Finite Group Databasebeta

← #537739

Subgroups — #537739

Order 512.

3764 conjugacy classes of subgroups.

The subgroup lattice has 737 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2409C211yesmaximal, characteristic
2C2×Q8×C22⋊C4C212yesmaximal, complemented
3C2×#2323C212yesmaximal, complemented
4#14538C212yesmaximal, complemented
5#14522C214yesmaximal, complemented
6#14402C212yesmaximal, complemented
7#14236C212yesmaximal, complemented
8#14143C212yesmaximal, complemented
9#14012C214yesmaximal
10#13976C212yesmaximal, complemented
11#13974C212yesmaximal, complemented
12#13969C212yesmaximal, complemented
13#13906C211yesmaximal, characteristic
14#13839C211yesmaximal, characteristic
15#13817C212yesmaximal, complemented
16C24×Q8C2211yescharacteristic
17C22×C22⋊Q8C2212yes—
18C22×C4×Q8C2212yes—
19C22×C4×Q8C2212yes—
20C22×C4×D4C2211yescharacteristic
21C22×#117C2211yescharacteristic
22#2409C4116yescomplemented
23C2×#395C2212yes—
24C2×#393C2212yes—
25Q8×C22⋊C4—216no—
26#2323—216no—
27C2×#391C2212yes—
28C2×#391C2214yes—
29C2×#391C2214yes—
30C2×#391C2212yes—
31C2×#391C2212yes—
32C2×#390C2212yes—
33C2×#389C2214yes—
34C2×#389C2212yes—
35C2×#388C2214yes—
36C2×#388C2214yes—
37C2×#388C2214yes—
38C2×#387C2214yes—
39C2×#386C2212yes—
40C2×#385C2212yes—
41C2×#384C2211yescharacteristic
42C2×#380C2212yes—
43C2×#380C2214yes—
44C2×#380C2212yes—
45C2×#380C2211yescharacteristic
46C2×C4×C4⋊C4C2212yes—
47C2×C4×C4⋊C4C2211yescharacteristic
48C2×C4×C22⋊C4C2212yes—
49C2×C4×C22⋊C4C2214yes—
50C2×C4×C22⋊C4C2211yescharacteristic
51#1444C2214yes—
52#1444C2214yes—
53#1443C2214yes—
54#1441C2211yescharacteristic
55#1441C2211yescharacteristic
56#1441C2214yes—
57#1440C2212yes—
58#1440C2214yes—
59#1439C2211yescharacteristic
60#1439C2211yescharacteristic
61#1439C2212yes—
62#1439C2212yes—
63#1439C2214yes—
64#1438C2212yes—
65#1437C2214yes—
66#1436C2214yes—
67#1436C2214yes—
68#1436C2214yes—
69#1436C2214yes—
70#1436C2214yes—
71#1436C2214yes—
72C4⋊#95C2212yes—
73C4⋊#95C2212yes—
74C4⋊#95C2212yes—
75#1434C2214yes—
76C4×#95C2212yes—
77C4×#95C2214yes—
78C23×Q8—28no—
79C23×Q8—28no—
80C23×Q8—42no—
81C23×Q8—42no—
82C23×Q8C2×C418yes—
83C23×Q8C2312yes—
84C23×Q8C2312yes—
85C23×D4C2311yescharacteristic
86C24×C4C2311yescharacteristic
87C24×C4C2311yescharacteristic
88C24×C4C2312yes—
89C2×C22⋊Q8—28no—
90C2×C22⋊Q8—28no—
91C2×C22⋊Q8C2×C4116yes—
92C2×C4×Q8—28no—
93C2×C4×Q8—28no—
94C2×C4×Q8—28no—
95C2×C4×Q8—28no—
96C2×C4×Q8—44no—
97C2×C4×Q8—44no—
98C2×C4×D4—42no—
99C2×C4×D4—42no—
100C2×C4×D4C2×C418yes—
101C2×#117—28no—
102C2×#117—42no—
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2311yescharacteristic
105C22×C4⋊C4C2311yescharacteristic
106C22×C4⋊C4C2314yes—
107C22×C4⋊C4C2314yes—
108C22×C4⋊C4C2314yes—
109C22×C4⋊C4C2312yes—
110C22×C4⋊C4C2312yes—
111C22×C4⋊C4C2312yes—
112C22×C4⋊C4C2314yes—
113C22×C4⋊C4C2312yes—
114C22×C4⋊C4C2311yescharacteristic
