Finite Group Databasebeta

← #4457515

Subgroups — #4457515

Order 512.

4405 conjugacy classes of subgroups.

The subgroup lattice has 570 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#543C211yesmaximal, characteristic, complemented
2C2×#2412C212yesmaximal
3C2×#2622C211yesmaximal, characteristic
4#16364C214yesmaximal, complemented
5#16263C214yesmaximal, complemented
6#16262C214yesmaximal, complemented
7#16251C214yesmaximal, complemented
8#14759C214yesmaximal, complemented
9#14758C214yesmaximal, complemented
10#13869C212yesmaximal, complemented
11C2×#1440C211yesmaximal, characteristic, complemented
12C2×#543—28no—
13C2×#543—216no—
14C22×#123C2212yes—
15C22×C22⋊Q8C2214yes—
16C22×C22⋊Q8C2214yes—
17C22×C22⋊Q8C2214yes—
18C22×#120C2211yescharacteristic
19C23×C22⋊C4C2211yescharacteristic
20C23×C22⋊C4C2212yes—
21C23×C22⋊C4C2212yes—
22#2622—28no—
23#2412—216no—
24C2×#397C2214yes—
25C2×#397C2214yes—
26C2×#395C2214yescomplemented
27C2×#394C2214yes—
28C2×#394C2214yes—
29C2×#393C2214yescomplemented
30C2×#386C2212yes—
31C2×#386C2214yes—
32C2×#386C2212yes—
33C2×#385C2212yes—
34C2×#385C2212yes—
35C2×#381C2212yes—
36C2×#381C2214yes—
37C2×#380C2214yes—
38C2×#379C2211yescharacteristic
39C22×#95C2214yes—
40#1446C2214yescomplemented
41#1446C2214yescomplemented
42#1444C2214yescomplemented
43#1444C2214yescomplemented
44#1444C2214yescomplemented
45#1442C2214yescomplemented
46#1442C2214yescomplemented
47#1442C2214yescomplemented
48#1441C2214yescomplemented
49#1441C2214yescomplemented
50#1441C2214yescomplemented
51#1440C2214yes—
52#1440C2214yes—
53#1440C2214yes—
54#1440C2214yes—
55#1439C2214yescomplemented
56#1439C2214yescomplemented
57#1439C2214yescomplemented
58C22⋊#95C2214yes—
59C22⋊#95C2214yes—
60C22⋊#95C2214yes—
61C26C2311yescharacteristic
62C23×Q8C2314yes—
63C23×D4C2311yescharacteristic
64C23×D4C2312yes—
65C24×C4C2314yes—
66C24×C4C2312yes—
67C24×C4C2312yes—
68C24×C4C2312yes—
69#543—2128no—
70C2×#123—28no—
71C2×#123—48no—
72C2×C22⋊Q8—216no—
73C2×C22⋊Q8—216no—
74C2×C22⋊Q8—216no—
75C2×C22⋊Q8—216no—
76C2×C22⋊Q8—216no—
77C2×C22⋊Q8—216no—
78C2×C22⋊Q8—216no—
79C2×C22⋊Q8—216no—
80C2×C22⋊Q8—216no—
81C2×#120—24no—
82C2×#120—28no—
83C22×C4⋊C4C2314yes—
84C22×C4⋊C4C2314yes—
85C22×C4⋊C4C2314yes—
86C22×C4⋊C4C2314yes—
87C22×C4⋊C4C2314yes—
88C22×C4⋊C4C2314yes—
89C22×C22⋊C4—28no—
90C22×C22⋊C4—28no—
91C22×C22⋊C4—28no—
92C22×C22⋊C4—28no—
93C22×C22⋊C4—48no—
94C22×C22⋊C4D418yes—
95C22×C22⋊C4D4116yes—
96C22×C22⋊C4C2312yes—
97C22×C22⋊C4C2312yes—
98C22×C22⋊C4C2311yescharacteristic
99C22×C22⋊C4C2312yes—
100C22×C22⋊C4C2314yes—
101C22×C22⋊C4C2314yes—
102C22×C22⋊C4C2314yes—
103C22×C22⋊C4C2314yes—
104C22×C22⋊C4C2314yes—
