Finite Group Databasebeta

← #732285

Subgroups — #732285

Order 512.

2950 conjugacy classes of subgroups.

The subgroup lattice has 759 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2562C211yesmaximal, characteristic
2C2×#2584C211yesmaximal, characteristic, complemented
3C2×#2398C211yesmaximal, characteristic
4C2×D4×C4⋊C4C211yesmaximal, characteristic
5#17074C212yesmaximal, complemented
6#17010C212yesmaximal, complemented
7#15472C214yesmaximal, complemented
8#15426C214yesmaximal, complemented
9#15183C212yesmaximal, complemented
10#15132C212yesmaximal, complemented
11#15129C212yesmaximal
12#14868C212yesmaximal
13#14283C211yesmaximal, characteristic, complemented
14#13926C211yesmaximal, characteristic, complemented
15#13907C212yesmaximal, complemented
16#13907C212yesmaximal, complemented
17#13783C211yesmaximal, characteristic, complemented
18C22×#125C2211yescharacteristic
19C22×C4×Q8C2211yescharacteristic
20C22×C4×D4C2211yescharacteristic
21C22×C4×D4C2212yes—
22C23×C4⋊C4C2212yes—
23#2584—28no—
24#2562—28no—
25#2398—28no—
26C2×#400C2212yes—
27C2×#400C2211yescharacteristic
28C2×#400C2211yescharacteristic
29C2×#398C2212yes—
30C2×#397C2212yes—
31C2×#397C2214yes—
32C2×#395C2212yes—
33C2×#394C2211yescharacteristic
34C2×#394C2212yes—
35C2×#394C2211yescharacteristic
36C2×#393C2212yes—
37D4×C4⋊C4—28no—
38C2×#391C2212yes—
39C2×#389C2212yes—
40C2×#389C2212yes—
41C2×#389C2211yescharacteristic
42C2×#389C2212yes—
43C2×#389C2211yescharacteristic
44C2×#389C2211yescharacteristic
45C2×#389C2211yescharacteristic
46C2×#389C2212yes—
47C2×#388C2214yes—
48C2×#385C2214yes—
49C2×#384C2211yescharacteristic
50C2×#380C2212yes—
51C2×#380C2212yes—
52C2×#380C2212yes—
53C2×#380C2214yes—
54C2×#380C2214yes—
55C2×C4×C4⋊C4C2211yescharacteristic
56C2×C4×C4⋊C4C2211yescharacteristic
57C2×C4×C22⋊C4C2212yes—
58C2×C4×C22⋊C4C2212yes—
59C2×#376C2211yescharacteristic
60#1445C2214yes—
61#1443C2214yes—
62#1443C2214yes—
63#1442C2214yes—
64#1441C2212yes—
65#1441C2212yes—
66#1441C2212yes—
67#1441C2212yes—
68#1441C2214yes—
69#1441C2214yes—
70#1441C2214yes—
71#1441C2214yes—
72#1441C2214yes—
73#1440C2214yes—
74#1440C2214yes—
75#1439C2214yes—
76#1436C2214yes—
77C4⋊#95C2212yes—
78C4⋊#95C2212yes—
79C4⋊#95C2212yes—
80C4⋊#95C2212yes—
81C4⋊#95C2212yes—
82C4⋊#95C2212yes—
83C4⋊#95C2212yes—
84#1434C2212yes—
85C22⋊#95C2214yes—
86C22⋊#95C2214yes—
87C23×Q8C2311yescharacteristic
88C23×D4C2311yescharacteristic
89C24×C4C2312yes—
90C24×C4C2314yes—
91C2×#125—24no—
92C2×#125—24no—
93C2×#125—24no—
94C2×C4×Q8—24no—
95C2×C4×Q8—24no—
96C2×C4×Q8—24no—
97C2×C4×D4—24no—
98C2×C4×D4—28no—
99C2×C4×D4—24no—
100C2×C4×D4—48no—
101C2×C4×D4D418yes—
102C22×C4⋊C4—28no—
103C22×C4⋊C4—28no—
104C22×C4⋊C4—28no—
105C22×C4⋊C4C2311yescharacteristic
106C22×C4⋊C4C2311yescharacteristic
107C22×C4⋊C4C2312yes—
108C22×C4⋊C4C2312yes—
109C22×C4⋊C4C2311yescharacteristic
110C22×C4⋊C4C2312yes—
111C22×C4⋊C4C2314yes—
