Finite Group Databasebeta

← #803735

Subgroups — #803735

Order 512.

3568 conjugacy classes of subgroups.

The subgroup lattice has 774 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2447C212yesmaximal, complemented
2C2×#2427C211yesmaximal, characteristic
3C2×#2422C212yesmaximal
4C2×D4×C4⋊C4C211yesmaximal, characteristic, complemented
5#17059C212yesmaximal, complemented
6#15539C214yesmaximal, complemented
7#15384C214yesmaximal, complemented
8#15199C211yesmaximal, characteristic, complemented
9#15193C211yesmaximal, characteristic, complemented
10#15135C211yesmaximal, characteristic, complemented
11#15069C212yesmaximal, complemented
12#14839C212yesmaximal, complemented
13#13997C212yesmaximal, complemented
14#13910C212yesmaximal, complemented
15#13880C212yesmaximal, complemented
16C4×#399C211yesmaximal, characteristic, complemented
17#13820C211yesmaximal, characteristic, complemented
18C22×#124C2211yescharacteristic
19C22×#123C2214yes—
20C22×C22⋊Q8C2211yescharacteristic
21C22×#121C2211yescharacteristic
22C22×C4×D4C2212yes—
23C22×C4×D4C2211yescharacteristic
24C23×C4⋊C4C2212yes—
25#2447—216no—
26#2427—28no—
27#2422—216no—
28C2×#399C2211yescharacteristic
29C2×#399C2212yes—
30C2×#397C2214yes—
31C2×#397C2214yes—
32C2×#397C2212yes—
33C2×#397C2214yes—
34C2×#394C2212yes—
35C2×#394C2214yes—
36D4×C4⋊C4—28no—
37C2×#391C2211yescharacteristic
38C2×#391C2212yes—
39C2×#391C2212yes—
40C2×#390C2212yes—
41C2×#389C2212yes—
42C2×#389C2212yes—
43C2×#389C2211yescharacteristic
44C2×#389C2212yes—
45C2×#386C2212yes—
46C2×#385C2214yes—
47C2×#385C2214yes—
48C2×#385C2214yes—
49C2×#384C2211yescharacteristic
50C2×#384C2212yes—
51C2×#384C2214yes—
52C2×#381C2214yes—
53C2×#380C2212yes—
54C2×#380C2212yes—
55C2×#380C2212yes—
56C2×C4×C4⋊C4C2211yescharacteristic
57C2×C4×C4⋊C4C2212yes—
58C2×C4×C4⋊C4C2212yes—
59C2×C4×C22⋊C4C2211yescharacteristic
60C2×C4×C22⋊C4C2212yes—
61C2×C4×C22⋊C4C2212yes—
62#1445C2214yes—
63#1445C2212yes—
64#1445C2211yescharacteristic
65#1445C2212yes—
66#1445C2212yes—
67#1445C2214yes—
68#1444C2214yes—
69#1444C2214yes—
70#1444C2212yes—
71#1444C2211yescharacteristic
72#1444C2212yes—
73#1444C2212yes—
74#1444C2214yes—
75#1444C2212yes—
76#1439C2212yes—
77#1439C2214yes—
78#1439C2212yes—
79#1439C2211yescharacteristic
80#1437C2214yes—
81#1437C2214yes—
82C4⋊#95C2211yescharacteristic
83C4⋊#95C2214yes—
84C4⋊#95C2214yes—
85C22⋊#95C2212yes—
86C4×#95C2211yescharacteristic
87C23×Q8C2311yescharacteristic
88C23×D4C2312yes—
89C23×D4C2311yescharacteristic
