Finite Group Databasebeta

← #544513

Subgroups — #544513

Order 512.

3238 conjugacy classes of subgroups.

The subgroup lattice has 759 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2392C211yesmaximal, characteristic
2C2×#2352C212yesmaximal, complemented
3#14634C212yesmaximal, complemented
4#14581C212yesmaximal, complemented
5#14572C212yesmaximal
6#14571C214yesmaximal, complemented
7#14340C211yesmaximal, characteristic, complemented
8#14315C214yesmaximal, complemented
9#14224C212yesmaximal, complemented
10#13885C212yesmaximal, complemented
11#13881C212yesmaximal, complemented
12#13876C212yesmaximal, complemented
13#13872C212yesmaximal, complemented
14#13872C212yesmaximal, complemented
15#13795C211yesmaximal, characteristic, complemented
16C22×C4×D4C2212yes—
17C22×C4×D4C2211yescharacteristic
18C22×#117C2211yescharacteristic
19C22×#117C2211yescharacteristic
20C23×C4⋊C4C2211yescharacteristic
21C23×C4⋊C4C2212yes—
22#2392C4116yescomplemented
23C2×#398C2211yescharacteristic
24C2×#397C2211yescharacteristic
25C2×#396C2212yes—
26C2×#394C2211yescharacteristic
27C2×#394C2212yes—
28C2×#392C2211yescharacteristic
29#2352—216no—
30C2×#388C2214yes—
31C2×#388C2214yes—
32C2×#387C2214yes—
33C2×#386C2214yes—
34C2×#386C2214yes—
35C2×#386C2214yes—
36C2×#384C2211yescharacteristic
37C2×#382C2212yes—
38C2×#382C2211yescharacteristic
39C2×#380C2212yes—
40C2×#380C2212yes—
41C2×#380C2212yes—
42C2×#380C2212yes—
43C2×C4×C4⋊C4C2212yes—
44C2×C4×C4⋊C4C2211yescharacteristic
45C2×C4×C22⋊C4C2214yes—
46C2×C4×C22⋊C4C2212yes—
47#1445C2214yes—
48#1443C2214yes—
49#1442C2214yes—
50#1442C2214yes—
51#1442C2214yes—
52#1441C2212yes—
53#1441C2212yes—
54#1441C2214yes—
55#1441C2214yes—
56#1439C2212yes—
57#1439C2212yes—
58#1439C2214yes—
59#1439C2214yes—
60#1439C2214yes—
61#1436C2214yes—
62#1436C2214yes—
63C4⋊#95C2212yes—
64C4⋊#95C2212yes—
65C4⋊#95C2212yes—
66C4⋊#95C2212yes—
67C4⋊#95C2212yes—
68C4⋊#95C2212yes—
69C4⋊#95C2212yes—
70#1434C2214yes—
71C22⋊#95C2214yes—
72C22⋊#95C2214yes—
73C22⋊#95C2214yes—
74C22⋊#95C2214yes—
75C22⋊#95C2212yes—
76C22⋊#95C2212yes—
77C23×D4C2312yes—
78C23×D4C2311yescharacteristic
79C24×C4C2312yes—
80C24×C4C2311yescharacteristic
81C24×C4C2314yes—
82C24×C4C2312yes—
83C2×C4×D4—28no—
84C2×C4×D4—28no—
85C2×C4×D4—42no—
86C2×C4×D4—42no—
87C2×C4×D4C2×C4116yescomplemented
88C2×C4×D4C2×C418yescomplemented
89C2×#117—42no—
90C2×#117—42no—
91C2×#117—42no—
92C2×#117—42no—
93C2×#117C2×C418yescomplemented
94C2×#117D418yescomplemented
95C22×C4⋊C4—24no—
96C22×C4⋊C4—28no—
97C22×C4⋊C4—44no—
98C22×C4⋊C4—44no—
99C22×C4⋊C4—42no—
100C22×C4⋊C4—42no—
101C22×C4⋊C4C2312yes—
102C22×C4⋊C4C2312yes—
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2312yes—
106C22×C4⋊C4C2312yes—
107C22×C4⋊C4C2314yes—
108C22×C4⋊C4C2312yes—
109C22×C4⋊C4C2312yes—
110C22×C4⋊C4C2312yes—
111C22×C4⋊C4C2311yescharacteristic
112C22×C4⋊C4C2311yescharacteristic
113C22×C4⋊C4C2311yescharacteristic
114C22×C22⋊C4C2314yes—
