Finite Group Databasebeta

← #719245

Subgroups — #719245

Order 512.

3248 conjugacy classes of subgroups.

The subgroup lattice has 776 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2422C212yesmaximal
2C2×D4×C4⋊C4C211yesmaximal, characteristic, complemented
3C2×#2324C211yesmaximal, characteristic
4#16816C212yesmaximal, complemented
5#16815C212yesmaximal, complemented
6#15398C214yesmaximal, complemented
7#15271C214yesmaximal, complemented
8#15183C212yesmaximal, complemented
9#15137C212yesmaximal, complemented
10#15069C212yesmaximal, complemented
11#14786C212yesmaximal, complemented
12#14340C211yesmaximal, characteristic, complemented
13#14150C211yesmaximal, characteristic, complemented
14#13941C211yesmaximal, characteristic, complemented
15#13916C212yesmaximal, complemented
16#13910C211yesmaximal, characteristic, complemented
17#13852C211yesmaximal, characteristic, complemented
18C22×#123C2214yes—
19C22×#121C2211yescharacteristic
20C22×C4×D4C2212yes—
21C22×C4×D4C2211yescharacteristic
22C22×#117C2211yescharacteristic
23C23×C4⋊C4C2212yes—
24#2422—216no—
25C2×#398C2214yes—
26C2×#397C2214yes—
27C2×#394C2214yes—
28D4×C4⋊C4—28no—
29#2324—28no—
30C2×#390C2211yescharacteristic
31C2×#390C2212yes—
32C2×#390C2212yes—
33C2×#389C2211yescharacteristic
34C2×#389C2212yes—
35C2×#389C2212yes—
36C2×#389C2214yes—
37C2×#388C2212yes—
38C2×#386C2212yes—
39C2×#385C2212yes—
40C2×#385C2214yes—
41C2×#385C2214yes—
42C2×#384C2211yescharacteristic
43C2×#384C2212yes—
44C2×#382C2211yescharacteristic
45C2×#381C2214yes—
46C2×#380C2212yes—
47C2×#380C2211yescharacteristic
48C2×#380C2212yes—
49C2×C4×C4⋊C4C2211yescharacteristic
50C2×C4×C4⋊C4C2211yescharacteristic
51C2×C4×C4⋊C4C2211yescharacteristic
52C2×C4×C22⋊C4C2212yes—
53#1445C2214yes—
54#1444C2214yes—
55#1444C2212yes—
56#1444C2212yes—
57#1444C2212yes—
58#1443C2214yes—
59#1442C2214yes—
60#1442C2212yes—
61#1441C2212yes—
62#1441C2212yes—
63#1441C2212yes—
64#1441C2211yescharacteristic
65#1441C2211yescharacteristic
66#1441C2214yes—
67#1440C2214yes—
68#1440C2214yes—
69#1440C2214yes—
70#1439C2212yes—
71#1439C2212yes—
72#1439C2212yes—
73#1439C2211yescharacteristic
74#1439C2211yescharacteristic
75#1439C2214yes—
76#1439C2214yes—
77#1436C2214yes—
78C4⋊#95C2212yes—
79C4⋊#95C2212yes—
80C4⋊#95C2212yes—
81C4⋊#95C2212yes—
82C22⋊#95C2214yes—
83C22⋊#95C2212yes—
84C22⋊#95C2212yes—
85C4×#95C2212yes—
86C23×D4C2311yescharacteristic
87C23×D4C2312yes—
88C24×C4C2311yescharacteristic
89C24×C4C2314yes—
90C24×C4C2312yes—
91C2×#123—216no—
92C2×#123—48no—
93C2×#123—48no—
94C2×#121—24no—
95C2×#121—24no—
96C2×#121D418yes—
97C2×C4×D4—28no—
98C2×C4×D4—28no—
99C2×C4×D4—28no—
100C2×C4×D4—24no—
101C2×C4×D4—24no—
102C2×C4×D4D418yes—
103C2×#117—24no—
104C2×#117—24no—
105C2×#117—24no—
106C22×C4⋊C4—28no—
107C22×C4⋊C4—48no—
108C22×C4⋊C4C2311yescharacteristic
109C22×C4⋊C4C2311yescharacteristic
110C22×C4⋊C4C2311yescharacteristic
111C22×C4⋊C4C2312yes—
