Finite Group Databasebeta

← #108342

Subgroups — #108342

Order 512.

970 conjugacy classes of subgroups.

The subgroup lattice has 541 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34145C211yesmaximal, characteristic
2C2×#1503C211yesmaximal, characteristic
3C2×#1493C211yesmaximal, characteristic
4#10852C211yesmaximal, characteristic
5C4×#340C211yesmaximal, characteristic
6#7901C211yesmaximal, characteristic
7#7890C211yesmaximal, characteristic
8C2×#533C2211yescharacteristic
9C2×#431C2211yescharacteristic
10C2×#423C2211yescharacteristic
11C22×C4×C8C2211yescharacteristic
12#2370—22no—
13#2369C412yescomplemented
14#2368—22no—
15#2365C412yescomplemented
16#2360—22no—
17#2296—21no—
18#2295—21no—
19#2295—22no—
20C42×Q8—21no—
21C4×#117—21no—
22C4×#117C2211yescharacteristic
23C2×C4×C4⋊C4C2211yescharacteristic
24C2×C4×C4⋊C4C2211yescharacteristic
25C2×#340—22no—
26C2×#339—21no—
27C2×#339—21no—
28#1503—22no—
29#1503C414yescomplemented
30C2×#328—22no—
31#1493—22no—
32#1493C414yescomplemented
33C2×Q8⋊C8—22no—
34#1279—22no—
35#1279—21no—
36#1279—21no—
37C22×M16—21no—
38C23×C8—21no—
39#533—21no—
40#533—21no—
41#533—21no—
42#533—21no—
43#533C2×C412yes—
44#533C2×C412yes—
45#533D412yes—
46#533Q812yes—
47C2×C4⋊Q8C2×C411yescharacteristic
48C2×C4⋊Q8C2×C411yescharacteristic
49C2×#125—21no—
50C2×C22⋊Q8—21no—
51C2×C22⋊Q8D412yes—
52C2×C4×Q8—22no—
53C2×C4×Q8—21no—
54C2×C4×Q8D412yes—
55C2×#117—21no—
56C2×#117D411yescharacteristic
57C2×#117D411yescharacteristic
58C22×C4⋊C4—21no—
59C22×C4⋊C4D412yes—
60C22×C42—21no—
61C22×C42—21no—
62C22×C42C2311yescharacteristic
63#431—21no—
64#431—21no—
65#431—21no—
66#431C2×C412yes—
67#423—21no—
68#423—21no—
69#423—21no—
70#423C2×C412yes—
71C2×C4⋊C8—21no—
72C2×C4⋊C8—21no—
73C2×C4⋊C8—21no—
74C2×C22⋊C8—21no—
75C2×C4×C8—21no—
76C2×C4×C8—21no—
77C2×C4×C8—21no—
78C2×C4×C8—21no—
79C2×C4×C8—21no—
80C2×C4×C8—21no—
81C2×C4×C8C2×C412yes—
82#391—24no—
83#391—22no—
84#389—24no—
85#389—22no—
86#389—22no—
87#389—22no—
88#387—24no—
89#387—24no—
90#387—24no—
91#385—24no—
92C4×C4⋊C4—24no—
93C4×C4⋊C4—22no—
94C4×C4⋊C4—22no—
95C4×C4⋊C4—22no—
96C4×C4⋊C4—22no—
97C4×C4⋊C4—22no—
98C4×C4⋊C4—21no—
99C4×C4⋊C4—21no—
100C4×C4⋊C4—21no—
101C4×C4⋊C4—21no—
102C4×C4⋊C4—21no—
103C4×C4⋊C4C2×C412yes—
104C4×C4⋊C4C2×C412yes—
105C4×C4⋊C4C2×C412yes—
106C4×C4⋊C4D412yes—
107C4×C4⋊C4D411yescharacteristic
108C4×C4⋊C4D411yescharacteristic
109C4×C22⋊C4—24no—
110C4×C22⋊C4—21no—
111C4×C22⋊C4—21no—
112#376—22no—
113#376—21no—
114C2×#95—21no—
115C2×#95—21no—
116C43—22no—
117C43—21no—
118#340—28no—
119#339—42no—
120#339—42no—
121#328—28no—
122Q8⋊C8—28no—
123C22×Q8—21no—
124C22×Q8C22⋊C412yes—
125C23×C4—22no—
126C23×C4—41no—
127C23×C4C2×D411yescharacteristic
128C23×C4C2×D411yescharacteristic
129C23×C4C2×D411yescharacteristic
130C2×M16—22no—
131C2×M16—41no—
132C2×M16—41no—
133C2×M16—41no—
134C2×M16—41no—
135C2×M16—41no—
136C22×C8—24no—
