Finite Group Databasebeta

← #11723444

Subgroups — #11723444

Order 864.

440 conjugacy classes of subgroups; only the 116 normal subgroups are shown.

normal subgroups only

This group has 440 conjugacy classes of subgroups — too many to draw. Only its 116 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#88857C211yesmaximal, characteristic
2C3×#3750C211yesmaximal, characteristic
3C32×C4×Dic3C211yesmaximal, characteristic
4#88807C211yesmaximal, characteristic
5#88806C211yesmaximal, characteristic
6Dic3⋊(C3⋊Dic3)C211yesmaximal, characteristic
7Dic3⋊(C3⋊Dic3)C211yesmaximal, characteristic
8#64079—31nomaximal
9#63867—34nomaximal
10C2×C33×C4C2211yescharacteristic
11C2×C3⋊(C3⋊Dic3)C2211yescharacteristic
12C2×C3⋊(C3⋊Dic3)C2211yescharacteristic
13C2×C3×C3⋊Dic3C2211yescharacteristic
14C2×C3×C3⋊Dic3C2211yescharacteristic
15C2×C32×Dic3C2211yescharacteristic
16C2×C32×Dic3C2211yescharacteristic
17C32×C42—31no—
18C4⋊(C3⋊Dic3)—31no—
19C4⋊(C3⋊Dic3)—31no—
20#3750—31no—
21#3750—34no—
22#3750—31no—
23#3750—31no—
24#3750—64no—
25#3750D311yescharacteristic
26C3×Dic3⋊C4—34no—
27C3×C4×Dic3D314yes—
28#3724—34no—
29Dic3⋊Dic3—34no—
30Dic3⋊Dic3—34no—
31Dic3⋊Dic3—34no—
32C22×C33C2311yescharacteristic
33C33×C4—21no—
34C3⋊(C3⋊Dic3)—21no—
35C3⋊(C3⋊Dic3)—21no—
36C3×C3⋊Dic3—21no—
37C3×C3⋊Dic3—21no—
38C32×Dic3—21no—
39C32×Dic3Q812yes—
40#889—91no—
41#870—94no—
42C2×C32×C4—24no—
43C2×C32×C4—31no—
44C2×C32×C4—31no—
45C2×C32×C4C2×D314yes—
46C2×C32×C4C2×D311yescharacteristic
47C2×C3⋊Dic3—31no—
48C2×C3⋊Dic3—31no—
49C2×C3⋊Dic3—34no—
50C2×C3⋊Dic3—34no—
51C2×C3⋊Dic3—64no—
52C2×C3⋊Dic3—64no—
53C2×C3⋊Dic3C2×D311yescharacteristic
54C2×C3⋊Dic3C2×D311yescharacteristic
55C2×C3×Dic3—34no—
56C2×C3×Dic3—34no—
57C2×C3×Dic3C2×D314yes—
58C2×C3×Dic3C2×D314yes—
59C2×C33C2×Q811yescharacteristic
60C2×C33#4111yescharacteristic
61C2×C33#4111yescharacteristic
62C3×C4⋊C4—91no—
63C3×C42—34no—
64C4⋊Dic3—94no—
65C4⋊Dic3—94no—
66C4⋊Dic3—91no—
67Dic3⋊C4—34no—
68Dic3⋊C4—94no—
69Dic3⋊C4—94no—
70Dic3⋊C4—94no—
71Dic3⋊C4—91no—
72Dic3⋊C4—91no—
73Dic3⋊C4—91no—
74Dic3⋊C4—91no—
75Dic3⋊C4—184no—
76C4×Dic3C3⋊D311yescharacteristic
77C22×C32—24no—
78C22×C32C22×D314yes—
79C22×C32C22×D311yescharacteristic
80C32×C4—21no—
81C32×C4—24no—
82C32×C4—32no—
83C32×C4—44no—
84C32×C4—61no—
85C3⋊Dic3—21no—
86C3⋊Dic3—21no—
87C3⋊Dic3—68no—
88C3⋊Dic3—64no—
89C3⋊Dic3—61no—
90C3⋊Dic3—61no—
91C3⋊Dic3—64no—
92C3⋊Dic3—124no—
