Finite Group Databasebeta

← #4432451

Subgroups — #4432451

Order 512.

2592 conjugacy classes of subgroups.

The subgroup lattice has 620 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2820C211yesmaximal, characteristic
2C2×#2584C211yesmaximal, characteristic, complemented
3C2×#2632C211yesmaximal, characteristic
4C2×#2351C211yesmaximal, characteristic, complemented
5#16564C212yesmaximal, complemented
6#16550C214yesmaximal, complemented
7#16518C212yesmaximal, complemented
8#15623C214yesmaximal, complemented
9#15615C214yesmaximal
10#14877C214yesmaximal, complemented
11#14870C212yesmaximal, complemented
12#14808C212yesmaximal, complemented
13#14296C211yesmaximal, characteristic, complemented
14#14289C211yesmaximal, characteristic, complemented
15#14153C211yesmaximal, characteristic
16C22×#125C2211yescharacteristic
17C22×#124C2211yescharacteristic
18C22×C4×Q8C2211yescharacteristic
19C22×C4×Q8C2212yes—
20#2820—28no—
21#2632—28no—
22#2584—28no—
23C2×#399C2212yes—
24C2×#399C2212yes—
25C2×#398C2212yes—
26C2×#398C2214yes—
27C2×#397C2212yes—
28C2×#396C2214yes—
29C2×#396C2214yes—
30C2×#396C2214yes—
31C2×#395C2212yes—
32C2×#395C2212yes—
33C2×#395C2212yes—
34C2×#394C2212yes—
35C2×#394C2212yes—
36C2×#393C2212yes—
37#2351—28no—
38C2×#391C2212yes—
39C2×#390C2212yes—
40C2×#389C2214yes—
41C2×#389C2214yes—
42C2×#389C2212yes—
43C2×#389C2212yes—
44C2×#389C2212yes—
45C2×#388C2212yes—
46C2×#388C2212yes—
47C2×#388C2214yes—
48C2×#383C2211yescharacteristic
49C2×#382C2212yes—
50C2×#382C2211yescharacteristic
51C2×C4×C4⋊C4C2212yes—
52C2×C4×C4⋊C4C2211yescharacteristic
53C2×C4×C4⋊C4C2211yescharacteristic
54#1445C2211yescharacteristic
55#1445C2211yescharacteristic
56#1445C2211yescharacteristic
57#1445C2211yescharacteristic
58#1444C2214yes—
59#1444C2212yes—
60#1442C2212yes—
61#1441C2214yes—
62#1441C2214yes—
63#1441C2214yes—
64#1441C2214yes—
65#1441C2214yes—
66#1441C2214yes—
67#1440C2214yes—
68#1439C2212yes—
69#1439C2214yes—
70#1439C2212yes—
71#1439C2214yes—
72#1439C2214yes—
73#1437C2214yes—
74#1437C2214yes—
75#1436C2214yes—
76#1436C2214yes—
77#1436C2214yes—
78C4⋊#95C2214yes—
79C23×Q8C2311yescharacteristic
80C23×Q8C2311yescharacteristic
81C23×D4C2311yescharacteristic
82C2×#125—24no—
83C2×#125—24no—
84C2×#125—24no—
85C2×#124—24no—
86C2×#124—24no—
87C2×#124—24no—
88C2×C4×Q8—28no—
89C2×C4×Q8—24no—
90C2×C4×Q8—24no—
91C2×C4×Q8—48no—
92C2×C4×Q8D418yes—
93C22×C4⋊C4C2312yes—
94C22×C4⋊C4C2312yes—
95C22×C4⋊C4C2314yes—
96C22×C4⋊C4C2311yescharacteristic
97C22×C4⋊C4C2311yescharacteristic
98C22×C4⋊C4C2312yes—
99C22×C4⋊C4C2314yes—
100C22×C4⋊C4C2314yes—
101C22×C4⋊C4C2312yes—
102C22×C4⋊C4C2312yes—
103C22×C4⋊C4C2311yescharacteristic
104C22×C4⋊C4C2314yes—
105C22×C4⋊C4C2312yes—
106C22×C4⋊C4C2312yes—
107C22×C4⋊C4C2314yes—
108C22×C4⋊C4C2314yes—
109C22×C4⋊C4C2312yes—
110C22×C22⋊C4C2312yes—
111C22×C22⋊C4C2312yes—
112C22×C22⋊C4C2312yes—
113C22×C22⋊C4C2312yes—
114C22×C22⋊C4C2314yes—
115C22×C22⋊C4C2314yes—
