Finite Group Databasebeta

← #740204

Subgroups — #740204

Order 512.

2802 conjugacy classes of subgroups.

The subgroup lattice has 765 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2542C211yesmaximal, characteristic
2C2×#2562C211yesmaximal, characteristic
3C2×#2642C211yesmaximal, characteristic, complemented
4C2×#2365C211yesmaximal, characteristic, complemented
5#17039C212yesmaximal, complemented
6#16899C212yesmaximal, complemented
7#15815C214yesmaximal
8#15789C214yesmaximal, complemented
9#15179C212yesmaximal, complemented
10#15179C212yesmaximal, complemented
11#15175C212yesmaximal, complemented
12#15103C212yesmaximal, complemented
13#14160C211yesmaximal, characteristic, complemented
14#13969C212yesmaximal, complemented
15#13969C212yesmaximal, complemented
16#13847C211yesmaximal, characteristic
17#13783C211yesmaximal, characteristic, complemented
18C22×C4⋊Q8C2211yescharacteristic
19C22×#125C2211yescharacteristic
20C22×C22⋊Q8C2212yes—
21C22×C22⋊Q8C2212yes—
22C22×C4×Q8C2212yes—
23C22×C4×D4C2211yescharacteristic
24#2642—28no—
25#2562—28no—
26#2542—28no—
27C2×#400C2212yes—
28C2×#400C2212yes—
29C2×#400C2214yes—
30C2×#400C2212yes—
31C2×#399C2212yes—
32C2×#399C2212yes—
33C2×#394C2212yes—
34#2365—28no—
35C2×#391C2212yes—
36C2×#391C2212yes—
37C2×#391C2212yes—
38C2×#390C2212yes—
39C2×#389C2212yes—
40C2×#389C2212yes—
41C2×#388C2212yes—
42C2×#388C2212yes—
43C2×#387C2214yes—
44C2×#384C2212yes—
45C2×#380C2211yescharacteristic
46C2×#380C2211yescharacteristic
47C2×#380C2211yescharacteristic
48C2×#380C2211yescharacteristic
49C2×C4×C4⋊C4C2211yescharacteristic
50C2×C4×C4⋊C4C2212yes—
51C2×C4×C4⋊C4C2211yescharacteristic
52C2×C4×C4⋊C4C2211yescharacteristic
53C2×C4×C22⋊C4C2212yes—
54C2×C4×C22⋊C4C2212yes—
55C2×#376C2211yescharacteristic
56#1445C2212yes—
57#1442C2214yes—
58#1442C2214yes—
59#1441C2214yes—
60#1441C2212yes—
61#1441C2214yes—
62#1441C2212yes—
63#1441C2214yes—
64#1441C2214yes—
65#1440C2214yes—
66#1440C2214yes—
67#1439C2211yescharacteristic
68#1439C2214yes—
69#1439C2214yes—
70#1439C2211yescharacteristic
71#1439C2211yescharacteristic
72#1439C2211yescharacteristic
73#1439C2214yes—
74#1439C2214yes—
75#1438C2212yes—
76#1437C2214yes—
77#1437C2214yes—
78#1436C2214yes—
79#1436C2214yes—
80#1436C2214yes—
81#1436C2214yes—
82C4⋊#95C2212yes—
83C4⋊#95C2212yes—
84C4⋊#95C2212yes—
85C4⋊#95C2212yes—
86#1434C2212yes—
87C23×Q8C2312yes—
88C23×D4C2311yescharacteristic
89C24×C4C2311yescharacteristic
90C24×C4C2311yescharacteristic
91C2×C4⋊Q8—24no—
92C2×C4⋊Q8—24no—
93C2×C4⋊Q8D418yes—
94C2×#125—24no—
95C2×#125—24no—
96C2×#125—24no—
97C2×C22⋊Q8—28no—
98C2×C22⋊Q8—28no—
99C2×C22⋊Q8—28no—
100C2×C22⋊Q8—28no—
101C2×C22⋊Q8—44no—
102C2×C22⋊Q8—44no—
103C2×C4×Q8—28no—
104C2×C4×Q8—44no—
105C2×C4×Q8—44no—
106C2×C4×D4—24no—
107C2×C4×D4—24no—
108C2×C4×D4—24no—
109C22×C4⋊C4C2312yes—
