Finite Group Databasebeta

← #391657

Subgroups — #391657

Order 512.

1470 conjugacy classes of subgroups.

The subgroup lattice has 651 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#392C211yesmaximal, characteristic
2#11298C212yesmaximal, complemented
3#11147C212yesmaximal, complemented
4#11147C211yesmaximal, characteristic, complemented
5#10964C212yesmaximal, complemented
6#10964C212yesmaximal, complemented
7#10964C212yesmaximal, complemented
8#10921C212yesmaximal, complemented
9#10921C211yesmaximal, characteristic, complemented
10C22×C4×D4C2212yes—
11C22×C4×D4C2211yescharacteristic
12C22×C22⋊C8C2212yes—
13C22×C22⋊C8C2212yes—
14C22×C22⋊C8C2212yes—
15C22×C22⋊C8C2211yescharacteristic
16C22×C22⋊C8C2211yescharacteristic
17C22×C22⋊C8C2212yes—
18C22×C22⋊C8C2212yes—
19C22×C4×C8C2211yescharacteristic
20C2×#392C411yescharacteristic
21C2×C4×C22⋊C4C2212yes—
22C2×C4×C22⋊C4C2211yescharacteristic
23C2×#336C2212yes—
24C2×#336C2212yes—
25C2×#336C2212yes—
26C2×#336C2212yes—
27C2×#336C2212yes—
28C2×#336C2212yes—
29C2×#336C2211yescharacteristic
30C2×#336C2212yes—
31#1444C411yescharacteristic
32#1439C412yes—
33#1439C411yescharacteristic
34#1438C411yescharacteristic
35#1438C412yes—
36C4×#95C2211yescharacteristic
37C23×D4C2×C411yescharacteristic
38C23×D4C2×C412yes—
39C24×C4C2311yescharacteristic
40C24×C4C2312yes—
41C23×C8C2312yes—
42C23×C8C2312yes—
43C23×C8C2312yes—
44C23×C8C2312yes—
45C2×C4×D4—42no—
46C2×C4×D4—42no—
47C2×C4×D4—42no—
48C2×C4×D4—42no—
49C2×C4×D4—42no—
50C2×C4×D4—42no—
51C2×C4×D4—42no—
52C2×C4×D4—41no—
53C2×C4×D4—42no—
54C2×C4×D4—41no—
55C22×C4⋊C4C2×C411yescharacteristic
56C22×C4⋊C4C2×C412yes—
57C22×C22⋊C4C2×C412yes—
58C22×C22⋊C4C2×C412yes—
59C22×C22⋊C4C2×C411yescharacteristic
60C22×C22⋊C4C2×C412yes—
61C22×C22⋊C4C2×C412yes—
62C22×C22⋊C4C2×C412yes—
63C22×C22⋊C4C2×C411yescharacteristic
64C22×C42C2311yescharacteristic
65C22×C42C2311yescharacteristic
66C22×C42C2312yes—
67C2×C22⋊C8—24no—
68C2×C22⋊C8—24no—
69C2×C22⋊C8—24no—
70C2×C22⋊C8—42no—
71C2×C22⋊C8—42no—
72C2×C22⋊C8—42no—
73C2×C22⋊C8—42no—
74C2×C22⋊C8—42no—
75C2×C22⋊C8—42no—
76C2×C22⋊C8—42no—
77C2×C22⋊C8—42no—
78C2×C22⋊C8—42no—
79C2×C22⋊C8—42no—
80C2×C22⋊C8—42no—
81C2×C22⋊C8—42no—
82C2×C22⋊C8—41no—
83C2×C22⋊C8—42no—
84C2×C22⋊C8—41no—
85C2×C22⋊C8—42no—
86C2×C4×C8—41no—
87C2×C4×C8—42no—
88#392—24no—
89C4×C22⋊C4—41no—
90C4×C22⋊C4—42no—
91C4×C22⋊C4—41no—
92C4×C22⋊C4—42no—
93C2×#95C2×C411yescharacteristic
94C2×#95C2×C412yes—
95C2×#95C2×C411yescharacteristic
96C2×#95C2×C411yescharacteristic
97C2×#95C2×C412yes—
98C2×#95C2×C411yescharacteristic
99C2×#95C2×C412yes—
100#336—42no—
101#336—42no—
102#336—42no—
103#336—42no—
104#336—42no—
105#336—42no—
106#336—41no—
107#336—42no—
108C25C22×C411yescharacteristic
109C25C22×C412yes—
110C22×D4—24no—
111C22×D4—24no—
112C22×D4—24no—
113C22×D4—22no—
114C22×D4—24no—
115C22×D4—42no—
