Finite Group Databasebeta

← #658209

Subgroups — #658209

Order 512.

2934 conjugacy classes of subgroups.

The subgroup lattice has 704 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2442C211yesmaximal, characteristic
2C22×#384C211yesmaximal, characteristic
3#15798C214yesmaximal, complemented
4#15766C214yesmaximal, complemented
5#15228C212yesmaximal, complemented
6#15133C212yesmaximal, complemented
7#15092C212yesmaximal
8#14821C212yesmaximal
9#14697C211yesmaximal, characteristic
10#14130C211yesmaximal, characteristic
11#14043C212yesmaximal
12#14005C212yesmaximal, complemented
13#13923C212yesmaximal
14#13909C212yesmaximal, complemented
15#13881C211yesmaximal, characteristic
16C22⋊C4×C4⋊C4C211yesmaximal, characteristic
17C4×#385C211yesmaximal, characteristic
18C22×#125C2211yescharacteristic
19C23×C4⋊C4C2212yes—
20C23×C4⋊C4C2211yescharacteristic
21C23×C4⋊C4C2212yes—
22C23×C4⋊C4C2211yescharacteristic
23C23×C42C2211yescharacteristic
24#2442—28no—
25C2×#400C2212yes—
26C2×#398C2212yes—
27C2×#398C2214yes—
28C2×#389C2211yescharacteristic
29C2×#389C2212yes—
30C2×#389C2211yescharacteristic
31C2×#389C2212yes—
32C2×#389C2214yes—
33C2×#385C2212yes—
34C2×#385C2212yes—
35C2×#385C2211yescharacteristic
36C2×#385C2211yescharacteristic
37C2×#385C2212yes—
38C2×#384—28no—
39C2×#384C2212yes—
40C2×#384C2212yes—
41C2×#384C2214yes—
42C2×#382C2211yescharacteristic
43C2×#382C2212yes—
44C2×#380C2211yescharacteristic
45C2×#380C2212yes—
46C2×#380C2211yescharacteristic
47C2×C4×C4⋊C4C2212yes—
48C2×C4×C4⋊C4C2212yes—
49C2×C4×C22⋊C4C2211yescharacteristic
50C2×C4×C22⋊C4C2212yes—
51C2×C4×C22⋊C4C2211yescharacteristic
52#1444C2212yes—
53#1444C2214yes—
54#1444C2214yes—
55#1444C2214yes—
56#1442C2212yes—
57#1442C2214yes—
58#1441C2214yes—
59#1441C2214yes—
60#1441C2214yes—
61#1441C2214yes—
62#1441C2214yes—
63#1441C2214yes—
64#1441C2214yes—
65#1441C2214yes—
66#1436C2214yes—
67#1436C2214yes—
68#1436C2214yes—
69#1436C2214yes—
70#1436C2212yes—
71#1436C2214yes—
72#1436C2212yes—
73C4⋊#95C2212yes—
74C4⋊#95C2212yes—
75C4⋊#95C2212yes—
76C4⋊#95C2212yes—
77C22⋊#95C2212yes—
78C22⋊#95C2212yes—
79C22⋊#95C2212yes—
80C22⋊#95C2211yescharacteristic
81C22⋊#95C2212yes—
82C22⋊#95C2211yescharacteristic
83C22⋊#95C2212yes—
84C4×#95C2212yes—
85C24×C4C2311yescharacteristic
86C24×C4C2311yescharacteristic
87C24×C4C2311yescharacteristic
88C24×C4C2312yes—
89C24×C4C2312yes—
90C2×#125—24no—
91C2×#125—44no—
92C22×C4⋊C4—28no—
93C22×C4⋊C4—28no—
94C22×C4⋊C4—24no—
95C22×C4⋊C4—28no—
96C22×C4⋊C4—44no—
97C22×C4⋊C4—44no—
98C22×C4⋊C4—48no—
99C22×C4⋊C4—44no—
100C22×C4⋊C4D418yes—
101C22×C4⋊C4C2312yes—
102C22×C4⋊C4C2314yes—
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2311yescharacteristic
105C22×C4⋊C4C2311yescharacteristic
