Finite Group Databasebeta

← #764117

Subgroups — #764117

Order 512.

3646 conjugacy classes of subgroups.

The subgroup lattice has 750 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#544C211yesmaximal, characteristic
2C2×#2567C211yesmaximal, characteristic, complemented
3C2×#2584C211yesmaximal, characteristic, complemented
4C2×#2396C211yesmaximal, characteristic
5#15850C212yesmaximal, complemented
6#15849C212yesmaximal, complemented
7#15671C212yesmaximal
8#15430C212yesmaximal
9#15318C214yesmaximal, complemented
10#15276C214yesmaximal
11#15218C212yesmaximal, complemented
12#15128C212yesmaximal, complemented
13#14160C211yesmaximal, characteristic, complemented
14#14155C211yesmaximal, characteristic, complemented
15#14130C212yesmaximal
16#14129C212yesmaximal
17#13777C211yesmaximal, characteristic
18C2×#544—28no—
19C2×#544—28no—
20C2×#544—28no—
21C22×C4⋊Q8C2211yescharacteristic
22C22×C4⋊Q8C2212yes—
23C22×C4⋊Q8C2211yescharacteristic
24C22×#125C2211yescharacteristic
25C22×#125C2212yes—
26C22×#125C2211yescharacteristic
27C22×C22⋊Q8C2212yes—
28C22×C22⋊Q8C2212yes—
29C22×C4×Q8C2211yescharacteristic
30C22×#117C2211yescharacteristic
31C22×#117C2211yescharacteristic
32C23×C4⋊C4C2211yescharacteristic
33#2584—28no—
34#2567—28no—
35#2396—28no—
36C2×#400C2211yescharacteristic
37C2×#399C2211yescharacteristic
38C2×#398C2212yes—
39C2×#398C2212yes—
40C2×#396C2212yes—
41C2×#395C2212yes—
42C2×#395C2214yes—
43C2×#395C2211yescharacteristic
44C2×#395C2214yes—
45C2×#393C2211yescharacteristic
46C2×#391C2212yes—
47C2×#391C2214yes—
48C2×#389C2212yes—
49C2×#389C2212yes—
50C2×#389C2212yes—
51C2×#389C2212yes—
52C2×#389C2214yes—
53C2×#385C2214yes—
54C2×#384C2212yes—
55C2×#380C2212yes—
56C2×C4×C4⋊C4C2212yes—
57C2×C4×C4⋊C4C2211yescharacteristic
58C2×C4×C4⋊C4C2211yescharacteristic
59C2×C4×C4⋊C4C2211yescharacteristic
60C2×C4×C22⋊C4C2212yes—
61#1446C2214yes—
62#1445C2212yes—
63#1445C2212yes—
64#1444C2214yes—
65#1444C2214yes—
66#1444C2214yes—
67#1444C2214yes—
68#1442C2214yes—
69#1441C2214yes—
70#1441C2214yes—
71#1441C2214yes—
72#1441C2214yes—
73#1439C2214yes—
74#1439C2214yes—
75#1437C2212yes—
76#1437C2212yes—
77#1436C2214yes—
78#1436C2214yes—
79#1436C2212yes—
80#1436C2212yes—
81#1436C2212yes—
82#1436C2212yes—
83C4⋊#95C2212yes—
84C4⋊#95C2212yes—
85C4⋊#95C2212yes—
86C4⋊#95C2212yes—
87C4⋊#95C2212yes—
88C22⋊#95C2214yes—
89C23×Q8C2311yescharacteristic
90C23×Q8C2312yes—
91C23×Q8C2312yes—
