Finite Group Databasebeta

← #723289

Subgroups — #723289

Order 512.

2998 conjugacy classes of subgroups.

The subgroup lattice has 732 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2538C211yesmaximal, characteristic
2C2×#2581C211yesmaximal, characteristic, complemented
3C2×#2349C211yesmaximal, characteristic
4C2×#2326C211yesmaximal, characteristic
5#16400C212yesmaximal, complemented
6#16264C212yesmaximal, complemented
7#15823C214yesmaximal, complemented
8#15769C214yesmaximal, complemented
9#15189C212yesmaximal, complemented
10#15137C212yesmaximal, complemented
11#14826C212yesmaximal, complemented
12#14758C212yesmaximal, complemented
13#14277C211yesmaximal, characteristic, complemented
14#14027C212yesmaximal
15#13926C211yesmaximal, characteristic, complemented
16#13920C212yesmaximal
17#13850C211yesmaximal, characteristic, complemented
18C22×#124C2211yescharacteristic
19C22×C22⋊Q8C2212yes—
20C22×C4×Q8C2211yescharacteristic
21C22×C4×Q8C2211yescharacteristic
22C22×C4×D4C2212yes—
23C22×C4×D4C2211yescharacteristic
24#2581—28no—
25#2538—28no—
26C2×#400C2212yes—
27C2×#398C2214yes—
28C2×#396C2214yes—
29C2×#395C2212yes—
30#2349—28no—
31#2326—28no—
32C2×#391C2211yescharacteristic
33C2×#391C2212yes—
34C2×#391C2212yes—
35C2×#390C2211yescharacteristic
36C2×#390C2212yes—
37C2×#389C2212yes—
38C2×#389C2212yes—
39C2×#389C2212yes—
40C2×#389C2212yes—
41C2×#388C2212yes—
42C2×#388C2212yes—
43C2×#387C2214yes—
44C2×#385C2212yes—
45C2×#385C2212yes—
46C2×#385C2214yes—
47C2×#384C2211yescharacteristic
48C2×#384C2211yescharacteristic
49C2×#382C2211yescharacteristic
50C2×#380C2212yes—
51C2×#380C2212yes—
52C2×#380C2214yes—
53C2×C4×C4⋊C4C2211yescharacteristic
54C2×#376C2211yescharacteristic
55C22×#95C2212yes—
56#1446C2211yescharacteristic
57#1446C2211yescharacteristic
58#1445C2211yescharacteristic
59#1445C2211yescharacteristic
60#1444C2212yes—
61#1444C2214yes—
62#1444C2212yes—
63#1443C2214yes—
64#1442C2214yes—
65#1442C2212yes—
66#1442C2212yes—
67#1441C2214yes—
68#1441C2214yes—
69#1441C2212yes—
70#1441C2212yes—
71#1441C2212yes—
72#1441C2214yes—
73#1441C2214yes—
74#1441C2212yes—
75#1440C2214yes—
76#1440C2214yes—
77#1439C2214yes—
78#1439C2214yes—
79C4⋊#95C2212yes—
80C4⋊#95C2212yes—
81C4⋊#95C2214yes—
82C22⋊#95C2214yes—
83C22⋊#95C2214yes—
84C22⋊#95C2214yes—
85C4×#95C2212yes—
86C4×#95C2212yes—
87C23×Q8C2311yescharacteristic
88C23×Q8C2311yescharacteristic
89C23×D4C2311yescharacteristic
90C24×C4C2314yes—
91C24×C4C2312yes—
92C2×#124—24no—
93C2×#124—24no—
94C2×#124Q818yescomplemented
95C2×C22⋊Q8—28no—
96C2×C22⋊Q8—48no—
97C2×C4×Q8—24no—
98C2×C4×Q8—24no—
99C2×C4×Q8—24no—
100C2×C4×Q8—24no—
101C2×C4×Q8—24no—
102C2×C4×Q8—24no—
103C2×C4×D4—28no—
104C2×C4×D4—28no—
105C2×C4×D4—24no—
106C2×C4×D4—24no—
107C2×C4×D4—44no—