90C24×C4C2311yescharacteristic
91C24×C4C2314yes—
92C24×C4C2312yes—
93C2×#124—24no—
94C2×#124—28no—
95C2×#123—216no—
96C2×#123—48no—
97C2×#123—48no—
98C2×C22⋊Q8—24no—
99C2×C22⋊Q8—28no—
100C2×#121—28no—
101C2×#121—24no—
102C2×C4×D4—28no—
103C2×C4×D4—28no—
104C2×C4×D4—28no—
105C2×C4×D4—28no—
106C2×C4×D4D418yes—
107C22×C4⋊C4—28no—
108C22×C4⋊C4—48no—
109C22×C4⋊C4C2312yes—
110C22×C4⋊C4C2312yes—
111C22×C4⋊C4C2311yescharacteristic
112C22×C4⋊C4C2311yescharacteristic
113C22×C4⋊C4C2312yes—
114C22×C4⋊C4C2311yescharacteristic
115C22×C4⋊C4C2314yes—
116C22×C4⋊C4C2314yes—
117C22×C4⋊C4C2312yes—
118C22×C4⋊C4C2312yes—
119C22×C4⋊C4C2314yes—
120C22×C4⋊C4C2312yes—
121C22×C4⋊C4C2314yes—
122C22×C4⋊C4C2314yes—
123C22×C4⋊C4C2312yes—
124C22×C4⋊C4C2312yes—
125C22×C4⋊C4C2311yescharacteristic
126C22×C22⋊C4C2312yes—
127C22×C22⋊C4C2314yes—
128C22×C22⋊C4C2312yes—
129C22×C22⋊C4C2312yes—
130C22×C22⋊C4C2312yes—
131C22×C22⋊C4C2312yes—
132C22×C22⋊C4C2311yescharacteristic
133C22×C22⋊C4C2312yes—
134C22×C22⋊C4C2311yescharacteristic
135C22×C22⋊C4C2314yes—
136C22×C22⋊C4C2312yes—
137C22×C22⋊C4C2314yes—
138C22×C22⋊C4C2314yes—
139C22×C42C2312yes—
140C22×C42C2311yescharacteristic
141C22×C42C2314yes—
142C22×C42C2311yescharacteristic
143C22×C42C2312yes—
144#399—24no—
145#399—44no—
146#397—216no—
147#397—28no—
148#397—216no—
149#397—216no—
150#394—216no—
151#394—28no—
152#391—28no—
153#391—24no—
154#391—28no—
155#390D4116yes—
156#389—28no—
157#389—28no—
158#389—24no—
159#389—28no—
160#386—28no—
161#385—216no—
162#385—216no—
163#385—216no—
164#384—24no—
165#384—28no—
166#384—48no—
167#381—216no—
168#380—28no—
169#380—28no—
170#380—28no—
171C4×C4⋊C4—28no—
172C4×C4⋊C4—24no—
173C4×C4⋊C4—28no—
174C4×C22⋊C4—28no—
175C4×C22⋊C4—24no—
176C4×C22⋊C4—28no—
177C2×#95C2314yes—
178C2×#95C2312yes—
179C2×#95C2314yes—
180C2×#95C2312yes—
181C2×#95C2312yes—
182C2×#95C2311yescharacteristic
183C2×#95C2312yes—
184C2×#95C2314yes—
185C2×#95C2314yes—
186C2×#95C2314yes—
187C2×#95C2314yes—
188C2×#95C2311yescharacteristic
189C2×#95C2312yes—
190C2×#95C2312yes—
191C2×#95C2314yes—
192C2×#95C2312yes—
193C2×#95C2314yes—
194C2×#95C2314yes—
195C2×#95C2312yes—
196C2×#95C2312yes—
197C2×#95C2311yescharacteristic
198C2×#95C2312yes—
199C2×#95C2312yes—
200C2×#95C2311yescharacteristic
201C25C2411yescharacteristic
202C25C2412yes—
203C22×Q8—22no—
204C22×Q8—24no—
205C22×Q8—24no—
206C22×Q8—22no—
207C22×Q8—22no—
208C22×D4—24no—
209C22×D4—28no—
210C22×D4—24no—
211C22×D4—24no—
212C22×D4—22no—
213C22×D4—22no—
214C22×D4—44no—
