Finite Group Databasebeta

← #207671

Subgroups — #207671

Order 512.

1116 conjugacy classes of subgroups.

The subgroup lattice has 637 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×#126C211yesmaximal, characteristic
2#20777C211yesmaximal, characteristic, complemented
3C4×#431C211yesmaximal, characteristic, complemented
4#9880C212yesmaximal, complemented
5#9651C212yesmaximal
6#9082C212yesmaximal, complemented
7#9080C212yesmaximal
8C4×#324C212yesmaximal, complemented
9C4×#323C212yesmaximal
10C22×#126C411yescharacteristic
11C2×#431C2212yes—
12C4×#126—21no—
13C4×#126—21no—
14C4×#126—21no—
15C4×#126—21no—
16C4×#126—21no—
17C4×#126—21no—
18C4×#126—21no—
19C4×#126—21no—
20C2×#388C411yescharacteristic
21C2×#388C2211yescharacteristic
22C2×#388C2211yescharacteristic
23C2×#387C411yescharacteristic
24C2×#387C2211yescharacteristic
25C2×#387C2211yescharacteristic
26C2×#383C411yescharacteristic
27C2×C4×C4⋊C4C411yescharacteristic
28C2×C4×C4⋊C4C411yescharacteristic
29C2×C4×C4⋊C4C2211yescharacteristic
30C2×C4×C22⋊C4C411yescharacteristic
31C2×C4×C22⋊C4C411yescharacteristic
32C2×C4×C22⋊C4C2211yescharacteristic
33C2×C43C2211yescharacteristic
34#1839—21no—
35#1831—21no—
36#1817C2212yes—
37#1797C2212yes—
38C4×C4⋊C8—21no—
39C4×C4⋊C8—21no—
40C4×C4⋊C8—21no—
41C4×C22⋊C8C2212yes—
42C4×C22⋊C8C2212yes—
43C4×C22⋊C8C2212yes—
44#1744—21no—
45#1744—21no—
46C42×C8—21no—
47C2×#324C2214yes—
48C2×#323C2214yes—
49#1280C2214yes—
50#1280C2214yes—
51C24×C4C2×C411yescharacteristic
52C2×#126—21no—
53C2×#126—21no—
54C2×#126—21no—
55C2×#126—21no—
56C2×#126—24no—
57C2×#126C818yescomplemented
58C22×C4⋊C4C2×C411yescharacteristic
59C22×C4⋊C4C2×C411yescharacteristic
60C22×C4⋊C4C2×C411yescharacteristic
61C22×C4⋊C4C2×C411yescharacteristic
62C22×C4⋊C4C2×C411yescharacteristic
63C22×C4⋊C4C2311yescharacteristic
64C22×C22⋊C4C2×C411yescharacteristic
65C22×C22⋊C4C2×C411yescharacteristic
66C22×C22⋊C4C2×C411yescharacteristic
67C22×C22⋊C4C2×C411yescharacteristic
68C22×C22⋊C4C2×C411yescharacteristic
69C22×C22⋊C4C2311yescharacteristic
70C22×C42C2×C411yescharacteristic
71C22×C42C2×C411yescharacteristic
72C22×C42C2×C411yescharacteristic
73C22×C42C2×C411yescharacteristic
74C22×C42C2×C411yescharacteristic
75C22×C42C2×C411yescharacteristic
76C22×C42C2×C411yescharacteristic
77C22×C42C2311yescharacteristic
78C22×C42C2311yescharacteristic
79C22×C42C2311yescharacteristic
80#431—28no—
81C2×C4⋊C8—22no—
82C2×C4⋊C8—22no—
83C2×C22⋊C8C2314yes—
84C2×C22⋊C8C2314yes—
85C2×C4×C8—22no—
86C2×C4×C8—22no—
87C2×C4×C8—22no—
88C2×C4×C8—22no—
89#388—21no—
90#388—21no—
91#388—21no—
92#388—21no—
93#388—21no—
94#388—21no—
95#388—21no—
96#388—21no—
97#388—21no—
98#388—21no—
99#388—21no—
100#388—21no—
101#387—21no—
102#387—21no—
103#387—21no—
104#387—21no—
105#387—21no—
106#387—21no—
107#387—21no—
108#387—21no—
109#387—21no—
110#387—21no—
111#387—21no—
112#387—21no—
113#383—21no—
114#383—21no—
115#383—21no—
116#383—21no—
117C4×C4⋊C4—21no—
118C4×C4⋊C4—21no—
119C4×C4⋊C4—21no—
120C4×C4⋊C4—21no—
121C4×C4⋊C4—21no—
122C4×C4⋊C4—21no—
123C4×C4⋊C4—21no—
124C4×C4⋊C4—21no—
125C4×C4⋊C4—21no—
126C4×C4⋊C4—21no—
127C4×C4⋊C4—21no—
128C4×C4⋊C4—21no—
129C4×C22⋊C4—21no—
