Finite Group Databasebeta

← #206970

Subgroups — #206970

Order 512.

1536 conjugacy classes of subgroups.

The subgroup lattice has 548 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2857C211yesmaximal, characteristic
2C4×#407C211yesmaximal, characteristic
3C4×#380C211yesmaximal, characteristic
4#9888C214yesmaximal, complemented
5#9885C214yesmaximal, complemented
6#8019C214yesmaximal
7C22×C4⋊Q8C411yescharacteristic, complemented
8C22×C22⋊Q8—21no—
9C23×C42C2211yescharacteristic
10C2×#407C2212yes—
11C2×C4×M16—21no—
12#2857—21no—
13#2857—21no—
14#2857—21no—
15#2857—21no—
16#2857C412yescomplemented
17#2857C412yescomplemented
18#2857C2212yes—
19#2857C2212yes—
20C2×#391—21no—
21C2×#389—21no—
22C2×#389C412yescomplemented
23C2×#384C411yescharacteristic, complemented
24C2×#380—21no—
25C2×#380—21no—
26C2×#380C2212yes—
27C2×C4×C4⋊C4—21no—
28C2×C4×C22⋊C4C2212yes—
29#1749—21no—
30#1749C2212yes—
31C4×C22⋊C8—21no—
32C4×C22⋊C8C2212yes—
33#1728—21no—
34C4⋊#95C2212yes—
35#1434C2212yes—
36C22⋊#95—21no—
37C4×#95—21no—
38#1294C22116yes—
39C23×Q8—21no—
40C24×C4D411yescharacteristic
41C24×C4D411yescharacteristic
42C24×C4C2311yescharacteristic
43C22×M16—22no—
44C2×C4⋊Q8—21no—
45C2×C4⋊Q8—21no—
46C2×C4⋊Q8—44no—
47C2×C4⋊Q8C2×C412yes—
48C2×C4⋊Q8C2×C412yes—
49C2×C22⋊Q8—22no—
50C2×C22⋊Q8—22no—
51C2×C22⋊Q8—22no—
52C2×C22⋊Q8—22no—
53C2×C22⋊Q8—48no—
54C22×C4⋊C4—21no—
55C22×C4⋊C4—22no—
56C22×C4⋊C4—21no—
57C22×C4⋊C4—21no—
58C22×C4⋊C4—22no—
59C22×C4⋊C4C2×C412yes—
60C22×C4⋊C4C2×C412yes—
61C22×C22⋊C4—22no—
62C22×C22⋊C4—22no—
63C22×C42—22no—
64C22×C42—44no—
65C22×C42C2×C411yescharacteristic
66C22×C42C2×C411yescharacteristic
67C22×C42C2×C411yescharacteristic
68C22×C42C2×C411yescharacteristic
69C22×C42D411yescharacteristic
70C22×C42D411yescharacteristic
71C22×C42D411yescharacteristic
72C22×C42D411yescharacteristic
73C22×C42D411yescharacteristic
74C22×C42D411yescharacteristic
75C22×C42C2311yescharacteristic
76C22×C42C2311yescharacteristic
77#407—24no—
78#407Q818yescomplemented
79C2×C22⋊C8—22no—
80C2×C22⋊C8C2314yes—
81C4×M16—44no—
82C2×#97—22no—
83C2×C4×C8—22no—
84#391—22no—
85#391—22no—
86#391—22no—
87#391—22no—
88#389—22no—
89#389—22no—
90#389—22no—
91#389—22no—
92#389—22no—
93#389—22no—
94#389C2×C414yes—
95#389C2×C414yes—
96#384—21no—
97#384—21no—
98#384C2×C412yes—
99#384C2×C412yes—
100#380—28no—
101#380—22no—
102#380—22no—
103#380—22no—
104#380—22no—
105#380—22no—
106#380—22no—
107#380C2×C414yes—
108#380C2314yes—
109C4×C4⋊C4—44no—
110C4×C22⋊C4—28no—
111C2×#95—22no—
112C2×#95—22no—
113C2×#95—22no—
114C2×#95—22no—
115C2×#95—22no—
116C2×#95C2314yes—
117#336—22no—
118#336—22no—
119C25C2×D411yescharacteristic
120C22×Q8—22no—
121C22×Q8—22no—
122C22×Q8—44no—
123C22×Q8—44no—
