Finite Group Databasebeta

← #556137

Subgroups — #556137

Order 512.

2866 conjugacy classes of subgroups.

The subgroup lattice has 608 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2313C211yesmaximal, characteristic
2C2×#2313C211yesmaximal, characteristic
3C22×#389C211yesmaximal, characteristic
4#14705C212yesmaximal
5#14502C214yesmaximal, complemented
6#14468C214yesmaximal, complemented
7#14406C214yesmaximal, complemented
8#14404C214yesmaximal, complemented
9#14263C212yesmaximal
10#14149C212yesmaximal
11#14130C212yesmaximal
12#13966C211yesmaximal, characteristic
13#13965C211yesmaximal, characteristic
14#13881C211yesmaximal, characteristic
15#13880C211yesmaximal, characteristic
16C23×C4⋊C4C2211yescharacteristic
17C23×C4⋊C4C2211yescharacteristic
18C23×C4⋊C4C2211yescharacteristic
19C23×C4⋊C4C2211yescharacteristic
20C23×C42C2211yescharacteristic
21#2313—28no—
22#2313C4116yescomplemented
23C2×#389—28no—
24C2×#389C2214yes—
25C2×#389C2214yes—
26C2×#389C2214yes—
27C2×#387C2212yes—
28C2×#387C2212yes—
29C2×#387C2212yes—
30C2×#387C2212yes—
31C2×#387C2214yes—
32C2×#387C2214yes—
33C2×#385C2212yes—
34C2×#385C2212yes—
35C2×#385C2212yes—
36C2×#385C2212yes—
37C2×#384C2212yes—
38C2×#382C2212yes—
39C2×#381C2212yes—
40C2×#381C2212yes—
41C2×#380C2211yescharacteristic
42C2×#380C2211yescharacteristic
43C2×#380C2212yes—
44C2×C4×C22⋊C4C2212yes—
45C2×C4×C22⋊C4C2211yescharacteristic
46C2×C4×C22⋊C4C2211yescharacteristic
47C22×#95C2211yescharacteristic
48C22×#95C2211yescharacteristic
49#1444C2214yes—
50#1444C2214yes—
51#1444C2214yes—
52#1444C2212yes—
53#1444C2214yes—
54#1444C2212yes—
55#1442C2214yes—
56#1442C2214yes—
57#1442C2212yes—
58#1442C2214yes—
59#1442C2212yes—
60#1442C2214yes—
61#1441C2214yes—
62#1441C2214yes—
63#1436C2214yes—
64#1436C2214yes—
65#1436C2214yes—
66#1436C2214yes—
67#1436C2214yes—
68#1436C2214yes—
69C4⋊#95C2214yes—
70C4⋊#95C2212yes—
71C4⋊#95C2212yes—
72C4⋊#95C2214yes—
73#1434C2214yes—
74C22⋊#95C2212yes—
75C22⋊#95C2212yes—
76C22⋊#95C2212yes—
77C22⋊#95C2212yes—
78C24×C4C2311yescharacteristic
79C24×C4C2311yescharacteristic
80C24×C4C2311yescharacteristic
81C24×C4C2311yescharacteristic
82C24×C4C2311yescharacteristic
83C24×C4C2311yescharacteristic
84C24×C4C2311yescharacteristic
85C22×C4⋊C4—24no—
86C22×C4⋊C4—24no—
87C22×C4⋊C4—24no—
88C22×C4⋊C4—44no—
89C22×C4⋊C4—44no—
90C22×C4⋊C4—44no—
91C22×C4⋊C4C2×C418yes—
92C22×C4⋊C4D418yes—
93C22×C4⋊C4Q818yes—
94C22×C4⋊C4C2312yes—
95C22×C4⋊C4C2314yes—
96C22×C4⋊C4C2312yes—
97C22×C4⋊C4C2314yes—
98C22×C4⋊C4C2311yescharacteristic
99C22×C4⋊C4C2311yescharacteristic
100C22×C4⋊C4C2314yes—
101C22×C4⋊C4C2314yes—
102C22×C4⋊C4C2311yescharacteristic
103C22×C4⋊C4C2311yescharacteristic
104C22×C4⋊C4C2314yes—
105C22×C4⋊C4C2314yes—
106C22×C4⋊C4C2314yes—
107C22×C4⋊C4C2312yes—
108C22×C4⋊C4C2312yes—
109C22×C22⋊C4C2312yes—
110C22×C22⋊C4C2312yes—
111C22×C22⋊C4C2312yes—
112C22×C22⋊C4C2312yes—
