Finite Group Databasebeta

← #387961

Subgroups — #387961

Order 512.

1668 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×#1923C212yesmaximal, complemented
3#11346C212yesmaximal, complemented
4#11042C212yesmaximal, complemented
5#10964C211yesmaximal, characteristic
6#10964C211yesmaximal, characteristic
7#10921C212yesmaximal, complemented
8#10834C212yesmaximal, 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×C4×C8C2212yes—
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#1923—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×#123C818yescomplemented
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×C4×C8—24no—
89C2×C4×C8—24no—
90C2×C4×C8—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×D4C2×C814yes—
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×C4C2×C814yes—
151C23×C4C22×C411yescharacteristic
152C23×C4C22×C411yescharacteristic
153C23×C4C22×C412yes—
154C23×C4C22×C411yescharacteristic
155C23×C4C22×C412yes—
156C23×C4C22×C411yescharacteristic
157C23×C4C22×C412yes—
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—22no—
195C2×C4⋊C4—24no—
196C2×C4⋊C4—24no—
197C2×C4⋊C4—42no—
198C2×C4⋊C4—41no—
199C2×C4⋊C4—41no—
200C2×C4⋊C4C2×C818yes—
201C2×C22⋊C4—24no—
202C2×C22⋊C4—24no—
203C2×C22⋊C4—22no—
204C2×C22⋊C4—22no—
205C2×C22⋊C4—24no—
206C2×C22⋊C4—24no—
207C2×C22⋊C4—41no—
208C2×C22⋊C4—41no—
209C2×C22⋊C4—42no—
210C2×C22⋊C4—41no—
211C2×C22⋊C4—41no—
212C2×C22⋊C4—41no—
213C2×C22⋊C4—41no—
214C2×C22⋊C4C22⋊C414yes—
215C2×C22⋊C4C22⋊C414yes—
216C2×C22⋊C4C2×C818yes—
217C2×C22⋊C4C2×C814yes—
218C2×C22⋊C4M1614yes—
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—
246C4×C8—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—
271C24C22×C812yes—
272C24C22×C812yes—
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—22no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—21no—
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—21no—
296C22×C4—21no—
297C22×C4—21no—
298C22×C4—21no—
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—21no—
313C22×C4—21no—
314C22×C4—28no—
315C22×C4—24no—
316C22×C4—24no—
317C22×C4—24no—
318C22×C4—24no—
319C22×C4—24no—
320C22×C4—24no—
321C22×C4—21no—
322C22×C4—21no—
323C22×C4—22no—
324C22×C4—21no—
325C22×C4—21no—
326C22×C4—21no—
327C22×C4—21no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—44no—
332C22×C4—44no—
333C22×C4—44no—
334C22×C4—48no—
335C22×C4—48no—
336C22×C4—44no—
337C22×C4—44no—
338C22×C4—44no—
339C22×C4—44no—
340C22×C4—44no—
341C22×C4—44no—
342C22×C4—42no—
343C22×C4—42no—
344C22×C4—42no—
345C22×C4—42no—
346C22×C4—41no—
347C22×C4—42no—
348C22×C4—41no—
349C22×C4—41no—
350C22×C4—42no—
351C22×C4—41no—
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—41no—
363C22×C4—42no—
364C22×C4—41no—
365C22×C4—41no—
366C22×C4—42no—
367C22×C4—41no—
368C22×C4C2×C22⋊C414yes—
369C22×C4C2×C22⋊C412yes—
370C22×C4C2×C22⋊C414yes—
371C22×C4C22×C814yes—
372C22×C4C22×C812yes—
373C22×C4C22×C814yes—
374C22×C4C2×M1614yes—
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—44no—
417C22⋊C4—48no—
418C22⋊C4—44no—
419C22⋊C4—44no—
420C22⋊C4C22⋊C8116yescomplemented
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—
477C23C2×C22⋊C814yes—
478C23C22×C22⋊C411yescharacteristic
479C23C4×#4112yes—
480C23#51811yescharacteristic
481C23#51811yescharacteristic
482C23#52012yes—
483C23C23×C811yescharacteristic
484C23C22×M1611yescharacteristic
485C23C2×#13112yes—
486C23#56811yescharacteristic
487C23#56811yescharacteristic
488C23#56812yes—
489D4—416no—
490C2×C4—24no—
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—22no—
501C2×C4—24no—
502C2×C4—24no—
503C2×C4—24no—
504C2×C4—24no—
505C2×C4—24no—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—24no—
509C2×C4—24no—
510C2×C4—416no—
511C2×C4—416no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—41no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—41no—
518C2×C4—42no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—44no—
532C2×C4—44no—
533C2×C4—44no—
534C2×C4—44no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—42no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—82no—
546C2×C4—82no—
547C2×C4—82no—
548C2×C4—82no—
549C2×C4C2×C22⋊C818yes—
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—44no—
577C22—44no—
578C22—42no—
579C22—44no—
580C22—42no—
581C22—42no—
582C22—42no—
583C22—44no—
584C22—44no—
585C22—42no—
586C22—81no—
587C22—81no—
588C22—81no—
589C22—82no—
590C22—81no—
591C22—82no—
592C22#230911yescharacteristic
593C22#233512yes—
594C22#233912yes—
595C22#237411yescharacteristic
596C22#237611yescharacteristic
597C22C22×C22⋊C811yescharacteristic
598C22#287911yescharacteristic
599C22#288012yes—
600C22C8×#4112yes—
601C22#296712yes—
602C22#296912yes—
603C22#297012yes—
604C22#297211yescharacteristic
605C22#297211yescharacteristic
606C22#297511yescharacteristic
607C22#297511yescharacteristic
608C22#297612yes—
609C22#297711yescharacteristic
610C22#297811yescharacteristic
611C22#298812yes—
612C22#299012yes—
613C22#299312yes—
614C22#299512yes—
615C4—28no—
616C4—44no—
617C4—42no—
618C4—44no—
619C4—42no—
620C4—44no—
621C4—81no—
622C4—81no—
623C4#270312yes—
624C4#270612yes—
625C4#271012yes—
626C4#271812yes—
627C2—44no—
628C2—82no—
629C2#1450611yescharacteristic
630C2#2178611yescharacteristic
631C2#2179211yescharacteristic
632C2#2235712yes—
633C2#2236312yes—
634C2#2246912yes—
635C2#2247212yes—
636C2#2329411yescharacteristic
637C2#2330411yescharacteristic
638C2#2331411yescharacteristic
639C2#2331611yescharacteristic
subgroupG/H
center, Frattini, cosocleC23×C4C24
commutatorC23C23×C8
socleC24C23×C4
prime radicalC23×D4C2×C4
LCSG ▷ C23 ▷ 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×#1923C2semidirect
3. #11346C2semidirect
4. #11042C2semidirect
7. #10921C2semidirect
8. #10834C2semidirect
9. #10712C2semidirect
50. C2×#123C8semidirect
420. C22⋊C4C22⋊C8semidirect

Maximal subgroups

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

subgroupindex
C2×C4×#1232
C2×#19232
#113462
#110422
#109642
#109212
#108342
#107122