Finite Group Databasebeta

← #6528943

Subgroups — #6528943

Order 512.

3132 conjugacy classes of subgroups.

The subgroup lattice has 637 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3507C211yesmaximal, characteristic, complemented
2C2×#3110C211yesmaximal, characteristic, complemented
3C2×#3109C211yesmaximal, characteristic
4C2×#2580C211yesmaximal, characteristic, complemented
5#24340C214yesmaximal, complemented
6#24339C214yesmaximal, complemented
7#24198C214yesmaximal, complemented
8#24197C214yesmaximal, complemented
9#24067C214yesmaximal, complemented
10#23936C214yesmaximal, complemented
11#21866C211yesmaximal, characteristic, complemented
12#21865C211yesmaximal, characteristic, complemented
13#21822C211yesmaximal, characteristic, complemented
14#3507—28no—
15#3507—28no—
16C2×#559C2212yescomplemented
17C2×#557C2212yescomplemented
18C2×#556—22no—
19C2×#556C2212yescomplemented
20C2×#554C2212yescomplemented
21C2×#552—22no—
22C2×#539—22no—
23C2×#536—21no—
24C2×#533C2211yescharacteristic
25C2×#532C2211yescharacteristic
26C2×#529C2214yes—
27C2×#520—21no—
28C22×C4×Q8C2211yescharacteristic
29C22×C4×D4C2211yescharacteristic
30#3110—28no—
31#3109—28no—
32C2×#484C2212yes—
33C2×#481C2212yes—
34C2×#476C2212yes—
35C2×#474C2212yes—
36C2×#461C2212yes—
37C2×#459C2212yes—
38C2×#450C2212yes—
39C2×#447C2212yes—
40C2×#442C2214yes—
41C2×#440C2214yes—
42C2×#432C2211yescharacteristic
43C2×#432C2211yescharacteristic
44#2580—22no—
45#2580—22no—
46#2580C2214yes—
47#2580C2214yes—
48C2×#399—21no—
49C2×#393—21no—
50C2×#390C2212yes—
51C2×#389C2212yes—
52C2×#385—22no—
53C2×#382C2211yescharacteristic
54#2059C2218yescomplemented
55#2042C2218yescomplemented
56#2025C2218yes—
57#2012C2218yes—
58#1993C2212yescomplemented
59#1993C2212yescomplemented
60#1985C2214yescomplemented
61#1985C2214yescomplemented
62#1961C2212yescomplemented
63#1960C2212yescomplemented
64#1880C2218yes—
65#1879C2218yes—
66#1874C2214yes—
67#1872C2214yes—
68#1865C2218yescomplemented
69#1862C2218yescomplemented
70#1817C2214yes—
71#1805C2214yescomplemented
72#1804C2214yescomplemented
73C22×#41—21no—
74C23×Q8D411yescharacteristic
75C23×D4C2311yescharacteristic
76C24×C4—21no—
77C22×SD16C2314yes—
78C22×D8C2314yes—
79#559—28no—
80#559—28no—
81#557—28no—
82#557—28no—
83#556—232no—
84#556—28no—
85#556—28no—
86#554—28no—
87#554—28no—
88#552—232no—
89#539—232no—
90#536—216no—
91#533—24no—
92#533D418yes—
93#532—24no—
94#532D418yes—
95C2×C4⋊Q8—22no—
96C2×C4⋊Q8C2311yescharacteristic
97C2×C4⋊Q8C2311yescharacteristic
98C2×C4⋊Q8C2311yescharacteristic
