Finite Group Databasebeta

← #988985

Subgroups — #988985

Order 512.

3578 conjugacy classes of subgroups.

The subgroup lattice has 728 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2683C212yesmaximal, complemented
2C2×#2538C212yesmaximal, complemented
3C2×#2452C212yesmaximal, complemented
4#17135C212yesmaximal, complemented
5#16556C212yesmaximal, complemented
6#16265C214yesmaximal, complemented
7#15951C212yesmaximal, complemented
8#15309C214yesmaximal, complemented
9#15141C212yesmaximal, complemented
10#15134C212yesmaximal, complemented
11#15128C211yesmaximal, characteristic, complemented
12#14154C211yesmaximal, characteristic
13#14025C212yesmaximal
14#14005C212yesmaximal, complemented
15C4×#395C211yesmaximal, characteristic, complemented
16C22×C4⋊Q8C2211yescharacteristic
17C22×#127C2211yescharacteristic
18C22×C22⋊Q8C2214yes—
19C22×C22⋊Q8C2214yes—
20C22×C4×Q8C2211yescharacteristic
21C22×C4×D4C2212yes—
22#2683—216no—
23#2538—216no—
24#2452—216no—
25C2×#396C2214yes—
26C2×#396C2214yes—
27C2×#396C2214yes—
28C2×#396C2214yes—
29C2×#396C2214yes—
30C2×#396C2214yes—
31C2×#395C2212yes—
32C2×#395C2211yescharacteristic
33C2×#393C2214yes—
34C2×#390C2212yes—
35C2×#390C2212yes—
36C2×#390C2212yes—
37C2×#390C2212yes—
38C2×#390C2212yes—
39C2×#390C2212yes—
40C2×#390C2212yes—
41C2×#389C2212yes—
42C2×#389C2212yes—
43C2×#389C2214yes—
44C2×#389C2212yes—
45C2×#389C2212yes—
46C2×#386C2214yes—
47C2×#386C2214yes—
48C2×#385C2214yes—
49C2×#384C2212yes—
50C2×#380C2214yes—
51C2×C4×C4⋊C4C2212yes—
52C2×C4×C4⋊C4C2211yescharacteristic
53C2×C4×C4⋊C4C2211yescharacteristic
54#1446C2212yes—
55#1446C2211yescharacteristic
56#1444C2212yes—
57#1444C2212yes—
58#1444C2211yescharacteristic
59#1444C2214yes—
60#1441C2212yes—
61#1441C2212yes—
62#1441C2211yescharacteristic
63#1441C2214yes—
64#1439C2214yes—
65#1439C2214yes—
66#1439C2214yes—
67#1439C2214yes—
68#1439C2214yes—
69#1438C2212yes—
70#1438C2212yes—
71#1438C2214yes—
72#1438C2214yes—
73#1436C2214yes—
74C4⋊#95C2211yescharacteristic
75C4⋊#95C2212yes—
76C4×#95C2211yescharacteristic
77C4×#95C2212yes—
78C23×Q8C2311yescharacteristic
79C23×Q8C2312yes—
80C23×D4C2312yes—
81C23×D4C2312yes—
82C23×D4C2312yes—
83C24×C4C2314yes—
84C2×C4⋊Q8—28no—
85C2×C4⋊Q8—42no—
86C2×#127—42no—
87C2×#127—42no—
88C2×#127Q818yescomplemented
89C2×C22⋊Q8—216no—
90C2×C22⋊Q8—216no—
91C2×C22⋊Q8—216no—
92C2×C22⋊Q8—216no—
93C2×C22⋊Q8—48no—
94C2×C22⋊Q8—48no—
95C2×C4×Q8—28no—
96C2×C4×Q8—42no—
97C2×C4×D4—28no—
98C2×C4×D4—28no—
99C2×C4×D4—44no—
100C22×C4⋊C4C2312yes—
101C22×C4⋊C4C2314yes—
102C22×C4⋊C4C2311yescharacteristic
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2314yes—
106C22×C4⋊C4C2312yes—
107C22×C4⋊C4C2314yes—
108C22×C4⋊C4C2311yescharacteristic
109C22×C4⋊C4C2312yes—
110C22×C4⋊C4C2314yes—
111C22×C4⋊C4C2312yes—
112C22×C4⋊C4C2312yes—
113C22×C4⋊C4C2314yes—
114C22×C4⋊C4C2314yes—
