Finite Group Databasebeta

← #818486

Subgroups — #818486

Order 512.

2982 conjugacy classes of subgroups.

The subgroup lattice has 734 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2538C211yesmaximal, characteristic
2C2×#2429C212yesmaximal, complemented
3C2×#2365C211yesmaximal, characteristic, complemented
4#16820C212yesmaximal, complemented
5#15620C214yesmaximal, complemented
6#15599C214yesmaximal, complemented
7#15185C211yesmaximal, characteristic, complemented
8#15177C211yesmaximal, characteristic
9#15163C212yesmaximal
10#15151C211yesmaximal, characteristic, complemented
11#14982C212yesmaximal
12#14226C212yesmaximal, complemented
13#13974C212yesmaximal, complemented
14#13969C212yesmaximal, complemented
15#13969C212yesmaximal, complemented
16C4×#394C211yesmaximal, characteristic
17Q8×#95C211yesmaximal, characteristic, complemented
18C22×C4⋊Q8C2211yescharacteristic
19C22×C22⋊Q8C2212yes—
20C22×C22⋊Q8C2212yes—
21C22×C4×Q8C2212yes—
22C22×C4×Q8C2211yescharacteristic
23C22×C4×Q8C2211yescharacteristic
24C22×C4×D4C2211yescharacteristic
25#2538—28no—
26#2429—216no—
27C2×#398C2212yes—
28C2×#398C2214yes—
29C2×#398C2212yes—
30C2×#396C2214yes—
31C2×#396C2214yes—
32C2×#394C2211yescharacteristic
33#2365—28no—
34C2×#391C2211yescharacteristic
35C2×#391C2212yes—
36C2×#391C2211yescharacteristic
37C2×#390C2212yes—
38C2×#389C2212yes—
39C2×#389C2214yes—
40C2×#389C2212yes—
41C2×#389C2212yes—
42C2×#388C2214yes—
43C2×#388C2214yes—
44C2×#387C2212yes—
45C2×#387C2212yes—
46C2×#384C2211yescharacteristic
47C2×#380C2212yes—
48C2×#380C2212yes—
49C2×C4×C4⋊C4C2211yescharacteristic
50C2×C4×C4⋊C4C2212yes—
51C2×C4×C4⋊C4C2211yescharacteristic
52C2×C4×C22⋊C4C2212yes—
53C2×C4×C22⋊C4C2212yes—
54#1445C2212yes—
55#1445C2212yes—
56#1445C2212yes—
57#1445C2212yes—
58#1444C2212yes—
59#1443C2214yes—
60#1442C2214yes—
61#1442C2211yescharacteristic
62#1441C2214yes—
63#1441C2214yes—
64#1441C2211yescharacteristic
65#1440C2212yes—
66#1440C2214yes—
67#1439C2212yes—
68#1439C2214yes—
69#1439C2214yes—
70#1439C2214yes—
71#1438C2211yescharacteristic
72#1438C2212yes—
73#1437C2214yes—
74#1437C2214yes—
75#1436C2214yes—
76#1436C2214yes—
77#1436C2214yes—
78C4⋊#95C2214yes—
79C4⋊#95C2211yescharacteristic
80C4⋊#95C2212yes—
81C4⋊#95C2212yes—
82C4⋊#95C2212yes—
83C4⋊#95C2212yes—
84C4×#95C2211yescharacteristic
85C4×#95C2212yes—
86C23×Q8C2311yescharacteristic
87C23×Q8C2311yescharacteristic
88C23×D4C2311yescharacteristic
89C24×C4C2312yes—
90C2×C4⋊Q8—24no—
91C2×C4⋊Q8—24no—
92C2×C4⋊Q8D418yescomplemented
93C2×C22⋊Q8—28no—
94C2×C22⋊Q8—28no—
95C2×C22⋊Q8—28no—
96C2×C22⋊Q8—28no—
97C2×C22⋊Q8—28no—
98C2×C22⋊Q8—44no—
99C2×C4×Q8—28no—
100C2×C4×Q8—24no—
101C2×C4×Q8—28no—
102C2×C4×Q8—44no—
103C2×C4×Q8—44no—
104C2×C4×Q8—44no—
105C2×C4×Q8D418yes—
106C2×C4×D4—24no—
107C2×C4×D4—44no—