112C22×C4⋊C4C2311yescharacteristic
113C22×C4⋊C4C2311yescharacteristic
114C22×C4⋊C4C2312yes—
115C22×C4⋊C4C2312yes—
116C22×C4⋊C4C2312yes—
117C22×C4⋊C4C2311yescharacteristic
118C22×C4⋊C4C2311yescharacteristic
119C22×C4⋊C4C2312yes—
120C22×C4⋊C4C2312yes—
121C22×C4⋊C4C2312yes—
122C22×C4⋊C4C2314yes—
123C22×C4⋊C4C2312yes—
124C22×C4⋊C4C2311yescharacteristic
125C22×C22⋊C4C2312yes—
126C22×C22⋊C4C2314yes—
127C22×C22⋊C4C2312yes—
128C22×C22⋊C4C2312yes—
129C22×C22⋊C4C2312yes—
130C22×C22⋊C4C2314yes—
131C22×C22⋊C4C2312yes—
132C22×C22⋊C4C2314yes—
133C22×C42C2311yescharacteristic
134C22×C42C2311yescharacteristic
135C22×C42C2312yes—
136C22×C42C2312yes—
137C22×C42C2312yes—
138C22×C42C2311yescharacteristic
139#400—28no—
140#400—24no—
141#400—24no—
142#398—28no—
143#397—28no—
144#397—48no—
145#395—28no—
146#394—28no—
147#394—24no—
148#394—24no—
149#393—28no—
150#391—44no—
151#389—24no—
152#389—24no—
153#389—24no—
154#389—24no—
155#389—28no—
156#389—28no—
157#389—28no—
158#389—28no—
159#388—216no—
160#385—216no—
161#384—24no—
162#380—28no—
163#380—28no—
164#380—28no—
165#380—48no—
166#380—48no—
167C4×C4⋊C4—24no—
168C4×C4⋊C4D418yes—
169C4×C22⋊C4—28no—
170C4×C22⋊C4—28no—
171#376—24no—
172C2×#95C2312yes—
173C2×#95C2312yes—
174C2×#95C2312yes—
175C2×#95C2311yescharacteristic
176C2×#95C2312yes—
177C2×#95C2311yescharacteristic
178C2×#95C2312yes—
179C2×#95C2312yes—
180C2×#95C2312yes—
181C2×#95C2312yes—
182C2×#95C2312yes—
183C2×#95C2312yes—
184C2×#95C2314yes—
185C2×#95C2314yes—
186C2×#95C2314yes—
187C2×#95C2314yes—
188C2×#95C2314yes—
189C2×#95C2314yes—
190C2×#95C2314yes—
191C2×#95C2314yes—
192C2×#95C2314yes—
193C2×#95C2314yes—
194C2×#95C2314yes—
195C2×#95C2314yes—
196C2×#95C2314yes—
197C2×#95C2314yes—
198C2×#95C2312yes—
199C25C2412yes—
200C22×Q8—22no—
201C22×Q8—22no—
202C22×Q8—22no—
203C22×Q8—22no—
204C22×Q8—22no—
205C22×Q8—41no—
206C22×Q8—41no—
207C22×D4—24no—
208C22×D4—24no—
209C22×D4—22no—
210C22×D4—22no—
211C22×D4C2×D414yes—
212C23×C4—24no—
213C23×C4—24no—
214C23×C4—24no—
215C23×C4—24no—
216C23×C4—24no—
217C23×C4—24no—
218C23×C4—28no—
219C23×C4—28no—
220C23×C4—28no—
221C23×C4—28no—
222C23×C4—44no—
223C23×C4—44no—
224C23×C4—44no—
225C23×C4C2×D418yes—
226C23×C4C2412yes—
227C23×C4C2412yes—
228C23×C4C2411yescharacteristic
229C23×C4C2411yescharacteristic
230C23×C4C2412yes—
231C23×C4C2414yes—
232C23×C4C2412yes—
233C23×C4C2414yes—
234C23×C4C2412yes—
235C23×C4C2412yes—
236C23×C4C2411yescharacteristic
237C23×C4C2411yescharacteristic
238C23×C4C2411yescharacteristic
239C23×C4C2411yescharacteristic
240C23×C4C2412yes—
241C23×C4C2411yescharacteristic
242#125—416no—
243C4×Q8—416no—
244C4×D4—232no—
245C4×D4—432no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—28no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4—22no—
260C2×C4⋊C4—24no—
261C2×C4⋊C4—24no—
262C2×C4⋊C4—22no—
263C2×C4⋊C4—22no—
264C2×C4⋊C4—24no—
265C2×C4⋊C4—24no—
266C2×C4⋊C4—24no—
