Finite Group Databasebeta

← #947940

Subgroups — #947940

Order 512.

2980 conjugacy classes of subgroups.

The subgroup lattice has 578 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2432C212yesmaximal
2#17193C212yesmaximal, complemented
3#17118C212yesmaximal, complemented
4#16681C214yesmaximal, complemented
5#16527C214yesmaximal, complemented
6#16110C211yesmaximal, characteristic, complemented
7#16093C212yesmaximal, complemented
8#15762C211yesmaximal, characteristic, complemented
9#15204C212yesmaximal, complemented
10#14763C214yesmaximal, complemented
11#14313C212yesmaximal, complemented
12#14202C214yesmaximal, complemented
13#13851C211yesmaximal, characteristic
14C23×#41C2211yescharacteristic
15C22×C4⋊Q8C2212yes—
16C22×#124C2212yes—
17C22×C22⋊Q8C2212yes—
18#2432—216no—
19C2×#397C2214yes—
20C2×#397C2212yes—
21C2×#395C2214yes—
22C2×#394C2214yes—
23C2×#394C2214yes—
24C2×#391C2212yes—
25C2×#390C2212yes—
26C2×#389C2212yes—
27C2×#389C2212yes—
28C2×#387C2214yes—
29C2×#385C2214yes—
30C2×#385C2214yes—
31C2×C4×C22⋊C4C2212yes—
32C2×C4×C22⋊C4C2211yescharacteristic
33C2×C4×C22⋊C4C2214yes—
34C2×C4×C22⋊C4C2211yescharacteristic
35#1446C2211yescharacteristic
36#1446C2211yescharacteristic
37#1446C2214yes—
38#1445C2212yes—
39#1444C2212yes—
40#1444C2214yes—
41#1444C2212yes—
42#1442C2212yes—
43#1442C2214yes—
44#1441C2214yes—
45#1441C2214yes—
46#1440C2214yes—
47#1440C2214yes—
48#1440C2214yes—
49#1439C2212yes—
50#1439C2212yes—
51#1439C2214yes—
52#1439C2214yes—
53#1439C2214yes—
54#1439C2214yes—
55#1439C2214yes—
56#1439C2212yes—
57#1439C2214yes—
58#1438C2214yes—
59#1438C2211yescharacteristic
60#1438C2211yescharacteristic
61#1436C2214yes—
62#1436C2214yes—
63#1436C2214yes—
64#1436C2214yes—
65C4⋊#95C2212yes—
66#1434C2214yes—
67#1434C2212yes—
68C22×#41—42no—
69C22×#41—42no—
70C22×#41—48no—
71C22×#41D418yes—
72C23×Q8C2311yescharacteristic
73C23×Q8C2312yes—
74C23×D4C2312yes—
75C23×D4C2311yescharacteristic
76C24×C4C2312yes—
77C24×C4C2311yescharacteristic
78C2×C4⋊Q8—28no—
79C2×C4⋊Q8—48no—
80C2×#124—28no—
81C2×#124—48no—
82C2×C22⋊Q8—28no—
83C2×C22⋊Q8—44no—
84C2×C22⋊Q8—44no—
85C22×C4⋊C4C2314yes—
86C22×C4⋊C4C2314yes—
87C22×C4⋊C4C2314yes—
88C22×C4⋊C4C2314yes—
89C22×C4⋊C4C2312yes—
90C22×C4⋊C4C2312yes—
91C22×C4⋊C4C2312yes—
92C22×C4⋊C4C2312yes—
93C22×C4⋊C4C2312yes—
94C22×C22⋊C4C2314yes—
95C22×C22⋊C4C2314yes—
96C22×C22⋊C4C2314yes—
97C22×C22⋊C4C2314yes—
98C22×C22⋊C4C2312yes—
99C22×C22⋊C4C2314yes—
100C22×C22⋊C4C2314yes—
101C22×C22⋊C4C2314yes—
102C22×C22⋊C4C2312yes—
103C22×C22⋊C4C2314yes—
104C22×C42C2312yes—
105C22×C42C2312yes—
106C22×C42C2312yes—
107C22×C42C2312yes—
108#397—216no—
109#397—28no—
110#395—48no—
111#394—216no—