92C24×C4C2311yescharacteristic
93C24×C4C2312yes—
94#544—2128no—
95C2×C4⋊Q8—28no—
96C2×C4⋊Q8—28no—
97C2×C4⋊Q8—24no—
98C2×C4⋊Q8—28no—
99C2×C4⋊Q8—24no—
100C2×C4⋊Q8—24no—
101C2×C4⋊Q8—24no—
102C2×C4⋊Q8—24no—
103C2×C4⋊Q8D418yes—
104C2×#125—28no—
105C2×#125—28no—
106C2×#125—24no—
107C2×#125—28no—
108C2×#125—24no—
109C2×#125—24no—
110C2×#125—24no—
111C2×#125—24no—
112C2×#125D418yes—
113C2×C22⋊Q8—28no—
114C2×C22⋊Q8—28no—
115C2×C22⋊Q8—28no—
116C2×C22⋊Q8—28no—
117C2×C22⋊Q8—28no—
118C2×C22⋊Q8—28no—
119C2×C4×Q8—24no—
120C2×C4×Q8—24no—
121C2×C4×Q8—24no—
122C2×#117—24no—
123C2×#117—24no—
124C2×#117—24no—
125C2×#117—24no—
126C2×#117—24no—
127C2×#117D418yes—
128C22×C4⋊C4—24no—
129C22×C4⋊C4—24no—
130C22×C4⋊C4—24no—
131C22×C4⋊C4C2312yes—
132C22×C4⋊C4C2311yescharacteristic
133C22×C4⋊C4C2312yes—
134C22×C4⋊C4C2312yes—
135C22×C4⋊C4C2312yes—
136C22×C4⋊C4C2312yes—
137C22×C4⋊C4C2312yes—
138C22×C4⋊C4C2312yes—
139C22×C4⋊C4C2314yes—
140C22×C4⋊C4C2312yes—
141C22×C4⋊C4C2312yes—
142C22×C4⋊C4C2311yescharacteristic
143C22×C4⋊C4C2311yescharacteristic
144C22×C4⋊C4C2311yescharacteristic
145C22×C4⋊C4C2311yescharacteristic
146C22×C4⋊C4C2311yescharacteristic
147C22×C4⋊C4C2311yescharacteristic
148C22×C4⋊C4C2312yes—
149C22×C4⋊C4C2312yes—
150C22×C4⋊C4C2312yes—
151C22×C4⋊C4C2312yes—
152C22×C4⋊C4C2314yes—
153C22×C4⋊C4C2314yes—
154C22×C4⋊C4C2314yes—
155C22×C22⋊C4C2312yes—
156C22×C22⋊C4C2312yes—
157C22×C22⋊C4C2312yes—
158C22×C22⋊C4C2311yescharacteristic
159C22×C22⋊C4C2311yescharacteristic
160C22×C22⋊C4C2314yes—
161C22×C42C2312yes—
162C22×C42C2312yes—
163C22×C42C2312yes—
164C22×C42C2311yescharacteristic
165C22×C42C2312yes—
166C22×C42C2312yes—
167C22×C42C2311yescharacteristic
168#400—24no—
169#399—24no—
170#398—28no—
171#398—28no—
172#396—28no—
173#395—216no—
174#395—28no—
175#395—24no—
176#395—48no—
177#393—24no—
178#391—28no—
179#391—48no—
180#389—28no—
181#389—28no—
182#389—28no—
183#389—216no—
184#389—28no—
185#385—48no—
186#384—28no—
187#380Q8116yescomplemented
188C4×C4⋊C4—24no—
189C4×C4⋊C4—24no—
190C4×C4⋊C4—28no—
191C4×C4⋊C4—24no—
192C4×C22⋊C4—28no—
193C2×#95C2312yes—
194C2×#95C2311yescharacteristic
195C2×#95C2312yes—
196C2×#95C2312yes—
197C2×#95C2312yes—
198C2×#95C2311yescharacteristic
199C2×#95C2314yes—
200C2×#95C2314yes—
201C2×#95C2314yes—
202C2×#95C2314yes—
203C2×#95C2314yes—
204C2×#95C2312yes—
205C2×#95C2312yes—
206C2×#95C2314yes—
207C2×#95C2314yes—
208C2×#95C2314yes—
209C2×#95C2314yes—
210C2×#95C2314yes—