115C22×C4⋊C4C2311yescharacteristic
116C22×C22⋊C4C2314yes—
117C22×C22⋊C4C2314yes—
118C22×C22⋊C4C2314yes—
119C22×C22⋊C4C2314yes—
120C22×C22⋊C4C2314yes—
121C22×C22⋊C4C2314yes—
122C22×C22⋊C4C2314yes—
123C22×C22⋊C4C2314yes—
124C22×C22⋊C4C2314yes—
125C22×C22⋊C4C2314yes—
126C22×C22⋊C4C2314yes—
127C22×C42C2311yescharacteristic
128C22×C42C2311yescharacteristic
129C22×C42C2312yes—
130C22×C42C2312yes—
131C22×C42C2311yescharacteristic
132C22×C42C2312yes—
133#396—216no—
134#396—216no—
135#396—216no—
136#396—216no—
137#396—216no—
138#396—48no—
139#395—28no—
140#395D418yescomplemented
141#393—216no—
142#390—28no—
143#390—28no—
144#390—28no—
145#390—28no—
146#390—28no—
147#390—44no—
148#390—44no—
149#389—28no—
150#389—28no—
151#389—216no—
152#389—44no—
153#389D4116yescomplemented
154#386—216no—
155#386—48no—
156#385—216no—
157#384—28no—
158#380—216no—
159C4×C4⋊C4—28no—
160C4×C4⋊C4—42no—
161C4×C4⋊C4—42no—
162C2×#95C2312yes—
163C2×#95C2314yes—
164C2×#95C2312yes—
165C2×#95C2314yes—
166C2×#95C2312yes—
167C2×#95C2314yes—
168C2×#95C2312yes—
169C2×#95C2312yes—
170C2×#95C2312yes—
171C2×#95C2312yes—
172C2×#95C2312yes—
173C2×#95C2311yescharacteristic
174C2×#95C2314yes—
175C2×#95C2314yes—
176C2×#95C2312yes—
177C2×#95C2311yescharacteristic
178C2×#95C2311yescharacteristic
179C2×#95C2314yes—
180C2×#95C2311yescharacteristic
181C2×#95C2312yes—
182C25C2414yes—
183C22×Q8—24no—
184C22×Q8—24no—
185C22×Q8—28no—
186C22×Q8—24no—
187C22×Q8—41no—
188C22×Q8—41no—
189C22×Q8—44no—
190C22×Q8—42no—
191C22×Q8—42no—
192C22×Q8C2×D414yes—
193C22×D4—24no—
194C22×D4—24no—
195C22×D4—24no—
196C22×D4—24no—
197C22×D4—24no—
198C22×D4—24no—
199C22×D4—24no—
200C22×D4—24no—
201C22×D4—42no—
202C22×D4—42no—
203C22×D4—42no—
204C22×D4—42no—
205C22×D4—42no—
206C22×D4—44no—
207C22×D4—44no—
208C22×D4—42no—
209C22×D4C2×Q818yes—
210C22×D4C2×Q818yes—
211C22×D4C2×Q818yes—
212C23×C4—28no—
213C23×C4—28no—
214C23×C4—28no—
215C23×C4—28no—
216C23×C4—44no—
217C23×C4—44no—
218C23×C4—44no—
219C23×C4C2414yes—
220C23×C4C2411yescharacteristic
221C23×C4C2412yes—
222C23×C4C2411yescharacteristic
223C23×C4C2414yes—
224C23×C4C2412yes—
225C23×C4C2412yes—
226C23×C4C2411yescharacteristic
227C23×C4C2414yes—
228C23×C4C2411yescharacteristic
229C23×C4C2411yescharacteristic
230C23×C4C2412yes—
231C23×C4C2411yescharacteristic
232C23×C4C2411yescharacteristic
233C4⋊Q8—416no—
234#127—416no—
235C22⋊Q8—464no—
236C22⋊Q8—464no—
237C4×Q8—416no—
238C4×D4—432no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—28no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—28no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—28no—
251C2×C4⋊C4—28no—
252C2×C4⋊C4—28no—
253C2×C4⋊C4—28no—
254C2×C4⋊C4—28no—
255C2×C4⋊C4—28no—
256C2×C4⋊C4—28no—
257C2×C4⋊C4—24no—
258C2×C4⋊C4—24no—
259C2×C4⋊C4—24no—
260C2×C4⋊C4—22no—
261C2×C4⋊C4—24no—
262C2×C4⋊C4—22no—
263C2×C4⋊C4—24no—
264C2×C4⋊C4—28no—
265C2×C4⋊C4—28no—
