Finite Group Databasebeta

← #719552

Subgroups — #719552

Order 512.

2922 conjugacy classes of subgroups.

The subgroup lattice has 712 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2552C211yesmaximal, characteristic, complemented
2C2×#2453C212yesmaximal, complemented
3C2×#2454C212yesmaximal
4C2×#2367C211yesmaximal, characteristic
5#17064C212yesmaximal, complemented
6#16984C212yesmaximal, complemented
7#15765C214yesmaximal, complemented
8#15513C214yesmaximal, complemented
9#15193C212yesmaximal, complemented
10#15148C212yesmaximal
11#15135C212yesmaximal, complemented
12#14969C212yesmaximal, complemented
13#14273C211yesmaximal, characteristic
14#14153C211yesmaximal, characteristic
15#13910C211yesmaximal, characteristic, complemented
16#13856C211yesmaximal, characteristic, complemented
17#13783C211yesmaximal, characteristic, complemented
18C22×#125C2212yes—
19C22×#124C2211yescharacteristic
20C22×C4×Q8C2211yescharacteristic
21C22×C4×D4C2211yescharacteristic
22#2552—28no—
23#2454—216no—
24#2453—216no—
25C2×#400C2214yes—
26C2×#399C2214yes—
27C2×#399C2214yes—
28C2×#398C2214yes—
29C2×#398C2214yes—
30C2×#396C2214yes—
31#2367—28no—
32C2×#391C2212yes—
33C2×#391C2211yescharacteristic
34C2×#391C2212yes—
35C2×#390C2211yescharacteristic
36C2×#390C2212yes—
37C2×#390C2212yes—
38C2×#389C2214yes—
39C2×#389C2212yes—
40C2×#389C2212yes—
41C2×#389C2212yes—
42C2×#389C2212yes—
43C2×#389C2212yes—
44C2×#388C2212yes—
45C2×#388C2212yes—
46C2×#387C2214yes—
47C2×#385C2212yes—
48C2×#385C2214yes—
49C2×#384C2212yes—
50C2×#384C2211yescharacteristic
51C2×#384C2212yes—
52C2×#380C2214yes—
53C2×C4×C4⋊C4C2212yes—
54C2×C4×C4⋊C4C2211yescharacteristic
55C2×C4×C4⋊C4C2211yescharacteristic
56C2×C4×C4⋊C4C2212yes—
57C2×C4×C4⋊C4C2211yescharacteristic
58C2×C4×C4⋊C4C2211yescharacteristic
59C2×C4×C22⋊C4C2212yes—
60C2×#376C2211yescharacteristic
61#1445C2214yes—
62#1445C2214yes—
63#1445C2214yes—
64#1444C2211yescharacteristic
65#1444C2212yes—
66#1444C2211yescharacteristic
67#1444C2211yescharacteristic
68#1444C2212yes—
69#1444C2212yes—
70#1444C2211yescharacteristic
71#1444C2212yes—
72#1444C2214yes—
73#1439C2212yes—
74#1439C2212yes—
75#1439C2214yes—
76#1439C2214yes—
77#1439C2214yes—
78#1439C2212yes—
79#1439C2212yes—
80#1439C2214yes—
81#1437C2214yes—
82#1436C2214yes—
83C4⋊#95C2212yes—
84C4⋊#95C2212yes—
85C4⋊#95C2212yes—
86#1434C2212yes—
87C23×Q8C2311yescharacteristic
88C23×D4C2311yescharacteristic
89C24×C4C2312yes—
90C2×#125—28no—
91C2×#125—28no—
92C2×#125—28no—
93C2×#124—24no—
94C2×#124—24no—
95C2×#124Q818yes—
96C2×C4×Q8—24no—
97C2×C4×Q8—28no—
98C2×C4×D4—24no—
99C2×C4×D4—28no—
100C22×C4⋊C4C2314yes—
101C22×C4⋊C4C2312yes—
102C22×C4⋊C4C2314yes—
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2314yes—
106C22×C4⋊C4C2311yescharacteristic
107C22×C4⋊C4C2312yes—
108C22×C4⋊C4C2311yescharacteristic
