Finite Group Databasebeta

← #240258

Subgroups — #240258

Order 512.

1154 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×#3101C211yesmaximal, characteristic
2#21318C211yesmaximal, characteristic, complemented
3C2×#1954C211yesmaximal, characteristic
4#20732C211yesmaximal, characteristic
5C2×#1564C211yesmaximal, characteristic, complemented
6C2×#1542C211yesmaximal, characteristic, complemented
7#10481C212yesmaximal, complemented
8#10480C212yesmaximal, complemented
9#10477C212yesmaximal, complemented
10#10476C212yesmaximal, complemented
11#8039C211yesmaximal, characteristic, complemented
12C2×#533C2211yescharacteristic
13C22×C4×D4C2211yescharacteristic
14#3101—24no—
15#3101—24no—
16C2×#483—21no—
17C2×#476C2212yes—
18C2×#461C2212yes—
19C2×#450—21no—
20C2×C4×SD16—22no—
21C2×#431C2211yescharacteristic
22C2×#431C2211yescharacteristic
23C22×C4⋊C8C2211yescharacteristic
24C22×C22⋊C8—21no—
25C22×C22⋊C8—21no—
26C22×C4×C8C2211yescharacteristic
27#2285C2211yescharacteristic
28#1991—21no—
29#1991—21no—
30#1990C2211yescharacteristic, complemented
31#1990C2211yescharacteristic, complemented
32#1990C2211yescharacteristic, complemented
33#1990C2211yescharacteristic, complemented
34#1954—21no—
35#1954—21no—
36#1954C2212yes—
37#1954C2212yes—
38#1823—21no—
39#1802C2212yes—
40#1802C2212yes—
41#1786—21no—
42C4×#102—22no—
43#1735—21no—
44#1735—21no—
45#1564—24no—
46C2×#335C2212yescomplemented
47C2×#334C2212yescomplemented
48#1542—24no—
49C2×#333C2212yescomplemented
50C2×#332C2212yescomplemented
51#1296C2212yescomplemented
52#1296C2212yescomplemented
53C23×D4D411yescharacteristic
54C24×C4—21no—
55C22×SD16—22no—
56C23×C8—21no—
57C23×C8D411yescharacteristic
58C23×C8D411yescharacteristic
59#533—22no—
60#533—22no—
61#533—22no—
62#533D414yes—
63C2×C4⋊Q8C2311yescharacteristic
64C2×C4⋊Q8C2311yescharacteristic
65C2×#125—21no—
66C2×C22⋊Q8—21no—
67C2×C22⋊Q8—21no—
68C2×C4×Q8—21no—
69C2×C4×Q8—21no—
70C2×C4×D4—21no—
71C2×C4×D4—22no—
72C2×C4×D4—22no—
73C2×C4×D4—21no—
74C2×C4×D4—22no—
75C2×C4×D4—22no—
76C2×C4×D4D412yes—
77C2×C4×D4C2312yes—
78C2×#117—21no—
79C2×#117—21no—
80C22×C4⋊C4D411yescharacteristic
81C22×C22⋊C4—21no—
82C22×C42C2311yescharacteristic
83#483—28no—
84#476—24no—
85#476—24no—
86#461—24no—
87#461—24no—
88#450—28no—
89C4×SD16—216no—
90#431—22no—
91#431—22no—
92#431—22no—
93#431D414yes—
94C2×#106—22no—
95C2×C4⋊C8—22no—
96C2×C4⋊C8—22no—
97C2×C4⋊C8—21no—
98C2×C4⋊C8C2312yes—
99C2×C4⋊C8C2312yes—
100C2×C4⋊C8C2312yes—
101C2×#103—22no—
102C2×#102—22no—
103C2×#102—22no—
104C2×#102—22no—
105C2×C22⋊C8—21no—
106C2×C22⋊C8—21no—
107C2×C22⋊C8—22no—
108C2×C22⋊C8—22no—
109C2×C22⋊C8—22no—
