Finite Group Databasebeta

← #876783

Subgroups — #876783

Order 512.

2518 conjugacy classes of subgroups.

The subgroup lattice has 569 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2851C212yesmaximal
2C2×#2552C212yesmaximal, complemented
3#16343C214yesmaximal
4#15401C214yesmaximal, complemented
5#15388C214yesmaximal, complemented
6#15228C212yesmaximal, complemented
7#15223C211yesmaximal, characteristic
8#15221C212yesmaximal, complemented
9#15216C211yesmaximal, characteristic
10#14405C214yesmaximal
11#13982C212yesmaximal, complemented
12#13876C212yesmaximal
13C4×#385C211yesmaximal, characteristic
14C22×#125C2212yes—
15C22×#125C2211yescharacteristic
16C23×C42C2211yescharacteristic
17#2851—216no—
18#2552—216no—
19C2×#398C2214yes—
20C2×#398C2214yes—
21C2×#389C2212yes—
22C2×#389C2214yes—
23C2×#389C2212yes—
24C2×#389C2212yes—
25C2×#388C2214yes—
26C2×#387C2214yes—
27C2×#387C2214yes—
28C2×#385C2214yes—
29C2×#385C2214yes—
30C2×#385C2212yes—
31C2×#384C2214yes—
32C2×#384C2212yes—
33C2×#382C2212yes—
34C2×#382C2211yescharacteristic
35C2×#380C2212yes—
36C2×#380C2212yes—
37C2×C4×C4⋊C4C2212yes—
38C2×C4×C4⋊C4C2212yes—
39C2×C4×C22⋊C4C2211yescharacteristic
40C2×C4×C22⋊C4C2211yescharacteristic
41#1445C2214yes—
42#1445C2214yes—
43#1445C2214yes—
44#1444C2214yes—
45#1444C2214yes—
46#1444C2214yes—
47#1443C2214yes—
48#1442C2214yes—
49#1442C2214yes—
50#1442C2214yes—
51#1442C2214yes—
52#1441C2214yes—
53#1441C2214yes—
54#1439C2214yes—
55#1439C2214yes—
56#1439C2214yes—
57#1436C2212yes—
58#1436C2214yes—
59#1436C2214yes—
60#1436C2212yes—
61C4⋊#95C2212yes—
62C4⋊#95C2212yes—
63C4⋊#95C2211yescharacteristic
64C4⋊#95C2212yes—
65C4⋊#95C2211yescharacteristic
66#1434C2214yes—
67C22⋊#95C2212yes—
68C4×#95C2212yes—
69C24×C4C2311yescharacteristic
70C24×C4C2312yes—
71C2×#125—28no—
72C2×#125—28no—
73C2×#125—28no—
74C2×#125—44no—
75C2×#125—42no—
76C22×C4⋊C4C2312yes—
77C22×C4⋊C4C2312yes—
78C22×C4⋊C4C2314yes—
79C22×C4⋊C4C2314yes—
80C22×C4⋊C4C2314yes—
81C22×C4⋊C4C2312yes—
82C22×C4⋊C4C2312yes—
83C22×C4⋊C4C2314yes—
84C22×C4⋊C4C2312yes—
85C22×C4⋊C4C2312yes—
86C22×C4⋊C4C2314yes—
87C22×C4⋊C4C2314yes—
88C22×C4⋊C4C2314yes—
89C22×C4⋊C4C2314yes—
90C22×C22⋊C4C2314yes—
91C22×C22⋊C4C2314yes—
92C22×C22⋊C4C2312yes—
93C22×C22⋊C4C2311yescharacteristic
94C22×C22⋊C4C2311yescharacteristic
95C22×C42—28no—
96C22×C42—42no—
97C22×C42C2311yescharacteristic
98C22×C42C2311yescharacteristic
99C22×C42C2311yescharacteristic
100C22×C42C2311yescharacteristic
101C22×C42C2314yes—
102C22×C42C2312yes—
103C22×C42C2312yes—
104#398—216no—
105#398—216no—
106#389—28no—
107#389—28no—
108#389—216no—
109#389D4116yes—
110#388—216no—
111#387—216no—
112#387—216no—
113#385—216no—
114#385—28no—
115#385—216no—
116#384—28no—
117#384—216no—
