Finite Group Databasebeta

← #387914

Subgroups — #387914

Order 512.

1466 conjugacy classes of subgroups.

The subgroup lattice has 659 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2395C211yesmaximal, characteristic
2C2×#1954C211yesmaximal, characteristic
3C2×#1870C211yesmaximal, characteristic
4#11202C212yesmaximal, complemented
5#11147C212yesmaximal, complemented
6#11147C212yesmaximal, complemented
7#10874C212yesmaximal, complemented
8#10715C212yesmaximal, complemented
9#10713C212yesmaximal, complemented
10C23×M16C2211yescharacteristic
11C22×C4×D4C2211yescharacteristic
12C22×#117C2211yescharacteristic
13C22×C4⋊C8C2211yescharacteristic
14C22×C22⋊C8C2212yes—
15C22×C22⋊C8C2212yes—
16C22×C22⋊C8C2212yes—
17C22×C22⋊C8C2211yescharacteristic
18C22×C4×C8C2211yescharacteristic
19#2395—24no—
20#2395—24no—
21C2×#400C411yescharacteristic
22C2×#399C411yescharacteristic
23C2×#398C411yescharacteristic
24C2×#397C412yes—
25C2×#397C411yescharacteristic
26C2×#394C412yes—
27C2×C4×C4⋊C4C2212yes—
28C2×C4×C22⋊C4C2211yescharacteristic
29C2×C4×C22⋊C4C2212yes—
30#1954—24no—
31#1870—24no—
32C2×#336C2212yes—
33C2×#336C2212yes—
34C2×#336C2212yes—
35C2×#336C2212yes—
36C2×#336C2212yes—
37C2×#336C2212yes—
38C2×#336C2212yes—
39C2×#336C2212yes—
40C2×#336C2212yes—
41C23×D4C2×C411yescharacteristic
42C24×C4C2311yescharacteristic
43C24×C4C2311yescharacteristic
44C22×M16—22no—
45C22×M16—24no—
46C22×M16—41no—
47C22×M16—41no—
48C22×M16—42no—
49C23×C8C2312yes—
50C23×C8C2312yes—
51C23×C8C2312yes—
52C23×C8C2312yes—
53C2×C4×D4—41no—
54C2×C4×D4—41no—
55C2×C4×D4—41no—
56C2×C4×D4—41no—
57C2×C4×D4D414yes—
58C2×C4×D4D414yes—
59C2×#117—22no—
60C2×#117—22no—
61C2×#117—41no—
62C2×#117—41no—
63C2×#117—41no—
64C2×#117—41no—
65C22×C4⋊C4C2×C412yes—
66C22×C4⋊C4C2×C412yes—
67C22×C4⋊C4C2×C411yescharacteristic
68C22×C4⋊C4C2×C411yescharacteristic
69C22×C4⋊C4C2×C411yescharacteristic
70C22×C22⋊C4C2×C412yes—
71C22×C22⋊C4C2×C412yes—
72C22×C22⋊C4C2×C412yes—
73C22×C22⋊C4C2×C412yes—
74C22×C22⋊C4C2×C411yescharacteristic
75C22×C22⋊C4C2×C411yescharacteristic
76C22×C42C2312yes—
77C22×C42C2312yes—
78C22×C42C2311yescharacteristic
79C2×C4⋊C8—22no—
80C2×C4⋊C8—41no—
81C2×C4⋊C8—41no—
82C2×C22⋊C8—24no—
83C2×C22⋊C8—22no—
84C2×C22⋊C8—24no—
85C2×C22⋊C8—24no—
86C2×C22⋊C8—24no—
87C2×C22⋊C8—24no—
88C2×C22⋊C8—24no—
89C2×C22⋊C8—42no—
90C2×C22⋊C8—42no—
91C2×C22⋊C8—42no—
92C2×C22⋊C8—41no—
93C2×C22⋊C8—41no—
94C2×C4×C8—22no—
95C2×C4×C8—41no—
96C2×C4×C8—41no—
97#400—24no—
98#399—24no—
99#398—24no—
100#397—28no—
101#397—24no—
102#394—28no—
103C4×C4⋊C4—24no—
104C4×C4⋊C4—24no—
105C4×C22⋊C4—24no—
106C4×C22⋊C4—24no—
107C4×C22⋊C4—22no—
108C4×C22⋊C4—22no—
109C2×#95C2×C412yes—
110C2×#95C2×C412yes—
111C2×#95C2×C412yes—
