Finite Group Databasebeta

← #803744

Subgroups — #803744

Order 512.

3090 conjugacy classes of subgroups.

The subgroup lattice has 755 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2448C212yesmaximal, complemented
2C2×#2426C211yesmaximal, characteristic
3C2×#2421C212yesmaximal, complemented
4C2×Q8×C4⋊C4C211yesmaximal, characteristic, complemented
5#17064C212yesmaximal, complemented
6#15808C214yesmaximal, complemented
7#15696C214yesmaximal, complemented
8#15244C211yesmaximal, characteristic, complemented
9#15148C211yesmaximal, characteristic, complemented
10#15136C212yesmaximal, complemented
11#15135C211yesmaximal, characteristic, complemented
12#15084C212yesmaximal, complemented
13#14231C212yesmaximal, complemented
14#13998C212yesmaximal
15#13954C212yesmaximal, complemented
16C4×#399C211yesmaximal, characteristic, complemented
17#13817C211yesmaximal, characteristic, complemented
18C22×#124C2211yescharacteristic
19C22×C22⋊Q8C2211yescharacteristic
20C22×#121C2211yescharacteristic
21C22×C4×Q8C2212yes—
22C22×C4×Q8C2211yescharacteristic
23#2448—216no—
24#2426—28no—
25#2421—216no—
26C2×#400C2214yes—
27C2×#399C2211yescharacteristic
28C2×#399C2212yes—
29C2×#399C2214yes—
30C2×#399C2212yes—
31C2×#397C2214yes—
32C2×#397C2214yes—
33C2×#396C2212yes—
34C2×#394C2212yes—
35Q8×C4⋊C4—28no—
36C2×#391C2212yes—
37C2×#390C2212yes—
38C2×#390C2211yescharacteristic
39C2×#390C2212yes—
40C2×#390C2212yes—
41C2×#390C2214yes—
42C2×#389C2212yes—
43C2×#389C2212yes—
44C2×#389C2214yes—
45C2×#389C2212yes—
46C2×#389C2211yescharacteristic
47C2×#388C2214yes—
48C2×#388C2214yes—
49C2×#388C2212yes—
50C2×#388C2214yes—
51C2×#387C2214yes—
52C2×#384C2211yescharacteristic
53C2×#384C2212yes—
54C2×#382C2212yes—
55C2×#380C2212yes—
56C2×C4×C4⋊C4C2211yescharacteristic
57C2×C4×C4⋊C4C2212yes—
58C2×C4×C4⋊C4C2212yes—
59C2×C4×C4⋊C4C2212yes—
60C2×C4×C22⋊C4C2211yescharacteristic
61C2×C4×C22⋊C4C2212yes—
62#1445C2212yes—
63#1445C2211yescharacteristic
64#1445C2212yes—
65#1444C2212yes—
66#1444C2211yescharacteristic
67#1444C2214yes—
68#1444C2214yes—
69#1439C2212yes—
70#1439C2211yescharacteristic
71#1439C2212yes—
72#1439C2212yes—
73#1439C2212yes—
74#1439C2214yes—
75#1439C2214yes—
76#1439C2212yes—
77#1439C2214yes—
78#1438C2214yes—
79#1437C2214yes—
80#1436C2214yes—
81C4⋊#95C2211yescharacteristic
82C4⋊#95C2214yes—
83C4⋊#95C2214yes—
84#1434C2212yes—
85C4×#95C2212yes—
86C4×#95C2211yescharacteristic
87C23×Q8C2311yescharacteristic
88C23×D4C2311yescharacteristic
89C23×D4C2312yes—
90C24×C4C2311yescharacteristic
91C2×#124—24no—
92C2×#124—28no—
93C2×C22⋊Q8—28no—
94C2×C22⋊Q8—24no—
95C2×#121—24no—
96C2×#121—28no—
97C2×C4×Q8—28no—
98C2×C4×Q8—28no—
99C2×C4×Q8—28no—
100C2×C4×Q8—28no—
101C2×C4×Q8D418yes—
102C22×C4⋊C4C2312yes—
103C22×C4⋊C4C2314yes—
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2311yescharacteristic
106C22×C4⋊C4C2314yes—
107C22×C4⋊C4C2312yes—
108C22×C4⋊C4C2312yes—
109C22×C4⋊C4C2312yes—
110C22×C4⋊C4C2311yescharacteristic
