Finite Group Databasebeta

← #687850

Subgroups — #687850

Order 512.

2526 conjugacy classes of subgroups.

The subgroup lattice has 725 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2567C211yesmaximal, characteristic
2C2×#2451C211yesmaximal, characteristic
3C2×#2351C211yesmaximal, characteristic
4C2×#2327C211yesmaximal, characteristic
5#16325C214yesmaximal
6#15969C214yesmaximal
7#15437C214yesmaximal
8#15240C211yesmaximal, characteristic
9#15237C211yesmaximal, characteristic
10#15231C211yesmaximal, characteristic
11#15230C211yesmaximal, characteristic
12#14790C214yesmaximal
13#14277C211yesmaximal, characteristic
14#14024C211yesmaximal, characteristic
15#14024C211yesmaximal, characteristic
16#13978C211yesmaximal, characteristic
17#13978C211yesmaximal, characteristic
18#13976C211yesmaximal, characteristic
19#13846C211yesmaximal, characteristic
20C22×C4⋊Q8C2211yescharacteristic
21C22×C4⋊Q8C2211yescharacteristic
22C22×C4×Q8C2211yescharacteristic
23C22×C4×Q8C2211yescharacteristic
24C22×C4×Q8C2211yescharacteristic
25C22×C4×Q8C2211yescharacteristic
26#2567—28no—
27#2451—28no—
28C2×#398C2212yes—
29C2×#395C2214yes—
30C2×#395C2212yes—
31#2351—28no—
32#2327—28no—
33C2×#391C2211yescharacteristic
34C2×#391C2212yes—
35C2×#391C2211yescharacteristic
36C2×#391C2211yescharacteristic
37C2×#391C2211yescharacteristic
38C2×#391C2211yescharacteristic
39C2×#391C2211yescharacteristic
40C2×#391C2211yescharacteristic
41C2×#391C2211yescharacteristic
42C2×#389C2212yes—
43C2×#389C2211yescharacteristic
44C2×#389C2211yescharacteristic
45C2×#389C2212yes—
46C2×#389C2214yes—
47C2×#389C2211yescharacteristic
48C2×#389C2211yescharacteristic
49C2×#387C2214yes—
50C2×#387C2214yes—
51C2×#387C2212yes—
52C2×#384C2211yescharacteristic
53C2×#384C2211yescharacteristic
54C2×#382C2211yescharacteristic
55C2×#382C2212yes—
56C2×#382C2211yescharacteristic
57C2×#382C2211yescharacteristic
58C2×#382C2211yescharacteristic
59C2×C4×C4⋊C4C2211yescharacteristic
60C2×C4×C4⋊C4C2212yes—
61C2×C4×C4⋊C4C2211yescharacteristic
62C2×#376C2211yescharacteristic
63#1446C2214yes—
64#1445C2212yes—
65#1445C2214yes—
66#1445C2214yes—
67#1444C2214yes—
68#1444C2214yes—
69#1443C2212yes—
70#1443C2214yes—
71#1442C2212yes—
72#1442C2214yes—
73#1442C2214yes—
74#1442C2214yes—
75#1442C2214yes—
76#1441C2212yes—
77#1441C2214yes—
78#1441C2214yes—
79#1441C2214yes—
80#1441C2214yes—
81#1441C2214yes—
82#1436C2214yes—
83#1436C2214yes—
84#1436C2214yes—
85#1436C2214yes—
86#1436C2214yes—
87C4⋊#95C2211yescharacteristic
88C4⋊#95C2211yescharacteristic
89C4⋊#95C2211yescharacteristic
90C4⋊#95C2211yescharacteristic
91C4×#95C2211yescharacteristic
92C4×#95C2211yescharacteristic
93C4×#95C2211yescharacteristic
94C4×#95C2211yescharacteristic
95C23×Q8C2311yescharacteristic
