Finite Group Databasebeta

← #519620

Subgroups — #519620

Order 512.

3386 conjugacy classes of subgroups.

The subgroup lattice has 712 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2536C211yesmaximal, characteristic
2C2×#2356C211yesmaximal, characteristic, complemented
3C2×C42×D4C211yesmaximal, characteristic
4#14707C211yesmaximal, characteristic, complemented
5#14702C212yesmaximal, complemented
6#14179C214yesmaximal, complemented
7#14173C214yesmaximal
8#13997C212yesmaximal, complemented
9#13994C212yesmaximal, complemented
10#13990C214yesmaximal, complemented
11#13905C211yesmaximal, characteristic
12#13816C214yesmaximal, complemented
13#13804C211yesmaximal, characteristic, complemented
14C4×#380C212yesmaximal, complemented
15C4×#391C211yesmaximal, characteristic, complemented
16C22×C22⋊Q8C2212yes—
17C22×C4×Q8C2211yescharacteristic
18C22×C4×D4C2212yes—
19C22×C4×D4C2211yescharacteristic
20C23×C42C2212yes—
21#2536C4116yescomplemented
22C2×#398C2212yes—
23C2×#394C2212yes—
24#2356—28no—
25C2×#391C2212yes—
26C2×#391C2211yescharacteristic
27C2×#391C2211yescharacteristic
28C2×#389C2212yes—
29C2×#388C2214yes—
30C2×#388C2214yes—
31C2×#387C2214yes—
32C2×#383C2212yes—
33C2×#382C2211yescharacteristic
34C2×#382C2211yescharacteristic
35C2×#380C2212yes—
36C2×#380C2212yes—
37C2×#380C2212yes—
38C42×D4—28no—
39C2×C4×C4⋊C4C2212yes—
40C2×C4×C4⋊C4C2211yescharacteristic
41C2×C4×C4⋊C4C2212yes—
42C2×C4×C4⋊C4C2212yes—
43C2×C4×C4⋊C4C2211yescharacteristic
44C2×C4×C22⋊C4C2214yes—
45C2×C4×C22⋊C4C2212yes—
46C2×C4×C22⋊C4C2212yes—
47C2×C4×C22⋊C4C2212yes—
48C2×#376C2212yes—
49C2×C43C2211yescharacteristic
50#1445C2212yes—
51#1445C2214yes—
52#1442C2214yes—
53#1440C2214yes—
54#1439C2214yes—
55#1438C2212yes—
56#1437C2214yes—
57#1437C2214yes—
58#1436C2214yes—
59#1436C2214yes—
60#1436C2214yes—
61#1436C2214yes—
62#1436C2214yes—
63#1436C2214yes—
64C4⋊#95C2214yes—
65C4⋊#95C2212yes—
66C4⋊#95C2212yes—
67C4⋊#95C2211yescharacteristic
68C4⋊#95C2211yescharacteristic
69#1434C2214yes—
70#1434C2214yes—
71#1434C2212yes—
72C22⋊#95C2214yes—
73C22⋊#95C2214yes—
74C22⋊#95C2214yes—
75C4×#95C2214yes—
76C4×#95C2214yes—
77C4×#95C2212yes—
78C23×Q8C2311yescharacteristic
79C23×D4C2311yescharacteristic
80C24×C4C2312yes—
81C24×C4C2312yes—
82C24×C4C2312yes—
83C2×C22⋊Q8—44no—
84C2×C22⋊Q8—44no—
85C2×C22⋊Q8C2×C4116yescomplemented
86C2×C4×Q8—24no—
87C2×C4×Q8—24no—
88C2×C4×Q8C2×C418yescomplemented
89C2×C4×D4—28no—
90C2×C4×D4—28no—
91C2×C4×D4—28no—
92C2×C4×D4—24no—
93C2×C4×D4—24no—
94C2×C4×D4C2×C418yes—
95C22×C4⋊C4C2311yescharacteristic
96C22×C4⋊C4C2312yes—
97C22×C4⋊C4C2312yes—
98C22×C4⋊C4C2311yescharacteristic
99C22×C4⋊C4C2311yescharacteristic
100C22×C4⋊C4C2312yes—
