Finite Group Databasebeta

← #917933

Subgroups — #917933

Order 512.

3812 conjugacy classes of subgroups.

The subgroup lattice has 742 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2501C211yesmaximal, characteristic
2C2×#2413C212yesmaximal, complemented
3C2×#2614C212yesmaximal, complemented
4C22×#395C211yesmaximal, characteristic, complemented
5#17104C214yesmaximal, complemented
6#16556C212yesmaximal, complemented
7#16522C212yesmaximal, complemented
8#16327C214yesmaximal, complemented
9#15662C212yesmaximal, complemented
10#15625C212yesmaximal, complemented
11#15069C212yesmaximal, complemented
12#14926C212yesmaximal
13#14912C212yesmaximal, complemented
14#13992C212yesmaximal, complemented
15#13870C211yesmaximal, characteristic, complemented
16C24×Q8C2211yescharacteristic
17C22×#123C2212yes—
18C22×#123C2212yes—
19C22×#123C2212yes—
20C22×#123C2212yes—
21C22×#121C2211yescharacteristic
22C22×#117C2211yescharacteristic
23C23×C4⋊C4C2212yes—
24C23×C4⋊C4C2211yescharacteristic
25#2614—216no—
26#2501—28no—
27#2413—216no—
28C2×#399C2212yes—
29C2×#399C2212yes—
30C2×#398C2214yes—
31C2×#396C2214yes—
32C2×#395—28no—
33C2×#395C2214yes—
34C2×#395C2212yes—
35C2×#395C2214yes—
36C2×#395C2212yes—
37C2×#395C2212yes—
38C2×#395C2212yes—
39C2×#393C2214yes—
40C2×#393C2214yes—
41C2×#390C2211yescharacteristic
42C2×#390C2211yescharacteristic
43C2×#389C2214yes—
44C2×#388C2214yes—
45C2×#386C2212yes—
46C2×#386C2212yes—
47C2×#385C2212yes—
48C2×#384C2211yescharacteristic
49C2×#384C2211yescharacteristic
50C2×#381C2212yes—
51C2×#381C2212yes—
52C2×#381C2211yescharacteristic
53C2×#381C2212yes—
54C2×C4×C22⋊C4C2211yescharacteristic
55C22×#95C2211yescharacteristic
56C22×#95C2212yes—
57#1444C2214yes—
58#1444C2214yes—
59#1444C2214yes—
60#1443C2214yes—
61#1442C2214yes—
62#1442C2214yes—
63#1441C2214yes—
64#1441C2212yes—
65#1441C2212yes—
66#1440C2214yes—
67#1439C2214yes—
68#1439C2214yes—
69#1439C2214yes—
70#1439C2214yes—
71#1439C2214yes—
72#1438C2214yes—
73#1436C2212yes—
74#1436C2214yes—
75#1436C2212yes—
76#1434C2214yes—
77C22⋊#95C2212yes—
78C22⋊#95C2212yes—
79C23×Q8—28no—
80C23×Q8—24no—
81C23×Q8—42no—
82C23×Q8—44no—
83C23×Q8—42no—
84C23×Q8C2312yes—
85C23×Q8C2312yes—
86C23×D4C2312yes—
87C23×D4C2311yescharacteristic
88C24×C4C2312yes—
89C24×C4C2311yescharacteristic
90C24×C4C2311yescharacteristic
91C24×C4C2312yes—
92C24×C4C2311yescharacteristic
93C2×#123—28no—
94C2×#123—28no—
95C2×#123—28no—
96C2×#123—28no—
97C2×#123—28no—
98C2×#123—28no—
99C2×#123—28no—
100C2×#123—44no—
101C2×#123—44no—
102C2×#123—44no—
103C2×#123—44no—
104C2×#123—44no—
105C2×#121—24no—
106C2×#121—42no—
107C2×#121—42no—
108C2×#117—28no—
109C2×#117—42no—
110C22×C4⋊C4—28no—
111C22×C4⋊C4—28no—
112C22×C4⋊C4—42no—
113C22×C4⋊C4—42no—
114C22×C4⋊C4—44no—
115C22×C4⋊C4D418yes—
116C22×C4⋊C4C2314yes—
117C22×C4⋊C4C2314yes—
118C22×C4⋊C4C2312yes—
