Finite Group Databasebeta

← #4320296

Subgroups — #4320296

Order 512.

2794 conjugacy classes of subgroups.

The subgroup lattice has 636 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#16823C211yesmaximal, characteristic, complemented
2#16706C212yesmaximal, complemented
3#16666C214yesmaximal, complemented
4#16254C211yesmaximal, characteristic, complemented
5#16133C214yesmaximal, complemented
6#15671C214yesmaximal, complemented
7#14887C214yesmaximal, complemented
8#14717C212yesmaximal
9#14717C212yesmaximal
10#14596C212yesmaximal, complemented
11#14449C212yesmaximal, complemented
12#13868C212yesmaximal
13#13794C211yesmaximal, characteristic
14C22×C22⋊Q8C2212yes—
15C22×C22⋊Q8C2212yes—
16C22×#117C2212yes—
17C23×C22⋊C4C2211yescharacteristic
18C23×C22⋊C4C2211yescharacteristic
19C23×C22⋊C4C2211yescharacteristic
20C2×#395C2214yescomplemented
21C2×#393C2212yes—
22C2×#393C2212yes—
23C2×#393C2212yes—
24C2×#393C2212yes—
25C2×#391C2214yescomplemented
26C2×#381C2212yes—
27C2×#379C2214yes—
28#1446C2214yescomplemented
29#1446C2214yescomplemented
30#1445C2214yescomplemented
31#1445C2214yescomplemented
32#1443C2214yescomplemented
33#1443C2214yescomplemented
34#1442C2214yescomplemented
35#1442C2214yescomplemented
36#1441C2214yescomplemented
37#1441C2214yescomplemented
38#1441C2214yescomplemented
39#1439C2212yes—
40#1439C2212yes—
41#1439C2212yes—
42#1439C2214yes—
43#1439C2212yes—
44#1439C2214yes—
45#1439C2214yes—
46#1439C2212yes—
47#1439C2212yes—
48#1439C2212yes—
49#1439C2212yes—
50#1439C2214yes—
51#1437C2214yes—
52#1437C2214yes—
53#1436C2214yescomplemented
54#1436C2214yescomplemented
55#1434C2214yescomplemented
56C22⋊#95C2211yescharacteristic
57C22⋊#95C2214yes—
58C22⋊#95C2212yes—
59C22⋊#95C2214yes—
60C22⋊#95C2214yes—
61C22⋊#95C2214yes—
62C22⋊#95C2214yes—
63C22⋊#95C2211yescharacteristic
64C22⋊#95C2211yescharacteristic
65C22⋊#95C2212yes—
66C22⋊#95C2211yescharacteristic
67C26C2311yescharacteristic
68C23×Q8C2312yes—
69C23×Q8C2312yes—
70C24×C4C2312yes—
71C24×C4C2312yes—
72C24×C4C2311yescharacteristic
73C24×C4C2311yescharacteristic
74C2×C22⋊Q8—28no—
75C2×C22⋊Q8—44no—
76C2×C22⋊Q8—48no—
77C2×C22⋊Q8—48no—
78C2×#117—48no—
79C2×#117—44no—
80C22×C4⋊C4C2312yes—
81C22×C4⋊C4C2312yes—
82C22×C4⋊C4C2314yes—
83C22×C4⋊C4C2312yes—
84C22×C4⋊C4C2314yes—
85C22×C4⋊C4C2312yes—
86C22×C22⋊C4—24no—
87C22×C22⋊C4—24no—
88C22×C22⋊C4—42no—
89C22×C22⋊C4—42no—
90C22×C22⋊C4—42no—
91C22×C22⋊C4—42no—
92C22×C22⋊C4—42no—
93C22×C22⋊C4—42no—
94C22×C22⋊C4Q818yescomplemented
