Finite Group Databasebeta

← #387963

Subgroups — #387963

Order 512.

1624 conjugacy classes of subgroups.

The subgroup lattice has 641 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×#123C211yesmaximal, characteristic
2C2×#1924C212yesmaximal, complemented
3#11346C212yesmaximal, complemented
4#11042C212yesmaximal, complemented
5#10964C211yesmaximal, characteristic
6#10964C211yesmaximal, characteristic
7#10922C212yesmaximal, complemented
8#10835C212yesmaximal, complemented
9#10712C212yesmaximal, complemented
10C22×#123C411yescharacteristic
11C22×C4×D4C2211yescharacteristic
12C23×C42C2211yescharacteristic
13C22×C4⋊C8C2212yes—
14C22×C22⋊C8C2212yes—
15C22×C22⋊C8C2212yes—
16C22×C22⋊C8C2211yescharacteristic
17C22×C22⋊C8C2212yes—
18C22×C22⋊C8C2211yescharacteristic
19C22×#97C2212yes—
20C4×#123—28no—
21C2×#388C412yes—
22C2×#387C412yes—
23C2×#386C411yescharacteristic
24C2×#385C411yescharacteristic
25C2×#381C411yescharacteristic
26C2×C4×C4⋊C4C2212yes—
27C2×C4×C22⋊C4C2212yes—
28C2×C4×C22⋊C4C2211yescharacteristic
29#1924—28no—
30C2×#336C2212yes—
31C2×#336C2212yes—
32C2×#336C2212yes—
33C2×#336C2212yes—
34C2×#336C2211yescharacteristic
35C2×#336C2212yes—
36C2×#336C2212yes—
37C2×#336C2211yescharacteristic
38C2×#336C2212yes—
39C23×D4C2×C411yescharacteristic
40C24×C4C2×C411yescharacteristic
41C24×C4C2×C411yescharacteristic
42C24×C4C2311yescharacteristic
43C24×C4C2311yescharacteristic
44C23×C8C2312yes—
45C23×C8C2312yes—
46C23×C8C2312yes—
47C23×C8C2312yes—
48C2×#123—24no—
49C2×#123—24no—
50C2×#123—24no—
51C2×C4×D4—24no—
52C2×C4×D4—42no—
53C2×C4×D4—41no—
54C2×C4×D4—41no—
55C22×C4⋊C4C2×C412yes—
56C22×C4⋊C4C2×C412yes—
57C22×C4⋊C4C2×C411yescharacteristic
58C22×C22⋊C4C2×C412yes—
59C22×C22⋊C4C2×C412yes—
60C22×C22⋊C4C2×C411yescharacteristic
61C22×C22⋊C4C2×C411yescharacteristic
62C22×C22⋊C4C2×C411yescharacteristic
63C22×C22⋊C4C2×C411yescharacteristic
64C22×C42—24no—
65C22×C42—42no—
66C22×C42—41no—
67C22×C42—41no—
68C22×C42C2×C412yes—
69C22×C42C2312yes—
70C22×C42C2311yescharacteristic
71C22×C42C2312yes—
72C2×C4⋊C8—24no—
73C2×C4⋊C8—24no—
74C2×C4⋊C8—42no—
75C2×C22⋊C8—24no—
76C2×C22⋊C8—24no—
77C2×C22⋊C8—24no—
78C2×C22⋊C8—24no—
79C2×C22⋊C8—24no—
80C2×C22⋊C8—24no—
81C2×C22⋊C8—41no—
82C2×C22⋊C8—42no—
83C2×C22⋊C8—42no—
84C2×C22⋊C8—42no—
85C2×C22⋊C8—42no—
86C2×C22⋊C8—41no—
87C2×C22⋊C8—42no—
88C2×#97—24no—
89C2×#97—24no—
90C2×#97—42no—
91#388—28no—
92#387—28no—
93#386—24no—
94#385—24no—
95#381—24no—
96C4×C4⋊C4—24no—
97C4×C4⋊C4—24no—
98C4×C22⋊C4—24no—
99C4×C22⋊C4—24no—
100C4×C22⋊C4D418yes—
101C2×#95C2×C412yes—
102C2×#95C2×C412yes—
103C2×#95C2×C412yes—
104C2×#95C2×C412yes—
105C2×#95C2×C412yes—
106#336—24no—
107#336—24no—
108#336—41no—
109#336—42no—
110#336—41no—
111#336—42no—
112#336—42no—
113#336—42no—
114#336—42no—