119C22×C4⋊C4C2312yes—
120C22×C4⋊C4C2314yes—
121C22×C4⋊C4C2312yes—
122C22×C4⋊C4C2314yes—
123C22×C4⋊C4C2312yes—
124C22×C4⋊C4C2312yes—
125C22×C4⋊C4C2314yes—
126C22×C4⋊C4C2311yescharacteristic
127C22×C4⋊C4C2311yescharacteristic
128C22×C4⋊C4C2311yescharacteristic
129C22×C4⋊C4C2311yescharacteristic
130C22×C4⋊C4C2311yescharacteristic
131C22×C22⋊C4C2312yes—
132C22×C22⋊C4C2314yes—
133C22×C22⋊C4C2312yes—
134C22×C22⋊C4C2312yes—
135C22×C22⋊C4C2312yes—
136C22×C22⋊C4C2314yes—
137C22×C22⋊C4C2312yes—
138C22×C22⋊C4C2312yes—
139C22×C22⋊C4C2312yes—
140C22×C22⋊C4C2314yes—
141C22×C22⋊C4C2312yes—
142C22×C22⋊C4C2314yes—
143C22×C42C2311yescharacteristic
144C22×C42C2312yes—
145C22×C42C2311yescharacteristic
146#399—28no—
147#399—28no—
148#398—216no—
149#396—216no—
150#395—28no—
151#395—28no—
152#395—28no—
153#395—28no—
154#395—216no—
155#395—216no—
156#393—216no—
157#393—216no—
158#390—24no—
159#390—24no—
160#389—216no—
161#388—216no—
162#386—28no—
163#386—44no—
164#385—28no—
165#384—24no—
166#384—24no—
167#381—28no—
168#381—28no—
169#381—42no—
170#381D4116yescomplemented
171C4×C22⋊C4—24no—
172C2×#95—28no—
173C2×#95—24no—
174C2×#95C2314yes—
175C2×#95C2314yes—
176C2×#95C2314yes—
177C2×#95C2314yes—
178C2×#95C2314yes—
179C2×#95C2312yes—
180C2×#95C2314yes—
181C2×#95C2312yes—
182C2×#95C2314yes—
183C2×#95C2314yes—
184C2×#95C2312yes—
185C2×#95C2314yes—
186C2×#95C2314yes—
187C2×#95C2314yes—
188C2×#95C2314yes—
189C2×#95C2312yes—
190C2×#95C2312yes—
191C2×#95C2314yes—
192C2×#95C2314yes—
193C2×#95C2314yes—
194C25C2412yes—
195C25C2411yescharacteristic
196C22×Q8—28no—
197C22×Q8—28no—
198C22×Q8—24no—
199C22×Q8—24no—
200C22×Q8—44no—
201C22×Q8—44no—
202C22×Q8—42no—
203C22×Q8—42no—
204C22×Q8—42no—
205C22×Q8—42no—
206C22×Q8—432no—
207C22×Q8—416no—
208C22×Q8—416no—
209C22×Q8—432no—
210C22×Q8—416no—
211C22×D4—24no—
212C22×D4—24no—
213C22×D4—24no—
214C22×D4—24no—
215C22×D4—24no—
216C22×D4—24no—
217C22×D4—24no—
218C22×D4—22no—
219C22×D4—24no—
220C22×D4—41no—
221C22×D4—41no—
222C22×D4—42no—
223C23×C4—24no—
224C23×C4—24no—
225C23×C4—24no—
226C23×C4—24no—
227C23×C4—24no—
228C23×C4—22no—
229C23×C4—22no—
230C23×C4—22no—
231C23×C4—24no—
232C23×C4—22no—
233C23×C4—22no—
234C23×C4—24no—
235C23×C4—22no—
236C23×C4—24no—
237C23×C4—24no—
238C23×C4—24no—
239C23×C4—24no—
240C23×C4—28no—
241C23×C4—24no—
242C23×C4—24no—
243C23×C4—24no—
244C23×C4—42no—
245C23×C4—44no—
246C23×C4C2×D414yes—
247C23×C4C2×D418yes—
248C23×C4C2×D414yes—
249C23×C4C2×D414yes—
250C23×C4C2414yes—
251C23×C4C2412yes—
252C23×C4C2412yes—
253C23×C4C2414yes—
254C23×C4C2412yes—
255C23×C4C2411yescharacteristic
256C23×C4C2411yescharacteristic
257C23×C4C2411yescharacteristic
258C23×C4C2411yescharacteristic
259C23×C4C2414yes—
260C23×C4C2411yescharacteristic
261C23×C4C2414yes—
262C23×C4C2411yescharacteristic
263#123—432no—
264#123—432no—
265#123—432no—
266#123—432no—