95C22×C22⋊C4C2312yes—
96C22×C22⋊C4C2312yes—
97C22×C22⋊C4C2312yes—
98C22×C22⋊C4C2314yes—
99C22×C22⋊C4C2314yes—
100C22×C22⋊C4C2314yes—
101C22×C22⋊C4C2314yes—
102C22×C22⋊C4C2314yes—
103C22×C22⋊C4C2311yescharacteristic
104C22×C22⋊C4C2314yes—
105C22×C22⋊C4C2311yescharacteristic
106C22×C22⋊C4C2312yes—
107C22×C22⋊C4C2311yescharacteristic
108C22×C22⋊C4C2311yescharacteristic
109C22×C42C2314yes—
110#395—48no—
111#393—28no—
112#393—44no—
113#393—44no—
114#393—44no—
115#391—48no—
116#381—44no—
117#379—48no—
118C2×#95C2312yes—
119C2×#95C2314yes—
120C2×#95C2314yes—
121C2×#95C2312yes—
122C2×#95C2314yes—
123C2×#95C2312yes—
124C2×#95C2312yes—
125C2×#95C2314yes—
126C2×#95C2314yes—
127C2×#95C2314yes—
128C2×#95C2314yes—
129C2×#95C2314yes—
130C2×#95C2314yes—
131C2×#95C2314yes—
132C2×#95C2314yes—
133C2×#95C2314yes—
134C2×#95C2314yes—
135C2×#95C2314yes—
136C2×#95C2314yes—
137C2×#95C2314yes—
138C2×#95C2314yes—
139C2×#95C2314yes—
140C2×#95C2314yes—
141C2×#95C2312yes—
142C2×#95C2312yes—
143C25—22no—
144C25—22no—
145C25—22no—
146C25—22no—
147C25—22no—
148C25—22no—
149C25—44no—
150C25—44no—
151C25C2×Q814yes—
152C25C2411yescharacteristic
153C25C2411yescharacteristic
154C25C2411yescharacteristic
155C22×Q8—24no—
156C22×Q8—24no—
157C22×Q8—28no—
158C22×Q8—44no—
159C22×Q8—42no—
160C22×Q8—44no—
161C22×Q8—42no—
162C22×Q8—42no—
163C22×Q8—42no—
164C22×Q8—44no—
165C23×C4—24no—
166C23×C4—24no—
167C23×C4—24no—
168C23×C4—22no—
169C23×C4—24no—
170C23×C4—44no—
171C23×C4—44no—
172C23×C4—41no—
173C23×C4—41no—
174C23×C4—42no—
175C23×C4—41no—
176C23×C4—41no—
177C23×C4—44no—
178C23×C4—44no—
179C23×C4—44no—
180C23×C4—44no—
181C23×C4C2×Q818yes—
182C23×C4#4114yes—
183C23×C4C2414yes—
184C23×C4C2414yes—
185C23×C4C2414yes—
186C23×C4C2412yes—
187C23×C4C2412yes—
188C23×C4C2412yes—
189C23×C4C2412yes—
190C23×C4C2412yes—
191C23×C4C2412yes—
192C23×C4C2414yes—
193C22⋊Q8—432no—
194C22⋊Q8—432no—
195#117—432no—
196C2×C4⋊C4—28no—
197C2×C4⋊C4—24no—
198C2×C4⋊C4—28no—
199C2×C4⋊C4—28no—
200C2×C4⋊C4—28no—
201C2×C4⋊C4—28no—
202C2×C4⋊C4—42no—
203C2×C4⋊C4—44no—
204C2×C4⋊C4—42no—
205C2×C4⋊C4—44no—
206C2×C4⋊C4—42no—
207C2×C4⋊C4—44no—
208C2×C4⋊C4—44no—
209C2×C4⋊C4—44no—
210C2×C22⋊C4—22no—
211C2×C22⋊C4—28no—
212C2×C22⋊C4—24no—
213C2×C22⋊C4—24no—
214C2×C22⋊C4—24no—
215C2×C22⋊C4—28no—
216C2×C22⋊C4—28no—
217C2×C22⋊C4—24no—
218C2×C22⋊C4—24no—
219C2×C22⋊C4—24no—
220C2×C22⋊C4—28no—
221C2×C22⋊C4—28no—
222C2×C22⋊C4—28no—
223C2×C22⋊C4—28no—
224C2×C22⋊C4—28no—
225C2×C22⋊C4—28no—
226C2×C22⋊C4—416no—
