Finite Group Databasebeta

← #99540

Subgroups — #99540

Order 512.

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

normal subgroups only

This group has 388 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#23369C211yesmaximal, characteristic
2C4×#491C211yesmaximal, characteristic
3#9873C212yesmaximal
4#7568C211yesmaximal, characteristic
5#7030C212yesmaximal
6C2×#491C411yescharacteristic
7C4×C4⋊Q8C2211yescharacteristic
8C4×#125C2211yescharacteristic
9#1992C411yescharacteristic
10#1960C411yescharacteristic
11#1939—22no—
12#1935—22no—
13#1916—22no—
14#1916—22no—
15#1836C411yescharacteristic
16#1831C2212yes—
17C4×#103—22no—
18#1744C2212yes—
19C4×#97C2211yescharacteristic
20#1279—22no—
21#1279—22no—
22#1271—22no—
23C2×C4⋊Q8C2×C411yescharacteristic
24C2×#125C2×C411yescharacteristic
25C2×C4×Q8—21no—
26#491—22no—
27#491C814yescomplemented
28C2×C4⋊C8—22no—
29C2×C4⋊C8D412yes—
30C2×C4⋊C8D412yes—
31C2×#103—22no—
32C2×#103—22no—
33C2×#97—21no—
34C2×#97C2×C411yescharacteristic
35C2×#97C2×C411yescharacteristic
36C2×C4×C8—22no—
37C2×C4×C8—21no—
38C2×C4×C8—22no—
39#391—21no—
40#389—22no—
41#389D412yes—
42#387—22no—
43#384C2×C411yescharacteristic
44#382C2×C411yescharacteristic
45C4×C4⋊C4—22no—
46C4×C4⋊C4—22no—
47C4×C4⋊C4D412yes—
48C43C2311yescharacteristic
49#340—22no—
50#340—22no—
51#336—22no—
52#336—22no—
53#336—22no—
54#336—22no—
55#336—22no—
56#336—22no—
57C22×Q8—21no—
58C22×C8—21no—
59C22×C8—21no—
60C22×C8—22no—
61C22×C8—22no—
62C22×C8—42no—
63C4⋊Q8—21no—
64C4⋊Q8—22no—
65C4⋊Q8C2×C812yes—
66#125—22no—
67#125C2×C812yes—
68#125M1612yes—
69C4×Q8—42no—
70C4×Q8—42no—
71C2×C4⋊C4—22no—
72C2×C4⋊C4—22no—
73C2×C4⋊C4—21no—
74C2×C4⋊C4—22no—
75C2×C4⋊C4—22no—
76C2×C4⋊C4—41no—
77C2×C4⋊C4C22⋊C412yes—
78C2×C4⋊C4C22⋊C412yes—
79C2×C42—22no—
80C2×C42—41no—
81C2×C42C22×C411yescharacteristic
82C2×C42C2×D411yescharacteristic
83C2×C42C2×D412yes—
84C2×C42C2×D411yescharacteristic
85C2×C42#4112yes—
86C4⋊C8—42no—
87C4⋊C8—42no—
88C4⋊C8—82no—
89#103—28no—
90#103—44no—
91#97—21no—
92#97—22no—
93#97—42no—
94#97C2×C812yes—
95C4×C8—24no—
96C4×C8—41no—
97C4×C8—41no—
98C4×C8—42no—
99C4×C8—82no—
100#95—22no—
101#95—22no—
102#95—22no—
103#95—42no—
104C2×Q8—22no—
105C2×Q8—42no—
106C2×Q8—42no—
107C2×Q8—41no—
108C22×C4—22no—
109C22×C4—22no—
110C22×C4—41no—
111C22×C4—41no—
112C22×C4C2×C22⋊C411yescharacteristic
113C22×C4C4×D412yes—
114C22×C4#12011yescharacteristic
115C22×C4#12112yes—
116C22×C4#12311yescharacteristic
117C2×C8—22no—
118C2×C8—22no—
119C2×C8—41no—
120C2×C8—41no—
121C2×C8—41no—
122C2×C8—41no—
123C2×C8—44no—
124C2×C8—42no—
125C2×C8—42no—
126C2×C8—42no—
127C2×C8—42no—
128C2×C8—42no—
129C2×C8—42no—
130C2×C8—82no—
131C2×C8—82no—
132C4⋊C4—22no—
133C4⋊C4—24no—
134C4⋊C4—24no—
135C4⋊C4—24no—
136C4⋊C4—42no—
137C4⋊C4—44no—
138C4⋊C4—42no—
139C4⋊C4—42no—
140C4⋊C4—82no—
141C4⋊C4—82no—
142C4⋊C4C22⋊C814yes—
143C42—21no—
144C42—21no—
145C42—22no—
146C42—22no—
147C42—41no—
148C42—42no—
149C42—42no—
150C42—42no—
151C42—81no—
152C42—81no—
153C42C22×C811yescharacteristic
154C42C2×M1611yescharacteristic
155C42#13112yes—
156C23#38611yescharacteristic
157Q8—44no—
158Q8—82no—
159C2×C4—22no—
160C2×C4—22no—
161C2×C4—21no—
162C2×C4—21no—
163C2×C4—24no—
164C2×C4—22no—
165C2×C4—22no—
166C2×C4—22no—
167C2×C4—22no—
168C2×C4—21no—
169C2×C4—42no—
170C2×C4—42no—
171C2×C4—42no—
172C2×C4—42no—
173C2×C4—42no—
174C2×C4—42no—
175C2×C4—81no—
176C2×C4—81no—
177C2×C4—82no—
178C2×C4—81no—
179C2×C4C2×C22⋊C811yescharacteristic
180C2×C4#40811yescharacteristic
181C2×C4C8×D412yes—
182C2×C4#43512yes—
183C2×C4#45411yescharacteristic
184C2×C4#45411yescharacteristic
185C2×C4#45512yes—
186C2×C4#45612yes—
187C8—42no—
188C8—42no—
189C8—84no—
190C8—82no—
191C8—82no—
192C22#186911yescharacteristic
193C22#188112yes—
194C22#188212yes—
195C22#188411yescharacteristic
196C22#188811yescharacteristic
197C4—21no—
198C4—21no—
199C4—22no—
200C4—22no—
201C4—42no—
202C4—41no—
203C4—44no—
204C4—82no—
205C4—82no—
206C4—161no—
207C4#166211yescharacteristic
208C4#166311yescharacteristic
209C4#166412yes—
210C2#895312yes—
211C2#896011yescharacteristic
212C2#899812yes—
213C2#900411yescharacteristic
214C2#900811yescharacteristic
subgroupG/H
centerC22×C4#120
commutatorC42C22×C8
Frattini, cosocleC43C23
socle, prime radicalC23#386
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C4 ◁ C43 ◁ G
LECSG ▷ C43 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
27. #491C8semidirect

Maximal subgroups

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

subgroupindex
#233692
C4×#4912
#98732
#75682
#70302