266C2×C4⋊C4—41no—
267C2×C4⋊C4—42no—
268C2×C4⋊C4—42no—
269C2×C4⋊C4—41no—
270C2×C4⋊C4—44no—
271C2×C4⋊C4—44no—
272C2×C4⋊C4—44no—
273C2×C4⋊C4—42no—
274C2×C4⋊C4—41no—
275C2×C4⋊C4—41no—
276C2×C4⋊C4—44no—
277C2×C4⋊C4—44no—
278C2×C4⋊C4—44no—
279C2×C4⋊C4—44no—
280C2×C4⋊C4C2×D418yes—
281C2×C4⋊C4C2×D4116yes—
282C2×C4⋊C4C2×D418yes—
283C2×C4⋊C4C2×D418yes—
284C2×C4⋊C4C2×D414yes—
285C2×C22⋊C4—28no—
286C2×C22⋊C4—28no—
287C2×C22⋊C4—28no—
288C2×C22⋊C4—28no—
289C2×C22⋊C4—28no—
290C2×C22⋊C4—28no—
291C2×C22⋊C4—28no—
292C2×C22⋊C4—28no—
293C2×C22⋊C4—28no—
294C2×C22⋊C4—28no—
295C2×C22⋊C4—28no—
296C2×C22⋊C4—28no—
297C2×C22⋊C4—28no—
298C2×C22⋊C4—28no—
299C2×C22⋊C4—28no—
300C2×C22⋊C4—28no—
301C2×C22⋊C4—28no—
302C2×C22⋊C4—44no—
303C2×C22⋊C4—44no—
304C2×C22⋊C4—44no—
305C2×C22⋊C4—44no—
306C2×C22⋊C4—44no—
307C2×C22⋊C4—44no—
308C2×C22⋊C4—44no—
309C2×C22⋊C4—44no—
310C2×C22⋊C4—44no—
311C2×C22⋊C4—44no—
312C2×C22⋊C4—44no—
313C2×C22⋊C4—44no—
314C2×C22⋊C4—44no—
315C2×C22⋊C4—44no—
316C2×C22⋊C4—44no—
317C2×C22⋊C4—44no—
318C2×C42—24no—
319C2×C42—24no—
320C2×C42—22no—
321C2×C42—22no—
322C2×C42—24no—
323C2×C42—24no—
324C2×C42—24no—
325C2×C42—22no—
326C2×C42—22no—
327C2×C42—24no—
328C2×C42—24no—
329C2×C42—24no—
330C2×C42—42no—
331C2×C42—42no—
332C2×C42C2×D418yes—
333C2×C42C2×Q814yes—
334#95—24no—
335#95—24no—
336#95—28no—
337#95—28no—
338#95—28no—
339#95—24no—
340#95—24no—
341#95—24no—
342#95—24no—
343#95—42no—
344#95—42no—
345#95—41no—
346#95—42no—
347#95—41no—
348#95—41no—
349#95—44no—
350#95—44no—
351#95C2×D4116yes—
352#95C2×D418yes—
353#95C2×D414yes—
354C24—24no—
355C24—24no—
356C24—24no—
357C24—24no—
358C24—24no—
359C24—24no—
360C24—24no—
361C24—24no—
362C24—24no—
363C24—24no—
364C24—24no—
365C24—24no—
366C24—24no—
367C24—24no—
368C24C22×Q818yes—
369C24C2511yescharacteristic
370C2×Q8—216no—
371C2×Q8—44no—
372C2×Q8—44no—
373C2×Q8—48no—
374C2×Q8—44no—
375C2×Q8—416no—
376C2×Q8—48no—
377C2×Q8—88no—
378C2×Q8—84no—
379C2×Q8—84no—
380C2×D4—216no—
381C2×D4—216no—
382C2×D4—216no—
383C2×D4—216no—
384C2×D4—48no—
385C2×D4—48no—
386C2×D4—48no—
387C2×D4—416no—
388C2×D4—48no—
389C2×D4—48no—
390C2×D4—48no—
391C2×D4—48no—
392C2×D4—84no—
393C2×D4—84no—
394C2×D4—84no—
395C2×D4—84no—
396C2×D4—84no—
397C2×D4—88no—
398C2×D4—84no—
399C22×C4—22no—
400C22×C4—21no—
401C22×C4—21no—
402C22×C4—22no—
403C22×C4—21no—
404C22×C4—22no—
405C22×C4—21no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—24no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—21no—
417C22×C4—21no—
418C22×C4—24no—
419C22×C4—21no—
420C22×C4—21no—
421C22×C4—21no—
422C22×C4—24no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—24no—
426C22×C4—24no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—24no—
433C22×C4—24no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—24no—
437C22×C4—21no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—21no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—24no—
