Finite Group Databasebeta

← #945684

Subgroups — #945684

Order 512.

2362 conjugacy classes of subgroups.

The subgroup lattice has 711 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2567C211yesmaximal, characteristic
2C2×#2550C211yesmaximal, characteristic
3C2×#2451C211yesmaximal, characteristic
4C2×#2660C211yesmaximal, characteristic
5#17085C214yesmaximal
6#16311C214yesmaximal
7#16293C214yesmaximal
8#15360C214yesmaximal
9#15240C211yesmaximal, characteristic
10#15238C211yesmaximal, characteristic
11#15234C211yesmaximal, characteristic
12#15231C211yesmaximal, characteristic
13#15143C211yesmaximal, characteristic
14#15143C211yesmaximal, characteristic
15#14261C211yesmaximal, characteristic
16#14049C211yesmaximal, characteristic
17#14000C211yesmaximal, characteristic
18#13978C211yesmaximal, characteristic
19#13846C211yesmaximal, characteristic
20C22×C4⋊Q8C2211yescharacteristic
21C22×C4⋊Q8C2211yescharacteristic
22C22×#125C2211yescharacteristic
23C22×C4×Q8C2211yescharacteristic
24#2660—28no—
25#2567—28no—
26#2550—28no—
27#2451—28no—
28C2×#398C2214yes—
29C2×#398C2212yes—
30C2×#398C2212yes—
31C2×#395C2214yes—
32C2×#395C2212yes—
33C2×#395C2214yes—
34C2×#395C2212yes—
35C2×#395C2214yes—
36C2×#395C2214yes—
37C2×#391C2212yes—
38C2×#391C2212yes—
39C2×#391C2211yescharacteristic
40C2×#391C2211yescharacteristic
41C2×#391C2211yescharacteristic
42C2×#391C2211yescharacteristic
43C2×#391C2211yescharacteristic
44C2×#391C2211yescharacteristic
45C2×#389C2212yes—
46C2×#389C2214yes—
47C2×#389C2211yescharacteristic
48C2×#389C2211yescharacteristic
49C2×#389C2211yescharacteristic
50C2×#389C2211yescharacteristic
51C2×#387C2212yes—
52C2×#387C2214yes—
53C2×#384C2211yescharacteristic
54C2×#384C2211yescharacteristic
55C2×#384C2211yescharacteristic
56C2×#384C2211yescharacteristic
57C2×#382C2211yescharacteristic
58C2×#382C2211yescharacteristic
59C2×#382C2211yescharacteristic
60C2×C4×C4⋊C4C2211yescharacteristic
61C2×C4×C4⋊C4C2211yescharacteristic
62C2×C4×C4⋊C4C2211yescharacteristic
63C2×#376C2211yescharacteristic
64#1445C2214yes—
65#1445C2212yes—
66#1445C2214yes—
67#1445C2214yes—
68#1444C2214yes—
69#1444C2214yes—
70#1443C2212yes—
71#1443C2214yes—
72#1442C2212yes—
73#1442C2214yes—
74#1442C2214yes—
75#1442C2214yes—
76#1442C2214yes—
77#1442C2214yes—
78#1442C2214yes—
79#1442C2214yes—
80#1441C2214yes—
81#1441C2214yes—
82#1441C2212yes—
83#1436C2214yes—
84#1436C2214yes—
85C4⋊#95C2211yescharacteristic
86C4⋊#95C2211yescharacteristic
87C4⋊#95C2211yescharacteristic
88C4⋊#95C2211yescharacteristic
89C4⋊#95C2211yescharacteristic
90C4⋊#95C2211yescharacteristic
91C4×#95C2211yescharacteristic
