Finite Group Databasebeta

← #4941755

Subgroups — #4941755

Order 512.

3522 conjugacy classes of subgroups.

The subgroup lattice has 594 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#563C211yesmaximal, characteristic
2C2×#2734C212yesmaximal, complemented
3C2×#2545C211yesmaximal, characteristic
4#16659C214yesmaximal, complemented
5#16608C212yesmaximal, complemented
6#15983C212yesmaximal
7#15279C214yesmaximal, complemented
8#15276C214yesmaximal, complemented
9#15262C212yesmaximal
10#15071C212yesmaximal, complemented
11#14931C214yesmaximal, complemented
12#14492C211yesmaximal, characteristic
13#14024C212yesmaximal, complemented
14C2×#563—216no—
15C2×#563—28no—
16C22×C4⋊Q8C2211yescharacteristic
17C22×C4⋊Q8C2212yes—
18C22×#126C2214yes—
19C22×#125C2212yes—
20C22×#124C2211yescharacteristic
21C22×#124C2211yescharacteristic
22C22×#123C2212yes—
23C22×C22⋊Q8C2214yes—
24C22×C4×Q8C2211yescharacteristic
25#2734—216no—
26#2545—28no—
27C2×#398C2214yes—
28C2×#396C2214yes—
29C2×#395C2214yes—
30C2×#395C2214yes—
31C2×#391C2211yescharacteristic
32C2×#391C2212yes—
33C2×#391C2212yes—
34C2×#391C2212yes—
35C2×#390C2211yescharacteristic
36C2×#389C2214yes—
37C2×#388C2214yes—
38C2×#387C2214yes—
39C2×#387C2214yes—
40C2×#385C2212yes—
41C2×#385C2212yes—
42C2×C4×C4⋊C4C2211yescharacteristic
43#1444C2214yes—
44#1444C2212yes—
45#1444C2214yes—
46#1442C2214yes—
47#1442C2214yes—
48#1442C2212yes—
49#1441C2212yes—
50#1441C2214yes—
51#1441C2214yes—
52#1441C2214yes—
53#1441C2214yes—
54#1441C2212yes—
55#1440C2212yes—
56#1440C2214yes—
57#1440C2212yes—
58#1439C2214yes—
59#1439C2214yes—
60#1439C2214yes—
61#1439C2214yes—
62#1439C2214yes—
63#1438C2212yes—
64#1437C2214yes—
65#1436C2214yes—
66#1436C2214yes—
67C4⋊#95C2212yes—
68#1434C2212yes—
69C22⋊#95C2214yes—
70C4×#95C2212yes—
71C23×Q8C2311yescharacteristic
72C23×Q8C2312yes—
73C23×Q8C2311yescharacteristic
74C23×D4C2311yescharacteristic
75C24×C4C2312yes—
76#563—2128no—
77C2×C4⋊Q8—28no—
78C2×C4⋊Q8—28no—
79C2×C4⋊Q8—28no—
80C2×C4⋊Q8—24no—
81C2×C4⋊Q8D4116yes—
82C2×#126—216no—
83C2×#126—216no—
84C2×#126—216no—
85C2×#125—216no—
86C2×#125—28no—
87C2×#124—28no—
88C2×#124—24no—
89C2×#124—42no—
90C2×#124—42no—
91C2×#124D418yes—
92C2×#123—28no—
93C2×#123—216no—
94C2×C22⋊Q8—216no—
95C2×C22⋊Q8—216no—
96C2×C22⋊Q8—216no—
97C2×C4×Q8—24no—
98C2×C4×Q8—42no—
99C2×C4×Q8—42no—
100C22×C4⋊C4C2314yes—
101C22×C4⋊C4C2312yes—
102C22×C4⋊C4C2314yes—
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2314yes—
105C22×C4⋊C4C2312yes—
106C22×C4⋊C4C2312yes—
107C22×C4⋊C4C2314yes—
108C22×C4⋊C4C2314yes—
109C22×C4⋊C4C2314yes—
110C22×C4⋊C4C2314yes—
111C22×C4⋊C4C2312yes—
112C22×C4⋊C4C2311yescharacteristic
113C22×C22⋊C4C2314yes—
114C22×C22⋊C4C2312yes—
115C22×C22⋊C4C2312yes—
116C22×C22⋊C4C2314yes—
117C22×C22⋊C4C2312yes—
118C22×C22⋊C4C2312yes—
