Finite Group Databasebeta

← #613739

Subgroups — #613739

Order 512.

2966 conjugacy classes of subgroups.

The subgroup lattice has 733 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2851C211yesmaximal, characteristic
2C2×#2584C211yesmaximal, characteristic, complemented
3C2×C4×C22⋊Q8C211yesmaximal, characteristic
4C2×C4×#117C211yesmaximal, characteristic
5#15204C214yesmaximal
6#14903C212yesmaximal, complemented
7#14901C212yesmaximal, complemented
8#14840C212yesmaximal
9#14763C212yesmaximal
10#14699C214yesmaximal, complemented
11#14160C211yesmaximal, characteristic, complemented
12#14046C212yesmaximal, complemented
13#14045C212yesmaximal, complemented
14#13954C211yesmaximal, characteristic, complemented
15#13951C212yesmaximal
16#13951C212yesmaximal
17#13925C211yesmaximal, characteristic, complemented
18C22×#125C2211yescharacteristic
19C22×C22⋊Q8C2211yescharacteristic
20C22×C4×Q8C2211yescharacteristic
21C22×#117C2212yes—
22C23×C42C2211yescharacteristic
23#2851—28no—
24#2584—28no—
25C2×#398C2212yes—
26C2×#395C2212yes—
27C4×C22⋊Q8—28no—
28C2×#391C2211yescharacteristic
29C2×#389C2211yescharacteristic
30C2×#389C2211yescharacteristic
31C2×#389C2211yescharacteristic
32C2×#389C2214yes—
33C2×#389C2212yes—
34C2×#389C2212yes—
35C2×#389C2214yes—
36C2×#389C2212yes—
37C2×#389C2212yes—
38C2×#387C2212yes—
39C2×#387C2212yes—
40C2×#387C2212yes—
41C2×#385C2212yes—
42C2×#385C2212yes—
43C2×#385C2212yes—
44C2×#382C2211yescharacteristic
45C2×#380C2211yescharacteristic
46C2×#380C2211yescharacteristic
47C2×#380C2211yescharacteristic
48C4×#117—28no—
49C2×C4×C4⋊C4C2211yescharacteristic
50C2×C4×C4⋊C4C2211yescharacteristic
51C2×C4×C4⋊C4C2212yes—
52C2×C4×C4⋊C4C2212yes—
53C2×C4×C4⋊C4C2211yescharacteristic
54C2×C4×C22⋊C4C2212yes—
55C2×C4×C22⋊C4C2212yes—
56C2×C4×C22⋊C4C2212yes—
57C2×#376C2212yes—
58C2×C43C2212yes—
59#1446C2212yes—
60#1445C2212yes—
61#1444C2212yes—
62#1444C2214yes—
63#1444C2212yes—
64#1442C2214yes—
65#1441C2212yes—
66#1441C2214yes—
67#1441C2212yes—
68#1441C2214yes—
69#1439C2214yes—
70#1439C2214yes—
71#1439C2214yes—
72#1439C2214yes—
73#1436C2214yes—
74#1436C2214yes—
75#1436C2214yes—
76#1436C2214yes—
77C4⋊#95C2214yes—
78C4⋊#95C2214yes—
79C4⋊#95C2212yes—
80C4⋊#95C2212yes—
81C4⋊#95C2214yes—
82C4⋊#95C2212yes—
83C4⋊#95C2212yes—
84C4⋊#95C2214yes—
85#1434C2214yes—
86#1434C2214yes—
87C23×Q8C2311yescharacteristic
88C24×C4C2311yescharacteristic
89C24×C4C2311yescharacteristic
90C24×C4C2311yescharacteristic
91C2×#125—24no—
92C2×#125—24no—
93C2×#125—24no—
94C2×C22⋊Q8—24no—
95C2×C22⋊Q8—28no—
96C2×C4×Q8—24no—
97C2×C4×Q8—24no—
98C2×C4×Q8—24no—
99C2×#117—28no—
100C2×#117—28no—
101C2×#117Q8116yescomplemented
102C22×C4⋊C4C2312yes—
103C22×C4⋊C4C2311yescharacteristic
