Finite Group Databasebeta

← #182647

Subgroups — #182647

Order 512.

1196 conjugacy classes of subgroups.

The subgroup lattice has 625 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#529C211yesmaximal, characteristic
2#20463C211yesmaximal, characteristic, complemented
3C2×#1521C212yesmaximal
4C4×#383C211yesmaximal, characteristic, complemented
5#8219C212yesmaximal, complemented
6#8218C212yesmaximal
7#8217C212yesmaximal, complemented
8#8216C212yesmaximal
9#8025C212yesmaximal, complemented
10C2×#529C2211yescharacteristic
11C2×#431C2212yes—
12C4×#125C411yescharacteristic, complemented
13C4×#125C411yescharacteristic, complemented
14C4×#126—21no—
15C4×#126—21no—
16C4×#124C411yescharacteristic, complemented
17C4×#124C411yescharacteristic, complemented
18C4×C22⋊Q8C2211yescharacteristic
19C4×C22⋊Q8C2211yescharacteristic
20C4×#123—21no—
21C4×#123—21no—
22C4×#121C2211yescharacteristic
23C4×#121C2211yescharacteristic
24#2298—21no—
25#2295C411yescharacteristic, complemented
26#2293C411yescharacteristic, complemented
27#2290—21no—
28#2283C2211yescharacteristic
29C2×#383C2212yes—
30C42×Q8C411yescharacteristic, complemented
31C42×D4—21no—
32C42×D4C411yescharacteristic, complemented
33C4×#117—21no—
34C2×C43C2211yescharacteristic
35#1767—21no—
36#1767—21no—
37#1749C2212yes—
38#1749C2212yes—
39#1744—21no—
40C4×#97—21no—
41#1521C4116yescomplemented
42C2×#328C2214yes—
43C2×#327C2214yes—
44#1434—21no—
45#1434—21no—
46#1434C2212yes—
47C4×#95—21no—
48C4×#95—21no—
49C4×#95C2212yes—
50#1294C2214yes—
51#1294C2214yes—
52#529—24no—
53#529C2×C418yes—
54C2×#126—21no—
55C2×#125C2×C411yescharacteristic
56C2×#124C2×C411yescharacteristic
57C2×#123—21no—
58C2×C22⋊Q8C2311yescharacteristic
59C2×#121C2311yescharacteristic
60C2×C4×Q8C2×C411yescharacteristic
61C2×C4×Q8C2×C411yescharacteristic
62C2×C4×Q8C2×C411yescharacteristic
63C2×C4×D4—21no—
64C2×C4×D4—21no—
65C2×C4×D4—21no—
66C2×C4×D4C2×C411yescharacteristic
67C2×C4×D4C2×C411yescharacteristic
68C2×C4×D4C2×C411yescharacteristic
69C2×#117—21no—
70C2×#117—21no—
71C22×C42—21no—
72C22×C42—22no—
73C22×C42D411yescharacteristic
74C22×C42D411yescharacteristic
75C22×C42D411yescharacteristic
76C22×C42D411yescharacteristic
77C22×C42C2311yescharacteristic
78C22×C42C2311yescharacteristic
79C22×C42C2311yescharacteristic
80#431—24no—
81#431C2×C418yescomplemented
82C2×C4⋊C8—22no—
83C2×C22⋊C8C2314yes—
84C2×#97—22no—
85C2×#97—22no—
86C2×C4×C8—22no—
87#391C2×C411yescharacteristic
88#391C2×C411yescharacteristic
89#390C2×C411yescharacteristic
90#390C2×C411yescharacteristic
91#389C2×C411yescharacteristic
92#389C2×C411yescharacteristic
93#389C2×C411yescharacteristic
94#389C2×C411yescharacteristic
95#388—21no—
96#388—21no—
97#388—21no—
98#388—21no—
99#388—21no—
100#388—21no—
101#388—21no—
102#388—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#386—21no—
112#386—21no—
113#385—21no—
114#385—21no—
115#383—21no—
116#383—24no—
117#383C2×C418yescomplemented
118#382C2×C411yescharacteristic
