Finite Group Databasebeta

← #761958

Subgroups — #761958

Order 512.

2524 conjugacy classes of subgroups.

The subgroup lattice has 735 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×C4⋊Q8C211yesmaximal, characteristic, complemented
2C2×C4×#126C211yesmaximal, characteristic
3#16965C212yesmaximal, complemented
4#16673C212yesmaximal
5#15212C212yesmaximal, complemented
6#15180C211yesmaximal, characteristic, complemented
7#15178C211yesmaximal, characteristic
8#15174C212yesmaximal
9#15151C211yesmaximal, characteristic, complemented
10#15150C211yesmaximal, characteristic
11#14903C212yesmaximal, complemented
12#14894C212yesmaximal
13#14249C212yesmaximal
14#14244C212yesmaximal, complemented
15#14021C212yesmaximal, complemented
16#14017C212yesmaximal, complemented
17#13997C212yesmaximal
18#13969C212yesmaximal
19#13755C211yesmaximal, characteristic, complemented
20C22×C4⋊Q8C2211yescharacteristic
21C22×#126C2211yescharacteristic
22C22×C22⋊Q8C2212yes—
23C22×C4×Q8C2212yes—
24C2×#398C2212yes—
25C2×#398C2212yes—
26C2×#397C2212yes—
27C2×#397C2212yes—
28C4×C4⋊Q8—28no—
29C4×#126—28no—
30C2×#391C2212yes—
31C2×#391C2212yes—
32C2×#391C2212yes—
33C2×#391C2212yes—
34C2×#389C2212yes—
35C2×#389C2212yes—
36C2×#389C2212yes—
37C2×#388C2211yescharacteristic
38C2×#388C2211yescharacteristic
39C2×#388C2211yescharacteristic
40C2×#387C2211yescharacteristic
41C2×#387C2211yescharacteristic
42C2×#387C2211yescharacteristic
43C2×#384C2211yescharacteristic
44C2×#383C2211yescharacteristic
45C2×#380C2212yes—
46C2×C4×C4⋊C4C2211yescharacteristic
47C2×C4×C4⋊C4C2212yes—
48C2×C4×C4⋊C4C2212yes—
49C2×C4×C4⋊C4C2212yes—
50C2×C4×C4⋊C4C2211yescharacteristic
51C2×C4×C4⋊C4C2211yescharacteristic
52C2×C4×C22⋊C4C2211yescharacteristic
53C2×C4×C22⋊C4C2211yescharacteristic
54C2×C4×C22⋊C4C2211yescharacteristic
55C2×#376C2212yes—
56C2×C43C2211yescharacteristic
57#1445C2212yes—
58#1445C2212yes—
59#1445C2212yes—
60#1445C2212yes—
61#1445C2214yes—
62#1444C2214yes—
63#1444C2212yes—
64#1444C2212yes—
65#1443C2214yes—
66#1442C2214yes—
67#1442C2214yes—
68#1442C2212yes—
69#1442C2212yes—
70#1441C2212yes—
71#1441C2214yes—
72#1441C2212yes—
73#1439C2214yes—
74#1439C2214yes—
75#1439C2212yes—
76#1439C2212yes—
77#1437C2214yes—
78#1437C2214yes—
79#1436C2214yes—
80#1436C2214yes—
81#1436C2214yes—
82#1436C2214yes—
83C4⋊#95C2211yescharacteristic
84C4⋊#95C2212yes—
85C4⋊#95C2211yescharacteristic
86C4⋊#95C2212yes—
87C4⋊#95C2214yes—
88C4⋊#95C2212yes—
89#1434C2212yes—
90#1434C2214yes—
91#1434C2212yes—
92C4×#95C2212yes—
93C23×Q8C2312yes—
94C24×C4C2311yescharacteristic
95C2×C4⋊Q8—28no—
96C2×C4⋊Q8—24no—
97C2×#126—24no—
98C2×#126—24no—
99C2×#126—24no—
100C2×C22⋊Q8—28no—
101C2×C22⋊Q8—28no—
102C2×C22⋊Q8—44no—
103C2×C4×Q8—28no—
104C2×C4×Q8—44no—
105C2×C4×Q8—44no—
106C22×C4⋊C4C2312yes—
107C22×C4⋊C4C2312yes—
108C22×C4⋊C4C2311yescharacteristic
109C22×C4⋊C4C2312yes—
110C22×C4⋊C4C2311yescharacteristic
111C22×C4⋊C4C2311yescharacteristic
