Finite Group Databasebeta

← #641339

Subgroups — #641339

Order 512.

2240 conjugacy classes of subgroups.

The subgroup lattice has 763 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2495C211yesmaximal, characteristic
2C2×#2429C211yesmaximal, characteristic
3#16872C214yesmaximal, complemented
4#16859C214yesmaximal
5#15239C211yesmaximal, characteristic, complemented
6#15236C211yesmaximal, characteristic
7#15188C211yesmaximal, characteristic, complemented
8#15182C211yesmaximal, characteristic
9#14983C214yesmaximal
10#14940C214yesmaximal, complemented
11#14703C211yesmaximal, characteristic, complemented
12#14276C211yesmaximal, characteristic, complemented
13#14274C211yesmaximal, characteristic, complemented
14#14258C211yesmaximal, characteristic
15#14258C211yesmaximal, characteristic
16#14003C211yesmaximal, characteristic, complemented
17#13976C211yesmaximal, characteristic, complemented
18C4×#391C211yesmaximal, characteristic, complemented
19#13761C211yesmaximal, characteristic
20C22×C4⋊Q8C2211yescharacteristic
21C22×C22⋊Q8C2211yescharacteristic
22C22×C22⋊Q8C2211yescharacteristic
23C22×C4×Q8C2211yescharacteristic
24C22×#117C2211yescharacteristic
25#2495—28no—
26#2429—28no—
27C2×#400C2212yes—
28C2×#398C2212yes—
29C2×#397C2212yes—
30C2×#396C2212yes—
31C2×#391C2211yescharacteristic
32C2×#391C2211yescharacteristic
33C2×#391C2211yescharacteristic
34C2×#391C2211yescharacteristic
35C2×#391C2211yescharacteristic
36C2×#391C2211yescharacteristic
37C2×#391C2211yescharacteristic
38C2×#391C2211yescharacteristic
39C2×#391C2211yescharacteristic
40C2×#389C2211yescharacteristic
41C2×#389C2212yes—
42C2×#389C2211yescharacteristic
43C2×#389C2211yescharacteristic
44C2×#388C2212yes—
45C2×#388C2212yes—
46C2×#384C2211yescharacteristic
47C2×#383C2212yes—
48C2×#382C2211yescharacteristic
49C2×#382C2211yescharacteristic
50C2×#380C2211yescharacteristic
51C2×#380C2211yescharacteristic
52C2×C4×C4⋊C4C2211yescharacteristic
53C2×C4×C22⋊C4C2211yescharacteristic
54C2×C4×C22⋊C4C2211yescharacteristic
55C2×#376C2211yescharacteristic
56C2×C43C2211yescharacteristic
57#1445C2214yes—
58#1445C2214yes—
59#1444C2214yes—
60#1443C2214yes—
61#1443C2214yes—
62#1442C2214yes—
63#1442C2214yes—
64#1442C2214yes—
65#1441C2214yes—
66#1441C2214yes—
67#1441C2214yes—
68#1441C2214yes—
69#1441C2214yes—
70#1441C2214yes—
71#1440C2214yes—
72#1440C2214yes—
73#1440C2214yes—
74#1439C2214yes—
75#1437C2212yes—
76#1437C2212yes—
77#1436C2212yes—
78#1436C2214yes—
79#1436C2214yes—
80#1436C2212yes—
81#1436C2214yes—
82C4⋊#95C2211yescharacteristic
83C4⋊#95C2211yescharacteristic
84C4⋊#95C2211yescharacteristic
85C4⋊#95C2211yescharacteristic
86C4⋊#95C2211yescharacteristic
87C4⋊#95C2211yescharacteristic
88#1434C2214yes—
89#1434C2214yes—
90C4×#95C2211yescharacteristic
91C4×#95C2214yes—
92C4×#95C2211yescharacteristic
93C23×Q8C2311yescharacteristic
94C23×Q8C2311yescharacteristic
95C24×C4C2311yescharacteristic
96C2×C4⋊Q8—24no—
97C2×C4⋊Q8—24no—
98C2×C4⋊Q8—42no—
