Finite Group Databasebeta

← #6495080

Subgroups — #6495080

Order 512.

3006 conjugacy classes of subgroups.

The subgroup lattice has 436 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2867C211yesmaximal, characteristic
2#36635C211yesmaximal, characteristic, complemented
3#33235C212yesmaximal, complemented
4#33231C212yesmaximal, complemented
5#30224C214yesmaximal, complemented
6#28756C214yesmaximal, complemented
7#28749C214yesmaximal, complemented
8#25860C214yesmaximal, complemented
9#25855C214yesmaximal, complemented
10#21810C214yesmaximal, complemented
11C4×#432C211yesmaximal, characteristic, complemented
12C22×#127—22no—
13C22×#127—22no—
14C22×#127C2212yes—
15C22×#127C2212yes—
16C22×#127C2212yes—
17C2×#490C2218yescomplemented
18C2×#486C2218yescomplemented
19C2×#432C2212yes—
20#2867—22no—
21#2867—22no—
22#2867C2212yes—
23#2867C2212yes—
24#2867C2212yes—
25#2867C2212yes—
26#2864C2212yescomplemented
27#2863—22no—
28#2862C2211yescharacteristic, complemented
29#2859C2211yescharacteristic, complemented
30#2857C2212yes—
31#2855—22no—
32#2851C2212yes—
33C4×C4⋊Q8C2212yescomplemented
34C4×#125C2212yescomplemented
35C4×C22⋊Q8—22no—
36#2295—22no—
37#2283C2212yes—
38C2×C43C2211yescharacteristic
39#2041C22116yescomplemented
40#2041C22116yescomplemented
41#1982C22116yescomplemented
42#1961C2218yescomplemented
43#1961C2218yescomplemented
44#1961C2218yescomplemented
45#1957C2218yescomplemented
46#1786C22116yes—
47#1767C2212yescomplemented
48C4×C4⋊C8C2212yescomplemented
49#1749C2212yes—
50C4×C22⋊C8C2212yes—
51#1745C2212yescomplemented
52C4×#97C2212yescomplemented
53C23×D4—24no—
54C23×D4—22no—
55C23×D4—22no—
56C23×D4—24no—
57C23×D4C2314yes—
58C2×C4⋊Q8C2312yes—
59C2×#127—24no—
60C2×#127—24no—
61C2×#127—22no—
62C2×#127—22no—
63C2×#127—24no—
64C2×#127—24no—
65C2×#127—24no—
66C2×#127—24no—
67C2×#127—24no—
68C2×#127—24no—
69C2×#127—24no—
70C2×#127—24no—
71C2×#127—216no—
72C2×#127—48no—
73C2×#127—48no—
74C2×#127—416no—
75C2×#127—416no—
76C2×#127C2314yes—
77C2×#127C2314yes—
78C2×#127C2314yes—
79C2×#127C2314yes—
80C2×#127C2314yes—
81C2×#127C2314yes—
82C2×#125C2312yes—
83C2×C22⋊Q8—22no—
84C2×C4×Q8—22no—
85C22×C42D412yes—
86C22×C42D412yes—
87C22×C42C2311yescharacteristic
88C22×C42C2311yescharacteristic
89C22×C42C2311yescharacteristic
90#490—416no—
91#486—416no—
92#432—28no—
93C2×C4⋊C8C2314yes—
94C2×#102C23132yes—
95C2×#102C23132yes—
96C2×C22⋊C8C2314yes—
97C2×#97C2314yes—
98C2×#97C2314yes—
99C2×C4×C8C2314yes—
100#391—22no—
101#391—22no—
102#391—22no—
103#391—22no—
104#389—22no—
105#389—22no—
106#389—22no—
107#389—22no—
108#389C2314yes—
109#389C2314yes—
110#387—24no—
111#387—24no—
112#387—24no—
113#387—24no—
114#385—24no—
115#384C2312yes—
116#383—22no—
117#382C2312yes—
118#382C2312yes—
119#382C2312yes—
120#381—22no—
121#380—22no—
122#380C2314yes—
123C4×C4⋊C4—24no—
124C4×C4⋊C4—22no—
125C4×C4⋊C4C2314yes—
126C4×C22⋊C4—22no—
127#376—22no—
128C43D412yes—
129C43D412yes—
130C43C2311yescharacteristic
131C43C2311yescharacteristic
132C43C2311yescharacteristic
133C43C2311yescharacteristic
134#336C2314yes—
135#336C2314yes—
136C25—24no—
137C22×Q8—22no—
138C22×D4—24no—