116C22×C22⋊C4C2314yes—
117C22×C22⋊C4C2312yes—
118C22×C22⋊C4C2312yes—
119C22×C22⋊C4C2314yes—
120C22×C42C2314yes—
121C22×C42C2312yes—
122C22×C42C2312yes—
123C22×C42C2312yes—
124C22×C42C2311yescharacteristic
125#399—28no—
126#399—28no—
127#398—28no—
128#398—48no—
129#397—28no—
130#396—216no—
131#396—48no—
132#396—48no—
133#395—28no—
134#395—28no—
135#395—28no—
136#394—28no—
137#394—28no—
138#393—28no—
139#391—44no—
140#390—44no—
141#389—216no—
142#389—28no—
143#389—28no—
144#389—28no—
145#389—216no—
146#388—28no—
147#388—28no—
148#388—48no—
149#383—24no—
150#382—28no—
151#382D418yes—
152C4×C4⋊C4—24no—
153C4×C4⋊C4—24no—
154C4×C4⋊C4—28no—
155C2×#95C2312yes—
156C2×#95C2312yes—
157C2×#95C2312yes—
158C2×#95C2312yes—
159C2×#95C2314yes—
160C2×#95C2312yes—
161C2×#95C2314yes—
162C2×#95C2314yes—
163C2×#95C2314yes—
164C2×#95C2314yes—
165C2×#95C2314yes—
166C2×#95C2311yescharacteristic
167C2×#95C2311yescharacteristic
168C2×#95C2314yes—
169C2×#95C2311yescharacteristic
170C2×#95C2311yescharacteristic
171C2×#95C2314yes—
172C2×#95C2314yes—
173C2×#95C2314yes—
174C2×#95C2314yes—
175C2×#95C2314yes—
176C2×#95C2314yes—
177C2×#95C2312yes—
178C2×#95C2312yes—
179C25C2412yes—
180C22×Q8—22no—
181C22×Q8—22no—
182C22×Q8—22no—
183C22×Q8—22no—
184C22×Q8—22no—
185C22×Q8—24no—
186C22×Q8—24no—
187C22×Q8—42no—
188C22×Q8—42no—
189C22×Q8C2×D414yes—
190C22×D4—22no—
191C22×D4—22no—
192C22×D4—22no—
193C22×D4—42no—
194C22×D4—42no—
195C23×C4C2412yes—
196C23×C4C2412yes—
197C23×C4C2412yes—
198C23×C4C2412yes—
199C23×C4C2412yes—
200C23×C4C2412yes—
201C23×C4C2412yes—
202C23×C4C2414yes—
203C23×C4C2412yes—
204C23×C4C2414yes—
205C23×C4C2411yescharacteristic
206C23×C4C2411yescharacteristic
207C23×C4C2411yescharacteristic
208C23×C4C2412yes—
209#125—416no—
210#124—416no—
211C4×Q8—232no—
212C4×Q8—432no—
213C2×C4⋊C4—24no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—24no—
216C2×C4⋊C4—24no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—28no—
219C2×C4⋊C4—28no—
220C2×C4⋊C4—24no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—24no—
224C2×C4⋊C4—24no—
225C2×C4⋊C4—24no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—28no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4—28no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—28no—
236C2×C4⋊C4—28no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—28no—
239C2×C4⋊C4—28no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—28no—
243C2×C4⋊C4—28no—
244C2×C4⋊C4—28no—
245C2×C4⋊C4—42no—
246C2×C4⋊C4—44no—
247C2×C4⋊C4—44no—
248C2×C4⋊C4—44no—
249C2×C4⋊C4—44no—
250C2×C4⋊C4—44no—
251C2×C4⋊C4—44no—
252C2×C4⋊C4—44no—
253C2×C4⋊C4—44no—
254C2×C4⋊C4—44no—
255C2×C4⋊C4—44no—
256C2×C4⋊C4—44no—
257C2×C4⋊C4C2×D418yes—
258C2×C4⋊C4C2×D414yes—
259C2×C4⋊C4C2×D418yes—
260C2×C22⋊C4—24no—
261C2×C22⋊C4—24no—
262C2×C22⋊C4—24no—
263C2×C22⋊C4—24no—
264C2×C22⋊C4—24no—
265C2×C22⋊C4—24no—
266C2×C22⋊C4—24no—
267C2×C22⋊C4—24no—
268C2×C22⋊C4—24no—
269C2×C22⋊C4—28no—
270C2×C22⋊C4—24no—
