Finite Group Databasebeta

← #4106330

Subgroups — #4106330

Order 512.

2522 conjugacy classes of subgroups.

The subgroup lattice has 575 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#17306C211yesmaximal, characteristic, complemented
2#17107C212yesmaximal, complemented
3#15834C214yesmaximal, complemented
4#15763C211yesmaximal, characteristic
5#15761C212yesmaximal, complemented
6#15506C212yesmaximal, complemented
7#15498C214yesmaximal, complemented
8#15107C212yesmaximal, complemented
9#14806C214yesmaximal, complemented
10#14473C212yesmaximal, complemented
11#14456C214yesmaximal, complemented
12#14176C212yesmaximal, complemented
13#14161C211yesmaximal, characteristic, complemented
14C22×C4⋊Q8C2211yescharacteristic
15C22×#124C2212yes—
16C22×C4×D4C2212yes—
17C2×#400C2212yes—
18C2×#394C2212yes—
19C2×#394C2212yes—
20C2×#394C2212yes—
21C2×#394C2214yes—
22C2×#394C2212yes—
23C2×#393C2214yes—
24C2×#392C2214yes—
25C2×#390C2212yes—
26C2×#388C2214yes—
27C2×#386C2214yes—
28C2×#386C2214yes—
29C2×#385C2214yes—
30C2×C4×C4⋊C4C2211yescharacteristic
31C2×C4×C4⋊C4C2211yescharacteristic
32#1446C2211yescharacteristic
33#1446C2211yescharacteristic
34#1446C2211yescharacteristic
35#1446C2211yescharacteristic
36#1446C2212yes—
37#1445C2212yes—
38#1445C2214yes—
39#1444C2212yes—
40#1443C2214yes—
41#1441C2212yes—
42#1441C2214yes—
43#1441C2214yes—
44#1440C2214yes—
45#1440C2214yes—
46#1440C2214yes—
47#1439C2214yes—
48#1439C2214yes—
49#1439C2214yes—
50#1439C2214yes—
51#1439C2214yes—
52#1438C2212yes—
53#1438C2212yes—
54#1437C2214yes—
55#1437C2214yes—
56#1437C2214yes—
57#1436C2212yes—
58#1436C2212yes—
59C4⋊#95C2214yes—
60C4⋊#95C2214yes—
61#1434C2212yes—
62#1434C2212yes—
63#1434C2214yes—
64C22⋊#95C2214yes—
65C22⋊#95C2214yes—
66C22⋊#95C2214yes—
67C23×Q8C2311yescharacteristic
68C23×Q8C2311yescharacteristic
69C23×D4C2312yes—
70C23×D4C2312yes—
71C23×D4C2312yes—
72C24×C4C2314yes—
73C2×C4⋊Q8—24no—
74C2×C4⋊Q8—42no—
75C2×C4⋊Q8—42no—
76C2×#124—28no—
77C2×#124—48no—
78C2×C4×D4—44no—
79C2×C4×D4—48no—
80C22×C4⋊C4C2312yes—
81C22×C4⋊C4C2312yes—
82C22×C4⋊C4C2312yes—
83C22×C4⋊C4C2314yes—
84C22×C4⋊C4C2312yes—
85C22×C4⋊C4C2312yes—
86C22×C4⋊C4C2312yes—
87C22×C22⋊C4C2314yes—
88C22×C22⋊C4C2314yes—
89C22×C22⋊C4C2314yes—
90C22×C22⋊C4C2314yes—
91C22×C22⋊C4C2314yes—
92C22×C22⋊C4C2314yes—
93C22×C22⋊C4C2314yes—
94C22×C22⋊C4C2314yes—
95C22×C22⋊C4C2314yes—
96C22×C22⋊C4C2314yes—
97C22×C22⋊C4C2314yes—
98C22×C42C2312yes—
99C22×C42C2311yescharacteristic
100C22×C42C2312yes—
101C22×C42C2312yes—
102#400—44no—
103#394—28no—
104#394—44no—
105#394—44no—
106#394—44no—
107#394—48no—
108#393—48no—
109#392—48no—
110#390—44no—
111#388—48no—
112#386—48no—
113#386—48no—
114#385—216no—
115C4×C4⋊C4—42no—
