Finite Group Databasebeta

← #6530957

Subgroups — #6530957

Order 512.

4058 conjugacy classes of subgroups.

The subgroup lattice has 642 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3499C211yesmaximal, characteristic
2C2×#3226C211yesmaximal, characteristic, complemented
3C22×#500C211yesmaximal, characteristic, complemented
4#35316C211yesmaximal, characteristic, complemented
5#25838C214yesmaximal, complemented
6#25813C214yesmaximal, complemented
7#25703C214yesmaximal, complemented
8#25673C214yesmaximal, complemented
9#24088C218yesmaximal, complemented
10#21833C211yesmaximal, characteristic, complemented
11#21401C211yesmaximal, characteristic, complemented
12#21373C211yesmaximal, characteristic, complemented
13C22×2−1+4—21no—
14#3499—28no—
15#3499—28no—
16C2×D4×Q8—21no—
17C2×D4×Q8C2212yes—
18C2×#547—22no—
19C2×#547—21no—
20C2×#547C2214yes—
21C2×#547C2212yes—
22C2×#536—21no—
23C2×#536—21no—
24C2×#536—22no—
25C2×#532C2212yes—
26C2×#532C2211yescharacteristic
27C22×C4⋊Q8C2212yes—
28C22×C4⋊Q8C2211yescharacteristic
29C22×C4⋊Q8C2211yescharacteristic
30C2×C4×#41C2211yescharacteristic
31#3226—28no—
32C2×#500—28no—
33C2×#500C2214yes—
34C2×#500C2214yes—
35C2×#500C2212yes—
36C2×#486C2212yes—
37C2×#480C2214yes—
38C2×#474C2214yes—
39C2×#431C2211yescharacteristic
40C22×#107C2214yes—
41C22×C4×C8C2211yescharacteristic
42#2857C2214yes—
43#2621—22no—
44#2547—21no—
45#2547C2212yes—
46#2546—21no—
47#2546C2212yes—
48#2486—22no—
49#2486C2214yes—
50#2310—21no—
51#2285C2211yescharacteristic
52#2057C22116yes—
53#2033C22116yes—
54#1990C2214yes—
55#1990C2212yes—
56#1988C2214yes—
57#1984C2212yes—
58#1957C2212yes—
59#1954C2212yes—
60#1893C2214yes—
61#1869C2214yes—
62#1862C22116yes—
63#1852C22116yes—
64#1823C2214yes—
65C4×#102C2214yes—
66C4×C22⋊C8C2214yes—
67C2×2−1+4—216no—
68C2×2−1+4—48no—
69C2×2−1+4—48no—
70C22×#41—24no—
71C22×#41—22no—
72C22×#41—22no—
73C22×#41—21no—
74C22×#41D411yescharacteristic
75C23×Q8—22no—
76C23×Q8—22no—
77C23×Q8—21no—
78C23×Q8—21no—
79C23×C8C2312yes—
80D4×Q8—216no—
81D4×Q8—28no—
82D4×Q8—28no—
83#547—216no—
84#547—232no—
85#547—28no—
86#547—216no—
87#547—28no—
88#547—216no—
89#536—216no—
90#536—232no—
91#536—48no—
92#532—28no—
93#532—28no—
94#532—24no—
95#532D418yes—
96C2×C4⋊Q8—28no—
97C2×C4⋊Q8—28no—
98C2×C4⋊Q8—22no—
99C2×C4⋊Q8—22no—
100C2×C4⋊Q8—21no—
101C2×C4⋊Q8—22no—
102C2×C4⋊Q8—21no—
103C2×C4⋊Q8—24no—
104C2×C4⋊Q8—21no—
105C2×C4⋊Q8—21no—
106C2×C4⋊Q8—24no—
107C2×C4⋊Q8—24no—
108C2×C4⋊Q8D418yes—
109C2×C4⋊Q8C2314yes—
110C2×C4⋊Q8C2312yes—
111C2×C4⋊Q8C2312yes—