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2311yescharacteristic
106C22×C4⋊C4C2311yescharacteristic
107C22×C4⋊C4C2311yescharacteristic
108C22×C4⋊C4C2312yes—
109C22×C4⋊C4C2311yescharacteristic
110C22×C4⋊C4C2311yescharacteristic
111C22×C4⋊C4C2312yes—
112C22×C4⋊C4C2314yes—
113C22×C4⋊C4C2314yes—
114C22×C4⋊C4C2312yes—
115C22×C4⋊C4C2312yes—
116C22×C4⋊C4C2314yes—
117C22×C4⋊C4C2314yes—
118C22×C4⋊C4C2311yescharacteristic
119C22×C4⋊C4C2312yes—
120C22×C4⋊C4C2312yes—
121C22×C4⋊C4C2312yes—
122C22×C22⋊C4C2312yes—
123C22×C22⋊C4C2314yes—
124C22×C22⋊C4C2312yes—
125C22×C22⋊C4C2312yes—
126C22×C22⋊C4C2312yes—
127C22×C42—24no—
128C22×C42—24no—
129C22×C42D418yes—
130C22×C42C2312yes—
131C22×C42C2312yes—
132C22×C42C2314yes—
133C22×C42C2311yescharacteristic
134C22×C42C2311yescharacteristic
135C22×C42C2311yescharacteristic
136C22×C42C2314yes—
137C22×C42C2312yes—
138C22×C42C2312yes—
139C22×C42C2311yescharacteristic
140#398—28no—
141#395—28no—
142#391—24no—
143#389—28no—
144#389—28no—
145#389—216no—
146#389—28no—
147#389—28no—
148#389—216no—
149#389—24no—
150#389—24no—
151#389—24no—
152#387—28no—
153#387—28no—
154#387—28no—
155#385—28no—
156#385—28no—
157#385—28no—
158#382—24no—
159#380—24no—
160#380—24no—
161#380—24no—
162C4×C4⋊C4—28no—
163C4×C4⋊C4—24no—
164C4×C4⋊C4—28no—
165C4×C4⋊C4—24no—
166C4×C4⋊C4D418yes—
167C4×C22⋊C4—28no—
168C4×C22⋊C4—28no—
169C4×C22⋊C4—28no—
170#376—28no—
171C2×#95C2314yes—
172C2×#95C2314yes—
173C2×#95C2312yes—
174C2×#95C2312yes—
175C2×#95C2312yes—
176C2×#95C2312yes—
177C2×#95C2314yes—
178C2×#95C2314yes—
179C2×#95C2314yes—
180C2×#95C2314yes—
181C2×#95C2312yes—
182C2×#95C2312yes—
183C2×#95C2312yes—
184C2×#95C2312yes—
185C2×#95C2312yes—
186C2×#95C2312yes—
187C2×#95C2314yes—
188C2×#95C2312yes—
189C2×#95C2312yes—
190C2×#95C2314yes—
191C2×#95C2314yes—
192C2×#95C2314yes—
193C2×#95C2314yes—
194C2×#95C2314yes—
195C2×#95C2314yes—
196C2×#95C2312yes—
197C43—28no—
198C25C2411yescharacteristic
199C22×Q8—22no—
200C22×Q8—22no—
201C22×Q8—24no—
202C22×Q8—22no—
203C22×Q8—42no—
204C23×C4—22no—
205C23×C4—24no—
206C23×C4—22no—
207C23×C4—22no—
208C23×C4—22no—
209C23×C4—24no—
210C23×C4—22no—
211C23×C4—22no—
212C23×C4—42no—
213C23×C4—41no—
214C23×C4—41no—
215C23×C4—41no—
216C23×C4—41no—
217C23×C4C2×D414yes—
218C23×C4C2×D414yes—
219C23×C4C2×D414yes—
220C23×C4C2×Q818yes—
221C23×C4C2412yes—
222C23×C4C2412yes—
223C23×C4C2412yes—
224C23×C4C2412yes—
225C23×C4C2412yes—
226C23×C4C2412yes—
227C23×C4C2411yescharacteristic
228C23×C4C2414yes—
229C23×C4C2412yes—
230C23×C4C2412yes—
231C23×C4C2411yescharacteristic
232C23×C4C2411yescharacteristic
233C23×C4C2411yescharacteristic
234C23×C4C2414yes—
235C23×C4C2411yescharacteristic
236C23×C4C2411yescharacteristic