130C4×C22⋊C4—21no—
131C4×C22⋊C4—21no—
132C4×C22⋊C4—21no—
133C4×C22⋊C4—21no—
134C4×C22⋊C4—21no—
135C4×C22⋊C4—21no—
136C4×C22⋊C4—21no—
137C4×C22⋊C4—21no—
138C4×C22⋊C4—21no—
139C4×C22⋊C4—21no—
140C4×C22⋊C4—21no—
141C2×#95C2×C411yescharacteristic
142C2×#95C2×C411yescharacteristic
143C2×#95C2×C411yescharacteristic
144C2×#95C2×C411yescharacteristic
145C2×#95C2×C411yescharacteristic
146C2×#95C2×C411yescharacteristic
147C2×#95C2×C411yescharacteristic
148C2×#95C2×C411yescharacteristic
149C2×#95C2×C411yescharacteristic
150C2×#95C2×C411yescharacteristic
151C2×#95C2311yescharacteristic
152C2×#95C2311yescharacteristic
153C43D411yescharacteristic
154C43D411yescharacteristic
155C43D411yescharacteristic
156C43D411yescharacteristic
157C43D411yescharacteristic
158C43D411yescharacteristic
159C43D411yescharacteristic
160C43D411yescharacteristic
161#336—22no—
162#336—22no—
163#336—22no—
164#336—22no—
165#336—22no—
166#336—22no—
167#324—28no—
168#323—28no—
169C25C22×C411yescharacteristic
170C23×C4—21no—
171C23×C4—21no—
172C23×C4—24no—
173C23×C4—42no—
174C23×C4—42no—
175C23×C4C22×C411yescharacteristic
176C23×C4C22×C411yescharacteristic
177C23×C4C22×C411yescharacteristic
178C23×C4C22×C411yescharacteristic
179C23×C4C22×C411yescharacteristic
180C23×C4C22×C411yescharacteristic
181C23×C4C22×C411yescharacteristic
182C23×C4C22×C411yescharacteristic
183C23×C4C22×C411yescharacteristic
184C23×C4C22×C411yescharacteristic
185C23×C4C22×C411yescharacteristic
186C23×C4C22×C411yescharacteristic
187C23×C4C22×C411yescharacteristic
188C23×C4C2411yescharacteristic
189C22×C8—24no—
190C22×C8—24no—
191#126—28no—
192#126—28no—
193#126—28no—
194#126—28no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—24no—
211C2×C4⋊C4—42no—
212C2×C4⋊C4—42no—
213C2×C4⋊C4C2×C814yes—
214C2×C4⋊C4C2×C814yes—
215C2×C4⋊C4C2×C814yes—
216C2×C22⋊C4—21no—
217C2×C22⋊C4—21no—
218C2×C22⋊C4—21no—
219C2×C22⋊C4—21no—
220C2×C22⋊C4—21no—
221C2×C22⋊C4—21no—
222C2×C22⋊C4—22no—
223C2×C22⋊C4—22no—
224C2×C22⋊C4—22no—
225C2×C22⋊C4—21no—
226C2×C22⋊C4—21no—
227C2×C22⋊C4—21no—
228C2×C22⋊C4—21no—
229C2×C22⋊C4—21no—
230C2×C22⋊C4—21no—
231C2×C22⋊C4—24no—
232C2×C22⋊C4—42no—
233C2×C22⋊C4—42no—
234C2×C22⋊C4C2×C814yes—
235C2×C22⋊C4C2×C814yes—
236C2×C22⋊C4C2×C814yes—
237C2×C42—24no—
238C2×C42—24no—
239C2×C42—21no—
240C2×C42—21no—
241C2×C42—21no—
242C2×C42—21no—
243C2×C42—21no—
244C2×C42—21no—
245C2×C42—24no—
246C2×C42—42no—
247C2×C42—42no—
248C2×C42—42no—
249C2×C42—42no—
250C2×C42—42no—
251C2×C42—42no—
252C2×C42C22⋊C411yescharacteristic
253C2×C42C22⋊C411yescharacteristic
254C2×C42C22⋊C411yescharacteristic
255C2×C42C22⋊C411yescharacteristic
256C2×C42C22⋊C411yescharacteristic
257C2×C42C22⋊C411yescharacteristic
258C2×C42C22⋊C411yescharacteristic
259C2×C42C22⋊C411yescharacteristic
260C2×C42C22⋊C411yescharacteristic
261C2×C42C22⋊C411yescharacteristic
262C2×C42C22⋊C411yescharacteristic
263C2×C42C22⋊C411yescharacteristic
264C2×C42C22⋊C411yescharacteristic
265C2×C42C22⋊C411yescharacteristic
266C2×C42C22⋊C411yescharacteristic
267C2×C42C22⋊C411yescharacteristic
268C2×C42C2×C814yes—
269C2×C42M1614yes—
270C2×C42C2×D411yescharacteristic
271C2×C42C2×D411yescharacteristic
