Finite Group Databasebeta

← #4320017

Subgroups — #4320017

Order 512.

2634 conjugacy classes of subgroups.

The subgroup lattice has 630 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#16824C211yesmaximal, characteristic, complemented
2#16707C212yesmaximal, complemented
3#16299C214yesmaximal, complemented
4#16254C211yesmaximal, characteristic, complemented
5#15998C214yesmaximal, complemented
6#15475C214yesmaximal, complemented
7#14772C214yesmaximal, complemented
8#14734C212yesmaximal
9#14717C212yesmaximal
10#14592C212yesmaximal, complemented
11#14448C212yesmaximal, complemented
12#13903C212yesmaximal
13#13794C211yesmaximal, characteristic
14C22×C22⋊Q8C2212yes—
15C23×C22⋊C4C2211yescharacteristic
16C23×C22⋊C4C2211yescharacteristic
17C23×C22⋊C4C2211yescharacteristic
18C2×#399C2212yes—
19C2×#399C2212yes—
20C2×#395C2214yescomplemented
21C2×#393C2212yes—
22C2×#393C2212yes—
23C2×#387C2214yescomplemented
24C2×#385C2214yes—
25C2×#381C2214yes—
26C2×#380C2212yes—
27C2×#380C2212yes—
28C2×#379C2214yes—
29C2×C4×C22⋊C4C2212yes—
30#1446C2214yescomplemented
31#1445C2214yescomplemented
32#1445C2214yescomplemented
33#1443C2214yescomplemented
34#1442C2214yescomplemented
35#1442C2214yescomplemented
36#1442C2214yescomplemented
37#1441C2214yescomplemented
38#1441C2214yescomplemented
39#1441C2214yescomplemented
40#1441C2214yescomplemented
41#1439C2212yes—
42#1439C2212yes—
43#1439C2212yes—
44#1439C2214yes—
45#1439C2212yes—
46#1439C2212yes—
47#1439C2214yes—
48#1439C2212yes—
49#1439C2214yes—
50#1439C2212yes—
51#1439C2212yes—
52#1439C2214yes—
53#1437C2214yes—
54#1437C2214yes—
55#1436C2214yescomplemented
56#1436C2214yescomplemented
57C4⋊#95C2214yescomplemented
58C22⋊#95C2211yescharacteristic
59C22⋊#95C2214yes—
60C22⋊#95C2211yescharacteristic
61C22⋊#95C2211yescharacteristic
62C22⋊#95C2212yes—
63C22⋊#95C2214yes—
64C22⋊#95C2212yes—
65C22⋊#95C2211yescharacteristic
66C22⋊#95C2214yes—
67C26C2311yescharacteristic
68C23×Q8C2312yes—
69C24×C4C2312yes—
70C24×C4C2312yes—
71C24×C4C2311yescharacteristic
72C24×C4C2311yescharacteristic
73C2×C22⋊Q8—28no—
74C2×C22⋊Q8—48no—
75C22×C4⋊C4C2312yes—
76C22×C4⋊C4C2314yes—
77C22×C4⋊C4C2312yes—
78C22×C4⋊C4C2314yes—
79C22×C4⋊C4C2314yes—
80C22×C4⋊C4C2312yes—
81C22×C4⋊C4C2312yes—
82C22×C4⋊C4C2312yes—
83C22×C22⋊C4—24no—
84C22×C22⋊C4—24no—
85C22×C22⋊C4—42no—
86C22×C22⋊C4—42no—
87C22×C22⋊C4—42no—
88C22×C22⋊C4—42no—
89C22×C22⋊C4—42no—
90C22×C22⋊C4—42no—
91C22×C22⋊C4Q818yescomplemented
92C22×C22⋊C4C2312yes—
93C22×C22⋊C4C2312yes—
94C22×C22⋊C4C2312yes—
95C22×C22⋊C4C2314yes—
96C22×C22⋊C4C2314yes—
97C22×C22⋊C4C2314yes—
98C22×C22⋊C4C2314yes—
99C22×C22⋊C4C2314yes—
100C22×C22⋊C4C2314yes—
101C22×C22⋊C4C2311yescharacteristic
102C22×C22⋊C4C2311yescharacteristic
103C22×C22⋊C4C2312yes—
104C22×C22⋊C4C2311yescharacteristic
105C22×C22⋊C4C2311yescharacteristic
106C22×C42C2314yes—
107#399—28no—
108#399—44no—
109#395—48no—
110#393—44no—
111#393—44no—
112#387—48no—
113#385—48no—
114#381—48no—
115#380—44no—
116#380—44no—
117#379—48no—
