Finite Group Databasebeta

← #517757

Subgroups — #517757

Order 512.

2966 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×#2628C211yesmaximal, characteristic
2C2×#2365C211yesmaximal, characteristic, complemented
3C2×#2274C211yesmaximal, characteristic
4#14320C212yesmaximal, complemented
5#14243C212yesmaximal
6#14216C214yesmaximal, complemented
7#14178C214yesmaximal
8#14172C212yesmaximal
9#14161C211yesmaximal, characteristic, complemented
10#14150C211yesmaximal, characteristic, complemented
11#14132C212yesmaximal, complemented
12#14130C212yesmaximal
13C4×#395C212yesmaximal, complemented
14#13852C211yesmaximal, characteristic
15#13821C212yesmaximal, complemented
16#13804C211yesmaximal, characteristic, complemented
17#13801C212yesmaximal
18C22×C4⋊Q8C2211yescharacteristic
19C22×C4×Q8C2212yes—
20C22×#117C2211yescharacteristic
21C22×#117C2211yescharacteristic
22C23×C4⋊C4C2211yescharacteristic
23#2628C4116yescomplemented
24C2×#400C2212yes—
25C2×#399C2212yes—
26C2×#395C2212yes—
27C2×#393C2212yes—
28#2365—28no—
29C2×#391C2212yes—
30C2×#389C2212yes—
31C2×#389C2212yes—
32C2×#389C2212yes—
33C2×#387C2214yes—
34C2×#385C2214yes—
35C2×#382C2211yescharacteristic
36C2×#380C2212yes—
37#2274—28no—
38C2×C4×C4⋊C4C2211yescharacteristic
39C2×C4×C4⋊C4C2211yescharacteristic
40C2×C4×C4⋊C4C2212yes—
41C2×C4×C4⋊C4C2211yescharacteristic
42C2×C4×C4⋊C4C2211yescharacteristic
43C2×C4×C4⋊C4C2212yes—
44C2×C4×C4⋊C4C2212yes—
45C2×C4×C4⋊C4C2211yescharacteristic
46C2×C4×C4⋊C4C2212yes—
47C2×C4×C4⋊C4C2214yes—
48C2×C4×C4⋊C4C2211yescharacteristic
49C2×C4×C22⋊C4C2212yes—
50C2×C4×C22⋊C4C2212yes—
51C2×C4×C22⋊C4C2212yes—
52#1446C2212yes—
53#1446C2212yes—
54#1445C2214yes—
55#1444C2214yes—
56#1444C2212yes—
57#1442C2214yes—
58#1442C2212yes—
59#1441C2211yescharacteristic
60#1441C2212yes—
61#1441C2211yescharacteristic
62#1441C2214yes—
63#1439C2211yescharacteristic
64#1439C2212yes—
65#1439C2211yescharacteristic
66#1437C2214yes—
67#1437C2214yes—
68#1436C2214yes—
69#1436C2214yes—
70#1436C2214yes—
71#1436C2214yes—
72#1436C2214yes—
73C4⋊#95C2212yes—
74C4⋊#95C2212yes—
75C4⋊#95C2212yes—
76C4⋊#95C2212yes—
77C4⋊#95C2214yes—
78#1434C2214yes—
79#1434C2212yes—
80#1434C2214yes—
81#1434C2214yes—
82C22⋊#95C2214yes—
83C4×#95C2212yes—
84C4×#95C2214yes—
85C4×#95C2212yes—
86C23×Q8C2312yes—
87C24×C4C2311yescharacteristic
88C24×C4C2312yes—
89C2×C4⋊Q8—24no—
90C2×C4⋊Q8—24no—
91C2×C4⋊Q8C2×C418yescomplemented
92C2×C4×Q8—28no—
93C2×C4×Q8—48no—
94C2×#117—24no—
95C2×#117—24no—
96C2×#117—24no—
97C2×#117—24no—
98C2×#117C2×C418yes—
99C2×#117Q818yescomplemented
100C22×C4⋊C4—24no—
101C22×C4⋊C4—24no—
102C22×C4⋊C4—24no—
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2311yescharacteristic
105C22×C4⋊C4C2314yes—
106C22×C4⋊C4C2312yes—
107C22×C4⋊C4C2314yes—
