Finite Group Databasebeta

← #360981

Subgroups — #360981

Order 512.

1766 conjugacy classes of subgroups.

The subgroup lattice has 640 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C4×D8C211yesmaximal, characteristic
2C2×#1926C212yesmaximal, complemented
3#13916C211yesmaximal, characteristic, complemented
4#13820C211yesmaximal, characteristic, complemented
5#11121C212yesmaximal, complemented
6#11022C212yesmaximal
7#10907C212yesmaximal, complemented
8#10861C211yesmaximal, characteristic, complemented
9#10797C211yesmaximal, characteristic, complemented
10#10788C211yesmaximal, characteristic, complemented
11#10788C211yesmaximal, characteristic, complemented
12C23×D8C2211yescharacteristic
13C22×C4×D4C2211yescharacteristic
14C22×C4×D4C2211yescharacteristic
15C2×C4×D8—28no—
16C2×C4×D8C418yescomplemented
17C22×#107C2211yescharacteristic
18C22×#106C2212yes—
19C22×#102C2212yes—
20C22×#102C2211yescharacteristic
21C22×#102C2212yes—
22C22×#102C2211yescharacteristic
23C22×C4×C8C2211yescharacteristic
24C2×#390C2212yes—
25C2×#390C2212yes—
26C2×#381—21no—
27C2×C4×C4⋊C4C2212yes—
28C2×C4×C22⋊C4—21no—
29#1926—28no—
30C2×#340C2212yes—
31C2×#340C2212yes—
32C2×#340C2212yes—
33C2×#336C2212yes—
34C2×#336C2211yescharacteristic
35C2×#336C2212yes—
36#1440—22no—
37#1437—22no—
38C4⋊#95C2211yescharacteristic
39C4⋊#95C2211yescharacteristic
40C4⋊#95C2212yes—
41C22⋊#95—21no—
42C22⋊#95—21no—
43C4×#95C2211yescharacteristic
44C23×D4C2311yescharacteristic
45C23×D4C2311yescharacteristic
46C24×C4—21no—
47C24×C4—21no—
48C22×D8—24no—
49C22×D8—24no—
50C22×D8—22no—
51C22×D8—22no—
52C22×D8C2×C414yes—
53C23×C8C2312yes—
54C23×C8C2311yescharacteristic
55C23×C8C2311yescharacteristic
56C2×C4×D4—24no—
57C2×C4×D4—24no—
58C2×C4×D4—22no—
59C2×C4×D4—22no—
60C2×C4×D4—42no—
61C2×C4×D4—42no—
62C2×C4×D4C2×C414yes—
63C2×C4×D4C2×C414yes—
64C22×C4⋊C4C2312yes—
65C22×C4⋊C4C2311yescharacteristic
66C22×C4⋊C4C2312yes—
67C22×C4⋊C4C2311yescharacteristic
68C22×C22⋊C4—21no—
69C22×C22⋊C4—21no—
70C22×C22⋊C4—22no—
71C22×C22⋊C4—22no—
72C22×C22⋊C4—22no—
73C22×C22⋊C4—22no—
74C22×C42—22no—
75C22×C42C2312yes—
76C22×C42C2311yescharacteristic
77C4×D8—232no—
78C2×#107—24no—
79C2×#107C2×C414yes—
80C2×#106—24no—
81C2×#106—24no—
82C2×#106—42no—
83C2×#102—24no—
84C2×#102—24no—
85C2×#102—24no—
86C2×#102—24no—
87C2×#102—42no—
88C2×#102—42no—
89C2×#102—42no—
90C2×#102C2×C414yes—
91C2×#102C2×C414yes—
92C2×#102D418yes—
93C2×C4×C8—24no—
94C2×C4×C8C2×C414yes—
95#390—24no—
96#390—24no—
97#390—24no—
98#390—42no—
99#381—44no—
100C4×C4⋊C4—24no—
101C4×C4⋊C4—42no—
102C4×C22⋊C4—44no—
103C2×#95—21no—
104C2×#95—21no—
105C2×#95—22no—
106C2×#95—22no—
107C2×#95—22no—
108C2×#95—21no—
109C2×#95—21no—
110C2×#95D411yescharacteristic
111C2×#95D411yescharacteristic
112C2×#95D412yes—
113#340—24no—
114#340—24no—
115#340—42no—
116#336—24no—
117#336—24no—
118#336—22no—
119C25—21no—
120C25—21no—
121C22×D4—22no—
122C22×D4—22no—
123C22×D4—21no—
124C22×D4—21no—
125C22×D4—21no—
126C22×D4—21no—