139C22×D4—24no—
140C22×D4—24no—
141C22×D4—24no—
142C22×D4—28no—
143C22×D4—28no—
144C22×D4—28no—
145C22×D4—24no—
146C22×D4—28no—
147C22×D4—216no—
148C22×D4—48no—
149C22×D4—48no—
150C22×D4—416no—
151C22×D4—416no—
152C22×D4—416no—
153C22×D4—416no—
154C22×D4—416no—
155C22×D4—416no—
156C22×D4—416no—
157C22×D4—48no—
158C22×D4—48no—
159C22×D4—416no—
160C22×D4C2418yes—
161C23×C4C2×D412yes—
162C23×C4C2×D411yescharacteristic
163C23×C4C2×D411yescharacteristic
164C23×C4C2×D412yes—
165C23×C4C2411yescharacteristic
166C22×C8C2418yes—
167C4⋊Q8—28no—
168#127—216no—
169#127—216no—
170#127—216no—
171#127—216no—
172#127—48no—
173#127—48no—
174#127—416no—
175#127—416no—
176#127—416no—
177#127—416no—
178#127—416no—
179#127—416no—
180#127—416no—
181#127—416no—
182#127—416no—
183#127—48no—
184#127—48no—
185#127—416no—
186#125—28no—
187C22⋊Q8—48no—
188C4×Q8—48no—
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⋊C4C2414yes—
196C2×C4⋊C4C2414yes—
197C2×C22⋊C4—24no—
198C2×C42—28no—
199C2×C42—28no—
200C2×C42—24no—
201C2×C42—24no—
202C2×C42—24no—
203C2×C42C2×D412yes—
204C2×C42C2×D412yes—
205C2×C42C2×D412yes—
206C2×C42C2×D412yes—
207C2×C42C2×D412yes—
208C2×C42C2×D412yes—
209C2×C42C2×D412yes—
210C2×C42C2×D412yes—
211C2×C42C2×D412yes—
212C2×C42C2×D411yescharacteristic
213C2×C42C2×D411yescharacteristic
214C2×C42C2×D412yes—
215C2×C42#4118yes—
216C2×C42C2411yescharacteristic
217C2×C42C2411yescharacteristic
218C2×C42C2411yescharacteristic
219C2×C42C2411yescharacteristic
220C2×C42C2411yescharacteristic
221C2×C42C2411yescharacteristic
222C4⋊C8—28no—
223#102—432no—
224#102—432no—
225C22⋊C8—28no—
226#97—28no—
227#97—28no—
228C4×C8—28no—
229#95—24no—
230#95—22no—
231#95—22no—
232#95—22no—
233#95—22no—
234#95—24no—
235#95—24no—
236#95—24no—
237#95—24no—
238C24—28no—
239C24—48no—
240C24—48no—
241C24—416no—
242C24—48no—
243C24—416no—
244C24C22×D411yescharacteristic
245C2×Q8—44no—
246C2×Q8—44no—
247C2×Q8—44no—
248C2×D4—216no—
249C2×D4—216no—
250C2×D4—216no—
251C2×D4—416no—
252C2×D4—416no—
253C2×D4—48no—
254C2×D4—48no—
255C2×D4—416no—
256C2×D4—416no—
257C2×D4—416no—
258C2×D4—416no—
259C2×D4—416no—
260C2×D4—416no—
261C2×D4—416no—
262C2×D4—48no—
263C2×D4—48no—
264C2×D4—48no—
265C2×D4—48no—
266C2×D4—48no—
267C2×D4—416no—
268C2×D4—416no—
269C2×D4—416no—
270C2×D4—416no—
271C2×D4—416no—
272C2×D4—416no—
273C2×D4—416no—
274C2×D4—832no—
275C2×D4—864no—
276C2×D4—832no—
277C2×D4—832no—
278C2×D4—864no—
279C22×C4—24no—
280C22×C4—24no—
281C22×C4—24no—
282C22×C4—24no—
283C22×C4—24no—
284C22×C4—24no—
285C22×C4—24no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—24no—
289C22×C4—24no—
290C22×C4#12414yes—
291C22×C4#12414yes—
292C22×C4AGL(1,ℤ/8)18yes—
293C22×C4C22×D411yescharacteristic
294C22×C4C22×D411yescharacteristic
295C22×C4C22×D411yescharacteristic
296C22×C4C22×D411yescharacteristic
297C22×C4C22×D412yes—
298C22×C4C22×D412yes—
299C22×C4C22×D412yes—
300C22×C4C22×D411yescharacteristic
301C22×C4C22×D412yes—
302C22×C4C2×#4114yes—
303C22×C4C2511yescharacteristic
304C2×C8—28no—
305C2×C8—28no—
306C2×C8—28no—
307C4⋊C4—28no—
308C4⋊C4—28no—
309C4⋊C4—48no—
310C4⋊C4—44no—
311C4⋊C4—44no—
