Finite Group Databasebeta

← #11723435

Subgroups — #11723435

Order 864.

548 conjugacy classes of subgroups; only the 132 normal subgroups are shown.

normal subgroups only

This group has 548 conjugacy classes of subgroups — too many to draw. Only its 132 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
1C2×C3×C4×C3⋊D3C211yesmaximal, characteristic
2C2×C32×Dic6C211yesmaximal, characteristic, complemented
3C4⋊(C3⋊(C3⋊Dic3))C211yesmaximal, characteristic, complemented
4#88806C212yesmaximal, complemented
5#88803C212yesmaximal
6#64155—31nomaximal
7#63856—34nomaximal
8C22×C3×C3⋊D3C2211yescharacteristic
9C2×C33×C4C2211yescharacteristic
10C2×C3⋊(C3⋊Dic3)C2212yes—
11C2×C3×C3⋊Dic3C2211yescharacteristic
12C3×C4×C3⋊D3—22no—
13C2×C32×Dic3C2212yes—
14C32×Dic6—22no—
15C2×C32×Q8—31no—
16C2×C4×C3⋊D3D311yescharacteristic
17C2×C3×C4×D3—34no—
18C2×C3×Dic6D314yescomplemented
19#3752—32no—
20C4⋊(C3⋊Dic3)—31no—
21C4⋊(C3⋊Dic3)—34no—
22C4⋊(C3⋊Dic3)—64no—
23#3750—32no—
24#3724—38no—
25(C2×D3)⋊Dic3—38no—
26C22×C33C2311yescharacteristic
27C2×C3×C3⋊D3—21no—
28C2×C3×C3⋊D3—21no—
29C2×C3×C3⋊D3Q812yescomplemented
30C33×C4D412yes—
31C3⋊(C3⋊Dic3)—22no—
32C3×C3⋊Dic3—21no—
33C32×Dic3—22no—
34#946—94no—
35#924—91no—
36C22×C3⋊D3C2×D311yescharacteristic
37C22×C3×D3—34no—
38C32×Q8—62no—
39C2×C32×C4—24no—
40C2×C32×C4—32no—
41C2×C32×C4C2×D314yes—
42C2×C32×C4C2×D311yescharacteristic
43C2×C3⋊Dic3—32no—
44C2×C3⋊Dic3—38no—
45C2×C3⋊Dic3—68no—
46C2×C3⋊Dic3C2×D311yescharacteristic
47C4×C3⋊D3—22no—
48C2×C3×Dic3—34no—
49C2×C3×Dic3C2×D318yes—
50C3×C4×D3—68no—
51C3×Dic6—28no—
52C2×C33C2×D411yescharacteristic
53C2×C33C2×Q811yescharacteristic
54C2×C33#4111yescharacteristic
55C3×C3⋊D3—22no—
56C2×C3×Q8—34no—
57C22×C3×C4—91no—
58C2×C4×D3—34no—
59C2×Dic6C3⋊D311yescharacteristic, complemented
60#217—92no—
61(C2×D3)⋊C4—98no—
62C4⋊Dic3—94no—
63C4⋊Dic3—91no—
64C4⋊Dic3—184no—
65Dic3⋊C4—98no—
66Dic3⋊C4—92no—
67C22×C32—24no—
68C22×C32C22×D314yes—
69C22×C32C22×D311yescharacteristic
70C2×C3⋊D3—21no—
71C2×C3⋊D3—21no—
72C2×C3⋊D3Dic612yescomplemented
73C2×C3×D3—38no—
74C2×C3×D3—64no—
75C2×C3×D3—64no—
76C32×C4—28no—
77C32×C4—62no—
78C32×C4#6718yes—
79C32×C4#6712yes—
80C3⋊Dic3—21no—
81C3⋊Dic3—62no—
82C3⋊Dic3—68no—
83C3⋊Dic3—128no—
84C3×Dic3—28no—
85C3×Dic3—64no—
86C22⋊Q8—271noSylow
87C33C22⋊Q811yescharacteristic, complemented, Sylow
88C23×C3—91no—
89C22×D3—34no—
90C3×Q8—68no—
91C2×C3×C4—24no—