108C22×C4⋊C4C2314yes—
109C22×C4⋊C4C2314yes—
110C22×C4⋊C4C2312yes—
111C22×C4⋊C4C2312yes—
112C22×C4⋊C4C2311yescharacteristic
113C22×C4⋊C4C2314yes—
114C22×C4⋊C4C2311yescharacteristic
115C22×C4⋊C4C2314yes—
116C22×C4⋊C4C2312yes—
117C22×C4⋊C4C2312yes—
118C22×C4⋊C4C2311yescharacteristic
119C22×C4⋊C4C2311yescharacteristic
120C22×C4⋊C4C2312yes—
121C22×C4⋊C4C2312yes—
122C22×C4⋊C4C2311yescharacteristic
123C22×C4⋊C4C2312yes—
124C22×C4⋊C4C2312yes—
125C22×C22⋊C4C2312yes—
126C22×C22⋊C4C2314yes—
127C22×C22⋊C4C2312yes—
128C22×C22⋊C4C2312yes—
129C22×C22⋊C4C2314yes—
130C22×C22⋊C4C2314yes—
131C22×C22⋊C4C2312yes—
132C22×C22⋊C4C2314yes—
133C22×C22⋊C4C2312yes—
134C22×C42C2312yes—
135C22×C42C2311yescharacteristic
136C22×C42C2314yes—
137C22×C42C2312yes—
138C22×C42C2311yescharacteristic
139C22×C42C2311yescharacteristic
140C22×C42C2312yes—
141C22×C42C2312yes—
142#398—216no—
143#398—28no—
144#398—44no—
145#396—216no—
146#396—216no—
147#394—24no—
148#391—24no—
149#391—28no—
150#391—24no—
151#390—28no—
152#389—28no—
153#389—216no—
154#389—28no—
155#389—28no—
156#388—216no—
157#388—48no—
158#387—28no—
159#387—28no—
160#384—24no—
161#380—28no—
162#380—28no—
163C4×C4⋊C4—24no—
164C4×C4⋊C4—24no—
165C4×C4⋊C4—28no—
166C4×C22⋊C4—28no—
167C4×C22⋊C4—44no—
168C2×#95C2311yescharacteristic
169C2×#95C2314yes—
170C2×#95C2314yes—
171C2×#95C2312yes—
172C2×#95C2312yes—
173C2×#95C2314yes—
174C2×#95C2314yes—
175C2×#95C2311yescharacteristic
176C2×#95C2314yes—
177C2×#95C2312yes—
178C2×#95C2312yes—
179C2×#95C2314yes—
180C2×#95C2312yes—
181C2×#95C2314yes—
182C2×#95C2312yes—
183C2×#95C2311yescharacteristic
184C2×#95C2312yes—
185C2×#95C2312yes—
186C2×#95C2312yes—
187C2×#95C2311yescharacteristic
188C2×#95C2312yes—
189C2×#95C2314yes—
190C2×#95C2314yes—
191C2×#95C2314yes—
192C2×#95C2314yes—
193C2×#95C2312yes—
194C2×#95C2312yes—
195C25C2412yes—
196C22×Q8—22no—
197C22×Q8—24no—
198C22×Q8—24no—
199C22×Q8—22no—
200C22×Q8—24no—
201C22×Q8—24no—
202C22×Q8C2×D414yes—
203C22×Q8C2×D414yes—
204C22×Q8C2×D414yes—
205C22×Q8#4114yes—
206C22×D4—24no—
207C22×D4—22no—
208C22×D4—22no—
209C22×D4—22no—
210C22×D4—42no—
211C23×C4—24no—
212C23×C4—24no—
213C23×C4—24no—
214C23×C4—24no—
215C23×C4—24no—
216C23×C4—24no—
217C23×C4—24no—
218C23×C4C2414yes—
219C23×C4C2411yescharacteristic
220C23×C4C2412yes—
221C23×C4C2412yes—
222C23×C4C2412yes—
223C23×C4C2412yes—
224C23×C4C2411yescharacteristic
225C23×C4C2412yes—
226C23×C4C2411yescharacteristic
227C23×C4C2414yes—
228C23×C4C2411yescharacteristic
229C23×C4C2412yes—
230C23×C4C2412yes—
231C23×C4C2411yescharacteristic
232C23×C4C2411yescharacteristic
233C23×C4C2411yescharacteristic
234C4⋊Q8—232no—
235C22⋊Q8—432no—
236C22⋊Q8—432no—
237C4×Q8—416no—
238C4×Q8—432no—
239C4×Q8—416no—
240C4×D4—416no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—28no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—28no—
