Finite Group Databasebeta

← #4420666

Subgroups — #4420666

Order 512.

3366 conjugacy classes of subgroups.

The subgroup lattice has 746 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2583C211yesmaximal, characteristic, complemented
2C2×#2406C211yesmaximal, characteristic
3C2×#2616C211yesmaximal, characteristic
4C2×#2341C211yesmaximal, characteristic
5#17069C212yesmaximal, complemented
6#17011C212yesmaximal, complemented
7#16399C212yesmaximal, complemented
8#16264C212yesmaximal, complemented
9#15300C214yesmaximal, complemented
10#15271C214yesmaximal, complemented
11#15158C212yesmaximal, complemented
12#14880C212yesmaximal, complemented
13#14840C212yesmaximal, complemented
14#14763C212yesmaximal, complemented
15#14660C211yesmaximal, characteristic, complemented
16#14355C211yesmaximal, characteristic, complemented
17#14339C211yesmaximal, characteristic, complemented
18C23×#41C2211yescharacteristic
19C22×#123C2212yes—
20C22×C22⋊Q8C2211yescharacteristic
21C22×C22⋊Q8C2212yes—
22C22×C4×Q8C2212yes—
23C23×C4⋊C4C2211yescharacteristic
24#2616—28no—
25#2583—28no—
26#2406—28no—
27C2×#398C2211yescharacteristic
28C2×#398C2211yescharacteristic
29C2×#398C2214yes—
30C2×#397C2214yes—
31C2×#395C2211yescharacteristic
32C2×#395C2211yescharacteristic
33C2×#395C2211yescharacteristic
34C2×#395C2212yes—
35C2×#395C2211yescharacteristic
36C2×#394C2212yes—
37C2×#394C2214yes—
38C2×#394C2211yescharacteristic
39C2×#394C2211yescharacteristic
40#2341—28no—
41C2×#390C2212yes—
42C2×#390C2214yes—
43C2×#389C2212yes—
44C2×#389C2212yes—
45C2×#389C2214yes—
46C2×#386C2212yes—
47C2×#385C2212yes—
48C2×#385C2212yes—
49C2×#385C2212yes—
50C2×#385C2212yes—
51C2×#381C2211yescharacteristic
52C2×#381C2212yes—
53C2×#380C2211yescharacteristic
54C2×#380C2211yescharacteristic
55C2×#380C2212yes—
56C2×#380C2212yes—
57C2×C4×C22⋊C4C2212yes—
58C2×C4×C22⋊C4C2211yescharacteristic
59#1446C2212yes—
60#1445C2212yes—
61#1444C2212yes—
62#1442C2212yes—
63#1441C2212yes—
64#1441C2212yes—
65#1441C2212yes—
66#1441C2212yes—
67#1441C2214yes—
68#1441C2214yes—
69#1441C2212yes—
70#1441C2212yes—
71#1441C2214yes—
72#1440C2214yes—
73#1440C2214yes—
74#1440C2214yes—
75#1439C2214yes—
76#1439C2214yes—
77#1439C2214yes—
78#1439C2214yes—
79#1439C2214yes—
80#1439C2214yes—
81#1436C2214yes—
82#1436C2214yes—
83C4⋊#95C2212yes—
84C4⋊#95C2214yes—
85#1434C2212yes—
86C22⋊#95C2212yes—
87C22×#41—24no—
88C22×#41—24no—
89C22×#41—48no—
90C22×#41D418yes—
91C23×Q8C2311yescharacteristic
92C23×Q8C2311yescharacteristic
93C23×D4C2312yes—
94C23×D4C2311yescharacteristic
95C24×C4C2311yescharacteristic
96C24×C4C2312yes—
97C24×C4C2311yescharacteristic
98C24×C4C2311yescharacteristic
99C2×#123—28no—
100C2×#123—48no—
101C2×C22⋊Q8—24no—
102C2×C22⋊Q8—28no—
103C2×C22⋊Q8—24no—
104C2×C22⋊Q8—24no—
105C2×C22⋊Q8—48no—
106C2×C4×Q8—28no—
107C2×C4×Q8—48no—
108C22×C4⋊C4—24no—
109C22×C4⋊C4—24no—
110C22×C4⋊C4—24no—
111C22×C4⋊C4C2314yes—
