Finite Group Databasebeta

← #688353

Subgroups — #688353

Order 512.

3558 conjugacy classes of subgroups.

The subgroup lattice has 703 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#544C211yesmaximal, characteristic
2C2×#2550C212yesmaximal, complemented
3C2×#2441C211yesmaximal, characteristic
4#16607C212yesmaximal, complemented
5#16603C212yesmaximal, complemented
6#15466C214yesmaximal
7#15276C214yesmaximal
8#15228C212yesmaximal, complemented
9#15218C212yesmaximal, complemented
10#15188C212yesmaximal, complemented
11#14929C212yesmaximal, complemented
12#14264C211yesmaximal, characteristic
13#14130C211yesmaximal, characteristic
14#14053C212yesmaximal, complemented
15#13881C211yesmaximal, characteristic
16#13844C211yesmaximal, characteristic
17#13801C211yesmaximal, characteristic
18C2×#544—216no—
19C2×#544—28no—
20C22×C4⋊Q8C2212yes—
21C22×C4⋊Q8C2212yes—
22C22×#125C2211yescharacteristic
23C22×#125C2212yes—
24C22×#125C2212yes—
25C22×C22⋊Q8C2214yes—
26C22×#117C2211yescharacteristic
27C22×#117C2211yescharacteristic
28C23×C4⋊C4C2211yescharacteristic
29#2550—216no—
30#2441—28no—
31C2×#399C2212yes—
32C2×#398C2214yes—
33C2×#396C2212yes—
34C2×#395C2214yes—
35C2×#391C2214yes—
36C2×#391C2214yes—
37C2×#389C2212yes—
38C2×#389C2214yes—
39C2×#389C2212yes—
40C2×#389C2212yes—
41C2×#389C2211yescharacteristic
42C2×#389C2211yescharacteristic
43C2×#387C2212yes—
44C2×#385C2212yes—
45C2×#384C2211yescharacteristic
46C2×#382C2212yes—
47C2×#380C2211yescharacteristic
48C2×#380C2211yescharacteristic
49C2×C4×C4⋊C4C2212yes—
50C2×C4×C4⋊C4C2212yes—
51C2×C4×C22⋊C4C2211yescharacteristic
52C2×C4×C22⋊C4C2211yescharacteristic
53#1446C2214yes—
54#1444C2211yescharacteristic
55#1444C2211yescharacteristic
56#1444C2212yes—
57#1444C2214yes—
58#1444C2212yes—
59#1444C2214yes—
60#1444C2214yes—
61#1444C2214yes—
62#1442C2211yescharacteristic
63#1442C2211yescharacteristic
64#1442C2212yes—
65#1442C2212yes—
66#1442C2212yes—
67#1442C2212yes—
68#1441C2214yes—
69#1441C2214yes—
70#1441C2214yes—
71#1441C2214yes—
72#1441C2214yes—
73#1439C2212yes—
74#1439C2212yes—
75#1439C2214yes—
76#1437C2214yes—
77#1436C2214yes—
78C4⋊#95C2212yes—
79C4⋊#95C2212yes—
80C4⋊#95C2212yes—
81C4⋊#95C2212yes—
82#1434C2212yes—
83C22⋊#95C2212yes—
84C22⋊#95C2214yes—
85C4×#95C2212yes—
86C4×#95C2212yes—
87C23×Q8C2314yes—
88C24×C4C2311yescharacteristic
89C24×C4C2312yes—
90#544—2128no—
91C2×C4⋊Q8—216no—
92C2×C4⋊Q8—28no—
93C2×C4⋊Q8—28no—
94C2×C4⋊Q8—28no—
95C2×C4⋊Q8D4116yes—
96C2×#125—216no—
97C2×#125—28no—
98C2×#125—28no—
99C2×#125—24no—
100C2×#125—28no—
101C2×#125—28no—
102C2×#125—28no—
103C2×C22⋊Q8—216no—
104C2×C22⋊Q8—216no—
105C2×C22⋊Q8—216no—
106C2×#117—24no—
107C2×#117—28no—
108C2×#117—28no—
109C2×#117D418yes—
110C22×C4⋊C4—24no—
111C22×C4⋊C4—28no—