116C22×D4—41no—
117C22×D4—42no—
118C22×D4—42no—
119C22×D4—42no—
120C22×D4—41no—
121C22×D4—42no—
122C23×C4—22no—
123C23×C4—22no—
124C23×C4—22no—
125C23×C4—22no—
126C23×C4—22no—
127C23×C4—22no—
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—21no—
141C23×C4—22no—
142C23×C4—21no—
143C23×C4—22no—
144C23×C4C22×C411yescharacteristic
145C23×C4C22×C412yes—
146C23×C4C22×C411yescharacteristic
147C23×C4C22×C412yes—
148C23×C4C22×C411yescharacteristic
149C23×C4C22×C411yescharacteristic
150C23×C4C22×C411yescharacteristic
151C23×C4C22×C412yes—
152C23×C4C2411yescharacteristic
153C22×C8—22no—
154C22×C8—22no—
155C22×C8—22no—
156C22×C8—22no—
157C22×C8—22no—
158C22×C8—22no—
159C22×C8—22no—
160C22×C8—22no—
161C22×C8—22no—
162C22×C8—22no—
163C22×C8—22no—
164C22×C8—22no—
165C22×C8—22no—
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#4114yes—
181C4×D4—48no—
182C4×D4—48no—
183C4×D4—48no—
184C4×D4—44no—
185C4×D4—48no—
186C4×D4—44no—
187C4×D4—48no—
188C2×C4⋊C4—41no—
189C2×C4⋊C4—42no—
190C2×C4⋊C4—42no—
191C2×C4⋊C4—42no—
192C2×C4⋊C4—42no—
193C2×C22⋊C4—24no—
194C2×C22⋊C4—24no—
195C2×C22⋊C4—24no—
196C2×C22⋊C4—24no—
197C2×C22⋊C4—22no—
198C2×C22⋊C4—41no—
199C2×C22⋊C4—41no—
200C2×C22⋊C4—42no—
201C2×C22⋊C4—42no—
202C2×C22⋊C4—42no—
203C2×C22⋊C4—42no—
204C2×C22⋊C4—42no—
205C2×C22⋊C4—42no—
206C2×C22⋊C4—42no—
207C2×C22⋊C4—42no—
208C2×C22⋊C4—42no—
209C2×C22⋊C4—41no—
210C2×C22⋊C4—42no—
211C2×C22⋊C4—42no—
212C2×C22⋊C4—41no—
213C2×C22⋊C4M1614yes—
214C2×C42—22no—
215C2×C42—22no—
216C2×C42—22no—
217C2×C42—22no—
218C2×C42—22no—
219C2×C42—22no—
220C2×C42—21no—
221C2×C42—22no—
222C2×C42—21no—
223C2×C42—22no—
224C2×C42—21no—
225C2×C42—22no—
226C2×C42—21no—
227C2×C42—22no—
228C22⋊C8—48no—
229C22⋊C8—48no—
230C22⋊C8—48no—
231C22⋊C8—44no—
232C22⋊C8—48no—
233C22⋊C8—44no—
234C22⋊C8—48no—
235C4×C8—44no—
236#95—22no—
237#95—24no—
238#95—41no—
239#95—41no—
240#95—42no—
241#95—41no—
242#95—42no—
243C24—21no—
244C24—22no—
245C24—22no—
246C24—21no—
247C24—21no—
248C24—22no—
249C24—22no—
250C24—22no—
251C24—22no—
252C24—22no—
253C24—22no—
254C24—22no—
255C24—22no—
256C24—21no—
257C24—22no—
258C24—22no—
259C24—22no—
260C24—22no—
261C24—22no—
262C24—22no—
263C24—21no—
264C24—21no—
265C24—22no—
266C24—22no—
267C24C2×M1612yes—
268C24#13114yes—
269C24C23×C411yescharacteristic
270C2×D4—48no—
271C2×D4—48no—
272C2×D4—48no—
273C2×D4—44no—
274C2×D4—48no—
275C2×D4—48no—
276C2×D4—44no—
277C2×D4—48no—
278C2×D4—48no—
279C2×D4—48no—
280C2×D4—44no—
281C2×D4—48no—
282C22×C4—21no—
283C22×C4—22no—
284C22×C4—21no—
285C22×C4—22no—
286C22×C4—21no—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—21no—
297C22×C4—21no—
298C22×C4—21no—
299C22×C4—22no—
300C22×C4—21no—
301C22×C4—22no—
302C22×C4—21no—
303C22×C4—22no—
304C22×C4—21no—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—21no—