92C4×#95C2211yescharacteristic
93C4×#95C2211yescharacteristic
94C4×#95C2211yescharacteristic
95C23×Q8C2311yescharacteristic
96C23×Q8C2312yes—
97C23×Q8C2312yes—
98C2×C4⋊Q8—24no—
99C2×C4⋊Q8—24no—
100C2×C4⋊Q8—24no—
101C2×C4⋊Q8—24no—
102C2×C4⋊Q8—42no—
103C2×C4⋊Q8—42no—
104C2×#125—24no—
105C2×#125—24no—
106C2×#125—42no—
107C2×C4×Q8—24no—
108C2×C4×Q8—24no—
109C2×C4×Q8—42no—
110C22×C4⋊C4C2312yes—
111C22×C4⋊C4C2311yescharacteristic
112C22×C4⋊C4C2311yescharacteristic
113C22×C4⋊C4C2311yescharacteristic
114C22×C4⋊C4C2311yescharacteristic
115C22×C4⋊C4C2312yes—
116C22×C4⋊C4C2312yes—
117C22×C4⋊C4C2312yes—
118C22×C4⋊C4C2312yes—
119C22×C4⋊C4C2312yes—
120C22×C4⋊C4C2312yes—
121C22×C4⋊C4C2311yescharacteristic
122C22×C4⋊C4C2311yescharacteristic
123C22×C4⋊C4C2311yescharacteristic
124C22×C4⋊C4C2311yescharacteristic
125C22×C4⋊C4C2312yes—
126C22×C4⋊C4C2312yes—
127C22×C4⋊C4C2314yes—
128C22×C4⋊C4C2314yes—
129C22×C4⋊C4C2314yes—
130C22×C4⋊C4C2312yes—
131C22×C4⋊C4C2311yescharacteristic
132C22×C4⋊C4C2314yes—
133C22×C42C2311yescharacteristic
134C22×C42C2311yescharacteristic
135C22×C42C2312yes—
136C22×C42C2311yescharacteristic
137C22×C42C2311yescharacteristic
138C22×C42C2311yescharacteristic
139C22×C42C2311yescharacteristic
140C22×C42C2311yescharacteristic
141C22×C42C2312yes—
142#398—216no—
143#398—28no—
144#398—28no—
145#395—216no—
146#395—216no—
147#395—28no—
148#395—28no—
149#395—48no—
150#395—48no—
151#391—24no—
152#391—24no—
153#391—24no—
154#391—24no—
155#391—28no—
156#391—28no—
157#391—42no—
158#391Q818yescomplemented
159#389—216no—
160#389—24no—
161#389—24no—
162#389—24no—
163#389—24no—
164#389—28no—
165#387—28no—
166#387—48no—
167#384—24no—
168#384—24no—
169#384—42no—
170#384D418yes—
171#382—24no—
172#382—24no—
173#382—24no—
174C4×C4⋊C4—24no—
175C4×C4⋊C4—24no—
176C4×C4⋊C4D418yes—
177#376—24no—
178C2×#95C2312yes—
179C2×#95C2312yes—
180C2×#95C2312yes—
181C2×#95C2312yes—
182C2×#95C2314yes—
183C2×#95C2314yes—
184C2×#95C2312yes—
185C2×#95C2312yes—
186C2×#95C2314yes—
187C2×#95C2314yes—
188C2×#95C2314yes—
189C2×#95C2312yes—
190C2×#95C2312yes—
191C2×#95C2312yes—
192C2×#95C2312yes—
193C2×#95C2314yes—
194C2×#95C2312yes—
195C2×#95C2312yes—
196C2×#95C2312yes—
197C2×#95C2312yes—
198C2×#95C2312yes—
199C2×#95C2312yes—
200C2×#95C2312yes—
201C2×#95C2312yes—
202C2×#95C2314yes—
203C2×#95C2312yes—
204C2×#95C2312yes—
205C2×#95C2314yes—
206C2×#95C2314yes—
207C2×#95C2312yes—
208C2×#95C2312yes—
209C2×#95C2314yes—
210C2×#95C2314yes—
211C2×#95C2311yescharacteristic
212C2×#95C2311yescharacteristic
213C2×#95C2314yes—
214C22×Q8—22no—
215C22×Q8—22no—
216C22×Q8—24no—
217C22×Q8—24no—
218C22×Q8—24no—