99C2×C22⋊Q8—24no—
100C2×C22⋊Q8—24no—
101C2×C22⋊Q8—24no—
102C2×C22⋊Q8—24no—
103C2×C22⋊Q8—42no—
104C2×C22⋊Q8—42no—
105C2×C4×Q8—24no—
106C2×C4×Q8—24no—
107C2×C4×Q8—42no—
108C2×#117—24no—
109C2×#117—42no—
110C2×#117—42no—
111C22×C4⋊C4C2311yescharacteristic
112C22×C4⋊C4C2311yescharacteristic
113C22×C4⋊C4C2311yescharacteristic
114C22×C4⋊C4C2311yescharacteristic
115C22×C4⋊C4C2311yescharacteristic
116C22×C4⋊C4C2312yes—
117C22×C4⋊C4C2312yes—
118C22×C4⋊C4C2312yes—
119C22×C4⋊C4C2312yes—
120C22×C4⋊C4C2312yes—
121C22×C4⋊C4C2311yescharacteristic
122C22×C4⋊C4C2312yes—
123C22×C4⋊C4C2312yes—
124C22×C4⋊C4C2312yes—
125C22×C4⋊C4C2312yes—
126C22×C4⋊C4C2314yes—
127C22×C22⋊C4C2312yes—
128C22×C22⋊C4C2312yes—
129C22×C22⋊C4C2312yes—
130C22×C22⋊C4C2311yescharacteristic
131C22×C22⋊C4C2312yes—
132C22×C22⋊C4C2312yes—
133C22×C22⋊C4C2312yes—
134C22×C22⋊C4C2311yescharacteristic
135C22×C42C2311yescharacteristic
136C22×C42C2312yes—
137C22×C42C2311yescharacteristic
138C22×C42C2312yes—
139C22×C42C2312yes—
140C22×C42C2311yescharacteristic
141C22×C42C2311yescharacteristic
142C22×C42C2311yescharacteristic
143C22×C42C2311yescharacteristic
144C22×C42C2312yes—
145#400—28no—
146#398—28no—
147#397—28no—
148#396—28no—
149#391—24no—
150#391—24no—
151#391—24no—
152#391—24no—
153#391—24no—
154#391—24no—
155#391—42no—
156#391—42no—
157#391—42no—
158#389—24no—
159#389—24no—
160#389—24no—
161#389—28no—
162#388—28no—
163#388—28no—
164#384—24no—
165#383—28no—
166#382—24no—
167#382—42no—
168#380—24no—
169#380—24no—
170C4×C4⋊C4—24no—
171C4×C22⋊C4—24no—
172C4×C22⋊C4D418yes—
173#376—42no—
174C2×#95C2312yes—
175C2×#95C2312yes—
176C2×#95C2312yes—
177C2×#95C2312yes—
178C2×#95C2312yes—
179C2×#95C2312yes—
180C2×#95C2312yes—
181C2×#95C2312yes—
182C2×#95C2312yes—
183C2×#95C2312yes—
184C2×#95C2314yes—
185C2×#95C2314yes—
186C2×#95C2312yes—
187C2×#95C2312yes—
188C2×#95C2314yes—
189C2×#95C2314yes—
190C2×#95C2314yes—
191C2×#95C2312yes—
192C2×#95C2314yes—
193C2×#95C2314yes—
194C2×#95C2314yes—
195C2×#95C2312yes—
196C2×#95C2312yes—
197C2×#95C2314yes—
198C2×#95C2314yes—
199C2×#95C2314yes—
200C2×#95C2314yes—
201C2×#95C2314yes—
202C2×#95C2314yes—
203C2×#95C2312yes—
204C2×#95C2312yes—
205C2×#95C2314yes—
206C2×#95C2311yescharacteristic
207C2×#95C2311yescharacteristic
208C43—42no—
209C25C2411yescharacteristic
210C22×Q8—22no—
211C22×Q8—22no—
212C22×Q8—22no—
213C22×Q8—22no—
214C22×Q8—22no—
215C22×Q8—22no—
216C22×Q8—22no—
217C22×Q8—22no—
218C22×Q8—22no—
219C22×Q8—22no—
220C22×Q8—22no—
221C22×Q8—22no—
222C22×Q8—22no—
223C22×Q8#4114yes—
224C23×C4—22no—
225C23×C4—22no—
226C23×C4—22no—
227C23×C4—22no—
228C23×C4—22no—
229C23×C4—22no—
230C23×C4C2×D414yes—
231C23×C4C2412yes—
232C23×C4C2412yes—
233C23×C4C2412yes—
234C23×C4C2412yes—
235C23×C4C2412yes—
236C23×C4C2412yes—
237C23×C4C2412yes—
238C23×C4C2411yescharacteristic
239C23×C4C2411yescharacteristic
240C23×C4C2411yescharacteristic
241C23×C4C2411yescharacteristic
242C23×C4C2412yes—
243C23×C4C2412yes—