211C2×#95C2314yes—
212C2×#95C2314yes—
213C2×#95C2314yes—
214C2×#95C2314yes—
215C2×#95C2314yes—
216C25C2411yescharacteristic
217C22×Q8—22no—
218C22×Q8—24no—
219C22×Q8—24no—
220C22×Q8—24no—
221C22×Q8—24no—
222C22×Q8—24no—
223C22×Q8—22no—
224C22×Q8—22no—
225C22×Q8—24no—
226C22×Q8—42no—
227C22×Q8—44no—
228C22×Q8—44no—
229C22×Q8—44no—
230C22×Q8—44no—
231C22×Q8C2×D418yes—
232C23×C4—22no—
233C23×C4—22no—
234C23×C4—24no—
235C23×C4—24no—
236C23×C4—24no—
237C23×C4—24no—
238C23×C4—44no—
239C23×C4—44no—
240C23×C4C2×D414yes—
241C23×C4C2×Q818yes—
242C23×C4C2412yes—
243C23×C4C2412yes—
244C23×C4C2412yes—
245C23×C4C2411yescharacteristic
246C23×C4C2412yes—
247C23×C4C2411yescharacteristic
248C23×C4C2412yes—
249C23×C4C2412yes—
250C23×C4C2414yes—
251C23×C4C2414yes—
252C23×C4C2412yes—
253C23×C4C2412yes—
254C23×C4C2411yescharacteristic
255C23×C4C2411yescharacteristic
256C23×C4C2411yescharacteristic
257C23×C4C2411yescharacteristic
258C4⋊Q8—264no—
259C4⋊Q8—232no—
260C4⋊Q8—232no—
261#125—264no—
262#125—232no—
263#125—232no—
264C22⋊Q8—264no—
265C22⋊Q8—264no—
266C4×Q8—416no—
267#117—232no—
268#117—232no—
269C2×C4⋊C4—232no—
270C2×C4⋊C4—24no—
271C2×C4⋊C4—24no—
272C2×C4⋊C4—24no—
273C2×C4⋊C4—22no—
274C2×C4⋊C4—22no—
275C2×C4⋊C4—22no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—22no—
279C2×C4⋊C4—22no—
280C2×C4⋊C4—22no—
281C2×C4⋊C4—22no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—24no—
286C2×C4⋊C4—24no—
287C2×C4⋊C4—24no—
288C2×C4⋊C4—24no—
289C2×C4⋊C4—24no—
290C2×C4⋊C4—24no—
291C2×C4⋊C4—24no—
292C2×C4⋊C4—24no—
293C2×C4⋊C4—24no—
294C2×C4⋊C4—24no—
295C2×C4⋊C4—24no—
296C2×C4⋊C4—24no—
297C2×C4⋊C4—24no—
298C2×C4⋊C4—24no—
299C2×C4⋊C4—24no—
300C2×C4⋊C4—24no—
301C2×C4⋊C4—24no—
302C2×C4⋊C4—24no—
303C2×C4⋊C4—24no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—24no—
306C2×C4⋊C4—24no—
307C2×C4⋊C4—24no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—24no—
311C2×C4⋊C4—24no—
312C2×C4⋊C4—24no—
313C2×C4⋊C4—28no—
314C2×C4⋊C4—24no—
315C2×C4⋊C4—24no—
316C2×C4⋊C4—24no—
317C2×C4⋊C4—28no—
318C2×C4⋊C4—28no—
319C2×C4⋊C4—28no—
320C2×C4⋊C4—28no—
321C2×C4⋊C4—42no—
322C2×C4⋊C4—42no—
323C2×C4⋊C4—44no—
324C2×C4⋊C4—42no—
325C2×C4⋊C4—44no—
326C2×C4⋊C4—44no—
327C2×C4⋊C4—44no—
328C2×C4⋊C4—44no—
329C2×C4⋊C4—44no—
330C2×C4⋊C4—44no—
331C2×C4⋊C4C2×D414yes—
332C2×C4⋊C4C2×D414yes—
333C2×C4⋊C4C2×D414yes—
334C2×C4⋊C4C2×D414yes—
335C2×C4⋊C4C2×D414yes—
336C2×C4⋊C4C2×D414yes—
337C2×C4⋊C4C2×D418yes—
338C2×C4⋊C4C2×D418yes—