219C22×Q8—24no—
220C22×Q8—22no—
221C22×Q8—22no—
222C22×Q8—22no—
223C22×Q8—22no—
224C22×Q8—42no—
225C22×Q8—42no—
226C22×Q8—44no—
227C22×Q8—44no—
228C22×Q8—44no—
229C22×Q8—44no—
230C22×Q8C2×Q814yes—
231C23×C4C2412yes—
232C23×C4C2412yes—
233C23×C4C2412yes—
234C23×C4C2412yes—
235C23×C4C2412yes—
236C23×C4C2412yes—
237C23×C4C2411yescharacteristic
238C23×C4C2411yescharacteristic
239C23×C4C2411yescharacteristic
240C23×C4C2412yes—
241C23×C4C2411yescharacteristic
242C23×C4C2411yescharacteristic
243C23×C4C2412yes—
244C23×C4C2412yes—
245C23×C4C2412yes—
246C23×C4C2412yes—
247C23×C4C2412yes—
248C23×C4C2411yescharacteristic
249C23×C4C2411yescharacteristic
250C4⋊Q8—416no—
251C4⋊Q8—416no—
252#125—416no—
253C4×Q8—416no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4—22no—
260C2×C4⋊C4—24no—
261C2×C4⋊C4—24no—
262C2×C4⋊C4—24no—
263C2×C4⋊C4—22no—
264C2×C4⋊C4—24no—
265C2×C4⋊C4—24no—
266C2×C4⋊C4—22no—
267C2×C4⋊C4—22no—
268C2×C4⋊C4—24no—
269C2×C4⋊C4—24no—
270C2×C4⋊C4—24no—
271C2×C4⋊C4—24no—
272C2×C4⋊C4—22no—
273C2×C4⋊C4—22no—
274C2×C4⋊C4—22no—
275C2×C4⋊C4—24no—
276C2×C4⋊C4—24no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—24no—
279C2×C4⋊C4—24no—
280C2×C4⋊C4—22no—
281C2×C4⋊C4—24no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—28no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—28no—
289C2×C4⋊C4—28no—
290C2×C4⋊C4—24no—
291C2×C4⋊C4—24no—
292C2×C4⋊C4—24no—
293C2×C4⋊C4—28no—
294C2×C4⋊C4—42no—
295C2×C4⋊C4—44no—
296C2×C4⋊C4—42no—
297C2×C4⋊C4—42no—
298C2×C4⋊C4—42no—
299C2×C4⋊C4—42no—
300C2×C4⋊C4—42no—
301C2×C4⋊C4—44no—
302C2×C4⋊C4—44no—
303C2×C4⋊C4—44no—
304C2×C4⋊C4—44no—
305C2×C4⋊C4—44no—
306C2×C4⋊C4—44no—
307C2×C4⋊C4—44no—
308C2×C4⋊C4—44no—
309C2×C4⋊C4—44no—
310C2×C4⋊C4—44no—
311C2×C4⋊C4C2×D414yes—
312C2×C4⋊C4C2×D414yes—
313C2×C4⋊C4C2×D414yes—
314C2×C4⋊C4C2×D414yes—
315C2×C4⋊C4C2×D4116yes—
316C2×C4⋊C4C2×Q814yes—
317C2×C42—22no—
318C2×C42—22no—
319C2×C42—22no—
320C2×C42—22no—
321C2×C42—22no—
322C2×C42—22no—
323C2×C42—22no—
324C2×C42—22no—
325C2×C42—22no—
326C2×C42—22no—
327C2×C42—24no—
328C2×C42—24no—
329C2×C42—22no—
330C2×C42—22no—
331C2×C42—22no—
332C2×C42—22no—
333C2×C42—22no—
334C2×C42—22no—
335C2×C42—24no—
336C2×C42—42no—
337C2×C42—44no—
338C2×C42C2×D414yes—
339C2×C42C2×D414yes—
340C2×C42C2×D414yes—
341C2×C42C2×D414yes—
342C2×C42C2×Q814yes—
343#95—24no—
344#95—24no—
345#95—24no—
346#95—24no—
347#95—24no—
348#95—28no—
349#95—28no—
350#95—24no—
351#95—24no—
352#95—24no—
353#95—24no—
354#95—24no—
355#95—28no—
356#95—28no—
357#95—24no—
358#95—24no—
359#95—24no—
360#95—24no—
361#95—24no—
362#95—28no—
363#95—28no—
364#95—24no—
365#95—24no—
366#95—24no—
367#95—24no—