105C22×C22⋊C4C2314yes—
106C22×C22⋊C4C2314yes—
107C22×C22⋊C4C2314yes—
108C22×C22⋊C4C2312yes—
109#397—216no—
110#397—216no—
111#395—48no—
112#394—216no—
113#394—216no—
114#393—48no—
115#386—28no—
116#386—28no—
117#386—216no—
118#385—28no—
119#385—28no—
120#381—28no—
121#381—48no—
122#380—216no—
123#379—24no—
124C2×#95—48no—
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×#95C2314yes—
142C2×#95C2314yes—
143C25—28no—
144C25—44no—
145C25—44no—
146C25C2×D418yes—
147C25C2×D414yes—
148C25C2411yescharacteristic
149C25C2412yes—
150C25C2411yescharacteristic
151C22×Q8—28no—
152C22×Q8—28no—
153C22×Q8—28no—
154C22×Q8—44no—
155C22×Q8—44no—
156C22×Q8—44no—
157C22×Q8—44no—
158C22×D4—24no—
159C22×D4—22no—
160C22×D4—24no—
161C22×D4—28no—
162C22×D4—24no—
163C22×D4—24no—
164C22×D4—44no—
165C22×D4—44no—
166C23×C4—24no—
167C23×C4—24no—
168C23×C4—24no—
169C23×C4—24no—
170C23×C4—24no—
171C23×C4—24no—
172C23×C4—28no—
173C23×C4—28no—
174C23×C4—28no—
175C23×C4—28no—
176C23×C4—44no—
177C23×C4—44no—
178C23×C4—44no—
179C23×C4—44no—
180C23×C4—44no—
181C23×C4—44no—
182C23×C4—44no—
183C23×C4—44no—
184C23×C4—44no—
185C23×C4C2×D418yes—
186C23×C4C2×D418yes—
187C23×C4C2×D418yes—
188C23×C4C2414yes—
189C23×C4C2414yes—
190C23×C4C2412yes—
191C23×C4C2411yescharacteristic
192C23×C4C2414yes—
193C23×C4C2414yes—
194C23×C4C2414yes—
195C23×C4C2414yes—
196#123—432no—
197C22⋊Q8—2128no—
198C22⋊Q8—2128no—
199C22⋊Q8—2128no—
200#120—416no—
201C2×C4⋊C4—28no—
202C2×C4⋊C4—28no—
203C2×C4⋊C4—28no—
204C2×C4⋊C4—28no—
205C2×C4⋊C4—28no—
206C2×C4⋊C4—28no—
207C2×C4⋊C4—28no—
208C2×C4⋊C4—28no—
209C2×C4⋊C4—28no—
210C2×C4⋊C4—28no—
211C2×C4⋊C4—28no—
212C2×C4⋊C4—28no—
213C2×C4⋊C4—28no—
214C2×C4⋊C4—44no—
215C2×C4⋊C4—44no—
216C2×C4⋊C4—44no—
217C2×C4⋊C4—44no—
218C2×C4⋊C4—44no—
219C2×C22⋊C4—232no—
220C2×C22⋊C4—264no—
221C2×C22⋊C4—28no—
222C2×C22⋊C4—28no—
223C2×C22⋊C4—28no—
224C2×C22⋊C4—28no—
225C2×C22⋊C4—28no—
226C2×C22⋊C4—28no—
227C2×C22⋊C4—24no—
228C2×C22⋊C4—28no—
229C2×C22⋊C4—24no—
230C2×C22⋊C4—28no—
231C2×C22⋊C4—24no—
232C2×C22⋊C4—24no—
233C2×C22⋊C4—22no—
234C2×C22⋊C4—24no—
235C2×C22⋊C4—24no—
236C2×C22⋊C4—24no—
237C2×C22⋊C4—28no—
238C2×C22⋊C4—28no—
239C2×C22⋊C4—28no—
240C2×C22⋊C4—28no—
241C2×C22⋊C4—28no—
242C2×C22⋊C4—28no—
243C2×C22⋊C4—28no—
244C2×C22⋊C4—432no—
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⋊C4—44no—
252C2×C22⋊C4—44no—
253C2×C22⋊C4—44no—
254C2×C22⋊C4C2×D4116yes—
255C2×C22⋊C4C2×D4116yes—
256C2×C22⋊C4C2×D4116yes—
257#95—28no—
258#95—28no—
259#95—28no—
260#95—28no—
261#95—28no—
262#95—28no—
263#95—28no—
264#95—28no—
265#95—28no—
266#95—28no—
267#95—44no—
268#95—44no—
269#95—44no—
270#95—44no—
271#95—44no—
272#95—44no—
