Finite Group Databasebeta

← #531842

Subgroups — #531842

Order 512.

3362 conjugacy classes of subgroups.

The subgroup lattice has 738 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2354C212yesmaximal, complemented
2C2×C4×#121C211yesmaximal, characteristic
3#14422C212yesmaximal, complemented
4#14418C212yesmaximal
5#14157C212yesmaximal, complemented
6#14076C214yesmaximal, complemented
7#14037C214yesmaximal, complemented
8#13884C212yesmaximal, complemented
9#13876C212yesmaximal, complemented
10#13871C212yesmaximal, complemented
11#13871C212yesmaximal, complemented
12#13825C212yesmaximal, complemented
13#13801C212yesmaximal, complemented
14C4×#380C211yesmaximal, characteristic, complemented
15#13756C211yesmaximal, characteristic, complemented
16C22×#121C2211yescharacteristic
17C22×C4×D4C2212yes—
18C22×C4×D4C2211yescharacteristic
19C22×#117C2212yes—
20C22×#117C2212yes—
21C23×C42C2211yescharacteristic
22#2354—216no—
23C4×#121C4116yescomplemented
24C2×#390C2214yes—
25C2×#390C2211yescharacteristic
26C2×#389C2211yescharacteristic
27C2×#388C2214yes—
28C2×#388C2214yes—
29C2×#388C2214yes—
30C2×#388C2212yes—
31C2×#387C2214yes—
32C2×#386C2212yes—
33C2×#384C2212yes—
34C2×#380C2211yescharacteristic
35C2×#380C2212yes—
36C2×#380C2212yes—
37C2×#380C2212yes—
38C2×#380C2211yescharacteristic
39C2×#380C2211yescharacteristic
40C2×C4×C4⋊C4C2214yes—
41C2×C4×C4⋊C4C2212yes—
42C2×C4×C4⋊C4C2212yes—
43C2×C4×C4⋊C4C2211yescharacteristic
44C2×C4×C22⋊C4C2212yes—
45C2×C4×C22⋊C4C2212yes—
46#1442C2214yes—
47#1442C2214yes—
48#1441C2214yes—
49#1440C2214yes—
50#1440C2214yes—
51#1439C2214yes—
52#1438C2214yes—
53#1438C2214yes—
54#1437C2214yes—
55#1437C2214yes—
56#1437C2214yes—
57#1437C2214yes—
58#1436C2214yes—
59#1436C2214yes—
60#1436C2214yes—
61#1436C2214yes—
62C4⋊#95C2212yes—
63C4⋊#95C2212yes—
64C4⋊#95C2212yes—
65C4⋊#95C2212yes—
66C4⋊#95C2211yescharacteristic
67C4⋊#95C2211yescharacteristic
68#1434C2214yes—
69#1434C2212yes—
70C22⋊#95C2212yes—
71C22⋊#95C2212yes—
72C22⋊#95C2212yes—
73C4×#95C2212yes—
74C4×#95C2212yes—
75C4×#95C2214yes—
76C4×#95C2212yes—
77C23×D4C2312yes—
78C23×D4C2311yescharacteristic
79C24×C4C2312yes—
80C24×C4C2311yescharacteristic
81C24×C4C2311yescharacteristic
82C24×C4C2311yescharacteristic
83C2×#121—28no—
84C2×#121C2×C418yes—
85C2×C4×D4—28no—
86C2×C4×D4—28no—
87C2×C4×D4—44no—
88C2×C4×D4C2×C4116yes—
89C2×C4×D4C2×C418yes—
90C2×#117—28no—
91C2×#117—28no—
92C2×#117—28no—
93C2×#117—28no—
94C2×#117—44no—
95C2×#117—44no—
96C22×C4⋊C4C2312yes—
97C22×C4⋊C4C2312yes—
98C22×C4⋊C4C2314yes—
99C22×C4⋊C4C2312yes—
100C22×C4⋊C4C2312yes—
101C22×C4⋊C4C2312yes—
102C22×C4⋊C4C2314yes—
103C22×C4⋊C4C2311yescharacteristic
104C22×C4⋊C4C2311yescharacteristic
105C22×C4⋊C4C2311yescharacteristic
106C22×C22⋊C4C2314yes—
107C22×C22⋊C4C2314yes—
108C22×C22⋊C4C2312yes—
