Finite Group Databasebeta

← #722780

Subgroups — #722780

Order 512.

2742 conjugacy classes of subgroups.

The subgroup lattice has 706 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2566C211yesmaximal, characteristic
2C2×D4×C4⋊C4C211yesmaximal, characteristic
3#16986C212yesmaximal, complemented
4#16986C212yesmaximal, complemented
5#15555C214yesmaximal, complemented
6#15392C214yesmaximal, complemented
7#15221C212yesmaximal, complemented
8#15133C212yesmaximal, complemented
9#15092C212yesmaximal, complemented
10#14786C212yesmaximal, complemented
11#14283C211yesmaximal, characteristic, complemented
12#14160C211yesmaximal, characteristic, complemented
13#14018C212yesmaximal
14#14018C212yesmaximal
15#13910C211yesmaximal, characteristic, complemented
16#13856C211yesmaximal, characteristic, complemented
17#13785C211yesmaximal, characteristic
18C22×#125C2211yescharacteristic
19C22×C4×D4C2211yescharacteristic
20C22×C4×D4C2212yes—
21C23×C4⋊C4C2212yes—
22#2566—28no—
23C2×#398C2214yes—
24C2×#396C2212yes—
25C2×#396C2212yes—
26D4×C4⋊C4—28no—
27C2×#390C2211yescharacteristic
28C2×#390C2211yescharacteristic
29C2×#390C2212yes—
30C2×#389C2212yes—
31C2×#389C2212yes—
32C2×#389C2212yes—
33C2×#385C2214yes—
34C2×#385C2212yes—
35C2×#385C2212yes—
36C2×#385C2214yes—
37C2×#385C2214yes—
38C2×#384C2211yescharacteristic
39C2×#384C2211yescharacteristic
40C2×#384C2212yes—
41C2×#382C2211yescharacteristic
42C2×#380C2212yes—
43C2×C4×C4⋊C4C2211yescharacteristic
44C2×C4×C4⋊C4C2211yescharacteristic
45C2×C4×C4⋊C4C2211yescharacteristic
46C2×C4×C4⋊C4C2212yes—
47C2×C4×C4⋊C4C2212yes—
48C2×#376C2211yescharacteristic
49#1445C2214yes—
50#1444C2212yes—
51#1444C2211yescharacteristic
52#1444C2211yescharacteristic
53#1444C2211yescharacteristic
54#1444C2212yes—
55#1444C2211yescharacteristic
56#1444C2214yes—
57#1444C2212yes—
58#1444C2212yes—
59#1442C2214yes—
60#1442C2214yes—
61#1441C2214yes—
62#1441C2214yes—
63#1441C2214yes—
64#1441C2214yes—
65#1441C2212yes—
66#1441C2212yes—
67#1441C2212yes—
68#1441C2212yes—
69#1439C2214yes—
70#1439C2214yes—
71#1439C2214yes—
72#1439C2214yes—
73#1437C2214yes—
74#1436C2214yes—
75C4⋊#95C2212yes—
76C4⋊#95C2212yes—
77C4⋊#95C2212yes—
78C4⋊#95C2212yes—
79C4⋊#95C2212yes—
80C4⋊#95C2212yes—
81C22⋊#95C2212yes—
82C22⋊#95C2212yes—
83C22⋊#95C2214yes—
84C22⋊#95C2214yes—
85C23×D4C2311yescharacteristic
86C24×C4C2314yes—
87C24×C4C2312yes—
88C2×#125—24no—
89C2×#125—24no—
90C2×#125—24no—
91C2×C4×D4—28no—
92C2×C4×D4—24no—
93C2×C4×D4—24no—
94C2×C4×D4—44no—
95C2×C4×D4—44no—
96C2×C4×D4D418yes—
97C22×C4⋊C4—28no—
98C22×C4⋊C4—48no—
99C22×C4⋊C4C2312yes—
100C22×C4⋊C4C2311yescharacteristic
101C22×C4⋊C4C2312yes—
102C22×C4⋊C4C2311yescharacteristic
103C22×C4⋊C4C2311yescharacteristic
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2311yescharacteristic
106C22×C4⋊C4C2311yescharacteristic
107C22×C4⋊C4C2312yes—
108C22×C4⋊C4C2314yes—
109C22×C4⋊C4C2311yescharacteristic
110C22×C4⋊C4C2312yes—
111C22×C4⋊C4C2312yes—
112C22×C4⋊C4C2312yes—
113C22×C4⋊C4C2314yes—
114C22×C4⋊C4C2312yes—
115C22×C4⋊C4C2312yes—
116C22×C4⋊C4C2312yes—
117C22×C4⋊C4C2311yescharacteristic
118C22×C4⋊C4C2312yes—
119C22×C4⋊C4C2312yes—
120C22×C4⋊C4C2314yes—
121C22×C22⋊C4C2312yes—
122C22×C22⋊C4C2312yes—
123C22×C22⋊C4C2312yes—
124C22×C22⋊C4C2312yes—
125C22×C22⋊C4C2312yes—
126C22×C22⋊C4C2314yes—
127C22×C22⋊C4C2314yes—