99C2×#127C2311yescharacteristic
100#529—216no—
101#529—216no—
102C2×#126—24no—
103C2×#126—24no—
104C2×#125—24no—
105C2×#125—21no—
106C2×#125C2314yes—
107C2×#124—22no—
108C2×#124—21no—
109C2×#124C2314yes—
110C2×#123—24no—
111C2×#123—24no—
112C2×C22⋊Q8—22no—
113C2×C22⋊Q8—22no—
114C2×C22⋊Q8—21no—
115C2×C22⋊Q8—24no—
116C2×C22⋊Q8C2312yes—
117C2×#121—21no—
118C2×#121C2312yes—
119#520—216no—
120C2×C4×Q8—22no—
121C2×C4×Q8—24no—
122C2×C4×Q8—24no—
123C2×C4×Q8—21no—
124C2×C4×Q8D412yes—
125C2×C4×Q8D412yes—
126C2×C4×Q8C2314yes—
127C2×C4×Q8C2312yes—
128C2×C4×Q8C2312yes—
129C2×C4×Q8C2312yes—
130C2×C4×D4—24no—
131C2×C4×D4—22no—
132C2×C4×D4—21no—
133C2×C4×D4—21no—
134C2×C4×D4—21no—
135C2×C4×D4—24no—
136C2×C4×D4—24no—
137C2×C4×D4C2312yes—
138C2×C4×D4C2312yes—
139C2×C4×D4C2314yes—
140C2×C4×D4C2312yes—
141C2×#117—21no—
142C2×#117—22no—
143C2×#117—24no—
144C22×C4⋊C4—22no—
145C22×C4⋊C4—22no—
146C22×C4⋊C4D411yescharacteristic
147C22×C4⋊C4D412yes—
148C22×C4⋊C4C2311yescharacteristic
149C22×C4⋊C4C2312yes—
150C22×C22⋊C4—21no—
151C22×C22⋊C4—22no—
152C22×C22⋊C4—22no—
153C22×C42C2312yes—
154C22×C42C2311yescharacteristic
155#484—28no—
156#481—28no—
157#476—28no—
158#474—28no—
159#461—28no—
160#459—28no—
161#450—28no—
162#447—28no—
163#442—216no—
164#440—216no—
165#432—24no—
166#432—24no—
167C2×#107C2314yes—
168C2×#106C2314yes—
169C2×C4⋊C8C2312yes—
170C2×C4⋊C8C2312yes—
171C2×#103C2314yes—
172C2×#103C2318yes—
173C2×#102C2314yes—
174C2×#102C2314yes—
175C2×#102C2314yes—
176C2×#102C2318yes—
177C2×C22⋊C8C2312yes—
178C2×C22⋊C8C2312yes—
179C2×#97C2312yes—
180C2×#97C2312yes—
181#399—24no—
182#399—24no—
183#393—24no—
184#393—24no—
185#390—24no—
186#390C2318yes—
187#389—24no—
188#389C2318yes—
189#385—28no—
190#385—28no—
191#382—21no—
192#382—21no—
193#382C2312yes—
194#382C2312yes—
195C2×#95—21no—
196C2×#95—22no—
197C2×#95—21no—
198#340C2318yes—
199#340C2318yes—
200#336C2318yes—
201#336C2318yes—
202C25—21no—
203C2×#41—28no—
204C2×#41—48no—
205C22×Q8—24no—
206C22×Q8—21no—
207C22×Q8—21no—
208C22×Q8—44no—
209C22×Q8C2×D412yes—
210C22×Q8C2×D412yes—
211C22×Q8C2412yes—
212C22×D4—22no—
213C22×D4—24no—
214C22×D4—21no—
215C22×D4—21no—
216C22×D4—21no—
217C22×D4—22no—
218C22×D4—22no—
219C22×D4—42no—
220C22×D4C2412yes—
221C22×D4C2412yes—
222C23×C4—22no—
223C23×C4—24no—
224C23×C4—22no—
225C23×C4—21no—
226C23×C4—22no—
227C23×C4—22no—
228C23×C4—44no—
229C23×C4—84no—
230C23×C4C2×D412yes—
231C23×C4C2×D411yescharacteristic
232C23×C4C2×D412yes—
233C23×C4C2×D411yescharacteristic
234C23×C4C2411yescharacteristic
235C2×SD16—28no—
236C2×SD16—28no—
237C2×SD16—44no—