271C2×C22⋊C4—28no—
272C2×C22⋊C4—24no—
273C2×C22⋊C4—24no—
274C2×C22⋊C4—24no—
275C2×C22⋊C4—24no—
276C2×C22⋊C4—28no—
277C2×C22⋊C4—28no—
278C2×C22⋊C4—28no—
279C2×C22⋊C4—44no—
280C2×C22⋊C4—44no—
281C2×C22⋊C4—44no—
282C2×C22⋊C4—44no—
283C2×C22⋊C4—44no—
284C2×C22⋊C4—44no—
285C2×C22⋊C4—44no—
286C2×C22⋊C4—44no—
287C2×C22⋊C4—44no—
288C2×C42—24no—
289C2×C42—24no—
290C2×C42—28no—
291C2×C42—24no—
292C2×C42—28no—
293C2×C42—24no—
294C2×C42—44no—
295C2×C42—44no—
296C2×C42—42no—
297C2×C42—42no—
298C2×C42C2×D418yes—
299C2×C42C2×D414yes—
300C2×C42C2×D414yes—
301C2×C42#4114yes—
302#95—22no—
303#95—22no—
304#95—24no—
305#95—28no—
306#95—22no—
307#95—22no—
308#95—24no—
309#95—24no—
310#95—24no—
311#95—24no—
312#95—28no—
313#95—28no—
314#95—28no—
315#95—28no—
316#95—28no—
317#95—28no—
318#95—44no—
319#95—44no—
320#95—44no—
321#95—44no—
322#95—44no—
323#95—44no—
324#95C2×D418yes—
325#95C2×D418yes—
326C24—22no—
327C24—22no—
328C24—22no—
329C24—22no—
330C24—22no—
331C24—24no—
332C24—22no—
333C24—22no—
334C24—24no—
335C24—24no—
336C24—24no—
337C24C2511yescharacteristic
338C2×Q8—28no—
339C2×Q8—28no—
340C2×Q8—216no—
341C2×Q8—28no—
342C2×Q8—216no—
343C2×Q8—48no—
344C2×Q8—44no—
345C2×Q8—84no—
346C2×Q8—82no—
347C2×Q8—82no—
348C2×D4—48no—
349C2×D4—44no—
350C2×D4—84no—
351C2×D4—82no—
352C2×D4—82no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—24no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—24no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—24no—
367C22×C4—24no—
368C22×C4—24no—
369C22×C4—24no—
370C22×C4—24no—
371C22×C4—24no—
372C22×C4—24no—
373C22×C4—24no—
374C22×C4—24no—
375C22×C4—24no—
376C22×C4—24no—
377C22×C4—24no—
378C22×C4—24no—
379C22×C4—24no—
380C22×C4—24no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—24no—
395C22×C4—24no—
396C22×C4—24no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—24no—
400C22×C4—22no—
401C22×C4—24no—
402C22×C4—24no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—24no—
406C22×C4—24no—
407C22×C4—24no—
408C22×C4—22no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4C22×D418yes—
413C22×C4C22×D414yes—
414C22×C4C22×D412yes—
415C22×C4C22×D412yes—
416C22×C4C22×D412yes—
417C22×C4C22×D412yes—
418C22×C4C22×D412yes—
419C22×C4C22×D412yes—
420C22×C4C22×D414yes—
421C22×C4C2×#4112yes—
422C22×C4C2×#4112yes—
423C22×C4C2×#4112yes—
424C22×C4H(5,𝔽2)14yes—
425C22×C4H(5,𝔽2)14yes—
426C22×C42−1+418yes—
427C22×C42−1+414yes—
428C4⋊C4—216no—
429C4⋊C4—216no—
430C4⋊C4—28no—
431C4⋊C4—48no—
432C4⋊C4—48no—
433C4⋊C4—416no—
434C4⋊C4—416no—
435C4⋊C4—44no—
436C4⋊C4—44no—
437C4⋊C4—48no—
438C4⋊C4—48no—
439C4⋊C4—416no—
440C4⋊C4—88no—
441C4⋊C4—88no—
442C4⋊C4—84no—
443C4⋊C4—84no—
444C4⋊C4—88no—
445C22⋊C4—416no—
446C22⋊C4—48no—
447C22⋊C4—48no—
448C22⋊C4—48no—
449C22⋊C4—48no—
450C22⋊C4—88no—
451C22⋊C4—88no—
452C22⋊C4—84no—
453C22⋊C4—84no—
454C22⋊C4—88no—
455C42—216no—
456C42—416no—
457C42—48no—
458C42—84no—
459C42#121116yes—
460C23—42no—
461C23—42no—