108C22×C4⋊C4C2311yescharacteristic
109C22×C4⋊C4C2312yes—
110C22×C4⋊C4C2311yescharacteristic
111C22×C4⋊C4C2312yes—
112C22×C4⋊C4C2312yes—
113C22×C4⋊C4C2311yescharacteristic
114C22×C4⋊C4C2311yescharacteristic
115C22×C4⋊C4C2311yescharacteristic
116C22×C4⋊C4C2311yescharacteristic
117C22×C4⋊C4C2314yes—
118C22×C4⋊C4C2311yescharacteristic
119C22×C22⋊C4C2312yes—
120C22×C22⋊C4C2312yes—
121C22×C22⋊C4C2311yescharacteristic
122C22×C22⋊C4C2311yescharacteristic
123C22×C22⋊C4C2312yes—
124C22×C22⋊C4C2314yes—
125C22×C42C2312yes—
126C22×C42C2312yes—
127C22×C42C2314yes—
128C22×C42C2314yes—
129C22×C42C2311yescharacteristic
130C22×C42C2312yes—
131C22×C42C2312yes—
132C22×C42C2312yes—
133C22×C42C2312yes—
134C22×C42C2312yes—
135C22×C42C2311yescharacteristic
136#400C2×C4116yescomplemented
137#399C2×C4116yes—
138#395C2×C4116yescomplemented
139#393C2×C4116yes—
140#391—28no—
141#389—28no—
142#389—28no—
143#389C2×C4116yescomplemented
144#387—216no—
145#385—48no—
146#382C2×C418yescomplemented
147#380C2×C4116yes—
148C4×C4⋊C4—28no—
149C4×C4⋊C4—24no—
150C4×C4⋊C4—24no—
151C4×C4⋊C4—28no—
152C4×C4⋊C4—24no—
153C4×C4⋊C4—28no—
154C4×C4⋊C4—24no—
155C4×C4⋊C4—24no—
156C4×C4⋊C4—28no—
157C4×C4⋊C4—48no—
158C4×C4⋊C4D418yes—
159C4×C22⋊C4—28no—
160C4×C22⋊C4—28no—
161C4×C22⋊C4—28no—
162C2×#95C2311yescharacteristic
163C2×#95C2312yes—
164C2×#95C2312yes—
165C2×#95C2312yes—
166C2×#95C2314yes—
167C2×#95C2314yes—
168C2×#95C2314yes—
169C2×#95C2312yes—
170C2×#95C2314yes—
171C2×#95C2311yescharacteristic
172C2×#95C2312yes—
173C2×#95C2312yes—
174C2×#95C2312yes—
175C2×#95C2312yes—
176C2×#95C2314yes—
177C2×#95C2314yes—
178C2×#95C2314yes—
179C2×#95C2312yes—
180C2×#95C2312yes—
181C2×#95C2314yes—
182C2×#95C2314yes—
183C2×#95C2314yes—
184C2×#95C2314yes—
185C2×#95C2314yes—
186C2×#95C2312yes—
187C2×#95C2314yes—
188C2×#95C2314yes—
189C2×#95C2312yes—
190C2×#95C2312yes—
191C25C2411yescharacteristic
192C22×Q8—24no—
193C22×Q8—24no—
194C22×Q8—44no—
195C22×Q8—44no—
196C22×Q8C22×C418yes—
197C23×C4—22no—
198C23×C4—22no—
199C23×C4—22no—
200C23×C4—22no—
201C23×C4—24no—
202C23×C4—24no—
203C23×C4—24no—
204C23×C4—44no—
205C23×C4—44no—
206C23×C4C22×C414yes—
207C23×C4C2×Q814yes—
208C23×C4#4114yes—
209C23×C4C2412yes—
210C23×C4C2412yes—
211C23×C4C2411yescharacteristic
212C23×C4C2414yes—
213C23×C4C2414yes—
214C23×C4C2411yescharacteristic
215C23×C4C2412yes—
216C23×C4C2412yes—
217C23×C4C2412yes—
218C23×C4C2412yes—
219C23×C4C2412yes—
220C23×C4C2411yescharacteristic
221C23×C4C2411yescharacteristic
222C23×C4C2411yescharacteristic
223C23×C4C2412yes—
224C23×C4C2411yescharacteristic
225C4⋊Q8—232no—
226C4×Q8—432no—
227#117—232no—
228#117—232no—
229C2×C4⋊C4—232no—
230C2×C4⋊C4—24no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—24no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—28no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—28no—
