Finite Group Databasebeta

← #1394557

Subgroups — #1394557

Order 512.

2778 conjugacy classes of subgroups.

The subgroup lattice has 726 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2538C211yesmaximal, characteristic
2C2×#2423C212yesmaximal, complemented
3C2×Q8×C4⋊C4C211yesmaximal, characteristic, complemented
4#17071C212yesmaximal
5#16992C212yesmaximal, complemented
6#16550C214yesmaximal, complemented
7#16515C214yesmaximal, complemented
8#15206C212yesmaximal, complemented
9#15196C211yesmaximal, characteristic, complemented
10#15181C211yesmaximal, characteristic
11#15145C211yesmaximal, characteristic, complemented
12#15131C212yesmaximal, complemented
13#15131C212yesmaximal, complemented
14#14312C212yesmaximal
15#13969C212yesmaximal, complemented
16#13935C211yesmaximal, characteristic, complemented
17C4×#394C211yesmaximal, characteristic
18C22×C22⋊Q8C2212yes—
19C22×C4×Q8C2211yescharacteristic
20C22×C4×Q8C2211yescharacteristic
21C22×C4×Q8C2211yescharacteristic
22C22×C4×D4C2211yescharacteristic
23#2538—28no—
24#2423—216no—
25C2×#399C2214yes—
26C2×#398C2214yes—
27C2×#398C2214yes—
28C2×#396C2214yes—
29C2×#396C2214yes—
30C2×#396C2214yes—
31C2×#394C2211yescharacteristic
32Q8×C4⋊C4—28no—
33C2×#391C2212yes—
34C2×#390C2212yes—
35C2×#389C2211yescharacteristic
36C2×#389C2212yes—
37C2×#389C2211yescharacteristic
38C2×#389C2212yes—
39C2×#389C2212yes—
40C2×#389C2212yes—
41C2×#389C2212yes—
42C2×#388C2214yes—
43C2×#387C2214yes—
44C2×#384C2211yescharacteristic
45C2×#384C2212yes—
46C2×#384C2211yescharacteristic
47C2×#384C2212yes—
48C2×#380C2212yes—
49C2×#380C2212yes—
50C2×#380C2212yes—
51C2×C4×C4⋊C4C2211yescharacteristic
52C2×C4×C4⋊C4C2212yes—
53C2×C4×C22⋊C4C2212yes—
54C2×C4×C22⋊C4C2212yes—
55#1444C2212yes—
56#1444C2212yes—
57#1444C2212yes—
58#1444C2212yes—
59#1442C2211yescharacteristic
60#1442C2212yes—
61#1441C2214yes—
62#1441C2214yes—
63#1441C2214yes—
64#1441C2214yes—
65#1441C2211yescharacteristic
66#1441C2212yes—
67#1441C2212yes—
68#1441C2214yes—
69#1441C2214yes—
70#1440C2212yes—
71#1439C2214yes—
72#1439C2214yes—
73#1439C2212yes—
74#1439C2214yes—
75#1439C2214yes—
76#1438C2211yescharacteristic
77#1438C2212yes—
78#1437C2214yes—
79#1436C2214yes—
80C4⋊#95C2212yes—
81C4⋊#95C2212yes—
82C4⋊#95C2211yescharacteristic
83C4⋊#95C2212yes—
84#1434C2212yes—
85C4×#95C2211yescharacteristic
86C23×Q8C2311yescharacteristic
87C23×D4C2311yescharacteristic
88C24×C4C2312yes—
89C2×C22⋊Q8—28no—
90C2×C22⋊Q8—28no—
91C2×C22⋊Q8—28no—
92C2×C4×Q8—24no—
93C2×C4×Q8—24no—
94C2×C4×Q8—24no—
95C2×C4×Q8—44no—
96C2×C4×Q8—44no—
97C2×C4×Q8D418yes—
98C2×C4×Q8D418yescomplemented
99C2×C4×D4—24no—
100C2×C4×D4—44no—
101C22×C4⋊C4C2314yes—
102C22×C4⋊C4C2311yescharacteristic
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2314yes—
