Finite Group Databasebeta

← #149696

Subgroups — #149696

Order 512.

350 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 350 conjugacy classes of subgroups — too many to draw. Only its 70 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#21691C211yesmaximal, characteristic
2#11478C212yesmaximal
3#7371C212yesmaximal
4#7355C212yesmaximal
5C2×#494C411yescharacteristic
6C2×#441—22no—
7C4×C4⋊Q8C411yescharacteristic
8C4×C4⋊Q8C2211yescharacteristic
9#1971—22no—
10#1956C411yescharacteristic
11#1956C2211yescharacteristic
12#1844C2211yescharacteristic
13C4×#97C411yescharacteristic
14#1461—22no—
15#1325C2214yes—
16#1319—22no—
17C22×Q16—22no—
18C2×C4⋊Q8C2×C411yescharacteristic
19C2×C4⋊Q8C2×C411yescharacteristic
20C2×C4×Q8—21no—
21C2×C4×Q8D412yes—
22#494—22no—
23#494C814yescomplemented
24#441—44no—
25#441—44no—
26#431—42no—
27C2×#106—22no—
28C2×#103—22no—
29C2×#103—22no—
30C2×#97—21no—
31C2×#97—21no—
32C2×C4×C8C2×C411yescharacteristic
33C2×C4×C8C2×C411yescharacteristic
34#391—21no—
35#391D412yes—
36#389—41no—
37#389—41no—
38#384C2×C411yescharacteristic
39#384C2311yescharacteristic
40C4×C4⋊C4—41no—
41C4×C4⋊C4—41no—
42C43C2×C411yescharacteristic
43#324—24no—
44#324—42no—
45C22×Q8—21no—
46C22×Q8C22⋊C412yes—
47C2×Q16—28no—
48C2×Q16—44no—
49C2×Q16—44no—
50C22×C8—21no—
51C22×C8—21no—
52C4⋊Q8—21no—
53C4⋊Q8—41no—
54C4⋊Q8—41no—
55C4⋊Q8C2×C812yes—
56C4⋊Q8C2×C812yes—
57C4⋊Q8M1612yes—
58C4×Q8—42no—
59C4×Q8—42no—
60C4×Q8—42no—
61C4×Q8—42no—
62C2×C4⋊C4—21no—
63C2×C4⋊C4—22no—
64C2×C4⋊C4—41no—
65C2×C4⋊C4—41no—
66C2×C4⋊C4—41no—
67C2×C4⋊C4—41no—
68C2×C4⋊C4—41no—
69C2×C4⋊C4C2×D412yes—
70C2×C42—21no—
71C2×C42—22no—
72C2×C42—21no—
73C2×C42—41no—
74C2×C42C22⋊C411yescharacteristic
75C2×C42C22⋊C411yescharacteristic
76C2×C42C22×C411yescharacteristic
77#106—44no—
78C4⋊C8—82no—
79#103—44no—
80#103—44no—
81C22⋊C8—44no—
82#97—24no—
83#97—42no—
84C4×C8—21no—
85C4×C8—41no—
86C4×C8—82no—
87C4×C8C2×C812yes—
88#95—42no—
89#95—42no—
90C2×Q8—22no—
91C2×Q8—22no—
92C2×Q8—42no—
93C2×Q8—42no—
94C2×Q8—42no—
95C2×Q8—42no—
96C2×Q8—41no—
97C2×Q8C22⋊C814yes—
98C22×C4—21no—
99C22×C4—22no—
100C22×C4—21no—
101C22×C4—22no—
102C22×C4—41no—
103C22×C4—41no—
104C22×C4#9911yescharacteristic
105C22×C4#9911yescharacteristic
106C22×C4C2×C22⋊C411yescharacteristic
107Q16—416no—
108C2×C8—22no—
109C2×C8—22no—
110C2×C8—22no—
111C2×C8—22no—
112C2×C8—41no—
113C2×C8—41no—
114C2×C8—84no—
115C4⋊C4—42no—
116C4⋊C4—42no—
117C4⋊C4—42no—
118C4⋊C4—42no—
119C4⋊C4—42no—
120C4⋊C4—42no—
121C4⋊C4—82no—
122C4⋊C4—82no—
123C4⋊C4—82no—
124C4⋊C4—82no—
125C42—22no—
126C42—21no—
127C42—41no—
128C42—42no—
129C42—42no—
130C42—42no—
131C42—42no—
132C42—81no—
133C42—81no—
134C42C22×C811yescharacteristic
135C42C2×M1611yescharacteristic
136C23C2×#9911yescharacteristic
137Q8—44no—
138Q8—44no—
139Q8—82no—
140Q8—82no—
141C2×C4—22no—
142C2×C4—22no—
143C2×C4—21no—
144C2×C4—21no—
145C2×C4—21no—
146C2×C4—21no—
147C2×C4—21no—
148C2×C4—24no—
149C2×C4—41no—
150C2×C4—42no—
151C2×C4—42no—
152C2×C4—41no—
153C2×C4—41no—
154C2×C4—41no—
155C2×C4—42no—
156C2×C4—41no—
157C2×C4—42no—
158C2×C4—81no—
159C2×C4—81no—
160C2×C4—82no—
161C2×C4—82no—
162C2×C4—81no—
163C2×C4#35312yes—
164C2×C4C2×C22⋊C811yescharacteristic
165C2×C4#41311yescharacteristic
166C8—24no—
167C8—42no—
168C8—164no—
169C22—21no—
170C22—21no—
171C22#145911yescharacteristic
172C22C2×#35311yescharacteristic
173C22#213211yescharacteristic
174C4—22no—
175C4—22no—
176C4—22no—
177C4—21no—
178C4—21no—
179C4—41no—
180C4—44no—
181C4—81no—
182C4—82no—
183C4—82no—
184C4—161no—
185C2—21no—
186C2—21no—
187C2#1147511yescharacteristic
188C2#1358611yescharacteristic
189C2#1359011yescharacteristic
subgroupG/H
center, socleC22C2×#353
commutatorC42C22×C8
Frattini, cosocle#384C23
prime radicalC23C2×#99
LCSG ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C42 ◁ C4×C4⋊Q8 ◁ G
LECSG ▷ #384 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
23. #494C8semidirect

Maximal subgroups

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

subgroupindex
#216912
#114782
#73712
#73552