119#382C2×C411yescharacteristic
120#381—21no—
121#380C2311yescharacteristic
122#380C2311yescharacteristic
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×C4⋊C4—21no—
130C4×C4⋊C4—21no—
131C4×C4⋊C4—21no—
132C4×C4⋊C4—21no—
133C4×C4⋊C4C2×C411yescharacteristic
134C4×C4⋊C4C2×C411yescharacteristic
135C4×C4⋊C4C2×C411yescharacteristic
136C4×C4⋊C4C2×C411yescharacteristic
137C4×C4⋊C4C2×C411yescharacteristic
138C4×C4⋊C4C2×C411yescharacteristic
139C4×C22⋊C4—21no—
140C4×C22⋊C4—21no—
141C4×C22⋊C4—21no—
142C4×C22⋊C4—21no—
143C4×C22⋊C4—21no—
144C4×C22⋊C4—21no—
145C4×C22⋊C4—21no—
146C4×C22⋊C4—21no—
147C4×C22⋊C4—21no—
148C4×C22⋊C4—21no—
149C4×C22⋊C4—21no—
150C4×C22⋊C4—21no—
151C4×C22⋊C4—21no—
152#376—21no—
153#376—21no—
154C2×#95—22no—
155C2×#95—22no—
156C2×#95—22no—
157C2×#95—22no—
158C2×#95—22no—
159C2×#95C2314yes—
160C43—21no—
161C43C2×C411yescharacteristic
162C43C2×C411yescharacteristic
163C43C2×C411yescharacteristic
164C43C2×C411yescharacteristic
165C43D411yescharacteristic
166C43D411yescharacteristic
167C43D411yescharacteristic
168C43D411yescharacteristic
169#336—22no—
170#336—22no—
171#328C2×C4116yescomplemented
172#327C2×C4116yes—
173C22×Q8C22×C411yescharacteristic
174C22×D4—21no—
175C22×D4C22×C411yescharacteristic
176C23×C4—21no—
177C23×C4—24no—
178C23×C4—21no—
179C23×C4—21no—
180C23×C4C2×D411yescharacteristic
181C23×C4C2×D411yescharacteristic
182C23×C4C2×D411yescharacteristic
183C23×C4C2×D411yescharacteristic
184C23×C4C2×D411yescharacteristic
185C23×C4C2×D411yescharacteristic
186C23×C4C2411yescharacteristic
187C22×C8—24no—
188#126—28no—
189#125—22no—
190#125C4214yescomplemented
191#124—22no—
192#124C4214yescomplemented
193#123—28no—
194C22⋊Q8—22no—
195C22⋊Q8C22×C414yes—
196#121—22no—
197#121C22×C414yes—
198C4×Q8—22no—
199C4×Q8—42no—
200C4×Q8—42no—
201C4×Q8C4214yescomplemented
202C4×D4—28no—
203C4×D4—22no—
204C4×D4—42no—
205C4×D4—42no—
206C4×D4—44no—
207C4×D4—44no—
208C4×D4C4214yescomplemented
209#117—28no—
210#117—44no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4C22×C411yescharacteristic
222C2×C4⋊C4C22×C411yescharacteristic
223C2×C4⋊C4C22×C411yescharacteristic
224C2×C4⋊C4C22×C411yescharacteristic
225C2×C4⋊C4C22×C411yescharacteristic
226C2×C4⋊C4C22×C411yescharacteristic
227C2×C22⋊C4—21no—
228C2×C22⋊C4—21no—
229C2×C22⋊C4—21no—
230C2×C22⋊C4—21no—
231C2×C22⋊C4—21no—
232C2×C22⋊C4—21no—
233C2×C22⋊C4—21no—
234C2×C22⋊C4—21no—
235C2×C22⋊C4—21no—
236C2×C22⋊C4—21no—
237C2×C22⋊C4—21no—
238C2×C22⋊C4—21no—
239C2×C22⋊C4—21no—
240C2×C42—21no—
241C2×C42—21no—
242C2×C42—21no—
243C2×C42—21no—
244C2×C42—21no—
245C2×C42—21no—
246C2×C42—21no—
247C2×C42—21no—
248C2×C42—21no—
249C2×C42—42no—
250C2×C42—42no—
251C2×C42—42no—
252C2×C42—42no—
253C2×C42—44no—
254C2×C42—42no—
255C2×C42—42no—
256C2×C42—44no—
257C2×C42—44no—
258C2×C42—44no—
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×C42C22⋊C411yescharacteristic
269C2×C42C22⋊C411yescharacteristic
270C2×C42C22⋊C411yescharacteristic
271C2×C42C22⋊C411yescharacteristic
