Finite Group Databasebeta

← #616605

Subgroups — #616605

Order 512.

2990 conjugacy classes of subgroups.

The subgroup lattice has 733 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2863C211yesmaximal, characteristic, complemented
2C2×C42×D4C211yesmaximal, characteristic
3#15246C214yesmaximal, complemented
4#15212C212yesmaximal, complemented
5#15203C212yesmaximal
6#15194C211yesmaximal, characteristic, complemented
7#15192C211yesmaximal, characteristic, complemented
8#15181C211yesmaximal, characteristic
9#15177C211yesmaximal, characteristic
10#14702C212yesmaximal, complemented
11#14697C212yesmaximal, complemented
12#14075C214yesmaximal, complemented
13#13997C212yesmaximal, complemented
14#13973C212yesmaximal, complemented
15#13959C212yesmaximal
16#13952C212yesmaximal, complemented
17#13755C211yesmaximal, characteristic, complemented
18C22×C22⋊Q8C2212yes—
19C22×C4×D4C2211yescharacteristic
20C22×C4×D4C2211yescharacteristic
21C22×C4×D4C2211yescharacteristic
22C23×C42C2212yes—
23#2863—28no—
24C2×#391C2211yescharacteristic
25C2×#391C2211yescharacteristic
26C2×#391C2211yescharacteristic
27C2×#391C2212yes—
28C2×#389C2211yescharacteristic
29C2×#389C2211yescharacteristic
30C2×#389C2211yescharacteristic
31C2×#389C2212yes—
32C2×#387C2212yes—
33C2×#387C2212yes—
34C2×#387C2212yes—
35C2×#385C2214yes—
36C2×#383C2212yes—
37C2×#380C2212yes—
38C2×#380C2212yes—
39C2×#380C2212yes—
40C42×D4—28no—
41C2×C4×C4⋊C4C2211yescharacteristic
42C2×C4×C4⋊C4C2211yescharacteristic
43C2×C4×C4⋊C4C2211yescharacteristic
44C2×C4×C4⋊C4C2212yes—
45C2×C4×C22⋊C4C2212yes—
46C2×C4×C22⋊C4C2212yes—
47C2×C4×C22⋊C4C2212yes—
48C2×#376C2212yes—
49C2×C43C2211yescharacteristic
50#1445C2212yes—
51#1445C2212yes—
52#1445C2214yes—
53#1444C2212yes—
54#1444C2212yes—
55#1443C2214yes—
56#1442C2212yes—
57#1442C2212yes—
58#1442C2214yes—
59#1441C2214yes—
60#1439C2214yes—
61#1439C2214yes—
62#1439C2214yes—
63#1439C2214yes—
64#1438C2212yes—
65#1438C2212yes—
66#1437C2214yes—
67#1437C2214yes—
68#1436C2214yes—
69#1436C2214yes—
70#1436C2214yes—
71#1436C2214yes—
72C4⋊#95C2211yescharacteristic
73C4⋊#95C2211yescharacteristic
74C4⋊#95C2212yes—
75C4⋊#95C2212yes—
76C4⋊#95C2212yes—
77C4⋊#95C2214yes—
78#1434C2212yes—
79#1434C2214yes—
80#1434C2212yes—
81C22⋊#95C2214yes—
82C22⋊#95C2214yes—
83C22⋊#95C2214yes—
84C4×#95C2212yes—
85C23×Q8C2311yescharacteristic
86C23×D4C2311yescharacteristic
87C24×C4C2312yes—
88C24×C4C2312yes—
89C24×C4C2312yes—
90C2×C22⋊Q8—28no—
91C2×C22⋊Q8—28no—
92C2×C22⋊Q8—44no—
93C2×C4×D4—28no—
94C2×C4×D4—28no—
95C2×C4×D4—24no—
96C2×C4×D4—24no—
97C2×C4×D4—24no—
98C2×C4×D4—24no—
99C2×C4×D4Q818yescomplemented
100C22×C4⋊C4C2311yescharacteristic
101C22×C4⋊C4C2311yescharacteristic
102C22×C4⋊C4C2312yes—
103C22×C4⋊C4C2311yescharacteristic
104C22×C4⋊C4C2311yescharacteristic
105C22×C4⋊C4C2312yes—
