Finite Group Databasebeta

← #847418

Subgroups — #847418

Order 512.

2858 conjugacy classes of subgroups.

The subgroup lattice has 613 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2583C211yesmaximal, characteristic
2C22×#384C211yesmaximal, characteristic
3#16563C212yesmaximal, complemented
4#16522C212yesmaximal, complemented
5#15790C212yesmaximal, complemented
6#15625C212yesmaximal, complemented
7#15438C214yesmaximal, complemented
8#15092C212yesmaximal
9#14728C212yesmaximal
10#14720C212yesmaximal
11#14695C212yesmaximal
12#14406C214yesmaximal, complemented
13#14043C212yesmaximal
14#14039C212yesmaximal
15#13923C211yesmaximal, characteristic
16C22×C22⋊Q8C2211yescharacteristic
17C23×C4⋊C4C2212yes—
18C23×C4⋊C4C2212yes—
19C23×C4⋊C4C2211yescharacteristic
20C23×C4⋊C4C2211yescharacteristic
21C23×C42C2211yescharacteristic
22#2583—28no—
23C2×#398C2211yescharacteristic
24C2×#398C2211yescharacteristic
25C2×#398C2214yes—
26C2×#395C2211yescharacteristic
27C2×#395C2211yescharacteristic
28C2×#395C2214yes—
29C2×#389C2212yes—
30C2×#389C2212yes—
31C2×#389C2214yes—
32C2×#387C2214yes—
33C2×#387C2214yes—
34C2×#385C2212yes—
35C2×#385C2212yes—
36C2×#385C2212yes—
37C2×#385C2212yes—
38C2×#385C2212yes—
39C2×#384—28no—
40C2×#384C2212yes—
41C2×#384C2212yes—
42C2×#384C2212yes—
43C2×#384C2212yes—
44C2×#380C2211yescharacteristic
45C2×C4×C22⋊C4C2211yescharacteristic
46C2×C4×C22⋊C4C2211yescharacteristic
47#1444C2214yes—
48#1444C2214yes—
49#1444C2214yes—
50#1444C2214yes—
51#1444C2214yes—
52#1444C2214yes—
53#1442C2214yes—
54#1442C2214yes—
55#1441C2214yes—
56#1441C2214yes—
57#1441C2212yes—
58#1441C2214yes—
59#1441C2214yes—
60#1441C2214yes—
61#1441C2212yes—
62#1441C2214yes—
63#1436C2214yes—
64#1436C2212yes—
65#1436C2212yes—
66#1436C2214yes—
67#1436C2214yes—
68#1436C2214yes—
69#1434C2214yes—
70C22⋊#95C2212yes—
71C22⋊#95C2212yes—
72C22⋊#95C2212yes—
73C22⋊#95C2212yes—
74C22⋊#95C2212yes—
75C22⋊#95C2212yes—
76C22⋊#95C2212yes—
77C23×Q8C2311yescharacteristic
78C24×C4C2311yescharacteristic
79C24×C4C2312yes—
80C24×C4C2312yes—
81C24×C4C2311yescharacteristic
82C24×C4C2311yescharacteristic
83C2×C22⋊Q8—24no—
84C2×C22⋊Q8—42no—
85C2×C22⋊Q8—42no—
86C22×C4⋊C4—28no—
87C22×C4⋊C4—28no—
88C22×C4⋊C4—24no—
89C22×C4⋊C4—48no—
90C22×C4⋊C4—48no—
91C22×C4⋊C4—42no—
92C22×C4⋊C4—42no—
93C22×C4⋊C4—42no—
94C22×C4⋊C4—42no—
95C22×C4⋊C4D418yes—
96C22×C4⋊C4C2312yes—
97C22×C4⋊C4C2312yes—
98C22×C4⋊C4C2314yes—
99C22×C4⋊C4C2314yes—
100C22×C4⋊C4C2314yes—
101C22×C4⋊C4C2314yes—
102C22×C4⋊C4C2314yes—
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2311yescharacteristic
106C22×C4⋊C4C2312yes—
107C22×C4⋊C4C2312yes—
108C22×C4⋊C4C2312yes—
109C22×C4⋊C4C2314yes—
110C22×C4⋊C4C2314yes—
111C22×C4⋊C4C2312yes—
112C22×C4⋊C4C2311yescharacteristic
113C22×C4⋊C4C2311yescharacteristic
114C22×C4⋊C4C2311yescharacteristic
115C22×C4⋊C4C2311yescharacteristic
116C22×C22⋊C4C2312yes—
117C22×C22⋊C4C2312yes—
118C22×C22⋊C4C2312yes—