238C2×D8—28no—
239C2×D8—28no—
240C2×D8—44no—
241C22×C8C2414yes—
242C22×C8C2414yes—
243C4⋊Q8—216no—
244C4⋊Q8—22no—
245C4⋊Q8—22no—
246C4⋊Q8—22no—
247C4⋊Q8C2×D414yes—
248C4⋊Q8C2×D414yes—
249C4⋊Q8C2×D414yes—
250#127—22no—
251#127C2×D414yes—
252#126—232no—
253#126—232no—
254#125—232no—
255#125—28no—
256#125—28no—
257#125—28no—
258#124—216no—
259#124—28no—
260#124—28no—
261#124—28no—
262#123—232no—
263#123—232no—
264C22⋊Q8—216no—
265C22⋊Q8—216no—
266C22⋊Q8—232no—
267C22⋊Q8—28no—
268C22⋊Q8—24no—
269C22⋊Q8C2×D418yes—
270#121—28no—
271#121—24no—
272#121C2×D418yes—
273C4×Q8—216no—
274C4×Q8—28no—
275C4×Q8—28no—
276C4×Q8—28no—
277C4×Q8—28no—
278C4×Q8—28no—
279C4×Q8—24no—
280C4×Q8—24no—
281C4×Q8—24no—
282C4×Q8—24no—
283C4×Q8—24no—
284C4×Q8C2×D418yes—
285C4×D4—232no—
286C4×D4—216no—
287C4×D4—28no—
288C4×D4—28no—
289C4×D4—28no—
290C4×D4—24no—
291C4×D4—24no—
292C4×D4—24no—
293C4×D4—24no—
294C4×D4—24no—
295C4×D4—44no—
296C4×D4—44no—
297C4×D4C2×D418yes—
298#117—216no—
299#117—232no—
300#117—28no—
301C2×C4⋊C4—216no—
302C2×C4⋊C4—24no—
303C2×C4⋊C4—28no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—24no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—24no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—21no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—24no—
313C2×C4⋊C4—22no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—21no—
316C2×C4⋊C4—22no—
317C2×C4⋊C4—28no—
318C2×C4⋊C4—44no—
319C2×C4⋊C4—84no—
320C2×C4⋊C4C2×D412yes—
321C2×C4⋊C4C2×D412yes—
322C2×C4⋊C4C2×D414yes—
323C2×C4⋊C4C2×D414yes—
324C2×C4⋊C4C2412yes—
325C2×C4⋊C4C2414yes—
326C2×C4⋊C4C2412yes—
327C2×C4⋊C4C2412yes—
328C2×C22⋊C4—22no—
329C2×C22⋊C4—22no—
330C2×C22⋊C4—24no—
331C2×C22⋊C4—24no—
332C2×C22⋊C4—22no—
333C2×C22⋊C4—24no—
334C2×C22⋊C4—22no—
335C2×C22⋊C4—22no—
336C2×C22⋊C4—28no—
337C2×C22⋊C4—28no—
338C2×C22⋊C4—48no—
339C2×C22⋊C4—44no—
340C2×C22⋊C4—84no—
341C2×C42—24no—
342C2×C42—22no—
343C2×C42—24no—
344C2×C42—24no—
345C2×C42C2×D414yes—
346C2×C42C2×D411yescharacteristic
347C2×C42C2×D411yescharacteristic
348C2×C42C2×D414yes—
349C2×C42C2×D414yes—
350C2×C42C2414yes—
351C2×C42C2411yescharacteristic
352C2×C42C2411yescharacteristic
353#107—28no—
354#106—28no—
355C4⋊C8—24no—
356C4⋊C8—24no—
357#103—28no—
358#103—48no—
359#102—28no—
360#102—28no—
361#102—28no—
362#102—48no—
363C22⋊C8—24no—
364C22⋊C8—24no—
365#97—24no—
366#97—24no—
367#95—22no—
368#95—22no—
369#95—22no—
370#95—22no—
371#95—28no—
372C24—24no—
373C24—22no—
374C24—22no—
375C24—44no—
376C24—42no—
377C24—42no—
378C24—44no—
379C24C22×D411yescharacteristic
380#41—432no—
