Finite Group Databasebeta

← #554248

Subgroups — #554248

Order 512.

2914 conjugacy classes of subgroups.

The subgroup lattice has 623 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2484C211yesmaximal, characteristic
2C2×#2317C211yesmaximal, characteristic
3C22×#389C211yesmaximal, characteristic
4#14695C212yesmaximal
5#14552C214yesmaximal, complemented
6#14548C214yesmaximal, complemented
7#14406C214yesmaximal, complemented
8#14394C214yesmaximal, complemented
9#14263C212yesmaximal
10#14128C211yesmaximal, characteristic
11#14043C212yesmaximal
12#13972C212yesmaximal
13#13958C211yesmaximal, characteristic
14#13870C211yesmaximal, characteristic
15#13776C211yesmaximal, characteristic
16C22×#117C2211yescharacteristic
17C22×#117C2211yescharacteristic
18C23×C4⋊C4C2211yescharacteristic
19C23×C4⋊C4C2211yescharacteristic
20C23×C4⋊C4C2211yescharacteristic
21C23×C4⋊C4C2211yescharacteristic
22C23×C42C2211yescharacteristic
23#2484C4116yescomplemented
24C2×#400C2212yes—
25C2×#398C2212yes—
26#2317—28no—
27C2×#389—28no—
28C2×#389C2212yes—
29C2×#389C2212yes—
30C2×#389C2214yes—
31C2×#389C2214yes—
32C2×#389C2214yes—
33C2×#387C2214yes—
34C2×#387C2214yes—
35C2×#385C2212yes—
36C2×#385C2212yes—
37C2×#385C2212yes—
38C2×#385C2212yes—
39C2×#384C2212yes—
40C2×#382C2212yes—
41C2×#381C2211yescharacteristic
42C2×#381C2212yes—
43C2×#381C2211yescharacteristic
44C2×#381C2212yes—
45C2×C4×C22⋊C4C2212yes—
46C22×#95C2211yescharacteristic
47C22×#95C2211yescharacteristic
48#1445C2212yes—
49#1445C2212yes—
50#1444C2212yes—
51#1444C2214yes—
52#1444C2214yes—
53#1444C2214yes—
54#1443C2214yes—
55#1442C2212yes—
56#1442C2214yes—
57#1442C2214yes—
58#1442C2214yes—
59#1441C2212yes—
60#1441C2212yes—
61#1441C2214yes—
62#1436C2214yes—
63#1436C2214yes—
64#1436C2214yes—
65#1436C2214yes—
66#1436C2214yes—
67#1436C2214yes—
68#1436C2214yes—
69#1436C2214yes—
70#1434C2214yes—
71#1434C2214yes—
72#1434C2214yes—
73C22⋊#95C2212yes—
74C22⋊#95C2212yes—
75C22⋊#95C2212yes—
76C22⋊#95C2212yes—
77C22⋊#95C2212yes—
78C24×C4C2311yescharacteristic
79C24×C4C2311yescharacteristic
80C24×C4C2311yescharacteristic
81C24×C4C2311yescharacteristic
82C24×C4C2311yescharacteristic
83C24×C4C2311yescharacteristic
84C24×C4C2311yescharacteristic
85C2×#117—24no—
86C2×#117—24no—
87C2×#117—24no—
88C2×#117—44no—
89C2×#117C2×C418yes—
90C22×C4⋊C4—24no—
91C22×C4⋊C4—24no—
92C22×C4⋊C4—24no—
93C22×C4⋊C4—24no—
94C22×C4⋊C4—44no—
95C22×C4⋊C4—44no—
96C22×C4⋊C4—44no—
97C22×C4⋊C4C2×C418yes—
98C22×C4⋊C4D418yes—
99C22×C4⋊C4C2314yes—
100C22×C4⋊C4C2311yescharacteristic
101C22×C4⋊C4C2312yes—
102C22×C4⋊C4C2314yes—
103C22×C4⋊C4C2311yescharacteristic
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2314yes—
106C22×C4⋊C4C2311yescharacteristic
107C22×C4⋊C4C2311yescharacteristic
108C22×C4⋊C4C2312yes—
109C22×C4⋊C4C2312yes—
110C22×C4⋊C4C2312yes—
111C22×C4⋊C4C2312yes—
112C22×C4⋊C4C2314yes—
113C22×C4⋊C4C2312yes—
114C22×C4⋊C4C2312yes—
115C22×C22⋊C4C2312yes—
116C22×C22⋊C4C2312yes—
117C22×C22⋊C4C2312yes—