127C22×D4—22no—
128C22×D4—21no—
129C22×D4—21no—
130C22×D4—22no—
131C22×D4—22no—
132C22×D4—21no—
133C22×D4—21no—
134C22×D4—22no—
135C22×D4—21no—
136C22×D4—21no—
137C22×D4C22×C412yes—
138C22×D4C22×C412yes—
139C22×D4C2×D414yes—
140C22×D4#4114yes—
141C23×C4—21no—
142C23×C4—22no—
143C23×C4—21no—
144C23×C4—24no—
145C23×C4—24no—
146C23×C4—22no—
147C23×C4—44no—
148C23×C4—44no—
149C23×C4—42no—
150C23×C4—42no—
151C23×C4—42no—
152C23×C4—42no—
153C23×C4—82no—
154C23×C4—82no—
155C23×C4C2×D412yes—
156C23×C4C2×D411yescharacteristic
157C23×C4C2×D412yes—
158C23×C4C2×D411yescharacteristic
159C23×C4C2411yescharacteristic
160C2×D8—216no—
161C2×D8—216no—
162C2×D8—28no—
163C2×D8—28no—
164C2×D8—28no—
165C22×C8—22no—
166C22×C8—22no—
167C22×C8—22no—
168C22×C8—21no—
169C22×C8—21no—
170C22×C8—22no—
171C22×C8—22no—
172C22×C8—22no—
173C22×C8—22no—
174C22×C8—22no—
175C22×C8—22no—
176C22×C8—22no—
177C22×C8C22×C412yes—
178C22×C8C22×C412yes—
179C22×C8C2×D414yes—
180C22×C8#4112yes—
181C22×C8#4112yes—
182C4×D4—28no—
183C4×D4—28no—
184C4×D4—44no—
185C4×D4—44no—
186C4×D4—48no—
187C4×D4—48no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4C22×C412yes—
205C2×C4⋊C4C22×C412yes—
206C2×C4⋊C4C2×D414yes—
207C2×C4⋊C4#4114yes—
208C2×C22⋊C4—28no—
209C2×C22⋊C4—24no—
210C2×C22⋊C4—24no—
211C2×C22⋊C4—28no—
212C2×C22⋊C4—44no—
213C2×C22⋊C4—44no—
214C2×C22⋊C4—44no—
215C2×C22⋊C4—44no—
216C2×C22⋊C4—44no—
217C2×C22⋊C4—44no—
218C2×C22⋊C4—44no—
219C2×C22⋊C4—44no—
220C2×C22⋊C4—82no—
221C2×C22⋊C4—82no—
222C2×C22⋊C4—82no—
223C2×C22⋊C4—82no—
224C2×C42—22no—
225C2×C42—22no—
226C2×C42—22no—
227C2×C42—22no—
228C2×C42—22no—
229C2×C42—22no—
230C2×C42—22no—
231C2×C42—22no—
232C2×C42—44no—
233C2×C42—44no—
234C2×C42—82no—
235C2×C42C22×C412yes—
236C2×C42#4112yes—
237#107—28no—
238#106—48no—
239#102—28no—
240#102—48no—
241#102—48no—
242#102C22⋊C4116yescomplemented
243C4×C8—28no—
244#95—24no—
245#95—22no—
246#95—22no—
247#95—44no—
248#95—44no—
249#95—42no—
250#95—42no—
251#95—82no—
252#95—81no—
253#95—81no—
254C24—22no—
255C24—22no—
256C24—24no—
257C24—24no—
258C24—42no—
259C24—42no—
260C24—41no—
261C24—41no—
262C24—41no—
263C24—41no—
264C24—42no—
265C24—42no—
266C24—41no—
267C24—41no—
268C24—41no—
269C24—41no—
270C24—42no—
271C24—42no—
272C24—41no—
273C24—41no—
274C24—41no—
275C24—41no—
276C24C22×D411yescharacteristic
277C2×D4—22no—
278C2×D4—22no—
279C2×D4—22no—
280C2×D4—22no—
281C2×D4—22no—
282C2×D4—22no—
283C2×D4—22no—
284C2×D4—22no—
285C2×D4—22no—
286C2×D4—22no—
287C2×D4—24no—
288C2×D4—24no—
289C2×D4—24no—
290C2×D4—24no—
291C2×D4—24no—
292C2×D4—24no—
293C2×D4—24no—
294C2×D4—24no—
295C2×D4—24no—
296C2×D4—24no—
297C2×D4—42no—
298C2×D4—42no—
299C2×D4—42no—
300C2×D4—42no—
301C2×D4—42no—
302C2×D4—42no—
303C2×D4—42no—
304C2×D4—42no—
305C2×D4—41no—
306C2×D4—41no—
307C2×D4—41no—
308C2×D4—41no—
309C2×D4—42no—
310C2×D4—42no—
311C2×D4C2×C22⋊C414yes—
312C2×D4#11714yes—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—24no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—21no—
