Finite Group Databasebeta

← #6465547

Subgroups — #6465547

Order 512.

4374 conjugacy classes of subgroups.

The subgroup lattice has 426 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3151C212yesmaximal, complemented
2C2×#3038C214yesmaximal, complemented
3C2×#2867C211yesmaximal, characteristic, complemented
4C4×#532C211yesmaximal, characteristic, complemented
5#28448C218yesmaximal, complemented
6#28441C214yesmaximal, complemented
7#28437C214yesmaximal, complemented
8#23430C214yesmaximal, complemented
9#21203C212yesmaximal, complemented
10C4×#423C211yesmaximal, characteristic, complemented
11C22×AGL(1,ℤ/8)C2218yes—
12C2×#532C2211yescharacteristic
13C22×#127—24no—
14C22×#127C2212yes—
15C22×#127C2212yes—
16C22×#127C2212yes—
17C2×C4×#41C2212yes—
18#3151—216no—
19C2×#496C2218yescomplemented
20C2×#493C2214yescomplemented
21C2×#492C2214yescomplemented
22#3038—232no—
23C2×#460C22116yescomplemented
24C2×#459C22116yescomplemented
25C2×#423C2212yes—
26C2×#418C2218yes—
27C2×C4×M16C2212yes—
28#2867—22no—
29#2867—22no—
30#2867C2214yes—
31#2867C2214yes—
32C4×C4⋊Q8C2212yescomplemented
33C4×#127C2212yescomplemented
34C4×#124C2214yescomplemented
35C4×#121—24no—
36#2297C2212yes—
37#2293—22no—
38#2285C2211yescharacteristic
39C42×D4—22no—
40C2×C43C2211yescharacteristic
41#2014C22116yescomplemented
42#2011C22116yescomplemented
43#1961C2218yescomplemented
44#1957C2218yescomplemented
45#1767C2214yescomplemented
46C4×C4⋊C8C2214yescomplemented
47#1728C2212yes—
48C22×#41C2312yes—
49C23×D4—24no—
50C23×D4—24no—
51C23×D4—24no—
52C23×D4C2314yes—
53C2×AGL(1,ℤ/8)—232no—
54C2×AGL(1,ℤ/8)—232no—
55C2×AGL(1,ℤ/8)—416no—
56C22×SD16C23116yes—
57C22×D8C23116yes—
58C22×M16C2314yes—
59#532—24no—
60#532D418yescomplemented
61C2×C4⋊Q8C2311yescharacteristic
62C2×#127—28no—
63C2×#127—216no—
64C2×#127—24no—
65C2×#127—24no—
66C2×#127—22no—
67C2×#127—22no—
68C2×#127—28no—
69C2×#127—28no—
70C2×#127—28no—
71C2×#127—28no—
72C2×#127—28no—
73C2×#127—48no—
74C2×#127—432no—
75C2×#127C2314yes—
76C2×#127C2314yes—
77C2×#127C2318yes—
78C2×#127C2311yescharacteristic
79C2×#127C2318yes—
80C2×#124C2312yes—
81C2×#121—22no—
82C4×#41—216no—
83C2×C4×Q8C2314yes—
84C2×C4×D4—24no—
85C2×C4×D4—22no—
86C2×C4×D4—24no—
87C2×C4×D4C2314yes—
88C2×#117—22no—
89C2×#117C2312yes—
90C22×C42—22no—
91C22×C42D414yes—
92C22×C42C2311yescharacteristic
93C22×C42C2311yescharacteristic
94C22×C42C2311yescharacteristic
95#496—232no—
96#493—216no—
97#492—216no—
98#460—264no—
99#459—264no—
100#423D4116yescomplemented
101C2×C4⋊C8C2318yes—
102#418—232no—
103C2×#102C23132yes—
104C4×M16—28no—
105C2×#97C2314yes—
106C2×C4×C8C2314yes—
107#391C2314yes—
108#390—24no—
109#390C2314yes—
110#389—24no—
111#388—28no—
112#386—24no—
113#384C2312yes—
114#382C2312yes—
115#380—22no—
116C4×C4⋊C4—24no—
117C4×C4⋊C4—22no—
118C4×C22⋊C4—24no—
119C4×C22⋊C4—28no—
120C43D412yes—
121C43D412yes—
122C43C2312yes—
123C43C2312yes—
124#336C2318yes—
125C25—24no—
126C2×#41—24no—
127C2×#41—28no—
128C2×#41—28no—
129C2×#41C2×D418yes—
130C22×Q8C2412yes—
131C22×D4—22no—
132C22×D4—22no—
133C22×D4—28no—
134C22×D4—216no—
135C22×D4—24no—
136C22×D4—28no—