118#382—28no—
119#382—42no—
120#380—28no—
121#380—28no—
122C4×C4⋊C4—28no—
123C4×C4⋊C4—28no—
124C4×C22⋊C4—42no—
125C4×C22⋊C4Q818yes—
126C2×#95C2312yes—
127C2×#95C2312yes—
128C2×#95C2314yes—
129C2×#95C2314yes—
130C2×#95C2312yes—
131C2×#95C2312yes—
132C2×#95C2314yes—
133C2×#95C2314yes—
134C2×#95C2314yes—
135C2×#95C2314yes—
136C2×#95C2314yes—
137C2×#95C2314yes—
138C2×#95C2314yes—
139C2×#95C2312yes—
140C2×#95C2312yes—
141C2×#95C2314yes—
142C2×#95C2314yes—
143C2×#95C2314yes—
144C2×#95C2314yes—
145C2×#95C2314yes—
146C2×#95C2314yes—
147C2×#95C2312yes—
148C2×#95C2314yes—
149C2×#95C2314yes—
150C2×#95C2312yes—
151C25C2411yescharacteristic
152C23×C4—24no—
153C23×C4—22no—
154C23×C4—22no—
155C23×C4—24no—
156C23×C4—24no—
157C23×C4—24no—
158C23×C4—24no—
159C23×C4—42no—
160C23×C4—42no—
161C23×C4—42no—
162C23×C4—42no—
163C23×C4C2×Q814yes—
164C23×C4C2414yes—
165C23×C4C2412yes—
166C23×C4C2414yes—
167C23×C4C2412yes—
168C23×C4C2414yes—
169C23×C4C2414yes—
170C23×C4C2412yes—
171C23×C4C2412yes—
172C23×C4C2412yes—
173C23×C4C2411yescharacteristic
174C23×C4C2412yes—
175C23×C4C2411yescharacteristic
176#125—432no—
177#125—416no—
178C2×C4⋊C4—24no—
179C2×C4⋊C4—24no—
180C2×C4⋊C4—24no—
181C2×C4⋊C4—24no—
182C2×C4⋊C4—24no—
183C2×C4⋊C4—24no—
184C2×C4⋊C4—28no—
185C2×C4⋊C4—28no—
186C2×C4⋊C4—28no—
187C2×C4⋊C4—28no—
188C2×C4⋊C4—28no—
189C2×C4⋊C4—28no—
190C2×C4⋊C4—24no—
191C2×C4⋊C4—24no—
192C2×C4⋊C4—24no—
193C2×C4⋊C4—28no—
194C2×C4⋊C4—28no—
195C2×C4⋊C4—24no—
196C2×C4⋊C4—24no—
197C2×C4⋊C4—28no—
198C2×C4⋊C4—24no—
199C2×C4⋊C4—24no—
200C2×C4⋊C4—24no—
201C2×C4⋊C4—28no—
202C2×C4⋊C4—28no—
203C2×C4⋊C4—44no—
204C2×C4⋊C4—44no—
205C2×C4⋊C4—42no—
206C2×C4⋊C4—44no—
207C2×C4⋊C4—42no—
208C2×C4⋊C4—44no—
209C2×C4⋊C4—44no—
210C2×C4⋊C4—44no—
211C2×C4⋊C4—44no—
212C2×C4⋊C4—44no—
213C2×C4⋊C4—44no—
214C2×C4⋊C4—44no—
215C2×C4⋊C4—44no—
216C2×C4⋊C4—44no—
217C2×C4⋊C4C2×D418yes—
218C2×C4⋊C4C2×D418yes—
219C2×C4⋊C4C2×D4116yes—
220C2×C22⋊C4—24no—
221C2×C22⋊C4—28no—
222C2×C22⋊C4—28no—
223C2×C22⋊C4—28no—
224C2×C22⋊C4—28no—
225C2×C22⋊C4—42no—
226C2×C22⋊C4—44no—
227C2×C22⋊C4—44no—
228C2×C22⋊C4—42no—
229C2×C22⋊C4—42no—
230C2×C22⋊C4C2×Q814yes—
231C2×C22⋊C4C2×Q814yes—
232C2×C22⋊C4#4118yes—
233C2×C42—24no—
234C2×C42—24no—
235C2×C42—24no—
236C2×C42—24no—
237C2×C42—24no—
238C2×C42—28no—
239C2×C42—24no—
240C2×C42—24no—
241C2×C42—28no—
242C2×C42—22no—
243C2×C42—22no—
244C2×C42—24no—
245C2×C42—416no—
246C2×C42—41no—
247C2×C42—41no—
248C2×C42—44no—
249C2×C42C2×D418yes—
250C2×C42C2×Q818yes—
251#95—28no—
252#95—24no—
253#95—24no—
254#95—28no—
255#95—28no—
256#95—28no—
257#95—28no—
258#95—28no—
259#95—28no—
260#95—24no—
261#95—24no—
262#95—28no—
263#95—28no—
264#95—42no—
265#95—44no—
266#95—44no—
267#95—44no—
268#95—44no—
269#95—42no—