267C2×C4⋊C4—24no—
268C2×C4⋊C4—24no—
269C2×C4⋊C4—28no—
270C2×C4⋊C4—28no—
271C2×C4⋊C4—24no—
272C2×C4⋊C4—24no—
273C2×C4⋊C4—22no—
274C2×C4⋊C4—22no—
275C2×C4⋊C4—22no—
276C2×C4⋊C4—24no—
277C2×C4⋊C4—24no—
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—24no—
285C2×C4⋊C4—24no—
286C2×C4⋊C4—24no—
287C2×C4⋊C4—28no—
288C2×C4⋊C4—42no—
289C2×C4⋊C4—42no—
290C2×C4⋊C4—44no—
291C2×C4⋊C4—432no—
292C2×C4⋊C4—42no—
293C2×C4⋊C4—44no—
294C2×C4⋊C4—44no—
295C2×C4⋊C4—44no—
296C2×C4⋊C4—42no—
297C2×C4⋊C4—44no—
298C2×C4⋊C4C2×D414yes—
299C2×C4⋊C4C2×D414yes—
300C2×C4⋊C4C2×D414yes—
301C2×C4⋊C4#4118yes—
302C2×C22⋊C4—24no—
303C2×C22⋊C4—24no—
304C2×C22⋊C4—24no—
305C2×C22⋊C4—24no—
306C2×C22⋊C4—24no—
307C2×C22⋊C4—24no—
308C2×C22⋊C4—28no—
309C2×C22⋊C4—28no—
310C2×C22⋊C4—24no—
311C2×C22⋊C4—28no—
312C2×C22⋊C4—24no—
313C2×C22⋊C4—24no—
314C2×C22⋊C4—28no—
315C2×C22⋊C4—24no—
316C2×C22⋊C4—28no—
317C2×C22⋊C4—24no—
318C2×C22⋊C4—24no—
319C2×C22⋊C4—28no—
320C2×C22⋊C4—42no—
321C2×C22⋊C4—42no—
322C2×C22⋊C4—44no—
323C2×C22⋊C4—44no—
324C2×C22⋊C4—44no—
325C2×C22⋊C4C2×D418yes—
326C2×C42—22no—
327C2×C42—22no—
328C2×C42—22no—
329C2×C42—22no—
330C2×C42—24no—
331C2×C42—22no—
332C2×C42—22no—
333C2×C42—24no—
334C2×C42—24no—
335C2×C42—24no—
336C2×C42—44no—
337C2×C42—42no—
338C2×C42—42no—
339C2×C42C2×D418yes—
340C2×C42C2×D414yes—
341C2×C42C2×D414yes—
342C2×C42#4114yes—
343#95—24no—
344#95—22no—
345#95—22no—
346#95—24no—
347#95—24no—
348#95—24no—
349#95—24no—
350#95—24no—
351#95—28no—
352#95—28no—
353#95—28no—
354#95—24no—
355#95—24no—
356#95—24no—
357#95—24no—
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#95—44no—
367#95—44no—
368#95—44no—
369#95C2×D418yes—
370C24—22no—
371C24—22no—
372C24—22no—
373C24—22no—
374C24—22no—
375C24—22no—
376C24—22no—
377C24—22no—
378C24—22no—
379C24—22no—
380C24—22no—
381C24—22no—
382C24C22×D414yes—
383C24H(5,𝔽2)14yes—
384C242−1+414yes—
385C24C2511yescharacteristic
386C2×Q8—28no—
387C2×Q8—44no—
388C2×Q8—44no—
389C2×Q8—44no—
390C2×Q8—44no—
391C2×Q8—44no—
392C2×Q8—44no—
393C2×D4—28no—
394C2×D4—28no—
395C2×D4—216no—
396C2×D4—216no—
397C2×D4—28no—
398C22×C4—22no—
399C22×C4—21no—
400C22×C4—24no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—21no—
404C22×C4—21no—
405C22×C4—21no—
406C22×C4—22no—
407C22×C4—21no—
408C22×C4—21no—
409C22×C4—21no—
410C22×C4—21no—
411C22×C4—22no—
412C22×C4—21no—
413C22×C4—21no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—21no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—21no—
424C22×C4—22no—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—21no—
428C22×C4—21no—
429C22×C4—216no—
430C22×C4—216no—
431C22×C4—216no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—24no—
435C22×C4—21no—
436C22×C4—21no—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—24no—
444C22×C4—22no—
445C22×C4—24no—
446C22×C4—21no—
447C22×C4—21no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—24no—
