Finite Group Databasebeta

← #722760

Subgroups — #722760

Order 512.

2720 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, complemented
2C2×#2628C211yesmaximal, characteristic
3C2×#2351C211yesmaximal, characteristic, complemented
4C2×#2323C211yesmaximal, characteristic
5#16567C212yesmaximal, complemented
6#16564C212yesmaximal
7#15633C214yesmaximal
8#15623C214yesmaximal, complemented
9#15137C212yesmaximal, complemented
10#15131C212yesmaximal
11#14870C212yesmaximal
12#14869C212yesmaximal, complemented
13#14155C211yesmaximal, characteristic, complemented
14#14150C211yesmaximal, characteristic, complemented
15#13969C212yesmaximal
16#13935C212yesmaximal, complemented
17#13852C211yesmaximal, characteristic
18C22×C4⋊Q8C2211yescharacteristic
19C22×C22⋊Q8C2211yescharacteristic
20C22×C4×Q8C2212yes—
21C22×C4×Q8C2211yescharacteristic
22C22×#117C2211yescharacteristic
23#2628—28no—
24#2567—28no—
25C2×#400C2212yes—
26C2×#399C2212yes—
27C2×#398C2214yes—
28C2×#396C2214yes—
29C2×#395C2212yes—
30C2×#395C2214yes—
31C2×#393C2212yes—
32#2351—28no—
33#2323—28no—
34C2×#391C2211yescharacteristic
35C2×#391C2212yes—
36C2×#391C2214yes—
37C2×#389C2212yes—
38C2×#389C2214yes—
39C2×#389C2211yescharacteristic
40C2×#389C2212yes—
41C2×#389C2212yes—
42C2×#389C2212yes—
43C2×#389C2212yes—
44C2×#389C2212yes—
45C2×#388C2212yes—
46C2×#388C2212yes—
47C2×#387C2212yes—
48C2×#384C2212yes—
49C2×#382C2212yes—
50C2×#382C2211yescharacteristic
51C2×#380C2211yescharacteristic
52C2×#380C2212yes—
53C2×C4×C4⋊C4C2211yescharacteristic
54C2×C4×C4⋊C4C2212yes—
55C2×C4×C4⋊C4C2211yescharacteristic
56C2×C4×C4⋊C4C2211yescharacteristic
57C2×C4×C22⋊C4C2212yes—
58#1445C2212yes—
59#1445C2212yes—
60#1444C2214yes—
61#1444C2212yes—
62#1442C2214yes—
63#1442C2212yes—
64#1441C2212yes—
65#1441C2214yes—
66#1441C2214yes—
67#1441C2214yes—
68#1441C2214yes—
69#1441C2214yes—
70#1441C2214yes—
71#1441C2211yescharacteristic
72#1441C2211yescharacteristic
73#1439C2212yes—
74#1439C2214yes—
75#1439C2214yes—
76#1439C2211yescharacteristic
77#1439C2211yescharacteristic
78#1437C2214yes—
79#1436C2214yes—
80#1436C2214yes—
81C4⋊#95C2214yes—
82C4⋊#95C2212yes—
83C4⋊#95C2212yes—
84C4⋊#95C2212yes—
85#1434C2214yes—
86C4×#95C2212yes—
87C23×Q8C2311yescharacteristic
88C23×Q8C2312yes—
89C24×C4C2311yescharacteristic
90C2×C4⋊Q8—24no—
91C2×C4⋊Q8—24no—
92C2×C4⋊Q8—24no—
93C2×C22⋊Q8—24no—
94C2×C22⋊Q8—24no—
95C2×C22⋊Q8Q818yescomplemented
96C2×C4×Q8—28no—
97C2×C4×Q8—28no—
98C2×C4×Q8—24no—
99C2×C4×Q8—24no—
100C2×C4×Q8—44no—
101C2×C4×Q8D418yes—
102C2×#117—24no—
103C2×#117—24no—
104C2×#117—24no—
105C22×C4⋊C4C2311yescharacteristic
106C22×C4⋊C4C2312yes—
107C22×C4⋊C4C2311yescharacteristic
108C22×C4⋊C4C2312yes—
109C22×C4⋊C4C2311yescharacteristic