368#95—24no—
369#95—28no—
370#95—42no—
371#95—42no—
372#95—44no—
373#95—44no—
374#95—44no—
375#95—44no—
376#95C2×D414yes—
377#95C2×D414yes—
378#95C2×Q8116yes—
379C24C2511yescharacteristic
380C2×Q8—28no—
381C2×Q8—28no—
382C2×Q8—28no—
383C2×Q8—48no—
384C2×Q8—48no—
385C2×Q8—48no—
386C2×Q8—48no—
387C2×Q8—44no—
388C2×Q8—44no—
389C2×Q8—44no—
390C2×Q8—44no—
391C2×Q8—88no—
392C2×Q8—88no—
393C2×Q8—84no—
394C2×Q8—84no—
395C2×Q8—88no—
396C2×Q8—88no—
397C22×C4—21no—
398C22×C4—21no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—21no—
406C22×C4—21no—
407C22×C4—21no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—24no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—24no—
416C22×C4—24no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—21no—
422C22×C4—21no—
423C22×C4—21no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—21no—
433C22×C4—22no—
434C22×C4—21no—
435C22×C4—22no—
436C22×C4—21no—
437C22×C4—21no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—24no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—24no—
459C22×C4—24no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—24no—
470C22×C4—22no—
471C22×C4—24no—
472C22×C4—24no—
473C22×C4—22no—
474C22×C4—22no—
475C22×C4—24no—
476C22×C4—24no—
477C22×C4—24no—
478C22×C4—22no—
479C22×C4—22no—
480C22×C4—24no—
481C22×C4—24no—
482C22×C4C22×D412yes—
483C22×C4C22×D414yes—
484C22×C4C22×D412yes—
485C22×C4C22×D412yes—
486C22×C4C22×D412yes—
487C22×C4C22×D412yes—
488C22×C4C22×D414yes—
489C22×C4C22×D414yes—
490C22×C4C22×D412yes—
491C22×C4C22×D412yes—
492C22×C4C22×D412yes—
493C22×C4C22×Q814yes—
494C22×C4C22×Q814yes—
495C22×C4C22×Q814yes—
496C22×C4C22×Q812yes—
497C22×C4H(5,𝔽2)14yes—
498C22×C4H(5,𝔽2)12yes—
499C22×C42−1+414yes—
500C22×C42−1+414yes—
501C22×C42−1+412yes—
502C22×C42−1+412yes—
503C22×C42−1+412yes—
504C4⋊C4—216no—
505C4⋊C4—28no—
506C4⋊C4—28no—
507C4⋊C4—28no—
508C4⋊C4—28no—
509C4⋊C4—44no—
510C4⋊C4—416no—
511C4⋊C4—48no—
512C4⋊C4—48no—
513C4⋊C4—48no—
514C4⋊C4—48no—
515C4⋊C4—48no—
516C4⋊C4—48no—
517C4⋊C4—44no—
518C4⋊C4—44no—
519C4⋊C4—44no—
520C4⋊C4—44no—
521C4⋊C4—88no—
522C4⋊C4—88no—
523C4⋊C4—84no—
524C4⋊C4—84no—
525C4⋊C4—84no—
526C4⋊C4—88no—
527C42—28no—
528C42—28no—
529C42—28no—
530C42—28no—
531C42—28no—
532C42—48no—
533C42—44no—
534C42—44no—
535C42—84no—
536C23C23×D411yescharacteristic
537C23C23×D411yescharacteristic
538C23C23×Q811yescharacteristic
539C23C2×H(5,𝔽2)11yescharacteristic
540C23C2×H(5,𝔽2)12yes—
541C23C2×2−1+411yescharacteristic
542C23C2×2−1+411yescharacteristic
543C23C2×2−1+411yescharacteristic
544C23C2×2−1+412yes—
545C23C2×2−1+412yes—
546C23#58512yes—
547Q8—416no—
548Q8—816no—