112#394—216no—
113#391—28no—
114#390—28no—
115#389—44no—
116#389—44no—
117#387—48no—
118#385—216no—
119#385—216no—
120C4×C22⋊C4—28no—
121C4×C22⋊C4—48no—
122C4×C22⋊C4—42no—
123C4×C22⋊C4—42no—
124C2×#95C2314yes—
125C2×#95C2314yes—
126C2×#95C2314yes—
127C2×#95C2314yes—
128C2×#95C2314yes—
129C2×#95C2314yes—
130C2×#95C2312yes—
131C2×#95C2314yes—
132C2×#95C2312yes—
133C2×#95C2314yes—
134C2×#95C2314yes—
135C2×#95C2314yes—
136C2×#95C2312yes—
137C2×#95C2314yes—
138C2×#95C2314yes—
139C2×#95C2314yes—
140C2×#95C2314yes—
141C2×#95C2311yescharacteristic
142C2×#95C2311yescharacteristic
143C2×#95C2311yescharacteristic
144C2×#95C2311yescharacteristic
145C2×#95C2314yes—
146C2×#95C2314yes—
147C2×#95C2312yes—
148C25C2412yes—
149C25C2411yescharacteristic
150C2×#41—416no—
151C2×#41—432no—
152C2×#41—432no—
153C2×#41—432no—
154C22×Q8—24no—
155C22×Q8—28no—
156C22×Q8—28no—
157C22×Q8—41no—
158C22×Q8—41no—
159C22×Q8—42no—
160C22×Q8—42no—
161C22×Q8—44no—
162C22×Q8C2×D414yes—
163C22×D4—28no—
164C22×D4—44no—
165C22×D4—42no—
166C22×D4—42no—
167C22×D4—41no—
168C22×D4—41no—
169C22×D4—44no—
170C22×D4C2×D418yes—
171C22×D4C2×D414yes—
172C23×C4—24no—
173C23×C4—24no—
174C23×C4—28no—
175C23×C4—22no—
176C23×C4—22no—
177C23×C4—44no—
178C23×C4—44no—
179C23×C4C2×D418yes—
180C23×C4C2×D414yes—
181C23×C4C2414yes—
182C23×C4C2414yes—
183C23×C4C2414yes—
184C23×C4C2412yes—
185C23×C4C2411yescharacteristic
186C23×C4C2414yes—
187C23×C4C2412yes—
188C23×C4C2412yes—
189C23×C4C2412yes—
190C23×C4C2411yescharacteristic
191C23×C4C2412yes—
192C4⋊Q8—432no—
193#124—432no—
194C22⋊Q8—432no—
195C2×C4⋊C4—28no—
196C2×C4⋊C4—24no—
197C2×C4⋊C4—28no—
198C2×C4⋊C4—28no—
199C2×C4⋊C4—28no—
200C2×C4⋊C4—24no—
201C2×C4⋊C4—28no—
202C2×C4⋊C4—28no—
203C2×C4⋊C4—28no—
204C2×C4⋊C4—28no—
205C2×C4⋊C4—28no—
206C2×C4⋊C4—42no—
207C2×C4⋊C4—44no—
208C2×C4⋊C4—42no—
209C2×C4⋊C4—42no—
210C2×C4⋊C4—44no—
211C2×C4⋊C4—44no—
212C2×C4⋊C4—44no—
213C2×C4⋊C4—42no—
214C2×C4⋊C4—42no—
215C2×C4⋊C4—42no—
216C2×C4⋊C4—42no—
217C2×C4⋊C4—44no—
218C2×C4⋊C4—44no—
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—28no—
226C2×C22⋊C4—28no—
227C2×C22⋊C4—28no—
228C2×C22⋊C4—28no—
229C2×C22⋊C4—24no—
230C2×C22⋊C4—24no—
231C2×C22⋊C4—28no—
232C2×C22⋊C4—24no—
233C2×C22⋊C4—28no—
234C2×C22⋊C4—44no—
235C2×C22⋊C4—44no—
236C2×C22⋊C4—44no—
237C2×C22⋊C4—42no—
238C2×C22⋊C4—44no—
239C2×C22⋊C4—44no—
240C2×C22⋊C4—44no—
241C2×C22⋊C4—44no—
242C2×C22⋊C4—44no—
243C2×C22⋊C4—44no—
244C2×C22⋊C4—44no—
245C2×C22⋊C4—44no—
246C2×C22⋊C4—42no—
247C2×C22⋊C4—44no—
248C2×C22⋊C4—42no—
249C2×C42—24no—
250C2×C42—24no—
251C2×C42—28no—
252C2×C42—24no—
253C2×C42—24no—
254C2×C42—44no—
255C2×C42—44no—
256C2×C42—44no—
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—24no—