93C3×Dic3—24no—
94C3×Dic3—64no—
95C3×Dic3—64no—
96C3×Dic3Dic618yes—
97#125—271noSylow
98C33#12511yescharacteristic, complemented, Sylow
99C2×C3×C4—24no—
100C2×C3×C4—34no—
101C2×C3×C4—34no—
102C2×C3×C4—91no—
103C2×C3×C4—91no—
104C2×C3×C4D3214yes—
105C2×C3×C4C2×C3⋊D311yescharacteristic
106C2×Dic3—34no—
107C2×Dic3—34no—
108C2×Dic3—94no—
109C2×Dic3—94no—
110C2×Dic3—91no—
111C2×Dic3—91no—
112C2×Dic3—184no—
113C2×Dic3—184no—
114C2×Dic3C2×C3⋊D311yescharacteristic
115C2×Dic3C2×C3⋊D311yescharacteristic
116C2×C32—24no—
117C2×C32—24no—
118C2×C32—24no—
119C2×C32C2×Dic614yes—
120C2×C32#23514yes—
121C2×C32#23514yes—
122C2×C32#23511yescharacteristic
123C2×C32#23711yescharacteristic
124C2×C32D3×Q811yescharacteristic
125C4⋊C4—91no—
126C4⋊C4—271no—
127C4⋊C4—271no—
128C4⋊C4—271no—
129C4⋊C4—271no—
130C4⋊C4—271no—
131C42—31no—
132C22×C3—24no—
133C22×C3C2×D3214yes—
134C22×C3C22×C3⋊D311yescharacteristic
135C3×C4—24no—
136C3×C4—21no—
137C3×C4—38no—
138C3×C4—44no—
139C3×C4—64no—
140C3×C4—181no—
141C3×C4—181no—
142Dic3—21no—
143Dic3—64no—
144Dic3—64no—
145Dic3—188no—
146Dic3—184no—
147Dic3—184no—
148Dic3—181no—
149Dic3—181no—
150Dic3—364no—
151Dic3#63412yes—
152C32—24no—
153C32#87014yescomplemented
154C32#88911yescharacteristic, complemented
155C2×C4—31no—
156C2×C4—31no—
157C2×C4—91no—
158C2×C4—91no—
159C2×C4—271no—
160C2×C4—271no—
161C2×C4D3×C3⋊D311yescharacteristic
162C2×C3—24no—
163C2×C3—24no—
164C2×C3—24no—
165C2×C3D3×Dic614yes—
166C2×C3#379814yes—
167C2×C3#380414yes—
168C2×C3C2×#63411yescharacteristic
169C2×C3#382811yescharacteristic
170C2×C3#382811yescharacteristic
171C22C2×D3×C3⋊D311yescharacteristic
172C4—21no—
173C4—32no—
174C4—61no—
175C4—181no—
176C4—181no—
177C4—541no—
178C4—541no—
179C3—24no—
180C3#6386714yescomplemented
181C3#6407911yescharacteristic, complemented
182C2D3×#63411yescharacteristic
183C2#8902011yescharacteristic
184C2#8903111yescharacteristic
subgroupG/H
center, FrattiniC22C2×D3×C3⋊D3
commutator, socle, prime radical, cosocleC22×C33C23
FittingC2×C33×C4C22
hypercenterC2×C4D3×C3⋊D3
hypocenterC33#125
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
98. C33#125semidirect
153. C32#870semidirect
154. C32#889semidirect
180. C3#63867semidirect
181. C3#64079semidirect

Maximal subgroups

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

subgroupindex
#888572
C3×#37502
C32×C4×Dic32
#888072
#888062
Dic3⋊(C3⋊Dic3)2
#640793
#638673