115C22×C22⋊C4C2312yes—
116C22×C22⋊C4C2312yes—
117C22×C22⋊C4C2312yes—
118C22×C22⋊C4C2314yes—
119C22×C22⋊C4C2314yes—
120C22×C22⋊C4C2314yes—
121C22×C22⋊C4C2312yes—
122C22×C22⋊C4C2311yescharacteristic
123C22×C22⋊C4C2311yescharacteristic
124C22×C22⋊C4C2312yes—
125C22×C22⋊C4C2312yes—
126C22×C42C2311yescharacteristic
127C22×C42C2311yescharacteristic
128C22×C42C2312yes—
129C22×C42C2314yes—
130C22×C42C2312yes—
131#398C2×C418yescomplemented
132#397C2×C418yescomplemented
133#396C2×C4116yescomplemented
134#394C2×C4116yescomplemented
135#394C2×C418yescomplemented
136#392C2×C418yes—
137#388—216no—
138#388—48no—
139#387—216no—
140#386—216no—
141#386—48no—
142#386—48no—
143#384—42no—
144#382—28no—
145#382—42no—
146#380—28no—
147#380—28no—
148#380—44no—
149#380—44no—
150C4×C4⋊C4—28no—
151C4×C4⋊C4C2×C418yescomplemented
152C4×C22⋊C4—216no—
153C4×C22⋊C4C2×C4116yescomplemented
154C2×#95C2314yes—
155C2×#95C2314yes—
156C2×#95C2314yes—
157C2×#95C2314yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2314yes—
161C2×#95C2314yes—
162C2×#95C2314yes—
163C2×#95C2312yes—
164C2×#95C2312yes—
165C2×#95C2314yes—
166C2×#95C2314yes—
167C2×#95C2314yes—
168C2×#95C2314yes—
169C2×#95C2314yes—
170C2×#95C2314yes—
171C2×#95C2314yes—
172C2×#95C2314yes—
173C2×#95C2311yescharacteristic
174C2×#95C2311yescharacteristic
175C2×#95C2312yes—
176C2×#95C2312yes—
177C25C2411yescharacteristic
178C25C2412yes—
179C22×D4—24no—
180C22×D4—24no—
181C22×D4—24no—
182C22×D4—24no—
183C22×D4—24no—
184C22×D4—24no—
185C22×D4—41no—
186C22×D4—41no—
187C22×D4—42no—
188C22×D4—42no—
189C22×D4C22×C418yes—
190C22×D4C22×C414yes—
191C23×C4—22no—
192C23×C4—22no—
193C23×C4—22no—
194C23×C4—22no—
195C23×C4—24no—
196C23×C4—24no—
197C23×C4—24no—
198C23×C4—24no—
199C23×C4—24no—
200C23×C4—24no—
201C23×C4—24no—
202C23×C4—28no—
203C23×C4—28no—
204C23×C4—44no—
205C23×C4—44no—
206C23×C4—42no—
207C23×C4—42no—
208C23×C4—44no—
209C23×C4—44no—
210C23×C4—44no—
211C23×C4—44no—
212C23×C4—44no—
213C23×C4C22×C418yes—
214C23×C4C22×C414yes—
215C23×C4C2×D414yes—
216C23×C4#4114yes—
217C23×C4C2414yes—
218C23×C4C2414yes—
219C23×C4C2414yes—
220C23×C4C2411yescharacteristic
221C23×C4C2411yescharacteristic
222C23×C4C2411yescharacteristic
223C23×C4C2412yes—
224C23×C4C2412yes—
225C23×C4C2412yes—
226C23×C4C2411yescharacteristic
227C23×C4C2414yes—
228C23×C4C2411yescharacteristic
229C23×C4C2411yescharacteristic
230C4×D4—264no—
231C4×D4—416no—
232#117—416no—
233#117—416no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—24no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—28no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—432no—
250C2×C4⋊C4—42no—
251C2×C4⋊C4—42no—
252C2×C4⋊C4—44no—
253C2×C4⋊C4—44no—
254C2×C4⋊C4—42no—
255C2×C4⋊C4—42no—
256C2×C4⋊C4—41no—
257C2×C4⋊C4—41no—
258C2×C4⋊C4—41no—
259C2×C4⋊C4—41no—
260C2×C4⋊C4—41no—
261C2×C4⋊C4—41no—
262C2×C4⋊C4—416no—
263C2×C4⋊C4—44no—
264C2×C4⋊C4—42no—
265C2×C4⋊C4—42no—
266C2×C4⋊C4—44no—
267C2×C4⋊C4C22×C418yes—
