Finite Group Databasebeta

← #239660

Subgroups — #239660

Order 512.

1422 conjugacy classes of subgroups.

The subgroup lattice has 545 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2542C211yesmaximal, characteristic
2#21833C211yesmaximal, characteristic, complemented
3#21170C211yesmaximal, characteristic, complemented
4#20732C211yesmaximal, characteristic
5C2×#1546C211yesmaximal, characteristic, complemented
6C2×#1486C211yesmaximal, characteristic
7#10421C214yesmaximal, complemented
8#10420C214yesmaximal, complemented
9#7892C211yesmaximal, characteristic, complemented
10C22×C4⋊Q8C2211yescharacteristic
11C22×C22⋊Q8—21no—
12C22×C22⋊Q8—21no—
13C22×C4×D4C2211yescharacteristic
14C2×#431C2211yescharacteristic
15C2×#431C2211yescharacteristic
16C2×#428—21no—
17C2×#423C2211yescharacteristic
18#2542—22no—
19#2542—22no—
20#2542—22no—
21#2542C2214yes—
22C2×#399—22no—
23C2×#390—21no—
24#2285C2211yescharacteristic
25C2×#384—21no—
26C2×C4×C4⋊C4C2211yescharacteristic
27#1990C2212yescomplemented
28#1990C2212yescomplemented
29#1988—22no—
30#1949—22no—
31#1948C2212yescomplemented
32#1947C2212yescomplemented
33#1823—22no—
34#1823—21no—
35#1802C2212yes—
36#1802C2212yes—
37#1786—21no—
38C4×#102—22no—
39#1736—21no—
40#1546—24no—
41C2×#332C2214yescomplemented
42#1486—24no—
43C2×D4⋊C8C2214yes—
44#1279C2214yescomplemented
45C23×Q8—21no—
46C23×D4D411yescharacteristic
47C24×C4—21no—
48C22×M16—21no—
49C2×C4⋊Q8—21no—
50C2×C4⋊Q8—21no—
51C2×C4⋊Q8—22no—
52C2×C4⋊Q8—22no—
53C2×C4⋊Q8—22no—
54C2×C4⋊Q8—22no—
55C2×C4⋊Q8D412yes—
56C2×C4⋊Q8C2312yes—
57C2×C22⋊Q8—24no—
58C2×C22⋊Q8—24no—
59C2×C22⋊Q8—24no—
60C2×C22⋊Q8—24no—
61C2×C22⋊Q8—44no—
62C2×C22⋊Q8—44no—
63C2×C22⋊Q8—44no—
64C2×C22⋊Q8—44no—
65C2×C4×D4—21no—
66C2×C4×D4—21no—
67C2×C4×D4—22no—
68C2×C4×D4—22no—
69C2×C4×D4—22no—
70C2×C4×D4—22no—
71C2×C4×D4D412yes—
72C2×C4×D4C2312yes—
73C2×#117—21no—
74C22×C4⋊C4—22no—
75C22×C4⋊C4—22no—
76C22×C4⋊C4—21no—
77C22×C4⋊C4D412yes—
78C22×C4⋊C4D411yescharacteristic
79C22×C4⋊C4D411yescharacteristic
80C22×C4⋊C4D411yescharacteristic
81C22×C22⋊C4—22no—
82C22×C22⋊C4—22no—
83C22×C22⋊C4—21no—
84C22×C42—21no—
85C22×C42C2311yescharacteristic
86#431—22no—
87#431—22no—
88#431—22no—
89#431—22no—
90#428—44no—
91C2×#107—22no—
92C2×#106—22no—
93#423—22no—
94#423D414yes—
95C2×C4⋊C8—21no—
96C2×C4⋊C8—21no—
97C2×C4⋊C8C2312yes—
98C2×C4⋊C8C2312yes—
99C2×#102—22no—
100C2×#102—22no—
101C2×C22⋊C8—21no—
102C2×C22⋊C8—21no—
103C2×C4×C8—21no—
104C2×C4×C8C2312yes—
105#399—28no—
106#399—28no—
107#390—24no—
108#390—24no—
109#384—24no—
110#384—24no—
111#384C2311yescharacteristic
112#384C2311yescharacteristic
113#382—21no—
114#380—21no—
115#380—21no—
