Finite Group Databasebeta

← #719932

Subgroups — #719932

Order 512.

3190 conjugacy classes of subgroups.

The subgroup lattice has 736 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2440C212yesmaximal, complemented
2C2×D4×C4⋊C4C211yesmaximal, characteristic
3C2×#2331C211yesmaximal, characteristic, complemented
4#17078C212yesmaximal, complemented
5#16820C212yesmaximal, complemented
6#15441C214yesmaximal, complemented
7#15276C214yesmaximal, complemented
8#15220C212yesmaximal, complemented
9#15177C212yesmaximal
10#15131C212yesmaximal, complemented
11#14812C212yesmaximal
12#14277C211yesmaximal, characteristic, complemented
13#13954C211yesmaximal, characteristic, complemented
14#13910C211yesmaximal, characteristic, complemented
15#13907C212yesmaximal, complemented
16#13905C211yesmaximal, characteristic
17#13847C211yesmaximal, characteristic
18C22×C4⋊Q8C2211yescharacteristic
19C22×C22⋊Q8C2214yes—
20C22×C4×D4C2211yescharacteristic
21C22×C4×D4C2212yes—
22C23×C4⋊C4C2212yes—
23#2440—216no—
24C2×#398C2214yes—
25C2×#396C2214yes—
26C2×#395C2214yes—
27D4×C4⋊C4—28no—
28#2331—28no—
29C2×#391C2212yes—
30C2×#391C2212yes—
31C2×#389C2211yescharacteristic
32C2×#389C2211yescharacteristic
33C2×#389C2212yes—
34C2×#389C2212yes—
35C2×#389C2212yes—
36C2×#389C2212yes—
37C2×#389C2214yes—
38C2×#387C2212yes—
39C2×#387C2212yes—
40C2×#385C2214yes—
41C2×#385C2214yes—
42C2×#384C2212yes—
43C2×#384C2211yescharacteristic
44C2×#382C2211yescharacteristic
45C2×#382C2211yescharacteristic
46C2×#380C2212yes—
47C2×#380C2212yes—
48C2×#380C2214yes—
49C2×#380C2212yes—
50C2×C4×C4⋊C4C2211yescharacteristic
51C2×C4×C4⋊C4C2211yescharacteristic
52C2×C4×C4⋊C4C2211yescharacteristic
53C2×C4×C22⋊C4C2212yes—
54C2×C4×C22⋊C4C2212yes—
55C2×#376C2211yescharacteristic
56#1445C2211yescharacteristic
57#1445C2211yescharacteristic
58#1445C2212yes—
59#1444C2212yes—
60#1444C2212yes—
61#1444C2214yes—
62#1444C2212yes—
63#1444C2212yes—
64#1443C2214yes—
65#1442C2212yes—
66#1442C2212yes—
67#1441C2214yes—
68#1441C2214yes—
69#1441C2214yes—
70#1441C2214yes—
71#1441C2214yes—
72#1439C2212yes—
73#1439C2212yes—
74#1439C2214yes—
75#1439C2214yes—
76#1439C2214yes—
77#1438C2211yescharacteristic
78#1438C2211yescharacteristic
79#1438C2212yes—
80#1437C2214yes—
81C4⋊#95C2212yes—
82C4⋊#95C2212yes—
83C4⋊#95C2212yes—
84C4⋊#95C2212yes—
85C22⋊#95C2214yes—
86C23×Q8C2312yes—
87C23×D4C2311yescharacteristic
88C24×C4C2312yes—
89C24×C4C2314yes—
90C2×C4⋊Q8—24no—
91C2×C4⋊Q8—24no—
92C2×C4⋊Q8D418yescomplemented
93C2×C22⋊Q8—216no—
94C2×C22⋊Q8—48no—
95C2×C22⋊Q8—48no—
96C2×C4×D4—24no—
97C2×C4×D4—28no—
98C2×C4×D4—24no—
99C2×C4×D4—48no—
100C2×C4×D4Q818yescomplemented
101C22×C4⋊C4—28no—
102C22×C4⋊C4—28no—
103C22×C4⋊C4—28no—
104C22×C4⋊C4C2314yes—
105C22×C4⋊C4C2312yes—
106C22×C4⋊C4C2311yescharacteristic
107C22×C4⋊C4C2312yes—