96C23×Q8C2311yescharacteristic
97C23×Q8C2312yes—
98C2×C4⋊Q8—24no—
99C2×C4⋊Q8—24no—
100C2×C4⋊Q8—24no—
101C2×C4⋊Q8—24no—
102C2×C4⋊Q8—42no—
103C2×C4⋊Q8Q818yescomplemented
104C2×C4×Q8—24no—
105C2×C4×Q8—24no—
106C2×C4×Q8—24no—
107C2×C4×Q8—24no—
108C2×C4×Q8—24no—
109C2×C4×Q8—24no—
110C2×C4×Q8—24no—
111C2×C4×Q8—24no—
112C2×C4×Q8—24no—
113C2×C4×Q8—42no—
114C2×C4×Q8—42no—
115C2×C4×Q8—42no—
116C22×C4⋊C4C2311yescharacteristic
117C22×C4⋊C4C2311yescharacteristic
118C22×C4⋊C4C2312yes—
119C22×C4⋊C4C2311yescharacteristic
120C22×C4⋊C4C2311yescharacteristic
121C22×C4⋊C4C2312yes—
122C22×C4⋊C4C2312yes—
123C22×C4⋊C4C2312yes—
124C22×C4⋊C4C2312yes—
125C22×C4⋊C4C2312yes—
126C22×C4⋊C4C2312yes—
127C22×C4⋊C4C2311yescharacteristic
128C22×C4⋊C4C2311yescharacteristic
129C22×C4⋊C4C2314yes—
130C22×C4⋊C4C2311yescharacteristic
131C22×C4⋊C4C2312yes—
132C22×C4⋊C4C2314yes—
133C22×C4⋊C4C2314yes—
134C22×C4⋊C4C2312yes—
135C22×C4⋊C4C2312yes—
136C22×C4⋊C4C2311yescharacteristic
137C22×C42C2312yes—
138C22×C42C2311yescharacteristic
139C22×C42C2311yescharacteristic
140C22×C42C2312yes—
141C22×C42C2312yes—
142C22×C42C2312yes—
143C22×C42C2311yescharacteristic
144C22×C42C2311yescharacteristic
145C22×C42C2311yescharacteristic
146C22×C42C2311yescharacteristic
147C22×C42C2311yescharacteristic
148#398—28no—
149#395—28no—
150#395—216no—
151#391—24no—
152#391—24no—
153#391—24no—
154#391—24no—
155#391—24no—
156#391—24no—
157#391—28no—
158#391—42no—
159#391D418yes—
160#389—28no—
161#389—28no—
162#389—24no—
163#389—24no—
164#389—24no—
165#389—24no—
166#389—216no—
167#387—28no—
168#387—216no—
169#387—216no—
170#384—24no—
171#384—24no—
172#382—24no—
173#382—24no—
174#382—24no—
175#382—28no—
176#382—42no—
177C4×C4⋊C4—24no—
178C4×C4⋊C4—28no—
179C4×C4⋊C4D418yes—
180#376—24no—
181C2×#95C2312yes—
182C2×#95C2314yes—
183C2×#95C2312yes—
184C2×#95C2312yes—
185C2×#95C2312yes—
186C2×#95C2312yes—
187C2×#95C2312yes—
188C2×#95C2312yes—
189C2×#95C2312yes—
190C2×#95C2312yes—
191C2×#95C2314yes—
192C2×#95C2314yes—
193C2×#95C2314yes—
194C2×#95C2312yes—
195C2×#95C2314yes—
196C2×#95C2314yes—
197C2×#95C2312yes—
198C2×#95C2312yes—
199C2×#95C2314yes—
200C2×#95C2312yes—
201C2×#95C2312yes—
202C2×#95C2312yes—
203C2×#95C2312yes—
204C2×#95C2312yes—
205C2×#95C2312yes—
206C2×#95C2312yes—
207C2×#95C2312yes—
208C2×#95C2312yes—
209C2×#95C2314yes—
210C2×#95C2314yes—
211C2×#95C2314yes—
212C2×#95C2314yes—
213C2×#95C2314yes—
214C2×#95C2314yes—
215C2×#95C2311yescharacteristic
216C2×#95C2311yescharacteristic
217C22×Q8—22no—
218C22×Q8—22no—
219C22×Q8—22no—
220C22×Q8—24no—
221C22×Q8—24no—
222C22×Q8—22no—
223C22×Q8—22no—
224C22×Q8—22no—
225C22×Q8—22no—
226C22×Q8—22no—
227C22×Q8—22no—
228C22×Q8—22no—
229C22×Q8—42no—
230C22×Q8—44no—
231C22×Q8—44no—