256C2×C4⋊C4—28no—
257C2×C4⋊C4—28no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4—22no—
260C2×C4⋊C4—24no—
261C2×C4⋊C4—28no—
262C2×C4⋊C4—28no—
263C2×C4⋊C4—28no—
264C2×C4⋊C4—24no—
265C2×C4⋊C4—28no—
266C2×C4⋊C4—24no—
267C2×C4⋊C4—24no—
268C2×C4⋊C4—28no—
269C2×C4⋊C4—28no—
270C2×C4⋊C4—22no—
271C2×C4⋊C4—24no—
272C2×C4⋊C4—24no—
273C2×C4⋊C4—41no—
274C2×C4⋊C4—42no—
275C2×C4⋊C4—44no—
276C2×C4⋊C4—44no—
277C2×C4⋊C4—44no—
278C2×C4⋊C4—44no—
279C2×C4⋊C4—42no—
280C2×C4⋊C4—42no—
281C2×C4⋊C4C2×D414yes—
282C2×C4⋊C4C2×D418yes—
283C2×C4⋊C4C2×D418yes—
284C2×C4⋊C4C2×D418yes—
285C2×C4⋊C4#4118yes—
286C2×C4⋊C4#4114yes—
287C2×C22⋊C4—24no—
288C2×C22⋊C4—24no—
289C2×C22⋊C4—24no—
290C2×C22⋊C4—24no—
291C2×C22⋊C4—28no—
292C2×C22⋊C4—24no—
293C2×C22⋊C4—28no—
294C2×C22⋊C4—28no—
295C2×C22⋊C4—28no—
296C2×C22⋊C4—24no—
297C2×C22⋊C4—28no—
298C2×C22⋊C4—28no—
299C2×C22⋊C4—28no—
300C2×C22⋊C4—24no—
301C2×C22⋊C4—24no—
302C2×C22⋊C4—24no—
303C2×C22⋊C4—24no—
304C2×C22⋊C4—28no—
305C2×C22⋊C4—42no—
306C2×C22⋊C4—42no—
307C2×C22⋊C4—42no—
308C2×C22⋊C4—44no—
309C2×C22⋊C4—42no—
310C2×C22⋊C4—42no—
311C2×C22⋊C4—44no—
312C2×C22⋊C4—44no—
313C2×C22⋊C4—44no—
314C2×C42—22no—
315C2×C42—22no—
316C2×C42—24no—
317C2×C42—24no—
318C2×C42—24no—
319C2×C42—22no—
320C2×C42—28no—
321C2×C42—24no—
322C2×C42—28no—
323C2×C42—24no—
324C2×C42—24no—
325C2×C42—24no—
326C2×C42—24no—
327C2×C42—44no—
328C2×C42—44no—
329C2×C42—44no—
330C2×C42—42no—
331C2×C42C2×D414yes—
332C2×C42C2×D414yes—
333C2×C42C2×D418yes—
334#95—24no—
335#95—24no—
336#95—22no—
337#95—24no—
338#95—22no—
339#95—24no—
340#95—24no—
341#95—28no—
342#95—24no—
343#95—24no—
344#95—28no—
345#95—28no—
346#95—28no—
347#95—28no—
348#95—22no—
349#95—24no—
350#95—28no—
351#95—28no—
352#95—22no—
353#95—24no—
354#95—44no—
355#95—42no—
356#95—42no—
357#95—44no—
358#95—44no—
359#95—42no—
360#95—44no—
361C24—22no—
362C24—22no—
363C24—22no—
364C24—22no—
365C24—22no—
366C24—22no—
367C24—22no—
368C24—22no—
369C24—22no—
370C24—22no—
371C24—22no—
372C24—22no—
373C24—22no—
374C24—22no—
375C242−1+414yes—
376C24C2511yescharacteristic
377C2×Q8—28no—
378C2×Q8—28no—
379C2×Q8—28no—
380C2×Q8—216no—
381C2×Q8—216no—
382C2×Q8—216no—
383C2×Q8—44no—
384C2×Q8—44no—
385C2×Q8—48no—
386C2×Q8#121116yes—
387C2×D4—28no—
388C2×D4—48no—
389C2×D4—48no—
390C2×D4—44no—
391C2×D4—44no—
392C22×C4—21no—
393C22×C4—21no—
394C22×C4—24no—
395C22×C4—22no—
396C22×C4—24no—
397C22×C4—21no—
398C22×C4—22no—
399C22×C4—21no—
400C22×C4—21no—
401C22×C4—22no—
402C22×C4—21no—
403C22×C4—22no—
404C22×C4—24no—
405C22×C4—24no—
406C22×C4—22no—
407C22×C4—21no—
408C22×C4—24no—
409C22×C4—22no—
410C22×C4—24no—
411C22×C4—22no—
412C22×C4—21no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—24no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—22no—