112C22×C4⋊C4C2312yes—
113C22×C4⋊C4C2312yes—
114C22×C4⋊C4C2312yes—
115C22×C4⋊C4C2311yescharacteristic
116C22×C4⋊C4C2311yescharacteristic
117C22×C4⋊C4C2314yes—
118C22×C4⋊C4C2311yescharacteristic
119C22×C4⋊C4C2312yes—
120C22×C4⋊C4C2314yes—
121C22×C4⋊C4C2314yes—
122C22×C22⋊C4C2311yescharacteristic
123C22×C22⋊C4C2311yescharacteristic
124C22×C22⋊C4C2311yescharacteristic
125C22×C22⋊C4C2311yescharacteristic
126C22×C22⋊C4C2311yescharacteristic
127C22×C22⋊C4C2314yes—
128C22×C22⋊C4C2311yescharacteristic
129C22×C22⋊C4C2314yes—
130C22×C42C2311yescharacteristic
131C22×C42C2311yescharacteristic
132C22×C42C2311yescharacteristic
133C22×C42C2314yes—
134C22×C42C2311yescharacteristic
135C22×C42C2311yescharacteristic
136C22×C42C2311yescharacteristic
137C22×C42C2311yescharacteristic
138C22×C42C2311yescharacteristic
139C22×C42C2311yescharacteristic
140C22×C42C2311yescharacteristic
141C22×C42C2314yes—
142#398—28no—
143#398—44no—
144#397—28no—
145#397—44no—
146#391—28no—
147#391—28no—
148#391—28no—
149#391—44no—
150#389—28no—
151#389—28no—
152#389—44no—
153#388—24no—
154#388—24no—
155#388—24no—
156#387—24no—
157#387—24no—
158#387—24no—
159#384—24no—
160#383—24no—
161#380—44no—
162C4×C4⋊C4—28no—
163C4×C4⋊C4—24no—
164C4×C4⋊C4—24no—
165C4×C4⋊C4—24no—
166C4×C4⋊C4—28no—
167C4×C4⋊C4—28no—
168C4×C22⋊C4—24no—
169C4×C22⋊C4—24no—
170C4×C22⋊C4—24no—
171#376—44no—
172C2×#95C2312yes—
173C2×#95C2314yes—
174C2×#95C2312yes—
175C2×#95C2312yes—
176C2×#95C2312yes—
177C2×#95C2314yes—
178C2×#95C2314yes—
179C2×#95C2311yescharacteristic
180C2×#95C2314yes—
181C2×#95C2314yes—
182C2×#95C2314yes—
183C2×#95C2314yes—
184C2×#95C2311yescharacteristic
185C2×#95C2311yescharacteristic
186C2×#95C2311yescharacteristic
187C2×#95C2314yes—
188C2×#95C2314yes—
189C2×#95C2314yes—
190C2×#95C2311yescharacteristic
191C2×#95C2311yescharacteristic
192C2×#95C2311yescharacteristic
193C2×#95C2311yescharacteristic
194C2×#95C2311yescharacteristic
195C2×#95C2311yescharacteristic
196C2×#95C2311yescharacteristic
197C2×#95C2311yescharacteristic
198C2×#95C2314yes—
199C2×#95C2314yes—
200C2×#95C2314yes—
201C2×#95C2314yes—
202C2×#95C2314yes—
203C2×#95C2314yes—
204C2×#95C2314yes—
205C43D418yes—
206C25C2411yescharacteristic
207C22×Q8—24no—
208C22×Q8—24no—
209C22×Q8—24no—
210C22×Q8—24no—
211C22×Q8—24no—
212C22×Q8—24no—
213C22×Q8#4118yes—
214C23×C4—22no—
215C23×C4—24no—
216C23×C4—24no—
217C23×C4—22no—
218C23×C4—22no—
219C23×C4C2414yes—
220C23×C4C2411yescharacteristic
221C23×C4C2411yescharacteristic
222C23×C4C2411yescharacteristic
223C23×C4C2414yes—
224C23×C4C2411yescharacteristic
225C23×C4C2411yescharacteristic
226C23×C4C2411yescharacteristic
227C23×C4C2411yescharacteristic
228C23×C4C2414yes—
229C23×C4C2411yescharacteristic
230C23×C4C2411yescharacteristic
231C23×C4C2414yes—
232C23×C4C2411yescharacteristic
233C23×C4C2411yescharacteristic
234C23×C4C2411yescharacteristic
235C23×C4C2411yescharacteristic
236C23×C4C2411yescharacteristic