108C22×C4⋊C4C2312yes—
109C22×C4⋊C4C2314yes—
110C22×C4⋊C4C2314yes—
111C22×C4⋊C4C2312yes—
112C22×C4⋊C4C2311yescharacteristic
113C22×C4⋊C4C2311yescharacteristic
114C22×C4⋊C4C2311yescharacteristic
115C22×C4⋊C4C2314yes—
116C22×C4⋊C4C2314yes—
117C22×C4⋊C4C2312yes—
118C22×C4⋊C4C2314yes—
119C22×C4⋊C4C2312yes—
120C22×C4⋊C4C2312yes—
121C22×C4⋊C4C2312yes—
122C22×C4⋊C4C2311yescharacteristic
123C22×C22⋊C4C2312yes—
124C22×C22⋊C4C2312yes—
125C22×C22⋊C4C2312yes—
126C22×C22⋊C4C2314yes—
127C22×C22⋊C4C2314yes—
128C22×C22⋊C4C2312yes—
129C22×C22⋊C4C2312yes—
130C22×C22⋊C4C2314yes—
131C22×C42C2311yescharacteristic
132C22×C42C2312yes—
133C22×C42C2311yescharacteristic
134C22×C42C2312yes—
135C22×C42C2312yes—
136C22×C42C2311yescharacteristic
137#398—216no—
138#396—48no—
139#395—216no—
140#391—28no—
141#391—28no—
142#389—24no—
143#389—24no—
144#389—28no—
145#389—28no—
146#389—28no—
147#389—48no—
148#389—44no—
149#387—28no—
150#387—28no—
151#385—216no—
152#385—216no—
153#384—28no—
154#384—24no—
155#382—24no—
156#382—24no—
157#380—28no—
158#380—28no—
159#380—48no—
160#380—44no—
161C4×C4⋊C4—24no—
162C4×C4⋊C4—24no—
163C4×C4⋊C4D418yes—
164C4×C22⋊C4—28no—
165C4×C22⋊C4—44no—
166#376—24no—
167C2×#95C2312yes—
168C2×#95C2314yes—
169C2×#95C2314yes—
170C2×#95C2312yes—
171C2×#95C2314yes—
172C2×#95C2312yes—
173C2×#95C2312yes—
174C2×#95C2311yescharacteristic
175C2×#95C2311yescharacteristic
176C2×#95C2314yes—
177C2×#95C2312yes—
178C2×#95C2311yescharacteristic
179C2×#95C2311yescharacteristic
180C2×#95C2312yes—
181C2×#95C2312yes—
182C2×#95C2312yes—
183C2×#95C2311yescharacteristic
184C2×#95C2312yes—
185C2×#95C2311yescharacteristic
186C2×#95C2314yes—
187C2×#95C2314yes—
188C2×#95C2314yes—
189C2×#95C2314yes—
190C2×#95C2314yes—
191C2×#95C2314yes—
192C2×#95C2312yes—
193C2×#95C2312yes—
194C2×#95C2312yes—
195C25C2412yes—
196C22×Q8—24no—
197C22×Q8—24no—
198C22×Q8—44no—
199C22×Q8—44no—
200C22×Q8C2×D418yes—
201C22×D4—24no—
202C22×D4—24no—
203C22×D4—22no—
204C22×D4—22no—
205C22×D4C2×Q814yes—
206C23×C4—24no—
207C23×C4—24no—
208C23×C4—24no—
209C23×C4—24no—
210C23×C4—24no—
211C23×C4—24no—
212C23×C4—28no—
213C23×C4—28no—
214C23×C4—28no—
215C23×C4—28no—
216C23×C4—44no—
217C23×C4—44no—
218C23×C4—44no—
219C23×C4C2×Q818yes—
220C23×C4C2414yes—
221C23×C4C2412yes—
222C23×C4C2411yescharacteristic
223C23×C4C2411yescharacteristic
224C23×C4C2414yes—
225C23×C4C2412yes—
226C23×C4C2411yescharacteristic
227C23×C4C2412yes—
228C23×C4C2411yescharacteristic
229C23×C4C2412yes—
230C23×C4C2412yes—
231C23×C4C2412yes—
232C23×C4C2412yes—
233C23×C4C2411yescharacteristic
234C23×C4C2411yescharacteristic
235C23×C4C2411yescharacteristic
236C4⋊Q8—232no—
237C22⋊Q8—464no—
238C4×D4—232no—
239C4×D4—432no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—28no—
254C2×C4⋊C4—28no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—28no—