106C22×C4⋊C4C2312yes—
107C22×C4⋊C4C2314yes—
108C22×C4⋊C4C2314yes—
109C22×C4⋊C4C2314yes—
110C22×C4⋊C4C2314yes—
111C22×C4⋊C4C2311yescharacteristic
112C22×C4⋊C4C2311yescharacteristic
113C22×C4⋊C4C2312yes—
114C22×C4⋊C4C2312yes—
115C22×C4⋊C4C2312yes—
116C22×C4⋊C4C2312yes—
117C22×C4⋊C4C2312yes—
118C22×C4⋊C4C2312yes—
119C22×C4⋊C4C2314yes—
120C22×C4⋊C4C2312yes—
121C22×C22⋊C4C2311yescharacteristic
122C22×C22⋊C4C2312yes—
123C22×C22⋊C4C2311yescharacteristic
124C22×C22⋊C4C2311yescharacteristic
125C22×C22⋊C4C2311yescharacteristic
126C22×C22⋊C4C2312yes—
127C22×C42—24no—
128C22×C42—28no—
129C22×C42C2312yes—
130C22×C42C2311yescharacteristic
131C22×C42C2311yescharacteristic
132C22×C42C2312yes—
133C22×C42C2312yes—
134C22×C42C2314yes—
135#400—28no—
136#398—28no—
137#398—48no—
138#389—24no—
139#389—24no—
140#389—28no—
141#389—48no—
142#389—44no—
143#385—28no—
144#385—24no—
145#385—28no—
146#385—24no—
147#385—44no—
148#384—28no—
149#384—28no—
150#384—216no—
151#382—24no—
152#382—44no—
153#380—28no—
154#380—24no—
155#380Q818yes—
156C4×C4⋊C4—28no—
157C4×C4⋊C4—44no—
158C4×C22⋊C4—24no—
159C4×C22⋊C4—44no—
160C4×C22⋊C4Q818yes—
161C2×#95C2311yescharacteristic
162C2×#95C2312yes—
163C2×#95C2312yes—
164C2×#95C2311yescharacteristic
165C2×#95C2312yes—
166C2×#95C2312yes—
167C2×#95C2312yes—
168C2×#95C2312yes—
169C2×#95C2312yes—
170C2×#95C2312yes—
171C2×#95C2312yes—
172C2×#95C2314yes—
173C2×#95C2312yes—
174C2×#95C2312yes—
175C2×#95C2314yes—
176C2×#95C2314yes—
177C2×#95C2314yes—
178C2×#95C2314yes—
179C2×#95C2314yes—
180C2×#95C2314yes—
181C2×#95C2314yes—
182C2×#95C2314yes—
183C2×#95C2314yes—
184C2×#95C2314yes—
185C2×#95C2314yes—
186C2×#95C2314yes—
187C2×#95C2312yes—
188C2×#95C2312yes—
189C25C2411yescharacteristic
190C23×C4—22no—
191C23×C4—22no—
192C23×C4—22no—
193C23×C4—22no—
194C23×C4—24no—
195C23×C4—24no—
196C23×C4—24no—
197C23×C4—24no—
198C23×C4—24no—
199C23×C4—24no—
200C23×C4—24no—
201C23×C4—24no—
202C23×C4—24no—
203C23×C4—24no—
204C23×C4—24no—
205C23×C4—44no—
206C23×C4—44no—
207C23×C4—44no—
208C23×C4—44no—
209C23×C4C2×D414yes—
210C23×C4C2×D418yes—
211C23×C4C2×Q814yes—
212C23×C4C2×Q814yes—
213C23×C4#4114yes—
214C23×C4#4114yes—
215C23×C4C2411yescharacteristic
216C23×C4C2412yes—
217C23×C4C2411yescharacteristic
218C23×C4C2412yes—
219C23×C4C2411yescharacteristic
220C23×C4C2411yescharacteristic
221C23×C4C2414yes—
222C23×C4C2412yes—
223C23×C4C2412yes—
224C23×C4C2414yes—
225C23×C4C2412yes—
226C23×C4C2412yes—
227C23×C4C2411yescharacteristic
228C23×C4C2412yes—
229C23×C4C2412yes—
230C23×C4C2411yescharacteristic
231#125—416no—
232C2×C4⋊C4—24no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—28no—
237C2×C4⋊C4—28no—
238C2×C4⋊C4—28no—
239C2×C4⋊C4—28no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—28no—