109C22×C22⋊C4C2312yes—
110C22×C22⋊C4C2312yes—
111C22×C22⋊C4C2312yes—
112C22×C22⋊C4C2312yes—
113C22×C22⋊C4C2314yes—
114C22×C22⋊C4C2314yes—
115C22×C22⋊C4C2314yes—
116C22×C22⋊C4C2312yes—
117C22×C22⋊C4C2312yes—
118C22×C42—44no—
119C22×C42C2×C418yes—
120C22×C42C2314yes—
121C22×C42C2314yes—
122C22×C42C2314yes—
123C22×C42C2311yescharacteristic
124C22×C42C2311yescharacteristic
125C22×C42C2312yes—
126C22×C42C2314yes—
127C22×C42C2311yescharacteristic
128C22×C42C2311yescharacteristic
129#390—216no—
130#390C2×C418yes—
131#389C2×C418yes—
132#388—216no—
133#388—216no—
134#388—48no—
135#388C2×C4116yes—
136#387—216no—
137#386C2×C4116yes—
138#384—28no—
139#380—24no—
140#380—28no—
141#380—28no—
142#380—44no—
143#380C2×C418yes—
144#380D418yescomplemented
145C4×C4⋊C4—28no—
146C4×C4⋊C4—216no—
147C4×C4⋊C4—44no—
148C4×C4⋊C4C2×C418yes—
149C4×C22⋊C4—44no—
150C4×C22⋊C4C2×C4116yes—
151C2×#95C2312yes—
152C2×#95C2312yes—
153C2×#95C2314yes—
154C2×#95C2314yes—
155C2×#95C2312yes—
156C2×#95C2314yes—
157C2×#95C2314yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2314yes—
161C2×#95C2312yes—
162C2×#95C2314yes—
163C2×#95C2312yes—
164C2×#95C2314yes—
165C2×#95C2314yes—
166C2×#95C2314yes—
167C2×#95C2314yes—
168C2×#95C2314yes—
169C2×#95C2312yes—
170C2×#95C2312yes—
171C2×#95C2312yes—
172C2×#95C2312yes—
173C25C2412yes—
174C25C2411yescharacteristic
175C22×D4—24no—
176C22×D4—24no—
177C22×D4—24no—
178C22×D4—24no—
179C22×D4—24no—
180C22×D4—42no—
181C22×D4—42no—
182C22×D4—42no—
183C22×D4—41no—
184C22×D4—41no—
185C22×D4C22×C418yes—
186C22×D4C22×C414yes—
187C23×C4—24no—
188C23×C4—24no—
189C23×C4—24no—
190C23×C4—24no—
191C23×C4—24no—
192C23×C4—24no—
193C23×C4—24no—
194C23×C4—24no—
195C23×C4—24no—
196C23×C4—22no—
197C23×C4—22no—
198C23×C4—42no—
199C23×C4—41no—
200C23×C4—41no—
201C23×C4—42no—
202C23×C4—42no—
203C23×C4C22×C418yes—
204C23×C4C22×C414yes—
205C23×C4C22×C414yes—
206C23×C4C22×C414yes—
207C23×C4C2×D414yes—
208C23×C4#4114yes—
209C23×C4C2414yes—
210C23×C4C2414yes—
211C23×C4C2412yes—
212C23×C4C2412yes—
213C23×C4C2411yescharacteristic
214C23×C4C2411yescharacteristic
215C23×C4C2414yes—
216C23×C4C2414yes—
217C23×C4C2411yescharacteristic
218C23×C4C2412yes—
219C23×C4C2411yescharacteristic
220C23×C4C2411yescharacteristic
221C23×C4C2411yescharacteristic
222#121—232no—
223C4×D4—264no—
224C4×D4—416no—
225#117—432no—
226#117—432no—
227C2×C4⋊C4—24no—
228C2×C4⋊C4—24no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4—24no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4—24no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—28no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—24no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—28no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—28no—
242C2×C4⋊C4—41no—
243C2×C4⋊C4—41no—
244C2×C4⋊C4—44no—