339C2×C4⋊C4C2×D418yes—
340C2×C4⋊C4C2×Q818yes—
341C2×C4⋊C4C2×Q818yes—
342C2×C22⋊C4—24no—
343C2×C22⋊C4—24no—
344C2×C22⋊C4—24no—
345C2×C22⋊C4—22no—
346C2×C22⋊C4—22no—
347C2×C22⋊C4—22no—
348C2×C22⋊C4—22no—
349C2×C22⋊C4—24no—
350C2×C22⋊C4—24no—
351C2×C22⋊C4—24no—
352C2×C22⋊C4—24no—
353C2×C22⋊C4—44no—
354C2×C22⋊C4—42no—
355C2×C22⋊C4—42no—
356C2×C22⋊C4—44no—
357C2×C22⋊C4C2×D414yes—
358C2×C22⋊C4C2×D414yes—
359C2×C22⋊C4C2×Q8116yes—
360C2×C42—24no—
361C2×C42—24no—
362C2×C42—24no—
363C2×C42—22no—
364C2×C42—22no—
365C2×C42—24no—
366C2×C42—24no—
367C2×C42—24no—
368C2×C42—24no—
369C2×C42—24no—
370C2×C42—24no—
371C2×C42—24no—
372C2×C42—24no—
373C2×C42—44no—
374C2×C42—44no—
375C2×C42C2×D414yes—
376C2×C42C2×D414yes—
377C2×C42C2×D414yes—
378C2×C42C2×D414yes—
379#95—24no—
380#95—24no—
381#95—24no—
382#95—22no—
383#95—24no—
384#95—24no—
385#95—22no—
386#95—24no—
387#95—28no—
388#95—28no—
389#95—28no—
390#95—28no—
391#95—28no—
392#95—28no—
393#95—28no—
394#95—44no—
395#95—44no—
396#95—44no—
397#95—44no—
398#95—44no—
399#95—44no—
400#95—44no—
401#95C2×Q8116yes—
402C24—21no—
403C24—21no—
404C24—22no—
405C24—22no—
406C24—22no—
407C24—22no—
408C24—22no—
409C24C22×D412yes—
410C24C22×Q814yes—
411C24C2511yescharacteristic
412C2×Q8—28no—
413C2×Q8—216no—
414C2×Q8—216no—
415C2×Q8—416no—
416C2×Q8—48no—
417C2×Q8—48no—
418C2×Q8—44no—
419C2×Q8—44no—
420C2×Q8—88no—
421C2×Q8—84no—
422C2×Q8—84no—
423C2×Q8—88no—
424C2×Q8—88no—
425C2×Q8—84no—
426C2×Q8—84no—
427C22×C4—21no—
428C22×C4—21no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—28no—
435C22×C4—28no—
436C22×C4—216no—
437C22×C4—216no—
438C22×C4—22no—
439C22×C4—24no—
440C22×C4—24no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—21no—
446C22×C4—22no—
447C22×C4—21no—
448C22×C4—22no—
449C22×C4—21no—
450C22×C4—22no—
451C22×C4—21no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—24no—
457C22×C4—24no—
458C22×C4—24no—
459C22×C4—24no—
460C22×C4—24no—
461C22×C4—24no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—24no—
466C22×C4—24no—
467C22×C4—24no—
468C22×C4—22no—
469C22×C4—24no—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—24no—
473C22×C4—24no—
474C22×C4—22no—
475C22×C4—22no—
476C22×C4—24no—
477C22×C4—24no—
478C22×C4—24no—
479C22×C4—24no—
480C22×C4—22no—
481C22×C4—22no—
482C22×C4—22no—
483C22×C4—24no—
484C22×C4—22no—
485C22×C4—24no—
486C22×C4—216no—
487C22×C4—22no—
488C22×C4—22no—
489C22×C4—22no—
490C22×C4—22no—
491C22×C4—22no—