119C22×C22⋊C4C2312yes—
120C22×C42—24no—
121C22×C42—28no—
122C22×C42C2312yes—
123C22×C42C2312yes—
124C22×C42C2312yes—
125C22×C42C2312yes—
126#398—24no—
127#398—24no—
128#398—48no—
129#395—24no—
130#395—24no—
131#395—48no—
132#389—28no—
133#389—28no—
134#389—48no—
135#387—216no—
136#387—48no—
137#385—28no—
138#385—28no—
139#385—44no—
140#385—44no—
141#385Q8116yescomplemented
142#384—28no—
143#384—28no—
144#384—28no—
145#384—28no—
146#380—42no—
147C4×C22⋊C4—24no—
148C4×C22⋊C4—42no—
149C2×#95C2314yes—
150C2×#95C2311yescharacteristic
151C2×#95C2311yescharacteristic
152C2×#95C2314yes—
153C2×#95C2312yes—
154C2×#95C2312yes—
155C2×#95C2314yes—
156C2×#95C2312yes—
157C2×#95C2312yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2314yes—
161C2×#95C2314yes—
162C2×#95C2314yes—
163C2×#95C2314yes—
164C2×#95C2314yes—
165C2×#95C2314yes—
166C2×#95C2314yes—
167C2×#95C2314yes—
168C2×#95C2314yes—
169C2×#95C2312yes—
170C2×#95C2314yes—
171C2×#95C2314yes—
172C2×#95C2312yes—
173C2×#95C2314yes—
174C25C2411yescharacteristic
175C22×Q8—22no—
176C22×Q8—41no—
177C22×Q8—41no—
178C22×Q8—44no—
179C23×C4—24no—
180C23×C4—22no—
181C23×C4—22no—
182C23×C4—24no—
183C23×C4—28no—
184C23×C4—24no—
185C23×C4—28no—
186C23×C4—24no—
187C23×C4—22no—
188C23×C4—22no—
189C23×C4—22no—
190C23×C4—22no—
191C23×C4—24no—
192C23×C4—24no—
193C23×C4—24no—
194C23×C4—44no—
195C23×C4—44no—
196C23×C4—44no—
197C23×C4—44no—
198C23×C4C2×D414yes—
199C23×C4C2×D414yes—
200C23×C4C2×D414yes—
201C23×C4C2×Q818yes—
202C23×C4C2412yes—
203C23×C4C2412yes—
204C23×C4C2414yes—
205C23×C4C2414yes—
206C23×C4C2411yescharacteristic
207C23×C4C2411yescharacteristic
208C23×C4C2411yescharacteristic
209C23×C4C2411yescharacteristic
210C23×C4C2412yes—
211C23×C4C2412yes—
212C23×C4C2414yes—
213C23×C4C2411yescharacteristic
214C23×C4C2411yescharacteristic
215C23×C4C2414yes—
216C22⋊Q8—416no—
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—24no—
225C2×C4⋊C4—24no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—24no—
228C2×C4⋊C4—24no—
229C2×C4⋊C4—28no—
230C2×C4⋊C4—28no—
231C2×C4⋊C4—28no—
232C2×C4⋊C4—28no—
233C2×C4⋊C4—28no—
234C2×C4⋊C4—28no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—24no—
237C2×C4⋊C4—416no—
238C2×C4⋊C4—41no—
239C2×C4⋊C4—41no—
240C2×C4⋊C4—41no—
241C2×C4⋊C4—41no—
242C2×C4⋊C4—41no—
243C2×C4⋊C4—41no—
244C2×C4⋊C4—41no—
245C2×C4⋊C4—41no—
246C2×C4⋊C4—44no—
247C2×C4⋊C4—44no—
248C2×C4⋊C4—44no—
249C2×C4⋊C4—44no—
250C2×C4⋊C4—42no—
251C2×C4⋊C4—42no—
252C2×C4⋊C4—42no—
253C2×C4⋊C4—42no—
254C2×C4⋊C4—42no—
255C2×C4⋊C4—42no—
256C2×C4⋊C4—41no—
257C2×C4⋊C4—41no—
258C2×C4⋊C4—432no—
259C2×C4⋊C4—432no—
260C2×C4⋊C4—416no—
261C2×C4⋊C4—42no—
262C2×C4⋊C4—44no—
263C2×C4⋊C4—44no—
264C2×C4⋊C4—44no—
265C2×C4⋊C4—44no—
266C2×C4⋊C4—44no—
267C2×C4⋊C4—44no—
268C2×C4⋊C4—44no—
269C2×C4⋊C4—44no—
270C2×C4⋊C4—44no—
271C2×C4⋊C4C2×D414yes—
272C2×C4⋊C4C2×D414yes—
273C2×C4⋊C4C2×D414yes—
274C2×C4⋊C4C2×D414yes—
275C2×C4⋊C4C2×Q8116yes—
276C2×C22⋊C4—24no—
277C2×C22⋊C4—24no—