232C22×Q8C2×D414yes—
233C22×Q8C2×Q814yes—
234C22×Q8C2×Q814yes—
235C22×Q8#4114yes—
236C23×C4C2412yes—
237C23×C4C2412yes—
238C23×C4C2412yes—
239C23×C4C2412yes—
240C23×C4C2412yes—
241C23×C4C2411yescharacteristic
242C23×C4C2412yes—
243C23×C4C2411yescharacteristic
244C23×C4C2411yescharacteristic
245C23×C4C2412yes—
246C23×C4C2411yescharacteristic
247C23×C4C2411yescharacteristic
248C23×C4C2412yes—
249C23×C4C2412yes—
250C23×C4C2412yes—
251C23×C4C2412yes—
252C23×C4C2412yes—
253C23×C4C2411yescharacteristic
254C23×C4C2411yescharacteristic
255C4⋊Q8—232no—
256C4⋊Q8—416no—
257C4×Q8—232no—
258C4×Q8—416no—
259C4×Q8—416no—
260C4×Q8—416no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—24no—
263C2×C4⋊C4—22no—
264C2×C4⋊C4—28no—
265C2×C4⋊C4—24no—
266C2×C4⋊C4—24no—
267C2×C4⋊C4—24no—
268C2×C4⋊C4—22no—
269C2×C4⋊C4—28no—
270C2×C4⋊C4—22no—
271C2×C4⋊C4—24no—
272C2×C4⋊C4—24no—
273C2×C4⋊C4—24no—
274C2×C4⋊C4—24no—
275C2×C4⋊C4—24no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—22no—
279C2×C4⋊C4—22no—
280C2×C4⋊C4—24no—
281C2×C4⋊C4—24no—
282C2×C4⋊C4—24no—
283C2×C4⋊C4—24no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—24no—
286C2×C4⋊C4—24no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—24no—
289C2×C4⋊C4—24no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—28no—
297C2×C4⋊C4—28no—
298C2×C4⋊C4—28no—
299C2×C4⋊C4—24no—
300C2×C4⋊C4—24no—
301C2×C4⋊C4—42no—
302C2×C4⋊C4—44no—
303C2×C4⋊C4—42no—
304C2×C4⋊C4—42no—
305C2×C4⋊C4—42no—
306C2×C4⋊C4—42no—
307C2×C4⋊C4—44no—
308C2×C4⋊C4—44no—
309C2×C4⋊C4—44no—
310C2×C4⋊C4—44no—
311C2×C4⋊C4—44no—
312C2×C4⋊C4—44no—
313C2×C4⋊C4C2×D414yes—
314C2×C4⋊C4C2×D414yes—
315C2×C4⋊C4C2×D414yes—
316C2×C4⋊C4C2×Q8116yes—
317C2×C4⋊C4#4114yes—
318C2×C42—22no—
319C2×C42—22no—
320C2×C42—24no—
321C2×C42—22no—
322C2×C42—24no—
323C2×C42—24no—
324C2×C42—22no—
325C2×C42—22no—
326C2×C42—22no—
327C2×C42—22no—
328C2×C42—22no—
329C2×C42—22no—
330C2×C42—22no—
331C2×C42—24no—
332C2×C42—24no—
333C2×C42—24no—
334C2×C42—24no—
335C2×C42—24no—
336C2×C42—24no—
337C2×C42—22no—
338C2×C42—22no—
339C2×C42—22no—
340C2×C42—22no—
341C2×C42—22no—
342C2×C42—22no—
343C2×C42—42no—
344C2×C42—42no—
345C2×C42—42no—
346C2×C42C2×D414yes—
347C2×C42C2×D414yes—
348C2×C42C2×D414yes—
349C2×C42C2×D414yes—
350C2×C42C2×Q814yes—
351#95—24no—
352#95—24no—
353#95—24no—
354#95—24no—
355#95—28no—
356#95—24no—
357#95—24no—
358#95—24no—
359#95—28no—
360#95—24no—
361#95—24no—
362#95—24no—
363#95—24no—
364#95—28no—
365#95—24no—
366#95—24no—
367#95—28no—
368#95—28no—
369#95—28no—
370#95—24no—
371#95—24no—
372#95—24no—
373#95—28no—
374#95—42no—
375#95—44no—
376#95—42no—
377#95—44no—
378#95—42no—
379#95—44no—
380#95—42no—
381#95—42no—
382#95—44no—
383#95—44no—
384#95C2×D414yes—
385#95C2×D414yes—
