Finite Group Databasebeta

← #544527

Subgroups — #544527

Order 512.

2990 conjugacy classes of subgroups.

The subgroup lattice has 744 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2398C211yesmaximal, characteristic
2C2×#2355C211yesmaximal, characteristic, complemented
3C2×D4×C4⋊C4C211yesmaximal, characteristic
4#14629C212yesmaximal, complemented
5#14575C214yesmaximal, complemented
6#14541C214yesmaximal, complemented
7#14354C211yesmaximal, characteristic, complemented
8#14318C212yesmaximal, complemented
9#14300C212yesmaximal
10#14176C212yesmaximal
11#13926C211yesmaximal, characteristic, complemented
12#13905C211yesmaximal, characteristic
13#13881C212yesmaximal, complemented
14#13880C212yesmaximal, complemented
15#13876C212yesmaximal, complemented
16#13875C212yesmaximal, complemented
17#13797C211yesmaximal, characteristic, complemented
18C22×C4×Q8C2211yescharacteristic
19C22×C4×D4C2211yescharacteristic
20C22×C4×D4C2212yes—
21C23×C4⋊C4C2212yes—
22#2398C4116yescomplemented
23C2×#400C2211yescharacteristic
24C2×#400C2211yescharacteristic
25C2×#397C2212yes—
26C2×#394C2211yescharacteristic
27C2×#394C2211yescharacteristic
28C2×#393C2212yes—
29#2355—28no—
30D4×C4⋊C4—28no—
31C2×#391C2212yes—
32C2×#391C2212yes—
33C2×#389C2212yes—
34C2×#388C2214yes—
35C2×#387C2212yes—
36C2×#387C2212yes—
37C2×#385C2214yes—
38C2×#385C2214yes—
39C2×#385C2214yes—
40C2×#384C2211yescharacteristic
41C2×#384C2211yescharacteristic
42C2×#384C2212yes—
43C2×#382C2211yescharacteristic
44C2×#382C2211yescharacteristic
45C2×#382C2212yes—
46C2×#380C2212yes—
47C2×#380C2212yes—
48C2×#380C2212yes—
49C2×C4×C4⋊C4C2211yescharacteristic
50C2×C4×C4⋊C4C2211yescharacteristic
51C2×C4×C4⋊C4C2212yes—
52C2×C4×C4⋊C4C2211yescharacteristic
53C2×C4×C4⋊C4C2211yescharacteristic
54C2×C4×C22⋊C4C2212yes—
55C2×C4×C22⋊C4C2212yes—
56C2×#376C2211yescharacteristic
57#1445C2212yes—
58#1444C2212yes—
59#1444C2214yes—
60#1444C2212yes—
61#1444C2214yes—
62#1444C2214yes—
63#1443C2214yes—
64#1442C2214yes—
65#1442C2214yes—
66#1442C2214yes—
67#1441C2212yes—
68#1441C2212yes—
69#1440C2214yes—
70#1439C2214yes—
71#1438C2212yes—
72#1437C2214yes—
73#1437C2214yes—
74#1436C2214yes—
75C4⋊#95C2212yes—
76C4⋊#95C2212yes—
77C4⋊#95C2212yes—
78C4⋊#95C2212yes—
79C4⋊#95C2212yes—
80#1434C2212yes—
81#1434C2214yes—
82#1434C2214yes—
83#1434C2212yes—
84C22⋊#95C2214yes—
85C22⋊#95C2214yes—
86C23×Q8C2311yescharacteristic
87C23×D4C2311yescharacteristic
88C24×C4C2312yes—
89C24×C4C2314yes—
90C2×C4×Q8—24no—
91C2×C4×Q8—24no—
92C2×C4×Q8C2×C418yescomplemented
93C2×C4×D4—24no—
94C2×C4×D4—28no—
95C2×C4×D4—24no—
96C2×C4×D4—48no—
97C2×C4×D4C2×C418yes—
98C22×C4⋊C4—28no—
99C22×C4⋊C4—44no—
100C22×C4⋊C4—44no—
101C22×C4⋊C4C2312yes—
102C22×C4⋊C4C2312yes—
103C22×C4⋊C4C2311yescharacteristic
104C22×C4⋊C4C2311yescharacteristic
105C22×C4⋊C4C2312yes—
106C22×C4⋊C4C2311yescharacteristic
107C22×C4⋊C4C2311yescharacteristic
108C22×C4⋊C4C2311yescharacteristic
