Finite Group Databasebeta

← #719571

Subgroups — #719571

Order 512.

3890 conjugacy classes of subgroups.

The subgroup lattice has 793 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2541C211yesmaximal, characteristic
2C2×#2445C212yesmaximal, complemented
3C2×#2366C211yesmaximal, characteristic, complemented
4#16526C214yesmaximal, complemented
5#15312C214yesmaximal, complemented
6#15260C214yesmaximal, complemented
7#15147C212yesmaximal, complemented
8#15127C212yesmaximal, complemented
9#14721C214yesmaximal, complemented
10#14272C211yesmaximal, characteristic, complemented
11#14150C211yesmaximal, characteristic, complemented
12#14018C212yesmaximal, complemented
13#13941C211yesmaximal, characteristic, complemented
14#13856C211yesmaximal, characteristic, complemented
15#13785C211yesmaximal, characteristic, complemented
16C22×#127C2212yes—
17C22×#127C2211yescharacteristic
18C22×#123C2214yes—
19C22×#121C2212yes—
20C22×#121C2212yes—
21C22×C4×D4C2211yescharacteristic
22C22×C4×D4C2212yes—
23C22×C4×D4C2212yes—
24#2541—28no—
25#2445—216no—
26C2×#397C2214yes—
27C2×#394C2214yes—
28C2×#394C2214yes—
29C2×#394C2214yes—
30C2×#394C2214yes—
31C2×#392C2214yes—
32#2366—28no—
33C2×#390C2211yescharacteristic
34C2×#390C2212yes—
35C2×#390C2212yes—
36C2×#390C2212yes—
37C2×#390C2212yes—
38C2×#390C2212yes—
39C2×#390C2212yes—
40C2×#390C2211yescharacteristic
41C2×#389C2212yes—
42C2×#386C2214yes—
43C2×#386C2214yes—
44C2×#385C2214yes—
45C2×#382C2211yescharacteristic
46C2×C4×C4⋊C4C2212yes—
47C2×C4×C4⋊C4C2211yescharacteristic
48C2×C4×C4⋊C4C2211yescharacteristic
49C2×C4×C4⋊C4C2211yescharacteristic
50C2×C4×C4⋊C4C2212yes—
51C2×#376C2211yescharacteristic
52#1444C2212yes—
53#1444C2212yes—
54#1444C2212yes—
55#1444C2212yes—
56#1444C2212yes—
57#1442C2212yes—
58#1440C2214yes—
59#1440C2214yes—
60#1439C2214yes—
61#1439C2212yes—
62#1439C2212yes—
63#1439C2214yes—
64#1439C2214yes—
65#1438C2214yes—
66#1438C2214yes—
67#1438C2214yes—
68#1438C2214yes—
69#1437C2214yes—
70#1436C2214yes—
71C4⋊#95C2212yes—
72C4⋊#95C2211yescharacteristic
73C4⋊#95C2212yes—
74C4⋊#95C2211yescharacteristic
75C22⋊#95C2214yes—
76C22⋊#95C2214yes—
77C22⋊#95C2214yes—
78C23×D4C2311yescharacteristic
79C23×D4C2312yes—
80C23×D4C2314yes—
81C23×D4C2314yes—
82C23×D4C2312yes—
83C23×D4C2312yes—
84C24×C4C2314yes—
85C24×C4C2312yes—
86C2×#127—28no—
87C2×#127—28no—
88C2×#127—24no—
89C2×#127—24no—
90C2×#127—44no—
91C2×#127D418yes—
92C2×#123—216no—
93C2×#123—48no—
94C2×#123—48no—
95C2×#121—28no—