108C2×C4×D4D418yes—
109C22×C4⋊C4C2312yes—
110C22×C4⋊C4C2312yes—
111C22×C4⋊C4C2311yescharacteristic
112C22×C4⋊C4C2311yescharacteristic
113C22×C4⋊C4C2312yes—
114C22×C4⋊C4C2314yes—
115C22×C4⋊C4C2314yes—
116C22×C4⋊C4C2312yes—
117C22×C4⋊C4C2312yes—
118C22×C4⋊C4C2311yescharacteristic
119C22×C4⋊C4C2312yes—
120C22×C4⋊C4C2312yes—
121C22×C4⋊C4C2314yes—
122C22×C4⋊C4C2312yes—
123C22×C4⋊C4C2312yes—
124C22×C4⋊C4C2314yes—
125C22×C22⋊C4C2314yes—
126C22×C22⋊C4C2314yes—
127C22×C22⋊C4C2312yes—
128C22×C22⋊C4C2312yes—
129C22×C22⋊C4C2312yes—
130C22×C22⋊C4C2314yes—
131C22×C22⋊C4C2312yes—
132C22×C22⋊C4C2312yes—
133C22×C42C2312yes—
134C22×C42C2312yes—
135C22×C42C2312yes—
136C22×C42C2311yescharacteristic
137C22×C42C2311yescharacteristic
138C22×C42C2311yescharacteristic
139#400—28no—
140#398—48no—
141#396—216no—
142#395—28no—
143#391—28no—
144#391—24no—
145#391—44no—
146#390—24no—
147#390—28no—
148#389—28no—
149#389—28no—
150#389—28no—
151#389—44no—
152#388—28no—
153#388—28no—
154#387—216no—
155#385—28no—
156#385—28no—
157#385—216no—
158#384—24no—
159#384—24no—
160#382D418yes—
161#380—28no—
162#380—28no—
163#380—48no—
164C4×C4⋊C4—24no—
165#376—24no—
166C2×#95—28no—
167C2×#95C2314yes—
168C2×#95C2314yes—
169C2×#95C2312yes—
170C2×#95C2314yes—
171C2×#95C2314yes—
172C2×#95C2312yes—
173C2×#95C2312yes—
174C2×#95C2312yes—
175C2×#95C2312yes—
176C2×#95C2311yescharacteristic
177C2×#95C2312yes—
178C2×#95C2311yescharacteristic
179C2×#95C2311yescharacteristic
180C2×#95C2311yescharacteristic
181C2×#95C2314yes—
182C2×#95C2311yescharacteristic
183C2×#95C2311yescharacteristic
184C2×#95C2314yes—
185C2×#95C2314yes—
186C2×#95C2312yes—
187C2×#95C2312yes—
188C2×#95C2312yes—
189C2×#95C2312yes—
190C2×#95C2314yes—
191C2×#95C2314yes—
192C2×#95C2314yes—
193C2×#95C2314yes—
194C2×#95C2314yes—
195C2×#95C2312yes—
196C2×#95C2312yes—
197C25C2412yes—
198C22×Q8—22no—
199C22×Q8—22no—
200C22×Q8—22no—
201C22×Q8—22no—
202C22×Q8—22no—
203C22×Q8—22no—
204C22×Q8—22no—
205C22×Q8—24no—
206C22×Q8—22no—
207C22×Q8—22no—
208C22×Q8—42no—
209C22×Q8C2×Q814yes—
210C22×D4—22no—
211C22×D4—22no—
212C22×D4—22no—
213C22×D4—22no—
214C22×D4C2×D414yes—
215C22×D4C2×Q814yes—
216C22×D4#4114yes—
217C23×C4—24no—
218C23×C4—24no—
219C23×C4—28no—
220C23×C4—28no—
221C23×C4—28no—
222C23×C4—44no—
223C23×C4—44no—
224C23×C4—44no—
225C23×C4—44no—
226C23×C4—44no—
227C23×C4—44no—
228C23×C4C2×D418yes—
229C23×C4C2412yes—
230C23×C4C2414yes—
231C23×C4C2411yescharacteristic
232C23×C4C2412yes—
233C23×C4C2412yes—
234C23×C4C2411yescharacteristic
235C23×C4C2412yes—
236C23×C4C2414yes—
237C23×C4C2411yescharacteristic
238C23×C4C2411yescharacteristic
239C23×C4C2412yes—
240C23×C4C2412yes—
241C23×C4C2412yes—
242C23×C4C2411yescharacteristic
243C23×C4C2411yescharacteristic
244C23×C4C2411yescharacteristic