110C2×C22⋊C8—22no—
111C2×C22⋊C8—44no—
112C2×C22⋊C8—44no—
113C2×C4×C8—22no—
114C2×C4×C8—21no—
115C2×C4×C8—22no—
116C2×C4×C8—21no—
117C2×C4×C8D411yescharacteristic
118C2×C4×C8D411yescharacteristic
119C2×C4×C8C2311yescharacteristic
120C2×C4×C8C2311yescharacteristic
121#384C2311yescharacteristic
122#384C2311yescharacteristic
123#382—21no—
124#380—21no—
125#380—21no—
126C4×C4⋊C4—21no—
127C4×C4⋊C4—21no—
128#340—21no—
129#340—21no—
130#340—21no—
131#340—21no—
132#340—21no—
133#340—21no—
134#340—21no—
135#340—21no—
136#340—22no—
137#340—22no—
138#335—24no—
139#334—24no—
140#333—24no—
141#332—24no—
142C25—21no—
143C22×Q8—21no—
144C22×Q8—21no—
145C22×D4—24no—
146C22×D4—22no—
147C22×D4—21no—
148C22×D4—22no—
149C22×D4—42no—
150C22×D4C2×D412yes—
151C23×C4—21no—
152C23×C4—22no—
153C23×C4—22no—
154C23×C4—42no—
155C23×C4—42no—
156C23×C4—42no—
157C23×C4—42no—
158C23×C4—42no—
159C23×C4—42no—
160C23×C4C2×D411yescharacteristic
161C23×C4C2×D411yescharacteristic
162C23×C4#4111yescharacteristic
163C2×SD16—28no—
164C2×SD16—44no—
165C2×SD16—44no—
166C22×C8—22no—
167C22×C8—22no—
168C22×C8—22no—
169C22×C8—22no—
170C22×C8—22no—
171C22×C8—22no—
172C22×C8—22no—
173C22×C8—22no—
174C22×C8—42no—
175C22×C8—42no—
176C22×C8—42no—
177C22×C8SD1614yescomplemented
178C22×C8C2×D411yescharacteristic
179C22×C8C2×D411yescharacteristic
180C22×C8C2×D411yescharacteristic
181C22×C8C2×D411yescharacteristic
182C4⋊Q8—21no—
183C4⋊Q8—21no—
184C4⋊Q8—21no—
185C4⋊Q8—21no—
186C4⋊Q8—21no—
187C4⋊Q8—21no—
188C4⋊Q8C2×D412yes—
189C4⋊Q8C2×D412yes—
190#125—24no—
191#125—42no—
192C22⋊Q8—24no—
193C22⋊Q8—24no—
194C22⋊Q8—42no—
195C22⋊Q8—42no—
196C4×Q8—24no—
197C4×Q8—24no—
198C4×Q8—42no—
199C4×Q8—42no—
200C4×D4—24no—
201C4×D4—24no—
202C4×D4—22no—
203C4×D4—22no—
204C4×D4—22no—
205C4×D4—22no—
206C4×D4—42no—
207C4×D4—42no—
208C4×D4—42no—
209C4×D4—42no—
210#117—24no—
211#117—42no—
212#117—42no—
213#117—42no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—21no—
227C2×C4⋊C4—41no—
228C2×C4⋊C4—41no—
229C2×C4⋊C4C2×D412yes—
230C2×C22⋊C4—22no—
231C2×C22⋊C4—22no—
232C2×C22⋊C4—41no—
233C2×C22⋊C4—42no—
234C2×C22⋊C4—42no—
235C2×C22⋊C4—41no—
236C2×C42—41no—
237C2×C42—41no—
238C2×C42C2×D411yescharacteristic
239C2×C42C2×D412yes—
240C2×C42C2×D412yes—
241C2×C42C2×D411yescharacteristic
242C2×C42#4112yes—
243C2×C42#4112yes—
244C2×C42#4111yescharacteristic
245C2×C42C2411yescharacteristic
246#106—28no—
247C4⋊C8—22no—
248C4⋊C8—22no—
249C4⋊C8—22no—
250C4⋊C8—22no—
251C4⋊C8—22no—
252C4⋊C8—42no—
253C4⋊C8C2×D414yes—
254#103—28no—
255#102—28no—
256#102—44no—
257#102—44no—
258C22⋊C8—24no—
259C22⋊C8—44no—
260C22⋊C8—44no—
261C22⋊C8—42no—
