Finite Group Databasebeta

← #536703

Subgroups — #536703

Order 512.

3266 conjugacy classes of subgroups.

The subgroup lattice has 745 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2535C211yesmaximal, characteristic
2C2×#2354C211yesmaximal, characteristic, complemented
3C2×#2354C211yesmaximal, characteristic, complemented
4#14567C214yesmaximal, complemented
5#14415C214yesmaximal, complemented
6#14193C214yesmaximal, complemented
7#14150C211yesmaximal, characteristic, complemented
8#14136C212yesmaximal, complemented
9#14071C214yesmaximal, complemented
10#13998C212yesmaximal, complemented
11#13974C212yesmaximal, complemented
12#13847C211yesmaximal, characteristic, complemented
13#13801C212yesmaximal, complemented
14#13785C211yesmaximal, characteristic, complemented
15#13779C211yesmaximal, characteristic, complemented
16C22×#121C2212yes—
17C22×C4×D4C2211yescharacteristic
18C22×C4×D4C2211yescharacteristic
19C22×#117C2212yes—
20C22×#117C2212yes—
21#2535C4116yescomplemented
22C2×#397C2212yes—
23C2×#392C2212yes—
24#2354—28no—
25#2354—28no—
26C2×#390C2211yescharacteristic
27C2×#390C2212yes—
28C2×#390C2212yes—
29C2×#390C2211yescharacteristic
30C2×#390C2214yes—
31C2×#389C2212yes—
32C2×#388C2214yes—
33C2×#388C2214yes—
34C2×#388C2214yes—
35C2×#386C2214yes—
36C2×#384C2211yescharacteristic
37C2×#382C2211yescharacteristic
38C2×#380C2212yes—
39C2×#380C2212yes—
40C2×#380C2212yes—
41C2×C4×C4⋊C4C2211yescharacteristic
42C2×C4×C4⋊C4C2211yescharacteristic
43C2×C4×C4⋊C4C2212yes—
44C2×C4×C4⋊C4C2212yes—
45C2×C4×C4⋊C4C2212yes—
46C2×C4×C22⋊C4C2212yes—
47C2×C4×C22⋊C4C2212yes—
48C2×#376C2211yescharacteristic
49#1445C2212yes—
50#1444C2212yes—
51#1443C2214yes—
52#1442C2212yes—
53#1440C2214yes—
54#1439C2212yes—
55#1439C2212yes—
56#1439C2214yes—
57#1438C2212yes—
58#1438C2214yes—
59#1438C2214yes—
60#1438C2214yes—
61#1437C2214yes—
62#1437C2214yes—
63#1437C2214yes—
64#1437C2214yes—
65#1437C2214yes—
66#1437C2214yes—
67#1436C2214yes—
68#1436C2214yes—
69#1436C2214yes—
70#1436C2214yes—
71C4⋊#95C2212yes—
72C4⋊#95C2212yes—
73C4⋊#95C2211yescharacteristic
74C4⋊#95C2211yescharacteristic
75#1434C2214yes—
76C22⋊#95C2214yes—
77C4×#95C2214yes—
78C23×D4C2311yescharacteristic
79C23×D4C2311yescharacteristic
80C23×D4C2314yes—
81C24×C4C2312yes—
82C24×C4C2312yes—
83C2×#121—44no—
84C2×#121—44no—
85C2×#121C2×C4116yes—
86C2×C4×D4—24no—
87C2×C4×D4—24no—
88C2×C4×D4—24no—
89C2×C4×D4—24no—
90C2×C4×D4C2×C418yes—
91C2×C4×D4C2×C418yes—
92C2×#117—28no—
93C2×#117—28no—
94C2×#117—28no—
95C2×#117—48no—
96C2×#117—44no—
97C22×C4⋊C4C2311yescharacteristic
98C22×C4⋊C4C2311yescharacteristic
99C22×C4⋊C4C2311yescharacteristic
100C22×C4⋊C4C2311yescharacteristic
101C22×C4⋊C4C2312yes—
102C22×C4⋊C4C2312yes—