96C2×#121—28no—
97C2×#121—48no—
98C2×#121—48no—
99C2×C4×D4—28no—
100C2×C4×D4—28no—
101C2×C4×D4—28no—
102C2×C4×D4—24no—
103C2×C4×D4—24no—
104C2×C4×D4—24no—
105C2×C4×D4—44no—
106C2×C4×D4—48no—
107C22×C4⋊C4C2311yescharacteristic
108C22×C4⋊C4C2311yescharacteristic
109C22×C4⋊C4C2311yescharacteristic
110C22×C4⋊C4C2311yescharacteristic
111C22×C4⋊C4C2312yes—
112C22×C4⋊C4C2311yescharacteristic
113C22×C4⋊C4C2312yes—
114C22×C4⋊C4C2312yes—
115C22×C4⋊C4C2312yes—
116C22×C4⋊C4C2314yes—
117C22×C4⋊C4C2311yescharacteristic
118C22×C4⋊C4C2312yes—
119C22×C4⋊C4C2311yescharacteristic
120C22×C22⋊C4C2312yes—
121C22×C22⋊C4C2314yes—
122C22×C22⋊C4C2314yes—
123C22×C22⋊C4C2314yes—
124C22×C22⋊C4C2314yes—
125C22×C22⋊C4C2314yes—
126C22×C22⋊C4C2314yes—
127C22×C22⋊C4C2314yes—
128C22×C22⋊C4C2314yes—
129C22×C22⋊C4C2314yes—
130C22×C22⋊C4C2314yes—
131C22×C22⋊C4C2314yes—
132C22×C22⋊C4C2314yes—
133C22×C22⋊C4C2314yes—
134C22×C42C2312yes—
135C22×C42C2312yes—
136C22×C42C2311yescharacteristic
137C22×C42C2311yescharacteristic
138C22×C42C2312yes—
139C22×C42C2311yescharacteristic
140#397—48no—
141#394—216no—
142#394—216no—
143#394—48no—
144#394—48no—
145#392—216no—
146#390—28no—
147#390—28no—
148#390—28no—
149#390—28no—
150#390—28no—
151#390—28no—
152#390—24no—
153#390D418yescomplemented
154#389—28no—
155#386—216no—
156#386—216no—
157#385—48no—
158#382—24no—
159C4×C4⋊C4—24no—
160C4×C4⋊C4—24no—
161C4×C4⋊C4—28no—
162C4×C4⋊C4—24no—
163C4×C4⋊C4—44no—
164#376—24no—
165C2×#95C2312yes—
166C2×#95C2312yes—
167C2×#95C2314yes—
168C2×#95C2314yes—
169C2×#95C2312yes—
170C2×#95C2312yes—
171C2×#95C2312yes—
172C2×#95C2312yes—
173C2×#95C2312yes—
174C2×#95C2312yes—
175C2×#95C2312yes—
176C2×#95C2314yes—
177C2×#95C2314yes—
178C2×#95C2314yes—
179C2×#95C2314yes—
180C2×#95C2314yes—
181C2×#95C2312yes—
182C2×#95C2312yes—
183C25C2414yes—
184C25C2412yes—
185C22×D4—28no—
186C22×D4—28no—
187C22×D4—28no—
188C22×D4—28no—
189C22×D4—22no—
190C22×D4—24no—
191C22×D4—22no—
192C22×D4—22no—
193C22×D4—24no—
194C22×D4—24no—
195C22×D4—24no—
196C22×D4—24no—
197C22×D4—22no—
198C22×D4—24no—
199C22×D4—24no—
200C22×D4—24no—
201C22×D4—24no—
202C22×D4—22no—
203C22×D4—28no—
204C22×D4—28no—
205C22×D4—22no—
206C22×D4—24no—
207C22×D4—44no—
208C22×D4—44no—
209C22×D4—44no—
210C22×D4—44no—
211C22×D4—44no—
212C22×D4—44no—
213C22×D4—44no—
214C22×D4—44no—
215C22×D4—44no—
216C22×D4—44no—
217C22×D4—44no—
218C22×D4—44no—
219C22×D4C2×D418yes—
220C22×D4C2×D414yes—
221C22×D4C2×D418yes—
222C22×D4C2×D418yes—
223C23×C4—28no—