444C22×C4—22no—
445C22×C4—24no—
446C22×C4—21no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—24no—
450C22×C4—24no—
451C22×C4—22no—
452C22×C4—24no—
453C22×C4—22no—
454C22×C4—21no—
455C22×C4—24no—
456C22×C4—24no—
457C22×C4—21no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—24no—
461C22×C4—21no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—24no—
466C22×C4—416no—
467C22×C4—416no—
468C22×C4—416no—
469C22×C4—416no—
470C22×C4—88no—
471C22×C4—88no—
472C22×C4—88no—
473C22×C4C22×D412yes—
474C22×C4C22×D414yes—
475C22×C4C22×D418yes—
476C22×C4C22×D418yes—
477C22×C4C22×D414yes—
478C22×C4C22×D412yes—
479C22×C4C22×D414yes—
480C22×C4C22×D414yes—
481C22×C4C22×D412yes—
482C22×C4C22×D414yes—
483C22×C4C22×Q812yes—
484C22×C4C22×Q812yes—
485C22×C4C22×Q812yes—
486C22×C4H(5,𝔽2)12yes—
487C22×C4H(5,𝔽2)14yes—
488C22×C4H(5,𝔽2)12yes—
489C22×C4H(5,𝔽2)12yes—
490C22×C42−1+412yes—
491C22×C42−1+414yes—
492C4⋊C4—216no—
493C4⋊C4—216no—
494C4⋊C4—216no—
495C4⋊C4—232no—
496C4⋊C4—44no—
497C4⋊C4—44no—
498C4⋊C4—48no—
499C4⋊C4—48no—
500C4⋊C4—44no—
501C4⋊C4—416no—
502C4⋊C4—416no—
503C4⋊C4—416no—
504C4⋊C4—48no—
505C4⋊C4—48no—
506C4⋊C4—88no—
507C4⋊C4—88no—
508C22⋊C4—416no—
509C22⋊C4—416no—
510C22⋊C4—416no—
511C22⋊C4—416no—
512C22⋊C4—416no—
513C22⋊C4—416no—
514C22⋊C4—88no—
515C22⋊C4—88no—
516C22⋊C4—88no—
517C22⋊C4—88no—
518C22⋊C4—88no—
519C42—28no—
520C42—216no—
521C42—48no—
522C42—44no—
523C42—44no—
524C42—84no—
525C23—28no—
526C23—28no—
527C23—28no—
528C23—28no—
529C23—28no—
530C23—28no—
531C23—28no—
532C23—44no—
533C23—44no—
534C23—44no—
535C23—44no—
536C23—44no—
537C23—44no—
538C23—44no—
539C23—44no—
540C23—44no—
541C23—44no—
542C23—44no—
543C23—44no—
544C23—44no—
545C23—44no—
546C23—44no—
547C23—44no—
548C23—44no—
549C23—44no—
550C23—44no—
551C23—44no—
552C23—44no—
553C23—44no—
554C23—44no—
555C23—44no—
556C23—44no—
557C23—44no—
558C23—44no—
559C23—44no—
560C23C23×D412yes—
561C23C23×D411yescharacteristic
562C23C23×Q811yescharacteristic
563C23C2×H(5,𝔽2)11yescharacteristic
564C23C2×H(5,𝔽2)11yescharacteristic
565C23C2×H(5,𝔽2)12yes—
566C23C2×H(5,𝔽2)11yescharacteristic
567C23C2×2−1+412yes—
568C23C2×2−1+411yescharacteristic
569C23#58511yescharacteristic
570C23#58512yes—
571Q8—416no—
572Q8—816no—
573D4—432no—
574D4—816no—
575D4—816no—
576C2×C4—22no—
577C2×C4—24no—
578C2×C4—24no—
579C2×C4—24no—
580C2×C4—24no—
581C2×C4—22no—
582C2×C4—24no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—24no—
587C2×C4—22no—
588C2×C4—28no—
589C2×C4—22no—
590C2×C4—24no—
591C2×C4—28no—
592C2×C4—24no—
593C2×C4—24no—
594C2×C4—24no—
595C2×C4—28no—
596C2×C4—28no—
597C2×C4—24no—
598C2×C4—28no—
599C2×C4—24no—
600C2×C4—28no—
601C2×C4—22no—
602C2×C4—22no—
603C2×C4—24no—
604C2×C4—24no—
605C2×C4—24no—
606C2×C4—24no—
607C2×C4—22no—
608C2×C4—42no—
609C2×C4—41no—
610C2×C4—41no—
