Finite Group Databasebeta

← #196798

Subgroups — #196798

Order 512.

1973 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
1#36621C211yesmaximal, characteristic
2C22×#376C211yesmaximal, characteristic, complemented
3#20714C211yesmaximal, characteristic, complemented
4C2×#1500C212yesmaximal, complemented
5#9149C212yesmaximal, complemented
6#9147C212yesmaximal
7#9145C212yesmaximal, complemented
8#9143C212yesmaximal, complemented
9#8013C212yesmaximal, complemented
10C2×#529C2211yescharacteristic
11C23×C42—21no—
12C23×C42C2211yescharacteristic
13C2×#431C2212yes—
14#2855C2211yescharacteristic
15#2854C2211yescharacteristic
16#2853—21no—
17#2852—22no—
18#2851C412yescomplemented
19#2850C412yescomplemented
20#2849—21no—
21#2848C2211yescharacteristic
22#2847C2211yescharacteristic
23C4×C22⋊Q8C412yescomplemented
24C4×#123—21no—
25C4×#121C412yescomplemented
26C4×#120—21no—
27#2290—22no—
28#2283C2211yescharacteristic
29C2×#376—21no—
30C2×#376—21no—
31C2×#376—28no—
32C2×#376C2212yes—
33C2×#376C2212yes—
34C22×#95—21no—
35C22×#95C2212yes—
36#1817C2212yes—
37#1797—22no—
38#1779C2212yes—
39C4×C22⋊C8—22no—
40#1500—28no—
41C2×#328C2214yes—
42C2×#327C2214yes—
43#1294C2214yes—
44#1294C2214yes—
45C24×C4—22no—
46C24×C4D411yescharacteristic
47C24×C4D411yescharacteristic
48C24×C4C2311yescharacteristic
49#529—24no—
50#529—24no—
51C2×#126—21no—
52C2×#125C2×C411yescharacteristic, complemented
53C2×#124C2×C411yescharacteristic, complemented
54C2×#123—21no—
55C2×#123—21no—
56C2×C22⋊Q8C2×C412yes—
57C2×C22⋊Q8C2311yescharacteristic
58C2×#121C2×C412yes—
59C2×#121C2311yescharacteristic
60C2×#120—21no—
61C2×C4×Q8C2×C411yescharacteristic, complemented
62C2×C4×D4—22no—
63C2×C4×D4—21no—
64C2×C4×D4C2×C411yescharacteristic, complemented
65C2×#117—21no—
66C22×C42—216no—
67C22×C42—28no—
68C22×C42—22no—
69C22×C42—22no—
70C22×C42—24no—
71C22×C42—24no—
72C22×C42C2×C412yes—
73C22×C42C2×C412yes—
74C22×C42D411yescharacteristic
75C22×C42D411yescharacteristic
76C22×C42D412yes—
77C22×C42D412yes—
78C22×C42C2311yescharacteristic
79C22×C42C2311yescharacteristic
80#431—24no—
81#431—24no—
82C2×C4⋊C8—22no—
83C2×C22⋊C8—24no—
84C2×C22⋊C8C2314yes—
85C2×C4×C8—22no—
86#391C2×C412yescomplemented
87#390C2×C412yescomplemented
88#389C2×C412yescomplemented
89#389C2×C412yescomplemented
90#388—22no—
91#388—22no—
92#388—22no—
93#388—22no—
94#387—22no—
95#387—22no—
96#387—22no—
97#387—22no—
98#386—22no—
99#386—22no—
100#386—22no—
101#386—22no—
102#386—21no—
103#386—21no—
104#385—22no—
105#385—22no—
106#385—22no—
107#385—22no—
108#385—21no—
109#385—21no—
110#383—21no—
111#382C2×C412yescomplemented
112#381—21no—
113#381—21no—
114#380C2×C412yes—
115#380C2×C412yes—
116#380C2311yescharacteristic
117#380C2311yescharacteristic
118#379—21no—
119C4×C4⋊C4—22no—