247C2×C4⋊C4—28no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—28no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—28no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—28no—
257C2×C4⋊C4—24no—
258C2×C4⋊C4—28no—
259C2×C4⋊C4—24no—
260C2×C4⋊C4—28no—
261C2×C4⋊C4—24no—
262C2×C4⋊C4—24no—
263C2×C4⋊C4—24no—
264C2×C4⋊C4—24no—
265C2×C4⋊C4—416no—
266C2×C4⋊C4—44no—
267C2×C4⋊C4—44no—
268C2×C4⋊C4—44no—
269C2×C4⋊C4—44no—
270C2×C4⋊C4—44no—
271C2×C4⋊C4—44no—
272C2×C4⋊C4—44no—
273C2×C4⋊C4—44no—
274C2×C4⋊C4—432no—
275C2×C4⋊C4—432no—
276C2×C4⋊C4—416no—
277C2×C4⋊C4—44no—
278C2×C4⋊C4—44no—
279C2×C4⋊C4—44no—
280C2×C4⋊C4—44no—
281C2×C4⋊C4—44no—
282C2×C4⋊C4C2×D414yes—
283C2×C4⋊C4C2×D414yes—
284C2×C4⋊C4C2×D418yes—
285C2×C4⋊C4C2×Q818yes—
286C2×C22⋊C4—22no—
287C2×C22⋊C4—22no—
288C2×C22⋊C4—24no—
289C2×C22⋊C4—22no—
290C2×C22⋊C4—24no—
291C2×C22⋊C4—22no—
292C2×C22⋊C4—22no—
293C2×C22⋊C4—22no—
294C2×C22⋊C4—24no—
295C2×C22⋊C4—22no—
296C2×C22⋊C4—22no—
297C2×C22⋊C4—24no—
298C2×C22⋊C4—22no—
299C2×C22⋊C4—22no—
300C2×C22⋊C4—24no—
301C2×C22⋊C4C2×Q814yes—
302C2×C22⋊C4C2×Q814yes—
303C2×C22⋊C4C2×Q818yes—
304C2×C42—232no—
305C2×C42—24no—
306C2×C42—24no—
307C2×C42—24no—
308C2×C42—24no—
309C2×C42—24no—
310C2×C42—24no—
311C2×C42—24no—
312C2×C42—22no—
313C2×C42—22no—
314C2×C42—24no—
315C2×C42—24no—
316C2×C42—44no—
317C2×C42—44no—
318C2×C42—44no—
319C2×C42C2×Q818yes—
320C2×C42#4118yes—
321#95—22no—
322#95—22no—
323#95—24no—
324#95—24no—
325#95—24no—
326#95—24no—
327#95—24no—
328#95—24no—
329#95—24no—
330#95—24no—
331#95—24no—
332#95—28no—
333#95—28no—
334#95—24no—
335#95—24no—
336#95—28no—
337#95—44no—
338#95—44no—
339#95—44no—
340#95—44no—
341#95—44no—
342#95—44no—
343#95—44no—
344#95—44no—
345#95—44no—
346#95—44no—
347#95C2×Q818yes—
348#95C2×Q818yes—
349C24—22no—
350C24—22no—
351C24—22no—
352C24—22no—
353C24C22×D412yes—
354C24C22×Q812yes—
355C24C22×Q812yes—
356C24C2×#4112yes—
357C24C2×#4112yes—
358C24H(5,𝔽2)12yes—
359C242−1+412yes—
360C24C2511yescharacteristic
361C22×C4—28no—
362C22×C4—28no—
363C22×C4—28no—
364C22×C4—28no—
365C22×C4—216no—
366C22×C4—216no—
367C22×C4—28no—
368C22×C4—28no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—28no—
380C22×C4—28no—
381C22×C4—216no—
382C22×C4—216no—
383C22×C4—216no—
384C22×C4—216no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—24no—
389C22×C4—24no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—24no—
399C22×C4—24no—
400C22×C4—24no—
401C22×C4—22no—
402C22×C4—24no—
403C22×C4—24no—
404C22×C4—22no—
405C22×C4—24no—
406C22×C4—24no—
407C22×C4—24no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—216no—
412C22×C4—21no—
413C22×C4—21no—
414C22×C4—21no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—21no—
418C22×C4—24no—
419C22×C4—21no—
