Finite Group Databasebeta

← #4464195

Subgroups — #4464195

Order 512.

1704 conjugacy classes of subgroups.

The subgroup lattice has 769 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#17298C211yesmaximal, characteristic, complemented
2#17270C212yesmaximal
3#17221C211yesmaximal, characteristic, complemented
4#17219C211yesmaximal, characteristic, complemented
5#17210C212yesmaximal, complemented
6#17206C212yesmaximal, complemented
7#17161C212yesmaximal
8#17119C212yesmaximal
9#17112C211yesmaximal, characteristic, complemented
10#17104C212yesmaximal
11#15165C211yesmaximal, characteristic
12#15161C211yesmaximal, characteristic
13#15056C212yesmaximal, complemented
14#15008C212yesmaximal, complemented
15#14893C212yesmaximal
16#14862C212yesmaximal, complemented
17#14816C212yesmaximal, complemented
18#14351C212yesmaximal
19#13844C211yesmaximal, characteristic
20C22×#117C2211yescharacteristic
21C2×#400C2212yes—
22C2×#400C2212yes—
23C2×#399C2212yes—
24C2×#399C2211yescharacteristic
25C2×#398C2212yes—
26C2×#398C2212yes—
27C2×#398C2212yes—
28C2×#396C2212yes—
29C2×#395C2212yes—
30C2×#395C2212yes—
31C2×#393C2211yescharacteristic
32C2×#391C2211yescharacteristic
33C2×#389C2212yes—
34C2×#389C2211yescharacteristic
35C2×C4×C4⋊C4C2212yes—
36C2×C4×C22⋊C4C2211yescharacteristic
37C2×C4×C22⋊C4C2211yescharacteristic
38#1446C2211yescharacteristic
39#1445C2212yes—
40#1445C2212yes—
41#1445C2211yescharacteristic
42#1444C2212yes—
43#1444C2212yes—
44#1444C2212yes—
45#1444C2212yes—
46#1444C2211yescharacteristic
47#1444C2211yescharacteristic
48#1444C2212yes—
49#1444C2212yes—
50#1444C2212yes—
51#1444C2211yescharacteristic
52#1444C2212yes—
53#1444C2212yes—
54#1443C2212yes—
55#1443C2212yes—
56#1443C2212yes—
57#1443C2212yes—
58#1443C2212yes—
59#1443C2212yes—
60#1442C2212yes—
61#1442C2212yes—
62#1442C2212yes—
63#1442C2212yes—
64#1442C2211yescharacteristic
65#1442C2211yescharacteristic
66#1442C2212yes—
67#1442C2211yescharacteristic
68#1441C2212yes—
69#1441C2212yes—
70#1441C2212yes—
71#1441C2212yes—
72#1441C2212yes—
73#1441C2212yes—
74#1441C2212yes—
75#1441C2212yes—
76#1441C2212yes—
77#1441C2212yes—
78#1441C2212yes—
79#1441C2212yes—
80#1441C2212yes—
81#1440C2212yes—
82#1440C2212yes—
83#1439C2211yescharacteristic
84#1439C2212yes—
85#1439C2212yes—
86#1439C2212yes—
87#1439C2212yes—
88#1439C2212yes—
89#1439C2212yes—
90#1439C2212yes—
91#1439C2212yes—
92#1439C2211yescharacteristic
93#1436C2212yes—
94#1436C2212yes—
95#1436C2212yes—
96#1436C2212yes—
97C4⋊#95C2211yescharacteristic
98C4⋊#95C2212yes—
99C4⋊#95C2211yescharacteristic
100#1434C2212yes—
101#1434C2212yes—
102#1434C2212yes—
103#1434C2212yes—
104C22⋊#95C2212yes—
105C22⋊#95C2212yes—
106C4×#95C2212yes—
107C23×Q8C2311yescharacteristic