112C22×C4⋊C4C2311yescharacteristic
113C22×C4⋊C4C2312yes—
114C22×C4⋊C4C2312yes—
115C22×C4⋊C4C2314yes—
116C22×C4⋊C4C2314yes—
117C22×C4⋊C4C2314yes—
118C22×C4⋊C4C2311yescharacteristic
119C22×C4⋊C4C2311yescharacteristic
120C22×C4⋊C4C2314yes—
121C22×C4⋊C4C2312yes—
122C22×C4⋊C4C2312yes—
123C22×C4⋊C4C2311yescharacteristic
124C22×C4⋊C4C2312yes—
125C22×C22⋊C4C2314yes—
126C22×C22⋊C4C2311yescharacteristic
127C22×C22⋊C4C2311yescharacteristic
128C22×C22⋊C4C2312yes—
129C22×C22⋊C4C2312yes—
130C22×C22⋊C4C2314yes—
131C22×C22⋊C4C2314yes—
132C22×C22⋊C4C2312yes—
133C22×C22⋊C4C2312yes—
134C22×C22⋊C4C2314yes—
135C22×C22⋊C4C2312yes—
136C22×C22⋊C4C2312yes—
137C22×C22⋊C4C2312yes—
138C22×C42C2311yescharacteristic
139C22×C42C2312yes—
140C22×C42C2312yes—
141C22×C42C2312yes—
142C22×C42C2311yescharacteristic
143#398—48no—
144#397—216no—
145#394—216no—
146#390—24no—
147#390—28no—
148#390—28no—
149#389—24no—
150#389—28no—
151#389—28no—
152#389—48no—
153#388—28no—
154#386—28no—
155#385—28no—
156#385—216no—
157#385—216no—
158#384—28no—
159#384—24no—
160#382—24no—
161#381—48no—
162#380—28no—
163#380—28no—
164#380—24no—
165C4×C4⋊C4—24no—
166C4×C4⋊C4—24no—
167C4×C4⋊C4—24no—
168C4×C22⋊C4—28no—
169C2×#95C2314yes—
170C2×#95C2312yes—
171C2×#95C2314yes—
172C2×#95C2312yes—
173C2×#95C2312yes—
174C2×#95C2312yes—
175C2×#95C2312yes—
176C2×#95C2311yescharacteristic
177C2×#95C2311yescharacteristic
178C2×#95C2314yes—
179C2×#95C2312yes—
180C2×#95C2312yes—
181C2×#95C2314yes—
182C2×#95C2312yes—
183C2×#95C2312yes—
184C2×#95C2312yes—
185C2×#95C2312yes—
186C2×#95C2314yes—
187C2×#95C2314yes—
188C2×#95C2312yes—
189C2×#95C2314yes—
190C2×#95C2314yes—
191C2×#95C2314yes—
192C2×#95C2314yes—
193C2×#95C2314yes—
194C25C2411yescharacteristic
195C25C2412yes—
196C22×D4—24no—
197C22×D4—24no—
198C22×D4—22no—
199C22×D4—22no—
200C22×D4—22no—
201C22×D4—22no—
202C22×D4—44no—
203C22×D4—44no—
204C22×D4C2×D414yes—
205C22×D4C2×D414yes—
206C22×D4C2×D418yes—
207C22×D4#4114yes—
208C23×C4—22no—
209C23×C4—22no—
210C23×C4—22no—
211C23×C4—24no—
212C23×C4—24no—
213C23×C4—22no—
214C23×C4—22no—
215C23×C4—22no—
216C23×C4—28no—
217C23×C4—28no—
218C23×C4—28no—
219C23×C4—28no—
220C23×C4—44no—
221C23×C4—44no—
222C23×C4—44no—
223C23×C4—44no—
224C23×C4—44no—
225C23×C4C2×D414yes—
226C23×C4C2×D418yes—
227C23×C4C2411yescharacteristic
228C23×C4C2414yes—
229C23×C4C2412yes—
230C23×C4C2411yescharacteristic
231C23×C4C2412yes—
232C23×C4C2414yes—
233C23×C4C2412yes—
234C23×C4C2411yescharacteristic
235C23×C4C2412yes—
236C23×C4C2412yes—
237C23×C4C2412yes—
238C23×C4C2412yes—
239C23×C4C2411yescharacteristic
240C23×C4C2411yescharacteristic
241C23×C4C2411yescharacteristic
242#123—464no—
243#121—232no—
244C4×D4—232no—
245C4×D4—432no—
246#117—416no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—28no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4—22no—
260C2×C4⋊C4—24no—
261C2×C4⋊C4—24no—
262C2×C4⋊C4—28no—
263C2×C4⋊C4—28no—
264C2×C4⋊C4—22no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4—24no—