106C22×C4⋊C4C2311yescharacteristic
107C22×C4⋊C4C2314yes—
108C22×C4⋊C4C2311yescharacteristic
109C22×C4⋊C4C2312yes—
110C22×C4⋊C4C2314yes—
111C22×C4⋊C4C2312yes—
112C22×C4⋊C4C2312yes—
113C22×C22⋊C4C2312yes—
114C22×C22⋊C4C2312yes—
115C22×C22⋊C4C2314yes—
116C22×C22⋊C4C2314yes—
117C22×C22⋊C4C2312yes—
118C22×C22⋊C4C2314yes—
119C22×C22⋊C4C2314yes—
120C22×C42—28no—
121C22×C42—44no—
122C22×C42—44no—
123C22×C42C2312yes—
124C22×C42C2314yes—
125C22×C42C2311yescharacteristic
126C22×C42C2311yescharacteristic
127C22×C42C2311yescharacteristic
128C22×C42C2311yescharacteristic
129C22×C42C2312yes—
130C22×C42C2312yes—
131C22×C42C2311yescharacteristic
132C22×C42C2311yescharacteristic
133C22×C42C2311yescharacteristic
134#391—24no—
135#391—24no—
136#391—24no—
137#391—44no—
138#389—24no—
139#389—24no—
140#389—24no—
141#389—44no—
142#387—28no—
143#387—28no—
144#387—28no—
145#385—216no—
146#383—28no—
147#380—28no—
148#380—28no—
149#380—44no—
150C4×C4⋊C4—28no—
151C4×C4⋊C4—24no—
152C4×C4⋊C4—24no—
153C4×C4⋊C4—24no—
154C4×C22⋊C4—28no—
155C4×C22⋊C4—28no—
156C4×C22⋊C4—28no—
157#376—28no—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2312yes—
161C2×#95C2312yes—
162C2×#95C2312yes—
163C2×#95C2312yes—
164C2×#95C2312yes—
165C2×#95C2314yes—
166C2×#95C2314yes—
167C2×#95C2312yes—
168C2×#95C2312yes—
169C2×#95C2314yes—
170C2×#95C2314yes—
171C2×#95C2312yes—
172C2×#95C2312yes—
173C2×#95C2312yes—
174C2×#95C2312yes—
175C2×#95C2312yes—
176C2×#95C2312yes—
177C2×#95C2312yes—
178C2×#95C2312yes—
179C2×#95C2312yes—
180C2×#95C2314yes—
181C2×#95C2314yes—
182C2×#95C2314yes—
183C2×#95C2314yes—
184C2×#95C2314yes—
185C2×#95C2314yes—
186C2×#95C2314yes—
187C43D418yes—
188C25C2412yes—
189C22×Q8—22no—
190C22×Q8—24no—
191C22×Q8—24no—
192C22×Q8—22no—
193C22×Q8—22no—
194C22×D4—24no—
195C22×D4—24no—
196C22×D4—22no—
197C22×D4—22no—
198C22×D4C2×Q814yes—
199C23×C4—24no—
200C23×C4—24no—
201C23×C4—24no—
202C23×C4—24no—
203C23×C4—24no—
204C23×C4—24no—
205C23×C4—24no—
206C23×C4—24no—
207C23×C4—24no—
208C23×C4—24no—
209C23×C4—42no—
210C23×C4—42no—
211C23×C4—42no—
212C23×C4—42no—
213C23×C4—42no—
214C23×C4—42no—
215C23×C4—42no—
216C23×C4—42no—
217C23×C4C2×Q818yes—
218C23×C4#4118yes—
219C23×C4#4118yes—
220C23×C4C2412yes—
221C23×C4C2412yes—
222C23×C4C2412yes—
223C23×C4C2412yes—
224C23×C4C2414yes—
225C23×C4C2411yescharacteristic
226C23×C4C2411yescharacteristic
227C23×C4C2412yes—
228C23×C4C2414yes—
229C23×C4C2412yes—
230C23×C4C2412yes—
231C23×C4C2411yescharacteristic
232C23×C4C2411yescharacteristic
233C23×C4C2411yescharacteristic
234C23×C4C2411yescharacteristic
235C23×C4C2411yescharacteristic
236C22⋊Q8—432no—
237C4×D4—232no—
238C4×D4—232no—
239C4×D4—232no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—28no—
255C2×C4⋊C4—28no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—24no—
