Finite Group Databasebeta

← #194162

Subgroups — #194162

Order 512.

1508 conjugacy classes of subgroups.

The subgroup lattice has 547 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3101C211yesmaximal, characteristic
2C2×#2354C211yesmaximal, characteristic
3#21860C211yesmaximal, characteristic, complemented
4#21312C211yesmaximal, characteristic, complemented
5C2×#1486C211yesmaximal, characteristic
6#8988C214yesmaximal, complemented
7#8976C214yesmaximal, complemented
8#8043C211yesmaximal, characteristic, complemented
9#7892C211yesmaximal, characteristic, complemented
10C2×#533C2211yescharacteristic
11C22×C4×D4C2211yescharacteristic
12C22×#117—21no—
13#3101—24no—
14#3101C418yescomplemented
15C2×#483—21no—
16C2×#476C2212yes—
17C2×#461C2212yes—
18C2×#450—21no—
19C2×C4×SD16—22no—
20C2×#431C2211yescharacteristic
21C2×#431C2211yescharacteristic
22C2×#423C2211yescharacteristic
23#2354—22no—
24#2354—22no—
25#2354—22no—
26#2354C2214yes—
27C2×#390—21no—
28C2×#388—22no—
29C2×#384C2211yescharacteristic
30C2×#380—21no—
31C2×C4×C4⋊C4—21no—
32C2×C4×C4⋊C4C2211yescharacteristic
33#1990C2212yescomplemented
34#1989—22no—
35#1988C2212yescomplemented
36#1987C2212yescomplemented
37#1956C2212yescomplemented
38#1872—22no—
39#1803—22no—
40#1736—21no—
41#1736—21no—
42#1486—24no—
43C2×D4⋊C8C2214yes—
44#1296C2214yescomplemented
45#1279C2214yescomplemented
46C23×D4D411yescharacteristic
47C24×C4—21no—
48C22×SD16—22no—
49C22×M16—21no—
50#533—22no—
51#533—22no—
52#533—22no—
53#533C2×C414yes—
54C2×C4⋊Q8C2311yescharacteristic
55C2×C4⋊Q8C2311yescharacteristic
56C2×#125—21no—
57C2×C22⋊Q8—21no—
58C2×C22⋊Q8—21no—
59C2×C4×Q8—21no—
60C2×C4×Q8—21no—
61C2×C4×D4—21no—
62C2×C4×D4—21no—
63C2×C4×D4—22no—
64C2×C4×D4—22no—
65C2×C4×D4C2×C414yes—
66C2×C4×D4D414yes—
67C2×C4×D4D412yes—
68C2×C4×D4C2312yes—
69C2×#117—24no—
70C2×#117—24no—
71C2×#117—21no—
72C2×#117—44no—
73C2×#117—44no—
74C22×C4⋊C4—21no—
75C22×C4⋊C4—21no—
76C22×C4⋊C4D411yescharacteristic
77C22×C4⋊C4D411yescharacteristic
78C22×C4⋊C4D411yescharacteristic
79C22×C4⋊C4D412yes—
80C22×C22⋊C4—21no—
81C22×C22⋊C4—22no—
82C22×C22⋊C4—22no—
83C22×C42—22no—
84C22×C42—21no—
85C22×C42C2311yescharacteristic
86#483—28no—
87#476—24no—
88#476C2×C418yescomplemented
89#461—24no—
90#461C2×C418yescomplemented
91#450—28no—
92C4×SD16—216no—
93#431—22no—
94#431—22no—
95#431—22no—
96#431C2×C414yes—
97C2×#106—22no—
98#423—22no—
99#423—22no—
100C2×C4⋊C8—21no—
101C2×C4⋊C8—21no—
102C2×C4⋊C8C2312yes—
103C2×C4⋊C8C2312yes—
104C2×#103—22no—
105C2×#103—24no—
106C2×#102—22no—
107C2×C22⋊C8—21no—
108C2×C22⋊C8—21no—
109C2×C4×C8—21no—
110C2×C4×C8C2312yes—
111#390—24no—
112#390—24no—
113#388—28no—
114#388—28no—
