Finite Group Databasebeta

← #4933892

Subgroups — #4933892

Order 512.

2270 conjugacy classes of subgroups.

The subgroup lattice has 551 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2724C211yesmaximal, characteristic
2C2×#2563C211yesmaximal, characteristic
3#16963C212yesmaximal, complemented
4#16831C214yesmaximal, complemented
5#15945C212yesmaximal, complemented
6#15783C212yesmaximal, complemented
7#15487C214yesmaximal, complemented
8#15371C214yesmaximal, complemented
9#15108C212yesmaximal, complemented
10#14767C214yesmaximal, complemented
11#14505C212yesmaximal
12#13968C212yesmaximal
13#13832C211yesmaximal, characteristic
14C22×C4⋊Q8C2211yescharacteristic
15C22×#123C2212yes—
16C22×C4×D4C2211yescharacteristic
17#2724—28no—
18#2563—28no—
19C2×#400C2212yes—
20C2×#398C2212yes—
21C2×#397C2214yes—
22C2×#393C2214yes—
23C2×#391C2212yes—
24C2×#390C2211yescharacteristic
25C2×#390C2211yescharacteristic
26C2×#389C2212yes—
27C2×#386C2212yes—
28C2×#385C2214yes—
29C2×#382C2211yescharacteristic
30C2×#382C2211yescharacteristic
31C2×#381C2212yes—
32C2×#380C2214yes—
33C2×C4×C4⋊C4C2211yescharacteristic
34C22×#95C2212yes—
35#1446C2214yes—
36#1446C2212yes—
37#1445C2214yes—
38#1443C2214yes—
39#1443C2214yes—
40#1443C2214yes—
41#1443C2214yes—
42#1443C2214yes—
43#1442C2212yes—
44#1442C2214yes—
45#1442C2212yes—
46#1442C2214yes—
47#1442C2214yes—
48#1442C2214yes—
49#1441C2212yes—
50#1441C2212yes—
51#1441C2214yes—
52#1441C2214yes—
53#1440C2214yes—
54#1440C2214yes—
55#1440C2214yes—
56#1437C2214yes—
57#1436C2214yes—
58#1436C2214yes—
59#1436C2212yes—
60#1436C2212yes—
61#1434C2212yes—
62#1434C2214yes—
63C22⋊#95C2214yes—
64C22⋊#95C2214yes—
65C22⋊#95C2212yes—
66C22⋊#95C2214yes—
67C22⋊#95C2212yes—
68C22⋊#95C2214yes—
69C23×Q8C2312yes—
70C23×D4C2311yescharacteristic
71C24×C4C2312yes—
72C24×C4C2312yes—
73C24×C4C2312yes—
74C2×C4⋊Q8—24no—
75C2×C4⋊Q8—42no—
76C2×C4⋊Q8—42no—
77C2×#123—28no—
78C2×#123—28no—
79C2×#123—28no—
80C2×C4×D4—24no—
81C2×C4×D4—42no—
82C2×C4×D4—42no—
83C22×C4⋊C4C2314yes—
84C22×C4⋊C4C2312yes—
85C22×C4⋊C4C2312yes—
86C22×C4⋊C4C2312yes—
87C22×C4⋊C4C2312yes—
88C22×C4⋊C4C2311yescharacteristic
89C22×C4⋊C4C2311yescharacteristic
90C22×C4⋊C4C2312yes—
91C22×C4⋊C4C2311yescharacteristic
92C22×C22⋊C4C2312yes—
93C22×C22⋊C4C2314yes—
94C22×C22⋊C4C2312yes—
95C22×C22⋊C4C2314yes—
96C22×C22⋊C4C2314yes—
97C22×C22⋊C4C2312yes—
98C22×C22⋊C4C2314yes—
99C22×C42C2311yescharacteristic
100C22×C42C2311yescharacteristic
101C22×C42C2312yes—
102C22×C42C2311yescharacteristic
103#400—44no—
