Finite Group Databasebeta

← #11723583

Subgroups — #11723583

Order 864.

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

normal subgroups only

This group has 444 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×#3832C211yesmaximal, characteristic, complemented
2#89014C211yesmaximal, characteristic, complemented
3#88863C211yesmaximal, characteristic, complemented
4C32×#215C211yesmaximal, characteristic, complemented
5#88796C211yesmaximal, characteristic, complemented
6#88789C211yesmaximal, characteristic, complemented
7C3⋊D3×C3⋊C8C211yesmaximal, characteristic, complemented
8#64125—31nomaximal
9#63947—34nomaximal
10C33×Q8C2211yescharacteristic
11#6308C2211yescharacteristic
12C3×C4⋊(C3⋊D3)—21no—
13C3×C4⋊(C3⋊D3)C2211yescharacteristic
14C3×C4×C3⋊D3—21no—
15C3×C4×C3⋊D3C2211yescharacteristic
16C32×D12C2211yescharacteristic
17#6290—21no—
18D3×C3⋊Dic3—21no—
19C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
20C32×C3⋊C8C2211yescharacteristic
21#3832D311yescharacteristic, complemented
22#3830—31no—
23C3×#239—34no—
24#3795—34no—
25C32×SD16—31no—
26#3756—31no—
27#3756—34no—
28#3756—64no—
29#3753—31no—
30#3744—31no—
31C8×C3⋊D3—31no—
32C3×#215D314yescomplemented
33#3716—34no—
34#3713—34no—
35D3×C3⋊C8—34no—
36C2×C3×C3⋊D3—21no—
37C2×C3×C3⋊D3D411yescharacteristic
38C2×C32×D3—21no—
39C33×C4—21no—
40C33×C4C2311yescharacteristic
41C3⋊(C3⋊Dic3)—21no—
42C3×C3⋊Dic3D411yescharacteristic
43#950—91no—
44#916—94no—
45C32×Q8—24no—
46C32×Q8C2×D314yes—
47C32×Q8C2×D311yescharacteristic
48C32×D4—31no—
49#638—61no—
50C2×C3⋊Dic3—61no—
51C4⋊(C3⋊D3)—21no—
52C4⋊(C3⋊D3)C2×D311yescharacteristic
53C4×C3⋊D3—21no—
54C4×C3⋊D3C2×D311yescharacteristic
55#634—31no—
56#634—34no—
57#634—64no—
58C3×D12—34no—
59C3×D12—64no—
60C3×D12C2×D314yes—
61C3×C4×D3—34no—
62C3×C4×D3—64no—
63#625—64no—
64D3×Dic3—64no—
65C32×C8—31no—
66C3⋊(C3⋊C8)—31no—
67C3⋊(C3⋊C8)—34no—
68C3⋊(C3⋊C8)—64no—
69C3×C3⋊C8C2×D314yes—
70C2×C33C2×D411yescharacteristic
71C3×C3⋊D3—21no—
72C3×C3⋊D3—41no—
73C32×D3—41no—
74C3×#41—91no—
75#239—34no—
76#237—94no—
77#235—91no—
78C3×SD16—34no—
79#216—94no—
80#216—91no—
81#216—184no—
82#215C3⋊D311yescharacteristic, complemented
83#214—91no—
84#213—94no—
85#213—91no—
86C2×C3⋊C8—91no—
87#204—94no—
88D3×C8—94no—
89C22×C32—61no—
90C2×C3⋊D3—21no—
91C2×C3⋊D3#6711yescharacteristic
92C2×C3×D3—24no—
93C2×C3×D3—34no—
94C2×C3×D3—64no—
95C32×C4—21no—
96C32×C4—24no—
97C32×C4—24no—
98C32×C4—44no—
99C32×C4C22×D314yes—
100C32×C4C22×D311yescharacteristic
101C3⋊Dic3—61no—
102C3⋊Dic3—64no—
103C3⋊Dic3—124no—
104C3⋊Dic3#6711yescharacteristic
105C3×Dic3—34no—