420C22×C4—22no—
421C22×C4—21no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—24no—
425C22×C4—21no—
426C22×C4—22no—
427C22×C4—21no—
428C22×C4—24no—
429C22×C4—22no—
430C22×C4—21no—
431C22×C4—22no—
432C22×C4—24no—
433C22×C4—24no—
434C22×C4—22no—
435C22×C4—24no—
436C22×C4—24no—
437C22×C4—24no—
438C22×C4—24no—
439C22×C4—24no—
440C22×C4—22no—
441C22×C4—21no—
442C22×C4—24no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—21no—
446C22×C4—22no—
447C22×C4—21no—
448C22×C4—22no—
449C22×C4—48no—
450C22×C4—48no—
451C22×C4—48no—
452C22×C4—48no—
453C22×C4—48no—
454C22×C4—88no—
455C22×C4—88no—
456C22×C4—88no—
457C22×C4—88no—
458C22×C4C4⋊Q8116yes—
459C22×C4C22×D414yes—
460C22×C4C22×D414yes—
461C22×C4C22×D412yes—
462C22×C4C22×D412yes—
463C22×C4C22×Q814yes—
464C22×C4C22×Q814yes—
465C22×C4C22×Q818yes—
466C22×C4C22×Q812yes—
467C22×C4C22×Q812yes—
468C22×C4C22×Q812yes—
469C22×C4C22×Q812yes—
470C22×C4C2×#4114yes—
471C22×C4C2×#4114yes—
472C22×C4C2×#4114yes—
473C22×C4C2×#4114yes—
474C22×C4C2×#4114yes—
475C22×C4H(5,𝔽2)14yes—
476C22×C4H(5,𝔽2)14yes—
477C22×C42−1+414yes—
478C4⋊C4—28no—
479C4⋊C4—28no—
480C4⋊C4—216no—
481C4⋊C4—48no—
482C4⋊C4—416no—
483C4⋊C4—416no—
484C4⋊C4—416no—
485C4⋊C4—416no—
486C4⋊C4—48no—
487C4⋊C4—48no—
488C4⋊C4—48no—
489C4⋊C4—44no—
490C4⋊C4—48no—
491C4⋊C4—44no—
492C4⋊C4—48no—
493C4⋊C4—48no—
494C4⋊C4—416no—
495C4⋊C4—84no—
496C4⋊C4—84no—
497C4⋊C4—88no—
498C22⋊C4—28no—
499C22⋊C4—28no—
500C22⋊C4—216no—
501C22⋊C4—44no—
502C22⋊C4—44no—
503C22⋊C4—48no—
504C42—216no—
505C42—216no—
506C42—216no—
507C42—44no—
508C42—44no—
509C42—88no—
510C23—22no—
511C23—22no—
512C23—24no—
513C23—24no—
514C23—24no—
515C23—24no—
516C23—22no—
517C23—22no—
518C23—24no—
519C23—24no—
520C23—24no—
521C23—24no—
522C23—24no—
523C23—24no—
524C23—24no—
525C23—24no—
526C23C2×C4⋊Q814yes—
527C23#53314yes—
528C23#54414yes—
529C23#54614yes—
530C23#54814yes—
531C23#55414yes—
532C23#55714yes—
533C23C23×D411yescharacteristic
534C23C23×Q811yescharacteristic
535C23C23×Q811yescharacteristic
536C23C22×#4112yes—
537C23C22×#4111yescharacteristic
538C23C22×#4111yescharacteristic
539C23C2×H(5,𝔽2)12yes—
540C23C2×H(5,𝔽2)11yescharacteristic
541C23C2×H(5,𝔽2)12yes—
542C23C2×2−1+411yescharacteristic
543C23C2×2−1+412yes—
544C2×C4—232no—
545C2×C4—232no—
546C2×C4—22no—
547C2×C4—24no—
548C2×C4—24no—
549C2×C4—22no—
550C2×C4—24no—
551C2×C4—24no—
552C2×C4—24no—
553C2×C4—24no—
554C2×C4—24no—
555C2×C4—24no—
556C2×C4—24no—
557C2×C4—24no—
558C2×C4—24no—
559C2×C4—24no—
560C2×C4—24no—
561C2×C4—24no—
562C2×C4—232no—
563C2×C4—24no—
564C2×C4—24no—
565C2×C4—24no—
566C2×C4—24no—
567C2×C4—24no—
568C2×C4—24no—
569C2×C4—24no—
570C2×C4—24no—
571C2×C4—28no—
572C2×C4—28no—
573C2×C4—24no—
574C2×C4—24no—
575C2×C4—24no—
576C2×C4—28no—
