
49C22×M16—42no—
50C23×C8C2312yes—
51C23×C8C2312yes—
52C23×C8C2312yes—
53C23×C8C2312yes—
54C2×C4×D4—24no—
55C2×C4×D4—24no—
56C2×C4×D4—24no—
57C2×C4×D4—24no—
58C2×C4×D4—42no—
59C2×C4×D4—42no—
60C2×C4×D4—41no—
61C2×C4×D4—41no—
62C2×C4×D4—42no—
63C2×C4×D4—42no—
64C22×C4⋊C4C2×C411yescharacteristic
65C22×C4⋊C4C2×C412yes—
66C22×C22⋊C4C2×C412yes—
67C22×C22⋊C4C2×C412yes—
68C22×C22⋊C4C2×C411yescharacteristic
69C22×C22⋊C4C2×C412yes—
70C22×C22⋊C4C2×C412yes—
71C22×C22⋊C4C2×C411yescharacteristic
72C22×C22⋊C4C2×C412yes—
73C22×C42C2311yescharacteristic
74C22×C42C2311yescharacteristic
75C22×C42C2312yes—
76C2×C22⋊C8—24no—
77C2×C22⋊C8—24no—
78C2×C22⋊C8—24no—
79C2×C22⋊C8—24no—
80C2×C22⋊C8—24no—
81C2×C22⋊C8—24no—
82C2×C22⋊C8—24no—
83C2×C22⋊C8—42no—
84C2×C22⋊C8—42no—
85C2×C22⋊C8—41no—
86C2×C22⋊C8—42no—
87C2×C22⋊C8—42no—
88C2×C22⋊C8—42no—
89C2×C22⋊C8—42no—
90C2×C22⋊C8—41no—
91C2×C22⋊C8D418yescomplemented
92C2×#97—42no—
93C2×#97—41no—
94#392—24no—
95C4×C22⋊C4—42no—
96C4×C22⋊C4—41no—
97C4×C22⋊C4—42no—
98C4×C22⋊C4—41no—
99C2×#95C2×C412yes—
100C2×#95C2×C411yescharacteristic
101C2×#95C2×C411yescharacteristic
102C2×#95C2×C412yes—
103C2×#95C2×C411yescharacteristic
104C2×#95C2×C412yes—
105C2×#95C2×C411yescharacteristic
106#336—24no—
107#336—24no—
108#336—24no—
109#336—24no—
110#336—42no—
111#336—42no—
112#336—41no—
113#336—42no—
114C25C22×C412yes—
115C25C22×C411yescharacteristic
116C22×D4—24no—
117C22×D4—24no—
118C22×D4—24no—
119C22×D4—22no—
120C22×D4—24no—
121C22×D4—41no—
122C22×D4—42no—
123C22×D4—42no—
124C22×D4—42no—
125C22×D4—41no—
126C22×D4—42no—
127C22×D4—42no—
128C23×C4—22no—
129C23×C4—22no—
130C23×C4—22no—
131C23×C4—22no—
132C23×C4—22no—
133C23×C4—22no—
134C23×C4—22no—
135C23×C4—22no—
136C23×C4—22no—
137C23×C4—22no—
138C23×C4—22no—
139C23×C4—22no—
140C23×C4—22no—
141C23×C4—22no—
142C23×C4—22no—
143C23×C4—22no—
144C23×C4—22no—
145C23×C4—22no—
146C23×C4—21no—
147C23×C4—21no—
148C23×C4C22×C411yescharacteristic
149C23×C4C22×C412yes—
150C23×C4C22×C411yescharacteristic
151C23×C4C22×C411yescharacteristic
152C23×C4C22×C411yescharacteristic
153C23×C4C22×C412yes—
154C23×C4C22×C411yescharacteristic
155C23×C4C22×C412yes—
156C23×C4C2×D414yes—
157C23×C4#4114yes—
158C23×C4C2411yescharacteristic
159C2×M16—48no—
160C2×M16—48no—
161C2×M16—48no—
162C2×M16—48no—
163C2×M16—48no—
164C2×M16—48no—
165C2×M16—48no—
166C22×C8—22no—
167C22×C8—22no—
168C22×C8—22no—
169C22×C8—22no—
170C22×C8—22no—
171C22×C8—22no—
172C22×C8—22no—
173C22×C8—22no—
174C22×C8—22no—
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—22no—
188C22×C8—22no—
189C22×C8—22no—
190C22×C8—22no—
191C22×C8C2×D414yes—