268C2×C4⋊C4C22×C414yes—
269C2×C4⋊C4C22×C414yes—
270C2×C4⋊C4C22×C414yes—
271C2×C4⋊C4C22×C418yes—
272C2×C4⋊C4C2×D418yes—
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—28no—
280C2×C22⋊C4—28no—
281C2×C22⋊C4—28no—
282C2×C22⋊C4—28no—
283C2×C22⋊C4—28no—
284C2×C22⋊C4—42no—
285C2×C22⋊C4—42no—
286C2×C22⋊C4—44no—
287C2×C22⋊C4—41no—
288C2×C22⋊C4—41no—
289C2×C22⋊C4—41no—
290C2×C22⋊C4—41no—
291C2×C22⋊C4—42no—
292C2×C22⋊C4—42no—
293C2×C22⋊C4—42no—
294C2×C22⋊C4—42no—
295C2×C22⋊C4—44no—
296C2×C22⋊C4—44no—
297C2×C22⋊C4—44no—
298C2×C22⋊C4—44no—
299C2×C22⋊C4—44no—
300C2×C22⋊C4—44no—
301C2×C22⋊C4C22×C418yes—
302C2×C22⋊C4C22×C418yes—
303C2×C22⋊C4C22×C418yes—
304C2×C22⋊C4C22×C414yes—
305C2×C22⋊C4C22×C414yes—
306C2×C22⋊C4C22×C418yes—
307C2×C22⋊C4C22×C418yes—
308C2×C22⋊C4C2×D418yes—
309C2×C42—24no—
310C2×C42—24no—
311C2×C42—22no—
312C2×C42—22no—
313C2×C42—22no—
314C2×C42—22no—
315C2×C42—28no—
316C2×C42—42no—
317C2×C42—42no—
318C2×C42—44no—
319C2×C42—44no—
320C2×C42C22×C414yes—
321C2×C42C22×C414yes—
322C2×C42C22×C418yes—
323C2×C42C2×D418yes—
324#95—28no—
325#95—28no—
326#95—28no—
327#95—28no—
328#95—28no—
329#95—44no—
330#95—44no—
331#95—44no—
332#95—44no—
333#95—44no—
334#95—44no—
335#95—42no—
336#95—42no—
337#95—44no—
338#95—44no—
339#95—44no—
340#95—44no—
341#95—44no—
342#95—44no—
343#95C22×C418yes—
344#95C22×C418yes—
345#95C22×C414yes—
346#95C22×C414yes—
347C24—22no—
348C24—22no—
349C24—21no—
350C24—22no—
351C24—21no—
352C24—21no—
353C24—22no—
354C24—22no—
355C24—22no—
356C24—22no—
357C24—21no—
358C24—22no—
359C24—22no—
360C24—22no—
361C24—22no—
362C24—22no—
363C24—22no—
364C24—22no—
365C24—22no—
366C24—22no—
367C24C23×C412yes—
368C24C23×C414yes—
369C24C22×D412yes—
370C24C2×#4112yes—
371C24H(5,𝔽2)14yes—
372C242−1+414yes—
373C24C2511yescharacteristic
374C2×D4—216no—
375C2×D4—216no—
376C2×D4—216no—
377C2×D4—44no—
378C2×D4—48no—
379C2×D4—48no—
380C2×D4—48no—
381C2×D4—48no—
382C2×D4—48no—
383C2×D4—48no—
384C2×D4—44no—
385C2×D4—44no—
386C22×C4—21no—
387C22×C4—21no—
388C22×C4—21no—
389C22×C4—21no—
390C22×C4—28no—
391C22×C4—28no—
392C22×C4—28no—
393C22×C4—28no—
394C22×C4—28no—
395C22×C4—28no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—21no—
405C22×C4—21no—
406C22×C4—22no—
407C22×C4—24no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—21no—
412C22×C4—21no—
413C22×C4—21no—
414C22×C4—21no—
415C22×C4—21no—
416C22×C4—21no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—24no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—24no—
431C22×C4—22no—
432C22×C4—24no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—216no—
437C22×C4—216no—
438C22×C4—216no—
439C22×C4—24no—
440C22×C4—24no—
441C22×C4—24no—
442C22×C4—24no—
443C22×C4—24no—
444C22×C4—24no—
445C22×C4—24no—
446C22×C4—24no—
447C22×C4—24no—
448C22×C4—24no—
449C22×C4—24no—
450C22×C4—24no—
451C22×C4—24no—
452C22×C4—24no—
453C22×C4—24no—
454C22×C4—24no—
455C22×C4—24no—
456C22×C4—24no—
457C22×C4—24no—