120C4×C4⋊C4C2×C412yescomplemented
121C4×C22⋊C4—22no—
122C4×C22⋊C4—21no—
123C4×C22⋊C4—22no—
124#376—28no—
125#376—28no—
126#376—24no—
127#376—21no—
128#376—28no—
129#376D418yescomplemented
130C2×#95—28no—
131C2×#95—22no—
132C2×#95—22no—
133C2×#95—22no—
134C2×#95—28no—
135C2×#95C2314yes—
136#336—24no—
137#328—28no—
138#327—28no—
139C25C2×D411yescharacteristic
140C22×Q8C22×C411yescharacteristic
141C22×D4—22no—
142C22×D4—21no—
143C22×D4C22×C411yescharacteristic
144C23×C4—28no—
145C23×C4—22no—
146C23×C4—22no—
147C23×C4—28no—
148C23×C4—216no—
149C23×C4—24no—
150C23×C4—24no—
151C23×C4—22no—
152C23×C4—22no—
153C23×C4—22no—
154C23×C4—22no—
155C23×C4—21no—
156C23×C4—42no—
157C23×C4—44no—
158C23×C4—48no—
159C23×C4—44no—
160C23×C4—44no—
161C23×C4C22⋊C412yes—
162C23×C4C22⋊C412yes—
163C23×C4C22⋊C412yes—
164C23×C4C22⋊C412yes—
165C23×C4C22×C412yes—
166C23×C4C2×D411yescharacteristic
167C23×C4C2×D411yescharacteristic
168C23×C4C2×D412yes—
169C23×C4C2×D411yescharacteristic
170C23×C4C2×D411yescharacteristic
171C23×C4C2×D411yescharacteristic
172C23×C4C2411yescharacteristic
173C22×C8—24no—
174#126—44no—
175#125—24no—
176#124—24no—
177#123—44no—
178#123—42no—
179#123—42no—
180C22⋊Q8—22no—
181C22⋊Q8—22no—
182C22⋊Q8—44no—
183#121—22no—
184#121—22no—
185#121—44no—
186#120—44no—
187C4×Q8—24no—
188C4×D4—24no—
189C4×D4—48no—
190C4×D4—42no—
191C4×D4—42no—
192#117—44no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4C22×C411yescharacteristic
201C2×C4⋊C4C22×C411yescharacteristic
202C2×C4⋊C4C22×C412yes—
203C2×C4⋊C4C22×C412yes—
204C2×C22⋊C4—22no—
205C2×C22⋊C4—22no—
206C2×C22⋊C4—22no—
207C2×C22⋊C4—21no—
208C2×C22⋊C4—21no—
209C2×C22⋊C4—21no—
210C2×C22⋊C4—21no—
211C2×C22⋊C4—21no—
212C2×C22⋊C4—22no—
213C2×C22⋊C4—21no—
214C2×C22⋊C4—22no—
215C2×C22⋊C4—22no—
216C2×C22⋊C4—21no—
217C2×C42—28no—
218C2×C42—264no—
219C2×C42—216no—
220C2×C42—216no—
221C2×C42—22no—
222C2×C42—22no—
223C2×C42—28no—
224C2×C42—22no—
225C2×C42—22no—
226C2×C42—22no—
227C2×C42—22no—
228C2×C42—28no—
229C2×C42—28no—
230C2×C42—28no—
231C2×C42—28no—
232C2×C42—21no—
233C2×C42—22no—
234C2×C42C22⋊C412yes—
235C2×C42C22⋊C412yes—
236C2×C42C22⋊C412yes—
237C2×C42C22⋊C412yes—
238C2×C42C22×C412yes—
239C2×C42C22×C411yescharacteristic
240C2×C42C22×C411yescharacteristic
241C2×C42C2×D412yes—
242C2×C42C2×D414yes—
243C2×C42C2×D411yescharacteristic
244C2×C42C2×D411yescharacteristic
245C2×C42#4114yes—
246C4⋊C8—44no—
247C22⋊C8—24no—
248C22⋊C8—24no—
249C22⋊C8—48no—
250C4×C8—44no—
251#95—28no—
252#95—28no—
253#95—22no—
254#95—22no—
255#95—22no—
256#95—22no—
257#95—22no—
258#95—22no—
259#95—22no—
260#95—22no—
261#95—22no—
262#95—28no—
263#95—24no—
264#95—21no—
265#95—21no—
266#95C2×D418yes—
267C24—21no—
268C24—21no—
269C24—22no—
270C24—24no—
271C24—24no—
272C24—21no—