113C22×C42—24no—
114C22×C42—44no—
115C22×C42C2314yes—
116C22×C42C2312yes—
117C22×C42C2312yes—
118C22×C42C2312yes—
119C22×C42C2312yes—
120#389—216no—
121#389—216no—
122#389—48no—
123#387—28no—
124#387—28no—
125#387—28no—
126#387—28no—
127#387C2×C4132yescomplemented
128#387C2×C4132yescomplemented
129#385—44no—
130#385—44no—
131#385—44no—
132#385—44no—
133#384—44no—
134#382—44no—
135#381—28no—
136#381C2×C4116yes—
137#380—24no—
138#380—24no—
139#380C2×C4116yes—
140C4×C22⋊C4—28no—
141C4×C22⋊C4C2×C418yes—
142C4×C22⋊C4C2×C418yes—
143C2×#95—24no—
144C2×#95—24no—
145C2×#95C2314yes—
146C2×#95C2314yes—
147C2×#95C2312yes—
148C2×#95C2312yes—
149C2×#95C2312yes—
150C2×#95C2312yes—
151C2×#95C2314yes—
152C2×#95C2312yes—
153C2×#95C2312yes—
154C2×#95C2312yes—
155C2×#95C2312yes—
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×#95C2314yes—
170C2×#95C2314yes—
171C2×#95C2312yes—
172C2×#95C2312yes—
173C25C2411yescharacteristic
174C23×C4—24no—
175C23×C4—24no—
176C23×C4—22no—
177C23×C4—22no—
178C23×C4—24no—
179C23×C4—24no—
180C23×C4—22no—
181C23×C4—22no—
182C23×C4—24no—
183C23×C4—24no—
184C23×C4—24no—
185C23×C4—24no—
186C23×C4—42no—
187C23×C4—42no—
188C23×C4—42no—
189C23×C4—42no—
190C23×C4—42no—
191C23×C4—42no—
192C23×C4—42no—
193C23×C4—42no—
194C23×C4—42no—
195C23×C4—42no—
196C23×C4C22×C414yes—
197C23×C4C22×C414yes—
198C23×C4C22×C414yes—
199C23×C4C2×D414yes—
200C23×C4C2×D414yes—
201C23×C4C2×D414yes—
202C23×C4C2×Q814yes—
203C23×C4C2×Q814yes—
204C23×C4C2×Q814yes—
205C23×C4C2412yes—
206C23×C4C2412yes—
207C23×C4C2412yes—
208C23×C4C2412yes—
209C23×C4C2412yes—
210C23×C4C2412yes—
211C23×C4C2414yes—
212C23×C4C2414yes—
213C23×C4C2412yes—
214C23×C4C2412yes—
215C23×C4C2411yescharacteristic
216C23×C4C2411yescharacteristic
217C23×C4C2412yes—
218C23×C4C2412yes—
219C2×C4⋊C4—28no—
220C2×C4⋊C4—24no—
221C2×C4⋊C4—24no—
222C2×C4⋊C4—28no—
223C2×C4⋊C4—24no—
224C2×C4⋊C4—24no—
225C2×C4⋊C4—28no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—28no—
231C2×C4⋊C4—28no—
232C2×C4⋊C4—42no—
233C2×C4⋊C4—44no—
234C2×C4⋊C4—42no—
235C2×C4⋊C4—48no—
236C2×C4⋊C4—44no—
237C2×C4⋊C4—44no—
238C2×C4⋊C4—416no—
239C2×C4⋊C4—416no—
240C2×C4⋊C4—416no—
241C2×C4⋊C4—42no—
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⋊C4—42no—
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⋊C4C4⋊C4164yescomplemented
256C2×C4⋊C4C22×C4116yes—
257C2×C4⋊C4C22×C418yes—
258C2×C4⋊C4C22×C418yes—
259C2×C4⋊C4C2×D418yes—
260C2×C4⋊C4C2×D418yes—
261C2×C4⋊C4C2×Q818yes—
262C2×C4⋊C4C2×Q818yes—
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—24no—
270C2×C22⋊C4—24no—
271C2×C22⋊C4—28no—
272C2×C22⋊C4C22×C418yes—
273C2×C22⋊C4C22×C418yes—
274C2×C42—28no—
275C2×C42—24no—
276C2×C42—24no—
277C2×C42—42no—
278C2×C42—42no—
279C2×C42—48no—
280C2×C42—42no—
281C2×C42—42no—
282C2×C42—416no—
283C2×C42—44no—
284C2×C42—44no—
285C2×C42C22×C418yes—
286C2×C42C22×C418yes—
287#95—28no—
288#95—24no—
289#95—24no—
