Finite Group Databasebeta

← #670349

Subgroups — #670349

Order 512.

2268 conjugacy classes of subgroups.

The subgroup lattice has 742 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2423C211yesmaximal, characteristic
2C2×#2347C211yesmaximal, characteristic
3#15918C214yesmaximal, complemented
4#15758C214yesmaximal
5#15237C211yesmaximal, characteristic, complemented
6#15236C211yesmaximal, characteristic
7#15233C211yesmaximal, characteristic
8#15153C211yesmaximal, characteristic, complemented
9#14951C214yesmaximal
10#14707C211yesmaximal, characteristic, complemented
11#14221C214yesmaximal, complemented
12#14138C211yesmaximal, characteristic
13#14048C211yesmaximal, characteristic, complemented
14#14000C211yesmaximal, characteristic, complemented
15#13978C211yesmaximal, characteristic, complemented
16#13911C211yesmaximal, characteristic, complemented
17#13879C211yesmaximal, characteristic
18C4×#391C211yesmaximal, characteristic, complemented
19C4×#388C211yesmaximal, characteristic
20C22×C22⋊Q8C2211yescharacteristic
21C22×C4×Q8C2211yescharacteristic
22C22×#117C2211yescharacteristic
23#2423—28no—
24C2×#400C2214yes—
25C2×#399C2212yes—
26C2×#398C2212yes—
27#2347—28no—
28C2×#391C2211yescharacteristic
29C2×#391C2211yescharacteristic
30C2×#391C2211yescharacteristic
31C2×#391C2211yescharacteristic
32C2×#391C2211yescharacteristic
33C2×#389C2211yescharacteristic
34C2×#389C2211yescharacteristic
35C2×#389C2211yescharacteristic
36C2×#388C2212yes—
37C2×#388C2212yes—
38C2×#388C2212yes—
39C2×#388C2212yes—
40C2×#387C2212yes—
41C2×#384C2211yescharacteristic
42C2×#384C2211yescharacteristic
43C2×#383C2212yes—
44C2×#382C2211yescharacteristic
45C2×#382C2211yescharacteristic
46C2×#382C2211yescharacteristic
47C2×#380C2211yescharacteristic
48C2×#380C2211yescharacteristic
49C2×C4×C4⋊C4C2211yescharacteristic
50C2×C4×C4⋊C4C2212yes—
51C2×C4×C4⋊C4C2211yescharacteristic
52C2×C4×C4⋊C4C2211yescharacteristic
53C2×C4×C22⋊C4C2211yescharacteristic
54C2×C4×C22⋊C4C2211yescharacteristic
55C2×C4×C22⋊C4C2211yescharacteristic
56C2×C43C2211yescharacteristic
57#1445C2214yes—
58#1443C2214yes—
59#1443C2214yes—
60#1442C2214yes—
61#1442C2214yes—
62#1442C2214yes—
63#1442C2214yes—
64#1441C2214yes—
65#1441C2214yes—
66#1440C2214yes—
67#1440C2214yes—
68#1440C2214yes—
69#1439C2214yes—
70#1437C2212yes—
71#1436C2214yes—
72#1436C2214yes—
73#1436C2214yes—
74#1436C2214yes—
75#1436C2214yes—
76#1436C2214yes—
77#1436C2214yes—
78#1436C2212yes—
79#1436C2214yes—
80C4⋊#95C2211yescharacteristic
81C4⋊#95C2211yescharacteristic
82C4⋊#95C2211yescharacteristic
83C4⋊#95C2211yescharacteristic
84C4⋊#95C2211yescharacteristic
85C4⋊#95C2211yescharacteristic
86C4⋊#95C2214yes—
87C4⋊#95C2211yescharacteristic
88C4⋊#95C2211yescharacteristic
89#1434C2212yes—
90C4×#95C2211yescharacteristic
91C4×#95C2214yes—
92C4×#95C2211yescharacteristic
