Finite Group Databasebeta

← #6500722

Subgroups — #6500722

Order 512.

2466 conjugacy classes of subgroups.

The subgroup lattice has 576 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2409C211yesmaximal, characteristic
2#34185C211yesmaximal, characteristic
3#27589C214yesmaximal, complemented
4#24875C218yesmaximal, complemented
5#24871C218yesmaximal, complemented
6#24010C214yesmaximal, complemented
7#21940C211yesmaximal, characteristic, complemented
8#21938C211yesmaximal, characteristic, complemented
9#21738C211yesmaximal, characteristic
10C2×#1787C211yesmaximal, characteristic, complemented
11C2×#1787C211yesmaximal, characteristic, complemented
12C24×Q8C2211yescharacteristic
13C22×C22⋊Q8C2212yes—
14C22×C4×D4C2211yescharacteristic
15C2×#485C2218yescomplemented
16C22×#103C2214yescomplemented
17C22×#103C2214yescomplemented
18C22×C22⋊C8C2211yescharacteristic
19C22×C22⋊C8C2211yescharacteristic
20#2409—22no—
21#2409—22no—
22#2409C2214yes—
23#2409C2214yes—
24C2×#395—21no—
25C2×#393—21no—
26#2353—22no—
27#2353—22no—
28#2352C2214yes—
29#2352C2214yes—
30#2313—21no—
31#2312—21no—
32#2308—21no—
33C2×#391C2212yes—
34C2×#386—21no—
35C2×#385—21no—
36#2283C2212yes—
37#2280C2212yes—
38C2×#380C2211yescharacteristic
39C2×#380C2211yescharacteristic
40C2×#380C2211yescharacteristic
41#2273—21no—
42#2097C2218yescomplemented
43#2074C2218yescomplemented
44#2036C2218yescomplemented
45#2035C2218yescomplemented
46#1928C2218yescomplemented
47#1927C2218yescomplemented
48#1873C2218yescomplemented
49#1869C2214yes—
50#1869C2214yes—
51#1853C2214yes—
52#1853C2214yescomplemented
53#1852C2214yes—
54#1852C2214yescomplemented
55#1817C2212yes—
56#1808C2214yescomplemented
57#1807C2214yescomplemented
58#1787C2214yes—
59#1787C2214yes—
60#1787C2214yescomplemented
61#1787C2214yescomplemented
62#1779C2212yes—
63C23×Q8—21no—
64C23×Q8—21no—
65C23×Q8—21no—
66C23×Q8—24no—
67C23×Q8—24no—
68C23×Q8—48no—
69C23×Q8C2312yes—
70C23×Q8C2312yes—
71C23×Q8C2312yes—
72C23×D4D411yescharacteristic
73C24×C4—21no—
74C24×C4C2311yescharacteristic
75C24×C4C2311yescharacteristic
76C23×C8C2312yes—
77C23×C8C2312yes—
78C2×C22⋊Q8—24no—
79C2×C22⋊Q8—48no—
80C2×C22⋊Q8C2318yes—
81C2×C4×D4—24no—
82C2×C4×D4—24no—
83C2×C4×D4D412yes—
84C2×C4×D4D412yes—
85C2×C4×D4C2312yes—
86C2×C4×D4C2312yes—
87C2×#117C2314yes—
88C22×C4⋊C4—22no—
89C22×C4⋊C4D411yescharacteristic
90C22×C4⋊C4C2312yes—
91C22×C4⋊C4C2312yes—
92C22×C4⋊C4C2312yes—
93C22×C22⋊C4—21no—
94C22×C22⋊C4—21no—
95C22×C22⋊C4—22no—
96C22×C22⋊C4—21no—
97C22×C22⋊C4D411yescharacteristic
98C22×C22⋊C4D411yescharacteristic
99C22×C42C2311yescharacteristic
100#485—416no—
101C2×#107C2318yes—
102C2×#106C2318yes—
103C2×#103—48no—
104C2×#103—48no—
105C2×#103C2318yes—
106C2×#103C2318yes—
107C2×#103C2318yes—
108C2×#103C2318yes—
109C2×C22⋊C8—24no—
110C2×C22⋊C8—24no—
111C2×C22⋊C8C2312yes—
112C2×C22⋊C8C2312yes—
113C2×C22⋊C8C2312yes—
114C2×C22⋊C8C2312yes—
