Finite Group Databasebeta

← #149776

Subgroups — #149776

Order 512.

464 conjugacy classes of subgroups; only the 62 normal subgroups are shown.

normal subgroups only

This group has 464 conjugacy classes of subgroups — too many to draw. Only its 62 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#11513C212yesmaximal
3#7353C214yesmaximal, complemented
4C22×C4⋊Q8C411yescharacteristic
5C2×#419—22no—
6C2×C4×M16C411yescharacteristic
7#2285C2211yescharacteristic
8#1960C412yes—
9#1960C2212yes—
10#1956—22no—
11#1464—22no—
12#1325—22no—
13#1324C2214yes—
14C23×Q8—21no—
15C22×M16—21no—
16C2×C4⋊Q8—24no—
17C2×C4⋊Q8—21no—
18C2×C4⋊Q8—41no—
19C2×C4⋊Q8—42no—
20C2×C4⋊Q8C2×C412yes—
21C2×#117D412yes—
22C22×C4⋊C4—41no—
23C22×C42C2×C411yescharacteristic
24#432—42no—
25#419—44no—
26#419—44no—
27C2×#103—24no—
28C2×#103—24no—
29C4×M16—42no—
30C2×#97C2×C412yes—
31C2×C4×C8—21no—
32#384D412yes—
33#382C2×C411yescharacteristic
34#382C2311yescharacteristic
35#380—41no—
36C4×C4⋊C4—41no—
37#324—24no—
38#323—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—41no—
48C23×C4C22⋊C411yescharacteristic
49C2×M16—22no—
50C2×M16—22no—
51C2×M16—42no—
52C2×M16—42no—
53C22×C8—22no—
54C4⋊Q8—24no—
55C4⋊Q8—22no—
56C4⋊Q8—42no—
57C4⋊Q8—48no—
58C4⋊Q8—42no—
59C4⋊Q8M1614yes—
60#117—42no—
61#117—42no—
62C2×C4⋊C4—42no—
63C2×C4⋊C4—48no—
64C2×C4⋊C4—42no—
65C2×C4⋊C4—42no—
66C2×C4⋊C4C22⋊C412yes—
67C2×C4⋊C4C2×D412yes—
68C2×C22⋊C4—22no—
69C2×C42—21no—
70C2×C42—22no—
71C2×C42—22no—
72C2×C42—22no—
73C2×C42C22⋊C411yescharacteristic
74C2×C42C22×C411yescharacteristic
75C4⋊C8—82no—
76#103—48no—
77#103—48no—
78C22⋊C8—44no—
79#97—42no—
80#97—82no—
81C4×C8—42no—
82#95—41no—
83#95—41no—
84C24#9911yescharacteristic
85C2×Q8—24no—
86C2×Q8—44no—
87C2×Q8—44no—
88C2×Q8—48no—
89C2×Q8—48no—
90C2×Q8—44no—
91C2×Q8—42no—
92C2×Q8—84no—
93C2×Q8—84no—
94C2×Q8—82no—
95C2×Q8—81no—
96C2×Q8—81no—
97C22×C4—22no—
98C22×C4—21no—
99C22×C4—21no—
100C22×C4—21no—
101C22×C4—21no—
102C22×C4—21no—
103C22×C4—24no—
104C22×C4—21no—
105C22×C4—22no—
106C22×C4—22no—
107C22×C4—44no—
108C22×C4—44no—
109C22×C4—42no—
110C22×C4—82no—
111C22×C4—82no—
112C22×C4#9911yescharacteristic
113C22×C4C2×C22⋊C411yescharacteristic
114M16—44no—
115M16—44no—
116C2×C8—24no—
117C2×C8—42no—
118C2×C8—42no—
119C2×C8—84no—
120C4⋊C4—42no—
121C4⋊C4—42no—
122C4⋊C4—44no—
123C4⋊C4—48no—
124C4⋊C4—44no—
125C4⋊C4—84no—
126C22⋊C4—82no—
127C42—24no—
128C42—21no—
129C42—21no—
130C42—42no—
131C42—42no—
132C42#10012yes—
133C42C2×M1612yes—
134C23—21no—
135C23—21no—
136C23—42no—
137C23#35512yes—
138C23C2×#9911yescharacteristic
139Q8—48no—
140Q8—84no—
141Q8—84no—
142Q8—84no—
143C2×C4—24no—
144C2×C4—22no—
145C2×C4—22no—
146C2×C4—22no—
147C2×C4—22no—
148C2×C4—21no—
149C2×C4—21no—
150C2×C4—21no—
151C2×C4—21no—
152C2×C4—22no—
153C2×C4—22no—
154C2×C4—48no—
155C2×C4—42no—
156C2×C4—42no—
157C2×C4—42no—
158C2×C4—44no—
159C2×C4—88no—
160C2×C4—82no—
161C2×C4—82no—
162C2×C4—82no—
163C2×C4—82no—
164C2×C4#35311yescharacteristic
165C2×C4#35311yescharacteristic
166C2×C4#40711yescharacteristic
167C2×C4C2×#10011yescharacteristic
168C8—44no—
169C8—164no—
170C22—22no—
171C22—22no—
172C22—42no—
173C22—42no—
174C22—42no—
175C22#146211yescharacteristic
176C22C2×#35311yescharacteristic
177C22C2×#35511yescharacteristic
178C4—24no—
179C4—21no—
180C4—21no—
181C4—21no—
182C4—21no—
183C4—42no—
184C4—82no—
185C4—82no—
186C4—84no—
187C2—21no—
188C2—21no—
189C2—42no—
190C2#1150511yescharacteristic
191C2#1358012yes—
subgroupG/H
center, socleC22C2×#353
commutatorC2×C42C22×C4
Frattini, cosocle#382C23
prime radicalC24#99
LCSG ▷ C2×C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4 ◁ C23×C4 ◁ #2285 ◁ G
LECSG ▷ #382 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #7353C2semidirect

Maximal subgroups

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

subgroupindex
#212022
#115132
#73532