Finite Group Databasebeta

← #1494770

Subgroups — #1494770

Order 512.

2374 conjugacy classes of subgroups.

The subgroup lattice has 553 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2724C211yesmaximal, characteristic
2C2×#2566C211yesmaximal, characteristic
3#15945C212yesmaximal, complemented
4#15881C212yesmaximal, complemented
5#15544C214yesmaximal, complemented
6#15470C214yesmaximal, complemented
7#15197C212yesmaximal, complemented
8#14833C214yesmaximal, complemented
9#14505C212yesmaximal
10#14242C212yesmaximal, complemented
11#13968C212yesmaximal
12#13896C214yesmaximal, complemented
13#13824C211yesmaximal, characteristic
14C22×#125C2211yescharacteristic
15C22×#123C2212yes—
16C22×C4×D4C2211yescharacteristic
17#2724—28no—
18#2566—28no—
19C2×#400C2212yes—
20C2×#400C2212yes—
21C2×#398C2212yes—
22C2×#397C2214yes—
23C2×#396C2214yes—
24C2×#390C2211yescharacteristic
25C2×#390C2211yescharacteristic
26C2×#389C2212yes—
27C2×#386C2212yes—
28C2×#385C2214yes—
29C2×#385C2214yes—
30C2×#384C2211yescharacteristic
31C2×#382C2211yescharacteristic
32C2×#382C2212yes—
33C2×#381C2212yes—
34C2×C4×C4⋊C4C2211yescharacteristic
35C2×C4×C22⋊C4C2214yes—
36C22×#95C2212yes—
37#1445C2212yes—
38#1445C2212yes—
39#1443C2214yes—
40#1443C2214yes—
41#1442C2214yes—
42#1442C2214yes—
43#1442C2212yes—
44#1442C2214yes—
45#1442C2214yes—
46#1442C2214yes—
47#1441C2214yes—
48#1441C2214yes—
49#1441C2212yes—
50#1440C2214yes—
51#1440C2214yes—
52#1440C2214yes—
53#1437C2214yes—
54#1436C2212yes—
55#1436C2214yes—
56#1436C2212yes—
57#1436C2214yes—
58#1436C2214yes—
59C4⋊#95C2214yes—
60C4⋊#95C2214yes—
61#1434C2214yes—
62#1434C2214yes—
63C22⋊#95C2214yes—
64C22⋊#95C2214yes—
65C22⋊#95C2214yes—
66C22⋊#95C2212yes—
67C22⋊#95C2212yes—
68C4×#95C2212yes—
69C23×D4C2311yescharacteristic
70C24×C4C2312yes—
71C24×C4C2312yes—
72C24×C4C2312yes—
73C2×#125—24no—
74C2×#125—42no—
75C2×#125—42no—
76C2×#123—28no—
77C2×#123—28no—
78C2×#123—28no—
79C2×C4×D4—24no—
80C2×C4×D4—42no—
81C2×C4×D4—42no—
82C22×C4⋊C4C2312yes—
83C22×C4⋊C4C2312yes—
84C22×C4⋊C4C2314yes—
85C22×C4⋊C4C2312yes—
86C22×C4⋊C4C2312yes—
87C22×C4⋊C4C2314yes—
88C22×C4⋊C4C2312yes—
89C22×C4⋊C4C2311yescharacteristic
90C22×C4⋊C4C2311yescharacteristic
91C22×C4⋊C4C2312yes—
92C22×C4⋊C4C2311yescharacteristic
93C22×C22⋊C4C2312yes—
94C22×C22⋊C4C2314yes—
95C22×C22⋊C4C2312yes—
96C22×C22⋊C4C2314yes—
97C22×C22⋊C4C2314yes—
98C22×C22⋊C4C2314yes—
99C22×C22⋊C4C2312yes—
100C22×C42C2311yescharacteristic
101C22×C42C2312yes—
102C22×C42C2314yes—
103C22×C42C2311yescharacteristic
