Finite Group Databasebeta

← #196838

Subgroups — #196838

Order 512.

1497 conjugacy classes of subgroups.

The subgroup lattice has 535 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36621C211yesmaximal, characteristic
2C2×#2283C211yesmaximal, characteristic, complemented
3#20714C211yesmaximal, characteristic, complemented
4C2×#1521C212yesmaximal, complemented
5#9155C212yesmaximal
6#9151C212yesmaximal, complemented
7#9149C212yesmaximal, complemented
8#9145C212yesmaximal, complemented
9#8025C212yesmaximal, complemented
10C2×#529C2211yescharacteristic
11C23×C42C2211yescharacteristic
12C2×#431C2212yes—
13#2855C2211yescharacteristic
14#2854C2211yescharacteristic
15#2853—21no—
16#2852—22no—
17#2851C412yescomplemented
18#2850C412yescomplemented
19#2849—21no—
20#2848C2211yescharacteristic
21#2847C2211yescharacteristic
22C4×C22⋊Q8C412yescomplemented
23C4×#123—21no—
24C4×#121C412yescomplemented
25C4×#120—21no—
26#2290—22no—
27#2283—28no—
28#2283C2211yescharacteristic
29C2×#383C2212yes—
30C2×#382—21no—
31C2×#381—21no—
32C2×#380C2212yes—
33C2×C4×C4⋊C4—21no—
34C2×C4×C22⋊C4—21no—
35C2×#376C2212yes—
36#1817C2212yes—
37#1797—22no—
38#1779C2212yes—
39C4×C22⋊C8—22no—
40#1521—28no—
41C2×#328C2214yes—
42C2×#327C2214yes—
43#1294C2214yes—
44#1294C2214yes—
45C24×C4D411yescharacteristic
46C24×C4D411yescharacteristic
47C24×C4C2311yescharacteristic
48#529—24no—
49#529—24no—
50C2×#126—21no—
51C2×#125C2×C411yescharacteristic
52C2×#124C2×C411yescharacteristic
53C2×#123—21no—
54C2×#123—21no—
55C2×C22⋊Q8C2×C412yes—
56C2×C22⋊Q8C2311yescharacteristic
57C2×#121C2×C412yes—
58C2×#121C2311yescharacteristic
59C2×#120—21no—
60C2×C4×Q8C2×C411yescharacteristic
61C2×C4×D4—22no—
62C2×C4×D4—21no—
63C2×C4×D4C2×C411yescharacteristic
64C2×#117—21no—
65C22×C4⋊C4—22no—
66C22×C22⋊C4—22no—
67C22×C42—24no—
68C22×C42—24no—
69C22×C42—22no—
70C22×C42C2×C412yes—
71C22×C42C2×C412yes—
72C22×C42D411yescharacteristic
73C22×C42D411yescharacteristic
74C22×C42D412yes—
75C22×C42D412yes—
76C22×C42C2311yescharacteristic
77C22×C42C2311yescharacteristic
78#431—24no—
79#431—24no—
80C2×C4⋊C8—22no—
81C2×C22⋊C8—24no—
82C2×C22⋊C8C2314yes—
83C2×C4×C8—22no—
84#391C2×C412yes—
85#390C2×C412yes—
86#389C2×C412yes—
87#389C2×C412yes—
88#388—22no—
89#388—22no—
90#388—22no—
91#388—22no—
92#387—22no—
93#387—22no—
94#387—22no—
95#387—22no—
96#386—22no—
97#386—22no—
98#386—22no—
99#386—22no—
100#386—21no—
101#386—21no—
102#385—22no—
103#385—22no—
104#385—22no—
105#385—22no—
106#385—21no—
107#385—21no—
108#383—21no—
109#383—24no—
110#383D418yescomplemented
111#382—28no—
112#382C2×C412yes—
113#381—28no—