112C22×C4⋊C4C2312yes—
113C22×C4⋊C4C2312yes—
114C22×C4⋊C4C2312yes—
115C22×C4⋊C4C2311yescharacteristic
116C22×C4⋊C4C2311yescharacteristic
117C22×C4⋊C4C2314yes—
118C22×C4⋊C4C2314yes—
119C22×C4⋊C4C2311yescharacteristic
120C22×C4⋊C4C2311yescharacteristic
121C22×C4⋊C4C2312yes—
122C22×C4⋊C4C2312yes—
123C22×C4⋊C4C2314yes—
124C22×C4⋊C4C2312yes—
125C22×C4⋊C4C2311yescharacteristic
126C22×C4⋊C4C2312yes—
127C22×C4⋊C4C2312yes—
128C22×C4⋊C4C2314yes—
129C22×C4⋊C4C2312yes—
130C22×C4⋊C4C2312yes—
131C22×C4⋊C4C2314yes—
132C22×C4⋊C4C2312yes—
133C22×C4⋊C4C2311yescharacteristic
134C22×C22⋊C4C2312yes—
135C22×C22⋊C4C2311yescharacteristic
136C22×C22⋊C4C2311yescharacteristic
137C22×C22⋊C4C2312yes—
138C22×C22⋊C4C2312yes—
139C22×C22⋊C4C2312yes—
140C22×C22⋊C4C2312yes—
141C22×C42C2311yescharacteristic
142C22×C42C2311yescharacteristic
143C22×C42C2312yes—
144C22×C42C2312yes—
145C22×C42C2312yes—
146C22×C42C2314yes—
147#399—28no—
148#398—216no—
149#396—28no—
150#395—216no—
151#391—216no—
152#391—48no—
153#389—24no—
154#389—24no—
155#389—216no—
156#389—28no—
157#389—28no—
158#389—28no—
159#387—28no—
160#385—44no—
161#384—24no—
162#382—28no—
163#380—24no—
164#380Q818yescomplemented
165C4×C4⋊C4—28no—
166C4×C4⋊C4—28no—
167C4×C22⋊C4—24no—
168C4×C22⋊C4Q818yescomplemented
169C2×#95C2314yes—
170C2×#95C2314yes—
171C2×#95C2314yes—
172C2×#95C2312yes—
173C2×#95C2314yes—
174C2×#95C2312yes—
175C2×#95C2312yes—
176C2×#95C2312yes—
177C2×#95C2314yes—
178C2×#95C2314yes—
179C2×#95C2312yes—
180C2×#95C2312yes—
181C2×#95C2311yescharacteristic
182C2×#95C2312yes—
183C2×#95C2311yescharacteristic
184C2×#95C2314yes—
185C2×#95C2312yes—
186C2×#95C2312yes—
187C2×#95C2314yes—
188C2×#95C2314yes—
189C2×#95C2314yes—
190C2×#95C2314yes—
191C2×#95C2314yes—
192C2×#95C2314yes—
193C2×#95C2311yescharacteristic
194C2×#95C2312yes—
195C2×#95C2311yescharacteristic
196C25C2411yescharacteristic
197C22×Q8—28no—
198C22×Q8—28no—
199C22×Q8—44no—
200C22×Q8—44no—
201C22×Q8—44no—
202C22×Q8—44no—
203C22×Q8C2×D4116yes—
204C23×C4—24no—
205C23×C4—24no—
206C23×C4—24no—
207C23×C4—28no—
208C23×C4—42no—
209C23×C4—44no—
210C23×C4—42no—
211C23×C4C2×D414yes—
212C23×C4C2×Q814yes—
213C23×C4C2×Q814yes—
214C23×C4C2411yescharacteristic
215C23×C4C2412yes—
216C23×C4C2411yescharacteristic
217C23×C4C2412yes—
218C23×C4C2412yes—
219C23×C4C2412yes—
220C23×C4C2412yes—
221C23×C4C2414yes—
222C23×C4C2414yes—
223C23×C4C2412yes—
224C23×C4C2412yes—
225C23×C4C2411yescharacteristic
226C23×C4C2411yescharacteristic
227C23×C4C2412yes—
228C23×C4C2411yescharacteristic
229C23×C4C2411yescharacteristic
230C4⋊Q8—264no—
231C4⋊Q8—264no—
232#125—264no—
233#125—264no—
234#125—416no—
235C22⋊Q8—2128no—
236#117—232no—
237#117—232no—
238C2×C4⋊C4—232no—
239C2×C4⋊C4—28no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—28no—
