Finite Group Databasebeta

← #195569

Subgroups — #195569

Order 512.

1390 conjugacy classes of subgroups.

The subgroup lattice has 537 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3206C211yesmaximal, characteristic
2C2×#2369C211yesmaximal, characteristic, complemented
3#21386C211yesmaximal, characteristic
4#21258C211yesmaximal, characteristic
5C2×#1548C211yesmaximal, characteristic, complemented
6#9063C214yesmaximal, complemented
7#9060C214yesmaximal, complemented
8#7901C211yesmaximal, characteristic, complemented
9#7892C211yesmaximal, characteristic, complemented
10C2×#532C2211yescharacteristic
11C22×C4⋊Q8C2211yescharacteristic
12#3206—24no—
13#3206C418yescomplemented
14C2×#492C2212yes—
15C2×#487C2212yes—
16C2×#461—22no—
17C2×#448—22no—
18C2×#431C2211yescharacteristic
19C2×#431C2211yescharacteristic
20C2×#423C2211yescharacteristic
21C2×#417—21no—
22C2×#417—21no—
23#2369—22no—
24#2369—22no—
25#2369—22no—
26#2369C2214yes—
27C2×#391—22no—
28C2×#389—21no—
29C2×#389—21no—
30C2×C4×C4⋊C4—21no—
31C2×C4×C4⋊C4—21no—
32C2×C4×C4⋊C4C2211yescharacteristic
33C2×C4×C4⋊C4C2211yescharacteristic
34#1988C2212yes—
35#1983—22no—
36#1982C2212yes—
37#1982C2212yes—
38#1981C2212yes—
39#1874—21no—
40#1874—21no—
41C4×#102—22no—
42#1548—24no—
43C2×#333C2214yes—
44#1279C2214yes—
45#1279C2214yes—
46C22×#41—21no—
47C23×Q8D412yes—
48C22×SD16—24no—
49C22×M16—21no—
50#532—22no—
51#532—22no—
52#532—22no—
53#532C2×C414yes—
54C2×C4⋊Q8—21no—
55C2×C4⋊Q8—21no—
56C2×C4⋊Q8—22no—
57C2×C4⋊Q8—22no—
58C2×C4⋊Q8C2×C414yes—
59C2×C4⋊Q8D412yes—
60C2×C4⋊Q8D414yes—
61C2×C4⋊Q8C2311yescharacteristic
62C2×C4⋊Q8C2312yes—
63C2×#127C2311yescharacteristic
64C2×#124—21no—
65C2×#121—21no—
66C2×#121—21no—
67C2×C4×D4—21no—
68C2×C4×D4—21no—
69C22×C4⋊C4—21no—
70C22×C4⋊C4—21no—
71C22×C4⋊C4—22no—
72C22×C4⋊C4D411yescharacteristic
73C22×C4⋊C4D411yescharacteristic
74C22×C4⋊C4D411yescharacteristic
75C22×C4⋊C4D411yescharacteristic
76C22×C42—21no—
77C22×C42—21no—
78C22×C42—22no—
79C22×C42C2311yescharacteristic
80#492—24no—
81#492C2×C418yescomplemented
82#487—24no—
83#487C2×C418yescomplemented
84#461—216no—
85#448—216no—
86#431—22no—
87#431—22no—
88#431—22no—
89#431C2×C414yes—
90#423—22no—
91#423—22no—
92C2×C4⋊C8—21no—
93C2×C4⋊C8—21no—
94C2×C4⋊C8C2312yes—
95C2×C4⋊C8C2312yes—
96#417—44no—
97#417—44no—
98C2×#103—24no—
99C2×#103—22no—
100C2×#103—22no—
101C2×#102—22no—
102C2×#102—22no—
103C2×#102—22no—
104C2×#102—22no—
105C2×C22⋊C8—21no—
106C2×C22⋊C8—21no—
107C2×C4×C8—21no—
108C2×C4×C8C2312yes—
109#391—28no—
110#391—28no—
111#389—24no—
112#389—24no—
113#389—24no—
114#389—24no—
115C4×C4⋊C4—24no—
116C4×C4⋊C4—24no—