273#95—44no—
274#95—44no—
275C24—24no—
276C24—24no—
277C24—24no—
278C24—24no—
279C24—24no—
280C24—24no—
281C24—24no—
282C24—24no—
283C24—216no—
284C24—232no—
285C24—22no—
286C24—21no—
287C24—24no—
288C24—24no—
289C24—24no—
290C24—48no—
291C24—48no—
292C24—416no—
293C24—416no—
294C24—88no—
295C24—88no—
296C24—88no—
297C24—88no—
298C24#120116yes—
299C24C22×D414yes—
300C24C22×D414yes—
301C24C22×D414yes—
302C24C22×D412yes—
303C24C22×D414yes—
304C242−1+418yes—
305C24C2511yescharacteristic
306C2×Q8—232no—
307C2×Q8—88no—
308C2×Q8—88no—
309C2×Q8—88no—
310C2×Q8—88no—
311C2×Q8—88no—
312C2×Q8—88no—
313C2×D4—48no—
314C2×D4—48no—
315C2×D4—48no—
316C2×D4—44no—
317C2×D4—48no—
318C2×D4—416no—
319C2×D4—88no—
320C2×D4—88no—
321C22×C4—24no—
322C22×C4—24no—
323C22×C4—24no—
324C22×C4—24no—
325C22×C4—24no—
326C22×C4—24no—
327C22×C4—24no—
328C22×C4—24no—
329C22×C4—24no—
330C22×C4—24no—
331C22×C4—24no—
332C22×C4—24no—
333C22×C4—24no—
334C22×C4—24no—
335C22×C4—24no—
336C22×C4—24no—
337C22×C4—24no—
338C22×C4—24no—
339C22×C4—24no—
340C22×C4—24no—
341C22×C4—24no—
342C22×C4—24no—
343C22×C4—24no—
344C22×C4—24no—
345C22×C4—216no—
346C22×C4—216no—
347C22×C4—216no—
348C22×C4—22no—
349C22×C4—21no—
350C22×C4—24no—
351C22×C4—24no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—24no—
356C22×C4—24no—
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—24no—
367C22×C4—24no—
368C22×C4—24no—
369C22×C4—24no—
370C22×C4—48no—
371C22×C4—48no—
372C22×C4—416no—
373C22×C4—416no—
374C22×C4—416no—
375C22×C4—416no—
376C22×C4—416no—
377C22×C4—416no—
378C22×C4—84no—
379C22×C4—84no—
380C22×C4—84no—
381C22×C4—84no—
382C22×C4—88no—
383C22×C4—88no—
384C22×C4—88no—
385C22×C4—88no—
386C22×C4—88no—
387C22×C4—88no—
388C22×C4—88no—
389C22×C4C22×D418yes—
390C22×C4C22×D418yes—
391C22×C4C22×D418yes—
392C22×C4H(5,𝔽2)18yes—
393C4⋊C4—232no—
394C4⋊C4—232no—
395C4⋊C4—232no—
396C4⋊C4—416no—
397C4⋊C4—88no—
398C4⋊C4—88no—
399C22⋊C4—216no—
400C22⋊C4—232no—
401C22⋊C4—232no—
402C22⋊C4—232no—
403C22⋊C4—416no—
404C22⋊C4—416no—
405C22⋊C4—48no—
406C22⋊C4—48no—
407C22⋊C4—44no—
408C22⋊C4—48no—
409C22⋊C4—88no—
410C22⋊C4—88no—
411C22⋊C4—88no—
412C23—24no—
413C23—24no—
414C23—28no—
415C23—28no—
416C23—28no—
417C23—28no—
418C23—28no—
419C23—28no—
420C23—28no—
421C23—28no—
422C23—24no—
423C23—24no—
424C23—24no—
425C23—24no—
426C23—24no—
427C23—24no—
428C23—28no—
429C23—28no—
430C23—28no—
431C23—28no—
432C23—232no—
433C23—432no—
434C23—44no—
435C23—44no—
436C23—44no—
437C23—44no—
438C23—44no—
439C23—44no—
440C23—44no—
441C23—44no—
442C23—41no—
443C23—44no—
444C23—42no—
445C23—42no—
446C23—42no—
447C23—44no—
448C23—44no—
449C23—44no—
450C23—44no—