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—21no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—21no—
324C22×C4—42no—
325C22×C4—42no—
326C22×C4—42no—
327C22×C4—42no—
328C22×C4—42no—
329C22×C4—42no—
330C22×C4—42no—
331C22×C4—42no—
332C22×C4—42no—
333C22×C4—42no—
334C22×C4—42no—
335C22×C4—42no—
336C22×C4—42no—
337C22×C4—42no—
338C22×C4—42no—
339C22×C4—42no—
340C22×C4—42no—
341C22×C4—42no—
342C22×C4—41no—
343C22×C4—42no—
344C22×C4—42no—
345C22×C4—42no—
346C22×C4—42no—
347C22×C4—42no—
348C22×C4—42no—
349C22×C4—42no—
350C22×C4—42no—
351C22×C4—42no—
352C22×C4—42no—
353C22×C4—42no—
354C22×C4—42no—
355C22×C4—42no—
356C22×C4—41no—
357C22×C4—42no—
358C22×C4—41no—
359C22×C4—42no—
360C22×C4—42no—
361C22×C4—42no—
362C22×C4—42no—
363C22×C4—41no—
364C22×C4—42no—
365C22×C4—42no—
366C22×C4—42no—
367C22×C4—42no—
368C22×C4C2×M1612yes—
369C22×C4C2×M1612yes—
370C22×C4#13114yes—
371C22×C4#13112yes—
372C22×C4C2×#4112yes—
373C22×C4H(5,𝔽2)11yescharacteristic
374C22×C4H(5,𝔽2)12yes—
375C22×C4H(5,𝔽2)12yes—
376C22×C4H(5,𝔽2)12yes—
377C22×C4H(5,𝔽2)12yes—
378C22×C4H(5,𝔽2)11yescharacteristic
379C22×C4H(5,𝔽2)12yes—
380C2×C8—24no—
381C2×C8—24no—
382C2×C8—24no—
383C2×C8—42no—
384C2×C8—42no—
385C2×C8—42no—
386C2×C8—42no—
387C2×C8—42no—
388C2×C8—42no—
389C2×C8—42no—
390C2×C8—42no—
391C2×C8—42no—
392C2×C8—42no—
393C2×C8—42no—
394C2×C8—42no—
395C2×C8—42no—
396C2×C8—42no—
397C2×C8—42no—
398C2×C8—42no—
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—
408C4⋊C4—44no—
409C4⋊C4—48no—
410C22⋊C4—44no—
411C22⋊C4—48no—
412C22⋊C4—48no—
413C22⋊C4—44no—
414C22⋊C4—48no—
415C22⋊C4—48no—
416C22⋊C4—48no—
417C42—41no—
418C42—41no—
419C42—42no—
420C42—42no—
421C42—42no—
422C42—42no—
423C42—41no—
424C42—42no—
425C42—41no—
426C42—42no—
427C23—22no—
428C23—24no—
429C23—22no—
430C23—24no—
431C23—24no—
432C23—24no—
433C23—22no—
434C23—24no—
435C23—24no—
436C23—24no—
437C23—24no—
438C23—24no—
439C23—42no—
440C23—42no—
441C23—42no—
442C23—42no—
443C23—41no—
444C23—42no—
445C23—42no—
446C23—42no—
447C23—41no—
448C23—42no—
449C23—41no—
450C23—42no—
451C23—42no—
452C23—42no—
453C23—42no—
454C23—42no—
455C23—41no—
456C23—42no—
457C23—42no—
458C23—42no—
459C23—42no—
460C23—42no—
461C23—41no—
462C23—42no—
463C23—42no—
464C23—42no—
465C23—42no—
466C23—42no—
467C23—42no—
468C23—42no—
469C23—42no—
470C23—42no—
471C23—41no—
472C23—42no—
473C23—42no—
474C23—42no—
475C23—42no—
476C23—42no—
477C23—41no—
478C23—42no—
479C23—42no—
480C23—42no—
481C23—42no—
482C23—42no—
483C23—42no—
484C23—41no—
485C23C4×#4111yescharacteristic
486C23#51812yes—
487C23#51811yescharacteristic
488C23#51811yescharacteristic
489C23#51812yes—
490C23C22×M1611yescharacteristic
491C23C2×#13111yescharacteristic
492C23C2×#13112yes—
493C23C2×#13112yes—
494C23#56811yescharacteristic
495C23#56811yescharacteristic
496D4—416no—
497D4—432no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—24no—