257C2×C4⋊C4—28no—
258C2×C4⋊C4—28no—
259C2×C4⋊C4—22no—
260C2×C4⋊C4—24no—
261C2×C4⋊C4—28no—
262C2×C4⋊C4—24no—
263C2×C4⋊C4—24no—
264C2×C4⋊C4—28no—
265C2×C4⋊C4—28no—
266C2×C4⋊C4—22no—
267C2×C4⋊C4—24no—
268C2×C4⋊C4—22no—
269C2×C4⋊C4—24no—
270C2×C4⋊C4—28no—
271C2×C4⋊C4—24no—
272C2×C4⋊C4—24no—
273C2×C4⋊C4—28no—
274C2×C4⋊C4—44no—
275C2×C4⋊C4—44no—
276C2×C4⋊C4—44no—
277C2×C4⋊C4—44no—
278C2×C4⋊C4—44no—
279C2×C4⋊C4—432no—
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⋊C4C2×D414yes—
288C2×C4⋊C4C2×D418yes—
289C2×C4⋊C4C2×D414yes—
290C2×C4⋊C4C2×D414yes—
291C2×C4⋊C4C2×D414yes—
292C2×C4⋊C4C2×Q814yes—
293C2×C4⋊C4#4114yes—
294C2×C22⋊C4—24no—
295C2×C22⋊C4—24no—
296C2×C22⋊C4—24no—
297C2×C22⋊C4—24no—
298C2×C22⋊C4—24no—
299C2×C22⋊C4—24no—
300C2×C22⋊C4—24no—
301C2×C22⋊C4—24no—
302C2×C22⋊C4—28no—
303C2×C22⋊C4—28no—
304C2×C22⋊C4—24no—
305C2×C22⋊C4—24no—
306C2×C22⋊C4—28no—
307C2×C22⋊C4—28no—
308C2×C22⋊C4—24no—
309C2×C22⋊C4—24no—
310C2×C22⋊C4—28no—
311C2×C22⋊C4—44no—
312C2×C22⋊C4—44no—
313C2×C22⋊C4—42no—
314C2×C22⋊C4—42no—
315C2×C22⋊C4—44no—
316C2×C22⋊C4—44no—
317C2×C22⋊C4C2×Q818yes—
318C2×C42—22no—
319C2×C42—22no—
320C2×C42—22no—
321C2×C42—24no—
322C2×C42—24no—
323C2×C42—22no—
324C2×C42—24no—
325C2×C42—22no—
326C2×C42—24no—
327C2×C42—44no—
328C2×C42—44no—
329C2×C42C2×D414yes—
330C2×C42C2×D418yes—
331C2×C42C2×D414yes—
332C2×C42C2×Q814yes—
333C2×C42#4114yes—
334#95—22no—
335#95—22no—
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—28no—
346#95—28no—
347#95—24no—
348#95—24no—
349#95—22no—
350#95—22no—
351#95—24no—
352#95—24no—
353#95—28no—
354#95—42no—
355#95—42no—
356#95—44no—
357#95—44no—
358#95—44no—
359#95—44no—
360#95—44no—
361#95C2×D418yes—
362C24—22no—
363C24—22no—
364C24—22no—
365C24—22no—
366C24—22no—
367C24—22no—
368C24—22no—
369C24—22no—
370C24—22no—
371C24—22no—
372C24—22no—
373C24—22no—
374C24C22×Q814yes—
375C24H(5,𝔽2)14yes—
376C242−1+414yes—
377C24C2511yescharacteristic
378C2×Q8—216no—
379C2×Q8—416no—
380C2×Q8—84no—
381C2×Q8—84no—
382C2×Q8—88no—
383C2×D4—28no—
384C2×D4—28no—
385C2×D4—216no—
386C2×D4—216no—
387C2×D4—28no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—24no—
391C22×C4—24no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—21no—
395C22×C4—21no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—21no—
404C22×C4—21no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—24no—
414C22×C4—24no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—24no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—24no—
427C22×C4—22no—
428C22×C4—216no—
429C22×C4—216no—
430C22×C4—216no—
431C22×C4—21no—
432C22×C4—21no—
433C22×C4—22no—
434C22×C4—24no—
435C22×C4—24no—
436C22×C4—24no—
437C22×C4—24no—
438C22×C4—24no—
439C22×C4—24no—
440C22×C4—24no—
441C22×C4—21no—
442C22×C4—21no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—24no—
