Finite Group Databasebeta

← #518208

Subgroups — #518208

Order 512.

2950 conjugacy classes of subgroups.

The subgroup lattice has 686 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2347C211yesmaximal, characteristic
2C2×#2346C211yesmaximal, characteristic
3C2×#2274C211yesmaximal, characteristic
4#14296C212yesmaximal, complemented
5#14264C211yesmaximal, characteristic
6#14150C212yesmaximal, complemented
7#14130C211yesmaximal, characteristic
8#14118C214yesmaximal
9#14112C214yesmaximal
10#14045C212yesmaximal, complemented
11#14005C212yesmaximal, complemented
12#13952C212yesmaximal, complemented
13#13880C211yesmaximal, characteristic
14#13856C212yesmaximal, complemented
15#13844C211yesmaximal, characteristic
16#13821C212yesmaximal, complemented
17C4⋊C42C212yesmaximal, complemented
18C22×#117C2211yescharacteristic
19C22×#117C2211yescharacteristic
20C23×C4⋊C4C2211yescharacteristic
21#2347—28no—
22#2346C4116yescomplemented
23C2×#389C2212yes—
24C2×#389C2211yescharacteristic
25C2×#389C2211yescharacteristic
26C2×#389C2212yes—
27C2×#389C2214yes—
28C2×#389C2212yes—
29C2×#388C2212yes—
30C2×#388C2212yes—
31C2×#388C2214yes—
32C2×#385C2212yes—
33C2×#385C2214yes—
34C2×#384C2212yes—
35C2×#383C2212yes—
36C2×#382C2212yes—
37C2×#380C2211yescharacteristic
38C2×#380C2211yescharacteristic
39#2274—28no—
40C2×C4×C4⋊C4C2212yes—
41C2×C4×C4⋊C4C2212yes—
42C2×C4×C4⋊C4C2212yes—
43C2×C4×C4⋊C4C2211yescharacteristic
44C2×C4×C4⋊C4C2211yescharacteristic
45C2×C4×C4⋊C4C2212yes—
46C2×C4×C4⋊C4C2212yes—
47C2×C4×C4⋊C4C2212yes—
48C2×C4×C4⋊C4C2214yes—
49C2×C4×C4⋊C4C2214yes—
50C2×C4×C4⋊C4C2212yes—
51C2×C4×C22⋊C4C2211yescharacteristic
52C2×C4×C22⋊C4C2214yes—
53C2×C4×C22⋊C4C2211yescharacteristic
54C2×#376C2212yes—
55#1444C2211yescharacteristic
56#1444C2211yescharacteristic
57#1444C2214yes—
58#1444C2214yes—
59#1444C2212yes—
60#1444C2212yes—
61#1444C2212yes—
62#1444C2212yes—
63#1442C2211yescharacteristic
64#1442C2211yescharacteristic
65#1442C2212yes—
66#1442C2212yes—
67#1441C2214yes—
68#1441C2214yes—
69#1439C2212yes—
70#1439C2212yes—
71#1439C2214yes—
72#1436C2214yes—
73#1436C2214yes—
74#1436C2214yes—
75#1436C2214yes—
76#1436C2214yes—
77#1436C2214yes—
78C4⋊#95C2214yes—
79C4⋊#95C2214yes—
80C4⋊#95C2212yes—
81C4⋊#95C2212yes—
82C4⋊#95C2212yes—
83#1434C2214yes—
84C22⋊#95C2212yes—
85C4×#95C2212yes—
86C24×C4C2311yescharacteristic
87C24×C4C2312yes—
88C2×#117—24no—
89C2×#117—28no—
90C2×#117C2×C418yes—
91C2×#117D418yes—
92C2×#117Q818yes—
93C22×C4⋊C4—24no—
94C22×C4⋊C4—28no—
95C22×C4⋊C4C2312yes—
96C22×C4⋊C4C2311yescharacteristic
97C22×C4⋊C4C2311yescharacteristic
98C22×C4⋊C4C2311yescharacteristic
99C22×C4⋊C4C2311yescharacteristic
100C22×C4⋊C4C2312yes—
101C22×C4⋊C4C2312yes—
102C22×C4⋊C4C2312yes—
103C22×C4⋊C4C2314yes—
104C22×C4⋊C4C2314yes—