325C22×C4—21no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—24no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—48no—
337C22×C4—48no—
338C22×C4—41no—
339C22×C4—41no—
340C22×C4—42no—
341C22×C4—42no—
342C22×C4—42no—
343C22×C4—42no—
344C22×C4—42no—
345C22×C4—41no—
346C22×C4—42no—
347C22×C4—41no—
348C22×C4—42no—
349C22×C4—42no—
350C22×C4—42no—
351C22×C4—48no—
352C22×C4—48no—
353C22×C4—42no—
354C22×C4—42no—
355C22×C4—42no—
356C22×C4—42no—
357C22×C4—42no—
358C22×C4—42no—
359C22×C4—42no—
360C22×C4—84no—
361C22×C4—84no—
362C22×C4—88no—
363C22×C4—88no—
364C22×C4—88no—
365C22×C4—88no—
366C22×C4C4×D412yes—
367C22×C4C4×D412yes—
368C22×C4#13514yes—
369C22×C4AGL(1,ℤ/8)12yes—
370C22×C4AGL(1,ℤ/8)12yes—
371C22×C4C23×C411yescharacteristic
372C22×C4C22×D412yes—
373C22×C4C2×#4112yes—
374C22×C4C2×#4111yescharacteristic
375C22×C4C2×#4111yescharacteristic
376C22×C4C2×#4112yes—
377C22×C4C2×#4111yescharacteristic
378C22×C4H(5,𝔽2)12yes—
379C22×C4H(5,𝔽2)11yescharacteristic
380C22×C42−1+411yescharacteristic
381D8—232no—
382C2×C8—22no—
383C2×C8—22no—
384C2×C8—22no—
385C2×C8—22no—
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×C8C2×C22⋊C414yes—
398C2×C8#11714yes—
399C4⋊C4—22no—
400C4⋊C4—24no—
401C4⋊C4—22no—
402C4⋊C4—22no—
403C4⋊C4—24no—
404C4⋊C4—24no—
405C4⋊C4—42no—
406C4⋊C4—42no—
407C4⋊C4—42no—
408C4⋊C4—42no—
409C4⋊C4—42no—
410C4⋊C4—42no—
411C4⋊C4—42no—
412C4⋊C4C2×C22⋊C414yes—
413C22⋊C4—48no—
414C22⋊C4—48no—
415C22⋊C4—416no—
416C22⋊C4—88no—
417C22⋊C4—88no—
418C22⋊C4—88no—
419C42—22no—
420C42—22no—
421C42—24no—
422C42—42no—
423C42—42no—
424C42—42no—
425C42—42no—
426C42—88no—
427C23—24no—
428C23—24no—
429C23—44no—
430C23—42no—
431C23—44no—
432C23—44no—
433C23—44no—
434C23—42no—
435C23—42no—
436C23—42no—
437C23—42no—
438C23—42no—
439C23—42no—
440C23—42no—
441C23—42no—
442C23—42no—
443C23—42no—
444C23—44no—
445C23—44no—
446C23—44no—
447C23—44no—
448C23—44no—
449C23—44no—
450C23—44no—
451C23—44no—
452C23—42no—
453C23—82no—
454C23—82no—
455C23—81no—
456C23—81no—
457C23—81no—
458C23—81no—
459C23—82no—
460C23—82no—
461C23—82no—
462C23—82no—
463C23—82no—
464C23—82no—
465C23—81no—
466C23—81no—
467C23—81no—
468C23—81no—
469C23—82no—
470C23—82no—
471C23—82no—
472C23—82no—
473C23C2×C4×D411yescharacteristic
474C23#52512yes—
475C23D4212yes—
476C23#54711yescharacteristic
477C23#55011yescharacteristic
478C23C2×#13512yes—
479C23C2×AGL(1,ℤ/8)11yescharacteristic
480C23C2×AGL(1,ℤ/8)11yescharacteristic
481C23#57612yes—
482C23#57711yescharacteristic
483C23#57711yescharacteristic
484D4—24no—
485D4—24no—
486D4—24no—
487D4—24no—
488D4—42no—
489D4—42no—
490D4—42no—
491D4—42no—
492D4—44no—
493D4—44no—
494D4—44no—
495D4—44no—
496C2×C4—22no—
497C2×C4—22no—
498C2×C4—24no—
499C2×C4—24no—
500C2×C4—24no—
501C2×C4—24no—
502C2×C4—22no—
503C2×C4—24no—
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—24no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—44no—