111C22×C4⋊C4C2312yes—
112C22×C4⋊C4C2314yes—
113C22×C4⋊C4C2312yes—
114C22×C4⋊C4C2312yes—
115C22×C4⋊C4C2314yes—
116C22×C4⋊C4C2312yes—
117C22×C4⋊C4C2311yescharacteristic
118C22×C4⋊C4C2311yescharacteristic
119C22×C22⋊C4C2312yes—
120C22×C22⋊C4C2312yes—
121C22×C22⋊C4C2312yes—
122C22×C22⋊C4C2312yes—
123C22×C22⋊C4C2311yescharacteristic
124C22×C22⋊C4C2312yes—
125C22×C22⋊C4C2311yescharacteristic
126C22×C22⋊C4C2312yes—
127C22×C22⋊C4C2314yes—
128C22×C22⋊C4C2314yes—
129C22×C22⋊C4C2312yes—
130C22×C22⋊C4C2312yes—
131C22×C22⋊C4C2314yes—
132C22×C22⋊C4C2314yes—
133C22×C22⋊C4C2314yes—
134C22×C42C2312yes—
135C22×C42C2314yes—
136C22×C42C2311yescharacteristic
137C22×C42C2312yes—
138C22×C42C2311yescharacteristic
139C22×C42C2312yes—
140C22×C42C2314yes—
141#400—216no—
142#399—24no—
143#399—28no—
144#399—216no—
145#399—44no—
146#397—216no—
147#397—216no—
148#396—28no—
149#394—28no—
150#391D4116yescomplemented
151#390—28no—
152#390—24no—
153#390—28no—
154#390—28no—
155#390—48no—
156#389—216no—
157#389—28no—
158#389—28no—
159#389—24no—
160#389—28no—
161#388—28no—
162#388—216no—
163#388—216no—
164#388—216no—
165#387—216no—
166#384—24no—
167#384—28no—
168#382—28no—
169#380—28no—
170C4×C4⋊C4—28no—
171C4×C4⋊C4—24no—
172C4×C4⋊C4—28no—
173C4×C4⋊C4—28no—
174C4×C22⋊C4—28no—
175C4×C22⋊C4—24no—
176C2×#95C2312yes—
177C2×#95C2311yescharacteristic
178C2×#95C2312yes—
179C2×#95C2314yes—
180C2×#95C2314yes—
181C2×#95C2314yes—
182C2×#95C2311yescharacteristic
183C2×#95C2312yes—
184C2×#95C2312yes—
185C2×#95C2314yes—
186C2×#95C2314yes—
187C2×#95C2311yescharacteristic
188C2×#95C2314yes—
189C2×#95C2312yes—
190C2×#95C2314yes—
191C2×#95C2312yes—
192C2×#95C2312yes—
193C2×#95C2311yescharacteristic
194C2×#95C2312yes—
195C2×#95C2312yes—
196C2×#95C2314yes—
197C2×#95C2314yes—
198C25C2412yes—
199C25C2411yescharacteristic
200C22×Q8—24no—
201C22×Q8—22no—
202C22×Q8—22no—
203C22×Q8C2×D414yes—
204C22×Q8C2×D418yes—
205C22×D4—24no—
206C22×D4—22no—
207C22×D4—24no—
208C22×D4—24no—
209C22×D4—24no—
210C22×D4—28no—
211C22×D4—22no—
212C22×D4—22no—
213C22×D4—44no—
214C22×D4—42no—
215C23×C4—22no—
216C23×C4—24no—
217C23×C4—24no—
218C23×C4—22no—
219C23×C4—22no—
220C23×C4C2411yescharacteristic
221C23×C4C2412yes—
222C23×C4C2414yes—
223C23×C4C2412yes—
224C23×C4C2411yescharacteristic
225C23×C4C2411yescharacteristic
226C23×C4C2411yescharacteristic
227C23×C4C2414yes—
228C23×C4C2412yes—
229C23×C4C2412yes—
230C23×C4C2412yes—
231C23×C4C2412yes—
232C23×C4C2412yes—
233C23×C4C2411yescharacteristic
234C23×C4C2411yescharacteristic
235#124—416no—
236C22⋊Q8—416no—
237#121—416no—
238C4×Q8—232no—
239C4×Q8—432no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—28no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—28no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—28no—
250C2×C4⋊C4—28no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—28no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4—24no—
258C2×C4⋊C4—24no—
259C2×C4⋊C4—28no—
260C2×C4⋊C4—28no—