237#125—416no—
238C22⋊Q8—232no—
239C4×Q8—416no—
240#117—264no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—24no—
259C2×C4⋊C4—22no—
260C2×C4⋊C4—24no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—22no—
263C2×C4⋊C4—24no—
264C2×C4⋊C4—24no—
265C2×C4⋊C4—28no—
266C2×C4⋊C4—24no—
267C2×C4⋊C4—22no—
268C2×C4⋊C4—24no—
269C2×C4⋊C4—24no—
270C2×C4⋊C4—22no—
271C2×C4⋊C4—22no—
272C2×C4⋊C4—28no—
273C2×C4⋊C4—28no—
274C2×C4⋊C4—28no—
275C2×C4⋊C4—28no—
276C2×C4⋊C4—24no—
277C2×C4⋊C4—24no—
278C2×C4⋊C4—28no—
279C2×C4⋊C4—28no—
280C2×C4⋊C4—44no—
281C2×C4⋊C4—44no—
282C2×C4⋊C4—42no—
283C2×C4⋊C4—42no—
284C2×C4⋊C4—42no—
285C2×C4⋊C4—42no—
286C2×C4⋊C4—42no—
287C2×C4⋊C4—44no—
288C2×C4⋊C4—44no—
289C2×C4⋊C4—44no—
290C2×C4⋊C4—44no—
291C2×C4⋊C4C2×D418yes—
292C2×C4⋊C4C2×Q818yes—
293C2×C4⋊C4C2×Q818yes—
294C2×C4⋊C4#4118yes—
295C2×C4⋊C4#4118yes—
296C2×C22⋊C4—24no—
297C2×C22⋊C4—28no—
298C2×C22⋊C4—28no—
299C2×C22⋊C4—24no—
300C2×C22⋊C4—24no—
301C2×C22⋊C4—24no—
302C2×C22⋊C4—24no—
303C2×C22⋊C4—24no—
304C2×C22⋊C4—42no—
305C2×C22⋊C4—42no—
306C2×C22⋊C4—44no—
307C2×C22⋊C4—42no—
308C2×C22⋊C4—42no—
309C2×C22⋊C4C2×Q8116yes—
310C2×C42—28no—
311C2×C42—22no—
312C2×C42—22no—
313C2×C42—22no—
314C2×C42—28no—
315C2×C42—22no—
316C2×C42—22no—
317C2×C42—22no—
318C2×C42—232no—
319C2×C42—24no—
320C2×C42—24no—
321C2×C42—28no—
322C2×C42—24no—
323C2×C42—24no—
324C2×C42—24no—
325C2×C42—28no—
326C2×C42—24no—
327C2×C42—24no—
328C2×C42—42no—
329C2×C42—42no—
330C2×C42—44no—
331C2×C42—42no—
332C2×C42—42no—
333C2×C42C2×D414yes—
334C2×C42C2×D414yes—
335C2×C42C2×D414yes—
336C2×C42C2×D418yes—
337C2×C42C2×D414yes—
338C2×C42C2×D414yes—
339C2×C42C2×Q8116yes—
340C2×C42#4114yes—
341#95—28no—
342#95—24no—
343#95—24no—
344#95—24no—
345#95—24no—
346#95—24no—
347#95—24no—
348#95—24no—
349#95—24no—
350#95—24no—
351#95—24no—
352#95—24no—
353#95—24no—
354#95—28no—
355#95—28no—
356#95—28no—
357#95—28no—
358#95—28no—
359#95—28no—
360#95—44no—
361#95—44no—
362#95—44no—
363#95—44no—
364#95—44no—
365#95—44no—
366#95C2×D418yes—
367C24—21no—
368C24—22no—
369C24—22no—
370C24—21no—
371C24—21no—
372C24—21no—
373C24—22no—
374C24—21no—
375C24—21no—
376C24C22×D412yes—
377C24C22×Q814yes—
378C24C2511yescharacteristic
379C2×Q8—28no—
380C2×Q8—48no—
381C2×Q8—48no—
382C2×Q8—44no—
383C2×Q8—44no—
384C22×C4—21no—
385C22×C4—21no—
386C22×C4—21no—
387C22×C4—21no—
388C22×C4—22no—
389C22×C4—24no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—21no—
393C22×C4—21no—
394C22×C4—21no—
395C22×C4—21no—
396C22×C4—24no—
397C22×C4—24no—
398C22×C4—24no—
399C22×C4—22no—
400C22×C4—28no—
401C22×C4—216no—
402C22×C4—28no—
403C22×C4—216no—
404C22×C4—21no—
405C22×C4—24no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—24no—
