Finite Group Databasebeta

← #160780

Subgroups — #160780

Order 512.

2762 conjugacy classes of subgroups.

The subgroup lattice has 623 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#390C211yesmaximal, characteristic
2#20715C211yesmaximal, characteristic, complemented
3C4×#407C211yesmaximal, characteristic, complemented
4#7626C214yesmaximal, complemented
5#7606C214yesmaximal, complemented
6#7572C214yesmaximal, complemented
7C24×D4C411yescharacteristic
8C23×C4⋊C4C411yescharacteristic
9C23×C22⋊C4—21no—
10C23×C22⋊C4—21no—
11C23×C42C2211yescharacteristic
12C2×#432—21no—
13C2×#407—21no—
14C2×C4×M16—21no—
15C2×#390—21no—
16C2×#390—21no—
17C2×#390—21no—
18C2×#390—21no—
19C2×#390—21no—
20C2×#390—21no—
21C2×#390C412yes—
22C2×#390C412yes—
23C2×#390C412yes—
24C2×#390C2212yes—
25C2×#390C2212yes—
26C2×#390C2212yes—
27#1817C2212yescomplemented
28#1797—21no—
29#1797C2212yescomplemented
30#1779—21no—
31#1749—21no—
32#1749—21no—
33#1749C2212yescomplemented
34#1749C2212yescomplemented
35C4×C22⋊C8—21no—
36C4×C22⋊C8C2212yescomplemented
37#1728C2212yescomplemented
38#1270C22116yescomplemented
39C26—21no—
40C23×D4—28no—
41C23×D4—21no—
42C23×D4—21no—
43C23×D4—21no—
44C23×D4—44no—
45C23×D4—44no—
46C23×D4C2×C412yes—
47C23×D4C2×C412yes—
48C23×D4C2×C412yes—
49C24×C4—21no—
50C24×C4C2×C411yescharacteristic
51C24×C4D411yescharacteristic
52C24×C4D411yescharacteristic
53C22×M16—22no—
54C22×C4⋊C4—21no—
55C22×C4⋊C4—21no—
56C22×C4⋊C4—21no—
57C22×C4⋊C4—44no—
58C22×C4⋊C4C2×C412yes—
59C22×C4⋊C4C2×C412yes—
60C22×C4⋊C4C2×C412yes—
61C22×C22⋊C4—22no—
62C22×C22⋊C4—22no—
63C22×C22⋊C4—22no—
64C22×C22⋊C4—22no—
65C22×C22⋊C4—22no—
66C22×C22⋊C4—22no—
67C22×C22⋊C4—22no—
68C22×C22⋊C4—22no—
69C22×C22⋊C4—22no—
70C22×C22⋊C4—22no—
71C22×C22⋊C4—22no—
72C22×C22⋊C4—22no—
73C22×C22⋊C4—48no—
74C22×C22⋊C4—48no—
75C22×C42—44no—
76C22×C42C2×C411yescharacteristic
77C22×C42C2×C411yescharacteristic
78C22×C42C2×C411yescharacteristic
79C22×C42D411yescharacteristic
80C22×C42D411yescharacteristic
81C22×C42D411yescharacteristic
82C22×C42D411yescharacteristic
83C22×C42D411yescharacteristic
84C22×C42D411yescharacteristic
85C22×C42C2311yescharacteristic
86C22×C42C2311yescharacteristic
87C22×C42C2311yescharacteristic
88#432—44no—
89C2×C4⋊C8—22no—
90#407—28no—
91C2×C22⋊C8—22no—
92C2×C22⋊C8—22no—
93C2×C22⋊C8—22no—
94C2×C22⋊C8—22no—
95C2×C22⋊C8—22no—
96C4×M16—44no—
97C2×#97—22no—
98C2×#97—22no—
99C2×C4×C8—22no—
100#390—22no—
101#390—22no—
102#390—22no—
103#390—22no—
104#390—22no—
105#390—22no—
106#390—22no—
107#390—22no—
108#390—22no—
109#390—22no—
110#390—22no—
111#390—22no—
112#390C2×C414yes—
113#390C2×C414yes—
114#390C2×C414yes—
115#390C2314yes—
116#336—22no—
117#336—22no—
118#336C2314yescomplemented
119#336C2314yescomplemented
120C25—28no—
121C25—22no—
122C25—22no—
123C25—22no—
124C25—44no—
125C25—44no—
126C25—44no—
127C25—44no—
128C25—44no—
129C25—44no—
130C25C22⋊C411yescharacteristic