104C22×C42C2311yescharacteristic
105#400—28no—
106#400—44no—
107#398—28no—
108#397—216no—
109#396—216no—
110#390—24no—
111#390D418yes—
112#389—28no—
113#386—28no—
114#385—216no—
115#385—216no—
116#384—24no—
117#382—24no—
118#382—44no—
119#381—44no—
120C4×C4⋊C4—24no—
121C4×C22⋊C4—216no—
122C2×#95—44no—
123C2×#95C2314yes—
124C2×#95C2312yes—
125C2×#95C2314yes—
126C2×#95C2314yes—
127C2×#95C2314yes—
128C2×#95C2314yes—
129C2×#95C2314yes—
130C2×#95C2314yes—
131C2×#95C2312yes—
132C2×#95C2314yes—
133C2×#95C2312yes—
134C2×#95C2314yes—
135C2×#95C2314yes—
136C2×#95C2312yes—
137C2×#95C2314yes—
138C2×#95C2314yes—
139C2×#95C2314yes—
140C2×#95C2314yes—
141C2×#95C2314yes—
142C2×#95C2314yes—
143C2×#95C2314yes—
144C2×#95C2314yes—
145C2×#95C2314yes—
146C2×#95C2314yes—
147C2×#95C2314yes—
148C2×#95C2312yes—
149C25C2412yes—
150C22×D4—41no—
151C22×D4—41no—
152C22×D4—42no—
153C22×D4C2×D414yes—
154C22×D4#4118yes—
155C23×C4—24no—
156C23×C4—24no—
157C23×C4—24no—
158C23×C4—24no—
159C23×C4—24no—
160C23×C4—28no—
161C23×C4—24no—
162C23×C4—42no—
163C23×C4—44no—
164C23×C4—44no—
165C23×C4—42no—
166C23×C4—42no—
167C23×C4—42no—
168C23×C4—42no—
169C23×C4—42no—
170C23×C4—42no—
171C23×C4—44no—
172C23×C4C2412yes—
173C23×C4C2414yes—
174C23×C4C2412yes—
175C23×C4C2414yes—
176C23×C4C2414yes—
177C23×C4C2412yes—
178C23×C4C2412yes—
179C23×C4C2414yes—
180C23×C4C2411yescharacteristic
181C23×C4C2411yescharacteristic
182C23×C4C2411yescharacteristic
183C23×C4C2412yes—
184#125—416no—
185#123—432no—
186C4×D4—416no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—28no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—24no—
191C2×C4⋊C4—24no—
192C2×C4⋊C4—24no—
193C2×C4⋊C4—24no—
194C2×C4⋊C4—28no—
195C2×C4⋊C4—28no—
196C2×C4⋊C4—24no—
197C2×C4⋊C4—24no—
198C2×C4⋊C4—24no—
199C2×C4⋊C4—24no—
200C2×C4⋊C4—24no—
201C2×C4⋊C4—24no—
202C2×C4⋊C4—28no—
203C2×C4⋊C4—24no—
204C2×C4⋊C4—24no—
205C2×C4⋊C4—41no—
206C2×C4⋊C4—41no—
207C2×C4⋊C4—41no—
208C2×C4⋊C4—41no—
209C2×C4⋊C4—42no—
210C2×C4⋊C4—42no—
211C2×C4⋊C4—41no—
212C2×C4⋊C4—41no—
213C2×C4⋊C4—44no—
214C2×C4⋊C4—44no—
215C2×C4⋊C4—44no—
216C2×C4⋊C4—42no—
217C2×C4⋊C4—42no—
218C2×C4⋊C4C2×D414yes—
219C2×C22⋊C4—28no—
220C2×C22⋊C4—28no—
221C2×C22⋊C4—24no—
222C2×C22⋊C4—28no—
223C2×C22⋊C4—24no—
224C2×C22⋊C4—24no—
225C2×C22⋊C4—24no—
226C2×C22⋊C4—28no—
227C2×C22⋊C4—28no—
228C2×C22⋊C4—28no—
229C2×C22⋊C4—28no—
230C2×C22⋊C4—24no—
231C2×C22⋊C4—28no—
232C2×C22⋊C4—24no—
233C2×C22⋊C4—44no—
234C2×C22⋊C4—42no—
235C2×C22⋊C4—42no—
236C2×C22⋊C4—44no—
237C2×C22⋊C4—44no—