577C2×C4—24no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—44no—
581C2×C4—44no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—44no—
586C2×C4—44no—
587C2×C4—42no—
588C2×C4—44no—
589C2×C4—42no—
590C2×C4—44no—
591C2×C4—44no—
592C2×C4—44no—
593C2×C4—42no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—42no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—42no—
600C2×C4—44no—
601C2×C4—41no—
602C2×C4—41no—
603C2×C4—44no—
604C2×C4—41no—
605C2×C4—42no—
606C2×C4—41no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—44no—
611C2×C4—44no—
612C2×C4—41no—
613C2×C4—41no—
614C2×C4—42no—
615C2×C4—41no—
616C2×C4—42no—
617C2×C4—41no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—44no—
621C2×C4—42no—
622C2×C4—44no—
623C2×C4—44no—
624C2×C4—42no—
625C2×C4—44no—
626C2×C4—42no—
627C2×C4—44no—
628C2×C4—44no—
629C2×C4—42no—
630C2×C4—44no—
631C2×C4—816no—
632C2×C4—816no—
633C2×C4#52514yes—
634C2×C4C2×C4⋊Q814yes—
635C2×C4C2×C4⋊Q814yes—
636C2×C4#53414yes—
637C2×C4#55218yes—
638C2×C4#55318yes—
639C2×C4#55418yes—
640C2×C4#55418yes—
641C22—24no—
642C22—24no—
643C22—24no—
644C22—24no—
645C22—28no—
646C22—28no—
647C22—28no—
648C22—24no—
649C22—28no—
650C22—24no—
651C22#255018yes—
652C22C2×#52511yescharacteristic
653C22C22×C4⋊Q811yescharacteristic
654C22C2×#53311yescharacteristic
655C22C2×#53411yescharacteristic
656C22C2×#54411yescharacteristic
657C22C2×#54611yescharacteristic
658C22C2×#54811yescharacteristic
659C22C2×#55212yes—
660C22C2×#55312yes—
661C22C2×#55412yes—
662C22C2×#55412yes—
663C22C2×#55411yescharacteristic
664C22C2×#55711yescharacteristic
665C22#348411yescharacteristic
666C22#348611yescharacteristic
667C22#349212yes—
668C22#349212yes—
669C22#350312yes—
670C22#350312yes—
671C22#350812yes—
672C22#351112yes—
673C22#354312yes—
674C22#355212yes—
675C4—24no—
676C4—28no—
677C4—28no—
678C4—44no—
679C4—44no—
680C4—44no—
681C4—48no—
682C4—44no—
683C4—81no—
684C4—84no—
685C4—81no—
686C4—82no—
687C4—81no—
688C4—81no—
689C4—82no—
690C4#258318yes—
691C2—28no—
692C2C2×#258311yescharacteristic
693C2C2×#255011yescharacteristic
694C2#3557611yescharacteristic
695C2#3616311yescharacteristic
696C2#3618912yes—
697C2#3658111yescharacteristic
698C2#3787412yes—
699C2#3946411yescharacteristic
700C2#3972111yescharacteristic
701C2#5930512yes—
702C2#5932112yes—
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. #15798C2semidirect
4. #15766C2semidirect
5. #15228C2semidirect
6. #15133C2semidirect
12. #14005C2semidirect
14. #13909C2semidirect

Maximal subgroups

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

subgroupindex
C2×#24422
C22×#3842
#157982
#157662
#152282
#151332
#150922
#148212
#146972
#141302
#140432
#140052
#139232
#139092
#138812
C22⋊C4×C4⋊C42
C4×#3852