458C22×C4—24no—
459C22×C4—24no—
460C22×C4—416no—
461C22×C4—416no—
462C22×C4—48no—
463C22×C4—48no—
464C22×C4—48no—
465C22×C4—48no—
466C22×C4—48no—
467C22×C4—88no—
468C22×C4—88no—
469C22×C4—88no—
470C22×C4—88no—
471C22×C4—88no—
472C22×C4—88no—
473C22×C4—84no—
474C22×C4—84no—
475C22×C4—88no—
476C22×C4C4×D4116yes—
477C22×C4C23×C414yes—
478C22×C4C23×C412yes—
479C22×C4C23×C412yes—
480C22×C4C23×C414yes—
481C22×C4C23×C412yes—
482C22×C4C23×C414yes—
483C22×C4C23×C412yes—
484C22×C4C23×C412yes—
485C22×C4C23×C412yes—
486C22×C4C22×D418yes—
487C22×C4C22×D412yes—
488C22×C4C22×D412yes—
489C22×C4C2×#4112yes—
490C22×C4C2×#4112yes—
491C22×C4H(5,𝔽2)12yes—
492C22×C4H(5,𝔽2)12yes—
493C22×C4H(5,𝔽2)12yes—
494C22×C42−1+412yes—
495C4⋊C4—216no—
496C4⋊C4—216no—
497C4⋊C4—48no—
498C4⋊C4—44no—
499C4⋊C4—44no—
500C4⋊C4—44no—
501C4⋊C4—48no—
502C4⋊C4—416no—
503C4⋊C4—48no—
504C4⋊C4—48no—
505C4⋊C4—48no—
506C4⋊C4—84no—
507C4⋊C4—84no—
508C22⋊C4—216no—
509C22⋊C4—216no—
510C22⋊C4—216no—
511C22⋊C4—48no—
512C22⋊C4—44no—
513C22⋊C4—44no—
514C22⋊C4—48no—
515C22⋊C4—48no—
516C22⋊C4—416no—
517C22⋊C4—88no—
518C22⋊C4—88no—
519C22⋊C4—88no—
520C42—216no—
521C42—28no—
522C42—28no—
523C42—48no—
524C42—88no—
525C23—24no—
526C23—22no—
527C23—24no—
528C23—22no—
529C23—24no—
530C23—24no—
531C23—24no—
532C23—24no—
533C23—24no—
534C23—24no—
535C23—22no—
536C23—22no—
537C23—22no—
538C23—22no—
539C23—24no—
540C23—24no—
541C23—24no—
542C23—24no—
543C23—24no—
544C23—24no—
545C23—24no—
546C23—24no—
547C23—24no—
548C23—24no—
549C23—24no—
550C23—24no—
551C23—24no—
552C23—24no—
553C23—24no—
554C23—24no—
555C23—42no—
556C23—42no—
557C23—42no—
558C23—42no—
559C23—42no—
560C23—42no—
561C23—42no—
562C23—42no—
563C23—42no—
564C23—41no—
565C23—41no—
566C23—42no—
567C23—41no—
568C23—42no—
569C23—42no—
570C23—42no—
571C23—41no—
572C23—42no—
573C23—42no—
574C23—42no—
575C23—42no—
576C23—42no—
577C23—42no—
578C23—42no—
579C23—42no—
580C23—42no—
581C23C2×C4×D414yes—
582C23#52018yes—
583C23C24×C411yescharacteristic
584C23C23×D411yescharacteristic
585C23C22×#4111yescharacteristic
586C23C2×H(5,𝔽2)11yescharacteristic
587C23C2×H(5,𝔽2)11yescharacteristic
588C23C2×H(5,𝔽2)11yescharacteristic
589C23C2×H(5,𝔽2)12yes—
590C23C2×2−1+411yescharacteristic
591C23C2×2−1+412yes—
592C23#58512yes—
593C23#58512yes—
594D4—416no—
595D4—432no—
596C2×C4—22no—
597C2×C4—22no—
598C2×C4—22no—
599C2×C4—22no—
600C2×C4—22no—
601C2×C4—22no—
602C2×C4—232no—
603C2×C4—24no—
604C2×C4—24no—
605C2×C4—24no—
606C2×C4—22no—
607C2×C4—22no—
608C2×C4—22no—
609C2×C4—22no—
610C2×C4—24no—
611C2×C4—24no—
612C2×C4—24no—
613C2×C4—24no—
614C2×C4—24no—
615C2×C4—24no—
616C2×C4—24no—
617C2×C4—24no—
618C2×C4—24no—
619C2×C4—24no—
620C2×C4—28no—
621C2×C4—28no—
622C2×C4—28no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—42no—
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—41no—
639C2×C4—41no—
640C2×C4—41no—
641C2×C4—41no—
642C2×C4—41no—
643C2×C4—42no—
644C2×C4—42no—