131C22×D4—28no—
132C22×D4—28no—
133C22×D4—28no—
134C22×D4—28no—
135C22×D4—28no—
136C22×D4—22no—
137C22×D4—22no—
138C22×D4—22no—
139C22×D4—24no—
140C22×D4—44no—
141C22×D4—44no—
142C22×D4—464no—
143C22×D4—44no—
144C22×D4—44no—
145C22×D4—44no—
146C22×D4—44no—
147C22×D4—44no—
148C22×D4—44no—
149C22×D4—44no—
150C22×D4—44no—
151C22×D4—44no—
152C22×D4—44no—
153C22×D4M1618yescomplemented
154C22×D4C22×C414yes—
155C23×C4—24no—
156C23×C4—24no—
157C23×C4—24no—
158C23×C4—24no—
159C23×C4—24no—
160C23×C4—24no—
161C23×C4—22no—
162C23×C4—22no—
163C23×C4—22no—
164C23×C4—42no—
165C23×C4—42no—
166C23×C4—42no—
167C23×C4—44no—
168C23×C4—44no—
169C23×C4—44no—
170C23×C4C22⋊C411yescharacteristic
171C23×C4C22⋊C411yescharacteristic
172C23×C4C22⋊C411yescharacteristic
173C23×C4C22⋊C411yescharacteristic
174C23×C4C22⋊C411yescharacteristic
175C23×C4C22⋊C411yescharacteristic
176C23×C4C22⋊C411yescharacteristic
177C23×C4C22⋊C411yescharacteristic
178C23×C4C22⋊C411yescharacteristic
179C23×C4C22×C411yescharacteristic
180C23×C4C22×C411yescharacteristic
181C23×C4C22×C411yescharacteristic
182C23×C4C2×D411yescharacteristic
183C23×C4C2×D411yescharacteristic
184C23×C4C2×D411yescharacteristic
185C23×C4C2×D411yescharacteristic
186C23×C4C2×D411yescharacteristic
187C23×C4C2×D411yescharacteristic
188C2×M16—28no—
189C2×M16—44no—
190C2×M16—44no—
191C22×C8—24no—
192C22×C8—24no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—44no—
197C2×C4⋊C4—44no—
198C2×C4⋊C4—44no—
199C2×C4⋊C4—44no—
200C2×C4⋊C4—44no—
201C2×C4⋊C4—44no—
202C2×C4⋊C4C22×C414yes—
203C2×C22⋊C4—24no—
204C2×C22⋊C4—24no—
205C2×C22⋊C4—24no—
206C2×C22⋊C4—24no—
207C2×C22⋊C4—24no—
208C2×C22⋊C4—24no—
209C2×C22⋊C4—24no—
210C2×C22⋊C4—24no—
211C2×C22⋊C4—48no—
212C2×C22⋊C4—48no—
213C2×C22⋊C4—48no—
214C2×C22⋊C4—48no—
215C2×C22⋊C4—48no—
216C2×C22⋊C4—48no—
217C2×C22⋊C4—48no—
218C2×C22⋊C4—48no—
219C2×C22⋊C4—48no—
220C2×C22⋊C4—48no—
221C2×C22⋊C4—48no—
222C2×C22⋊C4—48no—
223C2×C42—24no—
224C2×C42—24no—
225C2×C42—42no—
226C2×C42—42no—
227C2×C42—42no—
228C2×C42—42no—
229C2×C42—42no—
230C2×C42—42no—
231C2×C42—42no—
232C2×C42—42no—
233C2×C42—42no—
234C2×C42—42no—
235C2×C42C22⋊C411yescharacteristic
236C2×C42C22⋊C411yescharacteristic
237C2×C42C22⋊C411yescharacteristic
238C2×C42C22⋊C411yescharacteristic
239C2×C42C22⋊C411yescharacteristic
240C2×C42C22⋊C411yescharacteristic
241C2×C42C22×C411yescharacteristic
242C2×C42C22×C411yescharacteristic
243C2×C42C22×C411yescharacteristic
244C2×C42C2×D411yescharacteristic
245C2×C42C2×D411yescharacteristic
246C2×C42C2×D411yescharacteristic
247C2×C42C2×D411yescharacteristic
248C2×C42C2×D411yescharacteristic
249C2×C42C2×D411yescharacteristic
250C2×C42C2411yescharacteristic
251C4⋊C8—44no—
252C22⋊C8—28no—
253C22⋊C8—28no—
254C22⋊C8—44no—
255C22⋊C8—44no—
256C22⋊C8—44no—
257#97—44no—
258#97—44no—
259C4×C8—44no—
260C24—22no—
261C24—22no—
262C24—22no—
263C24—22no—
264C24—22no—
265C24—22no—