116C4×C4⋊C4—21no—
117C4×C4⋊C4—21no—
118C4×C4⋊C4—21no—
119C4×C4⋊C4—21no—
120C4×C4⋊C4D412yes—
121C4×C4⋊C4C2312yes—
122C2×#95—21no—
123#340—22no—
124#340—22no—
125#340—24no—
126#340—24no—
127#340—22no—
128#340—22no—
129#332—28no—
130D4⋊C8—28no—
131C25—21no—
132C22×Q8—22no—
133C22×Q8—22no—
134C22×Q8—42no—
135C22×Q8—42no—
136C22×Q8—42no—
137C22×Q8—42no—
138C22×Q8—42no—
139C22×Q8—42no—
140C22×D4—24no—
141C22×D4—21no—
142C22×D4—22no—
143C22×D4—22no—
144C22×D4—42no—
145C22×D4C2×D412yes—
146C23×C4—21no—
147C23×C4—21no—
148C23×C4—22no—
149C23×C4—21no—
150C23×C4—22no—
151C23×C4—22no—
152C23×C4—42no—
153C23×C4—42no—
154C23×C4—42no—
155C23×C4—42no—
156C23×C4—42no—
157C23×C4—42no—
158C23×C4C2×D411yescharacteristic
159C23×C4C2×D411yescharacteristic
160C23×C4C2×D411yescharacteristic
161C2×M16—24no—
162C2×M16—42no—
163C2×M16—42no—
164C22×C8—22no—
165C22×C8—22no—
166C22×C8—22no—
167C4⋊Q8—24no—
168C4⋊Q8—24no—
169C4⋊Q8—22no—
170C4⋊Q8—22no—
171C4⋊Q8—22no—
172C4⋊Q8—22no—
173C4⋊Q8—42no—
174C4⋊Q8—42no—
175C4⋊Q8—42no—
176C4⋊Q8—42no—
177C22⋊Q8—48no—
178C22⋊Q8—48no—
179C22⋊Q8—416no—
180C22⋊Q8—416no—
181C22⋊Q8—48no—
182C22⋊Q8—48no—
183C4×D4—24no—
184C4×D4—24no—
185C4×D4—22no—
186C4×D4—22no—
187C4×D4—22no—
188C4×D4—22no—
189C4×D4—42no—
190C4×D4—42no—
191C4×D4—42no—
192C4×D4—42no—
193#117—42no—
194#117—42no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—24no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—24no—
203C2×C4⋊C4—28no—
204C2×C4⋊C4—28no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—48no—
216C2×C4⋊C4—48no—
217C2×C4⋊C4—42no—
218C2×C4⋊C4—42no—
219C2×C4⋊C4D814yes—
220C2×C4⋊C4SD1614yes—
221C2×C4⋊C4C2×D412yes—
222C2×C4⋊C4C2×D414yes—
223C2×C4⋊C4C2×D412yes—
224C2×C4⋊C4C2×D412yes—
225C2×C22⋊C4—28no—
226C2×C22⋊C4—28no—
227C2×C22⋊C4—22no—
228C2×C22⋊C4—22no—
229C2×C22⋊C4—48no—
230C2×C22⋊C4—48no—
231C2×C22⋊C4—42no—
232C2×C22⋊C4—42no—
233C2×C22⋊C4—41no—
234C2×C42—22no—
235C2×C42—22no—
236C2×C42—44no—
237C2×C42—41no—
238C2×C42C2×D412yes—
239C2×C42C2×D411yescharacteristic
240C2×C42C2×D411yescharacteristic
241C2×C42C2×D411yescharacteristic
242C2×C42#4112yes—
243C2×C42#4112yes—
244C2×C42#4112yes—
245C2×C42C2411yescharacteristic
246#107—44no—
247#106—44no—
248C4⋊C8—24no—
249C4⋊C8—22no—
250C4⋊C8—22no—
251C4⋊C8—22no—
252C4⋊C8—42no—
253C4⋊C8C2×D414yes—
254#102—44no—
255#102—44no—
256C22⋊C8—42no—
257C22⋊C8—42no—
258C4×C8—22no—
259C4×C8—22no—
260C4×C8—42no—
261#95—22no—
262#95—22no—
263#95—41no—
264#95—41no—
265C24—24no—
266C24—22no—
267C24—42no—
268C24—42no—
269C24—42no—
270C24—42no—
271C24—42no—
272C24—42no—
273C24#12011yescharacteristic
274C2×Q8—42no—
275C2×Q8—42no—
276C2×Q8—42no—
277C2×Q8—42no—
278C2×Q8—42no—