93C23×Q8C2311yescharacteristic
94C24×C4C2311yescharacteristic
95C2×C22⋊Q8—24no—
96C2×C22⋊Q8—24no—
97C2×C22⋊Q8—42no—
98C2×C4×Q8—24no—
99C2×C4×Q8—24no—
100C2×C4×Q8—42no—
101C2×#117—24no—
102C2×#117—24no—
103C2×#117—24no—
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2312yes—
106C22×C4⋊C4C2311yescharacteristic
107C22×C4⋊C4C2311yescharacteristic
108C22×C4⋊C4C2311yescharacteristic
109C22×C4⋊C4C2311yescharacteristic
110C22×C4⋊C4C2311yescharacteristic
111C22×C4⋊C4C2312yes—
112C22×C4⋊C4C2312yes—
113C22×C4⋊C4C2312yes—
114C22×C4⋊C4C2312yes—
115C22×C4⋊C4C2311yescharacteristic
116C22×C4⋊C4C2311yescharacteristic
117C22×C4⋊C4C2312yes—
118C22×C4⋊C4C2312yes—
119C22×C4⋊C4C2314yes—
120C22×C22⋊C4C2312yes—
121C22×C22⋊C4C2312yes—
122C22×C22⋊C4C2312yes—
123C22×C22⋊C4C2311yescharacteristic
124C22×C22⋊C4C2311yescharacteristic
125C22×C22⋊C4C2312yes—
126C22×C22⋊C4C2312yes—
127C22×C22⋊C4C2312yes—
128C22×C42C2312yes—
129C22×C42C2311yescharacteristic
130C22×C42C2311yescharacteristic
131C22×C42C2311yescharacteristic
132C22×C42C2312yes—
133C22×C42C2311yescharacteristic
134C22×C42C2311yescharacteristic
135C22×C42C2311yescharacteristic
136C22×C42C2312yes—
137C22×C42C2312yes—
138C22×C42C2312yes—
139C22×C42C2312yes—
140#400—216no—
141#399—28no—
142#398—28no—
143#391—24no—
144#391—24no—
145#391—24no—
146#391—24no—
147#391—24no—
148#389—24no—
149#389—24no—
150#389—24no—
151#388—28no—
152#388—28no—
153#388—28no—
154#388—28no—
155#387—28no—
156#384—24no—
157#384—24no—
158#383—28no—
159#382—24no—
160#382—42no—
161#382—42no—
162#380—24no—
163#380—24no—
164C4×C4⋊C4—24no—
165C4×C4⋊C4—24no—
166C4×C4⋊C4—24no—
167C4×C4⋊C4—28no—
168C4×C22⋊C4—24no—
169C4×C22⋊C4—42no—
170C4×C22⋊C4D418yes—
171C2×#95C2312yes—
172C2×#95C2312yes—
173C2×#95C2312yes—
174C2×#95C2314yes—
175C2×#95C2314yes—
176C2×#95C2314yes—
177C2×#95C2314yes—
178C2×#95C2312yes—
179C2×#95C2312yes—
180C2×#95C2314yes—
181C2×#95C2314yes—
182C2×#95C2312yes—
183C2×#95C2312yes—
184C2×#95C2312yes—
185C2×#95C2314yes—
186C2×#95C2314yes—
187C2×#95C2314yes—
188C2×#95C2312yes—
189C2×#95C2312yes—
190C2×#95C2312yes—
191C2×#95C2312yes—
192C2×#95C2312yes—
193C2×#95C2312yes—
194C2×#95C2314yes—
195C2×#95C2312yes—
196C2×#95C2314yes—
197C2×#95C2314yes—
198C2×#95C2314yes—
199C2×#95C2314yes—
200C2×#95C2314yes—
201C2×#95C2311yescharacteristic
202C2×#95C2311yescharacteristic
203C2×#95C2312yes—
204C43—42no—
205C25C2411yescharacteristic
206C22×Q8—22no—
207C22×Q8—22no—
208C22×Q8—22no—
209C22×Q8—22no—
210C22×Q8—22no—
211C22×Q8—22no—
212C22×Q8#4114yes—
213C23×C4—22no—
214C23×C4—22no—
215C23×C4—22no—
216C23×C4—22no—
217C23×C4—22no—
218C23×C4—22no—
219C23×C4C2×D414yes—
220C23×C4C2412yes—
221C23×C4C2412yes—
222C23×C4C2412yes—
223C23×C4C2412yes—
224C23×C4C2412yes—
225C23×C4C2412yes—
226C23×C4C2411yescharacteristic
227C23×C4C2411yescharacteristic
