Finite Group Databasebeta

← #320562

Subgroups — #320562

Order 512.

884 conjugacy classes of subgroups; only the 190 normal subgroups are shown.

normal subgroups only

This group has 884 conjugacy classes of subgroups — too many to draw. Only its 190 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#33293C211yesmaximal, characteristic, complemented
2#28613C212yesmaximal, complemented
3#28509C212yesmaximal, complemented
4#21690C211yesmaximal, characteristic
5#10685C212yesmaximal, complemented
6#10305C212yesmaximal
7#9827C212yesmaximal, complemented
8#9820C212yesmaximal, complemented
9#8262C211yesmaximal, characteristic, complemented
10C2×#493C2211yescharacteristic
11C2×#492C2212yes—
12C2×#486C2212yes—
13C2×#476—24no—
14C2×#442—22no—
15#2868C2211yescharacteristic
16#2862C2211yescharacteristic
17C4×#127C2212yes—
18#2094—24no—
19#2055—24no—
20#1993—22no—
21#1990C2212yes—
22#1990C2212yes—
23#1990C2212yes—
24#1972—22no—
25#1957C2212yes—
26#1957C2212yes—
27#1957C2212yes—
28#1956C2212yes—
29#1877—24no—
30#1844C2212yes—
31#1844C2212yes—
32#1844C2211yescharacteristic
33#1836—22no—
34C4×C4⋊C8C2212yes—
35C4×#97C2211yescharacteristic
36C2×#322C2211yescharacteristic
37C2×C82C2211yescharacteristic
38#1273C2212yes—
39C22×SD16—24no—
40C22×D8—22no—
41C2×C4⋊Q8—21no—
42C2×C4⋊Q8—22no—
43C2×C4⋊Q8C2312yes—
44C2×#127C2312yes—
45C2×C4×D4—22no—
46C2×C4×D4—22no—
47C2×C4×D4—22no—
48#493—24no—
49#492—24no—
50#492—42no—
51#486—24no—
52#486—42no—
53#476—416no—
54#442—48no—
55C2×#106—24no—
56C2×#106—24no—
57C2×#106—22no—
58C2×C4⋊C8D414yes—
59C2×#103—24no—
60C2×#102—24no—
61C2×#102—24no—
62C2×#102—24no—
63C2×#102—24no—
64C2×#97D412yes—
65C2×#97D412yes—
66C2×C4×C8C2312yes—
67C2×C4×C8C2312yes—
68C2×C4×C8C2311yescharacteristic
69C2×C4×C8C2311yescharacteristic
70#391—22no—
71#391—22no—
72#390—22no—
73#390—22no—
74#390—22no—
75#384—21no—
76#384—22no—
77#384C2312yes—
78#384C2312yes—
79#382—21no—
80#382—21no—
81C43C2311yescharacteristic
82#340—24no—
83#340—24no—
84#340—24no—
85#336D414yes—
86#322—41no—
87C82—41no—
88C22×Q8—22no—
89C22×Q8—41no—
90C22×D4—22no—
91C22×D4—22no—
92C22×D4—22no—
93C23×C4—42no—
94C2×SD16—48no—
95C2×SD16—48no—
96C2×SD16—84no—
97C2×D8—28no—
98C2×D8—44no—
99C2×D8—44no—
100C22×C8C2×D414yes—
101C22×C8C2×D412yes—
102C22×C8C2×D412yes—
103C22×C8C2×D414yes—
104C4⋊Q8—22no—
105C4⋊Q8—22no—
106C4⋊Q8—22no—
107C4⋊Q8—22no—
108C4⋊Q8—44no—
109C4⋊Q8—48no—
110#127—22no—
111#127—22no—
112#127—22no—
113#127#4114yes—
114C4×D4—48no—
115C4×D4—48no—
116C4×D4—84no—
117C2×C4⋊C4—22no—
118C2×C4⋊C4—22no—
119C2×C4⋊C4—22no—
120C2×C4⋊C4—22no—
121C2×C4⋊C4—22no—
122C2×C4⋊C4—22no—
123C2×C4⋊C4—22no—
124C2×C4⋊C4—22no—
125C2×C4⋊C4—41no—
126C2×C4⋊C4—42no—
127C2×C22⋊C4—42no—
128C2×C22⋊C4—42no—
129C2×C22⋊C4—42no—
130C2×C42C2×D412yes—
131C2×C42C2×D412yes—
132C2×C42C2×D411yescharacteristic
133C2×C42C2×D411yescharacteristic
134C2×C42C2411yescharacteristic
135#106—48no—
136#106—44no—
137#106—48no—
138C4⋊C8—28no—
139#103—84no—
140#102—48no—
141#102—48no—
142#102—48no—
143#102—48no—
144#97—42no—
145#97SD1618yescomplemented
146C4×C8—22no—
