Finite Group Databasebeta

← #141001

Subgroups — #141001

Order 512.

335 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 335 conjugacy classes of subgroups — too many to draw. Only its 76 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#33232C211yesmaximal, characteristic
2#12659C211yesmaximal, characteristic
3#12595C211yesmaximal, characteristic
4C2×#1356C214yesmaximal
5C2×#491C412yes—
6C2×#491C2212yes—
7#2868C411yescharacteristic
8#2859C411yescharacteristic
9C2×#364—21no—
10C2×#347C2212yes—
11C2×#347C2212yes—
12#1960—21no—
13#1960—21no—
14#1956—21no—
15#1956—21no—
16C4×#97C2211yescharacteristic
17#1377—21no—
18#1377—21no—
19#1376—21no—
20#1356—28no—
21C2×C4⋊Q8—21no—
22C2×C4⋊Q8—21no—
23C2×C4⋊Q8C2×C412yes—
24C2×#125C2×C412yes—
25C2×M32—22no—
26C2×M32—22no—
27#491—24no—
28#491—24no—
29#491—44no—
30C2×#103—22no—
31C2×#103—22no—
32C2×#103—22no—
33C2×#103—22no—
34C2×#97C2×C411yescharacteristic
35C2×#97D411yescharacteristic
36C2×#97D411yescharacteristic
37C2×#97C2311yescharacteristic
38C2×C4×C8D411yescharacteristic
39C2×C4×C8D411yescharacteristic
40#389—42no—
41#384—21no—
42#384—42no—
43#382—21no—
44C43C2×C411yescharacteristic
45#364—24no—
46#347—24no—
47#347—22no—
48#347D414yes—
49C22×Q8—22no—
50C22×C8C22⋊C411yescharacteristic
51C22×C8C22⋊C411yescharacteristic
52C22×C8C2×D411yescharacteristic
53C22×C8C2×D411yescharacteristic
54C4⋊Q8—24no—
55C4⋊Q8—42no—
56C4⋊Q8—42no—
57C4⋊Q8—42no—
58C4⋊Q8—42no—
59C4⋊Q8—42no—
60#125—24no—
61#125—42no—
62C2×C4⋊C4—22no—
63C2×C4⋊C4—42no—
64C2×C4⋊C4—42no—
65C2×C4⋊C4—42no—
66C2×C4⋊C4—42no—
67C2×C4⋊C4—42no—
68C2×C42—22no—
69C2×C42C22⋊C411yescharacteristic
70C2×C42C22⋊C411yescharacteristic
71C2×C42C22×C411yescharacteristic
72M32—24no—
73M32—24no—
74M32—42no—
75M32—42no—
76C2×C16—42no—
77C2×C16—42no—
78#103—44no—
79#103—44no—
80#103—44no—
81#103—42no—
82#103—42no—
83#97—22no—
84#97—41no—
85#97—41no—
86#97C2×D412yes—
87#97#4112yes—
88C4×C8—22no—
89C4×C8—41no—
90#95—42no—
91C2×Q8—42no—
92C2×Q8—42no—
93C2×Q8—42no—
94C2×Q8—42no—
95C2×Q8—42no—
96C2×Q8—42no—
97C22×C4—21no—
98C22×C4—21no—
99C22×C4—42no—
100C22×C4—42no—
101C22×C4#10012yes—
102C22×C4C2×C22⋊C411yescharacteristic
103C2×C8—21no—
104C2×C8—21no—
105C2×C8—21no—
106C2×C8—21no—
107C2×C8—21no—
108C2×C8—21no—
109C2×C8—21no—
110C2×C8—21no—
111C2×C8—21no—
112C2×C8#10012yes—
113C2×C8#12112yes—
114C2×C8#12411yescharacteristic
115C2×C8#12711yescharacteristic
116C4⋊C4—42no—
117C4⋊C4—42no—
118C4⋊C4—84no—
119C4⋊C4—84no—
120C4⋊C4—84no—
121C4⋊C4—84no—
122C4⋊C4—84no—
123C42—21no—
124C42—21no—
125C42—21no—
126C42—21no—
127C42—44no—
128C42#10412yes—
129C42C4×D412yes—
130C16—44no—
131C16—44no—
132C23C2×#10011yescharacteristic
133Q8—82no—
134Q8—82no—
135Q8—82no—
136Q8—82no—
137C2×C4—24no—
138C2×C4—22no—
139C2×C4—22no—
140C2×C4—22no—
141C2×C4—42no—
142C2×C4—42no—
143C2×C4—84no—
144C2×C4—82no—
145C2×C4—82no—
146C2×C4—82no—
147C2×C4—82no—
148C2×C4#39011yescharacteristic
149C2×C4C2×#10011yescharacteristic
150C2×C4C2×#10411yescharacteristic
151C2×C4#42111yescharacteristic
152C2×C4#42111yescharacteristic
153C2×C4#42111yescharacteristic
154C8—22no—
155C8—21no—
156C8—21no—
157C8—22no—
158C8—41no—
159C22—22no—
160C22#152711yescharacteristic
161C22#153311yescharacteristic
162C22#196511yescharacteristic
163C4—24no—
164C4—42no—
165C4—42no—
166C4—162no—
167C4—162no—
168C4#153011yescharacteristic
169C4#153611yescharacteristic
170C4#196311yescharacteristic
171C4#198011yescharacteristic
172C2—21no—
173C2—21no—
174C2#968911yescharacteristic
175C2#1191612yes—
subgroupG/H
center, socleC22#1533
commutatorC2×C42C22×C4
Frattini, cosocleC2×#97C23
prime radicalC23C2×#100
LCSG ▷ C2×C42 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4 ◁ C4×#97 ◁ G
LECSG ▷ C2×#97 ▷ C2×C4 ▷ C2 ▷ C1

Maximal subgroups

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

subgroupindex
#332322
#126592
#125952
C2×#13562