549Q8—816no—
550C2×C4—22no—
551C2×C4—24no—
552C2×C4—22no—
553C2×C4—24no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—24no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—24no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—24no—
567C2×C4—24no—
568C2×C4—24no—
569C2×C4—24no—
570C2×C4—24no—
571C2×C4—24no—
572C2×C4—24no—
573C2×C4—24no—
574C2×C4—24no—
575C2×C4—28no—
576C2×C4—28no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—24no—
583C2×C4—24no—
584C2×C4—28no—
585C2×C4—24no—
586C2×C4—24no—
587C2×C4—24no—
588C2×C4—24no—
589C2×C4—24no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—44no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—44no—
624C2×C4—44no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4—42no—
632C2×C4—42no—
633C2×C4—42no—
634C2×C4—42no—
635C2×C4—42no—
636C2×C4—44no—
637C2×C4—44no—
638C2×C4—44no—
639C2×C4—42no—
640C2×C4—42no—
641C2×C4—42no—
642C2×C4—42no—
643C2×C4—42no—
644C2×C4—44no—
645C2×C4—44no—
646C2×C4—44no—
647C2×C4—44no—
648C2×C4—44no—
649C2×C4#53614yes—
650C2×C4#53614yes—
651C2×C4#53614yes—
652C2×C4#54414yes—
653C2×C4D4214yes—
654C2×C4D4×Q814yes—
655C2×C4D4×Q814yes—
656C2×C4#56314yes—
657C22C2×#53611yescharacteristic
658C22C2×#53611yescharacteristic
659C22C2×#53611yescharacteristic
660C22#344911yescharacteristic
661C22C2×#54411yescharacteristic
662C22C2×D4211yescharacteristic
663C22C2×D4×Q811yescharacteristic
664C22C2×D4×Q811yescharacteristic
665C22#348612yes—
666C22#349612yes—
667C22#349811yescharacteristic
668C22#349911yescharacteristic
669C22#350212yes—
670C22#350312yes—
671C22C2×#56311yescharacteristic
672C22#354412yes—
673C22#354412yes—
674C22#355212yes—
675C22#355212yes—
676C22#355512yes—
677C22#355512yes—
678C22#355812yes—
679C22#356312yes—
680C4—24no—
681C4—42no—
682C4—42no—
683C4—44no—
684C4—44no—
685C4—44no—
686C4—42no—
687C4—44no—
688C4—44no—
689C4—42no—
690C4—42no—
691C4—44no—
692C4—44no—
693C4—44no—
694C4—82no—
695C4—82no—
696C4—82no—
697C4—82no—
698C4#267718yes—
699C2#3730311yescharacteristic
700C2#3763511yescharacteristic
701C2#3767511yescharacteristic
702C2#3935511yescharacteristic
703C2#3961211yescharacteristic
704C2#3961711yescharacteristic
705C2#4243612yes—
706C2#4282512yes—
707C2#4489412yes—
708C2C2×#267711yescharacteristic
709C2#5303812yes—
subgroupG/H
center, commutator, Frattini, socle, prime radical, cosocleC24C25
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
158. #391Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#25672
C2×#25502
C2×#24512
C2×#26602
#170852
#163112
#162932
#153602
#152402
#152382
#152342
#152312
#151432
#142612
#140492
#140002
#139782
#138462