492C22×C4—22no—
493C22×C4—22no—
494C22×C4—22no—
495C22×C4—22no—
496C22×C4—22no—
497C22×C4—22no—
498C22×C4—22no—
499C22×C4—22no—
500C22×C4—22no—
501C22×C4—88no—
502C22×C4—84no—
503C22×C4—84no—
504C22×C4—88no—
505C22×C4C4⋊Q8116yes—
506C22×C4C22×D414yes—
507C22×C4C22×D414yes—
508C22×C4C22×D412yes—
509C22×C4C22×D412yes—
510C22×C4C22×D414yes—
511C22×C4C22×D414yes—
512C22×C4C22×D412yes—
513C22×C4C22×D412yes—
514C22×C4C22×D412yes—
515C22×C4C22×D412yes—
516C22×C4C22×D412yes—
517C22×C4C22×D412yes—
518C22×C4C22×D412yes—
519C22×C4C22×D412yes—
520C22×C4C22×D412yes—
521C22×C4C22×D412yes—
522C22×C4C22×Q818yes—
523C22×C4C22×Q818yes—
524C22×C4C22×Q814yes—
525C22×C4C22×Q814yes—
526C22×C4H(5,𝔽2)14yes—
527C22×C42−1+414yes—
528C22×C42−1+414yes—
529C4⋊C4—216no—
530C4⋊C4—216no—
531C4⋊C4—216no—
532C4⋊C4—216no—
533C4⋊C4—28no—
534C4⋊C4—28no—
535C4⋊C4—28no—
536C4⋊C4—28no—
537C4⋊C4—28no—
538C4⋊C4—28no—
539C4⋊C4—216no—
540C4⋊C4—216no—
541C4⋊C4—216no—
542C4⋊C4—216no—
543C4⋊C4—216no—
544C4⋊C4—48no—
545C4⋊C4—48no—
546C4⋊C4—48no—
547C4⋊C4—416no—
548C4⋊C4—48no—
549C4⋊C4—44no—
550C4⋊C4—88no—
551C4⋊C4—88no—
552C4⋊C4—88no—
553C22⋊C4—216no—
554C22⋊C4—28no—
555C22⋊C4—28no—
556C22⋊C4—416no—
557C22⋊C4—48no—
558C22⋊C4—48no—
559C42—216no—
560C42—28no—
561C42—48no—
562C42—48no—
563C42—48no—
564C42—84no—
565C42#120116yes—
566C23—22no—
567C23—22no—
568C23—24no—
569C23—24no—
570C23—24no—
571C23—24no—
572C23—24no—
573C23—24no—
574C23—24no—
575C23—24no—
576C23—41no—
577C23—41no—
578C23—42no—
579C23—42no—
580C23—41no—
581C23—41no—
582C23—42no—
583C23—42no—
584C23—42no—
585C23—42no—
586C23C2×C4⋊Q814yes—
587C23C23×D411yescharacteristic
588C23C23×D411yescharacteristic
589C23C23×D411yescharacteristic
590C23C23×Q812yes—
591C23C2×H(5,𝔽2)12yes—
592C23C2×2−1+412yes—
593C23C2×2−1+412yes—
594C23#58512yes—
595C23#58512yes—
596Q8—416no—
597Q8—816no—
598Q8—816no—
599C2×C4—22no—
600C2×C4—22no—
601C2×C4—24no—
602C2×C4—24no—
603C2×C4—22no—
604C2×C4—22no—
605C2×C4—24no—
606C2×C4—24no—
607C2×C4—24no—
608C2×C4—24no—
609C2×C4—22no—
610C2×C4—22no—
611C2×C4—232no—
612C2×C4—24no—
613C2×C4—22no—
614C2×C4—24no—
615C2×C4—22no—
616C2×C4—24no—
617C2×C4—24no—
618C2×C4—28no—
619C2×C4—28no—
620C2×C4—24no—
621C2×C4—24no—
622C2×C4—24no—
623C2×C4—24no—
624C2×C4—24no—
625C2×C4—28no—
626C2×C4—24no—
627C2×C4—28no—
628C2×C4—28no—
629C2×C4—24no—
630C2×C4—24no—
631C2×C4—28no—
632C2×C4—28no—
633C2×C4—28no—
634C2×C4—28no—
635C2×C4—24no—
636C2×C4—24no—