245C2×C4⋊C4—42no—
246C2×C4⋊C4—44no—
247C2×C4⋊C4—44no—
248C2×C4⋊C4C22×C418yes—
249C2×C4⋊C4C22×C414yes—
250C2×C4⋊C4C22×C414yes—
251C2×C4⋊C4C22×C414yes—
252C2×C4⋊C4C2×D418yes—
253C2×C4⋊C4#4118yes—
254C2×C4⋊C4#4118yes—
255C2×C22⋊C4—24no—
256C2×C22⋊C4—24no—
257C2×C22⋊C4—24no—
258C2×C22⋊C4—24no—
259C2×C22⋊C4—24no—
260C2×C22⋊C4—24no—
261C2×C22⋊C4—24no—
262C2×C22⋊C4—24no—
263C2×C22⋊C4—28no—
264C2×C22⋊C4—24no—
265C2×C22⋊C4—28no—
266C2×C22⋊C4—28no—
267C2×C22⋊C4—28no—
268C2×C22⋊C4—28no—
269C2×C22⋊C4—28no—
270C2×C22⋊C4—42no—
271C2×C22⋊C4—42no—
272C2×C22⋊C4—44no—
273C2×C22⋊C4—44no—
274C2×C22⋊C4—42no—
275C2×C22⋊C4—42no—
276C2×C22⋊C4—42no—
277C2×C22⋊C4—42no—
278C2×C22⋊C4—44no—
279C2×C22⋊C4—44no—
280C2×C22⋊C4—44no—
281C2×C22⋊C4—44no—
282C2×C22⋊C4—44no—
283C2×C22⋊C4—44no—
284C2×C22⋊C4—44no—
285C2×C22⋊C4C22×C418yes—
286C2×C22⋊C4C22×C418yes—
287C2×C22⋊C4C22×C418yes—
288C2×C22⋊C4C22×C418yes—
289C2×C22⋊C4C22×C418yes—
290C2×C22⋊C4C2×D418yes—
291C2×C42—24no—
292C2×C42—24no—
293C2×C42—24no—
294C2×C42—24no—
295C2×C42—28no—
296C2×C42—28no—
297C2×C42—28no—
298C2×C42—28no—
299C2×C42—28no—
300C2×C42—28no—
301C2×C42—28no—
302C2×C42—416no—
303C2×C42—44no—
304C2×C42—44no—
305C2×C42—42no—
306C2×C42—42no—
307C2×C42—44no—
308C2×C42—44no—
309C2×C42—44no—
310C2×C42C22×C418yes—
311C2×C42C22×C414yes—
312C2×C42C22×C414yes—
313C2×C42C22×C414yes—
314C2×C42C22×C414yes—
315#95—28no—
316#95—28no—
317#95—28no—
318#95—24no—
319#95—28no—
320#95—28no—
321#95—28no—
322#95—28no—
323#95—24no—
324#95—24no—
325#95—42no—
326#95—44no—
327#95—44no—
328#95—42no—
329#95—44no—
330#95—44no—
331#95—44no—
332#95—44no—
333#95C22×C418yes—
334#95C22×C418yes—
335#95C22×C418yes—
336#95C2×D418yes—
337C24—22no—
338C24—22no—
339C24—21no—
340C24—22no—
341C24—21no—
342C24—21no—
343C24—21no—
344C24—22no—
345C24—22no—
346C24—22no—
347C24—22no—
348C24—22no—
349C24—22no—
350C24—22no—
351C24—22no—
352C24—22no—
353C24—22no—
354C24—22no—
355C24—22no—
356C24—22no—
357C24—22no—
358C24—22no—
359C24C23×C414yes—
360C24C23×C412yes—
361C24C22×D412yes—
362C24C2×#4112yes—
363C24C2511yescharacteristic
364C2×D4—216no—
365C2×D4—216no—
366C2×D4—28no—
367C2×D4—44no—
368C2×D4—44no—
369C2×D4—48no—
370C2×D4—48no—
371C2×D4—48no—
372C2×D4—48no—
373C2×D4—48no—
374C2×D4—48no—
375C2×D4—48no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—21no—
386C22×C4—21no—
387C22×C4—28no—
388C22×C4—28no—
389C22×C4—216no—
390C22×C4—216no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—24no—
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—21no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—24no—
407C22×C4—24no—
408C22×C4—22no—
409C22×C4—24no—
410C22×C4—22no—
411C22×C4—24no—
412C22×C4—22no—
413C22×C4—21no—
414C22×C4—24no—
415C22×C4—22no—
416C22×C4—21no—
417C22×C4—21no—
418C22×C4—22no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—24no—