255C2×C4⋊C4—28no—
256C2×C4⋊C4—28no—
257C2×C4⋊C4—24no—
258C2×C4⋊C4—44no—
259C2×C4⋊C4—42no—
260C2×C4⋊C4—44no—
261C2×C4⋊C4—42no—
262C2×C4⋊C4—44no—
263C2×C4⋊C4—44no—
264C2×C4⋊C4C22×C414yes—
265C2×C4⋊C4C22×C414yes—
266C2×C4⋊C4C22×C418yes—
267C2×C4⋊C4C22×C418yes—
268C2×C4⋊C4C22×C418yes—
269C2×C4⋊C4C22×C4116yes—
270C2×C4⋊C4C2×D414yes—
271C2×C4⋊C4C2×D414yes—
272C2×C4⋊C4C2×Q814yes—
273C2×C4⋊C4C2×Q814yes—
274C2×C4⋊C4#4114yes—
275C2×C4⋊C4#4114yes—
276C2×C22⋊C4—22no—
277C2×C22⋊C4—22no—
278C2×C22⋊C4—24no—
279C2×C22⋊C4—22no—
280C2×C22⋊C4—22no—
281C2×C22⋊C4—24no—
282C2×C22⋊C4—24no—
283C2×C22⋊C4—24no—
284C2×C22⋊C4—24no—
285C2×C22⋊C4—24no—
286C2×C22⋊C4—44no—
287C2×C22⋊C4—42no—
288C2×C22⋊C4—42no—
289C2×C22⋊C4—44no—
290C2×C22⋊C4C22×C414yes—
291C2×C22⋊C4C22×C414yes—
292C2×C22⋊C4C22×C4116yes—
293C2×C22⋊C4C2×Q818yes—
294C2×C42—22no—
295C2×C42—24no—
296C2×C42—22no—
297C2×C42—24no—
298C2×C42—28no—
299C2×C42—24no—
300C2×C42—24no—
301C2×C42—24no—
302C2×C42—28no—
303C2×C42—24no—
304C2×C42—24no—
305C2×C42—24no—
306C2×C42—28no—
307C2×C42—28no—
308C2×C42—24no—
309C2×C42—24no—
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×C42C22×C414yes—
319C2×C42C2×D418yes—
320C2×C42C2×D414yes—
321C2×C42C2×Q818yes—
322C2×C42#4114yes—
323#95—24no—
324#95—24no—
325#95—24no—
326#95—24no—
327#95—28no—
328#95—28no—
329#95—24no—
330#95—24no—
331#95—22no—
332#95—28no—
333#95—22no—
334#95—28no—
335#95—24no—
336#95—24no—
337#95—44no—
338#95—44no—
339#95—44no—
340#95—44no—
341#95—44no—
342#95—42no—
343#95—42no—
344#95—44no—
345#95—44no—
346#95—44no—
347#95C22×C4116yes—
348#95C22×C418yes—
349#95C22×C418yes—
350#95C22×C4116yes—
351#95C2×D418yes—
352C24—21no—
353C24—21no—
354C24—21no—
355C24—21no—
356C24—22no—
357C24—22no—
358C24—22no—
359C24—22no—
360C24C23×C412yes—
361C24C22×Q812yes—
362C24C2×#4112yes—
363C24C2511yescharacteristic
364C2×Q8—216no—
365C2×Q8—416no—
366C2×Q8—48no—
367C2×Q8—48no—
368C2×Q8—416no—
369C22×C4—21no—
370C22×C4—21no—
371C22×C4—21no—
372C22×C4—21no—
373C22×C4—24no—
374C22×C4—22no—
375C22×C4—24no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—28no—
380C22×C4—28no—
381C22×C4—28no—
382C22×C4—28no—
383C22×C4—28no—
384C22×C4—28no—
385C22×C4—21no—
386C22×C4—21no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—24no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—24no—
394C22×C4—22no—
395C22×C4—24no—
396C22×C4—22no—
397C22×C4—21no—
398C22×C4—21no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—24no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—24no—
408C22×C4—24no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—24no—
415C22×C4—24no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—24no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—24no—