117C4×C4⋊C4—24no—
118C4×C4⋊C4—24no—
119C4×C4⋊C4—21no—
120C4×C4⋊C4—21no—
121C4×C4⋊C4—21no—
122C4×C4⋊C4—21no—
123C4×C4⋊C4D412yes—
124C4×C4⋊C4D412yes—
125C4×C4⋊C4C2312yes—
126C4×C4⋊C4C2312yes—
127C2×#95—22no—
128C2×#95—22no—
129C2×#95—21no—
130C2×#95—21no—
131#340—22no—
132#340—22no—
133#333—28no—
134C2×#41—24no—
135C2×#41—42no—
136C2×#41—42no—
137C2×#41—42no—
138C2×#41—42no—
139C2×#41—42no—
140C22×Q8—22no—
141C22×Q8—24no—
142C22×Q8—21no—
143C22×Q8—82no—
144C22×Q8—82no—
145C22×Q8C22⋊C418yes—
146C22×Q8C2×D414yes—
147C22×D4—21no—
148C22×D4—21no—
149C22×D4—21no—
150C22×D4—41no—
151C23×C4—22no—
152C23×C4—21no—
153C23×C4—21no—
154C23×C4—21no—
155C23×C4—21no—
156C23×C4—41no—
157C23×C4C2×D411yescharacteristic
158C23×C4C2×D411yescharacteristic
159C23×C4C2×D411yescharacteristic
160C2×SD16—216no—
161C2×SD16—84no—
162C2×SD16—84no—
163C2×M16—42no—
164C2×M16—42no—
165C2×M16—42no—
166C22×C8—22no—
167C22×C8—22no—
168C22×C8—22no—
169C4⋊Q8—24no—
170C4⋊Q8—24no—
171C4⋊Q8—24no—
172C4⋊Q8—24no—
173C4⋊Q8—24no—
174C4⋊Q8—21no—
175C4⋊Q8—22no—
176C4⋊Q8—22no—
177C4⋊Q8—21no—
178C4⋊Q8—21no—
179C4⋊Q8C22⋊C418yes—
180C4⋊Q8C22×C412yes—
181C4⋊Q8C22×C414yes—
182C4⋊Q8C2×D414yes—
183#127—21no—
184#127—21no—
185#127—21no—
186#127C22×C412yes—
187#124—24no—
188#124—42no—
189#121—24no—
190#121—24no—
191#121—42no—
192#121—42no—
193C4×D4—24no—
194C4×D4—24no—
195C4×D4—42no—
196C4×D4—42no—
197C2×C4⋊C4—24no—
198C2×C4⋊C4—24no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—24no—
204C2×C4⋊C4—24no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—24no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—28no—
216C2×C4⋊C4—42no—
217C2×C4⋊C4—84no—
218C2×C4⋊C4SD1614yescomplemented
219C2×C4⋊C4C2×D412yes—
220C2×C4⋊C4C2×D412yes—
221C2×C4⋊C4C2×D412yes—
222C2×C4⋊C4C2×D412yes—
223C2×C22⋊C4—41no—
224C2×C22⋊C4—41no—
225C2×C42—22no—
226C2×C42—22no—
227C2×C42—22no—
228C2×C42—22no—
229C2×C42—28no—
230C2×C42—44no—
231C2×C42—42no—
232C2×C42—42no—
233C2×C42—84no—
234C2×C42C22×C412yes—
235C2×C42C2×D411yescharacteristic
236C2×C42C2×D412yes—
237C2×C42C2×D411yescharacteristic
238C2×C42C2×D411yescharacteristic
239C2×C42#4112yes—
240C2×C42#4112yes—
241C2×C42C2411yescharacteristic
242C4⋊C8—24no—
243C4⋊C8—22no—
244C4⋊C8—22no—
245C4⋊C8—22no—
246C4⋊C8—22no—
247C4⋊C8—42no—
248#103—216no—
249#103—44no—
250#103—44no—
251#102—44no—
252#102—44no—
253#102—44no—
254#102—44no—
255C22⋊C8—24no—
256C22⋊C8—42no—
257C4×C8—22no—
258C4×C8—42no—
259C4×C8C22×C414yes—
260#95—28no—
261#95—28no—
262#95—22no—
263#95—22no—
264#95—22no—
265#95—22no—
266C24—41no—
267C24#12011yescharacteristic
268#41—48no—
269#41—48no—
270#41—84no—
271#41—84no—