422C22×C4—22no—
423C22×C4—28no—
424C22×C4—216no—
425C22×C4—216no—
426C22×C4—216no—
427C22×C4—28no—
428C22×C4—24no—
429C22×C4—24no—
430C22×C4—24no—
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—24no—
442C22×C4—44no—
443C22×C4—44no—
444C22×C4—44no—
445C22×C4—44no—
446C22×C4—48no—
447C22×C4—48no—
448C22×C4—48no—
449C22×C4—48no—
450C22×C4—48no—
451C22×C4—48no—
452C22×C4—48no—
453C22×C4—48no—
454C22×C4C4×D4116yes—
455C22×C4C23×C412yes—
456C22×C4C23×C412yes—
457C22×C4C23×C414yes—
458C22×C4C23×C414yes—
459C22×C4C23×C414yes—
460C22×C4C23×C412yes—
461C22×C4C23×C412yes—
462C22×C4C23×C412yes—
463C22×C4C23×C412yes—
464C22×C4C22×D418yes—
465C22×C4C22×D412yes—
466C22×C4C22×D412yes—
467C22×C4C2×#4118yes—
468C22×C4C2×#4114yes—
469C22×C4C2×#4114yes—
470C22×C4C2×#4114yes—
471C22×C4C2×#4112yes—
472C22×C4C2×#4112yes—
473C22×C4C2×#4114yes—
474C22×C4H(5,𝔽2)12yes—
475C22×C4H(5,𝔽2)14yes—
476C22×C42−1+412yes—
477C4⋊C4—216no—
478C4⋊C4—216no—
479C4⋊C4—216no—
480C4⋊C4—44no—
481C4⋊C4—44no—
482C4⋊C4—416no—
483C4⋊C4—48no—
484C4⋊C4—48no—
485C4⋊C4—88no—
486C4⋊C4#117116yescomplemented
487C22⋊C4—216no—
488C22⋊C4—216no—
489C22⋊C4—216no—
490C22⋊C4—216no—
491C22⋊C4—48no—
492C22⋊C4—48no—
493C22⋊C4—416no—
494C22⋊C4—416no—
495C22⋊C4—48no—
496C22⋊C4—88no—
497C22⋊C4—88no—
498C22⋊C4—88no—
499C42—216no—
500C42—28no—
501C42—28no—
502C42—44no—
503C42—44no—
504C42—416no—
505C42—416no—
506C42—416no—
507C42—88no—
508C23—22no—
509C23—22no—
510C23—22no—
511C23—22no—
512C23—22no—
513C23—22no—
514C23—24no—
515C23—24no—
516C23—24no—
517C23—24no—
518C23—22no—
519C23—22no—
520C23—22no—
521C23—22no—
522C23—24no—
523C23—24no—
524C23—24no—
525C23—24no—
526C23—24no—
527C23—24no—
528C23—24no—
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—42no—
546C23—42no—
547C23—42no—
548C23—42no—
549C23—42no—
550C23—42no—
551C23—42no—
552C23—42no—
553C23—42no—
554C23—42no—
555C23—42no—
556C23—42no—
557C23—42no—
558C23—42no—
559C23—42no—
560C23—42no—
561C23—42no—
562C23—42no—
563C23—42no—
564C23—42no—
565C23C2×C4×D414yes—
566C23C24×C411yescharacteristic
567C23C23×D411yescharacteristic
568C23C22×#4112yes—
569C23C22×#4111yescharacteristic
570C23C22×#4112yes—
571C23C2×H(5,𝔽2)12yes—
572C23C2×H(5,𝔽2)11yescharacteristic
573C23C2×2−1+411yescharacteristic
574C23#58512yes—
575C23#58511yescharacteristic
576C23#58511yescharacteristic
577D4—432no—
578D4—416no—
579C2×C4—24no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—24no—
587C2×C4—24no—
588C2×C4—24no—
589C2×C4—24no—
590C2×C4—24no—
591C2×C4—24no—
592C2×C4—24no—
593C2×C4—24no—
594C2×C4—24no—
595C2×C4—24no—
596C2×C4—24no—
597C2×C4—232no—
598C2×C4—22no—
599C2×C4—24no—
600C2×C4—24no—
601C2×C4—24no—
602C2×C4—22no—
603C2×C4—22no—
604C2×C4—24no—
605C2×C4—24no—
606C2×C4—22no—
607C2×C4—24no—
608C2×C4—22no—
609C2×C4—22no—