312C4⋊C4—48no—
313C4⋊C4—44no—
314C4⋊C4—48no—
315C22⋊C4—48no—
316C42—24no—
317C42—24no—
318C42—24no—
319C42—24no—
320C42—24no—
321C42—24no—
322C42—24no—
323C42—24no—
324C42—24no—
325C42—22no—
326C42—22no—
327C42—48no—
328C42#12414yes—
329C42#12414yes—
330C42AGL(1,ℤ/8)18yes—
331C42C2×#4114yes—
332C42C2×#4114yes—
333C42H(5,𝔽2)14yes—
334C42H(5,𝔽2)14yes—
335C23—21no—
336C23—21no—
337C23—22no—
338C23—48no—
339C23—48no—
340C23—48no—
341C23—416no—
342C23—416no—
343C23—832no—
344C23—832no—
345C23—816no—
346C23—816no—
347C23—816no—
348C23C2×#12412yes—
349C23C2×AGL(1,ℤ/8)14yes—
350C23C23×D411yescharacteristic
351Q8—88no—
352D4—432no—
353D4—832no—
354D4—832no—
355D4—832no—
356D4—816no—
357D4—816no—
358D4—816no—
359D4—816no—
360D4—816no—
361D4—832no—
362C2×C4—216no—
363C2×C4—28no—
364C2×C4—22no—
365C2×C4—22no—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—22no—
369C2×C4—22no—
370C2×C4—22no—
371C2×C4—22no—
372C2×C4—22no—
373C2×C4—22no—
374C2×C4—22no—
375C2×C4—28no—
376C2×C4—216no—
377C2×C4—28no—
378C2×C4—44no—
379C2×C4—44no—
380C2×C4—44no—
381C2×C4—44no—
382C2×C4—44no—
383C2×C4—44no—
384C2×C4C2×#12412yes—
385C2×C4C2×#12412yes—
386C2×C4C2×#12412yes—
387C2×C4C2×#12412yes—
388C2×C4C2×#12412yes—
389C2×C4#53514yes—
390C2×C4#53514yes—
391C2×C4C2×AGL(1,ℤ/8)14yes—
392C2×C4C2×AGL(1,ℤ/8)14yes—
393C2×C4C2×AGL(1,ℤ/8)14yes—
394C2×C4C2×AGL(1,ℤ/8)14yes—
395C2×C4C2×AGL(1,ℤ/8)14yes—
396C2×C4C22×#4112yes—
397C2×C4C2×H(5,𝔽2)12yes—
398C2×C4C2×H(5,𝔽2)12yes—
399C8—48no—
400C22—24no—
401C22—24no—
402C22—22no—
403C22—22no—
404C22—816no—
405C22—816no—
406C22—88no—
407C22—88no—
408C22—88no—
409C22—1632no—
410C22#313814yes—
411C22C22×#12411yescharacteristic
412C22C2×#53511yescharacteristic
413C22C2×#53511yescharacteristic
414C22C22×AGL(1,ℤ/8)12yes—
415C22C22×AGL(1,ℤ/8)12yes—
416C4—24no—
417C4—24no—
418C4—24no—
419C4—22no—
420C4—22no—
421C4—22no—
422C4—22no—
423C4—84no—
424C4—84no—
425C4#313814yes—
426C4#321718yes—
427C4#350814yes—
428C2—24no—
429C2—168no—
430C2#3528511yescharacteristic
431C2C2×#313811yescharacteristic
432C2C2×#313811yescharacteristic
433C2C2×#321712yes—
434C2#6079812yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#2867C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C43 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #36635C2semidirect
3. #33235C2semidirect
4. #33231C2semidirect
5. #30224C2semidirect
6. #28756C2semidirect
7. #28749C2semidirect
8. #25860C2semidirect
9. #25855C2semidirect
10. #21810C2semidirect
11. C4×#432C2semidirect
17. C2×#490C22semidirect
18. C2×#486C22semidirect
26. #2864C22semidirect
28. #2862C22semidirect
29. #2859C22semidirect
33. C4×C4⋊Q8C22semidirect
34. C4×#125C22semidirect
39. #2041C22semidirect
40. #2041C22semidirect
41. #1982C22semidirect
42. #1961C22semidirect
43. #1961C22semidirect
44. #1961C22semidirect
45. #1957C22semidirect
47. #1767C22semidirect
48. C4×C4⋊C8C22semidirect
51. #1745C22semidirect
52. C4×#97C22semidirect

Maximal subgroups

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

subgroupindex
C2×#28672
#366352
#332352
#332312
#302242
#287562
#287492
#258602
#258552
#218102
C4×#4322