115#384—21no—
116#384—21no—
117#384D412yes—
118#384C2312yes—
119#380—24no—
120#380—24no—
121C4×C4⋊C4—24no—
122C4×C4⋊C4—24no—
123C4×C4⋊C4—21no—
124C4×C4⋊C4—21no—
125C4×C4⋊C4D412yes—
126C4×C4⋊C4C2312yes—
127C2×#95—22no—
128C2×#95—21no—
129#340—24no—
130#340—22no—
131#340—22no—
132#340—22no—
133#340—22no—
134D4⋊C8—28no—
135C25—21no—
136C22×Q8—21no—
137C22×Q8—21no—
138C22×D4—21no—
139C22×D4—22no—
140C22×D4—42no—
141C22×D4—42no—
142C22×D4C22⋊C414yes—
143C22×D4C2×D412yes—
144C23×C4—21no—
145C23×C4—21no—
146C23×C4—24no—
147C23×C4—22no—
148C23×C4—24no—
149C23×C4—22no—
150C23×C4—22no—
151C23×C4—21no—
152C23×C4—42no—
153C23×C4—42no—
154C23×C4—42no—
155C23×C4—42no—
156C23×C4C2×D411yescharacteristic
157C23×C4C2×D411yescharacteristic
158C23×C4C2×D411yescharacteristic
159C2×SD16—28no—
160C2×SD16—44no—
161C2×SD16—44no—
162C2×M16—42no—
163C2×M16—42no—
164C2×M16—42no—
165C22×C8—22no—
166C22×C8—22no—
167C22×C8—22no—
168C4⋊Q8—21no—
169C4⋊Q8—21no—
170C4⋊Q8—21no—
171C4⋊Q8—21no—
172C4⋊Q8—21no—
173C4⋊Q8—21no—
174C4⋊Q8C22×C412yes—
175C4⋊Q8C22×C412yes—
176#125—24no—
177#125—42no—
178C22⋊Q8—24no—
179C22⋊Q8—24no—
180C22⋊Q8—42no—
181C22⋊Q8—42no—
182C4×Q8—24no—
183C4×Q8—24no—
184C4×Q8—42no—
185C4×Q8—42no—
186C4×D4—24no—
187C4×D4—24no—
188C4×D4—24no—
189C4×D4—24no—
190C4×D4—24no—
191C4×D4—22no—
192C4×D4—22no—
193C4×D4C22⋊C418yes—
194C4×D4C22×C414yes—
195C4×D4C2×D414yes—
196#117—24no—
197#117—48no—
198#117—416no—
199#117—42no—
200#117—48no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—44no—
218C2×C4⋊C4—41no—
219C2×C4⋊C4—41no—
220C2×C4⋊C4—42no—
221C2×C4⋊C4—42no—
222C2×C4⋊C4—82no—
223C2×C4⋊C4C22⋊C414yes—
224C2×C4⋊C4SD1614yescomplemented
225C2×C4⋊C4C2×D412yes—
226C2×C4⋊C4C2×D412yes—
227C2×C4⋊C4C2×D412yes—
228C2×C4⋊C4C2×D414yes—
229C2×C22⋊C4—24no—
230C2×C22⋊C4—22no—
231C2×C22⋊C4—24no—
232C2×C22⋊C4—22no—
233C2×C22⋊C4—28no—
234C2×C22⋊C4—28no—
235C2×C22⋊C4—48no—
236C2×C22⋊C4—41no—
237C2×C22⋊C4—84no—
238C2×C42—28no—
239C2×C42—22no—
240C2×C42—22no—
241C2×C42—48no—
242C2×C42—44no—
243C2×C42—41no—
244C2×C42C22×C412yes—
245C2×C42C2×D411yescharacteristic
246C2×C42C2×D412yes—
247C2×C42C2×D411yescharacteristic
248C2×C42C2×D411yescharacteristic
249C2×C42#4112yes—
250C2×C42#4112yes—
251C2×C42C2411yescharacteristic
252#106—28no—
253C4⋊C8—22no—
254C4⋊C8—22no—
255C4⋊C8—22no—
256C4⋊C8—42no—
257C4⋊C8—42no—
258C4⋊C8C22×C414yes—
259#103—28no—
260#103—84no—
261#102—28no—
262C22⋊C8—24no—
263C22⋊C8—42no—
264C4×C8—24no—
265C4×C8—22no—
266C4×C8—22no—
267#95—28no—
268#95—22no—
269#95—22no—
270C24—22no—
271C24—24no—
272C24—42no—
273C24—42no—
274C24—42no—
275C24—42no—
276C24—42no—
277C24—42no—
278C24#12011yescharacteristic
279C2×Q8—22no—
280C2×Q8—22no—
