Finite Group Databasebeta

← #238677

Subgroups — #238677

Order 512.

1338 conjugacy classes of subgroups.

The subgroup lattice has 542 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2538C211yesmaximal, characteristic
2#21981C211yesmaximal, characteristic, complemented
3#20732C211yesmaximal, characteristic
4#20647C211yesmaximal, characteristic, complemented
5C2×#1486C211yesmaximal, characteristic
6C2×#1479C211yesmaximal, characteristic, complemented
7#10436C214yesmaximal, complemented
8#10435C214yesmaximal, complemented
9#8043C211yesmaximal, characteristic, complemented
10C22×C22⋊Q8—21no—
11C22×C4×Q8C2211yescharacteristic
12C22×C4×D4C2211yescharacteristic
13C2×#431C2211yescharacteristic
14C2×#431C2211yescharacteristic
15C2×#423C2211yescharacteristic
16C2×#419—21no—
17#2538—22no—
18#2538—22no—
19#2538—22no—
20#2538C2214yes—
21C2×#396—22no—
22C2×#390—21no—
23C2×#389—21no—
24#2285C2211yescharacteristic
25C2×#384C2211yescharacteristic
26C2×#380—21no—
27#1948—22no—
28#1853—22no—
29#1844C2212yescomplemented
30#1834C2212yescomplemented
31#1823—21no—
32#1805—22no—
33#1803C2212yescomplemented
34#1803C2212yescomplemented
35#1802C2212yes—
36#1802C2212yes—
37#1786—21no—
38C4×#102—22no—
39#1736—21no—
40#1486—24no—
41#1479—24no—
42C2×Q8⋊C8C2214yescomplemented
43C2×D4⋊C8C2214yes—
44#1296C2214yescomplemented
45C23×Q8D411yescharacteristic
46C23×D4D411yescharacteristic
47C24×C4—21no—
48C22×M16—21no—
49C2×C22⋊Q8—24no—
50C2×C22⋊Q8—24no—
51C2×C22⋊Q8—44no—
52C2×C22⋊Q8—44no—
53C2×C4×Q8—21no—
54C2×C4×Q8—21no—
55C2×C4×Q8—22no—
56C2×C4×Q8—22no—
57C2×C4×Q8—22no—
58C2×C4×Q8—22no—
59C2×C4×Q8D412yes—
60C2×C4×Q8C2312yes—
61C2×C4×D4—21no—
62C2×C4×D4—21no—
63C2×C4×D4—22no—
64C2×C4×D4—22no—
65C2×C4×D4—22no—
66C2×C4×D4—22no—
67C2×C4×D4D412yes—
68C2×C4×D4C2312yes—
69C2×#117—21no—
70C22×C4⋊C4—22no—
71C22×C4⋊C4—21no—
72C22×C4⋊C4—21no—
73C22×C4⋊C4—21no—
74C22×C4⋊C4D411yescharacteristic
75C22×C4⋊C4D411yescharacteristic
76C22×C4⋊C4D412yes—
77C22×C22⋊C4—22no—
78C22×C22⋊C4—21no—
79C22×C22⋊C4—22no—
80C22×C42—21no—
81C22×C42C2311yescharacteristic
82#431—22no—
83#431—22no—
84#431—22no—
85#431D414yes—
86C2×#106—24no—
87#423—22no—
88#423—22no—
89C2×C4⋊C8—21no—
90C2×C4⋊C8—21no—
91C2×C4⋊C8C2312yes—
92C2×C4⋊C8C2312yes—
93#419—44no—
94C2×#103—22no—
95C2×#103—22no—
96C2×#102—22no—
97C2×#102—22no—
98C2×C22⋊C8—21no—
99C2×C22⋊C8—21no—
100C2×C4×C8—21no—
101C2×C4×C8C2312yes—
102#396—28no—
103#396—28no—
104#390—24no—
105#390—24no—
106#389—24no—
107#389—24no—
108#384—21no—
109#384—21no—
110#384D412yes—
111#384C2311yescharacteristic
112#384C2311yescharacteristic
113#384C2312yes—
114#382—21no—
115#380—24no—