421C22×C4—24no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—24no—
427C22×C4—22no—
428C22×C4—21no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—21no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—21no—
435C22×C4—24no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—24no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—24no—
442C22×C4—24no—
443C22×C4—22no—
444C22×C4—24no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—21no—
449C22×C4—22no—
450C22×C4—24no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—24no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—24no—
460C22×C4—22no—
461C22×C4—24no—
462C22×C4—21no—
463C22×C4—48no—
464C22×C4—48no—
465C22×C4—48no—
466C22×C4—48no—
467C22×C4—48no—
468C22×C4—48no—
469C22×C4—48no—
470C22×C4C22×D412yes—
471C22×C4C22×D414yes—
472C22×C4C22×D414yes—
473C22×C4C22×D412yes—
474C22×C4C22×D414yes—
475C22×C4C22×D414yes—
476C22×C4C22×D412yes—
477C22×C4C22×D412yes—
478C22×C4C22×D412yes—
479C22×C4C22×D412yes—
480C22×C4C2×#4112yes—
481C22×C4C2×#4118yes—
482C22×C4C2×#4114yes—
483C22×C4C2×#4112yes—
484C22×C4C2×#4112yes—
485C22×C4C2×#4112yes—
486C22×C4C2×#4114yes—
487C22×C4H(5,𝔽2)12yes—
488C22×C4H(5,𝔽2)14yes—
489C22×C42−1+414yes—
490C22×C42−1+412yes—
491C4⋊C4—216no—
492C4⋊C4—216no—
493C4⋊C4—28no—
494C4⋊C4—216no—
495C4⋊C4—28no—
496C4⋊C4—48no—
497C4⋊C4—44no—
498C4⋊C4—416no—
499C4⋊C4—416no—
500C4⋊C4—44no—
501C4⋊C4—48no—
502C4⋊C4—48no—
503C4⋊C4—416no—
504C4⋊C4—44no—
505C4⋊C4—48no—
506C4⋊C4—416no—
507C4⋊C4—84no—
508C22⋊C4—216no—
509C22⋊C4—416no—
510C22⋊C4—416no—
511C22⋊C4—48no—
512C22⋊C4—48no—
513C22⋊C4—48no—
514C22⋊C4—416no—
515C22⋊C4—48no—
516C22⋊C4—88no—
517C42—28no—
518C42—28no—
519C42—48no—
520C42—48no—
521C42—416no—
522C42—48no—
523C42—44no—
524C42—48no—
525C23—24no—
526C23—24no—
527C23—24no—
528C23—24no—
529C23—24no—
530C23—24no—
531C23—24no—
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—
553C23—42no—
554C23—42no—
555C23—42no—
556C23—42no—
557C23—42no—
558C23—42no—
559C23—42no—
560C23C23×D411yescharacteristic
561C23C23×D411yescharacteristic
562C23C22×#4111yescharacteristic
563C23C22×#4112yes—
564C23C22×#4111yescharacteristic
565C23C2×H(5,𝔽2)12yes—
566C23C2×H(5,𝔽2)11yescharacteristic
567C23C2×2−1+411yescharacteristic
568C23C2×2−1+412yes—
569C23C2×2−1+412yes—
570C23#58511yescharacteristic
571Q8—232no—
572Q8—416no—
573D4—416no—
574C2×C4—22no—
575C2×C4—24no—
576C2×C4—22no—
577C2×C4—24no—
578C2×C4—28no—
579C2×C4—22no—
580C2×C4—28no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—24no—
586C2×C4—22no—
587C2×C4—24no—
588C2×C4—22no—
589C2×C4—24no—
590C2×C4—22no—
591C2×C4—24no—
592C2×C4—28no—
593C2×C4—28no—
594C2×C4—22no—
595C2×C4—28no—
596C2×C4—24no—
597C2×C4—24no—
598C2×C4—24no—
599C2×C4—22no—
600C2×C4—24no—
601C2×C4—22no—
602C2×C4—28no—
603C2×C4—24no—