447C22×C4—24no—
448C22×C4—24no—
449C22×C4—24no—
450C22×C4—24no—
451C22×C4—24no—
452C22×C4—21no—
453C22×C4—21no—
454C22×C4—24no—
455C22×C4—22no—
456C22×C4—24no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—24no—
460C22×C4—416no—
461C22×C4—416no—
462C22×C4—416no—
463C22×C4—48no—
464C22×C4—48no—
465C22×C4—48no—
466C22×C4—48no—
467C22×C4—416no—
468C22×C4—88no—
469C22×C4—88no—
470C22×C4—88no—
471C22×C4C22×D412yes—
472C22×C4C22×D414yes—
473C22×C4C22×D412yes—
474C22×C4C22×D414yes—
475C22×C4C22×D414yes—
476C22×C4C22×D414yes—
477C22×C4C22×D412yes—
478C22×C4C22×D412yes—
479C22×C4C22×D412yes—
480C22×C4C22×D412yes—
481C22×C4C22×Q814yes—
482C22×C4C22×Q812yes—
483C22×C4C22×Q812yes—
484C22×C4C22×Q812yes—
485C22×C4C2×#4114yes—
486C22×C4C2×#4112yes—
487C22×C4C2×#4112yes—
488C22×C4C2×#4112yes—
489C22×C4C2×#4112yes—
490C22×C4H(5,𝔽2)14yes—
491C22×C4H(5,𝔽2)12yes—
492C22×C42−1+414yes—
493C22×C42−1+412yes—
494C4⋊C4—216no—
495C4⋊C4—216no—
496C4⋊C4—28no—
497C4⋊C4—28no—
498C4⋊C4—28no—
499C4⋊C4—416no—
500C4⋊C4—416no—
501C4⋊C4—48no—
502C4⋊C4—416no—
503C4⋊C4—416no—
504C4⋊C4—48no—
505C4⋊C4—44no—
506C4⋊C4—48no—
507C4⋊C4—84no—
508C4⋊C4—84no—
509C4⋊C4—84no—
510C4⋊C4—88no—
511C4⋊C4—88no—
512C4⋊C4#121116yes—
513C22⋊C4—216no—
514C22⋊C4—216no—
515C22⋊C4—216no—
516C22⋊C4—48no—
517C22⋊C4—48no—
518C22⋊C4—416no—
519C22⋊C4—416no—
520C22⋊C4—88no—
521C42—28no—
522C42—48no—
523C42—44no—
524C42—48no—
525C42—84no—
526C42C22⋊Q8116yescomplemented
527C23—24no—
528C23—24no—
529C23—24no—
530C23—24no—
531C23—24no—
532C23—24no—
533C23—24no—
534C23—24no—
535C23—24no—
536C23—24no—
537C23—24no—
538C23—24no—
539C23—24no—
540C23—24no—
541C23—24no—
542C23—24no—
543C23—24no—
544C23—24no—
545C23—42no—
546C23—42no—
547C23—42no—
548C23—42no—
549C23—42no—
550C23—42no—
551C23—42no—
552C23—42no—
553C23—42no—
554C23—42no—
555C23—42no—
556C23—42no—
557C23—42no—
558C23—42no—
559C23—42no—
560C23—42no—
561C23#54418yes—
562C23C23×D411yescharacteristic
563C23C23×D411yescharacteristic
564C23C23×Q811yescharacteristic
565C23C22×#4111yescharacteristic
566C23C22×#4111yescharacteristic
567C23C2×H(5,𝔽2)11yescharacteristic
568C23C2×H(5,𝔽2)12yes—
569C23C2×H(5,𝔽2)12yes—
570C23C2×2−1+411yescharacteristic
571C23C2×2−1+412yes—
572C23C2×2−1+412yes—
573Q8—816no—
574D4—232no—
575C2×C4—24no—
576C2×C4—22no—
577C2×C4—24no—
578C2×C4—28no—
579C2×C4—22no—
580C2×C4—24no—
581C2×C4—24no—
582C2×C4—22no—
583C2×C4—24no—
584C2×C4—22no—
585C2×C4—24no—
586C2×C4—28no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—24no—
590C2×C4—24no—
591C2×C4—24no—
592C2×C4—24no—
593C2×C4—24no—
594C2×C4—24no—
595C2×C4—24no—
596C2×C4—24no—
597C2×C4—28no—
598C2×C4—24no—
599C2×C4—24no—
600C2×C4—24no—
601C2×C4—24no—
602C2×C4—24no—
603C2×C4—24no—
604C2×C4—24no—
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—41no—
