Finite Group Databasebeta

← #6508620

Subgroups — #6508620

Order 512.

3138 conjugacy classes of subgroups.

The subgroup lattice has 609 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3501C211yesmaximal, characteristic
2C2×#3095C212yesmaximal, complemented
3C2×#2485C211yesmaximal, characteristic
4#24109C218yesmaximal, complemented
5#24064C212yesmaximal, complemented
6#24063C212yesmaximal, complemented
7#24053C218yesmaximal, complemented
8#24006C212yesmaximal, complemented
9#24003C212yesmaximal, complemented
10#21765C212yesmaximal, complemented
11#21742C211yesmaximal, characteristic
12#3501—216no—
13C2×#554—24no—
14C2×#554C2214yes—
15C2×D4×Q8C2214yes—
16C2×#544—21no—
17C2×#543—21no—
18C2×#543—21no—
19C2×#533C2212yes—
20C22×C22⋊Q8C2212yes—
21C22×C22⋊Q8C2212yes—
22C2×#518—21no—
23C22×C4×D4C2211yescharacteristic
24C22×#117C2211yescharacteristic
25#3095—216no—
26C2×#484C2214yes—
27C2×#485C2218yescomplemented
28C2×#483C2214yes—
29C2×#428C2214yescomplemented
30C2×#419C2214yescomplemented
31C2×#407C2212yes—
32#2485—24no—
33#2485C2218yes—
34C2×#397—21no—
35C2×#394—21no—
36C2×#389C2211yescharacteristic
37C2×#389C2211yescharacteristic
38C2×#388—21no—
39C2×#387—21no—
40C2×#380C2211yescharacteristic
41C2×#380C2211yescharacteristic
42C2×#380C2211yescharacteristic
43#2058C2214yescomplemented
44#2057C2214yescomplemented
45#2043C2214yescomplemented
46#2043C2214yescomplemented
47#2036C2218yescomplemented
48#2035C2218yescomplemented
49#1894C2218yescomplemented
50#1872C2214yes—
51#1872C2214yes—
52#1870C2218yes—
53#1865C2218yescomplemented
54#1863C2218yescomplemented
55#1853C2214yescomplemented
56#1852C2214yescomplemented
57#1823C2214yescomplemented
58#1817C2212yes—
59#1787C2214yescomplemented
60#1787C2214yescomplemented
61#1787C2214yescomplemented
62#1779C2212yes—
63C23×Q8C2312yes—
64C23×Q8C2312yes—
65C23×D4D411yescharacteristic
66C24×C4C2311yescharacteristic
67C24×C4C2311yescharacteristic
68C22×M16C2314yes—
69#554—264no—
70#554—216no—
71#554—216no—
72D4×Q8—216no—
73D4×Q8—216no—
74#544—216no—
75#543—216no—
76#543—216no—
77#533—216no—
78C2×C4⋊Q8—24no—
79C2×C4⋊Q8C2312yes—
80C2×C4⋊Q8C2312yes—
81C2×#125—24no—
82C2×#125—22no—
83C2×C22⋊Q8—22no—
84C2×C22⋊Q8—22no—
85C2×C22⋊Q8—22no—
86C2×C22⋊Q8—22no—
87C2×C22⋊Q8—28no—
88C2×C22⋊Q8—22no—
89C2×C22⋊Q8—216no—
90C2×C22⋊Q8—24no—
91C2×C22⋊Q8—24no—
92C2×C22⋊Q8—24no—
93C2×C22⋊Q8—24no—
94C2×C22⋊Q8D4116yes—
95C2×C22⋊Q8C2314yes—
96C2×C22⋊Q8C2314yes—
97C2×C22⋊Q8C2314yes—
98#518—216no—
99C2×C4×Q8—22no—
100C2×C4×Q8C2314yes—
101C2×C4×D4—24no—
102C2×C4×D4—24no—
103C2×C4×D4—28no—
104C2×C4×D4D414yes—
105C2×C4×D4C2314yes—
106C2×#117—22no—
107C2×#117—22no—
108C2×#117—28no—
109C2×#117C2314yes—
110C22×C4⋊C4—22no—
111C22×C4⋊C4—22no—
112C22×C4⋊C4—21no—
113C22×C4⋊C4—21no—
114C22×C4⋊C4D411yescharacteristic
115C22×C4⋊C4C2311yescharacteristic
116C22×C4⋊C4C2311yescharacteristic
117C22×C4⋊C4C2312yes—
118C22×C4⋊C4C2312yes—
119C22×C4⋊C4C2311yescharacteristic
120C22×C4⋊C4C2311yescharacteristic
121C22×C22⋊C4—22no—
