Finite Group Databasebeta

← #239655

Subgroups — #239655

Order 512.

1426 conjugacy classes of subgroups.

The subgroup lattice has 544 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2548C211yesmaximal, characteristic
2#21803C211yesmaximal, characteristic
3#21170C211yesmaximal, characteristic, complemented
4#20734C211yesmaximal, characteristic, complemented
5C2×#1545C211yesmaximal, characteristic
6C2×#1487C211yesmaximal, characteristic, complemented
7#10411C214yesmaximal, complemented
8#10410C214yesmaximal, complemented
9#7892C211yesmaximal, characteristic, complemented
10C22×#127C2211yescharacteristic
11C22×#124—21no—
12C22×C4×Q8C2211yescharacteristic
13C2×#431C2211yescharacteristic
14C2×#431C2211yescharacteristic
15C2×#428—21no—
16C2×#423C2211yescharacteristic
17#2548—22no—
18#2548—22no—
19#2548—22no—
20#2548C2214yes—
21C2×#394—22no—
22C2×#390—21no—
23C2×#390—21no—
24C2×#390—21no—
25#2285C2211yescharacteristic
26C2×C4×C4⋊C4C2211yescharacteristic
27#1982—22no—
28#1957C2212yes—
29#1957C2212yes—
30#1949—22no—
31#1948C2212yescomplemented
32#1947C2212yescomplemented
33#1824—21no—
34#1803C2212yescomplemented
35#1803C2212yescomplemented
36#1787—21no—
37#1786—22no—
38C4×#103—22no—
39#1736—21no—
40#1545—24no—
41C2×#331C2214yes—
42#1487—24no—
43C2×Q8⋊C8C2214yescomplemented
44#1279C2214yescomplemented
45C23×Q8D411yescharacteristic
46C23×D4—21no—
47C23×D4—21no—
48C23×D4D412yes—
49C22×M16—21no—
50C2×#127—22no—
51C2×#127—21no—
52C2×#127—21no—
53C2×#127—22no—
54C2×#127—22no—
55C2×#127—22no—
56C2×#127D412yes—
57C2×#127C2312yes—
58C2×#124—24no—
59C2×#124—24no—
60C2×#124—48no—
61C2×C4×Q8—21no—
62C2×C4×Q8—21no—
63C2×C4×Q8—22no—
64C2×C4×Q8—22no—
65C2×C4×Q8—22no—
66C2×C4×Q8—22no—
67C2×C4×Q8D412yes—
68C2×C4×Q8C2312yes—
69C2×#117—21no—
70C22×C4⋊C4—21no—
71C22×C4⋊C4D411yescharacteristic
72C22×C4⋊C4D411yescharacteristic
73C22×C4⋊C4D411yescharacteristic
74C22×C22⋊C4—22no—
75C22×C22⋊C4—22no—
76C22×C22⋊C4—22no—
77C22×C22⋊C4—22no—
78C22×C42—21no—
79C22×C42—21no—
80C22×C42C2311yescharacteristic
81#431—22no—
82#431—22no—
83#431—22no—
84#431—22no—
85#428—44no—
86C2×#107—22no—
87C2×#106—22no—
88#423—22no—
89#423D414yes—
90C2×C4⋊C8—21no—
91C2×C4⋊C8—21no—
92C2×C4⋊C8C2312yes—
93C2×C4⋊C8C2312yes—
94C2×#103—22no—
95C2×#103—22no—
96C2×#102—24no—
97C2×#102—24no—
98C2×C22⋊C8—21no—
99C2×C22⋊C8—21no—
100C2×C4×C8—21no—
101C2×C4×C8C2312yes—
102#394—28no—
103#394—28no—
104#390—24no—
105#390—24no—
106#390—24no—
107#390—24no—
108#390—24no—
109#390—24no—
110#384C2311yescharacteristic
111#384C2311yescharacteristic
112#382—21no—
113#380—21no—
114#380—21no—
115C4×C4⋊C4—21no—
116C4×C4⋊C4—21no—