112C22×C4⋊C4C2311yescharacteristic
113C22×C4⋊C4C2311yescharacteristic
114C22×C4⋊C4C2312yes—
115C22×C4⋊C4C2312yes—
116C22×C4⋊C4C2311yescharacteristic
117C22×C4⋊C4C2311yescharacteristic
118C22×C4⋊C4C2312yes—
119C22×C4⋊C4C2311yescharacteristic
120C22×C4⋊C4C2312yes—
121C22×C4⋊C4C2314yes—
122C22×C4⋊C4C2312yes—
123C22×C4⋊C4C2312yes—
124C22×C4⋊C4C2312yes—
125C22×C4⋊C4C2312yes—
126C22×C4⋊C4C2312yes—
127C22×C4⋊C4C2314yes—
128C22×C22⋊C4C2312yes—
129C22×C22⋊C4C2314yes—
130C22×C22⋊C4C2312yes—
131C22×C22⋊C4C2312yes—
132C22×C22⋊C4C2312yes—
133C22×C22⋊C4C2314yes—
134C22×C22⋊C4C2314yes—
135C22×C22⋊C4C2312yes—
136C22×C22⋊C4C2312yes—
137C22×C22⋊C4C2312yes—
138C22×C22⋊C4C2314yes—
139C22×C22⋊C4C2312yes—
140C22×C22⋊C4C2312yes—
141C22×C42C2314yes—
142C22×C42C2312yes—
143#398—24no—
144#398—24no—
145#398—48no—
146#397—48no—
147#395—28no—
148#395—24no—
149#395—24no—
150#395—24no—
151#395—24no—
152#394—216no—
153#394—28no—
154#394—24no—
155#394—24no—
156#390—216no—
157#390—28no—
158#389—28no—
159#389—28no—
160#389—48no—
161#386—28no—
162#385—28no—
163#385—28no—
164#385—28no—
165#385—28no—
166#381—24no—
167#381—44no—
168#380—24no—
169#380—28no—
170#380—24no—
171#380—28no—
172C4×C22⋊C4—28no—
173C4×C22⋊C4D418yes—
174C2×#95C2311yescharacteristic
175C2×#95C2312yes—
176C2×#95C2311yescharacteristic
177C2×#95C2314yes—
178C2×#95C2312yes—
179C2×#95C2312yes—
180C2×#95C2312yes—
181C2×#95C2312yes—
182C2×#95C2314yes—
183C2×#95C2312yes—
184C2×#95C2312yes—
185C2×#95C2314yes—
186C2×#95C2312yes—
187C2×#95C2312yes—
188C2×#95C2314yes—
189C2×#95C2314yes—
190C2×#95C2314yes—
191C2×#95C2314yes—
192C2×#95C2314yes—
193C2×#95C2314yes—
194C2×#95C2314yes—
195C2×#95C2314yes—
196C2×#95C2314yes—
197C2×#95C2311yescharacteristic
198C2×#95C2311yescharacteristic
199C25C2412yes—
200C25C2411yescharacteristic
201C2×#41—232no—
202C2×#41—432no—
203C2×#41—432no—
204C2×#41—432no—
205C22×Q8—22no—
206C22×Q8—22no—
207C22×Q8—22no—
208C22×Q8—22no—
209C22×Q8—22no—
210C22×Q8—28no—
211C22×Q8—44no—
212C22×Q8C2×D414yes—
213C22×D4—24no—
214C22×D4—24no—
215C22×D4—22no—
216C22×D4—22no—
217C22×D4—28no—
218C22×D4—44no—
219C22×D4—44no—
220C22×D4C2×D418yes—
221C22×D4C2×D414yes—
222C23×C4—22no—
223C23×C4—24no—
224C23×C4—24no—
225C23×C4—28no—
226C23×C4—22no—
227C23×C4—22no—
228C23×C4—22no—
229C23×C4—22no—
230C23×C4—22no—
231C23×C4—22no—
232C23×C4—22no—
233C23×C4—41no—
234C23×C4—41no—
235C23×C4—44no—
236C23×C4—44no—
237C23×C4—44no—
238C23×C4C2×D418yes—
239C23×C4C2×D414yes—
240C23×C4C2×D414yes—
241C23×C4#4114yes—
242C23×C4C2412yes—
243C23×C4C2412yes—
244C23×C4C2412yes—
245C23×C4C2414yes—
246C23×C4C2414yes—
247C23×C4C2412yes—
248C23×C4C2411yescharacteristic
249C23×C4C2411yescharacteristic
250C23×C4C2412yes—
251C23×C4C2411yescharacteristic
252C23×C4C2411yescharacteristic
253C23×C4C2411yescharacteristic
254C23×C4C2411yescharacteristic
255C23×C4C2412yes—
256C23×C4C2412yes—
257#123—432no—