110C22×C4⋊C4C2311yescharacteristic
111C22×C4⋊C4C2312yes—
112C22×C4⋊C4C2311yescharacteristic
113C22×C4⋊C4C2311yescharacteristic
114C22×C4⋊C4C2311yescharacteristic
115C22×C4⋊C4C2314yes—
116C22×C4⋊C4C2314yes—
117C22×C4⋊C4C2312yes—
118C22×C4⋊C4C2312yes—
119C22×C4⋊C4C2311yescharacteristic
120C22×C4⋊C4C2311yescharacteristic
121C22×C4⋊C4C2312yes—
122C22×C4⋊C4C2314yes—
123C22×C4⋊C4C2314yes—
124C22×C4⋊C4C2311yescharacteristic
125C22×C22⋊C4C2311yescharacteristic
126C22×C22⋊C4C2311yescharacteristic
127C22×C22⋊C4C2312yes—
128C22×C22⋊C4C2312yes—
129C22×C22⋊C4C2314yes—
130C22×C22⋊C4C2314yes—
131C22×C22⋊C4C2312yes—
132C22×C22⋊C4C2312yes—
133C22×C22⋊C4C2312yes—
134C22×C22⋊C4C2312yes—
135C22×C22⋊C4C2312yes—
136C22×C22⋊C4C2312yes—
137C22×C42C2312yes—
138C22×C42C2312yes—
139C22×C42C2314yes—
140C22×C42C2312yes—
141C22×C42C2312yes—
142C22×C42C2311yescharacteristic
143#400—216no—
144#400—28no—
145#400—28no—
146#400—28no—
147#399—28no—
148#399—28no—
149#394—28no—
150#391—28no—
151#391—28no—
152#391—28no—
153#390—28no—
154#389—28no—
155#389—28no—
156#388—28no—
157#388—28no—
158#387—216no—
159#384—28no—
160#380—24no—
161#380—24no—
162#380—24no—
163#380—24no—
164C4×C4⋊C4—24no—
165C4×C4⋊C4—28no—
166C4×C4⋊C4—24no—
167C4×C4⋊C4D418yes—
168C4×C22⋊C4—44no—
169C4×C22⋊C4—44no—
170#376—24no—
171C2×#95C2312yes—
172C2×#95C2311yescharacteristic
173C2×#95C2311yescharacteristic
174C2×#95C2312yes—
175C2×#95C2314yes—
176C2×#95C2314yes—
177C2×#95C2314yes—
178C2×#95C2314yes—
179C2×#95C2312yes—
180C2×#95C2312yes—
181C2×#95C2312yes—
182C2×#95C2312yes—
183C2×#95C2314yes—
184C2×#95C2314yes—
185C2×#95C2312yes—
186C2×#95C2314yes—
187C2×#95C2314yes—
188C2×#95C2314yes—
189C2×#95C2314yes—
190C2×#95C2314yes—
191C2×#95C2314yes—
192C2×#95C2314yes—
193C2×#95C2312yes—
194C2×#95C2312yes—
195C2×#95C2312yes—
196C2×#95C2312yes—
197C2×#95C2312yes—
198C25C2411yescharacteristic
199C25C2411yescharacteristic
200C22×Q8—24no—
201C22×Q8—24no—
202C22×Q8—24no—
203C22×Q8—24no—
204C22×Q8—24no—
205C22×Q8—24no—
206C22×Q8C2×D418yes—
207C22×D4—22no—
208C22×D4—22no—
209C22×D4—22no—
210C22×D4—22no—
211C22×D4—22no—
212C22×D4—41no—
213C22×D4—41no—
214C23×C4—22no—
215C23×C4—22no—
216C23×C4—22no—
217C23×C4—22no—
218C23×C4—22no—
219C23×C4—22no—
220C23×C4—24no—
221C23×C4—24no—
222C23×C4—24no—
223C23×C4—24no—
224C23×C4C2412yes—
225C23×C4C2412yes—
226C23×C4C2412yes—
227C23×C4C2412yes—
228C23×C4C2412yes—
229C23×C4C2411yescharacteristic
230C23×C4C2411yescharacteristic
231C23×C4C2414yes—
232C23×C4C2412yes—
233C23×C4C2412yes—
234C23×C4C2414yes—
235C23×C4C2412yes—
236C23×C4C2411yescharacteristic
237C23×C4C2411yescharacteristic
238C23×C4C2411yescharacteristic
239C4⋊Q8—232no—
240#125—416no—
241C22⋊Q8—432no—
242C22⋊Q8—432no—
243C4×Q8—432no—
244C4×D4—416no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4—28no—
258C2×C4⋊C4—28no—
259C2×C4⋊C4—28no—
