Finite Group Databasebeta

← #100096

Subgroups — #100096

Order 512.

428 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 428 conjugacy classes of subgroups — too many to draw. Only its 96 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#23363C211yesmaximal, characteristic
2#21691C211yesmaximal, characteristic
3#7605C211yesmaximal, characteristic
4#7604C212yesmaximal
5#7019C212yesmaximal
6C2×#494C411yescharacteristic
7C2×#441—22no—
8C4×C4⋊Q8C2211yescharacteristic
9C4×C4⋊Q8C2211yescharacteristic
10#1984—22no—
11#1971—22no—
12#1956C411yescharacteristic
13#1956C411yescharacteristic
14#1916—24no—
15#1844C411yescharacteristic
16#1839C2212yes—
17#1839C2212yes—
18C4×#97C2211yescharacteristic
19#1272—22no—
20#1271—24no—
21C22×Q16—22no—
22C2×C4⋊Q8C2×C411yescharacteristic
23C2×C4⋊Q8C2×C411yescharacteristic
24C2×C4×Q8—21no—
25C2×C4×Q8D412yes—
26#494—22no—
27#494C814yescomplemented
28#441—44no—
29#441—44no—
30C2×#106—22no—
31C2×C4⋊C8—22no—
32C2×#103—22no—
33C2×#103—22no—
34C2×#97—21no—
35C2×#97—21no—
36C2×C4×C8—22no—
37C2×C4×C8C2×C411yescharacteristic
38C2×C4×C8C2×C411yescharacteristic
39#391—21no—
40#391D412yes—
41#389—22no—
42#389—22no—
43#384C2×C411yescharacteristic
44#384C2×C411yescharacteristic
45C4×C4⋊C4—22no—
46C4×C4⋊C4—22no—
47C43C2311yescharacteristic
48#336—24no—
49#336—22no—
50#336—22no—
51#336—22no—
52#336—22no—
53#336—22no—
54#336D414yes—
55C22×Q8—21no—
56C22×Q8C22⋊C412yes—
57C2×Q16—28no—
58C2×Q16—44no—
59C2×Q16—44no—
60C22×C8—21no—
61C22×C8—21no—
62C22×C8—24no—
63C22×C8—42no—
64C4⋊Q8—21no—
65C4⋊Q8—22no—
66C4⋊Q8—22no—
67C4⋊Q8C2×C812yes—
68C4⋊Q8C2×C812yes—
69C4⋊Q8M1612yes—
70C4×Q8—42no—
71C4×Q8—42no—
72C4×Q8—42no—
73C4×Q8—42no—
74C2×C4⋊C4—22no—
75C2×C4⋊C4—22no—
76C2×C4⋊C4—21no—
77C2×C4⋊C4—22no—
78C2×C4⋊C4—22no—
79C2×C4⋊C4—22no—
80C2×C4⋊C4—41no—
81C2×C4⋊C4C22⋊C412yes—
82C2×C42—22no—
83C2×C42—41no—
84C2×C42C22×C411yescharacteristic
85C2×C42C2×D411yescharacteristic
86C2×C42C2×D411yescharacteristic
87C2×C42C2×D412yes—
88C2×C42#4112yes—
89#106—44no—
90C4⋊C8—82no—
91#103—44no—
92#103—44no—
93#97—24no—
94#97—42no—
95C4×C8—21no—
96C4×C8—22no—
97C4×C8—82no—
98C4×C8C2×C812yes—
99#95—24no—
100#95—42no—
101C2×Q8—22no—
102C2×Q8—22no—
103C2×Q8—42no—
104C2×Q8—42no—
105C2×Q8—42no—
106C2×Q8—42no—
107C2×Q8—41no—
108C2×Q8C22⋊C814yes—
109C22×C4—22no—
110C22×C4—22no—
111C22×C4—41no—
112C22×C4—41no—
113C22×C4C2×C22⋊C411yescharacteristic
114C22×C4C4×D412yes—
115C22×C4#12011yescharacteristic
116C22×C4#12112yes—
117C22×C4#12311yescharacteristic
118Q16—416no—
119C2×C8—22no—
120C2×C8—22no—
121C2×C8—22no—
122C2×C8—22no—
123C2×C8—41no—
124C2×C8—41no—
125C2×C8—48no—
126C2×C8—44no—
127C2×C8—82no—
128C2×C8—82no—
129C2×C8—82no—
130C4⋊C4—24no—
131C4⋊C4—24no—
132C4⋊C4—24no—
133C4⋊C4—42no—
134C4⋊C4—42no—
135C4⋊C4—42no—
136C4⋊C4—82no—
137C4⋊C4—82no—
138C4⋊C4—82no—
139C4⋊C4—82no—
140C42—22no—
141C42—41no—
142C42—42no—
143C42—42no—
144C42—42no—
145C42—42no—
146C42—81no—
147C42—81no—
148C42C22×C811yescharacteristic
149C42C2×M1611yescharacteristic
150C42#13112yes—
151C23#38611yescharacteristic
152Q8—44no—
153Q8—44no—
154Q8—82no—
155Q8—82no—
156C2×C4—21no—
157C2×C4—21no—
158C2×C4—22no—
159C2×C4—22no—
160C2×C4—22no—
161C2×C4—24no—
162C2×C4—22no—
163C2×C4—21no—
164C2×C4—22no—
165C2×C4—22no—
166C2×C4—22no—
167C2×C4—42no—
168C2×C4—42no—
169C2×C4—42no—
170C2×C4—42no—
171C2×C4—44no—
172C2×C4—81no—
173C2×C4—81no—
174C2×C4—81no—
175C2×C4—82no—
176C2×C4C2×C22⋊C811yescharacteristic
177C2×C4#40811yescharacteristic
178C2×C4C8×D412yes—
179C2×C4#43512yes—
180C2×C4#45312yes—
181C8—24no—
182C8—42no—
183C8—84no—
184C8—162no—
185C22#186911yescharacteristic
186C22#188411yescharacteristic
187C22#188711yescharacteristic
188C22#190212yes—
189C22#190512yes—
190C4—22no—
191C4—22no—
192C4—22no—
193C4—21no—
194C4—21no—
195C4—42no—
196C4—41no—
197C4—44no—
198C4—82no—
199C4—82no—
200C4—161no—
201C2#895911yescharacteristic
202C2#908812yes—
203C2#941612yes—
204C2#942111yescharacteristic
205C2#942911yescharacteristic
subgroupG/H
center, socle, prime radicalC23#386
commutatorC42C22×C8
Frattini, cosocleC43C23
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C43 ◁ G
LECSG ▷ C43 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
27. #494C8semidirect

Maximal subgroups

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

subgroupindex
#233632
#216912
#76052
#76042
#70192