108C24×C4C2311yescharacteristic
109C2×#117—42no—
110C2×#117—44no—
111C22×C4⋊C4C2311yescharacteristic
112C22×C4⋊C4C2312yes—
113C22×C4⋊C4C2312yes—
114C22×C4⋊C4C2312yes—
115C22×C4⋊C4C2312yes—
116C22×C4⋊C4C2312yes—
117C22×C4⋊C4C2311yescharacteristic
118C22×C4⋊C4C2312yes—
119C22×C4⋊C4C2312yes—
120C22×C4⋊C4C2312yes—
121C22×C4⋊C4C2312yes—
122C22×C4⋊C4C2312yes—
123C22×C4⋊C4C2312yes—
124C22×C4⋊C4C2312yes—
125C22×C4⋊C4C2312yes—
126C22×C4⋊C4C2312yes—
127C22×C4⋊C4C2311yescharacteristic
128C22×C4⋊C4C2311yescharacteristic
129C22×C4⋊C4C2312yes—
130C22×C4⋊C4C2311yescharacteristic
131C22×C22⋊C4C2311yescharacteristic
132C22×C22⋊C4C2311yescharacteristic
133C22×C22⋊C4C2312yes—
134C22×C22⋊C4C2312yes—
135C22×C22⋊C4C2312yes—
136C22×C22⋊C4C2312yes—
137C22×C22⋊C4C2312yes—
138C22×C22⋊C4C2312yes—
139C22×C42C2312yes—
140C22×C42C2311yescharacteristic
141C22×C42C2311yescharacteristic
142C22×C42C2312yes—
143#400—44no—
144#400—44no—
145#399—44no—
146#399—42no—
147#398—44no—
148#398—44no—
149#398—44no—
150#396—44no—
151#395—44no—
152#395—44no—
153#393—42no—
154#391—42no—
155#389—42no—
156#389—44no—
157C4×C4⋊C4—44no—
158C4×C22⋊C4—42no—
159C4×C22⋊C4Q818yescomplemented
160C2×#95C2312yes—
161C2×#95C2312yes—
162C2×#95C2312yes—
163C2×#95C2312yes—
164C2×#95C2312yes—
165C2×#95C2312yes—
166C2×#95C2312yes—
167C2×#95C2312yes—
168C2×#95C2312yes—
169C2×#95C2312yes—
170C2×#95C2312yes—
171C2×#95C2312yes—
172C2×#95C2311yescharacteristic
173C2×#95C2311yescharacteristic
174C2×#95C2311yescharacteristic
175C2×#95C2312yes—
176C2×#95C2312yes—
177C2×#95C2312yes—
178C2×#95C2312yes—
179C2×#95C2312yes—
180C2×#95C2312yes—
181C2×#95C2312yes—
182C2×#95C2312yes—
183C2×#95C2312yes—
184C2×#95C2311yescharacteristic
185C2×#95C2312yes—
186C2×#95C2312yes—
187C2×#95C2312yes—
188C2×#95C2312yes—
189C2×#95C2312yes—
190C2×#95C2312yes—
191C2×#95C2312yes—
192C2×#95C2312yes—
193C2×#95C2312yes—
194C2×#95C2312yes—
195C2×#95C2312yes—
196C2×#95C2312yes—
197C2×#95C2312yes—
198C2×#95C2312yes—
199C2×#95C2312yes—
200C2×#95C2312yes—
201C2×#95C2312yes—
202C2×#95C2312yes—
203C2×#95C2312yes—
204C2×#95C2312yes—
205C2×#95C2312yes—
206C2×#95C2312yes—
207C2×#95C2312yes—
208C2×#95C2312yes—
209C2×#95C2311yescharacteristic
210C2×#95C2311yescharacteristic
211C2×#95C2311yescharacteristic
212C2×#95C2311yescharacteristic
213C25C2411yescharacteristic
214C22×Q8—22no—
215C22×Q8—42no—
216C22×Q8—41no—
217C22×Q8—41no—
218C22×Q8—42no—
219C23×C4—22no—
220C23×C4—22no—
221C23×C4—42no—
222C23×C4—42no—
223C23×C4C2×Q814yes—
224C23×C4C2412yes—
225C23×C4C2412yes—
226C23×C4C2412yes—
227C23×C4C2412yes—
228C23×C4C2412yes—
229C23×C4C2412yes—
230C23×C4C2412yes—
231C23×C4C2412yes—
232C23×C4C2412yes—
233C23×C4C2412yes—
234C23×C4C2412yes—
235C23×C4C2411yescharacteristic