278C2×C22⋊C4—24no—
279C2×C22⋊C4—24no—
280C2×C22⋊C4—24no—
281C2×C22⋊C4—24no—
282C2×C22⋊C4—24no—
283C2×C22⋊C4—24no—
284C2×C22⋊C4—24no—
285C2×C22⋊C4—24no—
286C2×C22⋊C4C2×Q818yes—
287C2×C22⋊C4C2×Q818yes—
288C2×C42—232no—
289C2×C42—24no—
290C2×C42—24no—
291C2×C42—28no—
292C2×C42—24no—
293C2×C42—24no—
294C2×C42—42no—
295C2×C42—44no—
296C2×C42—44no—
297C2×C42#4118yes—
298#95—22no—
299#95—22no—
300#95—24no—
301#95—24no—
302#95—24no—
303#95—24no—
304#95—28no—
305#95—28no—
306#95—28no—
307#95—28no—
308#95—28no—
309#95—28no—
310#95—28no—
311#95—28no—
312#95—42no—
313#95—42no—
314#95—44no—
315#95—44no—
316#95—44no—
317#95—44no—
318#95—44no—
319#95—44no—
320#95—44no—
321#95—44no—
322#95C2×Q8116yes—
323C24—22no—
324C24—24no—
325C24—22no—
326C24C22×D412yes—
327C24C22×Q814yes—
328C24H(5,𝔽2)12yes—
329C24H(5,𝔽2)12yes—
330C242−1+414yes—
331C24C2511yescharacteristic
332C2×Q8—44no—
333C2×Q8—84no—
334C2×Q8—84no—
335C2×Q8—84no—
336C22×C4—216no—
337C22×C4—216no—
338C22×C4—232no—
339C22×C4—216no—
340C22×C4—216no—
341C22×C4—28no—
342C22×C4—216no—
343C22×C4—24no—
344C22×C4—22no—
345C22×C4—24no—
346C22×C4—21no—
347C22×C4—21no—
348C22×C4—21no—
349C22×C4—21no—
350C22×C4—24no—
351C22×C4—22no—
352C22×C4—24no—
353C22×C4—28no—
354C22×C4—28no—
355C22×C4—232no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—24no—
359C22×C4—24no—
360C22×C4—24no—
361C22×C4—24no—
362C22×C4—24no—
363C22×C4—24no—
364C22×C4—24no—
365C22×C4—21no—
366C22×C4—21no—
367C22×C4—24no—
368C22×C4—24no—
369C22×C4—21no—
370C22×C4—24no—
371C22×C4—24no—
372C22×C4—21no—
373C22×C4—21no—
374C22×C4—21no—
375C22×C4—21no—
376C22×C4—21no—
377C22×C4—24no—
378C22×C4—24no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—24no—
382C22×C4—28no—
383C22×C4—28no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—24no—
387C22×C4—24no—
388C22×C4—24no—
389C22×C4—24no—
390C22×C4—24no—
391C22×C4—24no—
392C22×C4—24no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—24no—
396C22×C4—24no—
397C22×C4—24no—
398C22×C4—24no—
399C22×C4—24no—
400C22×C4—24no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—24no—
404C22×C4—24no—
405C22×C4—24no—
406C22×C4—24no—
407C22×C4—24no—
408C22×C4—48no—
409C22×C4—48no—
410C22×C4—48no—
411C22×C4—48no—
412C22×C4—48no—
413C22×C4—84no—
414C22×C4—84no—
415C22×C4—84no—
416C22×C4—84no—
417C22×C4—88no—
418C22×C4—88no—
419C22×C4C22×D412yes—
420C22×C4C22×D412yes—
421C22×C4C22×D412yes—
422C22×C4C22×D412yes—
423C22×C4C22×D412yes—
424C22×C4C22×D412yes—
425C22×C4C22×Q814yes—
426C22×C4C22×Q818yes—
427C22×C4C22×Q814yes—
428C22×C4C22×Q818yes—
429C22×C4C2×#4114yes—
430C22×C4C2×#4118yes—
431C22×C4H(5,𝔽2)18yes—
432C22×C42−1+414yes—
433C4⋊C4—232no—
434C4⋊C4—44no—
435C4⋊C4—44no—
436C4⋊C4—44no—
437C4⋊C4—44no—
438C4⋊C4—416no—
439C4⋊C4—416no—
440C4⋊C4—416no—
441C4⋊C4—416no—
442C4⋊C4—48no—
443C4⋊C4—48no—
444C4⋊C4—48no—
445C4⋊C4—44no—
446C4⋊C4—48no—
447C4⋊C4—48no—
448C4⋊C4—84no—
449C4⋊C4—84no—
450C4⋊C4—88no—
