Finite Group Databasebeta

← #235783

Subgroups — #235783

Order 512.

1578 conjugacy classes of subgroups.

The subgroup lattice has 457 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3208C211yesmaximal, characteristic
2C22×#486C211yesmaximal, characteristic
3#35320C211yesmaximal, characteristic, complemented
4#21260C211yesmaximal, characteristic
5C2×#1546C211yesmaximal, characteristic, complemented
6C2×#1493C211yesmaximal, characteristic, complemented
7#10053C218yesmaximal, complemented
8#7890C211yesmaximal, characteristic, complemented
9C2×#533C2211yescharacteristic
10C22×C4⋊Q8C2211yescharacteristic
11C22×#127C2211yescharacteristic
12#3208—24no—
13#3208—24no—
14C2×#492—21no—
15C2×#486—24no—
16C2×#486—21no—
17C2×#486—22no—
18C2×#486—24no—
19C2×#486C2214yes—
20C2×#460C2214yes—
21C2×#450—22no—
22C2×#431C2211yescharacteristic
23C2×#423C2211yescharacteristic
24C2×#418—22no—
25C22×#102—22no—
26C22×C4×C8C2211yescharacteristic
27#2862—21no—
28#2862—21no—
29#2642—22no—
30#2568—22no—
31#2568C2214yescomplemented
32#2503—22no—
33#2503—22no—
34#2310—22no—
35C4×#117C2211yescharacteristic
36#1982—22no—
37#1982C2214yes—
38#1546—24no—
39C2×#332C2214yescomplemented
40#1493—24no—
41C2×Q8⋊C8C2214yescomplemented
42#1279C2214yescomplemented
43C23×Q8—21no—
44C23×D4—21no—
45C23×D4—21no—
46C23×D4—21no—
47C22×SD16—24no—
48C22×M16—21no—
49C23×C8—21no—
50#533—22no—
51#533—22no—
52#533—22no—
53#533—22no—
54C2×C4⋊Q8—21no—
55C2×C4⋊Q8—21no—
56C2×C4⋊Q8—22no—
57C2×C4⋊Q8—22no—
58C2×C4⋊Q8—22no—
59C2×C4⋊Q8—22no—
60C2×C4⋊Q8D411yescharacteristic
61C2×C4⋊Q8D411yescharacteristic
62C2×C4⋊Q8D412yes—
63C2×C4⋊Q8C2312yes—
64C2×#127—21no—
65C2×#127—22no—
66C2×#127—21no—
67C2×#127—22no—
68C2×#127—21no—
69C2×#127—22no—
70C2×#127D414yes—
71C2×#127C2312yes—
72C2×#125—21no—
73C2×C22⋊Q8—21no—
74C2×C22⋊Q8D412yes—
75C2×C4×Q8—21no—
76C2×C4×Q8C2312yes—
77C2×#117—21no—
78C2×#117D412yes—
79C22×C4⋊C4—21no—
80C22×C4⋊C4D412yes—
81C22×C42C2311yescharacteristic
82#492—28no—
83#486—216no—
84#486—28no—
85#486—28no—
86#486—28no—
87#460—28no—
88#460—28no—
89#450—216no—
90#431—22no—
91#431D414yes—
92#423—22no—
93#423—22no—
94C2×C4⋊C8—21no—
95C2×C4⋊C8C2312yes—
96C2×C4⋊C8C2312yes—
97#418—48no—
98C2×#102—216no—
99C2×#102—28no—
100C2×#102—24no—
101C2×#102—28no—
102C2×C22⋊C8—21no—
103C2×C4×C8—21no—
104C2×C4×C8—22no—
105C2×C4×C8—21no—
106C2×C4×C8D414yes—
107C2×C4×C8C2312yes—
108#400—28no—
109#400—28no—
110#391—24no—
111#391—24no—
112#391—24no—
113#391—24no—
114#391—28no—
115#382—24no—
116#382—24no—
117#381—28no—
118C4×C4⋊C4D412yes—
119C4×C4⋊C4C2312yes—
120C4×C22⋊C4—21no—
121C4×C22⋊C4—21no—
122#376—21no—
123C43—21no—
124#332—28no—
125Q8⋊C8—28no—
126C25—41no—
127C22×Q8—21no—
128C22×Q8—22no—
129C22×Q8—22no—
130C22×Q8—42no—
131C22×Q8—42no—
132C22×Q8—44no—
133C22×Q8—44no—
134C22×Q8C2×D412yes—
135C22×D4—24no—
136C22×D4—24no—
137C22×D4—24no—
138C22×D4—22no—
139C22×D4—22no—
140C22×D4—22no—
141C22×D4—48no—
142C22×D4—41no—