267C2×C4⋊C4—22no—
268C2×C4⋊C4—22no—
269C2×C4⋊C4—22no—
270C2×C4⋊C4—22no—
271C2×C4⋊C4—24no—
272C2×C4⋊C4—24no—
273C2×C4⋊C4—28no—
274C2×C4⋊C4—28no—
275C2×C4⋊C4—28no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—24no—
279C2×C4⋊C4—24no—
280C2×C4⋊C4—28no—
281C2×C4⋊C4—24no—
282C2×C4⋊C4—432no—
283C2×C4⋊C4—44no—
284C2×C4⋊C4—44no—
285C2×C4⋊C4—42no—
286C2×C4⋊C4—44no—
287C2×C4⋊C4—44no—
288C2×C4⋊C4—44no—
289C2×C4⋊C4—44no—
290C2×C4⋊C4—44no—
291C2×C4⋊C4—44no—
292C2×C4⋊C4C2×D414yes—
293C2×C4⋊C4C2×D414yes—
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—28no—
302C2×C22⋊C4—28no—
303C2×C22⋊C4—28no—
304C2×C22⋊C4—28no—
305C2×C22⋊C4—24no—
306C2×C22⋊C4—28no—
307C2×C22⋊C4—28no—
308C2×C22⋊C4—22no—
309C2×C22⋊C4—22no—
310C2×C22⋊C4—22no—
311C2×C22⋊C4—22no—
312C2×C22⋊C4—22no—
313C2×C22⋊C4—22no—
314C2×C22⋊C4—28no—
315C2×C22⋊C4—28no—
316C2×C22⋊C4—24no—
317C2×C22⋊C4—24no—
318C2×C22⋊C4—24no—
319C2×C22⋊C4—24no—
320C2×C22⋊C4—44no—
321C2×C22⋊C4—42no—
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⋊C4C2×D418yes—
329C2×C22⋊C4C2×D418yes—
330C2×C42—22no—
331C2×C42—24no—
332C2×C42—24no—
333C2×C42—24no—
334C2×C42—22no—
335C2×C42—22no—
336C2×C42—24no—
337C2×C42—24no—
338C2×C42—24no—
339C2×C42—24no—
340C2×C42—44no—
341C2×C42C2×D414yes—
342C2×C42#4114yes—
343C2×C42#4114yes—
344#95—24no—
345#95—24no—
346#95—24no—
347#95—24no—
348#95—24no—
349#95—28no—
350#95—28no—
351#95—24no—
352#95—28no—
353#95—22no—
354#95—22no—
355#95—24no—
356#95—24no—
357#95—24no—
358#95—28no—
359#95—24no—
360#95—24no—
361#95—44no—
362#95—42no—
363#95—44no—
364#95—44no—
365#95—44no—
366#95—44no—
367#95—44no—
368#95—44no—
369C24—22no—
370C24—22no—
371C24—24no—
372C24—24no—
373C24—24no—
374C24—22no—
375C24—22no—
376C24—24no—
377C24—21no—
378C24—21no—
379C24—21no—
380C24—21no—
381C24—22no—
382C24—22no—
383C24—22no—
384C24—21no—
385C24—21no—
386C24—22no—
387C24C22×D412yes—
388C24C22×D414yes—
389C24C22×D414yes—
390C24C2×#4114yes—
391C24C2511yescharacteristic
392C2×D4—216no—
393C2×D4—28no—
394C2×D4—28no—
395C2×D4—28no—
396C2×D4—28no—
397C2×D4—28no—
398C2×D4—28no—
399C2×D4—416no—
400C2×D4—84no—
401C2×D4—84no—
402C2×D4—88no—
403C2×D4#121116yes—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—21no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—21no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—21no—
422C22×C4—21no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—24no—
427C22×C4—24no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—24no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—21no—
436C22×C4—21no—
437C22×C4—22no—
438C22×C4—24no—
439C22×C4—22no—
440C22×C4—28no—
441C22×C4—216no—
442C22×C4—28no—
443C22×C4—28no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—21no—
447C22×C4—22no—
448C22×C4—24no—
449C22×C4—24no—
450C22×C4—24no—
451C22×C4—21no—
452C22×C4—24no—
453C22×C4—22no—
454C22×C4—21no—
455C22×C4—21no—
456C22×C4—21no—
457C22×C4—21no—