451C4⋊C4—84no—
452C4⋊C4—88no—
453C22⋊C4—216no—
454C22⋊C4—216no—
455C22⋊C4—48no—
456C22⋊C4—48no—
457C42—216no—
458C42—216no—
459C42—84no—
460C42—84no—
461C23—22no—
462C23—22no—
463C23—28no—
464C23—24no—
465C23—24no—
466C23—24no—
467C23—28no—
468C23—28no—
469C23—24no—
470C23—24no—
471C23—28no—
472C23#53414yes—
473C23#54418yes—
474C23D4×Q818yes—
475C23Q8214yes—
476C23#56314yes—
477C23C23×D411yescharacteristic
478C23C23×Q812yes—
479C23C22×#4112yes—
480C23C2×H(5,𝔽2)11yescharacteristic
481C23C2×H(5,𝔽2)11yescharacteristic
482C23C2×H(5,𝔽2)14yes—
483C23C2×2−1+412yes—
484C23C2×2−1+412yes—
485Q8—88no—
486C2×C4—264no—
487C2×C4—232no—
488C2×C4—22no—
489C2×C4—28no—
490C2×C4—22no—
491C2×C4—22no—
492C2×C4—22no—
493C2×C4—28no—
494C2×C4—28no—
495C2×C4—28no—
496C2×C4—28no—
497C2×C4—24no—
498C2×C4—24no—
499C2×C4—28no—
500C2×C4—24no—
501C2×C4—24no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—28no—
507C2×C4—24no—
508C2×C4—24no—
509C2×C4—28no—
510C2×C4—28no—
511C2×C4—24no—
512C2×C4—24no—
513C2×C4—24no—
514C2×C4—24no—
515C2×C4—28no—
516C2×C4—28no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—44no—
522C2×C4—44no—
523C2×C4—44no—
524C2×C4—44no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—44no—
534C2×C4—44no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—44no—
539C2×C4—42no—
540C2×C4—44no—
541C2×C4—44no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—44no—
545C2×C4—44no—
546C2×C4—44no—
547C2×C4—44no—
548C2×C4—44no—
549C2×C4—44no—
550C2×C4—44no—
551C2×C4—44no—
552C2×C4—44no—
553C2×C4—44no—
554C2×C4—44no—
555C2×C4—44no—
556C2×C4—44no—
557C2×C4—44no—
558C2×C4—44no—
559C2×C4—88no—
560C2×C4—88no—
561C2×C4—816no—
562C2×C4#52514yes—
563C2×C4#53614yes—
564C2×C4#554116yes—
565C22—24no—
566C22—28no—
567C22—28no—
568C22—28no—
569C22—24no—
570C22—216no—
571C22—28no—
572C22#273418yes—
573C22C2×#52511yescharacteristic
574C22C2×#53411yescharacteristic
575C22C2×#53611yescharacteristic
576C22C2×#54412yes—
577C22C2×D4×Q812yes—
578C22C2×#55414yes—
579C22C2×Q8211yescharacteristic
580C22#348412yes—
581C22#349212yes—
582C22#349214yes—
583C22#349212yes—
584C22#350314yes—
585C22C2×#56311yescharacteristic
586C22#354312yes—
587C22#354312yes—
588C22#355214yes—
589C4—216no—
590C4—24no—
591C4—24no—
592C4—42no—
593C4—42no—
594C4—42no—
595C4—42no—
596C4—44no—
597C4—44no—
598C4—48no—
599C4—84no—
600C4—82no—
601C4—82no—
602C4—84no—
603C2—28no—
604C2#3505711yescharacteristic
605C2#3654612yes—
606C2#3723211yescharacteristic
607C2#3787414yes—
608C2#3959212yes—
609C2#4632612yes—
610C2C2×#273411yescharacteristic
611C2#5932212yes—
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
3. #16563C2semidirect
4. #16522C2semidirect
5. #15790C2semidirect
6. #15625C2semidirect
7. #15438C2semidirect
12. #14406C2semidirect
141. #385Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#25832
C22×#3842
#165632
#165222
#157902
#156252
#154382
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