258C2×C4⋊C4—24no—
259C2×C4⋊C4—24no—
260C2×C4⋊C4—28no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—24no—
263C2×C4⋊C4—22no—
264C2×C4⋊C4—28no—
265C2×C4⋊C4—42no—
266C2×C4⋊C4—42no—
267C2×C4⋊C4—44no—
268C2×C4⋊C4—44no—
269C2×C4⋊C4—44no—
270C2×C4⋊C4—44no—
271C2×C4⋊C4C2×Q814yes—
272C2×C4⋊C4#4118yes—
273C2×C22⋊C4—24no—
274C2×C22⋊C4—24no—
275C2×C22⋊C4—24no—
276C2×C22⋊C4—24no—
277C2×C22⋊C4—24no—
278C2×C22⋊C4—24no—
279C2×C22⋊C4—24no—
280C2×C22⋊C4—24no—
281C2×C22⋊C4—28no—
282C2×C22⋊C4—28no—
283C2×C22⋊C4—28no—
284C2×C22⋊C4—28no—
285C2×C22⋊C4—28no—
286C2×C22⋊C4—28no—
287C2×C22⋊C4—44no—
288C2×C22⋊C4—44no—
289C2×C22⋊C4—44no—
290C2×C22⋊C4—44no—
291C2×C22⋊C4—44no—
292C2×C22⋊C4—44no—
293C2×C22⋊C4C2×Q818yes—
294C2×C42—24no—
295C2×C42—24no—
296C2×C42—24no—
297C2×C42—22no—
298C2×C42—22no—
299C2×C42—24no—
300C2×C42—24no—
301C2×C42—24no—
302C2×C42—28no—
303C2×C42—28no—
304C2×C42—24no—
305C2×C42—24no—
306C2×C42—432no—
307C2×C42—42no—
308C2×C42—42no—
309C2×C42—42no—
310C2×C42—42no—
311C2×C42—42no—
312C2×C42—41no—
313C2×C42—41no—
314C2×C42—44no—
315C2×C42C2×D414yes—
316C2×C42C2×D414yes—
317C2×C42C2×D414yes—
318C2×C42C2×D414yes—
319C2×C42C2×D414yes—
320C2×C42C2×D414yes—
321C2×C42C2×D414yes—
322C2×C42C2×Q814yes—
323C2×C42#4114yes—
324#95—24no—
325#95—24no—
326#95—24no—
327#95—24no—
328#95—24no—
329#95—24no—
330#95—24no—
331#95—24no—
332#95—24no—
333#95—24no—
334#95—24no—
335#95—24no—
336#95—24no—
337#95—28no—
338#95—24no—
339#95—28no—
340#95—28no—
341#95—28no—
342#95—28no—
343#95—28no—
344#95—42no—
345#95—42no—
346#95—44no—
347#95—44no—
348#95—44no—
349#95—44no—
350#95—44no—
351#95—44no—
352#95—44no—
353C24—22no—
354C24—22no—
355C24—22no—
356C24—22no—
357C24—22no—
358C24—22no—
359C24—22no—
360C24—22no—
361C24—22no—
362C24—22no—
363C24—22no—
364C24—22no—
365C24C22×Q814yes—
366C24C2×#4114yes—
367C24C2×#4114yes—
368C24C2511yescharacteristic
369C2×Q8—44no—
370C2×Q8—44no—
371C2×Q8—44no—
372C2×Q8—48no—
373C2×Q8—48no—
374C2×D4—28no—
375C2×D4—28no—
376C2×D4—216no—
377C2×D4—216no—
378C2×D4—28no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—21no—
383C22×C4—21no—
384C22×C4—21no—
385C22×C4—21no—
386C22×C4—22no—
387C22×C4—21no—
388C22×C4—21no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—24no—
395C22×C4—22no—
396C22×C4—24no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—24no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—24no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—21no—
412C22×C4—21no—
413C22×C4—24no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—24no—
418C22×C4—21no—
419C22×C4—21no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—216no—
423C22×C4—216no—
424C22×C4—216no—
425C22×C4—216no—
426C22×C4—216no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—24no—
432C22×C4—24no—
433C22×C4—24no—
434C22×C4—24no—