110C22×C4⋊C4C2311yescharacteristic
111C22×C4⋊C4C2311yescharacteristic
112C22×C4⋊C4C2314yes—
113C22×C4⋊C4C2314yes—
114C22×C4⋊C4C2312yes—
115C22×C4⋊C4C2312yes—
116C22×C4⋊C4C2312yes—
117C22×C4⋊C4C2311yescharacteristic
118C22×C4⋊C4C2311yescharacteristic
119C22×C4⋊C4C2312yes—
120C22×C4⋊C4C2314yes—
121C22×C4⋊C4C2314yes—
122C22×C4⋊C4C2314yes—
123C22×C4⋊C4C2314yes—
124C22×C4⋊C4C2312yes—
125C22×C22⋊C4C2311yescharacteristic
126C22×C22⋊C4C2311yescharacteristic
127C22×C22⋊C4C2312yes—
128C22×C22⋊C4C2312yes—
129C22×C22⋊C4C2314yes—
130C22×C22⋊C4C2312yes—
131C22×C22⋊C4C2312yes—
132C22×C42C2311yescharacteristic
133C22×C42C2312yes—
134C22×C42C2312yes—
135C22×C42C2314yes—
136C22×C42C2312yes—
137C22×C42C2312yes—
138C22×C42C2311yescharacteristic
139#400—28no—
140#399—28no—
141#398—48no—
142#396—48no—
143#395—216no—
144#395—28no—
145#393—28no—
146#391—28no—
147#391—24no—
148#391—48no—
149#389—28no—
150#389—24no—
151#389—28no—
152#389—28no—
153#389—28no—
154#389—28no—
155#389—216no—
156#389—44no—
157#388—28no—
158#388—28no—
159#387—28no—
160#384—28no—
161#382—28no—
162#382D418yes—
163#380—24no—
164#380—28no—
165C4×C4⋊C4—24no—
166C4×C4⋊C4—24no—
167C4×C4⋊C4—24no—
168C4×C4⋊C4—28no—
169C4×C22⋊C4—44no—
170C2×#95C2314yes—
171C2×#95C2312yes—
172C2×#95C2314yes—
173C2×#95C2314yes—
174C2×#95C2314yes—
175C2×#95C2314yes—
176C2×#95C2312yes—
177C2×#95C2314yes—
178C2×#95C2312yes—
179C2×#95C2312yes—
180C2×#95C2312yes—
181C2×#95C2312yes—
182C2×#95C2314yes—
183C2×#95C2314yes—
184C2×#95C2311yescharacteristic
185C2×#95C2311yescharacteristic
186C2×#95C2312yes—
187C2×#95C2312yes—
188C2×#95C2312yes—
189C2×#95C2314yes—
190C2×#95C2312yes—
191C2×#95C2314yes—
192C2×#95C2314yes—
193C2×#95C2314yes—
194C2×#95C2312yes—
195C2×#95C2312yes—
196C2×#95C2312yes—
197C2×#95C2312yes—
198C25C2411yescharacteristic
199C22×Q8—22no—
200C22×Q8—24no—
201C22×Q8—24no—
202C22×Q8—22no—
203C22×Q8—24no—
204C22×Q8—22no—
205C22×Q8—22no—
206C22×Q8—44no—
207C22×Q8—44no—
208C22×Q8C2×D414yes—
209C22×Q8C2×Q814yes—
210C22×Q8#4114yes—
211C23×C4—22no—
212C23×C4—22no—
213C23×C4—22no—
214C23×C4—22no—
215C23×C4—22no—
216C23×C4—22no—
217C23×C4C2×Q814yes—
218C23×C4C2411yescharacteristic
219C23×C4C2414yes—
220C23×C4C2412yes—
221C23×C4C2412yes—
222C23×C4C2411yescharacteristic
223C23×C4C2412yes—
224C23×C4C2414yes—
225C23×C4C2412yes—
226C23×C4C2411yescharacteristic
227C23×C4C2412yes—
228C23×C4C2412yes—
229C23×C4C2412yes—
230C23×C4C2411yescharacteristic
231C23×C4C2411yescharacteristic
232C23×C4C2412yes—
233C23×C4C2411yescharacteristic
234C4⋊Q8—416no—
235C22⋊Q8—232no—
236C4×Q8—232no—
237C4×Q8—432no—
238#117—416no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4—28no—
258C2×C4⋊C4—24no—
259C2×C4⋊C4—24no—
260C2×C4⋊C4—24no—