227C2×C22⋊C4—41no—
228C2×C22⋊C4—41no—
229C2×C22⋊C4—42no—
230C2×C22⋊C4—41no—
231C2×C22⋊C4—42no—
232C2×C22⋊C4—41no—
233C2×C22⋊C4—41no—
234C2×C22⋊C4—41no—
235C2×C22⋊C4—42no—
236C2×C22⋊C4—42no—
237C2×C22⋊C4—42no—
238C2×C22⋊C4—41no—
239C2×C22⋊C4—41no—
240C2×C22⋊C4—44no—
241C2×C22⋊C4—44no—
242C2×C22⋊C4—44no—
243C2×C22⋊C4—416no—
244C2×C22⋊C4—416no—
245C2×C22⋊C4—44no—
246C2×C22⋊C4—44no—
247C2×C22⋊C4—44no—
248C2×C22⋊C4—44no—
249C2×C22⋊C4—44no—
250C2×C22⋊C4—44no—
251C2×C22⋊C4C2×Q814yes—
252C2×C22⋊C4C2×Q814yes—
253C2×C22⋊C4C2×Q818yes—
254C2×C22⋊C4#4114yes—
255C2×C42—28no—
256C2×C42—44no—
257C2×C42—44no—
258#95—24no—
259#95—28no—
260#95—44no—
261#95—44no—
262#95—44no—
263#95—44no—
264#95—44no—
265#95—44no—
266#95—44no—
267#95—42no—
268#95—44no—
269#95—42no—
270#95—44no—
271#95—44no—
272#95—44no—
273#95—44no—
274#95—44no—
275#95—42no—
276#95—42no—
277#95—42no—
278#95—44no—
279#95—44no—
280#95—44no—
281#95—44no—
282#95—44no—
283C24—28no—
284C24—28no—
285C24—28no—
286C24—28no—
287C24—21no—
288C24—21no—
289C24—21no—
290C24—21no—
291C24—24no—
292C24—24no—
293C24—24no—
294C24—21no—
295C24—21no—
296C24—21no—
297C24—21no—
298C24—21no—
299C24—21no—
300C24—21no—
301C24—21no—
302C24—24no—
303C24—24no—
304C24—24no—
305C24—28no—
306C24—28no—
307C24—28no—
308C24—416no—
309C24—84no—
310C24—84no—
311C24—88no—
312C24—84no—
313C24—84no—
314C24—84no—
315C24—88no—
316C24—84no—
317C24—88no—
318C24C22×Q812yes—
319C24C22×Q812yes—
320C24C22×Q812yes—
321C24C2×#4112yes—
322C24C2×#4112yes—
323C24H(5,𝔽2)12yes—
324C24H(5,𝔽2)12yes—
325C24H(5,𝔽2)12yes—
326C242−1+412yes—
327C24C2511yescharacteristic
328C2×Q8—216no—
329C2×Q8—48no—
330C2×Q8—84no—
331C2×Q8—84no—
332C2×Q8—84no—
333C2×Q8—88no—
334C2×Q8—88no—
335C2×Q8—88no—
336C2×Q8—88no—
337C2×Q8—84no—
338C22×C4—216no—
339C22×C4—24no—
340C22×C4—24no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—24no—
344C22×C4—24no—
345C22×C4—24no—
346C22×C4—24no—
347C22×C4—24no—
348C22×C4—24no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—24no—
353C22×C4—24no—
354C22×C4—24no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—24no—
359C22×C4—24no—
360C22×C4—24no—
361C22×C4—24no—
362C22×C4—24no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—24no—
366C22×C4—24no—
367C22×C4—24no—
368C22×C4—24no—
369C22×C4—24no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—24no—
374C22×C4—24no—
375C22×C4—24no—
376C22×C4—24no—
377C22×C4—24no—
378C22×C4—24no—
379C22×C4—24no—
380C22×C4—22no—
381C22×C4—24no—
382C22×C4—24no—
383C22×C4—24no—