244C23×C4C2412yes—
245C23×C4C2412yes—
246C23×C4C2412yes—
247C23×C4C2411yescharacteristic
248C23×C4C2411yescharacteristic
249C4⋊Q8—416no—
250C22⋊Q8—416no—
251C22⋊Q8—416no—
252C4×Q8—416no—
253#117—416no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—24no—
259C2×C4⋊C4—24no—
260C2×C4⋊C4—24no—
261C2×C4⋊C4—24no—
262C2×C4⋊C4—22no—
263C2×C4⋊C4—22no—
264C2×C4⋊C4—22no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4—24no—
267C2×C4⋊C4—24no—
268C2×C4⋊C4—24no—
269C2×C4⋊C4—24no—
270C2×C4⋊C4—24no—
271C2×C4⋊C4—22no—
272C2×C4⋊C4—24no—
273C2×C4⋊C4—24no—
274C2×C4⋊C4—22no—
275C2×C4⋊C4—22no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—22no—
279C2×C4⋊C4—22no—
280C2×C4⋊C4—22no—
281C2×C4⋊C4—22no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—24no—
284C2×C4⋊C4—24no—
285C2×C4⋊C4—28no—
286C2×C4⋊C4—44no—
287C2×C4⋊C4—42no—
288C2×C4⋊C4—42no—
289C2×C4⋊C4—42no—
290C2×C4⋊C4—42no—
291C2×C4⋊C4—42no—
292C2×C4⋊C4—42no—
293C2×C4⋊C4—42no—
294C2×C4⋊C4—44no—
295C2×C4⋊C4—44no—
296C2×C4⋊C4—44no—
297C2×C4⋊C4—44no—
298C2×C4⋊C4#4114yes—
299C2×C22⋊C4—24no—
300C2×C22⋊C4—24no—
301C2×C22⋊C4—24no—
302C2×C22⋊C4—24no—
303C2×C22⋊C4—24no—
304C2×C22⋊C4—24no—
305C2×C22⋊C4—24no—
306C2×C22⋊C4—24no—
307C2×C22⋊C4—24no—
308C2×C22⋊C4—24no—
309C2×C22⋊C4—42no—
310C2×C22⋊C4—42no—
311C2×C22⋊C4—44no—
312C2×C22⋊C4—42no—
313C2×C22⋊C4—42no—
314C2×C22⋊C4—42no—
315C2×C22⋊C4—42no—
316C2×C22⋊C4—44no—
317C2×C22⋊C4C2×D414yes—
318C2×C22⋊C4C2×D414yes—
319C2×C42—22no—
320C2×C42—22no—
321C2×C42—22no—
322C2×C42—22no—
323C2×C42—22no—
324C2×C42—22no—
325C2×C42—22no—
326C2×C42—22no—
327C2×C42—24no—
328C2×C42—24no—
329C2×C42—22no—
330C2×C42—24no—
331C2×C42—22no—
332C2×C42—22no—
333C2×C42—22no—
334C2×C42—22no—
335C2×C42—22no—
336C2×C42—24no—
337C2×C42—24no—
338C2×C42—28no—
339C2×C42—42no—
340C2×C42—42no—
341C2×C42—42no—
342C2×C42—42no—
343C2×C42—42no—
344C2×C42C2×D414yes—
345C2×C42C2×D414yes—
346C2×C42#4114yes—
347C2×C42#4114yes—
348#95—24no—
349#95—24no—
350#95—24no—
351#95—24no—
352#95—24no—
353#95—24no—
354#95—24no—
355#95—24no—
356#95—24no—
357#95—24no—
358#95—24no—
359#95—28no—
360#95—28no—
361#95—28no—
362#95—28no—
363#95—28no—
364#95—24no—
365#95—24no—
366#95—28no—
367#95—44no—
368#95—44no—
369#95—44no—
370#95—44no—
371#95—42no—
372#95—42no—
373#95—44no—
374#95—42no—
375#95—42no—
376#95—44no—
377#95—44no—
378#95—44no—
379#95—44no—
380#95C2×D414yes—
381#95C2×D414yes—
382C24—21no—
383C24—21no—
384C24—21no—
385C24—21no—
386C24—21no—
387C24—21no—
388C24—22no—
389C24—22no—
390C24—22no—
391C24—22no—
392C24C22×D412yes—
393C24C2511yescharacteristic
394C2×Q8—28no—
395C2×Q8—28no—
396C2×Q8—28no—
397C2×Q8—44no—
398C2×Q8—44no—
399C2×Q8—44no—
400C2×Q8—44no—
401C2×Q8—44no—
402C2×Q8—44no—
403C2×Q8—44no—
404C2×Q8—44no—
405C2×Q8—44no—
406C2×Q8—44no—
407C2×Q8—44no—
408C22×C4—21no—
409C22×C4—21no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—21no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—21no—
419C22×C4—21no—