272C2×Q8—24no—
273C2×Q8—28no—
274C2×Q8—22no—
275C2×Q8—44no—
276C2×Q8—44no—
277C2×Q8—41no—
278C2×Q8—41no—
279C2×Q8—41no—
280C2×Q8—41no—
281C2×Q8—41no—
282C2×Q8—82no—
283C2×Q8—82no—
284C2×Q8—88no—
285C2×Q8—88no—
286C2×Q8C2×C22⋊C418yes—
287C2×D4—22no—
288C2×D4—22no—
289C2×D4—22no—
290C2×D4—42no—
291C2×D4—42no—
292C2×D4—44no—
293C2×D4—41no—
294C2×D4—41no—
295C2×D4—41no—
296C2×D4—41no—
297C2×D4—41no—
298C2×D4—41no—
299C2×D4—42no—
300C2×D4—41no—
301C2×D4—42no—
302C2×D4—82no—
303C2×D4—82no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—24no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—24no—
310C22×C4—28no—
311C22×C4—24no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—41no—
315C22×C4—41no—
316C22×C4—42no—
317C22×C4—42no—
318C22×C4—42no—
319C22×C4—44no—
320C22×C4—44no—
321C22×C4—42no—
322C22×C4—42no—
323C22×C4—42no—
324C22×C4—42no—
325C22×C4—42no—
326C22×C4—41no—
327C22×C4—42no—
328C22×C4—42no—
329C22×C4—44no—
330C22×C4—82no—
331C22×C4—82no—
332C22×C4C2×C22⋊C412yes—
333C22×C4C4×D412yes—
334C22×C4C4×D412yes—
335C22×C4#12011yescharacteristic
336C22×C4#12011yescharacteristic
337C22×C4#12011yescharacteristic
338C22×C4#12112yes—
339C22×C4#12112yes—
340C22×C4#12312yes—
341C22×C4C2×SD1612yes—
342C22×C4#13512yes—
343C22×C4AGL(1,ℤ/8)12yes—
344C22×C4#13712yes—
345C22×C4C22×D411yescharacteristic
346C22×C4C22×D411yescharacteristic
347C22×C4C22×D411yescharacteristic
348SD16—816no—
349M16—84no—
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—24no—
360C4⋊C4—24no—
361C4⋊C4—24no—
362C4⋊C4—24no—
363C4⋊C4—24no—
364C4⋊C4—22no—
365C4⋊C4—22no—
366C4⋊C4—28no—
367C4⋊C4—28no—
368C4⋊C4—24no—
369C4⋊C4—22no—
370C4⋊C4—22no—
371C4⋊C4—22no—
372C4⋊C4—22no—
373C4⋊C4—22no—
374C4⋊C4—22no—
375C4⋊C4—42no—
376C4⋊C4—41no—
377C4⋊C4—41no—
378C4⋊C4—42no—
379C4⋊C4—88no—
380C4⋊C4C2×SD1614yes—
381C4⋊C4#13714yes—
382C22⋊C4—44no—
383C22⋊C4—44no—
384C42—44no—
385C42—42no—
386C42—42no—
387C42—44no—
388C42—44no—
389C42—88no—
390C42C2×C22⋊C412yes—
391C42C4×D412yes—
392C42C4×D412yes—
393C42#12112yes—
394C42#12112yes—
395C42#12312yes—
396C42C23×C411yescharacteristic
397C42C22×D411yescharacteristic
398C42C2×#4111yescharacteristic
399C42C2×#4111yescharacteristic
400C23—21no—
401C23—21no—
402C23—21no—
403C23—21no—
404C23—42no—
405C23—81no—
406C23—81no—
407C23—81no—
408C23—81no—
409C23—81no—
410C23—81no—
411C23#38612yes—
412C23#44812yes—
413C23#44912yes—
414C23C2×#12011yescharacteristic
415Q8—48no—
416Q8—42no—
417Q8—42no—
418Q8—84no—
419Q8—88no—
420Q8—81no—
421Q8—81no—
422D4—44no—
423D4—42no—
424D4—42no—
425D4—44no—
426D4—82no—
427D4—88no—
428D4—81no—
429D4—81no—
430D4—82no—
431C2×C4—21no—