261C2×C4⋊C4—24no—
262C2×C4⋊C4—22no—
263C2×C4⋊C4—28no—
264C2×C4⋊C4—24no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4—22no—
267C2×C4⋊C4—28no—
268C2×C4⋊C4—24no—
269C2×C4⋊C4—24no—
270C2×C4⋊C4—22no—
271C2×C4⋊C4—24no—
272C2×C4⋊C4—28no—
273C2×C4⋊C4—22no—
274C2×C4⋊C4—28no—
275C2×C4⋊C4—28no—
276C2×C4⋊C4—24no—
277C2×C4⋊C4—28no—
278C2×C4⋊C4—28no—
279C2×C4⋊C4—42no—
280C2×C4⋊C4—44no—
281C2×C4⋊C4—44no—
282C2×C4⋊C4—44no—
283C2×C4⋊C4—44no—
284C2×C4⋊C4—44no—
285C2×C4⋊C4—44no—
286C2×C4⋊C4—44no—
287C2×C4⋊C4—44no—
288C2×C4⋊C4—44no—
289C2×C4⋊C4—44no—
290C2×C4⋊C4—44no—
291C2×C4⋊C4C2×D418yes—
292C2×C4⋊C4C2×D414yes—
293C2×C4⋊C4C2×D418yes—
294C2×C4⋊C4C2×Q814yes—
295C2×C4⋊C4C2×Q818yes—
296C2×C22⋊C4—24no—
297C2×C22⋊C4—24no—
298C2×C22⋊C4—22no—
299C2×C22⋊C4—22no—
300C2×C22⋊C4—22no—
301C2×C22⋊C4—22no—
302C2×C22⋊C4—22no—
303C2×C22⋊C4—22no—
304C2×C22⋊C4—24no—
305C2×C22⋊C4—24no—
306C2×C22⋊C4—24no—
307C2×C22⋊C4—24no—
308C2×C22⋊C4—28no—
309C2×C22⋊C4—42no—
310C2×C22⋊C4—42no—
311C2×C22⋊C4—42no—
312C2×C22⋊C4—44no—
313C2×C22⋊C4—44no—
314C2×C22⋊C4—44no—
315C2×C22⋊C4C2×Q818yes—
316C2×C42—24no—
317C2×C42—24no—
318C2×C42—22no—
319C2×C42—22no—
320C2×C42—28no—
321C2×C42—24no—
322C2×C42—24no—
323C2×C42—22no—
324C2×C42—28no—
325C2×C42—24no—
326C2×C42—24no—
327C2×C42—24no—
328C2×C42—24no—
329C2×C42—24no—
330C2×C42—44no—
331C2×C42—44no—
332C2×C42C2×D418yes—
333C2×C42C2×D414yes—
334C2×C42C2×D414yes—
335C2×C42#4114yes—
336#95—22no—
337#95—22no—
338#95—24no—
339#95—24no—
340#95—24no—
341#95—28no—
342#95—28no—
343#95—24no—
344#95—24no—
345#95—24no—
346#95—28no—
347#95—28no—
348#95—28no—
349#95—28no—
350#95—28no—
351#95—24no—
352#95—28no—
353#95—24no—
354#95—24no—
355#95—42no—
356#95—42no—
357#95—42no—
358#95—44no—
359#95—44no—
360#95—44no—
361#95—44no—
362#95C2×D418yes—
363#95C2×D418yes—
364C24—21no—
365C24—21no—
366C24—21no—
367C24—21no—
368C24—22no—
369C24—21no—
370C24—22no—
371C24—22no—
372C24—21no—
373C24—22no—
374C24C22×Q812yes—
375C24C2511yescharacteristic
376C2×Q8—28no—
377C2×Q8—28no—
378C2×Q8—28no—
379C2×Q8—28no—
380C2×Q8—28no—
381C2×Q8—28no—
382C2×Q8—416no—
383C2×Q8—48no—
384C2×Q8—88no—
385C2×Q8—84no—
386C2×Q8—84no—
387C2×Q8C22⋊Q8116yes—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—21no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—21no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—24no—
406C22×C4—22no—
407C22×C4—21no—
408C22×C4—21no—
409C22×C4—24no—
410C22×C4—22no—
411C22×C4—21no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—24no—
416C22×C4—24no—
417C22×C4—22no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—24no—
428C22×C4—21no—
429C22×C4—22no—
430C22×C4—24no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—24no—
434C22×C4—28no—