119C22×C22⋊C4C2314yes—
120C22×C22⋊C4C2314yes—
121C22×C22⋊C4C2312yes—
122C22×C42C2311yescharacteristic
123C22×C42C2311yescharacteristic
124C22×C42C2312yes—
125C22×C42C2311yescharacteristic
126C22×C42C2312yes—
127C22×C42C2312yes—
128#398—216no—
129#396—216no—
130#395—216no—
131#395—216no—
132#391—28no—
133#391—24no—
134#391—44no—
135#391—44no—
136#390—24no—
137#389—216no—
138#388—216no—
139#387—216no—
140#387—216no—
141#385—44no—
142#385—44no—
143C4×C4⋊C4—24no—
144C2×#95C2314yes—
145C2×#95C2314yes—
146C2×#95C2314yes—
147C2×#95C2314yes—
148C2×#95C2314yes—
149C2×#95C2314yes—
150C2×#95C2314yes—
151C2×#95C2314yes—
152C2×#95C2314yes—
153C2×#95C2314yes—
154C2×#95C2314yes—
155C2×#95C2314yes—
156C2×#95C2312yes—
157C2×#95C2314yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2312yes—
161C2×#95C2312yes—
162C2×#95C2314yes—
163C2×#95C2314yes—
164C25C2412yes—
165C22×Q8—24no—
166C22×Q8—24no—
167C22×Q8—22no—
168C22×Q8—24no—
169C22×Q8—41no—
170C22×Q8—41no—
171C22×Q8—42no—
172C22×Q8—41no—
173C22×Q8—41no—
174C22×Q8—42no—
175C22×Q8—42no—
176C22×Q8—42no—
177C22×Q8—42no—
178C22×Q8—42no—
179C22×Q8C2×D418yes—
180C22×Q8C2×D414yes—
181C22×Q8C2×D418yes—
182C22×D4—24no—
183C22×D4—41no—
184C22×D4—41no—
185C22×D4—42no—
186C22×D4C2×D414yes—
187C23×C4—24no—
188C23×C4—28no—
189C23×C4—44no—
190C23×C4—42no—
191C23×C4—42no—
192C23×C4C2412yes—
193C23×C4C2414yes—
194C23×C4C2412yes—
195C23×C4C2412yes—
196C23×C4C2414yes—
197C23×C4C2414yes—
198C23×C4C2411yescharacteristic
199C23×C4C2411yescharacteristic
200C23×C4C2412yes—
201C23×C4C2414yes—
202C23×C4C2412yes—
203C23×C4C2411yescharacteristic
204C4⋊Q8—264no—
205C4⋊Q8—416no—
206#126—2128no—
207#125—264no—
208#124—232no—
209#124—416no—
210#123—264no—
211C22⋊Q8—2128no—
212C4×Q8—416no—
213C2×C4⋊C4—28no—
214C2×C4⋊C4—24no—
215C2×C4⋊C4—28no—
216C2×C4⋊C4—28no—
217C2×C4⋊C4—28no—
218C2×C4⋊C4—28no—
219C2×C4⋊C4—28no—
220C2×C4⋊C4—28no—
221C2×C4⋊C4—28no—
222C2×C4⋊C4—28no—
223C2×C4⋊C4—24no—
224C2×C4⋊C4—28no—
225C2×C4⋊C4—28no—
226C2×C4⋊C4—28no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—24no—
229C2×C4⋊C4—28no—
230C2×C4⋊C4—28no—
231C2×C4⋊C4—28no—
232C2×C4⋊C4—24no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—28no—
236C2×C4⋊C4—24no—
237C2×C4⋊C4—28no—
238C2×C4⋊C4—42no—
239C2×C4⋊C4—42no—
240C2×C4⋊C4—41no—
241C2×C4⋊C4—41no—
242C2×C4⋊C4—42no—
243C2×C4⋊C4—44no—
244C2×C4⋊C4—44no—
245C2×C4⋊C4—44no—
246C2×C4⋊C4—44no—
247C2×C4⋊C4—42no—
248C2×C4⋊C4C2×D4116yes—
249C2×C4⋊C4C2×D4116yes—
250C2×C22⋊C4—28no—
251C2×C22⋊C4—24no—
252C2×C22⋊C4—28no—
253C2×C22⋊C4—28no—
254C2×C22⋊C4—28no—
255C2×C22⋊C4—28no—
256C2×C22⋊C4—28no—
257C2×C22⋊C4—24no—
258C2×C22⋊C4—24no—
259C2×C22⋊C4—28no—
260C2×C22⋊C4—24no—
261C2×C22⋊C4—24no—
262C2×C22⋊C4—28no—
263C2×C22⋊C4—28no—
264C2×C22⋊C4—28no—
265C2×C22⋊C4—44no—
266C2×C22⋊C4—44no—
267C2×C22⋊C4—42no—