192C22×C8C2×D414yes—
193C22×C8#4114yes—
194C4×D4—44no—
195C4×D4—48no—
196C4×D4—44no—
197C4×D4—48no—
198C4×D4—48no—
199C4×D4—48no—
200C4×D4—48no—
201C2×C4⋊C4—24no—
202C2×C4⋊C4—24no—
203C2×C4⋊C4—41no—
204C2×C4⋊C4—42no—
205C2×C4⋊C4—42no—
206C2×C22⋊C4—24no—
207C2×C22⋊C4—24no—
208C2×C22⋊C4—22no—
209C2×C22⋊C4—24no—
210C2×C22⋊C4—24no—
211C2×C22⋊C4—22no—
212C2×C22⋊C4—24no—
213C2×C22⋊C4—41no—
214C2×C22⋊C4—42no—
215C2×C22⋊C4—41no—
216C2×C22⋊C4—42no—
217C2×C22⋊C4—42no—
218C2×C22⋊C4—42no—
219C2×C22⋊C4—42no—
220C2×C22⋊C4—42no—
221C2×C22⋊C4—42no—
222C2×C22⋊C4—42no—
223C2×C22⋊C4—42no—
224C2×C22⋊C4—41no—
225C2×C22⋊C4—41no—
226C2×C22⋊C4—42no—
227C2×C42—22no—
228C2×C42—22no—
229C2×C42—22no—
230C2×C42—22no—
231C2×C42—22no—
232C2×C42—22no—
233C2×C42—22no—
234C2×C42—22no—
235C2×C42—21no—
236C2×C42—21no—
237C2×C42—22no—
238C2×C42—21no—
239C2×C42—22no—
240C2×C42—21no—
241C22⋊C8—48no—
242C22⋊C8—44no—
243C22⋊C8—48no—
244C22⋊C8—48no—
245C22⋊C8—48no—
246C22⋊C8—44no—
247#97—44no—
248#95—41no—
249#95—42no—
250#95—42no—
251#95—42no—
252#95—41no—
253#95—41no—
254#95M1614yescomplemented
255C24—22no—
256C24—22no—
257C24—22no—
258C24—21no—
259C24—21no—
260C24—22no—
261C24—21no—
262C24—22no—
263C24—22no—
264C24—22no—
265C24—22no—
266C24—22no—
267C24—21no—
268C24—22no—
269C24—22no—
270C24—22no—
271C24—22no—
272C24—21no—
273C24—21no—
274C24—22no—
275C24—22no—
276C24—22no—
277C24—22no—
278C24—22no—
279C24C4×D414yes—
280C24#13112yes—
281C24C23×C411yescharacteristic
282C2×D4—48no—
283C2×D4—48no—
284C2×D4—48no—
285C2×D4—48no—
286C2×D4—44no—
287C2×D4—44no—
288C2×D4—48no—
289C2×D4—48no—
290C2×D4—48no—
291C2×D4—48no—
292C2×D4—48no—
293C2×D4—44no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—21no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—21no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—24no—
311C22×C4—24no—
312C22×C4—24no—
313C22×C4—24no—
314C22×C4—24no—
315C22×C4—24no—
316C22×C4—24no—
317C22×C4—24no—
318C22×C4—24no—
319C22×C4—24no—
320C22×C4—24no—
321C22×C4—24no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—21no—
326C22×C4—21no—
327C22×C4—22no—
328C22×C4—21no—
329C22×C4—22no—
330C22×C4—21no—
331C22×C4—21no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—21no—
335C22×C4—22no—
336C22×C4—21no—
337C22×C4—21no—
338C22×C4—21no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—21no—
343C22×C4—22no—
344C22×C4—21no—
345C22×C4—22no—
346C22×C4—21no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—42no—
350C22×C4—42no—
351C22×C4—42no—
352C22×C4—42no—
353C22×C4—42no—
354C22×C4—42no—
355C22×C4—42no—
356C22×C4—42no—
357C22×C4—42no—
358C22×C4—42no—
359C22×C4—42no—