435C22×C4—24no—
436C22×C4—24no—
437C22×C4—24no—
438C22×C4—24no—
439C22×C4—24no—
440C22×C4—24no—
441C22×C4—22no—
442C22×C4—24no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—24no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—24no—
450C22×C4—24no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—24no—
454C22×C4—48no—
455C22×C4—48no—
456C22×C4—48no—
457C22×C4—48no—
458C22×C4—48no—
459C22×C4—48no—
460C22×C4—48no—
461C22×C4—48no—
462C22×C4—48no—
463C22×C4—48no—
464C22×C4—48no—
465C22×C4—48no—
466C22×C4—48no—
467C22×C4—48no—
468C22×C4—48no—
469C22×C4—48no—
470C22×C4C22×D412yes—
471C22×C4C22×D412yes—
472C22×C4C22×D412yes—
473C22×C4C22×D412yes—
474C22×C4C22×D412yes—
475C22×C4C22×D412yes—
476C22×C4C22×D412yes—
477C22×C4C22×Q814yes—
478C22×C4C22×Q812yes—
479C22×C4C22×Q812yes—
480C22×C4C22×Q812yes—
481C22×C4C2×#4118yes—
482C22×C4C2×#4114yes—
483C22×C4C2×#4114yes—
484C22×C4C2×#4112yes—
485C22×C4C2×#4112yes—
486C22×C4C2×#4112yes—
487C22×C4C2×#4114yes—
488C22×C4C2×#4114yes—
489C22×C4C2×#4114yes—
490C22×C4H(5,𝔽2)12yes—
491C22×C4H(5,𝔽2)12yes—
492C4⋊C4—28no—
493C4⋊C4—28no—
494C4⋊C4—28no—
495C4⋊C4—216no—
496C4⋊C4—416no—
497C4⋊C4—48no—
498C4⋊C4—48no—
499C4⋊C4—48no—
500C4⋊C4—44no—
501C4⋊C4—44no—
502C4⋊C4—44no—
503C4⋊C4—48no—
504C4⋊C4—88no—
505C22⋊C4—216no—
506C22⋊C4—216no—
507C22⋊C4—216no—
508C22⋊C4—416no—
509C22⋊C4—416no—
510C22⋊C4—88no—
511C22⋊C4—88no—
512C42—28no—
513C42—28no—
514C42—28no—
515C42—28no—
516C42—44no—
517C42—44no—
518C42—416no—
519C42—48no—
520C42—48no—
521C42—48no—
522C42C22⋊Q8116yescomplemented
523C23—24no—
524C23—24no—
525C23—24no—
526C23—24no—
527C23—24no—
528C23—24no—
529C23—24no—
530C23—24no—
531C23—24no—
532C23—24no—
533C23—24no—
534C23—24no—
535C23—24no—
536C23—24no—
537C23—24no—
538C23—24no—
539C23—24no—
540C23—24no—
541C23—42no—
542C23—42no—
543C23—42no—
544C23—42no—
545C23—42no—
546C23—42no—
547C23—42no—
548C23—42no—
549C23—42no—
550C23—42no—
551C23—42no—
552C23—42no—
553C23—42no—
554C23—42no—
555C23—42no—
556C23—42no—
557C23#53318yes—
558C23C23×D411yescharacteristic
559C23C23×Q811yescharacteristic
560C23C22×#4111yescharacteristic
561C23C22×#4112yes—
562C23C22×#4112yes—
563C23C22×#4112yes—
564C23C2×H(5,𝔽2)11yescharacteristic
565C23C2×H(5,𝔽2)11yescharacteristic
566C23#58511yescharacteristic
567C23#58512yes—
568C23#58511yescharacteristic
569Q8—88no—
570D4—232no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—24no—
575C2×C4—24no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—24no—
582C2×C4—24no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—24no—
586C2×C4—24no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—24no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—24no—
593C2×C4—24no—
594C2×C4—22no—
595C2×C4—22no—
596C2×C4—22no—
597C2×C4—22no—
598C2×C4—28no—
599C2×C4—24no—