384C22×C4—24no—
385C22×C4—24no—
386C22×C4—24no—
387C22×C4—24no—
388C22×C4—24no—
389C22×C4—24no—
390C22×C4—24no—
391C22×C4—24no—
392C22×C4—416no—
393C22×C4—44no—
394C22×C4—48no—
395C22×C4—48no—
396C22×C4—48no—
397C22×C4—48no—
398C22×C4—416no—
399C22×C4—44no—
400C22×C4—48no—
401C22×C4—88no—
402C22×C4—84no—
403C22×C4—84no—
404C22×C4—84no—
405C22×C4—84no—
406C22×C4—88no—
407C22×C4—84no—
408C22×C4—84no—
409C22×C4C22×Q814yes—
410C22×C4C22×Q814yes—
411C22×C4C2×#4114yes—
412C22×C4C2×#4114yes—
413C22×C4H(5,𝔽2)18yes—
414C4⋊C4—48no—
415C4⋊C4—416no—
416C4⋊C4—48no—
417C4⋊C4—48no—
418C4⋊C4—416no—
419C4⋊C4—48no—
420C22⋊C4—44no—
421C22⋊C4—48no—
422C22⋊C4—44no—
423C22⋊C4—44no—
424C22⋊C4—48no—
425C22⋊C4—48no—
426C22⋊C4—48no—
427C22⋊C4—416no—
428C22⋊C4—416no—
429C22⋊C4—416no—
430C22⋊C4—44no—
431C22⋊C4—88no—
432C22⋊C4—88no—
433C22⋊C4—88no—
434C42—416no—
435C23—232no—
436C23—22no—
437C23—22no—
438C23—22no—
439C23—22no—
440C23—22no—
441C23—22no—
442C23—24no—
443C23—22no—
444C23—22no—
445C23—22no—
446C23—22no—
447C23—22no—
448C23—22no—
449C23—22no—
450C23—22no—
451C23—22no—
452C23—22no—
453C23—22no—
454C23—22no—
455C23—28no—
456C23—28no—
457C23—28no—
458C23—24no—
459C23—28no—
460C23—24no—
461C23—24no—
462C23—24no—
463C23—28no—
464C23—24no—
465C23—28no—
466C23—42no—
467C23—42no—
468C23—42no—
469C23—42no—
470C23—44no—
471C23—44no—
472C23—42no—
473C23—42no—
474C23—42no—
475C23—42no—
476C23—42no—
477C23—42no—
478C23—44no—
479C23—44no—
480C23—42no—
481C23—42no—
482C23—44no—
483C23—44no—
484C23—816no—
485C23—816no—
486C23—816no—
487C23—816no—
488C23—816no—
489C23—816no—
490C23—88no—
491C23—88no—
492C23#53314yes—
493C23#54314yes—
494C23#54414yes—
495C23C23×Q811yescharacteristic
496C23C22×#4111yescharacteristic
497C23C22×#4111yescharacteristic
498C23C2×H(5,𝔽2)11yescharacteristic
499C23C2×H(5,𝔽2)11yescharacteristic
500C23C2×H(5,𝔽2)11yescharacteristic
501C23C2×H(5,𝔽2)14yes—
502C23C2×2−1+411yescharacteristic
503C23#58514yes—
504Q8—816no—
505Q8—816no—
506C2×C4—24no—
507C2×C4—24no—
508C2×C4—24no—
509C2×C4—24no—
510C2×C4—28no—
511C2×C4—24no—
512C2×C4—24no—
513C2×C4—28no—
514C2×C4—28no—
515C2×C4—28no—
516C2×C4—28no—
517C2×C4—28no—
518C2×C4—432no—
519C2×C4—44no—
520C2×C4—44no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—44no—
525C2×C4—44no—
526C2×C4—44no—
527C2×C4—44no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—44no—
531C2×C4—44no—
532C2×C4—44no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—44no—
537C2×C4—44no—
538C2×C4—44no—