258C22⋊Q8—432no—
259C22⋊Q8—416no—
260C4×Q8—432no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—22no—
263C2×C4⋊C4—22no—
264C2×C4⋊C4—22no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4—22no—
267C2×C4⋊C4—22no—
268C2×C4⋊C4—22no—
269C2×C4⋊C4—28no—
270C2×C4⋊C4—24no—
271C2×C4⋊C4—24no—
272C2×C4⋊C4—22no—
273C2×C4⋊C4—24no—
274C2×C4⋊C4—24no—
275C2×C4⋊C4—28no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—24no—
278C2×C4⋊C4—24no—
279C2×C4⋊C4—24no—
280C2×C4⋊C4—24no—
281C2×C4⋊C4—22no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—24no—
287C2×C4⋊C4—28no—
288C2×C4⋊C4—28no—
289C2×C4⋊C4—28no—
290C2×C4⋊C4—28no—
291C2×C4⋊C4—28no—
292C2×C4⋊C4—24no—
293C2×C4⋊C4—24no—
294C2×C4⋊C4—28no—
295C2×C4⋊C4—24no—
296C2×C4⋊C4—24no—
297C2×C4⋊C4—42no—
298C2×C4⋊C4—42no—
299C2×C4⋊C4—44no—
300C2×C4⋊C4—44no—
301C2×C4⋊C4—44no—
302C2×C4⋊C4—44no—
303C2×C4⋊C4—44no—
304C2×C4⋊C4—416no—
305C2×C4⋊C4—44no—
306C2×C4⋊C4—44no—
307C2×C22⋊C4—28no—
308C2×C22⋊C4—24no—
309C2×C22⋊C4—28no—
310C2×C22⋊C4—24no—
311C2×C22⋊C4—24no—
312C2×C22⋊C4—24no—
313C2×C22⋊C4—24no—
314C2×C22⋊C4—24no—
315C2×C22⋊C4—24no—
316C2×C22⋊C4—24no—
317C2×C22⋊C4—28no—
318C2×C22⋊C4—24no—
319C2×C22⋊C4—24no—
320C2×C22⋊C4—28no—
321C2×C22⋊C4—28no—
322C2×C22⋊C4—24no—
323C2×C22⋊C4—28no—
324C2×C22⋊C4—24no—
325C2×C22⋊C4—28no—
326C2×C22⋊C4—42no—
327C2×C22⋊C4—42no—
328C2×C22⋊C4—44no—
329C2×C22⋊C4—44no—
330C2×C22⋊C4—44no—
331C2×C22⋊C4—44no—
332C2×C22⋊C4—44no—
333C2×C22⋊C4—44no—
334C2×C22⋊C4—42no—
335C2×C22⋊C4—42no—
336C2×C22⋊C4—44no—
337C2×C22⋊C4—42no—
338C2×C22⋊C4—42no—
339C2×C22⋊C4—44no—
340C2×C22⋊C4—44no—
341C2×C22⋊C4C2×D418yes—
342C2×C42—28no—
343C2×C42—28no—
344C2×C42—44no—
345C2×C42—44no—
346C2×C42C2×D418yes—
347#95—24no—
348#95—24no—
349#95—28no—
350#95—24no—
351#95—22no—
352#95—22no—
353#95—24no—
354#95—24no—
355#95—24no—
356#95—24no—
357#95—24no—
358#95—24no—
359#95—28no—
360#95—28no—
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#95—44no—
370#95C2×D414yes—
371#95C2×D414yes—
372C24—21no—
373C24—24no—
374C24—24no—
375C24—21no—
376C24—21no—
377C24—21no—
378C24—22no—
379C24—22no—
380C24—22no—
381C24—24no—
382C24—24no—
383C24—22no—
384C24—24no—
385C24—24no—
386C24—22no—
387C24—22no—
388C24C22×D414yes—
389C24C22×D412yes—
390C24C22×D412yes—
391C24C2×#4112yes—
392C24C2511yescharacteristic
393#41—4128no—
394C2×Q8—216no—
395C2×Q8—216no—
396C2×Q8—28no—
397C2×Q8—216no—
398C2×Q8—48no—
399C2×Q8—44no—
400C2×Q8—84no—
401C2×Q8—84no—
402C2×D4—216no—
403C2×D4—28no—
404C2×D4—216no—
405C2×D4—48no—
406C2×D4—48no—
407C2×D4—416no—
408C2×D4—416no—
409C2×D4—48no—
410C2×D4—48no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—24no—
414C22×C4—24no—
415C22×C4—24no—
416C22×C4—24no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—28no—
420C22×C4—21no—
421C22×C4—21no—
422C22×C4—21no—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—22no—
428C22×C4—24no—
429C22×C4—24no—
430C22×C4—21no—
431C22×C4—22no—