124C22×Q8—44no—
125C23×C4—24no—
126C23×C4—24no—
127C23×C4—24no—
128C23×C4—22no—
129C23×C4—22no—
130C23×C4—24no—
131C23×C4—42no—
132C23×C4—42no—
133C23×C4—42no—
134C23×C4—42no—
135C23×C4—42no—
136C23×C4C22⋊C411yescharacteristic
137C23×C4C22⋊C411yescharacteristic
138C23×C4C22⋊C411yescharacteristic
139C23×C4C22⋊C411yescharacteristic
140C23×C4C22⋊C411yescharacteristic
141C23×C4C22⋊C411yescharacteristic
142C23×C4C22⋊C411yescharacteristic
143C23×C4C22⋊C411yescharacteristic
144C23×C4C22×C411yescharacteristic
145C23×C4C22×C411yescharacteristic
146C23×C4C2×D411yescharacteristic
147C23×C4C2×D411yescharacteristic
148C23×C4C2×D411yescharacteristic
149C23×C4C2×D411yescharacteristic
150C23×C4C2×D411yescharacteristic
151C23×C4C2×D411yescharacteristic
152C23×C4C2×D411yescharacteristic
153C23×C4C2×Q814yes—
154C23×C4#4114yes—
155C23×C4C2411yescharacteristic
156C2×M16—28no—
157C2×M16—44no—
158C2×M16—44no—
159C22×C8—24no—
160C4⋊Q8—44no—
161C4⋊Q8—44no—
162C4⋊Q8—44no—
163C4⋊Q8—44no—
164C22⋊Q8—48no—
165C22⋊Q8—48no—
166C22⋊Q8—48no—
167C22⋊Q8—48no—
168C2×C4⋊C4—28no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—24no—
172C2×C4⋊C4—24no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—44no—
180C2×C4⋊C4—44no—
181C2×C4⋊C4—48no—
182C2×C4⋊C4—44no—
183C2×C4⋊C4—44no—
184C2×C4⋊C4—44no—
185C2×C4⋊C4—44no—
186C2×C4⋊C4—44no—
187C2×C4⋊C4C22×C414yes—
188C2×C4⋊C4C22×C414yes—
189C2×C22⋊C4—28no—
190C2×C22⋊C4—24no—
191C2×C22⋊C4—24no—
192C2×C22⋊C4—44no—
193C2×C22⋊C4—48no—
194C2×C22⋊C4—44no—
195C2×C42—24no—
196C2×C42—24no—
197C2×C42—24no—
198C2×C42—42no—
199C2×C42—42no—
200C2×C42—42no—
201C2×C42—42no—
202C2×C42—42no—
203C2×C42—42no—
204C2×C42—42no—
205C2×C42—42no—
206C2×C42—42no—
207C2×C42—44no—
208C2×C42—44no—
209C2×C42—44no—
210C2×C42C22⋊C411yescharacteristic
211C2×C42C22⋊C411yescharacteristic
212C2×C42C22⋊C411yescharacteristic
213C2×C42C22⋊C411yescharacteristic
214C2×C42C22⋊C411yescharacteristic
215C2×C42C22⋊C411yescharacteristic
216C2×C42C22⋊C411yescharacteristic
217C2×C42C22⋊C411yescharacteristic
218C2×C42C22×C411yescharacteristic
219C2×C42C22×C411yescharacteristic
220C2×C42C22×C411yescharacteristic
221C2×C42C22×C411yescharacteristic
222C2×C42C2×D411yescharacteristic
223C2×C42C2×D411yescharacteristic
224C2×C42C2×D411yescharacteristic
225C2×C42C2×D411yescharacteristic
226C22⋊C8—28no—
227C22⋊C8—24no—
228C22⋊C8C2×Q818yescomplemented
229#97—44no—
230C4×C8—44no—
231#95—28no—
232#95—24no—
233#95—24no—
234#95—24no—
235#95—24no—
236#95—28no—
237#95—44no—
238#95—44no—
239C24—22no—
240C24—22no—
241C24—22no—
242C24—22no—
243C24—41no—
244C24—41no—
245C24C2×C22⋊C411yescharacteristic
246C24C2×C22⋊C411yescharacteristic
247C24#12412yes—
248C24C4⋊Q812yes—