116C4×C4⋊C4—42no—
117C2×#95C2312yes—
118C2×#95C2314yes—
119C2×#95C2314yes—
120C2×#95C2314yes—
121C2×#95C2314yes—
122C2×#95C2314yes—
123C2×#95C2314yes—
124C2×#95C2314yes—
125C2×#95C2312yes—
126C2×#95C2312yes—
127C2×#95C2314yes—
128C2×#95C2312yes—
129C2×#95C2312yes—
130C2×#95C2314yes—
131C2×#95C2311yescharacteristic
132C2×#95C2314yes—
133C2×#95C2314yes—
134C2×#95C2314yes—
135C2×#95C2314yes—
136C2×#95C2312yes—
137C2×#95C2314yes—
138C2×#95C2314yes—
139C2×#95C2311yescharacteristic
140C2×#95C2311yescharacteristic
141C2×#95C2311yescharacteristic
142C25C2414yes—
143C22×Q8—22no—
144C22×Q8—44no—
145C22×Q8—41no—
146C22×Q8—41no—
147C22×Q8—41no—
148C22×Q8—41no—
149C22×Q8—44no—
150C22×Q8#4114yes—
151C22×D4—28no—
152C22×D4—28no—
153C22×D4—28no—
154C22×D4—24no—
155C22×D4—24no—
156C22×D4—24no—
157C22×D4—42no—
158C22×D4—44no—
159C22×D4—42no—
160C22×D4—44no—
161C22×D4—42no—
162C22×D4—44no—
163C22×D4—42no—
164C22×D4—42no—
165C22×D4—42no—
166C23×C4—28no—
167C23×C4—44no—
168C23×C4—44no—
169C23×C4—44no—
170C23×C4—44no—
171C23×C4—44no—
172C23×C4—44no—
173C23×C4C2412yes—
174C23×C4C2414yes—
175C23×C4C2412yes—
176C23×C4C2414yes—
177C23×C4C2414yes—
178C23×C4C2411yescharacteristic
179C23×C4C2412yes—
180C23×C4C2412yes—
181C23×C4C2412yes—
182C23×C4C2412yes—
183C23×C4C2411yescharacteristic
184C23×C4C2411yescharacteristic
185C4⋊Q8—416no—
186#124—432no—
187C4×D4—432no—
188C2×C4⋊C4—24no—
189C2×C4⋊C4—24no—
190C2×C4⋊C4—24no—
191C2×C4⋊C4—24no—
192C2×C4⋊C4—24no—
193C2×C4⋊C4—24no—
194C2×C4⋊C4—28no—
195C2×C4⋊C4—24no—
196C2×C4⋊C4—42no—
197C2×C4⋊C4—42no—
198C2×C4⋊C4—44no—
199C2×C4⋊C4—42no—
200C2×C4⋊C4—42no—
201C2×C4⋊C4—42no—
202C2×C4⋊C4—42no—
203C2×C4⋊C4—44no—
204C2×C4⋊C4—42no—
205C2×C4⋊C4—44no—
206C2×C4⋊C4—42no—
207C2×C4⋊C4#4118yes—
208C2×C22⋊C4—28no—
209C2×C22⋊C4—28no—
210C2×C22⋊C4—28no—
211C2×C22⋊C4—28no—
212C2×C22⋊C4—28no—
213C2×C22⋊C4—28no—
214C2×C22⋊C4—28no—
215C2×C22⋊C4—28no—
216C2×C22⋊C4—28no—
217C2×C22⋊C4—28no—
218C2×C22⋊C4—28no—
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—44no—
226C2×C22⋊C4—44no—
227C2×C22⋊C4—44no—
228C2×C22⋊C4—44no—
229C2×C22⋊C4—44no—
230C2×C22⋊C4—44no—
231C2×C22⋊C4—44no—
232C2×C22⋊C4—44no—
233C2×C22⋊C4—44no—
234C2×C22⋊C4—44no—
235C2×C22⋊C4—44no—
236C2×C22⋊C4—44no—
237C2×C22⋊C4—44no—
238C2×C22⋊C4—44no—
239C2×C22⋊C4—44no—
240C2×C22⋊C4—44no—
241C2×C42—22no—
242C2×C42—22no—
243C2×C42—22no—
244C2×C42—24no—
245C2×C42—24no—
246C2×C42—28no—
247C2×C42—24no—
248C2×C42—44no—
249C2×C42—44no—
250#95—28no—
251#95—28no—
252#95—24no—
253#95—28no—
254#95—42no—
255#95—44no—
256#95—41no—
257#95—41no—
258#95—41no—
259#95—41no—
260#95—44no—
261#95—44no—
262#95—44no—
263#95—44no—
264#95—44no—
265#95—44no—
266#95—44no—
267#95—44no—
268#95—44no—
269#95—44no—
