Finite Group Databasebeta

← #149778

Subgroups — #149778

Order 512.

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

normal subgroups only

This group has 514 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×#1956C211yesmaximal, characteristic
2#11512C212yesmaximal
3#7355C214yesmaximal, complemented
4C22×C4⋊Q8C411yescharacteristic
5C22×#103—22no—
6C22×C4×C8C411yescharacteristic
7C2×#384C2211yescharacteristic
8#1956—22no—
9#1956C412yes—
10#1956C2212yes—
11#1463—22no—
12#1325—22no—
13#1325C2214yes—
14C23×Q8—21no—
15C23×C8—21no—
16C2×C4⋊Q8—24no—
17C2×C4⋊Q8—21no—
18C2×C4⋊Q8—41no—
19C2×C4⋊Q8—42no—
20C2×C4⋊Q8C2×C412yes—
21C22×C4⋊C4—41no—
22C22×C4⋊C4—41no—
23C22×C4⋊C4D412yes—
24C22×C42C2×C411yescharacteristic
25C2×#103—24no—
26C2×#103—24no—
27C2×#103—44no—
28C2×#103—44no—
29#407—42no—
30C2×C4×C8—21no—
31C2×C4×C8—42no—
32C2×C4×C8C2×C412yes—
33#384—41no—
34#384C2×C411yescharacteristic
35#384D412yes—
36#384C2311yescharacteristic
37#324—24no—
38#324—42no—
39C22×Q8—24no—
40C22×Q8—22no—
41C22×Q8—42no—
42C22×Q8—44no—
43C22×Q8—42no—
44C22×Q8—42no—
45C22×Q8—41no—
46C23×C4—21no—
47C23×C4—22no—
48C23×C4—41no—
49C23×C4C22⋊C411yescharacteristic
50C2×M16—82no—
51C22×C8—22no—
52C22×C8—22no—
53C22×C8—22no—
54C22×C8—42no—
55C22×C8—42no—
56C4⋊Q8—24no—
57C4⋊Q8—22no—
58C4⋊Q8—42no—
59C4⋊Q8—48no—
60C4⋊Q8—42no—
61C4⋊Q8M1614yescomplemented
62C2×C4⋊C4—22no—
63C2×C4⋊C4—42no—
64C2×C4⋊C4—48no—
65C2×C4⋊C4—42no—
66C2×C4⋊C4—42no—
67C2×C4⋊C4—42no—
68C2×C4⋊C4—41no—
69C2×C4⋊C4—42no—
70C2×C4⋊C4—41no—
71C2×C4⋊C4—84no—
72C2×C4⋊C4C22⋊C412yes—
73C2×C4⋊C4C2×D412yes—
74C2×C42—21no—
75C2×C42—22no—
76C2×C42—22no—
77C2×C42C22⋊C411yescharacteristic
78C2×C42C22×C411yescharacteristic
79#103—48no—
80#103—48no—
81C22⋊C8—44no—
82C22⋊C8—82no—
83C4×C8—42no—
84C4×C8—42no—
85C24#9911yescharacteristic
86C2×Q8—24no—
87C2×Q8—44no—
88C2×Q8—44no—
89C2×Q8—48no—
90C2×Q8—48no—
91C2×Q8—44no—
92C2×Q8—42no—
93C2×Q8—84no—
94C2×Q8—84no—
95C2×Q8—82no—
96C2×Q8—82no—
97C22×C4—24no—
98C22×C4—24no—
99C22×C4—24no—
100C22×C4—21no—
101C22×C4—21no—
102C22×C4—21no—
103C22×C4—21no—
104C22×C4—22no—
105C22×C4—21no—
106C22×C4—21no—
107C22×C4—22no—
108C22×C4—22no—
109C22×C4—44no—
110C22×C4—42no—
111C22×C4—42no—
112C22×C4—42no—
113C22×C4—82no—
114C22×C4—81no—
115C22×C4—81no—
116C22×C4—84no—
117C22×C4#9911yescharacteristic
118C22×C4C2×C22⋊C411yescharacteristic
119M16—164no—
120C2×C8—24no—
121C2×C8—44no—
122C2×C8—44no—
123C2×C8—42no—
124C2×C8—42no—
125C2×C8—84no—
126C4⋊C4—42no—
127C4⋊C4—42no—
128C4⋊C4—44no—
129C4⋊C4—48no—
130C4⋊C4—44no—
131C4⋊C4—82no—
132C4⋊C4—82no—
133C4⋊C4—84no—
134C4⋊C4—82no—
135C42—24no—
136C42—21no—
137C42—21no—
138C42#10012yes—
139C42C2×M1612yes—
140C23—21no—
141C23—22no—
142C23—22no—
143C23—21no—
144C23#35311yescharacteristic
145C23#35311yescharacteristic
146C23C2×#9911yescharacteristic
147Q8—48no—
148Q8—84no—
149Q8—88no—
150C2×C4—22no—
151C2×C4—22no—
152C2×C4—22no—
153C2×C4—22no—
154C2×C4—216no—
155C2×C4—22no—
156C2×C4—22no—
157C2×C4—22no—
158C2×C4—22no—
159C2×C4—21no—
160C2×C4—21no—
161C2×C4—21no—
162C2×C4—21no—
163C2×C4—22no—
164C2×C4—22no—
165C2×C4—42no—
166C2×C4—42no—
167C2×C4—44no—
168C2×C4—84no—
169C2×C4—84no—
170C2×C4—82no—
171C2×C4—82no—
172C2×C4—84no—
173C2×C4—82no—
174C2×C4—82no—
175C2×C4—162no—
176C2×C4#35512yes—
177C2×C4#40711yescharacteristic
178C2×C4C2×#10011yescharacteristic
179C8—44no—
180C8—164no—
181C22—24no—
182C22—24no—
183C22—21no—
184C22—21no—
185C22—21no—
186C22—21no—
187C22—21no—
188C22—21no—
189C22#140912yes—
190C22#141112yes—
191C22#146211yescharacteristic
192C22C2×#35311yescharacteristic
193C22C2×#35511yescharacteristic
194C4—24no—
195C4—24no—
196C4—21no—
197C4—22no—
198C4—21no—
199C4—82no—
200C4—82no—
201C4—84no—
202C2—21no—
203C2—21no—
204C2—22no—
205C2—22no—
206C2#1150511yescharacteristic
207C2C2×#140911yescharacteristic
208C2C2×#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. #7355C2semidirect
61. C4⋊Q8M16semidirect

Maximal subgroups

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

subgroupindex
C2×#19562
#115122
#73552