109C22×C4⋊C4C2311yescharacteristic
110C22×C4⋊C4C2311yescharacteristic
111C22×C4⋊C4C2311yescharacteristic
112C22×C4⋊C4C2314yes—
113C22×C4⋊C4C2312yes—
114C22×C4⋊C4C2312yes—
115C22×C4⋊C4C2312yes—
116C22×C4⋊C4C2314yes—
117C22×C4⋊C4C2314yes—
118C22×C22⋊C4C2314yes—
119C22×C22⋊C4C2312yes—
120C22×C22⋊C4C2312yes—
121C22×C22⋊C4C2312yes—
122C22×C22⋊C4C2312yes—
123C22×C22⋊C4C2314yes—
124C22×C22⋊C4C2314yes—
125C22×C22⋊C4C2312yes—
126C22×C42C2312yes—
127C22×C42C2311yescharacteristic
128C22×C42C2311yescharacteristic
129C22×C42C2312yes—
130C22×C42C2312yes—
131C22×C42C2312yes—
132C22×C42C2312yes—
133C22×C42C2311yescharacteristic
134#400C2×C418yescomplemented
135#400C2×C418yescomplemented
136#397C2×C4116yescomplemented
137#394C2×C418yes—
138#394C2×C418yes—
139#393C2×C4116yescomplemented
140#391—28no—
141#391—44no—
142#389—28no—
143#388—48no—
144#387—28no—
145#387—28no—
146#385—216no—
147#385—48no—
148#385—48no—
149#384—24no—
150#384—24no—
151#384—44no—
152#382—24no—
153#382—24no—
154#382—44no—
155#380—28no—
156#380—44no—
157#380—44no—
158C4×C4⋊C4—24no—
159C4×C4⋊C4—24no—
160C4×C4⋊C4—28no—
161C4×C4⋊C4—24no—
162C4×C4⋊C4D418yes—
163C4×C22⋊C4C2×C4116yescomplemented
164C4×C22⋊C4C2×C4116yescomplemented
165#376C2×C418yescomplemented
166C2×#95C2312yes—
167C2×#95C2312yes—
168C2×#95C2312yes—
169C2×#95C2314yes—
170C2×#95C2314yes—
171C2×#95C2312yes—
172C2×#95C2312yes—
173C2×#95C2312yes—
174C2×#95C2312yes—
175C2×#95C2314yes—
176C2×#95C2314yes—
177C2×#95C2314yes—
178C2×#95C2314yes—
179C2×#95C2314yes—
180C2×#95C2314yes—
181C2×#95C2314yes—
182C2×#95C2314yes—
183C2×#95C2314yes—
184C2×#95C2314yes—
185C2×#95C2314yes—
186C2×#95C2312yes—
187C2×#95C2312yes—
188C2×#95C2312yes—
189C2×#95C2311yescharacteristic
190C2×#95C2311yescharacteristic
191C2×#95C2312yes—
192C2×#95C2312yes—
193C2×#95C2312yes—
194C25C2412yes—
195C22×Q8—22no—
196C22×Q8—22no—
197C22×Q8—22no—
198C22×Q8—22no—
199C22×Q8—41no—
200C22×Q8—41no—
201C22×Q8C22×C414yes—
202C22×D4—24no—
203C22×D4—24no—
204C22×D4—22no—
205C22×D4—22no—
206C22×D4C22×C414yes—
207C23×C4—24no—
208C23×C4—24no—
209C23×C4—24no—
210C23×C4—24no—
211C23×C4—24no—
212C23×C4—24no—
213C23×C4—28no—
214C23×C4—28no—
215C23×C4—44no—
216C23×C4—44no—
217C23×C4—44no—
218C23×C4—44no—
219C23×C4—44no—
220C23×C4C22×C418yes—
221C23×C4C2412yes—
222C23×C4C2412yes—
223C23×C4C2414yes—
224C23×C4C2414yes—
225C23×C4C2411yescharacteristic
226C23×C4C2412yes—
227C23×C4C2412yes—
228C23×C4C2411yescharacteristic
229C23×C4C2411yescharacteristic
230C23×C4C2412yes—
231C23×C4C2412yes—
232C23×C4C2412yes—
233C23×C4C2411yescharacteristic
234C23×C4C2411yescharacteristic
235C23×C4C2411yescharacteristic
236C23×C4C2411yescharacteristic
237C4×Q8—232no—
238C4×D4—232no—
239C4×D4—432no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—28no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—24no—
258C2×C4⋊C4—24no—
259C2×C4⋊C4—24no—
260C2×C4⋊C4—24no—
261C2×C4⋊C4—24no—
262C2×C4⋊C4—24no—