128C22×C22⋊C4C2314yes—
129C22×C42C2312yes—
130C22×C42C2311yescharacteristic
131C22×C42C2311yescharacteristic
132C22×C42C2312yes—
133C22×C42C2312yes—
134C22×C42C2311yescharacteristic
135#398—48no—
136#396—28no—
137#396—28no—
138#390—24no—
139#390—24no—
140#390Q8116yes—
141#389—28no—
142#389—44no—
143#389—44no—
144#385—28no—
145#385—28no—
146#385—216no—
147#385—48no—
148#385—48no—
149#384—24no—
150#384—24no—
151#384—44no—
152#382—24no—
153#380—28no—
154C4×C4⋊C4—24no—
155C4×C4⋊C4—24no—
156C4×C4⋊C4—24no—
157C4×C4⋊C4—44no—
158C4×C4⋊C4—44no—
159#376—24no—
160C2×#95C2312yes—
161C2×#95C2312yes—
162C2×#95C2314yes—
163C2×#95C2311yescharacteristic
164C2×#95C2311yescharacteristic
165C2×#95C2312yes—
166C2×#95C2312yes—
167C2×#95C2312yes—
168C2×#95C2314yes—
169C2×#95C2311yescharacteristic
170C2×#95C2311yescharacteristic
171C2×#95C2314yes—
172C2×#95C2314yes—
173C2×#95C2312yes—
174C2×#95C2312yes—
175C2×#95C2314yes—
176C2×#95C2314yes—
177C2×#95C2314yes—
178C2×#95C2314yes—
179C2×#95C2314yes—
180C2×#95C2314yes—
181C2×#95C2314yes—
182C2×#95C2314yes—
183C2×#95C2314yes—
184C2×#95C2312yes—
185C2×#95C2312yes—
186C25C2412yes—
187C22×D4—22no—
188C22×D4—22no—
189C22×D4—24no—
190C22×D4C2×D414yes—
191C22×D4C2×Q818yes—
192C23×C4—24no—
193C23×C4—24no—
194C23×C4—28no—
195C23×C4—28no—
196C23×C4—44no—
197C23×C4—44no—
198C23×C4—44no—
199C23×C4—44no—
200C23×C4—44no—
201C23×C4—44no—
202C23×C4—44no—
203C23×C4C2×D418yes—
204C23×C4C2411yescharacteristic
205C23×C4C2412yes—
206C23×C4C2412yes—
207C23×C4C2411yescharacteristic
208C23×C4C2412yes—
209C23×C4C2414yes—
210C23×C4C2412yes—
211C23×C4C2414yes—
212C23×C4C2412yes—
213C23×C4C2412yes—
214C23×C4C2411yescharacteristic
215C23×C4C2412yes—
216C23×C4C2411yescharacteristic
217C23×C4C2411yescharacteristic
218C23×C4C2411yescharacteristic
219C23×C4C2411yescharacteristic
220#125—416no—
221C4×D4—232no—
222C4×D4—432no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—24no—
227C2×C4⋊C4—24no—
228C2×C4⋊C4—24no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4—28no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—28no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—24no—
258C2×C4⋊C4—24no—
259C2×C4⋊C4—28no—
260C2×C4⋊C4—432no—
261C2×C4⋊C4—44no—
262C2×C4⋊C4—44no—
263C2×C4⋊C4—41no—
264C2×C4⋊C4—41no—
265C2×C4⋊C4—42no—
266C2×C4⋊C4—42no—
267C2×C4⋊C4—42no—
268C2×C4⋊C4—42no—
269C2×C4⋊C4—42no—
270C2×C4⋊C4—42no—
271C2×C4⋊C4—42no—
272C2×C4⋊C4—42no—
273C2×C4⋊C4—42no—
274C2×C4⋊C4—42no—
275C2×C4⋊C4—44no—
276C2×C4⋊C4—42no—
277C2×C4⋊C4—44no—
278C2×C4⋊C4—44no—
279C2×C4⋊C4—44no—
280C2×C4⋊C4—44no—
281C2×C4⋊C4—44no—
282C2×C4⋊C4—44no—
283C2×C4⋊C4C2×D414yes—
284C2×C4⋊C4C2×Q818yes—
285C2×C22⋊C4—24no—
286C2×C22⋊C4—24no—
287C2×C22⋊C4—28no—
288C2×C22⋊C4—24no—
289C2×C22⋊C4—24no—
290C2×C22⋊C4—24no—
291C2×C22⋊C4—24no—
292C2×C22⋊C4—24no—
293C2×C22⋊C4—24no—
294C2×C22⋊C4—44no—
295C2×C22⋊C4—44no—
296C2×C22⋊C4—42no—
297C2×C22⋊C4—42no—
298C2×C22⋊C4—44no—
299C2×C22⋊C4—44no—
300C2×C22⋊C4—44no—
301C2×C22⋊C4—44no—
302C2×C22⋊C4—44no—
303C2×C22⋊C4—44no—
304C2×C22⋊C4C2×D418yes—
305C2×C22⋊C4C2×Q8116yes—
306C2×C22⋊C4C2×Q8116yes—
307C2×C42—24no—
308C2×C42—24no—
309C2×C42—24no—
310C2×C42—22no—
311C2×C42—22no—
312C2×C42—24no—
313C2×C42—24no—
314C2×C42—24no—
315C2×C42—24no—
316C2×C42—24no—
317C2×C42—42no—
318C2×C42—42no—
319C2×C42C2×D414yes—