260C2×C4⋊C4—22no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—24no—
263C2×C4⋊C4—22no—
264C2×C4⋊C4—22no—
265C2×C4⋊C4—24no—
266C2×C4⋊C4—24no—
267C2×C4⋊C4—24no—
268C2×C4⋊C4—22no—
269C2×C4⋊C4—22no—
270C2×C4⋊C4—28no—
271C2×C4⋊C4—28no—
272C2×C4⋊C4—24no—
273C2×C4⋊C4—24no—
274C2×C4⋊C4—22no—
275C2×C4⋊C4—22no—
276C2×C4⋊C4—28no—
277C2×C4⋊C4—42no—
278C2×C4⋊C4—44no—
279C2×C4⋊C4—44no—
280C2×C4⋊C4—44no—
281C2×C4⋊C4—44no—
282C2×C4⋊C4—44no—
283C2×C4⋊C4—44no—
284C2×C4⋊C4C2×D414yes—
285C2×C4⋊C4C2×D418yes—
286C2×C4⋊C4C2×D418yes—
287C2×C4⋊C4C2×D414yes—
288C2×C4⋊C4#4118yes—
289C2×C22⋊C4—24no—
290C2×C22⋊C4—24no—
291C2×C22⋊C4—24no—
292C2×C22⋊C4—24no—
293C2×C22⋊C4—22no—
294C2×C22⋊C4—22no—
295C2×C22⋊C4—22no—
296C2×C22⋊C4—22no—
297C2×C22⋊C4—28no—
298C2×C22⋊C4—28no—
299C2×C22⋊C4—24no—
300C2×C22⋊C4—24no—
301C2×C22⋊C4—22no—
302C2×C22⋊C4—22no—
303C2×C22⋊C4—28no—
304C2×C22⋊C4—28no—
305C2×C22⋊C4—24no—
306C2×C22⋊C4—24no—
307C2×C22⋊C4—24no—
308C2×C22⋊C4—24no—
309C2×C22⋊C4—42no—
310C2×C22⋊C4—42no—
311C2×C22⋊C4—42no—
312C2×C22⋊C4—42no—
313C2×C22⋊C4—42no—
314C2×C22⋊C4—42no—
315C2×C22⋊C4—44no—
316C2×C22⋊C4—44no—
317C2×C22⋊C4—42no—
318C2×C22⋊C4—42no—
319C2×C22⋊C4—42no—
320C2×C22⋊C4—42no—
321C2×C22⋊C4—44no—
322C2×C22⋊C4—44no—
323C2×C42—28no—
324C2×C42—24no—
325C2×C42—24no—
326C2×C42—24no—
327C2×C42—24no—
328C2×C42—24no—
329C2×C42—24no—
330C2×C42—24no—
331C2×C42—44no—
332C2×C42—44no—
333C2×C42—44no—
334C2×C42—42no—
335C2×C42—42no—
336C2×C42C2×D418yes—
337C2×C42C2×D414yes—
338C2×C42C2×D414yes—
339C2×C42#4114yes—
340#95—22no—
341#95—22no—
342#95—24no—
343#95—24no—
344#95—28no—
345#95—24no—
346#95—24no—
347#95—24no—
348#95—24no—
349#95—24no—
350#95—28no—
351#95—28no—
352#95—28no—
353#95—28no—
354#95—28no—
355#95—28no—
356#95—28no—
357#95—44no—
358#95—44no—
359#95—44no—
360#95—44no—
361#95—44no—
362#95—42no—
363#95—42no—
364#95—42no—
365#95—42no—
366#95C2×D418yes—
367C24—21no—
368C24—21no—
369C24—21no—
370C24—21no—
371C24—22no—
372C24—22no—
373C24—21no—
374C24—21no—
375C24—22no—
376C24—22no—
377C24—22no—
378C24—22no—
379C24—21no—
380C24—21no—
381C24—22no—
382C24—22no—
383C24—21no—
384C24—21no—
385C24—21no—
386C24—21no—
387C242−1+412yes—
388C242−1+412yes—
389C24C2511yescharacteristic
390C2×Q8—216no—
391C2×Q8—216no—
392C2×Q8—216no—
393C2×Q8—48no—
394C2×Q8—48no—
395C2×Q8—48no—
396C2×Q8—48no—
397C2×D4—28no—
398C2×D4—44no—
399C2×D4—44no—
400C2×D4—44no—
401C2×D4—44no—
402C2×D4—44no—
403C2×D4—44no—
404C22×C4—22no—
405C22×C4—21no—
406C22×C4—21no—
407C22×C4—22no—
408C22×C4—21no—
409C22×C4—21no—
410C22×C4—22no—
411C22×C4—24no—
412C22×C4—24no—
413C22×C4—24no—
414C22×C4—24no—
415C22×C4—24no—
416C22×C4—24no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—24no—
422C22×C4—24no—
423C22×C4—24no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—24no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—24no—
432C22×C4—24no—