104#398—28no—
105#397—216no—
106#393—216no—
107#391—44no—
108#390—24no—
109#390D418yes—
110#389—28no—
111#386—28no—
112#385—216no—
113#382—24no—
114#382—24no—
115#381—44no—
116#380—216no—
117C4×C4⋊C4—24no—
118C2×#95—44no—
119C2×#95C2312yes—
120C2×#95C2314yes—
121C2×#95C2314yes—
122C2×#95C2312yes—
123C2×#95C2314yes—
124C2×#95C2314yes—
125C2×#95C2314yes—
126C2×#95C2314yes—
127C2×#95C2314yes—
128C2×#95C2314yes—
129C2×#95C2312yes—
130C2×#95C2314yes—
131C2×#95C2312yes—
132C2×#95C2314yes—
133C2×#95C2314yes—
134C2×#95C2314yes—
135C2×#95C2314yes—
136C2×#95C2314yes—
137C2×#95C2314yes—
138C2×#95C2314yes—
139C2×#95C2314yes—
140C2×#95C2314yes—
141C2×#95C2314yes—
142C2×#95C2314yes—
143C2×#95C2314yes—
144C2×#95C2312yes—
145C2×#95C2314yes—
146C2×#95C2314yes—
147C25C2412yes—
148C22×Q8—24no—
149C22×Q8—24no—
150C22×Q8—42no—
151C22×Q8—44no—
152C22×Q8—44no—
153C22×D4—41no—
154C22×D4—41no—
155C22×D4—42no—
156C22×D4C2×D414yes—
157C22×D4#4118yes—
158C23×C4—24no—
159C23×C4—24no—
160C23×C4—24no—
161C23×C4—24no—
162C23×C4—24no—
163C23×C4—24no—
164C23×C4—28no—
165C23×C4—42no—
166C23×C4—44no—
167C23×C4—44no—
168C23×C4—42no—
169C23×C4—42no—
170C23×C4—42no—
171C23×C4—42no—
172C23×C4—42no—
173C23×C4—42no—
174C23×C4—44no—
175C23×C4C2412yes—
176C23×C4C2414yes—
177C23×C4C2412yes—
178C23×C4C2414yes—
179C23×C4C2414yes—
180C23×C4C2412yes—
181C23×C4C2414yes—
182C23×C4C2412yes—
183C23×C4C2412yes—
184C23×C4C2411yescharacteristic
185C23×C4C2411yescharacteristic
186C23×C4C2411yescharacteristic
187C4⋊Q8—416no—
188#123—432no—
189C4×D4—416no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—28no—
192C2×C4⋊C4—24no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—24no—
195C2×C4⋊C4—24no—
196C2×C4⋊C4—28no—
197C2×C4⋊C4—28no—
198C2×C4⋊C4—24no—
199C2×C4⋊C4—24no—
200C2×C4⋊C4—24no—
201C2×C4⋊C4—24no—
202C2×C4⋊C4—42no—
203C2×C4⋊C4—44no—
204C2×C4⋊C4—41no—
205C2×C4⋊C4—41no—
206C2×C4⋊C4—41no—
207C2×C4⋊C4—41no—
208C2×C4⋊C4—42no—
209C2×C4⋊C4—41no—
210C2×C4⋊C4—41no—
211C2×C4⋊C4—44no—
212C2×C4⋊C4—42no—
213C2×C4⋊C4—42no—
214C2×C4⋊C4C2×D414yes—
215C2×C22⋊C4—24no—
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—24no—
223C2×C22⋊C4—28no—
224C2×C22⋊C4—24no—
225C2×C22⋊C4—28no—
226C2×C22⋊C4—44no—
227C2×C22⋊C4—44no—
228C2×C22⋊C4—42no—
229C2×C22⋊C4—42no—
230C2×C22⋊C4—44no—
231C2×C22⋊C4—44no—
232C2×C22⋊C4—44no—
233C2×C22⋊C4—44no—
234C2×C22⋊C4—42no—
235C2×C22⋊C4C2×D4116yes—
236C2×C42—22no—
237C2×C42—22no—
238C2×C42—22no—
239C2×C42—22no—
240C2×C42—42no—
241C2×C42—42no—
242C2×C42—42no—
243C2×C42—44no—
