Finite Group Databasebeta

← #102822

Subgroups — #102822

Order 512.

645 conjugacy classes of subgroups; only the 104 normal subgroups are shown.

normal subgroups only

This group has 645 conjugacy classes of subgroups — too many to draw. Only its 104 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#30239C211yesmaximal, characteristic
2#27973C212yesmaximal, complemented
3#11371C211yesmaximal, characteristic, complemented
4#9680C212yesmaximal, complemented
5#7016C211yesmaximal, characteristic, complemented
6C2×#466—22no—
7#2856C2212yes—
8C2×#384C2211yescharacteristic
9#2093—22no—
10#2071—24no—
11#2011—22no—
12#1957—22no—
13#1874—24no—
14#1834—22no—
15C4×C22⋊C8C2212yes—
16C2×#340—22no—
17C2×#336C2211yescharacteristic
18C2×#327—24no—
19C4⋊#95C2211yescharacteristic
20C22×D8—22no—
21C23×C8D412yes—
22C2×#127D412yes—
23C2×#121—22no—
24C2×#121—22no—
25C2×#121—22no—
26C22×C4⋊C4—21no—
27C22×C4⋊C4—21no—
28C22×C4⋊C4—22no—
29C22×C42C2311yescharacteristic
30#466—44no—
31#466—44no—
32C2×#107—22no—
33C2×#102—24no—
34C2×#102—28no—
35C2×C22⋊C8D414yes—
36C2×C4×C8—22no—
37#390—22no—
38#390—22no—
39#390—22no—
40#384—22no—
41#384—21no—
42#384D412yes—
43#384D412yes—
44C2×#95—22no—
45#340—28no—
46#340—44no—
47#336—22no—
48#336—22no—
49#327—48no—
50C22×D4—22no—
51C22×D4—22no—
52C22×D4—22no—
53C22×D4—42no—
54C23×C4—41no—
55C23×C4—41no—
56C23×C4C2×D411yescharacteristic
57C23×C4C2×D412yes—
58C2×D8—44no—
59C2×D8—82no—
60C2×D8—82no—
61C22×C8—22no—
62C22×C8—22no—
63C22×C8—24no—
64C22×C8—42no—
65C22×C8—42no—
66C22×C8D814yes—
67#127—44no—
68#121—44no—
69#121—44no—
70#121—44no—
71#121—44no—
72#121—44no—
73#121—44no—
74C2×C4⋊C4—22no—
75C2×C4⋊C4—24no—
76C2×C4⋊C4—22no—
77C2×C4⋊C4—22no—
78C2×C4⋊C4—22no—
79C2×C4⋊C4—22no—
80C2×C4⋊C4—42no—
81C2×C4⋊C4—42no—
82C2×C4⋊C4—42no—
83C2×C4⋊C4—42no—
84C2×C4⋊C4—41no—
85C2×C4⋊C4—41no—
86C2×C4⋊C4—42no—
87C2×C4⋊C4—84no—
88C2×C22⋊C4—42no—
89C2×C22⋊C4—42no—
90C2×C22⋊C4—42no—
91C2×C42—42no—
92C2×C42C2×D412yes—
93C2×C42C2×D411yescharacteristic
94C2×C42#4111yescharacteristic
95#107—44no—
96#102—48no—
97#102—88no—
98C22⋊C8—24no—
99C22⋊C8—24no—
100C4×C8—44no—
101#95—44no—
102C24—42no—
103C24#12711yescharacteristic
104C2×D4—44no—
105C2×D4—44no—
106C2×D4—44no—
107C2×D4—44no—
108C2×D4—42no—
109C2×D4—42no—
110C2×D4—44no—
111C2×D4—44no—
112C2×D4—82no—
113C2×D4—82no—
114C2×D4—82no—
115C2×D4—82no—
116C2×D4—82no—
117C2×D4—82no—
118C2×D4—82no—
119C2×D4—82no—
120C22×C4—21no—
121C22×C4—21no—
122C22×C4—22no—
123C22×C4—42no—
124C22×C4—42no—
125C22×C4—42no—
126C22×C4—42no—
127C22×C4—42no—
128C22×C4—81no—
129C22×C4—84no—
130C22×C4—82no—
131C22×C4—81no—