272C2×C42C22⋊C411yescharacteristic
273C2×C42C22⋊C411yescharacteristic
274C2×C42C22⋊C411yescharacteristic
275C2×C42C22×C411yescharacteristic
276C2×C42C22×C411yescharacteristic
277C2×C42C22×C411yescharacteristic
278C2×C42C22×C411yescharacteristic
279C2×C42C22×C411yescharacteristic
280C2×C42C22×C411yescharacteristic
281C2×C42C22×C414yes—
282C2×C42C2×D411yescharacteristic
283C2×C42C2×D411yescharacteristic
284C2×C42C2×D411yescharacteristic
285C2×C42C2×D411yescharacteristic
286C2×C42C2×D411yescharacteristic
287C2×C42C2×D411yescharacteristic
288C2×C42#4114yes—
289C4⋊C8—28no—
290C22⋊C8—24no—
291C22⋊C8C22×C418yes—
292#97—44no—
293#97—44no—
294C4×C8—28no—
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—28no—
314#95—28no—
315#95—21no—
316#95—21no—
317#95—24no—
318#95—44no—
319#95—44no—
320#95—44no—
321#95C22×C418yes—
322C24—21no—
323C24C22×D411yescharacteristic
324C2×Q8—21no—
325C2×Q8—41no—
326C2×Q8—41no—
327C2×Q8C2×C4212yes—
328C2×D4—24no—
329C2×D4—21no—
330C2×D4—41no—
331C2×D4—41no—
332C2×D4—42no—
333C2×D4—42no—
334C2×D4C2×C4212yes—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—21no—
341C22×C4—21no—
342C22×C4—21no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—24no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—21no—
350C22×C4—21no—
351C22×C4—21no—
352C22×C4—21no—
353C22×C4—28no—
354C22×C4—41no—
355C22×C4—41no—
356C22×C4—41no—
357C22×C4—41no—
358C22×C4—41no—
359C22×C4—41no—
360C22×C4—41no—
361C22×C4—41no—
362C22×C4—42no—
363C22×C4—42no—
364C22×C4—42no—
365C22×C4—42no—
366C22×C4—42no—
367C22×C4—42no—
368C22×C4—44no—
369C22×C4—44no—
370C22×C4—44no—
371C22×C4—44no—
372C22×C4—44no—
373C22×C4—44no—
374C22×C4—42no—
375C22×C4—42no—
376C22×C4C2×C22⋊C411yescharacteristic
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×C4C4×D414yes—
389C22×C4C4×D414yes—
390C22×C4C23×C411yescharacteristic
391C22×C4C23×C412yes—
392C22×C4C22×D411yescharacteristic
393C22×C4C2×#4112yes—
394C2×C8—28no—
395C2×C8—44no—
396C2×C8—44no—
397C4⋊C4—21no—
398C4⋊C4—21no—
399C4⋊C4—24no—
400C4⋊C4—24no—
401C4⋊C4—24no—
402C4⋊C4—24no—
403C4⋊C4—41no—
404C4⋊C4—41no—
405C4⋊C4—41no—
406C4⋊C4—41no—
407C4⋊C4—42no—
408C4⋊C4—42no—
409C4⋊C4—42no—
410C4⋊C4—42no—
411C4⋊C4—42no—
412C4⋊C4—42no—
413C4⋊C4C2×C4212yes—
414C4⋊C4C2×C4212yes—
415C22⋊C4—24no—
416C22⋊C4—24no—
417C22⋊C4—24no—
418C22⋊C4—24no—
419C22⋊C4—24no—
420C22⋊C4—42no—
421C22⋊C4—42no—
422C22⋊C4—42no—
423C22⋊C4—42no—
424C22⋊C4—42no—
425C22⋊C4—42no—
426C22⋊C4—42no—
427C22⋊C4—42no—
428C42—24no—
429C42—22no—
430C42—22no—
431C42—22no—
432C42—22no—
433C42—22no—
434C42—22no—
435C42—22no—
436C42—22no—
437C42—41no—
438C42—41no—
439C42—41no—
440C42—41no—
441C42—41no—
442C42—41no—
443C42—41no—
444C42—41no—
445C42—41no—
446C42—41no—
447C42—41no—
448C42—41no—
449C42—41no—
450C42—41no—
451C42—41no—
452C42—41no—
453C42—42no—
454C42—42no—
455C42—42no—
456C42—42no—
457C42—42no—
458C42—42no—
459C42—42no—
460C42—42no—
461C42—88no—
462C42C2×C4212yes—
463C42C2×C4212yes—
464C42#11712yes—