101C22×C4⋊C4C2312yes—
102C22×C4⋊C4C2311yescharacteristic
103C22×C4⋊C4C2314yes—
104C22×C4⋊C4C2314yes—
105C22×C22⋊C4C2312yes—
106C22×C22⋊C4C2314yes—
107C22×C22⋊C4C2314yes—
108C22×C22⋊C4C2312yes—
109C22×C22⋊C4C2312yes—
110C22×C22⋊C4C2314yes—
111C22×C22⋊C4C2314yes—
112C22×C42—28no—
113C22×C42—28no—
114C22×C42—44no—
115C22×C42C2312yes—
116C22×C42C2314yes—
117C22×C42C2311yescharacteristic
118C22×C42C2312yes—
119C22×C42C2312yes—
120C22×C42C2312yes—
121C22×C42C2312yes—
122C22×C42C2312yes—
123C22×C42C2311yescharacteristic
124C22×C42C2312yes—
125C22×C42C2314yes—
126C22×C42C2311yescharacteristic
127#398C2×C4116yescomplemented
128#394C2×C4116yes—
129#391—24no—
130#391—28no—
131#391—24no—
132#389—44no—
133#388—216no—
134#388C2×C4132yescomplemented
135#387—216no—
136#383—28no—
137#382—24no—
138#382C2×C418yescomplemented
139#380—28no—
140#380—44no—
141#380C2×C4116yescomplemented
142C4×C4⋊C4—28no—
143C4×C4⋊C4—28no—
144C4×C4⋊C4—24no—
145C4×C4⋊C4—44no—
146C4×C4⋊C4D418yes—
147C4×C22⋊C4—216no—
148C4×C22⋊C4—28no—
149C4×C22⋊C4—28no—
150C4×C22⋊C4—28no—
151#376—28no—
152C2×#95C2314yes—
153C2×#95C2314yes—
154C2×#95C2314yes—
155C2×#95C2314yes—
156C2×#95C2314yes—
157C2×#95C2314yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2314yes—
161C2×#95C2314yes—
162C2×#95C2314yes—
163C2×#95C2312yes—
164C2×#95C2312yes—
165C2×#95C2312yes—
166C2×#95C2312yes—
167C2×#95C2312yes—
168C2×#95C2314yes—
169C2×#95C2314yes—
170C2×#95C2314yes—
171C2×#95C2314yes—
172C2×#95C2312yes—
173C2×#95C2314yes—
174C2×#95C2312yes—
175C2×#95C2311yescharacteristic
176C2×#95C2311yescharacteristic
177C43—24no—
178C25C2412yes—
179C22×Q8—22no—
180C22×Q8—22no—
181C22×Q8—22no—
182C22×Q8—22no—
183C22×Q8—22no—
184C22×Q8—22no—
185C22×Q8C22×C414yes—
186C22×D4—24no—
187C22×D4—24no—
188C22×D4—22no—
189C22×D4—22no—
190C22×D4C22×C414yes—
191C23×C4—24no—
192C23×C4—24no—
193C23×C4—24no—
194C23×C4—24no—
195C23×C4—24no—
196C23×C4—24no—
197C23×C4—24no—
198C23×C4—24no—
199C23×C4—24no—
200C23×C4—24no—
201C23×C4—24no—
202C23×C4—24no—
203C23×C4—24no—
204C23×C4—24no—
205C23×C4—24no—
206C23×C4—42no—
207C23×C4—42no—
208C23×C4—42no—
209C23×C4—42no—
210C23×C4C22×C418yes—
211C23×C4#4118yes—
212C23×C4C2414yes—
213C23×C4C2412yes—
214C23×C4C2412yes—
215C23×C4C2414yes—
216C23×C4C2412yes—
217C23×C4C2412yes—
218C23×C4C2412yes—
219C23×C4C2412yes—
220C23×C4C2412yes—
221C23×C4C2411yescharacteristic
222C23×C4C2411yescharacteristic
223C23×C4C2412yes—
224C23×C4C2412yes—
225C23×C4C2411yescharacteristic
226C22⋊Q8—432no—
227C4×Q8—232no—
228C4×D4—264no—
229C4×D4—232no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—24no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—28no—
249C2×C4⋊C4—44no—