115C22×C22⋊C4C2312yes—
116C22×C22⋊C4C2312yes—
117C22×C22⋊C4C2312yes—
118C22×C22⋊C4C2314yes—
119C22×C22⋊C4C2314yes—
120C22×C22⋊C4C2312yes—
121C22×C22⋊C4C2312yes—
122C22×C22⋊C4C2314yes—
123C22×C22⋊C4C2311yescharacteristic
124C22×C22⋊C4C2311yescharacteristic
125C22×C42C2312yes—
126C22×C42C2314yes—
127C22×C42C2314yes—
128C22×C42C2311yescharacteristic
129C22×C42C2312yes—
130C22×C42C2312yes—
131C22×C42C2314yes—
132#395C2×C4116yes—
133#393C2×C4116yes—
134#391—28no—
135#391—216no—
136#391—216no—
137#391C2×C4116yes—
138#391D4116yescomplemented
139#390—44no—
140#389—28no—
141#389—216no—
142#388—216no—
143#388—216no—
144#388—48no—
145#387—216no—
146#386C2×C4116yes—
147#385C2×C4116yes—
148#384—42no—
149#380—28no—
150#380—28no—
151#380—216no—
152#380C2×C418yes—
153C4×C4⋊C4—28no—
154C4×C4⋊C4—42no—
155C4×C22⋊C4—28no—
156C4×C22⋊C4—216no—
157C4×C22⋊C4—42no—
158C2×#95C2312yes—
159C2×#95C2312yes—
160C2×#95C2314yes—
161C2×#95C2312yes—
162C2×#95C2314yes—
163C2×#95C2314yes—
164C2×#95C2314yes—
165C2×#95C2314yes—
166C2×#95C2312yes—
167C2×#95C2314yes—
168C2×#95C2314yes—
169C2×#95C2312yes—
170C2×#95C2312yes—
171C2×#95C2312yes—
172C2×#95C2314yes—
173C2×#95C2314yes—
174C2×#95C2314yes—
175C2×#95C2311yescharacteristic
176C2×#95C2311yescharacteristic
177C2×#95C2312yes—
178C2×#95C2314yes—
179C2×#95C2314yes—
180C2×#95C2314yes—
181C2×#95C2314yes—
182C25C2411yescharacteristic
183C25C2411yescharacteristic
184C22×Q8—264no—
185C22×Q8—24no—
186C22×Q8—28no—
187C22×Q8—24no—
188C22×Q8—28no—
189C22×Q8—24no—
190C22×Q8—24no—
191C22×Q8—42no—
192C22×Q8—42no—
193C22×Q8—416no—
194C22×Q8—416no—
195C22×Q8—432no—
196C22×Q8—416no—
197C22×Q8C22×C418yes—
198C22×Q8C22×C418yes—
199C22×Q8C2×D418yes—
200C22×Q8#4118yes—
201C22×D4—22no—
202C22×D4—22no—
203C22×D4—22no—
204C22×D4—22no—
205C22×D4—41no—
206C22×D4—41no—
207C22×D4C22×C414yes—
208C23×C4—24no—
209C23×C4—28no—
210C23×C4—24no—
211C23×C4—24no—
212C23×C4—28no—
213C23×C4—24no—
214C23×C4—22no—
215C23×C4—22no—
216C23×C4—22no—
217C23×C4—22no—
218C23×C4—42no—
219C23×C4—42no—
220C23×C4C22×C414yes—
221C23×C4C22×C414yes—
222C23×C4C22×C418yes—
223C23×C4C2414yes—
224C23×C4C2412yes—
225C23×C4C2414yes—
226C23×C4C2411yescharacteristic
227C23×C4C2411yescharacteristic
228C23×C4C2411yescharacteristic
229C23×C4C2412yes—
230C23×C4C2414yes—
231C23×C4C2412yes—
232C23×C4C2412yes—
233C23×C4C2414yes—
234C23×C4C2411yescharacteristic
235C23×C4C2411yescharacteristic
236C22⋊Q8—264no—
237C4×Q8—432no—
238C4×Q8—432no—
239C4×D4—416no—
240#117—416no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—28no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—28no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—28no—
251C2×C4⋊C4—28no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—28no—
254C2×C4⋊C4—28no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—28no—
257C2×C4⋊C4—24no—