637C2×C4—42no—
638C2×C4—42no—
639C2×C4—42no—
640C2×C4—41no—
641C2×C4—44no—
642C2×C4—42no—
643C2×C4—41no—
644C2×C4—42no—
645C2×C4—44no—
646C2×C4—44no—
647C2×C4—44no—
648C2×C4—44no—
649C2×C4—42no—
650C2×C4—42no—
651C2×C4—44no—
652C2×C4—42no—
653C2×C4—42no—
654C2×C4—44no—
655C2×C4—44no—
656C2×C4—42no—
657C2×C4—44no—
658C2×C4—42no—
659C2×C4—42no—
660C2×C4—41no—
661C2×C4—42no—
662C2×C4—41no—
663C2×C4—42no—
664C2×C4—44no—
665C2×C4—42no—
666C2×C4—44no—
667C2×C4—44no—
668C2×C4—44no—
669C2×C4—42no—
670C2×C4—42no—
671C2×C4—42no—
672C2×C4—42no—
673C2×C4—42no—
674C2×C4—42no—
675C2×C4—42no—
676C2×C4—42no—
677C2×C4—42no—
678C2×C4—42no—
679C2×C4—42no—
680C2×C4—42no—
681C2×C4—42no—
682C2×C4—42no—
683C2×C4—42no—
684C2×C4—816no—
685C2×C4C2×#12014yes—
686C2×C4C2×#12014yes—
687C2×C4C2×#12014yes—
688C2×C4C2×C4⋊Q814yes—
689C2×C4C2×C4⋊Q814yes—
690C2×C4#53614yes—
691C2×C4D4×Q818yes—
692C2×C4D4×Q818yes—
693C22—24no—
694C22—28no—
695C22—44no—
696C22—42no—
697C22—44no—
698C22—44no—
699C22—44no—
700C22—42no—
701C22—44no—
702C22C22×#12011yescharacteristic
703C22C22×C4⋊Q811yescharacteristic
704C22C2×#53611yescharacteristic
705C22C2×D4×Q812yes—
706C22C2×D4×Q812yes—
707C22#348611yescharacteristic
708C22#348711yescharacteristic
709C22#348812yes—
710C22#348912yes—
711C22#348912yes—
712C22#349612yes—
713C22#349811yescharacteristic
714C22#350011yescharacteristic
715C22#350112yes—
716C22#350212yes—
717C22#350312yes—
718C22#350412yes—
719C22#354412yes—
720C22#355512yes—
721C22#355912yes—
722C22#356512yes—
723C4—24no—
724C4—24no—
725C4—24no—
726C4—48no—
727C4—44no—
728C4—42no—
729C4—44no—
730C4—48no—
731C4—44no—
732C4—44no—
733C4—44no—
734C4—42no—
735C4—44no—
736C4—82no—
737C4—82no—
738C4#243418yes—
739C2—44no—
740C2C2×#243411yescharacteristic
741C2#3551311yescharacteristic
742C2#3556612yes—
743C2#3565911yescharacteristic
744C2#3728112yes—
745C2#4489412yes—
746C2#4494112yes—
747C2#4511012yes—
748C2#4527412yes—
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×#2567C2semidirect
3. C2×#2584C2semidirect
5. #15850C2semidirect
6. #15849C2semidirect
9. #15318C2semidirect
11. #15218C2semidirect
12. #15128C2semidirect
13. #14160C2semidirect
14. #14155C2semidirect
187. #380Q8semidirect

Maximal subgroups

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

subgroupindex
C22×#5442
C2×#25672
C2×#25842
C2×#23962
#158502
#158492
#156712
#154302
#153182
#152762
#152182
#151282
#141602
#141552
#141302
#141292
#137772