118C4×C22⋊C4—44no—
119C2×#95C2314yes—
120C2×#95C2314yes—
121C2×#95C2312yes—
122C2×#95C2314yes—
123C2×#95C2312yes—
124C2×#95C2314yes—
125C2×#95C2312yes—
126C2×#95C2314yes—
127C2×#95C2314yes—
128C2×#95C2312yes—
129C2×#95C2314yes—
130C2×#95C2314yes—
131C2×#95C2314yes—
132C2×#95C2314yes—
133C2×#95C2314yes—
134C2×#95C2314yes—
135C2×#95C2314yes—
136C2×#95C2314yes—
137C2×#95C2314yes—
138C2×#95C2314yes—
139C2×#95C2314yes—
140C2×#95C2312yes—
141C2×#95C2312yes—
142C2×#95C2314yes—
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—44no—
157C22×Q8—42no—
158C22×Q8—42no—
159C22×Q8—44no—
160C23×C4—24no—
161C23×C4—24no—
162C23×C4—24no—
163C23×C4—22no—
164C23×C4—24no—
165C23×C4—42no—
166C23×C4—41no—
167C23×C4—41no—
168C23×C4—41no—
169C23×C4—41no—
170C23×C4—44no—
171C23×C4—44no—
172C23×C4—44no—
173C23×C4—44no—
174C23×C4—44no—
175C23×C4—44no—
176C23×C4C2×Q818yes—
177C23×C4#4114yes—
178C23×C4C2414yes—
179C23×C4C2414yes—
180C23×C4C2414yes—
181C23×C4C2412yes—
182C23×C4C2412yes—
183C23×C4C2412yes—
184C23×C4C2412yes—
185C23×C4C2412yes—
186C23×C4C2412yes—
187C23×C4C2414yes—
188C22⋊Q8—432no—
189C2×C4⋊C4—24no—
190C2×C4⋊C4—28no—
191C2×C4⋊C4—28no—
192C2×C4⋊C4—28no—
193C2×C4⋊C4—28no—
194C2×C4⋊C4—24no—
195C2×C4⋊C4—28no—
196C2×C4⋊C4—44no—
197C2×C4⋊C4—44no—
198C2×C4⋊C4—44no—
199C2×C4⋊C4—44no—
200C2×C4⋊C4—42no—
201C2×C4⋊C4—42no—
202C2×C4⋊C4—42no—
203C2×C4⋊C4—42no—
204C2×C4⋊C4—44no—
205C2×C4⋊C4—44no—
206C2×C4⋊C4—44no—
207C2×C4⋊C4—44no—
208C2×C4⋊C4—42no—
209C2×C22⋊C4—22no—
210C2×C22⋊C4—28no—
211C2×C22⋊C4—24no—
212C2×C22⋊C4—24no—
213C2×C22⋊C4—24no—
214C2×C22⋊C4—28no—
215C2×C22⋊C4—28no—
216C2×C22⋊C4—24no—
217C2×C22⋊C4—24no—
218C2×C22⋊C4—24no—
219C2×C22⋊C4—28no—
220C2×C22⋊C4—28no—
221C2×C22⋊C4—28no—
222C2×C22⋊C4—28no—
223C2×C22⋊C4—28no—
224C2×C22⋊C4—28no—
225C2×C22⋊C4—416no—
226C2×C22⋊C4—41no—
227C2×C22⋊C4—41no—
228C2×C22⋊C4—42no—
229C2×C22⋊C4—42no—
230C2×C22⋊C4—41no—
231C2×C22⋊C4—41no—
232C2×C22⋊C4—41no—
233C2×C22⋊C4—41no—
234C2×C22⋊C4—42no—
235C2×C22⋊C4—42no—
236C2×C22⋊C4—42no—
237C2×C22⋊C4—41no—
238C2×C22⋊C4—41no—
239C2×C22⋊C4—44no—
240C2×C22⋊C4—44no—
241C2×C22⋊C4—44no—
242C2×C22⋊C4—416no—
243C2×C22⋊C4—416no—
244C2×C22⋊C4—44no—
245C2×C22⋊C4—44no—
246C2×C22⋊C4—44no—
247C2×C22⋊C4—44no—
248C2×C22⋊C4—44no—
249C2×C22⋊C4—44no—
250C2×C22⋊C4C2×Q814yes—
251C2×C22⋊C4C2×Q814yes—
252C2×C22⋊C4C2×Q818yes—
253C2×C22⋊C4#4114yes—
254C2×C42—44no—
255C2×C42—44no—
256C2×C42—44no—
257#95—24no—
258#95—28no—
259#95—28no—
260#95—28no—
261#95—44no—
262#95—44no—
263#95—44no—
264#95—44no—
265#95—42no—
266#95—42no—
267#95—44no—
268#95—44no—
269#95—44no—
270#95—44no—
271#95—44no—
272#95—44no—
273#95—44no—
274#95—44no—
275#95—42no—
276#95—42no—
277#95—44no—
278#95—44no—
279#95—44no—
280#95—42no—
281C24—28no—
282C24—28no—
283C24—28no—
284C24—28no—
285C24—21no—
286C24—21no—
287C24—21no—
288C24—21no—
289C24—24no—
290C24—24no—
291C24—24no—
292C24—21no—
293C24—21no—
294C24—21no—
295C24—21no—
296C24—21no—
297C24—21no—
298C24—21no—
299C24—21no—
300C24—24no—
301C24—24no—
302C24—24no—