237C4⋊Q8—232no—
238#126—232no—
239C22⋊Q8—432no—
240C4×Q8—432no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—28no—
253C2×C4⋊C4—28no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—28no—
257C2×C4⋊C4—24no—
258C2×C4⋊C4—24no—
259C2×C4⋊C4—22no—
260C2×C4⋊C4—22no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—28no—
263C2×C4⋊C4—28no—
264C2×C4⋊C4—24no—
265C2×C4⋊C4—24no—
266C2×C4⋊C4—24no—
267C2×C4⋊C4—24no—
268C2×C4⋊C4—24no—
269C2×C4⋊C4—24no—
270C2×C4⋊C4—24no—
271C2×C4⋊C4—24no—
272C2×C4⋊C4—22no—
273C2×C4⋊C4—24no—
274C2×C4⋊C4—24no—
275C2×C4⋊C4—22no—
276C2×C4⋊C4—24no—
277C2×C4⋊C4—28no—
278C2×C4⋊C4—44no—
279C2×C4⋊C4—44no—
280C2×C4⋊C4—44no—
281C2×C4⋊C4—44no—
282C2×C4⋊C4—44no—
283C2×C4⋊C4#4118yes—
284C2×C4⋊C4#4114yes—
285C2×C22⋊C4—22no—
286C2×C22⋊C4—22no—
287C2×C22⋊C4—22no—
288C2×C22⋊C4—22no—
289C2×C22⋊C4—22no—
290C2×C22⋊C4—22no—
291C2×C22⋊C4—22no—
292C2×C22⋊C4—22no—
293C2×C22⋊C4—22no—
294C2×C22⋊C4—22no—
295C2×C22⋊C4—22no—
296C2×C22⋊C4—28no—
297C2×C22⋊C4—28no—
298C2×C22⋊C4—22no—
299C2×C22⋊C4—24no—
300C2×C22⋊C4—24no—
301C2×C22⋊C4—22no—
302C2×C22⋊C4—28no—
303C2×C22⋊C4—44no—
304C2×C22⋊C4—44no—
305C2×C22⋊C4—44no—
306C2×C22⋊C4#4114yes—
307C2×C42—22no—
308C2×C42—22no—
309C2×C42—24no—
310C2×C42—24no—
311C2×C42—22no—
312C2×C42—22no—
313C2×C42—22no—
314C2×C42—24no—
315C2×C42—24no—
316C2×C42—22no—
317C2×C42—28no—
318C2×C42—22no—
319C2×C42—22no—
320C2×C42—22no—
321C2×C42—28no—
322C2×C42—24no—
323C2×C42—24no—
324C2×C42—22no—
325C2×C42—22no—
326C2×C42—28no—
327C2×C42—44no—
328C2×C42—44no—
329C2×C42—44no—
330C2×C42C2×D414yes—
331C2×C42C2×D414yes—
332C2×C42C2×D414yes—
333C2×C42C2×D414yes—
334C2×C42C2×D414yes—
335C2×C42C2×D414yes—
336C2×C42C2×D414yes—
337#95—22no—
338#95—22no—
339#95—22no—
340#95—22no—
341#95—22no—
342#95—22no—
343#95—22no—
344#95—22no—
345#95—22no—
346#95—22no—
347#95—22no—
348#95—22no—
349#95—28no—
350#95—28no—
351#95—28no—
352#95—28no—
353#95—28no—
354#95—28no—
355#95—28no—
356#95—24no—
357#95—28no—
358#95—24no—
359#95—28no—
360#95—24no—
361#95—24no—
362#95—28no—
363#95—44no—
364#95—44no—
365#95—44no—
366#95—44no—
367#95—44no—
368#95—44no—
369#95—44no—
370C24—21no—
371C24—21no—
372C24—22no—
373C24—22no—
374C24—21no—
375C24—21no—
376C24—21no—
377C24—22no—
378C24—22no—
379C24—21no—
380C24C2×#4112yes—
381C24C2511yescharacteristic
382C2×Q8—216no—
383C2×Q8—216no—
384C2×Q8—216no—
385C2×Q8—48no—
386C2×Q8—48no—
387C2×Q8—48no—
388C2×Q8—48no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—21no—
393C22×C4—21no—
394C22×C4—21no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—21no—
399C22×C4—21no—
400C22×C4—21no—
401C22×C4—21no—
402C22×C4—24no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—24no—
407C22×C4—24no—
408C22×C4—21no—
409C22×C4—21no—
410C22×C4—21no—
411C22×C4—21no—
412C22×C4—21no—
413C22×C4—21no—
414C22×C4—21no—
415C22×C4—24no—
416C22×C4—24no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—21no—
420C22×C4—21no—