433C22×C4—24no—
434C22×C4—24no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—21no—
439C22×C4—21no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—21no—
446C22×C4—21no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—21no—
450C22×C4—21no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—24no—
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—22no—
475C22×C4—24no—
476C22×C4—22no—
477C22×C4—22no—
478C22×C4—22no—
479C22×C4—44no—
480C22×C4—44no—
481C22×C4—44no—
482C22×C4—44no—
483C22×C4—44no—
484C22×C4—44no—
485C22×C4—48no—
486C22×C4—48no—
487C22×C4—48no—
488C22×C4—48no—
489C22×C4C22×D414yes—
490C22×C4C22×D418yes—
491C22×C4C22×D414yes—
492C22×C4C22×D412yes—
493C22×C4C22×D412yes—
494C22×C4C22×D412yes—
495C22×C4C22×D412yes—
496C22×C4C22×D412yes—
497C22×C4C22×D412yes—
498C22×C4C2×#4118yes—
499C22×C4C2×#4112yes—
500C22×C4C2×#4112yes—
501C22×C4C2×#4112yes—
502C22×C4C2×#4114yes—
503C22×C4H(5,𝔽2)14yes—
504C22×C42−1+414yes—
505C4⋊C4—28no—
506C4⋊C4—216no—
507C4⋊C4—216no—
508C4⋊C4—28no—
509C4⋊C4—28no—
510C4⋊C4—216no—
511C4⋊C4—416no—
512C4⋊C4—416no—
513C4⋊C4—44no—
514C4⋊C4—416no—
515C4⋊C4—44no—
516C4⋊C4—48no—
517C4⋊C4—48no—
518C4⋊C4—44no—
519C4⋊C4—44no—
520C4⋊C4—88no—
521C22⋊C4—416no—
522C22⋊C4—416no—
523C22⋊C4—48no—
524C22⋊C4—48no—
525C22⋊C4—48no—
526C22⋊C4—48no—
527C22⋊C4—48no—
528C22⋊C4—48no—
529C22⋊C4—44no—
530C22⋊C4—44no—
531C22⋊C4—84no—
532C22⋊C4—84no—
533C42—48no—
534C42—48no—
535C42—416no—
536C42—48no—
537C42—48no—
538C42#121116yes—
539C23—22no—
540C23—22no—
541C23—22no—
542C23—22no—
543C23—22no—
544C23—22no—
545C23—24no—
546C23—24no—
547C23—24no—
548C23—24no—
549C23—41no—
550C23—41no—
551C23—41no—
552C23—41no—
553C23—41no—
554C23—41no—
555C23—42no—
556C23—42no—
557C23—42no—
558C23—42no—
559C23—42no—
560C23—42no—
561C23—41no—
562C23—41no—
563C23—41no—
564C23—41no—
565C23—41no—
566C23—41no—
567C23—41no—
568C23—41no—
569C23—41no—
570C23—41no—
571C23—42no—
572C23—42no—
573C23—42no—
574C23—42no—
575C23—42no—
576C23—42no—
577C23—42no—
578C23—42no—
579C23—42no—
580C23—42no—
581C23—42no—
582C23—42no—
583C23—42no—
584C23—42no—
585C23C23×D411yescharacteristic
586C23C23×D411yescharacteristic
587C23C22×#4111yescharacteristic
588C23C22×#4112yes—
589C23C2×H(5,𝔽2)12yes—
590C23C2×2−1+412yes—
591C23C2×2−1+411yescharacteristic
592C23C2×2−1+411yescharacteristic
593C23#58512yes—
594C23#58511yescharacteristic
595C23#58511yescharacteristic
596Q8—432no—
597D4—416no—
598C2×C4—24no—
599C2×C4—22no—
600C2×C4—22no—
601C2×C4—22no—
602C2×C4—22no—
603C2×C4—22no—
604C2×C4—22no—
605C2×C4—22no—
606C2×C4—22no—
607C2×C4—24no—
608C2×C4—28no—
609C2×C4—24no—
610C2×C4—24no—
611C2×C4—28no—
612C2×C4—28no—
613C2×C4—28no—
614C2×C4—24no—
615C2×C4—24no—
616C2×C4—24no—
617C2×C4—24no—
618C2×C4—24no—
619C2×C4—28no—
620C2×C4—28no—
621C2×C4—24no—
622C2×C4—24no—
623C2×C4—44no—