420C22×C4—21no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—21no—
430C22×C4—21no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—21no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—21no—
437C22×C4—21no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—24no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—24no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—28no—
458C22×C4—28no—
459C22×C4—28no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—22no—
473C22×C4—22no—
474C22×C4—22no—
475C22×C4—22no—
476C22×C4—22no—
477C22×C4—22no—
478C22×C4—22no—
479C22×C4—22no—
480C22×C4—22no—
481C22×C4—22no—
482C22×C4—22no—
483C22×C4—22no—
484C22×C4—22no—
485C22×C4—24no—
486C22×C4—22no—
487C22×C4—22no—
488C22×C4—24no—
489C22×C4—24no—
490C22×C4—24no—
491C22×C4—22no—
492C22×C4—22no—
493C22×C4—24no—
494C22×C4—24no—
495C22×C4—24no—
496C22×C4—24no—
497C22×C4—24no—
498C22×C4—24no—
499C22×C4—44no—
500C22×C4—44no—
501C22×C4—44no—
502C22×C4—44no—
503C22×C4C22×D412yes—
504C22×C4C22×D412yes—
505C22×C4C22×D412yes—
506C22×C4C22×D412yes—
507C22×C4C22×D412yes—
508C22×C4C22×D412yes—
509C22×C4C2×#4114yes—
510C22×C4C2×#4114yes—
511C22×C4C2×#4114yes—
512C22×C4C2×#4114yes—
513C22×C4C2×#4112yes—
514C22×C4C2×#4112yes—
515C22×C4C2×#4112yes—
516C22×C4C2×#4112yes—
517C22×C4H(5,𝔽2)12yes—
518C22×C4H(5,𝔽2)12yes—
519C22×C42−1+414yes—
520C22×C42−1+414yes—
521C4⋊C4—28no—
522C4⋊C4—48no—
523C4⋊C4—48no—
524C4⋊C4—48no—
525C4⋊C4—48no—
526C4⋊C4—48no—
527C4⋊C4—48no—
528C4⋊C4—48no—
529C4⋊C4—44no—
530C4⋊C4—44no—
531C4⋊C4—44no—
532C4⋊C4—44no—
533C4⋊C4—44no—
534C4⋊C4—84no—
535C4⋊C4—84no—
536C4⋊C4—88no—
537C22⋊C4—28no—
538C22⋊C4—44no—
539C22⋊C4—48no—
540C22⋊C4—48no—
541C22⋊C4—48no—
542C22⋊C4—48no—
543C22⋊C4—48no—
544C22⋊C4—84no—
545C42—28no—
546C42—28no—
547C42—28no—
548C42—28no—
549C42—48no—
550C42—48no—
551C42—48no—
552C42—48no—
553C42—44no—
554C42—44no—
555C23—22no—
556C23—22no—
557C23—22no—
558C23—22no—
559C23—22no—
560C23—22no—
561C23—22no—
562C23—41no—
563C23—41no—
564C23—42no—
565C23—42no—
566C23—42no—
567C23—41no—
568C23—41no—
569C23—42no—
570C23—42no—
571C23—42no—
572C23—42no—
573C23—42no—
574C23—42no—
575C23—42no—
576C23—42no—
577C23—42no—
578C23C23×D411yescharacteristic
579C23C22×#4111yescharacteristic
580C23C22×#4111yescharacteristic
581C23C22×#4111yescharacteristic
582C23C22×#4111yescharacteristic
583C23C2×H(5,𝔽2)11yescharacteristic
584C23C2×H(5,𝔽2)11yescharacteristic
585C23C2×2−1+412yes—
586C23C2×2−1+412yes—
587C23#58512yes—
588C23#58512yes—
589Q8—416no—
590Q8—88no—
591C2×C4—22no—
592C2×C4—22no—
593C2×C4—24no—
594C2×C4—24no—
595C2×C4—24no—
596C2×C4—22no—
597C2×C4—22no—
598C2×C4—22no—
599C2×C4—22no—
600C2×C4—24no—
601C2×C4—24no—
602C2×C4—24no—
603C2×C4—24no—
604C2×C4—22no—
605C2×C4—22no—
606C2×C4—22no—
607C2×C4—24no—
608C2×C4—24no—
609C2×C4—22no—
610C2×C4—22no—
611C2×C4—24no—
612C2×C4—24no—
613C2×C4—24no—
614C2×C4—24no—
615C2×C4—22no—
616C2×C4—22no—
617C2×C4—22no—
618C2×C4—24no—
619C2×C4—28no—
620C2×C4—24no—