303C24—28no—
304C24—28no—
305C24—28no—
306C24—416no—
307C24—84no—
308C24—84no—
309C24—88no—
310C24—84no—
311C24—88no—
312C24—88no—
313C24—84no—
314C24—84no—
315C24—84no—
316C24C22×Q812yes—
317C24C22×Q812yes—
318C24C22×Q812yes—
319C24C2×#4112yes—
320C24C2×#4112yes—
321C24H(5,𝔽2)12yes—
322C24H(5,𝔽2)12yes—
323C24H(5,𝔽2)12yes—
324C242−1+412yes—
325C24C2511yescharacteristic
326C2×Q8—48no—
327C2×Q8—84no—
328C2×Q8—84no—
329C2×Q8—88no—
330C2×Q8—88no—
331C22×C4—24no—
332C22×C4—24no—
333C22×C4—24no—
334C22×C4—24no—
335C22×C4—24no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—24no—
339C22×C4—24no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—24no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—24no—
348C22×C4—24no—
349C22×C4—24no—
350C22×C4—24no—
351C22×C4—24no—
352C22×C4—24no—
353C22×C4—216no—
354C22×C4—24no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—24no—
358C22×C4—24no—
359C22×C4—24no—
360C22×C4—24no—
361C22×C4—24no—
362C22×C4—24no—
363C22×C4—24no—
364C22×C4—24no—
365C22×C4—24no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—24no—
369C22×C4—24no—
370C22×C4—22no—
371C22×C4—24no—
372C22×C4—24no—
373C22×C4—24no—
374C22×C4—24no—
375C22×C4—22no—
376C22×C4—24no—
377C22×C4—24no—
378C22×C4—24no—
379C22×C4—24no—
380C22×C4—24no—
381C22×C4—24no—
382C22×C4—24no—
383C22×C4—24no—
384C22×C4—24no—
385C22×C4—44no—
386C22×C4—48no—
387C22×C4—416no—
388C22×C4—48no—
389C22×C4—44no—
390C22×C4—48no—
391C22×C4—88no—
392C22×C4—88no—
393C22×C4—84no—
394C22×C4—84no—
395C22×C4—84no—
396C22×C4—84no—
397C22×C4—88no—
398C22×C4—84no—
399C22×C4—84no—
400C22×C4—84no—
401C22×C4—84no—
402C22×C4C22×Q814yes—
403C22×C4C22×Q814yes—
404C22×C4C2×#4114yes—
405C22×C4C2×#4114yes—
406C22×C4H(5,𝔽2)18yes—
407C4⋊C4—216no—
408C4⋊C4—416no—
409C4⋊C4—84no—
410C4⋊C4—88no—
411C4⋊C4—88no—
412C4⋊C4—84no—
413C4⋊C4—84no—
414C4⋊C4—84no—
415C22⋊C4—44no—
416C22⋊C4—48no—
417C22⋊C4—44no—
418C22⋊C4—44no—
419C22⋊C4—416no—
420C22⋊C4—416no—
421C22⋊C4—48no—
422C22⋊C4—48no—
423C22⋊C4—48no—
424C22⋊C4—416no—
425C22⋊C4—44no—
426C22⋊C4—88no—
427C22⋊C4—88no—
428C22⋊C4—88no—
429C42—88no—
430C23—232no—
431C23—22no—
432C23—22no—
433C23—22no—
434C23—22no—
435C23—22no—
436C23—22no—
437C23—24no—
438C23—22no—
439C23—22no—
440C23—22no—
441C23—22no—
442C23—22no—
443C23—22no—
444C23—22no—
445C23—22no—
446C23—22no—
447C23—22no—
448C23—22no—
449C23—22no—
450C23—28no—
451C23—28no—
452C23—28no—
453C23—24no—
454C23—28no—
455C23—24no—
456C23—28no—
457C23—28no—
458C23—24no—
459C23—24no—
460C23—24no—
461C23—42no—
462C23—42no—
463C23—42no—
464C23—42no—
465C23—44no—
466C23—44no—
467C23—42no—
468C23—42no—
469C23—44no—
470C23—44no—
471C23—44no—
472C23—44no—
473C23—42no—
474C23—42no—
475C23—42no—
476C23—42no—
477C23—42no—
478C23—42no—
479C23—816no—
480C23—816no—
481C23—816no—
482C23—816no—
483C23—816no—
484C23—88no—
485C23—816no—
486C23—88no—
487C23#53314yes—
488C23#54314yes—
489C23#54414yes—
490C23C23×Q811yescharacteristic
491C23C22×#4111yescharacteristic
492C23C22×#4111yescharacteristic