320C2×C42C2×Q818yes—
321C2×C42#4114yes—
322C2×C42#4114yes—
323#95—22no—
324#95—22no—
325#95—24no—
326#95—28no—
327#95—24no—
328#95—24no—
329#95—24no—
330#95—22no—
331#95—22no—
332#95—24no—
333#95—28no—
334#95—44no—
335#95—44no—
336#95—44no—
337#95—44no—
338#95—42no—
339#95—42no—
340#95—42no—
341#95—42no—
342#95—44no—
343#95—44no—
344#95—44no—
345#95—44no—
346#95—44no—
347#95—44no—
348#95—44no—
349C24—22no—
350C24—22no—
351C24—22no—
352C24—24no—
353C24—24no—
354C24—22no—
355C24—22no—
356C24—24no—
357C24—22no—
358C24C22×D414yes—
359C24C22×Q818yes—
360C24C2511yescharacteristic
361C2×D4—28no—
362C2×D4—28no—
363C2×D4—216no—
364C2×D4—216no—
365C2×D4C4⋊Q8116yes—
366C22×C4—21no—
367C22×C4—22no—
368C22×C4—21no—
369C22×C4—21no—
370C22×C4—21no—
371C22×C4—22no—
372C22×C4—21no—
373C22×C4—21no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—24no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—24no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—24no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—24no—
393C22×C4—24no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—24no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—24no—
401C22×C4—24no—
402C22×C4—21no—
403C22×C4—21no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—216no—
409C22×C4—22no—
410C22×C4—21no—
411C22×C4—21no—
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—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—24no—
426C22×C4—24no—
427C22×C4—24no—
428C22×C4—24no—
429C22×C4—24no—
430C22×C4—22no—
431C22×C4—21no—
432C22×C4—22no—
433C22×C4—21no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—24no—
440C22×C4—24no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—48no—
446C22×C4—48no—
447C22×C4—416no—
448C22×C4—416no—
449C22×C4—416no—
450C22×C4—416no—
451C22×C4—88no—
452C22×C4—88no—
453C22×C4—88no—
454C22×C4—88no—
455C22×C4—88no—
456C22×C4C22×D414yes—
457C22×C4C22×D412yes—
458C22×C4C22×D412yes—
459C22×C4C22×D412yes—
460C22×C4C22×Q814yes—
461C22×C4C22×Q814yes—
462C22×C4C22×Q818yes—
463C22×C4C22×Q814yes—
464C22×C4C2×#4112yes—
465C22×C4C2×#4112yes—
466C22×C4C2×#4112yes—
467C22×C4C2×#4112yes—
468C22×C4C2×#4112yes—
469C22×C4C2×#4112yes—
470C22×C4H(5,𝔽2)14yes—
471C22×C4H(5,𝔽2)12yes—
472C22×C4H(5,𝔽2)14yes—
473C22×C42−1+412yes—
474C4⋊C4—28no—
475C4⋊C4—28no—
476C4⋊C4—216no—
477C4⋊C4—44no—
478C4⋊C4—44no—
479C4⋊C4—48no—
480C4⋊C4—416no—
481C4⋊C4—44no—
482C4⋊C4—48no—
483C4⋊C4—48no—
484C4⋊C4—48no—
485C4⋊C4—48no—
486C4⋊C4—48no—
487C4⋊C4—48no—
488C4⋊C4—44no—
489C4⋊C4—44no—
490C4⋊C4—48no—
491C4⋊C4—84no—
492C4⋊C4—84no—
493C4⋊C4—84no—
494C4⋊C4—88no—
495C4⋊C4—88no—
496C22⋊C4—216no—
497C22⋊C4—416no—
498C22⋊C4—416no—
499C22⋊C4—48no—
500C22⋊C4—48no—
501C22⋊C4—416no—
502C22⋊C4—84no—
503C22⋊C4—84no—
504C42—216no—
505C42—48no—
506C42—44no—
507C42—44no—
508C42—48no—
509C42#123116yes—
510C23—24no—
511C23—24no—
512C23—28no—
513C23—28no—
514C23—24no—
515C23—24no—
516C23—24no—
517C23—24no—
518C23—28no—
519C23—28no—
520C23—28no—
521C23—28no—
522C23—44no—
523C23—44no—
524C23—42no—
525C23—42no—
526C23—42no—
527C23—42no—
528C23—44no—
529C23—44no—
530C23—44no—
531C23—44no—
532C23C2×C4⋊Q818yes—
533C23C23×D411yescharacteristic
534C23C23×Q812yes—
535C23C22×#4111yescharacteristic
536C23C22×#4111yescharacteristic
537C23C2×H(5,𝔽2)12yes—
538C23C2×H(5,𝔽2)12yes—
539C23C2×H(5,𝔽2)11yescharacteristic