249C2×C4⋊C4—28no—
250C2×C4⋊C4—28no—
251C2×C4⋊C4—28no—
252C2×C4⋊C4—28no—
253C2×C4⋊C4—28no—
254C2×C4⋊C4—28no—
255C2×C4⋊C4—28no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—24no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4—24no—
260C2×C4⋊C4—24no—
261C2×C4⋊C4—28no—
262C2×C4⋊C4—24no—
263C2×C4⋊C4—28no—
264C2×C4⋊C4—24no—
265C2×C4⋊C4—24no—
266C2×C4⋊C4—22no—
267C2×C4⋊C4—24no—
268C2×C4⋊C4—24no—
269C2×C4⋊C4—28no—
270C2×C4⋊C4—24no—
271C2×C4⋊C4—24no—
272C2×C4⋊C4—24no—
273C2×C4⋊C4—24no—
274C2×C4⋊C4—28no—
275C2×C4⋊C4—24no—
276C2×C4⋊C4—42no—
277C2×C4⋊C4—42no—
278C2×C4⋊C4—42no—
279C2×C4⋊C4—44no—
280C2×C4⋊C4—42no—
281C2×C4⋊C4—42no—
282C2×C4⋊C4—44no—
283C2×C4⋊C4—42no—
284C2×C4⋊C4—44no—
285C2×C4⋊C4—44no—
286C2×C4⋊C4—44no—
287C2×C4⋊C4—44no—
288C2×C4⋊C4C2×D418yes—
289C2×C4⋊C4C2×D414yes—
290C2×C4⋊C4C2×D4116yes—
291C2×C4⋊C4C2×D418yes—
292C2×C4⋊C4C2×D414yes—
293C2×C4⋊C4C2×Q818yes—
294C2×C22⋊C4—24no—
295C2×C22⋊C4—24no—
296C2×C22⋊C4—24no—
297C2×C22⋊C4—28no—
298C2×C22⋊C4—22no—
299C2×C22⋊C4—22no—
300C2×C22⋊C4—22no—
301C2×C22⋊C4—22no—
302C2×C22⋊C4—24no—
303C2×C22⋊C4—42no—
304C2×C22⋊C4—42no—
305C2×C22⋊C4—42no—
306C2×C22⋊C4—41no—
307C2×C22⋊C4—41no—
308C2×C22⋊C4—42no—
309C2×C22⋊C4—44no—
310C2×C22⋊C4C2×D418yes—
311C2×C22⋊C4C2×Q818yes—
312C2×C22⋊C4C2×Q818yes—
313C2×C42—28no—
314C2×C42—24no—
315C2×C42—24no—
316C2×C42—24no—
317C2×C42—24no—
318C2×C42—22no—
319C2×C42—24no—
320C2×C42—22no—
321C2×C42—24no—
322C2×C42—28no—
323C2×C42—44no—
324C2×C42—44no—
325C2×C42C2×D418yes—
326C2×C42C2×D418yes—
327C2×C42C2×Q818yes—
328#95—22no—
329#95—22no—
330#95—24no—
331#95—24no—
332#95—28no—
333#95—24no—
334#95—28no—
335#95—28no—
336#95—24no—
337#95—24no—
338#95—28no—
339#95—28no—
340#95—28no—
341#95—24no—
342#95—24no—
343#95—44no—
344#95—44no—
345#95—42no—
346#95—42no—
347#95—44no—
348#95—44no—
349#95—44no—
350#95—44no—
351#95—44no—
352#95C2×Q814yes—
353#95C2×Q814yes—
354#95C2×Q818yes—
355C24—22no—
356C24—21no—
357C24—21no—
358C24—22no—
359C24—22no—
360C24—21no—
361C24—22no—
362C24—21no—
363C24C22×D412yes—
364C24C22×Q812yes—
365C24C22×Q812yes—
366C24C2511yescharacteristic
367C2×Q8—232no—
368C2×Q8—416no—
369C2×Q8—416no—
370C2×Q8—88no—
371C2×Q8—88no—
372C2×Q8—88no—
373C2×Q8—88no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—24no—
378C22×C4—22no—
379C22×C4—28no—
380C22×C4—28no—
381C22×C4—216no—
382C22×C4—216no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—21no—
386C22×C4—24no—
387C22×C4—21no—
388C22×C4—22no—
389C22×C4—24no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—24no—
394C22×C4—24no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—24no—
398C22×C4—24no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—24no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—24no—
413C22×C4—22no—