106C22×C4⋊C4C2312yes—
107C22×C4⋊C4C2311yescharacteristic
108C22×C4⋊C4C2312yes—
109C22×C4⋊C4C2311yescharacteristic
110C22×C4⋊C4C2311yescharacteristic
111C22×C4⋊C4C2311yescharacteristic
112C22×C4⋊C4C2312yes—
113C22×C4⋊C4C2314yes—
114C22×C4⋊C4C2312yes—
115C22×C4⋊C4C2312yes—
116C22×C4⋊C4C2314yes—
117C22×C4⋊C4C2312yes—
118C22×C4⋊C4C2314yes—
119C22×C4⋊C4C2311yescharacteristic
120C22×C4⋊C4C2312yes—
121C22×C4⋊C4C2312yes—
122C22×C4⋊C4C2312yes—
123C22×C22⋊C4C2314yes—
124C22×C22⋊C4C2312yes—
125C22×C22⋊C4C2312yes—
126C22×C22⋊C4C2314yes—
127C22×C22⋊C4C2312yes—
128C22×C22⋊C4C2312yes—
129C22×C22⋊C4C2314yes—
130C22×C22⋊C4C2314yes—
131C22×C22⋊C4C2312yes—
132C22×C42C2311yescharacteristic
133C22×C42C2312yes—
134C22×C42C2311yescharacteristic
135C22×C42C2311yescharacteristic
136C22×C42C2312yes—
137C22×C42C2312yes—
138C22×C42C2312yes—
139#399—216no—
140#398—216no—
141#398—48no—
142#396—216no—
143#396—48no—
144#396—48no—
145#394Q818yescomplemented
146#391—28no—
147#390—28no—
148#389—24no—
149#389—28no—
150#389—24no—
151#389—28no—
152#389—28no—
153#389—28no—
154#389—44no—
155#388—216no—
156#387—216no—
157#384—24no—
158#384—28no—
159#384—28no—
160#384—24no—
161#380—28no—
162#380—28no—
163#380—44no—
164C4×C4⋊C4—24no—
165C4×C4⋊C4—28no—
166C4×C22⋊C4—28no—
167C4×C22⋊C4—44no—
168C2×#95C2314yes—
169C2×#95C2314yes—
170C2×#95C2312yes—
171C2×#95C2312yes—
172C2×#95C2314yes—
173C2×#95C2314yes—
174C2×#95C2314yes—
175C2×#95C2311yescharacteristic
176C2×#95C2312yes—
177C2×#95C2311yescharacteristic
178C2×#95C2312yes—
179C2×#95C2312yes—
180C2×#95C2312yes—
181C2×#95C2314yes—
182C2×#95C2311yescharacteristic
183C2×#95C2314yes—
184C2×#95C2314yes—
185C2×#95C2314yes—
186C2×#95C2314yes—
187C2×#95C2312yes—
188C2×#95C2312yes—
189C2×#95C2314yes—
190C2×#95C2312yes—
191C2×#95C2311yescharacteristic
192C25C2412yes—
193C22×Q8—24no—
194C22×Q8—24no—
195C22×Q8C2×D414yes—
196C22×Q8C2×D414yes—
197C22×Q8#4114yes—
198C22×D4—22no—
199C22×D4—22no—
200C22×D4—42no—
201C22×D4—42no—
202C22×D4C2×Q814yes—
203C23×C4—24no—
204C23×C4—24no—
205C23×C4—24no—
206C23×C4—24no—
207C23×C4—24no—
208C23×C4—24no—
209C23×C4—24no—
210C23×C4C2414yes—
211C23×C4C2412yes—
212C23×C4C2411yescharacteristic
213C23×C4C2414yes—
214C23×C4C2412yes—
215C23×C4C2411yescharacteristic
216C23×C4C2412yes—
217C23×C4C2411yescharacteristic
218C23×C4C2412yes—
219C23×C4C2412yes—
220C23×C4C2411yescharacteristic
221C23×C4C2412yes—
222C23×C4C2412yes—
223C23×C4C2411yescharacteristic
224C23×C4C2411yescharacteristic
225C23×C4C2411yescharacteristic
226C22⋊Q8—432no—
227C4×Q8—232no—
228C4×Q8—416no—
229C4×Q8—416no—
230C4×D4—416no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—24no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—28no—