249C24C22×D411yescharacteristic
250C2×Q8—44no—
251C2×Q8—44no—
252C2×Q8—44no—
253C2×Q8—44no—
254C2×Q8—44no—
255C2×Q8—44no—
256C2×Q8—816no—
257C22×C4—22no—
258C22×C4—22no—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—22no—
262C22×C4—22no—
263C22×C4—22no—
264C22×C4—22no—
265C22×C4—22no—
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—22no—
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—22no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—28no—
287C22×C4—24no—
288C22×C4—24no—
289C22×C4—48no—
290C22×C4—48no—
291C22×C4—41no—
292C22×C4—41no—
293C22×C4—41no—
294C22×C4—41no—
295C22×C4—41no—
296C22×C4—41no—
297C22×C4—41no—
298C22×C4—41no—
299C22×C4—41no—
300C22×C4—41no—
301C22×C4—41no—
302C22×C4—41no—
303C22×C4—41no—
304C22×C4—41no—
305C22×C4—41no—
306C22×C4—41no—
307C22×C4—416no—
308C22×C4—416no—
309C22×C4—44no—
310C22×C4—44no—
311C22×C4—44no—
312C22×C4—44no—
313C22×C4—44no—
314C22×C4—44no—
315C22×C4—44no—
316C22×C4—44no—
317C22×C4—44no—
318C22×C4—44no—
319C22×C4—44no—
320C22×C4—44no—
321C22×C4#10414yes—
322C22×C4C2×C22⋊C411yescharacteristic
323C22×C4C2×C22⋊C411yescharacteristic
324C22×C4C2×C22⋊C411yescharacteristic
325C22×C4C2×C22⋊C411yescharacteristic
326C22×C4C2×C22⋊C411yescharacteristic
327C22×C4C2×C22⋊C411yescharacteristic
328C22×C4C2×C22⋊C411yescharacteristic
329C22×C4C2×C22⋊C411yescharacteristic
330C22×C4C2×C22⋊C411yescharacteristic
331C22×C4C2×C22⋊C411yescharacteristic
332C22×C4C4×Q814yes—
333C22×C4C4×Q814yes—
334C22×C4C22⋊Q814yes—
335C22×C4C22⋊Q814yes—
336C22×C4#12412yes—
337C22×C4C4⋊Q812yes—
338C22×C4AGL(1,ℤ/8)14yes—
339C22×C4#13714yes—
340C22×C4C23×C411yescharacteristic
341C22×C4C22×D411yescharacteristic
342C22×C4C22×Q812yes—
343C22×C4C2×#4112yes—
344M16—416no—
345C2×C8—28no—
346C2×C8—44no—
347C2×C8—44no—
348C4⋊C4—44no—
349C4⋊C4—44no—
350C4⋊C4—44no—
351C4⋊C4—44no—
352C4⋊C4—416no—
353C4⋊C4—44no—
354C4⋊C4—48no—
355C4⋊C4—44no—
356C4⋊C4—44no—
357C4⋊C4—44no—
358C4⋊C4—44no—
359C4⋊C4—44no—
360C4⋊C4—48no—
361C22⋊C4—416no—
362C22⋊C4—48no—
363C22⋊C4—48no—
364C42—22no—
365C42—22no—
366C42—22no—
367C42—22no—
368C42—22no—
369C42—22no—
370C42—22no—
371C42—22no—
372C42—22no—
373C42—41no—
374C42—41no—
375C42—41no—
376C42—41no—
377C42—41no—
378C42—41no—
379C42—88no—
380C42#9914yes—
381C23—21no—
382C23—21no—
383C23—21no—
384C23—21no—
385C23—21no—
386C23—21no—
387C23—21no—
388C23—21no—
389C23—21no—
390C23—21no—
391C23—24no—
392C23—21no—
393C23—21no—
394C23—21no—
395C23—21no—
396C23—24no—
397C23—24no—
398C23—24no—
399C23—42no—
400C23—42no—
401C23—44no—
402C23—44no—
403C23—44no—
404C23—44no—
405C23#39112yes—
406C23#39112yes—
407C23C2×#10412yes—
408C23#42112yes—
409C23C22×C22⋊C411yescharacteristic
410C23C2×#12411yescharacteristic