258C2×C4⋊C4—41no—
259C2×C4⋊C4—41no—
260C2×C4⋊C4—41no—
261C2×C4⋊C4—41no—
262C2×C4⋊C4—44no—
263C2×C4⋊C4—44no—
264C2×C4⋊C4—44no—
265C2×C4⋊C4—44no—
266C2×C4⋊C4—42no—
267C2×C4⋊C4—42no—
268C2×C4⋊C4—44no—
269C2×C4⋊C4—42no—
270C2×C4⋊C4C22×C418yes—
271C2×C4⋊C4C22×C4116yes—
272C2×C4⋊C4C22×C414yes—
273C2×C4⋊C4C2×D418yes—
274C2×C22⋊C4—24no—
275C2×C22⋊C4—24no—
276C2×C22⋊C4—24no—
277C2×C22⋊C4—28no—
278C2×C22⋊C4—24no—
279C2×C22⋊C4—28no—
280C2×C22⋊C4—24no—
281C2×C22⋊C4—24no—
282C2×C22⋊C4—28no—
283C2×C22⋊C4—24no—
284C2×C22⋊C4—24no—
285C2×C22⋊C4—28no—
286C2×C22⋊C4—24no—
287C2×C22⋊C4—24no—
288C2×C22⋊C4—28no—
289C2×C22⋊C4—44no—
290C2×C22⋊C4—42no—
291C2×C22⋊C4—42no—
292C2×C22⋊C4—42no—
293C2×C22⋊C4—41no—
294C2×C22⋊C4—41no—
295C2×C22⋊C4—41no—
296C2×C22⋊C4—41no—
297C2×C22⋊C4—42no—
298C2×C22⋊C4—44no—
299C2×C22⋊C4—44no—
300C2×C22⋊C4C22×C4116yes—
301C2×C22⋊C4C22×C418yes—
302C2×C22⋊C4C22×C414yes—
303C2×C22⋊C4C22×C414yes—
304C2×C42—24no—
305C2×C42—24no—
306C2×C42—22no—
307C2×C42—22no—
308C2×C42—28no—
309C2×C42—28no—
310C2×C42—24no—
311C2×C42—28no—
312C2×C42—28no—
313C2×C42—24no—
314C2×C42—24no—
315C2×C42—28no—
316C2×C42—24no—
317C2×C42—44no—
318C2×C42—44no—
319C2×C42—42no—
320C2×C42—44no—
321C2×C42—42no—
322C2×C42—44no—
323C2×C42C22×C414yes—
324C2×C42C2×D418yes—
325#95—24no—
326#95—24no—
327#95—28no—
328#95—28no—
329#95—28no—
330#95—28no—
331#95—28no—
332#95—28no—
333#95—24no—
334#95—28no—
335#95—24no—
336#95—24no—
337#95—44no—
338#95—42no—
339#95—41no—
340#95—41no—
341#95—42no—
342#95—44no—
343#95—44no—
344#95C22×C418yes—
345#95C22×C4116yes—
346#95C22×C4116yes—
347#95C2×D4116yes—
348#95C2×D4116yes—
349C24—22no—
350C24—22no—
351C24—21no—
352C24—21no—
353C24—22no—
354C24—21no—
355C24—21no—
356C24—22no—
357C24—22no—
358C24—22no—
359C24—21no—
360C24—21no—
361C24—21no—
362C24—22no—
363C24—21no—
364C24—21no—
365C24—21no—
366C24—22no—
367C24C23×C412yes—
368C24C23×C412yes—
369C24H(5,𝔽2)12yes—
370C242−1+412yes—
371C24C2511yescharacteristic
372C2×Q8—216no—
373C2×Q8—232no—
374C2×Q8—216no—
375C2×Q8—216no—
376C2×Q8—216no—
377C2×Q8—216no—
378C2×Q8—216no—
379C2×Q8—48no—
380C2×Q8—48no—
381C2×Q8—416no—
382C2×Q8—48no—
383C2×Q8—4128no—
384C2×Q8C4×D4132yes—
385C2×D4—28no—
386C2×D4—28no—
387C2×D4—44no—
388C2×D4—44no—
389C2×D4—44no—
390C2×D4—44no—
391C2×D4—44no—
392C22×C4—24no—
393C22×C4—24no—
394C22×C4—21no—
395C22×C4—21no—
396C22×C4—21no—
397C22×C4—24no—
398C22×C4—24no—
399C22×C4—21no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—24no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—24no—
408C22×C4—24no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—21no—
412C22×C4—24no—
413C22×C4—21no—
414C22×C4—22no—
415C22×C4—24no—
416C22×C4—24no—
417C22×C4—22no—
418C22×C4—21no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—21no—
422C22×C4—21no—
423C22×C4—21no—
424C22×C4—24no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—24no—
