Finite Group Databasebeta

← #149759

Subgroups — #149759

Order 512.

405 conjugacy classes of subgroups; only the 56 normal subgroups are shown.

normal subgroups only

This group has 405 conjugacy classes of subgroups — too many to draw. Only its 56 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#20729C211yesmaximal, characteristic
2#11522C212yesmaximal, complemented
3#7351C212yesmaximal, complemented
4#7342C212yesmaximal
5C2×#518C411yescharacteristic
6#2921C411yescharacteristic
7#2921C2211yescharacteristic
8#2165—22no—
9#1822C411yescharacteristic
10#1822C2211yescharacteristic
11#1800—22no—
12#1783C411yescharacteristic
13#1783C2211yescharacteristic
14C4×#100—22no—
15#1327—22no—
16#1320C2214yes—
17C23×D4C2×C411yescharacteristic
18#518—42no—
19#518—42no—
20C2×C4×D4—22no—
21C2×C4×D4—21no—
22C2×C4×D4—21no—
23C2×#117C2×C411yescharacteristic
24C2×#117C2×C411yescharacteristic
25C22×C22⋊C4—22no—
26#433—42no—
27#433—42no—
28#423—21no—
29#423D412yes—
30C2×#100—22no—
31C2×#100—22no—
32#407C2×C411yescharacteristic
33#407C2×C411yescharacteristic
34#407C2×C411yescharacteristic
35#407C2311yescharacteristic
36C4×M16—21no—
37C4×M16D412yes—
38#364—42no—
39#349—24no—
40#347—42no—
41#343—22no—
42#343—22no—
43C25—21no—
44C22×D4—21no—
45C22×D4—41no—
46C22×D4—41no—
47C22×D4—44no—
48C22×D4M1612yes—
49C23×C4—22no—
50C23×C4—22no—
51C23×C4C22×C411yescharacteristic
52C2×M16—21no—
53C2×M16—21no—
54C2×M16C22⋊C412yes—
55C2×M16C2×D412yes—
56C4×D4—42no—
57C4×D4—42no—
58C4×D4—48no—
59C4×D4—42no—
60C4×D4—42no—
61#117—22no—
62#117—22no—
63#117—41no—
64#117—41no—
65C2×C4⋊C4—21no—
66C2×C4⋊C4—21no—
67C2×C22⋊C4—21no—
68C2×C22⋊C4—22no—
69C2×C22⋊C4—22no—
70C2×C22⋊C4—48no—
71C2×C22⋊C4M1612yes—
72C2×C42—21no—
73C2×C42C22⋊C411yescharacteristic
74C2×C42C22⋊C411yescharacteristic
75M32—44no—
76C4⋊C8—42no—
77C4⋊C8—42no—
78#100—44no—
79#100—44no—
80C22⋊C8—41no—
81C22⋊C8—41no—
82C22⋊C8—41no—
83C22⋊C8—41no—
84#97—22no—
85#97—41no—
86C4×C8—22no—
87C4×C8—41no—
88C24—22no—
89C24—42no—
90C24—44no—
91C24—162no—
92C24C2×M1611yescharacteristic
93C2×D4—24no—
94C2×D4—42no—
95C2×D4—42no—
96C2×D4—48no—
97C2×D4—44no—
98C2×D4—44no—
99C2×D4—44no—
100C2×D4—41no—
101C2×D4—41no—
102C22×C4—21no—
103C22×C4—21no—
104C22×C4—21no—
105C22×C4—21no—
106C22×C4—22no—
107C22×C4—42no—
108C22×C4—44no—
109C22×C4—44no—
110C22×C4—162no—
111C22×C4—162no—
112C22×C4C2×C22⋊C411yescharacteristic
113C22×C4C2×M1611yescharacteristic
114M16—42no—
115M16—42no—
116M16—42no—
117M16—42no—
118C2×C8—22no—
119C2×C8—22no—
120C2×C8—41no—
121C2×C8—41no—
122C4⋊C4—24no—
123C4⋊C4—41no—
124C4⋊C4—41no—
125C22⋊C4—24no—
126C22⋊C4—44no—
127C22⋊C4—44no—
128C22⋊C4—42no—
129C42—21no—
130C42—21no—
131C42—42no—
132C42#9912yes—
133C42#10012yes—
134C16—84no—
135C23—24no—
136C23—21no—
137C23—42no—
138C23—41no—
139C23—48no—
140C23—42no—
141C23—81no—
142C23—82no—
143C23—82no—
144C23—164no—
145C23—168no—
146C23#40711yescharacteristic
147D4—44no—
148D4—44no—
149D4—44no—
150D4—44no—
151C2×C4—22no—
152C2×C4—22no—
153C2×C4—22no—
154C2×C4—21no—
155C2×C4—42no—
156C2×C4—42no—
157C2×C4—41no—
158C2×C4—42no—
159C2×C4—81no—
160C2×C4—81no—
161C2×C4—82no—
162C2×C4—164no—
163C2×C4—164no—
164C2×C4C2×#9911yescharacteristic
165C2×C4C2×#10011yescharacteristic
166C8—42no—
167C8—42no—
168C8—81no—
169C8—81no—
170C22—21no—
171C22—41no—
172C22—42no—
173C22—41no—
174C22—44no—
175C22—82no—
176C22—82no—
177C22—84no—
178C22—82no—
179C22—84no—
180C22—168no—
181C22#146211yescharacteristic
182C4—21no—
183C4—42no—
184C4—42no—
185C4—41no—
186C4—82no—
187C4#212012yes—
188C2—21no—
189C2—41no—
190C2—81no—
191C2—84no—
192C2#1150111yescharacteristic
subgroupG/H
center, socleC2#11501
commutatorC23×C4C22×C4
Frattini, cosocle#407C23
prime radicalC23×D4C2×C4
LCSG ▷ C23×C4 ▷ C23 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C42 ◁ #2921 ◁ G
LECSG ▷ #407 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #11522C2semidirect
3. #7351C2semidirect

Maximal subgroups

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

subgroupindex
#207292
#115222
#73512
#73422