Finite Group Databasebeta

← #713498

Subgroups — #713498

Order 512.

2544 conjugacy classes of subgroups.

The subgroup lattice has 734 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2354C211yesmaximal, characteristic
2C2×#2316C211yesmaximal, characteristic, complemented
3#16982C212yesmaximal, complemented
4#16980C212yesmaximal, complemented
5#15992C214yesmaximal, complemented
6#15326C214yesmaximal
7#15220C212yesmaximal, complemented
8#15166C212yesmaximal, complemented
9#15137C212yesmaximal, complemented
10#15108C212yesmaximal, complemented
11#14264C211yesmaximal, characteristic, complemented
12#14129C212yesmaximal, complemented
13#14030C212yesmaximal, complemented
14#13951C211yesmaximal, characteristic, complemented
15#13881C211yesmaximal, characteristic, complemented
16#13872C211yesmaximal, characteristic
17#13839C211yesmaximal, characteristic
18C22×C4×D4C2211yescharacteristic
19C22×#117C2211yescharacteristic
20C22×#117C2211yescharacteristic
21C22×#117C2211yescharacteristic
22C23×C4⋊C4C2211yescharacteristic
23C2×#398C2214yes—
24#2354—28no—
25#2316—28no—
26C2×#390C2212yes—
27C2×#389C2212yes—
28C2×#389C2212yes—
29C2×#389C2212yes—
30C2×#389C2212yes—
31C2×#389C2212yes—
32C2×#389C2212yes—
33C2×#388C2212yes—
34C2×#388C2212yes—
35C2×#386C2212yes—
36C2×#384C2211yescharacteristic
37C2×#382C2212yes—
38C2×#380C2214yes—
39C2×#380C2212yes—
40C2×#380C2211yescharacteristic
41C2×#380C2211yescharacteristic
42C2×#380C2211yescharacteristic
43C2×C4×C4⋊C4C2212yes—
44C2×C4×C4⋊C4C2211yescharacteristic
45C2×C4×C22⋊C4C2211yescharacteristic
46C2×C4×C22⋊C4C2212yes—
47#1444C2214yes—
48#1444C2212yes—
49#1443C2214yes—
50#1443C2214yes—
51#1442C2214yes—
52#1442C2212yes—
53#1442C2212yes—
54#1442C2212yes—
55#1442C2214yes—
56#1441C2211yescharacteristic
57#1441C2212yes—
58#1441C2211yescharacteristic
59#1441C2214yes—
60#1441C2214yes—
61#1440C2212yes—
62#1439C2211yescharacteristic
63#1439C2211yescharacteristic
64#1439C2212yes—
65#1439C2214yes—
66#1439C2214yes—
67#1439C2214yes—
68#1439C2214yes—
69#1439C2214yes—
70#1439C2214yes—
71#1438C2212yes—
72#1436C2214yes—
73#1436C2214yes—
74#1436C2214yes—
75C4⋊#95C2212yes—
76C4⋊#95C2212yes—
77C4⋊#95C2212yes—
78C4⋊#95C2212yes—
79C4⋊#95C2212yes—
80#1434C2212yes—
81#1434C2214yes—
82C22⋊#95C2212yes—
83C22⋊#95C2214yes—
84C4×#95C2212yes—
85C23×D4C2311yescharacteristic
86C24×C4C2311yescharacteristic
87C24×C4C2311yescharacteristic
88C24×C4C2312yes—
89C2×C4×D4—24no—
90C2×C4×D4—24no—
91C2×C4×D4Q818yes—
92C2×#117—24no—
93C2×#117—24no—
94C2×#117—44no—
95C2×#117—44no—
96C2×#117—44no—
97C2×#117D418yes—
98C22×C4⋊C4—24no—
99C22×C4⋊C4—28no—
100C22×C4⋊C4C2312yes—
101C22×C4⋊C4C2311yescharacteristic
102C22×C4⋊C4C2312yes—
103C22×C4⋊C4C2311yescharacteristic