238C2×C22⋊C4—42no—
239C2×C22⋊C4C2×D4116yes—
240C2×C42—22no—
241C2×C42—22no—
242C2×C42—22no—
243C2×C42—22no—
244C2×C42—28no—
245C2×C42—28no—
246C2×C42—42no—
247C2×C42—42no—
248C2×C42—42no—
249C2×C42—44no—
250C2×C42—44no—
251C2×C42C2×D414yes—
252#95—24no—
253#95—28no—
254#95—28no—
255#95—28no—
256#95—24no—
257#95—28no—
258#95—28no—
259#95—28no—
260#95—28no—
261#95—28no—
262#95—28no—
263#95—24no—
264#95—44no—
265#95—44no—
266#95—44no—
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—44no—
277#95#4118yes—
278C24—24no—
279C24—22no—
280C24—22no—
281C24—24no—
282C24—24no—
283C24—24no—
284C24—22no—
285C24—22no—
286C24C22×D414yes—
287C24C2×#4114yes—
288C24C2×#4114yes—
289C24C2511yescharacteristic
290C2×D4—44no—
291C2×D4—44no—
292C2×D4—48no—
293C2×D4—48no—
294C2×D4#124116yes—
295C22×C4—21no—
296C22×C4—21no—
297C22×C4—24no—
298C22×C4—22no—
299C22×C4—24no—
300C22×C4—21no—
301C22×C4—21no—
302C22×C4—22no—
303C22×C4—24no—
304C22×C4—24no—
305C22×C4—24no—
306C22×C4—24no—
307C22×C4—22no—
308C22×C4—21no—
309C22×C4—21no—
310C22×C4—22no—
311C22×C4—24no—
312C22×C4—22no—
313C22×C4—24no—
314C22×C4—24no—
315C22×C4—24no—
316C22×C4—24no—
317C22×C4—22no—
318C22×C4—24no—
319C22×C4—24no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—24no—
323C22×C4—22no—
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—22no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—24no—
338C22×C4—24no—
339C22×C4—22no—
340C22×C4—24no—
341C22×C4—24no—
342C22×C4—24no—
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—48no—
354C22×C4—48no—
355C22×C4—416no—
356C22×C4—48no—
357C22×C4—48no—
358C22×C4—48no—
359C22×C4—48no—
360C22×C4—88no—
361C22×C4—84no—
362C22×C4—88no—
363C22×C4—84no—
364C22×C4—84no—
365C22×C4—84no—
366C22×C4—84no—
367C22×C4—84no—
368C22×C4—84no—
369C22×C4—88no—
370C22×C4C22×D414yes—
371C22×C4C22×D412yes—
372C22×C4C22×D414yes—
373C22×C4C2×#4118yes—
374C22×C4C2×#4114yes—
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—44no—
383C4⋊C4—44no—
384C4⋊C4—416no—
385C4⋊C4—48no—
386C4⋊C4—48no—
387C4⋊C4—84no—
388C4⋊C4—88no—
389C4⋊C4—84no—
390C22⋊C4—232no—
391C22⋊C4—416no—
392C22⋊C4—48no—
393C22⋊C4—48no—
394C22⋊C4—416no—
395C22⋊C4—88no—
396C22⋊C4—84no—
397C42—28no—
398C42—44no—
399C42—48no—
400C42—44no—
401C42—88no—
402C23—28no—
403C23—28no—
404C23—28no—
405C23—24no—
406C23—24no—
407C23—28no—
408C23—24no—
409C23—24no—
410C23—24no—
411C23—24no—
412C23—28no—
413C23—28no—
414C23—42no—
415C23—44no—
416C23—44no—
417C23—44no—
418C23—42no—
419C23—42no—
420C23—42no—