435C22×C4—28no—
436C22×C4—28no—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—21no—
442C22×C4—24no—
443C22×C4—24no—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—21no—
447C22×C4—22no—
448C22×C4—24no—
449C22×C4—24no—
450C22×C4—24no—
451C22×C4—24no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
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—24no—
466C22×C4—44no—
467C22×C4—44no—
468C22×C4—44no—
469C22×C4—44no—
470C22×C4C22×D418yes—
471C22×C4C22×D414yes—
472C22×C4C22×D412yes—
473C22×C4C22×D412yes—
474C22×C4C22×D412yes—
475C22×C4C22×D412yes—
476C22×C4C22×D414yes—
477C22×C4C22×D412yes—
478C22×C4C22×D412yes—
479C22×C4C22×Q814yes—
480C22×C4C22×Q812yes—
481C22×C4C22×Q814yes—
482C22×C4C22×Q812yes—
483C22×C4C2×#4112yes—
484C22×C4C2×#4112yes—
485C22×C4C2×#4114yes—
486C22×C4C2×#4112yes—
487C22×C4C2×#4112yes—
488C22×C4H(5,𝔽2)14yes—
489C22×C4H(5,𝔽2)12yes—
490C22×C42−1+414yes—
491C22×C42−1+414yes—
492C22×C42−1+412yes—
493C4⋊C4—28no—
494C4⋊C4—28no—
495C4⋊C4—216no—
496C4⋊C4—28no—
497C4⋊C4—28no—
498C4⋊C4—216no—
499C4⋊C4—216no—
500C4⋊C4—48no—
501C4⋊C4—416no—
502C4⋊C4—416no—
503C4⋊C4—44no—
504C4⋊C4—44no—
505C4⋊C4—48no—
506C4⋊C4—48no—
507C4⋊C4—44no—
508C4⋊C4—88no—
509C4⋊C4—84no—
510C4⋊C4—88no—
511C4⋊C4—88no—
512C4⋊C4—88no—
513C22⋊C4—216no—
514C22⋊C4—48no—
515C22⋊C4—48no—
516C22⋊C4—44no—
517C22⋊C4—44no—
518C22⋊C4—88no—
519C22⋊C4—84no—
520C42—216no—
521C42—416no—
522C42—44no—
523C42—48no—
524C42—48no—
525C42—84no—
526C42#121116yes—
527C23—22no—
528C23—22no—
529C23—22no—
530C23—24no—
531C23—24no—
532C23—41no—
533C23—41no—
534C23—42no—
535C23—41no—
536C23—41no—
537C23—41no—
538C23—41no—
539C23—42no—
540C23—41no—
541C23—42no—
542C23—41no—
543C23—42no—
544C23—42no—
545C23—42no—
546C23—42no—
547C23—42no—
548C23—42no—
549C23—42no—
550C23C23×D411yescharacteristic
551C23C23×D411yescharacteristic
552C23C23×Q811yescharacteristic
553C23C22×#4111yescharacteristic
554C23C22×#4111yescharacteristic
555C23C2×H(5,𝔽2)11yescharacteristic
556C23C2×H(5,𝔽2)12yes—
557C23C2×2−1+411yescharacteristic
558C23C2×2−1+412yes—
559C23C2×2−1+412yes—
560C23#58512yes—
561Q8—232no—
562Q8—816no—
563C2×C4—24no—
564C2×C4—24no—
565C2×C4—24no—
566C2×C4—24no—
567C2×C4—24no—
568C2×C4—24no—
569C2×C4—24no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—24no—
573C2×C4—22no—
574C2×C4—24no—
575C2×C4—22no—
576C2×C4—24no—
577C2×C4—24no—
578C2×C4—24no—
579C2×C4—22no—
580C2×C4—28no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—24no—
584C2×C4—24no—
585C2×C4—28no—
586C2×C4—24no—
587C2×C4—24no—
588C2×C4—28no—
589C2×C4—24no—
590C2×C4—28no—
591C2×C4—24no—
592C2×C4—24no—
593C2×C4—24no—
594C2×C4—24no—
595C2×C4—24no—
596C2×C4—24no—
597C2×C4—41no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—41no—