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2311yescharacteristic
106C22×C4⋊C4C2311yescharacteristic
107C22×C4⋊C4C2311yescharacteristic
108C22×C4⋊C4C2312yes—
109C22×C4⋊C4C2314yes—
110C22×C4⋊C4C2311yescharacteristic
111C22×C4⋊C4C2311yescharacteristic
112C22×C4⋊C4C2314yes—
113C22×C4⋊C4C2312yes—
114C22×C4⋊C4C2314yes—
115C22×C4⋊C4C2314yes—
116C22×C22⋊C4C2311yescharacteristic
117C22×C22⋊C4C2311yescharacteristic
118C22×C22⋊C4C2312yes—
119C22×C22⋊C4C2312yes—
120C22×C22⋊C4C2312yes—
121C22×C22⋊C4C2312yes—
122C22×C22⋊C4C2312yes—
123C22×C22⋊C4C2314yes—
124C22×C22⋊C4C2312yes—
125C22×C22⋊C4C2312yes—
126C22×C22⋊C4C2312yes—
127C22×C22⋊C4C2312yes—
128C22×C42C2312yes—
129C22×C42C2312yes—
130C22×C42C2312yes—
131C22×C42C2311yescharacteristic
132C22×C42C2312yes—
133C22×C42C2312yes—
134#398—48no—
135#390—28no—
136#389—28no—
137#389—28no—
138#389—28no—
139#389—28no—
140#389—44no—
141#389—44no—
142#388—28no—
143#388—28no—
144#386—44no—
145#384—24no—
146#382—28no—
147#380—24no—
148#380—24no—
149#380—24no—
150#380—28no—
151#380—216no—
152C4×C4⋊C4—24no—
153C4×C4⋊C4—28no—
154C4×C22⋊C4—24no—
155C4×C22⋊C4—44no—
156C2×#95C2312yes—
157C2×#95C2314yes—
158C2×#95C2312yes—
159C2×#95C2314yes—
160C2×#95C2314yes—
161C2×#95C2314yes—
162C2×#95C2311yescharacteristic
163C2×#95C2312yes—
164C2×#95C2314yes—
165C2×#95C2311yescharacteristic
166C2×#95C2314yes—
167C2×#95C2312yes—
168C2×#95C2312yes—
169C2×#95C2312yes—
170C2×#95C2314yes—
171C2×#95C2312yes—
172C2×#95C2312yes—
173C2×#95C2314yes—
174C2×#95C2314yes—
175C2×#95C2314yes—
176C2×#95C2314yes—
177C2×#95C2314yes—
178C2×#95C2314yes—
179C2×#95C2314yes—
180C2×#95C2312yes—
181C2×#95C2312yes—
182C2×#95C2314yes—
183C25C2411yescharacteristic
184C25C2411yescharacteristic
185C22×D4—22no—
186C22×D4—22no—
187C22×D4—22no—
188C22×D4—22no—
189C22×D4—41no—
190C22×D4—41no—
191C22×D4C2×Q814yes—
192C23×C4—24no—
193C23×C4—24no—
194C23×C4—24no—
195C23×C4—22no—
196C23×C4—22no—
197C23×C4—24no—
198C23×C4—24no—
199C23×C4—28no—
200C23×C4—42no—
201C23×C4—44no—
202C23×C4—42no—
203C23×C4C2×D414yes—
204C23×C4C2×Q814yes—
205C23×C4C2×Q814yes—
206C23×C4#4114yes—
207C23×C4C2412yes—
208C23×C4C2412yes—
209C23×C4C2414yes—
210C23×C4C2412yes—
211C23×C4C2412yes—
212C23×C4C2412yes—
213C23×C4C2412yes—
214C23×C4C2412yes—
215C23×C4C2412yes—
216C23×C4C2411yescharacteristic
217C23×C4C2411yescharacteristic
218C23×C4C2414yes—
219C23×C4C2411yescharacteristic
220C23×C4C2411yescharacteristic
221C23×C4C2411yescharacteristic
222C4×D4—232no—
223#117—416no—
224#117—416no—
225#117—416no—
226C2×C4⋊C4—232no—
227C2×C4⋊C4—24no—
228C2×C4⋊C4—24no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—28no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—28no—
247C2×C4⋊C4—28no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—42no—
250C2×C4⋊C4—42no—