117C4×C4⋊C4—21no—
118C4×C4⋊C4—21no—
119C4×C4⋊C4D412yes—
120C4×C4⋊C4C2312yes—
121C2×#95—21no—
122#340—22no—
123#340—22no—
124#340—22no—
125#340—22no—
126#331—28no—
127Q8⋊C8—28no—
128C25—22no—
129C22×Q8—24no—
130C22×Q8—21no—
131C22×Q8—22no—
132C22×Q8—22no—
133C22×Q8—42no—
134C22×Q8C2×D412yes—
135C22×D4—22no—
136C22×D4—22no—
137C22×D4—22no—
138C22×D4—24no—
139C22×D4—24no—
140C22×D4—22no—
141C22×D4—22no—
142C22×D4—44no—
143C22×D4—44no—
144C22×D4—42no—
145C22×D4—42no—
146C22×D4—42no—
147C22×D4—42no—
148C22×D4—42no—
149C22×D4—42no—
150C22×D4—42no—
151C22×D4—42no—
152C22×D4—82no—
153C22×D4—82no—
154C22×D4C2×D414yes—
155C23×C4—21no—
156C23×C4—21no—
157C23×C4—21no—
158C23×C4—21no—
159C23×C4C2×D411yescharacteristic
160C23×C4C2×D411yescharacteristic
161C23×C4C2×D411yescharacteristic
162C2×M16—24no—
163C2×M16—42no—
164C2×M16—42no—
165C22×C8—22no—
166C22×C8—22no—
167C22×C8—22no—
168#127—24no—
169#127—24no—
170#127—22no—
171#127—22no—
172#127—22no—
173#127—22no—
174#127—42no—
175#127—42no—
176#127—42no—
177#127—42no—
178#124—416no—
179#124—48no—
180#124—48no—
181C4×Q8—24no—
182C4×Q8—24no—
183C4×Q8—22no—
184C4×Q8—22no—
185C4×Q8—22no—
186C4×Q8—22no—
187C4×Q8—42no—
188C4×Q8—42no—
189C4×Q8—42no—
190C4×Q8—42no—
191#117—42no—
192#117—42no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—21no—
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—42no—
211C2×C4⋊C4—42no—
212C2×C4⋊C4C2×D412yes—
213C2×C4⋊C4C2×D412yes—
214C2×C4⋊C4C2×D412yes—
215C2×C22⋊C4—28no—
216C2×C22⋊C4—28no—
217C2×C22⋊C4—28no—
218C2×C22⋊C4—28no—
219C2×C22⋊C4—48no—
220C2×C22⋊C4—48no—
221C2×C22⋊C4—41no—
222C2×C22⋊C4—84no—
223C2×C22⋊C4—84no—
224C2×C42—22no—
225C2×C42—22no—
226C2×C42—22no—
227C2×C42—22no—
228C2×C42—44no—
229C2×C42—42no—
230C2×C42—42no—
231C2×C42—41no—
232C2×C42C2×D412yes—
233C2×C42C2×D411yescharacteristic
234C2×C42C2×D411yescharacteristic
235C2×C42C2×D411yescharacteristic
236C2×C42#4112yes—
237C2×C42#4112yes—
238C2×C42#4112yes—
239C2×C42C2411yescharacteristic
240#107—44no—
241#106—44no—
242C4⋊C8—24no—
243C4⋊C8—22no—
244C4⋊C8—22no—
245C4⋊C8—22no—
246C4⋊C8—42no—
247C4⋊C8C2×D414yes—
248#103—44no—
249#103—44no—
250#102—84no—
251#102—84no—
252C22⋊C8—42no—
253C22⋊C8—42no—
254C4×C8—22no—
255C4×C8—22no—
256C4×C8—42no—
257#95—22no—
258#95—22no—
259#95—41no—
260#95—41no—
261C24—24no—
262C24—44no—
263C24—44no—
264C24—44no—
265C24—44no—
266C24—44no—
267C24—44no—
268C24—44no—
269C24#12011yescharacteristic
270C2×Q8—24no—
271C2×Q8—22no—
272C2×Q8—22no—
273C2×Q8—22no—
274C2×Q8—22no—
275C2×Q8—22no—
276C2×Q8—48no—
277C2×Q8—48no—
278C2×Q8—42no—
279C2×Q8—42no—
280C2×Q8AGL(1,ℤ/8)14yes—
281C2×D4—24no—