270#95—42no—
271#95—42no—
272#95—42no—
273#95—42no—
274#95—44no—
275C24—24no—
276C24—24no—
277C24—24no—
278C24—24no—
279C24—24no—
280C24—24no—
281C24—24no—
282C24—24no—
283C24—24no—
284C24—24no—
285C24—24no—
286C24—24no—
287C24—24no—
288C24—24no—
289C24H(5,𝔽2)18yes—
290C24C2511yescharacteristic
291C2×Q8—48no—
292C2×Q8—44no—
293C2×Q8—44no—
294C2×Q8—84no—
295C2×Q8—84no—
296C2×Q8—84no—
297C2×Q8—84no—
298C2×Q8—84no—
299C2×D4—216no—
300C2×D4—416no—
301C2×D4—48no—
302C2×D4—416no—
303C2×D4—48no—
304C2×D4—416no—
305C2×D4—48no—
306C2×D4—48no—
307C2×D4—84no—
308C2×D4—84no—
309C2×D4—84no—
310C2×D4—88no—
311C2×D4—88no—
312C2×D4—88no—
313C2×D4—84no—
314C22×C4—21no—
315C22×C4—21no—
316C22×C4—21no—
317C22×C4—21no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—24no—
321C22×C4—21no—
322C22×C4—21no—
323C22×C4—24no—
324C22×C4—24no—
325C22×C4—24no—
326C22×C4—22no—
327C22×C4—24no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—24no—
332C22×C4—24no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—24no—
336C22×C4—22no—
337C22×C4—24no—
338C22×C4—22no—
339C22×C4—24no—
340C22×C4—24no—
341C22×C4—24no—
342C22×C4—24no—
343C22×C4—24no—
344C22×C4—22no—
345C22×C4—24no—
346C22×C4—24no—
347C22×C4—24no—
348C22×C4—24no—
349C22×C4—24no—
350C22×C4—24no—
351C22×C4—24no—
352C22×C4—24no—
353C22×C4—24no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—24no—
357C22×C4—24no—
358C22×C4—24no—
359C22×C4—24no—
360C22×C4—24no—
361C22×C4—24no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—24no—
365C22×C4—24no—
366C22×C4—22no—
367C22×C4—24no—
368C22×C4—24no—
369C22×C4—24no—
370C22×C4—416no—
371C22×C4—88no—
372C22×C4—88no—
373C22×C4—88no—
374C22×C4—88no—
375C22×C4—88no—
376C22×C4—88no—
377C22×C4C2×#4114yes—
378C22×C4C2×#4114yes—
379C22×C4C2×#4112yes—
380C22×C4C2×#4114yes—
381C22×C4C2×#4114yes—
382C22×C4H(5,𝔽2)12yes—
383C22×C4H(5,𝔽2)14yes—
384C22×C4H(5,𝔽2)12yes—
385C22×C42−1+414yes—
386C4⋊C4—48no—
387C4⋊C4—48no—
388C4⋊C4—48no—
389C4⋊C4—48no—
390C4⋊C4—48no—
391C4⋊C4—88no—
392C4⋊C4—84no—
393C22⋊C4—416no—
394C22⋊C4—416no—
395C22⋊C4—416no—
396C22⋊C4—416no—
397C22⋊C4—416no—
398C22⋊C4—88no—
399C22⋊C4—88no—
400C22⋊C4—88no—
401C22⋊C4—88no—
402C22⋊C4—88no—
403C22⋊C4—88no—
404C42—28no—
405C42—48no—
406C42—48no—
407C42—84no—
408C23—28no—
409C23—28no—
410C23—28no—
411C23—28no—
412C23—28no—
413C23—28no—
414C23—28no—
415C23—44no—
416C23—44no—
417C23—44no—
418C23—44no—
419C23—44no—
420C23—44no—
421C23—44no—
422C23—44no—
423C23—44no—
424C23—44no—
425C23—44no—
426C23—44no—
427C23—44no—
428C23—44no—
429C23—44no—
430C23—44no—
431C23—44no—
432C23—44no—
433C23—44no—
434C23—44no—
435C23—44no—
436C23—44no—
437C23—44no—
438C23—44no—
439C23—44no—
440C23—44no—
441C23—44no—
442C23—44no—
