Finite Group Databasebeta

← #149983

Subgroups — #149983

Order 512.

266 conjugacy classes of subgroups; only the 58 normal subgroups are shown.

normal subgroups only

This group has 266 conjugacy classes of subgroups — too many to draw. Only its 58 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#28725C211yesmaximal, characteristic
2#11532C211yesmaximal, characteristic
3#11478C211yesmaximal, characteristic
4#7358C214yesmaximal
5C2×#491C412yes—
6#2859C411yescharacteristic
7C4×C4⋊Q8C2211yescharacteristic
8#2074—21no—
9#2074—21no—
10#1992C2212yes—
11#1803—21no—
12#1803—21no—
13#1767C411yescharacteristic
14#1467—21no—
15#1461—21no—
16#1325C2212yes—
17#1325C2212yes—
18#1319—21no—
19#1319—21no—
20C2×C4⋊Q8C2×C411yescharacteristic
21C2×#125C2×C412yes—
22C2×C4×Q8D411yescharacteristic
23C2×C4×Q8D411yescharacteristic
24#491—44no—
25#432—41no—
26#431—41no—
27C2×C4⋊C8—21no—
28C2×#103—22no—
29C2×#103—22no—
30C2×#97C2×C412yes—
31#391D411yescharacteristic
32#391D411yescharacteristic
33#389—41no—
34#389—41no—
35#389—41no—
36#384—21no—
37#384C2311yescharacteristic
38#382—21no—
39C4×C4⋊C4—41no—
40C43C2×C411yescharacteristic
41#340—22no—
42#340—22no—
43#336—21no—
44#324—22no—
45#324—22no—
46#324—41no—
47#324—41no—
48C22×Q8C22⋊C411yescharacteristic
49C22×Q8C22⋊C411yescharacteristic
50C22×C8—22no—
51C4⋊Q8—22no—
52C4⋊Q8—22no—
53#125—24no—
54#125—42no—
55C4×Q8—41no—
56C4×Q8—41no—
57C4×Q8—41no—
58C4×Q8—41no—
59C2×C4⋊C4—22no—
60C2×C4⋊C4—21no—
61C2×C4⋊C4—21no—
62C2×C4⋊C4—41no—
63C2×C4⋊C4—41no—
64C2×C4⋊C4—42no—
65C2×C4⋊C4—41no—
66C2×C4⋊C4—41no—
67C2×C4⋊C4C2×D411yescharacteristic
68C2×C4⋊C4C2×D411yescharacteristic
69C2×C42—21no—
70C2×C42—21no—
71C2×C42—21no—
72C2×C42—21no—
73C2×C42C22⋊C411yescharacteristic
74C2×C42C22⋊C411yescharacteristic
75C2×C42C22×C411yescharacteristic
76C4⋊C8—41no—
77C4⋊C8—41no—
78C4⋊C8—81no—
79C4⋊C8—81no—
80#103—44no—
81#103—44no—
82C22⋊C8—42no—
83C22⋊C8—42no—
84#97—42no—
85#97—81no—
86C4×C8—81no—
87#95—41no—
88#95—41no—
89#95—42no—
90C2×Q8—21no—
91C2×Q8—21no—
92C2×Q8—21no—
93C2×Q8—41no—
94C2×Q8—41no—
95C2×Q8—41no—
96C2×Q8—41no—
97C2×Q8#10012yes—
98C22×C4—21no—
99C22×C4—21no—
100C22×C4—21no—
101C22×C4—21no—
102C22×C4—21no—
103C22×C4—21no—
104C22×C4—42no—
105C22×C4#9911yescharacteristic
106C22×C4#9911yescharacteristic
107C22×C4C2×C22⋊C411yescharacteristic
108C2×C8—42no—
109C2×C8—42no—
110C2×C8—42no—
111C2×C8—82no—
112C2×C8—82no—
113C4⋊C4—44no—
114C4⋊C4—41no—
115C4⋊C4—41no—
116C4⋊C4—42no—
117C4⋊C4—42no—
118C4⋊C4—81no—
119C4⋊C4—81no—
120C4⋊C4—84no—
121C4⋊C4—82no—
122C4⋊C4—82no—
123C4⋊C4—82no—
124C42—21no—
125C42—21no—
126C42—21no—
127C42—41no—
128C42—42no—
129C42—41no—
130C42—41no—
131C42—41no—
132C42—41no—
133C42—81no—
134C42#13112yes—
135C23C2×#9911yescharacteristic
136Q8—42no—
137Q8—42no—
138Q8—81no—
139Q8—81no—
140C2×C4—21no—
141C2×C4—21no—
142C2×C4—22no—
143C2×C4—21no—
144C2×C4—22no—
145C2×C4—22no—
146C2×C4—41no—
147C2×C4—41no—
148C2×C4—41no—
149C2×C4—41no—
150C2×C4—41no—
151C2×C4—41no—
152C2×C4—41no—
153C2×C4—41no—
154C2×C4—41no—
155C2×C4—41no—
156C2×C4—41no—
157C2×C4—41no—
158C2×C4—41no—
159C2×C4—81no—
160C2×C4—81no—
161C2×C4—82no—
162C2×C4—82no—
163C2×C4—82no—
164C2×C4#35411yescharacteristic
165C2×C4#35411yescharacteristic
166C2×C4#40811yescharacteristic
167C2×C4C2×#10011yescharacteristic
168C8—82no—
169C8—162no—
170C8—162no—
171C22—21no—
172C22—21no—
173C22#147111yescharacteristic
174C22C2×#35411yescharacteristic
175C22#213511yescharacteristic
176C4—21no—
177C4—21no—
178C4—21no—
179C4—42no—
180C4—41no—
181C4—42no—
182C4—81no—
183C4—81no—
184C4—81no—
185C4—81no—
186C4—162no—
187C4#141312yes—
188C2—21no—
189C2—21no—
190C2#1156411yescharacteristic
191C2#1359611yescharacteristic
192C2C2×#141311yescharacteristic
subgroupG/H
center, socleC22C2×#354
commutatorC2×C42C22×C4
Frattini, cosocle#384C23
prime radicalC23C2×#99
LCSG ▷ C2×C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C42 ◁ C4×C4⋊Q8 ◁ G
LECSG ▷ #384 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1

Maximal subgroups

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

subgroupindex
#287252
#115322
#114782
#73582