137C22×D4—28no—
138C22×D4—216no—
139C22×D4—28no—
140C22×D4—28no—
141C22×D4—28no—
142C22×D4—216no—
143C22×D4—416no—
144C22×D4—432no—
145C22×D4—432no—
146C22×D4—432no—
147C22×D4—416no—
148C22×D4—416no—
149C22×D4C2418yes—
150C22×D4C2412yes—
151C23×C4—22no—
152C23×C4—24no—
153C23×C4C2×D412yes—
154C23×C4C2×D412yes—
155C23×C4C2×D412yes—
156C23×C4C2411yescharacteristic
157AGL(1,ℤ/8)—4128no—
158C2×SD16—232no—
159C2×SD16—232no—
160C2×SD16—416no—
161C2×D8—232no—
162C2×D8—232no—
163C2×D8—416no—
164C2×M16—28no—
165C2×M16—28no—
166C2×M16C2×D4116yes—
167C22×C8C2418yes—
168C4⋊Q8—22no—
169C4⋊Q8C2×D414yescomplemented
170#127—232no—
171#127—28no—
172#127—28no—
173#127—22no—
174#127—28no—
175#127—28no—
176#127—416no—
177#127—416no—
178#127—432no—
179#127—432no—
180#127—432no—
181#127—416no—
182#127—48no—
183#127—48no—
184#127—432no—
185#127C2×D414yescomplemented
186#124—24no—
187#124C2×D418yescomplemented
188#121—216no—
189C4×Q8—216no—
190C4×D4—216no—
191C4×D4—216no—
192C4×D4—416no—
193C4×D4—416no—
194#117—28no—
195#117—48no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—24no—
198C2×C4⋊C4—24no—
199C2×C4⋊C4C2414yes—
200C2×C22⋊C4—24no—
201C2×C22⋊C4—22no—
202C2×C22⋊C4—24no—
203C2×C22⋊C4—28no—
204C2×C42—24no—
205C2×C42—216no—
206C2×C42—22no—
207C2×C42—24no—
208C2×C42—48no—
209C2×C42C2×D412yes—
210C2×C42C2×D414yes—
211C2×C42C2×D414yes—
212C2×C42C2×D414yes—
213C2×C42C2×D412yes—
214C2×C42C2×D414yes—
215C2×C42C2×D411yescharacteristic
216C2×C42C2×D411yescharacteristic
217C2×C42C2×D418yes—
218C2×C42C2×D414yes—
219C2×C42#4114yes—
220C2×C42C2412yes—
221C2×C42C2412yes—
222C2×C42C2411yescharacteristic
223C2×C42C2411yescharacteristic
224C4⋊C8C2×D4132yescomplemented
225#102—264no—
226#97—28no—
227C4×C8—28no—
228#95—24no—
229#95—28no—
230C24—22no—
231C24—28no—
232C24—432no—
233C24—48no—
234C24—48no—
235C24—48no—
236C24C22×D411yescharacteristic
237#41—232no—
238#41—416no—
239C2×Q8—22no—
240C2×Q8—24no—
241C2×Q8—24no—
242C2×Q8C22×D414yes—
243C2×D4—28no—
244C2×D4—28no—
245C2×D4—24no—
246C2×D4—216no—
247C2×D4—28no—
248C2×D4—22no—
249C2×D4—28no—
250C2×D4—24no—
251C2×D4—216no—
252C2×D4—432no—
253C2×D4—432no—
254C2×D4—432no—
255C2×D4—432no—
256C2×D4—44no—
257C2×D4—44no—
258C2×D4—416no—
259C2×D4—416no—
260C2×D4—48no—
261C2×D4—416no—
262C2×D4—416no—
263C2×D4—48no—
264C2×D4—48no—
265C2×D4—432no—
266C2×D4—432no—
267C2×D4—48no—
268C2×D4—432no—
269C2×D4—432no—
270C2×D4—432no—
271C2×D4—864no—
272C2×D4—864no—
273C2×D4—864no—
274C2×D4C22×D414yes—
275C22×C4—24no—
276C22×C4—28no—
277C22×C4—28no—
278C22×C4—28no—
279C22×C4—22no—
280C22×C4—28no—
281C22×C4—24no—
282C22×C4—44no—
283C22×C4—44no—
284C22×C4—48no—
285C22×C4—416no—
286C22×C4#12118yes—
287C22×C4#12718yes—
288C22×C4C22×D412yes—
289C22×C4C22×D412yes—
290C22×C4C22×D414yes—
291C22×C4C22×D411yescharacteristic
292C22×C4C22×D412yes—
293C22×C4C22×D412yes—
294C22×C4C22×D412yes—
295C22×C4C22×D414yes—
296C22×C4C2×#4112yes—
297C22×C4H(5,𝔽2)14yes—
298C22×C4C2511yescharacteristic
299SD16—464no—
300D8—464no—
301M16—232no—
302C2×C8—28no—
303C2×C8—28no—
304C2×C8C22×D4116yes—
305C4⋊C4—28no—
306C4⋊C4—28no—
307C4⋊C4—48no—
308C4⋊C4—48no—