109C22×C4⋊C4C2311yescharacteristic
110C22×C4⋊C4C2312yes—
111C22×C4⋊C4C2314yes—
112C22×C4⋊C4C2312yes—
113C22×C4⋊C4C2312yes—
114C22×C4⋊C4C2314yes—
115C22×C4⋊C4C2312yes—
116C22×C4⋊C4C2311yescharacteristic
117C22×C4⋊C4C2311yescharacteristic
118C22×C4⋊C4C2312yes—
119C22×C4⋊C4C2314yes—
120C22×C4⋊C4C2311yescharacteristic
121C22×C22⋊C4C2312yes—
122C22×C22⋊C4C2312yes—
123C22×C22⋊C4C2312yes—
124C22×C22⋊C4C2312yes—
125C22×C22⋊C4C2314yes—
126C22×C22⋊C4C2314yes—
127C22×C22⋊C4C2314yes—
128C22×C22⋊C4C2314yes—
129C22×C22⋊C4C2312yes—
130C22×C42C2311yescharacteristic
131C22×C42C2312yes—
132C22×C42C2311yescharacteristic
133C22×C42C2312yes—
134C22×C42C2312yes—
135C22×C42C2312yes—
136C22×C42C2312yes—
137C22×C42C2311yescharacteristic
138#400—216no—
139#399—216no—
140#399—48no—
141#398—216no—
142#398—216no—
143#396—216no—
144#391—24no—
145#391—28no—
146#391—28no—
147#390—24no—
148#390—28no—
149#390—28no—
150#389—216no—
151#389—28no—
152#389—28no—
153#389—28no—
154#389—28no—
155#389D4116yes—
156#388—28no—
157#388—28no—
158#387—216no—
159#385—28no—
160#385—216no—
161#384—28no—
162#384—24no—
163#384—44no—
164#380—216no—
165C4×C4⋊C4—24no—
166C4×C4⋊C4—28no—
167C4×C4⋊C4—28no—
168C4×C4⋊C4—24no—
169C4×C4⋊C4—24no—
170C4×C4⋊C4D418yes—
171C4×C22⋊C4—28no—
172#376—24no—
173C2×#95C2312yes—
174C2×#95C2312yes—
175C2×#95C2314yes—
176C2×#95C2314yes—
177C2×#95C2314yes—
178C2×#95C2314yes—
179C2×#95C2312yes—
180C2×#95C2314yes—
181C2×#95C2314yes—
182C2×#95C2311yescharacteristic
183C2×#95C2312yes—
184C2×#95C2311yescharacteristic
185C2×#95C2314yes—
186C2×#95C2312yes—
187C2×#95C2312yes—
188C2×#95C2311yescharacteristic
189C2×#95C2311yescharacteristic
190C2×#95C2312yes—
191C2×#95C2312yes—
192C2×#95C2312yes—
193C2×#95C2314yes—
194C2×#95C2314yes—
195C2×#95C2312yes—
196C2×#95C2314yes—
197C25C2412yes—
198C22×Q8—22no—
199C22×Q8—24no—
200C22×Q8—22no—
201C22×Q8—24no—
202C22×Q8C2×Q814yes—
203C22×D4—22no—
204C22×D4—24no—
205C22×D4—22no—
206C22×D4—24no—
207C22×D4C2×Q814yes—
208C23×C4—24no—
209C23×C4—24no—
210C23×C4—28no—
211C23×C4—44no—
212C23×C4—42no—
213C23×C4C2414yes—
214C23×C4C2411yescharacteristic
215C23×C4C2412yes—
216C23×C4C2412yes—
217C23×C4C2411yescharacteristic
218C23×C4C2412yes—
219C23×C4C2414yes—
220C23×C4C2412yes—
221C23×C4C2412yes—
222C23×C4C2412yes—
223C23×C4C2411yescharacteristic
224C23×C4C2411yescharacteristic
225C23×C4C2412yes—
226C23×C4C2411yescharacteristic
227C23×C4C2411yescharacteristic
228C23×C4C2411yescharacteristic
229#125—432no—
230#124—232no—
231C4×Q8—416no—
232C4×D4—416no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—24no—
237C2×C4⋊C4—28no—
238C2×C4⋊C4—28no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—28no—
241C2×C4⋊C4—28no—
242C2×C4⋊C4—28no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—28no—
246C2×C4⋊C4—28no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—28no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—28no—