493C23C2×H(5,𝔽2)11yescharacteristic
494C23C2×H(5,𝔽2)11yescharacteristic
495C23C2×H(5,𝔽2)11yescharacteristic
496C23C2×H(5,𝔽2)14yes—
497C23C2×2−1+411yescharacteristic
498C23#58514yes—
499Q8—816no—
500C2×C4—24no—
501C2×C4—24no—
502C2×C4—24no—
503C2×C4—24no—
504C2×C4—28no—
505C2×C4—24no—
506C2×C4—24no—
507C2×C4—28no—
508C2×C4—28no—
509C2×C4—28no—
510C2×C4—28no—
511C2×C4—28no—
512C2×C4—44no—
513C2×C4—44no—
514C2×C4—42no—
515C2×C4—44no—
516C2×C4—44no—
517C2×C4—44no—
518C2×C4—44no—
519C2×C4—44no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—44no—
524C2×C4—44no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—44no—
530C2×C4—44no—
531C2×C4—44no—
532C2×C4—44no—
533C2×C4—44no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—44no—
537C2×C4—44no—
538C2×C4—44no—
539C2×C4—44no—
540C2×C4—44no—
541C2×C4—44no—
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—44no—
551C2×C4—44no—
552C2×C4—44no—
553C2×C4—44no—
554C2×C4—44no—
555C2×C4—42no—
556C2×C4—44no—
557C2×C4—44no—
558C2×C4—816no—
559C2×C4—88no—
560C2×C4—816no—
561C2×C4—88no—
562C22—28no—
563C22—28no—
564C22—28no—
565C22—24no—
566C22—24no—
567C22—24no—
568C22—44no—
569C22—44no—
570C22—44no—
571C22—44no—
572C22—44no—
573C22—44no—
574C22—44no—
575C22—44no—
576C22—42no—
577C22—42no—
578C22—42no—
579C22—44no—
580C22—44no—
581C22—44no—
582C22—44no—
583C22—44no—
584C22—44no—
585C22—44no—
586C22—42no—
587C22—42no—
588C22—42no—
589C22—832no—
590C22C2×#53311yescharacteristic
591C22C2×#54311yescharacteristic
592C22C2×#54411yescharacteristic
593C22#349012yes—
594C22#349011yescharacteristic
595C22#349011yescharacteristic
596C22#349211yescharacteristic
597C22#349211yescharacteristic
598C22#350114yes—
599C22#351012yes—
600C22#351414yes—
601C22#353312yes—
602C22#353414yes—
603C22#353514yes—
604C22#354312yes—
605C22#354312yes—
606C22#354512yes—
607C4—48no—
608C4—44no—
609C4—44no—
610C4—44no—
611C4—44no—
612C4—84no—
613C4—84no—
614C4—84no—
615C4—82no—
616C4—82no—
617C2—44no—
618C2—44no—
619C2—44no—
620C2#3505211yescharacteristic
621C2#3570512yes—
622C2#3713412yes—
623C2#4061312yes—
624C2#4638711yescharacteristic
625C2#4685812yes—
626C2#4703511yescharacteristic
627C2#4788312yes—
628C2#4897112yes—
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. #16824C2semidirect
2. #16707C2semidirect
3. #16299C2semidirect
4. #16254C2semidirect
5. #15998C2semidirect
6. #15475C2semidirect
7. #14772C2semidirect
10. #14592C2semidirect
11. #14448C2semidirect
20. C2×#395C22semidirect
23. C2×#387C22semidirect
30. #1446C22semidirect
31. #1445C22semidirect
32. #1445C22semidirect
33. #1443C22semidirect
34. #1442C22semidirect
35. #1442C22semidirect
36. #1442C22semidirect
37. #1441C22semidirect
38. #1441C22semidirect
39. #1441C22semidirect
40. #1441C22semidirect
55. #1436C22semidirect
56. #1436C22semidirect
57. C4⋊#95C22semidirect
91. C22×C22⋊C4Q8semidirect

Maximal subgroups

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

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