261C2×C4⋊C4—24no—
262C2×C4⋊C4—24no—
263C2×C4⋊C4—28no—
264C2×C4⋊C4—28no—
265C2×C4⋊C4—24no—
266C2×C4⋊C4—42no—
267C2×C4⋊C4—44no—
268C2×C4⋊C4—44no—
269C2×C4⋊C4—42no—
270C2×C4⋊C4—42no—
271C2×C4⋊C4—42no—
272C2×C4⋊C4—44no—
273C2×C4⋊C4—44no—
274C2×C4⋊C4—44no—
275C2×C4⋊C4—42no—
276C2×C4⋊C4—44no—
277C2×C4⋊C4—42no—
278C2×C4⋊C4—44no—
279C2×C4⋊C4C2×D418yes—
280C2×C4⋊C4C2×D418yes—
281C2×C4⋊C4C2×D414yes—
282C2×C4⋊C4#4114yes—
283C2×C4⋊C4#4118yes—
284C2×C22⋊C4—24no—
285C2×C22⋊C4—28no—
286C2×C22⋊C4—22no—
287C2×C22⋊C4—24no—
288C2×C22⋊C4—28no—
289C2×C22⋊C4—24no—
290C2×C22⋊C4—24no—
291C2×C22⋊C4—28no—
292C2×C22⋊C4—28no—
293C2×C22⋊C4—24no—
294C2×C22⋊C4—24no—
295C2×C22⋊C4—22no—
296C2×C22⋊C4—22no—
297C2×C22⋊C4—24no—
298C2×C22⋊C4—28no—
299C2×C22⋊C4—28no—
300C2×C22⋊C4—24no—
301C2×C22⋊C4—28no—
302C2×C22⋊C4—28no—
303C2×C22⋊C4—24no—
304C2×C22⋊C4—28no—
305C2×C22⋊C4—22no—
306C2×C22⋊C4—24no—
307C2×C22⋊C4—22no—
308C2×C22⋊C4—24no—
309C2×C22⋊C4—28no—
310C2×C22⋊C4—24no—
311C2×C22⋊C4—28no—
312C2×C22⋊C4—42no—
313C2×C22⋊C4—44no—
314C2×C22⋊C4—42no—
315C2×C22⋊C4—44no—
316C2×C22⋊C4—41no—
317C2×C22⋊C4—44no—
318C2×C22⋊C4—44no—
319C2×C22⋊C4—44no—
320C2×C22⋊C4—44no—
321C2×C22⋊C4—44no—
322C2×C22⋊C4—44no—
323C2×C22⋊C4—44no—
324C2×C42—28no—
325C2×C42—24no—
326C2×C42—24no—
327C2×C42—24no—
328C2×C42—24no—
329C2×C42—28no—
330C2×C42—24no—
331C2×C42—24no—
332C2×C42—28no—
333C2×C42—28no—
334C2×C42—22no—
335C2×C42—24no—
336C2×C42—28no—
337C2×C42—44no—
338C2×C42—44no—
339C2×C42C2×D414yes—
340C2×C42C2×D418yes—
341C2×C42C2×D418yes—
342#95—24no—
343#95—22no—
344#95—22no—
345#95—24no—
346#95—24no—
347#95—24no—
348#95—28no—
349#95—24no—
350#95—28no—
351#95—28no—
352#95—28no—
353#95—24no—
354#95—28no—
355#95—28no—
356#95—28no—
357#95—22no—
358#95—24no—
359#95—22no—
360#95—42no—
361#95—42no—
362#95C2×D4116yes—
363#95C2×D4116yes—
364C24—22no—
365C24—21no—
366C24—21no—
367C24—22no—
368C24—21no—
369C24—21no—
370C24—21no—
371C24—22no—
372C24—21no—
373C24—22no—
374C24—24no—
375C24—22no—
376C24—22no—
377C24—24no—
378C24—22no—
379C24—22no—
380C24—22no—
381C24—22no—
382C24—24no—
383C24—24no—
384C24—22no—
385C24H(5,𝔽2)12yes—
386C24C2511yescharacteristic
387C2×Q8—28no—
388C2×Q8—28no—
389C2×Q8—216no—
390C2×Q8—216no—
391C2×Q8#127116yes—
392C2×D4—48no—
393C2×D4—416no—
394C2×D4—48no—
395C2×D4—44no—
396C2×D4—44no—
397C2×D4—44no—
398C2×D4—48no—
399C2×D4—48no—
400C2×D4—84no—
401C2×D4—88no—
402C22×C4—22no—
403C22×C4—24no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—24no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—24no—
415C22×C4—22no—
416C22×C4—24no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—21no—
420C22×C4—24no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—24no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—24no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—24no—
436C22×C4—22no—
437C22×C4—21no—
438C22×C4—22no—
439C22×C4—21no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—24no—