115C2×C22⋊C8C2318yes—
116#395—24no—
117#395—24no—
118#393—24no—
119#393—24no—
120#391—24no—
121#391C2318yes—
122#388—24no—
123#388—24no—
124#387—24no—
125#387—24no—
126#386—24no—
127#386—24no—
128#386—24no—
129#386—24no—
130#385—24no—
131#385—24no—
132#383—22no—
133#382C2314yes—
134#381—22no—
135#381—22no—
136#380—21no—
137#380—21no—
138#380—21no—
139#380—21no—
140#380—21no—
141#380—21no—
142#380C2312yes—
143#380C2312yes—
144#380C2312yes—
145#380C2312yes—
146#380C2314yes—
147#380C2312yes—
148#380C2312yes—
149#379—22no—
150C4×C4⋊C4C2314yes—
151C4×C22⋊C4—22no—
152C4×C22⋊C4—22no—
153C4×C22⋊C4—22no—
154C4×C22⋊C4—24no—
155C4×C22⋊C4—24no—
156#376—22no—
157C2×#95—21no—
158C2×#95—21no—
159C2×#95—21no—
160C2×#95—22no—
161C2×#95—22no—
162#340C2318yes—
163#340C2318yes—
164#336C2318yes—
165C25C2×D411yescharacteristic
166C25C2×D411yescharacteristic
167C22×Q8—28no—
168C22×Q8—22no—
169C22×Q8—22no—
170C22×Q8—22no—
171C22×Q8—28no—
172C22×Q8—28no—
173C22×Q8—24no—
174C22×Q8—24no—
175C22×Q8—24no—
176C22×Q8—24no—
177C22×Q8—24no—
178C22×Q8—24no—
179C22×Q8—44no—
180C22×Q8—44no—
181C22×Q8—44no—
182C22×Q8—48no—
183C22×Q8—432no—
184C22×Q8—432no—
185C22×Q8—48no—
186C22×Q8—48no—
187C22×Q8—44no—
188C22×Q8—44no—
189C22×Q8—44no—
190C22×Q8C2414yes—
191C22×D4—24no—
192C22×D4—24no—
193C22×D4—24no—
194C22×D4C2×D412yes—
195C22×D4C2×D412yes—
196C23×C4—22no—
197C23×C4—22no—
198C23×C4—22no—
199C23×C4—22no—
200C23×C4—22no—
201C23×C4—22no—
202C23×C4—22no—
203C23×C4—22no—
204C23×C4—44no—
205C23×C4—44no—
206C23×C4—48no—
207C23×C4—44no—
208C23×C4C2×D411yescharacteristic
209C23×C4C2×D411yescharacteristic
210C23×C4C2×D411yescharacteristic
211C23×C4C2×D412yes—
212C23×C4C2×D411yescharacteristic
213C23×C4C2×D411yescharacteristic
214C23×C4C2×D411yescharacteristic
215C23×C4#4114yes—
216C23×C4#4114yes—
217C23×C4C2412yes—
218C23×C4C2411yescharacteristic
219C23×C4C2411yescharacteristic
220C23×C4C2411yescharacteristic
221C22×C8—24no—
222C22×C8—24no—
223C22×C8—44no—
224C22×C8—44no—
225C22×C8C2414yes—
226C22×C8C2414yes—
227C22⋊Q8—216no—
228C22⋊Q8—48no—
229C22⋊Q8—416no—
230C4×D4—28no—
231C4×D4—28no—
232C4×D4—24no—
233C4×D4—24no—
234C4×D4—24no—
235C4×D4—24no—
236#117—48no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—28no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—44no—
243C2×C4⋊C4—44no—
244C2×C4⋊C4—44no—
245C2×C4⋊C4—48no—
246C2×C4⋊C4C2×D412yes—
247C2×C4⋊C4C2×D412yes—
248C2×C4⋊C4C2414yes—
249C2×C4⋊C4C2414yes—
250C2×C4⋊C4C2414yes—
251C2×C22⋊C4—24no—
252C2×C22⋊C4—24no—
253C2×C22⋊C4—28no—
254C2×C22⋊C4—22no—
255C2×C22⋊C4—22no—
256C2×C22⋊C4—22no—
257C2×C22⋊C4—22no—
258C2×C22⋊C4—22no—
259C2×C22⋊C4—22no—
260C2×C22⋊C4—24no—
261C2×C22⋊C4—24no—
262C2×C22⋊C4—48no—
263C2×C22⋊C4—44no—
264C2×C22⋊C4—44no—
265C2×C22⋊C4—44no—
266C2×C22⋊C4C2×D412yes—
267C2×C22⋊C4C2×D412yes—
268C2×C22⋊C4C2×D412yes—
269C2×C22⋊C4C2×D412yes—
270C2×C42—22no—
271C2×C42—22no—