236C23×C4C2411yescharacteristic
237C23×C4C2411yescharacteristic
238C23×C4C2412yes—
239C23×C4C2411yescharacteristic
240C23×C4C2411yescharacteristic
241C23×C4C2411yescharacteristic
242#117—416no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—42no—
254C2×C4⋊C4—42no—
255C2×C4⋊C4—42no—
256C2×C4⋊C4—42no—
257C2×C4⋊C4—42no—
258C2×C4⋊C4—41no—
259C2×C4⋊C4—42no—
260C2×C4⋊C4—41no—
261C2×C4⋊C4—42no—
262C2×C4⋊C4—41no—
263C2×C4⋊C4—42no—
264C2×C4⋊C4—42no—
265C2×C4⋊C4—42no—
266C2×C4⋊C4—42no—
267C2×C4⋊C4—42no—
268C2×C4⋊C4—42no—
269C2×C4⋊C4—42no—
270C2×C4⋊C4—42no—
271C2×C4⋊C4—42no—
272C2×C4⋊C4—42no—
273C2×C4⋊C4—42no—
274C2×C4⋊C4—42no—
275C2×C4⋊C4—42no—
276C2×C4⋊C4—42no—
277C2×C4⋊C4—42no—
278C2×C4⋊C4—41no—
279C2×C4⋊C4—42no—
280C2×C4⋊C4—42no—
281C2×C4⋊C4—41no—
282C2×C4⋊C4—42no—
283C2×C4⋊C4—42no—
284C2×C4⋊C4—42no—
285C2×C4⋊C4—42no—
286C2×C4⋊C4—42no—
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—41no—
295C2×C4⋊C4—42no—
296C2×C4⋊C4—42no—
297C2×C4⋊C4—42no—
298C2×C4⋊C4—42no—
299C2×C22⋊C4—24no—
300C2×C22⋊C4—22no—
301C2×C22⋊C4—22no—
302C2×C22⋊C4—22no—
303C2×C22⋊C4—22no—
304C2×C22⋊C4—42no—
305C2×C22⋊C4—42no—
306C2×C22⋊C4—42no—
307C2×C22⋊C4—42no—
308C2×C22⋊C4—41no—
309C2×C22⋊C4—41no—
310C2×C22⋊C4—42no—
311C2×C22⋊C4—42no—
312C2×C22⋊C4—42no—
313C2×C22⋊C4—42no—
314C2×C22⋊C4—42no—
315C2×C22⋊C4—42no—
316C2×C22⋊C4—42no—
317C2×C22⋊C4—42no—
318C2×C22⋊C4—42no—
319C2×C22⋊C4—42no—
320C2×C22⋊C4—42no—
321C2×C22⋊C4—42no—
322C2×C22⋊C4C2×Q818yes—
323C2×C42—24no—
324C2×C42—24no—
325C2×C42—22no—
326C2×C42—22no—
327C2×C42—42no—
328C2×C42—42no—
329C2×C42—42no—
330C2×C42—42no—
331C2×C42—42no—
332C2×C42C2×Q818yes—
333#95—24no—
334#95—24no—
335#95—24no—
336#95—24no—
337#95—24no—
338#95—24no—
339#95—24no—
340#95—24no—
341#95—24no—
342#95—24no—
343#95—22no—
344#95—22no—
345#95—24no—
346#95—24no—
347#95—42no—
348#95—42no—
349#95—42no—
350#95—42no—
351#95—42no—
352#95—42no—
353#95—42no—
354#95—42no—
355#95—42no—
356#95—42no—
357#95—42no—
358#95—42no—
359#95—42no—
360#95—42no—
361#95—42no—
362#95—42no—
363#95—41no—
364#95—41no—
365#95—42no—
366#95—42no—
367#95—41no—
368#95—42no—
369#95—42no—
370#95—42no—
371#95—42no—
372#95—42no—
373#95—42no—
374#95—42no—
375#95—42no—
376#95—42no—
377#95—41no—
378#95—42no—
379#95—42no—
380#95—42no—
381#95—42no—
382#95—42no—
383#95—42no—
384#95C2×Q814yes—
385#95C2×Q814yes—
386C24—21no—
387C24—21no—
388C24—22no—
389C24—21no—
390C24—21no—
391C24—21no—
392C24—21no—
393C24—22no—
394C24—22no—
395C24—22no—
396C24C22×Q812yes—
397C24C2511yescharacteristic
398C2×Q8—44no—
399C2×Q8—82no—
400C2×Q8—84no—
401C2×Q8—82no—
402C2×Q8—84no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—21no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—21no—
413C22×C4—22no—
414C22×C4—21no—