410C22×C4—22no—
411C22×C4—21no—
412C22×C4—24no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—22no—
428C22×C4—28no—
429C22×C4—28no—
430C22×C4—28no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—24no—
442C22×C4—24no—
443C22×C4—24no—
444C22×C4—24no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—24no—
458C22×C4—24no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—44no—
468C22×C4—44no—
469C22×C4—44no—
470C22×C4—48no—
471C22×C4—48no—
472C22×C4—44no—
473C22×C4—44no—
474C22×C4—44no—
475C22×C4—44no—
476C22×C4C4⋊Q8116yes—
477C22×C4C22×D412yes—
478C22×C4C22×D412yes—
479C22×C4C22×D412yes—
480C22×C4C22×D412yes—
481C22×C4C22×D412yes—
482C22×C4C22×D418yes—
483C22×C4C22×D412yes—
484C22×C4C22×D412yes—
485C22×C4C22×D412yes—
486C22×C4C22×D412yes—
487C22×C4C22×Q818yes—
488C22×C4C22×Q818yes—
489C22×C4C22×Q814yes—
490C22×C4C22×Q814yes—
491C22×C4C2×#4114yes—
492C22×C4C2×#4114yes—
493C22×C4C2×#4112yes—
494C22×C4C2×#4112yes—
495C22×C4C2×#4118yes—
496C22×C4C2×#4114yes—
497C22×C4C2×#4112yes—
498C22×C4C2×#4114yes—
499C4⋊C4—28no—
500C4⋊C4—216no—
501C4⋊C4—216no—
502C4⋊C4—416no—
503C4⋊C4—416no—
504C4⋊C4—44no—
505C4⋊C4—48no—
506C4⋊C4—48no—
507C4⋊C4—48no—
508C4⋊C4—44no—
509C4⋊C4—48no—
510C4⋊C4—48no—
511C4⋊C4—44no—
512C4⋊C4—44no—
513C4⋊C4—48no—
514C4⋊C4—44no—
515C4⋊C4—44no—
516C4⋊C4—88no—
517C4⋊C4—88no—
518C4⋊C4C22⋊Q8132yescomplemented
519C22⋊C4—216no—
520C22⋊C4—232no—
521C22⋊C4—48no—
522C22⋊C4—48no—
523C22⋊C4—48no—
524C42—28no—
525C42—28no—
526C42—28no—
527C42—232no—
528C42—48no—
529C42—416no—
530C42—48no—
531C42—48no—
532C42—48no—
533C42#121116yes—
534C23—22no—
535C23—24no—
536C23—22no—
537C23—22no—
538C23—24no—
539C23—22no—
540C23—22no—
541C23—22no—
542C23—24no—
543C23—24no—
544C23—24no—
545C23—24no—
546C23—42no—
547C23—42no—
548C23—42no—
549C23—42no—
550C23—41no—
551C23—41no—
552C23—41no—
553C23—41no—
554C23—42no—
555C23—42no—
556C23C2×C4⋊Q814yes—
557C23C23×D411yescharacteristic
558C23C23×D411yescharacteristic
559C23C23×Q812yes—
560C23C22×#4111yescharacteristic
561C23C22×#4112yes—
562C23C22×#4112yes—
563C23#58512yes—
564C23#58511yescharacteristic
565C23#58511yescharacteristic
566C23#58511yescharacteristic
567C23#58511yescharacteristic
568Q8—416no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—24no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—24no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—24no—
580C2×C4—28no—
581C2×C4—24no—
582C2×C4—28no—
583C2×C4—24no—
584C2×C4—24no—
585C2×C4—28no—
586C2×C4—22no—
587C2×C4—28no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—28no—
592C2×C4—28no—
593C2×C4—232no—
594C2×C4—22no—
595C2×C4—28no—
596C2×C4—24no—
597C2×C4—24no—
598C2×C4—22no—
599C2×C4—22no—
600C2×C4—24no—