452C22×C4—24no—
453C22×C4—24no—
454C22×C4—24no—
455C22×C4—22no—
456C22×C4—24no—
457C22×C4—22no—
458C22×C4—24no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—24no—
463C22×C4—22no—
464C22×C4—24no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—24no—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—22no—
471C22×C4—24no—
472C22×C4—22no—
473C22×C4—22no—
474C22×C4—24no—
475C22×C4—22no—
476C22×C4—22no—
477C22×C4—22no—
478C22×C4—22no—
479C22×C4—24no—
480C22×C4—24no—
481C22×C4—22no—
482C22×C4—416no—
483C22×C4—416no—
484C22×C4—416no—
485C22×C4—48no—
486C22×C4—48no—
487C22×C4—48no—
488C22×C4—48no—
489C22×C4—416no—
490C22×C4—88no—
491C22×C4—88no—
492C22×C4—88no—
493C22×C4C22×D414yes—
494C22×C4C22×D414yes—
495C22×C4C22×D412yes—
496C22×C4C22×D412yes—
497C22×C4C22×D412yes—
498C22×C4C22×D412yes—
499C22×C4C22×D414yes—
500C22×C4C22×D412yes—
501C22×C4C22×D412yes—
502C22×C4C2×#4112yes—
503C22×C4C2×#4112yes—
504C22×C4C2×#4118yes—
505C22×C4C2×#4112yes—
506C22×C4C2×#4114yes—
507C22×C4H(5,𝔽2)12yes—
508C22×C42−1+412yes—
509C22×C42−1+414yes—
510C4⋊C4—28no—
511C4⋊C4—28no—
512C4⋊C4—48no—
513C4⋊C4—48no—
514C4⋊C4—48no—
515C4⋊C4—44no—
516C4⋊C4—44no—
517C4⋊C4—44no—
518C4⋊C4—48no—
519C4⋊C4—48no—
520C4⋊C4—44no—
521C4⋊C4—44no—
522C4⋊C4—416no—
523C4⋊C4—48no—
524C4⋊C4—48no—
525C4⋊C4—84no—
526C4⋊C4—84no—
527C4⋊C4—88no—
528C4⋊C4—84no—
529C4⋊C4#124116yes—
530C22⋊C4—216no—
531C22⋊C4—48no—
532C22⋊C4—48no—
533C22⋊C4—416no—
534C22⋊C4—48no—
535C22⋊C4—48no—
536C22⋊C4—416no—
537C22⋊C4—88no—
538C42—48no—
539C42—44no—
540C42—44no—
541C42—48no—
542C42—84no—
543C42#121116yes—
544C23—24no—
545C23—24no—
546C23—24no—
547C23—24no—
548C23—24no—
549C23—24no—
550C23—24no—
551C23—24no—
552C23—24no—
553C23—24no—
554C23—24no—
555C23—24no—
556C23—24no—
557C23—24no—
558C23—24no—
559C23—24no—
560C23—24no—
561C23—24no—
562C23—42no—
563C23—42no—
564C23—42no—
565C23—42no—
566C23—42no—
567C23—42no—
568C23—42no—
569C23—42no—
570C23—42no—
571C23—42no—
572C23—42no—
573C23—42no—
574C23—42no—
575C23—42no—
576C23—42no—
577C23—42no—
578C23#53718yes—
579C23C23×D411yescharacteristic
580C23C23×D411yescharacteristic
581C23C22×#4111yescharacteristic
582C23C22×#4112yes—
583C23C2×H(5,𝔽2)12yes—
584C23C2×H(5,𝔽2)11yescharacteristic
585C23C2×2−1+412yes—
586C23C2×2−1+412yes—
587C23C2×2−1+411yescharacteristic
588C23#58511yescharacteristic
589C23#58511yescharacteristic
590Q8—416no—
591D4—232no—
592C2×C4—24no—
593C2×C4—22no—
594C2×C4—24no—
595C2×C4—22no—
596C2×C4—22no—
597C2×C4—24no—
598C2×C4—22no—
599C2×C4—22no—
600C2×C4—22no—
601C2×C4—24no—
602C2×C4—22no—
603C2×C4—22no—
604C2×C4—28no—
605C2×C4—28no—
606C2×C4—24no—
607C2×C4—24no—
608C2×C4—24no—
609C2×C4—24no—
610C2×C4—24no—
611C2×C4—22no—
612C2×C4—22no—
613C2×C4—22no—
614C2×C4—22no—
615C2×C4—24no—
616C2×C4—22no—
617C2×C4—22no—
618C2×C4—41no—
619C2×C4—41no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—41no—
625C2×C4—41no—