122C22×C22⋊C4—22no—
123C22×C22⋊C4—21no—
124C22×C22⋊C4—21no—
125C22×C22⋊C4—21no—
126C22×C22⋊C4—21no—
127C22×C22⋊C4D411yescharacteristic
128C22×C22⋊C4D411yescharacteristic
129C22×C42—21no—
130C22×C42C2311yescharacteristic
131#485—232no—
132#484—216no—
133#483—216no—
134#428—216no—
135C2×#107C2318yes—
136C2×#106C2318yes—
137#419—216no—
138C2×#103C2318yes—
139C2×#103C2318yes—
140C2×#103C2318yes—
141C2×#103C2318yes—
142#407—28no—
143C2×C22⋊C8C2314yes—
144C2×C22⋊C8C2314yes—
145C2×C22⋊C8C2314yes—
146C2×C22⋊C8C2314yes—
147#397—28no—
148#394—28no—
149#389—22no—
150#389—22no—
151#389C2314yes—
152#389C2314yes—
153#388—28no—
154#387—28no—
155#380—22no—
156#380—22no—
157#380—22no—
158#380C2314yes—
159#380C2314yes—
160#380C2314yes—
161C2×#95—21no—
162C2×#95—21no—
163C2×#95—21no—
164C2×#95—21no—
165C2×#95—21no—
166#340C2318yes—
167#340C2318yes—
168#336C2314yes—
169#336C2314yes—
170C25C2×D411yescharacteristic
171C25C2×D411yescharacteristic
172C22×Q8—28no—
173C22×Q8—22no—
174C22×Q8—22no—
175C22×Q8—22no—
176C22×Q8—24no—
177C22×Q8—44no—
178C22×Q8—44no—
179C22×Q8—44no—
180C22×Q8—44no—
181C22×Q8C2×D418yes—
182C22×Q8C2414yes—
183C22×D4—24no—
184C22×D4—24no—
185C22×D4—24no—
186C22×D4C2×D414yes—
187C23×C4—22no—
188C23×C4—22no—
189C23×C4—22no—
190C23×C4—22no—
191C23×C4—24no—
192C23×C4—21no—
193C23×C4—21no—
194C23×C4—21no—
195C23×C4—21no—
196C23×C4—42no—
197C23×C4—42no—
198C23×C4—42no—
199C23×C4—42no—
200C23×C4C2×D411yescharacteristic
201C23×C4C2×D411yescharacteristic
202C23×C4C2×D411yescharacteristic
203C23×C4C2×D412yes—
204C23×C4C2×D411yescharacteristic
205C23×C4C2×D412yes—
206C23×C4C2×D418yes—
207C23×C4C2412yes—
208C23×C4C2411yescharacteristic
209C23×C4C2412yes—
210C2×M16—28no—
211C2×M16—44no—
212C2×M16—44no—
213C22×C8C2418yes—
214C4⋊Q8—232no—
215C4⋊Q8—28no—
216C4⋊Q8—28no—
217#125—216no—
218#125—232no—
219C22⋊Q8—216no—
220C22⋊Q8—216no—
221C22⋊Q8—216no—
222C22⋊Q8—232no—
223C22⋊Q8—232no—
224C22⋊Q8—232no—
225C22⋊Q8—232no—
226C22⋊Q8—28no—
227C22⋊Q8—216no—
228C22⋊Q8—216no—
229C22⋊Q8—28no—
230C22⋊Q8—28no—
231C22⋊Q8—28no—
232C22⋊Q8C2×D4116yes—
233C22⋊Q8C2×D4116yes—
234C4×Q8—216no—
235C4×Q8—28no—
236C4×Q8—28no—
237C4×D4—232no—
238C4×D4—232no—
239C4×D4—216no—
240C4×D4—28no—
241C4×D4—28no—
242#117—216no—
243#117—28no—
244#117—28no—
245#117—48no—
246C2×C4⋊C4—216no—
247C2×C4⋊C4—216no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—24no—
259C2×C4⋊C4—28no—
260C2×C4⋊C4—24no—
261C2×C4⋊C4—24no—
262C2×C4⋊C4—24no—
263C2×C4⋊C4—21no—
264C2×C4⋊C4—21no—
265C2×C4⋊C4—21no—
266C2×C4⋊C4—21no—
267C2×C4⋊C4—24no—
268C2×C4⋊C4—42no—
269C2×C4⋊C4—42no—
270C2×C4⋊C4—44no—
271C2×C4⋊C4—82no—
272C2×C4⋊C4C2×D414yes—
273C2×C4⋊C4C2×D418yes—
274C2×C4⋊C4C2412yes—
275C2×C4⋊C4C2412yes—
276C2×C4⋊C4C2412yes—
277C2×C4⋊C4C2412yes—
278C2×C4⋊C4C2414yes—
279C2×C22⋊C4—216no—
280C2×C22⋊C4—216no—
281C2×C22⋊C4—24no—
282C2×C22⋊C4—24no—
283C2×C22⋊C4—24no—
284C2×C22⋊C4—24no—
285C2×C22⋊C4—24no—
286C2×C22⋊C4—24no—
287C2×C22⋊C4—24no—
288C2×C22⋊C4—24no—
289C2×C22⋊C4—24no—