267#121—416no—
268#117—416no—
269C2×C4⋊C4—28no—
270C2×C4⋊C4—28no—
271C2×C4⋊C4—28no—
272C2×C4⋊C4—28no—
273C2×C4⋊C4—28no—
274C2×C4⋊C4—22no—
275C2×C4⋊C4—24no—
276C2×C4⋊C4—24no—
277C2×C4⋊C4—24no—
278C2×C4⋊C4—24no—
279C2×C4⋊C4—24no—
280C2×C4⋊C4—28no—
281C2×C4⋊C4—24no—
282C2×C4⋊C4—28no—
283C2×C4⋊C4—24no—
284C2×C4⋊C4—28no—
285C2×C4⋊C4—24no—
286C2×C4⋊C4—28no—
287C2×C4⋊C4—28no—
288C2×C4⋊C4—416no—
289C2×C4⋊C4—41no—
290C2×C4⋊C4—41no—
291C2×C4⋊C4—41no—
292C2×C4⋊C4—41no—
293C2×C4⋊C4—41no—
294C2×C4⋊C4—41no—
295C2×C4⋊C4—41no—
296C2×C4⋊C4—41no—
297C2×C4⋊C4—44no—
298C2×C4⋊C4—44no—
299C2×C4⋊C4—44no—
300C2×C4⋊C4—42no—
301C2×C4⋊C4—41no—
302C2×C4⋊C4—41no—
303C2×C4⋊C4—44no—
304C2×C4⋊C4—42no—
305C2×C4⋊C4—42no—
306C2×C4⋊C4—42no—
307C2×C4⋊C4—42no—
308C2×C4⋊C4—44no—
309C2×C4⋊C4—44no—
310C2×C4⋊C4—432no—
311C2×C4⋊C4C2×D414yes—
312C2×C4⋊C4C2×D414yes—
313C2×C4⋊C4C2×D414yes—
314C2×C4⋊C4C2×D414yes—
315C2×C22⋊C4—24no—
316C2×C22⋊C4—24no—
317C2×C22⋊C4—24no—
318C2×C22⋊C4—24no—
319C2×C22⋊C4—24no—
320C2×C22⋊C4—24no—
321C2×C22⋊C4—24no—
322C2×C22⋊C4—24no—
323C2×C22⋊C4—24no—
324C2×C22⋊C4—28no—
325C2×C22⋊C4—24no—
326C2×C22⋊C4—24no—
327C2×C22⋊C4—24no—
328C2×C22⋊C4—24no—
329C2×C22⋊C4—24no—
330C2×C22⋊C4—28no—
331C2×C22⋊C4—24no—
332C2×C22⋊C4—28no—
333C2×C22⋊C4—24no—
334C2×C22⋊C4—28no—
335C2×C22⋊C4—28no—
336C2×C22⋊C4—28no—
337C2×C22⋊C4—44no—
338C2×C22⋊C4—42no—
339C2×C22⋊C4—42no—
340C2×C22⋊C4—44no—
341C2×C22⋊C4—42no—
342C2×C22⋊C4—42no—
343C2×C22⋊C4—42no—
344C2×C22⋊C4—42no—
345C2×C22⋊C4—44no—
346C2×C22⋊C4—44no—
347C2×C22⋊C4—44no—
348C2×C22⋊C4—44no—
349C2×C22⋊C4C2×D418yes—
350C2×C22⋊C4C2×D418yes—
351C2×C42—28no—
352C2×C42—22no—
353C2×C42—22no—
354C2×C42—24no—
355C2×C42—42no—
356C2×C42—42no—
357#95—24no—
358#95—24no—
359#95—28no—
360#95—28no—
361#95—24no—
362#95—28no—
363#95—28no—
364#95—28no—
365#95—28no—
366#95—28no—
367#95—28no—
368#95—28no—
369#95—28no—
370#95—44no—
371#95—44no—
372#95—42no—
373#95—42no—
374#95—44no—
375#95C2×D4116yes—
376#95C2×D4116yes—
377C24—22no—
378C24—24no—
379C24—22no—
380C24—22no—
381C24—22no—
382C24—22no—
383C24—22no—
384C24—22no—
385C24—22no—
386C24—22no—
387C24—24no—
388C24—24no—
389C24—24no—
390C24—24no—
391C24C22×D412yes—
392C24C22×D414yes—
393C24H(5,𝔽2)12yes—
394C24H(5,𝔽2)12yes—
395C24H(5,𝔽2)14yes—
396C24C2511yescharacteristic
397C2×Q8—48no—
398C2×Q8—48no—
399C2×Q8—416no—
400C2×Q8—416no—
401C2×Q8—48no—
402C2×Q8—48no—
403C2×Q8—416no—
404C2×Q8—416no—
405C2×Q8—4128no—
406C2×Q8—48no—
407C2×Q8—48no—
408C2×D4—44no—
409C2×D4—44no—
410C2×D4—48no—
411C2×D4—48no—
412C2×D4—48no—
413C2×D4—48no—
414C2×D4—48no—
415C2×D4—48no—
416C2×D4—48no—
417C2×D4—44no—
418C2×D4—84no—
419C2×D4—84no—
420C22×C4—21no—
421C22×C4—21no—
422C22×C4—22no—
423C22×C4—216no—
424C22×C4—28no—
425C22×C4—28no—
426C22×C4—28no—
427C22×C4—216no—
428C22×C4—216no—
429C22×C4—216no—
430C22×C4—216no—
431C22×C4—216no—
