Finite Group Databasebeta

← #200885

Subgroups — #200885

Order 512.

1395 conjugacy classes of subgroups; only the 246 normal subgroups are shown.

normal subgroups only

This group has 1395 conjugacy classes of subgroups — too many to draw. Only its 246 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60798C211yesmaximal, characteristic
2#34364C211yesmaximal, characteristic, complemented
3#17741C211yesmaximal, characteristic, complemented
4#17558C212yesmaximal, complemented
5#9428C214yesmaximal, complemented
6#9425C214yesmaximal, complemented
7#7976C212yesmaximal, complemented
8#3444C2211yescharacteristic
9C22×#127C2211yescharacteristic
10#3217—24no—
11#3208C414yescomplemented
12#3205C414yescomplemented
13#3151C2212yes—
14#3138C2212yes—
15#3038—22no—
16#3003—22no—
17#2920C2212yes—
18#2920C2211yescharacteristic
19#2380C2214yes—
20#2366—22no—
21D4×C22⋊C4—24no—
22#2328—24no—
23#2308—22no—
24#2283C2212yes—
25#1555C2214yes—
26#1545—24no—
27#1515C2214yescomplemented
28#1514C2214yescomplemented
29#1513—24no—
30#1500—24no—
31#1288C2218yescomplemented
32C23×D4—22no—
33C23×D4D412yes—
34C2×AGL(1,ℤ/8)—24no—
35#533C2×C412yescomplemented
36#533C2311yescharacteristic
37#532C2×C412yescomplemented
38#532C2311yescharacteristic
39C2×#127—22no—
40C2×#127—22no—
41C2×#127—22no—
42C2×#127C2×C414yes—
43C2×#127D412yes—
44C2×#127D414yes—
45C2×#127C2312yes—
46#529—22no—
47#529—22no—
48#529—22no—
49#525—21no—
50#525—22no—
51C4×#41—21no—
52C4×#41—22no—
53C2×C4×D4—28no—
54C22×C22⋊C4—24no—
55C22×C42C2311yescharacteristic
56#496—24no—
57#493C2×C414yescomplemented
58#492C2×C414yescomplemented
59#490—24no—
60#486C2×C418yescomplemented
61#460—28no—
62#459—28no—
63#450—28no—
64#447—28no—
65#432—22no—
66#432—22no—
67#432C2314yes—
68#431—22no—
69#431—22no—
70#431C2×C414yescomplemented
71#423—21no—
72#423—22no—
73#418—24no—
74#407—21no—
75#407—22no—
76C4×M16—22no—
77C4×M16C2312yes—
78#390—28no—
79#386—28no—
80#386—28no—
81#386—216no—
82#383C2314yes—
83#382—22no—
84#381—22no—
85#380D414yes—
86#379—24no—
87C4×C4⋊C4—22no—
88C4×C22⋊C4—22no—
89#376D414yes—
90#331—28no—
91#328—28no—
92#327—28no—
93C25—22no—
94C2×#41—22no—
95C2×#41—21no—
96C22×D4—24no—
97C22×D4—24no—
98C22×D4—24no—
99C22×D4—22no—
100C22×D4—24no—
101C22×D4—24no—
102C22×D4—24no—
103C22×D4—24no—
104C22×D4—88no—
105C22×D4—84no—
106C22×D4C22⋊C418yes—
107C22×D4C2×D414yes—
108C23×C4—28no—
109C23×C4C2×D411yescharacteristic
110C23×C4C2×D412yes—
111AGL(1,ℤ/8)—84no—
112AGL(1,ℤ/8)—84no—
113C2×SD16—28no—
114C2×D8—28no—
115C2×M16—22no—
116C2×M16—42no—
117C4⋊Q8C22×C412yes—
118C4⋊Q8C22×C411yescharacteristic
119#127—24no—
120#127—24no—
121#127—24no—
122#127—24no—
123#127—22no—
124#127—22no—
125#127—24no—
126#127C22⋊C418yes—
127#127C22×C411yescharacteristic
128#127C22×C414yes—
129#127C2×D414yes—
130#126—42no—
131#125—22no—
132#125—21no—
133#124—22no—
134#124—21no—
135#123—41no—
136#123—42no—
137C22⋊Q8—22no—
138C22⋊Q8—22no—
139C22⋊Q8—22no—
140#121—22no—
141#121—22no—
142#121—22no—
143#120—41no—
144C4×Q8—22no—
145C4×Q8—22no—
146C4×Q8—22no—
147C4×D4—22no—
148C4×D4—22no—
149C4×D4—22no—
150C4×D4—42no—
151C4×D4—42no—
152C4×D4—42no—
153C4×D4—816no—
154#117—22no—
155#117—21no—
156#117—42no—
157#117—41no—
158C2×C4⋊C4—24no—
159C2×C22⋊C4—24no—
160C2×C22⋊C4—28no—
161C2×C22⋊C4—216no—
162C2×C22⋊C4—88no—
163C2×C42—24no—
164C2×C42—41no—
165C2×C42C22×C412yes—
166C2×C42C2×D411yescharacteristic
167C2×C42C2×D412yes—
168C2×C42C2×D412yes—
169C2×C42#4112yes—
170C2×C42#4112yes—
171C2×C42C2411yescharacteristic
172C4⋊C8—24no—
173C4⋊C8—24no—
174C4⋊C8—42no—
175#102—216no—
176C22⋊C8—24no—