386#95C2×D4116yes—
387C24C2511yescharacteristic
388C2×Q8—28no—
389C2×Q8—28no—
390C2×Q8—28no—
391C2×Q8—28no—
392C2×Q8—28no—
393C2×Q8—28no—
394C2×Q8—28no—
395C2×Q8—28no—
396C2×Q8—28no—
397C2×Q8—48no—
398C2×Q8—48no—
399C2×Q8—44no—
400C2×Q8—44no—
401C2×Q8—44no—
402C2×Q8—44no—
403C2×Q8—88no—
404C2×Q8—84no—
405C2×Q8—88no—
406C2×Q8C22⋊Q8116yes—
407C22×C4—21no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—21no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—21no—
418C22×C4—21no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—21no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—21no—
427C22×C4—21no—
428C22×C4—22no—
429C22×C4—21no—
430C22×C4—21no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—21no—
435C22×C4—21no—
436C22×C4—21no—
437C22×C4—22no—
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—24no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—24no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—24no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—22no—
473C22×C4—22no—
474C22×C4—22no—
475C22×C4—22no—
476C22×C4—22no—
477C22×C4—22no—
478C22×C4—22no—
479C22×C4—22no—
480C22×C4—24no—
481C22×C4—24no—
482C22×C4—22no—
483C22×C4—22no—
484C22×C4—22no—
485C22×C4—24no—
486C22×C4—22no—
487C22×C4—22no—
488C22×C4—24no—
489C22×C4—24no—
490C22×C4—24no—
491C22×C4—24no—
492C22×C4—24no—
493C22×C4C22×D412yes—
494C22×C4C22×D414yes—
495C22×C4C22×D412yes—
496C22×C4C22×D412yes—
497C22×C4C22×D412yes—
498C22×C4C22×D414yes—
499C22×C4C22×D412yes—
500C22×C4C22×D414yes—
501C22×C4C22×D412yes—
502C22×C4C22×D412yes—
503C22×C4C22×D412yes—
504C22×C4C22×Q814yes—
505C22×C4C22×Q814yes—
506C22×C4C22×Q814yes—
507C22×C4C22×Q812yes—
508C22×C4C2×#4114yes—
509C22×C4C2×#4114yes—
510C22×C4C2×#4112yes—
511C22×C4C2×#4112yes—
512C22×C4H(5,𝔽2)12yes—
513C22×C42−1+414yes—
514C22×C42−1+414yes—
515C22×C42−1+412yes—
516C4⋊C4—216no—
517C4⋊C4—216no—
518C4⋊C4—232no—
519C4⋊C4—28no—
520C4⋊C4—28no—
521C4⋊C4—28no—
522C4⋊C4—44no—
523C4⋊C4—48no—
524C4⋊C4—48no—
525C4⋊C4—48no—
526C4⋊C4—48no—
527C4⋊C4—48no—
528C4⋊C4—44no—
529C4⋊C4—48no—
530C4⋊C4—44no—
531C4⋊C4—44no—
532C4⋊C4—44no—
533C4⋊C4—84no—
534C4⋊C4—88no—
535C4⋊C4—84no—
536C4⋊C4—88no—
537C42—216no—
538C42—28no—
539C42—28no—
540C42—28no—
541C42—28no—
542C42—28no—
543C42—48no—
544C42—48no—
545C42—48no—
546C42—44no—
547C42—44no—
548C23C23×D411yescharacteristic
549C23C23×D411yescharacteristic
550C23C23×Q811yescharacteristic
551C23C22×#4111yescharacteristic
552C23C22×#4111yescharacteristic
553C23C2×H(5,𝔽2)11yescharacteristic
554C23C2×2−1+412yes—
555C23C2×2−1+411yescharacteristic
556C23C2×2−1+412yes—
557C23#58512yes—
558C23#58512yes—
559Q8—232no—
560Q8—416no—
561Q8—816no—
562C2×C4—24no—
563C2×C4—24no—
564C2×C4—22no—
565C2×C4—24no—
566C2×C4—24no—