266#95—24no—
267#95—42no—
268#95—44no—
269#95—41no—
270#95—41no—
271#95—41no—
272#95—41no—
273#95—44no—
274#95—44no—
275#95—44no—
276#95—44no—
277#95—44no—
278#95—44no—
279#95—44no—
280#95#4118yescomplemented
281C24—22no—
282C24—24no—
283C24—22no—
284C24—22no—
285C24—21no—
286C24—21no—
287C24—24no—
288C24—24no—
289C24—24no—
290C24—22no—
291C24—24no—
292C24—24no—
293C24—22no—
294C24—22no—
295C24—24no—
296C24C22×D414yes—
297C24C22×D412yes—
298C24C2511yescharacteristic
299#41—4128no—
300C2×Q8—216no—
301C2×Q8—44no—
302C2×Q8—48no—
303C2×Q8—48no—
304C2×Q8—48no—
305C2×Q8—48no—
306C2×Q8—48no—
307C2×Q8—88no—
308C2×Q8—88no—
309C2×D4—216no—
310C2×D4—48no—
311C2×D4—48no—
312C2×D4—48no—
313C2×D4—44no—
314C2×D4—416no—
315C2×D4—416no—
316C2×D4—48no—
317C2×D4—48no—
318C22×C4—24no—
319C22×C4—24no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—24no—
323C22×C4—24no—
324C22×C4—24no—
325C22×C4—24no—
326C22×C4—24no—
327C22×C4—22no—
328C22×C4—24no—
329C22×C4—21no—
330C22×C4—21no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—24no—
334C22×C4—24no—
335C22×C4—24no—
336C22×C4—24no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—24no—
340C22×C4—22no—
341C22×C4—24no—
342C22×C4—24no—
343C22×C4—216no—
344C22×C4—216no—
345C22×C4—28no—
346C22×C4—24no—
347C22×C4—24no—
348C22×C4—24no—
349C22×C4—22no—
350C22×C4—24no—
351C22×C4—24no—
352C22×C4—24no—
353C22×C4—24no—
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—22no—
371C22×C4—24no—
372C22×C4—48no—
373C22×C4—48no—
374C22×C4—416no—
375C22×C4—416no—
376C22×C4—48no—
377C22×C4—48no—
378C22×C4C22×D412yes—
379C22×C4C22×D412yes—
380C22×C4C22×D414yes—
381C22×C4C2×#4114yes—
382C22×C4C2×#4118yes—
383C22×C4H(5,𝔽2)18yes—
384C22×C4H(5,𝔽2)12yes—
385C22×C4H(5,𝔽2)12yes—
386C22×C42−1+414yes—
387C22×C42−1+414yes—
388C4⋊C4—416no—
389C4⋊C4—48no—
390C4⋊C4—48no—
391C4⋊C4—416no—
392C4⋊C4—48no—
393C4⋊C4—416no—
394C4⋊C4—416no—
395C4⋊C4—84no—
396C4⋊C4—84no—
397C22⋊C4—416no—
398C22⋊C4—48no—
399C22⋊C4—416no—
400C22⋊C4—416no—
401C22⋊C4—416no—
402C22⋊C4—48no—
403C22⋊C4—416no—
404C22⋊C4—416no—
405C22⋊C4—416no—
406C22⋊C4—416no—
407C42—48no—
408C42—48no—
409C42—84no—
410C42—84no—
411C23—24no—
412C23—24no—
413C23—24no—
414C23—22no—
415C23—24no—
416C23—24no—
417C23—28no—
418C23—28no—
419C23—24no—
420C23—42no—
421C23—44no—
422C23—42no—
423C23—41no—
424C23—44no—
425C23—44no—
426C23—44no—
427C23—44no—
428C23—44no—
429C23—44no—
430C23—42no—
431C23—41no—
432C23—44no—
433C23—42no—
434C23—44no—
435C23—42no—
436C23—42no—
437C23—44no—
438C23—44no—
439C23—44no—
440C23—44no—
441C23—42no—
442C23—41no—
443C23—42no—
444C23—41no—
445C23—44no—
446C23—44no—
447C23—44no—
448C23C23×D411yescharacteristic