443C22×C4—24no—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—22no—
447C22×C4—24no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—24no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—21no—
455C22×C4—22no—
456C22×C4—24no—
457C22×C4—21no—
458C22×C4—22no—
459C22×C4—21no—
460C22×C4—21no—
461C22×C4—21no—
462C22×C4—24no—
463C22×C4—22no—
464C22×C4—24no—
465C22×C4—24no—
466C22×C4—22no—
467C22×C4—21no—
468C22×C4—24no—
469C22×C4—22no—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—44no—
473C22×C4—44no—
474C22×C4—44no—
475C22×C4—48no—
476C22×C4—48no—
477C22×C4C22×D414yes—
478C22×C4C22×D414yes—
479C22×C4C22×D418yes—
480C22×C4C22×D412yes—
481C22×C4C22×D414yes—
482C22×C4C22×D418yes—
483C22×C4C22×D414yes—
484C22×C4C22×D412yes—
485C22×C4C22×D412yes—
486C22×C4C22×D414yes—
487C22×C4C2×#4114yes—
488C22×C4C2×#4118yes—
489C22×C4C2×#4112yes—
490C22×C4C2×#4114yes—
491C22×C42−1+412yes—
492C22×C42−1+412yes—
493C22×C42−1+412yes—
494C4⋊C4—216no—
495C4⋊C4—216no—
496C4⋊C4—28no—
497C4⋊C4—48no—
498C4⋊C4—416no—
499C4⋊C4—48no—
500C4⋊C4—44no—
501C4⋊C4—48no—
502C4⋊C4—416no—
503C4⋊C4—48no—
504C4⋊C4—44no—
505C4⋊C4—48no—
506C4⋊C4—88no—
507C4⋊C4—84no—
508C4⋊C4—88no—
509C4⋊C4—84no—
510C4⋊C4#123116yes—
511C22⋊C4—416no—
512C22⋊C4—48no—
513C22⋊C4—48no—
514C22⋊C4—416no—
515C22⋊C4—44no—
516C22⋊C4—416no—
517C22⋊C4—48no—
518C22⋊C4—48no—
519C22⋊C4—48no—
520C22⋊C4—48no—
521C22⋊C4—44no—
522C22⋊C4—48no—
523C22⋊C4—88no—
524C22⋊C4—88no—
525C22⋊C4—84no—
526C42—216no—
527C42—28no—
528C42—216no—
529C42—416no—
530C42—48no—
531C42—44no—
532C42—88no—
533C23—24no—
534C23—22no—
535C23—24no—
536C23—22no—
537C23—22no—
538C23—41no—
539C23—41no—
540C23—41no—
541C23—42no—
542C23—42no—
543C23—41no—
544C23—42no—
545C23—42no—
546C23—41no—
547C23—42no—
548C23—42no—
549C23—41no—
550C23—42no—
551C23—41no—
552C23—42no—
553C23—41no—
554C23—42no—
555C23—42no—
556C23—42no—
557C23—42no—
558C23—42no—
559C23—44no—
560C23—44no—
561C23—42no—
562C23—44no—
563C23—44no—
564C23—44no—
565C23—42no—
566C23—44no—
567C23—42no—
568C23—44no—
569C23—42no—
570C23—44no—
571C23—44no—
572C23—42no—
573C23—44no—
574C23—42no—
575C23—42no—
576C23—42no—
577C23—44no—
578C23—44no—
579C23C23×D411yescharacteristic
580C23C23×D412yes—
581C23C22×#4112yes—
582C23C22×#4111yescharacteristic
583C23C2×H(5,𝔽2)11yescharacteristic
584C23C2×2−1+411yescharacteristic
585C23C2×2−1+411yescharacteristic
586C23C2×2−1+411yescharacteristic
587C23#58512yes—
588C23#58512yes—
589C23#58511yescharacteristic
590Q8—232no—
591D4—816no—
592D4—88no—
593C2×C4—22no—
594C2×C4—24no—
595C2×C4—24no—
596C2×C4—28no—
597C2×C4—28no—
598C2×C4—24no—
599C2×C4—22no—
600C2×C4—24no—
601C2×C4—24no—
602C2×C4—28no—
603C2×C4—24no—
604C2×C4—28no—
605C2×C4—22no—
606C2×C4—24no—
607C2×C4—24no—
608C2×C4—28no—
609C2×C4—24no—
610C2×C4—24no—
611C2×C4—24no—
612C2×C4—28no—
613C2×C4—24no—
614C2×C4—24no—
615C2×C4—24no—
616C2×C4—28no—
617C2×C4—22no—
618C2×C4—24no—
619C2×C4—22no—
620C2×C4—42no—