262C22⋊C8—44no—
263C22⋊C8—44no—
264C4×C8—24no—
265C4×C8—22no—
266C4×C8—22no—
267C4×C8—21no—
268C4×C8—21no—
269C4×C8—21no—
270C4×C8—21no—
271C4×C8—42no—
272#95—41no—
273#95—41no—
274C24—24no—
275C24—22no—
276C24—42no—
277C24—42no—
278C24—42no—
279C24—42no—
280C24—42no—
281C24—42no—
282C24#12111yescharacteristic
283C2×Q8—22no—
284C2×Q8—22no—
285C2×Q8—41no—
286C2×Q8—41no—
287C2×Q8—42no—
288C2×Q8—42no—
289C2×Q8—81no—
290C2×Q8—81no—
291C2×D4—24no—
292C2×D4—22no—
293C2×D4—22no—
294C2×D4—22no—
295C2×D4—22no—
296C2×D4—22no—
297C2×D4—48no—
298C2×D4—48no—
299C2×D4—42no—
300C2×D4—42no—
301C2×D4#13514yes—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—41no—
306C22×C4—41no—
307C22×C4—41no—
308C22×C4—41no—
309C22×C4—42no—
310C22×C4—42no—
311C22×C4—42no—
312C22×C4—42no—
313C22×C4—42no—
314C22×C4—42no—
315C22×C4—42no—
316C22×C4—42no—
317C22×C4—42no—
318C22×C4—44no—
319C22×C4—44no—
320C22×C4—44no—
321C22×C4—41no—
322C22×C4—41no—
323C22×C4—84no—
324C22×C4—84no—
325C22×C4—84no—
326C22×C4—84no—
327C22×C4#12012yes—
328C22×C4#12111yescharacteristic
329C22×C4#12111yescharacteristic
330C22×C4#12112yes—
331C22×C4#12112yes—
332C22×C4#12111yescharacteristic
333C22×C4#12312yes—
334C22×C4#12312yes—
335C22×C4#12412yes—
336C22×C4C2×SD1612yes—
337C22×C4#13512yes—
338C22×C4AGL(1,ℤ/8)12yes—
339C22×C4#13712yes—
340C22×C4C22×D411yescharacteristic
341C22×C4C22×D411yescharacteristic
342C22×C4C2×#4111yescharacteristic
343SD16—416no—
344C2×C8—21no—
345C2×C8—21no—
346C2×C8—21no—
347C2×C8—21no—
348C2×C8—28no—
349C2×C8—24no—
350C2×C8—21no—
351C2×C8—21no—
352C2×C8—21no—
353C2×C8—21no—
354C2×C8—21no—
355C2×C8—21no—
356C2×C8—42no—
357C2×C8—42no—
358C2×C8—44no—
359C2×C8—42no—
360C2×C8—42no—
361C2×C8—42no—
362C2×C8—42no—
363C2×C8—42no—
364C2×C8—42no—
365C2×C8—84no—
366C2×C8C2×SD1612yes—
367C2×C8C2×SD1612yes—
368C4⋊C4—22no—
369C4⋊C4—22no—
370C4⋊C4—22no—
371C4⋊C4—22no—
372C4⋊C4—22no—
373C4⋊C4—22no—
374C4⋊C4—41no—
375C4⋊C4—41no—
376C4⋊C4—41no—
377C4⋊C4—41no—
378C4⋊C4—42no—
379C4⋊C4—44no—
380C4⋊C4—44no—
381C4⋊C4—42no—
382C4⋊C4—41no—
383C4⋊C4—41no—
384C4⋊C4—41no—
385C4⋊C4—41no—
386C4⋊C4—42no—
387C4⋊C4—81no—
388C4⋊C4—81no—
389C22⋊C4—44no—
390C22⋊C4—42no—
391C22⋊C4—42no—
392C22⋊C4—44no—
393C22⋊C4—82no—
394C42—44no—
395C42—82no—
396C42#12012yes—
397C42#12112yes—
398C42#12112yes—
399C42#12312yes—
400C42#12312yes—
401C42#12412yes—
402C42C22×D411yescharacteristic
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#39412yes—
425C23#46112yes—
426C23#46312yes—
427C23C2×#12111yescharacteristic
428Q8—44no—
429Q8—82no—
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—42no—
446C2×C4—42no—
447C2×C4—42no—