282C2×D4—24no—
283C2×D4—44no—
284C2×D4—44no—
285C2×D4—44no—
286C2×D4—44no—
287C2×D4—42no—
288C2×D4—42no—
289C2×D4—42no—
290C2×D4—42no—
291C2×D4—42no—
292C2×D4—42no—
293C2×D4—44no—
294C2×D4—42no—
295C2×D4—42no—
296C2×D4—44no—
297C2×D4—44no—
298C2×D4—44no—
299C2×D4—44no—
300C2×D4—44no—
301C2×D4—44no—
302C2×D4—82no—
303C2×D4—82no—
304C2×D4—88no—
305C2×D4—88no—
306C2×D4—88no—
307C2×D4—88no—
308C2×D4—84no—
309C2×D4—84no—
310C2×D4—84no—
311C2×D4—84no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—24no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—41no—
320C22×C4—41no—
321C22×C4—42no—
322C22×C4—42no—
323C22×C4—42no—
324C22×C4—42no—
325C22×C4—42no—
326C22×C4—42no—
327C22×C4—42no—
328C22×C4—42no—
329C22×C4—42no—
330C22×C4—42no—
331C22×C4—42no—
332C22×C4—41no—
333C22×C4—41no—
334C22×C4#12011yescharacteristic
335C22×C4#12011yescharacteristic
336C22×C4#12011yescharacteristic
337C22×C4#12112yes—
338C22×C4#12112yes—
339C22×C4#12112yes—
340C22×C4#12312yes—
341C22×C4#12312yes—
342C22×C4#12412yes—
343C22×C4#13512yes—
344C22×C4#13512yes—
345C22×C4AGL(1,ℤ/8)12yes—
346C22×C4#13712yes—
347C22×C4C22×D411yescharacteristic
348C22×C4C22×D411yescharacteristic
349C22×C4C22×D411yescharacteristic
350M16—48no—
351C2×C8—24no—
352C2×C8—42no—
353C2×C8—42no—
354C2×C8—42no—
355C2×C8—42no—
356C2×C8—42no—
357C2×C8—42no—
358C2×C8—42no—
359C2×C8—42no—
360C4⋊C4—22no—
361C4⋊C4—22no—
362C4⋊C4—22no—
363C4⋊C4—22no—
364C4⋊C4—22no—
365C4⋊C4—22no—
366C4⋊C4—42no—
367C4⋊C4—42no—
368C4⋊C4—42no—
369C4⋊C4—42no—
370C4⋊C4—42no—
371C4⋊C4—44no—
372C4⋊C4—42no—
373C4⋊C4—42no—
374C4⋊C4—41no—
375C4⋊C4—41no—
376C4⋊C4—42no—
377C4⋊C4—41no—
378C4⋊C4—41no—
379C4⋊C4—42no—
380C22⋊C4—416no—
381C22⋊C4—416no—
382C22⋊C4—88no—
383C22⋊C4—88no—
384C22⋊C4—82no—
385C42—44no—
386C42—42no—
387C42—42no—
388C42—44no—
389C42—44no—
390C42—82no—
391C42#12112yes—
392C42#12112yes—
393C42#12112yes—
394C42#12312yes—
395C42#12312yes—
396C42#12412yes—
397C42C22×D411yescharacteristic
398C42C2×#4111yescharacteristic
399C42C2×#4111yescharacteristic
400C42C2×#4111yescharacteristic
401C23—21no—
402C23—21no—
403C23—21no—
404C23—21no—
405C23—44no—
406C23—44no—
407C23—44no—
408C23—44no—
409C23—44no—
410C23—44no—
411C23—44no—
412C23—88no—
413C23—88no—
414C23—88no—
415C23—88no—
416C23—88no—
417C23—88no—
418C23—88no—
419C23#39412yes—
420C23#44912yes—
421C23#45212yes—
422C23C2×#12011yescharacteristic
423Q8—24no—
424Q8—24no—
425Q8—48no—
426Q8—42no—
427Q8—42no—
428D4—48no—
429D4—84no—
430D4—88no—
431D4—88no—
432D4—84no—
433D4—82no—
434D4—82no—
435D4—82no—
436D4—82no—
437D4—84no—
438D4—88no—
439C2×C4—21no—
440C2×C4—21no—
441C2×C4—21no—