115C25C22×C411yescharacteristic
116C25C22×C411yescharacteristic
117C22×D4—22no—
118C22×D4—22no—
119C22×D4—41no—
120C22×D4—41no—
121C22×D4—41no—
122C22×D4—41no—
123C22×D4M1614yes—
124C23×C4—22no—
125C23×C4—22no—
126C23×C4—22no—
127C23×C4—22no—
128C23×C4—22no—
129C23×C4—21no—
130C23×C4—21no—
131C23×C4—22no—
132C23×C4—22no—
133C23×C4—22no—
134C23×C4—24no—
135C23×C4—22no—
136C23×C4—22no—
137C23×C4—22no—
138C23×C4—22no—
139C23×C4—21no—
140C23×C4—21no—
141C23×C4—22no—
142C23×C4—22no—
143C23×C4—42no—
144C23×C4—41no—
145C23×C4—41no—
146C23×C4—41no—
147C23×C4—41no—
148C23×C4—41no—
149C23×C4—41no—
150C23×C4M1614yes—
151C23×C4C22×C411yescharacteristic
152C23×C4C22×C411yescharacteristic
153C23×C4C22×C412yes—
154C23×C4C22×C412yes—
155C23×C4C22×C411yescharacteristic
156C23×C4C22×C412yes—
157C23×C4C22×C411yescharacteristic
158C23×C4C22×C412yes—
159C23×C4C2×D414yes—
160C23×C4C2411yescharacteristic
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—22no—
181C22×C8—22no—
182C22×C8—22no—
183C22×C8—22no—
184C22×C8—22no—
185C22×C8—22no—
186C22×C8—22no—
187C22×C8#4114yes—
188C22×C8#4114yes—
189#123—232no—
190C4×D4—48no—
191C4×D4—48no—
192C2×C4⋊C4—24no—
193C2×C4⋊C4—24no—
194C2×C4⋊C4—24no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—24no—
197C2×C4⋊C4—24no—
198C2×C4⋊C4—42no—
199C2×C4⋊C4—41no—
200C2×C4⋊C4—41no—
201C2×C22⋊C4—24no—
202C2×C22⋊C4—22no—
203C2×C22⋊C4—22no—
204C2×C22⋊C4—24no—
205C2×C22⋊C4—24no—
206C2×C22⋊C4—22no—
207C2×C22⋊C4—22no—
208C2×C22⋊C4—24no—
209C2×C22⋊C4—24no—
210C2×C22⋊C4—41no—
211C2×C22⋊C4—41no—
212C2×C22⋊C4—42no—
213C2×C22⋊C4—41no—
214C2×C22⋊C4—41no—
215C2×C22⋊C4—41no—
216C2×C22⋊C4—41no—
217C2×C22⋊C4C22⋊C414yes—
218C2×C22⋊C4C22⋊C414yes—
219C2×C42—22no—
220C2×C42—22no—
221C2×C42—22no—
222C2×C42—22no—
223C2×C42—24no—
224C2×C42—22no—
225C2×C42—22no—
226C2×C42—22no—
227C2×C42—22no—
228C2×C42—21no—
229C2×C42—21no—
230C2×C42—22no—
231C2×C42—22no—
232C2×C42—22no—
233C2×C42—22no—
234C2×C42—42no—
235C2×C42—42no—
236C2×C42—48no—
237C2×C42—48no—
238C2×C42C2×D414yes—
239C2×C42C2×D414yes—
240C4⋊C8—48no—
241C22⋊C8—44no—
242C22⋊C8—48no—
243C22⋊C8—48no—
244C22⋊C8—48no—
245C22⋊C8—44no—
246#97—48no—
247#95—24no—
248#95—24no—
249#95—24no—
250#95—24no—
251#95C22⋊C418yes—
252C24—22no—
253C24—21no—
254C24—21no—
255C24—22no—
256C24—22no—
257C24—21no—
258C24—21no—
259C24—22no—
260C24—21no—
261C24—22no—
262C24—21no—
263C24—21no—
264C24—22no—
265C24—21no—
266C24—22no—
267C24—22no—
268C24—21no—
269C24—21no—
270C24C2×C22⋊C412yes—
271C24C2×M1612yes—
272C24C2×M1612yes—
273C24C2×M1612yes—
274C24C23×C411yescharacteristic
275C2×D4—28no—
276C2×D4—44no—
277C2×D4—44no—
278C2×D4—44no—
279C2×D4—44no—
280C2×D4—44no—
281C2×D4—44no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—21no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—21no—