567C2×C4—24no—
568C2×C4—24no—
569C2×C4—24no—
570C2×C4—22no—
571C2×C4—24no—
572C2×C4—24no—
573C2×C4—24no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—24no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—24no—
586C2×C4—24no—
587C2×C4—24no—
588C2×C4—24no—
589C2×C4—24no—
590C2×C4—24no—
591C2×C4—22no—
592C2×C4—22no—
593C2×C4—22no—
594C2×C4—24no—
595C2×C4—22no—
596C2×C4—28no—
597C2×C4—22no—
598C2×C4—24no—
599C2×C4—24no—
600C2×C4—28no—
601C2×C4—24no—
602C2×C4—24no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—42no—
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—42no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—44no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4—42no—
632C2×C4—44no—
633C2×C4—42no—
634C2×C4—42no—
635C2×C4—42no—
636C2×C4—42no—
637C2×C4—42no—
638C2×C4—42no—
639C2×C4—42no—
640C2×C4—42no—
641C2×C4—42no—
642C2×C4—42no—
643C2×C4—44no—
644C2×C4—42no—
645C2×C4—42no—
646C2×C4—42no—
647C2×C4—44no—
648C2×C4—42no—
649C2×C4—44no—
650C2×C4—42no—
651C2×C4—44no—
652C2×C4—44no—
653C2×C4—42no—
654C2×C4—42no—
655C2×C4—44no—
656C2×C4—42no—
657C2×C4—42no—
658C2×C4—42no—
659C2×C4—44no—
660C2×C4—44no—
661C2×C4—42no—
662C2×C4C2×C22⋊Q818yes—
663C2×C4C2×C22⋊Q814yes—
664C2×C4#52514yes—
665C2×C4#53614yes—
666C2×C4#53914yes—
667C2×C4D4214yes—
668C2×C4#54714yes—
669C2×C4D4×Q814yes—
670C2×C4#55414yes—
671C22C22×C22⋊Q811yescharacteristic
672C22C2×#52511yescharacteristic
673C22C2×#53611yescharacteristic
674C22#344911yescharacteristic
675C22C2×#53911yescharacteristic
676C22C2×D4211yescharacteristic
677C22C2×#54711yescharacteristic
678C22C2×D4×Q811yescharacteristic
679C22C2×#55411yescharacteristic
680C22#348612yes—
681C22#349911yescharacteristic
682C22#350011yescharacteristic
683C22#350012yes—
684C22#350212yes—
685C22#350212yes—
686C22#351512yes—
687C22#351612yes—
688C22#351812yes—
689C22#351812yes—
690C22#355212yes—
691C22#355512yes—
692C22#356212yes—
693C22#356312yes—
694C4—24no—
695C4—28no—
696C4—42no—
697C4—42no—
698C4—44no—
699C4—44no—
700C4—44no—
701C4—44no—
702C4—42no—
703C4—44no—
704C4—42no—
705C4—44no—
706C4—42no—
707C4—82no—
708C4—82no—
709C4—82no—
710C4—82no—
711C4—82no—
712C4#244918yes—
713C2C2×#244911yescharacteristic
714C2#3520011yescharacteristic
715C2#3566911yescharacteristic
716C2#3571112yes—
717C2#3767511yescharacteristic
718C2#3935711yescharacteristic
719C2#3961211yescharacteristic
720C2#3961911yescharacteristic
721C2#4290112yes—
722C2#4517712yes—
723C2#4527812yes—
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
103. C2×C4⋊Q8Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#25672
C2×#24512
C2×#23512
C2×#23272
#163252
#159692
#154372
#152402
#152372
#152312
#152302
#147902
#142772
#140242
#139782
#139762
#138462