600C2×C4—24no—
601C2×C4—28no—
602C2×C4—28no—
603C2×C4—24no—
604C2×C4—24no—
605C2×C4—24no—
606C2×C4—24no—
607C2×C4—432no—
608C2×C4—432no—
609C2×C4—42no—
610C2×C4—44no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—44no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—44no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—44no—
629C2×C4—432no—
630C2×C4—42no—
631C2×C4—42no—
632C2×C4—42no—
633C2×C4—42no—
634C2×C4—42no—
635C2×C4—42no—
636C2×C4—44no—
637C2×C4—42no—
638C2×C4—42no—
639C2×C4—42no—
640C2×C4—42no—
641C2×C4—42no—
642C2×C4—42no—
643C2×C4—44no—
644C2×C4—44no—
645C2×C4—44no—
646C2×C4—44no—
647C2×C4—44no—
648C2×C4—44no—
649C2×C4—44no—
650C2×C4—44no—
651C2×C4—44no—
652C2×C4—44no—
653C2×C4—44no—
654C2×C4—44no—
655C2×C4—44no—
656C2×C4—44no—
657C2×C4C2×C22⋊Q814yes—
658C2×C4C2×C22⋊Q814yes—
659C2×C4C2×C22⋊Q814yes—
660C2×C4#52514yes—
661C2×C4#52514yes—
662C2×C4#53214yes—
663C2×C4#53514yes—
664C2×C4#55118yes—
665C2×C4#55318yes—
666C22—28no—
667C22—28no—
668C22—28no—
669C22—44no—
670C22—44no—
671C22—44no—
672C22—44no—
673C22—44no—
674C22—44no—
675C22—44no—
676C22—44no—
677C22—44no—
678C22—44no—
679C22—44no—
680C22—44no—
681C22C22×C22⋊Q811yescharacteristic
682C22C2×#52511yescharacteristic
683C22C2×#52511yescharacteristic
684C22C2×#53211yescharacteristic
685C22C2×#53312yes—
686C22#344412yes—
687C22C2×#53511yescharacteristic
688C22#344911yescharacteristic
689C22#344911yescharacteristic
690C22C2×#55112yes—
691C22C2×#55312yes—
692C22#347512yes—
693C22#347512yes—
694C22Q8×#4112yes—
695C22#348012yes—
696C22#350111yescharacteristic
697C22#350111yescharacteristic
698C22#350512yes—
699C22#350512yes—
700C22#350611yescharacteristic
701C22#350611yescharacteristic
702C22#351012yes—
703C22#351212yes—
704C4—24no—
705C4—24no—
706C4—24no—
707C4—24no—
708C4—42no—
709C4—44no—
710C4—48no—
711C4—44no—
712C4—42no—
713C4—44no—
714C4—82no—
715C4—82no—
716C4—84no—
717C4—82no—
718C4—82no—
719C4#260318yes—
720C2—48no—
721C2C2×#260311yescharacteristic
722C2#3491111yescharacteristic
723C2#3494711yescharacteristic
724C2#3510211yescharacteristic
725C2#3517211yescharacteristic
726C2#3534211yescharacteristic
727C2#3627712yes—
728C2#3627812yes—
729C2#3639212yes—
730C2#3639312yes—
731C2#3662511yescharacteristic
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
1. C2×#2863C2semidirect
3. #15246C2semidirect
4. #15212C2semidirect
6. #15194C2semidirect
7. #15192C2semidirect
10. #14702C2semidirect
11. #14697C2semidirect
12. #14075C2semidirect
13. #13997C2semidirect
14. #13973C2semidirect
16. #13952C2semidirect
17. #13755C2semidirect
99. C2×C4×D4Q8semidirect
522. C42C22⋊Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#28632
C2×C42×D42
#152462
#152122
#152032
#151942
#151922
#151812
#151772
#147022
#146972
#140752
#139972
#139732
#139592
#139522
#137552