432C22×C4—21no—
433C22×C4—21no—
434C22×C4—21no—
435C22×C4—21no—
436C22×C4—216no—
437C22×C4—216no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—24no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—24no—
447C22×C4—22no—
448C22×C4—21no—
449C22×C4—21no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—21no—
453C22×C4—21no—
454C22×C4—24no—
455C22×C4—24no—
456C22×C4—22no—
457C22×C4—24no—
458C22×C4—24no—
459C22×C4—24no—
460C22×C4—22no—
461C22×C4—24no—
462C22×C4—24no—
463C22×C4—24no—
464C22×C4—24no—
465C22×C4—24no—
466C22×C4—24no—
467C22×C4—24no—
468C22×C4—24no—
469C22×C4—24no—
470C22×C4—24no—
471C22×C4—24no—
472C22×C4—22no—
473C22×C4—22no—
474C22×C4—24no—
475C22×C4—24no—
476C22×C4—22no—
477C22×C4—22no—
478C22×C4—24no—
479C22×C4—22no—
480C22×C4—24no—
481C22×C4—22no—
482C22×C4—48no—
483C22×C4—48no—
484C22×C4—48no—
485C22×C4—48no—
486C22×C4—416no—
487C22×C4—416no—
488C22×C4—44no—
489C22×C4—44no—
490C22×C4—44no—
491C22×C4—44no—
492C22×C4—44no—
493C22×C4—44no—
494C22×C4—48no—
495C22×C4—44no—
496C22×C4—84no—
497C22×C4—84no—
498C22×C4#121116yes—
499C22×C4C22×D414yes—
500C22×C4C22×D412yes—
501C22×C4C22×D412yes—
502C22×C4C22×D412yes—
503C22×C4C22×D412yes—
504C22×C4C22×D414yes—
505C22×C4C22×D414yes—
506C22×C4C2×#4112yes—
507C22×C4C2×#4112yes—
508C22×C4H(5,𝔽2)12yes—
509C22×C4H(5,𝔽2)18yes—
510C22×C42−1+412yes—
511C22×C42−1+418yes—
512C4⋊C4—48no—
513C4⋊C4—48no—
514C4⋊C4—416no—
515C4⋊C4—416no—
516C4⋊C4—48no—
517C4⋊C4—48no—
518C4⋊C4—44no—
519C4⋊C4—44no—
520C4⋊C4—44no—
521C4⋊C4—44no—
522C4⋊C4—44no—
523C4⋊C4—48no—
524C4⋊C4—416no—
525C4⋊C4—48no—
526C4⋊C4—48no—
527C4⋊C4—84no—
528C4⋊C4—84no—
529C22⋊C4—216no—
530C22⋊C4—48no—
531C22⋊C4—48no—
532C22⋊C4—416no—
533C22⋊C4—48no—
534C22⋊C4—48no—
535C22⋊C4—48no—
536C22⋊C4—48no—
537C22⋊C4—416no—
538C22⋊C4—88no—
539C22⋊C4—84no—
540C22⋊C4—84no—
541C22⋊C4—88no—
542C42—48no—
543C42—416no—
544C23—24no—
545C23—24no—
546C23—24no—
547C23—22no—
548C23—22no—
549C23—22no—
550C23—22no—
551C23—24no—
552C23—22no—
553C23—22no—
554C23—24no—
555C23—24no—
556C23—24no—
557C23—24no—
558C23—24no—
559C23—24no—
560C23—24no—
561C23—24no—
562C23—24no—
563C23—44no—
564C23—44no—
565C23—44no—
566C23—44no—
567C23—44no—
568C23—44no—
569C23—42no—
570C23—42no—
571C23—44no—
572C23—42no—
573C23—42no—
574C23—44no—
575C23—44no—
576C23—44no—
577C23—44no—
578C23—44no—
579C23—44no—
580C23—44no—
581C23—44no—
582C23—44no—
583C23C2×#12114yes—
584C23C23×D411yescharacteristic
585C23C23×D411yescharacteristic
586C23C22×#4111yescharacteristic
587C23C2×H(5,𝔽2)14yes—
588C23C2×H(5,𝔽2)11yescharacteristic
589C23C2×2−1+414yes—
590C23C2×2−1+411yescharacteristic
591C23#58511yescharacteristic
592C23#58511yescharacteristic
593Q8—232no—
594Q8—88no—
595D4—432no—
596D4—416no—
597C2×C4—24no—
598C2×C4—24no—
599C2×C4—24no—
600C2×C4—24no—
601C2×C4—24no—
602C2×C4—24no—
603C2×C4—28no—
604C2×C4—24no—
605C2×C4—24no—
606C2×C4—24no—
607C2×C4—24no—