465C42#11712yes—
466C42C4×D414yes—
467C42C4×D414yes—
468C23—21no—
469C23—21no—
470C23—21no—
471C23—21no—
472C23—22no—
473C23—41no—
474C23—41no—
475C23—41no—
476C23—41no—
477C23—41no—
478C23—41no—
479C23C22×C22⋊C411yescharacteristic
480C23C2×C4×D412yes—
481C23#57512yes—
482C23#57512yes—
483Q8—42no—
484Q8—42no—
485D4—42no—
486D4—42no—
487D4—48no—
488C2×C4—28no—
489C2×C4—21no—
490C2×C4—21no—
491C2×C4—21no—
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—28no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—22no—
525C2×C4—44no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—44no—
529C2×C4—44no—
530C2×C4—48no—
531C2×C4—44no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—48no—
539C2×C4—48no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—48no—
553C2×C4—84no—
554C2×C4—84no—
555C2×C4—84no—
556C2×C4—84no—
557C2×C4—84no—
558C2×C4—84no—
559C2×C4C4×C22⋊C412yes—
560C2×C4C4×C22⋊C412yes—
561C2×C4C4×C22⋊C412yes—
562C2×C4C4×C22⋊C412yes—
563C2×C4#41012yes—
564C2×C4#41012yes—
565C2×C4#42112yes—
566C2×C4#42112yes—
567C2×C4C22×C4211yescharacteristic
568C2×C4C2×#11711yescharacteristic
569C2×C4C2×C4×D412yes—
570C8—48no—
571C22—22no—
572C22—22no—
573C22—22no—
574C22—22no—
575C22—22no—
576C22—22no—
577C22—42no—
578C22—42no—
579C22—42no—
580C22—81no—
581C22—81no—
582C22—81no—
583C22—81no—
584C22C2×C4×C22⋊C411yescharacteristic
585C22C2×#41011yescharacteristic
586C22C2×#41011yescharacteristic
587C22#289611yescharacteristic
588C22#289611yescharacteristic
589C22C2×#42111yescharacteristic
590C22C2×#42111yescharacteristic
591C22#294314yes—
592C4—22no—
593C4—21no—
594C4—21no—
595C4—22no—
596C4—22no—
597C4—22no—
598C4—22no—
599C4—22no—
600C4—22no—
601C4—22no—
602C4—22no—
603C4—42no—
604C4—41no—
605C4—41no—
606C4—42no—
607C4—41no—
608C4—41no—
609C4—42no—
610C4—88no—
611C4—81no—
612C4—81no—
613C4—81no—
614C4—81no—
615C2—24no—
616C2—82no—
617C2#1757911yescharacteristic
618C2#1758011yescharacteristic
619C2#1759311yescharacteristic
620C2#1759411yescharacteristic
621C2C4×#41011yescharacteristic
622C2C4×#42111yescharacteristic
623C2#2041111yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC2×C4C22×C42
Frattini, cosocleC23×C4C24
prime radicalC2×#121C23
LCSG ▷ C2×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C2×C43 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #20463C2semidirect
4. C4×#383C2semidirect
5. #8219C2semidirect
7. #8217C2semidirect
9. #8025C2semidirect
12. C4×#125C4semidirect
13. C4×#125C4semidirect
16. C4×#124C4semidirect
17. C4×#124C4semidirect
25. #2295C4semidirect
26. #2293C4semidirect
30. C42×Q8C4semidirect
32. C42×D4C4semidirect
41. #1521C4semidirect
81. #431C2×C4semidirect
117. #383C2×C4semidirect
171. #328C2×C4semidirect
190. #125C42semidirect
192. #124C42semidirect
201. C4×Q8C42semidirect
208. C4×D4C42semidirect

Maximal subgroups

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

subgroupindex
C4×#5292
#204632
C2×#15212
C4×#3832
#82192
#82182
#82172
#82162
#80252