279C2×Q8—42no—
280C2×Q8—44no—
281C2×Q8—44no—
282C2×Q8—44no—
283C2×Q8—84no—
284C2×Q8—84no—
285C2×Q8—84no—
286C2×Q8—84no—
287C2×D4—24no—
288C2×D4—22no—
289C2×D4—22no—
290C2×D4—22no—
291C2×D4—22no—
292C2×D4—22no—
293C2×D4—48no—
294C2×D4—48no—
295C2×D4—42no—
296C2×D4—42no—
297C2×D4#13714yes—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—24no—
303C22×C4—24no—
304C22×C4—24no—
305C22×C4—22no—
306C22×C4—41no—
307C22×C4—41no—
308C22×C4—42no—
309C22×C4—42no—
310C22×C4—42no—
311C22×C4—42no—
312C22×C4—42no—
313C22×C4—44no—
314C22×C4—44no—
315C22×C4—42no—
316C22×C4—44no—
317C22×C4—42no—
318C22×C4—42no—
319C22×C4—42no—
320C22×C4—42no—
321C22×C4—42no—
322C22×C4—42no—
323C22×C4—42no—
324C22×C4—44no—
325C22×C4—44no—
326C22×C4—44no—
327C22×C4—41no—
328C22×C4—41no—
329C22×C4—84no—
330C22×C4—84no—
331C22×C4—84no—
332C22×C4—84no—
333C22×C4#12011yescharacteristic
334C22×C4#12011yescharacteristic
335C22×C4#12011yescharacteristic
336C22×C4#12112yes—
337C22×C4#12112yes—
338C22×C4#12112yes—
339C22×C4#12312yes—
340C22×C4#12312yes—
341C22×C4#12412yes—
342C22×C4C2×D812yes—
343C22×C4C2×SD1612yes—
344C22×C4AGL(1,ℤ/8)12yes—
345C22×C4AGL(1,ℤ/8)12yes—
346C22×C4C22×D411yescharacteristic
347C22×C4C22×D411yescharacteristic
348C22×C4C22×D411yescharacteristic
349M16—48no—
350C2×C8—24no—
351C2×C8—42no—
352C2×C8—42no—
353C2×C8—42no—
354C2×C8—42no—
355C2×C8—42no—
356C2×C8—42no—
357C2×C8—42no—
358C2×C8—42no—
359C4⋊C4—22no—
360C4⋊C4—22no—
361C4⋊C4—24no—
362C4⋊C4—24no—
363C4⋊C4—22no—
364C4⋊C4—24no—
365C4⋊C4—22no—
366C4⋊C4—24no—
367C4⋊C4—416no—
368C4⋊C4—416no—
369C4⋊C4—42no—
370C4⋊C4—42no—
371C4⋊C4—42no—
372C4⋊C4—42no—
373C4⋊C4—42no—
374C4⋊C4—44no—
375C4⋊C4—44no—
376C4⋊C4—41no—
377C4⋊C4—41no—
378C4⋊C4—41no—
379C4⋊C4—41no—
380C4⋊C4—42no—
381C4⋊C4—42no—
382C4⋊C4C2×D814yes—
383C4⋊C4C2×SD1614yes—
384C22⋊C4—416no—
385C22⋊C4—416no—
386C22⋊C4—42no—
387C22⋊C4—42no—
388C22⋊C4—44no—
389C22⋊C4—82no—
390C42—44no—
391C42—44no—
392C42—82no—
393C42#12112yes—
394C42#12112yes—
395C42#12112yes—
396C42#12312yes—
397C42#12312yes—
398C42#12412yes—
399C42C22×D411yescharacteristic
400C42C2×#4111yescharacteristic
401C42C2×#4111yescharacteristic
402C42C2×#4111yescharacteristic
403C23—21no—
404C23—21no—
405C23—21no—
406C23—21no—
407C23—24no—
408C23—48no—
409C23—48no—
410C23—48no—
411C23—42no—
412C23—42no—
413C23—42no—
414C23—42no—
415C23—42no—
416C23—42no—
417C23—84no—
418C23—84no—
419C23—84no—
420C23—84no—
421C23#39412yes—
422C23#44712yes—
423C23#45012yes—
424C23C2×#12011yescharacteristic
425Q8—82no—
426Q8—82no—
427Q8—82no—
428Q8—82no—
429Q8—88no—
430D4—24no—
431D4—24no—
432D4—48no—
433D4—42no—
434D4—42no—
435C2×C4—21no—
436C2×C4—21no—
