Finite Group Databasebeta

← #11723561

Subgroups — #11723561

Order 864.

460 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 460 conjugacy classes of subgroups — too many to draw. Only its 84 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
1C3×#3830C211yesmaximal, characteristic, complemented
2#89018C211yesmaximal, characteristic, complemented
3#88860C211yesmaximal, characteristic, complemented
4C32×#214C211yesmaximal, characteristic, complemented
5#88798C211yesmaximal, characteristic, complemented
6#88797C211yesmaximal, characteristic, complemented
7C3⋊D3×C3⋊C8C211yesmaximal, characteristic, complemented
8#64125—31nomaximal
9#63935—34nomaximal
10C33×D4C2211yescharacteristic
11C4⋊(C3⋊(C3⋊D3))C2211yescharacteristic
12C3×#638—21no—
13C2×C3×C3⋊Dic3—21no—
14C3×C4×C3⋊D3C2211yescharacteristic
15C3×#634C2211yescharacteristic
16C32×Dic6C2211yescharacteristic
17#6292—21no—
18#6289—21no—
19C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
20C32×C3⋊C8C2211yescharacteristic
21#3832—31no—
22#3830D311yescharacteristic, complemented
23C3×#237—34no—
24#3799—34no—
25C32×SD16—31no—
26#3756—31no—
27#3753—31no—
28#3753—34no—
29#3753—64no—
30#3744—31no—
31C8×C3⋊D3—31no—
32C3×#214D314yescomplemented
33#3718—34no—
34#3717—34no—
35D3×C3⋊C8—34no—
36C22×C33—21no—
37C2×C3⋊(C3⋊D3)—21no—
38C2×C3×C3⋊D3D411yescharacteristic
39C33×C4C2311yescharacteristic
40C3×C3⋊Dic3—21no—
41C3×C3⋊Dic3D411yescharacteristic
42C32×Dic3—21no—
43#950—91no—
44#916—94no—
45C32×Q8—31no—
46C32×D4—24no—
47C32×D4C2×D314yes—
48C32×D4C2×D311yescharacteristic
49#638—21no—
50C2×C3⋊Dic3—21no—
51C4⋊(C3⋊D3)—31no—
52C4⋊(C3⋊D3)—34no—
53C4⋊(C3⋊D3)—64no—
54C4⋊(C3⋊D3)—61no—
55C4×C3⋊D3—61no—
56C4×C3⋊D3C2×D311yescharacteristic
57#634C2×D311yescharacteristic
58C3×#67—64no—
59C2×C3×Dic3—64no—
60C3×C4×D3—34no—
61C3×Dic6—34no—
62C3×Dic6C2×D314yes—
63#626—64no—
64#624—64no—
65C32×C8—31no—
66C3⋊(C3⋊C8)—31no—
67C3⋊(C3⋊C8)—34no—
68C3⋊(C3⋊C8)—64no—
69C3×C3⋊C8C2×D314yes—
70C2×C33—41no—
71C2×C33C2×D411yescharacteristic
72C3⋊(C3⋊D3)—41no—
73C3×C3⋊D3—21no—
74C3×#41—91no—
75#239—94no—
76#237—34no—
77#235—91no—
78C3×SD16—34no—
79#216—94no—
80#216—91no—
81#215—91no—
82#214C3⋊D311yescharacteristic, complemented
83#213—94no—
84#213—91no—
85#213—184no—
86C2×C3⋊C8—91no—
87#204—94no—
88D3×C8—94no—
89C22×C32—21no—
90C22×C32—24no—
91C22×C32—44no—
92C2×C3⋊D3—61no—
93C2×C3⋊D3—64no—
94C2×C3⋊D3—124no—
95C2×C3⋊D3#6711yescharacteristic
96C2×C3×D3—34no—
97C32×C4—24no—
98C32×C4—61no—
99C32×C4C22×D314yes—
100C32×C4C22×D311yescharacteristic
101C3⋊Dic3—21no—
102C3⋊Dic3#6711yescharacteristic
103C3×Dic3—24no—
104C3×Dic3—34no—
105C3×Dic3—64no—
106#135—271noSylow