626C2×C4—41no—
627C2×C4—41no—
628C2×C4—41no—
629C2×C4—41no—
630C2×C4—44no—
631C2×C4—44no—
632C2×C4—432no—
633C2×C4—41no—
634C2×C4—41no—
635C2×C4—42no—
636C2×C4—42no—
637C2×C4—44no—
638C2×C4—42no—
639C2×C4—42no—
640C2×C4—44no—
641C2×C4—42no—
642C2×C4—44no—
643C2×C4—42no—
644C2×C4—42no—
645C2×C4—42no—
646C2×C4—44no—
647C2×C4—44no—
648C2×C4—41no—
649C2×C4—41no—
650C2×C4—42no—
651C2×C4—41no—
652C2×C4—41no—
653C2×C4—42no—
654C2×C4—44no—
655C2×C4—42no—
656C2×C4—42no—
657C2×C4—42no—
658C2×C4—44no—
659C2×C4—42no—
660C2×C4—42no—
661C2×C4—42no—
662C2×C4—42no—
663C2×C4—42no—
664C2×C4—44no—
665C2×C4—42no—
666C2×C4—42no—
667C2×C4—42no—
668C2×C4—44no—
669C2×C4—44no—
670C2×C4—44no—
671C2×C4—42no—
672C2×C4—42no—
673C2×C4—44no—
674C2×C4—44no—
675C2×C4—44no—
676C2×C4—42no—
677C2×C4—42no—
678C2×C4—44no—
679C2×C4—42no—
680C2×C4—44no—
681C2×C4—42no—
682C2×C4—42no—
683C2×C4—832no—
684C2×C4C2×#12114yes—
685C2×C4C2×#12114yes—
686C2×C4C2×#12114yes—
687C2×C4C2×#12418yes—
688C2×C4C2×#12414yes—
689C2×C4#53614yes—
690C2×C4#54718yes—
691C2×C4#55618yes—
692C22—28no—
693C22—28no—
694C22—28no—
695C22—44no—
696C22—44no—
697C22—44no—
698C22—44no—
699C22—44no—
700C22—44no—
701C22—44no—
702C22—44no—
703C22—44no—
704C22—44no—
705C22—44no—
706C22—44no—
707C22C22×#12111yescharacteristic
708C22C22×#12411yescharacteristic
709C22C2×#53611yescharacteristic
710C22C2×#53712yes—
711C22C2×#54712yes—
712C22C2×#55612yes—
713C22#348411yescharacteristic
714C22#348611yescharacteristic
715C22#348811yescharacteristic
716C22#348811yescharacteristic
717C22#348911yescharacteristic
718C22#348911yescharacteristic
719C22#349212yes—
720C22#349211yescharacteristic
721C22#349311yescharacteristic
722C22#351312yes—
723C22#351612yes—
724C22#351912yes—
725C22#352312yes—
726C22#354412yes—
727C22#355212yes—
728C22#355912yes—
729C22#356312yes—
730C4—28no—
731C4—24no—
732C4—24no—
733C4—44no—
734C4—42no—
735C4—42no—
736C4—44no—
737C4—84no—
738C4—81no—
739C4—81no—
740C4—82no—
741C4—82no—
742C4—84no—
743C4—82no—
744C4—82no—
745C4#254718yes—
746C2—48no—
747C2C2×#254711yescharacteristic
748C2#3522311yescharacteristic
749C2#3551811yescharacteristic
750C2#3590311yescharacteristic
751C2#3591011yescharacteristic
752C2#3804212yes—
753C2#3804312yes—
754C2#4248211yescharacteristic
755C2#4281411yescharacteristic
756C2#4498412yes—
757C2#4516512yes—
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
2. C2×#2584C2semidirect
5. #17074C2semidirect
6. #17010C2semidirect
7. #15472C2semidirect
8. #15426C2semidirect
9. #15183C2semidirect
10. #15132C2semidirect
13. #14283C2semidirect
14. #13926C2semidirect
15. #13907C2semidirect
16. #13907C2semidirect
17. #13783C2semidirect

Maximal subgroups

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

subgroupindex
C2×#25622
C2×#25842
C2×#23982
C2×D4×C4⋊C42
#170742
#170102
#154722
#154262
#151832
#151322
#151292
#148682
#142832
#139262
#139072
#137832