415C22×C4—21no—
416C22×C4—21no—
417C22×C4—21no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—21no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—21no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—28no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
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—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—22no—
486C22×C4—22no—
487C22×C4—22no—
488C22×C4—22no—
489C22×C4—22no—
490C22×C4—22no—
491C22×C4—22no—
492C22×C4—22no—
493C22×C4—22no—
494C22×C4—22no—
495C22×C4—22no—
496C22×C4—22no—
497C22×C4—22no—
498C22×C4—22no—
499C22×C4—22no—
500C22×C4—22no—
501C22×C4—22no—
502C22×C4—22no—
503C22×C4—22no—
504C22×C4—22no—
505C22×C4—22no—
506C22×C4—22no—
507C22×C4—48no—
508C22×C4—44no—
509C22×C4—44no—
510C22×C4—48no—
511C22×C4C22×Q812yes—
512C22×C4C22×Q814yes—
513C22×C4C22×Q814yes—
514C22×C4C22×Q812yes—
515C22×C4H(5,𝔽2)14yes—
516C22×C4H(5,𝔽2)12yes—
517C22×C4H(5,𝔽2)12yes—
518C22×C4H(5,𝔽2)12yes—
519C22×C42−1+412yes—
520C22×C42−1+412yes—
521C22×C42−1+412yes—
522C4⋊C4—48no—
523C4⋊C4—44no—
524C4⋊C4—48no—
525C4⋊C4—44no—
526C4⋊C4—48no—
527C4⋊C4—44no—
528C4⋊C4—84no—
529C4⋊C4—84no—
530C4⋊C4—84no—
531C4⋊C4—82no—
532C4⋊C4—84no—
533C4⋊C4—84no—
534C4⋊C4—82no—
535C4⋊C4—84no—
536C4⋊C4—84no—
537C4⋊C4—84no—
538C4⋊C4—84no—
539C4⋊C4—84no—
540C4⋊C4—84no—
541C4⋊C4—84no—
542C22⋊C4—48no—
543C22⋊C4—44no—
544C22⋊C4—44no—
545C22⋊C4—48no—
546C22⋊C4—84no—
547C22⋊C4—84no—
548C22⋊C4—84no—
549C22⋊C4—84no—
550C42—48no—
551C42—48no—
552C42—82no—
553C42—82no—
554C23—22no—
555C23—24no—
556C23—24no—
557C23—22no—
558C23—22no—
559C23—41no—
560C23—41no—
561C23—42no—
562C23—41no—
563C23—41no—
564C23—42no—
565C23—41no—
566C23—41no—
567C23—42no—
568C23—42no—
569C23—42no—
570C23—41no—
571C23—41no—
572C23—42no—
573C23—42no—
574C23—42no—
575C23—42no—
576C23—42no—
577C23C23×Q811yescharacteristic
578C23C2×H(5,𝔽2)11yescharacteristic
579C23C2×H(5,𝔽2)12yes—
580C23C2×H(5,𝔽2)11yescharacteristic
581C23C2×H(5,𝔽2)11yescharacteristic
582C23C2×2−1+411yescharacteristic
583C23C2×2−1+411yescharacteristic
584C23C2×2−1+411yescharacteristic
585C23#58512yes—
586C23#58512yes—
587C23#58512yes—
588Q8—88no—
589C2×C4—24no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—24no—
593C2×C4—22no—
594C2×C4—24no—
595C2×C4—24no—
596C2×C4—24no—
597C2×C4—24no—
598C2×C4—24no—
599C2×C4—24no—
600C2×C4—24no—
601C2×C4—24no—
602C2×C4—24no—
603C2×C4—24no—
604C2×C4—22no—
605C2×C4—22no—
606C2×C4—22no—
607C2×C4—22no—
608C2×C4—24no—
609C2×C4—24no—
610C2×C4—24no—
611C2×C4—24no—
612C2×C4—416no—
613C2×C4—41no—
614C2×C4—42no—
615C2×C4—41no—
616C2×C4—41no—
617C2×C4—41no—
618C2×C4—41no—
619C2×C4—41no—
620C2×C4—41no—
621C2×C4—41no—
622C2×C4—41no—