614C2×C4—42no—
615C2×C4—41no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—41no—
620C2×C4—41no—
621C2×C4—44no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—44no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—44no—
628C2×C4—432no—
629C2×C4—44no—
630C2×C4—41no—
631C2×C4—41no—
632C2×C4—42no—
633C2×C4—44no—
634C2×C4—44no—
635C2×C4—44no—
636C2×C4—44no—
637C2×C4—41no—
638C2×C4—44no—
639C2×C4—41no—
640C2×C4—42no—
641C2×C4—44no—
642C2×C4—44no—
643C2×C4—44no—
644C2×C4—42no—
645C2×C4—41no—
646C2×C4—44no—
647C2×C4—41no—
648C2×C4—44no—
649C2×C4—44no—
650C2×C4—42no—
651C2×C4—42no—
652C2×C4—44no—
653C2×C4—44no—
654C2×C4—44no—
655C2×C4—42no—
656C2×C4—44no—
657C2×C4—42no—
658C2×C4—832no—
659C2×C4C2×#12118yes—
660C2×C4C2×#12114yes—
661C2×C4C2×C22⋊Q814yes—
662C2×C4C2×C22⋊Q814yes—
663C2×C4C2×C22⋊Q814yes—
664C2×C4#53714yes—
665C2×C4D4×Q814yes—
666C2×C4#55014yes—
667C2×C4#55414yes—
668C2×C4#55614yes—
669C22—28no—
670C22—28no—
671C22—28no—
672C22—44no—
673C22—44no—
674C22—44no—
675C22—44no—
676C22—44no—
677C22—44no—
678C22—44no—
679C22—44no—
680C22—44no—
681C22—44no—
682C22—44no—
683C22—44no—
684C22C22×#12111yescharacteristic
685C22C22×C22⋊Q811yescharacteristic
686C22C2×#53711yescharacteristic
687C22C2×#54412yes—
688C22C2×D4×Q811yescharacteristic
689C22C2×#55011yescharacteristic
690C22C2×#55411yescharacteristic
691C22C2×#55611yescharacteristic
692C22#348412yes—
693C22#348411yescharacteristic
694C22#348611yescharacteristic
695C22#348612yes—
696C22#349212yes—
697C22#349212yes—
698C22#349811yescharacteristic
699C22#349911yescharacteristic
700C22#350312yes—
701C22#351912yes—
702C22#352312yes—
703C22#353612yes—
704C22#353612yes—
705C22#354412yes—
706C22#355212yes—
707C4—28no—
708C4—24no—
709C4—24no—
710C4—44no—
711C4—44no—
712C4—44no—
713C4—42no—
714C4—44no—
715C4—42no—
716C4—44no—
717C4—81no—
718C4—82no—
719C4—84no—
720C4—84no—
721C4—81no—
722C4#253818yes—
723C2—48no—
724C2C2×#253811yescharacteristic
725C2#3522411yescharacteristic
726C2#3553211yescharacteristic
727C2#3569111yescharacteristic
728C2#3570011yescharacteristic
729C2#3768612yes—
730C2#3809912yes—
731C2#4242811yescharacteristic
732C2#4279411yescharacteristic
733C2#4498412yes—
734C2#4535612yes—
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×#2440C2semidirect
3. C2×#2331C2semidirect
4. #17078C2semidirect
5. #16820C2semidirect
6. #15441C2semidirect
7. #15276C2semidirect
8. #15220C2semidirect
10. #15131C2semidirect
12. #14277C2semidirect
13. #13954C2semidirect
14. #13910C2semidirect
15. #13907C2semidirect
92. C2×C4⋊Q8D4semidirect
100. C2×C4×D4Q8semidirect
526. C42C22⋊Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#24402
C2×D4×C4⋊C42
C2×#23312
#170782
#168202
#154412
#152762
#152202
#151772
#151312
#148122
#142772
#139542
#139102
#139072
#139052
#138472