224C23×C4—24no—
225C23×C4—28no—
226C23×C4—28no—
227C23×C4—24no—
228C23×C4—24no—
229C23×C4—44no—
230C23×C4—44no—
231C23×C4—44no—
232C23×C4—44no—
233C23×C4—44no—
234C23×C4—44no—
235C23×C4C2412yes—
236C23×C4C2414yes—
237C23×C4C2412yes—
238C23×C4C2412yes—
239C23×C4C2412yes—
240C23×C4C2412yes—
241C23×C4C2412yes—
242C23×C4C2412yes—
243C23×C4C2412yes—
244C23×C4C2412yes—
245C23×C4C2411yescharacteristic
246C23×C4C2411yescharacteristic
247C23×C4C2411yescharacteristic
248#127—232no—
249#127—432no—
250#123—464no—
251#121—432no—
252#121—432no—
253C4×D4—432no—
254C4×D4—432no—
255C4×D4—416no—
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—24no—
264C2×C4⋊C4—24no—
265C2×C4⋊C4—24no—
266C2×C4⋊C4—24no—
267C2×C4⋊C4—22no—
268C2×C4⋊C4—28no—
269C2×C4⋊C4—22no—
270C2×C4⋊C4—22no—
271C2×C4⋊C4—24no—
272C2×C4⋊C4—24no—
273C2×C4⋊C4—22no—
274C2×C4⋊C4—22no—
275C2×C4⋊C4—24no—
276C2×C4⋊C4—24no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—28no—
279C2×C4⋊C4—24no—
280C2×C4⋊C4—24no—
281C2×C4⋊C4—24no—
282C2×C4⋊C4—24no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—42no—
285C2×C4⋊C4—42no—
286C2×C4⋊C4—42no—
287C2×C4⋊C4—44no—
288C2×C4⋊C4—42no—
289C2×C4⋊C4—42no—
290C2×C4⋊C4C2×D414yes—
291C2×C22⋊C4—28no—
292C2×C22⋊C4—28no—
293C2×C22⋊C4—28no—
294C2×C22⋊C4—28no—
295C2×C22⋊C4—28no—
296C2×C22⋊C4—24no—
297C2×C22⋊C4—28no—
298C2×C22⋊C4—28no—
299C2×C22⋊C4—24no—
300C2×C22⋊C4—24no—
301C2×C22⋊C4—28no—
302C2×C22⋊C4—28no—
303C2×C22⋊C4—28no—
304C2×C22⋊C4—28no—
305C2×C22⋊C4—28no—
306C2×C22⋊C4—28no—
307C2×C22⋊C4—28no—
308C2×C22⋊C4—28no—
309C2×C22⋊C4—28no—
310C2×C22⋊C4—28no—
311C2×C22⋊C4—44no—
312C2×C22⋊C4—44no—
313C2×C22⋊C4—44no—
314C2×C22⋊C4—44no—
315C2×C22⋊C4—44no—
316C2×C22⋊C4—44no—
317C2×C22⋊C4—44no—
318C2×C22⋊C4—44no—
319C2×C22⋊C4—44no—
320C2×C22⋊C4—44no—
321C2×C22⋊C4—44no—
322C2×C22⋊C4—44no—
323C2×C22⋊C4—44no—
324C2×C22⋊C4—44no—
325C2×C22⋊C4—44no—
326C2×C22⋊C4—44no—
327C2×C22⋊C4—44no—
328C2×C22⋊C4—44no—
329C2×C22⋊C4—44no—
330C2×C22⋊C4—44no—
331C2×C22⋊C4—44no—
332C2×C22⋊C4C2×D4116yes—
333C2×C42—22no—
334C2×C42—22no—
335C2×C42—22no—
336C2×C42—22no—
337C2×C42—24no—
338C2×C42—24no—
339C2×C42—24no—
340C2×C42—24no—
341C2×C42—24no—
342C2×C42—24no—
343C2×C42—24no—
344C2×C42—44no—
345C2×C42C2×D414yes—
346C2×C42C2×D414yes—
347C2×C42#4114yes—
348C2×C42#4114yes—
349C2×C42#4114yes—
350#95—24no—
351#95—24no—
352#95—24no—
353#95—24no—
354#95—24no—
355#95—28no—
356#95—24no—
357#95—24no—
358#95—24no—
359#95—24no—
360#95—28no—
361#95—42no—
362#95—42no—
363#95—44no—