604C2×C4—24no—
605C2×C4—24no—
606C2×C4—42no—
607C2×C4—41no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—44no—
611C2×C4—44no—
612C2×C4—41no—
613C2×C4—44no—
614C2×C4—41no—
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—44no—
624C2×C4—44no—
625C2×C4—44no—
626C2×C4—42no—
627C2×C4—44no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4—41no—
632C2×C4—42no—
633C2×C4—42no—
634C2×C4—42no—
635C2×C4—42no—
636C2×C4—44no—
637C2×C4—42no—
638C2×C4—44no—
639C2×C4—42no—
640C2×C4—42no—
641C2×C4—44no—
642C2×C4—42no—
643C2×C4—42no—
644C2×C4—42no—
645C2×C4—42no—
646C2×C4—44no—
647C2×C4—41no—
648C2×C4—44no—
649C2×C4—41no—
650C2×C4—44no—
651C2×C4—44no—
652C2×C4—42no—
653C2×C4—42no—
654C2×C4—44no—
655C2×C4—42no—
656C2×C4—42no—
657C2×C4—42no—
658C2×C4—816no—
659C2×C4C2×#12118yes—
660C2×C4C2×#12114yes—
661C2×C4#53214yes—
662C2×C4#53414yes—
663C2×C4#54618yes—
664C2×C4#54814yes—
665C2×C4#55014yes—
666C2×C4#55318yes—
667C22—44no—
668C22—44no—
669C22—44no—
670C22—44no—
671C22—44no—
672C22—44no—
673C22—44no—
674C22—82no—
675C22—82no—
676C22—82no—
677C22—82no—
678C22—82no—
679C22—82no—
680C22—82no—
681C22—82no—
682C22C22×#12111yescharacteristic
683C22C2×#53211yescharacteristic
684C22C2×#53411yescharacteristic
685C22C2×#54612yes—
686C22C2×#54811yescharacteristic
687C22C2×#55011yescharacteristic
688C22C2×#55312yes—
689C22#348011yescharacteristic
690C22#348212yes—
691C22#348611yescharacteristic
692C22#348612yes—
693C22#348811yescharacteristic
694C22#349711yescharacteristic
695C22#349911yescharacteristic
696C22#351612yes—
697C22#352012yes—
698C22#352012yes—
699C22#352211yescharacteristic
700C22#352312yes—
701C22#352312yes—
702C22#353812yes—
703C22#354112yes—
704C22#354812yes—
705C4—28no—
706C4—24no—
707C4—24no—
708C4—44no—
709C4—42no—
710C4—48no—
711C4—42no—
712C4—44no—
713C4—44no—
714C4—42no—
715C4—82no—
716C4—82no—
717C4—82no—
718C4—81no—
719C4—84no—
720C4#254318yes—
721C2—84no—
722C2C2×#254311yescharacteristic
723C2#3570011yescharacteristic
724C2#3590911yescharacteristic
725C2#3650811yescharacteristic
726C2#3657611yescharacteristic
727C2#3912111yescharacteristic
728C2#4291611yescharacteristic
729C2#4508912yes—
730C2#4524912yes—
731C2#5904012yes—
732C2#5940212yes—
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×#2429C2semidirect
3. C2×#2365C2semidirect
4. #16820C2semidirect
5. #15620C2semidirect
6. #15599C2semidirect
7. #15185C2semidirect
10. #15151C2semidirect
12. #14226C2semidirect
13. #13974C2semidirect
14. #13969C2semidirect
15. #13969C2semidirect
17. Q8×#95C2semidirect
92. C2×C4⋊Q8D4semidirect

Maximal subgroups

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

subgroupindex
C2×#25382
C2×#24292
C2×#23652
#168202
#156202
#155992
#151852
#151772
#151632
#151512
#149822
#142262
#139742
#139692
C4×#3942
Q8×#952