458C22×C4—24no—
459C22×C4—24no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—24no—
463C22×C4—24no—
464C22×C4—24no—
465C22×C4—24no—
466C22×C4—24no—
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—24no—
476C22×C4—22no—
477C22×C4—22no—
478C22×C4—24no—
479C22×C4—22no—
480C22×C4—48no—
481C22×C4—48no—
482C22×C4—416no—
483C22×C4—416no—
484C22×C4—416no—
485C22×C4—416no—
486C22×C4—416no—
487C22×C4—44no—
488C22×C4—44no—
489C22×C4—44no—
490C22×C4—44no—
491C22×C4—416no—
492C22×C4—88no—
493C22×C4—88no—
494C22×C4—88no—
495C22×C4C22×D412yes—
496C22×C4C22×D414yes—
497C22×C4C22×D414yes—
498C22×C4C22×D412yes—
499C22×C4C22×D412yes—
500C22×C4C22×D412yes—
501C22×C4C22×D412yes—
502C22×C4C2×#4112yes—
503C22×C4C2×#4112yes—
504C22×C4C2×#4112yes—
505C22×C4C2×#4112yes—
506C22×C4C2×#4112yes—
507C22×C4C2×#4112yes—
508C22×C4C2×#4112yes—
509C22×C4H(5,𝔽2)14yes—
510C22×C4H(5,𝔽2)14yes—
511C22×C42−1+414yes—
512C22×C42−1+412yes—
513C22×C42−1+412yes—
514C4⋊C4—216no—
515C4⋊C4—28no—
516C4⋊C4—28no—
517C4⋊C4—28no—
518C4⋊C4—416no—
519C4⋊C4—416no—
520C4⋊C4—416no—
521C4⋊C4—44no—
522C4⋊C4—48no—
523C4⋊C4—48no—
524C4⋊C4—44no—
525C4⋊C4—44no—
526C4⋊C4—44no—
527C4⋊C4—48no—
528C4⋊C4—84no—
529C4⋊C4—84no—
530C4⋊C4—88no—
531C22⋊C4—216no—
532C22⋊C4—216no—
533C22⋊C4—216no—
534C22⋊C4—416no—
535C22⋊C4—48no—
536C22⋊C4—416no—
537C22⋊C4—416no—
538C22⋊C4—416no—
539C22⋊C4—44no—
540C22⋊C4—44no—
541C22⋊C4—84no—
542C22⋊C4—84no—
543C22⋊C4—84no—
544C42—48no—
545C42—44no—
546C42—48no—
547C42—84no—
548C42#123116yescomplemented
549C23—24no—
550C23—24no—
551C23—24no—
552C23—24no—
553C23—24no—
554C23—24no—
555C23—28no—
556C23—24no—
557C23—24no—
558C23—28no—
559C23—24no—
560C23—24no—
561C23—28no—
562C23—28no—
563C23—22no—
564C23—22no—
565C23—22no—
566C23—24no—
567C23—24no—
568C23—44no—
569C23—44no—
570C23—44no—
571C23—44no—
572C23—44no—
573C23—44no—
574C23—44no—
575C23—44no—
576C23—41no—
577C23—41no—
578C23—42no—
579C23—41no—
580C23—41no—
581C23—41no—
582C23—41no—
583C23—42no—
584C23—41no—
585C23—41no—
586C23—42no—
587C23—42no—
588C23—42no—
589C23—42no—
590C23—42no—
591C23—42no—
592C23—42no—
593C23—42no—
594C23C2×#12118yes—
595C23C23×D411yescharacteristic
596C23C23×D411yescharacteristic
597C23C22×#4111yescharacteristic
598C23C22×#4111yescharacteristic
599C23C22×#4111yescharacteristic
600C23C2×H(5,𝔽2)12yes—
601C23C2×H(5,𝔽2)12yes—
602C23C2×2−1+411yescharacteristic
603C23C2×2−1+411yescharacteristic
604C23C2×2−1+412yes—
605C23#58512yes—
606D4—232no—
607D4—816no—
608C2×C4—24no—
609C2×C4—22no—
610C2×C4—22no—
611C2×C4—24no—
612C2×C4—22no—
613C2×C4—22no—
614C2×C4—24no—
615C2×C4—24no—
616C2×C4—22no—
617C2×C4—24no—
618C2×C4—24no—
619C2×C4—22no—
620C2×C4—22no—
621C2×C4—24no—
622C2×C4—24no—
623C2×C4—24no—
624C2×C4—24no—
625C2×C4—28no—
626C2×C4—24no—
627C2×C4—24no—
628C2×C4—24no—
629C2×C4—24no—
630C2×C4—24no—
631C2×C4—24no—
632C2×C4—24no—
633C2×C4—432no—
634C2×C4—42no—
635C2×C4—41no—
636C2×C4—41no—
637C2×C4—44no—
638C2×C4—42no—