118C22×C22⋊C4C2312yes—
119C22×C42—24no—
120C22×C42—44no—
121C22×C42C2314yes—
122C22×C42C2312yes—
123C22×C42C2312yes—
124C22×C42C2314yes—
125#400C2×C4116yescomplemented
126#398C2×C4116yescomplemented
127#389—28no—
128#389—28no—
129#389—216no—
130#389—216no—
131#389—48no—
132#387—48no—
133#387C2×C4132yescomplemented
134#385—28no—
135#385—28no—
136#385—44no—
137#385C2×C4116yes—
138#384—28no—
139#382—28no—
140#381—24no—
141#381—44no—
142#381C2×C418yes—
143#381C2×C4116yes—
144C4×C22⋊C4—44no—
145C2×#95—24no—
146C2×#95—24no—
147C2×#95C2314yes—
148C2×#95C2314yes—
149C2×#95C2314yes—
150C2×#95C2314yes—
151C2×#95C2312yes—
152C2×#95C2312yes—
153C2×#95C2312yes—
154C2×#95C2312yes—
155C2×#95C2314yes—
156C2×#95C2312yes—
157C2×#95C2312yes—
158C2×#95C2312yes—
159C2×#95C2312yes—
160C2×#95C2314yes—
161C2×#95C2314yes—
162C2×#95C2314yes—
163C2×#95C2314yes—
164C2×#95C2314yes—
165C2×#95C2314yes—
166C2×#95C2314yes—
167C2×#95C2314yes—
168C2×#95C2314yes—
169C2×#95C2314yes—
170C2×#95C2312yes—
171C2×#95C2314yes—
172C2×#95C2314yes—
173C2×#95C2314yes—
174C2×#95C2312yes—
175C25C2411yescharacteristic
176C23×C4—22no—
177C23×C4—22no—
178C23×C4—24no—
179C23×C4—24no—
180C23×C4—24no—
181C23×C4—24no—
182C23×C4—22no—
183C23×C4—22no—
184C23×C4—22no—
185C23×C4—24no—
186C23×C4—24no—
187C23×C4—24no—
188C23×C4—24no—
189C23×C4—24no—
190C23×C4—22no—
191C23×C4—24no—
192C23×C4—24no—
193C23×C4—42no—
194C23×C4—42no—
195C23×C4—42no—
196C23×C4—42no—
197C23×C4—42no—
198C23×C4—42no—
199C23×C4—42no—
200C23×C4C22×C414yes—
201C23×C4C22×C414yes—
202C23×C4C22×C414yes—
203C23×C4C2×D414yes—
204C23×C4C2×D414yes—
205C23×C4C2×D414yes—
206C23×C4#4114yes—
207C23×C4C2412yes—
208C23×C4C2412yes—
209C23×C4C2414yes—
210C23×C4C2412yes—
211C23×C4C2412yes—
212C23×C4C2412yes—
213C23×C4C2414yes—
214C23×C4C2412yes—
215C23×C4C2412yes—
216C23×C4C2412yes—
217C23×C4C2412yes—
218C23×C4C2411yescharacteristic
219C23×C4C2411yescharacteristic
220C23×C4C2412yes—
221#117—232no—
222#117—416no—
223C2×C4⋊C4—232no—
224C2×C4⋊C4—24no—
225C2×C4⋊C4—24no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—24no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—28no—
233C2×C4⋊C4—28no—
234C2×C4⋊C4—28no—
235C2×C4⋊C4—28no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—28no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—416no—
244C2×C4⋊C4—42no—
245C2×C4⋊C4—42no—
246C2×C4⋊C4—42no—
247C2×C4⋊C4—42no—
248C2×C4⋊C4—44no—
249C2×C4⋊C4—44no—
250C2×C4⋊C4—44no—
251C2×C4⋊C4—42no—
252C2×C4⋊C4—42no—
253C2×C4⋊C4—416no—
254C2×C4⋊C4—416no—
255C2×C4⋊C4—44no—
256C2×C4⋊C4—44no—
257C2×C4⋊C4—44no—
258C2×C4⋊C4—44no—
259C2×C4⋊C4—44no—
260C2×C4⋊C4—44no—
261C2×C4⋊C4—44no—
262C2×C4⋊C4C22×C418yes—
263C2×C4⋊C4C22×C418yes—
264C2×C4⋊C4C22×C414yes—
265C2×C4⋊C4C22×C414yes—
266C2×C4⋊C4C22×C4116yes—
267C2×C4⋊C4C2×D418yes—
268C2×C4⋊C4C2×D418yes—
269C2×C22⋊C4—24no—
270C2×C22⋊C4—24no—
271C2×C22⋊C4—24no—
272C2×C22⋊C4—24no—
273C2×C22⋊C4—24no—
274C2×C22⋊C4—24no—
275C2×C22⋊C4—24no—
276C2×C22⋊C4—24no—
277C2×C22⋊C4—24no—
278C2×C22⋊C4—24no—