422C22×C4—24no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—24no—
428C22×C4—24no—
429C22×C4—216no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
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—84no—
445C22×C4—84no—
446C22×C4—88no—
447C22×C4—88no—
448C22×C4C4×Q8116yescomplemented
449C22×C4C23×C412yes—
450C22×C4C23×C418yes—
451C22×C4C23×C418yes—
452C22×C4C23×C412yes—
453C22×C4C23×C412yes—
454C22×C4C23×C412yes—
455C22×C4C23×C412yes—
456C22×C4C23×C412yes—
457C22×C4C22×D414yes—
458C22×C4C22×D412yes—
459C22×C4C22×D412yes—
460C22×C4C22×D414yes—
461C22×C4C22×D412yes—
462C22×C4C22×Q814yes—
463C22×C4C22×Q812yes—
464C22×C4C22×Q814yes—
465C22×C4C22×Q812yes—
466C22×C4C2×#4114yes—
467C22×C4C2×#4112yes—
468C22×C4C2×#4112yes—
469C22×C4C2×#4112yes—
470C22×C4C2×#4114yes—
471C22×C4C2×#4112yes—
472C22×C4C2×#4112yes—
473C22×C4C2×#4112yes—
474C22×C4C2×#4112yes—
475C22×C4H(5,𝔽2)14yes—
476C22×C42−1+414yes—
477C4⋊C4—216no—
478C4⋊C4—28no—
479C4⋊C4—28no—
480C4⋊C4—28no—
481C4⋊C4—216no—
482C4⋊C4—216no—
483C4⋊C4—216no—
484C4⋊C4—28no—
485C4⋊C4—28no—
486C4⋊C4—28no—
487C4⋊C4—416no—
488C4⋊C4—48no—
489C4⋊C4—416no—
490C4⋊C4—416no—
491C4⋊C4—44no—
492C4⋊C4C22⋊Q8116yes—
493C22⋊C4—28no—
494C22⋊C4—28no—
495C22⋊C4—216no—
496C22⋊C4—416no—
497C22⋊C4—48no—
498C22⋊C4—48no—
499C42—28no—
500C42—216no—
501C42—48no—
502C42—416no—
503C42—416no—
504C42—48no—
505C42—48no—
506C42—84no—
507C42—84no—
508C42—84no—
509C42C4×D4116yes—
510C23—22no—
511C23—22no—
512C23—24no—
513C23—24no—
514C23—22no—
515C23—22no—
516C23—22no—
517C23—22no—
518C23—24no—
519C23—24no—
520C23—24no—
521C23—24no—
522C23—41no—
523C23—41no—
524C23—42no—
525C23—41no—
526C23—41no—
527C23—42no—
528C23—42no—
529C23—42no—
530C23—42no—
531C23—42no—
532C23C2×C4×Q814yes—
533C23C24×C411yescharacteristic
534C23C23×D411yescharacteristic
535C23C23×Q811yescharacteristic
536C23C22×#4111yescharacteristic
537C23C22×#4111yescharacteristic
538C23C22×#4111yescharacteristic
539C23C22×#4111yescharacteristic
540C23C2×H(5,𝔽2)12yes—
541C23C2×2−1+412yes—
542C23#58512yes—
543C23#58512yes—
544Q8—432no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—24no—
548C2×C4—24no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—24no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—24no—
557C2×C4—24no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—28no—
561C2×C4—24no—
562C2×C4—28no—
563C2×C4—232no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—28no—
567C2×C4—28no—
568C2×C4—28no—
569C2×C4—28no—
570C2×C4—24no—
571C2×C4—28no—
572C2×C4—24no—
573C2×C4—28no—
574C2×C4—24no—
575C2×C4—24no—
576C2×C4—24no—
577C2×C4—24no—
578C2×C4—24no—
579C2×C4—24no—
580C2×C4—24no—
581C2×C4—24no—
582C2×C4—24no—
583C2×C4—28no—
584C2×C4—24no—
585C2×C4—28no—
586C2×C4—24no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—44no—