137C22×C8—22no—
138C22×C8—42no—
139C22×C8—41no—
140C22×C8—41no—
141C22×C8—41no—
142C22×C8—41no—
143C4⋊Q8—21no—
144C4⋊Q8—21no—
145C4⋊Q8—21no—
146C4⋊Q8—21no—
147C4⋊Q8C4212yes—
148C4⋊Q8C4212yes—
149C4⋊Q8C4⋊C412yes—
150C4⋊Q8C4⋊C412yes—
151#125—22no—
152#125—22no—
153#125—22no—
154#125—22no—
155C22⋊Q8—22no—
156C22⋊Q8—22no—
157C22⋊Q8—22no—
158C22⋊Q8—22no—
159C22⋊Q8—22no—
160C22⋊Q8—22no—
161C22⋊Q8C22⋊C414yes—
162C22⋊Q8C4⋊C414yes—
163C4×Q8—22no—
164C4×Q8—22no—
165C4×Q8—22no—
166C4×Q8—22no—
167C4×Q8—22no—
168C4×Q8—22no—
169C4×Q8—44no—
170C4×Q8—44no—
171C4×Q8C22⋊C414yes—
172C4×Q8C4⋊C414yes—
173#117—22no—
174#117—22no—
175#117—22no—
176#117—22no—
177#117—21no—
178#117—21no—
179#117—21no—
180#117—21no—
181#117—21no—
182#117—21no—
183#117SD1612yes—
184#117Q1612yes—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—24no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—24no—
193C2×C4⋊C4—24no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—24no—
196C2×C4⋊C4—42no—
197C2×C4⋊C4—42no—
198C2×C4⋊C4—41no—
199C2×C4⋊C4C22⋊C412yes—
200C2×C4⋊C4C22⋊C412yes—
201C2×C4⋊C4C22⋊C412yes—
202C2×C4⋊C4C22⋊C412yes—
203C2×C4⋊C4C22⋊C412yes—
204C2×C22⋊C4—22no—
205C2×C22⋊C4—21no—
206C2×C22⋊C4—21no—
207C2×C42—22no—
208C2×C42—21no—
209C2×C42—24no—
210C2×C42—22no—
211C2×C42—22no—
212C2×C42—21no—
213C2×C42—21no—
214C2×C42—21no—
215C2×C42—22no—
216C2×C42—24no—
217C2×C42—22no—
218C2×C42—41no—
219C2×C42—41no—
220C2×C42—41no—
221C2×C42—41no—
222C2×C42—41no—
223C2×C42—41no—
224C2×C42—41no—
225C2×C42—41no—
226C2×C42—41no—
227C2×C42C22×C411yescharacteristic
228C2×C42C22×C411yescharacteristic
229C2×C42C22×C411yescharacteristic
230C2×C42C2×D411yescharacteristic
231C2×C42C2×D411yescharacteristic
232C2×C42C2×D411yescharacteristic
233C2×C42C2×D411yescharacteristic
234C2×C42C2×Q811yescharacteristic
235C2×C42#4111yescharacteristic
236C2×C42#4111yescharacteristic
237C2×C42#4111yescharacteristic
238C2×C42#4111yescharacteristic
239C4⋊C8—22no—
240C4⋊C8—22no—
241C4⋊C8—22no—
242C4⋊C8—41no—
243C4⋊C8—41no—
244C4⋊C8—41no—
245C22⋊C8—22no—
246C22⋊C8—41no—
247C4×C8—22no—
248C4×C8—22no—
249C4×C8—22no—
250C4×C8—41no—
251C4×C8—41no—
252C4×C8—41no—
253#95—24no—
254#95—24no—
255#95—24no—
256#95—24no—
257#95—22no—
258#95—22no—
259#95—21no—
260#95—21no—
261#95—22no—
262#95—22no—
263#95—41no—
264C24#12011yescharacteristic
265C2×Q8—22no—
266C2×Q8—22no—
267C2×Q8—22no—
268C2×Q8—44no—
269C2×Q8—42no—
270C2×Q8—42no—
271C2×Q8#9514yes—
272C22×C4—22no—
273C22×C4—22no—
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—42no—
281C22×C4—44no—
282C22×C4—44no—
283C22×C4—42no—
284C22×C4—42no—
285C22×C4—42no—
286C22×C4—82no—
287C22×C4—82no—
288C22×C4—81no—
289C22×C4—81no—
290C22×C4C2×C22⋊C411yescharacteristic
291C22×C4C2×C22⋊C411yescharacteristic
292C22×C4C2×C22⋊C411yescharacteristic
293C22×C4C2×C22⋊C411yescharacteristic
294C22×C4C2×C4⋊C411yescharacteristic
295C22×C4C4×D411yescharacteristic
