Finite Group Databasebeta

← #11723609

Subgroups — #11723609

Order 864.

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

normal subgroups only

This group has 684 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
1C22×C3⋊(C3⋊Dic3)C211yesmaximal, characteristic, complemented
2C2×C3×#638C211yesmaximal, characteristic
3C32×#217C211yesmaximal, characteristic, complemented
4Dic3⋊(C3⋊Dic3)C212yesmaximal, complemented
5#88803C212yesmaximal, complemented
6#64095—31nomaximal
7#63962—34nomaximal
8C23×C33C2211yescharacteristic
9C22×C3×C3⋊D3C2211yescharacteristic
10C2×C3⋊(C3⋊Dic3)—22no—
11C2×C3⋊(C3⋊Dic3)C2212yes—
12C3×#638—22no—
13C2×C3×C3⋊Dic3C2211yescharacteristic
14C2×C32×Dic3C2212yes—
15C2×#638D311yescharacteristic
16C22×C3⋊Dic3—31no—
17C22×C3⋊Dic3—34no—
18C22×C3⋊Dic3—64no—
19C2×C3×#67—34no—
20C32×C22⋊C4—31no—
21#3752—32no—
22C4⋊(C3⋊Dic3)—32no—
23C3×#217D314yescomplemented
24Dic3⋊Dic3—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#886—94no—
36C23×C32—24no—
37C23×C32C2×D314yes—
38C23×C32C2×D311yescharacteristic
39C22×C3⋊D3C2×D311yescharacteristic
40C22×C3×D3—34no—
41C2×C32×C4—32no—
42#638—22no—
43C2×C3⋊Dic3—32no—
44C2×C3⋊Dic3—38no—
45C2×C3⋊Dic3—68no—
46C2×C3⋊Dic3—616no—
47C2×C3⋊Dic3—68no—
48C2×C3⋊Dic3—62no—
49C2×C3⋊Dic3C2×D311yescharacteristic
50C3×#67—68no—
51C2×C3×Dic3—34no—
52C2×C3×Dic3C2×D318yes—
53C2×C33—22no—
54C2×C33C2×D411yescharacteristic
55C2×C33#4112yes—
56C3×C3⋊D3—41no—
57C2×C3×D4—91no—
58C2×#67—34no—
59C22×Dic3—94no—
60C22×Dic3—91no—
61C22×Dic3—184no—
62C3×C22⋊C4—34no—
63#217—92no—
64#217C3⋊D311yescharacteristic, complemented
65(C2×D3)⋊C4—98no—
66C4⋊Dic3—98no—
67Dic3⋊C4—92no—
68C22×C32—216no—
69C22×C32—28no—
70C22×C32—22no—
71C22×C32—28no—
72C22×C32—24no—
73C22×C32D1218yes—
74C22×C32#6712yes—
75C22×C32C22×D314yes—
76C22×C32C22×D311yescharacteristic
77C2×C3⋊D3—22no—
78C2×C3⋊D3—21no—
79C2×C3×D3—64no—
80C2×C3×D3—68no—
81C32×C4—62no—
82C3⋊Dic3—21no—
83C3⋊Dic3—616no—
84C3⋊Dic3—68no—
85C3⋊Dic3—62no—
86C3×Dic3—28no—
87C3×Dic3—64no—
88#123—271noSylow
89C33#12311yescharacteristic, complemented, Sylow
90C23×C3—24no—
91C23×C3—91no—
92C23×C3D3214yes—
93C23×C3C2×C3⋊D311yescharacteristic
94C22×D3—34no—
95C3×D4—182no—
96C2×C3×C4—38no—
97C2×C3×C4—91no—
98#67—68no—
99C2×Dic3—34no—
100C2×Dic3—98no—
101C2×Dic3—92no—
102C2×Dic3—188no—
103C2×Dic3—1816no—
104C2×Dic3—188no—
105C2×Dic3—182no—
106C2×Dic3C2×C3⋊D312yes—
107C2×C32—216no—
108C2×C32—28no—
109C2×C32—22no—
110C2×C32—28no—
111C2×C32—24no—
112C2×C32C2×D1214yes—
113C2×C32#23718yes—
114C2×C32#23712yes—
115C2×C32C2×#6711yescharacteristic
116C3⋊D3—41no—
117C3×D3—124no—
118C2×D4—91no—
119C22×C4—271no—
120C4⋊C4—272no—
121C22⋊C4—31no—
122C22⋊C4—272no—
123C22×C3—216no—
124C22×C3—28no—
125C22×C3—22no—
126C22×C3—28no—
127C22×C3—24no—
128C22×C3—181no—
129C22×C3—182no—
130C22×C3#62618yes—
131C22×C3C4⋊(C3⋊D3)12yes—
132C22×C3C2×D3214yes—
133C22×C3C22×C3⋊D311yescharacteristic
134C2×D3—68no—
135C2×D3—64no—
136C3×C4—68no—
137C3×C4—181no—
138Dic3—22no—
139Dic3—64no—
140Dic3—1816no—
141Dic3—188no—
142Dic3—182no—
143C32—24no—
144C32#88614yescomplemented
145C32#93511yescharacteristic, complemented
146C23—91no—
147C23D3×C3⋊D311yescharacteristic
148D4—182no—
149C2×C4—32no—
150C2×C4—91no—
151C2×C4—272no—
152C2×C4—542no—
153C2×C3—216no—
154C2×C3—22no—
155C2×C3—28no—
156C2×C3—28no—
157C2×C3—24no—
158C2×C3—361no—
159C2×C3#380518yes—
160C2×C3C2×#62614yes—
161C2×C3C2×C4⋊(C3⋊D3)11yescharacteristic
162C2×C3#383012yes—
163D3—124no—
164C22—22no—
165C22—182no—
166C22—181no—
167C22#629212yes—
168C22C2×D3×C3⋊D311yescharacteristic
169C4—62no—
170C4—181no—
171C4—542no—
172C3—24no—
173C3#6396214yescomplemented
174C3#6409511yescharacteristic, complemented
175C2—22no—
176C2—361no—
177C2#8902912yes—
178C2C2×#629211yescharacteristic
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
1. C22×C3⋊(C3⋊Dic3)C2semidirect
3. C32×#217C2semidirect
4. Dic3⋊(C3⋊Dic3)C2semidirect
5. #88803C2semidirect
23. C3×#217D3semidirect
64. #217C3⋊D3semidirect
89. C33#123semidirect
144. C32#886semidirect
145. C32#935semidirect
173. C3#63962semidirect
174. C3#64095semidirect

Maximal subgroups

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

subgroupindex
C22×C3⋊(C3⋊Dic3)2
C2×C3×#6382
C32×#2172
Dic3⋊(C3⋊Dic3)2
#888032
#640953
#639623