259C2×C4⋊C4—24no—
260C2×C4⋊C4—22no—
261C2×C4⋊C4—24no—
262C2×C4⋊C4—28no—
263C2×C4⋊C4—24no—
264C2×C4⋊C4—28no—
265C2×C4⋊C4—44no—
266C2×C4⋊C4—44no—
267C2×C4⋊C4—41no—
268C2×C4⋊C4—41no—
269C2×C4⋊C4—42no—
270C2×C4⋊C4—42no—
271C2×C4⋊C4—44no—
272C2×C4⋊C4—42no—
273C2×C4⋊C4—42no—
274C2×C4⋊C4—44no—
275C2×C4⋊C4—42no—
276C2×C4⋊C4—42no—
277C2×C4⋊C4—42no—
278C2×C4⋊C4—42no—
279C2×C4⋊C4—42no—
280C2×C4⋊C4—44no—
281C2×C4⋊C4—44no—
282C2×C4⋊C4—42no—
283C2×C4⋊C4—44no—
284C2×C4⋊C4—44no—
285C2×C4⋊C4C2×D414yes—
286C2×C4⋊C4C2×D418yes—
287C2×C4⋊C4C2×D4116yes—
288C2×C4⋊C4C2×D414yes—
289C2×C4⋊C4C2×D418yes—
290C2×C22⋊C4—24no—
291C2×C22⋊C4—24no—
292C2×C22⋊C4—24no—
293C2×C22⋊C4—24no—
294C2×C22⋊C4—24no—
295C2×C22⋊C4—28no—
296C2×C22⋊C4—24no—
297C2×C22⋊C4—28no—
298C2×C22⋊C4—24no—
299C2×C22⋊C4—28no—
300C2×C22⋊C4—28no—
301C2×C22⋊C4—24no—
302C2×C22⋊C4—28no—
303C2×C22⋊C4—42no—
304C2×C22⋊C4—44no—
305C2×C22⋊C4—44no—
306C2×C22⋊C4—44no—
307C2×C22⋊C4—44no—
308C2×C22⋊C4—44no—
309C2×C22⋊C4—44no—
310C2×C22⋊C4—44no—
311C2×C22⋊C4—44no—
312C2×C22⋊C4—44no—
313C2×C22⋊C4C2×Q818yes—
314C2×C22⋊C4C2×Q818yes—
315C2×C42—24no—
316C2×C42—24no—
317C2×C42—24no—
318C2×C42—22no—
319C2×C42—24no—
320C2×C42—24no—
321C2×C42—22no—
322C2×C42—28no—
323C2×C42—24no—
324C2×C42—24no—
325C2×C42—24no—
326C2×C42—24no—
327C2×C42—42no—
328C2×C42—44no—
329C2×C42—42no—
330C2×C42C2×D418yes—
331C2×C42C2×D418yes—
332C2×C42C2×D414yes—
333C2×C42C2×Q814yes—
334C2×C42#4114yes—
335#95—22no—
336#95—24no—
337#95—24no—
338#95—24no—
339#95—22no—
340#95—24no—
341#95—24no—
342#95—24no—
343#95—28no—
344#95—28no—
345#95—28no—
346#95—24no—
347#95—28no—
348#95—28no—
349#95—28no—
350#95—28no—
351#95—22no—
352#95—22no—
353#95—24no—
354#95—24no—
355#95—44no—
356#95—44no—
357#95C2×D418yes—
358#95C2×D4116yes—
359C24—22no—
360C24—24no—
361C24—22no—
362C24—22no—
363C24—22no—
364C24—24no—
365C24—22no—
366C24—24no—
367C24—22no—
368C24—24no—
369C24C22×Q814yes—
370C24C2511yescharacteristic
371C2×Q8—28no—
372C2×Q8—216no—
373C2×Q8—44no—
374C2×Q8—44no—
375C2×Q8—48no—
376C2×D4—28no—
377C2×D4—216no—
378C2×D4—44no—
379C2×D4—44no—
380C2×D4—48no—
381C22×C4—21no—
382C22×C4—21no—
383C22×C4—22no—
384C22×C4—21no—
385C22×C4—22no—
386C22×C4—24no—
387C22×C4—21no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—24no—
391C22×C4—21no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—21no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—24no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—24no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—24no—
406C22×C4—24no—
407C22×C4—22no—
408C22×C4—24no—
409C22×C4—22no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—24no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—21no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—24no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—21no—