293C22×C4—21no—
294C22×C4—21no—
295C22×C4—21no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—21no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—21no—
315C22×C4—21no—
316C22×C4—28no—
317C22×C4—24no—
318C22×C4—24no—
319C22×C4—24no—
320C22×C4—24no—
321C22×C4—24no—
322C22×C4—24no—
323C22×C4—21no—
324C22×C4—21no—
325C22×C4—22no—
326C22×C4—21no—
327C22×C4—21no—
328C22×C4—21no—
329C22×C4—21no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—44no—
334C22×C4—44no—
335C22×C4—44no—
336C22×C4—48no—
337C22×C4—48no—
338C22×C4—44no—
339C22×C4—44no—
340C22×C4—44no—
341C22×C4—44no—
342C22×C4—44no—
343C22×C4—44no—
344C22×C4—42no—
345C22×C4—42no—
346C22×C4—42no—
347C22×C4—42no—
348C22×C4—41no—
349C22×C4—42no—
350C22×C4—41no—
351C22×C4—41no—
352C22×C4—42no—
353C22×C4—41no—
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—41no—
365C22×C4—42no—
366C22×C4—41no—
367C22×C4—41no—
368C22×C4—42no—
369C22×C4—41no—
370C22×C4C2×C22⋊C414yes—
371C22×C4C2×C22⋊C414yes—
372C22×C4C2×C22⋊C412yes—
373C22×C4C2×M1614yes—
374C22×C4C2×M1612yes—
375C22×C4#13114yes—
376C22×C4C22×D412yes—
377C22×C4C2×#4112yes—
378C22×C4C2×#4112yes—
379C22×C4H(5,𝔽2)12yes—
380C22×C4H(5,𝔽2)11yescharacteristic
381C22×C4H(5,𝔽2)11yescharacteristic
382C22×C4H(5,𝔽2)12yes—
383C22×C42−1+412yes—
384C2×C8—24no—
385C2×C8—24no—
386C2×C8—24no—
387C2×C8—24no—
388C2×C8—24no—
389C2×C8—24no—
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—
408C2×C8—42no—
409C2×C8—42no—
410C2×C8—42no—
411C2×C8—42no—
412C4⋊C4—216no—
413C4⋊C4—48no—
414C4⋊C4—44no—
415C22⋊C4—216no—
416C22⋊C4—28no—
417C22⋊C4—44no—
418C22⋊C4—48no—
419C22⋊C4—44no—
420C22⋊C4—44no—
421C42—24no—
422C42—24no—
423C42—24no—
424C42—24no—
425C42—48no—
426C42—42no—
427C42—42no—
428C42—42no—
429C42—42no—
430C42—42no—
431C42—42no—
432C23—22no—
433C23—24no—
434C23—22no—
435C23—24no—
436C23—24no—
437C23—24no—
438C23—22no—
439C23—22no—
440C23—22no—
441C23—22no—
442C23—24no—
443C23—24no—
444C23—22no—
445C23—24no—
446C23—22no—
447C23—22no—
448C23—24no—
449C23—42no—
450C23—41no—
451C23—41no—
452C23—42no—
453C23—42no—
454C23—42no—
455C23—41no—
456C23—41no—
457C23—42no—
458C23—42no—
459C23—41no—
460C23—42no—
461C23—42no—
462C23—41no—
463C23—42no—
464C23—41no—
465C23—41no—
466C23—42no—
467C23—42no—
468C23—42no—
469C23—41no—
470C23—41no—
471C23—42no—
472C23—41no—
473C23—41no—
474C23—42no—
475C23—42no—
476C23—42no—
477C23#40714yes—
478C23C22×C22⋊C411yescharacteristic
479C23C4×#4112yes—
480C23#51811yescharacteristic
481C23#51811yescharacteristic
482C23#52012yes—
483C23C22×M1611yescharacteristic
484C23C22×M1611yescharacteristic
485C23C2×#13112yes—
486C23#56811yescharacteristic
487C23#56811yescharacteristic
488C23#56812yes—
489D4—416no—
490C2×C4—22no—
491C2×C4—24no—
492C2×C4—24no—
493C2×C4—24no—
494C2×C4—24no—