540C23C2×2−1+411yescharacteristic
541C23#58512yes—
542C23#58511yescharacteristic
543C23#58511yescharacteristic
544D4—232no—
545C2×C4—24no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—24no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—24no—
556C2×C4—24no—
557C2×C4—24no—
558C2×C4—28no—
559C2×C4—24no—
560C2×C4—24no—
561C2×C4—24no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—24no—
565C2×C4—24no—
566C2×C4—28no—
567C2×C4—28no—
568C2×C4—24no—
569C2×C4—24no—
570C2×C4—24no—
571C2×C4—24no—
572C2×C4—24no—
573C2×C4—24no—
574C2×C4—432no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—44no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—42no—
584C2×C4—44no—
585C2×C4—44no—
586C2×C4—44no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—44no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—44no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—41no—
603C2×C4—41no—
604C2×C4—44no—
605C2×C4—44no—
606C2×C4—41no—
607C2×C4—42no—
608C2×C4—44no—
609C2×C4—41no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—41no—
614C2×C4—44no—
615C2×C4—42no—
616C2×C4—41no—
617C2×C4—42no—
618C2×C4—44no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—44no—
625C2×C4—44no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4—42no—
632C2×C4—42no—
633C2×C4—44no—
634C2×C4—44no—
635C2×C4—832no—
636C2×C4C2×#12314yes—
637C2×C4C2×#12314yes—
638C2×C4C2×#12314yes—
639C2×C4C2×C4⋊Q814yes—
640C2×C4#53514yes—
641C2×C4#55418yes—
642C2×C4#55418yes—
643C22—28no—
644C22—28no—
645C22—28no—
646C22—48no—
647C22—48no—
648C22—48no—
649C22—44no—
650C22—48no—
651C22—44no—
652C22—44no—
653C22—44no—
654C22C22×#12311yescharacteristic
655C22C22×C4⋊Q811yescharacteristic
656C22C2×#53511yescharacteristic
657C22#346012yes—
658C22C2×#55412yes—
659C22C2×#55412yes—
660C22#348411yescharacteristic
661C22#348511yescharacteristic
662C22#348811yescharacteristic
663C22#348911yescharacteristic
664C22#350112yes—
665C22#350112yes—
666C22#350312yes—
667C22#350412yes—
668C22#350611yescharacteristic
669C22#350612yes—
670C22#350711yescharacteristic
671C22#351111yescharacteristic
672C22#351211yescharacteristic
673C22#353712yes—
674C22#353912yes—
675C22#354112yes—
676C22#354212yes—
677C4—24no—
678C4—24no—
679C4—44no—
680C4—44no—
681C4—48no—
682C4—44no—
683C4—42no—
684C4—42no—
685C4—44no—
686C4—82no—
687C4—82no—
688C4—81no—
689C4—82no—
690C4—81no—
691C4—84no—
692C4#250218yes—
693C2—48no—
694C2C2×#250211yescharacteristic
695C2#3515211yescharacteristic
696C2#3551711yescharacteristic
697C2#3592811yescharacteristic
698C2#3595011yescharacteristic
699C2#3968812yes—
700C2#3969212yes—
701C2#4248011yescharacteristic
702C2#4267911yescharacteristic
703C2#4496612yes—
704C2#4536212yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC23×D4C23
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #16986C2semidirect
4. #16986C2semidirect
5. #15555C2semidirect
6. #15392C2semidirect
7. #15221C2semidirect
8. #15133C2semidirect
9. #15092C2semidirect
10. #14786C2semidirect
11. #14283C2semidirect
12. #14160C2semidirect
15. #13910C2semidirect
16. #13856C2semidirect

Maximal subgroups

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

subgroupindex
C2×#25662
C2×D4×C4⋊C42
#169862
#155552
#153922
#152212
#151332
#150922
#147862
#142832
#141602
#140182
#139102
#138562
#137852