423C22×C4—22no—
424C22×C4—21no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—24no—
430C22×C4—22no—
431C22×C4—21no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—24no—
436C22×C4—24no—
437C22×C4—22no—
438C22×C4—24no—
439C22×C4—21no—
440C22×C4—22no—
441C22×C4—21no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—24no—
447C22×C4—24no—
448C22×C4—24no—
449C22×C4—21no—
450C22×C4—22no—
451C22×C4—21no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—48no—
456C22×C4—416no—
457C22×C4—48no—
458C22×C4—84no—
459C22×C4—88no—
460C22×C4C22×D414yes—
461C22×C4C22×D414yes—
462C22×C4C22×D418yes—
463C22×C4C22×D414yes—
464C22×C4C22×D412yes—
465C22×C4C22×D412yes—
466C22×C4C22×D414yes—
467C22×C4C22×D418yes—
468C22×C4C22×D412yes—
469C22×C4C22×D414yes—
470C22×C4C22×Q814yes—
471C22×C4C22×Q812yes—
472C22×C4C22×Q812yes—
473C22×C4C22×Q812yes—
474C22×C4C2×#4112yes—
475C22×C4C2×#4112yes—
476C22×C4C2×#4112yes—
477C22×C4H(5,𝔽2)12yes—
478C22×C42−1+412yes—
479C22×C42−1+414yes—
480C4⋊C4—28no—
481C4⋊C4—216no—
482C4⋊C4—416no—
483C4⋊C4—44no—
484C4⋊C4—44no—
485C4⋊C4—48no—
486C4⋊C4—416no—
487C4⋊C4—48no—
488C4⋊C4—44no—
489C4⋊C4—44no—
490C4⋊C4—48no—
491C4⋊C4—48no—
492C4⋊C4—48no—
493C4⋊C4—48no—
494C4⋊C4—416no—
495C4⋊C4—416no—
496C4⋊C4—48no—
497C4⋊C4—84no—
498C4⋊C4—88no—
499C4⋊C4—88no—
500C4⋊C4#127116yes—
501C22⋊C4—216no—
502C22⋊C4—216no—
503C22⋊C4—48no—
504C22⋊C4—416no—
505C22⋊C4—48no—
506C22⋊C4—84no—
507C22⋊C4—88no—
508C22⋊C4—88no—
509C22⋊C4—88no—
510C42—216no—
511C42—48no—
512C42—48no—
513C42—44no—
514C42—48no—
515C42—44no—
516C42—48no—
517C42C22⋊Q8116yes—
518C23—24no—
519C23—28no—
520C23—24no—
521C23—28no—
522C23—24no—
523C23—42no—
524C23—42no—
525C23—42no—
526C23—42no—
527C23—44no—
528C23—42no—
529C23—44no—
530C23—44no—
531C23—42no—
532C23—42no—
533C23—44no—
534C23—44no—
535C23—42no—
536C23—44no—
537C23—44no—
538C23—44no—
539C23—44no—
540C23—44no—
541C23C23×D411yescharacteristic
542C23C23×D412yes—
543C23C23×Q811yescharacteristic
544C23C22×#4111yescharacteristic
545C23C2×H(5,𝔽2)11yescharacteristic
546C23C2×2−1+412yes—
547C23C2×2−1+411yescharacteristic
548C23#58511yescharacteristic
549C23#58511yescharacteristic
550C23#58512yes—
551C23#58512yes—
552Q8—416no—
553D4—416no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—24no—
558C2×C4—28no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—28no—
563C2×C4—24no—
564C2×C4—22no—
565C2×C4—24no—
566C2×C4—22no—
567C2×C4—24no—
568C2×C4—24no—
569C2×C4—24no—
570C2×C4—24no—
571C2×C4—24no—
572C2×C4—28no—
573C2×C4—28no—
574C2×C4—24no—
575C2×C4—28no—
576C2×C4—24no—
577C2×C4—24no—
578C2×C4—22no—
579C2×C4—24no—
580C2×C4—24no—
581C2×C4—28no—
582C2×C4—24no—
583C2×C4—24no—
584C2×C4—22no—
585C2×C4—24no—
586C2×C4—41no—
587C2×C4—42no—
588C2×C4—41no—
589C2×C4—42no—