601C2×C4—44no—
602C2×C4—42no—
603C2×C4—41no—
604C2×C4—41no—
605C2×C4—42no—
606C2×C4—44no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—44no—
613C2×C4—44no—
614C2×C4—44no—
615C2×C4—44no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—44no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—416no—
623C2×C4—41no—
624C2×C4—41no—
625C2×C4—44no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—44no—
629C2×C4—44no—
630C2×C4—42no—
631C2×C4—42no—
632C2×C4—41no—
633C2×C4—41no—
634C2×C4—44no—
635C2×C4—42no—
636C2×C4—42no—
637C2×C4—44no—
638C2×C4—42no—
639C2×C4—41no—
640C2×C4—44no—
641C2×C4—42no—
642C2×C4—42no—
643C2×C4—44no—
644C2×C4—41no—
645C2×C4—42no—
646C2×C4—42no—
647C2×C4—42no—
648C2×C4—44no—
649C2×C4—42no—
650C2×C4—42no—
651C2×C4—44no—
652C2×C4C2×#12114yes—
653C2×C4C2×#12114yes—
654C2×C4C2×#12114yes—
655C2×C4C2×C22⋊Q818yes—
656C2×C4C2×C22⋊Q814yes—
657C2×C4#53714yes—
658C2×C4#54614yes—
659C2×C4D4×Q814yes—
660C2×C4#55214yes—
661C2×C4#55414yes—
662C22—44no—
663C22—42no—
664C22—42no—
665C22—42no—
666C22—44no—
667C22—81no—
668C22—81no—
669C22—81no—
670C22—82no—
671C22—81no—
672C22—82no—
673C22C22×#12111yescharacteristic
674C22C22×C22⋊Q811yescharacteristic
675C22C2×#53711yescharacteristic
676C22#346012yes—
677C22C2×#54611yescharacteristic
678C22C2×D4×Q811yescharacteristic
679C22C2×#55211yescharacteristic
680C22C2×#55411yescharacteristic
681C22#348611yescharacteristic
682C22#348611yescharacteristic
683C22#348812yes—
684C22#349811yescharacteristic
685C22#349912yes—
686C22#350011yescharacteristic
687C22#350212yes—
688C22#350312yes—
689C22#350912yes—
690C22#351812yes—
691C22#352212yes—
692C22#353612yes—
693C22#354112yes—
694C22#355212yes—
695C22#356312yes—
696C4—24no—
697C4—28no—
698C4—24no—
699C4—44no—
700C4—44no—
701C4—42no—
702C4—44no—
703C4—44no—
704C4—48no—
705C4—44no—
706C4—82no—
707C4—81no—
708C4—81no—
709C4—84no—
710C4—82no—
711C4#241718yes—
712C2—82no—
713C2C2×#241711yescharacteristic
714C2#3525611yescharacteristic
715C2#3566711yescharacteristic
716C2#3571111yescharacteristic
717C2#3574811yescharacteristic
718C2#3827312yes—
719C2#3969112yes—
720C2#4279011yescharacteristic
721C2#4279411yescharacteristic
722C2#4527212yes—
723C2#4530312yes—
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
1. C2×#2567C2semidirect
3. C2×#2351C2semidirect
5. #16567C2semidirect
8. #15623C2semidirect
9. #15137C2semidirect
12. #14869C2semidirect
13. #14155C2semidirect
14. #14150C2semidirect
16. #13935C2semidirect
95. C2×C22⋊Q8Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#25672
C2×#26282
C2×#23512
C2×#23232
#165672
#165642
#156332
#156232
#151372
#151312
#148702
#148692
#141552
#141502
#139692
#139352
#138522