462C23—42no—
463C23—44no—
464C23—42no—
465C23—42no—
466C23—42no—
467C23—42no—
468C23—44no—
469C23—42no—
470C23—42no—
471C23—42no—
472C23—44no—
473C23—42no—
474C23—44no—
475C23—44no—
476C23—44no—
477C23—44no—
478C23—44no—
479C23—44no—
480C23—44no—
481C23—44no—
482C23—44no—
483C23C23×D411yescharacteristic
484C23C23×D411yescharacteristic
485C23C22×#4111yescharacteristic
486C23C2×H(5,𝔽2)12yes—
487C23C2×H(5,𝔽2)12yes—
488C23C2×2−1+412yes—
489C23C2×2−1+414yes—
490C23#58512yes—
491Q8—232no—
492Q8—88no—
493D4—88no—
494C2×C4—24no—
495C2×C4—24no—
496C2×C4—24no—
497C2×C4—24no—
498C2×C4—22no—
499C2×C4—24no—
500C2×C4—24no—
501C2×C4—22no—
502C2×C4—24no—
503C2×C4—24no—
504C2×C4—24no—
505C2×C4—24no—
506C2×C4—24no—
507C2×C4—22no—
508C2×C4—24no—
509C2×C4—22no—
510C2×C4—24no—
511C2×C4—24no—
512C2×C4—28no—
513C2×C4—28no—
514C2×C4—28no—
515C2×C4—24no—
516C2×C4—24no—
517C2×C4—28no—
518C2×C4—24no—
519C2×C4—28no—
520C2×C4—28no—
521C2×C4—28no—
522C2×C4—44no—
523C2×C4—44no—
524C2×C4—42no—
525C2×C4—44no—
526C2×C4—44no—
527C2×C4—42no—
528C2×C4—44no—
529C2×C4—44no—
530C2×C4—44no—
531C2×C4—44no—
532C2×C4—44no—
533C2×C4—44no—
534C2×C4—44no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—44no—
541C2×C4—44no—
542C2×C4—44no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—44no—
549C2×C4—42no—
550C2×C4—44no—
551C2×C4—44no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—44no—
560C2×C4—44no—
561C2×C4—44no—
562C2×C4—44no—
563C2×C4—44no—
564C2×C4—44no—
565C2×C4C2×#12114yes—
566C2×C4C2×#12114yes—
567C2×C4C2×#12114yes—
568C2×C4#53718yes—
569C22—84no—
570C22—84no—
571C22—82no—
572C22—82no—
573C22—82no—
574C22—82no—
575C22—82no—
576C22—84no—
577C22—84no—
578C22—82no—
579C22—82no—
580C22C22×#12111yescharacteristic
581C22C2×#53712yes—
582C22#346012yes—
583C22#348411yescharacteristic
584C22#348611yescharacteristic
585C22#348612yes—
586C22#348812yes—
587C22#349212yes—
588C22#349811yescharacteristic
589C22#349912yes—
590C22#350011yescharacteristic
591C22#350912yes—
592C22#353614yes—
593C22#355214yes—
594C22#355414yes—
595C22#355514yes—
596C4—24no—
597C4—24no—
598C4—28no—
599C4—24no—
600C4—48no—
601C4—44no—
602C4—48no—
603C4—44no—
604C4—82no—
605C4—82no—
606C4—82no—
607C4—82no—
608C4—82no—
609C4—82no—
610C2—162no—
611C2#3522311yescharacteristic
612C2#3525611yescharacteristic
613C2#3566711yescharacteristic
614C2#4109914yes—
615C2#4279412yes—
616C2#4279512yes—
617C2#5068912yes—
618C2#5069212yes—
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×#2584C2semidirect
4. C2×#2351C2semidirect
5. #16564C2semidirect
6. #16550C2semidirect
7. #16518C2semidirect
8. #15623C2semidirect
10. #14877C2semidirect
11. #14870C2semidirect
12. #14808C2semidirect
13. #14296C2semidirect
14. #14289C2semidirect

Maximal subgroups

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

subgroupindex
C2×#28202
C2×#25842
C2×#26322
C2×#23512
#165642
#165502
#165182
#156232
#156152
#148772
#148702
#148082
#142962
#142892
#141532