147C4×C8—22no—
148C4×C8—22no—
149C4×C8—21no—
150C4×C8—22no—
151C4×C8—22no—
152C4×C8—21no—
153#95—41no—
154#95—41no—
155#95—42no—
156C24—42no—
157C2×Q8—42no—
158C2×Q8—42no—
159C2×Q8—44no—
160C2×Q8—44no—
161C2×Q8—84no—
162C2×Q8—82no—
163C2×D4—24no—
164C2×D4—24no—
165C2×D4—24no—
166C2×D4—42no—
167C2×D4—42no—
168C2×D4—42no—
169C2×D4—44no—
170C2×D4—44no—
171C2×D4—44no—
172C2×D4—44no—
173C2×D4—44no—
174C2×D4—44no—
175C22×C4—42no—
176C22×C4—42no—
177C22×C4—42no—
178C22×C4—84no—
179C22×C4—84no—
180C22×C4—84no—
181C22×C4#12711yescharacteristic
182C22×C4#12711yescharacteristic
183C22×C4#12712yes—
184C22×C4C22×D411yescharacteristic
185C22×C4C22×D412yes—
186SD16—816no—
187D8—416no—
188C2×C8—22no—
189C2×C8—22no—
190C2×C8—24no—
191C2×C8—22no—
192C2×C8—22no—
193C2×C8—24no—
194C2×C8—24no—
195C2×C8—24no—
196C2×C8—24no—
197C2×C8C2×SD1614yes—
198C2×C8C2×SD1614yes—
199C2×C8AGL(1,ℤ/8)18yes—
200C4⋊C4—42no—
201C4⋊C4—42no—
202C4⋊C4—42no—
203C4⋊C4—42no—
204C4⋊C4—44no—
205C4⋊C4—44no—
206C4⋊C4—44no—
207C4⋊C4—44no—
208C4⋊C4—44no—
209C4⋊C4—44no—
210C4⋊C4—84no—
211C4⋊C4—82no—
212C22⋊C4—84no—
213C22⋊C4—84no—
214C22⋊C4—84no—
215C42—24no—
216C42—22no—
217C42—24no—
218C42C2×SD1614yes—
219C42C2×#4112yes—
220C42H(5,𝔽2)11yescharacteristic
221C422−1+411yescharacteristic
222C23—44no—
223C23—82no—
224C23—82no—
225C23—82no—
226C23—82no—
227C23—82no—
228C23—82no—
229C23C2×#12711yescharacteristic
230Q8—84no—
231Q8—84no—
232Q8—164no—
233D4—48no—
234D4—48no—
235D4—48no—
236D4—84no—
237D4—84no—
238D4—84no—
239C2×C4—22no—
240C2×C4—22no—
241C2×C4—22no—
242C2×C4—21no—
243C2×C4—21no—
244C2×C4—22no—
245C2×C4—82no—
246C2×C4—82no—
247C2×C4—82no—
248C2×C4—82no—
249C2×C4—82no—
250C2×C4—82no—
251C2×C4—82no—
252C2×C4—82no—
253C2×C4—82no—
254C2×C4—168no—
255C2×C4#49214yes—
256C2×C4#52511yescharacteristic
257C2×C4#53711yescharacteristic
258C2×C4#54612yes—
259C2×C4#54812yes—
260C2×C4C22×SD1612yes—
261C2×C4C2×AGL(1,ℤ/8)14yes—
262C2×C4#57512yes—
263C2×C4#57714yes—
264C8—24no—
265C8—24no—
266C8—28no—
267C8—44no—
268C22—84no—
269C22—84no—
270C22—84no—
271C22—162no—
272C22—162no—
273C22—162no—
274C22—162no—
275C22#254011yescharacteristic
276C22C2×#49211yescharacteristic
277C22#315112yes—
278C22#315311yescharacteristic
279C22#315412yes—
280C4—24no—
281C4—41no—
282C4—41no—
283C4—162no—
284C4—162no—
285C4—162no—
286C4#310112yes—
287C4#324612yes—
288C4#330614yes—
289C4#337314yes—
290C2—164no—
291C2#1962112yes—
292C2#1972411yescharacteristic
293C2#1974711yescharacteristic
294C2#2837011yescharacteristic
295C2#2843012yes—
subgroupG/H
center, socleC23C2×#127
commutator, Frattini, cosocleC2×C42C24
prime radicalC2×#493C22
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C43 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #33293C2semidirect
2. #28613C2semidirect
3. #28509C2semidirect
5. #10685C2semidirect
7. #9827C2semidirect
8. #9820C2semidirect
9. #8262C2semidirect
145. #97SD16semidirect

Maximal subgroups

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

subgroupindex
#332932
#286132
#285092
#216902
#106852
#103052
#98272
#98202
#82622