273C24C2×C22⋊C412yes—
274C24#13514yes—
275C24C22×D411yescharacteristic
276C2×Q8—22no—
277C2×Q8—42no—
278C2×D4—22no—
279C2×D4—42no—
280C2×D4—41no—
281C2×D4—41no—
282C2×D4—42no—
283C2×D4—42no—
284C2×D4—44no—
285C2×D4—44no—
286C2×D4—44no—
287C22×C4—216no—
288C22×C4—216no—
289C22×C4—21no—
290C22×C4—21no—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—24no—
294C22×C4—24no—
295C22×C4—24no—
296C22×C4—24no—
297C22×C4—24no—
298C22×C4—24no—
299C22×C4—24no—
300C22×C4—232no—
301C22×C4—216no—
302C22×C4—21no—
303C22×C4—21no—
304C22×C4—28no—
305C22×C4—21no—
306C22×C4—21no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—24no—
310C22×C4—24no—
311C22×C4—24no—
312C22×C4—24no—
313C22×C4—24no—
314C22×C4—24no—
315C22×C4—24no—
316C22×C4—24no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—21no—
320C22×C4—21no—
321C22×C4—21no—
322C22×C4—42no—
323C22×C4—42no—
324C22×C4—42no—
325C22×C4—42no—
326C22×C4—42no—
327C22×C4—42no—
328C22×C4—48no—
329C22×C4—48no—
330C22×C4—48no—
331C22×C4—48no—
332C22×C4—416no—
333C22×C4—432no—
334C22×C4—48no—
335C22×C4—432no—
336C22×C4—416no—
337C22×C4—44no—
338C22×C4—44no—
339C22×C4—42no—
340C22×C4—44no—
341C22×C4#10418yes—
342C22×C4C2×C22⋊C412yes—
343C22×C4C2×C22⋊C412yes—
344C22×C4C2×C22⋊C411yescharacteristic
345C22×C4C2×C22⋊C411yescharacteristic
346C22×C4C2×C22⋊C411yescharacteristic
347C22×C4C2×C22⋊C412yes—
348C22×C4C2×C22⋊C411yescharacteristic
349C22×C4#12012yes—
350C22×C4#12012yes—
351C22×C4#12312yes—
352C22×C4#12312yes—
353C22×C4C23×C411yescharacteristic
354C22×C4C22×D411yescharacteristic
355C22×C4C22×D412yes—
356C22×C4C2×#4112yes—
357C2×C8—48no—
358C2×C8—44no—
359C4⋊C4—22no—
360C4⋊C4—22no—
361C4⋊C4—42no—
362C4⋊C4—42no—
363C4⋊C4—44no—
364C4⋊C4—44no—
365C4⋊C4—42no—
366C4⋊C4—42no—
367C4⋊C4—42no—
368C4⋊C4—42no—
369C4⋊C4—44no—
370C22⋊C4—44no—
371C22⋊C4—44no—
372C22⋊C4—41no—
373C22⋊C4—41no—
374C22⋊C4—42no—
375C22⋊C4—42no—
376C22⋊C4—44no—
377C22⋊C4—44no—
378C22⋊C4—44no—
379C22⋊C4—42no—
380C22⋊C4—42no—
381C22⋊C4—42no—
382C22⋊C4—42no—
383C22⋊C4—44no—
384C42—232no—
385C42—232no—
386C42—24no—
387C42—24no—
388C42—24no—
389C42—24no—
390C42—24no—
391C42—24no—
392C42—42no—
393C42—44no—
394C42C4×D414yes—
395C42C4×D414yes—
396C42#12114yes—
397C42#12114yes—
398C23—24no—
399C23—24no—
400C23—24no—
401C23—216no—
402C23—24no—
403C23—24no—
404C23—22no—
405C23—22no—
406C23—22no—
407C23—22no—
408C23—22no—
409C23—24no—
410C23—216no—
411C23—22no—
412C23—42no—
413C23—41no—
414C23—41no—
415C23—42no—
416C23—41no—
417C23#41014yes—
418C23#41614yes—
419C23C2×#10414yes—
420C23C22×C22⋊C411yescharacteristic
421C23C2×#12011yescharacteristic
422C23C2×#12311yescharacteristic
423C23C2×#13512yes—
424C23#57512yes—
425Q8—82no—
426D4—82no—
427D4—82no—
428D4—82no—
429D4—88no—
430C2×C4—24no—
431C2×C4—216no—