437C2×C4—21no—
438C2×C4—21no—
439C2×C4—21no—
440C2×C4—21no—
441C2×C4—24no—
442C2×C4—24no—
443C2×C4—24no—
444C2×C4—24no—
445C2×C4—42no—
446C2×C4—42no—
447C2×C4—48no—
448C2×C4—48no—
449C2×C4—42no—
450C2×C4—42no—
451C2×C4—42no—
452C2×C4—42no—
453C2×C4—42no—
454C2×C4—44no—
455C2×C4—44no—
456C2×C4—42no—
457C2×C4—44no—
458C2×C4—42no—
459C2×C4—84no—
460C2×C4—88no—
461C2×C4—82no—
462C2×C4—82no—
463C2×C4—82no—
464C2×C4—82no—
465C2×C4—88no—
466C2×C4—81no—
467C2×C4—81no—
468C2×C4—81no—
469C2×C4—81no—
470C2×C4—81no—
471C2×C4—81no—
472C2×C4#39412yes—
473C2×C4#39412yes—
474C2×C4#39412yes—
475C2×C4#44712yes—
476C2×C4#45012yes—
477C2×C4#45914yes—
478C2×C4#46014yes—
479C2×C4#48614yes—
480C2×C4#49014yes—
481C2×C4C2×#12111yescharacteristic
482C2×C4C2×#12111yescharacteristic
483C2×C4C2×#12111yescharacteristic
484C2×C4C2×#12311yescharacteristic
485C2×C4C2×#12311yescharacteristic
486C2×C4C2×#12411yescharacteristic
487C2×C4C22×D811yescharacteristic
488C2×C4C22×SD1611yescharacteristic
489C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
490C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
491C2×C4C2×#13712yes—
492C2×C4#57512yes—
493C8—44no—
494C8—82no—
495C8—82no—
496C22—22no—
497C22—22no—
498C22—22no—
499C22—22no—
500C22—22no—
501C22—22no—
502C22—44no—
503C22—44no—
504C22—44no—
505C22—816no—
506C22—82no—
507C22—82no—
508C22—82no—
509C22—82no—
510C22#204114yes—
511C22C2×#39411yescharacteristic
512C22C2×#44711yescharacteristic
513C22C2×#45011yescharacteristic
514C22#300411yescharacteristic
515C22#300511yescharacteristic
516C22#302011yescharacteristic
517C22#302011yescharacteristic
518C4—22no—
519C4—21no—
520C4—21no—
521C4—22no—
522C4—44no—
523C4—44no—
524C4—82no—
525C4—84no—
526C4—161no—
527C4—161no—
528C4#204114yes—
529C4C2×#45911yescharacteristic
530C4C2×#46011yescharacteristic
531C4#305612yes—
532C4#309512yes—
533C4C2×#48611yescharacteristic
534C4C2×#49011yescharacteristic
535C2—24no—
536C2—84no—
537C2#1875611yescharacteristic
538C2#1875711yescharacteristic
539C2#1877611yescharacteristic
540C2#1877711yescharacteristic
541C2#2348111yescharacteristic
542C2#2349411yescharacteristic
543C2C2×#204111yescharacteristic
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. #21833C2semidirect
3. #21170C2semidirect
5. C2×#1546C2semidirect
7. #10421C2semidirect
8. #10420C2semidirect
9. #7892C2semidirect
27. #1990C22semidirect
28. #1990C22semidirect
31. #1948C22semidirect
32. #1947C22semidirect
41. C2×#332C22semidirect
44. #1279C22semidirect

Maximal subgroups

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

subgroupindex
C2×#25422
#218332
#211702
#207322
C2×#15462
C2×#14862
#104212
#104202
#78922