114#381—21no—
115#381—21no—
116#380—28no—
117#380C2×C412yes—
118#380C2×C412yes—
119#380C2311yescharacteristic
120#380C2311yescharacteristic
121#379—21no—
122C4×C4⋊C4—28no—
123C4×C4⋊C4—22no—
124C4×C4⋊C4C2×C412yes—
125C4×C22⋊C4—28no—
126C4×C22⋊C4—22no—
127C4×C22⋊C4—21no—
128C4×C22⋊C4—22no—
129#376—21no—
130#376—28no—
131C2×#95—22no—
132C2×#95—22no—
133C2×#95—22no—
134C2×#95C2314yes—
135#336—24no—
136#328—28no—
137#327—28no—
138C25C2×D411yescharacteristic
139C22×Q8C22×C411yescharacteristic
140C22×D4—22no—
141C22×D4—21no—
142C22×D4C22×C411yescharacteristic
143C23×C4—24no—
144C23×C4—22no—
145C23×C4—22no—
146C23×C4—22no—
147C23×C4—22no—
148C23×C4—22no—
149C23×C4—22no—
150C23×C4—24no—
151C23×C4—21no—
152C23×C4—44no—
153C23×C4—44no—
154C23×C4—42no—
155C23×C4C22⋊C412yes—
156C23×C4C22⋊C412yes—
157C23×C4C22⋊C412yes—
158C23×C4C22⋊C412yes—
159C23×C4C22×C412yes—
160C23×C4C2×D411yescharacteristic
161C23×C4C2×D411yescharacteristic
162C23×C4C2×D411yescharacteristic
163C23×C4C2×D412yes—
164C23×C4C2×D411yescharacteristic
165C23×C4C2×D411yescharacteristic
166C23×C4C2411yescharacteristic
167C22×C8—24no—
168#126—44no—
169#125—24no—
170#124—24no—
171#123—44no—
172#123—42no—
173#123—42no—
174C22⋊Q8—22no—
175C22⋊Q8—22no—
176C22⋊Q8—44no—
177#121—22no—
178#121—22no—
179#121—44no—
180#120—44no—
181C4×Q8—24no—
182C4×D4—24no—
183C4×D4—48no—
184C4×D4—42no—
185C4×D4—42no—
186#117—44no—
187C2×C4⋊C4—28no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—48no—
196C2×C4⋊C4C22×C411yescharacteristic
197C2×C4⋊C4C22×C411yescharacteristic
198C2×C4⋊C4C22×C412yes—
199C2×C4⋊C4C22×C412yes—
200C2×C22⋊C4—28no—
201C2×C22⋊C4—22no—
202C2×C22⋊C4—22no—
203C2×C22⋊C4—22no—
204C2×C22⋊C4—21no—
205C2×C22⋊C4—21no—
206C2×C22⋊C4—21no—
207C2×C22⋊C4—21no—
208C2×C22⋊C4—21no—
209C2×C22⋊C4—22no—
210C2×C22⋊C4—21no—
211C2×C22⋊C4—22no—
212C2×C22⋊C4—22no—
213C2×C22⋊C4—21no—
214C2×C22⋊C4—44no—
215C2×C22⋊C4—44no—
216C2×C42—22no—
217C2×C42—22no—
218C2×C42—22no—
219C2×C42—22no—
220C2×C42—22no—
221C2×C42—22no—
222C2×C42—28no—
223C2×C42—28no—
224C2×C42—28no—
225C2×C42—28no—
226C2×C42—28no—
227C2×C42—21no—
228C2×C42—22no—
229C2×C42—48no—
230C2×C42C22⋊C412yes—
231C2×C42C22⋊C412yes—
232C2×C42C22⋊C412yes—
233C2×C42C22⋊C412yes—
234C2×C42C22×C412yes—
235C2×C42C22×C411yescharacteristic
236C2×C42C22×C411yescharacteristic
237C2×C42C2×D412yes—
238C2×C42C2×D414yes—
239C2×C42C2×D411yescharacteristic
240C2×C42C2×D411yescharacteristic
241C2×C42#4114yes—
242C4⋊C8—44no—
243C22⋊C8—24no—
244C22⋊C8—24no—
245C22⋊C8—48no—
246C4×C8—44no—
247#95—28no—
248#95—22no—
249#95—22no—
250#95—22no—
251#95—22no—
252#95—22no—
