Finite Group Databasebeta

← #11723605

Subgroups — #11723605

Order 864.

604 conjugacy classes of subgroups; only the 128 normal subgroups are shown.

normal subgroups only

This group has 604 conjugacy classes of subgroups — too many to draw. Only its 128 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×#638C211yesmaximal, characteristic
2C22×C32×Dic3C211yesmaximal, characteristic, complemented
3#88864C211yesmaximal, characteristic, complemented
4#88806C212yesmaximal, complemented
5#88803C212yesmaximal, complemented
6#64138—31nomaximal
7#63961—34nomaximal
8C23×C33C2211yescharacteristic
9C22×C3×C3⋊D3C2211yescharacteristic
10C2×C3⋊(C3⋊Dic3)C2212yes—
11C3×#638—22no—
12C2×C3×C3⋊Dic3C2211yescharacteristic
13C2×C32×Dic3—22no—
14C2×C32×Dic3C2212yes—
15C22×C32×C4—31no—
16C2×#638D311yescharacteristic
17C2×C3×#67—34no—
18C22×C3×Dic3D314yescomplemented
19#3757—31no—
20#3757—34no—
21#3757—64no—
22#3752—32no—
23#3750—32no—
24#3724—38no—
25(C2×D3)⋊Dic3—38no—
26C22×C33—22no—
27C22×C33D412yes—
28C22×C33C2311yescharacteristic
29C2×C3×C3⋊D3—21no—
30C2×C3×C3⋊D3—22no—
31C3⋊(C3⋊Dic3)—22no—
32C3×C3⋊Dic3—21no—
33C32×Dic3—22no—
34#935—91no—
35#929—94no—
36C23×C32—24no—
37C23×C32C2×D314yes—
38C23×C32C2×D311yescharacteristic
39C22×C3⋊D3C2×D311yescharacteristic
40C22×C3×D3—34no—
41C2×C32×C4—32no—
42C2×C32×C4—62no—
43#638—22no—
44C2×C3⋊Dic3—32no—
45C2×C3⋊Dic3—38no—
46C2×C3⋊Dic3—68no—
47C2×C3⋊Dic3C2×D311yescharacteristic
48C3×#67—68no—
49C2×C3×Dic3—28no—
50C2×C3×Dic3—34no—
51C2×C3×Dic3C2×D318yes—
52C2×C33—22no—
53C2×C33C2×D411yescharacteristic
54C2×C33#4112yes—
55C3×C3⋊D3—41no—
56C2×C3×D4—91no—
57C22×C3×C4—34no—
58C2×#67—34no—
59C22×Dic3C3⋊D311yescharacteristic, complemented
60#217—94no—
61#217—91no—
62#217—92no—
63#217—184no—
64(C2×D3)⋊C4—98no—
65Dic3⋊C4—98no—
66Dic3⋊C4—92no—
67C22×C32—22no—
68C22×C32—28no—
69C22×C32—24no—
70C22×C32—28no—
71C22×C32—48no—
72C22×C32#6718yes—
73C22×C32#6712yes—
74C22×C32C22×D314yes—
75C22×C32C22×D311yescharacteristic
76C2×C3⋊D3—22no—
77C2×C3⋊D3—21no—
78C2×C3×D3—68no—
79C2×C3×D3—64no—
80C32×C4—62no—
81C3⋊Dic3—21no—
82C3⋊Dic3—62no—
83C3⋊Dic3—68no—
84C3⋊Dic3—128no—
85C3×Dic3—28no—
86C3×Dic3—64no—
87#123—271noSylow
88C33#12311yescharacteristic, complemented, Sylow
89C23×C3—24no—
90C23×C3—91no—
91C23×C3D3214yes—
92C23×C3C2×C3⋊D311yescharacteristic
93C22×D3—34no—
94C3×D4—182no—
95C2×C3×C4—38no—
96C2×C3×C4—68no—
97C2×C3×C4—91no—
98#67—68no—
99C2×Dic3—22no—
100C2×Dic3—34no—
101C2×Dic3—98no—
102C2×Dic3—92no—
103C2×Dic3—188no—
104C2×Dic3C2×C3⋊D312yes—
105C2×C32—22no—
106C2×C32—28no—
107C2×C32—24no—
108C2×C32—28no—
109C2×C32—48no—
110C2×C32#23518yes—
111C2×C32#23712yes—
112C2×C32C2×#6714yes—
113C2×C32C2×#6711yescharacteristic
114C3⋊D3—41no—
115C3×D3—124no—
116C2×D4—91no—
117C22×C4—31no—
118C4⋊C4—272no—
119C22⋊C4—272no—
120C22⋊C4—271no—
121C22×C3—28no—
122C22×C3—22no—
123C22×C3—28no—
124C22×C3—24no—
125C22×C3—48no—
126C22×C3—182no—
127C22×C3—181no—
128C22×C3#62518yes—
129C22×C3#63812yes—
130C22×C3C2×D3214yes—
131C22×C3C22×C3⋊D311yescharacteristic
132C2×D3—68no—
133C2×D3—64no—
134C3×C4—68no—
135C3×C4—181no—
136Dic3—22no—
137Dic3—64no—
138Dic3—188no—
139Dic3—182no—
140Dic3—368no—
141C32—24no—
142C32#92914yescomplemented
143C32#93511yescharacteristic, complemented
144C23—91no—
145C23D3×C3⋊D311yescharacteristic
146D4—182no—
147C2×C4—32no—
148C2×C4—62no—
149C2×C4—91no—
150C2×C4—272no—
151C2×C3—22no—
152C2×C3—28no—
153C2×C3—28no—
154C2×C3—24no—
155C2×C3—48no—
156C2×C3—361no—
157C2×C3#380418yes—
158C2×C3C2×#62514yes—
159C2×C3#382812yes—
160C2×C3C2×#63811yescharacteristic
161D3—124no—
162C22—22no—
163C22—182no—
164C22—181no—
165C22#629012yes—
166C22C2×D3×C3⋊D311yescharacteristic
167C4—62no—
168C4—181no—
169C4—542no—
170C3—24no—
171C3#6396114yescomplemented
172C3#6413811yescharacteristic, complemented
173C2—22no—
174C2—361no—
175C2#8903112yes—
176C2C2×#629011yescharacteristic
subgroupG/H
center, FrattiniC22C2×D3×C3⋊D3
commutator, socle, cosocleC22×C33C23
FittingC23×C33C22
prime radicalC2×C3×#638C2
hypercenterC23D3×C3⋊D3
hypocenterC33#123
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C23

Complements

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

NH ≅ G/Ntype
2. C22×C32×Dic3C2semidirect
3. #88864C2semidirect
4. #88806C2semidirect
5. #88803C2semidirect
18. C22×C3×Dic3D3semidirect
59. C22×Dic3C3⋊D3semidirect
88. C33#123semidirect
142. C32#929semidirect
143. C32#935semidirect
171. C3#63961semidirect
172. C3#64138semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#6382
C22×C32×Dic32
#888642
#888062
#888032
#641383
#639613