244C2×C4⋊C4—28no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—28no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—28no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—28no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4—24no—
258C2×C4⋊C4—28no—
259C2×C4⋊C4—24no—
260C2×C4⋊C4—24no—
261C2×C4⋊C4—24no—
262C2×C4⋊C4—24no—
263C2×C4⋊C4—28no—
264C2×C4⋊C4—28no—
265C2×C4⋊C4—41no—
266C2×C4⋊C4—42no—
267C2×C4⋊C4—44no—
268C2×C4⋊C4—44no—
269C2×C4⋊C4—42no—
270C2×C4⋊C4—44no—
271C2×C4⋊C4—44no—
272C2×C4⋊C4—44no—
273C2×C4⋊C4—44no—
274C2×C4⋊C4—44no—
275C2×C4⋊C4—44no—
276C2×C4⋊C4—44no—
277C2×C4⋊C4—44no—
278C2×C4⋊C4—44no—
279C2×C4⋊C4—44no—
280C2×C4⋊C4—44no—
281C2×C4⋊C4—44no—
282C2×C4⋊C4C2×D414yes—
283C2×C4⋊C4C2×D414yes—
284C2×C4⋊C4C2×D418yes—
285C2×C4⋊C4C2×D418yes—
286C2×C4⋊C4C2×Q814yes—
287C2×C22⋊C4—24no—
288C2×C22⋊C4—24no—
289C2×C22⋊C4—24no—
290C2×C22⋊C4—28no—
291C2×C22⋊C4—28no—
292C2×C22⋊C4—24no—
293C2×C22⋊C4—28no—
294C2×C22⋊C4—24no—
295C2×C22⋊C4—24no—
296C2×C22⋊C4—28no—
297C2×C22⋊C4—28no—
298C2×C22⋊C4—42no—
299C2×C22⋊C4—42no—
300C2×C22⋊C4—42no—
301C2×C22⋊C4—42no—
302C2×C22⋊C4—42no—
303C2×C22⋊C4—42no—
304C2×C22⋊C4—42no—
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⋊C4C2×Q818yes—
313C2×C22⋊C4C2×Q818yes—
314C2×C42—24no—
315C2×C42—22no—
316C2×C42—24no—
317C2×C42—22no—
318C2×C42—24no—
319C2×C42—24no—
320C2×C42—22no—
321C2×C42—24no—
322C2×C42—24no—
323C2×C42—24no—
324C2×C42—24no—
325C2×C42—44no—
326C2×C42—44no—
327C2×C42C2×D414yes—
328C2×C42C2×D414yes—
329C2×C42C2×D418yes—
330C2×C42C2×D418yes—
331#95—24no—
332#95—24no—
333#95—24no—
334#95—22no—
335#95—22no—
336#95—24no—
337#95—28no—
338#95—28no—
339#95—28no—
340#95—28no—
341#95—28no—
342#95—28no—
343#95—24no—
344#95—22no—
345#95—24no—
346#95—44no—
347#95—42no—
348#95—42no—
349#95—42no—
350#95—44no—
351#95—44no—
352#95—44no—
353#95—44no—
354#95C2×Q814yes—
355C24—22no—
356C24—22no—
357C24—22no—
358C24—22no—
359C24—22no—
360C24—22no—
361C24—22no—
362C24—22no—
363C24—22no—
364C24—22no—
365C24—22no—
366C24—22no—
367C24—22no—
368C24—22no—
369C24C22×Q814yes—
370C24C2511yescharacteristic
371C2×Q8—28no—
372C2×Q8—28no—
373C2×Q8—216no—
374C2×Q8—216no—
375C2×Q8#121116yes—
376C2×D4—28no—
377C2×D4—48no—
378C2×D4—44no—
379C2×D4—44no—
380C2×D4—48no—
381C22×C4—21no—
382C22×C4—21no—
383C22×C4—22no—
384C22×C4—24no—
385C22×C4—21no—
386C22×C4—24no—
387C22×C4—21no—
388C22×C4—22no—
389C22×C4—21no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—21no—
393C22×C4—22no—
394C22×C4—21no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—24no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—24no—
405C22×C4—24no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—21no—
411C22×C4—21no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—21no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—24no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—24no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—21no—