639C2×C4—42no—
640C2×C4—41no—
641C2×C4—41no—
642C2×C4—42no—
643C2×C4—42no—
644C2×C4—42no—
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—416no—
656C2×C4—42no—
657C2×C4—44no—
658C2×C4—41no—
659C2×C4—41no—
660C2×C4—42no—
661C2×C4—42no—
662C2×C4—44no—
663C2×C4—44no—
664C2×C4—44no—
665C2×C4—44no—
666C2×C4—44no—
667C2×C4—41no—
668C2×C4—44no—
669C2×C4—41no—
670C2×C4—42no—
671C2×C4—42no—
672C2×C4—44no—
673C2×C4—42no—
674C2×C4—42no—
675C2×C4—42no—
676C2×C4—42no—
677C2×C4—41no—
678C2×C4—44no—
679C2×C4—42no—
680C2×C4—41no—
681C2×C4—42no—
682C2×C4—44no—
683C2×C4—42no—
684C2×C4—42no—
685C2×C4—42no—
686C2×C4—44no—
687C2×C4—44no—
688C2×C4—44no—
689C2×C4—44no—
690C2×C4—44no—
691C2×C4—44no—
692C2×C4—832no—
693C2×C4C2×#12114yes—
694C2×C4C2×#12314yes—
695C2×C4C2×#12314yes—
696C2×C4C2×#12314yes—
697C2×C4#53614yes—
698C2×C4#54714yes—
699C2×C4#54714yes—
700C2×C4#55214yes—
701C2×C4#55214yes—
702C22—28no—
703C22—28no—
704C22—28no—
705C22—44no—
706C22—48no—
707C22—48no—
708C22—44no—
709C22—48no—
710C22—44no—
711C22—48no—
712C22—44no—
713C22—42no—
714C22—44no—
715C22—42no—
716C22—42no—
717C22—44no—
718C22—81no—
719C22—82no—
720C22—82no—
721C22—81no—
722C22—81no—
723C22—81no—
724C22C22×#12111yescharacteristic
725C22C22×#12311yescharacteristic
726C22C2×#53611yescharacteristic
727C22#346012yes—
728C22C2×#54711yescharacteristic
729C22C2×#54711yescharacteristic
730C22C2×#55211yescharacteristic
731C22C2×#55211yescharacteristic
732C22#348411yescharacteristic
733C22#348412yes—
734C22#348511yescharacteristic
735C22#348912yes—
736C22#349212yes—
737C22#349811yescharacteristic
738C22#349911yescharacteristic
739C22#350612yes—
740C22#351112yes—
741C22#351512yes—
742C22#351912yes—
743C22#355012yes—
744C22#355212yes—
745C22#355412yes—
746C22#355912yes—
747C4—24no—
748C4—24no—
749C4—44no—
750C4—44no—
751C4—44no—
752C4—42no—
753C4—44no—
754C4—44no—
755C4—81no—
756C4—84no—
757C4—82no—
758C4—84no—
759C4—82no—
760C4—81no—
761C4#248618yes—
762C2—48no—
763C2—82no—
764C2C2×#248611yescharacteristic
765C2#3515211yescharacteristic
766C2#3553011yescharacteristic
767C2#3569111yescharacteristic
768C2#3602011yescharacteristic
769C2#3779412yes—
770C2#3827412yes—
771C2#4242711yescharacteristic
772C2#4265111yescharacteristic
773C2#4491012yes—
774C2#4525512yes—
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×D4×C4⋊C4C2semidirect
4. #16816C2semidirect
5. #16815C2semidirect
6. #15398C2semidirect
7. #15271C2semidirect
8. #15183C2semidirect
9. #15137C2semidirect
10. #15069C2semidirect
11. #14786C2semidirect
12. #14340C2semidirect
13. #14150C2semidirect
14. #13941C2semidirect
15. #13916C2semidirect
16. #13910C2semidirect
17. #13852C2semidirect
548. C42#123semidirect

Maximal subgroups

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

subgroupindex
C2×#24222
C2×D4×C4⋊C42
C2×#23242
#168162
#168152
#153982
#152712
#151832
#151372
#150692
#147862
#143402
#141502
#139412
#139162
#139102
#138522