116#380—24no—
117#380—21no—
118#380—21no—
119C4×C4⋊C4—21no—
120C4×C4⋊C4—21no—
121C2×#95—22no—
122#340—22no—
123#340—22no—
124#340—22no—
125#340—22no—
126#340—24no—
127Q8⋊C8—28no—
128D4⋊C8—28no—
129C25—21no—
130C22×Q8—21no—
131C22×Q8—22no—
132C22×Q8—22no—
133C22×Q8—22no—
134C22×Q8—22no—
135C22×Q8—22no—
136C22×Q8SD1614yes—
137C22×Q8C2×D412yes—
138C22×D4—24no—
139C22×D4—21no—
140C22×D4—22no—
141C22×D4—22no—
142C22×D4—42no—
143C22×D4C2×D412yes—
144C23×C4—21no—
145C23×C4—22no—
146C23×C4—21no—
147C23×C4—21no—
148C23×C4—22no—
149C23×C4—22no—
150C23×C4—42no—
151C23×C4—42no—
152C23×C4—42no—
153C23×C4—42no—
154C23×C4—42no—
155C23×C4—42no—
156C23×C4C2×D411yescharacteristic
157C23×C4C2×D411yescharacteristic
158C23×C4C2×D411yescharacteristic
159C2×M16—42no—
160C2×M16—42no—
161C2×M16—42no—
162C22×C8—22no—
163C22×C8—22no—
164C22×C8—22no—
165C22⋊Q8—48no—
166C22⋊Q8—416no—
167C22⋊Q8—48no—
168C4×Q8—24no—
169C4×Q8—24no—
170C4×Q8—22no—
171C4×Q8—22no—
172C4×Q8—22no—
173C4×Q8—22no—
174C4×Q8—42no—
175C4×Q8—42no—
176C4×Q8—42no—
177C4×Q8—42no—
178C4×D4—24no—
179C4×D4—24no—
180C4×D4—22no—
181C4×D4—22no—
182C4×D4—22no—
183C4×D4—22no—
184C4×D4—42no—
185C4×D4—42no—
186C4×D4—42no—
187C4×D4—42no—
188#117—42no—
189#117—42no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—28no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—44no—
211C2×C4⋊C4—48no—
212C2×C4⋊C4—42no—
213C2×C4⋊C4—42no—
214C2×C4⋊C4—42no—
215C2×C4⋊C4—42no—
216C2×C4⋊C4—82no—
217C2×C4⋊C4C2×D412yes—
218C2×C4⋊C4C2×D412yes—
219C2×C4⋊C4C2×D414yes—
220C2×C22⋊C4—28no—
221C2×C22⋊C4—22no—
222C2×C22⋊C4—22no—
223C2×C22⋊C4—28no—
224C2×C22⋊C4—48no—
225C2×C22⋊C4—42no—
226C2×C22⋊C4—42no—
227C2×C22⋊C4—41no—
228C2×C22⋊C4—84no—
229C2×C42—22no—
230C2×C42—22no—
231C2×C42—42no—
232C2×C42—42no—
233C2×C42—41no—
234C2×C42C2×D412yes—
235C2×C42C2×D411yescharacteristic
236C2×C42C2×D411yescharacteristic
237C2×C42C2×D411yescharacteristic
238C2×C42#4112yes—
239C2×C42#4112yes—
240C2×C42#4112yes—
241C2×C42C2411yescharacteristic
242#106—84no—
243C4⋊C8—24no—
244C4⋊C8—22no—
245C4⋊C8—22no—
246C4⋊C8—22no—
247C4⋊C8—22no—
248C4⋊C8—42no—
249#103—44no—
250#103—44no—
251#102—44no—
252#102—44no—
253C22⋊C8—24no—
254C22⋊C8—42no—
255C4×C8—22no—
256C4×C8—42no—
257C4×C8C2×D414yes—
258#95—28no—
259#95—41no—
260#95—41no—
261C24—24no—
262C24—22no—
263C24—42no—
264C24—42no—
265C24—42no—
266C24—42no—
267C24—42no—
268C24—42no—
269C24#12011yescharacteristic
270C2×Q8—22no—
271C2×Q8—24no—
272C2×Q8—22no—
273C2×Q8—22no—
274C2×Q8—22no—
275C2×Q8—22no—
276C2×Q8—28no—
277C2×Q8—28no—
278C2×Q8—42no—
279C2×Q8—42no—
280C2×Q8—44no—
281C2×Q8—44no—
282C2×Q8C2×SD1614yes—
283C2×D4—24no—