426C22×C4—24no—
427C22×C4—24no—
428C22×C4—24no—
429C22×C4—22no—
430C22×C4—24no—
431C22×C4—24no—
432C22×C4—22no—
433C22×C4—24no—
434C22×C4—24no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—24no—
439C22×C4—24no—
440C22×C4—24no—
441C22×C4—24no—
442C22×C4—22no—
443C22×C4—21no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—24no—
447C22×C4—24no—
448C22×C4—24no—
449C22×C4—21no—
450C22×C4—22no—
451C22×C4—24no—
452C22×C4—48no—
453C22×C4—48no—
454C22×C4—48no—
455C22×C4—48no—
456C22×C4—48no—
457C22×C4—48no—
458C22×C4—48no—
459C22×C4C22×D414yes—
460C22×C4C22×D412yes—
461C22×C4C22×D414yes—
462C22×C4C22×D412yes—
463C22×C4C22×D414yes—
464C22×C4C22×D414yes—
465C22×C4C22×D412yes—
466C22×C4C22×D412yes—
467C22×C4C22×D412yes—
468C22×C4C22×D412yes—
469C22×C4C22×Q814yes—
470C22×C4C22×Q812yes—
471C22×C4C22×Q812yes—
472C22×C4C22×Q812yes—
473C22×C4C2×#4114yes—
474C22×C4C2×#4112yes—
475C22×C4H(5,𝔽2)14yes—
476C22×C4H(5,𝔽2)14yes—
477C22×C42−1+412yes—
478C22×C42−1+412yes—
479C22×C42−1+414yes—
480C4⋊C4—216no—
481C4⋊C4—28no—
482C4⋊C4—28no—
483C4⋊C4—28no—
484C4⋊C4—48no—
485C4⋊C4—44no—
486C4⋊C4—48no—
487C4⋊C4—416no—
488C4⋊C4—48no—
489C4⋊C4—44no—
490C4⋊C4—48no—
491C4⋊C4—48no—
492C4⋊C4—416no—
493C4⋊C4—44no—
494C4⋊C4—48no—
495C4⋊C4—88no—
496C4⋊C4—88no—
497C4⋊C4—84no—
498C4⋊C4—84no—
499C4⋊C4—84no—
500C4⋊C4—84no—
501C4⋊C4—88no—
502C22⋊C4—216no—
503C22⋊C4—48no—
504C22⋊C4—48no—
505C22⋊C4—48no—
506C22⋊C4—48no—
507C22⋊C4—416no—
508C22⋊C4—88no—
509C22⋊C4—88no—
510C22⋊C4—88no—
511C42—216no—
512C42—28no—
513C42—28no—
514C42—48no—
515C42—48no—
516C42—44no—
517C42—48no—
518C23—24no—
519C23—24no—
520C23—24no—
521C23—24no—
522C23—24no—
523C23—24no—
524C23—24no—
525C23—42no—
526C23—42no—
527C23—42no—
528C23—42no—
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—
553C23C23×D411yescharacteristic
554C23C23×D411yescharacteristic
555C23C23×Q811yescharacteristic
556C23C22×#4111yescharacteristic
557C23C2×H(5,𝔽2)12yes—
558C23C2×H(5,𝔽2)12yes—
559C23C2×2−1+412yes—
560C23C2×2−1+411yescharacteristic
561C23C2×2−1+411yescharacteristic
562C23#58511yescharacteristic
563C23#58512yes—
564Q8—232no—
565D4—416no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—24no—
569C2×C4—24no—
570C2×C4—28no—
571C2×C4—28no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—24no—
578C2×C4—22no—
579C2×C4—24no—
580C2×C4—22no—
581C2×C4—24no—
582C2×C4—22no—
583C2×C4—24no—
584C2×C4—24no—
585C2×C4—22no—
586C2×C4—24no—
587C2×C4—24no—
588C2×C4—28no—
589C2×C4—22no—
590C2×C4—24no—
591C2×C4—24no—
592C2×C4—22no—
593C2×C4—24no—
594C2×C4—24no—
595C2×C4—24no—
596C2×C4—24no—
597C2×C4—24no—
598C2×C4—41no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—41no—