105C22×C4⋊C4C2312yes—
106C22×C4⋊C4C2312yes—
107C22×C4⋊C4C2312yes—
108C22×C4⋊C4C2314yes—
109C22×C4⋊C4C2314yes—
110C22×C4⋊C4C2314yes—
111C22×C4⋊C4C2311yescharacteristic
112C22×C4⋊C4C2311yescharacteristic
113C22×C22⋊C4C2312yes—
114C22×C22⋊C4C2311yescharacteristic
115C22×C22⋊C4C2311yescharacteristic
116C22×C22⋊C4C2312yes—
117C22×C22⋊C4C2312yes—
118C22×C22⋊C4C2312yes—
119C22×C22⋊C4C2312yes—
120C22×C42C2311yescharacteristic
121C22×C42C2311yescharacteristic
122C22×C42C2312yes—
123C22×C42C2314yes—
124C22×C42C2312yes—
125C22×C42C2314yes—
126C22×C42C2314yes—
127C22×C42C2312yes—
128C22×C42C2312yes—
129C22×C42C2312yes—
130#389—24no—
131#389—24no—
132#389—28no—
133#389—28no—
134#389—48no—
135#389C2×C4116yescomplemented
136#388—28no—
137#388—28no—
138#388C2×C4132yes—
139#385—48no—
140#385—44no—
141#384—28no—
142#383—28no—
143#382C2×C4116yescomplemented
144#380—24no—
145#380C2×C418yes—
146C4×C4⋊C4—24no—
147C4×C4⋊C4—24no—
148C4×C4⋊C4—28no—
149C4×C4⋊C4—28no—
150C4×C4⋊C4—28no—
151C4×C4⋊C4—28no—
152C4×C4⋊C4—28no—
153C4×C4⋊C4—28no—
154C4×C4⋊C4—48no—
155C4×C4⋊C4—48no—
156C4×C4⋊C4C2×C4116yescomplemented
157C4×C22⋊C4—24no—
158C4×C22⋊C4—216no—
159C4×C22⋊C4C2×C418yes—
160#376C2×C4116yescomplemented
161C2×#95C2312yes—
162C2×#95C2312yes—
163C2×#95C2312yes—
164C2×#95C2314yes—
165C2×#95C2312yes—
166C2×#95C2314yes—
167C2×#95C2314yes—
168C2×#95C2314yes—
169C2×#95C2312yes—
170C2×#95C2312yes—
171C2×#95C2312yes—
172C2×#95C2312yes—
173C2×#95C2312yes—
174C2×#95C2314yes—
175C2×#95C2311yescharacteristic
176C2×#95C2311yescharacteristic
177C2×#95C2312yes—
178C2×#95C2312yes—
179C2×#95C2314yes—
180C2×#95C2314yes—
181C2×#95C2314yes—
182C2×#95C2314yes—
183C2×#95C2314yes—
184C2×#95C2314yes—
185C2×#95C2314yes—
186C2×#95C2312yes—
187C2×#95C2311yescharacteristic
188C2×#95C2311yescharacteristic
189C25C2411yescharacteristic
190C23×C4—24no—
191C23×C4—24no—
192C23×C4—24no—
193C23×C4—28no—
194C23×C4—42no—
195C23×C4—44no—
196C23×C4—42no—
197C23×C4C22×C414yes—
198C23×C4C2×D414yes—
199C23×C4C2×Q814yes—
200C23×C4C2412yes—
201C23×C4C2412yes—
202C23×C4C2411yescharacteristic
203C23×C4C2411yescharacteristic
204C23×C4C2412yes—
205C23×C4C2414yes—
206C23×C4C2412yes—
207C23×C4C2412yes—
208C23×C4C2414yes—
209C23×C4C2411yescharacteristic
210C23×C4C2411yescharacteristic
211C23×C4C2412yes—
212C23×C4C2412yes—
213C23×C4C2411yescharacteristic
214C23×C4C2412yes—
215C23×C4C2411yescharacteristic
216#117—232no—
217#117C4⋊C4164yescomplemented
218C2×C4⋊C4—232no—
219C2×C4⋊C4—24no—
220C2×C4⋊C4—24no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—28no—
224C2×C4⋊C4—24no—
225C2×C4⋊C4—24no—
226C2×C4⋊C4—24no—
227C2×C4⋊C4—24no—
228C2×C4⋊C4—24no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4—24no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—24no—