284C2×D4—22no—
285C2×D4—22no—
286C2×D4—22no—
287C2×D4—22no—
288C2×D4—22no—
289C2×D4—48no—
290C2×D4—48no—
291C2×D4—42no—
292C2×D4—42no—
293C2×D4#13714yes—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—24no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—24no—
301C22×C4—41no—
302C22×C4—41no—
303C22×C4—42no—
304C22×C4—42no—
305C22×C4—42no—
306C22×C4—44no—
307C22×C4—44no—
308C22×C4—42no—
309C22×C4—42no—
310C22×C4—42no—
311C22×C4—42no—
312C22×C4—44no—
313C22×C4—42no—
314C22×C4—42no—
315C22×C4—42no—
316C22×C4—42no—
317C22×C4—42no—
318C22×C4—42no—
319C22×C4—42no—
320C22×C4—44no—
321C22×C4—44no—
322C22×C4—44no—
323C22×C4—41no—
324C22×C4—41no—
325C22×C4—84no—
326C22×C4—84no—
327C22×C4—84no—
328C22×C4—84no—
329C22×C4#12011yescharacteristic
330C22×C4#12011yescharacteristic
331C22×C4#12011yescharacteristic
332C22×C4#12112yes—
333C22×C4#12112yes—
334C22×C4#12112yes—
335C22×C4#12312yes—
336C22×C4#12312yes—
337C22×C4#12412yes—
338C22×C4C2×SD1612yes—
339C22×C4#13512yes—
340C22×C4AGL(1,ℤ/8)12yes—
341C22×C4AGL(1,ℤ/8)12yes—
342C22×C4C22×D411yescharacteristic
343C22×C4C22×D411yescharacteristic
344C22×C4C22×D411yescharacteristic
345M16—84no—
346C2×C8—24no—
347C2×C8—42no—
348C2×C8—42no—
349C2×C8—42no—
350C2×C8—42no—
351C2×C8—42no—
352C2×C8—42no—
353C2×C8—42no—
354C2×C8—42no—
355C4⋊C4—22no—
356C4⋊C4—22no—
357C4⋊C4—22no—
358C4⋊C4—22no—
359C4⋊C4—44no—
360C4⋊C4—44no—
361C4⋊C4—416no—
362C4⋊C4—42no—
363C4⋊C4—42no—
364C4⋊C4—42no—
365C4⋊C4—42no—
366C4⋊C4—42no—
367C4⋊C4—42no—
368C4⋊C4—44no—
369C4⋊C4—42no—
370C4⋊C4—42no—
371C4⋊C4—44no—
372C4⋊C4—44no—
373C4⋊C4—41no—
374C4⋊C4—41no—
375C4⋊C4—41no—
376C4⋊C4—41no—
377C4⋊C4—42no—
378C4⋊C4—42no—
379C4⋊C4—82no—
380C22⋊C4—416no—
381C22⋊C4—42no—
382C22⋊C4—42no—
383C22⋊C4—44no—
384C22⋊C4—88no—
385C22⋊C4—82no—
386C42—42no—
387C42—42no—
388C42—44no—
389C42—82no—
390C42#12112yes—
391C42#12112yes—
392C42#12112yes—
393C42#12312yes—
394C42#12312yes—
395C42#12412yes—
396C42C22×D411yescharacteristic
397C42C2×#4111yescharacteristic
398C42C2×#4111yescharacteristic
399C42C2×#4111yescharacteristic
400C23—21no—
401C23—21no—
402C23—21no—
403C23—21no—
404C23—24no—
405C23—48no—
406C23—48no—
407C23—48no—
408C23—42no—
409C23—42no—
410C23—42no—
411C23—42no—
412C23—42no—
413C23—42no—
414C23—84no—
415C23—84no—
416C23—84no—
417C23—84no—
418C23#39412yes—
419C23#44912yes—
420C23#45012yes—
421C23C2×#12011yescharacteristic
422Q8—24no—
423Q8—24no—
424Q8—42no—
425Q8—42no—
426Q8—48no—
427D4—24no—
428D4—24no—
429D4—48no—
430D4—42no—
431D4—42no—
432C2×C4—21no—
433C2×C4—21no—
434C2×C4—21no—
435C2×C4—21no—
436C2×C4—21no—
437C2×C4—21no—
438C2×C4—24no—
439C2×C4—24no—