414C22×C4—24no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—22no—
420C22×C4—24no—
421C22×C4—24no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—24no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—216no—
432C22×C4—21no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—24no—
436C22×C4—21no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—21no—
441C22×C4—21no—
442C22×C4—24no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—24no—
446C22×C4—21no—
447C22×C4—21no—
448C22×C4—24no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—88no—
453C22×C4—88no—
454C22×C4—84no—
455C22×C4—84no—
456C22×C4C4⋊Q8116yes—
457C22×C4C22×D414yes—
458C22×C4C22×D418yes—
459C22×C4C22×D414yes—
460C22×C4C22×D414yes—
461C22×C4C22×D418yes—
462C22×C4C22×D414yes—
463C22×C4C22×D412yes—
464C22×C4C22×D412yes—
465C22×C4C22×D414yes—
466C22×C4C22×Q814yes—
467C22×C4C22×Q814yes—
468C22×C4C22×Q818yes—
469C22×C4C22×Q812yes—
470C22×C4C22×Q812yes—
471C22×C4C22×Q812yes—
472C22×C4C22×Q812yes—
473C22×C4H(5,𝔽2)12yes—
474C22×C42−1+412yes—
475C22×C42−1+414yes—
476C4⋊C4—216no—
477C4⋊C4—28no—
478C4⋊C4—28no—
479C4⋊C4—232no—
480C4⋊C4—232no—
481C4⋊C4—216no—
482C4⋊C4—28no—
483C4⋊C4—232no—
484C4⋊C4—216no—
485C4⋊C4—48no—
486C4⋊C4—48no—
487C4⋊C4—48no—
488C4⋊C4—48no—
489C4⋊C4—416no—
490C4⋊C4—48no—
491C4⋊C4—48no—
492C4⋊C4—44no—
493C4⋊C4—44no—
494C4⋊C4—88no—
495C4⋊C4—84no—
496C4⋊C4—84no—
497C4⋊C4#120116yes—
498C22⋊C4—216no—
499C22⋊C4—216no—
500C22⋊C4—48no—
501C22⋊C4—48no—
502C22⋊C4—44no—
503C22⋊C4—44no—
504C22⋊C4—84no—
505C42—216no—
506C42—216no—
507C42—216no—
508C42—44no—
509C42—44no—
510C42—88no—
511C23—22no—
512C23—22no—
513C23—24no—
514C23—24no—
515C23—24no—
516C23—22no—
517C23—22no—
518C23—22no—
519C23—22no—
520C23—22no—
521C23—22no—
522C23—22no—
523C23—24no—
524C23—22no—
525C23—42no—
526C23—42no—
527C23—42no—
528C23—42no—
529C23—42no—
530C23—42no—
531C23—42no—
532C23—42no—
533C23C2×C4⋊Q814yes—
534C23C23×D411yescharacteristic
535C23C23×D412yes—
536C23C23×Q811yescharacteristic
537C23C23×Q811yescharacteristic
538C23C2×H(5,𝔽2)11yescharacteristic
539C23C2×2−1+412yes—
540C23C2×2−1+411yescharacteristic
541C23#58512yes—
542C23#58512yes—
543C23#58511yescharacteristic
544C23#58511yescharacteristic
545Q8—832no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—24no—
549C2×C4—24no—
550C2×C4—28no—
551C2×C4—24no—
552C2×C4—28no—
553C2×C4—24no—
554C2×C4—232no—
555C2×C4—24no—
556C2×C4—24no—
557C2×C4—24no—
558C2×C4—28no—
559C2×C4—24no—
560C2×C4—24no—
561C2×C4—24no—
562C2×C4—24no—
563C2×C4—28no—
564C2×C4—24no—
565C2×C4—28no—
566C2×C4—24no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—24no—
570C2×C4—28no—
571C2×C4—24no—
572C2×C4—24no—
573C2×C4—24no—
574C2×C4—28no—
575C2×C4—24no—
576C2×C4—28no—
577C2×C4—24no—