495C2×C4—24no—
496C2×C4—22no—
497C2×C4—24no—
498C2×C4—24no—
499C2×C4—24no—
500C2×C4—24no—
501C2×C4—24no—
502C2×C4—24no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—24no—
506C2×C4—24no—
507C2×C4—416no—
508C2×C4—416no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—41no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—41no—
515C2×C4—42no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—42no—
527C2×C4—42no—
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—41no—
537C2×C4—41no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—42no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—82no—
547C2×C4—82no—
548C2×C4—82no—
549C2×C4—82no—
550C2×C4#53512yes—
551C2×C4#53512yes—
552C2×C4#53812yes—
553C2×C4#53912yes—
554C2×C4#54012yes—
555C2×C4#54212yes—
556C2×C4#54612yes—
557C2×C4#54612yes—
558C2×C4#54712yes—
559C2×C4#54712yes—
560C2×C4#55111yescharacteristic
561C2×C4#55312yes—
562C2×C4#55911yescharacteristic
563C2×C4#56011yescharacteristic
564C2×C4#56212yes—
565C2×C4#56311yescharacteristic
566C8—44no—
567C8—44no—
568C8—82no—
569C8—82no—
570C22—24no—
571C22—28no—
572C22—42no—
573C22—42no—
574C22—42no—
575C22—42no—
576C22—42no—
577C22—44no—
578C22—44no—
579C22—42no—
580C22—44no—
581C22—42no—
582C22—42no—
583C22—42no—
584C22—44no—
585C22—44no—
586C22—81no—
587C22—81no—
588C22—81no—
589C22—82no—
590C22—81no—
591C22—82no—
592C22#230911yescharacteristic
593C22#233512yes—
594C22#233912yes—
595C22#237411yescharacteristic
596C22#237611yescharacteristic
597C22C2×#40711yescharacteristic
598C22#287911yescharacteristic
599C22#288012yes—
600C22#296512yes—
601C22#296712yes—
602C22#296912yes—
603C22#297012yes—
604C22#297311yescharacteristic
605C22#297311yescharacteristic
606C22#297311yescharacteristic
607C22#297311yescharacteristic
608C22#297612yes—
609C22#297711yescharacteristic
610C22#297811yescharacteristic
611C22#298912yes—
612C22#299012yes—
613C22#299312yes—
614C22#299512yes—
615C4—44no—
616C4—42no—
617C4—44no—
618C4—42no—
619C4—44no—
620C4—44no—
621C4—81no—
622C4—81no—
623C4#270312yes—
624C4#270612yes—
625C4#271012yes—
626C4#271812yes—
627C2—44no—
628C2—82no—
629C2#1450611yescharacteristic
630C2#2179011yescharacteristic
631C2#2179211yescharacteristic
632C2#2236712yes—
633C2#2236912yes—
634C2#2247112yes—
635C2#2247212yes—
636C2#2329611yescharacteristic
637C2#2330211yescharacteristic
638C2#2332211yescharacteristic
639C2#2332411yescharacteristic
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
2. C2×#1924C2semidirect
3. #11346C2semidirect
4. #11042C2semidirect
7. #10922C2semidirect
8. #10835C2semidirect
9. #10712C2semidirect

Maximal subgroups

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

subgroupindex
C2×C4×#1232
C2×#19242
#113462
#110422
#109642
#109222
#108352
#107122