611C2×C4—41no—
612C2×C4—44no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—44no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—44no—
619C2×C4—44no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—44no—
623C2×C4—42no—
624C2×C4—44no—
625C2×C4—41no—
626C2×C4—41no—
627C2×C4—41no—
628C2×C4—44no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4—44no—
632C2×C4—42no—
633C2×C4—41no—
634C2×C4—42no—
635C2×C4—41no—
636C2×C4—42no—
637C2×C4—44no—
638C2×C4—41no—
639C2×C4—44no—
640C2×C4—42no—
641C2×C4—42no—
642C2×C4—44no—
643C2×C4—42no—
644C2×C4—44no—
645C2×C4—42no—
646C2×C4—44no—
647C2×C4—41no—
648C2×C4—42no—
649C2×C4—44no—
650C2×C4—44no—
651C2×C4—42no—
652C2×C4—41no—
653C2×C4—42no—
654C2×C4—42no—
655C2×C4—832no—
656C2×C4#53718yes—
657C2×C4#54314yes—
658C2×C4D4218yes—
659C2×C4D4214yes—
660C2×C4D4×Q814yes—
661C2×C4D4×Q818yes—
662C2×C4#56314yes—
663C22—48no—
664C22—48no—
665C22—48no—
666C22—48no—
667C22—48no—
668C22—48no—
669C22—48no—
670C22—84no—
671C22—84no—
672C22—84no—
673C22—84no—
674C22—84no—
675C22—84no—
676C22—84no—
677C22—84no—
678C22C2×#53712yes—
679C22C2×#54311yescharacteristic
680C22C2×D4211yescharacteristic
681C22C2×D4212yes—
682C22C2×D4×Q812yes—
683C22C2×D4×Q811yescharacteristic
684C22#348812yes—
685C22#348812yes—
686C22#349611yescharacteristic
687C22#349712yes—
688C22#349712yes—
689C22#349811yescharacteristic
690C22#350111yescharacteristic
691C22#350112yes—
692C22C2×#56311yescharacteristic
693C22#353611yescharacteristic
694C22#353711yescharacteristic
695C22#354612yes—
696C22#354812yes—
697C22#355012yes—
698C22#355111yescharacteristic
699C22#355712yes—
700C22#356111yescharacteristic
701C4—28no—
702C4—24no—
703C4—48no—
704C4—42no—
705C4—42no—
706C4—48no—
707C4—44no—
708C4—44no—
709C4—42no—
710C4—42no—
711C4—82no—
712C4—84no—
713C4—81no—
714C4#268318yescomplemented
715C2—88no—
716C2#3934112yes—
717C2#3936711yescharacteristic
718C2#4249011yescharacteristic
719C2#4279912yes—
720C2#4494812yes—
721C2#4524411yescharacteristic
722C2#4632911yescharacteristic
723C2C2×#268311yescharacteristic
724C2#5309911yescharacteristic
725C2#5899711yescharacteristic
726C2#5981712yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC22×#127C22
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#2683C2semidirect
2. C2×#2538C2semidirect
3. C2×#2452C2semidirect
4. #17135C2semidirect
5. #16556C2semidirect
6. #16265C2semidirect
7. #15951C2semidirect
8. #15309C2semidirect
9. #15141C2semidirect
10. #15134C2semidirect
11. #15128C2semidirect
14. #14005C2semidirect
15. C4×#395C2semidirect
88. C2×#127Q8semidirect
140. #395D4semidirect
153. #389D4semidirect
714. C4#2683semidirect

Maximal subgroups

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

subgroupindex
C2×#26832
C2×#25382
C2×#24522
#171352
#165562
#162652
#159512
#153092
#151412
#151342
#151282
#141542
#140252
#140052
C4×#3952