270#95—44no—
271#95—44no—
272#95—42no—
273#95—44no—
274#95C2×D4116yes—
275#95C2×Q818yes—
276C24—22no—
277C24—21no—
278C24—21no—
279C24—22no—
280C24—22no—
281C24—22no—
282C24—22no—
283C24C22×Q812yes—
284C24C2×#4114yes—
285C24C2511yescharacteristic
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—22no—
294C22×C4—24no—
295C22×C4—24no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—24no—
299C22×C4—21no—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—24no—
303C22×C4—22no—
304C22×C4—24no—
305C22×C4—24no—
306C22×C4—21no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—24no—
311C22×C4—24no—
312C22×C4—24no—
313C22×C4—24no—
314C22×C4—24no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—24no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—28no—
325C22×C4—216no—
326C22×C4—24no—
327C22×C4—22no—
328C22×C4—24no—
329C22×C4—22no—
330C22×C4—24no—
331C22×C4—24no—
332C22×C4—22no—
333C22×C4—24no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—24no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—24no—
341C22×C4—24no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—24no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—24no—
348C22×C4—24no—
349C22×C4—24no—
350C22×C4—24no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—24no—
354C22×C4—48no—
355C22×C4—48no—
356C22×C4—48no—
357C22×C4—48no—
358C22×C4—48no—
359C22×C4—48no—
360C22×C4—48no—
361C22×C4—48no—
362C22×C4—44no—
363C22×C4—44no—
364C22×C4C22×D418yes—
365C22×C4C22×D414yes—
366C22×C4C22×D414yes—
367C22×C4C22×D418yes—
368C22×C4C22×D414yes—
369C22×C4C22×Q812yes—
370C22×C4C22×Q814yes—
371C22×C4C22×Q814yes—
372C22×C4C22×Q812yes—
373C22×C4C2×#4114yes—
374C22×C4C2×#4114yes—
375C22×C4H(5,𝔽2)12yes—
376C22×C42−1+412yes—
377C22×C42−1+414yes—
378C4⋊C4—216no—
379C4⋊C4—216no—
380C4⋊C4—416no—
381C4⋊C4—48no—
382C4⋊C4—416no—
383C4⋊C4—48no—
384C4⋊C4—416no—
385C4⋊C4—416no—
386C4⋊C4—416no—
387C4⋊C4—416no—
388C4⋊C4—48no—
389C4⋊C4—48no—
390C4⋊C4—88no—
391C4⋊C4—88no—
392C22⋊C4—44no—
393C22⋊C4—416no—
394C22⋊C4—416no—
395C22⋊C4—48no—
396C22⋊C4#125116yescomplemented
397C42—216no—
398C42—216no—
399C42—416no—
400C42—44no—
401C42—44no—
402C42—44no—
403C42—44no—
404C23—22no—
405C23—22no—
406C23—24no—
407C23—24no—
408C23—24no—
409C23—24no—
410C23—24no—
411C23—24no—
412C23—24no—
413C23—24no—
414C23—42no—
415C23—41no—
416C23—42no—
417C23—42no—
418C23—41no—
419C23—42no—
420C23—41no—
421C23—41no—
422C23—42no—
423C23—42no—
424C23C2×#12514yes—
425C23C23×D412yes—
426C23C23×Q811yescharacteristic
427C23C22×#4112yes—
428C23C2×H(5,𝔽2)11yescharacteristic
429C23C2×2−1+411yescharacteristic
430C23C2×2−1+412yes—
431C23#58512yes—
432C23#58512yes—
433C23#58512yes—
434C2×C4—24no—
435C2×C4—24no—
436C2×C4—24no—