244C2×C42C2×D414yes—
245#95—24no—
246#95—28no—
247#95—28no—
248#95—28no—
249#95—28no—
250#95—24no—
251#95—28no—
252#95—28no—
253#95—28no—
254#95—28no—
255#95—44no—
256#95—42no—
257#95—42no—
258#95—44no—
259#95—44no—
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—44no—
271#95—42no—
272#95—44no—
273C24—22no—
274C24—22no—
275C24—24no—
276C24—24no—
277C24—24no—
278C24—24no—
279C24—22no—
280C24—22no—
281C24C22×D414yes—
282C24C2×#4114yes—
283C24C2×#4114yes—
284C24C2511yescharacteristic
285C2×Q8—48no—
286C2×Q8—48no—
287C2×Q8—88no—
288C2×Q8—84no—
289C2×Q8—88no—
290C2×D4—44no—
291C2×D4—44no—
292C2×D4—48no—
293C2×D4—48no—
294C2×D4#124116yes—
295C22×C4—21no—
296C22×C4—21no—
297C22×C4—21no—
298C22×C4—21no—
299C22×C4—22no—
300C22×C4—22no—
301C22×C4—21no—
302C22×C4—22no—
303C22×C4—21no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—24no—
308C22×C4—24no—
309C22×C4—24no—
310C22×C4—24no—
311C22×C4—24no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—24no—
315C22×C4—22no—
316C22×C4—24no—
317C22×C4—24no—
318C22×C4—24no—
319C22×C4—24no—
320C22×C4—24no—
321C22×C4—24no—
322C22×C4—24no—
323C22×C4—24no—
324C22×C4—24no—
325C22×C4—24no—
326C22×C4—24no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—24no—
332C22×C4—24no—
333C22×C4—22no—
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—22no—
343C22×C4—24no—
344C22×C4—24no—
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—24no—
355C22×C4—48no—
356C22×C4—416no—
357C22×C4—48no—
358C22×C4—48no—
359C22×C4—48no—
360C22×C4—48no—
361C22×C4—48no—
362C22×C4—84no—
363C22×C4—88no—
364C22×C4—84no—
365C22×C4—84no—
366C22×C4—84no—
367C22×C4—84no—
368C22×C4—84no—
369C22×C4—88no—
370C22×C4—84no—
371C22×C4—88no—
372C22×C4C22×D414yes—
373C22×C4C22×D414yes—
374C22×C4C22×D412yes—
375C22×C4C2×#4114yes—
376C22×C4H(5,𝔽2)12yes—
377C22×C4H(5,𝔽2)12yes—
378C22×C42−1+414yes—
379C4⋊C4—44no—
380C4⋊C4—48no—
381C4⋊C4—48no—
382C4⋊C4—48no—
383C4⋊C4—44no—
384C4⋊C4—44no—
385C4⋊C4—416no—
386C4⋊C4—84no—
387C4⋊C4—84no—
388C22⋊C4—416no—
389C22⋊C4—48no—
390C22⋊C4—48no—
391C22⋊C4—416no—
392C22⋊C4—88no—
393C22⋊C4—84no—
394C22⋊C4—88no—
395C42—28no—
396C42—44no—
397C42—44no—
398C42—84no—
399C23—24no—
400C23—28no—
401C23—24no—
402C23—28no—
403C23—24no—
404C23—24no—
405C23—24no—
406C23—24no—
407C23—28no—
408C23—28no—
409C23—28no—
410C23—28no—
411C23—42no—
412C23—44no—
413C23—42no—
414C23—42no—
415C23—42no—
416C23—44no—
417C23—44no—
418C23—44no—
419C23—44no—
420C23—44no—
421C23C2×#12418yes—
422C23C23×D411yescharacteristic
423C23C22×#4112yes—