215C22×D4—42no—
216C22×D4C2×D414yes—
217C22×D4C2×D418yes—
218C23×C4—28no—
219C23×C4—24no—
220C23×C4—24no—
221C23×C4—28no—
222C23×C4—28no—
223C23×C4—28no—
224C23×C4—22no—
225C23×C4—24no—
226C23×C4—24no—
227C23×C4—28no—
228C23×C4—22no—
229C23×C4—22no—
230C23×C4—44no—
231C23×C4—44no—
232C23×C4—44no—
233C23×C4—44no—
234C23×C4C2×D418yes—
235C23×C4C2411yescharacteristic
236C23×C4C2412yes—
237C23×C4C2411yescharacteristic
238C23×C4C2412yes—
239C23×C4C2414yes—
240C23×C4C2411yescharacteristic
241C23×C4C2412yes—
242C23×C4C2414yes—
243C23×C4C2412yes—
244C23×C4C2411yescharacteristic
245C23×C4C2412yes—
246C23×C4C2412yes—
247C23×C4C2412yes—
248C23×C4C2411yescharacteristic
249C23×C4C2411yescharacteristic
250#124—416no—
251#123—464no—
252C22⋊Q8—416no—
253#121—416no—
254C4×D4—232no—
255C4×D4—432no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4—24no—
258C2×C4⋊C4—24no—
259C2×C4⋊C4—24no—
260C2×C4⋊C4—24no—
261C2×C4⋊C4—28no—
262C2×C4⋊C4—24no—
263C2×C4⋊C4—24no—
264C2×C4⋊C4—24no—
265C2×C4⋊C4—24no—
266C2×C4⋊C4—22no—
267C2×C4⋊C4—24no—
268C2×C4⋊C4—28no—
269C2×C4⋊C4—28no—
270C2×C4⋊C4—22no—
271C2×C4⋊C4—24no—
272C2×C4⋊C4—24no—
273C2×C4⋊C4—28no—
274C2×C4⋊C4—22no—
275C2×C4⋊C4—24no—
276C2×C4⋊C4—28no—
277C2×C4⋊C4—28no—
278C2×C4⋊C4—24no—
279C2×C4⋊C4—24no—
280C2×C4⋊C4—22no—
281C2×C4⋊C4—24no—
282C2×C4⋊C4—24no—
283C2×C4⋊C4—28no—
284C2×C4⋊C4—28no—
285C2×C4⋊C4—432no—
286C2×C4⋊C4—44no—
287C2×C4⋊C4—44no—
288C2×C4⋊C4—42no—
289C2×C4⋊C4—44no—
290C2×C4⋊C4—44no—
291C2×C4⋊C4—44no—
292C2×C4⋊C4—44no—
293C2×C4⋊C4—42no—
294C2×C4⋊C4—44no—
295C2×C4⋊C4—44no—
296C2×C4⋊C4—44no—
297C2×C4⋊C4—42no—
298C2×C4⋊C4—44no—
299C2×C4⋊C4C2×D418yes—
300C2×C4⋊C4C2×D414yes—
301C2×C4⋊C4#4118yes—
302C2×C22⋊C4—24no—
303C2×C22⋊C4—24no—
304C2×C22⋊C4—28no—
305C2×C22⋊C4—28no—
306C2×C22⋊C4—24no—
307C2×C22⋊C4—28no—
308C2×C22⋊C4—28no—
309C2×C22⋊C4—28no—
310C2×C22⋊C4—28no—
311C2×C22⋊C4—24no—
312C2×C22⋊C4—28no—
313C2×C22⋊C4—24no—
314C2×C22⋊C4—28no—
315C2×C22⋊C4—28no—
316C2×C22⋊C4—24no—
317C2×C22⋊C4—28no—
318C2×C22⋊C4—22no—
319C2×C22⋊C4—24no—
320C2×C22⋊C4—28no—
321C2×C22⋊C4—22no—
322C2×C22⋊C4—22no—
323C2×C22⋊C4—24no—
324C2×C22⋊C4—28no—
325C2×C22⋊C4—22no—
326C2×C22⋊C4—24no—
327C2×C22⋊C4—24no—
328C2×C22⋊C4—22no—
329C2×C22⋊C4—41no—
330C2×C22⋊C4—44no—
331C2×C22⋊C4—44no—
332C2×C22⋊C4—44no—
333C2×C22⋊C4C2×D418yes—
334C2×C22⋊C4C2×D4116yes—
335C2×C22⋊C4C2×D4116yes—
336C2×C42—24no—
337C2×C42—24no—
338C2×C42—24no—
339C2×C42—24no—
340C2×C42—24no—
341C2×C42—24no—
342C2×C42—22no—
343C2×C42—24no—
344C2×C42—28no—
345C2×C42—28no—