263C2×C4⋊C4—24no—
264C2×C4⋊C4—24no—
265C2×C4⋊C4—24no—
266C2×C4⋊C4—432no—
267C2×C4⋊C4—44no—
268C2×C4⋊C4—44no—
269C2×C4⋊C4—42no—
270C2×C4⋊C4—44no—
271C2×C4⋊C4—42no—
272C2×C4⋊C4—44no—
273C2×C4⋊C4—44no—
274C2×C4⋊C4—44no—
275C2×C4⋊C4—44no—
276C2×C4⋊C4—44no—
277C2×C4⋊C4—44no—
278C2×C4⋊C4—44no—
279C2×C4⋊C4—42no—
280C2×C4⋊C4C22×C414yes—
281C2×C4⋊C4C22×C414yes—
282C2×C4⋊C4C22×C414yes—
283C2×C4⋊C4C22×C414yes—
284C2×C4⋊C4C2×D414yes—
285C2×C4⋊C4C2×D414yes—
286C2×C22⋊C4—24no—
287C2×C22⋊C4—24no—
288C2×C22⋊C4—24no—
289C2×C22⋊C4—24no—
290C2×C22⋊C4—24no—
291C2×C22⋊C4—24no—
292C2×C22⋊C4—24no—
293C2×C22⋊C4—24no—
294C2×C22⋊C4—28no—
295C2×C22⋊C4—28no—
296C2×C22⋊C4—28no—
297C2×C22⋊C4—28no—
298C2×C22⋊C4—42no—
299C2×C22⋊C4—42no—
300C2×C22⋊C4—44no—
301C2×C22⋊C4—44no—
302C2×C22⋊C4—44no—
303C2×C22⋊C4—44no—
304C2×C22⋊C4—44no—
305C2×C22⋊C4C22×C418yes—
306C2×C22⋊C4C22×C418yes—
307C2×C22⋊C4C22×C418yes—
308C2×C22⋊C4C22×C418yes—
309C2×C22⋊C4C22×C418yes—
310C2×C42—22no—
311C2×C42—22no—
312C2×C42—22no—
313C2×C42—22no—
314C2×C42—24no—
315C2×C42—24no—
316C2×C42—24no—
317C2×C42—24no—
318C2×C42—24no—
319C2×C42—24no—
320C2×C42—44no—
321C2×C42—42no—
322C2×C42—44no—
323C2×C42—42no—
324C2×C42—42no—
325C2×C42C22×C414yes—
326C2×C42C22×C414yes—
327C2×C42C22×C418yes—
328C2×C42C22×C414yes—
329C2×C42C2×D418yes—
330C2×C42C2×D414yes—
331C2×C42#4114yes—
332#95—24no—
333#95—24no—
334#95—24no—
335#95—24no—
336#95—28no—
337#95—24no—
338#95—24no—
339#95—24no—
340#95—28no—
341#95—44no—
342#95—44no—
343#95—44no—
344#95—44no—
345#95—44no—
346#95—44no—
347#95—44no—
348#95—44no—
349#95—44no—
350#95—44no—
351#95—44no—
352#95C22×C418yes—
353#95C22×C414yes—
354#95C22×C414yes—
355#95C22×C418yes—
356#95C22×C418yes—
357#95C22×C418yes—
358#95C22×C418yes—
359#95C2×D418yes—
360C24—22no—
361C24—22no—
362C24—22no—
363C24—22no—
364C24—22no—
365C24—22no—
366C24—22no—
367C24—22no—
368C24—22no—
369C24—22no—
370C24—22no—
371C24—22no—
372C24C23×C414yes—
373C24H(5,𝔽2)14yes—
374C242−1+414yes—
375C24C2511yescharacteristic
376C2×Q8—28no—
377C2×Q8—28no—
378C2×Q8—44no—
379C2×Q8—44no—
380C2×Q8—44no—
381C2×Q8—44no—
382C2×Q8—44no—
383C2×D4—28no—
384C2×D4—28no—
385C2×D4—216no—
386C2×D4—216no—
387C2×D4—28no—
388C22×C4—21no—
389C22×C4—21no—
390C22×C4—22no—
391C22×C4—21no—
392C22×C4—22no—
393C22×C4—21no—
394C22×C4—21no—
395C22×C4—21no—
396C22×C4—21no—
397C22×C4—21no—
398C22×C4—21no—
399C22×C4—21no—
400C22×C4—21no—
401C22×C4—22no—
402C22×C4—21no—
403C22×C4—21no—
404C22×C4—21no—
405C22×C4—22no—
406C22×C4—21no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—24no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—21no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—21no—
435C22×C4—21no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—216no—
439C22×C4—216no—
440C22×C4—216no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—24no—
444C22×C4—24no—
445C22×C4—24no—
446C22×C4—24no—
447C22×C4—24no—
448C22×C4—24no—
