Finite Group Databasebeta

← #149809

Subgroups — #149809

Order 512.

422 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 422 conjugacy classes of subgroups — too many to draw. Only its 66 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
1C2×#1845C211yesmaximal, characteristic
2#11512C212yesmaximal
3#7370C214yesmaximal, complemented
4C22×#107—22no—
5C22×C4×C8C411yescharacteristic
6C2×#384C411yescharacteristic
7C2×#384C2211yescharacteristic
8#1845—22no—
9#1845C412yes—
10#1845C2212yes—
11#1463—22no—
12#1325—22no—
13#1325C2214yes—
14C23×C8—21no—
15C22×C4⋊C4—21no—
16C22×C4⋊C4—41no—
17C22×C4⋊C4—41no—
18C22×C4⋊C4D412yes—
19C22×C42C2×C411yescharacteristic
20C2×#107—24no—
21C2×#107—24no—
22C2×#107—44no—
23C2×#107—44no—
24#407—42no—
25C2×C4×C8—21no—
26C2×C4×C8—24no—
27C2×C4×C8C2×C412yes—
28#384—21no—
29#384—41no—
30#384—41no—
31#384C2×C412yes—
32#384C2×C411yescharacteristic
33#384D412yes—
34#384C2311yescharacteristic
35#324—24no—
36#324—42no—
37C23×C4—21no—
38C23×C4—22no—
39C23×C4—41no—
40C23×C4C22⋊C411yescharacteristic
41C2×M16—82no—
42C22×C8—24no—
43C22×C8—24no—
44C22×C8—22no—
45C22×C8—22no—
46C22×C8—22no—
47C2×C4⋊C4—22no—
48C2×C4⋊C4—22no—
49C2×C4⋊C4—42no—
50C2×C4⋊C4—42no—
51C2×C4⋊C4—42no—
52C2×C4⋊C4—42no—
53C2×C4⋊C4—41no—
54C2×C4⋊C4—42no—
55C2×C4⋊C4—41no—
56C2×C4⋊C4—42no—
57C2×C4⋊C4—42no—
58C2×C4⋊C4—41no—
59C2×C4⋊C4—84no—
60C2×C4⋊C4—84no—
61C2×C4⋊C4C22⋊C412yes—
62C2×C4⋊C4C2×D412yes—
63C2×C42—21no—
64C2×C42—22no—
65C2×C42—22no—
66C2×C42C22⋊C411yescharacteristic
67C2×C42C22×C411yescharacteristic
68#107—48no—
69#107—48no—
70C22⋊C8—44no—
71C22⋊C8—82no—
72C4×C8—24no—
73C4×C8—22no—
74C4×C8M1614yescomplemented
75C24#9911yescharacteristic
76C22×C4—24no—
77C22×C4—24no—
78C22×C4—24no—
79C22×C4—21no—
80C22×C4—21no—
81C22×C4—21no—
82C22×C4—21no—
83C22×C4—22no—
84C22×C4—21no—
85C22×C4—21no—
86C22×C4—22no—
87C22×C4—22no—
88C22×C4—42no—
89C22×C4—42no—
90C22×C4—42no—
91C22×C4—84no—
92C22×C4—84no—
93C22×C4—81no—
94C22×C4—81no—
95C22×C4#9911yescharacteristic
96C22×C4C2×C22⋊C411yescharacteristic
97M16—164no—
98C2×C8—24no—
99C2×C8—28no—
100C2×C8—28no—
101C2×C8—24no—
102C2×C8—24no—
103C2×C8—84no—
104C4⋊C4—42no—
105C4⋊C4—42no—
106C4⋊C4—44no—
107C4⋊C4—82no—
108C4⋊C4—82no—
109C4⋊C4—82no—
110C4⋊C4—84no—
111C4⋊C4—82no—
112C4⋊C4—84no—
113C4⋊C4—84no—
114C42—24no—
115C42—21no—
116C42—21no—
117C42#10112yes—
118C42C2×M1612yes—
119C23—21no—
120C23—22no—
121C23—22no—
122C23—21no—
123C23#35311yescharacteristic
124C23#35311yescharacteristic
125C23C2×#9911yescharacteristic
126C2×C4—22no—
127C2×C4—22no—
128C2×C4—22no—
129C2×C4—22no—
130C2×C4—216no—
131C2×C4—22no—
132C2×C4—22no—
133C2×C4—22no—
134C2×C4—22no—
135C2×C4—21no—
136C2×C4—21no—
137C2×C4—21no—
138C2×C4—21no—
139C2×C4—22no—
140C2×C4—22no—
141C2×C4—42no—
142C2×C4—42no—
143C2×C4—84no—
144C2×C4—82no—
145C2×C4—82no—
146C2×C4—84no—
147C2×C4—82no—
148C2×C4—162no—
149C2×C4—162no—
150C2×C4—162no—
151C2×C4#35612yes—
152C2×C4#40711yescharacteristic
153C2×C4C2×#10111yescharacteristic
154C8—28no—
155C8—164no—
156C22—24no—
157C22—24no—
158C22—21no—
159C22—21no—
160C22—21no—
161C22—21no—
162C22—21no—
163C22—21no—
164C22#140812yes—
165C22#141112yes—
166C22#146311yescharacteristic
167C22C2×#35311yescharacteristic
168C22C2×#35611yescharacteristic
169C4—24no—
170C4—24no—
171C4—21no—
172C4—22no—
173C4—21no—
174C4—82no—
175C4—82no—
176C4—162no—
177C2—21no—
178C2—21no—
179C2—22no—
180C2—22no—
181C2#1151211yescharacteristic
182C2C2×#140811yescharacteristic
183C2C2×#141111yescharacteristic
subgroupG/H
center, socleC22C2×#353
commutatorC2×C42C22×C4
Frattini, cosocle#384C23
prime radicalC24#99
LCSG ▷ C2×C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C24 ◁ C23×C4 ◁ C2×#384 ◁ G
LECSG ▷ #384 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #7370C2semidirect
74. C4×C8M16semidirect

Maximal subgroups

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

subgroupindex
C2×#18452
#115122
#73702