143C22×D4—41no—
144C22×D4—41no—
145C22×D4—42no—
146C22×D4—42no—
147C22×D4—42no—
148C22×D4—44no—
149C22×D4—44no—
150C22×D4—84no—
151C23×C4—22no—
152C23×C4C2×D411yescharacteristic
153C23×C4C2×D411yescharacteristic
154C23×C4C2×D411yescharacteristic
155C2×SD16—216no—
156C2×SD16—88no—
157C2×M16—42no—
158C2×M16—44no—
159C22×C8—24no—
160C22×C8—22no—
161C22×C8—22no—
162C22×C8—22no—
163C22×C8—44no—
164C4⋊Q8—24no—
165C4⋊Q8—22no—
166C4⋊Q8—22no—
167C4⋊Q8—24no—
168C4⋊Q8—22no—
169C4⋊Q8—22no—
170C4⋊Q8—22no—
171C4⋊Q8—22no—
172C4⋊Q8—22no—
173C4⋊Q8—22no—
174C4⋊Q8—42no—
175C4⋊Q8—42no—
176C4⋊Q8—42no—
177C4⋊Q8—42no—
178#127—24no—
179#127—24no—
180#127—24no—
181#127—22no—
182#127—22no—
183#127—22no—
184#127—42no—
185#127—42no—
186#127—42no—
187#127C2×D414yes—
188#125—42no—
189#125—42no—
190C22⋊Q8—24no—
191C22⋊Q8—24no—
192C22⋊Q8—42no—
193C22⋊Q8—42no—
194C4×Q8—22no—
195C4×Q8—22no—
196C4×Q8—22no—
197C4×Q8—22no—
198C4×Q8—42no—
199C4×Q8—42no—
200#117—24no—
201#117—24no—
202#117—42no—
203#117—42no—
204C2×C4⋊C4—24no—
205C2×C4⋊C4—24no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—42no—
215C2×C4⋊C4—42no—
216C2×C4⋊C4C2×D414yes—
217C2×C4⋊C4C2×D414yes—
218C2×C4⋊C4C2×D412yes—
219C2×C22⋊C4—22no—
220C2×C22⋊C4—22no—
221C2×C42—22no—
222C2×C42—22no—
223C2×C42—22no—
224C2×C42—22no—
225C2×C42C2×D411yescharacteristic
226C2×C42C2×D411yescharacteristic
227C2×C42C2×D412yes—
228C2×C42C2×D412yes—
229C2×C42C2×D412yes—
230C2×C42C2×D411yescharacteristic
231C2×C42#4112yes—
232C2×C42C2411yescharacteristic
233C4⋊C8—22no—
234C4⋊C8—22no—
235C4⋊C8—22no—
236C4⋊C8—42no—
237C4⋊C8C2×D414yes—
238#102—232no—
239#102—216no—
240#102—416no—
241C22⋊C8—24no—
242C4×C8—24no—
243C4×C8—24no—
244C4×C8—22no—
245C4×C8C2×D414yes—
246#95—216no—
247#95—216no—
248#95—22no—
249#95—22no—
250C24—42no—
251C24—44no—
252C24—84no—
253C24—84no—
254C24—84no—
255C24#12011yescharacteristic
256C2×Q8—22no—
257C2×Q8—22no—
258C2×Q8—28no—
259C2×Q8—42no—
260C2×Q8—42no—
261C2×Q8—44no—
262C2×Q8—44no—
263C2×Q8—42no—
264C2×Q8—44no—
265C2×Q8—48no—
266C2×Q8—44no—
267C2×Q8—88no—
268C2×Q8—84no—
269C2×Q8—84no—
270C2×D4—24no—
271C2×D4—24no—
272C2×D4—24no—
273C2×D4—48no—
274C2×D4—416no—
275C2×D4—42no—
276C2×D4—42no—
277C2×D4—42no—
278C2×D4—44no—
279C2×D4—44no—
280C2×D4—48no—
281C2×D4—44no—
282C2×D4—44no—
283C2×D4—44no—
284C2×D4—416no—
285C2×D4—88no—
286C2×D4—88no—
287C2×D4—84no—
288C2×D4—88no—
289C2×D4—84no—
290C2×D4—84no—
291C2×D4—84no—
292C2×D4—84no—
293C22×C4—24no—
294C22×C4—24no—
295C22×C4—24no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—24no—
299C22×C4—44no—
300C22×C4—48no—
301C22×C4#12012yes—
302C22×C4#12012yes—
303C22×C4#12011yescharacteristic
304C22×C4#12011yescharacteristic
305C22×C4#12011yescharacteristic
306C22×C4#12112yes—
307C22×C4#12112yes—
308C22×C4#12112yes—
309C22×C4#12712yes—
310C22×C4#13514yes—
311C22×C4AGL(1,ℤ/8)14yes—
312C22×C4C22×D411yescharacteristic
313C22×C4C22×D411yescharacteristic
314C22×C4C22×D411yescharacteristic