421C23—44no—
422C23—44no—
423C23—44no—
424C23C2×#12418yes—
425C23C23×D411yescharacteristic
426C23C22×#4112yes—
427C23C22×#4112yes—
428C23C2×H(5,𝔽2)11yescharacteristic
429C23C2×H(5,𝔽2)11yescharacteristic
430C23C2×2−1+412yes—
431C23#58514yes—
432C23#58512yes—
433D4—416no—
434C2×C4—22no—
435C2×C4—22no—
436C2×C4—24no—
437C2×C4—24no—
438C2×C4—24no—
439C2×C4—24no—
440C2×C4—28no—
441C2×C4—28no—
442C2×C4—28no—
443C2×C4—28no—
444C2×C4—28no—
445C2×C4—22no—
446C2×C4—22no—
447C2×C4—24no—
448C2×C4—28no—
449C2×C4—24no—
450C2×C4—24no—
451C2×C4—42no—
452C2×C4—44no—
453C2×C4—42no—
454C2×C4—41no—
455C2×C4—42no—
456C2×C4—44no—
457C2×C4—44no—
458C2×C4—44no—
459C2×C4—44no—
460C2×C4—41no—
461C2×C4—42no—
462C2×C4—42no—
463C2×C4—44no—
464C2×C4—44no—
465C2×C4—44no—
466C2×C4—42no—
467C2×C4—44no—
468C2×C4—44no—
469C2×C4—44no—
470C2×C4—42no—
471C2×C4—42no—
472C2×C4—44no—
473C2×C4—42no—
474C2×C4—44no—
475C2×C4—42no—
476C2×C4—44no—
477C2×C4—44no—
478C2×C4—44no—
479C2×C4—44no—
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—42no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—44no—
496C2×C4—44no—
497C2×C4—44no—
498C2×C4—44no—
499C2×C4—816no—
500C2×C4—816no—
501C2×C4—816no—
502C2×C4C2×#12414yes—
503C2×C4#54718yes—
504C22—28no—
505C22—216no—
506C22—44no—
507C22—44no—
508C22—44no—
509C22—48no—
510C22—48no—
511C22—44no—
512C22—44no—
513C22—44no—
514C22—48no—
515C22C22×#12411yescharacteristic
516C22C2×#54712yes—
517C22#347514yes—
518C22#348111yescharacteristic
519C22#348411yescharacteristic
520C22#348511yescharacteristic
521C22#350012yes—
522C22#350512yes—
523C22#350512yes—
524C22#351514yes—
525C22#352114yes—
526C22#352212yes—
527C22#353312yes—
528C22#355011yescharacteristic
529C22#355112yes—
530C22#355314yes—
531C4—24no—
532C4—28no—
533C4—42no—
534C4—48no—
535C4—44no—
536C4—42no—
537C4—82no—
538C4—82no—
539C4—84no—
540C4—84no—
541C4—84no—
542C4—82no—
543C2—48no—
544C2#3537711yescharacteristic
545C2#3616412yes—
546C2#4349512yes—
547C2#4369814yes—
548C2#4907712yes—
549C2#5197811yescharacteristic
550C2#5907011yescharacteristic
551C2#6001812yes—
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. #15945C2semidirect
4. #15881C2semidirect
5. #15544C2semidirect
6. #15470C2semidirect
7. #15197C2semidirect
8. #14833C2semidirect
10. #14242C2semidirect
12. #13896C2semidirect

Maximal subgroups

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

subgroupindex
C2×#27242
C2×#25662
#159452
#158812
#155442
#154702
#151972
#148332
#145052
#142422
#139682
#138962
#138242