430C22×C4—22no—
431C22×C4—24no—
432C22×C4—24no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—24no—
439C22×C4—22no—
440C22×C4—24no—
441C22×C4—22no—
442C22×C4—216no—
443C22×C4—232no—
444C22×C4—216no—
445C22×C4—28no—
446C22×C4—28no—
447C22×C4—24no—
448C22×C4—22no—
449C22×C4—24no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—24no—
454C22×C4—24no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—24no—
458C22×C4—22no—
459C22×C4—24no—
460C22×C4—24no—
461C22×C4—48no—
462C22×C4—48no—
463C22×C4—416no—
464C22×C4—48no—
465C22×C4—44no—
466C22×C4—44no—
467C22×C4—48no—
468C22×C4—48no—
469C22×C4—44no—
470C22×C4—44no—
471C22×C4C23×C412yes—
472C22×C4C23×C418yes—
473C22×C4C23×C412yes—
474C22×C4C23×C412yes—
475C22×C4C23×C418yes—
476C22×C4C23×C412yes—
477C22×C4C23×C412yes—
478C22×C4C22×D414yes—
479C22×C4C22×D418yes—
480C22×C4C22×D414yes—
481C22×C4C22×D418yes—
482C22×C4C22×D414yes—
483C22×C4C2×#4118yes—
484C22×C4C2×#4114yes—
485C22×C4H(5,𝔽2)12yes—
486C22×C4H(5,𝔽2)12yes—
487C22×C42−1+414yes—
488C22×C42−1+412yes—
489C22×C42−1+412yes—
490C4⋊C4—216no—
491C4⋊C4—232no—
492C4⋊C4—216no—
493C4⋊C4—44no—
494C4⋊C4—44no—
495C4⋊C4—44no—
496C4⋊C4—416no—
497C4⋊C4—416no—
498C4⋊C4—48no—
499C4⋊C4—84no—
500C4⋊C4—88no—
501C4⋊C4—84no—
502C22⋊C4—216no—
503C22⋊C4—216no—
504C22⋊C4—232no—
505C22⋊C4—44no—
506C22⋊C4—44no—
507C22⋊C4—48no—
508C22⋊C4—416no—
509C22⋊C4—48no—
510C22⋊C4—48no—
511C22⋊C4—88no—
512C42—28no—
513C42—216no—
514C42—416no—
515C42—416no—
516C42—48no—
517C42—84no—
518C42—88no—
519C23—24no—
520C23—22no—
521C23—22no—
522C23—24no—
523C23—24no—
524C23—24no—
525C23—24no—
526C23—24no—
527C23—22no—
528C23—22no—
529C23—22no—
530C23—22no—
531C23—24no—
532C23—22no—
533C23—22no—
534C23—22no—
535C23—24no—
536C23—41no—
537C23—42no—
538C23—41no—
539C23—42no—
540C23—42no—
541C23—42no—
542C23—42no—
543C23—42no—
544C23—41no—
545C23—41no—
546C23—41no—
547C23—42no—
548C23—41no—
549C23—42no—
550C23—42no—
551C23—42no—
552C23—42no—
553C23—42no—
554C23—42no—
555C23—42no—
556C23—42no—
557C23—41no—
558C23—41no—
559C23—41no—
560C23—41no—
561C23—42no—
562C23—41no—
563C23—41no—
564C23#52014yes—
565C23C24×C411yescharacteristic
566C23C23×D412yes—
567C23C22×#4112yes—
568C23C2×H(5,𝔽2)11yescharacteristic
569C23C2×H(5,𝔽2)11yescharacteristic
570C23C2×H(5,𝔽2)11yescharacteristic
571C23C2×2−1+411yescharacteristic
572C23C2×2−1+411yescharacteristic
573C23C2×2−1+411yescharacteristic
574C23C2×2−1+412yes—
575C23#58512yes—
576Q8—264no—
577Q8—432no—
578D4—416no—
579C2×C4—28no—
580C2×C4—22no—
581C2×C4—28no—
582C2×C4—22no—
583C2×C4—28no—
584C2×C4—28no—
585C2×C4—28no—
586C2×C4—28no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—28no—
592C2×C4—24no—
593C2×C4—24no—
594C2×C4—22no—
595C2×C4—22no—
596C2×C4—22no—
597C2×C4—28no—
598C2×C4—24no—
599C2×C4—28no—
600C2×C4—24no—
601C2×C4—24no—
602C2×C4—28no—
603C2×C4—24no—
604C2×C4—24no—