112C2×C4⋊Q8C2311yescharacteristic
113C2×C4⋊Q8C2312yes—
114C2×C4⋊Q8C2312yes—
115C2×#127C2311yescharacteristic
116C2×#124—22no—
117C2×#124—22no—
118C2×#124—21no—
119C2×#123—24no—
120C2×#123—24no—
121C2×#123—24no—
122C2×#123—24no—
123C2×C22⋊Q8—24no—
124C2×C22⋊Q8—22no—
125C2×C22⋊Q8—24no—
126C2×C22⋊Q8—22no—
127C2×C22⋊Q8—24no—
128C2×#121—22no—
129C2×#121—21no—
130C2×#121C2314yes—
131C2×#121C2312yes—
132C4×#41—24no—
133C4×#41—24no—
134C2×C4×Q8D411yescharacteristic
135C2×C4×Q8C2311yescharacteristic
136C2×C4×D4—22no—
137C2×C4×D4—21no—
138C2×C4×D4D412yes—
139C2×C4×D4D411yescharacteristic
140C2×C4×D4C2314yes—
141C2×C4×D4C2312yes—
142C2×C4×D4C2312yes—
143C2×C4×D4C2311yescharacteristic
144C2×#117—21no—
145C2×#117D411yescharacteristic
146C2×#117D412yes—
147C22×C4⋊C4—22no—
148C22×C4⋊C4—21no—
149C22×C4⋊C4—21no—
150C22×C4⋊C4C2312yes—
151C22×C4⋊C4C2314yes—
152C22×C4⋊C4C2312yes—
153C22×C42C2312yes—
154C22×C42C2311yescharacteristic
155#500—216no—
156#500—216no—
157#500—28no—
158#486—28no—
159#480—216no—
160#474—216no—
161#431—24no—
162C2×#107—216no—
163C2×#107C2318yes—
164C2×#107C23116yes—
165C2×C4⋊C8C2312yes—
166C2×#102C2318yes—
167C2×#102C2318yes—
168C2×C22⋊C8C2312yes—
169C2×C22⋊C8C2318yes—
170C2×C4×C8—24no—
171C2×C4×C8C2312yes—
172C2×C4×C8C2312yes—
173C2×C4×C8C2312yes—
174#398—24no—
175#398—24no—
176#397—24no—
177#397—24no—
178#391—22no—
179#391—22no—
180#391—24no—
181#389—24no—
182#389C2318yes—
183#384C2312yes—
184#382—21no—
185#381—24no—
186#380—24no—
187#380—21no—
188#380C2312yes—
189#380C2318yes—
190C4×C4⋊C4—21no—
191C4×C4⋊C4C2312yes—
192#340C2318yes—
193#340C23116yes—
194#336C2318yes—
1952−1+4—4128no—
196C2×#41—216no—
197C2×#41—216no—
198C2×#41—232no—
199C2×#41—24no—
200C2×#41—28no—
201C2×#41—24no—
202C2×#41—48no—
203C2×#41—416no—
204C2×#41—48no—
205C2×#41—416no—
206C2×#41—48no—
207C2×#41—48no—
208C2×#41—44no—
209C2×#41—44no—
210C2×#41D818yes—
211C22×Q8—216no—
212C22×Q8—216no—
213C22×Q8—28no—
214C22×Q8—24no—
215C22×Q8—22no—
216C22×Q8—22no—
217C22×Q8—21no—
218C22×Q8—22no—
219C22×Q8—22no—
220C22×Q8—48no—
221C22×Q8—48no—
222C22×Q8—48no—
223C22×Q8—44no—
224C22×Q8—48no—
225C22×Q8—44no—
226C22×Q8—44no—
227C22×Q8—44no—
228C22×Q8—44no—
229C22×Q8C2×D411yescharacteristic
230C22×D4—24no—
231C22×D4—22no—
232C22×D4—21no—
233C22×D4—21no—
234C22×D4C2×D412yes—
235C22×D4C2×D411yescharacteristic
236C22×D4C2412yes—
237C23×C4—24no—
238C23×C4—22no—
239C23×C4—24no—
240C23×C4—22no—
241C23×C4—22no—
242C23×C4C2×D412yes—
243C23×C4C2×D411yescharacteristic
244C23×C4C2×D412yes—