432C2×C4—21no—
433C2×C4—21no—
434C2×C4—21no—
435C2×C4—21no—
436C2×C4—21no—
437C2×C4—24no—
438C2×C4—24no—
439C2×C4—24no—
440C2×C4—24no—
441C2×C4—28no—
442C2×C4—42no—
443C2×C4—42no—
444C2×C4—48no—
445C2×C4—48no—
446C2×C4—416no—
447C2×C4—42no—
448C2×C4—42no—
449C2×C4—42no—
450C2×C4—42no—
451C2×C4—44no—
452C2×C4—44no—
453C2×C4—42no—
454C2×C4—42no—
455C2×C4—42no—
456C2×C4—42no—
457C2×C4—42no—
458C2×C4—42no—
459C2×C4—42no—
460C2×C4—84no—
461C2×C4—84no—
462C2×C4—81no—
463C2×C4—81no—
464C2×C4—88no—
465C2×C4#38612yes—
466C2×C4#38612yes—
467C2×C4#38612yes—
468C2×C4C4×SD1614yes—
469C2×C4#44014yes—
470C2×C4#44812yes—
471C2×C4#44912yes—
472C2×C4#46014yes—
473C2×C4#46414yes—
474C2×C4C22×C22⋊C411yescharacteristic
475C2×C4C2×C4×D411yescharacteristic
476C2×C4C2×C4×D411yescharacteristic
477C2×C4C2×#12111yescharacteristic
478C2×C4C2×#12111yescharacteristic
479C2×C4C2×#12311yescharacteristic
480C2×C4C22×SD1611yescharacteristic
481C2×C4C2×#13511yescharacteristic
482C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
483C2×C4C2×#13711yescharacteristic
484C2×C4C2×#13712yes—
485C2×C4#57512yes—
486C8—44no—
487C8—82no—
488C8—82no—
489C22—22no—
490C22—22no—
491C22—22no—
492C22—22no—
493C22—22no—
494C22—22no—
495C22—82no—
496C22—82no—
497C22—82no—
498C22—161no—
499C22—161no—
500C22—161no—
501C22—161no—
502C22#188014yes—
503C22C2×#38611yescharacteristic
504C22C2×#44811yescharacteristic
505C22C2×#44911yescharacteristic
506C22#300411yescharacteristic
507C22#300511yescharacteristic
508C22#301911yescharacteristic
509C22#301911yescharacteristic
510C4—22no—
511C4—21no—
512C4—21no—
513C4—22no—
514C4—44no—
515C4—44no—
516C4—48no—
517C4—82no—
518C4—82no—
519C4—82no—
520C4#188014yes—
521C4C2×C4×SD1611yescharacteristic
522C4C2×#44011yescharacteristic
523C4C4×#13712yes—
524C4C2×#46011yescharacteristic
525C4C2×#46411yescharacteristic
526C4#303712yes—
527C2—24no—
528C2—162no—
529C2#1874811yescharacteristic
530C2#1875211yescharacteristic
531C2#1876811yescharacteristic
532C2#1877211yescharacteristic
533C2C2×#188011yescharacteristic
534C2#2092611yescharacteristic
535C2#2095311yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radicalC2×#532C22
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
2. C2×#2369C2semidirect
5. C2×#1548C2semidirect
6. #9063C2semidirect
7. #9060C2semidirect
8. #7901C2semidirect
9. #7892C2semidirect
13. #3206C4semidirect
81. #492C2×C4semidirect
83. #487C2×C4semidirect
218. C2×C4⋊C4SD16semidirect

Maximal subgroups

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

subgroupindex
C2×#32062
C2×#23692
#213862
#212582
C2×#15482
#90632
#90602
#79012
#78922