645C2×C4—41no—
646C2×C4—41no—
647C2×C4—41no—
648C2×C4—44no—
649C2×C4—44no—
650C2×C4—44no—
651C2×C4—432no—
652C2×C4—44no—
653C2×C4—44no—
654C2×C4—44no—
655C2×C4—44no—
656C2×C4—44no—
657C2×C4—44no—
658C2×C4—44no—
659C2×C4—44no—
660C2×C4—44no—
661C2×C4—44no—
662C2×C4—44no—
663C2×C4—44no—
664C2×C4—44no—
665C2×C4—44no—
666C2×C4—44no—
667C2×C4—44no—
668C2×C4—44no—
669C2×C4—44no—
670C2×C4—44no—
671C2×C4—44no—
672C2×C4—44no—
673C2×C4—832no—
674C2×C4—816no—
675C2×C4C2×C4×D414yes—
676C2×C4C2×C4×D414yes—
677C2×C4#51814yes—
678C2×C4#52014yes—
679C2×C4#53414yes—
680C2×C4#53714yes—
681C2×C4#53814yes—
682C2×C4#53914yes—
683C22—24no—
684C22—28no—
685C22—28no—
686C22—28no—
687C22—28no—
688C22—44no—
689C22—42no—
690C22—42no—
691C22—44no—
692C22—44no—
693C22—44no—
694C22—44no—
695C22—44no—
696C22—44no—
697C22—44no—
698C22—42no—
699C22—44no—
700C22—44no—
701C22—44no—
702C22—44no—
703C22—42no—
704C22—44no—
705C22—42no—
706C22—44no—
707C22—42no—
708C22—44no—
709C22C22×C4×D411yescharacteristic
710C22C2×#51811yescharacteristic
711C22C2×#52011yescharacteristic
712C22C2×#52012yes—
713C22#342812yes—
714C22C2×#53411yescharacteristic
715C22C2×#53711yescharacteristic
716C22C2×#53811yescharacteristic
717C22C2×#53911yescharacteristic
718C22#348812yes—
719C22#348912yes—
720C22#349412yes—
721C22#349512yes—
722C22#353612yes—
723C22#353711yescharacteristic
724C22#353811yescharacteristic
725C22#355112yes—
726C22#355111yescharacteristic
727C22#355111yescharacteristic
728C22#355812yes—
729C22#355912yes—
730C22#356112yes—
731C22#356312yes—
732C4—24no—
733C4—44no—
734C4—42no—
735C4—42no—
736C4—42no—
737C4—42no—
738C4—44no—
739C4—44no—
740C4—48no—
741C4—84no—
742C4—84no—
743C4—84no—
744C4#232518yes—
745C2—44no—
746C2—48no—
747C2C2×#232511yescharacteristic
748C2#3420212yes—
749C2#3436011yescharacteristic
750C2#3436911yescharacteristic
751C2#3438212yes—
752C2#4295012yes—
753C2#4295211yescharacteristic
754C2#4295612yes—
755C2#4295811yescharacteristic
756C2#4298011yescharacteristic
757C2#4298211yescharacteristic
subgroupG/H
center, Frattini, socle, cosocleC24C25
commutatorC23C24×C4
prime radicalC2×#392C22
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×#2352C2semidirect
3. #14634C2semidirect
4. #14581C2semidirect
6. #14571C2semidirect
7. #14340C2semidirect
8. #14315C2semidirect
9. #14224C2semidirect
10. #13885C2semidirect
11. #13881C2semidirect
12. #13876C2semidirect
13. #13872C2semidirect
14. #13872C2semidirect
15. #13795C2semidirect
22. #2392C4semidirect
87. C2×C4×D4C2×C4semidirect
88. C2×C4×D4C2×C4semidirect
93. C2×#117C2×C4semidirect
94. C2×#117D4semidirect
131. #398C2×C4semidirect
132. #397C2×C4semidirect
133. #396C2×C4semidirect
134. #394C2×C4semidirect
135. #394C2×C4semidirect
151. C4×C4⋊C4C2×C4semidirect
153. C4×C22⋊C4C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#23922
C2×#23522
#146342
#145812
#145722
#145712
#143402
#143152
#142242
#138852
#138812
#138762
#138722
#137952