103C22×C4⋊C4C2311yescharacteristic
104C22×C4⋊C4C2314yes—
105C22×C4⋊C4C2311yescharacteristic
106C22×C4⋊C4C2312yes—
107C22×C4⋊C4C2312yes—
108C22×C22⋊C4C2314yes—
109C22×C22⋊C4C2314yes—
110C22×C22⋊C4C2312yes—
111C22×C22⋊C4C2312yes—
112C22×C22⋊C4C2312yes—
113C22×C22⋊C4C2314yes—
114C22×C22⋊C4C2314yes—
115C22×C22⋊C4C2312yes—
116C22×C22⋊C4C2314yes—
117C22×C22⋊C4C2312yes—
118C22×C22⋊C4C2314yes—
119C22×C22⋊C4C2312yes—
120C22×C22⋊C4C2314yes—
121C22×C22⋊C4C2314yes—
122C22×C42C2311yescharacteristic
123C22×C42C2312yes—
124C22×C42C2314yes—
125C22×C42C2314yes—
126C22×C42C2311yescharacteristic
127C22×C42C2312yes—
128C22×C42C2314yes—
129C22×C42C2311yescharacteristic
130#397C2×C4116yes—
131#392C2×C4116yes—
132#390—28no—
133#390—28no—
134#390—24no—
135#390—24no—
136#390—48no—
137#389—44no—
138#388—216no—
139#388—216no—
140#388—48no—
141#386C2×C4132yes—
142#384D418yescomplemented
143#382C2×C418yes—
144#380—28no—
145#380—28no—
146#380C2×C4116yes—
147C4×C4⋊C4—28no—
148C4×C4⋊C4—28no—
149C4×C4⋊C4—24no—
150C4×C4⋊C4—24no—
151C4×C4⋊C4—44no—
152C4×C22⋊C4—28no—
153C4×C22⋊C4—44no—
154#376—24no—
155C2×#95C2312yes—
156C2×#95C2312yes—
157C2×#95C2312yes—
158C2×#95C2312yes—
159C2×#95C2314yes—
160C2×#95C2312yes—
161C2×#95C2312yes—
162C2×#95C2314yes—
163C2×#95C2314yes—
164C2×#95C2314yes—
165C2×#95C2314yes—
166C2×#95C2314yes—
167C2×#95C2314yes—
168C2×#95C2314yes—
169C2×#95C2314yes—
170C2×#95C2312yes—
171C2×#95C2314yes—
172C2×#95C2312yes—
173C2×#95C2312yes—
174C2×#95C2314yes—
175C2×#95C2312yes—
176C25C2412yes—
177C25C2412yes—
178C22×D4—22no—
179C22×D4—22no—
180C22×D4—22no—
181C22×D4—22no—
182C22×D4—22no—
183C22×D4—22no—
184C22×D4—22no—
185C22×D4—22no—
186C22×D4—42no—
187C22×D4—42no—
188C22×D4—44no—
189C22×D4—44no—
190C22×D4—44no—
191C22×D4—44no—
192C22×D4—44no—
193C22×D4—44no—
194C22×D4C22×C414yes—
195C22×D4C22×C414yes—
196C22×D4C22×C4116yes—
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—44no—
206C23×C4—44no—
207C23×C4C22×C418yes—
208C23×C4C22×C418yes—
209C23×C4C2412yes—
210C23×C4C2412yes—
211C23×C4C2412yes—
212C23×C4C2412yes—
213C23×C4C2412yes—
214C23×C4C2414yes—
215C23×C4C2414yes—
216C23×C4C2412yes—
217C23×C4C2412yes—
218C23×C4C2412yes—
219C23×C4C2411yescharacteristic
220C23×C4C2411yescharacteristic
221C23×C4C2411yescharacteristic
222#121—432no—
223C4×D4—232no—
224C4×D4—232no—
225#117—432no—
226#117—432no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—24no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—28no—
244C2×C4⋊C4—44no—
245C2×C4⋊C4—42no—
246C2×C4⋊C4—44no—
247C2×C4⋊C4—44no—
248C2×C4⋊C4—44no—
249C2×C4⋊C4C22×C414yes—
250C2×C4⋊C4C22×C414yes—
251C2×C4⋊C4C22×C418yes—
252C2×C4⋊C4C2×D418yes—
253C2×C4⋊C4C2×D418yes—
254C2×C4⋊C4C2×D418yes—