315SD16—816no—
316M16—84no—
317C2×C8—24no—
318C2×C8—24no—
319C2×C8—44no—
320C2×C8—48no—
321C2×C8—44no—
322C2×C8—42no—
323C2×C8—44no—
324C4⋊C4—24no—
325C4⋊C4—22no—
326C4⋊C4—24no—
327C4⋊C4—22no—
328C4⋊C4—24no—
329C4⋊C4—24no—
330C4⋊C4—42no—
331C4⋊C4—42no—
332C4⋊C4—42no—
333C4⋊C4—42no—
334C4⋊C4—44no—
335C4⋊C4—44no—
336C4⋊C4—42no—
337C4⋊C4—42no—
338C4⋊C4—44no—
339C4⋊C4—42no—
340C22⋊C4—44no—
341C22⋊C4—44no—
342C42—42no—
343C42—42no—
344C42—44no—
345C42—44no—
346C42—44no—
347C42#12012yes—
348C42#12012yes—
349C42#12112yes—
350C42#12112yes—
351C42#12112yes—
352C42#12712yes—
353C42C22×D411yescharacteristic
354C42C22×D411yescharacteristic
355C42C22×D411yescharacteristic
356C42C2×#4111yescharacteristic
357C23—22no—
358C23—22no—
359C23—44no—
360C23—88no—
361C23—88no—
362C23—84no—
363C23—84no—
364C23—84no—
365C23—88no—
366C23—1616no—
367C23#39212yes—
368C23#44914yes—
369C23C2×#12011yescharacteristic
370Q8—44no—
371Q8—44no—
372Q8—82no—
373Q8—82no—
374Q8—84no—
375Q8—84no—
376Q8—88no—
377D4—48no—
378D4—48no—
379D4—48no—
380D4—84no—
381D4—88no—
382D4—82no—
383D4—82no—
384D4—88no—
385D4—88no—
386D4—84no—
387C2×C4—22no—
388C2×C4—22no—
389C2×C4—24no—
390C2×C4—24no—
391C2×C4—24no—
392C2×C4—24no—
393C2×C4—42no—
394C2×C4—42no—
395C2×C4—44no—
396C2×C4—44no—
397C2×C4—48no—
398C2×C4—44no—
399C2×C4—44no—
400C2×C4—44no—
401C2×C4—44no—
402C2×C4—88no—
403C2×C4#39212yes—
404C2×C4#39212yes—
405C2×C4#39212yes—
406C2×C4#44914yes—
407C2×C4#45414yes—
408C2×C4#46318yes—
409C2×C4#49514yes—
410C2×C4#49614yes—
411C2×C4C2×#12011yescharacteristic
412C2×C4C2×#12011yescharacteristic
413C2×C4C2×#12111yescharacteristic
414C2×C4C2×#12111yescharacteristic
415C2×C4C2×#12111yescharacteristic
416C2×C4C2×#12711yescharacteristic
417C2×C4C2×#13512yes—
418C2×C4C2×AGL(1,ℤ/8)12yes—
419C2×C4#57714yes—
420C8—44no—
421C8—44no—
422C8—82no—
423C22—22no—
424C22—22no—
425C22—28no—
426C22—84no—
427C22—84no—
428C22—84no—
429C22—1616no—
430C22—168no—
431C22#201214yes—
432C22C2×#39211yescharacteristic
433C22C2×#44912yes—
434C22#301012yes—
435C22C2×#45412yes—
436C4—22no—
437C4—21no—
438C4—21no—
439C4—82no—
440C4—82no—
441C4—84no—
442C4—84no—
443C4#200314yes—
444C4#201214yes—
445C4#300912yes—
446C4C2×#46312yes—
447C4#305712yes—
448C4C2×#49511yescharacteristic
449C4C2×#49611yescharacteristic
450C2—24no—
451C2—164no—
452C2#1864414yes—
453C2#2338811yescharacteristic
454C2C2×#200311yescharacteristic
455C2C2×#201211yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC2×C42C24
prime radicalC22×#127C22
LCSG ▷ C2×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
3. #35320C2semidirect
5. C2×#1546C2semidirect
6. C2×#1493C2semidirect
7. #10053C2semidirect
8. #7890C2semidirect
31. #2568C22semidirect
39. C2×#332C22semidirect
41. C2×Q8⋊C8C22semidirect
42. #1279C22semidirect

Maximal subgroups

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

subgroupindex
C2×#32082
C22×#4862
#353202
#212602
C2×#15462
C2×#14932
#100532
#78902