268C2×C22⋊C4—42no—
269C2×C22⋊C4—42no—
270C2×C22⋊C4—42no—
271C2×C22⋊C4—44no—
272C2×C22⋊C4—44no—
273C2×C22⋊C4C2×D4116yes—
274C2×C42—22no—
275C2×C42—24no—
276C2×C42—24no—
277C2×C42—24no—
278C2×C42—22no—
279C2×C42—22no—
280C2×C42—24no—
281C2×C42—22no—
282C2×C42—24no—
283C2×C42—24no—
284C2×C42—24no—
285C2×C42—42no—
286C2×C42—42no—
287C2×C42—42no—
288C2×C42C2×D414yes—
289C2×C42C2×D418yes—
290#95—24no—
291#95—28no—
292#95—28no—
293#95—28no—
294#95—28no—
295#95—28no—
296#95—28no—
297#95—28no—
298#95—28no—
299#95—28no—
300#95—28no—
301#95—28no—
302#95—28no—
303#95—44no—
304#95—44no—
305#95—44no—
306#95—42no—
307#95—42no—
308#95—44no—
309#95—44no—
310C24—24no—
311C24—22no—
312C24—22no—
313C24—24no—
314C24—24no—
315C24—24no—
316C24—24no—
317C24—24no—
318C24C22×D414yes—
319C24C2511yescharacteristic
320C2×Q8—216no—
321C2×Q8—44no—
322C2×Q8—44no—
323C2×Q8—48no—
324C2×Q8—48no—
325C2×Q8—48no—
326C2×Q8—48no—
327C2×Q8—44no—
328C2×Q8—44no—
329C2×Q8—44no—
330C2×Q8—84no—
331C2×Q8—84no—
332C2×Q8—84no—
333C2×Q8—84no—
334C2×Q8—84no—
335C2×Q8—84no—
336C2×Q8#120116yes—
337C2×D4—28no—
338C2×D4—44no—
339C2×D4—44no—
340C2×D4—84no—
341C2×D4—84no—
342C22×C4—22no—
343C22×C4—21no—
344C22×C4—21no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—24no—
350C22×C4—24no—
351C22×C4—21no—
352C22×C4—22no—
353C22×C4—24no—
354C22×C4—24no—
355C22×C4—24no—
356C22×C4—21no—
357C22×C4—24no—
358C22×C4—24no—
359C22×C4—24no—
360C22×C4—24no—
361C22×C4—24no—
362C22×C4—21no—
363C22×C4—21no—
364C22×C4—24no—
365C22×C4—24no—
366C22×C4—24no—
367C22×C4—24no—
368C22×C4—24no—
369C22×C4—24no—
370C22×C4—24no—
371C22×C4—24no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—24no—
375C22×C4—24no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—24no—
379C22×C4—22no—
380C22×C4—24no—
381C22×C4—24no—
382C22×C4—216no—
383C22×C4—24no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—24no—
387C22×C4—24no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—24no—
393C22×C4—24no—
394C22×C4—24no—
395C22×C4—24no—
396C22×C4—24no—
397C22×C4—24no—
398C22×C4—24no—
399C22×C4—24no—
400C22×C4—88no—
401C22×C4—88no—
402C22×C4—84no—
403C22×C4—84no—
404C22×C4C22×D418yes—
405C22×C4C22×D414yes—
406C22×C4C22×D418yes—
407C22×C4C22×D414yes—
408C22×C4C22×D414yes—
409C22×C4C22×D412yes—
410C22×C4C22×D414yes—
411C22×C4C22×D414yes—
412C22×C4H(5,𝔽2)12yes—
413C22×C4H(5,𝔽2)12yes—
414C22×C42−1+418yes—
415C22×C42−1+414yes—
416C4⋊C4—232no—
417C4⋊C4—216no—
418C4⋊C4—232no—
419C4⋊C4—216no—
420C4⋊C4—232no—
421C4⋊C4—232no—
422C4⋊C4—48no—
423C4⋊C4—48no—
424C4⋊C4—44no—
425C4⋊C4—416no—
426C4⋊C4—84no—
427C4⋊C4—88no—
428C4⋊C4—88no—
429C22⋊C4—232no—
430C22⋊C4—216no—
431C22⋊C4—232no—
432C22⋊C4—48no—
433C22⋊C4—416no—
434C22⋊C4—48no—
435C22⋊C4—48no—
436C22⋊C4—48no—
437C22⋊C4—88no—
438C42—28no—
439C42—28no—
440C42—216no—