346C2×C42—44no—
347C2×C42C2×D414yes—
348C2×C42C2×D418yes—
349#95—22no—
350#95—24no—
351#95—24no—
352#95—22no—
353#95—24no—
354#95—24no—
355#95—24no—
356#95—28no—
357#95—28no—
358#95—24no—
359#95—28no—
360#95—28no—
361#95—28no—
362#95—28no—
363#95—24no—
364#95—28no—
365#95—24no—
366#95—28no—
367#95—24no—
368#95—28no—
369#95—24no—
370#95—22no—
371#95—42no—
372#95#4114yes—
373C24—22no—
374C24—24no—
375C24—22no—
376C24—24no—
377C24—24no—
378C24—22no—
379C24—22no—
380C24—22no—
381C24—22no—
382C24—22no—
383C24—21no—
384C24—21no—
385C24—22no—
386C24—22no—
387C24—22no—
388C24—21no—
389C24—21no—
390C24—21no—
391C24—21no—
392C24C22×D418yes—
393C24C22×D414yes—
394C24H(5,𝔽2)12yes—
395C24C2511yescharacteristic
396C2×Q8—44no—
397C2×Q8—44no—
398C2×Q8—44no—
399C2×Q8—48no—
400C2×Q8—48no—
401C2×D4—28no—
402C2×D4—28no—
403C2×D4—216no—
404C2×D4—216no—
405C2×D4—416no—
406C2×D4—48no—
407C2×D4—48no—
408C2×D4—84no—
409C2×D4—88no—
410C2×D4#127116yes—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—24no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—22no—
427C22×C4—21no—
428C22×C4—24no—
429C22×C4—24no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—24no—
439C22×C4—24no—
440C22×C4—21no—
441C22×C4—22no—
442C22×C4—216no—
443C22×C4—216no—
444C22×C4—22no—
445C22×C4—24no—
446C22×C4—21no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—24no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—21no—
455C22×C4—21no—
456C22×C4—22no—
457C22×C4—24no—
458C22×C4—24no—
459C22×C4—24no—
460C22×C4—21no—
461C22×C4—22no—
462C22×C4—24no—
463C22×C4—22no—
464C22×C4—21no—
465C22×C4—24no—
466C22×C4—24no—
467C22×C4—22no—
468C22×C4—24no—
469C22×C4—24no—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—21no—
473C22×C4—22no—
474C22×C4—22no—
475C22×C4—22no—
476C22×C4—21no—
477C22×C4—21no—
478C22×C4—22no—
479C22×C4—22no—
480C22×C4—21no—
481C22×C4—24no—
482C22×C4—22no—
483C22×C4—48no—
484C22×C4—416no—
485C22×C4—416no—
486C22×C4—416no—
487C22×C4—416no—
488C22×C4—416no—
489C22×C4—416no—
490C22×C4—44no—
491C22×C4—44no—
492C22×C4—44no—
493C22×C4—48no—
494C22×C4—48no—
495C22×C4—88no—
496C22×C4—88no—
497C22×C4—88no—
498C22×C4C22×D412yes—
499C22×C4C22×D414yes—
500C22×C4C22×D418yes—
501C22×C4C22×D414yes—
502C22×C4C22×D414yes—
503C22×C4C22×D412yes—
504C22×C4C22×D412yes—
505C22×C4C22×D414yes—
506C22×C4C2×#4118yes—
507C22×C4C2×#4112yes—
508C22×C4C2×#4114yes—
509C22×C4C2×#4114yes—
510C22×C4H(5,𝔽2)14yes—
511C22×C42−1+414yes—
512C22×C42−1+412yes—
513C22×C42−1+412yes—