112C2×#95C2×C412yes—
113C2×#95C2×C411yescharacteristic
114C2×#95C2×C411yescharacteristic
115#336—24no—
116#336—24no—
117#336—24no—
118#336—42no—
119#336—42no—
120#336—42no—
121#336—42no—
122#336—42no—
123#336—42no—
124C25C22×C411yescharacteristic
125C25C22×C411yescharacteristic
126C22×D4—22no—
127C22×D4—22no—
128C22×D4—41no—
129C22×D4—41no—
130C22×D4—41no—
131C22×D4—41no—
132C22×D4C22⋊C414yes—
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—21no—
144C23×C4—21no—
145C23×C4—22no—
146C23×C4—22no—
147C23×C4—22no—
148C23×C4—22no—
149C23×C4C22×C412yes—
150C23×C4C22×C412yes—
151C23×C4C22×C412yes—
152C23×C4C22×C411yescharacteristic
153C23×C4C22×C411yescharacteristic
154C23×C4C22×C411yescharacteristic
155C23×C4C22×C412yes—
156C23×C4C22×C411yescharacteristic
157C23×C4C2×D412yes—
158C23×C4C2×D412yes—
159C23×C4C2×D412yes—
160C23×C4C2×D412yes—
161C23×C4C2411yescharacteristic
162C2×M16—44no—
163C2×M16—48no—
164C2×M16—48no—
165C2×M16—44no—
166C2×M16—44no—
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×C8—22no—
192C22×C8—22no—
193C22×C8#4114yes—
194C22×C8#4114yes—
195C4×D4—44no—
196C4×D4—44no—
197C4×D4—44no—
198C4×D4—44no—
199#117—44no—
200#117—44no—
201#117—44no—
202#117—44no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—24no—
206C2×C4⋊C4—24no—
207C2×C4⋊C4—24no—
208C2×C4⋊C4—24no—
209C2×C4⋊C4—24no—
210C2×C4⋊C4—42no—
211C2×C4⋊C4—41no—
212C2×C4⋊C4—41no—
213C2×C4⋊C4—41no—
214C2×C4⋊C4—41no—
215C2×C4⋊C4—41no—
216C2×C4⋊C4—41no—
217C2×C4⋊C4C22⋊C414yes—
218C2×C22⋊C4—24no—
219C2×C22⋊C4—24no—
220C2×C22⋊C4—24no—
221C2×C22⋊C4—24no—
222C2×C22⋊C4—24no—
223C2×C22⋊C4—24no—
224C2×C22⋊C4—42no—
225C2×C22⋊C4—42no—
226C2×C22⋊C4—42no—
227C2×C22⋊C4—42no—
228C2×C22⋊C4—42no—
229C2×C22⋊C4—42no—
230C2×C22⋊C4—41no—
231C2×C22⋊C4—41no—
232C2×C22⋊C4—41no—
233C2×C22⋊C4—41no—
234C2×C22⋊C4C22⋊C414yes—
235C2×C22⋊C4C22⋊C414yes—
236C2×C42—21no—
237C2×C42—21no—
238C2×C42—21no—
239C2×C42—21no—
240C2×C42—22no—
241C2×C42—22no—
242C2×C42—22no—
243C2×C42—22no—
244C2×C42—22no—
245C2×C42—22no—
246C2×C42—22no—
247C2×C42—22no—
248C2×C42—22no—
249C2×C42—22no—
250C2×C42—22no—
251C2×C42—22no—
252C2×C42C2×D412yes—
253C2×C42C2×D412yes—
254C4⋊C8—44no—
255C22⋊C8—48no—
256C22⋊C8—48no—
257C22⋊C8—48no—
258C22⋊C8—44no—
259C4×C8—44no—
260#95—24no—
261#95—22no—
262#95—24no—
263#95—22no—
264#95—24no—
265#95—24no—
266C24—22no—
267C24—22no—
268C24—21no—
269C24—21no—
270C24—21no—
271C24—21no—
272C24—22no—
273C24—22no—
274C24—22no—
275C24—22no—
276C24—21no—
277C24—21no—
278C24—21no—
279C24—21no—
280C24—21no—
281C24—21no—
282C24—22no—
283C24—22no—
284C24—21no—
285C24—21no—
286C24C2×C22⋊C412yes—
287C24C2×C22⋊C412yes—
288C24C23×C411yescharacteristic
289C2×D4—28no—
290C2×D4—44no—
291C2×D4—44no—