290#95—24no—
291#95—24no—
292#95—24no—
293#95—24no—
294#95—24no—
295#95—24no—
296#95—24no—
297#95—24no—
298#95—28no—
299#95—28no—
300#95—44no—
301#95—44no—
302#95—44no—
303#95—44no—
304#95—44no—
305#95—44no—
306#95—44no—
307#95—44no—
308#95—44no—
309#95—44no—
310#95C22×C418yes—
311#95C22×C418yes—
312#95C22×C4116yes—
313#95C22×C4116yes—
314#95C22×C4116yes—
315C24—22no—
316C24—22no—
317C24—22no—
318C24—22no—
319C24C23×C412yes—
320C24C22×D412yes—
321C24C22×Q812yes—
322C24H(5,𝔽2)14yes—
323C242−1+414yes—
324C24C2511yescharacteristic
325C22×C4—216no—
326C22×C4—28no—
327C22×C4—28no—
328C22×C4—216no—
329C22×C4—24no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—24no—
333C22×C4—24no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—24no—
337C22×C4—28no—
338C22×C4—28no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—24no—
342C22×C4—24no—
343C22×C4—24no—
344C22×C4—24no—
345C22×C4—24no—
346C22×C4—24no—
347C22×C4—22no—
348C22×C4—24no—
349C22×C4—24no—
350C22×C4—24no—
351C22×C4—24no—
352C22×C4—22no—
353C22×C4—24no—
354C22×C4—24no—
355C22×C4—24no—
356C22×C4—24no—
357C22×C4—24no—
358C22×C4—24no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—24no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—24no—
371C22×C4—24no—
372C22×C4—24no—
373C22×C4—24no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—24no—
377C22×C4—24no—
378C22×C4—24no—
379C22×C4—24no—
380C22×C4—24no—
381C22×C4—22no—
382C22×C4—24no—
383C22×C4—24no—
384C22×C4—24no—
385C22×C4—24no—
386C22×C4—22no—
387C22×C4—44no—
388C22×C4—44no—
389C22×C4—48no—
390C22×C4—48no—
391C22×C4—44no—
392C22×C4—44no—
393C22×C4—48no—
394C22×C4—48no—
395C22×C4—416no—
396C22×C4—416no—
397C22×C4—44no—
398C22×C4—44no—
399C22×C4—44no—
400C22×C4—44no—
401C22×C4—44no—
402C22×C4—44no—
403C22×C4—44no—
404C22×C4—44no—
405C22×C4—44no—
406C22×C4—44no—
407C22×C4—88no—
408C22×C4—88no—
409C22×C4C2×C4⋊C4116yes—
410C22×C4C2×C4⋊C4116yes—
411C22×C4C2×C4⋊C4116yes—
412C22×C4C23×C418yes—
413C22×C4C23×C418yes—
414C22×C4C23×C414yes—
415C22×C4C23×C412yes—
416C22×C4C23×C412yes—
417C22×C4C23×C414yes—
418C22×C4C22×D414yes—
419C22×C4C22×D414yes—
420C22×C4C22×D412yes—
421C22×C4C22×D412yes—
422C22×C4C22×Q814yes—
423C22×C4C22×Q812yes—
424C22×C4C22×Q812yes—
425C22×C4C22×Q814yes—
426C22×C4H(5,𝔽2)14yes—
427C22×C42−1+414yes—
428C4⋊C4—416no—
429C4⋊C4—416no—
430C4⋊C4—48no—
431C4⋊C4—48no—
432C4⋊C4—416no—
433C4⋊C4—82no—
434C4⋊C4—82no—
435C4⋊C4—82no—
436C4⋊C4—82no—
437C4⋊C4—88no—
438C4⋊C4—88no—
439C4⋊C4—88no—
440C4⋊C4—88no—
441C4⋊C4C2×C4⋊C4132yes—
442C4⋊C4C2×C4⋊C4132yes—
443C22⋊C4—216no—
444C22⋊C4—216no—
445C22⋊C4—48no—
446C22⋊C4—48no—
447C42—48no—
448C42—48no—
449C42—416no—
450C42—84no—
451C42—84no—
452C23—24no—
453C23—24no—
454C23—22no—
455C23—22no—
456C23—22no—
457C23—22no—
458C23—24no—
459C23—24no—
460C23—28no—
461C23—28no—
462C23—28no—
463C23—28no—
464C23C22×C4⋊C414yes—
465C23#52018yes—