381C2×Q8—24no—
382C2×Q8—24no—
383C2×Q8—24no—
384C2×Q8—24no—
385C2×Q8—22no—
386C2×Q8—24no—
387C2×Q8—24no—
388C2×Q8—44no—
389C2×Q8—44no—
390C2×Q8—416no—
391C2×Q8—44no—
392C2×Q8—82no—
393C2×Q8C22×D414yes—
394C2×D4—24no—
395C2×D4—28no—
396C2×D4—24no—
397C2×D4—24no—
398C2×D4—22no—
399C2×D4—22no—
400C2×D4—22no—
401C2×D4—24no—
402C2×D4—22no—
403C2×D4—24no—
404C2×D4—22no—
405C2×D4—22no—
406C2×D4—48no—
407C2×D4—48no—
408C2×D4—42no—
409C2×D4—44no—
410C2×D4—48no—
411C2×D4—42no—
412C2×D4—82no—
413C2×D4C22×D414yes—
414C2×D42−1+414yes—
415C22×C4—24no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—216no—
425C22×C4—24no—
426C22×C4—28no—
427C22×C4—24no—
428C22×C4—22no—
429C22×C4—42no—
430C22×C4—42no—
431C22×C4—42no—
432C22×C4—48no—
433C22×C4—44no—
434C22×C4—44no—
435C22×C4—44no—
436C22×C4—42no—
437C22×C4—44no—
438C22×C4—44no—
439C22×C4—44no—
440C22×C4—84no—
441C22×C4—84no—
442C22×C4—84no—
443C22×C4—84no—
444C22×C4—816no—
445C22×C4#12014yes—
446C22×C4#12014yes—
447C22×C4C22×D414yes—
448C22×C4C22×D414yes—
449C22×C4C22×D411yescharacteristic
450C22×C4C22×D411yescharacteristic
451C22×C4C22×D411yescharacteristic
452C22×C4C22×D411yescharacteristic
453C22×C4C22×D411yescharacteristic
454C22×C4C22×D412yes—
455C22×C4C22×D412yes—
456C22×C4C2511yescharacteristic
457SD16—416no—
458D8—416no—
459C2×C8—24no—
460C2×C8—24no—
461C2×C8—24no—
462C2×C8—24no—
463C2×C8—24no—
464C2×C8—24no—
465C4⋊C4—216no—
466C4⋊C4—216no—
467C4⋊C4—216no—
468C4⋊C4—24no—
469C4⋊C4—24no—
470C4⋊C4—24no—
471C4⋊C4—28no—
472C4⋊C4—28no—
473C4⋊C4—24no—
474C4⋊C4—28no—
475C4⋊C4—24no—
476C4⋊C4—24no—
477C4⋊C4—28no—
478C4⋊C4—22no—
479C4⋊C4—22no—
480C4⋊C4—22no—
481C4⋊C4—22no—
482C4⋊C4—28no—
483C4⋊C4—44no—
484C4⋊C4—42no—
485C4⋊C4—88no—
486C4⋊C4C22×D414yes—
487C4⋊C4C22×D414yes—
488C22⋊C4—216no—
489C22⋊C4—216no—
490C22⋊C4—216no—
491C22⋊C4—28no—
492C22⋊C4—28no—
493C22⋊C4—28no—
494C22⋊C4—416no—
495C22⋊C4—44no—
496C22⋊C4—44no—
497C22⋊C4—88no—
498C42—216no—
499C42—28no—
500C42—28no—
501C42—24no—
502C42—24no—
503C42#12014yes—
504C42#12014yes—
505C42C22×D412yes—
506C42C22×D412yes—
507C42C22×D412yes—
508C42C22×D412yes—
509C23—22no—
510C23—21no—
511C23—21no—
512C23—24no—
513C23—24no—
514C23—48no—
515C23—416no—
516C23—44no—
517C23—44no—
518C23—42no—
519C23—42no—
520C23—42no—
521C23—42no—
522C23—44no—
523C23—44no—
524C23—84no—
525C23—88no—
526C23—84no—
527C23C2×#12012yes—
528C23#57514yes—
529C23C23×D411yescharacteristic
530Q8—28no—
531Q8—48no—
532Q8—48no—
533Q8—48no—
534Q8—44no—
535Q8—84no—
536D4—24no—
537D4—28no—
538D4—24no—
539D4—48no—
540D4—48no—
541D4—416no—
542D4—44no—