421C22×C4—24no—
422C22×C4—24no—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—24no—
427C22×C4—21no—
428C22×C4—24no—
429C22×C4—22no—
430C22×C4—21no—
431C22×C4—24no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—21no—
435C22×C4—21no—
436C22×C4—21no—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—24no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—24no—
443C22×C4—24no—
444C22×C4—24no—
445C22×C4—24no—
446C22×C4—24no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—24no—
450C22×C4—21no—
451C22×C4—21no—
452C22×C4—21no—
453C22×C4—21no—
454C22×C4—21no—
455C22×C4—24no—
456C22×C4—24no—
457C22×C4—24no—
458C22×C4—24no—
459C22×C4—21no—
460C22×C4—24no—
461C22×C4—24no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—21no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—24no—
468C22×C4—21no—
469C22×C4—44no—
470C22×C4—44no—
471C22×C4—44no—
472C22×C4—48no—
473C22×C4—48no—
474C22×C4C22×D412yes—
475C22×C4C22×D412yes—
476C22×C4C22×D412yes—
477C22×C4C22×D412yes—
478C22×C4C22×D412yes—
479C22×C4C22×D412yes—
480C22×C4C22×D412yes—
481C22×C4C2×#4112yes—
482C22×C4C2×#4112yes—
483C22×C4C2×#4118yes—
484C22×C4C2×#4118yes—
485C22×C4C2×#4112yes—
486C22×C4C2×#4112yes—
487C22×C4C2×#4112yes—
488C22×C4C2×#4114yes—
489C22×C4C2×#4114yes—
490C22×C4H(5,𝔽2)12yes—
491C22×C4H(5,𝔽2)12yes—
492C22×C4H(5,𝔽2)12yes—
493C22×C42−1+414yes—
494C22×C42−1+414yes—
495C22×C42−1+412yes—
496C4⋊C4—28no—
497C4⋊C4—28no—
498C4⋊C4—216no—
499C4⋊C4—216no—
500C4⋊C4—28no—
501C4⋊C4—216no—
502C4⋊C4—416no—
503C4⋊C4—416no—
504C4⋊C4—48no—
505C4⋊C4—48no—
506C4⋊C4—44no—
507C4⋊C4—44no—
508C4⋊C4—44no—
509C4⋊C4—48no—
510C4⋊C4—48no—
511C4⋊C4—88no—
512C22⋊C4—28no—
513C22⋊C4—28no—
514C22⋊C4—28no—
515C22⋊C4—416no—
516C22⋊C4—44no—
517C22⋊C4—44no—
518C22⋊C4—44no—
519C22⋊C4—88no—
520C42—28no—
521C42—28no—
522C42—28no—
523C42—28no—
524C42—28no—
525C42—28no—
526C42—28no—
527C42—416no—
528C42—416no—
529C42—44no—
530C42—44no—
531C42—44no—
532C23—22no—
533C23—24no—
534C23—24no—
535C23—22no—
536C23—22no—
537C23—41no—
538C23—41no—
539C23—41no—
540C23—41no—
541C23—41no—
542C23—41no—
543C23—42no—
544C23—42no—
545C23—42no—
546C23—42no—
547C23—41no—
548C23—41no—
549C23—42no—
550C23—42no—
551C23—42no—
552C23—42no—
553C23—42no—
554C23—42no—
555C23C23×D411yescharacteristic
556C23C22×#4112yes—
557C23C22×#4111yescharacteristic
558C23C22×#4111yescharacteristic
559C23C22×#4112yes—
560C23C2×H(5,𝔽2)11yescharacteristic
561C23C2×H(5,𝔽2)11yescharacteristic
562C23C2×H(5,𝔽2)11yescharacteristic
563C23C2×2−1+412yes—
564C23C2×2−1+412yes—
565C23C2×2−1+411yescharacteristic
566Q8—432no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—22no—
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—22no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—28no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—22no—
593C2×C4—28no—
594C2×C4—28no—
595C2×C4—28no—
596C2×C4—28no—
597C2×C4—24no—
598C2×C4—24no—
599C2×C4—22no—
