Finite Group Databasebeta

← #10592741

Subgroups — #10592741

Order 576.

360 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 360 conjugacy classes of subgroups — too many to draw. Only its 97 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
1C32×#121C211yesmaximal, characteristic
2C2×#3754C211yesmaximal, characteristic, complemented
3#64133C211yesmaximal, characteristic, complemented
4C2×#3748C211yesmaximal, characteristic, complemented
5#63659C211yesmaximal, characteristic, complemented
6#63637C211yesmaximal, characteristic, complemented
7#63634C211yesmaximal, characteristic, complemented
8#5042—34nomaximal
9C2×C32×D4—21no—
10C2×C32×D4C2211yescharacteristic
11C22×C32×C4C2211yescharacteristic
12C2×#634C2211yescharacteristic
13C32×C4⋊C4C2211yescharacteristic
14C32×C22⋊C4—21no—
15#3757—21no—
16#3754—22no—
17C4⋊(C3⋊Dic3)C2211yescharacteristic
18#3750—21no—
19#3748—22no—
20C2×C3⋊(C3⋊C8)C2211yescharacteristic
21C2×C3⋊(C3⋊C8)C2211yescharacteristic
22C3×#121D314yes—
23C2×#214—34no—
24#924—34no—
25C2×#208—34no—
26#832—34no—
27#810—34no—
28#807—34no—
29C23×C32—21no—
30C23×C32D411yescharacteristic
31C32×D4—21no—
32C32×D4—21no—
33C32×D4—42no—
34C2×C32×C4—21no—
35C2×C32×C4—21no—
36C2×C32×C4—21no—
37C2×C32×C4D411yescharacteristic
38C2×C32×C4C2311yescharacteristic
39C2×C3⋊Dic3—21no—
40C2×C3⋊Dic3—21no—
41#634—21no—
42#634—21no—
43C3⋊(C3⋊C8)—21no—
44C3⋊(C3⋊C8)D412yescomplemented
45#468—91noSylow
46C2×C3×D4—24no—
47C2×C3×D4C2×D314yes—
48C22×C3×C4C2×D314yes—
49C2×Dic6—34no—
50C3×C4⋊C4C2×D314yes—
51C3×C22⋊C4—24no—
52#217—64no—
53#214—68no—
54C4⋊Dic3—34no—
55Dic3⋊C4—64no—
56#208—68no—
57C2×C3⋊C8—34no—
58C2×C3⋊C8—34no—
59C22×C32—21no—
60C22×C32—21no—
61C22×C32—21no—
62C22×C32—41no—
63C22×C32—41no—
64C22×C32—41no—
65C22×C32C2×D411yescharacteristic
66C32×C4—21no—
67C32×C4—41no—
68C32×C4C2×D411yescharacteristic
69C32×C4#4111yescharacteristic
70C3⋊Dic3—41no—
71C3⋊Dic3—41no—
72C2×SD16—91no—
73C2×M16—91no—
74C22⋊Q8—91no—
75#121C3⋊D311yescharacteristic
76#107—91no—
77#103—91no—
78#102—91no—
79C23×C3—24no—
80C23×C3#6714yes—
81C3×D4—24no—
82C3×D4—24no—
83C3×D4—48no—
84C2×C3×C4—24no—
85C2×C3×C4—24no—
86C2×C3×C4—24no—
87C2×C3×C4#6714yes—
88C2×C3×C4C22×D314yes—
89C2×Dic3—64no—
90C2×Dic3—64no—
91Dic6—64no—
92Dic6—64no—
93C3⋊C8—38no—
94C3⋊C8—64no—
95C2×C32—41no—
96C2×C32—81no—
97C2×C32#12111yescharacteristic
98C2×C32AGL(1,ℤ/8)11yescharacteristic
99C2×C32#13711yescharacteristic
100C2×Q8—91no—
101C2×D4—21no—
102C2×D4C2×C3⋊D311yescharacteristic
103C22×C4C2×C3⋊D311yescharacteristic
104SD16—182no—
105M16—182no—
106C2×C8—91no—
107C2×C8—91no—
108C4⋊C4—91no—
109C4⋊C4—181no—
110C4⋊C4C2×C3⋊D311yescharacteristic
111C22⋊C4—21no—
112C22⋊C4—181no—
113C22×C3—24no—
114C22×C3—24no—
115C22×C3—24no—
116C22×C3—44no—
117C22×C3—44no—
118C22×C3—44no—
119C22×C3C2×#6714yes—
120C3×C4—24no—
121C3×C4—44no—
122C3×C4D3×D414yes—
123C3×C4#23714yes—
124Dic3—124no—
125Dic3—124no—
126C32#46811yescharacteristic, complemented, Sylow
127C23—21no—
128C23#63811yescharacteristic
129Q8—181no—
130Q8—181no—
131D4—21no—
132D4—21no—
133D4—42no—
134C2×C4—21no—
135C2×C4—21no—
136C2×C4—21no—
137C2×C4—181no—
138C2×C4—181no—
139C2×C4#63811yescharacteristic
140C2×C4C22×C3⋊D311yescharacteristic
141C8—92no—
142C8—181no—
143C2×C3—44no—
144C2×C3—84no—
145C2×C3#93214yes—
146C2×C3#93914yes—
147C2×C3#95114yes—
148C22—21no—
149C22—21no—
150C22—21no—
151C22—41no—
152C22—41no—
153C22—41no—
154C22C2×#63811yescharacteristic
155C4—21no—
156C4—41no—
157C4—361no—
158C4—361no—
159C4D4×C3⋊D311yescharacteristic
160C4#383011yescharacteristic
161C3#504214yescomplemented
162C2—41no—
163C2—81no—
164C2#6414111yescharacteristic
165C2#6414811yescharacteristic
166C2#6416011yescharacteristic
subgroupG/H
centerC22C2×#638
commutator, cosocleC2×C32×C4C23
FrattiniC2×C4C22×C3⋊D3
Fitting, prime radicalC32×#121C2
socleC22×C32C2×D4
hypercenter#121C3⋊D3
hypocenterC32#468
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C22×C4 ◁ #121

Complements

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

NH ≅ G/Ntype
2. C2×#3754C2semidirect
3. #64133C2semidirect
4. C2×#3748C2semidirect
5. #63659C2semidirect
6. #63637C2semidirect
7. #63634C2semidirect
44. C3⋊(C3⋊C8)D4semidirect
126. C32#468semidirect
161. C3#5042semidirect

Maximal subgroups

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

subgroupindex
C32×#1212
C2×#37542
#641332
C2×#37482
#636592
#636372
#636342
#50423