514C22×C42−1+412yes—
515C4⋊C4—28no—
516C4⋊C4—216no—
517C4⋊C4—48no—
518C4⋊C4—416no—
519C4⋊C4—416no—
520C4⋊C4—48no—
521C4⋊C4—416no—
522C4⋊C4—48no—
523C4⋊C4—48no—
524C4⋊C4—44no—
525C4⋊C4—48no—
526C4⋊C4—44no—
527C4⋊C4—84no—
528C4⋊C4—88no—
529C4⋊C4—88no—
530C4⋊C4—84no—
531C4⋊C4#123116yescomplemented
532C22⋊C4—216no—
533C22⋊C4—216no—
534C22⋊C4—232no—
535C22⋊C4—416no—
536C22⋊C4—44no—
537C22⋊C4—416no—
538C22⋊C4—416no—
539C22⋊C4—48no—
540C22⋊C4—48no—
541C22⋊C4—48no—
542C22⋊C4—48no—
543C22⋊C4—48no—
544C22⋊C4—44no—
545C42—28no—
546C42—216no—
547C42—416no—
548C42—48no—
549C42—44no—
550C23—24no—
551C23—24no—
552C23—28no—
553C23—24no—
554C23—28no—
555C23—28no—
556C23—28no—
557C23—24no—
558C23—28no—
559C23—24no—
560C23—28no—
561C23—24no—
562C23—24no—
563C23—22no—
564C23—22no—
565C23—24no—
566C23—22no—
567C23—44no—
568C23—42no—
569C23—44no—
570C23—44no—
571C23—42no—
572C23—44no—
573C23—42no—
574C23—44no—
575C23—44no—
576C23—42no—
577C23—41no—
578C23—41no—
579C23—41no—
580C23—41no—
581C23—42no—
582C23—42no—
583C23—42no—
584C23—42no—
585C23—41no—
586C23—42no—
587C23—42no—
588C23—41no—
589C23—41no—
590C23—42no—
591C23—42no—
592C23—42no—
593C23—42no—
594C23—41no—
595C23C2×#12718yes—
596C23C23×D411yescharacteristic
597C23C23×D412yes—
598C23C22×#4112yes—
599C23C22×#4111yescharacteristic
600C23C2×H(5,𝔽2)12yes—
601C23C2×H(5,𝔽2)11yescharacteristic
602C23C2×2−1+411yescharacteristic
603C23C2×2−1+411yescharacteristic
604C23C2×2−1+412yes—
605C23C2×2−1+411yescharacteristic
606C23#58511yescharacteristic
607Q8—88no—
608D4—232no—
609D4—816no—
610C2×C4—22no—
611C2×C4—24no—
612C2×C4—24no—
613C2×C4—24no—
614C2×C4—28no—
615C2×C4—24no—
616C2×C4—22no—
617C2×C4—24no—
618C2×C4—24no—
619C2×C4—24no—
620C2×C4—24no—
621C2×C4—24no—
622C2×C4—28no—
623C2×C4—24no—
624C2×C4—22no—
625C2×C4—24no—
626C2×C4—28no—
627C2×C4—28no—
628C2×C4—24no—
629C2×C4—24no—
630C2×C4—28no—
631C2×C4—24no—
632C2×C4—22no—
633C2×C4—24no—
634C2×C4—24no—
635C2×C4—22no—
636C2×C4—432no—
637C2×C4—42no—
638C2×C4—44no—
639C2×C4—42no—
640C2×C4—42no—
641C2×C4—42no—
642C2×C4—42no—
643C2×C4—42no—
644C2×C4—42no—
645C2×C4—42no—
646C2×C4—44no—
647C2×C4—44no—
648C2×C4—42no—
649C2×C4—44no—
650C2×C4—42no—
651C2×C4—42no—
652C2×C4—41no—
653C2×C4—41no—
654C2×C4—42no—
655C2×C4—41no—
656C2×C4—41no—
657C2×C4—42no—
658C2×C4—44no—
659C2×C4—42no—
660C2×C4—41no—
661C2×C4—42no—
662C2×C4—41no—
663C2×C4—42no—
664C2×C4—42no—
665C2×C4—42no—
666C2×C4—42no—
667C2×C4—42no—
668C2×C4—42no—
669C2×C4—44no—
670C2×C4—44no—
671C2×C4—44no—
672C2×C4—44no—