250C2×C4⋊C4—44no—
251C2×C4⋊C4—44no—
252C2×C4⋊C4—44no—
253C2×C4⋊C4C22×C414yes—
254C2×C4⋊C4C22×C414yes—
255C2×C4⋊C4C22×C418yes—
256C2×C4⋊C4C22×C418yes—
257C2×C4⋊C4C22×C4116yes—
258C2×C4⋊C4C2×D418yes—
259C2×C4⋊C4#4114yes—
260C2×C22⋊C4—24no—
261C2×C22⋊C4—28no—
262C2×C22⋊C4—28no—
263C2×C22⋊C4—24no—
264C2×C22⋊C4—24no—
265C2×C22⋊C4—24no—
266C2×C22⋊C4—24no—
267C2×C22⋊C4—24no—
268C2×C22⋊C4—24no—
269C2×C22⋊C4—28no—
270C2×C22⋊C4—28no—
271C2×C22⋊C4—28no—
272C2×C22⋊C4—44no—
273C2×C22⋊C4—44no—
274C2×C22⋊C4—44no—
275C2×C22⋊C4—44no—
276C2×C22⋊C4—44no—
277C2×C22⋊C4C22×C418yes—
278C2×C22⋊C4C22×C4116yes—
279C2×C22⋊C4C22×C4116yes—
280C2×C22⋊C4#4118yes—
281C2×C42—24no—
282C2×C42—24no—
283C2×C42—22no—
284C2×C42—24no—
285C2×C42—24no—
286C2×C42—24no—
287C2×C42—24no—
288C2×C42—24no—
289C2×C42—22no—
290C2×C42—24no—
291C2×C42—22no—
292C2×C42—28no—
293C2×C42—24no—
294C2×C42—24no—
295C2×C42—28no—
296C2×C42—28no—
297C2×C42—22no—
298C2×C42—24no—
299C2×C42—24no—
300C2×C42—22no—
301C2×C42—24no—
302C2×C42—28no—
303C2×C42—24no—
304C2×C42—432no—
305C2×C42—42no—
306C2×C42—42no—
307C2×C42—42no—
308C2×C42—44no—
309C2×C42—44no—
310C2×C42—44no—
311C2×C42C22×C418yes—
312C2×C42C22×C414yes—
313C2×C42C2×D418yes—
314C2×C42C2×D414yes—
315C2×C42#4114yes—
316C2×C42#4114yes—
317#95—24no—
318#95—28no—
319#95—24no—
320#95—24no—
321#95—28no—
322#95—24no—
323#95—28no—
324#95—28no—
325#95—28no—
326#95—24no—
327#95—28no—
328#95—28no—
329#95—28no—
330#95—28no—
331#95—28no—
332#95—28no—
333#95—44no—
334#95—44no—
335#95—44no—
336#95—44no—
337#95C22×C4116yes—
338#95C22×C418yes—
339#95C22×C418yes—
340#95C2×D414yes—
341#95C2×D414yes—
342C24—22no—
343C24—22no—
344C24—22no—
345C24—22no—
346C24—22no—
347C24—22no—
348C24—22no—
349C24—22no—
350C24—22no—
351C24—22no—
352C24—22no—
353C24—22no—
354C24C23×C414yes—
355C24C2×#4114yes—
356C24C2×#4114yes—
357C24C2511yescharacteristic
358C2×Q8—28no—
359C2×Q8—28no—
360C2×Q8—28no—
361C2×Q8—44no—
362C2×Q8—44no—
363C2×Q8—44no—
364C2×Q8—44no—
365C2×D4—216no—
366C2×D4—28no—
367C2×D4—28no—
368C2×D4—216no—
369C2×D4—28no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—216no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—24no—
392C22×C4—24no—
393C22×C4—24no—
394C22×C4—24no—
395C22×C4—22no—
396C22×C4—24no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
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—22no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—24no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—22no—
421C22×C4—216no—
422C22×C4—216no—
423C22×C4—216no—
424C22×C4—216no—
425C22×C4—216no—
426C22×C4—24no—
427C22×C4—24no—
428C22×C4—22no—
429C22×C4—24no—
430C22×C4—22no—
431C22×C4—24no—
432C22×C4—24no—