364#95—44no—
365#95—44no—
366#95—44no—
367#95—44no—
368C24—24no—
369C24—22no—
370C24—22no—
371C24—24no—
372C24—22no—
373C24—22no—
374C24—24no—
375C24—24no—
376C24—24no—
377C24—24no—
378C24—24no—
379C24—24no—
380C24—24no—
381C24—24no—
382C24—24no—
383C24—24no—
384C24—24no—
385C24—24no—
386C24—22no—
387C24—24no—
388C24—22no—
389C24—24no—
390C24—24no—
391C24—24no—
392C24C22×D418yes—
393C24C22×D414yes—
394C24C2511yescharacteristic
395C2×D4—216no—
396C2×D4—216no—
397C2×D4—216no—
398C2×D4—28no—
399C2×D4—216no—
400C2×D4—28no—
401C2×D4—28no—
402C2×D4—216no—
403C2×D4—48no—
404C2×D4—48no—
405C2×D4—416no—
406C2×D4—416no—
407C2×D4—48no—
408C2×D4—48no—
409C2×D4—416no—
410C2×D4—416no—
411C2×D4—416no—
412C2×D4—416no—
413C2×D4—416no—
414C2×D4—416no—
415C2×D4—44no—
416C2×D4—48no—
417C2×D4—44no—
418C2×D4—48no—
419C2×D4—48no—
420C2×D4—44no—
421C2×D4—44no—
422C2×D4—48no—
423C2×D4—88no—
424C2×D4—88no—
425C2×D4—88no—
426C2×D4—88no—
427C2×D4—88no—
428C2×D4—88no—
429C2×D4—88no—
430C2×D4—88no—
431C2×D4—88no—
432C2×D4—88no—
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—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
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—24no—
455C22×C4—24no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—22no—
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—22no—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—22no—
473C22×C4—22no—
474C22×C4—24no—
475C22×C4—22no—
476C22×C4—24no—
477C22×C4—24no—
478C22×C4—22no—
479C22×C4—22no—
480C22×C4—22no—
481C22×C4—24no—
482C22×C4—24no—
483C22×C4—24no—
484C22×C4—24no—
485C22×C4—22no—
486C22×C4—22no—
487C22×C4—24no—
488C22×C4—24no—
489C22×C4—24no—
490C22×C4—22no—
491C22×C4—22no—
492C22×C4—24no—
493C22×C4—22no—
494C22×C4—22no—
495C22×C4—416no—
496C22×C4—416no—
497C22×C4—416no—
498C22×C4—48no—
499C22×C4—48no—
500C22×C4—48no—
501C22×C4—88no—
502C22×C4—88no—
503C22×C4—88no—
504C22×C4—88no—
505C22×C4—88no—
506C22×C4—88no—
507C22×C4C22×D414yes—
508C22×C4C22×D414yes—
509C22×C4C22×D412yes—
510C22×C4C22×D412yes—
511C22×C4C22×D412yes—
512C22×C4C22×D412yes—
513C22×C4C2×#4114yes—
514C22×C4C2×#4112yes—
515C22×C4C2×#4112yes—
516C22×C4C2×#4112yes—
517C22×C4C2×#4112yes—
518C22×C4C2×#4112yes—
519C22×C4C2×#4112yes—
520C22×C4C2×#4112yes—
521C22×C4H(5,𝔽2)14yes—
522C22×C4H(5,𝔽2)14yes—
523C22×C4H(5,𝔽2)14yes—
524C22×C4H(5,𝔽2)12yes—
525C22×C42−1+412yes—
526C4⋊C4—28no—
527C4⋊C4—28no—
528C4⋊C4—28no—
529C4⋊C4—216no—
530C4⋊C4—28no—
531C4⋊C4—48no—
532C4⋊C4—48no—