602C2×C4—41no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—41no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—44no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—44no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—44no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—44no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—41no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—44no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—44no—
631C2×C4—44no—
632C2×C4—44no—
633C2×C4—44no—
634C2×C4—41no—
635C2×C4—42no—
636C2×C4—44no—
637C2×C4—44no—
638C2×C4—42no—
639C2×C4—41no—
640C2×C4—44no—
641C2×C4—44no—
642C2×C4—44no—
643C2×C4—42no—
644C2×C4—44no—
645C2×C4—44no—
646C2×C4—44no—
647C2×C4—44no—
648C2×C4—42no—
649C2×C4—44no—
650C2×C4—816no—
651C2×C4C2×#12118yes—
652C2×C4C2×#12114yes—
653C2×C4#53414yes—
654C2×C4#53614yes—
655C2×C4#53718yes—
656C2×C4#53918yes—
657C2×C4D4×Q814yes—
658C2×C4D4×Q814yes—
659C22—44no—
660C22—44no—
661C22—44no—
662C22—44no—
663C22—44no—
664C22—44no—
665C22—44no—
666C22—82no—
667C22—82no—
668C22—82no—
669C22—82no—
670C22—82no—
671C22—82no—
672C22—82no—
673C22—82no—
674C22C22×#12111yescharacteristic
675C22C2×#53411yescharacteristic
676C22C2×#53611yescharacteristic
677C22C2×#53712yes—
678C22C2×#53912yes—
679C22C2×D4×Q811yescharacteristic
680C22C2×D4×Q811yescharacteristic
681C22Q8×#4111yescharacteristic
682C22#348711yescharacteristic
683C22#348812yes—
684C22#348911yescharacteristic
685C22#349711yescharacteristic
686C22#349811yescharacteristic
687C22#350112yes—
688C22#350312yes—
689C22#351712yes—
690C22#352311yescharacteristic
691C22#354112yes—
692C22#354212yes—
693C22#354412yes—
694C22#354812yes—
695C22#355512yes—
696C22#356512yes—
697C4—24no—
698C4—24no—
699C4—28no—
700C4—44no—
701C4—42no—
702C4—44no—
703C4—42no—
704C4—44no—
705C4—42no—
706C4—44no—
707C4—84no—
708C4—81no—
709C4—82no—
710C4—84no—
711C4—82no—
712C4#261818yes—
713C2—84no—
714C2C2×#261811yescharacteristic
715C2#3602911yescharacteristic
716C2Q8×#12111yescharacteristic
717C2#3959511yescharacteristic
718C2#3959812yes—
719C2#3961711yescharacteristic
720C2#3965812yes—
721C2#4279812yes—
722C2#4284912yes—
723C2#4289111yescharacteristic
724C2#4290911yescharacteristic
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
2. C2×#2423C2semidirect
3. C2×Q8×C4⋊C4C2semidirect
5. #16992C2semidirect
6. #16550C2semidirect
7. #16515C2semidirect
8. #15206C2semidirect
9. #15196C2semidirect
11. #15145C2semidirect
12. #15131C2semidirect
13. #15131C2semidirect
15. #13969C2semidirect
16. #13935C2semidirect
98. C2×C4×Q8D4semidirect
145. #394Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#25382
C2×#24232
C2×Q8×C4⋊C42
#170712
#169922
#165502
#165152
#152062
#151962
#151812
#151452
#151312
#143122
#139692
#139352
C4×#3942