251C2×C4⋊C4—42no—
252C2×C4⋊C4—44no—
253C2×C4⋊C4—44no—
254C2×C4⋊C4—44no—
255C2×C4⋊C4—44no—
256C2×C4⋊C4—44no—
257C2×C4⋊C4—44no—
258C2×C4⋊C4—44no—
259C2×C4⋊C4—44no—
260C2×C4⋊C4—44no—
261C2×C4⋊C4—44no—
262C2×C4⋊C4—44no—
263C2×C4⋊C4C2×D414yes—
264C2×C4⋊C4C2×D414yes—
265C2×C4⋊C4C2×Q814yes—
266C2×C4⋊C4#4118yes—
267C2×C22⋊C4—22no—
268C2×C22⋊C4—22no—
269C2×C22⋊C4—22no—
270C2×C22⋊C4—22no—
271C2×C22⋊C4—24no—
272C2×C22⋊C4—24no—
273C2×C22⋊C4—24no—
274C2×C22⋊C4—24no—
275C2×C22⋊C4—24no—
276C2×C22⋊C4—24no—
277C2×C22⋊C4—28no—
278C2×C22⋊C4—42no—
279C2×C22⋊C4—42no—
280C2×C22⋊C4—42no—
281C2×C22⋊C4—42no—
282C2×C22⋊C4—42no—
283C2×C22⋊C4—42no—
284C2×C22⋊C4—44no—
285C2×C22⋊C4—44no—
286C2×C22⋊C4—42no—
287C2×C22⋊C4—42no—
288C2×C22⋊C4—42no—
289C2×C22⋊C4—42no—
290C2×C22⋊C4—44no—
291C2×C22⋊C4—44no—
292C2×C22⋊C4—42no—
293C2×C22⋊C4—44no—
294C2×C22⋊C4—44no—
295C2×C22⋊C4—42no—
296C2×C22⋊C4C2×D418yes—
297C2×C22⋊C4C2×Q814yes—
298C2×C22⋊C4C2×Q814yes—
299C2×C42—22no—
300C2×C42—22no—
301C2×C42—24no—
302C2×C42—24no—
303C2×C42—24no—
304C2×C42—24no—
305C2×C42—24no—
306C2×C42—24no—
307C2×C42—24no—
308C2×C42—24no—
309C2×C42—24no—
310C2×C42—44no—
311C2×C42—44no—
312C2×C42—42no—
313C2×C42C2×D418yes—
314C2×C42C2×Q814yes—
315#95—24no—
316#95—24no—
317#95—24no—
318#95—28no—
319#95—28no—
320#95—22no—
321#95—22no—
322#95—28no—
323#95—24no—
324#95—28no—
325#95—28no—
326#95—28no—
327#95—24no—
328#95—24no—
329#95—44no—
330#95—42no—
331#95—44no—
332#95—42no—
333#95—44no—
334#95—44no—
335#95—44no—
336#95—44no—
337#95—42no—
338#95—42no—
339#95—44no—
340#95—44no—
341#95—44no—
342C24—22no—
343C24—22no—
344C24—21no—
345C24—21no—
346C24—22no—
347C24—21no—
348C24—22no—
349C24—21no—
350C24—21no—
351C24—22no—
352C24—21no—
353C24—22no—
354C24—21no—
355C24—21no—
356C24—22no—
357C24—22no—
358C24—21no—
359C24—21no—
360C24C22×D412yes—
361C24C22×Q812yes—
362C24C22×Q812yes—
363C24C2×#4112yes—
364C24C2511yescharacteristic
365C2×D4—28no—
366C2×D4—28no—
367C2×D4—44no—
368C2×D4—44no—
369C2×D4—44no—
370C2×D4—44no—
371C2×D4—44no—
372C22×C4—21no—
373C22×C4—21no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—21no—
377C22×C4—21no—
378C22×C4—28no—
379C22×C4—28no—
380C22×C4—216no—
381C22×C4—216no—
382C22×C4—21no—
383C22×C4—21no—
384C22×C4—24no—
385C22×C4—22no—
386C22×C4—21no—
387C22×C4—21no—
388C22×C4—24no—
389C22×C4—24no—
390C22×C4—22no—
391C22×C4—24no—
392C22×C4—24no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—24no—
398C22×C4—22no—
399C22×C4—24no—
400C22×C4—24no—
401C22×C4—24no—
402C22×C4—24no—
403C22×C4—22no—
404C22×C4—216no—
405C22×C4—28no—
406C22×C4—216no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—24no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—24no—
417C22×C4—24no—
418C22×C4—22no—
419C22×C4—24no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
