Finite Group Databasebeta

← #125033

Subgroups — #125033

Order 512.

295 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 295 conjugacy classes of subgroups — too many to draw. Only its 68 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#21635C211yesmaximal, characteristic
2#12897C212yesmaximal, complemented
3#7133C212yesmaximal, complemented
4#7105C212yesmaximal
5C2×C8×Q8C411yescharacteristic
6C2×C8×D4C411yescharacteristic
7#2398C411yescharacteristic
8#1985C2211yescharacteristic
9#1955C2211yescharacteristic
10#1924—22no—
11#1832C411yescharacteristic
12C8×C22⋊C4—22no—
13#1726C2211yescharacteristic
14#1377C2214yes—
15#1376—22no—
16#1375—22no—
17C23×C8—21no—
18C2×C4×Q8C2×C411yescharacteristic
19C2×C4×D4C2×C411yescharacteristic
20C2×M32—24no—
21C8×Q8—42no—
22C8×D4—42no—
23C2×C4⋊C8—21no—
24C2×C4⋊C8C2×C411yescharacteristic
25C2×C4⋊C8C2×C411yescharacteristic
26C2×C22⋊C8—21no—
27C2×C22⋊C8—22no—
28C2×#97D412yes—
29C2×#97C2311yescharacteristic
30C2×C4×C8—21no—
31C2×C4×C8C2×C411yescharacteristic
32C2×C4×C8D412yes—
33#400—21no—
34#397—21no—
35#394—21no—
36#393—21no—
37C4×C22⋊C4—22no—
38#376C2×C411yescharacteristic
39#336—22no—
40#336—22no—
41C22×Q8M1611yescharacteristic
42C22×D4M1611yescharacteristic
43C23×C4—21no—
44C22×C8—21no—
45C22×C8—21no—
46C22×C8—44no—
47C22×C8—42no—
48C22×C8C22⋊C411yescharacteristic
49C22×C8C22⋊C411yescharacteristic
50C22×C8C2×D412yes—
51C4×Q8—41no—
52C4×Q8—41no—
53C4×D4—41no—
54C4×D4—41no—
55C2×C4⋊C4—21no—
56C2×C4⋊C4M1611yescharacteristic
57C2×C4⋊C4M1611yescharacteristic
58C2×C22⋊C4—22no—
59C2×C22⋊C4—22no—
60C2×C22⋊C4—21no—
61C2×C42—21no—
62C2×C42C22⋊C412yes—
63C2×C42C22×C411yescharacteristic
64M32—44no—
65M32—44no—
66C2×C16—44no—
67C4⋊C8—41no—
68C4⋊C8—41no—
69C4⋊C8—42no—
70C22⋊C8—42no—
71C22⋊C8—44no—
72#97—24no—
73#97—41no—
74C4×C8—22no—
75C4×C8—42no—
76C4×C8—42no—
77#95—22no—
78#95—22no—
79#95—21no—
80#95—21no—
81C24—21no—
82C2×Q8—42no—
83C2×Q8—41no—
84C2×D4—41no—
85C2×D4—42no—
86C22×C4—21no—
87C22×C4—22no—
88C22×C4—21no—
89C22×C4—21no—
90C22×C4—41no—
91C22×C4—41no—
92C22×C4—42no—
93C22×C4—42no—
94C22×C4C2×C22⋊C411yescharacteristic
95C22×C4C2×M1611yescharacteristic
96C22×C4C2×M1611yescharacteristic
97C2×C8—22no—
98C2×C8—22no—
99C2×C8—22no—
100C2×C8—21no—
101C2×C8—22no—
102C2×C8—41no—
103C2×C8—41no—
104C2×C8—42no—
105C2×C8—42no—
106C2×C8—84no—
107C2×C8#10012yes—
108C2×C8#13514yes—
109C4⋊C4—41no—
110C4⋊C4—41no—
111C4⋊C4—42no—
112C22⋊C4—44no—
113C22⋊C4—42no—
114C22⋊C4—44no—
115C42—21no—
116C42—21no—
117C42—22no—
118C42—22no—
119C42—41no—
120C42—41no—
121C16—84no—
122C23—22no—
123C23—42no—
124C23—41no—
125C23—41no—
126C23—42no—
127C23#40711yescharacteristic
128Q8—42no—
129Q8—42no—
130D4—42no—
131D4—42no—
132C2×C4—22no—
133C2×C4—21no—
134C2×C4—21no—
135C2×C4—41no—
136C2×C4—41no—
137C2×C4—41no—
138C2×C4—41no—
139C2×C4—42no—
140C2×C4—42no—
141C2×C4—42no—
142C2×C4—41no—
143C2×C4—41no—
144C2×C4—42no—
145C2×C4—42no—
146C2×C4—81no—
147C2×C4—82no—
148C2×C4—81no—
149C2×C4#41011yescharacteristic
150C2×C4C2×#10011yescharacteristic
151C2×C4#41612yes—
152C2×C4#42112yes—
153C8—24no—
154C8—22no—
155C8—41no—
156C8—81no—
157C8—81no—
158C22—21no—
159C22—21no—
160C22—44no—
161C22—42no—
162C22—82no—
163C22—81no—
164C22—81no—
165C22#146611yescharacteristic
166C22#148812yes—
167C4—42no—
168C4—41no—
169C4—41no—
170C4—82no—
171C4—81no—
172C4—81no—
173C4#152112yes—
174C4#153311yescharacteristic
175C4#154311yescharacteristic
176C2—22no—
177C2—82no—
178C2#764911yescharacteristic
179C2#1153711yescharacteristic
180C2#1153711yescharacteristic
subgroupG/H
centerC2×C4#410
commutatorC2×C42C22×C4
Frattini, cosocleC2×#97C23
socleC22#1466
prime radicalC22×D4M16
LCSG ▷ C2×C42 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C22×C8 ◁ #1726 ◁ G
LECSG ▷ C2×#97 ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #12897C2semidirect
3. #7133C2semidirect

Maximal subgroups

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

subgroupindex
#216352
#128972
#71332
#71052