449C22×C4—24no—
450C22×C4—24no—
451C22×C4—24no—
452C22×C4—24no—
453C22×C4—24no—
454C22×C4—24no—
455C22×C4—22no—
456C22×C4—24no—
457C22×C4—22no—
458C22×C4—24no—
459C22×C4—24no—
460C22×C4—24no—
461C22×C4—24no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—24no—
465C22×C4—416no—
466C22×C4—416no—
467C22×C4—48no—
468C22×C4—48no—
469C22×C4—48no—
470C22×C4—48no—
471C22×C4—88no—
472C22×C4—88no—
473C22×C4—88no—
474C22×C4—88no—
475C22×C4—88no—
476C22×C4C23×C414yes—
477C22×C4C23×C414yes—
478C22×C4C23×C414yes—
479C22×C4C23×C412yes—
480C22×C4C23×C412yes—
481C22×C4C23×C412yes—
482C22×C4C23×C412yes—
483C22×C4C23×C412yes—
484C22×C4C23×C412yes—
485C22×C4C23×C412yes—
486C22×C4C22×D414yes—
487C22×C4C22×D414yes—
488C22×C4C22×D412yes—
489C22×C4C22×D412yes—
490C22×C4C22×D412yes—
491C22×C4C2×#4112yes—
492C22×C4C2×#4112yes—
493C22×C4C2×#4112yes—
494C22×C4H(5,𝔽2)12yes—
495C22×C4H(5,𝔽2)14yes—
496C22×C42−1+412yes—
497C22×C42−1+414yes—
498C4⋊C4—28no—
499C4⋊C4—28no—
500C4⋊C4—28no—
501C4⋊C4—28no—
502C4⋊C4—28no—
503C4⋊C4—28no—
504C4⋊C4—48no—
505C4⋊C4—416no—
506C4⋊C4—48no—
507C4⋊C4—44no—
508C4⋊C4—44no—
509C4⋊C4—48no—
510C4⋊C4—48no—
511C4⋊C4—84no—
512C4⋊C4—84no—
513C4⋊C4—88no—
514C4⋊C4—88no—
515C22⋊C4—216no—
516C22⋊C4—216no—
517C22⋊C4—216no—
518C22⋊C4—216no—
519C22⋊C4—48no—
520C22⋊C4—416no—
521C22⋊C4—88no—
522C22⋊C4—88no—
523C42—28no—
524C42—28no—
525C42—216no—
526C42—48no—
527C42—48no—
528C42—84no—
529C42—84no—
530C42C4×D4116yes—
531C23—24no—
532C23—24no—
533C23—24no—
534C23—24no—
535C23—24no—
536C23—24no—
537C23—24no—
538C23—24no—
539C23—24no—
540C23—24no—
541C23—24no—
542C23—24no—
543C23—24no—
544C23—24no—
545C23—24no—
546C23—24no—
547C23—24no—
548C23—24no—
549C23—42no—
550C23—42no—
551C23—42no—
552C23—42no—
553C23—42no—
554C23—42no—
555C23—42no—
556C23—42no—
557C23—42no—
558C23—42no—
559C23—42no—
560C23—42no—
561C23—42no—
562C23—42no—
563C23—42no—
564C23—42no—
565C23#52018yes—
566C23C24×C411yescharacteristic
567C23C23×D411yescharacteristic
568C23C22×#4111yescharacteristic
569C23C2×H(5,𝔽2)12yes—
570C23C2×H(5,𝔽2)12yes—
571C23C2×H(5,𝔽2)11yescharacteristic
572C23C2×2−1+412yes—
573C23C2×2−1+412yes—
574C23C2×2−1+411yescharacteristic
575C23#58511yescharacteristic
576C23#58511yescharacteristic
577Q8—416no—
578D4—232no—
579C2×C4—22no—
580C2×C4—24no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—24no—
591C2×C4—22no—
592C2×C4—22no—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—22no—
596C2×C4—22no—
597C2×C4—22no—
598C2×C4—24no—
599C2×C4—24no—
600C2×C4—24no—
601C2×C4—24no—
602C2×C4—24no—
603C2×C4—24no—
604C2×C4—24no—
605C2×C4—24no—
606C2×C4—24no—
607C2×C4—24no—
608C2×C4—22no—
609C2×C4—22no—
610C2×C4—24no—
611C2×C4—24no—
612C2×C4—24no—
613C2×C4—24no—
614C2×C4—24no—
615C2×C4—24no—
616C2×C4—24no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—41no—
622C2×C4—41no—
623C2×C4—41no—
624C2×C4—41no—
625C2×C4—41no—
626C2×C4—41no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—41no—