433C22×C4—22no—
434C22×C4—24no—
435C22×C4—24no—
436C22×C4—24no—
437C22×C4—24no—
438C22×C4—48no—
439C22×C4—48no—
440C22×C4—48no—
441C22×C4—48no—
442C22×C4—48no—
443C22×C4—48no—
444C22×C4—48no—
445C22×C4—48no—
446C22×C4—48no—
447C22×C4—48no—
448C22×C4—48no—
449C22×C4—48no—
450C22×C4—48no—
451C22×C4—48no—
452C22×C4—48no—
453C22×C4C23×C418yes—
454C22×C4C23×C414yes—
455C22×C4C23×C414yes—
456C22×C4C23×C414yes—
457C22×C4C23×C412yes—
458C22×C4C23×C412yes—
459C22×C4C23×C412yes—
460C22×C4C22×D414yes—
461C22×C4C22×D412yes—
462C22×C4C22×D412yes—
463C22×C4C22×D414yes—
464C22×C4C22×D412yes—
465C22×C4C2×#4114yes—
466C22×C4C2×#4114yes—
467C22×C4C2×#4112yes—
468C22×C4C2×#4112yes—
469C22×C4C2×#4114yes—
470C22×C4C2×#4114yes—
471C22×C4C2×#4114yes—
472C22×C4C2×#4114yes—
473C22×C4C2×#4112yes—
474C22×C4C2×#4112yes—
475C22×C4C2×#4112yes—
476C22×C4H(5,𝔽2)12yes—
477C22×C4H(5,𝔽2)12yes—
478C22×C42−1+414yes—
479C4⋊C4—28no—
480C4⋊C4—28no—
481C4⋊C4—216no—
482C4⋊C4—216no—
483C4⋊C4—216no—
484C4⋊C4—28no—
485C4⋊C4—416no—
486C4⋊C4—48no—
487C4⋊C4—44no—
488C4⋊C4—88no—
489C22⋊C4—216no—
490C22⋊C4—232no—
491C22⋊C4—216no—
492C22⋊C4—216no—
493C22⋊C4—416no—
494C22⋊C4—416no—
495C22⋊C4—88no—
496C42—216no—
497C42—216no—
498C42—28no—
499C42—216no—
500C42—48no—
501C42—48no—
502C42—48no—
503C42—416no—
504C42—44no—
505C42—88no—
506C42—84no—
507C42C4×D4116yes—
508C23—24no—
509C23—24no—
510C23—24no—
511C23—24no—
512C23—24no—
513C23—24no—
514C23—24no—
515C23—24no—
516C23—24no—
517C23—24no—
518C23—24no—
519C23—24no—
520C23—24no—
521C23—24no—
522C23—24no—
523C23—24no—
524C23—24no—
525C23—24no—
526C23—42no—
527C23—42no—
528C23—42no—
529C23—42no—
530C23—42no—
531C23—42no—
532C23—42no—
533C23—42no—
534C23—42no—
535C23—42no—
536C23—42no—
537C23—42no—
538C23—42no—
539C23—42no—
540C23—42no—
541C23—42no—
542C23C4×#4118yes—
543C23C24×C411yescharacteristic
544C23C23×D411yescharacteristic
545C23C22×#4111yescharacteristic
546C23C22×#4112yes—
547C23C22×#4112yes—
548C23C22×#4111yescharacteristic
549C23C22×#4111yescharacteristic
550C23C2×H(5,𝔽2)11yescharacteristic
551C23C2×H(5,𝔽2)11yescharacteristic
552C23C2×2−1+412yes—
553C23#58512yes—
554Q8—416no—
555D4—232no—
556C2×C4—24no—
557C2×C4—24no—
558C2×C4—24no—
559C2×C4—24no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—24no—
563C2×C4—24no—
564C2×C4—24no—
565C2×C4—24no—
566C2×C4—24no—
567C2×C4—24no—
568C2×C4—24no—
569C2×C4—24no—
570C2×C4—24no—
571C2×C4—24no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—24no—
575C2×C4—24no—
576C2×C4—24no—
577C2×C4—24no—
578C2×C4—24no—
579C2×C4—28no—
580C2×C4—28no—
581C2×C4—24no—
582C2×C4—24no—
583C2×C4—24no—
584C2×C4—28no—
585C2×C4—24no—
586C2×C4—28no—
587C2×C4—24no—
588C2×C4—24no—