621C2×C4—41no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—41no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—41no—
630C2×C4—41no—
631C2×C4—42no—
632C2×C4—42no—
633C2×C4—42no—
634C2×C4—42no—
635C2×C4—42no—
636C2×C4—42no—
637C2×C4—42no—
638C2×C4—42no—
639C2×C4—42no—
640C2×C4—42no—
641C2×C4—42no—
642C2×C4—42no—
643C2×C4—44no—
644C2×C4—42no—
645C2×C4—42no—
646C2×C4—42no—
647C2×C4—42no—
648C2×C4—42no—
649C2×C4—42no—
650C2×C4—416no—
651C2×C4—42no—
652C2×C4—42no—
653C2×C4—42no—
654C2×C4—42no—
655C2×C4—42no—
656C2×C4—42no—
657C2×C4—42no—
658C2×C4—42no—
659C2×C4—42no—
660C2×C4—42no—
661C2×C4—42no—
662C2×C4—44no—
663C2×C4—44no—
664C2×C4—42no—
665C2×C4—42no—
666C2×C4—44no—
667C2×C4—42no—
668C2×C4—42no—
669C2×C4—42no—
670C2×C4—42no—
671C2×C4—42no—
672C2×C4—42no—
673C2×C4—42no—
674C2×C4—44no—
675C2×C4—42no—
676C2×C4—42no—
677C2×C4—42no—
678C2×C4—42no—
679C2×C4—42no—
680C2×C4—44no—
681C2×C4—44no—
682C2×C4—44no—
683C2×C4—42no—
684C2×C4—42no—
685C2×C4—44no—
686C2×C4—44no—
687C2×C4—44no—
688C2×C4—42no—
689C2×C4—42no—
690C2×C4#52914yes—
691C2×C4#53214yes—
692C2×C4#53614yes—
693C2×C4#53814yes—
694C2×C4#54614yes—
695C2×C4#54614yes—
696C2×C4#55314yes—
697C2×C4#55314yes—
698C22—42no—
699C22—42no—
700C22—42no—
701C22—42no—
702C22—42no—
703C22—42no—
704C22—42no—
705C22—82no—
706C22—82no—
707C22—82no—
708C22—82no—
709C22C2×#52911yescharacteristic
710C22C2×#53211yescharacteristic
711C22C2×#53611yescharacteristic
712C22#344911yescharacteristic
713C22C2×#53811yescharacteristic
714C22#346112yes—
715C22#346112yes—
716C22#346112yes—
717C22C2×#54611yescharacteristic
718C22C2×#54611yescharacteristic
719C22C2×#55311yescharacteristic
720C22C2×#55311yescharacteristic
721C22#349512yes—
722C22#349911yescharacteristic
723C22#350011yescharacteristic
724C22#351612yes—
725C22#351812yes—
726C22#352112yes—
727C22#352312yes—
728C22#353812yes—
729C22#353912yes—
730C22#354712yes—
731C22#354712yes—
732C4—24no—
733C4—42no—
734C4—44no—
735C4—42no—
736C4—42no—
737C4—44no—
738C4—44no—
739C4—44no—
740C4—42no—
741C4—44no—
742C4—82no—
743C4—82no—
744C4—82no—
745C4—82no—
746C4—82no—
747C4—82no—
748C4—82no—
749C4#255918yes—
750C2—82no—
751C2#3463111yescharacteristic
752C2C2×#255911yescharacteristic
753C2#3521712yes—
754C2#3531811yescharacteristic
755C2#3751912yes—
756C2#3808111yescharacteristic
757C2#3809211yescharacteristic
758C2#3821512yes—
759C2#3894611yescharacteristic
760C2#3895811yescharacteristic
761C2#3925712yes—
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
3. #16872C2semidirect
5. #15239C2semidirect
7. #15188C2semidirect
10. #14940C2semidirect
11. #14703C2semidirect
12. #14276C2semidirect
13. #14274C2semidirect
16. #14003C2semidirect
17. #13976C2semidirect
18. C4×#391C2semidirect

Maximal subgroups

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

subgroupindex
C2×#24952
C2×#24292
#168722
#168592
#152392
#152362
#151882
#151822
#149832
#149402
#147032
#142762
#142742
#142582
#140032
#139762
C4×#3912
#137612