290C2×C22⋊C4—44no—
291C2×C22⋊C4—44no—
292C2×C22⋊C4—82no—
293C2×C22⋊C4—82no—
294C2×C22⋊C4C2×D414yes—
295C2×C22⋊C4C2×D414yes—
296C2×C42—24no—
297C2×C42—24no—
298C2×C42—24no—
299C2×C42—44no—
300C2×C42C2×D412yes—
301C2×C42C2412yes—
302#107—216no—
303#106—216no—
304#103—216no—
305#103—216no—
306#103—48no—
307#103—48no—
308C22⋊C8—28no—
309C22⋊C8—28no—
310C22⋊C8—44no—
311C22⋊C8—44no—
312#95—24no—
313#95—24no—
314#95—24no—
315#95—24no—
316#95—24no—
317C24—22no—
318C24—22no—
319C24—22no—
320C24—22no—
321C24—41no—
322C24—41no—
323C24—41no—
324C24—41no—
325C24—41no—
326C24—41no—
327C24—41no—
328C24—41no—
329C24#12014yes—
330C24#13714yes—
331C24C22×D412yes—
332C24C22×D412yes—
333C24C22×D411yescharacteristic
334C2×Q8—28no—
335C2×Q8—28no—
336C2×Q8—28no—
337C2×Q8—24no—
338C2×Q8—28no—
339C2×Q8—28no—
340C2×Q8—416no—
341C2×Q8—82no—
342C2×Q8—82no—
343C2×Q8—82no—
344C2×Q8—82no—
345C2×Q8—82no—
346C2×Q8—82no—
347C2×Q8—88no—
348C2×Q8—816no—
349C2×Q8C22×D418yes—
350C2×D4—232no—
351C2×D4—28no—
352C2×D4—28no—
353C2×D4—28no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—24no—
359C22×C4—24no—
360C22×C4—24no—
361C22×C4—24no—
362C22×C4—28no—
363C22×C4—28no—
364C22×C4—22no—
365C22×C4—28no—
366C22×C4—28no—
367C22×C4—22no—
368C22×C4—24no—
369C22×C4—24no—
370C22×C4—24no—
371C22×C4—24no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—416no—
377C22×C4—42no—
378C22×C4—42no—
379C22×C4—41no—
380C22×C4—41no—
381C22×C4—41no—
382C22×C4—42no—
383C22×C4—42no—
384C22×C4—41no—
385C22×C4—41no—
386C22×C4—41no—
387C22×C4—42no—
388C22×C4—42no—
389C22×C4—42no—
390C22×C4—42no—
391C22×C4—42no—
392C22×C4—42no—
393C22×C4—42no—
394C22×C4—42no—
395C22×C4—42no—
396C22×C4—42no—
397C22×C4—42no—
398C22×C4—42no—
399C22×C4—88no—
400C22×C4—82no—
401C22×C4—82no—
402C22×C4—82no—
403C22×C4—82no—
404C22×C4—82no—
405C22×C4—82no—
406C22×C4—82no—
407C22×C4—82no—
408C22×C4#12014yes—
409C22×C4#12018yes—
410C22×C4#13714yes—
411C22×C4C22×D412yes—
412C22×C4C22×D412yes—
413C22×C4C22×D412yes—
414C22×C4C22×D411yescharacteristic
415C22×C4C22×D414yes—
416C22×C4C22×D414yes—
417C22×C4C22×D414yes—
418C22×C4C22×D412yes—
419C22×C4C2511yescharacteristic
420M16—416no—
421C2×C8—28no—
422C2×C8—28no—
423C2×C8—28no—
424C4⋊C4—28no—
425C4⋊C4—216no—
426C4⋊C4—28no—
427C4⋊C4—28no—
428C4⋊C4—216no—
429C4⋊C4—28no—
430C4⋊C4—216no—
431C4⋊C4—216no—
432C4⋊C4—28no—
433C4⋊C4—28no—
434C4⋊C4—24no—
435C4⋊C4—24no—
436C4⋊C4—24no—
437C4⋊C4—24no—
438C4⋊C4—24no—
439C4⋊C4—42no—
440C4⋊C4—42no—
441C4⋊C4—42no—
442C4⋊C4—42no—
443C4⋊C4—48no—
444C4⋊C4—84no—
445C4⋊C4C22×D418yes—
446C22⋊C4—216no—
447C22⋊C4—216no—
448C22⋊C4—216no—
449C22⋊C4—28no—
450C22⋊C4—28no—
451C22⋊C4—48no—
452C22⋊C4—48no—
453C22⋊C4—84no—
454C22⋊C4—84no—
455C42—216no—
456C42—24no—
457C42—24no—
458C42—48no—
459C23—21no—
460C23—21no—
461C23—21no—
462C23—21no—
463C23—22no—
464C23—22no—
465C23—22no—
466C23—22no—
467C23—22no—
468C23—22no—
469C23—22no—
470C23—22no—
471C23—22no—
472C23—22no—
473C23—22no—
474C23—22no—
475C23—28no—
476C23—28no—
477C23—44no—