533C4⋊C4—416no—
534C4⋊C4—44no—
535C4⋊C4—48no—
536C4⋊C4—44no—
537C4⋊C4—48no—
538C4⋊C4—44no—
539C22⋊C4—416no—
540C22⋊C4—416no—
541C22⋊C4—416no—
542C22⋊C4—416no—
543C22⋊C4—416no—
544C22⋊C4—416no—
545C22⋊C4—416no—
546C22⋊C4—416no—
547C22⋊C4—48no—
548C22⋊C4—88no—
549C22⋊C4—88no—
550C22⋊C4—88no—
551C22⋊C4—88no—
552C22⋊C4—88no—
553C42—28no—
554C42—28no—
555C42—48no—
556C42—48no—
557C42—48no—
558C42#123116yescomplemented
559C23—28no—
560C23—24no—
561C23—28no—
562C23—24no—
563C23—28no—
564C23—24no—
565C23—28no—
566C23—28no—
567C23—28no—
568C23—28no—
569C23—28no—
570C23—28no—
571C23—42no—
572C23—44no—
573C23—42no—
574C23—44no—
575C23—44no—
576C23—44no—
577C23—44no—
578C23—42no—
579C23—44no—
580C23—42no—
581C23—44no—
582C23—44no—
583C23—42no—
584C23—42no—
585C23—44no—
586C23—42no—
587C23—42no—
588C23—44no—
589C23—44no—
590C23—44no—
591C23—44no—
592C23—44no—
593C23—44no—
594C23—44no—
595C23—44no—
596C23—44no—
597C23—44no—
598C23—44no—
599C23—44no—
600C23—44no—
601C23—44no—
602C23—44no—
603C23—44no—
604C23—44no—
605C23—44no—
606C23—44no—
607C23—44no—
608C23—44no—
609C23—44no—
610C23—44no—
611C23—44no—
612C23—44no—
613C23—44no—
614C23—44no—
615C23—44no—
616C23—44no—
617C23C23×D411yescharacteristic
618C23C23×D411yescharacteristic
619C23C22×#4111yescharacteristic
620C23C22×#4111yescharacteristic
621C23C22×#4111yescharacteristic
622C23C2×H(5,𝔽2)11yescharacteristic
623C23C2×H(5,𝔽2)12yes—
624C23C2×H(5,𝔽2)12yes—
625C23C2×H(5,𝔽2)12yes—
626C23C2×2−1+411yescharacteristic
627C23#58512yes—
628D4—432no—
629D4—432no—
630D4—416no—
631D4—816no—
632D4—832no—
633D4—832no—
634C2×C4—24no—
635C2×C4—22no—
636C2×C4—22no—
637C2×C4—22no—
638C2×C4—22no—
639C2×C4—24no—
640C2×C4—24no—
641C2×C4—24no—
642C2×C4—24no—
643C2×C4—24no—
644C2×C4—24no—
645C2×C4—24no—
646C2×C4—24no—
647C2×C4—24no—
648C2×C4—24no—
649C2×C4—24no—
650C2×C4—28no—
651C2×C4—24no—
652C2×C4—24no—
653C2×C4—24no—
654C2×C4—24no—
655C2×C4—24no—
656C2×C4—24no—
657C2×C4—24no—
658C2×C4—24no—
659C2×C4—42no—
660C2×C4—42no—
661C2×C4—42no—
662C2×C4—42no—
663C2×C4—42no—
664C2×C4—42no—
665C2×C4—42no—
666C2×C4—42no—
667C2×C4—42no—
668C2×C4—42no—
669C2×C4—42no—
670C2×C4—42no—
671C2×C4—42no—
672C2×C4—42no—
673C2×C4—42no—
674C2×C4—42no—
675C2×C4—42no—
676C2×C4—44no—
677C2×C4—42no—
678C2×C4—42no—
679C2×C4—42no—
680C2×C4—42no—
681C2×C4—42no—
682C2×C4—42no—
683C2×C4—44no—
684C2×C4—42no—
685C2×C4—42no—
686C2×C4—42no—
687C2×C4—44no—
688C2×C4—42no—
689C2×C4—42no—
690C2×C4—42no—
691C2×C4—42no—
692C2×C4—42no—
693C2×C4—44no—