228C23×C4C2411yescharacteristic
229C23×C4C2412yes—
230C23×C4C2412yes—
231C23×C4C2412yes—
232C23×C4C2412yes—
233C23×C4C2412yes—
234C23×C4C2412yes—
235C23×C4C2411yescharacteristic
236C23×C4C2411yescharacteristic
237C23×C4C2411yescharacteristic
238C22⋊Q8—416no—
239C4×Q8—416no—
240#117—232no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4—24no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4—24no—
260C2×C4⋊C4—22no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—24no—
263C2×C4⋊C4—22no—
264C2×C4⋊C4—22no—
265C2×C4⋊C4—28no—
266C2×C4⋊C4—28no—
267C2×C4⋊C4—22no—
268C2×C4⋊C4—22no—
269C2×C4⋊C4—22no—
270C2×C4⋊C4—28no—
271C2×C4⋊C4—22no—
272C2×C4⋊C4—24no—
273C2×C4⋊C4—24no—
274C2×C4⋊C4—28no—
275C2×C4⋊C4—42no—
276C2×C4⋊C4—42no—
277C2×C4⋊C4—42no—
278C2×C4⋊C4—44no—
279C2×C4⋊C4—44no—
280C2×C4⋊C4—44no—
281C2×C4⋊C4—44no—
282C2×C4⋊C4#4114yes—
283C2×C4⋊C4#4114yes—
284C2×C22⋊C4—24no—
285C2×C22⋊C4—24no—
286C2×C22⋊C4—24no—
287C2×C22⋊C4—24no—
288C2×C22⋊C4—24no—
289C2×C22⋊C4—24no—
290C2×C22⋊C4—24no—
291C2×C22⋊C4—24no—
292C2×C22⋊C4—24no—
293C2×C22⋊C4—24no—
294C2×C22⋊C4—24no—
295C2×C22⋊C4—24no—
296C2×C22⋊C4—42no—
297C2×C22⋊C4—42no—
298C2×C22⋊C4—42no—
299C2×C22⋊C4—42no—
300C2×C22⋊C4—44no—
301C2×C22⋊C4—44no—
302C2×C22⋊C4C2×D414yes—
303C2×C22⋊C4C2×D414yes—
304C2×C42—24no—
305C2×C42—24no—
306C2×C42—24no—
307C2×C42—22no—
308C2×C42—22no—
309C2×C42—22no—
310C2×C42—22no—
311C2×C42—22no—
312C2×C42—22no—
313C2×C42—24no—
314C2×C42—24no—
315C2×C42—22no—
316C2×C42—22no—
317C2×C42—22no—
318C2×C42—24no—
319C2×C42—22no—
320C2×C42—22no—
321C2×C42—22no—
322C2×C42—22no—
323C2×C42—22no—
324C2×C42—24no—
325C2×C42—22no—
326C2×C42—24no—
327C2×C42—24no—
328C2×C42—42no—
329C2×C42—42no—
330C2×C42—42no—
331C2×C42—42no—
332C2×C42—42no—
333C2×C42—44no—
334C2×C42—44no—
335C2×C42C2×D414yes—
336C2×C42C2×D414yes—
337C2×C42#4114yes—
338#95—24no—
339#95—24no—
340#95—24no—
341#95—24no—
342#95—24no—
343#95—24no—
344#95—24no—
345#95—24no—
346#95—24no—
347#95—24no—
348#95—24no—
349#95—28no—
350#95—24no—
351#95—28no—
352#95—28no—
353#95—28no—
354#95—28no—
355#95—28no—
356#95—28no—
357#95—28no—
358#95—28no—
359#95—24no—
360#95—44no—
361#95—44no—
362#95—44no—
363#95—42no—
364#95—44no—
365#95—42no—
366#95—44no—
367#95—44no—
368#95C2×D414yes—
369#95C2×D414yes—
370#95#4118yes—
371C24—22no—
372C24—21no—
373C24—21no—
374C24—21no—
375C24—21no—
376C24—21no—
377C24—22no—
378C24—21no—
379C24—22no—
380C24—22no—
381C24C22×D412yes—
382C24C2511yescharacteristic
383C2×Q8—28no—
384C2×Q8—28no—
385C2×Q8—28no—
386C2×Q8—44no—
387C2×Q8—44no—
388C2×Q8—44no—
389C2×Q8—44no—
390C22×C4—21no—
391C22×C4—21no—
392C22×C4—21no—
393C22×C4—21no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—21no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—24no—
404C22×C4—21no—
405C22×C4—22no—
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—22no—