478C23—44no—
479C23—44no—
480C23—44no—
481C23—82no—
482C23—82no—
483C23—82no—
484C23—82no—
485C23—82no—
486C23—82no—
487C23—82no—
488C23—82no—
489C23C2×#12014yes—
490C23C2×#12012yes—
491C23C2×#13714yes—
492C23C2×#13712yes—
493C23C2×#13712yes—
494C23C23×D411yescharacteristic
495Q8—216no—
496Q8—48no—
497Q8—88no—
498Q8—88no—
499Q8—88no—
500D4—232no—
501C2×C4—22no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—28no—
507C2×C4—22no—
508C2×C4—28no—
509C2×C4—28no—
510C2×C4—28no—
511C2×C4—22no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—28no—
515C2×C4—28no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—48no—
520C2×C4—44no—
521C2×C4—44no—
522C2×C4—44no—
523C2×C4—44no—
524C2×C4—44no—
525C2×C4—48no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—84no—
539C2×C4—82no—
540C2×C4—82no—
541C2×C4—88no—
542C2×C4—88no—
543C2×C4—88no—
544C2×C4—88no—
545C2×C4—84no—
546C2×C4—84no—
547C2×C4—84no—
548C2×C4—84no—
549C2×C4C2×#12012yes—
550C2×C4C2×#12014yes—
551C2×C4C2×#13714yes—
552C2×C4C23×D412yes—
553C2×C4#58512yes—
554C2×C4#58512yes—
555C8—48no—
556C22—22no—
557C22—22no—
558C22—22no—
559C22—22no—
560C22—22no—
561C22—22no—
562C22—24no—
563C22—24no—
564C22—24no—
565C22—24no—
566C22—24no—
567C22—24no—
568C22—42no—
569C22—42no—
570C22—42no—
571C22—42no—
572C22—42no—
573C22—42no—
574C22—42no—
575C22—42no—
576C22—88no—
577C22—88no—
578C22#300414yes—
579C22C22×#12011yescharacteristic
580C22#348511yescharacteristic
581C22#348511yescharacteristic
582C22C22×#13711yescharacteristic
583C22C22×#13711yescharacteristic
584C22#358511yescharacteristic
585C22#358611yescharacteristic
586C4—22no—
587C4—24no—
588C4—24no—
589C4—42no—
590C4—48no—
591C4—42no—
592C4—48no—
593C4—82no—
594C4—82no—
595C4—82no—
596C4—82no—
597C4#348514yes—
598C2—24no—
599C2—44no—
600C2—44no—
601C2#3544611yescharacteristic
602C2C2×#300411yescharacteristic
603C2#6037111yescharacteristic
604C2#6170511yescharacteristic
605C2#6170811yescharacteristic
606C2#6172111yescharacteristic
607C2#6172311yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×D4D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C4×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#3095C2semidirect
4. #24109C2semidirect
5. #24064C2semidirect
6. #24063C2semidirect
7. #24053C2semidirect
8. #24006C2semidirect
9. #24003C2semidirect
10. #21765C2semidirect
27. C2×#485C22semidirect
29. C2×#428C22semidirect
30. C2×#419C22semidirect
43. #2058C22semidirect
44. #2057C22semidirect
45. #2043C22semidirect
46. #2043C22semidirect
47. #2036C22semidirect
48. #2035C22semidirect
49. #1894C22semidirect
53. #1865C22semidirect
54. #1863C22semidirect
55. #1853C22semidirect
56. #1852C22semidirect
57. #1823C22semidirect
59. #1787C22semidirect
60. #1787C22semidirect
61. #1787C22semidirect

Maximal subgroups

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

subgroupindex
C2×#35012
C2×#30952
C2×#24852
#241092
#240642
#240632
#240532
#240062
#240032
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