448C2×C4—42no—
449C2×C4—42no—
450C2×C4—42no—
451C2×C4—42no—
452C2×C4—84no—
453C2×C4—81no—
454C2×C4—81no—
455C2×C4—81no—
456C2×C4—81no—
457C2×C4—88no—
458C2×C4—82no—
459C2×C4—82no—
460C2×C4—82no—
461C2×C4—82no—
462C2×C4—81no—
463C2×C4—81no—
464C2×C4—81no—
465C2×C4—81no—
466C2×C4—81no—
467C2×C4—81no—
468C2×C4#39412yes—
469C2×C4#39412yes—
470C2×C4#39412yes—
471C2×C4#46014yes—
472C2×C4#46112yes—
473C2×C4#46312yes—
474C2×C4#46414yes—
475C2×C4#46514yes—
476C2×C4#46814yes—
477C2×C4C2×#12011yescharacteristic
478C2×C4C2×#12111yescharacteristic
479C2×C4C2×#12111yescharacteristic
480C2×C4C2×#12311yescharacteristic
481C2×C4C2×#12311yescharacteristic
482C2×C4C2×#12411yescharacteristic
483C2×C4C22×SD1611yescharacteristic
484C2×C4C2×#13512yes—
485C2×C4C2×#13511yescharacteristic
486C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
487C2×C4C2×#13711yescharacteristic
488C2×C4#57512yes—
489C8—22no—
490C8—22no—
491C8—44no—
492C8—41no—
493C8—41no—
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#205514yes—
511C22C2×#39411yescharacteristic
512C22C2×#46111yescharacteristic
513C22C2×#46311yescharacteristic
514C22#305011yescharacteristic
515C22#305411yescharacteristic
516C22#306711yescharacteristic
517C22#306711yescharacteristic
518C4—22no—
519C4—21no—
520C4—21no—
521C4—22no—
522C4—82no—
523C4—82no—
524C4—82no—
525C4—161no—
526C4—161no—
527C4#205514yes—
528C4#300212yes—
529C4C2×#46011yescharacteristic
530C4C2×#46411yescharacteristic
531C4C2×#46511yescharacteristic
532C4C2×#46811yescharacteristic
533C4#309112yes—
534C2—24no—
535C2—84no—
536C2#1910511yescharacteristic
537C2#1911011yescharacteristic
538C2#1913711yescharacteristic
539C2#1914211yescharacteristic
540C2#2347911yescharacteristic
541C2#2353011yescharacteristic
542C2C2×#205511yescharacteristic
subgroupG/H
centerC22×C4#121
commutator, Frattini, cosocleC2×C42C24
socleC23C2×#121
prime radicalC23×D4D4
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C4 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21318C2semidirect
5. C2×#1564C2semidirect
6. C2×#1542C2semidirect
7. #10481C2semidirect
8. #10480C2semidirect
9. #10477C2semidirect
10. #10476C2semidirect
11. #8039C2semidirect
30. #1990C22semidirect
31. #1990C22semidirect
32. #1990C22semidirect
33. #1990C22semidirect
46. C2×#335C22semidirect
47. C2×#334C22semidirect
49. C2×#333C22semidirect
50. C2×#332C22semidirect
51. #1296C22semidirect
52. #1296C22semidirect
177. C22×C8SD16semidirect

Maximal subgroups

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

subgroupindex
C2×#31012
#213182
C2×#19542
#207322
C2×#15642
C2×#15422
#104812
#104802
#104772
#104762
#80392