Finite Group Databasebeta

← #141000

Subgroups — #141000

Order 512.

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

normal subgroups only

This group has 413 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#33231C211yesmaximal, characteristic
2#12659C211yesmaximal, characteristic, complemented
3#12595C211yesmaximal, characteristic, complemented
4C2×#1355C214yesmaximal, complemented
5C2×#490C412yes—
6C2×#490C2212yes—
7#2867C411yescharacteristic
8#2859C411yescharacteristic
9C2×#364—21no—
10C2×#347C2212yes—
11C2×#347C2212yes—
12#1961—21no—
13#1961—21no—
14#1957—21no—
15#1957—21no—
16C4×#97C2211yescharacteristic
17#1377—21no—
18#1377—21no—
19#1376—21no—
20#1355—28no—
21C2×#127—21no—
22C2×#127—21no—
23C2×#127—42no—
24C2×#127C2×C412yes—
25C2×#125C2×C412yes—
26C2×M32—22no—
27C2×M32—22no—
28#490—24no—
29#490—24no—
30#490—44no—
31C2×#102—22no—
32C2×#102—22no—
33C2×#102—22no—
34C2×#102—22no—
35C2×#97C2×C411yescharacteristic
36C2×#97D411yescharacteristic
37C2×#97D411yescharacteristic
38C2×#97C2311yescharacteristic
39C2×C4×C8D411yescharacteristic
40C2×C4×C8D411yescharacteristic
41#389—42no—
42#384—21no—
43#382—21no—
44C43C2×C411yescharacteristic
45#364—24no—
46#347—24no—
47#347—22no—
48#347D414yescomplemented
49C22×D4—22no—
50C22×D4—42no—
51C22×D4—42no—
52C22×D4—42no—
53C22×C8C22⋊C411yescharacteristic
54C22×C8C22⋊C411yescharacteristic
55C22×C8C2×D411yescharacteristic
56C22×C8C2×D411yescharacteristic
57#127—24no—
58#127—42no—
59#127—42no—
60#127—42no—
61#127—42no—
62#127—42no—
63#127—88no—
64#125—24no—
65#125—42no—
66C2×C4⋊C4—22no—
67C2×C4⋊C4—42no—
68C2×C4⋊C4—42no—
69C2×C42—22no—
70C2×C42C22⋊C411yescharacteristic
71C2×C42C22⋊C411yescharacteristic
72C2×C42C22×C411yescharacteristic
73M32—24no—
74M32—24no—
75M32—42no—
76M32—42no—
77C2×C16—42no—
78C2×C16—42no—
79#102—44no—
80#102—44no—
81#102—44no—
82#102—42no—
83#102—42no—
84#97—22no—
85#97—41no—
86#97—41no—
87#97C2×D412yes—
88#97#4112yes—
89C4×C8—22no—
90C4×C8—41no—
91#95—42no—
92C24—42no—
93C2×D4—42no—
94C2×D4—42no—
95C2×D4—42no—
96C2×D4—42no—
97C2×D4—42no—
98C2×D4—42no—
99C2×D4—84no—
100C2×D4—84no—
101C2×D4—84no—
102C2×D4—88no—
103C2×D4—88no—
104C2×D4—88no—
105C22×C4—21no—
106C22×C4—21no—
107C22×C4—42no—
108C22×C4#10012yes—
109C22×C4C2×C22⋊C411yescharacteristic
110C2×C8—21no—
111C2×C8—21no—
112C2×C8—21no—
113C2×C8—21no—
114C2×C8—21no—
115C2×C8—21no—
116C2×C8—21no—
117C2×C8—21no—
118C2×C8—21no—
119C2×C8#10012yes—
120C2×C8#12112yes—
121C2×C8#12411yescharacteristic
122C2×C8#12711yescharacteristic
123C4⋊C4—42no—
124C4⋊C4—42no—
125C4⋊C4—84no—
126C4⋊C4—84no—
127C42—21no—
128C42—21no—
129C42—21no—
130C42—21no—
131C42—44no—
132C42#10412yes—
133C42C4×D412yes—
134C16—44no—
135C16—44no—
136C23—88no—
137C23—82no—
138C23—82no—
139C23—82no—
140C23C2×#10011yescharacteristic
141D4—82no—
142D4—82no—
143D4—82no—
144D4—82no—
145D4—168no—
146D4—168no—
147D4—168no—
148C2×C4—24no—
149C2×C4—22no—
150C2×C4—22no—
151C2×C4—22no—
152C2×C4—42no—
153C2×C4—42no—
154C2×C4—84no—
155C2×C4—82no—
156C2×C4#39011yescharacteristic
157C2×C4C2×#10011yescharacteristic
158C2×C4C2×#10411yescharacteristic
159C2×C4#42111yescharacteristic
160C2×C4#42111yescharacteristic
161C2×C4#42111yescharacteristic
162C8—22no—
163C8—21no—
164C8—21no—
165C8—22no—
166C8—41no—
167C22—22no—
168C22—164no—
169C22—164no—
170C22—164no—
171C22—162no—
172C22#152711yescharacteristic
173C22#153311yescharacteristic
174C22#196511yescharacteristic
175C4—24no—
176C4—42no—
177C4—42no—
178C4—162no—
179C4#153011yescharacteristic
180C4#153611yescharacteristic
181C4#196311yescharacteristic
182C4#198011yescharacteristic
183C2—21no—
184C2—21no—
185C2—322no—
186C2#968911yescharacteristic
187C2#1191512yes—
subgroupG/H
center, socleC22#1533
commutatorC2×C42C22×C4
Frattini, cosocleC2×#97C23
prime radical#2867C4
LCSG ▷ C2×C42 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4 ◁ C4×#97 ◁ G
LECSG ▷ C2×#97 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #12659C2semidirect
3. #12595C2semidirect
4. C2×#1355C2semidirect
48. #347D4semidirect

Maximal subgroups

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

subgroupindex
#332312
#126592
#125952
C2×#13552