539C2×C4—44no—
540C2×C4—44no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—44no—
544C2×C4—44no—
545C2×C4—44no—
546C2×C4—44no—
547C2×C4—44no—
548C2×C4—44no—
549C2×C4—44no—
550C2×C4—42no—
551C2×C4—44no—
552C2×C4—44no—
553C2×C4—44no—
554C2×C4—44no—
555C2×C4—44no—
556C2×C4—44no—
557C2×C4—44no—
558C2×C4—42no—
559C2×C4—44no—
560C2×C4—44no—
561C2×C4—44no—
562C2×C4—44no—
563C2×C4—44no—
564C2×C4—44no—
565C2×C4—88no—
566C2×C4—816no—
567C2×C4—88no—
568C22—28no—
569C22—28no—
570C22—28no—
571C22—24no—
572C22—24no—
573C22—24no—
574C22—44no—
575C22—44no—
576C22—44no—
577C22—44no—
578C22—44no—
579C22—44no—
580C22—44no—
581C22—44no—
582C22—44no—
583C22—44no—
584C22—44no—
585C22—44no—
586C22—44no—
587C22—42no—
588C22—42no—
589C22—42no—
590C22—44no—
591C22—44no—
592C22—42no—
593C22—42no—
594C22—42no—
595C22—832no—
596C22C2×#53311yescharacteristic
597C22C2×#54311yescharacteristic
598C22C2×#54411yescharacteristic
599C22#349012yes—
600C22#349011yescharacteristic
601C22#349011yescharacteristic
602C22#349211yescharacteristic
603C22#349211yescharacteristic
604C22#350114yes—
605C22#351012yes—
606C22#351414yes—
607C22#353212yes—
608C22#353212yes—
609C22#353312yes—
610C22#354214yes—
611C22#354512yes—
612C22#356114yes—
613C4—44no—
614C4—44no—
615C4—44no—
616C4—44no—
617C4—48no—
618C4—84no—
619C4—82no—
620C4—84no—
621C4—84no—
622C4—82no—
623C2—44no—
624C2—44no—
625C2—44no—
626C2#3505211yescharacteristic
627C2#3570512yes—
628C2#3674712yes—
629C2#4061012yes—
630C2#4699611yescharacteristic
631C2#4703511yescharacteristic
632C2#4714112yes—
633C2#4788212yes—
634C2#4892212yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC26C23
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #16823C2semidirect
2. #16706C2semidirect
3. #16666C2semidirect
4. #16254C2semidirect
5. #16133C2semidirect
6. #15671C2semidirect
7. #14887C2semidirect
10. #14596C2semidirect
11. #14449C2semidirect
20. C2×#395C22semidirect
25. C2×#391C22semidirect
28. #1446C22semidirect
29. #1446C22semidirect
30. #1445C22semidirect
31. #1445C22semidirect
32. #1443C22semidirect
33. #1443C22semidirect
34. #1442C22semidirect
35. #1442C22semidirect
36. #1441C22semidirect
37. #1441C22semidirect
38. #1441C22semidirect
53. #1436C22semidirect
54. #1436C22semidirect
55. #1434C22semidirect
94. C22×C22⋊C4Q8semidirect

Maximal subgroups

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

subgroupindex
#168232
#167062
#166662
#162542
#161332
#156712
#148872
#147172
#145962
#144492
#138682
#137942