415C22×C4—21no—
416C22×C4—22no—
417C22×C4—21no—
418C22×C4—22no—
419C22×C4—21no—
420C22×C4—22no—
421C22×C4—21no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—21no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—24no—
430C22×C4—21no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—24no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—24no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—28no—
443C22×C4—28no—
444C22×C4—28no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—24no—
453C22×C4—24no—
454C22×C4—24no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—24no—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—22no—
471C22×C4—24no—
472C22×C4—24no—
473C22×C4—24no—
474C22×C4—24no—
475C22×C4—24no—
476C22×C4—24no—
477C22×C4—44no—
478C22×C4—44no—
479C22×C4—44no—
480C22×C4—44no—
481C22×C4C22×D412yes—
482C22×C4C22×D412yes—
483C22×C4C22×D412yes—
484C22×C4C22×D412yes—
485C22×C4C22×D412yes—
486C22×C4C22×D412yes—
487C22×C4C2×#4114yes—
488C22×C4C2×#4114yes—
489C22×C4C2×#4114yes—
490C22×C4C2×#4114yes—
491C22×C4C2×#4114yes—
492C22×C4C2×#4114yes—
493C22×C4C2×#4114yes—
494C22×C4C2×#4112yes—
495C22×C4C2×#4112yes—
496C22×C4C2×#4112yes—
497C22×C4C2×#4112yes—
498C22×C4H(5,𝔽2)12yes—
499C22×C42−1+414yes—
500C22×C42−1+412yes—
501C4⋊C4—28no—
502C4⋊C4—28no—
503C4⋊C4—48no—
504C4⋊C4—48no—
505C4⋊C4—48no—
506C4⋊C4—44no—
507C4⋊C4—48no—
508C4⋊C4—48no—
509C4⋊C4—48no—
510C4⋊C4—44no—
511C4⋊C4—44no—
512C4⋊C4—44no—
513C4⋊C4—44no—
514C4⋊C4—84no—
515C4⋊C4—84no—
516C4⋊C4—88no—
517C22⋊C4—216no—
518C22⋊C4—28no—
519C22⋊C4—28no—
520C22⋊C4—48no—
521C22⋊C4—48no—
522C22⋊C4—48no—
523C22⋊C4—84no—
524C22⋊C4—84no—
525C42—216no—
526C42—28no—
527C42—28no—
528C42—28no—
529C42—48no—
530C42—48no—
531C42—48no—
532C42—44no—
533C42—44no—
534C42—44no—
535C42—84no—
536C42—84no—
537C23—22no—
538C23—22no—
539C23—22no—
540C23—22no—
541C23—22no—
542C23—22no—
543C23—22no—
544C23—41no—
545C23—41no—
546C23—41no—
547C23—41no—
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
561C23C22×#4111yescharacteristic
562C23C22×#4111yescharacteristic
563C23C22×#4111yescharacteristic
564C23C22×#4111yescharacteristic
565C23C22×#4112yes—
566C23C2×H(5,𝔽2)11yescharacteristic
567C23C2×2−1+412yes—
568C23C2×2−1+411yescharacteristic
569C23#58512yes—
570C23#58512yes—
571Q8—416no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—24no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—24no—
578C2×C4—24no—
579C2×C4—24no—
580C2×C4—22no—
581C2×C4—24no—
582C2×C4—24no—
583C2×C4—24no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—24no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—24no—
592C2×C4—24no—
593C2×C4—24no—
594C2×C4—24no—
595C2×C4—24no—
596C2×C4—24no—
597C2×C4—24no—
598C2×C4—28no—
599C2×C4—22no—
600C2×C4—22no—
601C2×C4—24no—
602C2×C4—24no—
603C2×C4—24no—