600C2×C4—22no—
601C2×C4—28no—
602C2×C4—41no—
603C2×C4—41no—
604C2×C4—44no—
605C2×C4—44no—
606C2×C4—41no—
607C2×C4—41no—
608C2×C4—41no—
609C2×C4—41no—
610C2×C4—44no—
611C2×C4—44no—
612C2×C4—44no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—44no—
618C2×C4—41no—
619C2×C4—41no—
620C2×C4—41no—
621C2×C4—41no—
622C2×C4—41no—
623C2×C4—41no—
624C2×C4—44no—
625C2×C4—44no—
626C2×C4—41no—
627C2×C4—44no—
628C2×C4—41no—
629C2×C4—41no—
630C2×C4—41no—
631C2×C4—44no—
632C2×C4—41no—
633C2×C4—41no—
634C2×C4—41no—
635C2×C4—41no—
636C2×C4—41no—
637C2×C4—41no—
638C2×C4—41no—
639C2×C4—41no—
640C2×C4—44no—
641C2×C4—42no—
642C2×C4—44no—
643C2×C4—42no—
644C2×C4—44no—
645C2×C4—44no—
646C2×C4—44no—
647C2×C4—44no—
648C2×C4—44no—
649C2×C4—44no—
650C2×C4—44no—
651C2×C4—44no—
652C2×C4—42no—
653C2×C4—44no—
654C2×C4—42no—
655C2×C4—42no—
656C2×C4—42no—
657C2×C4—44no—
658C2×C4—42no—
659C2×C4—42no—
660C2×C4—88no—
661C2×C4#52514yes—
662C2×C4#52514yes—
663C2×C4#53214yes—
664C2×C4#53514yes—
665C2×C4#53614yes—
666C2×C4#53614yes—
667C2×C4#53714yes—
668C2×C4#55118yes—
669C2×C4#55318yes—
670C22—44no—
671C22—44no—
672C22—42no—
673C22—42no—
674C22—42no—
675C22—81no—
676C22—82no—
677C22—82no—
678C22—81no—
679C22—81no—
680C22—81no—
681C22C2×#52511yescharacteristic
682C22C2×#52511yescharacteristic
683C22C2×#53211yescharacteristic
684C22C2×#53511yescharacteristic
685C22C2×#53611yescharacteristic
686C22C2×#53611yescharacteristic
687C22C2×#53711yescharacteristic
688C22C2×#55112yes—
689C22C2×#55312yes—
690C22#348212yes—
691C22#348212yes—
692C22#348212yes—
693C22#348212yes—
694C22#350811yescharacteristic
695C22#350811yescharacteristic
696C22#351111yescharacteristic
697C22#351111yescharacteristic
698C22#352012yes—
699C22#352212yes—
700C22#352312yes—
701C22#352312yes—
702C22#355212yes—
703C22#355212yes—
704C4—24no—
705C4—24no—
706C4—24no—
707C4—24no—
708C4—48no—
709C4—48no—
710C4—42no—
711C4—42no—
712C4—42no—
713C4—42no—
714C4—42no—
715C4—81no—
716C4—84no—
717C4—81no—
718C4—81no—
719C4—84no—
720C4—81no—
721C4#260318yes—
722C2—82no—
723C2C2×#260311yescharacteristic
724C2#3497111yescharacteristic
725C2#3497911yescharacteristic
726C2#3524111yescharacteristic
727C2#3524711yescharacteristic
728C2#3529211yescharacteristic
729C2#3761511yescharacteristic
730C2#5947912yes—
731C2#5948112yes—
732C2#5969512yes—
733C2#5969712yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC25C24
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×C4×C4⋊Q8C2semidirect
3. #16965C2semidirect
5. #15212C2semidirect
6. #15180C2semidirect
9. #15151C2semidirect
11. #14903C2semidirect
14. #14244C2semidirect
15. #14021C2semidirect
16. #14017C2semidirect
19. #13755C2semidirect

Maximal subgroups

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

subgroupindex
C2×C4×C4⋊Q82
C2×C4×#1262
#169652
#166732
#152122
#151802
#151782
#151742
#151512
#151502
#149032
#148942
#142492
#142442
#140212
#140172
#139972
#139692
#137552