309C22⋊C4—216no—
310C22⋊C4—44no—
311C22⋊C4—48no—
312C22⋊C4—416no—
313C42—24no—
314C42—28no—
315C42—28no—
316C42—28no—
317C42—24no—
318C42—44no—
319C42—48no—
320C42#12118yes—
321C42#12714yes—
322C42#12714yes—
323C42AGL(1,ℤ/8)116yescomplemented
324C42C22×D412yes—
325C42C22×D412yes—
326C42C22×D418yes—
327C42C2×#4112yes—
328C42C2×#4112yes—
329C23—24no—
330C23—24no—
331C23—42no—
332C23—42no—
333C23—48no—
334C23—48no—
335C23—48no—
336C23—48no—
337C23—432no—
338C23—832no—
339C23—832no—
340C23—832no—
341C23—816no—
342C23C2×#12112yes—
343C23C2×#12712yes—
344C23#53512yes—
345C23C23×D411yescharacteristic
346Q8—28no—
347Q8—44no—
348D4—28no—
349D4—416no—
350D4—416no—
351D4—44no—
352D4—416no—
353D4—416no—
354D4—864no—
355D4—832no—
356D4—832no—
357D4—832no—
358D4—832no—
359D4—816no—
360C2×C4—216no—
361C2×C4—24no—
362C2×C4—24no—
363C2×C4—24no—
364C2×C4—24no—
365C2×C4—216no—
366C2×C4—216no—
367C2×C4—24no—
368C2×C4—42no—
369C2×C4—42no—
370C2×C4—416no—
371C2×C4—44no—
372C2×C4—48no—
373C2×C4—816no—
374C2×C4C2×#12114yes—
375C2×C4C2×#12114yes—
376C2×C4C2×#12112yes—
377C2×C4C2×#12714yes—
378C2×C4C2×#12714yes—
379C2×C4C2×#12712yes—
380C2×C4#53512yes—
381C2×C4D4218yes—
382C2×C4#55018yes—
383C2×C4C2×AGL(1,ℤ/8)18yes—
384C2×C4C2×AGL(1,ℤ/8)18yes—
385C2×C4C2×AGL(1,ℤ/8)18yes—
386C2×C4C23×D411yescharacteristic
387C2×C4C23×D412yes—
388C2×C4C22×#4111yescharacteristic
389C2×C4C2×H(5,𝔽2)12yes—
390C8—216no—
391C22—24no—
392C22—24no—
393C22—24no—
394C22—44no—
395C22—48no—
396C22—84no—
397C22—84no—
398C22—88no—
399C22—816no—
400C22—816no—
401C22—816no—
402C22—1632no—
403C22#254114yes—
404C22C22×#12111yescharacteristic
405C22C22×#12711yescharacteristic
406C22C2×#53511yescharacteristic
407C22C22×AGL(1,ℤ/8)14yes—
408C4—28no—
409C4—24no—
410C4—24no—
411C4—44no—
412C4—84no—
413C4#254114yes—
414C4#303818yescomplemented
415C4#315118yescomplemented
416C4C2×D4212yes—
417C4C2×#55012yes—
418C2—24no—
419C2—84no—
420C2—168no—
421C2C2×#254111yescharacteristic
422C2C2×#303812yes—
423C2C2×#315112yes—
424C2#6079812yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
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
1. C2×#3151C2semidirect
2. C2×#3038C2semidirect
3. C2×#2867C2semidirect
4. C4×#532C2semidirect
5. #28448C2semidirect
6. #28441C2semidirect
7. #28437C2semidirect
8. #23430C2semidirect
9. #21203C2semidirect
10. C4×#423C2semidirect
19. C2×#496C22semidirect
20. C2×#493C22semidirect
21. C2×#492C22semidirect
23. C2×#460C22semidirect
24. C2×#459C22semidirect
32. C4×C4⋊Q8C22semidirect
33. C4×#127C22semidirect
34. C4×#124C22semidirect
41. #2014C22semidirect
42. #2011C22semidirect
43. #1961C22semidirect
44. #1957C22semidirect
45. #1767C22semidirect
46. C4×C4⋊C8C22semidirect
60. #532D4semidirect
100. #423D4semidirect
169. C4⋊Q8C2×D4semidirect
185. #127C2×D4semidirect
187. #124C2×D4semidirect
224. C4⋊C8C2×D4semidirect
323. C42AGL(1,ℤ/8)semidirect
414. C4#3038semidirect
415. C4#3151semidirect

Maximal subgroups

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

subgroupindex
C2×#31512
C2×#30382
C2×#28672
C4×#5322
#284482
#284412
#284372
#234302
#212032
C4×#4232