501C2×C4—24no—
502C2×C4—24no—
503C2×C4—22no—
504C2×C4—24no—
505C2×C4—24no—
506C2×C4—24no—
507C2×C4—41no—
508C2×C4—42no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—42no—
520C2×C4—41no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—41no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—41no—
531C2×C4—42no—
532C2×C4—41no—
533C2×C4—42no—
534C2×C4—41no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—41no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—82no—
545C2×C4—82no—
546C2×C4—82no—
547C2×C4—82no—
548C2×C4—82no—
549C2×C4—82no—
550C2×C4—82no—
551C2×C4—82no—
552C2×C4—82no—
553C2×C4—82no—
554C2×C4—82no—
555C2×C4—82no—
556C2×C4#53812yes—
557C2×C4#53812yes—
558C2×C4#53812yes—
559C2×C4#54012yes—
560C2×C4#54012yes—
561C2×C4#54012yes—
562C2×C4#56011yescharacteristic
563C2×C4#56012yes—
564C2×C4#56012yes—
565C2×C4#56012yes—
566C2×C4#56011yescharacteristic
567C2×C4#56012yes—
568C2×C4#56011yescharacteristic
569C2×C4#56012yes—
570C2×C4H(3,𝔽4)11yescharacteristic
571C2×C4H(3,𝔽4)12yes—
572C8—44no—
573C8—82no—
574C8—82no—
575C8—82no—
576C22—44no—
577C22—42no—
578C22—44no—
579C22—44no—
580C22—44no—
581C22—44no—
582C22—42no—
583C22—44no—
584C22—44no—
585C22—44no—
586C22—44no—
587C22—42no—
588C22—82no—
589C22—82no—
590C22—82no—
591C22—82no—
592C22—81no—
593C22—81no—
594C22—82no—
595C22—81no—
596C22—82no—
597C22—82no—
598C22—82no—
599C22—82no—
600C22—81no—
601C22—82no—
602C22#233311yescharacteristic
603C22#233312yes—
604C22#237511yescharacteristic
605C22#237512yes—
606C22#238011yescharacteristic
607C22#296511yescharacteristic
608C22#296612yes—
609C22#296711yescharacteristic
610C22#297311yescharacteristic
611C22#297412yes—
612C22#297412yes—
613C22#297511yescharacteristic
614C22#297612yes—
615C22#297612yes—
616C22#297612yes—
617C22#297611yescharacteristic
618C22#297711yescharacteristic
619C22#297712yes—
620C22#297712yes—
621C22#297912yes—
622C22#297911yescharacteristic
623C22#298111yescharacteristic
624C22#298112yes—
625C4—42no—
626C4—44no—
627C4—42no—
628C4—42no—
629C4—82no—
630C4—81no—
631C4—81no—
632C4—82no—
633C4#282512yes—
634C4#282812yes—
635C4#282812yes—
636C4#282812yes—
637C2—84no—
638C2—82no—
639C2#1465111yescharacteristic
640C2#2233011yescharacteristic
641C2#2233312yes—
642C2#2233412yes—
643C2#2233711yescharacteristic
644C2#2330711yescharacteristic
645C2#2330812yes—
646C2#2331012yes—
647C2#2331211yescharacteristic
648C2#2335611yescharacteristic
649C2#2335811yescharacteristic
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
2. #11298C2semidirect
3. #11147C2semidirect
4. #11147C2semidirect
5. #10964C2semidirect
6. #10964C2semidirect
7. #10964C2semidirect
8. #10921C2semidirect
9. #10921C2semidirect

Maximal subgroups

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

subgroupindex
C4×#3922
#112982
#111472
#109642
#109212