623C2×C4—41no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—42no—
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—42no—
644C2×C4—42no—
645C2×C4—42no—
646C2×C4—42no—
647C2×C4—42no—
648C2×C4—42no—
649C2×C4—42no—
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—42no—
659C2×C4—42no—
660C2×C4—42no—
661C2×C4—42no—
662C2×C4—42no—
663C2×C4—42no—
664C2×C4—42no—
665C2×C4—42no—
666C2×C4—42no—
667C2×C4—42no—
668C2×C4—42no—
669C2×C4—42no—
670C2×C4—42no—
671C2×C4—42no—
672C2×C4—42no—
673C2×C4—42no—
674C2×C4—42no—
675C2×C4—42no—
676C2×C4—42no—
677C2×C4—42no—
678C2×C4—42no—
679C2×C4—42no—
680C2×C4—42no—
681C2×C4—42no—
682C2×C4—42no—
683C2×C4—42no—
684C2×C4—42no—
685C2×C4—42no—
686C2×C4—42no—
687C2×C4—42no—
688C2×C4—42no—
689C2×C4—42no—
690C2×C4—42no—
691C2×C4—42no—
692C2×C4—42no—
693C2×C4—42no—
694C2×C4—42no—
695C2×C4—42no—
696C2×C4—42no—
697C2×C4—42no—
698C2×C4—42no—
699C2×C4—42no—
700C2×C4—42no—
701C2×C4—42no—
702C2×C4—42no—
703C2×C4#54414yes—
704C22—44no—
705C22—42no—
706C22—42no—
707C22—44no—
708C22—42no—
709C22—81no—
710C22—81no—
711C22—81no—
712C22—81no—
713C22—82no—
714C22—82no—
715C22C2×#54411yescharacteristic
716C22#345812yes—
717C22#350112yes—
718C22#350311yescharacteristic
719C22#350311yescharacteristic
720C22#352911yescharacteristic
721C22#352911yescharacteristic
722C22#353112yes—
723C22#353211yescharacteristic
724C22#353311yescharacteristic
725C22#353512yes—
726C22#353711yescharacteristic
727C22#353811yescharacteristic
728C22#354112yes—
729C22#354212yes—
730C22#354612yes—
731C22#355011yescharacteristic
732C22#355011yescharacteristic
733C22#355312yes—
734C22#355712yes—
735C22#355912yes—
736C22#356012yes—
737C22#356212yes—
738C4—24no—
739C4—42no—
740C4—44no—
741C4—42no—
742C4—42no—
743C4—44no—
744C4—42no—
745C4—44no—
746C4—42no—
747C4—82no—
748C4—82no—
749C4—82no—
750C4—82no—
751C4—82no—
752C4—82no—
753C4—82no—
754C4—82no—
755C4—82no—
756C2—82no—
757C2#3669111yescharacteristic
758C2#3674811yescharacteristic
759C2#3675312yes—
760C2#3713712yes—
761C2#3730811yescharacteristic
762C2#5240511yescharacteristic
763C2#5242311yescharacteristic
764C2#5243812yes—
765C2#5245712yes—
766C2#5274911yescharacteristic
767C2#5276211yescharacteristic
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. #17298C2semidirect
3. #17221C2semidirect
4. #17219C2semidirect
5. #17210C2semidirect
6. #17206C2semidirect
9. #17112C2semidirect
13. #15056C2semidirect
14. #15008C2semidirect
16. #14862C2semidirect
17. #14816C2semidirect
159. C4×C22⋊C4Q8semidirect

Maximal subgroups

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

subgroupindex
#172982
#172702
#172212
#172192
#172102
#172062
#171612
#171192
#171122
#171042
#151652
#151612
#150562
#150082
#148932
#148622
#148162
#143512
#138442