Finite Group Databasebeta

← #149576

Subgroups — #149576

Order 512.

474 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 474 conjugacy classes of subgroups — too many to draw. Only its 82 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#21202C211yesmaximal, characteristic
2#11447C212yesmaximal
3#7355C214yesmaximal, complemented
4C22×C4⋊Q8C411yescharacteristic
5C2×#419—22no—
6C2×C4×M16C411yescharacteristic
7#2285C2211yescharacteristic
8#1960—22no—
9#1956C412yes—
10#1956C2212yes—
11#1458—22no—
12#1325C2214yes—
13#1324—22no—
14C23×Q8—21no—
15C22×M16—21no—
16C2×C4⋊Q8—21no—
17C2×C4⋊Q8—22no—
18C2×C4⋊Q8—41no—
19C2×C4⋊Q8—42no—
20C2×C4⋊Q8C814yescomplemented
21C2×C4⋊Q8C2×C412yes—
22C2×#117D412yes—
23C22×C4⋊C4—41no—
24C22×C42C2×C411yescharacteristic
25#431—42no—
26#419—44no—
27#419—44no—
28C2×#103—24no—
29C2×#103—24no—
30C4×M16—42no—
31C2×#97—21no—
32C2×C4×C8C2×C412yes—
33#384C2×C411yescharacteristic
34#384C2311yescharacteristic
35#382D412yes—
36#380—41no—
37C4×C4⋊C4—41no—
38#324—24no—
39#323—42no—
40C22×Q8—24no—
41C22×Q8—22no—
42C22×Q8—44no—
43C22×Q8—42no—
44C22×Q8—42no—
45C22×Q8—41no—
46C22×Q8—42no—
47C23×C4—21no—
48C23×C4—41no—
49C23×C4C22⋊C411yescharacteristic
50C2×M16—24no—
51C2×M16—42no—
52C2×M16—42no—
53C22×C8—22no—
54C4⋊Q8—24no—
55C4⋊Q8—22no—
56C4⋊Q8—44no—
57C4⋊Q8—42no—
58C4⋊Q8—44no—
59C4⋊Q8—42no—
60C4⋊Q8C2×C814yes—
61#117—42no—
62#117—42no—
63C2×C4⋊C4—44no—
64C2×C4⋊C4—42no—
65C2×C4⋊C4—44no—
66C2×C4⋊C4—42no—
67C2×C4⋊C4—41no—
68C2×C4⋊C4—41no—
69C2×C4⋊C4C22⋊C412yes—
70C2×C4⋊C4C2×D412yes—
71C2×C22⋊C4—22no—
72C2×C42—21no—
73C2×C42—22no—
74C2×C42—22no—
75C2×C42C22⋊C411yescharacteristic
76C2×C42C2×C812yes—
77C2×C42M1612yes—
78C2×C42C22×C411yescharacteristic
79C4⋊C8—82no—
80#103—48no—
81#103—48no—
82C22⋊C8—44no—
83#97—42no—
84C4×C8—42no—
85C4×C8—82no—
86#95—42no—
87C24#9911yescharacteristic
88C2×Q8—24no—
89C2×Q8—44no—
90C2×Q8—44no—
91C2×Q8—48no—
92C2×Q8—48no—
93C2×Q8—44no—
94C2×Q8—42no—
95C2×Q8—84no—
96C2×Q8—84no—
97C2×Q8—82no—
98C2×Q8—82no—
99C22×C4—21no—
100C22×C4—22no—
101C22×C4—22no—
102C22×C4—22no—
103C22×C4—22no—
104C22×C4—21no—
105C22×C4—22no—
106C22×C4—22no—
107C22×C4—42no—
108C22×C4—42no—
109C22×C4—44no—
110C22×C4—42no—
111C22×C4—81no—
112C22×C4—81no—
113C22×C4—82no—
114C22×C4C22⋊C812yes—
115C22×C4#9911yescharacteristic
116C22×C4#10012yes—
117C22×C4C2×C22⋊C411yescharacteristic
118M16—48no—
119C2×C8—24no—
120C2×C8—42no—
121C2×C8—42no—
122C2×C8—84no—
123C4⋊C4—42no—
124C4⋊C4—42no—
125C4⋊C4—44no—
126C4⋊C4—44no—
127C4⋊C4—48no—
128C4⋊C4—82no—
129C4⋊C4—82no—
130C22⋊C4—82no—
131C42—22no—
132C42—22no—
133C42—21no—
134C42—21no—
135C42—42no—
136C42—42no—
137C42C22⋊C812yes—
138C42C22×C811yescharacteristic
139C42C2×M1611yescharacteristic
140C23—21no—
141C23—21no—
142C23—41no—
143C23—41no—
144C23#32312yes—
145C23C2×#9911yescharacteristic
146Q8—48no—
147Q8—84no—
148Q8—88no—
149C2×C4—22no—
150C2×C4—22no—
151C2×C4—21no—
152C2×C4—21no—
153C2×C4—21no—
154C2×C4—21no—
155C2×C4—24no—
156C2×C4—24no—
157C2×C4—22no—
158C2×C4—44no—
159C2×C4—44no—
160C2×C4—41no—
161C2×C4—41no—
162C2×C4—42no—
163C2×C4—42no—
164C2×C4—44no—
165C2×C4—84no—
166C2×C4—84no—
167C2×C4—82no—
168C2×C4—82no—
169C2×C4—82no—
170C2×C4—82no—
171C2×C4#32312yes—
172C2×C4#35312yes—
173C2×C4#35512yes—
174C2×C4C2×C22⋊C811yescharacteristic
175C2×C4C2×#10011yescharacteristic
176C8—44no—
177C8—164no—
178C22—21no—
179C22—21no—
180C22—22no—
181C22—42no—
182C22—42no—
183C22—41no—
184C22—41no—
185C22C2×#32311yescharacteristic
186C22C2×#35311yescharacteristic
187C22C2×#35511yescharacteristic
188C4—22no—
189C4—22no—
190C4—21no—
191C4—21no—
192C4—42no—
193C4—82no—
194C4—82no—
195C4—84no—
196C4#131814yes—
197C2—21no—
198C2—21no—
199C2—42no—
200C2C2×#131811yescharacteristic
201C2#1358611yescharacteristic
202C2#1359211yescharacteristic
subgroupG/H
center, socleC22C2×#353
commutatorC42C22×C8
Frattini, cosocle#384C23
prime radicalC24#99
LCSG ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4 ◁ C23×C4 ◁ #2285 ◁ G
LECSG ▷ #384 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #7355C2semidirect
20. C2×C4⋊Q8C8semidirect

Maximal subgroups

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

subgroupindex
#212022
#114472
#73552