245#124—232no—
246C22⋊Q8—432no—
247C4×Q8—416no—
248C4×Q8—416no—
249C4×D4—232no—
250C4×D4—432no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4—24no—
258C2×C4⋊C4—24no—
259C2×C4⋊C4—22no—
260C2×C4⋊C4—24no—
261C2×C4⋊C4—24no—
262C2×C4⋊C4—22no—
263C2×C4⋊C4—24no—
264C2×C4⋊C4—28no—
265C2×C4⋊C4—24no—
266C2×C4⋊C4—24no—
267C2×C4⋊C4—28no—
268C2×C4⋊C4—24no—
269C2×C4⋊C4—24no—
270C2×C4⋊C4—22no—
271C2×C4⋊C4—22no—
272C2×C4⋊C4—28no—
273C2×C4⋊C4—28no—
274C2×C4⋊C4—24no—
275C2×C4⋊C4—24no—
276C2×C4⋊C4—24no—
277C2×C4⋊C4—28no—
278C2×C4⋊C4—24no—
279C2×C4⋊C4—24no—
280C2×C4⋊C4—24no—
281C2×C4⋊C4—42no—
282C2×C4⋊C4—44no—
283C2×C4⋊C4—44no—
284C2×C4⋊C4—44no—
285C2×C4⋊C4—44no—
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⋊C4—44no—
293C2×C4⋊C4C2×D418yes—
294C2×C4⋊C4C2×D414yes—
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—28no—
306C2×C22⋊C4—24no—
307C2×C22⋊C4—24no—
308C2×C22⋊C4—44no—
309C2×C22⋊C4—44no—
310C2×C22⋊C4—44no—
311C2×C22⋊C4—44no—
312C2×C22⋊C4—44no—
313C2×C22⋊C4—44no—
314C2×C22⋊C4C2×D418yes—
315C2×C22⋊C4C2×Q818yes—
316C2×C22⋊C4C2×Q818yes—
317C2×C42—22no—
318C2×C42—22no—
319C2×C42—24no—
320C2×C42—24no—
321C2×C42—24no—
322C2×C42—24no—
323C2×C42—24no—
324C2×C42—22no—
325C2×C42—24no—
326C2×C42—24no—
327C2×C42—24no—
328C2×C42—24no—
329C2×C42—22no—
330C2×C42—22no—
331C2×C42C2×D414yes—
332C2×C42C2×D414yes—
333C2×C42C2×Q814yes—
334C2×C42#4114yes—
335#95—22no—
336#95—28no—
337#95—22no—
338#95—22no—
339#95—22no—
340#95—22no—
341#95—22no—
342#95—28no—
343#95—24no—
344#95—24no—
345#95—28no—
346#95—28no—
347#95—28no—
348#95—24no—
349#95—24no—
350#95—24no—
351#95—24no—
352#95—44no—
353#95—42no—
354#95—42no—
355#95—42no—
356#95—42no—
357#95—44no—
358#95—44no—
359#95—44no—
360#95—44no—
361#95—44no—
362#95—44no—
363#95C2×D418yes—
364#95C2×D418yes—
365C24—22no—
366C24—22no—
367C24—24no—
368C24—22no—
369C24—24no—
370C24—22no—
371C24—24no—
372C24—24no—
373C24C22×D414yes—
374C24C22×Q814yes—
375C24C2×#4114yes—
376C24C2511yescharacteristic
377C2×Q8—28no—
378C2×Q8—28no—
379C2×Q8—28no—
380C2×Q8—28no—
381C2×Q8—48no—
382C2×Q8—48no—
383C2×Q8—44no—
384C2×Q8—44no—
385C2×Q8—44no—
386C2×Q8—44no—
387C2×Q8—44no—
388C2×Q8—44no—
389C2×D4—28no—
390C2×D4—28no—
391C2×D4—28no—
392C2×D4—28no—
393C2×D4—28no—
394C2×D4—28no—
395C2×D4C22⋊Q8116yes—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—21no—
403C22×C4—21no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—21no—
409C22×C4—21no—
410C22×C4—22no—
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—24no—
421C22×C4—24no—
422C22×C4—22no—
423C22×C4—24no—
424C22×C4—22no—
425C22×C4—24no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—24no—
431C22×C4—22no—
432C22×C4—24no—
433C22×C4—24no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—216no—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—24no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—21no—