604C2×C4—28no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—41no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—41no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—41no—
615C2×C4—41no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—44no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—44no—
630C2×C4—42no—
631C2×C4—42no—
632C2×C4—42no—
633C2×C4—42no—
634C2×C4—42no—
635C2×C4—416no—
636C2×C4—42no—
637C2×C4—42no—
638C2×C4—42no—
639C2×C4—42no—
640C2×C4—42no—
641C2×C4—42no—
642C2×C4—42no—
643C2×C4—42no—
644C2×C4—42no—
645C2×C4—42no—
646C2×C4—42no—
647C2×C4—44no—
648C2×C4—44no—
649C2×C4—44no—
650C2×C4—42no—
651C2×C4—42no—
652C2×C4—42no—
653C2×C4—42no—
654C2×C4—42no—
655C2×C4—42no—
656C2×C4—42no—
657C2×C4—42no—
658C2×C4—44no—
659C2×C4—44no—
660C2×C4—42no—
661C2×C4—44no—
662C2×C4—44no—
663C2×C4—44no—
664C2×C4—44no—
665C2×C4—42no—
666C2×C4—44no—
667C2×C4—42no—
668C2×C4#52514yes—
669C2×C4#52914yes—
670C2×C4#53214yes—
671C2×C4#53614yes—
672C2×C4#54214yes—
673C2×C4#54614yes—
674C2×C4#54714yes—
675C2×C4#55214yes—
676C2×C4#55314yes—
677C22—42no—
678C22—42no—
679C22—42no—
680C22—42no—
681C22—42no—
682C22—42no—
683C22—42no—
684C22—82no—
685C22—82no—
686C22—82no—
687C22—82no—
688C22C2×#52511yescharacteristic
689C22C2×#52911yescharacteristic
690C22C2×#53211yescharacteristic
691C22C2×#53611yescharacteristic
692C22C2×#54211yescharacteristic
693C22C2×#54611yescharacteristic
694C22C2×#54711yescharacteristic
695C22C2×#55211yescharacteristic
696C22C2×#55311yescharacteristic
697C22#347512yes—
698C22#348012yes—
699C22#348212yes—
700C22#348212yes—
701C22#350011yescharacteristic
702C22#350011yescharacteristic
703C22#350612yes—
704C22#350712yes—
705C22#351512yes—
706C22#351612yes—
707C22#352112yes—
708C22#352412yes—
709C22#354112yes—
710C22#355512yes—
711C4—24no—
712C4—24no—
713C4—44no—
714C4—44no—
715C4—42no—
716C4—42no—
717C4—44no—
718C4—42no—
719C4—44no—
720C4—44no—
721C4—44no—
722C4—82no—
723C4—82no—
724C4—82no—
725C4—82no—
726C4—82no—
727C4—82no—
728C4#247018yes—
729C2—82no—
730C2C2×#247011yescharacteristic
731C2#3497911yescharacteristic
732C2#3562311yescharacteristic
733C2#3618411yescharacteristic
734C2#3641911yescharacteristic
735C2#3648412yes—
736C2#3808511yescharacteristic
737C2#3949011yescharacteristic
738C2#4021612yes—
739C2#5950812yes—
740C2#5985712yes—
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. #15918C2semidirect
5. #15237C2semidirect
8. #15153C2semidirect
10. #14707C2semidirect
11. #14221C2semidirect
13. #14048C2semidirect
14. #14000C2semidirect
15. #13978C2semidirect
16. #13911C2semidirect
18. C4×#391C2semidirect

Maximal subgroups

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

subgroupindex
C2×#24232
C2×#23472
#159182
#157582
#152372
#152362
#152332
#151532
#149512
#147072
#142212
#141382
#140482
#140002
#139782
#139112
#138792
C4×#3912
C4×#3882