432C22×C4—24no—
433C22×C4—24no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—24no—
437C22×C4—24no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—24no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—24no—
446C22×C4—21no—
447C22×C4—24no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—21no—
451C22×C4—21no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—24no—
455C22×C4—24no—
456C22×C4—24no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—21no—
460C22×C4—21no—
461C22×C4—21no—
462C22×C4—21no—
463C22×C4—21no—
464C22×C4—24no—
465C22×C4—24no—
466C22×C4—24no—
467C22×C4—24no—
468C22×C4—24no—
469C22×C4—24no—
470C22×C4—24no—
471C22×C4—22no—
472C22×C4—24no—
473C22×C4—22no—
474C22×C4—24no—
475C22×C4—24no—
476C22×C4—24no—
477C22×C4—24no—
478C22×C4—28no—
479C22×C4—24no—
480C22×C4—22no—
481C22×C4—22no—
482C22×C4—22no—
483C22×C4—24no—
484C22×C4—22no—
485C22×C4—24no—
486C22×C4—22no—
487C22×C4—22no—
488C22×C4—22no—
489C22×C4—24no—
490C22×C4—24no—
491C22×C4—24no—
492C22×C4—24no—
493C22×C4—44no—
494C22×C4—48no—
495C22×C4—48no—
496C22×C4—416no—
497C22×C4—416no—
498C22×C4—48no—
499C22×C4—48no—
500C22×C4—48no—
501C22×C4—48no—
502C22×C4—48no—
503C22×C4—48no—
504C22×C4—44no—
505C22×C4—44no—
506C22×C4—48no—
507C22×C4—44no—
508C22×C4—44no—
509C22×C4—48no—
510C22×C4C22×D412yes—
511C22×C4C22×D414yes—
512C22×C4C22×D418yes—
513C22×C4C22×D414yes—
514C22×C4C22×D412yes—
515C22×C4C22×D412yes—
516C22×C4C22×D418yes—
517C22×C4C22×D412yes—
518C22×C4C22×D412yes—
519C22×C4C22×D412yes—
520C22×C42−1+418yes—
521C22×C42−1+414yes—
522C4⋊C4—44no—
523C4⋊C4—44no—
524C4⋊C4—44no—
525C4⋊C4—44no—
526C4⋊C4—416no—
527C4⋊C4—416no—
528C4⋊C4—416no—
529C4⋊C4—48no—
530C4⋊C4—416no—
531C4⋊C4—48no—
532C4⋊C4—48no—
533C4⋊C4—48no—
534C4⋊C4—48no—
535C4⋊C4—44no—
536C4⋊C4—416no—
537C22⋊C4—216no—
538C22⋊C4—216no—
539C22⋊C4—416no—
540C22⋊C4—416no—
541C22⋊C4—48no—
542C22⋊C4—48no—
543C22⋊C4—48no—
544C22⋊C4—48no—
545C22⋊C4—48no—
546C22⋊C4—416no—
547C22⋊C4—48no—
548C22⋊C4—88no—
549C42—48no—
550C42—44no—
551C42—44no—
552C23—22no—
553C23—22no—
554C23—24no—
555C23—28no—
556C23—28no—
557C23—28no—
558C23—24no—
559C23—24no—
560C23—24no—
561C23—28no—
562C23—24no—
563C23—42no—
564C23—42no—
565C23—42no—
566C23—42no—
567C23—42no—
568C23—42no—
569C23—42no—
570C23—42no—
571C23—42no—
572C23—42no—
573C23—44no—
574C23—44no—
575C23—44no—
576C23—44no—
577C23—42no—
578C23—44no—
579C23—44no—
580C23—44no—
581C23—44no—
582C23—44no—
583C23—44no—
584C23—44no—
585C23—44no—
586C23#53414yes—
587C23#53518yes—
588C23D4214yes—
589C23D4218yes—
590C23#56014yes—
591C23C23×D411yescharacteristic
592C23C23×D412yes—
593C23C2×H(5,𝔽2)11yescharacteristic
594C23C2×H(5,𝔽2)11yescharacteristic
595C23C2×H(5,𝔽2)12yes—
596C23C2×2−1+412yes—
597C23C2×2−1+414yes—
598C23#58512yes—
599Q8—432no—
600Q8—432no—
601D4—88no—
602D4—816no—
603C2×C4—22no—
604C2×C4—22no—