292C2×D4—44no—
293C2×D4—44no—
294C2×D4—44no—
295C2×D4—44no—
296C22×C4—21no—
297C22×C4—21no—
298C22×C4—21no—
299C22×C4—21no—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—21no—
305C22×C4—21no—
306C22×C4—21no—
307C22×C4—21no—
308C22×C4—21no—
309C22×C4—21no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—24no—
327C22×C4—24no—
328C22×C4—24no—
329C22×C4—24no—
330C22×C4—24no—
331C22×C4—24no—
332C22×C4—24no—
333C22×C4—24no—
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—42no—
356C22×C4—42no—
357C22×C4—42no—
358C22×C4—42no—
359C22×C4—42no—
360C22×C4—42no—
361C22×C4—42no—
362C22×C4—42no—
363C22×C4—41no—
364C22×C4—41no—
365C22×C4—42no—
366C22×C4—42no—
367C22×C4—41no—
368C22×C4—41no—
369C22×C4—41no—
370C22×C4—41no—
371C22×C4—42no—
372C22×C4—42no—
373C22×C4—41no—
374C22×C4—41no—
375C22×C4C2×C22⋊C412yes—
376C22×C4C2×C22⋊C412yes—
377C22×C4C2×C22⋊C412yes—
378C22×C4C2×C22⋊C412yes—
379C22×C4#13114yes—
380C22×C4#13112yes—
381C22×C4#13112yes—
382C22×C4C22×D411yescharacteristic
383C22×C4C22×D411yescharacteristic
384C22×C4C2×#4112yes—
385C22×C4C2×#4112yes—
386C22×C4H(5,𝔽2)11yescharacteristic
387C22×C4H(5,𝔽2)11yescharacteristic
388C22×C4H(5,𝔽2)11yescharacteristic
389C22×C4H(5,𝔽2)11yescharacteristic
390C22×C4H(5,𝔽2)12yes—
391C22×C42−1+412yes—
392M16—416no—
393C2×C8—24no—
394C2×C8—24no—
395C2×C8—24no—
396C2×C8—24no—
397C2×C8—24no—
398C2×C8—24no—
399C2×C8—42no—
400C2×C8—42no—
401C2×C8—42no—
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—
423C4⋊C4—44no—
424C4⋊C4—44no—
425C4⋊C4—48no—
426C22⋊C4—44no—
427C22⋊C4—44no—
428C22⋊C4—48no—
429C22⋊C4—48no—
430C22⋊C4—48no—
431C22⋊C4—48no—
432C42—22no—
433C42—22no—
434C42—22no—
435C42—22no—
436C42—42no—
437C42—42no—
438C42—42no—
439C42—42no—
440C42—42no—
441C42—42no—
442C42—42no—
443C42—42no—
444C23—22no—
445C23—22no—
446C23—24no—
447C23—24no—
448C23—22no—
449C23—22no—
450C23—22no—
451C23—22no—
452C23—24no—
453C23—24no—
454C23—42no—
455C23—42no—
456C23—41no—
457C23—41no—
458C23—41no—
459C23—41no—
460C23—42no—
461C23—42no—
462C23—42no—
463C23—42no—
464C23—42no—
465C23—42no—
466C23—42no—
467C23—42no—
468C23—41no—
469C23—41no—
470C23—41no—
471C23—41no—
472C23—41no—
473C23—41no—
474C23—41no—
475C23—41no—
476C23—41no—
477C23—41no—
478C23—41no—
479C23—41no—
480C23—42no—
481C23—42no—
482C23—42no—
483C23—42no—
484C23—42no—
485C23—42no—
486C23—42no—
487C23—42no—
488C23—42no—
489C23—42no—
490C23C22×C22⋊C411yescharacteristic
491C23C4×#4112yes—
492C23#51811yescharacteristic
493C23#51811yescharacteristic
494C23#52012yes—
495C23C2×#13111yescharacteristic
496C23C2×#13111yescharacteristic
497C23C2×#13112yes—
498C23#56811yescharacteristic
499C23#56811yescharacteristic
500C23#56812yes—