255C2×C4⋊C4#4114yes—
256C2×C22⋊C4—24no—
257C2×C22⋊C4—24no—
258C2×C22⋊C4—24no—
259C2×C22⋊C4—24no—
260C2×C22⋊C4—28no—
261C2×C22⋊C4—28no—
262C2×C22⋊C4—28no—
263C2×C22⋊C4—24no—
264C2×C22⋊C4—24no—
265C2×C22⋊C4—28no—
266C2×C22⋊C4—24no—
267C2×C22⋊C4—24no—
268C2×C22⋊C4—28no—
269C2×C22⋊C4—28no—
270C2×C22⋊C4—28no—
271C2×C22⋊C4—28no—
272C2×C22⋊C4—44no—
273C2×C22⋊C4—42no—
274C2×C22⋊C4—42no—
275C2×C22⋊C4—42no—
276C2×C22⋊C4—42no—
277C2×C22⋊C4—42no—
278C2×C22⋊C4—42no—
279C2×C22⋊C4—44no—
280C2×C22⋊C4—44no—
281C2×C22⋊C4—44no—
282C2×C22⋊C4—44no—
283C2×C22⋊C4—42no—
284C2×C22⋊C4—42no—
285C2×C22⋊C4—44no—
286C2×C22⋊C4—44no—
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⋊C4—44no—
294C2×C22⋊C4C22×C418yes—
295C2×C22⋊C4C22×C418yes—
296C2×C22⋊C4C22×C4116yes—
297C2×C22⋊C4C22×C4116yes—
298C2×C42—22no—
299C2×C42—22no—
300C2×C42—28no—
301C2×C42—28no—
302C2×C42—24no—
303C2×C42—24no—
304C2×C42—28no—
305C2×C42—22no—
306C2×C42—24no—
307C2×C42—22no—
308C2×C42—28no—
309C2×C42—22no—
310C2×C42—24no—
311C2×C42—44no—
312C2×C42—44no—
313C2×C42—44no—
314C2×C42—44no—
315C2×C42—44no—
316C2×C42—44no—
317C2×C42C22×C414yes—
318C2×C42C2×D414yes—
319C2×C42#4114yes—
320C2×C42#4114yes—
321#95—28no—
322#95—28no—
323#95—24no—
324#95—24no—
325#95—24no—
326#95—24no—
327#95—24no—
328#95—24no—
329#95—44no—
330#95—44no—
331#95—44no—
332#95—44no—
333#95—42no—
334#95—42no—
335#95—44no—
336#95—44no—
337#95—44no—
338#95—44no—
339#95C22×C4116yes—
340#95C22×C418yes—
341#95C22×C418yes—
342C24—22no—
343C24—22no—
344C24—22no—
345C24—22no—
346C24—22no—
347C24—22no—
348C24—22no—
349C24—22no—
350C24—22no—
351C24—24no—
352C24—22no—
353C24—22no—
354C24—22no—
355C24—22no—
356C24—22no—
357C24—22no—
358C24—24no—
359C24—22no—
360C24—22no—
361C24—24no—
362C24—24no—
363C24—22no—
364C24—22no—
365C24—22no—
366C24C23×C414yes—
367C24C23×C414yes—
368C24C2511yescharacteristic
369C2×D4—28no—
370C2×D4—28no—
371C2×D4—28no—
372C2×D4—28no—
373C2×D4—48no—
374C2×D4—44no—
375C2×D4—44no—
376C2×D4—48no—
377C2×D4—416no—
378C2×D4—416no—
379C2×D4—416no—
380C2×D4—44no—
381C2×D4—44no—
382C2×D4—44no—
383C2×D4—44no—
384C2×D4—88no—
385C2×D4—88no—
386C2×D4—88no—
387C2×D4—88no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—24no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—24no—
406C22×C4—24no—
407C22×C4—22no—
408C22×C4—24no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—22no—
413C22×C4—24no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—24no—
422C22×C4—24no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—22no—
427C22×C4—24no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—216no—
431C22×C4—216no—
432C22×C4—216no—