451C22×C4—24no—
452C22×C4—22no—
453C22×C4—24no—
454C22×C4—24no—
455C22×C4—24no—
456C22×C4—21no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—21no—
461C22×C4—21no—
462C22×C4—24no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—24no—
469C22×C4—22no—
470C22×C4—22no—
471C22×C4—24no—
472C22×C4—24no—
473C22×C4—48no—
474C22×C4—48no—
475C22×C4—416no—
476C22×C4—416no—
477C22×C4—416no—
478C22×C4—88no—
479C22×C4—88no—
480C22×C4—88no—
481C22×C4—88no—
482C22×C4—88no—
483C22×C4—88no—
484C22×C4C22×D418yes—
485C22×C4C22×D414yes—
486C22×C4C22×D412yes—
487C22×C4C22×D412yes—
488C22×C4C22×D412yes—
489C22×C4C22×D412yes—
490C22×C4C22×D412yes—
491C22×C4C22×D412yes—
492C22×C4C22×Q812yes—
493C22×C4C22×Q812yes—
494C22×C4C22×Q814yes—
495C22×C4C22×Q812yes—
496C22×C4C2×#4112yes—
497C22×C4C2×#4112yes—
498C22×C4C2×#4112yes—
499C22×C4C2×#4112yes—
500C22×C4H(5,𝔽2)14yes—
501C22×C4H(5,𝔽2)12yes—
502C22×C42−1+414yes—
503C22×C42−1+412yes—
504C4⋊C4—28no—
505C4⋊C4—216no—
506C4⋊C4—48no—
507C4⋊C4—416no—
508C4⋊C4—416no—
509C4⋊C4—48no—
510C4⋊C4—48no—
511C4⋊C4—48no—
512C4⋊C4—44no—
513C4⋊C4—44no—
514C4⋊C4—48no—
515C4⋊C4—84no—
516C4⋊C4—84no—
517C4⋊C4—84no—
518C4⋊C4—88no—
519C4⋊C4—88no—
520C22⋊C4—216no—
521C22⋊C4—216no—
522C22⋊C4—216no—
523C22⋊C4—416no—
524C22⋊C4—416no—
525C22⋊C4—84no—
526C22⋊C4—84no—
527C22⋊C4—88no—
528C42—28no—
529C42—48no—
530C42—48no—
531C42—44no—
532C42—48no—
533C42#121116yes—
534C23—24no—
535C23—24no—
536C23—24no—
537C23—24no—
538C23—24no—
539C23—24no—
540C23—24no—
541C23—24no—
542C23—24no—
543C23—24no—
544C23—28no—
545C23—28no—
546C23—28no—
547C23—28no—
548C23—44no—
549C23—44no—
550C23—44no—
551C23—44no—
552C23—44no—
553C23—44no—
554C23—44no—
555C23—44no—
556C23C2×C22⋊Q818yes—
557C23C23×D411yescharacteristic
558C23C23×D411yescharacteristic
559C23C23×Q811yescharacteristic
560C23C22×#4111yescharacteristic
561C23C22×#4111yescharacteristic
562C23C2×H(5,𝔽2)11yescharacteristic
563C23C2×H(5,𝔽2)12yes—
564C23C2×2−1+411yescharacteristic
565C23C2×2−1+412yes—
566C23#58512yes—
567C23#58512yes—
568Q8—416no—
569Q8—416no—
570D4—232no—
571C2×C4—24no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—24no—
575C2×C4—24no—
576C2×C4—24no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—24no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—24no—
583C2×C4—28no—
584C2×C4—24no—
585C2×C4—24no—
586C2×C4—24no—
587C2×C4—24no—
588C2×C4—24no—
589C2×C4—24no—
590C2×C4—28no—
591C2×C4—24no—
592C2×C4—28no—
593C2×C4—24no—
594C2×C4—24no—
595C2×C4—24no—
596C2×C4—24no—
597C2×C4—24no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—41no—
605C2×C4—41no—
606C2×C4—42no—
607C2×C4—44no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—44no—