432C2×C4—24no—
433C2×C4—24no—
434C2×C4—22no—
435C2×C4—22no—
436C2×C4—21no—
437C2×C4—21no—
438C2×C4—21no—
439C2×C4—21no—
440C2×C4—22no—
441C2×C4—22no—
442C2×C4—24no—
443C2×C4—22no—
444C2×C4—24no—
445C2×C4—216no—
446C2×C4—24no—
447C2×C4—216no—
448C2×C4—22no—
449C2×C4—22no—
450C2×C4—22no—
451C2×C4—22no—
452C2×C4—22no—
453C2×C4—22no—
454C2×C4—42no—
455C2×C4—42no—
456C2×C4—48no—
457C2×C4—48no—
458C2×C4—48no—
459C2×C4—48no—
460C2×C4—464no—
461C2×C4—416no—
462C2×C4—416no—
463C2×C4—48no—
464C2×C4—42no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—42no—
468C2×C4—48no—
469C2×C4—42no—
470C2×C4—42no—
471C2×C4—42no—
472C2×C4—42no—
473C2×C4—41no—
474C2×C4—42no—
475C2×C4—41no—
476C2×C4—42no—
477C2×C4—42no—
478C2×C4—42no—
479C2×C4—41no—
480C2×C4—41no—
481C2×C4—41no—
482C2×C4—84no—
483C2×C4#38612yes—
484C2×C4#38612yes—
485C2×C4#38612yes—
486C2×C4#38612yes—
487C2×C4C2×#10414yes—
488C2×C4C2×#10414yes—
489C2×C4C2×C4×D412yes—
490C2×C4C2×#12112yes—
491C8—84no—
492C22—216no—
493C22—216no—
494C22—24no—
495C22—24no—
496C22—24no—
497C22—24no—
498C22—24no—
499C22—24no—
500C22—21no—
501C22—21no—
502C22—24no—
503C22—21no—
504C22—21no—
505C22—24no—
506C22—82no—
507C22—82no—
508C22—81no—
509C22—81no—
510C22—81no—
511C22#150018yescomplemented
512C22C2×#38611yescharacteristic
513C22C2×#41012yes—
514C22C2×#41611yescharacteristic
515C22#289611yescharacteristic
516C22C22×#10412yes—
517C22#300212yes—
518C22#309112yes—
519C4—22no—
520C4—22no—
521C4—22no—
522C4—24no—
523C4—22no—
524C4—22no—
525C4—416no—
526C4—42no—
527C4—42no—
528C4—82no—
529C4—82no—
530C4—81no—
531C4—81no—
532C4—81no—
533C4#189614yes—
534C2—22no—
535C2—22no—
536C2—28no—
537C2—161no—
538C2C2×#150012yes—
539C2#1754112yes—
540C2#2090811yescharacteristic
541C2C2×#189611yescharacteristic
542C2#2098311yescharacteristic
subgroupG/H
centerC22×C4C2×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
socleC23C22×C22⋊C4
prime radical#2847C22
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C4 ◁ C23×C42 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C22×#376C2semidirect
3. #20714C2semidirect
4. C2×#1500C2semidirect
5. #9149C2semidirect
7. #9145C2semidirect
8. #9143C2semidirect
9. #8013C2semidirect
18. #2851C4semidirect
19. #2850C4semidirect
23. C4×C22⋊Q8C4semidirect
25. C4×#121C4semidirect
52. C2×#125C2×C4semidirect
53. C2×#124C2×C4semidirect
61. C2×C4×Q8C2×C4semidirect
64. C2×C4×D4C2×C4semidirect
86. #391C2×C4semidirect
87. #390C2×C4semidirect
88. #389C2×C4semidirect
89. #389C2×C4semidirect
111. #382C2×C4semidirect
120. C4×C4⋊C4C2×C4semidirect
129. #376D4semidirect
511. C22#1500semidirect

Maximal subgroups

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

subgroupindex
#366212
C22×#3762
#207142
C2×#15002
#91492
#91472
#91452
#91432
#80132