624C2×C4—42no—
625C2×C4—44no—
626C2×C4—44no—
627C2×C4—44no—
628C2×C4—44no—
629C2×C4—42no—
630C2×C4—44no—
631C2×C4—44no—
632C2×C4—44no—
633C2×C4—42no—
634C2×C4—44no—
635C2×C4—44no—
636C2×C4—42no—
637C2×C4—44no—
638C2×C4—44no—
639C2×C4—41no—
640C2×C4—41no—
641C2×C4—42no—
642C2×C4—42no—
643C2×C4—41no—
644C2×C4—41no—
645C2×C4—42no—
646C2×C4—41no—
647C2×C4—41no—
648C2×C4—44no—
649C2×C4—42no—
650C2×C4—42no—
651C2×C4—42no—
652C2×C4—42no—
653C2×C4—42no—
654C2×C4—42no—
655C2×C4—42no—
656C2×C4—42no—
657C2×C4—42no—
658C2×C4—42no—
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—44no—
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—42no—
677C2×C4—42no—
678C2×C4—42no—
679C2×C4—42no—
680C2×C4—42no—
681C2×C4—42no—
682C2×C4—88no—
683C2×C4—88no—
684C2×C4C2×#12114yes—
685C2×C4C2×#12114yes—
686C2×C4C2×#12114yes—
687C2×C4#53214yes—
688C2×C4#53514yes—
689C2×C4#54618yes—
690C2×C4#55918yes—
691C22—42no—
692C22—42no—
693C22—42no—
694C22—42no—
695C22—44no—
696C22—44no—
697C22—42no—
698C22—42no—
699C22—44no—
700C22—44no—
701C22—81no—
702C22—81no—
703C22—81no—
704C22—81no—
705C22—81no—
706C22—81no—
707C22—82no—
708C22—82no—
709C22—81no—
710C22—81no—
711C22—82no—
712C22—82no—
713C22C22×#12111yescharacteristic
714C22C2×#53211yescharacteristic
715C22C2×#53511yescharacteristic
716C22#346112yes—
717C22C2×#54612yes—
718C22C2×#55912yes—
719C22#348511yescharacteristic
720C22#348611yescharacteristic
721C22#348611yescharacteristic
722C22#348711yescharacteristic
723C22#348911yescharacteristic
724C22#348911yescharacteristic
725C22#349511yescharacteristic
726C22#349511yescharacteristic
727C22#349712yes—
728C22#351512yes—
729C22#351612yes—
730C22#352112yes—
731C22#352312yes—
732C22#353812yes—
733C22#353912yes—
734C22#354112yes—
735C22#354112yes—
736C4—24no—
737C4—24no—
738C4—44no—
739C4—44no—
740C4—48no—
741C4—44no—
742C4—48no—
743C4—81no—
744C4—81no—
745C4—82no—
746C4—82no—
747C4—82no—
748C4—82no—
749C4—82no—
750C4#254418yes—
751C2—82no—
752C2—82no—
753C2C2×#254411yescharacteristic
754C2#3515311yescharacteristic
755C2#3566111yescharacteristic
756C2#3585611yescharacteristic
757C2#3594911yescharacteristic
758C2#3927112yes—
759C2#3927412yes—
760C2#4267911yescharacteristic
761C2#4280811yescharacteristic
762C2#4524312yes—
763C2#4524912yes—
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
3. C2×#2642C2semidirect
4. C2×#2365C2semidirect
5. #17039C2semidirect
6. #16899C2semidirect
8. #15789C2semidirect
9. #15179C2semidirect
10. #15179C2semidirect
11. #15175C2semidirect
12. #15103C2semidirect
13. #14160C2semidirect
14. #13969C2semidirect
15. #13969C2semidirect
17. #13783C2semidirect

Maximal subgroups

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

subgroupindex
C2×#25422
C2×#25622
C2×#26422
C2×#23652
#170392
#168992
#158152
#157892
#151792
#151752
#151032
#141602
#139692
#138472
#137832