281C2×Q8—42no—
282C2×Q8—42no—
283C2×Q8—41no—
284C2×Q8—41no—
285C2×Q8—42no—
286C2×Q8—42no—
287C2×D4—22no—
288C2×D4—22no—
289C2×D4—22no—
290C2×D4—22no—
291C2×D4—22no—
292C2×D4—24no—
293C2×D4—48no—
294C2×D4—42no—
295C2×D4—42no—
296C2×D4—48no—
297C2×D4C2×C22⋊C414yes—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—24no—
303C22×C4—24no—
304C22×C4—24no—
305C22×C4—24no—
306C22×C4—28no—
307C22×C4—22no—
308C22×C4—24no—
309C22×C4—41no—
310C22×C4—41no—
311C22×C4—42no—
312C22×C4—42no—
313C22×C4—42no—
314C22×C4—42no—
315C22×C4—42no—
316C22×C4—42no—
317C22×C4—42no—
318C22×C4—42no—
319C22×C4—48no—
320C22×C4—48no—
321C22×C4—44no—
322C22×C4—44no—
323C22×C4—48no—
324C22×C4—48no—
325C22×C4—42no—
326C22×C4—42no—
327C22×C4—44no—
328C22×C4—44no—
329C22×C4—41no—
330C22×C4—41no—
331C22×C4—44no—
332C22×C4—42no—
333C22×C4C2×C22⋊C412yes—
334C22×C4C4×D412yes—
335C22×C4C4×D412yes—
336C22×C4#12011yescharacteristic
337C22×C4#12011yescharacteristic
338C22×C4#12011yescharacteristic
339C22×C4#12112yes—
340C22×C4#12112yes—
341C22×C4#12312yes—
342C22×C4C2×SD1612yes—
343C22×C4#13512yes—
344C22×C4AGL(1,ℤ/8)12yes—
345C22×C4#13712yes—
346C22×C4C22×D411yescharacteristic
347C22×C4C22×D411yescharacteristic
348C22×C4C22×D411yescharacteristic
349SD16—416no—
350M16—84no—
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—24no—
363C4⋊C4—24no—
364C4⋊C4—22no—
365C4⋊C4—22no—
366C4⋊C4—22no—
367C4⋊C4—22no—
368C4⋊C4—22no—
369C4⋊C4—22no—
370C4⋊C4—44no—
371C4⋊C4—44no—
372C4⋊C4—41no—
373C4⋊C4—41no—
374C4⋊C4—41no—
375C4⋊C4—41no—
376C4⋊C4—42no—
377C4⋊C4—42no—
378C4⋊C4—44no—
379C4⋊C4—44no—
380C4⋊C4—44no—
381C4⋊C4—44no—
382C4⋊C4—42no—
383C4⋊C4—82no—
384C4⋊C4C2×C22⋊C414yes—
385C4⋊C4C2×SD1614yes—
386C22⋊C4—24no—
387C22⋊C4—24no—
388C22⋊C4—28no—
389C22⋊C4—416no—
390C22⋊C4—44no—
391C22⋊C4—88no—
392C42—44no—
393C42—416no—
394C42—44no—
395C42—44no—
396C42C2×C22⋊C412yes—
397C42C4×D412yes—
398C42C4×D412yes—
399C42#12112yes—
400C42#12112yes—
401C42#12312yes—
402C42C23×C411yescharacteristic
403C42C22×D411yescharacteristic
404C42C2×#4111yescharacteristic
405C42C2×#4111yescharacteristic
406C23—21no—
407C23—21no—
408C23—21no—
409C23—21no—
410C23—24no—
411C23—48no—
412C23—48no—
413C23—48no—
414C23—42no—
415C23—42no—
416C23—42no—
417C23—42no—
418C23—42no—
419C23—42no—
420C23—84no—
421C23—84no—
422C23—84no—
423C23—84no—
424C23#38612yes—
425C23#45012yes—
426C23#45212yes—
427C23C2×#12011yescharacteristic
428Q8—44no—
429Q8—44no—
430Q8—82no—
431Q8—82no—
432D4—24no—
433D4—24no—
434D4—48no—
435D4—42no—
436D4—42no—
437C2×C4—21no—
438C2×C4—21no—
439C2×C4—21no—
440C2×C4—21no—
441C2×C4—21no—
442C2×C4—21no—
443C2×C4—24no—
444C2×C4—24no—
445C2×C4—24no—
446C2×C4—24no—