514C2×C4—44no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—44no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—44no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—44no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—816no—
531C2×C4—816no—
532C2×C4—82no—
533C2×C4—82no—
534C2×C4—82no—
535C2×C4—82no—
536C2×C4—82no—
537C2×C4—82no—
538C2×C4—82no—
539C2×C4—82no—
540C2×C4—82no—
541C2×C4—82no—
542C2×C4—82no—
543C2×C4—82no—
544C2×C4—82no—
545C2×C4#44214yes—
546C2×C4C22×C22⋊C411yescharacteristic
547C2×C4C2×#11711yescharacteristic
548C2×C4C2×#11711yescharacteristic
549C2×C4C4×#4111yescharacteristic
550C2×C4#51811yescharacteristic
551C2×C4#52011yescharacteristic
552C2×C4#52512yes—
553C2×C4#52512yes—
554C2×C4#52912yes—
555C2×C4#53812yes—
556C2×C4#53912yes—
557C2×C4#54612yes—
558C2×C4#54612yes—
559C2×C4#54712yes—
560C2×C4#55312yes—
561C2×C4#55511yescharacteristic
562C2×C4#55611yescharacteristic
563C2×C4#55912yes—
564C8—24no—
565C8—24no—
566C8—82no—
567C22—44no—
568C22—44no—
569C22—44no—
570C22—44no—
571C22—44no—
572C22—44no—
573C22—82no—
574C22—82no—
575C22—82no—
576C22—82no—
577C22—82no—
578C22—82no—
579C22—82no—
580C22—82no—
581C22—84no—
582C22—84no—
583C22—84no—
584C22—84no—
585C22—84no—
586C22—84no—
587C22—84no—
588C22—84no—
589C22#229011yescharacteristic
590C22D4×C22⋊C411yescharacteristic
591C22#235411yescharacteristic
592C22#245512yes—
593C22#245612yes—
594C22C4×#13511yescharacteristic
595C22C2×#44211yescharacteristic
596C22#294811yescharacteristic
597C22#295211yescharacteristic
598C22#310512yes—
599C22#310612yes—
600C22#311812yes—
601C22#311912yes—
602C22#326412yes—
603C22#326412yes—
604C22#328212yes—
605C22#328412yes—
606C22#333912yes—
607C22#335411yescharacteristic
608C22#335411yescharacteristic
609C22#339612yes—
610C22#339711yescharacteristic
611C22#339711yescharacteristic
612C4—24no—
613C4—44no—
614C4—42no—
615C4—44no—
616C4—82no—
617C4—82no—
618C4—84no—
619C4—162no—
620C4#228011yescharacteristic
621C4#231811yescharacteristic
622C4#233911yescharacteristic
623C4#235711yescharacteristic
624C4#245912yes—
625C4#246612yes—
626C2—84no—
627C2—84no—
628C2#1388611yescharacteristic
629C2#2111811yescharacteristic
630C2#2113411yescharacteristic
631C2#2243611yescharacteristic
632C2#2244911yescharacteristic
633C2#2281011yescharacteristic
634C2#2281311yescharacteristic
635C2#2654712yes—
636C2#2654912yes—
637C2#2658512yes—
638C2#2659312yes—
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC23×D8C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C4×#95 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. C2×#1926C2semidirect
3. #13916C2semidirect
4. #13820C2semidirect
5. #11121C2semidirect
7. #10907C2semidirect
8. #10861C2semidirect
9. #10797C2semidirect
10. #10788C2semidirect
11. #10788C2semidirect
16. C2×C4×D8C4semidirect
242. #102C22⋊C4semidirect

Maximal subgroups

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

subgroupindex
C22×C4×D82
C2×#19262
#139162
#138202
#111212
#110222
#109072
#108612
#107972
#107882