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—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—24no—
438C22×C4—24no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—24no—
445C22×C4—24no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—24no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—44no—
453C22×C4—44no—
454C22×C4—48no—
455C22×C4—88no—
456C22×C4—88no—
457C22×C4—84no—
458C22×C4—84no—
459C22×C4C22⋊Q8116yes—
460C22×C4C22×D414yes—
461C22×C4C22×D414yes—
462C22×C4C22×D412yes—
463C22×C4C22×D412yes—
464C22×C4C22×Q812yes—
465C22×C4C22×Q812yes—
466C22×C4C22×Q812yes—
467C22×C4C22×Q812yes—
468C22×C4C22×Q812yes—
469C22×C4C2×#4118yes—
470C22×C4C2×#4112yes—
471C22×C4C2×#4112yes—
472C22×C4C2×#4114yes—
473C22×C4H(5,𝔽2)12yes—
474C22×C4H(5,𝔽2)14yes—
475C22×C42−1+412yes—
476C4⋊C4—216no—
477C4⋊C4—28no—
478C4⋊C4—216no—
479C4⋊C4—28no—
480C4⋊C4—28no—
481C4⋊C4—48no—
482C4⋊C4—44no—
483C4⋊C4—44no—
484C4⋊C4—44no—
485C4⋊C4—44no—
486C4⋊C4—48no—
487C4⋊C4—88no—
488C4⋊C4—84no—
489C4⋊C4—88no—
490C4⋊C4—88no—
491C4⋊C4—88no—
492C22⋊C4—28no—
493C22⋊C4—28no—
494C22⋊C4—48no—
495C22⋊C4—48no—
496C22⋊C4—48no—
497C22⋊C4—48no—
498C22⋊C4—48no—
499C22⋊C4—48no—
500C22⋊C4—84no—
501C22⋊C4—88no—
502C22⋊C4—84no—
503C22⋊C4—84no—
504C42—28no—
505C42—48no—
506C42—48no—
507C42—48no—
508C42—48no—
509C42—48no—
510C23—22no—
511C23—22no—
512C23—22no—
513C23—22no—
514C23—24no—
515C23—24no—
516C23—22no—
517C23—22no—
518C23—24no—
519C23—22no—
520C23—22no—
521C23—22no—
522C23—22no—
523C23—24no—
524C23—22no—
525C23—22no—
526C23—22no—
527C23—24no—
528C23—24no—
529C23—42no—
530C23—42no—
531C23—42no—
532C23—42no—
533C23—42no—
534C23—42no—
535C23—42no—
536C23—42no—
537C23—41no—
538C23—41no—
539C23—42no—
540C23—41no—
541C23—41no—
542C23—42no—
543C23—41no—
544C23—41no—
545C23—41no—
546C23—41no—
547C23—42no—
548C23—42no—
549C23—42no—
550C23—42no—
551C23—42no—
552C23—42no—
553C23—42no—
554C23—42no—
555C23C2×C22⋊Q814yes—
556C23C23×D411yescharacteristic
557C23C23×Q811yescharacteristic
558C23C22×#4111yescharacteristic
559C23C22×#4112yes—
560C23C2×H(5,𝔽2)12yes—
561C23C2×H(5,𝔽2)11yescharacteristic
562C23C2×2−1+411yescharacteristic
563C23#58511yescharacteristic
564C23#58511yescharacteristic
565C23#58512yes—
566C23#58512yes—
567D4—416no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—24no—
573C2×C4—24no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—232no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—28no—
582C2×C4—24no—
583C2×C4—24no—
584C2×C4—24no—
585C2×C4—24no—
586C2×C4—24no—
587C2×C4—24no—
588C2×C4—24no—
589C2×C4—24no—
590C2×C4—28no—
591C2×C4—28no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—42no—
595C2×C4—44no—
596C2×C4—42no—
597C2×C4—41no—
598C2×C4—44no—
599C2×C4—41no—
600C2×C4—44no—