433C22×C4—216no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—24no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—24no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—416no—
454C22×C4—48no—
455C22×C4—48no—
456C22×C4—416no—
457C22×C4—48no—
458C22×C4—48no—
459C22×C4—48no—
460C22×C4—48no—
461C22×C4C23×C418yes—
462C22×C4C23×C414yes—
463C22×C4C23×C414yes—
464C22×C4C23×C412yes—
465C22×C4C23×C412yes—
466C22×C4C23×C412yes—
467C22×C4C22×D418yes—
468C22×C4C22×D412yes—
469C22×C4C22×D412yes—
470C22×C4C22×D412yes—
471C22×C4C2×#4114yes—
472C22×C4C2×#4114yes—
473C22×C4C2×#4112yes—
474C22×C4C2×#4112yes—
475C22×C4C2×#4112yes—
476C22×C4C2×#4112yes—
477C22×C4C2×#4112yes—
478C22×C4H(5,𝔽2)14yes—
479C22×C4H(5,𝔽2)12yes—
480C22×C4H(5,𝔽2)12yes—
481C4⋊C4—28no—
482C4⋊C4—28no—
483C4⋊C4—28no—
484C4⋊C4—216no—
485C4⋊C4—216no—
486C4⋊C4—28no—
487C4⋊C4—48no—
488C4⋊C4—48no—
489C4⋊C4—44no—
490C4⋊C4—44no—
491C4⋊C4—88no—
492C22⋊C4—216no—
493C22⋊C4—216no—
494C22⋊C4—416no—
495C22⋊C4—416no—
496C22⋊C4—48no—
497C22⋊C4—416no—
498C22⋊C4—416no—
499C22⋊C4—48no—
500C22⋊C4—48no—
501C22⋊C4—48no—
502C22⋊C4—88no—
503C22⋊C4—88no—
504C22⋊C4—88no—
505C22⋊C4—88no—
506C42—28no—
507C42—416no—
508C42—416no—
509C42—48no—
510C42—48no—
511C42—44no—
512C42—88no—
513C42C4×D4116yes—
514C23—24no—
515C23—24no—
516C23—24no—
517C23—24no—
518C23—24no—
519C23—24no—
520C23—24no—
521C23—24no—
522C23—28no—
523C23—24no—
524C23—28no—
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—
542C23—42no—
543C23—42no—
544C23—42no—
545C23—44no—
546C23—42no—
547C23—44no—
548C23—42no—
549C23—44no—
550C23—42no—
551C23—44no—
552C23—42no—
553C23—42no—
554C23—42no—
555C23—42no—
556C23—44no—
557C23—44no—
558C23—42no—
559C23—42no—
560C23—42no—
561C23—42no—
562C23—44no—
563C23—42no—
564C23—42no—
565C23—44no—
566C23—42no—
567C23—44no—
568C23—42no—
569C23—42no—
570C23—44no—
571C23—42no—
572C23C24×C411yescharacteristic
573C23C23×D411yescharacteristic
574C23C22×#4111yescharacteristic
575C23C22×#4111yescharacteristic
576C23C22×#4111yescharacteristic
577C23C2×H(5,𝔽2)11yescharacteristic
578C23C2×H(5,𝔽2)11yescharacteristic
579C23C2×H(5,𝔽2)12yes—
580C23#58512yes—
581C23#58512yes—
582C23#58512yes—
583D4—416no—
584D4—416no—
585D4—832no—
586C2×C4—24no—
587C2×C4—24no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—24no—
593C2×C4—24no—
594C2×C4—24no—
595C2×C4—24no—
596C2×C4—24no—
597C2×C4—24no—
598C2×C4—28no—
599C2×C4—28no—
600C2×C4—28no—
601C2×C4—28no—
602C2×C4—24no—
603C2×C4—28no—
604C2×C4—24no—
605C2×C4—24no—
606C2×C4—28no—
607C2×C4—28no—
608C2×C4—24no—
609C2×C4—24no—
610C2×C4—24no—
611C2×C4—28no—
612C2×C4—24no—
613C2×C4—432no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—44no—
628C2×C4—44no—
629C2×C4—44no—
630C2×C4—44no—
631C2×C4—44no—
632C2×C4—42no—
633C2×C4—42no—