578C2×C4—24no—
579C2×C4—24no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—42no—
583C2×C4—44no—
584C2×C4—42no—
585C2×C4—44no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—44no—
589C2×C4—44no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—44no—
594C2×C4—44no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—44no—
598C2×C4—42no—
599C2×C4—41no—
600C2×C4—42no—
601C2×C4—41no—
602C2×C4—44no—
603C2×C4—42no—
604C2×C4—44no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—44no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—41no—
613C2×C4—41no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—44no—
619C2×C4—41no—
620C2×C4—42no—
621C2×C4—41no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—44no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—41no—
629C2×C4—41no—
630C2×C4—44no—
631C2×C4—44no—
632C2×C4—816no—
633C2×C4C2×#12018yes—
634C2×C4C2×#12014yes—
635C2×C4C2×C4⋊Q814yes—
636C2×C4C2×C4⋊Q814yes—
637C2×C4#53614yes—
638C2×C4D4×Q818yes—
639C2×C4D4×Q818yes—
640C22—24no—
641C22—24no—
642C22—24no—
643C22—44no—
644C22—44no—
645C22—44no—
646C22—42no—
647C22—42no—
648C22—42no—
649C22—44no—
650C22—42no—
651C22C22×#12011yescharacteristic
652C22C22×C4⋊Q811yescharacteristic
653C22C2×#53611yescharacteristic
654C22#345812yes—
655C22C2×D4×Q812yes—
656C22C2×D4×Q812yes—
657C22#348512yes—
658C22#348712yes—
659C22#348811yescharacteristic
660C22#348911yescharacteristic
661C22#349612yes—
662C22#349712yes—
663C22#350011yescharacteristic
664C22#350011yescharacteristic
665C22#350111yescharacteristic
666C22#350211yescharacteristic
667C22#350311yescharacteristic
668C22#350412yes—
669C22#350411yescharacteristic
670C22#355912yes—
671C22#356312yes—
672C22#356512yes—
673C22#356512yes—
674C4—24no—
675C4—28no—
676C4—48no—
677C4—44no—
678C4—48no—
679C4—44no—
680C4—44no—
681C4—44no—
682C4—42no—
683C4—44no—
684C4—42no—
685C4—81no—
686C4—81no—
687C4—82no—
688C4—82no—
689C4#243418yes—
690C2—44no—
691C2C2×#243411yescharacteristic
692C2#3533011yescharacteristic
693C2#3556611yescharacteristic
694C2#3575011yescharacteristic
695C2#3575711yescharacteristic
696C2#3769112yes—
697C2#3936512yes—
698C2#4294411yescharacteristic
699C2#4297811yescharacteristic
700C2#4509912yes—
701C2#4511012yes—
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
2. C2×#2550C2semidirect
4. #16607C2semidirect
5. #16603C2semidirect
8. #15228C2semidirect
9. #15218C2semidirect
10. #15188C2semidirect
11. #14929C2semidirect
14. #14053C2semidirect
164. #380Q8semidirect
168. C4×C22⋊C4Q8semidirect

Maximal subgroups

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

subgroupindex
C22×#5442
C2×#25502
C2×#24412
#166072
#166032
#154662
#152762
#152282
#152182
#151882
#149292
#142642
#141302
#140532
#138812
#138442
#138012