447C2×C4—42no—
448C2×C4—42no—
449C2×C4—48no—
450C2×C4—48no—
451C2×C4—42no—
452C2×C4—42no—
453C2×C4—42no—
454C2×C4—42no—
455C2×C4—416no—
456C2×C4—44no—
457C2×C4—44no—
458C2×C4—44no—
459C2×C4—44no—
460C2×C4—44no—
461C2×C4—42no—
462C2×C4—42no—
463C2×C4—44no—
464C2×C4—42no—
465C2×C4—44no—
466C2×C4—42no—
467C2×C4—42no—
468C2×C4—88no—
469C2×C4—84no—
470C2×C4—81no—
471C2×C4—81no—
472C2×C4—81no—
473C2×C4—81no—
474C2×C4#38612yes—
475C2×C4#38612yes—
476C2×C4#38612yes—
477C2×C4C4×SD1614yes—
478C2×C4#44014yes—
479C2×C4#45012yes—
480C2×C4#45212yes—
481C2×C4#46114yes—
482C2×C4#46314yes—
483C2×C4C22×C22⋊C411yescharacteristic
484C2×C4C2×C4×D411yescharacteristic
485C2×C4C2×C4×D411yescharacteristic
486C2×C4C2×#12111yescharacteristic
487C2×C4C2×#12111yescharacteristic
488C2×C4C2×#12311yescharacteristic
489C2×C4C22×SD1611yescharacteristic
490C2×C4C2×#13511yescharacteristic
491C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
492C2×C4C2×#13711yescharacteristic
493C2×C4#57512yes—
494C2×C4#57512yes—
495C8—44no—
496C8—82no—
497C8—82no—
498C22—22no—
499C22—22no—
500C22—22no—
501C22—22no—
502C22—22no—
503C22—22no—
504C22—44no—
505C22—44no—
506C22—44no—
507C22—816no—
508C22—82no—
509C22—82no—
510C22—82no—
511C22—82no—
512C22#187714yes—
513C22C2×#38611yescharacteristic
514C22C2×#45011yescharacteristic
515C22C2×#45211yescharacteristic
516C22#300511yescharacteristic
517C22#300511yescharacteristic
518C22#301911yescharacteristic
519C22#302011yescharacteristic
520C4—22no—
521C4—21no—
522C4—21no—
523C4—22no—
524C4—44no—
525C4—44no—
526C4—84no—
527C4—82no—
528C4—82no—
529C4—82no—
530C4#187714yes—
531C4#289612yes—
532C4C2×C4×SD1611yescharacteristic
533C4C2×#44011yescharacteristic
534C4C2×#46111yescharacteristic
535C4C2×#46311yescharacteristic
536C4#309312yes—
537C2—24no—
538C2—84no—
539C2#1875711yescharacteristic
540C2#1876311yescharacteristic
541C2#1876711yescharacteristic
542C2#1877311yescharacteristic
543C2#2091011yescharacteristic
544C2C2×#187711yescharacteristic
545C2#2094311yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radicalC23×D4D4
LCSG ▷ 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. #21860C2semidirect
4. #21312C2semidirect
6. #8988C2semidirect
7. #8976C2semidirect
8. #8043C2semidirect
9. #7892C2semidirect
14. #3101C4semidirect
33. #1990C22semidirect
35. #1988C22semidirect
36. #1987C22semidirect
37. #1956C22semidirect
44. #1296C22semidirect
45. #1279C22semidirect
88. #476C2×C4semidirect
90. #461C2×C4semidirect
224. C2×C4⋊C4SD16semidirect

Maximal subgroups

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

subgroupindex
C2×#31012
C2×#23542
#218602
#213122
C2×#14862
#89882
#89762
#80432
#78922