451C23—44no—
452C23—44no—
453C23—832no—
454C23—832no—
455C23—832no—
456C23C2×#12018yes—
457C23C2×#12014yes—
458C23#53618yes—
459C23#536116yes—
460C23C23×D412yes—
461C23C23×D411yescharacteristic
462C23C2×H(5,𝔽2)14yes—
463C23C2×2−1+414yes—
464C23#58514yes—
465Q8—832no—
466D4—816no—
467D4—88no—
468C2×C4—28no—
469C2×C4—28no—
470C2×C4—28no—
471C2×C4—28no—
472C2×C4—28no—
473C2×C4—28no—
474C2×C4—28no—
475C2×C4—28no—
476C2×C4—28no—
477C2×C4—28no—
478C2×C4—28no—
479C2×C4—28no—
480C2×C4—44no—
481C2×C4—44no—
482C2×C4—44no—
483C2×C4—44no—
484C2×C4—44no—
485C2×C4—44no—
486C2×C4—44no—
487C2×C4—44no—
488C2×C4—44no—
489C2×C4—44no—
490C2×C4—44no—
491C2×C4—44no—
492C2×C4—44no—
493C2×C4—44no—
494C2×C4—44no—
495C2×C4—44no—
496C2×C4—42no—
497C2×C4—41no—
498C2×C4—44no—
499C2×C4—44no—
500C2×C4—44no—
501C2×C4—42no—
502C2×C4—44no—
503C2×C4—42no—
504C2×C4—42no—
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—44no—
514C2×C4—44no—
515C2×C4—44no—
516C2×C4—44no—
517C2×C4—44no—
518C2×C4—816no—
519C2×C4—816no—
520C2×C4—816no—
521C2×C4—832no—
522C22—28no—
523C22—216no—
524C22—216no—
525C22—216no—
526C22—28no—
527C22—28no—
528C22—28no—
529C22—48no—
530C22—48no—
531C22—48no—
532C22—48no—
533C22—48no—
534C22—48no—
535C22—832no—
536C22—82no—
537C22—81no—
538C22—84no—
539C22—84no—
540C22—84no—
541C22#261418yes—
542C22C22×#12011yescharacteristic
543C22C2×#53614yes—
544C22C2×#53612yes—
545C22#348412yes—
546C22#348512yes—
547C22#349614yes—
548C22#349614yes—
549C22#353514yes—
550C22#353614yes—
551C22#356114yes—
552C22#356414yes—
553C4—48no—
554C4—48no—
555C4—48no—
556C4—48no—
557C4—82no—
558C4—81no—
559C4—84no—
560C4—84no—
561C2—28no—
562C2—48no—
563C2—161no—
564C2C2×#261411yescharacteristic
565C2#3534712yes—
566C2#3674014yes—
567C2#4048714yes—
568C2#4051114yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC22×#120C22
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C22×#543C2semidirect
4. #16364C2semidirect
5. #16263C2semidirect
6. #16262C2semidirect
7. #16251C2semidirect
8. #14759C2semidirect
9. #14758C2semidirect
10. #13869C2semidirect
11. C2×#1440C2semidirect
26. C2×#395C22semidirect
29. C2×#393C22semidirect
40. #1446C22semidirect
41. #1446C22semidirect
42. #1444C22semidirect
43. #1444C22semidirect
44. #1444C22semidirect
45. #1442C22semidirect
46. #1442C22semidirect
47. #1442C22semidirect
48. #1441C22semidirect
49. #1441C22semidirect
50. #1441C22semidirect
55. #1439C22semidirect
56. #1439C22semidirect
57. #1439C22semidirect

Maximal subgroups

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

subgroupindex
C22×#5432
C2×#24122
C2×#26222
#163642
#162632
#162622
#162512
#147592
#147582
#138692
C2×#14402