253#95—22no—
254#95—22no—
255#95—22no—
256#95—22no—
257#95—28no—
258#95—28no—
259#95—24no—
260#95—21no—
261#95—21no—
262#95C2×D418yes—
263C24—21no—
264C24—21no—
265C24—22no—
266C24—21no—
267C24—42no—
268C24—42no—
269C24C2×C22⋊C412yes—
270C24#13514yes—
271C24C22×D411yescharacteristic
272C2×Q8—22no—
273C2×Q8—42no—
274C2×D4—22no—
275C2×D4—42no—
276C2×D4—44no—
277C2×D4—44no—
278C2×D4—41no—
279C2×D4—41no—
280C2×D4—42no—
281C2×D4—42no—
282C2×D4—44no—
283C22×C4—21no—
284C22×C4—21no—
285C22×C4—21no—
286C22×C4—21no—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—24no—
290C22×C4—24no—
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—24no—
296C22×C4—24no—
297C22×C4—24no—
298C22×C4—24no—
299C22×C4—24no—
300C22×C4—24no—
301C22×C4—24no—
302C22×C4—24no—
303C22×C4—24no—
304C22×C4—24no—
305C22×C4—24no—
306C22×C4—24no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—21no—
310C22×C4—21no—
311C22×C4—21no—
312C22×C4—28no—
313C22×C4—42no—
314C22×C4—42no—
315C22×C4—42no—
316C22×C4—42no—
317C22×C4—42no—
318C22×C4—42no—
319C22×C4—42no—
320C22×C4—42no—
321C22×C4—48no—
322C22×C4—48no—
323C22×C4—48no—
324C22×C4—48no—
325C22×C4—48no—
326C22×C4—48no—
327C22×C4—44no—
328C22×C4—48no—
329C22×C4—48no—
330C22×C4—44no—
331C22×C4—44no—
332C22×C4—42no—
333C22×C4C2×C22⋊C412yes—
334C22×C4C2×C22⋊C411yescharacteristic
335C22×C4C2×C22⋊C411yescharacteristic
336C22×C4C2×C22⋊C411yescharacteristic
337C22×C4C2×C22⋊C411yescharacteristic
338C22×C4C2×C22⋊C412yes—
339C22×C4C2×C22⋊C412yes—
340C22×C4#12012yes—
341C22×C4#12012yes—
342C22×C4#12312yes—
343C22×C4#12312yes—
344C22×C4C23×C411yescharacteristic
345C22×C4C22×D411yescharacteristic
346C22×C4C22×D412yes—
347C22×C4C2×#4112yes—
348C2×C8—48no—
349C2×C8—44no—
350C4⋊C4—22no—
351C4⋊C4—22no—
352C4⋊C4—42no—
353C4⋊C4—42no—
354C4⋊C4—416no—
355C4⋊C4—44no—
356C4⋊C4—42no—
357C4⋊C4—42no—
358C4⋊C4—42no—
359C4⋊C4—42no—
360C4⋊C4—44no—
361C4⋊C4—44no—
362C22⋊C4—44no—
363C22⋊C4—416no—
364C22⋊C4—44no—
365C22⋊C4—44no—
366C22⋊C4—41no—
367C22⋊C4—41no—
368C22⋊C4—42no—
369C22⋊C4—42no—
370C22⋊C4—44no—
371C22⋊C4—42no—
372C22⋊C4—42no—
373C22⋊C4—42no—
374C22⋊C4—42no—
375C22⋊C4—44no—
376C22⋊C4—44no—
377C42—24no—
378C42—24no—
379C42—24no—
380C42—24no—
381C42—24no—
382C42—24no—
383C42—416no—
384C42—44no—
385C42—42no—
386C42C4×D414yes—
387C42C4×D414yes—
388C42#12114yes—
389C42#12114yes—
390C23—22no—
391C23—22no—
392C23—24no—
393C23—22no—
394C23—22no—
395C23—22no—
396C23—22no—
397C23—22no—
398C23—24no—
399C23—22no—
400C23—48no—
401C23—42no—
402C23—42no—
403C23—42no—
404C23—42no—
405C23—48no—
406C23—42no—
407C23—41no—
408C23—41no—
409C23—42no—