437C2×C4—24no—
438C2×C4—24no—
439C2×C4—24no—
440C2×C4—28no—
441C2×C4—22no—
442C2×C4—24no—
443C2×C4—24no—
444C2×C4—24no—
445C2×C4—28no—
446C2×C4—24no—
447C2×C4—24no—
448C2×C4—28no—
449C2×C4—28no—
450C2×C4—24no—
451C2×C4—24no—
452C2×C4—24no—
453C2×C4—28no—
454C2×C4—28no—
455C2×C4—24no—
456C2×C4—24no—
457C2×C4—432no—
458C2×C4—42no—
459C2×C4—41no—
460C2×C4—44no—
461C2×C4—42no—
462C2×C4—42no—
463C2×C4—42no—
464C2×C4—42no—
465C2×C4—42no—
466C2×C4—44no—
467C2×C4—42no—
468C2×C4—44no—
469C2×C4—42no—
470C2×C4—42no—
471C2×C4—44no—
472C2×C4—44no—
473C2×C4—41no—
474C2×C4—44no—
475C2×C4—42no—
476C2×C4—42no—
477C2×C4—44no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—44no—
481C2×C4—416no—
482C2×C4—44no—
483C2×C4—42no—
484C2×C4—44no—
485C2×C4—44no—
486C2×C4—44no—
487C2×C4—42no—
488C2×C4—44no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—44no—
492C2×C4—44no—
493C2×C4—44no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—44no—
498C2×C4—44no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—44no—
502C2×C4—42no—
503C2×C4—44no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—44no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—44no—
510C2×C4C2×#12518yes—
511C2×C4#53618yes—
512C2×C4#54414yes—
513C2×C4D4214yes—
514C2×C4D4×Q818yes—
515C2×C4#56314yes—
516C22—28no—
517C22—24no—
518C22—42no—
519C22—44no—
520C22—44no—
521C22—42no—
522C22—44no—
523C22—44no—
524C22—44no—
525C22C22×#12511yescharacteristic
526C22C2×#53612yes—
527C22C2×#54411yescharacteristic
528C22#345811yescharacteristic
529C22C2×D4211yescharacteristic
530C22C2×D4×Q812yes—
531C22D4×#4112yes—
532C22D4×#4112yes—
533C22#348512yes—
534C22#348712yes—
535C22#350412yes—
536C22#350612yes—
537C22#350712yes—
538C22#351512yes—
539C22#351512yes—
540C22C2×#56311yescharacteristic
541C22#353912yes—
542C22#355711yescharacteristic
543C22#355711yescharacteristic
544C22#356512yes—
545C22#356512yes—
546C4—28no—
547C4—44no—
548C4—48no—
549C4—44no—
550C4—44no—
551C4—42no—
552C4—48no—
553C4—82no—
554C4—84no—
555C4—82no—
556C4—84no—
557C4#267718yes—
558C2—44no—
559C2#3518111yescharacteristic
560C2D4×#12512yes—
561C2#3742711yescharacteristic
562C2#3769112yes—
563C2#3985912yes—
564C2#3988012yes—
565C2#4570912yes—
566C2C2×#267711yescharacteristic
567C2#5899912yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC25C24
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2552C2semidirect
4. #15401C2semidirect
5. #15388C2semidirect
6. #15228C2semidirect
8. #15221C2semidirect
11. #13982C2semidirect
396. C22⋊C4#125semidirect

Maximal subgroups

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

subgroupindex
C2×#28512
C2×#25522
#163432
#154012
#153882
#152282
#152232
#152212
#152162
#144052
#139822
#138762
C4×#3852