296C22×C4C4×D411yescharacteristic
297C22×C4C4×D411yescharacteristic
298C22×C4C4×D411yescharacteristic
299C22×C4#12011yescharacteristic
300C22×C4#12011yescharacteristic
301C22×C4#12011yescharacteristic
302C22×C4#12111yescharacteristic
303C22×C4#12111yescharacteristic
304C22×C4C22⋊Q811yescharacteristic
305C22×C4C22⋊Q811yescharacteristic
306C22×C4#12311yescharacteristic
307C22×C4#12311yescharacteristic
308C22×C4C2×SD1611yescharacteristic
309C22×C4C2×Q1611yescharacteristic
310C22×C4#13511yescharacteristic
311C22×C4#13511yescharacteristic
312C22×C4AGL(1,ℤ/8)11yescharacteristic
313C22×C4#13711yescharacteristic
314C22×C4#13711yescharacteristic
315C22×C4#13711yescharacteristic
316M16—42no—
317M16—42no—
318M16—81no—
319M16—81no—
320C2×C8—24no—
321C2×C8—44no—
322C2×C8—44no—
323C2×C8—42no—
324C2×C8—42no—
325C2×C8—42no—
326C2×C8—42no—
327C2×C8—82no—
328C2×C8—81no—
329C2×C8—81no—
330C4⋊C4—22no—
331C4⋊C4—22no—
332C4⋊C4—22no—
333C4⋊C4—22no—
334C4⋊C4—22no—
335C4⋊C4—22no—
336C4⋊C4—22no—
337C4⋊C4—24no—
338C4⋊C4—24no—
339C4⋊C4—24no—
340C4⋊C4—24no—
341C4⋊C4—22no—
342C4⋊C4—22no—
343C4⋊C4—24no—
344C4⋊C4—22no—
345C4⋊C4—22no—
346C4⋊C4—22no—
347C4⋊C4—22no—
348C4⋊C4—44no—
349C4⋊C4—44no—
350C4⋊C4—42no—
351C4⋊C4—48no—
352C4⋊C4—84no—
353C4⋊C4#9514yes—
354C4⋊C4#10314yes—
355C4⋊C4#10414yes—
356C22⋊C4—24no—
357C22⋊C4—24no—
358C22⋊C4—22no—
359C22⋊C4—22no—
360C22⋊C4—41no—
361C22⋊C4—41no—
362C42—24no—
363C42—24no—
364C42—22no—
365C42—22no—
366C42—48no—
367C42—42no—
368C42—42no—
369C42—42no—
370C42—42no—
371C42—41no—
372C42—41no—
373C42—41no—
374C42—41no—
375C42—41no—
376C42—41no—
377C42—42no—
378C42—42no—
379C42—82no—
380C42—84no—
381C42—81no—
382C42—81no—
383C42C2×C4211yescharacteristic
384C42C2×C22⋊C411yescharacteristic
385C42C2×C4⋊C411yescharacteristic
386C42C2×C4⋊C411yescharacteristic
387C42#11711yescharacteristic
388C42#11711yescharacteristic
389C42C4×D411yescharacteristic
390C42C4×D411yescharacteristic
391C42C4×D411yescharacteristic
392C42C4×D411yescharacteristic
393C42#12111yescharacteristic
394C42#12111yescharacteristic
395C42C22⋊Q811yescharacteristic
396C42C22⋊Q811yescharacteristic
397C42#12311yescharacteristic
398C42#12311yescharacteristic
399C23—22no—
400C23—22no—
401C23#37911yescharacteristic
402C23#38511yescharacteristic
403C23#38611yescharacteristic
404C23#44811yescharacteristic
405C23#44911yescharacteristic
406C23#45111yescharacteristic
407C23#45211yescharacteristic
408Q8—44no—
409Q8—44no—
410Q8—42no—
411Q8—42no—
412Q8—82no—
413C2×C4—22no—
414C2×C4—22no—
415C2×C4—22no—
416C2×C4—22no—
417C2×C4—22no—
418C2×C4—21no—
419C2×C4—21no—
420C2×C4—21no—
421C2×C4—21no—
422C2×C4—24no—
423C2×C4—24no—
424C2×C4—24no—
425C2×C4—24no—
426C2×C4—41no—
427C2×C4—41no—
428C2×C4—41no—
429C2×C4—41no—
430C2×C4—42no—
431C2×C4—42no—
432C2×C4—44no—
433C2×C4—42no—
434C2×C4—42no—
435C2×C4—44no—
436C2×C4—44no—
437C2×C4—42no—
438C2×C4—42no—
439C2×C4—42no—