630C2×C4—41no—
631C2×C4—42no—
632C2×C4—44no—
633C2×C4—42no—
634C2×C4—44no—
635C2×C4—42no—
636C2×C4—42no—
637C2×C4—42no—
638C2×C4—44no—
639C2×C4—42no—
640C2×C4—44no—
641C2×C4—42no—
642C2×C4—42no—
643C2×C4—432no—
644C2×C4—44no—
645C2×C4—42no—
646C2×C4—42no—
647C2×C4—44no—
648C2×C4—42no—
649C2×C4—44no—
650C2×C4—42no—
651C2×C4—42no—
652C2×C4—44no—
653C2×C4—42no—
654C2×C4—44no—
655C2×C4—44no—
656C2×C4—44no—
657C2×C4—44no—
658C2×C4—44no—
659C2×C4—44no—
660C2×C4—44no—
661C2×C4—44no—
662C2×C4—44no—
663C2×C4—44no—
664C2×C4—44no—
665C2×C4—44no—
666C2×C4—44no—
667C2×C4—44no—
668C2×C4—832no—
669C2×C4C2×C4×D414yes—
670C2×C4C2×C4×D414yes—
671C2×C4C2×C4×D414yes—
672C2×C4#52014yes—
673C2×C4#52018yes—
674C2×C4#53514yes—
675C2×C4#53614yes—
676C2×C4#54218yes—
677C22—28no—
678C22—28no—
679C22—28no—
680C22—44no—
681C22—44no—
682C22—44no—
683C22—44no—
684C22—44no—
685C22—44no—
686C22—44no—
687C22—44no—
688C22—44no—
689C22—44no—
690C22—44no—
691C22—44no—
692C22C22×C4×D411yescharacteristic
693C22C2×#52012yes—
694C22C2×#52012yes—
695C22C2×#52011yescharacteristic
696C22#342811yescharacteristic
697C22C2×#53511yescharacteristic
698C22C2×#53611yescharacteristic
699C22C2×#54212yes—
700C22#348411yescharacteristic
701C22#348611yescharacteristic
702C22#348811yescharacteristic
703C22#348911yescharacteristic
704C22#349212yes—
705C22#349411yescharacteristic
706C22#349511yescharacteristic
707C22#354312yes—
708C22#354412yes—
709C22#355212yes—
710C22#355212yes—
711C22#355812yes—
712C22#355912yes—
713C22#356112yes—
714C22#356312yes—
715C4—24no—
716C4—24no—
717C4—24no—
718C4—44no—
719C4—44no—
720C4—44no—
721C4—42no—
722C4—42no—
723C4—42no—
724C4—44no—
725C4—44no—
726C4—84no—
727C4—82no—
728C4—82no—
729C4—84no—
730C4#232618yes—
731C2—48no—
732C2C2×#232611yescharacteristic
733C2#3420011yescharacteristic
734C2#3420211yescharacteristic
735C2#3438212yes—
736C2#3438412yes—
737C2#4248211yescharacteristic
738C2#4248411yescharacteristic
739C2#4251212yes—
740C2#4251412yes—
741C2#4281211yescharacteristic
742C2#4281411yescharacteristic
subgroupG/H
center, Frattini, socle, cosocleC24C25
commutatorC23C24×C4
prime radicalC23×D4C23
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. C2×#2355C2semidirect
4. #14629C2semidirect
5. #14575C2semidirect
6. #14541C2semidirect
7. #14354C2semidirect
8. #14318C2semidirect
11. #13926C2semidirect
13. #13881C2semidirect
14. #13880C2semidirect
15. #13876C2semidirect
16. #13875C2semidirect
17. #13797C2semidirect
22. #2398C4semidirect
92. C2×C4×Q8C2×C4semidirect
134. #400C2×C4semidirect
135. #400C2×C4semidirect
136. #397C2×C4semidirect
139. #393C2×C4semidirect
163. C4×C22⋊C4C2×C4semidirect
164. C4×C22⋊C4C2×C4semidirect
165. #376C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#23982
C2×#23552
C2×D4×C4⋊C42
#146292
#145752
#145412
#143542
#143182
#143002
#141762
#139262
#139052
#138812
#138802
#138762
#138752
#137972