443C23C22×#4112yes—
444C23C22×#4111yescharacteristic
445C23C2×H(5,𝔽2)11yescharacteristic
446C23C2×H(5,𝔽2)11yescharacteristic
447C23C2×H(5,𝔽2)14yes—
448C23C2×H(5,𝔽2)12yes—
449C23C2×2−1+412yes—
450C23#58512yes—
451Q8—88no—
452Q8—88no—
453D4—432no—
454D4—816no—
455D4—816no—
456C2×C4—24no—
457C2×C4—22no—
458C2×C4—22no—
459C2×C4—24no—
460C2×C4—24no—
461C2×C4—24no—
462C2×C4—22no—
463C2×C4—22no—
464C2×C4—22no—
465C2×C4—24no—
466C2×C4—24no—
467C2×C4—24no—
468C2×C4—24no—
469C2×C4—24no—
470C2×C4—24no—
471C2×C4—28no—
472C2×C4—28no—
473C2×C4—28no—
474C2×C4—24no—
475C2×C4—42no—
476C2×C4—42no—
477C2×C4—42no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—44no—
481C2×C4—44no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—44no—
485C2×C4—44no—
486C2×C4—44no—
487C2×C4—42no—
488C2×C4—44no—
489C2×C4—44no—
490C2×C4—44no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—44no—
495C2×C4—44no—
496C2×C4—44no—
497C2×C4—44no—
498C2×C4—44no—
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—42no—
508C2×C4—44no—
509C2×C4—44no—
510C2×C4—42no—
511C2×C4—44no—
512C2×C4—44no—
513C2×C4—44no—
514C2×C4—44no—
515C2×C4—44no—
516C2×C4—44no—
517C2×C4—44no—
518C2×C4—42no—
519C2×C4—832no—
520C2×C4#55514yes—
521C2×C4H(3,𝔽4)14yes—
522C22—48no—
523C22—48no—
524C22—48no—
525C22—48no—
526C22—48no—
527C22—48no—
528C22—48no—
529C22—84no—
530C22—84no—
531C22—84no—
532C22—84no—
533C22—84no—
534C22—84no—
535C22—84no—
536C22—84no—
537C22C2×#55511yescharacteristic
538C22#348012yes—
539C22#349011yescharacteristic
540C22#349012yes—
541C22#350512yes—
542C22#351914yes—
543C22#352212yes—
544C22#352214yes—
545C22C2×H(3,𝔽4)11yescharacteristic
546C22#353514yes—
547C22#353612yes—
548C22#353712yes—
549C22#353712yes—
550C22#354214yes—
551C22#354411yescharacteristic
552C22#354511yescharacteristic
553C4—24no—
554C4—24no—
555C4—44no—
556C4—42no—
557C4—44no—
558C4—44no—
559C4—44no—
560C4—84no—
561C4—82no—
562C4—84no—
563C4—84no—
564C4—82no—
565C2—88no—
566C2#4407514yes—
567C2#4537812yes—
568C2#4564512yes—
569C2#4609911yescharacteristic
570C2#4638411yescharacteristic
571C2#4697212yes—
572C2#5750912yes—
573C2#5988511yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radical#15763C2
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #17306C2semidirect
2. #17107C2semidirect
3. #15834C2semidirect
5. #15761C2semidirect
6. #15506C2semidirect
7. #15498C2semidirect
8. #15107C2semidirect
9. #14806C2semidirect
10. #14473C2semidirect
11. #14456C2semidirect
12. #14176C2semidirect
13. #14161C2semidirect

Maximal subgroups

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

subgroupindex
#173062
#171072
#158342
#157632
#157612
#155062
#154982
#151072
#148062
#144732
#144562
#141762
#141612