237C2×C4⋊C4—28no—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—28no—
240C2×C4⋊C4—28no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—28no—
243C2×C4⋊C4—28no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—28no—
248C2×C4⋊C4—44no—
249C2×C4⋊C4—42no—
250C2×C4⋊C4—42no—
251C2×C4⋊C4—44no—
252C2×C4⋊C4—42no—
253C2×C4⋊C4—44no—
254C2×C4⋊C4—44no—
255C2×C4⋊C4—44no—
256C2×C4⋊C4—44no—
257C2×C4⋊C4—44no—
258C2×C4⋊C4—44no—
259C2×C4⋊C4C22×C418yes—
260C2×C4⋊C4C22×C4116yes—
261C2×C4⋊C4C22×C414yes—
262C2×C4⋊C4C22×C414yes—
263C2×C4⋊C4C2×D414yes—
264C2×C4⋊C4C2×D414yes—
265C2×C4⋊C4C2×Q814yes—
266C2×C4⋊C4C2×Q814yes—
267C2×C4⋊C4#4118yes—
268C2×C22⋊C4—24no—
269C2×C22⋊C4—24no—
270C2×C22⋊C4—24no—
271C2×C22⋊C4—24no—
272C2×C22⋊C4—24no—
273C2×C22⋊C4—22no—
274C2×C22⋊C4—22no—
275C2×C22⋊C4—22no—
276C2×C22⋊C4—22no—
277C2×C22⋊C4—24no—
278C2×C22⋊C4—42no—
279C2×C22⋊C4—41no—
280C2×C22⋊C4—41no—
281C2×C22⋊C4—42no—
282C2×C22⋊C4—44no—
283C2×C22⋊C4C22×C418yes—
284C2×C22⋊C4C22×C418yes—
285C2×C22⋊C4C22×C418yes—
286C2×C22⋊C4C2×D418yes—
287C2×C22⋊C4C2×Q818yes—
288C2×C42—24no—
289C2×C42—24no—
290C2×C42—24no—
291C2×C42—24no—
292C2×C42—28no—
293C2×C42—28no—
294C2×C42—22no—
295C2×C42—22no—
296C2×C42—24no—
297C2×C42—24no—
298C2×C42—28no—
299C2×C42—28no—
300C2×C42—24no—
301C2×C42—24no—
302C2×C42—44no—
303C2×C42—44no—
304C2×C42—44no—
305C2×C42—44no—
306C2×C42—44no—
307C2×C42—44no—
308C2×C42C22×C4116yes—
309C2×C42C22×C418yes—
310C2×C42C22×C418yes—
311C2×C42C2×D418yes—
312C2×C42C2×Q818yes—
313C2×C42#4118yes—
314#95—28no—
315#95—28no—
316#95—22no—
317#95—22no—
318#95—24no—
319#95—24no—
320#95—24no—
321#95—24no—
322#95—24no—
323#95—24no—
324#95—24no—
325#95—28no—
326#95—28no—
327#95—24no—
328#95—28no—
329#95—24no—
330#95—24no—
331#95—42no—
332#95—44no—
333#95—44no—
334#95—44no—
335#95—44no—
336#95—44no—
337#95C22×C414yes—
338#95C22×C418yes—
339#95C22×C4116yes—
340#95C22×C414yes—
341#95C22×C4116yes—
342C24—22no—
343C24—22no—
344C24—22no—
345C24—21no—
346C24—22no—
347C24—21no—
348C24—21no—
349C24—21no—
350C24C23×C412yes—
351C24C22×D412yes—
352C24C22×Q812yes—
353C24C2511yescharacteristic
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—24no—
359C22×C4—22no—
360C22×C4—24no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—216no—
367C22×C4—28no—
368C22×C4—28no—
369C22×C4—216no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—24no—
373C22×C4—21no—
374C22×C4—21no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—24no—
378C22×C4—24no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—24no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—24no—
388C22×C4—22no—
389C22×C4—24no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—24no—
393C22×C4—22no—
394C22×C4—24no—
395C22×C4—24no—
396C22×C4—22no—