543D4—42no—
544D4—42no—
545D4—84no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—28no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—24no—
554C2×C4—24no—
555C2×C4—24no—
556C2×C4—24no—
557C2×C4—24no—
558C2×C4—24no—
559C2×C4—24no—
560C2×C4—24no—
561C2×C4—24no—
562C2×C4—24no—
563C2×C4—24no—
564C2×C4—44no—
565C2×C4—416no—
566C2×C4—44no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—48no—
570C2×C4—44no—
571C2×C4—44no—
572C2×C4—44no—
573C2×C4—42no—
574C2×C4—44no—
575C2×C4—44no—
576C2×C4—88no—
577C2×C4—88no—
578C2×C4—84no—
579C2×C4—816no—
580C2×C4—88no—
581C2×C4—84no—
582C2×C4C2×#12012yes—
583C2×C4C2×#12012yes—
584C2×C4C2×#12012yes—
585C2×C4C2×#12012yes—
586C2×C4C2×#12012yes—
587C2×C4C23×D411yescharacteristic
588C2×C4C23×D411yescharacteristic
589C2×C4C2×2−1+412yes—
590C2×C4#58512yes—
591C8—44no—
592C8—44no—
593C22—22no—
594C22—24no—
595C22—24no—
596C22—22no—
597C22—44no—
598C22—44no—
599C22—48no—
600C22—44no—
601C22—44no—
602C22—816no—
603C22—84no—
604C22—82no—
605C22—84no—
606C22—84no—
607C22—82no—
608C22#300514yes—
609C22C22×#12011yescharacteristic
610C22#348611yescharacteristic
611C22#348711yescharacteristic
612C22C2×#57512yes—
613C22#358512yes—
614C4—22no—
615C4—22no—
616C4—22no—
617C4—22no—
618C4—22no—
619C4—44no—
620C4—44no—
621C4—44no—
622C4—44no—
623C4—44no—
624C4—84no—
625C4—82no—
626C4#348912yes—
627C4#348912yes—
628C2—24no—
629C2—84no—
630C2—84no—
631C2#3565911yescharacteristic
632C2C2×#300511yescharacteristic
633C2#6036911yescharacteristic
634C2#6183112yes—
635C2#6189512yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#3109C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C4×Q8 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#3507C2semidirect
2. C2×#3110C2semidirect
4. C2×#2580C2semidirect
5. #24340C2semidirect
6. #24339C2semidirect
7. #24198C2semidirect
8. #24197C2semidirect
9. #24067C2semidirect
10. #23936C2semidirect
11. #21866C2semidirect
12. #21865C2semidirect
13. #21822C2semidirect
16. C2×#559C22semidirect
17. C2×#557C22semidirect
19. C2×#556C22semidirect
20. C2×#554C22semidirect
54. #2059C22semidirect
55. #2042C22semidirect
58. #1993C22semidirect
59. #1993C22semidirect
60. #1985C22semidirect
61. #1985C22semidirect
62. #1961C22semidirect
63. #1960C22semidirect
68. #1865C22semidirect
69. #1862C22semidirect
71. #1805C22semidirect
72. #1804C22semidirect

Maximal subgroups

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

subgroupindex
C2×#35072
C2×#31102
C2×#31092
C2×#25802
#243402
#243392
#241982
#241972
#240672
#239362
#218662
#218652
#218222