441C42—48no—
442C42—44no—
443C42—84no—
444C23—24no—
445C23—24no—
446C23—24no—
447C23—28no—
448C23—28no—
449C23—44no—
450C23—44no—
451C23—44no—
452C23—44no—
453C23—44no—
454C23—44no—
455C23—42no—
456C23—42no—
457C23—42no—
458C23—44no—
459C23—42no—
460C23—44no—
461C23—44no—
462C23—44no—
463C23—44no—
464C23—44no—
465C23C23×D411yescharacteristic
466C23C23×D412yes—
467C23C2×H(5,𝔽2)11yescharacteristic
468C23C2×H(5,𝔽2)11yescharacteristic
469C23C2×2−1+412yes—
470C23C2×2−1+414yes—
471C23#58512yes—
472C23#58512yes—
473Q8—416no—
474Q8—816no—
475Q8—88no—
476D4—88no—
477C2×C4—22no—
478C2×C4—24no—
479C2×C4—22no—
480C2×C4—24no—
481C2×C4—24no—
482C2×C4—24no—
483C2×C4—24no—
484C2×C4—24no—
485C2×C4—24no—
486C2×C4—24no—
487C2×C4—24no—
488C2×C4—28no—
489C2×C4—28no—
490C2×C4—22no—
491C2×C4—22no—
492C2×C4—28no—
493C2×C4—28no—
494C2×C4—28no—
495C2×C4—24no—
496C2×C4—24no—
497C2×C4—28no—
498C2×C4—28no—
499C2×C4—28no—
500C2×C4—24no—
501C2×C4—28no—
502C2×C4—24no—
503C2×C4—24no—
504C2×C4—28no—
505C2×C4—42no—
506C2×C4—44no—
507C2×C4—44no—
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—44no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—44no—
522C2×C4—41no—
523C2×C4—42no—
524C2×C4—41no—
525C2×C4—44no—
526C2×C4—44no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—44no—
532C2×C4—44no—
533C2×C4—44no—
534C2×C4—44no—
535C2×C4—44no—
536C2×C4—44no—
537C2×C4—44no—
538C2×C4—42no—
539C2×C4—44no—
540C2×C4—42no—
541C2×C4—44no—
542C2×C4—44no—
543C2×C4—44no—
544C2×C4—816no—
545C2×C4C2×#12014yes—
546C2×C4C2×#12018yes—
547C22—44no—
548C22—44no—
549C22—48no—
550C22—44no—
551C22—48no—
552C22—84no—
553C22—84no—
554C22—84no—
555C22—84no—
556C22C22×#12011yescharacteristic
557C22#348411yescharacteristic
558C22#348511yescharacteristic
559C22#348614yes—
560C22#348912yes—
561C22#349612yes—
562C22#349612yes—
563C22#349714yes—
564C22#349912yes—
565C22#352911yescharacteristic
566C22#353812yes—
567C22#354412yes—
568C22#355011yescharacteristic
569C22#355514yes—
570C22#355912yes—
571C22#356414yes—
572C4—24no—
573C4—28no—
574C4—48no—
575C4—48no—
576C4—42no—
577C4—48no—
578C4—44no—
579C4—44no—
580C4—42no—
581C4—82no—
582C4—82no—
583C4—84no—
584C2—84no—
585C2#3534711yescharacteristic
586C2#3568712yes—
587C2#4622811yescharacteristic
588C2#4631212yes—
589C2#5089212yes—
590C2#5154211yescharacteristic
591C2#5201514yes—
592C2#5207912yes—
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
2. C2×#2734C2semidirect
4. #16659C2semidirect
5. #16608C2semidirect
7. #15279C2semidirect
8. #15276C2semidirect
10. #15071C2semidirect
11. #14931C2semidirect
13. #14024C2semidirect

Maximal subgroups

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

subgroupindex
C22×#5632
C2×#27342
C2×#25452
#166592
#166082
#159832
#152792
#152762
#152622
#150712
#149312
#144922
#140242