410C23—41no—
411C23#41614yes—
412C23C22×C22⋊C411yescharacteristic
413C23C2×#12011yescharacteristic
414C23C2×#12311yescharacteristic
415C23C2×#13512yes—
416C23#57512yes—
417Q8—82no—
418D4—82no—
419D4—88no—
420D4—82no—
421D4—82no—
422C2×C4—24no—
423C2×C4—24no—
424C2×C4—24no—
425C2×C4—24no—
426C2×C4—24no—
427C2×C4—22no—
428C2×C4—22no—
429C2×C4—22no—
430C2×C4—21no—
431C2×C4—21no—
432C2×C4—21no—
433C2×C4—21no—
434C2×C4—22no—
435C2×C4—22no—
436C2×C4—22no—
437C2×C4—24no—
438C2×C4—22no—
439C2×C4—22no—
440C2×C4—22no—
441C2×C4—22no—
442C2×C4—22no—
443C2×C4—42no—
444C2×C4—42no—
445C2×C4—42no—
446C2×C4—42no—
447C2×C4—48no—
448C2×C4—48no—
449C2×C4—48no—
450C2×C4—48no—
451C2×C4—48no—
452C2×C4—48no—
453C2×C4—42no—
454C2×C4—42no—
455C2×C4—48no—
456C2×C4—48no—
457C2×C4—42no—
458C2×C4—42no—
459C2×C4—42no—
460C2×C4—42no—
461C2×C4—41no—
462C2×C4—42no—
463C2×C4—41no—
464C2×C4—42no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—48no—
468C2×C4—41no—
469C2×C4—41no—
470C2×C4—41no—
471C2×C4—84no—
472C2×C4—88no—
473C2×C4—88no—
474C2×C4#38612yes—
475C2×C4#38612yes—
476C2×C4#38612yes—
477C2×C4#38612yes—
478C2×C4#41014yes—
479C2×C4#42114yes—
480C2×C4C2×C4×D412yes—
481C2×C4C2×#12112yes—
482C8—84no—
483C22—24no—
484C22—24no—
485C22—21no—
486C22—21no—
487C22—21no—
488C22—21no—
489C22—24no—
490C22—24no—
491C22—24no—
492C22—48no—
493C22—48no—
494C22—42no—
495C22—42no—
496C22—42no—
497C22—42no—
498C22—82no—
499C22—81no—
500C22—82no—
501C22—81no—
502C22—81no—
503C22C2×#38611yescharacteristic
504C22C2×#41012yes—
505C22C2×#41611yescharacteristic
506C22#289611yescharacteristic
507C22C2×#42112yes—
508C22#300212yes—
509C22#309112yes—
510C4—22no—
511C4—22no—
512C4—24no—
513C4—22no—
514C4—22no—
515C4—22no—
516C4—24no—
517C4—22no—
518C4—42no—
519C4—88no—
520C4—82no—
521C4—81no—
522C4—82no—
523C4—81no—
524C4—81no—
525C2—22no—
526C2—22no—
527C2—44no—
528C2—161no—
529C2#1756712yes—
530C2#1759412yes—
531C2#2090811yescharacteristic
532C2#2097911yescharacteristic
533C2#2098311yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radical#2847C22
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C23×C42 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2283C2semidirect
3. #20714C2semidirect
4. C2×#1521C2semidirect
6. #9151C2semidirect
7. #9149C2semidirect
8. #9145C2semidirect
9. #8025C2semidirect
17. #2851C4semidirect
18. #2850C4semidirect
22. C4×C22⋊Q8C4semidirect
24. C4×#121C4semidirect
110. #383D4semidirect

Maximal subgroups

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

subgroupindex
#366212
C2×#22832
#207142
C2×#15212
#91552
#91512
#91492
#91452
#80252