266C24—28no—
267C24—28no—
268C24—24no—
269C24—28no—
270C24—44no—
271C24—44no—
272C24—44no—
273C24—44no—
274C24—432no—
275C24—44no—
276C24—44no—
277C24—44no—
278C24—44no—
279C24—432no—
280C24—44no—
281C24—44no—
282C24—432no—
283C24—44no—
284C24—44no—
285C24—44no—
286C24—44no—
287C24—44no—
288C24—44no—
289C24—44no—
290C24—44no—
291C24—816no—
292C24—816no—
293C24—816no—
294C24—816no—
295C24#10414yes—
296C24C2×C22⋊C411yescharacteristic
297C24C2×C22⋊C411yescharacteristic
298C24C2×C22⋊C411yescharacteristic
299C2×D4—28no—
300C2×D4—28no—
301C2×D4—24no—
302C2×D4—28no—
303C2×D4—44no—
304C2×D4—44no—
305C2×D4—432no—
306C2×D4—432no—
307C2×D4—432no—
308C2×D4—44no—
309C2×D4—44no—
310C2×D4—432no—
311C2×D4—432no—
312C2×D4—416no—
313C2×D4—416no—
314C2×D4—44no—
315C2×D4—44no—
316C2×D4—44no—
317C2×D4—44no—
318C2×D4C2×M1618yes—
319C22×C4—232no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—24no—
358C22×C4—41no—
359C22×C4—41no—
360C22×C4—41no—
361C22×C4—41no—
362C22×C4—41no—
363C22×C4—41no—
364C22×C4—41no—
365C22×C4—41no—
366C22×C4—41no—
367C22×C4—41no—
368C22×C4—41no—
369C22×C4—41no—
370C22×C4—416no—
371C22×C4—416no—
372C22×C4—44no—
373C22×C4—44no—
374C22×C4—44no—
375C22×C4—44no—
376C22×C4—44no—
377C22×C4—44no—
378C22×C4—44no—
379C22×C4—44no—
380C22×C4—44no—
381C22×C4—816no—
382C22×C4#9914yes—
383C22×C4#10014yes—
384C22×C4C2×C22⋊C411yescharacteristic
385C22×C4C2×C22⋊C411yescharacteristic
386C22×C4C2×C22⋊C411yescharacteristic
387C22×C4C2×C22⋊C411yescharacteristic
388C22×C4C2×C22⋊C411yescharacteristic
389C22×C4C2×C22⋊C411yescharacteristic
390C22×C4C2×C22⋊C411yescharacteristic
391C22×C4C2×C22⋊C411yescharacteristic
392C22×C4C2×C22⋊C411yescharacteristic
393C22×C4C2×M1614yes—
394C22×C4AGL(1,ℤ/8)14yes—
395C22×C4AGL(1,ℤ/8)14yes—
396C22×C4C23×C411yescharacteristic
397C22×C4C22×D411yescharacteristic
398C22×C4C22×D411yescharacteristic
399M16—416no—
400C2×C8—28no—
401C2×C8—44no—
402C2×C8—44no—
403C2×C8—44no—
404C2×C8—44no—
405C2×C8—44no—
406C4⋊C4—44no—
407C4⋊C4—44no—
408C4⋊C4—44no—
409C4⋊C4—44no—
410C22⋊C4—48no—
411C22⋊C4—48no—
412C22⋊C4—48no—
413C22⋊C4—48no—
414C22⋊C4—48no—
415C22⋊C4—48no—
416C22⋊C4—48no—
417C22⋊C4—48no—
418C42—22no—
419C42—22no—
420C42—22no—
421C42—22no—
422C42—22no—
423C42—22no—
424C42—22no—
425C42—22no—
426C42—41no—
427C42—41no—
428C42—41no—
429C42—41no—
430C42—41no—
431C42—41no—
432C42—41no—
433C42—41no—
434C23—28no—
435C23—28no—
436C23—28no—
437C23—28no—
438C23—21no—
439C23—21no—
440C23—21no—
441C23—21no—
442C23—21no—
443C23—21no—
444C23—21no—
445C23—21no—
446C23—21no—
447C23—21no—
448C23—21no—
449C23—21no—
450C23—28no—
451C23—28no—
452C23—28no—
453C23—28no—
454C23—416no—
455C23—416no—
456C23—416no—
457C23—416no—
458C23—416no—
459C23—416no—
460C23—416no—
461C23—416no—
462C23—44no—
463C23—44no—
464C23—44no—
465C23—44no—
466C23—44no—
467C23—44no—
468C23—416no—
469C23—8128no—
470C23—88no—
471C23—88no—
472C23—88no—
473C23—88no—