694C2×C4—42no—
695C2×C4—42no—
696C2×C4—44no—
697C2×C4—44no—
698C2×C4—44no—
699C2×C4—44no—
700C2×C4—44no—
701C2×C4—44no—
702C2×C4—44no—
703C2×C4—42no—
704C2×C4—42no—
705C2×C4—44no—
706C2×C4—816no—
707C2×C4—832no—
708C2×C4C2×#12314yes—
709C2×C4C2×#12314yes—
710C2×C4C2×#12314yes—
711C2×C4#53214yes—
712C2×C4#54214yes—
713C2×C4D4214yes—
714C2×C4#54714yes—
715C2×C4#54714yes—
716C2×C4#55314yes—
717C22—48no—
718C22—48no—
719C22—44no—
720C22—48no—
721C22—48no—
722C22—48no—
723C22—44no—
724C22—48no—
725C22—44no—
726C22—48no—
727C22—48no—
728C22—48no—
729C22—84no—
730C22—84no—
731C22—84no—
732C22—82no—
733C22—82no—
734C22—84no—
735C22—84no—
736C22—82no—
737C22—84no—
738C22—84no—
739C22—84no—
740C22—84no—
741C22—84no—
742C22—82no—
743C22C22×#12311yescharacteristic
744C22C2×#53211yescharacteristic
745C22C2×#54211yescharacteristic
746C22#346012yes—
747C22C2×D4211yescharacteristic
748C22C2×#54711yescharacteristic
749C22C2×#54711yescharacteristic
750C22C2×#55311yescharacteristic
751C22#348411yescharacteristic
752C22#348411yescharacteristic
753C22#348412yes—
754C22#348411yescharacteristic
755C22#348511yescharacteristic
756C22#348812yes—
757C22#350612yes—
758C22#350612yes—
759C22#350812yes—
760C22#350812yes—
761C22#350812yes—
762C22#353712yes—
763C22#354212yes—
764C22#354812yes—
765C22#355412yes—
766C4—24no—
767C4—24no—
768C4—44no—
769C4—44no—
770C4—44no—
771C4—44no—
772C4—44no—
773C4—44no—
774C4—84no—
775C4—82no—
776C4—82no—
777C4—82no—
778C4#242218yes—
779C2—88no—
780C2—84no—
781C2C2×#242211yescharacteristic
782C2#3515211yescharacteristic
783C2#3555411yescharacteristic
784C2#3593012yes—
785C2#3779212yes—
786C2#3981812yes—
787C2#4252412yes—
788C2#4493111yescharacteristic
789C2#4493211yescharacteristic
790C2#4535411yescharacteristic
791C2#4535511yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC2×#2541C2
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×#2445C2semidirect
3. C2×#2366C2semidirect
4. #16526C2semidirect
5. #15312C2semidirect
6. #15260C2semidirect
7. #15147C2semidirect
8. #15127C2semidirect
9. #14721C2semidirect
10. #14272C2semidirect
11. #14150C2semidirect
12. #14018C2semidirect
13. #13941C2semidirect
14. #13856C2semidirect
15. #13785C2semidirect
153. #390D4semidirect
558. C42#123semidirect

Maximal subgroups

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

subgroupindex
C2×#25412
C2×#24452
C2×#23662
#165262
#153122
#152602
#151472
#151272
#147212
#142722
#141502
#140182
#139412
#138562
#137852