605C2×C4—28no—
606C2×C4—24no—
607C2×C4—24no—
608C2×C4—24no—
609C2×C4—28no—
610C2×C4—24no—
611C2×C4—24no—
612C2×C4—432no—
613C2×C4—41no—
614C2×C4—41no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—41no—
619C2×C4—41no—
620C2×C4—42no—
621C2×C4—44no—
622C2×C4—42no—
623C2×C4—44no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—44no—
629C2×C4—44no—
630C2×C4—44no—
631C2×C4—44no—
632C2×C4—44no—
633C2×C4—42no—
634C2×C4—42no—
635C2×C4—42no—
636C2×C4—42no—
637C2×C4—44no—
638C2×C4—42no—
639C2×C4—41no—
640C2×C4—41no—
641C2×C4—42no—
642C2×C4—416no—
643C2×C4—416no—
644C2×C4—44no—
645C2×C4—42no—
646C2×C4—44no—
647C2×C4—42no—
648C2×C4—44no—
649C2×C4—44no—
650C2×C4—44no—
651C2×C4—42no—
652C2×C4—42no—
653C2×C4—42no—
654C2×C4—44no—
655C2×C4—42no—
656C2×C4—42no—
657C2×C4—44no—
658C2×C4C2×C4×D4116yes—
659C2×C4C2×C4×D418yes—
660C2×C4#51814yes—
661C2×C4#51914yes—
662C2×C4D4214yes—
663C2×C4#54618yes—
664C2×C4#55114yes—
665C22—24no—
666C22—28no—
667C22—44no—
668C22—42no—
669C22—42no—
670C22—42no—
671C22—42no—
672C22—42no—
673C22—44no—
674C22—44no—
675C22—42no—
676C22—44no—
677C22—42no—
678C22—42no—
679C22—42no—
680C22—44no—
681C22—81no—
682C22—81no—
683C22—81no—
684C22—82no—
685C22—82no—
686C22—81no—
687C22C22×C4×D412yes—
688C22C2×#51811yescharacteristic
689C22C2×#51911yescharacteristic
690C22C2×#52011yescharacteristic
691C22C4×2−1+412yes—
692C22C2×D4211yescharacteristic
693C22C2×#54612yes—
694C22C2×#55111yescharacteristic
695C22#348612yes—
696C22#348812yes—
697C22#349712yes—
698C22#349912yes—
699C22#351612yes—
700C22#351812yes—
701C22#352212yes—
702C22#352312yes—
703C22#353611yescharacteristic
704C22#353611yescharacteristic
705C22#353811yescharacteristic
706C22#353811yescharacteristic
707C22#354112yes—
708C22#355011yescharacteristic
709C22#355111yescharacteristic
710C4—24no—
711C4—216no—
712C4—44no—
713C4—42no—
714C4—42no—
715C4—42no—
716C4—44no—
717C4—44no—
718C4—48no—
719C4—48no—
720C4—84no—
721C4—82no—
722C4#235818yes—
723C2—44no—
724C2—82no—
725C2C2×#235811yescharacteristic
726C2#3429412yes—
727C2#3431112yes—
728C2#3438011yescharacteristic
729C2#3441311yescharacteristic
730C2#4524411yescharacteristic
731C2#4524912yes—
732C2#4530312yes—
733C2#4530811yescharacteristic
734C2#4532111yescharacteristic
735C2#4535111yescharacteristic
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×Q8×C22⋊C4C2semidirect
3. C2×#2323C2semidirect
4. #14538C2semidirect
5. #14522C2semidirect
6. #14402C2semidirect
7. #14236C2semidirect
8. #14143C2semidirect
10. #13976C2semidirect
11. #13974C2semidirect
12. #13969C2semidirect
15. #13817C2semidirect
22. #2409C4semidirect
138. #391D4semidirect

Maximal subgroups

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

subgroupindex
C2×#24092
C2×Q8×C22⋊C42
C2×#23232
#145382
#145222
#144022
#142362
#141432
#140122
#139762
#139742
#139692
#139062
#138392
#138172