442C2×C4—21no—
443C2×C4—21no—
444C2×C4—21no—
445C2×C4—24no—
446C2×C4—24no—
447C2×C4—24no—
448C2×C4—42no—
449C2×C4—42no—
450C2×C4—48no—
451C2×C4—42no—
452C2×C4—42no—
453C2×C4—42no—
454C2×C4—42no—
455C2×C4—42no—
456C2×C4—42no—
457C2×C4—42no—
458C2×C4—42no—
459C2×C4—42no—
460C2×C4—42no—
461C2×C4—42no—
462C2×C4—84no—
463C2×C4—84no—
464C2×C4—84no—
465C2×C4—81no—
466C2×C4—81no—
467C2×C4—81no—
468C2×C4—81no—
469C2×C4—81no—
470C2×C4—81no—
471C2×C4#39412yes—
472C2×C4#39412yes—
473C2×C4#39412yes—
474C2×C4#44912yes—
475C2×C4#45212yes—
476C2×C4#46314yes—
477C2×C4#46414yes—
478C2×C4#48814yes—
479C2×C4#48914yes—
480C2×C4C2×#12111yescharacteristic
481C2×C4C2×#12111yescharacteristic
482C2×C4C2×#12111yescharacteristic
483C2×C4C2×#12311yescharacteristic
484C2×C4C2×#12311yescharacteristic
485C2×C4C2×#12411yescharacteristic
486C2×C4C2×#13511yescharacteristic
487C2×C4C2×#13511yescharacteristic
488C2×C4C2×AGL(1,ℤ/8)12yes—
489C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
490C2×C4C2×#13711yescharacteristic
491C2×C4#57512yes—
492C8—44no—
493C8—82no—
494C8—82no—
495C22—22no—
496C22—22no—
497C22—22no—
498C22—22no—
499C22—22no—
500C22—22no—
501C22—84no—
502C22—84no—
503C22—84no—
504C22—84no—
505C22—84no—
506C22—84no—
507C22—84no—
508C22—1616no—
509C22#204214yes—
510C22C2×#39411yescharacteristic
511C22C2×#44911yescharacteristic
512C22C2×#45211yescharacteristic
513C22#300311yescharacteristic
514C22#300511yescharacteristic
515C22#301911yescharacteristic
516C22#301911yescharacteristic
517C4—22no—
518C4—21no—
519C4—21no—
520C4—22no—
521C4—44no—
522C4—82no—
523C4—82no—
524C4—82no—
525C4—161no—
526C4—161no—
527C4#204214yes—
528C4C2×#46311yescharacteristic
529C4C2×#46411yescharacteristic
530C4#305512yes—
531C4#309412yes—
532C4C2×#48811yescharacteristic
533C4C2×#48911yescharacteristic
534C2—24no—
535C2—164no—
536C2#1875011yescharacteristic
537C2#1875111yescharacteristic
538C2#1877011yescharacteristic
539C2#1877111yescharacteristic
540C2#2348011yescharacteristic
541C2#2349311yescharacteristic
542C2C2×#204211yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC2×C42C24
prime radicalC22×#127C22
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #21170C2semidirect
4. #20734C2semidirect
6. C2×#1487C2semidirect
7. #10411C2semidirect
8. #10410C2semidirect
9. #7892C2semidirect
31. #1948C22semidirect
32. #1947C22semidirect
34. #1803C22semidirect
35. #1803C22semidirect
43. C2×Q8⋊C8C22semidirect
44. #1279C22semidirect

Maximal subgroups

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

subgroupindex
C2×#25482
#218032
#211702
#207342
C2×#15452
C2×#14872
#104112
#104102
#78922