92C2×C3×C4—38no—
93C2×C3×C4—91no—
94C2×C3×C4—182no—
95C2×C3×C4D3214yes—
96C2×C3×C4C2×C3⋊D311yescharacteristic
97C2×Dic3—34no—
98C2×Dic3—98no—
99C2×Dic3—92no—
100C2×Dic3—188no—
101C2×Dic3C2×C3⋊D312yes—
102C4×D3—68no—
103Dic6—22no—
104C2×C32—24no—
105C2×C32—24no—
106C2×C32—24no—
107C2×C32C2×Dic611yescharacteristic
108C2×C32#23511yescharacteristic
109C2×C32D3×Q814yes—
110C2×C32#23914yes—
111C2×C32C2×#6714yes—
112C2×C32C2×#6711yescharacteristic
113C3⋊D3—22no—
114C3×D3—68no—
115C2×Q8—31no—
116C22×C4—91no—
117C4⋊C4—272no—
118C4⋊C4—271no—
119C22⋊C4—272no—
120C22×C3—24no—
121C22×C3—92no—
122C22×C3—181no—
123C22×C3—181no—
124C22×C3C2×D3214yes—
125C22×C3C22×C3⋊D311yescharacteristic
126C2×D3—38no—
127C2×D3—64no—
128C2×D3—64no—
129C3×C4—28no—
130C3×C4—68no—
131C3×C4—181no—
132C3×C4#62518yes—
133C3×C4#63812yes—
134Dic3—22no—
135Dic3—64no—
136Dic3—188no—
137Dic3—182no—
138Dic3—368no—
139C32—24no—
140C32#92411yescharacteristic, complemented
141C32#94614yescomplemented
142C23—91no—
143Q8—62no—
144C2×C4—32no—
145C2×C4—91no—
146C2×C4—182no—
147C2×C4—272no—
148C2×C4D3×C3⋊D311yescharacteristic
149C2×C3—24no—
150C2×C3—24no—
151C2×C3—24no—
152C2×C3—182no—
153C2×C3D3×Dic614yes—
154C2×C3#379914yes—
155C2×C3C2×#62514yes—
156C2×C3Q8×C3⋊D311yescharacteristic
157C2×C3#383211yescharacteristic
158C2×C3C2×#63811yescharacteristic
159D3—68no—
160C22—92no—
161C22—181no—
162C22—181no—
163C22C2×D3×C3⋊D311yescharacteristic
164C4—62no—
165C4—181no—
166C4—542no—
167C4#629012yes—
168C3—24no—
169C3#6385614yescomplemented
170C3#6415511yescharacteristic, complemented
171C2—182no—
172C2C3⋊D3×Dic611yescharacteristic
173C2#8901811yescharacteristic
174C2C2×#629011yescharacteristic
subgroupG/H
center, FrattiniC22C2×D3×C3⋊D3
commutator, socle, cosocleC22×C33C23
FittingC2×C33×C4C22
prime radicalC22×C3×C3⋊D3C22
hypercenterC2×C4D3×C3⋊D3
hypocenterC33C22⋊Q8
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
2. C2×C32×Dic6C2semidirect
3. C4⋊(C3⋊(C3⋊Dic3))C2semidirect
4. #88806C2semidirect
18. C2×C3×Dic6D3semidirect
29. C2×C3×C3⋊D3Q8semidirect
59. C2×Dic6C3⋊D3semidirect
72. C2×C3⋊D3Dic6semidirect
87. C33C22⋊Q8semidirect
140. C32#924semidirect
141. C32#946semidirect
169. C3#63856semidirect
170. C3#64155semidirect

Maximal subgroups

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

subgroupindex
C2×C3×C4×C3⋊D32
C2×C32×Dic62
C4⋊(C3⋊(C3⋊Dic3))2
#888062
#888032
#641553
#638563