411C23C2×C4⋊Q811yescharacteristic
412C23C2×AGL(1,ℤ/8)12yes—
413C23C2×#13712yes—
414Q8—88no—
415Q8—88no—
416C2×C4—28no—
417C2×C4—21no—
418C2×C4—21no—
419C2×C4—21no—
420C2×C4—21no—
421C2×C4—21no—
422C2×C4—21no—
423C2×C4—21no—
424C2×C4—21no—
425C2×C4—21no—
426C2×C4—21no—
427C2×C4—21no—
428C2×C4—21no—
429C2×C4—21no—
430C2×C4—21no—
431C2×C4—21no—
432C2×C4—21no—
433C2×C4—24no—
434C2×C4—28no—
435C2×C4—28no—
436C2×C4—24no—
437C2×C4—28no—
438C2×C4—28no—
439C2×C4—21no—
440C2×C4—21no—
441C2×C4—21no—
442C2×C4—21no—
443C2×C4—21no—
444C2×C4—21no—
445C2×C4—44no—
446C2×C4—44no—
447C2×C4—44no—
448C2×C4—42no—
449C2×C4—42no—
450C2×C4—44no—
451C2×C4—44no—
452C2×C4—44no—
453C2×C4—44no—
454C2×C4—44no—
455C2×C4—44no—
456C2×C4—44no—
457C2×C4—44no—
458C2×C4—48no—
459C2×C4—48no—
460C2×C4—48no—
461C2×C4—44no—
462C2×C4—44no—
463C2×C4—44no—
464C2×C4—44no—
465C2×C4—44no—
466C2×C4—44no—
467C2×C4—84no—
468C2×C4—84no—
469C2×C4—84no—
470C2×C4—88no—
471C2×C4—88no—
472C2×C4—84no—
473C2×C4#39112yes—
474C2×C4#39112yes—
475C2×C4C2×#9912yes—
476C2×C4C2×#9912yes—
477C2×C4C2×#9912yes—
478C2×C4#41012yes—
479C2×C4#41812yes—
480C2×C4#41812yes—
481C2×C4#41912yes—
482C2×C4#41912yes—
483C2×C4C2×#10412yes—
484C2×C4C2×#10412yes—
485C2×C4C2×C4×Q812yes—
486C2×C4C2×C22⋊Q812yes—
487C8—48no—
488C22—21no—
489C22—21no—
490C22—22no—
491C22—21no—
492C22—21no—
493C22—22no—
494C22—22no—
495C22—22no—
496C22—22no—
497C22—22no—
498C22—22no—
499C22—22no—
500C22—22no—
501C22—22no—
502C22—22no—
503C22—22no—
504C22—22no—
505C22—48no—
506C22—48no—
507C22—41no—
508C22—41no—
509C22—41no—
510C22—41no—
511C22#198614yes—
512C22C2×#39111yescharacteristic
513C22C22×#9911yescharacteristic
514C22C2×#41011yescharacteristic
515C22C2×#41811yescharacteristic
516C22C2×#41911yescharacteristic
517C22C22×#10411yescharacteristic
518C22C2×#42111yescharacteristic
519C22#314112yes—
520C22#316512yes—
521C4—22no—
522C4—22no—
523C4—22no—
524C4—22no—
525C4—22no—
526C4—22no—
527C4—22no—
528C4—22no—
529C4—41no—
530C4—41no—
531C4—88no—
532C4—84no—
533C4—84no—
534C4#150714yes—
535C4#199614yes—
536C2—22no—
537C2—22no—
538C2—24no—
539C2—42no—
540C2C2×#150711yescharacteristic
541C2#1754511yescharacteristic
542C2#1757811yescharacteristic
543C2#1758311yescharacteristic
544C2C2×#198611yescharacteristic
545C2#2132411yescharacteristic
546C2C2×#199611yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
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
4. #9888C2semidirect
5. #9885C2semidirect
7. C22×C4⋊Q8C4semidirect
16. #2857C4semidirect
17. #2857C4semidirect
22. C2×#389C4semidirect
23. C2×#384C4semidirect
78. #407Q8semidirect
228. C22⋊C8C2×Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#28572
C4×#4072
C4×#3802
#98882
#98852
#80192