272C2×C42—24no—
273C2×C42—24no—
274C2×C42C2×D412yes—
275C2×C42C2412yes—
276#107—48no—
277#106—48no—
278#103—216no—
279#103—216no—
280#103—48no—
281#103—48no—
282C22⋊C8—24no—
283C22⋊C8—24no—
284C22⋊C8—24no—
285C22⋊C8—24no—
286C22⋊C8—48no—
287#95—22no—
288#95—22no—
289#95—22no—
290#95—22no—
291#95—22no—
292#95—22no—
293#95—28no—
294#95—28no—
295#95—24no—
296#95—24no—
297#95—24no—
298#95—24no—
299C24—22no—
300C24—22no—
301C24—22no—
302C24—42no—
303C24—42no—
304C24—42no—
305C24—42no—
306C24#12314yes—
307C24#13514yes—
308C24#13714yes—
309C24C22×D412yes—
310C24C22×D411yescharacteristic
311C24C22×D411yescharacteristic
312C24C22×D411yescharacteristic
313C2×Q8—28no—
314C2×Q8—24no—
315C2×Q8—216no—
316C2×Q8—216no—
317C2×Q8—24no—
318C2×Q8—24no—
319C2×Q8—24no—
320C2×Q8—24no—
321C2×Q8—48no—
322C2×Q8—48no—
323C2×Q8—416no—
324C2×Q8—416no—
325C2×Q8—416no—
326C2×Q8—416no—
327C2×Q8—44no—
328C2×Q8—48no—
329C2×Q8—48no—
330C2×Q8—432no—
331C2×Q8—432no—
332C2×Q8—432no—
333C2×Q8—44no—
334C2×Q8—44no—
335C2×Q8—44no—
336C2×Q8H(5,𝔽2)18yes—
337C2×D4—24no—
338C2×D4—24no—
339C2×D4—24no—
340C2×D4—24no—
341C2×D4—24no—
342C2×D4—24no—
343C2×D4—416no—
344C22×C4—24no—
345C22×C4—24no—
346C22×C4—24no—
347C22×C4—28no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—24no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—28no—
357C22×C4—24no—
358C22×C4—24no—
359C22×C4—24no—
360C22×C4—24no—
361C22×C4—48no—
362C22×C4—48no—
363C22×C4—42no—
364C22×C4—42no—
365C22×C4—42no—
366C22×C4—42no—
367C22×C4—44no—
368C22×C4—42no—
369C22×C4—48no—
370C22×C4—48no—
371C22×C4—44no—
372C22×C4—432no—
373C22×C4—44no—
374C22×C4—44no—
375C22×C4—44no—
376C22×C4—44no—
377C22×C4—44no—
378C22×C4—44no—
379C22×C4—44no—
380C22×C4—44no—
381C22×C4—84no—
382C22×C4—84no—
383C22×C4—82no—
384C22×C4—82no—
385C22×C4—82no—
386C22×C4—82no—
387C22×C4#12314yes—
388C22×C4#12314yes—
389C22×C4#12314yes—
390C22×C4C22×D412yes—
391C22×C4C22×D412yes—
392C22×C4C22×D412yes—
393C22×C4C22×D411yescharacteristic
394C22×C4C22×D411yescharacteristic
395C22×C4C22×D411yescharacteristic
396C22×C4C2×#4112yes—
397C22×C4C2×#4112yes—
398C22×C4C2×#4112yes—
399C22×C4C2×#4112yes—
400C22×C4C2×#4112yes—
401C22×C4C2×#4112yes—
402C22×C4C2511yescharacteristic
403C2×C8—24no—
404C2×C8—24no—
405C2×C8—28no—
406C2×C8—28no—
407C2×C8—84no—
408C2×C8—84no—
409C4⋊C4—24no—
410C4⋊C4—24no—
411C4⋊C4—28no—
412C4⋊C4—28no—
413C4⋊C4—28no—
414C4⋊C4—416no—
415C4⋊C4—44no—
416C4⋊C4—82no—
417C4⋊C4—82no—
418C22⋊C4—24no—
419C22⋊C4—24no—
420C22⋊C4—24no—
421C22⋊C4—24no—
422C22⋊C4—416no—
423C22⋊C4—44no—
424C22⋊C4—44no—
425C22⋊C4—44no—
426C22⋊C4—44no—
427C22⋊C4—44no—
428C22⋊C4—44no—
429C22⋊C4—84no—
430C22⋊C4—84no—
431C42—24no—
432C42—22no—
433C42—22no—
434C42—84no—
435C42—84no—
436C23—22no—
437C23—22no—
438C23—22no—
439C23—22no—
440C23—22no—
441C23—22no—
442C23—24no—
443C23—22no—
444C23—24no—
445C23—22no—
446C23—22no—
447C23—44no—
448C23—44no—