608C2×C4—28no—
609C2×C4—28no—
610C2×C4—28no—
611C2×C4—24no—
612C2×C4—24no—
613C2×C4—22no—
614C2×C4—24no—
615C2×C4—22no—
616C2×C4—22no—
617C2×C4—22no—
618C2×C4—22no—
619C2×C4—22no—
620C2×C4—416no—
621C2×C4—432no—
622C2×C4—416no—
623C2×C4—41no—
624C2×C4—41no—
625C2×C4—41no—
626C2×C4—41no—
627C2×C4—41no—
628C2×C4—41no—
629C2×C4—41no—
630C2×C4—41no—
631C2×C4—44no—
632C2×C4—44no—
633C2×C4—42no—
634C2×C4—42no—
635C2×C4—41no—
636C2×C4—41no—
637C2×C4—42no—
638C2×C4—42no—
639C2×C4—44no—
640C2×C4—42no—
641C2×C4—42no—
642C2×C4—44no—
643C2×C4—44no—
644C2×C4—44no—
645C2×C4—42no—
646C2×C4—42no—
647C2×C4—42no—
648C2×C4—42no—
649C2×C4—42no—
650C2×C4—44no—
651C2×C4—42no—
652C2×C4—44no—
653C2×C4—42no—
654C2×C4—42no—
655C2×C4—42no—
656C2×C4—42no—
657C2×C4—42no—
658C2×C4—42no—
659C2×C4—44no—
660C2×C4—42no—
661C2×C4—42no—
662C2×C4—42no—
663C2×C4—44no—
664C2×C4—42no—
665C2×C4—44no—
666C2×C4—44no—
667C2×C4—44no—
668C2×C4—44no—
669C2×C4—44no—
670C2×C4—44no—
671C2×C4—44no—
672C2×C4—44no—
673C2×C4—44no—
674C2×C4—44no—
675C2×C4—44no—
676C2×C4—44no—
677C2×C4—88no—
678C2×C4C2×#12114yes—
679C2×C4C2×#12114yes—
680C2×C4#53418yes—
681C2×C4#53618yes—
682C22—24no—
683C22—24no—
684C22—24no—
685C22—44no—
686C22—44no—
687C22—44no—
688C22—44no—
689C22—44no—
690C22—44no—
691C22—44no—
692C22—44no—
693C22—44no—
694C22—44no—
695C22—44no—
696C22—44no—
697C22—44no—
698C22—84no—
699C22—84no—
700C22—84no—
701C22—84no—
702C22C22×#12111yescharacteristic
703C22C2×#53412yes—
704C22C2×#53612yes—
705C22#348412yes—
706C22#348811yescharacteristic
707C22#348811yescharacteristic
708C22#348911yescharacteristic
709C22#348911yescharacteristic
710C22#349012yes—
711C22#349211yescharacteristic
712C22#349311yescharacteristic
713C22#349912yes—
714C22#350912yes—
715C22#354114yes—
716C22#354814yes—
717C22#355214yes—
718C22#355414yes—
719C4—28no—
720C4—24no—
721C4—24no—
722C4—28no—
723C4—42no—
724C4—42no—
725C4—44no—
726C4—82no—
727C4—84no—
728C4—81no—
729C4—81no—
730C4—82no—
731C4—82no—
732C4—82no—
733C4—84no—
734C2—44no—
735C2—84no—
736C2#3498111yescharacteristic
737C2#3531611yescharacteristic
738C2#3551811yescharacteristic
739C2#4058912yes—
740C2#4247912yes—
741C2#4291612yes—
742C2#4291712yes—
743C2#4788812yes—
744C2#4959412yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC23×#41C22
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×#2583C2semidirect
5. #17069C2semidirect
6. #17011C2semidirect
7. #16399C2semidirect
8. #16264C2semidirect
9. #15300C2semidirect
10. #15271C2semidirect
11. #15158C2semidirect
12. #14880C2semidirect
13. #14840C2semidirect
14. #14763C2semidirect
15. #14660C2semidirect
16. #14355C2semidirect
17. #14339C2semidirect

Maximal subgroups

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

subgroupindex
C2×#25832
C2×#24062
C2×#26162
C2×#23412
#170692
#170112
#163992
#162642
#153002
#152712
#151582
#148802
#148402
#147632
#146602
#143552
#143392