621C2×C4—44no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—44no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4—42no—
632C2×C4—42no—
633C2×C4—44no—
634C2×C4—42no—
635C2×C4—44no—
636C2×C4—42no—
637C2×C4—42no—
638C2×C4—41no—
639C2×C4—41no—
640C2×C4—41no—
641C2×C4—44no—
642C2×C4—42no—
643C2×C4—42no—
644C2×C4—41no—
645C2×C4—41no—
646C2×C4—42no—
647C2×C4—42no—
648C2×C4—41no—
649C2×C4—42no—
650C2×C4—42no—
651C2×C4—44no—
652C2×C4—44no—
653C2×C4—44no—
654C2×C4—42no—
655C2×C4—42no—
656C2×C4—42no—
657C2×C4—41no—
658C2×C4—42no—
659C2×C4—44no—
660C2×C4—41no—
661C2×C4—42no—
662C2×C4—44no—
663C2×C4—42no—
664C2×C4—44no—
665C2×C4—42no—
666C2×C4—44no—
667C2×C4—42no—
668C2×C4—42no—
669C2×C4—44no—
670C2×C4—42no—
671C2×C4—41no—
672C2×C4—88no—
673C2×C4C2×#12318yes—
674C2×C4C2×#12314yes—
675C2×C4C2×#12718yes—
676C2×C4C2×#12714yes—
677C2×C4#53614yes—
678C2×C4#54718yes—
679C2×C4#54718yes—
680C2×C4#54814yes—
681C22—42no—
682C22—44no—
683C22—42no—
684C22—44no—
685C22—42no—
686C22—81no—
687C22—81no—
688C22—82no—
689C22—81no—
690C22—81no—
691C22—82no—
692C22—82no—
693C22—82no—
694C22—82no—
695C22—84no—
696C22—84no—
697C22—82no—
698C22—82no—
699C22—84no—
700C22—82no—
701C22—82no—
702C22—84no—
703C22C22×#12311yescharacteristic
704C22C22×#12711yescharacteristic
705C22C2×#53611yescharacteristic
706C22C2×#54712yes—
707C22C2×#54712yes—
708C22C2×#54811yescharacteristic
709C22D4×#4112yes—
710C22#348012yes—
711C22#348711yescharacteristic
712C22#348712yes—
713C22#348811yescharacteristic
714C22#348812yes—
715C22#349712yes—
716C22#350011yescharacteristic
717C22#351312yes—
718C22#351612yes—
719C22#351811yescharacteristic
720C22#352112yes—
721C22#352211yescharacteristic
722C22#354411yescharacteristic
723C22#355511yescharacteristic
724C22#355712yes—
725C22#356512yes—
726C4—28no—
727C4—28no—
728C4—24no—
729C4—44no—
730C4—42no—
731C4—48no—
732C4—44no—
733C4—44no—
734C4—81no—
735C4—82no—
736C4—81no—
737C4—81no—
738C4—84no—
739C4—82no—
740C4#250118yes—
741C2—82no—
742C2—162no—
743C2C2×#250111yescharacteristic
744C2#3597611yescharacteristic
745C2#3606611yescharacteristic
746C2#3646511yescharacteristic
747C2#3657811yescharacteristic
748C2#3912511yescharacteristic
749C2#4287511yescharacteristic
750C2#4501612yes—
751C2#4524212yes—
752C2#5907212yes—
753C2#5911812yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC22×#121C22
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×#2448C2semidirect
3. C2×#2421C2semidirect
4. C2×Q8×C4⋊C4C2semidirect
5. #17064C2semidirect
6. #15808C2semidirect
7. #15696C2semidirect
8. #15244C2semidirect
9. #15148C2semidirect
10. #15136C2semidirect
11. #15135C2semidirect
12. #15084C2semidirect
13. #14231C2semidirect
15. #13954C2semidirect
16. C4×#399C2semidirect
17. #13817C2semidirect
150. #391D4semidirect

Maximal subgroups

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

subgroupindex
C2×#24482
C2×#24262
C2×#24212
C2×Q8×C4⋊C42
#170642
#158082
#156962
#152442
#151482
#151362
#151352
#150842
#142312
#139982
#139542
C4×#3992
#138172