397C22×C4—24no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—24no—
405C22×C4—24no—
406C22×C4—21no—
407C22×C4—21no—
408C22×C4—216no—
409C22×C4—21no—
410C22×C4—21no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—24no—
416C22×C4—24no—
417C22×C4—24no—
418C22×C4—21no—
419C22×C4—22no—
420C22×C4—24no—
421C22×C4—22no—
422C22×C4—21no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—88no—
430C22×C4—88no—
431C22×C4—84no—
432C22×C4—84no—
433C22×C4C2×C4⋊C4116yes—
434C22×C4C23×C414yes—
435C22×C4C23×C418yes—
436C22×C4C23×C412yes—
437C22×C4C23×C414yes—
438C22×C4C23×C414yes—
439C22×C4C23×C414yes—
440C22×C4C23×C412yes—
441C22×C4C22×D414yes—
442C22×C4C22×D414yes—
443C22×C4C22×D412yes—
444C22×C4C22×D412yes—
445C22×C4C22×Q812yes—
446C22×C4C22×Q814yes—
447C22×C4C22×Q814yes—
448C22×C4C22×Q812yes—
449C22×C4C2×#4114yes—
450C22×C4C2×#4114yes—
451C22×C4C2×#4118yes—
452C22×C4C2×#4114yes—
453C22×C4C2×#4114yes—
454C22×C4H(5,𝔽2)12yes—
455C22×C4H(5,𝔽2)14yes—
456C22×C42−1+412yes—
457C22×C42−1+414yes—
458C4⋊C4—216no—
459C4⋊C4—216no—
460C4⋊C4—28no—
461C4⋊C4—28no—
462C4⋊C4—216no—
463C4⋊C4—216no—
464C4⋊C4—416no—
465C4⋊C4—48no—
466C4⋊C4—48no—
467C4⋊C4—48no—
468C4⋊C4—416no—
469C4⋊C4—416no—
470C4⋊C4—416no—
471C4⋊C4—44no—
472C4⋊C4—44no—
473C4⋊C4—88no—
474C4⋊C4C2×C4⋊C4116yes—
475C4⋊C4C2×C4⋊C4116yes—
476C22⋊C4—216no—
477C22⋊C4—216no—
478C22⋊C4—48no—
479C22⋊C4—44no—
480C22⋊C4—44no—
481C22⋊C4—84no—
482C22⋊C4C2×C4⋊C4132yes—
483C42—216no—
484C42—232no—
485C42—216no—
486C42—44no—
487C42—44no—
488C42—88no—
489C42—88no—
490C42—84no—
491C42—84no—
492C42C2×C4⋊C4132yes—
493C23—24no—
494C23—24no—
495C23—22no—
496C23—22no—
497C23—22no—
498C23—22no—
499C23—24no—
500C23—22no—
501C23—22no—
502C23—22no—
503C23—22no—
504C23—24no—
505C23—22no—
506C23—22no—
507C23—42no—
508C23—42no—
509C23—42no—
510C23—42no—
511C23—42no—
512C23—42no—
513C23—42no—
514C23—42no—
515C23C22×C4⋊C414yes—
516C23C24×C411yescharacteristic
517C23C23×D411yescharacteristic
518C23C23×Q811yescharacteristic
519C23C22×#4112yes—
520C23C22×#4112yes—
521C23C2×H(5,𝔽2)12yes—
522C23C2×H(5,𝔽2)11yescharacteristic
523C23C2×2−1+412yes—
524C23C2×2−1+411yescharacteristic
525C23#58511yescharacteristic
526C23#58511yescharacteristic
527C2×C4—24no—
528C2×C4—24no—
529C2×C4—28no—
530C2×C4—24no—
531C2×C4—24no—
532C2×C4—24no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—24no—
536C2×C4—24no—
537C2×C4—24no—
538C2×C4—24no—
539C2×C4—24no—
540C2×C4—24no—
541C2×C4—24no—
542C2×C4—28no—
543C2×C4—24no—
544C2×C4—24no—
545C2×C4—24no—
546C2×C4—24no—
547C2×C4—232no—
548C2×C4—24no—
549C2×C4—28no—
550C2×C4—28no—
551C2×C4—24no—
552C2×C4—28no—
553C2×C4—24no—
554C2×C4—24no—
555C2×C4—24no—
556C2×C4—28no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—24no—
560C2×C4—24no—
561C2×C4—24no—
562C2×C4—24no—
563C2×C4—24no—