440C2×C4—42no—
441C2×C4—42no—
442C2×C4—42no—
443C2×C4—84no—
444C2×C4—84no—
445C2×C4—82no—
446C2×C4—84no—
447C2×C4—84no—
448C2×C4—82no—
449C2×C4C2×#9511yescharacteristic
450C2×C4C4×C22⋊C411yescharacteristic
451C2×C4C4×C22⋊C411yescharacteristic
452C2×C4#38011yescharacteristic
453C2×C4#38011yescharacteristic
454C2×C4#38111yescharacteristic
455C2×C4#38511yescharacteristic
456C2×C4#38511yescharacteristic
457C2×C4#38511yescharacteristic
458C2×C4#38611yescharacteristic
459C2×C4#38611yescharacteristic
460C2×C4#38611yescharacteristic
461C2×C4C2×#10311yescharacteristic
462C2×C4#41611yescharacteristic
463C2×C4#41711yescharacteristic
464C2×C4#41911yescharacteristic
465C2×C4C2×#10412yes—
466C2×C4#42112yes—
467C2×C4C4×SD1611yescharacteristic
468C2×C4C4×SD1611yescharacteristic
469C2×C4C4×Q1611yescharacteristic
470C2×C4C4×Q1611yescharacteristic
471C2×C4#44011yescharacteristic
472C2×C4#44011yescharacteristic
473C2×C4#44111yescharacteristic
474C2×C4#44111yescharacteristic
475C2×C4#44811yescharacteristic
476C2×C4#44911yescharacteristic
477C2×C4#45111yescharacteristic
478C2×C4#45211yescharacteristic
479C2×C4#46011yescharacteristic
480C2×C4#46211yescharacteristic
481C2×C4#46411yescharacteristic
482C2×C4#46411yescharacteristic
483C2×C4#47511yescharacteristic
484C2×C4#47711yescharacteristic
485C2×C4#47911yescharacteristic
486C2×C4#47911yescharacteristic
487C8—44no—
488C8—82no—
489C8—82no—
490C22—21no—
491C22—21no—
492C22—21no—
493C22—21no—
494C22—22no—
495C22—22no—
496C22—22no—
497C22—22no—
498C22C22⋊#9511yescharacteristic
499C22#178711yescharacteristic
500C22#178811yescharacteristic
501C22#178912yes—
502C22#179412yes—
503C22#186311yescharacteristic
504C22#186511yescharacteristic
505C22#187811yescharacteristic
506C22#188011yescharacteristic
507C4—21no—
508C4—21no—
509C4—21no—
510C4—21no—
511C4—21no—
512C4—21no—
513C4—44no—
514C4—44no—
515C4—44no—
516C4—44no—
517C4—84no—
518C4—82no—
519C4—82no—
520C4C4×#10411yescharacteristic
521C4#175911yescharacteristic
522C4C4×#10311yescharacteristic
523C4#176311yescharacteristic
524C4#180311yescharacteristic
525C4#180511yescharacteristic
526C4#180612yes—
527C4#186311yescharacteristic
528C4#186511yescharacteristic
529C4#187811yescharacteristic
530C4#188011yescharacteristic
531C2—21no—
532C2—21no—
533C2—21no—
534C2—21no—
535C2#843812yes—
536C2#843912yes—
537C2#1075211yescharacteristic
538C2#1078911yescharacteristic
539C2#1079011yescharacteristic
subgroupG/H
centerC22×C4#120
commutatorC42C2×C42
Frattini, cosocleC22×C42C23
socleC23#379
prime radicalC24#120
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C4 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
13. #2369C4semidirect
15. #2365C4semidirect
29. #1503C4semidirect
32. #1493C4semidirect

Maximal subgroups

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

subgroupindex
#341452
C2×#15032
C2×#14932
#108522
C4×#3402
#79012
#78902