272C2×C42C2×D411yescharacteristic
273C2×C42C2×D411yescharacteristic
274C2×C42C2×D411yescharacteristic
275C2×C42C2×D411yescharacteristic
276C2×C42C2×D411yescharacteristic
277C2×C42C2×D411yescharacteristic
278C2×C42C2×D411yescharacteristic
279C2×C42C2×D411yescharacteristic
280C2×C42C2×D411yescharacteristic
281C2×C42C2×D411yescharacteristic
282C4⋊C8—28no—
283C4⋊C8—44no—
284C22⋊C8—24no—
285C22⋊C8—24no—
286C22⋊C8—24no—
287C22⋊C8#4118yes—
288C4×C8—28no—
289C4×C8—44no—
290C4×C8—44no—
291C4×C8—44no—
292#95—21no—
293#95—21no—
294#95—21no—
295#95—21no—
296#95—21no—
297#95—21no—
298#95—21no—
299#95—21no—
300#95—21no—
301#95—21no—
302#95—21no—
303#95—21no—
304#95—21no—
305#95—21no—
306#95—21no—
307#95—21no—
308#95—21no—
309#95—21no—
310#95—21no—
311#95—21no—
312#95—21no—
313#95—21no—
314#95—21no—
315#95—21no—
316C24—21no—
317C24—21no—
318C24—22no—
319C24—22no—
320C24—41no—
321C24—41no—
322C24—41no—
323C24—41no—
324C24C22×C812yes—
325C24C23×C411yescharacteristic
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—21no—
331C22×C4—21no—
332C22×C4—21no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—21no—
336C22×C4—21no—
337C22×C4—21no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—21no—
343C22×C4—21no—
344C22×C4—21no—
345C22×C4—21no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—41no—
351C22×C4—41no—
352C22×C4—41no—
353C22×C4—41no—
354C22×C4—41no—
355C22×C4—41no—
356C22×C4—41no—
357C22×C4—41no—
358C22×C4—42no—
359C22×C4—42no—
360C22×C4—41no—
361C22×C4—41no—
362C22×C4—41no—
363C22×C4—41no—
364C22×C4—42no—
365C22×C4—42no—
366C22×C4—42no—
367C22×C4—42no—
368C22×C4—41no—
369C22×C4—41no—
370C22×C4—41no—
371C22×C4—41no—
372C22×C4—41no—
373C22×C4—41no—
374C22×C4—41no—
375C22×C4—41no—
376C22×C4—88no—
377C22×C4C2×C22⋊C411yescharacteristic
378C22×C4C2×C22⋊C411yescharacteristic
379C22×C4C2×C22⋊C411yescharacteristic
380C22×C4C2×C22⋊C411yescharacteristic
381C22×C4C2×C22⋊C411yescharacteristic
382C22×C4C2×C22⋊C411yescharacteristic
383C22×C4C2×C22⋊C411yescharacteristic
384C22×C4C2×C22⋊C411yescharacteristic
385C22×C4C2×C22⋊C411yescharacteristic
386C22×C4C2×C22⋊C411yescharacteristic
387C22×C4C2×C22⋊C411yescharacteristic
388C22×C4C2×C22⋊C411yescharacteristic
389C22×C4C22×C812yes—
390C22×C4C22×C812yes—
391C22×C4C22×C812yes—
392C22×C4C22×C812yes—
393C22×C4C22×C812yes—
394C22×C4C22×C812yes—
395C22×C4C2×M1612yes—
396C22×C4C2×M1612yes—
397C22×C4C2×M1612yes—
398C22×C4#13114yes—
399C22×C4C22×D411yescharacteristic
400C22×C4C22×D411yescharacteristic
401C22×C4C2×#4114yes—
402C2×C8—28no—
403C2×C8—44no—
404C2×C8—44no—
405C2×C8—44no—
406C2×C8—44no—
407C2×C8—44no—
408C4⋊C4—24no—
409C4⋊C4—24no—
410C4⋊C4—24no—
411C4⋊C4—24no—
412C4⋊C4—24no—
413C4⋊C4—24no—
414C4⋊C4—42no—
415C4⋊C4—42no—
416C4⋊C4—42no—
417C4⋊C4—42no—
418C4⋊C4—42no—
419C4⋊C4—42no—
420C22⋊C4—24no—
421C22⋊C4—24no—
422C22⋊C4—24no—
423C22⋊C4—24no—
424C22⋊C4—24no—
425C22⋊C4—24no—
426C22⋊C4—42no—
427C22⋊C4—42no—
428C22⋊C4—42no—
429C22⋊C4—42no—
430C22⋊C4—42no—
431C22⋊C4—42no—
432C42—22no—
433C42—22no—
434C42—22no—
435C42—22no—
436C42—22no—
437C42—22no—
438C42—22no—
439C42—22no—
440C42—22no—
441C42—22no—
442C42—22no—
443C42—22no—