501D4—416no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—22no—
511C2×C4—24no—
512C2×C4—24no—
513C2×C4—24no—
514C2×C4—24no—
515C2×C4—24no—
516C2×C4—24no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—42no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—44no—
533C2×C4—44no—
534C2×C4—44no—
535C2×C4—44no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—42no—
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#40814yes—
564C2×C4#53411yescharacteristic
565C2×C4#53411yescharacteristic
566C2×C4#53411yescharacteristic
567C2×C4#53411yescharacteristic
568C2×C4#53812yes—
569C2×C4#53912yes—
570C2×C4#54012yes—
571C2×C4#54212yes—
572C2×C4#54612yes—
573C2×C4#54612yes—
574C2×C4#54712yes—
575C2×C4#54712yes—
576C2×C4#55111yescharacteristic
577C2×C4#55312yes—
578C2×C4#55911yescharacteristic
579C2×C4#56011yescharacteristic
580C2×C4#56212yes—
581C2×C4#56311yescharacteristic
582C8—44no—
583C8—44no—
584C8—82no—
585C8—82no—
586C22—42no—
587C22—42no—
588C22—44no—
589C22—44no—
590C22—44no—
591C22—44no—
592C22—42no—
593C22—42no—
594C22—42no—
595C22—42no—
596C22—81no—
597C22—81no—
598C22—81no—
599C22—81no—
600C22—81no—
601C22—81no—
602C22—82no—
603C22—82no—
604C22—82no—
605C22—82no—
606C22—81no—
607C22—81no—
608C22#230811yescharacteristic
609C22#233512yes—
610C22#233912yes—
611C22#237411yescharacteristic
612C22#237611yescharacteristic
613C22C2×#40811yescharacteristic
614C22#287911yescharacteristic
615C22#288012yes—
616C22#296612yes—
617C22#296712yes—
618C22#296912yes—
619C22#296912yes—
620C22#297611yescharacteristic
621C22#297611yescharacteristic
622C22#297611yescharacteristic
623C22#297611yescharacteristic
624C22#297911yescharacteristic
625C22#298011yescharacteristic
626C22#298112yes—
627C22#299312yes—
628C22#299312yes—
629C22#299412yes—
630C22#299512yes—
631C4—24no—
632C4—42no—
633C4—42no—
634C4—44no—
635C4—42no—
636C4—82no—
637C4—82no—
638C4—82no—
639C4#269711yescharacteristic
640C4#270112yes—
641C4#270812yes—
642C4#270911yescharacteristic
643C4#271711yescharacteristic
644C4#272411yescharacteristic
645C2—82no—
646C2—82no—
647C2#1450511yescharacteristic
648C2#2174111yescharacteristic
649C2#2174511yescharacteristic
650C2#2236012yes—
651C2#2236612yes—
652C2#2247212yes—
653C2#2247412yes—
654C2#2329511yescharacteristic
655C2#2330311yescharacteristic
656C2#2331711yescharacteristic
657C2#2332511yescharacteristic
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
4. #11202C2semidirect
5. #11147C2semidirect
6. #11147C2semidirect
7. #10874C2semidirect
8. #10715C2semidirect
9. #10713C2semidirect

Maximal subgroups

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

subgroupindex
C2×#23952
C2×#19542
C2×#18702
#112022
#111472
#108742
#107152
#107132