474C23—88no—
475C23—88no—
476C23—88no—
477C23—88no—
478C23—88no—
479C23—88no—
480C23—88no—
481C23—88no—
482C23#40712yes—
483C23C2×#9912yes—
484C23C2×#10012yes—
485C23#41812yes—
486C23#41812yes—
487C23C2×#10412yes—
488C23C2×#10412yes—
489C23C2×#10412yes—
490C23#42112yes—
491C23C22×C22⋊C411yescharacteristic
492D4—48no—
493D4—48no—
494D4—416no—
495D4—416no—
496D4—416no—
497C2×C4—28no—
498C2×C4—28no—
499C2×C4—28no—
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—21no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—28no—
515C2×C4—28no—
516C2×C4—28no—
517C2×C4—28no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—44no—
531C2×C4—44no—
532C2×C4—44no—
533C2×C4—44no—
534C2×C4—44no—
535C2×C4—44no—
536C2×C4—44no—
537C2×C4—44no—
538C2×C4—44no—
539C2×C4—44no—
540C2×C4—44no—
541C2×C4—44no—
542C2×C4—44no—
543C2×C4—44no—
544C2×C4—88no—
545C2×C4—88no—
546C2×C4—88no—
547C2×C4#40712yes—
548C2×C4C2×#9912yes—
549C2×C4C2×#9912yes—
550C2×C4C2×#10012yes—
551C2×C4C2×#10012yes—
552C2×C4#41812yes—
553C2×C4#41812yes—
554C2×C4C22×M1612yes—
555C2×C4C2×AGL(1,ℤ/8)12yes—
556C2×C4C2×AGL(1,ℤ/8)12yes—
557C8—48no—
558C8—84no—
559C22—232no—
560C22—22no—
561C22—22no—
562C22—22no—
563C22—22no—
564C22—22no—
565C22—22no—
566C22—22no—
567C22—22no—
568C22—22no—
569C22—22no—
570C22—22no—
571C22—22no—
572C22—22no—
573C22—22no—
574C22—22no—
575C22—22no—
576C22—22no—
577C22—22no—
578C22—48no—
579C22—48no—
580C22—48no—
581C22—832no—
582C22—832no—
583C22—832no—
584C22—84no—
585C22—84no—
586C22—84no—
587C22—84no—
588C22#149014yes—
589C22#150714yes—
590C22#152714yes—
591C22C2×#40711yescharacteristic
592C22C22×#9911yescharacteristic
593C22C22×#10011yescharacteristic
594C22C2×#41811yescharacteristic
595C22C2×#41811yescharacteristic
596C22C22×#10411yescharacteristic
597C22C2×#42111yescharacteristic
598C4—22no—
599C4—22no—
600C4—22no—
601C4—22no—
602C4—22no—
603C4—22no—
604C4—22no—
605C4—41no—
606C4—41no—
607C4—41no—
608C4—41no—
609C4—84no—
610C4#146214yes—
611C2—24no—
612C2—24no—
613C2—24no—
614C2—88no—
615C2C2×#146211yescharacteristic
616C2C2×#149011yescharacteristic
617C2#1744811yescharacteristic
618C2C2×#150711yescharacteristic
619C2#1757711yescharacteristic
620C2C2×#152711yescharacteristic
621C2#1774111yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC24×D4C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C23×C42 ◁ G
LECSG ▷ C2×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #20715C2semidirect
3. C4×#407C2semidirect
4. #7626C2semidirect
5. #7606C2semidirect
6. #7572C2semidirect
27. #1817C22semidirect
29. #1797C22semidirect
33. #1749C22semidirect
34. #1749C22semidirect
36. C4×C22⋊C8C22semidirect
37. #1728C22semidirect
38. #1270C22semidirect
118. #336C23semidirect
119. #336C23semidirect
153. C22×D4M16semidirect

Maximal subgroups

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

subgroupindex
C22×#3902
#207152
C4×#4072
#76262
#76062
#75722