444C42—22no—
445C42—22no—
446C42—22no—
447C42—22no—
448C42—41no—
449C42—41no—
450C42—41no—
451C42—41no—
452C42—41no—
453C42—41no—
454C42—41no—
455C42—41no—
456C42—42no—
457C42—42no—
458C42—42no—
459C42—42no—
460C42—42no—
461C42—42no—
462C42C22⋊C814yes—
463C42C22⋊C814yes—
464C42C22⋊C814yes—
465C42C22⋊C814yes—
466C23—21no—
467C23—21no—
468C23—21no—
469C23—21no—
470C23—22no—
471C23—24no—
472C23—42no—
473C23—42no—
474C23—42no—
475C23—42no—
476C23—41no—
477C23—41no—
478C23—41no—
479C23—41no—
480C23—41no—
481C23—42no—
482C23—41no—
483C23—82no—
484C23—82no—
485C23—82no—
486C23—82no—
487C23C22×C22⋊C411yescharacteristic
488C23C4×#4112yes—
489C23C23×C811yescharacteristic
490C23C22×M1611yescharacteristic
491C23C2×#13112yes—
492C2×C4—21no—
493C2×C4—21no—
494C2×C4—21no—
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—24no—
512C2×C4—24no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—28no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—24no—
526C2×C4—21no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—44no—
532C2×C4—44no—
533C2×C4—44no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—42no—
542C2×C4—41no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—84no—
558C2×C4—84no—
559C2×C4—82no—
560C2×C4—82no—
561C2×C4—82no—
562C2×C4—82no—
563C2×C4C2×C22⋊C812yes—
564C2×C4C2×C22⋊C812yes—
565C2×C4C2×C22⋊C812yes—
566C2×C4C2×C22⋊C812yes—
567C2×C4C2×C22⋊C812yes—
568C2×C4C2×C22⋊C812yes—
569C2×C4#40812yes—
570C2×C4#40812yes—
571C2×C4#41012yes—
572C2×C4#41012yes—
573C2×C4#41312yes—
574C2×C4#41312yes—
575C2×C4#52512yes—
576C2×C4#53212yes—
577C8—48no—
578C8—84no—
579C22—22no—
580C22—22no—
581C22—21no—
582C22—21no—
583C22—21no—
584C22—21no—
585C22—22no—
586C22—22no—
587C22—42no—
588C22—42no—
589C22—42no—
590C22—84no—
591C22—84no—
592C22—81no—
593C22—81no—
594C22—81no—
595C22—81no—
596C22#229711yescharacteristic
597C22C22×C22⋊C811yescharacteristic
598C22C2×#40811yescharacteristic
599C22C2×#41011yescharacteristic
600C22C2×#41011yescharacteristic
601C22C2×#41311yescharacteristic
602C22C2×#41311yescharacteristic
603C22C8×#4112yes—
604C22#296512yes—
605C4—22no—
606C4—22no—
607C4—22no—
608C4—22no—
609C4—22no—
610C4—22no—
611C4—22no—
612C4—22no—
613C4—22no—
614C4—22no—
615C4—22no—
616C4—42no—
617C4—41no—
618C4—42no—
619C4—42no—
620C4—41no—
621C4—81no—
622C4—81no—
623C4—81no—
624C4—81no—
625C4#147314yes—
626C2—22no—
627C2—22no—
628C2—82no—
629C2#1734911yescharacteristic
630C2#1734911yescharacteristic
631C2#1737611yescharacteristic
632C2C2×#147311yescharacteristic
633C2#2137311yescharacteristic
634C2#2142111yescharacteristic
635C2#2142611yescharacteristic
subgroupG/H
centerC22×C4C2×C22⋊C4
commutatorC23C23×C8
Frattini, cosocleC23×C4C24
socleC23C22×C22⋊C4
prime radicalC25C22×C4
LCSG ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C2×C43 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. #20777C2semidirect
3. C4×#431C2semidirect
4. #9880C2semidirect
6. #9082C2semidirect
8. C4×#324C2semidirect
57. C2×#126C8semidirect

Maximal subgroups

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

subgroupindex
C2×C4×#1262
#207772
C4×#4312
#98802
#96512
#90822
#90802
C4×#3242
C4×#3232