177C22⋊C8—24no—
178C22⋊C8—42no—
179#97—22no—
180#97—24no—
181C4×C8—42no—
182C4×C8C22×C414yes—
183#95—24no—
184#95—24no—
185#95—24no—
186#95C2×D418yes—
187C24—24no—
188C24—28no—
189C24—48no—
190C24—88no—
191C24#12011yescharacteristic
192#41—42no—
193#41—42no—
194#41—84no—
195C2×Q8—21no—
196C2×Q8—22no—
197C2×D4—21no—
198C2×D4—28no—
199C2×D4—22no—
200C2×D4—28no—
201C2×D4—24no—
202C2×D4—28no—
203C2×D4—28no—
204C2×D4—28no—
205C2×D4—416no—
206C2×D4—44no—
207C2×D4—44no—
208C2×D4—41no—
209C2×D4—42no—
210C2×D4—88no—
211C2×D4—88no—
212C2×D4—84no—
213C2×D4—88no—
214C2×D4—88no—
215C2×D4—88no—
216C2×D4—88no—
217C2×D4—88no—
218C2×D4C2×C22⋊C418yes—
219C22×C4—28no—
220C22×C4—44no—
221C22×C4—42no—
222C22×C4—41no—
223C22×C4—42no—
224C22×C4—88no—
225C22×C4—816no—
226C22×C4C2×C22⋊C412yes—
227C22×C4C4×D412yes—
228C22×C4C4×D412yes—
229C22×C4#12011yescharacteristic
230C22×C4#12012yes—
231C22×C4#12112yes—
232C22×C4#12112yes—
233C22×C4#12312yes—
234C22×C4C22×D411yescharacteristic
235C22×C4C22×D412yes—
236SD16—88no—
237D8—88no—
238M16—44no—
239M16—84no—
240C2×C8—24no—
241C2×C8—44no—
242C4⋊C4—22no—
243C4⋊C4—22no—
244C4⋊C4—22no—
245C4⋊C4—48no—
246C4⋊C4—42no—
247C4⋊C4—42no—
248C4⋊C4—42no—
249C22⋊C4—48no—
250C22⋊C4—41no—
251C22⋊C4—42no—
252C22⋊C4—42no—
253C22⋊C4—42no—
254C22⋊C4—42no—
255C22⋊C4—88no—
256C22⋊C4—816no—
257C42—48no—
258C42—41no—
259C42—42no—
260C42C2×C22⋊C412yes—
261C42C4×D412yes—
262C42C4×D412yes—
263C42#12112yes—
264C42#12112yes—
265C42#12311yescharacteristic
266C42#12311yescharacteristic
267C42C23×C411yescharacteristic
268C42C22×D411yescharacteristic
269C42C2×#4112yes—
270C23—22no—
271C23—28no—
272C23—42no—
273C23—416no—
274C23—416no—
275C23—44no—
276C23—44no—
277C23—41no—
278C23—88no—
279C23—88no—
280C23—1616no—
281C23#38612yes—
282C23C2×#12011yescharacteristic
283Q8—42no—
284Q8—82no—
285D4—48no—
286D4—416no—
287D4—42no—
288D4—416no—
289D4—84no—
290D4—88no—
291D4—88no—
292D4—88no—
293D4—82no—
294D4—84no—
295D4—82no—
296C2×C4—22no—
297C2×C4—22no—
298C2×C4—24no—
299C2×C4—24no—
300C2×C4—24no—
301C2×C4—42no—
302C2×C4—42no—
303C2×C4—42no—
304C2×C4—42no—
305C2×C4—42no—
306C2×C4—48no—
307C2×C4—48no—
308C2×C4—48no—
309C2×C4—42no—
310C2×C4—41no—
311C2×C4—81no—
312C2×C4—81no—
313C2×C4—84no—
314C2×C4—1616no—
315C2×C4#38612yes—
316C2×C4#38612yes—
317C2×C4#38612yes—
318C2×C4C22×C22⋊C411yescharacteristic
319C2×C4C2×C4×D412yes—
320C2×C4C2×#12112yes—
321C2×C4C2×#12311yescharacteristic
322C8—44no—
323C8—84no—
324C22—22no—
325C22—22no—
326C22—22no—
327C22—22no—
328C22—44no—
329C22—416no—
330C22—48no—
331C22—88no—
332C22—88no—
333C22—82no—
334C22—81no—
335C22—1616no—
336C22C2×#38611yescharacteristic
337C22#301912yes—
338C4—22no—
339C4—22no—
340C4—22no—
341C4—21no—
342C4—21no—
343C4—22no—
344C4—88no—
345C4—82no—
346C4—81no—
347C4#190414yes—
348C2—22no—
349C2—42no—
350C2—88no—
351C2—161no—
352C2#2095311yescharacteristic
353C2C2×#190412yes—
subgroupG/H
center, socleC22C2×#386
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radical#60798C2
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #34364C2semidirect
3. #17741C2semidirect
4. #17558C2semidirect
5. #9428C2semidirect
6. #9425C2semidirect
7. #7976C2semidirect
11. #3208C4semidirect
12. #3205C4semidirect
27. #1515C22semidirect
28. #1514C22semidirect
31. #1288C22semidirect
35. #533C2×C4semidirect
37. #532C2×C4semidirect
57. #493C2×C4semidirect
58. #492C2×C4semidirect
60. #486C2×C4semidirect
70. #431C2×C4semidirect

Maximal subgroups

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

subgroupindex
#607982
#343642
#177412
#175582
#94282
#94252
#79762