132C22×C4—81no—
133C22×C4—82no—
134C22×C4—81no—
135C22×C4#12012yes—
136C22×C4#12011yescharacteristic
137C22×C4#12111yescharacteristic
138C22×C4#12112yes—
139C22×C4C2×D812yes—
140C22×C4#13514yes—
141C22×C4AGL(1,ℤ/8)12yes—
142C22×C4AGL(1,ℤ/8)14yes—
143D8—88no—
144C2×C8—24no—
145C2×C8—42no—
146C2×C8—42no—
147C2×C8—44no—
148C2×C8—44no—
149C2×C8—44no—
150C2×C8—44no—
151C2×C8—82no—
152C4⋊C4—48no—
153C4⋊C4—44no—
154C4⋊C4—44no—
155C4⋊C4—44no—
156C4⋊C4—44no—
157C4⋊C4—44no—
158C4⋊C4—82no—
159C4⋊C4—84no—
160C4⋊C4—82no—
161C22⋊C4—84no—
162C22⋊C4—84no—
163C22⋊C4—84no—
164C42—24no—
165C42—41no—
166C42—41no—
167C23—21no—
168C23—22no—
169C23—21no—
170C23—82no—
171C23—82no—
172C23—82no—
173C23—82no—
174C23—82no—
175C23—82no—
176C23—82no—
177C23#39211yescharacteristic
178C23#49311yescharacteristic
179C23#49512yes—
180C23#49611yescharacteristic
181C23#49612yes—
182D4—88no—
183D4—84no—
184D4—84no—
185D4—164no—
186D4—162no—
187D4—162no—
188D4—162no—
189D4—162no—
190C2×C4—21no—
191C2×C4—22no—
192C2×C4—22no—
193C2×C4—22no—
194C2×C4—22no—
195C2×C4—21no—
196C2×C4—22no—
197C2×C4—22no—
198C2×C4—22no—
199C2×C4—22no—
200C2×C4—44no—
201C2×C4—44no—
202C2×C4—82no—
203C2×C4—84no—
204C2×C4—84no—
205C2×C4—82no—
206C2×C4—82no—
207C2×C4—82no—
208C2×C4—82no—
209C2×C4—162no—
210C2×C4—161no—
211C2×C4—162no—
212C2×C4—162no—
213C2×C4—161no—
214C2×C4#44712yes—
215C2×C4#44914yes—
216C2×C4#45511yescharacteristic
217C2×C4#45611yescharacteristic
218C2×C4#45812yes—
219C2×C4#45912yes—
220C2×C4#46314yes—
221C8—44no—
222C8—84no—
223C22—22no—
224C22—22no—
225C22—22no—
226C22—22no—
227C22—21no—
228C22—21no—
229C22—21no—
230C22—21no—
231C22—162no—
232C22—162no—
233C22—162no—
234C22—162no—
235C22—162no—
236C22—162no—
237C22—162no—
238C22#170911yescharacteristic
239C22#171712yes—
240C22#172111yescharacteristic
241C22#200711yescharacteristic
242C22#200812yes—
243C22#201111yescharacteristic
244C22#201212yes—
245C22#202211yescharacteristic
246C4—24no—
247C4—24no—
248C4—22no—
249C4—162no—
250C4—162no—
251C4—162no—
252C4#160712yes—
253C4#161012yes—
254C2—21no—
255C2—21no—
256C2—22no—
257C2—322no—
258C2#995012yes—
259C2#997011yescharacteristic
260C2#1019612yes—
261C2#1021811yescharacteristic
262C2#1030711yescharacteristic
subgroupG/H
center, socleC23#392
commutator, Frattini, cosocleC22×C42C23
prime radical#30239C2
LCSG ▷ C22×C42 ▷ C23 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #27973C2semidirect
3. #11371C2semidirect
4. #9680C2semidirect
5. #7016C2semidirect

Maximal subgroups

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

subgroupindex
#302392
#279732
#113712
#96802
#70162