601C2×C4—44no—
602C2×C4—44no—
603C2×C4—44no—
604C2×C4—44no—
605C2×C4—44no—
606C2×C4—416no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—44no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—44no—
621C2×C4—44no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—44no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4—44no—
632C2×C4—42no—
633C2×C4—42no—
634C2×C4—44no—
635C2×C4—44no—
636C2×C4—42no—
637C2×C4—42no—
638C2×C4—42no—
639C2×C4—42no—
640C2×C4—44no—
641C2×C4—44no—
642C2×C4—42no—
643C2×C4—42no—
644C2×C4—42no—
645C2×C4—44no—
646C2×C4—42no—
647C2×C4—44no—
648C2×C4—42no—
649C2×C4—42no—
650C2×C4—42no—
651C2×C4—42no—
652C2×C4—816no—
653C2×C4C2×C22⋊Q814yes—
654C2×C4C2×C22⋊Q814yes—
655C2×C4#52514yes—
656C2×C4#53414yes—
657C2×C4#54414yes—
658C2×C4#55318yes—
659C2×C4#55418yes—
660C22—24no—
661C22—24no—
662C22—24no—
663C22—44no—
664C22—44no—
665C22—44no—
666C22—44no—
667C22—42no—
668C22—42no—
669C22—42no—
670C22—42no—
671C22—42no—
672C22—42no—
673C22—44no—
674C22—42no—
675C22—44no—
676C22—82no—
677C22—82no—
678C22—81no—
679C22—81no—
680C22—81no—
681C22—81no—
682C22C22×C22⋊Q811yescharacteristic
683C22C2×#52511yescharacteristic
684C22C2×#53411yescharacteristic
685C22C2×#54411yescharacteristic
686C22#345811yescharacteristic
687C22C2×#55312yes—
688C22C2×#55412yes—
689C22#348811yescharacteristic
690C22#348911yescharacteristic
691C22#349312yes—
692C22#349411yescharacteristic
693C22#349511yescharacteristic
694C22#350011yescharacteristic
695C22#350011yescharacteristic
696C22#350412yes—
697C22#351412yes—
698C22#351812yes—
699C22#352112yes—
700C22#352412yes—
701C22#353912yes—
702C22#353912yes—
703C22#354612yes—
704C22#354712yes—
705C4—24no—
706C4—24no—
707C4—42no—
708C4—42no—
709C4—44no—
710C4—48no—
711C4—44no—
712C4—82no—
713C4—82no—
714C4—84no—
715C4—82no—
716C4—82no—
717C4—82no—
718C4—82no—
719C4#257918yes—
720C2—44no—
721C2—82no—
722C2C2×#257911yescharacteristic
723C2#3525311yescharacteristic
724C2#3562011yescharacteristic
725C2#3607511yescharacteristic
726C2#3607711yescharacteristic
727C2#3810712yes—
728C2#3896712yes—
729C2#4298511yescharacteristic
730C2#4299211yescharacteristic
731C2#4539612yes—
732C2#4539812yes—
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×#2316C2semidirect
3. #16982C2semidirect
4. #16980C2semidirect
5. #15992C2semidirect
7. #15220C2semidirect
8. #15166C2semidirect
9. #15137C2semidirect
10. #15108C2semidirect
11. #14264C2semidirect
12. #14129C2semidirect
13. #14030C2semidirect
14. #13951C2semidirect
15. #13881C2semidirect

Maximal subgroups

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

subgroupindex
C2×#23542
C2×#23162
#169822
#169802
#159922
#153262
#152202
#151662
#151372
#151082
#142642
#141292
#140302
#139512
#138812
#138722
#138392