Finite Group Databasebeta

← #11723340

Subgroups — #11723340

Order 864.

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

normal subgroups only

This group has 432 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⋊D3×Dic6C211yesmaximal, characteristic, complemented
2#89014C211yesmaximal, characteristic
3#88853C211yesmaximal, characteristic, complemented
4C3×#3743C211yesmaximal, characteristic, complemented
5C32×#204C211yesmaximal, characteristic, complemented
6#88799C211yesmaximal, characteristic, complemented
7#88794C211yesmaximal, characteristic, complemented
8#64124—31nomaximal
9#63800—34nomaximal
10#6308C2211yescharacteristic
11#6308C2211yescharacteristic
12C3×C4×C3⋊D3C2211yescharacteristic
13C32×D12C2211yescharacteristic
14C32×Dic6C2211yescharacteristic
15#6293—21no—
16#6290—21no—
17Dic3×C3⋊D3—21no—
18D3×C3⋊Dic3—21no—
19C33×C8C2211yescharacteristic
20C3×C3⋊(C3⋊C8)C2211yescharacteristic
21Q8×C3⋊D3—31no—
22#3830—31no—
23#3795—34no—
24D3×Dic6—34no—
25C32×SD16—31no—
26#3756—31no—
27#3754—31no—
28#3746—31no—
29#3746—34no—
30#3746—64no—
31#3743D311yescharacteristic, complemented
32C3×#204D314yescomplemented
33C3×ΓL(1,𝔽25)—34no—
34#3719—34no—
35#3716—34no—
36C2×C3×C3⋊D3D411yescharacteristic
37C2×C32×D3—21no—
38C33×C4C2311yescharacteristic
39C3⋊(C3⋊Dic3)—21no—
40C3⋊(C3⋊Dic3)—21no—
41C3×C3⋊Dic3D411yescharacteristic
42C32×Dic3—21no—
43#915—94no—
44#909—91no—
45C32×Q8—31no—
46C32×D4—31no—
47#638—61no—
48C2×C3⋊Dic3—61no—
49C4×C3⋊D3—61no—
50C4×C3⋊D3C2×D311yescharacteristic
51#634—31no—
52#634—31no—
53#634—34no—
54#634—34no—
55#634—64no—
56#634—64no—
57#634—61no—
58C3×D12C2×D314yes—
59C3×C4×D3—34no—
60C3×Dic6C2×D314yes—
61#627—64no—
62#625—64no—
63D3×Dic3—64no—
64D3×Dic3—64no—
65C32×C8—24no—
66C32×C8C2×D314yes—
67C32×C8C2×D311yescharacteristic
68C3⋊(C3⋊C8)C2×D311yescharacteristic
69C3×C3⋊C8—34no—
70C2×C33C2×D411yescharacteristic
71C3×C3⋊D3—21no—
72C32×D3—41no—
73D3×Q8—94no—
74#237—94no—
75#235—91no—
76C2×Dic6—91no—
77C3×SD16—34no—
78C3×M16—91no—
79#216—94no—
80#214—94no—
81Dic12—91no—
82Dic12—94no—
83Dic12—91no—
84Dic12—184no—
85#204—91no—
86#204C3⋊D311yescharacteristic, complemented
87ΓL(1,𝔽25)—34no—
88C22×C32—61no—
89C2×C3⋊D3D1211yescharacteristic
90C2×C3×D3—24no—
91C2×C3×D3—34no—
92C32×C4—24no—
93C32×C4—61no—
94C32×C4C22×D314yes—
95C32×C4C22×D311yescharacteristic
96C3⋊Dic3—61no—
97C3⋊Dic3—61no—
98C3⋊Dic3—64no—
99C3⋊Dic3—64no—
100C3⋊Dic3—124no—
101C3⋊Dic3—124no—
102C3⋊Dic3D1211yescharacteristic
103C3×Dic3—24no—
104C3×Dic3—34no—
105#137—271noSylow
106C33#13711yescharacteristic, complemented, Sylow
107C3×Q8—34no—
108C3×D4—34no—
109C2×C3×C4—91no—
110#67—181no—
111#67—184no—
112C2×Dic3—181no—
113C2×Dic3—184no—
114D12C2×C3⋊D311yescharacteristic
115C4×D3—34no—
116C4×D3—184no—
117C4×D3—181no—
118Dic6—94no—
119Dic6—94no—
120Dic6—91no—
121Dic6—91no—
122Dic6—184no—
123Dic6—184no—
124Dic6—181no—
125Dic6—184no—
126Dic6C2×C3⋊D311yescharacteristic
127C3×C8—24no—
128C3×C8—91no—
129C3×C8D3214yes—
130C3×C8C2×C3⋊D311yescharacteristic
131C3⋊C8—34no—
132C2×C32—24no—
133C2×C32—121no—
134C2×C32C2×D1211yescharacteristic
135C2×C32D3×D414yes—
136C3⋊D3—21no—
137C3×D3—44no—
138C3×D3—64no—
139#41—271no—
140C2×Q8—271no—
141Q16—271no—
142Q16—271no—
143SD16—31no—
144SD16—271no—
145M16—91no—
146C22×C3—64no—
147C22×C3—91no—
148C2×D3—21no—
149C2×D3—34no—
150C3×C4—24no—
151C3×C4—64no—
152C3×C4—91no—
153C3×C4C2×D3214yes—
154C3×C4C22×C3⋊D311yescharacteristic
155Dic3—21no—
156Dic3—34no—
157Dic3—184no—
158Dic3—184no—
159Dic3—181no—
160Dic3—181no—
161Dic3—364no—
162Dic3—364no—
163C32—24no—
164C32#90911yescharacteristic, complemented
165C32#91514yescomplemented
166Q8—31no—
167Q8—271no—
168Q8—271no—
169Q8—541no—
170D4—31no—
171D4—541no—
172C2×C4—91no—
173C2×C4—541no—
174C2×C4—541no—
175C8—91no—
176C8D3×C3⋊D311yescharacteristic
177C2×C3—24no—
178C2×C3—124no—
179C2×C3—181no—
180C2×C3D3×D1214yes—
181C2×C3D4×C3⋊D311yescharacteristic
182D3—41no—
183D3—64no—
184C22—61no—
185C22—91no—
186C4—61no—
187C4—91no—
188C4—541no—
189C4—541no—
190C4C2×D3×C3⋊D311yescharacteristic
191C3—24no—
192C3#6380014yescomplemented
193C3#6412411yescharacteristic, complemented
194C2—121no—
195C2—181no—
196C2C3⋊D3×D1211yescharacteristic
subgroupG/H
centerC2C3⋊D3×D12
commutator, cosocleC33×C4C23
FrattiniC4C2×D3×C3⋊D3
FittingC33×C8C22
socleC2×C33C2×D4
prime radical#89014C2
hypercenterC8D3×C3⋊D3
hypocenterC33#137
LCSG ▷ C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C2 ◁ C4 ◁ C8

Complements

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

NH ≅ G/Ntype
1. C3⋊D3×Dic6C2semidirect
3. #88853C2semidirect
4. C3×#3743C2semidirect
5. C32×#204C2semidirect
6. #88799C2semidirect
7. #88794C2semidirect
31. #3743D3semidirect
32. C3×#204D3semidirect
86. #204C3⋊D3semidirect
106. C33#137semidirect
164. C32#909semidirect
165. C32#915semidirect
192. C3#63800semidirect
193. C3#64124semidirect

Maximal subgroups

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

subgroupindex
C3⋊D3×Dic62
#890142
#888532
C3×#37432
C32×#2042
#887992
#887942
#641243
#638003