605C2×C4—24no—
606C2×C4—24no—
607C2×C4—22no—
608C2×C4—22no—
609C2×C4—22no—
610C2×C4—22no—
611C2×C4—22no—
612C2×C4—28no—
613C2×C4—22no—
614C2×C4—22no—
615C2×C4—28no—
616C2×C4—24no—
617C2×C4—24no—
618C2×C4—24no—
619C2×C4—24no—
620C2×C4—24no—
621C2×C4—24no—
622C2×C4—28no—
623C2×C4—28no—
624C2×C4—28no—
625C2×C4—28no—
626C2×C4—28no—
627C2×C4—28no—
628C2×C4—28no—
629C2×C4—28no—
630C2×C4—28no—
631C2×C4—416no—
632C2×C4—432no—
633C2×C4—416no—
634C2×C4—416no—
635C2×C4—432no—
636C2×C4—44no—
637C2×C4—42no—
638C2×C4—42no—
639C2×C4—42no—
640C2×C4—42no—
641C2×C4—44no—
642C2×C4—42no—
643C2×C4—42no—
644C2×C4—44no—
645C2×C4—41no—
646C2×C4—44no—
647C2×C4—41no—
648C2×C4—44no—
649C2×C4—41no—
650C2×C4—41no—
651C2×C4—42no—
652C2×C4—42no—
653C2×C4—44no—
654C2×C4—44no—
655C2×C4—44no—
656C2×C4—41no—
657C2×C4—41no—
658C2×C4—41no—
659C2×C4—41no—
660C2×C4—44no—
661C2×C4—44no—
662C2×C4—44no—
663C2×C4—44no—
664C2×C4—42no—
665C2×C4—42no—
666C2×C4—44no—
667C2×C4—44no—
668C2×C4—42no—
669C2×C4—44no—
670C2×C4—42no—
671C2×C4—42no—
672C2×C4—42no—
673C2×C4—42no—
674C2×C4—42no—
675C2×C4—44no—
676C2×C4—44no—
677C2×C4—42no—
678C2×C4—44no—
679C2×C4—44no—
680C2×C4—44no—
681C2×C4—44no—
682C2×C4#53614yes—
683C22—28no—
684C22—216no—
685C22—24no—
686C22—24no—
687C22—28no—
688C22—28no—
689C22—28no—
690C22—82no—
691C22—82no—
692C22—84no—
693C22—84no—
694C22—82no—
695C22—82no—
696C22—82no—
697C22—82no—
698C22—82no—
699C22—84no—
700C22—84no—
701C22#266918yes—
702C22C2×#53411yescharacteristic
703C22C2×#53512yes—
704C22C2×#53611yescharacteristic
705C22#344911yescharacteristic
706C22C2×D4212yes—
707C22C2×D4211yescharacteristic
708C22#348612yes—
709C22#348914yes—
710C22#349914yes—
711C22C2×#56011yescharacteristic
712C22#353514yes—
713C22#353512yes—
714C22#353512yes—
715C22#354412yes—
716C22#354414yes—
717C22#354412yes—
718C4—24no—
719C4—44no—
720C4—48no—
721C4—42no—
722C4—42no—
723C4—42no—
724C4—42no—
725C4—42no—
726C4—48no—
727C4—44no—
728C4—48no—
729C4—84no—
730C4—82no—
731C2—28no—
732C2—162no—
733C2#3725811yescharacteristic
734C2#3744111yescharacteristic
735C2#3765312yes—
736C2#3933712yes—
737C2#4229714yes—
738C2#4475812yes—
739C2C2×#266911yescharacteristic
740C2#5291312yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC22×#121C22
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×#2413C2semidirect
3. C2×#2614C2semidirect
4. C22×#395C2semidirect
5. #17104C2semidirect
6. #16556C2semidirect
7. #16522C2semidirect
8. #16327C2semidirect
9. #15662C2semidirect
10. #15625C2semidirect
11. #15069C2semidirect
13. #14912C2semidirect
14. #13992C2semidirect
15. #13870C2semidirect
170. #381D4semidirect

Maximal subgroups

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

subgroupindex
C2×#25012
C2×#24132
C2×#26142
C22×#3952
#171042
#165562
#165222
#163272
#156622
#156252
#150692
#149262
#149122
#139922
#138702