360C22×C4—42no—
361C22×C4—42no—
362C22×C4—42no—
363C22×C4—42no—
364C22×C4—42no—
365C22×C4—42no—
366C22×C4—42no—
367C22×C4—42no—
368C22×C4—42no—
369C22×C4—42no—
370C22×C4—42no—
371C22×C4—41no—
372C22×C4—42no—
373C22×C4—42no—
374C22×C4—42no—
375C22×C4—41no—
376C22×C4—42no—
377C22×C4—42no—
378C22×C4—41no—
379C22×C4—42no—
380C22×C4—41no—
381C22×C4C4×D414yes—
382C22×C4C2×M1612yes—
383C22×C4C2×M1614yes—
384C22×C4C22×D412yes—
385C22×C4C2×#4112yes—
386C22×C4C2×#4112yes—
387C22×C4H(5,𝔽2)12yes—
388C22×C4H(5,𝔽2)12yes—
389C22×C4H(5,𝔽2)12yes—
390C22×C4H(5,𝔽2)11yescharacteristic
391C22×C4H(5,𝔽2)11yescharacteristic
392M16—432no—
393C2×C8—24no—
394C2×C8—24no—
395C2×C8—24no—
396C2×C8—24no—
397C2×C8—24no—
398C2×C8—24no—
399C2×C8—24no—
400C2×C8—24no—
401C2×C8—24no—
402C2×C8—42no—
403C2×C8—42no—
404C2×C8—42no—
405C2×C8—42no—
406C2×C8—42no—
407C2×C8—42no—
408C2×C8—42no—
409C2×C8—42no—
410C2×C8—42no—
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—
421C4⋊C4—44no—
422C4⋊C4—48no—
423C22⋊C4—48no—
424C22⋊C4—48no—
425C22⋊C4—44no—
426C22⋊C4—48no—
427C22⋊C4—44no—
428C22⋊C4—48no—
429C22⋊C4—48no—
430C42—42no—
431C42—42no—
432C42—42no—
433C42—42no—
434C42—41no—
435C42—42no—
436C42—41no—
437C42—41no—
438C42—42no—
439C42—41no—
440C23—24no—
441C23—24no—
442C23—24no—
443C23—24no—
444C23—24no—
445C23—24no—
446C23—24no—
447C23—24no—
448C23—22no—
449C23—22no—
450C23—22no—
451C23—24no—
452C23—42no—
453C23—42no—
454C23—41no—
455C23—42no—
456C23—41no—
457C23—42no—
458C23—42no—
459C23—41no—
460C23—42no—
461C23—42no—
462C23—42no—
463C23—42no—
464C23—42no—
465C23—42no—
466C23—42no—
467C23—42no—
468C23—42no—
469C23—42no—
470C23—42no—
471C23—42no—
472C23—41no—
473C23—42no—
474C23—42no—
475C23—41no—
476C23—42no—
477C23—42no—
478C23—42no—
479C23—42no—
480C23—42no—
481C23—42no—
482C23—42no—
483C23—42no—
484C23—42no—
485C23—42no—
486C23—42no—
487C23—42no—
488C23—42no—
489C23—42no—
490C23—41no—
491C23—42no—
492C23—42no—
493C23—41no—
494C23—41no—
495C23—42no—
496C23—42no—
497C23—42no—
498C23C2×C4×D412yes—
499C23C4×#4111yescharacteristic
500C23#51811yescharacteristic
501C23#51812yes—
502C23#51811yescharacteristic
503C23C22×M1611yescharacteristic
504C23C2×#13111yescharacteristic
505C23#56811yescharacteristic
506C23#56812yes—
507C23#56811yescharacteristic
508C23#56812yes—
509D4—432no—
510D4—416no—
511C2×C4—24no—
512C2×C4—24no—
513C2×C4—24no—
514C2×C4—24no—
515C2×C4—24no—
516C2×C4—24no—
517C2×C4—24no—
518C2×C4—22no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—41no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—41no—
530C2×C4—44no—
531C2×C4—44no—
532C2×C4—44no—
533C2×C4—44no—