245C23×C4C2×D411yescharacteristic
246C23×C4C2412yes—
247C23×C4C2411yescharacteristic
248C22×C8—24no—
249C22×C8—28no—
250C22×C8C2414yes—
251C22×C8C2414yes—
252C4⋊Q8—216no—
253C4⋊Q8—216no—
254C4⋊Q8—28no—
255C4⋊Q8—28no—
256C4⋊Q8—28no—
257C4⋊Q8—28no—
258C4⋊Q8—22no—
259C4⋊Q8—24no—
260C4⋊Q8—24no—
261C4⋊Q8—24no—
262C4⋊Q8—24no—
263C4⋊Q8—24no—
264C4⋊Q8—24no—
265C4⋊Q8—48no—
266C4⋊Q8—44no—
267C4⋊Q8—44no—
268C4⋊Q8C2×D414yes—
269C4⋊Q8C2×D418yes—
270C4⋊Q8C2×D418yes—
271#127—22no—
272#127C2×D414yes—
273#124—216no—
274#124—28no—
275#124—48no—
276#123—232no—
277#123—232no—
278#123—232no—
279#123—416no—
280C22⋊Q8—216no—
281C22⋊Q8—216no—
282C22⋊Q8—232no—
283C22⋊Q8—232no—
284C22⋊Q8—416no—
285#121—216no—
286#121—28no—
287#121—28no—
288#121—24no—
289#121—28no—
290#121C2×D418yes—
291C4×Q8—24no—
292C4×Q8—22no—
293C4×Q8—22no—
294C4×D4—216no—
295C4×D4—28no—
296C4×D4—28no—
297C4×D4—24no—
298C4×D4—24no—
299C4×D4—28no—
300C4×D4—28no—
301C4×D4—24no—
302C4×D4—22no—
303C4×D4—22no—
304C4×D4—24no—
305C4×D4C2×D418yes—
306#117—24no—
307#117—28no—
308#117—44no—
309C2×C4⋊C4—216no—
310C2×C4⋊C4—28no—
311C2×C4⋊C4—21no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4—24no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—24no—
316C2×C4⋊C4—22no—
317C2×C4⋊C4—24no—
318C2×C4⋊C4—28no—
319C2×C4⋊C4—28no—
320C2×C4⋊C4—24no—
321C2×C4⋊C4—24no—
322C2×C4⋊C4—22no—
323C2×C4⋊C4—22no—
324C2×C4⋊C4—22no—
325C2×C4⋊C4—22no—
326C2×C4⋊C4—28no—
327C2×C4⋊C4—22no—
328C2×C4⋊C4—44no—
329C2×C4⋊C4C2×D411yescharacteristic
330C2×C4⋊C4C2×D412yes—
331C2×C4⋊C4C2×D411yescharacteristic
332C2×C4⋊C4C2×D412yes—
333C2×C4⋊C4C2×D418yes—
334C2×C4⋊C4C2414yes—
335C2×C4⋊C4C2414yes—
336C2×C4⋊C4C2412yes—
337C2×C4⋊C4C2414yes—
338C2×C22⋊C4—24no—
339C2×C22⋊C4—22no—
340C2×C22⋊C4—24no—
341C2×C22⋊C4—22no—
342C2×C22⋊C4—22no—
343C2×C22⋊C4—28no—
344C2×C22⋊C4C2×D412yes—
345C2×C22⋊C4C2×D412yes—
346C2×C22⋊C4C2×D412yes—
347C2×C42—24no—
348C2×C42—24no—
349C2×C42—22no—
350C2×C42C2×D412yes—
351C2×C42C2×D412yes—
352C2×C42C2×D411yescharacteristic
353C2×C42C2×D411yescharacteristic
354C2×C42C2×D414yes—
355C2×C42C2×D414yes—
356C2×C42C2412yes—
357C2×C42C2411yescharacteristic
358C2×C42C2411yescharacteristic
359#107—232no—
360#107—216no—
361C4⋊C8—24no—
362#102—216no—
363#102—216no—
364C22⋊C8—24no—
365C22⋊C8—216no—
366C4×C8—24no—
367C4×C8—24no—
368C4×C8—24no—
369#95—22no—
370#95—22no—
371#95—28no—
372#95—28no—
373C24—22no—
374C24C22×D412yes—
375C24C22×D411yescharacteristic
376#41—232no—
377#41—464no—
378#41—4128no—
379#41—464no—
380#41—432no—
381C2×Q8—216no—
382C2×Q8—22no—
