Finite Group Databasebeta

← #152258

Subgroups — #152258

Order 512.

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

normal subgroups only

This group has 319 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#22277C211yesmaximal, characteristic
2C2×#1411C212yesmaximal
3#12804C212yesmaximal
4#7505C212yesmaximal
5C2×#494C2211yescharacteristic
6#2327C411yescharacteristic
7#2327C2211yescharacteristic
8C2×#356C2214yes—
9#1960C411yescharacteristic
10#1960C2211yescharacteristic
11#1878—22no—
12#1878—22no—
13#1835C411yescharacteristic
14#1801—22no—
15#1746C411yescharacteristic
16#1411—24no—
17#1394—22no—
18C22×Q16—22no—
19C2×C4⋊Q8C2×C411yescharacteristic
20C2×C4⋊Q8C2311yescharacteristic
21C2×C4×Q8—21no—
22C2×C4×Q8D412yes—
23#494—22no—
24#494D412yes—
25#494Q812yes—
26#423—42no—
27C2×#103—22no—
28C2×#103—22no—
29C2×#101—24no—
30C2×#97C2×C411yescharacteristic
31C2×C4×C8C2×C411yescharacteristic
32#391—21no—
33#391D412yes—
34#387—42no—
35#387—42no—
36#382C2×C411yescharacteristic
37#382C2×C411yescharacteristic
38#376C2×C411yescharacteristic
39#356—24no—
40#356—24no—
41#343—42no—
42#340—22no—
43#336—21no—
44#336—21no—
45C22×Q8—21no—
46C22×Q8C2×D412yes—
47C2×Q16—24no—
48C2×Q16—24no—
49C2×Q16—82no—
50C2×Q16—82no—
51C2×M16—44no—
52C22×C8—21no—
53C22×C8—21no—
54C4⋊Q8—21no—
55C4⋊Q8—41no—
56C4⋊Q8—41no—
57C4⋊Q8C4⋊C412yes—
58C4⋊Q8C2×D411yescharacteristic
59C4⋊Q8C2×Q811yescharacteristic
60C4⋊Q8#4112yes—
61C4×Q8—82no—
62C4×Q8—82no—
63C2×C4⋊C4—21no—
64C2×C4⋊C4—22no—
65C2×C4⋊C4—41no—
66C2×C4⋊C4—41no—
67C2×C4⋊C4—41no—
68C2×C4⋊C4C22⋊C412yes—
69C2×C42—22no—
70C2×C42—41no—
71C2×C42C22⋊C411yescharacteristic
72C2×C42C22⋊C411yescharacteristic
73C2×C42C22×C411yescharacteristic
74C4⋊C8—84no—
75#103—82no—
76#103—82no—
77#101—44no—
78#101—44no—
79#97—41no—
80C4×C8—21no—
81C4×C8C4⋊C412yes—
82#95—22no—
83#95—41no—
84#95—42no—
85#95—41no—
86#95—42no—
87C2×Q8—22no—
88C2×Q8—22no—
89C2×Q8—42no—
90C2×Q8—42no—
91C2×Q8—42no—
92C2×Q8—42no—
93C2×Q8—41no—
94C2×Q8#12112yes—
95C2×Q8C22⋊Q812yes—
96C22×C4—21no—
97C22×C4—22no—
98C22×C4—21no—
99C22×C4—22no—
100C22×C4—41no—
101C22×C4—41no—
102C22×C4#9912yes—
103C22×C4C2×C22⋊C411yescharacteristic
104Q16—84no—
105Q16—84no—
106M16—84no—
107M16—84no—
108C2×C8—22no—
109C2×C8—41no—
110C2×C8—41no—
111C2×C8—41no—
112C2×C8—41no—
113C2×C8—41no—
114C2×C8—84no—
115C4⋊C4—41no—
116C4⋊C4—41no—
117C4⋊C4—42no—
118C4⋊C4—42no—
119C4⋊C4—42no—
120C4⋊C4—42no—
121C4⋊C4—42no—
122C4⋊C4—82no—
123C4⋊C4—82no—
124C42—21no—
125C42—41no—
126C42—41no—
127C42—44no—
128C42—82no—
129C42C2×C4⋊C411yescharacteristic
130C42#11711yescharacteristic
131C23C2×#9911yescharacteristic
132Q8—42no—
133Q8—42no—
134Q8—44no—
135Q8—82no—
136Q8—82no—
137C2×C4—21no—
138C2×C4—21no—
139C2×C4—21no—
140C2×C4—21no—
141C2×C4—22no—
142C2×C4—22no—
143C2×C4—22no—
144C2×C4—42no—
145C2×C4—42no—
146C2×C4—42no—
147C2×C4—42no—
148C2×C4—42no—
149C2×C4—42no—
150C2×C4—42no—
151C2×C4—41no—
152C2×C4—81no—
153C2×C4—81no—
154C2×C4—82no—
155C2×C4—82no—
156C2×C4—81no—
157C2×C4#38011yescharacteristic
158C2×C4#41011yescharacteristic
159C8—42no—
160C8—81no—
161C8—164no—
162C22—22no—
163C22#179911yescharacteristic
164C22#212711yescharacteristic
165C22#212811yescharacteristic
166C4—21no—
167C4—21no—
168C4—41no—
169C4—41no—
170C4—42no—
171C4—42no—
172C4—42no—
173C4—81no—
174C4—82no—
175C4—82no—
176C4—161no—
177C2—21no—
178C2—21no—
179C2#1279911yescharacteristic
180C2#1361212yes—
subgroupG/H
center, socleC22#2127
commutatorC2×C42C22×C4
Frattini, cosocleC2×C4⋊Q8C23
prime radicalC23C2×#99
LCSG ▷ C2×C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23 ◁ C2×C42 ◁ #2327 ◁ G
LECSG ▷ C2×C4⋊Q8 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1

Maximal subgroups

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

subgroupindex
#222772
C2×#14112
#128042
#75052