534C2×C4—44no—
535C2×C4—44no—
536C2×C4—44no—
537C2×C4—44no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—42no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—41no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—82no—
566C2×C4—82no—
567C2×C4—82no—
568C2×C4—82no—
569C2×C4#53412yes—
570C2×C4#53512yes—
571C2×C4#53812yes—
572C2×C4#53812yes—
573C2×C4#53812yes—
574C2×C4#54012yes—
575C2×C4D4211yescharacteristic
576C2×C4#54612yes—
577C2×C4#54612yes—
578C2×C4#54612yes—
579C2×C4#55111yescharacteristic
580C2×C4#55312yes—
581C2×C4#55312yes—
582C2×C4#56012yes—
583C2×C4#56011yescharacteristic
584C2×C4#56011yescharacteristic
585C8—44no—
586C8—44no—
587C8—44no—
588C8—82no—
589C22—44no—
590C22—44no—
591C22—44no—
592C22—42no—
593C22—42no—
594C22—44no—
595C22—42no—
596C22—44no—
597C22—44no—
598C22—44no—
599C22—44no—
600C22—44no—
601C22—82no—
602C22—82no—
603C22—82no—
604C22—82no—
605C22—82no—
606C22—81no—
607C22—82no—
608C22—82no—
609C22—81no—
610C22—81no—
611C22—82no—
612C22—82no—
613C22—82no—
614C22—81no—
615C22#232112yes—
616C22#233311yescharacteristic
617C22#235811yescharacteristic
618C22#236412yes—
619C22#238011yescharacteristic
620C22#293012yes—
621C22#293212yes—
622C22#293312yes—
623C22D4×M1612yes—
624C22#296712yes—
625C22#296811yescharacteristic
626C22#297111yescharacteristic
627C22#297411yescharacteristic
628C22#297512yes—
629C22#297511yescharacteristic
630C22#297612yes—
631C22#297611yescharacteristic
632C22#297812yes—
633C22#297812yes—
634C22#297811yescharacteristic
635C22#298011yescharacteristic
636C22#298012yes—
637C22#298011yescharacteristic
638C4—44no—
639C4—42no—
640C4—44no—
641C4—82no—
642C4—81no—
643C4—81no—
644C4—81no—
645C4—81no—
646C4#267812yes—
647C4#268812yes—
648C4#270312yes—
649C4#277812yes—
650C2—84no—
651C2—82no—
652C2#1453311yescharacteristic
653C2#2203512yes—
654C2#2203812yes—
655C2#2233111yescharacteristic
656C2#2233811yescharacteristic
657C2#2285111yescharacteristic
658C2#2285211yescharacteristic
659C2#2312112yes—
660C2#2312412yes—
661C2#2335711yescharacteristic
662C2#2335911yescharacteristic
subgroupG/H
center, Frattini, cosocleC23×C4C24
commutator, socleC24C23×C4
prime radicalC2×#392C4
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
1. C2×#1870C2semidirect
2. C2×#1870C2semidirect
4. #11298C2semidirect
5. #11147C2semidirect
6. #10964C2semidirect
7. #10922C2semidirect
8. #10922C2semidirect
9. #10715C2semidirect
91. C2×C22⋊C8D4semidirect
254. #95M16semidirect

Maximal subgroups

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

subgroupindex
C2×#18702
C4×#3922
#112982
#111472
#109642
#109222
#107152