449C23C22×#4112yes—
450C23C2×H(5,𝔽2)11yescharacteristic
451C23C2×H(5,𝔽2)11yescharacteristic
452C23C2×H(5,𝔽2)14yes—
453C23C2×2−1+412yes—
454C23C2×2−1+412yes—
455C23#58512yes—
456Q8—416no—
457Q8—816no—
458D4—432no—
459D4—416no—
460C2×C4—24no—
461C2×C4—24no—
462C2×C4—24no—
463C2×C4—24no—
464C2×C4—24no—
465C2×C4—28no—
466C2×C4—28no—
467C2×C4—24no—
468C2×C4—22no—
469C2×C4—24no—
470C2×C4—28no—
471C2×C4—28no—
472C2×C4—28no—
473C2×C4—28no—
474C2×C4—28no—
475C2×C4—24no—
476C2×C4—24no—
477C2×C4—28no—
478C2×C4—432no—
479C2×C4—416no—
480C2×C4—44no—
481C2×C4—42no—
482C2×C4—44no—
483C2×C4—44no—
484C2×C4—42no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—44no—
491C2×C4—42no—
492C2×C4—44no—
493C2×C4—44no—
494C2×C4—44no—
495C2×C4—44no—
496C2×C4—44no—
497C2×C4—44no—
498C2×C4—44no—
499C2×C4—44no—
500C2×C4—42no—
501C2×C4—44no—
502C2×C4—44no—
503C2×C4—44no—
504C2×C4—44no—
505C2×C4—44no—
506C2×C4—44no—
507C2×C4—44no—
508C2×C4—44no—
509C2×C4—44no—
510C2×C4—44no—
511C2×C4—44no—
512C2×C4—42no—
513C2×C4—44no—
514C2×C4—44no—
515C2×C4—44no—
516C2×C4—44no—
517C2×C4—44no—
518C2×C4—44no—
519C2×C4—42no—
520C2×C4—44no—
521C2×C4#53418yes—
522C2×C4#53414yes—
523C22—44no—
524C22—42no—
525C22—48no—
526C22—48no—
527C22—48no—
528C22—44no—
529C22—42no—
530C22—44no—
531C22—42no—
532C22—82no—
533C22—84no—
534C22—84no—
535C22—82no—
536C22—82no—
537C22—84no—
538C22—82no—
539C22—84no—
540C22C2×#53411yescharacteristic
541C22C2×#53412yes—
542C22#346012yes—
543C22D4×#4112yes—
544C22#348311yescharacteristic
545C22#349912yes—
546C22#351114yes—
547C22#352014yes—
548C22#352911yescharacteristic
549C22#353212yes—
550C22#353212yes—
551C22#353812yes—
552C22#354114yes—
553C22#355011yescharacteristic
554C22#355011yescharacteristic
555C22#355414yes—
556C4—24no—
557C4—28no—
558C4—48no—
559C4—42no—
560C4—44no—
561C4—44no—
562C4—44no—
563C4—84no—
564C4—82no—
565C4—84no—
566C4—84no—
567C2—84no—
568C2—82no—
569C2#3738611yescharacteristic
570C2#3743611yescharacteristic
571C2#3995614yes—
572C2#4117412yes—
573C2#4204312yes—
574C2#4437712yes—
575C2#5216711yescharacteristic
576C2#5240612yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC23×#41C22
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #17193C2semidirect
3. #17118C2semidirect
4. #16681C2semidirect
5. #16527C2semidirect
6. #16110C2semidirect
7. #16093C2semidirect
8. #15762C2semidirect
9. #15204C2semidirect
10. #14763C2semidirect
11. #14313C2semidirect
12. #14202C2semidirect
280. #95#41semidirect

Maximal subgroups

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

subgroupindex
C2×#24322
#171932
#171182
#166812
#165272
#161102
#160932
#157622
#152042
#147632
#143132
#142022
#138512