601C2×C4—24no—
602C2×C4—24no—
603C2×C4—28no—
604C2×C4—24no—
605C2×C4—28no—
606C2×C4—416no—
607C2×C4—416no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—44no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—44no—
617C2×C4—42no—
618C2×C4—41no—
619C2×C4—44no—
620C2×C4—41no—
621C2×C4—41no—
622C2×C4—41no—
623C2×C4—44no—
624C2×C4—42no—
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—42no—
637C2×C4—42no—
638C2×C4—42no—
639C2×C4—42no—
640C2×C4—42no—
641C2×C4—42no—
642C2×C4—44no—
643C2×C4—44no—
644C2×C4—44no—
645C2×C4—44no—
646C2×C4—42no—
647C2×C4—42no—
648C2×C4—42no—
649C2×C4—42no—
650C2×C4—42no—
651C2×C4—42no—
652C2×C4—42no—
653C2×C4—42no—
654C2×C4—42no—
655C2×C4—44no—
656C2×C4—42no—
657C2×C4—42no—
658C2×C4—44no—
659C2×C4—42no—
660C2×C4C2×#12114yes—
661C2×C4C2×#12114yes—
662C2×C4C2×#12114yes—
663C2×C4C2×C22⋊Q8116yes—
664C2×C4C2×C22⋊Q818yes—
665C2×C4#52514yes—
666C2×C4C2×C4⋊Q814yes—
667C2×C4C2×C4⋊Q814yes—
668C2×C4#53214yes—
669C2×C4#53318yes—
670C22—24no—
671C22—24no—
672C22—24no—
673C22—42no—
674C22—44no—
675C22—42no—
676C22—44no—
677C22—44no—
678C22—42no—
679C22—42no—
680C22—44no—
681C22C22×#12111yescharacteristic
682C22C22×C22⋊Q812yes—
683C22C2×#52511yescharacteristic
684C22C22×C4⋊Q811yescharacteristic
685C22C2×#53211yescharacteristic
686C22C2×#53312yes—
687C22#344911yescharacteristic
688C22#344911yescharacteristic
689C22#344911yescharacteristic
690C22D4×#4112yes—
691C22#347512yes—
692C22#347512yes—
693C22Q8×#4112yes—
694C22Q8×#4112yes—
695C22#348012yes—
696C22#348511yescharacteristic
697C22#348711yescharacteristic
698C22#349311yescharacteristic
699C22#349311yescharacteristic
700C22#350412yes—
701C22#350412yes—
702C22#352112yes—
703C22#352412yes—
704C4—24no—
705C4—24no—
706C4—216no—
707C4—24no—
708C4—24no—
709C4—42no—
710C4—48no—
711C4—44no—
712C4—44no—
713C4—82no—
714C4—82no—
715C4—82no—
716C4—82no—
717C4—82no—
718C4—82no—
719C4#285718yes—
720C2—44no—
721C2#3473711yescharacteristic
722C2#3491111yescharacteristic
723C2#3495311yescharacteristic
724C2#3515011yescharacteristic
725C2#3521211yescharacteristic
726C2#3533511yescharacteristic
727C2#3621912yes—
728C2#3628412yes—
729C2#3643112yes—
730C2#3647812yes—
731C2C2×#285711yescharacteristic
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×#2584C2semidirect
6. #14903C2semidirect
7. #14901C2semidirect
10. #14699C2semidirect
11. #14160C2semidirect
12. #14046C2semidirect
13. #14045C2semidirect
14. #13954C2semidirect
17. #13925C2semidirect
101. C2×#117Q8semidirect
518. C4⋊C4C22⋊Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#28512
C2×#25842
C2×C4×C22⋊Q82
C2×C4×#1172
#152042
#149032
#149012
#148402
#147632
#146992
#141602
#140462
#140452
#139542
#139512
#139252