383C2×Q8—28no—
384C2×Q8—28no—
385C2×Q8—24no—
386C2×Q8—22no—
387C2×Q8—48no—
388C2×Q8—464no—
389C2×Q8—464no—
390C2×Q8—44no—
391C2×Q8—44no—
392C2×Q8—48no—
393C2×Q8—44no—
394C2×Q8—42no—
395C2×Q8—44no—
396C2×Q8—44no—
397C2×Q8—44no—
398C2×Q8—42no—
399C2×Q8—44no—
400C2×Q8—44no—
401C2×Q8—44no—
402C2×Q8—44no—
403C2×Q8—432no—
404C2×Q8—816no—
405C2×Q8C2×D814yes—
406C2×D4—24no—
407C2×D4—24no—
408C2×D4—216no—
409C2×D4—28no—
410C2×D4—22no—
411C2×D4—24no—
412C2×D4—24no—
413C2×D4—22no—
414C2×D4—22no—
415C2×D4—28no—
416C2×D4—48no—
417C2×D4—48no—
418C2×D4—44no—
419C2×D4—44no—
420C2×D4—42no—
421C2×D4—42no—
422C2×D4—44no—
423C2×D4C2×D818yes—
424C2×D4C2×D814yes—
425C2×D4C22×D414yes—
426C22×C4—24no—
427C22×C4—24no—
428C22×C4—24no—
429C22×C4—24no—
430C22×C4—216no—
431C22×C4—216no—
432C22×C4—28no—
433C22×C4—24no—
434C22×C4—22no—
435C22×C4—24no—
436C22×C4—28no—
437C22×C4—28no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—28no—
441C22×C4—24no—
442C22×C4—24no—
443C22×C4—48no—
444C22×C4—48no—
445C22×C4—48no—
446C22×C4—44no—
447C22×C4—44no—
448C22×C4—48no—
449C22×C4—44no—
450C22×C4—44no—
451C22×C4—44no—
452C22×C4—44no—
453C22×C4—44no—
454C22×C4#12014yes—
455C22×C4#12014yes—
456C22×C4C2×D818yes—
457C22×C4C2×D814yes—
458C22×C4AGL(1,ℤ/8)14yes—
459C22×C4C22×D412yes—
460C22×C4C22×D412yes—
461C22×C4C22×D411yescharacteristic
462C22×C4C22×D411yescharacteristic
463C22×C4C22×D412yes—
464C22×C4C22×D412yes—
465C22×C4C22×D412yes—
466C22×C4C22×D411yescharacteristic
467C22×C4C22×D411yescharacteristic
468C22×C4C22×D411yescharacteristic
469C22×C4C2511yescharacteristic
470C2×C8—216no—
471C2×C8—24no—
472C2×C8—24no—
473C2×C8—24no—
474C2×C8—28no—
475C2×C82−1+418yescomplemented
476C4⋊C4—216no—
477C4⋊C4—216no—
478C4⋊C4—24no—
479C4⋊C4—28no—
480C4⋊C4—28no—
481C4⋊C4—28no—
482C4⋊C4—24no—
483C4⋊C4—24no—
484C4⋊C4—22no—
485C4⋊C4—22no—
486C4⋊C4—22no—
487C4⋊C4—24no—
488C4⋊C4—24no—
489C4⋊C4—24no—
490C4⋊C4—28no—
491C4⋊C4—44no—
492C4⋊C4—44no—
493C4⋊C4—44no—
494C4⋊C4—44no—
495C4⋊C4—416no—
496C4⋊C4—44no—
497C4⋊C4C22×D414yes—
498C4⋊C4C22×D418yes—
499C22⋊C4—216no—
500C22⋊C4—216no—
501C22⋊C4—24no—
502C22⋊C4—28no—
503C22⋊C4—28no—
504C22⋊C4—24no—
505C22⋊C4—24no—
506C22⋊C4—44no—
507C22⋊C4—416no—
508C42—24no—
509C42—24no—
510C42—22no—
511C42—22no—
512C42—44no—
513C42#12014yes—
514C42#12014yes—
515C42C22×D412yes—
516C42C22×D412yes—
517C42C22×D412yes—
518C42C22×D412yes—
519C23—22no—
520C23—22no—
521C23—22no—
522C23—24no—
523C23—22no—
524C23—24no—
525C23—24no—
526C23—44no—
527C23—42no—
528C23—42no—
529C23—44no—