424C23C2×H(5,𝔽2)11yescharacteristic
425C23C2×H(5,𝔽2)11yescharacteristic
426C23C2×2−1+412yes—
427C23#58512yes—
428C23#58514yes—
429C23#58512yes—
430Q8—816no—
431D4—416no—
432C2×C4—22no—
433C2×C4—22no—
434C2×C4—22no—
435C2×C4—24no—
436C2×C4—24no—
437C2×C4—24no—
438C2×C4—22no—
439C2×C4—24no—
440C2×C4—22no—
441C2×C4—24no—
442C2×C4—24no—
443C2×C4—24no—
444C2×C4—28no—
445C2×C4—24no—
446C2×C4—24no—
447C2×C4—42no—
448C2×C4—42no—
449C2×C4—42no—
450C2×C4—44no—
451C2×C4—42no—
452C2×C4—44no—
453C2×C4—44no—
454C2×C4—44no—
455C2×C4—44no—
456C2×C4—42no—
457C2×C4—42no—
458C2×C4—44no—
459C2×C4—42no—
460C2×C4—42no—
461C2×C4—44no—
462C2×C4—44no—
463C2×C4—44no—
464C2×C4—42no—
465C2×C4—44no—
466C2×C4—44no—
467C2×C4—44no—
468C2×C4—44no—
469C2×C4—44no—
470C2×C4—44no—
471C2×C4—44no—
472C2×C4—44no—
473C2×C4—44no—
474C2×C4—44no—
475C2×C4—44no—
476C2×C4—44no—
477C2×C4—44no—
478C2×C4—44no—
479C2×C4—44no—
480C2×C4—44no—
481C2×C4—42no—
482C2×C4—44no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—44no—
486C2×C4—44no—
487C2×C4—42no—
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—44no—
497C2×C4—816no—
498C2×C4—816no—
499C2×C4—816no—
500C2×C4C2×#12414yes—
501C2×C4#56314yes—
502C22—28no—
503C22—216no—
504C22—44no—
505C22—44no—
506C22—44no—
507C22—44no—
508C22—48no—
509C22—44no—
510C22—44no—
511C22—48no—
512C22—48no—
513C22C22×#12411yescharacteristic
514C22#348411yescharacteristic
515C22#348511yescharacteristic
516C22#348512yes—
517C22#348912yes—
518C22#350512yes—
519C22#350512yes—
520C22#351514yes—
521C22#352114yes—
522C22C2×#56311yescharacteristic
523C22#353312yes—
524C22#354012yes—
525C22#354511yescharacteristic
526C22#356112yes—
527C22#356514yes—
528C22#356614yes—
529C4—24no—
530C4—24no—
531C4—44no—
532C4—42no—
533C4—44no—
534C4—84no—
535C4—82no—
536C4—82no—
537C4—82no—
538C4—84no—
539C4—84no—
540C4—84no—
541C2—48no—
542C2#3537711yescharacteristic
543C2#3604212yes—
544C2#4569612yes—
545C2#4613611yescharacteristic
546C2#4878112yes—
547C2#4905912yes—
548C2#4939914yes—
549C2#5205411yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC23×D4C23
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #16963C2semidirect
4. #16831C2semidirect
5. #15945C2semidirect
6. #15783C2semidirect
7. #15487C2semidirect
8. #15371C2semidirect
9. #15108C2semidirect
10. #14767C2semidirect

Maximal subgroups

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

subgroupindex
C2×#27242
C2×#25632
#169632
#168312
#159452
#157832
#154872
#153712
#151082
#147672
#145052
#139682
#138322