589C2×C4—24no—
590C2×C4—24no—
591C2×C4—28no—
592C2×C4—24no—
593C2×C4—24no—
594C2×C4—24no—
595C2×C4—432no—
596C2×C4—432no—
597C2×C4—44no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—44no—
615C2×C4—44no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—44no—
619C2×C4—42no—
620C2×C4—44no—
621C2×C4—44no—
622C2×C4—42no—
623C2×C4—432no—
624C2×C4—44no—
625C2×C4—42no—
626C2×C4—44no—
627C2×C4—42no—
628C2×C4—44no—
629C2×C4—44no—
630C2×C4—44no—
631C2×C4—44no—
632C2×C4—42no—
633C2×C4—44no—
634C2×C4—44no—
635C2×C4—44no—
636C2×C4C2×C4×D414yes—
637C2×C4C2×C4×D414yes—
638C2×C4C2×C4×D414yes—
639C2×C4C4×#4114yes—
640C2×C4#51814yes—
641C2×C4#52518yes—
642C2×C4#54614yes—
643C2×C4#55014yes—
644C2×C4#55218yes—
645C2×C4#55314yes—
646C2×C4#55514yes—
647C22—28no—
648C22—28no—
649C22—28no—
650C22—44no—
651C22—44no—
652C22—44no—
653C22—44no—
654C22—44no—
655C22—44no—
656C22—44no—
657C22—44no—
658C22—44no—
659C22—44no—
660C22—44no—
661C22—44no—
662C22C22×C4×D411yescharacteristic
663C22C2×C4×#4112yes—
664C22C2×C4×#4111yescharacteristic
665C22C2×#51811yescharacteristic
666C22C4×2−1+412yes—
667C22C2×#52512yes—
668C22#344411yescharacteristic
669C22#344411yescharacteristic
670C22C2×#54611yescharacteristic
671C22C2×#55011yescharacteristic
672C22C2×#55212yes—
673C22C2×#55311yescharacteristic
674C22C2×#55511yescharacteristic
675C22#347512yes—
676C22#347512yes—
677C22#348212yes—
678C22#348912yes—
679C22#350512yes—
680C22#350911yescharacteristic
681C22#351011yescharacteristic
682C22#351112yes—
683C22#351412yes—
684C22#352012yes—
685C4—28no—
686C4—28no—
687C4—24no—
688C4—24no—
689C4—48no—
690C4—44no—
691C4—44no—
692C4—44no—
693C4—44no—
694C4—44no—
695C4—44no—
696C4—82no—
697C4—84no—
698C4#236318yes—
699C2—48no—
700C2C4×#52511yescharacteristic
701C2C2×#236311yescharacteristic
702C2C4×#55212yes—
703C2#3426811yescharacteristic
704C2#3432712yes—
705C2#3615612yes—
706C2#3619511yescharacteristic
707C2#3620911yescharacteristic
708C2#3650412yes—
709C2#3650911yescharacteristic
710C2#3653611yescharacteristic
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×#2356C2semidirect
4. #14707C2semidirect
5. #14702C2semidirect
6. #14179C2semidirect
8. #13997C2semidirect
9. #13994C2semidirect
10. #13990C2semidirect
12. #13816C2semidirect
13. #13804C2semidirect
14. C4×#380C2semidirect
15. C4×#391C2semidirect
21. #2536C4semidirect
85. C2×C22⋊Q8C2×C4semidirect
88. C2×C4×Q8C2×C4semidirect
127. #398C2×C4semidirect
134. #388C2×C4semidirect
138. #382C2×C4semidirect
141. #380C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#25362
C2×#23562
C2×C42×D42
#147072
#147022
#141792
#141732
#139972
#139942
#139902
#139052
#138162
#138042
C4×#3802
C4×#3912