530C23C2×#12012yes—
531C23C22×D814yes—
532C23C22×D812yes—
533C23C2×AGL(1,ℤ/8)12yes—
534C23C23×D411yescharacteristic
535Q8—28no—
536Q8—432no—
537Q8—48no—
538Q8—416no—
539Q8—416no—
540Q8—88no—
541Q8—88no—
542D4—216no—
543D4—28no—
544D4—28no—
545D4—432no—
546D4—416no—
547D4—44no—
548D4—48no—
549D4—48no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—24no—
555C2×C4—24no—
556C2×C4—216no—
557C2×C4—28no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—28no—
561C2×C4—24no—
562C2×C4—24no—
563C2×C4—24no—
564C2×C4—432no—
565C2×C4—432no—
566C2×C4—416no—
567C2×C4—44no—
568C2×C4—42no—
569C2×C4—44no—
570C2×C4—42no—
571C2×C4—44no—
572C2×C4—44no—
573C2×C4—416no—
574C2×C4—48no—
575C2×C4—44no—
576C2×C4—44no—
577C2×C4—44no—
578C2×C4—44no—
579C2×C4—44no—
580C2×C4—44no—
581C2×C4—88no—
582C2×C4#447116yes—
583C2×C4C2×#12012yes—
584C2×C4C2×#12012yes—
585C2×C4C2×#12012yes—
586C2×C4C2×#12012yes—
587C2×C4C2×#12012yes—
588C2×C4C22×D814yes—
589C2×C4C22×D812yes—
590C2×C4C22×D812yes—
591C2×C4C2×AGL(1,ℤ/8)12yes—
592C2×C4C2×AGL(1,ℤ/8)12yes—
593C2×C4#57814yes—
594C2×C4C23×D411yescharacteristic
595C2×C4C23×D411yescharacteristic
596C2×C4C2×2−1+414yes—
597C8—28no—
598C8—44no—
599C22—24no—
600C22—24no—
601C22—22no—
602C22—22no—
603C22—28no—
604C22—24no—
605C22—44no—
606C22—42no—
607C22—42no—
608C22—44no—
609C22—44no—
610C22—44no—
611C22—84no—
612C22—84no—
613C22C2×#44714yes—
614C22C22×#12011yescharacteristic
615C22#349912yes—
616C22C23×D811yescharacteristic
617C22C22×AGL(1,ℤ/8)11yescharacteristic
618C22C2×#57812yes—
619C4—22no—
620C4—24no—
621C4—22no—
622C4—42no—
623C4—48no—
624C4—44no—
625C4—42no—
626C4—84no—
627C4—84no—
628C4C2×#44714yes—
629C4C2×#44714yes—
630C4#301114yes—
631C4#348612yes—
632C4#349912yes—
633C2—24no—
634C2—44no—
635C2—84no—
636C2#3566611yescharacteristic
637C2C22×#44711yescharacteristic
638C2C2×#301111yescharacteristic
639C2#6247112yes—
640C2#6247912yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#3499C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×#41 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#3226C2semidirect
3. C22×#500C2semidirect
4. #35316C2semidirect
5. #25838C2semidirect
6. #25813C2semidirect
7. #25703C2semidirect
8. #25673C2semidirect
9. #24088C2semidirect
10. #21833C2semidirect
11. #21401C2semidirect
12. #21373C2semidirect
475. C2×C82−1+4semidirect

Maximal subgroups

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

subgroupindex
C2×#34992
C2×#32262
C22×#5002
#353162
#258382
#258132
#257032
#256732
#240882
#218332
#214012
#213732