279C2×C22⋊C4C22×C418yes—
280C2×C22⋊C4C22×C418yes—
281C2×C42—24no—
282C2×C42—24no—
283C2×C42—24no—
284C2×C42—28no—
285C2×C42—28no—
286C2×C42—42no—
287C2×C42—42no—
288C2×C42—44no—
289C2×C42—416no—
290C2×C42—44no—
291C2×C42—44no—
292C2×C42—44no—
293C2×C42C22×C418yes—
294#95—28no—
295#95—24no—
296#95—24no—
297#95—24no—
298#95—24no—
299#95—24no—
300#95—24no—
301#95—24no—
302#95—24no—
303#95—28no—
304#95—28no—
305#95—28no—
306#95—28no—
307#95—44no—
308#95—44no—
309#95—44no—
310#95—44no—
311#95—44no—
312#95—44no—
313#95—44no—
314#95—44no—
315#95—44no—
316#95—44no—
317#95C22×C418yes—
318#95C22×C4116yes—
319#95C22×C4116yes—
320#95C22×C418yes—
321#95C22×C4116yes—
322C24—22no—
323C24—22no—
324C24—22no—
325C24—22no—
326C24C23×C412yes—
327C24C22×D412yes—
328C24C2×#4112yes—
329C24H(5,𝔽2)14yes—
330C242−1+414yes—
331C24C2511yescharacteristic
332C22×C4—28no—
333C22×C4—28no—
334C22×C4—216no—
335C22×C4—28no—
336C22×C4—216no—
337C22×C4—28no—
338C22×C4—28no—
339C22×C4—216no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—216no—
349C22×C4—28no—
350C22×C4—28no—
351C22×C4—216no—
352C22×C4—28no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—24no—
361C22×C4—24no—
362C22×C4—24no—
363C22×C4—24no—
364C22×C4—24no—
365C22×C4—22no—
366C22×C4—24no—
367C22×C4—22no—
368C22×C4—24no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—24no—
372C22×C4—24no—
373C22×C4—24no—
374C22×C4—24no—
375C22×C4—24no—
376C22×C4—24no—
377C22×C4—22no—
378C22×C4—24no—
379C22×C4—24no—
380C22×C4—24no—
381C22×C4—24no—
382C22×C4—24no—
383C22×C4—28no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—24no—
388C22×C4—24no—
389C22×C4—24no—
390C22×C4—24no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—24no—
395C22×C4—24no—
396C22×C4—22no—
397C22×C4—24no—
398C22×C4—24no—
399C22×C4—24no—
400C22×C4—24no—
401C22×C4—24no—
402C22×C4—24no—
403C22×C4—22no—
404C22×C4—24no—
405C22×C4—22no—
406C22×C4—44no—
407C22×C4—44no—
408C22×C4—44no—
409C22×C4—44no—
410C22×C4—416no—
411C22×C4—44no—
412C22×C4—416no—
413C22×C4—44no—
414C22×C4—44no—
415C22×C4—44no—
416C22×C4—48no—
417C22×C4—44no—
418C22×C4—44no—
419C22×C4—44no—
420C22×C4—88no—
421C22×C4—88no—
422C22×C4C4×D4116yes—
423C22×C4C23×C418yes—
424C22×C4C23×C414yes—
425C22×C4C23×C414yes—
426C22×C4C23×C414yes—
427C22×C4C23×C412yes—
428C22×C4C23×C412yes—
429C22×C4C23×C414yes—
430C22×C4C22×D414yes—
431C22×C4C22×D414yes—
432C22×C4C22×D412yes—
433C22×C4C22×D412yes—
434C22×C4C2×#4112yes—
435C22×C4C2×#4112yes—
436C22×C4H(5,𝔽2)14yes—
437C22×C42−1+414yes—
438C22×C42−1+414yes—
439C4⋊C4—216no—
440C4⋊C4—216no—
441C4⋊C4—28no—
442C4⋊C4—28no—
443C4⋊C4—232no—
444C4⋊C4—48no—
445C4⋊C4—48no—
446C4⋊C4—416no—
447C4⋊C4—416no—
448C4⋊C4—44no—
449C4⋊C4—44no—
450C4⋊C4—84no—
451C4⋊C4—84no—
452C4⋊C4—84no—
453C4⋊C4—84no—
454C4⋊C4—88no—
455C22⋊C4—216no—
456C22⋊C4—216no—
457C22⋊C4—48no—
458C22⋊C4—48no—
459C42—216no—
460C42—416no—
461C42—48no—
462C42—88no—
463C23—24no—
464C23—24no—
465C23—22no—
466C23—22no—
467C23—22no—
468C23—22no—
469C23—24no—