634C2×C4—44no—
635C2×C4—44no—
636C2×C4—44no—
637C2×C4—42no—
638C2×C4—44no—
639C2×C4—44no—
640C2×C4—432no—
641C2×C4—42no—
642C2×C4—42no—
643C2×C4—42no—
644C2×C4—42no—
645C2×C4—42no—
646C2×C4—42no—
647C2×C4—42no—
648C2×C4—42no—
649C2×C4—42no—
650C2×C4—42no—
651C2×C4—44no—
652C2×C4—42no—
653C2×C4—42no—
654C2×C4—42no—
655C2×C4—42no—
656C2×C4—42no—
657C2×C4—42no—
658C2×C4—42no—
659C2×C4—44no—
660C2×C4C2×C4×D414yes—
661C2×C4C2×C4×D414yes—
662C2×C4C2×C4×D414yes—
663C2×C4C4×#4114yes—
664C2×C4#51814yes—
665C2×C4#54614yes—
666C2×C4#55014yes—
667C2×C4#55314yes—
668C2×C4#55514yes—
669C22—44no—
670C22—44no—
671C22—44no—
672C22—44no—
673C22—44no—
674C22—48no—
675C22—44no—
676C22—44no—
677C22—44no—
678C22—44no—
679C22—44no—
680C22—48no—
681C22—82no—
682C22—82no—
683C22—82no—
684C22—82no—
685C22—84no—
686C22—82no—
687C22—82no—
688C22—84no—
689C22—82no—
690C22—82no—
691C22—82no—
692C22—82no—
693C22—82no—
694C22—82no—
695C22C22×C4×D411yescharacteristic
696C22C2×C4×#4111yescharacteristic
697C22C2×#51811yescharacteristic
698C22#342812yes—
699C22C4×H(5,𝔽2)12yes—
700C22C2×#54611yescharacteristic
701C22C2×#55011yescharacteristic
702C22C2×#55311yescharacteristic
703C22C2×#55511yescharacteristic
704C22#348411yescharacteristic
705C22#348511yescharacteristic
706C22#348711yescharacteristic
707C22#348711yescharacteristic
708C22#349312yes—
709C22#349312yes—
710C22#349412yes—
711C22#350612yes—
712C22#351712yes—
713C22#352112yes—
714C22#354012yes—
715C22#354212yes—
716C22#354612yes—
717C22#356212yes—
718C4—24no—
719C4—24no—
720C4—44no—
721C4—44no—
722C4—44no—
723C4—48no—
724C4—48no—
725C4—44no—
726C4—44no—
727C4—82no—
728C4—82no—
729C4—82no—
730C4#236318yes—
731C2—84no—
732C2—84no—
733C2C2×#236311yescharacteristic
734C2#3420111yescharacteristic
735C2#3426511yescharacteristic
736C2#3430112yes—
737C2#3447812yes—
738C2#4490712yes—
739C2#4490912yes—
740C2#4493311yescharacteristic
741C2#4493511yescharacteristic
742C2#4526911yescharacteristic
743C2#4538811yescharacteristic
subgroupG/H
center, Frattini, socle, cosocleC24C25
commutatorC23C24×C4
prime radicalC2×#2535C2
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×#2354C2semidirect
3. C2×#2354C2semidirect
4. #14567C2semidirect
5. #14415C2semidirect
6. #14193C2semidirect
7. #14150C2semidirect
8. #14136C2semidirect
9. #14071C2semidirect
10. #13998C2semidirect
11. #13974C2semidirect
12. #13847C2semidirect
13. #13801C2semidirect
14. #13785C2semidirect
15. #13779C2semidirect
21. #2535C4semidirect
142. #384D4semidirect

Maximal subgroups

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

subgroupindex
C2×#25352
C2×#23542
#145672
#144152
#141932
#141502
#141362
#140712
#139982
#139742
#138472
#138012
#137852
#137792