590C2×C4—41no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—41no—
594C2×C4—44no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—44no—
598C2×C4—44no—
599C2×C4—42no—
600C2×C4—41no—
601C2×C4—42no—
602C2×C4—41no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—44no—
608C2×C4—44no—
609C2×C4—44no—
610C2×C4—42no—
611C2×C4—44no—
612C2×C4—44no—
613C2×C4—44no—
614C2×C4—44no—
615C2×C4—44no—
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—42no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4—816no—
632C2×C4C2×C4×D414yes—
633C2×C4C2×C4×D414yes—
634C2×C4C2×C4×D414yes—
635C2×C4C2×C4×Q814yes—
636C2×C4C2×C4×Q814yes—
637C2×C4C4×#4114yes—
638C2×C4C2×C22⋊Q818yes—
639C2×C4C2×C22⋊Q814yes—
640C2×C4#52514yes—
641C2×C4#52914yes—
642C2×C4#53314yes—
643C22—28no—
644C22—24no—
645C22—42no—
646C22—42no—
647C22—42no—
648C22—42no—
649C22—42no—
650C22—42no—
651C22—44no—
652C22—44no—
653C22—44no—
654C22C22×C4×D411yescharacteristic
655C22C22×C4×Q811yescharacteristic
656C22C2×C4×#4111yescharacteristic
657C22C4×H(5,𝔽2)12yes—
658C22C4×2−1+412yes—
659C22C22×C22⋊Q811yescharacteristic
660C22C2×#52511yescharacteristic
661C22C2×#52911yescharacteristic
662C22C2×#53311yescharacteristic
663C22#348411yescharacteristic
664C22#348711yescharacteristic
665C22#349412yes—
666C22#349512yes—
667C22#349512yes—
668C22#349911yescharacteristic
669C22#350011yescharacteristic
670C22#350312yes—
671C22#350412yes—
672C22#350612yes—
673C22#351312yes—
674C22#351812yes—
675C22#352012yes—
676C22#352412yes—
677C4—28no—
678C4—24no—
679C4—24no—
680C4—24no—
681C4—44no—
682C4—44no—
683C4—44no—
684C4—44no—
685C4—42no—
686C4—48no—
687C4—42no—
688C4—48no—
689C4—44no—
690C4—82no—
691C4—82no—
692C4C4×C22⋊Q818yes—
693C2—44no—
694C2C2×C4×C22⋊Q811yescharacteristic
695C2#3426511yescharacteristic
696C2#3427212yes—
697C2#3427611yescharacteristic
698C2#3428412yes—
699C2#3573011yescharacteristic
700C2#3573911yescharacteristic
701C2#3581512yes—
702C2#3582812yes—
703C2#3603011yescharacteristic
704C2#3604011yescharacteristic
subgroupG/H
center, Frattini, socle, cosocleC24C25
commutatorC23C24×C4
prime radicalC25C24
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×#2365C2semidirect
4. #14320C2semidirect
6. #14216C2semidirect
9. #14161C2semidirect
10. #14150C2semidirect
11. #14132C2semidirect
13. C4×#395C2semidirect
15. #13821C2semidirect
16. #13804C2semidirect
23. #2628C4semidirect
91. C2×C4⋊Q8C2×C4semidirect
99. C2×#117Q8semidirect
136. #400C2×C4semidirect
138. #395C2×C4semidirect
143. #389C2×C4semidirect
146. #382C2×C4semidirect
448. C22×C4C4×Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#26282
C2×#23652
C2×#22742
#143202
#142432
#142162
#141782
#141722
#141612
#141502
#141322
#141302
C4×#3952
#138522
#138212
#138042
#138012