449C23—48no—
450C23—48no—
451C23—48no—
452C23—44no—
453C23—82no—
454C23—82no—
455C23—84no—
456C23#485116yescomplemented
457C23C2×#12312yes—
458C23C2×#12312yes—
459C23C2×#12312yes—
460C23C2×#13512yes—
461C23C2×#13512yes—
462C23C2×#13512yes—
463C23C2×#13712yes—
464C23C2×#13712yes—
465C23C2×#13712yes—
466C23C23×D411yescharacteristic
467Q8—28no—
468Q8—28no—
469Q8—416no—
470Q8—416no—
471Q8—416no—
472Q8—44no—
473Q8—44no—
474D4—48no—
475D4—48no—
476C2×C4—22no—
477C2×C4—22no—
478C2×C4—22no—
479C2×C4—24no—
480C2×C4—24no—
481C2×C4—24no—
482C2×C4—24no—
483C2×C4—24no—
484C2×C4—24no—
485C2×C4—24no—
486C2×C4—22no—
487C2×C4—22no—
488C2×C4—22no—
489C2×C4—24no—
490C2×C4—28no—
491C2×C4—22no—
492C2×C4—22no—
493C2×C4—44no—
494C2×C4—44no—
495C2×C4—44no—
496C2×C4—416no—
497C2×C4—416no—
498C2×C4—416no—
499C2×C4—44no—
500C2×C4—44no—
501C2×C4—44no—
502C2×C4—44no—
503C2×C4—44no—
504C2×C4—44no—
505C2×C4—44no—
506C2×C4—44no—
507C2×C4—44no—
508C2×C4—44no—
509C2×C4—44no—
510C2×C4—44no—
511C2×C4—44no—
512C2×C4—44no—
513C2×C4—84no—
514C2×C4—82no—
515C2×C4—82no—
516C2×C4—88no—
517C2×C4—164no—
518C2×C4—164no—
519C2×C4C2×#12312yes—
520C2×C4C2×#12312yes—
521C2×C4C2×#12312yes—
522C2×C4C22×#4111yescharacteristic
523C2×C4C22×#4111yescharacteristic
524C2×C4C2×H(5,𝔽2)14yes—
525C8—44no—
526C8—44no—
527C22—22no—
528C22—22no—
529C22—22no—
530C22—22no—
531C22—22no—
532C22—22no—
533C22—24no—
534C22—24no—
535C22—24no—
536C22—24no—
537C22—24no—
538C22—24no—
539C22—416no—
540C22—42no—
541C22—42no—
542C22—42no—
543C22—44no—
544C22—44no—
545C22—88no—
546C22C2×#48514yes—
547C22C2×#48514yes—
548C22C2×#48514yes—
549C22C22×#12311yescharacteristic
550C22#348412yes—
551C22C22×#13511yescharacteristic
552C22C22×#13711yescharacteristic
553C22#358612yes—
554C4—22no—
555C4—22no—
556C4—22no—
557C4—42no—
558C4—48no—
559C4—42no—
560C4—42no—
561C4—84no—
562C4—84no—
563C4—84no—
564C4#349012yes—
565C4#349012yes—
566C2—24no—
567C2—24no—
568C2—24no—
569C2—82no—
570C2#3535011yescharacteristic
571C2C22×#48511yescharacteristic
572C2#6051911yescharacteristic
573C2#6156412yes—
574C2#6157412yes—
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
3. #27589C2semidirect
4. #24875C2semidirect
5. #24871C2semidirect
6. #24010C2semidirect
7. #21940C2semidirect
8. #21938C2semidirect
10. C2×#1787C2semidirect
11. C2×#1787C2semidirect
15. C2×#485C22semidirect
16. C22×#103C22semidirect
17. C22×#103C22semidirect
42. #2097C22semidirect
43. #2074C22semidirect
44. #2036C22semidirect
45. #2035C22semidirect
46. #1928C22semidirect
47. #1927C22semidirect
48. #1873C22semidirect
52. #1853C22semidirect
54. #1852C22semidirect
56. #1808C22semidirect
57. #1807C22semidirect
60. #1787C22semidirect
61. #1787C22semidirect
456. C23#485semidirect

Maximal subgroups

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

subgroupindex
C2×#24092
#341852
#275892
#248752
#248712
#240102
#219402
#219382
#217382
C2×#17872