673C2×C4—44no—
674C2×C4—42no—
675C2×C4—41no—
676C2×C4—44no—
677C2×C4—42no—
678C2×C4—41no—
679C2×C4—42no—
680C2×C4—44no—
681C2×C4—44no—
682C2×C4—44no—
683C2×C4—44no—
684C2×C4—42no—
685C2×C4—42no—
686C2×C4—42no—
687C2×C4—42no—
688C2×C4—41no—
689C2×C4—42no—
690C2×C4—832no—
691C2×C4—88no—
692C2×C4C2×#12318yes—
693C2×C4C2×#12314yes—
694C2×C4C2×#12714yes—
695C2×C4#53614yes—
696C2×C4#54718yes—
697C2×C4#54714yes—
698C2×C4#54718yes—
699C2×C4#54718yes—
700C22—28no—
701C22—28no—
702C22—28no—
703C22—44no—
704C22—48no—
705C22—48no—
706C22—44no—
707C22—44no—
708C22—48no—
709C22—44no—
710C22—48no—
711C22—42no—
712C22—44no—
713C22—42no—
714C22—42no—
715C22—44no—
716C22—82no—
717C22—81no—
718C22—81no—
719C22—82no—
720C22—81no—
721C22—81no—
722C22C22×#12311yescharacteristic
723C22C22×#12711yescharacteristic
724C22C2×#53611yescharacteristic
725C22C2×#54711yescharacteristic
726C22C2×#54712yes—
727C22C2×#54712yes—
728C22C2×#54712yes—
729C22#348112yes—
730C22#348412yes—
731C22#348411yescharacteristic
732C22#348811yescharacteristic
733C22#348812yes—
734C22#349912yes—
735C22#349911yescharacteristic
736C22#351511yescharacteristic
737C22#351612yes—
738C22#352012yes—
739C22#352211yescharacteristic
740C22#352212yes—
741C22#354411yescharacteristic
742C22#355012yes—
743C22#355212yes—
744C22#355511yescharacteristic
745C4—24no—
746C4—28no—
747C4—28no—
748C4—44no—
749C4—44no—
750C4—42no—
751C4—48no—
752C4—81no—
753C4—84no—
754C4—82no—
755C4—82no—
756C4—81no—
757C4—81no—
758C4—82no—
759C4#250118yes—
760C2—48no—
761C2—82no—
762C2C2×#250111yescharacteristic
763C2#3592111yescharacteristic
764C2#3599611yescharacteristic
765C2#3617911yescharacteristic
766C2#3623711yescharacteristic
767C2#3892111yescharacteristic
768C2#4242711yescharacteristic
769C2#4487512yes—
770C2#4507712yes—
771C2#5906212yes—
772C2#5907012yes—
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
1. C2×#2447C2semidirect
4. C2×D4×C4⋊C4C2semidirect
5. #17059C2semidirect
6. #15539C2semidirect
7. #15384C2semidirect
8. #15199C2semidirect
9. #15193C2semidirect
10. #15135C2semidirect
11. #15069C2semidirect
12. #14839C2semidirect
13. #13997C2semidirect
14. #13910C2semidirect
15. #13880C2semidirect
16. C4×#399C2semidirect
17. #13820C2semidirect
531. C4⋊C4#123semidirect

Maximal subgroups

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

subgroupindex
C2×#24472
C2×#24272
C2×#24222
C2×D4×C4⋊C42
#170592
#155392
#153842
#151992
#151932
#151352
#150692
#148392
#139972
#139102
#138802
C4×#3992
#138202