470C23—24no—
471C23—28no—
472C23—28no—
473C23—28no—
474C23—28no—
475C23C2×C4×D414yes—
476C23#52018yes—
477C23#53414yes—
478C23#53614yes—
479C23#53918yes—
480C23C24×C411yescharacteristic
481C23C23×D411yescharacteristic
482C23C22×#4111yescharacteristic
483C23C2×H(5,𝔽2)12yes—
484C23C2×H(5,𝔽2)12yes—
485C23C2×2−1+412yes—
486C23C2×2−1+412yes—
487C23C2×2−1+412yes—
488C23#58512yes—
489C2×C4—232no—
490C2×C4—232no—
491C2×C4—24no—
492C2×C4—24no—
493C2×C4—24no—
494C2×C4—24no—
495C2×C4—22no—
496C2×C4—22no—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—24no—
500C2×C4—24no—
501C2×C4—24no—
502C2×C4—24no—
503C2×C4—24no—
504C2×C4—24no—
505C2×C4—24no—
506C2×C4—24no—
507C2×C4—24no—
508C2×C4—28no—
509C2×C4—28no—
510C2×C4—28no—
511C2×C4—28no—
512C2×C4—28no—
513C2×C4—28no—
514C2×C4—24no—
515C2×C4—24no—
516C2×C4—24no—
517C2×C4—24no—
518C2×C4—24no—
519C2×C4—24no—
520C2×C4—24no—
521C2×C4—28no—
522C2×C4—416no—
523C2×C4—416no—
524C2×C4—416no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—44no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—44no—
535C2×C4—44no—
536C2×C4—44no—
537C2×C4—44no—
538C2×C4—44no—
539C2×C4—44no—
540C2×C4—44no—
541C2×C4—44no—
542C2×C4—44no—
543C2×C4—44no—
544C2×C4—44no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—44no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—44no—
556C2×C4—44no—
557C2×C4—44no—
558C2×C4—44no—
559C2×C4—44no—
560C2×C4—44no—
561C2×C4—44no—
562C2×C4—44no—
563C2×C4—44no—
564C2×C4—44no—
565C2×C4—44no—
566C2×C4—44no—
567C2×C4—88no—
568C2×C4—88no—
569C2×C4C2×C4×D414yes—
570C2×C4C2×C4×D414yes—
571C2×C4#52018yes—
572C22—24no—
573C22—24no—
574C22—24no—
575C22—24no—
576C22—24no—
577C22—216no—
578C22—216no—
579C22—24no—
580C22#232218yes—
581C22C22×C4×D411yescharacteristic
582C22C2×#52012yes—
583C22C2×#52012yes—
584C22C4×2−1+412yes—
585C22C2×#53411yescharacteristic
586C22C2×#53611yescharacteristic
587C22C2×#53912yes—
588C22#348411yescharacteristic
589C22#348511yescharacteristic
590C22#348611yescharacteristic
591C22#348611yescharacteristic
592C22#349212yes—
593C22#350712yes—
594C22#353814yes—
595C22#354214yes—
596C22#354414yes—
597C22#356314yes—
598C4—24no—
599C4—24no—
600C4—28no—
601C4—44no—
602C4—44no—
603C4—44no—
604C4—44no—
605C4—44no—
606C4—48no—
607C4—44no—
608C4—82no—
609C4—82no—
610C4—82no—
611C4—84no—
612C2—28no—
613C2C2×#232211yescharacteristic
614C2#3420011yescharacteristic
615C2#3426611yescharacteristic
616C2#3446514yes—
617C2#3670411yescharacteristic
618C2#3670811yescharacteristic
619C2#3682414yes—
620C2#3695911yescharacteristic
621C2#3725811yescharacteristic
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. #14552C2semidirect
6. #14548C2semidirect
7. #14406C2semidirect
8. #14394C2semidirect
23. #2484C4semidirect
125. #400C2×C4semidirect
126. #398C2×C4semidirect
133. #387C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#24842
C2×#23172
C22×#3892
#146952
#145522
#145482
#144062
#143942
#142632
#141282
#140432
#139722
#139582
#138702
#137762