107C33#13511yescharacteristic, complemented, Sylow
108C3×Q8—34no—
109C3×Q8—91no—
110C3×D4—24no—
111C3×D4—181no—
112C3×D4D3214yes—
113C3×D4C2×C3⋊D311yescharacteristic
114C2×C3×C4—91no—
115C2×C3×C4—181no—
116#67—64no—
117#67—181no—
118C2×Dic3—64no—
119D12—94no—
120D12—91no—
121D12—184no—
122D12—184no—
123C4×D3—34no—
124C4×D3—181no—
125C4×D3—184no—
126Dic6—34no—
127Dic6C2×C3⋊D311yescharacteristic
128C3×C8—34no—
129C3⋊C8—94no—
130C3⋊C8—91no—
131C3⋊C8—184no—
132C3⋊C8C2×C3⋊D311yescharacteristic
133C2×C32—24no—
134C2×C32—41no—
135C2×C32—44no—
136C2×C32—84no—
137C2×C32D3×D414yes—
138C2×C32C2×#6711yescharacteristic
139C3⋊D3—21no—
140C3⋊D3—121no—
141C3⋊D3—124no—
142C3⋊D3—244no—
143C3×D3—64no—
144#41—91no—
145#41—271no—
146Q16—271no—
147SD16—31no—
148SD16—271no—
149D8—271no—
150C2×C8—271no—
151C22×C3—24no—
152C22×C3—21no—
153C22×C3—44no—
154C22×C3—91no—
155C2×D3—34no—
156C2×D3—184no—
157C2×D3—181no—
158C2×D3—364no—
159C3×C4—24no—
160C3×C4—64no—
161C3×C4—91no—
162C3×C4—181no—
163C3×C4C2×D3214yes—
164C3×C4C22×C3⋊D311yescharacteristic
165Dic3—21no—
166Dic3—34no—
167Dic3—64no—
168C32—24no—
169C32#91614yescomplemented
170C32#95011yescharacteristic, complemented
171Q8—31no—
172Q8—91no—
173D4—181no—
174D4—271no—
175D4—541no—
176D4D3×C3⋊D311yescharacteristic
177C2×C4—91no—
178C2×C4—181no—
179C2×C4—541no—
180C8—31no—
181C8—271no—
182C2×C3—24no—
183C2×C3—41no—
184C2×C3—44no—
185C2×C3—84no—
186C2×C3—181no—
187C2×C3D3×#6714yes—
188C2×C3D4×C3⋊D311yescharacteristic
189D3—64no—
190D3—361no—
191D3—364no—
192D3—724no—
193C22—21no—
194C22—91no—
195C22—541no—
196C4—61no—
197C4—91no—
198C4—181no—
199C4C2×D3×C3⋊D311yescharacteristic
200C3—24no—
201C3#6393514yescomplemented
202C3#6412511yescharacteristic, complemented
203C2—41no—
204C2—181no—
205C2—1081no—
206C2C3⋊D3×#6711yescharacteristic
subgroupG/H
centerC2C3⋊D3×#67
commutator, cosocleC33×C4C23
FrattiniC4C2×D3×C3⋊D3
FittingC33×D4C22
socleC2×C33C2×D4
hypercenterD4D3×C3⋊D3
hypocenterC33#135
LCSG ▷ C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C2 ◁ C4 ◁ D4

Complements

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

NH ≅ G/Ntype
1. C3×#3830C2semidirect
2. #89018C2semidirect
3. #88860C2semidirect
4. C32×#214C2semidirect
5. #88798C2semidirect
6. #88797C2semidirect
7. C3⋊D3×C3⋊C8C2semidirect
22. #3830D3semidirect
32. C3×#214D3semidirect
82. #214C3⋊D3semidirect
107. C33#135semidirect
169. C32#916semidirect
170. C32#950semidirect
201. C3#63935semidirect
202. C3#64125semidirect

Maximal subgroups

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

subgroupindex
C3×#38302
#890182
#888602
C32×#2142
#887982
#887972
C3⋊D3×C3⋊C82
#641253
#639353