440C2×C4—24no—
441C2×C4—42no—
442C2×C4—42no—
443C2×C4—42no—
444C2×C4—42no—
445C2×C4—48no—
446C2×C4—42no—
447C2×C4—42no—
448C2×C4—44no—
449C2×C4—42no—
450C2×C4—42no—
451C2×C4—44no—
452C2×C4—42no—
453C2×C4—44no—
454C2×C4—42no—
455C2×C4—88no—
456C2×C4—84no—
457C2×C4—84no—
458C2×C4—82no—
459C2×C4—82no—
460C2×C4—82no—
461C2×C4—82no—
462C2×C4—88no—
463C2×C4—81no—
464C2×C4—81no—
465C2×C4—81no—
466C2×C4—81no—
467C2×C4—81no—
468C2×C4—81no—
469C2×C4#39412yes—
470C2×C4#39412yes—
471C2×C4#39412yes—
472C2×C4#44912yes—
473C2×C4#45012yes—
474C2×C4#46514yes—
475C2×C4#46914yes—
476C2×C4#48114yes—
477C2×C4#48214yes—
478C2×C4C2×#12111yescharacteristic
479C2×C4C2×#12111yescharacteristic
480C2×C4C2×#12111yescharacteristic
481C2×C4C2×#12311yescharacteristic
482C2×C4C2×#12311yescharacteristic
483C2×C4C2×#12411yescharacteristic
484C2×C4C22×SD1611yescharacteristic
485C2×C4C2×#13511yescharacteristic
486C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
487C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
488C2×C4C2×#13712yes—
489C2×C4#57512yes—
490C8—44no—
491C8—82no—
492C8—82no—
493C22—22no—
494C22—22no—
495C22—22no—
496C22—22no—
497C22—22no—
498C22—22no—
499C22—44no—
500C22—44no—
501C22—44no—
502C22—816no—
503C22—82no—
504C22—82no—
505C22—82no—
506C22—82no—
507C22#203414yes—
508C22C2×#39411yescharacteristic
509C22C2×#45011yescharacteristic
510C22C2×#44911yescharacteristic
511C22#300411yescharacteristic
512C22#300511yescharacteristic
513C22#301911yescharacteristic
514C22#302011yescharacteristic
515C4—22no—
516C4—21no—
517C4—21no—
518C4—22no—
519C4—44no—
520C4—84no—
521C4—82no—
522C4—82no—
523C4—161no—
524C4—161no—
525C4#203414yes—
526C4C2×#46511yescharacteristic
527C4C2×#46911yescharacteristic
528C4#305612yes—
529C4C2×#48211yescharacteristic
530C4C2×#48111yescharacteristic
531C4#309512yes—
532C2—24no—
533C2—84no—
534C2#1875211yescharacteristic
535C2#1876211yescharacteristic
536C2#1876711yescharacteristic
537C2#1877711yescharacteristic
538C2C2×#203411yescharacteristic
539C2#2348111yescharacteristic
540C2#2349911yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC2×C42C24
prime radicalC23×D4D4
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
2. #21981C2semidirect
4. #20647C2semidirect
6. C2×#1479C2semidirect
7. #10436C2semidirect
8. #10435C2semidirect
9. #8043C2semidirect
29. #1844C22semidirect
30. #1834C22semidirect
33. #1803C22semidirect
34. #1803C22semidirect
42. C2×Q8⋊C8C22semidirect
44. #1296C22semidirect

Maximal subgroups

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

subgroupindex
C2×#25382
#219812
#207322
#206472
C2×#14862
C2×#14792
#104362
#104352
#80432