106#135—271noSylow
107C33#13511yescharacteristic, complemented, Sylow
108C3×Q8—24no—
109C3×Q8D3214yes—
110C3×Q8C2×C3⋊D311yescharacteristic
111C3×D4—34no—
112C3×D4—91no—
113C3×D4—181no—
114C2×C3×C4—91no—
115C2×C3×C4—181no—
116#67—181no—
117#67—184no—
118C2×Dic3—184no—
119D12—34no—
120D12—64no—
121D12C2×C3⋊D311yescharacteristic
122C4×D3—34no—
123C4×D3—64no—
124C4×D3—181no—
125Dic6—94no—
126Dic6—91no—
127Dic6—184no—
128C3×C8—34no—
129C3⋊C8—94no—
130C3⋊C8—91no—
131C3⋊C8—184no—
132C3⋊C8C2×C3⋊D311yescharacteristic
133C2×C32—24no—
134C2×C32—121no—
135C2×C32D3×D414yes—
136C2×C32C2×#6711yescharacteristic
137C3⋊D3—21no—
138C3⋊D3—41no—
139C3×D3—44no—
140C3×D3—64no—
141C3×D3—124no—
142#41—91no—
143#41—271no—
144Q16—271no—
145SD16—31no—
146SD16—271no—
147D8—271no—
148C2×C8—271no—
149C22×C3—64no—
150C22×C3—91no—
151C22×C3—181no—
152C2×D3—21no—
153C2×D3—34no—
154C2×D3—64no—
155C3×C4—24no—
156C3×C4—21no—
157C3×C4—24no—
158C3×C4—44no—
159C3×C4—91no—
160C3×C4C2×D3214yes—
161C3×C4C22×C3⋊D311yescharacteristic
162Dic3—34no—
163Dic3—184no—
164Dic3—181no—
165Dic3—364no—
166C32—24no—
167C32#91614yescomplemented
168C32#95011yescharacteristic, complemented
169Q8—271no—
170Q8D3×C3⋊D311yescharacteristic
171D4—31no—
172D4—91no—
173D4—181no—
174D4—541no—
175C2×C4—91no—
176C2×C4—181no—
177C2×C4—541no—
178C8—31no—
179C8—271no—
180C2×C3—24no—
181C2×C3—124no—
182C2×C3—181no—
183C2×C3—361no—
184C2×C3D3×#6714yes—
185C2×C3D4×C3⋊D311yescharacteristic
186D3—41no—
187D3—64no—
188D3—124no—
189C22—61no—
190C22—91no—
191C22—181no—
192C4—21no—
193C4—91no—
194C4—541no—
195C4C2×D3×C3⋊D311yescharacteristic
196C3—24no—
197C3#6394714yescomplemented
198C3#6412511yescharacteristic, complemented
199C2—121no—
200C2—181no—
201C2—361no—
202C2C3⋊D3×#6711yescharacteristic
subgroupG/H
centerC2C3⋊D3×#67
commutator, cosocleC33×C4C23
FrattiniC4C2×D3×C3⋊D3
FittingC33×Q8C22
socleC2×C33C2×D4
hypercenterQ8D3×C3⋊D3
hypocenterC33#135
LCSG ▷ C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C2 ◁ C4 ◁ Q8

Complements

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

NH ≅ G/Ntype
1. C3×#3832C2semidirect
2. #89014C2semidirect
3. #88863C2semidirect
4. C32×#215C2semidirect
5. #88796C2semidirect
6. #88789C2semidirect
7. C3⋊D3×C3⋊C8C2semidirect
21. #3832D3semidirect
32. C3×#215D3semidirect
82. #215C3⋊D3semidirect
107. C33#135semidirect
167. C32#916semidirect
168. C32#950semidirect
197. C3#63947semidirect
198. C3#64125semidirect

Maximal subgroups

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

subgroupindex
C3×#38322
#890142
#888632
C32×#2152
#887962
#887892
C3⋊D3×C3⋊C82
#641253
#639473