590C2×C4—44no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—44no—
594C2×C4—41no—
595C2×C4—42no—
596C2×C4—41no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—44no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—44no—
603C2×C4—44no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—44no—
607C2×C4—42no—
608C2×C4—44no—
609C2×C4—41no—
610C2×C4—41no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—44no—
614C2×C4—41no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—44no—
620C2×C4—42no—
621C2×C4—44no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—41no—
625C2×C4—42no—
626C2×C4—44no—
627C2×C4—42no—
628C2×C4—44no—
629C2×C4—42no—
630C2×C4—44no—
631C2×C4—41no—
632C2×C4—42no—
633C2×C4—42no—
634C2×C4—44no—
635C2×C4—42no—
636C2×C4—41no—
637C2×C4—42no—
638C2×C4—42no—
639C2×C4—42no—
640C2×C4—816no—
641C2×C4C2×C22⋊Q814yes—
642C2×C4C2×C22⋊Q814yes—
643C2×C4C2×C22⋊Q814yes—
644C2×C4C2×#12718yes—
645C2×C4C2×#12714yes—
646C2×C4#53614yes—
647C2×C4#54718yes—
648C2×C4D4×Q818yes—
649C22—44no—
650C22—48no—
651C22—44no—
652C22—48no—
653C22—44no—
654C22—82no—
655C22—84no—
656C22—84no—
657C22—82no—
658C22—82no—
659C22—82no—
660C22C22×C22⋊Q811yescharacteristic
661C22C22×#12711yescharacteristic
662C22C2×#53611yescharacteristic
663C22#346011yescharacteristic
664C22#346111yescharacteristic
665C22C2×#54712yes—
666C22C2×D4×Q812yes—
667C22#348511yescharacteristic
668C22#348512yes—
669C22#348711yescharacteristic
670C22#348712yes—
671C22#348812yes—
672C22#348811yescharacteristic
673C22#348912yes—
674C22#348911yescharacteristic
675C22#350311yescharacteristic
676C22#350412yes—
677C22#350411yescharacteristic
678C22#351212yes—
679C22#355412yes—
680C22#355512yes—
681C22#355712yes—
682C22#355712yes—
683C4—24no—
684C4—24no—
685C4—28no—
686C4—44no—
687C4—44no—
688C4—48no—
689C4—44no—
690C4—44no—
691C4—42no—
692C4—42no—
693C4—81no—
694C4—82no—
695C4—81no—
696C4—82no—
697C4—84no—
698C4#254218yes—
699C2—84no—
700C2C2×#254211yescharacteristic
701C2#3525711yescharacteristic
702C2#3563911yescharacteristic
703C2#3606011yescharacteristic
704C2#3606611yescharacteristic
705C2#3779812yes—
706C2#3779912yes—
707C2#4267811yescharacteristic
708C2#4288411yescharacteristic
709C2#4513912yes—
710C2#4542412yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC23×D4C23
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×#2552C2semidirect
2. C2×#2453C2semidirect
5. #17064C2semidirect
6. #16984C2semidirect
7. #15765C2semidirect
8. #15513C2semidirect
9. #15193C2semidirect
11. #15135C2semidirect
12. #14969C2semidirect
15. #13910C2semidirect
16. #13856C2semidirect
17. #13783C2semidirect

Maximal subgroups

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

subgroupindex
C2×#25522
C2×#24532
C2×#24542
C2×#23672
#170642
#169842
#157652
#155132
#151932
#151482
#151352
#149692
#142732
#141532
#139102
#138562
#137832