564C2×C4—28no—
565C2×C4—24no—
566C2×C4—28no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—42no—
570C2×C4—44no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—44no—
578C2×C4—42no—
579C2×C4—44no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—44no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—44no—
588C2×C4—44no—
589C2×C4—42no—
590C2×C4—44no—
591C2×C4—42no—
592C2×C4—44no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—41no—
598C2×C4—42no—
599C2×C4—41no—
600C2×C4—42no—
601C2×C4—44no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—41no—
606C2×C4—41no—
607C2×C4—44no—
608C2×C4—42no—
609C2×C4—44no—
610C2×C4—42no—
611C2×C4—44no—
612C2×C4—42no—
613C2×C4—816no—
614C2×C4C22×C4⋊C414yes—
615C2×C4C22×C4⋊C418yes—
616C2×C4C22×C4⋊C418yes—
617C2×C4C22×C4⋊C414yes—
618C2×C4C4×#4118yes—
619C2×C4#52018yes—
620C2×C4#52514yes—
621C2×C4#52514yes—
622C2×C4#53318yes—
623C22—24no—
624C22—24no—
625C22—24no—
626C22—44no—
627C22—44no—
628C22—42no—
629C22—42no—
630C22—44no—
631C22—42no—
632C22—44no—
633C22—42no—
634C22C23×C4⋊C411yescharacteristic
635C22C2×C4×#4112yes—
636C22C2×#52012yes—
637C22C4×H(5,𝔽2)11yescharacteristic
638C22C4×2−1+411yescharacteristic
639C22C2×#52511yescharacteristic
640C22C2×#52511yescharacteristic
641C22C2×#53312yes—
642C22#348811yescharacteristic
643C22#348911yescharacteristic
644C22#349212yes—
645C22#349512yes—
646C22#349811yescharacteristic
647C22#349911yescharacteristic
648C22#350312yes—
649C22#350311yescharacteristic
650C22#350411yescharacteristic
651C22#350612yes—
652C22#351112yes—
653C22#351312yes—
654C22#351812yes—
655C22#351912yes—
656C22#352312yes—
657C4—28no—
658C4—28no—
659C4—28no—
660C4—24no—
661C4—44no—
662C4—48no—
663C4—44no—
664C4—42no—
665C4—42no—
666C4—48no—
667C4—44no—
668C4—82no—
669C4—82no—
670C4—81no—
671C4—81no—
672C4#229618yes—
673C2—44no—
674C2C2×#229611yescharacteristic
675C2#3418311yescharacteristic
676C2#3425011yescharacteristic
677C2#3430412yes—
678C2#3432112yes—
679C2#3568911yescharacteristic
680C2#3574712yes—
681C2#3581512yes—
682C2#3599511yescharacteristic
683C2#3600711yescharacteristic
684C2#3608911yescharacteristic
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
4. #14296C2semidirect
6. #14150C2semidirect
10. #14045C2semidirect
11. #14005C2semidirect
12. #13952C2semidirect
14. #13856C2semidirect
16. #13821C2semidirect
17. C4⋊C42C2semidirect
22. #2346C4semidirect
135. #389C2×C4semidirect
143. #382C2×C4semidirect
156. C4×C4⋊C4C2×C4semidirect
160. #376C2×C4semidirect
217. #117C4⋊C4semidirect

Maximal subgroups

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

subgroupindex
C2×#23472
C2×#23462
C2×#22742
#142962
#142642
#141502
#141302
#141182
#141122
#140452
#140052
#139522
#138802
#138562
#138442
#138212
C4⋊C422