611C2×C4—42no—
612C2×C4—41no—
613C2×C4—41no—
614C2×C4—42no—
615C2×C4—44no—
616C2×C4—42no—
617C2×C4—44no—
618C2×C4—42no—
619C2×C4—44no—
620C2×C4—42no—
621C2×C4—44no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—41no—
625C2×C4—41no—
626C2×C4—44no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—44no—
631C2×C4—41no—
632C2×C4—42no—
633C2×C4—41no—
634C2×C4—42no—
635C2×C4—44no—
636C2×C4—42no—
637C2×C4—42no—
638C2×C4—42no—
639C2×C4—42no—
640C2×C4—44no—
641C2×C4—41no—
642C2×C4—42no—
643C2×C4—42no—
644C2×C4—44no—
645C2×C4—41no—
646C2×C4—42no—
647C2×C4—42no—
648C2×C4—42no—
649C2×C4—44no—
650C2×C4—42no—
651C2×C4—44no—
652C2×C4—42no—
653C2×C4—44no—
654C2×C4—44no—
655C2×C4—42no—
656C2×C4—42no—
657C2×C4—42no—
658C2×C4—816no—
659C2×C4—832no—
660C2×C4C2×#12114yes—
661C2×C4C2×#12114yes—
662C2×C4C2×#12114yes—
663C2×C4C2×C22⋊Q814yes—
664C2×C4#53714yes—
665C2×C4#54614yes—
666C2×C4D4×Q814yes—
667C2×C4#55214yes—
668C2×C4#55414yes—
669C22—28no—
670C22—28no—
671C22—28no—
672C22—48no—
673C22—48no—
674C22—48no—
675C22—44no—
676C22—44no—
677C22—44no—
678C22—44no—
679C22—48no—
680C22C22×#12111yescharacteristic
681C22C22×C22⋊Q811yescharacteristic
682C22C2×#53711yescharacteristic
683C22#346112yes—
684C22C2×#54611yescharacteristic
685C22C2×D4×Q811yescharacteristic
686C22C2×#55211yescharacteristic
687C22C2×#55411yescharacteristic
688C22#348411yescharacteristic
689C22#348511yescharacteristic
690C22#348812yes—
691C22#348912yes—
692C22#349412yes—
693C22#349911yescharacteristic
694C22#350011yescharacteristic
695C22#350312yes—
696C22#350412yes—
697C22#352012yes—
698C22#352112yes—
699C22#354612yes—
700C22#355412yes—
701C22#355512yes—
702C22#356112yes—
703C4—24no—
704C4—24no—
705C4—48no—
706C4—44no—
707C4—44no—
708C4—42no—
709C4—44no—
710C4—44no—
711C4—42no—
712C4—82no—
713C4—81no—
714C4—82no—
715C4—81no—
716C4—84no—
717C4—82no—
718C4#241718yes—
719C2—48no—
720C2C2×#241711yescharacteristic
721C2#3530811yescharacteristic
722C2#3551411yescharacteristic
723C2#3573911yescharacteristic
724C2#3603111yescharacteristic
725C2#3822412yes—
726C2#3967512yes—
727C2#4242911yescharacteristic
728C2#4264911yescharacteristic
729C2#4508412yes—
730C2#4542412yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC23×D4C23
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×#2581C2semidirect
5. #16400C2semidirect
6. #16264C2semidirect
7. #15823C2semidirect
8. #15769C2semidirect
9. #15189C2semidirect
10. #15137C2semidirect
11. #14826C2semidirect
12. #14758C2semidirect
13. #14277C2semidirect
15. #13926C2semidirect
17. #13850C2semidirect
94. C2×#124Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#25382
C2×#25812
C2×#23492
C2×#23262
#164002
#162642
#158232
#157692
#151892
#151372
#148262
#147582
#142772
#140272
#139262
#139202
#138502