610C2×C4—28no—
611C2×C4—28no—
612C2×C4—28no—
613C2×C4—28no—
614C2×C4—28no—
615C2×C4—28no—
616C2×C4—416no—
617C2×C4—416no—
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—44no—
627C2×C4—42no—
628C2×C4—44no—
629C2×C4—44no—
630C2×C4—42no—
631C2×C4—42no—
632C2×C4—42no—
633C2×C4—44no—
634C2×C4—44no—
635C2×C4—42no—
636C2×C4—44no—
637C2×C4—42no—
638C2×C4—432no—
639C2×C4—44no—
640C2×C4—44no—
641C2×C4—44no—
642C2×C4—44no—
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×C4C2×#11718yes—
654C2×C4C2×#11714yes—
655C2×C4C2×C4×D414yes—
656C2×C4C2×C4×D414yes—
657C2×C4C4×#4114yes—
658C2×C4#51814yes—
659C2×C4#52014yes—
660C2×C4#54618yes—
661C2×C4#55318yes—
662C22—24no—
663C22—24no—
664C22—24no—
665C22—42no—
666C22—42no—
667C22—42no—
668C22—44no—
669C22—44no—
670C22—42no—
671C22—44no—
672C22—44no—
673C22—44no—
674C22—44no—
675C22—44no—
676C22—44no—
677C22—44no—
678C22—44no—
679C22—44no—
680C22—82no—
681C22—82no—
682C22—82no—
683C22—82no—
684C22—82no—
685C22—82no—
686C22—82no—
687C22—82no—
688C22C22×#11711yescharacteristic
689C22C22×C4×D411yescharacteristic
690C22C2×C4×#4111yescharacteristic
691C22C2×#51811yescharacteristic
692C22C2×#52011yescharacteristic
693C22#342811yescharacteristic
694C22#342811yescharacteristic
695C22C2×#54612yes—
696C22D4×#4112yes—
697C22C2×#55312yes—
698C22#347512yes—
699C22#348212yes—
700C22#348312yes—
701C22#349611yescharacteristic
702C22#349711yescharacteristic
703C22#350612yes—
704C22#350612yes—
705C22#350611yescharacteristic
706C22#350711yescharacteristic
707C22#352112yes—
708C22#352412yes—
709C22#354112yes—
710C22#354212yes—
711C4—24no—
712C4—28no—
713C4—24no—
714C4—24no—
715C4—42no—
716C4—44no—
717C4—42no—
718C4—44no—
719C4—48no—
720C4—48no—
721C4—84no—
722C4—84no—
723C4#236418yes—
724C2—44no—
725C2—84no—
726C2D4×#11711yescharacteristic
727C2#3412911yescharacteristic
728C2C2×#236411yescharacteristic
729C2#3418611yescharacteristic
730C2#3420611yescharacteristic
731C2#3424411yescharacteristic
732C2#3436211yescharacteristic
733C2#5981012yes—
734C2#5981112yes—
735C2#5985712yes—
736C2#5985812yes—
subgroupG/H
center, Frattini, socle, cosocleC24C25
commutatorC23C24×C4
prime radicalC22×#121C22
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. C2×#2354C2semidirect
3. #14422C2semidirect
5. #14157C2semidirect
6. #14076C2semidirect
7. #14037C2semidirect
8. #13884C2semidirect
9. #13876C2semidirect
10. #13871C2semidirect
11. #13871C2semidirect
12. #13825C2semidirect
13. #13801C2semidirect
14. C4×#380C2semidirect
15. #13756C2semidirect
23. C4×#121C4semidirect
144. #380D4semidirect
486. C4⋊C4#117semidirect

Maximal subgroups

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

subgroupindex
C2×#23542
C2×C4×#1212
#144222
#144182
#141572
#140762
#140372
#138842
#138762
#138712
#138252
#138012
C4×#3802
#137562