Finite Group Databasebeta

← #331077

Subgroups — #331077

Order 512.

471 conjugacy classes of subgroups; only the 110 normal subgroups are shown.

normal subgroups only

This group has 471 conjugacy classes of subgroups — too many to draw. Only its 110 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#30226C211yesmaximal, characteristic, complemented
2#22237C211yesmaximal, characteristic
3#10632C212yesmaximal, complemented
4#10369C211yesmaximal, characteristic, complemented
5#10366C211yesmaximal, characteristic
6#10299C211yesmaximal, characteristic, complemented
7#10289C211yesmaximal, characteristic
8#10226C212yesmaximal
9#9292C212yesmaximal, complemented
10#9291C212yesmaximal
11#8802C211yesmaximal, characteristic, complemented
12#3229C2212yes—
13#3226C2212yes—
14#3135C2211yescharacteristic
15#2961C2211yescharacteristic
16#2932C2211yescharacteristic
17#2868C2211yescharacteristic
18#2864C2211yescharacteristic
19#2043—22no—
20#2043—22no—
21#1994C2211yescharacteristic
22#1962C2211yescharacteristic
23#1960C2212yes—
24#1960C2212yes—
25#1929—22no—
26#1866—22no—
27#1849C2211yescharacteristic
28#1745C2211yescharacteristic
29#1546C2212yes—
30#1546C2212yes—
31#1518C2212yes—
32#1517C2212yes—
33#1286C2212yes—
34#1286C2212yes—
35#1281C2212yes—
36#1276C2211yescharacteristic
37#1276C2211yescharacteristic
38#1275C2212yes—
39C2×#131—21no—
40#533C2311yescharacteristic
41#532C2311yescharacteristic
42C2×C4⋊Q8—22no—
43C2×C4⋊Q8C2312yes—
44#500—22no—
45#498—22no—
46#491—21no—
47#490—21no—
48#489—22no—
49#487—22no—
50#486—22no—
51#483—42no—
52#480—42no—
53#475—42no—
54#474—42no—
55#446—42no—
56#435—42no—
57#431C2312yes—
58#431C2312yes—
59C2×#108—22no—
60#424—21no—
61#423C2311yescharacteristic
62#423C2311yescharacteristic
63C2×#104—22no—
64#416—22no—
65C2×#103—22no—
66C2×#103—22no—
67C2×#103—22no—
68C2×#103—22no—
69#408—21no—
70#405—21no—
71C2×#97C2311yescharacteristic
72C2×#97C2311yescharacteristic
73#389—22no—
74#387—42no—
75#384—42no—
76#382C2312yes—
77C43C2311yescharacteristic
78#339—22no—
79#336D412yes—
80#336D412yes—
81#332—42no—
82#332—42no—
83#328—42no—
84#327—42no—
85C2×#41—21no—
86C22×Q8—22no—
87#131—82no—
88C2×M16—21no—
89C2×M16—21no—
90C2×M16—41no—
91C22×C8—41no—
92C22×C8C2×D411yescharacteristic
93C22×C8C2×D411yescharacteristic
94C22×C8C2×D412yes—
95C4⋊Q8—22no—
96C4⋊Q8—22no—
97C4⋊Q8—22no—
98C4⋊Q8—22no—
99C4⋊Q8—82no—
100C4⋊Q8—82no—
101C4⋊Q8#4111yescharacteristic
102C4⋊Q8#4112yes—
103#127#4111yescharacteristic
104#125—21no—
105#124—21no—
106C22⋊Q8—22no—
107#121—22no—
108C4×Q8—22no—
109C4×D4—22no—
110#117—21no—
111C2×C4⋊C4—22no—
112C2×C4⋊C4—22no—
113C2×C4⋊C4—22no—
114C2×C4⋊C4—44no—
115C2×C4⋊C4—42no—
116C2×C42—22no—
117C2×C42C2×D412yes—
