Finite Group Databasebeta

← #11723580

Subgroups — #11723580

Order 864.

444 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 444 conjugacy classes of subgroups — too many to draw. Only its 92 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
1C32×#239C211yesmaximal, characteristic, complemented
2#89021C211yesmaximal, characteristic, complemented
3#88863C211yesmaximal, characteristic, complemented
4C3×#3755C211yesmaximal, characteristic, complemented
5#88794C211yesmaximal, characteristic, complemented
6#88789C211yesmaximal, characteristic, complemented
7D3×C3⋊(C3⋊C8)C211yesmaximal, characteristic, complemented
8#64159—31nomaximal
9#63947—34nomaximal
10C33×Q8C2211yescharacteristic
11#6308C2211yescharacteristic
12C3×C4⋊(C3⋊D3)C2211yescharacteristic
13C32×D12—21no—
14C32×D12C2211yescharacteristic
15C32×C4×D3—21no—
16C32×C4×D3C2211yescharacteristic
17#6290—21no—
18Dic3×C3⋊D3—21no—
19C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
20C3×C3⋊(C3⋊C8)C2211yescharacteristic
21C32×#41—31no—
22#3828—31no—
23C3×#239D314yescomplemented
24#3795—34no—
25#3756—31no—
26#3756—34no—
27#3756—64no—
28#3755D311yescharacteristic, complemented
29#3754—31no—
30#3753—31no—
31C2×C3⋊(C3⋊C8)—31no—
32C3×#215—34no—
33#3716—34no—
34#3713—34no—
35D3×C3⋊C8—34no—
36C2×C3×C3⋊D3—21no—
37C2×C32×D3—21no—
38C2×C32×D3D411yescharacteristic
39C33×C4—21no—
40C33×C4C2311yescharacteristic
41C3⋊(C3⋊Dic3)—21no—
42C32×Dic3D411yescharacteristic
43#950—94no—
44#916—91no—
45C32×Q8—24no—
46C32×Q8C2×D314yes—
47C32×Q8C2×D311yescharacteristic
48C32×D4—31no—
49C32×D4—61no—
50C2×C32×C4—31no—
51C2×C32×C4—61no—
52#638—61no—
53C4⋊(C3⋊D3)C2×D311yescharacteristic
54C4×C3⋊D3—61no—
55#634—31no—
56#634—34no—
57#634—64no—
58C3×D12—24no—
59C3×D12—34no—
60C3×D12C2×D314yes—
61C3×C4×D3—24no—
62C3×C4×D3C2×D314yes—
63#625—64no—
64D3×Dic3—64no—
65C3⋊(C3⋊C8)—31no—
66C3⋊(C3⋊C8)—34no—
67C3⋊(C3⋊C8)—64no—
68C3⋊(C3⋊C8)C2×D311yescharacteristic
69C3×C3⋊C8—34no—
70C2×C33C2×D411yescharacteristic
71C3×C3⋊D3—41no—
72C32×D3—21no—
73C32×D3—41no—
74C3×#41—34no—
75#239C3⋊D311yescharacteristic, complemented
76#237—91no—
77#235—94no—
78C3×SD16—91no—
79#216—94no—
80#216—91no—
81#216—184no—
82#215—34no—
83#214—94no—
84#213—94no—
85#213—91no—
86C2×C3⋊C8—94no—
87#204—91no—
88D3×C8—91no—
89C22×C32—31no—
90C22×C32—61no—
91C2×C3⋊D3—21no—
92C2×C3×D3—24no—
93C2×C3×D3—64no—
94C2×C3×D3#6714yes—
95C32×C4—21no—
96C32×C4—24no—
97C32×C4—24no—
98C32×C4—31no—
99C32×C4—44no—
100C32×C4C22×D314yes—
101C32×C4C22×D311yescharacteristic
102C3⋊Dic3—61no—
103C3⋊Dic3—64no—
104C3⋊Dic3—124no—
105C3×Dic3#6714yes—
106#135—271noSylow
107C33#13511yescharacteristic, complemented, Sylow
108C3×Q8—24no—
109C3×Q8D3214yes—
110C3×Q8C2×C3⋊D311yescharacteristic
111C3×D4—34no—
112C3×D4—64no—
113C3×D4—91no—
114C2×C3×C4—34no—
115C2×C3×C4—64no—
116#67—184no—
117#67—181no—
118C2×Dic3—181no—
119D12—21no—
120D12—34no—
121D12C2×C3⋊D311yescharacteristic
122C4×D3—21no—
123C4×D3—184no—
124C4×D3C2×C3⋊D311yescharacteristic
125Dic6—94no—
126Dic6—91no—
127Dic6—184no—
128C3×C8—91no—
129C3⋊C8—34no—
130C3⋊C8—94no—
131C3⋊C8—91no—
132C3⋊C8—184no—
133C2×C32—24no—
134C2×C32—61no—
135C2×C32—121no—
136C2×C32D3×D411yescharacteristic
137C2×C32C2×#6714yes—
138C3⋊D3—41no—
139C3×D3—24no—
140C3×D3—44no—
141C3×D3—124no—
142#41—31no—
143#41—271no—
144Q16—271no—
145SD16—91no—
146SD16—271no—
147D8—271no—
148C2×C8—271no—
149C22×C3—34no—
150C22×C3—64no—
151C22×C3—181no—
152C2×D3—21no—
153C2×D3—64no—
154C2×D3#63811yescharacteristic
155C3×C4—24no—
156C3×C4—21no—
157C3×C4—24no—
158C3×C4—34no—
159C3×C4—44no—
160C3×C4C2×D3214yes—
161C3×C4C22×C3⋊D311yescharacteristic
162Dic3—184no—
163Dic3—181no—
164Dic3—364no—
165Dic3#63811yescharacteristic
166C32—24no—
167C32#91611yescharacteristic, complemented
168C32#95014yescomplemented
169Q8—271no—
170Q8D3×C3⋊D311yescharacteristic
171D4—31no—
172D4—61no—
173D4—91no—
174D4—541no—
175C2×C4—31no—
176C2×C4—61no—
177C2×C4—541no—
178C8—91no—
179C8—271no—
180C2×C3—24no—
181C2×C3—64no—
182C2×C3—124no—
183C2×C3—361no—
184C2×C3D3×#6714yes—
185C2×C3C2×#63811yescharacteristic
186D3—21no—
187D3—41no—
188D3—124no—
189C22—31no—
190C22—61no—
191C22—181no—
192C4—21no—
193C4—31no—
194C4—541no—
195C4C2×D3×C3⋊D311yescharacteristic
196C3—24no—
197C3#6394714yescomplemented
198C3#6415911yescharacteristic, complemented
199C2—61no—
200C2—121no—
201C2—361no—
202C2D3×#63811yescharacteristic
subgroupG/H
centerC2D3×#638
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. C32×#239C2semidirect
2. #89021C2semidirect
3. #88863C2semidirect
4. C3×#3755C2semidirect
5. #88794C2semidirect
6. #88789C2semidirect
7. D3×C3⋊(C3⋊C8)C2semidirect
23. C3×#239D3semidirect
28. #3755D3semidirect
75. #239C3⋊D3semidirect
107. C33#135semidirect
167. C32#916semidirect
168. C32#950semidirect
197. C3#63947semidirect
198. C3#64159semidirect

Maximal subgroups

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

subgroupindex
C32×#2392
#890212
#888632
C3×#37552
#887942
#887892
D3×C3⋊(C3⋊C8)2
#641593
#639473