466C23#53514yes—
467C23#53614yes—
468C23#54418yes—
469C23C24×C411yescharacteristic
470C23C23×D411yescharacteristic
471C23C23×Q811yescharacteristic
472C23C2×H(5,𝔽2)12yes—
473C23C2×H(5,𝔽2)12yes—
474C23C2×2−1+412yes—
475C23C2×2−1+412yes—
476C23#58512yes—
477C23#58512yes—
478C2×C4—232no—
479C2×C4—28no—
480C2×C4—24no—
481C2×C4—24no—
482C2×C4—28no—
483C2×C4—24no—
484C2×C4—24no—
485C2×C4—28no—
486C2×C4—24no—
487C2×C4—22no—
488C2×C4—22no—
489C2×C4—24no—
490C2×C4—22no—
491C2×C4—22no—
492C2×C4—24no—
493C2×C4—24no—
494C2×C4—28no—
495C2×C4—28no—
496C2×C4—28no—
497C2×C4—28no—
498C2×C4—28no—
499C2×C4—28no—
500C2×C4—28no—
501C2×C4—24no—
502C2×C4—24no—
503C2×C4—28no—
504C2×C4—28no—
505C2×C4—416no—
506C2×C4—416no—
507C2×C4—416no—
508C2×C4—416no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—44no—
514C2×C4—44no—
515C2×C4—44no—
516C2×C4—44no—
517C2×C4—44no—
518C2×C4—44no—
519C2×C4—44no—
520C2×C4—44no—
521C2×C4—44no—
522C2×C4—44no—
523C2×C4—42no—
524C2×C4—42no—
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—42no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—44no—
542C2×C4—44no—
543C2×C4—44no—
544C2×C4—44no—
545C2×C4—44no—
546C2×C4—44no—
547C2×C4—44no—
548C2×C4—44no—
549C2×C4—48no—
550C2×C4—48no—
551C2×C4—88no—
552C2×C4—88no—
553C2×C4C22×C4⋊C418yes—
554C2×C4C22×C4⋊C414yes—
555C2×C4C22×C4⋊C414yes—
556C2×C4C22×C4⋊C418yes—
557C22—24no—
558C22—216no—
559C22—24no—
560C22—216no—
561C22—24no—
562C22—24no—
563C22—24no—
564C22—24no—
565C22#231718yes—
566C22C23×C4⋊C411yescharacteristic
567C22C2×#52012yes—
568C22C4×H(5,𝔽2)12yes—
569C22C4×2−1+412yes—
570C22C2×#53511yescharacteristic
571C22C2×#53611yescharacteristic
572C22C2×#54412yes—
573C22#348411yescharacteristic
574C22#348511yescharacteristic
575C22#348611yescharacteristic
576C22#348711yescharacteristic
577C22#350112yes—
578C22#350212yes—
579C22#353414yes—
580C22#353514yes—
581C22#355214yes—
582C22#355314yes—
583C4—28no—
584C4—24no—
585C4—24no—
586C4—28no—
587C4—44no—
588C4—48no—
589C4—48no—
590C4—44no—
591C4—82no—
592C4—82no—
593C4—82no—
594C4—82no—
595C4—82no—
596C4—82no—
597C2—28no—
598C2C2×#231711yescharacteristic
599C2#3424811yescharacteristic
600C2#3424911yescharacteristic
601C2#3441714yes—
602C2#3711811yescharacteristic
603C2#3712011yescharacteristic
604C2#3713514yes—
605C2#3718511yescharacteristic
606C2#3723311yescharacteristic
subgroupG/H
center, Frattini, socle, cosocleC24C25
commutatorC23C24×C4
prime radicalC25C24
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
5. #14502C2semidirect
6. #14468C2semidirect
7. #14406C2semidirect
8. #14404C2semidirect
22. #2313C4semidirect
127. #387C2×C4semidirect
128. #387C2×C4semidirect
255. C2×C4⋊C4C4⋊C4semidirect

Maximal subgroups

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

subgroupindex
C2×#23132
C22×#3892
#147052
#145022
#144682
#144062
#144042
#142632
#141492
#141302
#139662
#139652
#138812
#138802