118C2×C42C2411yescharacteristic
119#108—82no—
120#107—44no—
121#106—44no—
122C4⋊C8—22no—
123C4⋊C8—22no—
124C4⋊C8—22no—
125C4⋊C8—22no—
126C4⋊C8—42no—
127#104—82no—
128#103—24no—
129#103—44no—
130#103—82no—
131#103—82no—
132#103—82no—
133#103—82no—
134#102—24no—
135#102—44no—
136C22⋊C8—22no—
137C22⋊C8—22no—
138C22⋊C8—42no—
139#97—21no—
140#97—21no—
141#97—22no—
142#97—41no—
143C4×C8—22no—
144C4×C8—22no—
145C4×C8—41no—
146#95—22no—
147#95—22no—
148#95—42no—
149#41—42no—
150#41—42no—
151C2×Q8—21no—
152C2×Q8—22no—
153C2×Q8—42no—
154C2×Q8—42no—
155C2×Q8—42no—
156C2×Q8—42no—
157C2×Q8—42no—
158C2×Q8—42no—
159C2×D4—21no—
160C2×D4—22no—
161C2×D4—41no—
162C22×C4—22no—
163C22×C4—41no—
164C22×C4—42no—
165C22×C4—42no—
166C22×C4C22×D411yescharacteristic
167C22×C4H(5,𝔽2)12yes—
168M16—42no—
169M16—42no—
170M16—82no—
171C2×C8—22no—
172C2×C8—21no—
173C2×C8—21no—
174C2×C8—22no—
175C2×C8—22no—
176C2×C8—22no—
177C2×C8—22no—
178C2×C8—41no—
179C2×C8—41no—
180C2×C8—42no—
181C2×C8—42no—
182C2×C8—82no—
183C4⋊C4—22no—
184C4⋊C4—22no—
185C4⋊C4—22no—
186C4⋊C4—42no—
187C4⋊C4—42no—
188C4⋊C4—42no—
189C4⋊C4—42no—
190C4⋊C4—42no—
191C4⋊C4—42no—
192C4⋊C4—42no—
193C4⋊C4—84no—
194C4⋊C4—88no—
195C22⋊C4—41no—
196C22⋊C4—42no—
197C42—22no—
198C42—22no—
199C42—44no—
200C42—41no—
201C42C2×#4111yescharacteristic
202C42C2×#4111yescharacteristic
203C42H(5,𝔽2)12yes—
204C23—41no—
205C23#53511yescharacteristic
206Q8—42no—
207Q8—82no—
208Q8—82no—
209Q8—82no—
210Q8—82no—
211Q8—82no—
212D4—42no—
213D4—82no—
214D4—82no—
215C2×C4—24no—
216C2×C4—42no—
217C2×C4—42no—
218C2×C4—42no—
219C2×C4—42no—
220C2×C4—41no—
221C2×C4—81no—
222C2×C4—81no—
223C2×C4—84no—
224C2×C4—82no—
225C2×C4—82no—
226C2×C4—82no—
227C2×C4—82no—
228C2×C4#52511yescharacteristic
229C2×C4#53511yescharacteristic
230C2×C4#53812yes—
231C2×C4#53812yes—
232C2×C4#57612yes—
233C2×C4#57712yes—
234C8—42no—
235C8—41no—
236C8—41no—
237C8—82no—
238C8—82no—
239C8—81no—
240C8—81no—
241C22—22no—
242C22—81no—
243C22—82no—
244C22#263111yescharacteristic
245C22#322412yes—
246C4—22no—
247C4—22no—
248C4—44no—
249C4—81no—
250C4—82no—
251C4—162no—
252C4—162no—
253C4#311912yes—
254C4#322011yescharacteristic
255C4#322111yescharacteristic
256C2—22no—
257C2—161no—
258C2#3005311yescharacteristic
259C2#3005812yes—
subgroupG/H
center, socleC22#2631
commutator, Frattini, cosocleC2×C42C24
prime radical#532C23
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #30226C2semidirect
3. #10632C2semidirect
4. #10369C2semidirect
6. #10299C2semidirect
9. #9292C2semidirect
11. #8802C2semidirect

Maximal subgroups

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

subgroupindex
#302262
#222372
#106322
#103692
#103662
#102992
#102892
#102262
#92922
#92912
#88022