Finite Group Databasebeta

← #11723582

Subgroups — #11723582

Order 864.

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

normal subgroups only

This group has 452 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#89022C211yesmaximal, characteristic, complemented
3#88862C211yesmaximal, characteristic, complemented
4C3×#3756C211yesmaximal, characteristic, complemented
5#88792C211yesmaximal, characteristic, complemented
6#88790C211yesmaximal, characteristic, complemented
7D3×C3⋊(C3⋊C8)C211yesmaximal, characteristic, complemented
8#64159—31nomaximal
9#63949—34nomaximal
10C33×Q8C2211yescharacteristic
11C4⋊(C3⋊(C3⋊D3))C2211yescharacteristic
12C3×#634C2211yescharacteristic
13C32×D12—21no—
14C32×D12C2211yescharacteristic
15C32×C4×D3—21no—
16C32×C4×D3C2211yescharacteristic
17#6291—21no—
18#6289—21no—
19C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
20C3×C3⋊(C3⋊C8)C2211yescharacteristic
21C32×#41—31no—
22#3828—31no—
23C3×#239D314yescomplemented
24#3799—34no—
25#3756D311yescharacteristic, complemented
26#3755—31no—
27#3755—34no—
28#3755—64no—
29#3754—31no—
30#3753—31no—
31C2×C3⋊(C3⋊C8)—31no—
32C3×#216—34no—
33#3715—34no—
34#3714—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#919—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)—31no—
54C4⋊(C3⋊D3)—34no—
55C4⋊(C3⋊D3)—64no—
56C4×C3⋊D3—61no—
57#634C2×D311yescharacteristic
58C3×D12—24no—
59C3×D12C2×D314yes—
60C3×C4×D3—24no—
61C3×C4×D3C2×D314yes—
62C3×Dic6—34no—
63#626—64no—
64#624—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#239—91no—
76#239C3⋊D311yescharacteristic, complemented
77#235—94no—
78C3×Q16—91no—
79#216—34no—
80#215—94no—
81#215—91no—
82#215—91no—
83#215—184no—
84#214—94no—
85#213—94no—
86C2×C3⋊C8—94no—
87D24—91no—
88D3×C8—91no—
89C22×C32—31no—
90C22×C32—61no—
91C2×C3⋊D3—61no—
92C2×C3⋊D3—64no—
93C2×C3⋊D3—124no—
94C2×C3×D3—24no—
95C2×C3×D3#6714yes—
96C32×C4—21no—
97C32×C4—24no—
98C32×C4—24no—
99C32×C4—31no—
100C32×C4—44no—
101C32×C4C22×D314yes—
102C32×C4C22×D311yescharacteristic
103C3⋊Dic3—21no—
104C3×Dic3—64no—
105C3×Dic3#6714yes—
106#135—271noSylow
107C33#13511yescharacteristic, complemented, Sylow
108C3×Q8—24no—
109C3×Q8—91no—
110C3×Q8D3214yes—
111C3×Q8C2×C3⋊D311yescharacteristic
112C3×D4—34no—
113C3×D4—64no—
114C2×C3×C4—34no—
115C2×C3×C4—64no—
116#67—184no—
117D12—21no—
118D12—94no—
119D12—91no—
120D12—184no—
121D12—181no—
122D12C2×C3⋊D311yescharacteristic
123C4×D3—21no—
124C4×D3—184no—
125C4×D3—181no—
126C4×D3C2×C3⋊D311yescharacteristic
127Dic6—34no—
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—121no—
139C3⋊D3—124no—
140C3⋊D3—244no—
141C3×D3—24no—
142C3×D3—44no—
143#41—31no—
144#41—271no—
145Q16—91no—
146SD16—271no—
147SD16—271no—
148D8—271no—
149C2×C8—271no—
150C22×C3—34no—
151C22×C3—64no—
152C2×D3—21no—
153C2×D3—184no—
154C2×D3—181no—
155C2×D3—364no—
156C2×D3#63811yescharacteristic
157C3×C4—24no—
158C3×C4—21no—
159C3×C4—24no—
160C3×C4—34no—
161C3×C4—44no—
162C3×C4—181no—
163C3×C4C2×D3214yes—
164C3×C4C22×C3⋊D311yescharacteristic
165Dic3—64no—
166Dic3#63811yescharacteristic
167C32—24no—
168C32#91911yescharacteristic, complemented
169C32#95014yescomplemented
170Q8—91no—
171Q8D3×C3⋊D311yescharacteristic
172D4—31no—
173D4—61no—
174D4—271no—
175D4—541no—
176C2×C4—31no—
177C2×C4—61no—
178C2×C4—541no—
179C8—91no—
180C8—271no—
181C2×C3—24no—
182C2×C3—64no—
183C2×C3—124no—
184C2×C3D3×#6714yes—
185C2×C3C2×#63811yescharacteristic
186D3—21no—
187D3—41no—
188D3—361no—
189D3—364no—
190D3—724no—
191C22—31no—
192C22—61no—
193C22—541no—
194C4—21no—
195C4—31no—
196C4—181no—
197C4C2×D3×C3⋊D311yescharacteristic
198C3—24no—
199C3#6394914yescomplemented
200C3#6415911yescharacteristic, complemented
201C2—61no—
202C2—121no—
203C2—1081no—
204C2D3×#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. #89022C2semidirect
3. #88862C2semidirect
4. C3×#3756C2semidirect
5. #88792C2semidirect
6. #88790C2semidirect
7. D3×C3⋊(C3⋊C8)C2semidirect
23. C3×#239D3semidirect
25. #3756D3semidirect
76. #239C3⋊D3semidirect
107. C33#135semidirect
168. C32#919semidirect
169. C32#950semidirect
199. C3#63949semidirect
200. C3#64159semidirect

Maximal subgroups

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

subgroupindex
C32×#2392
#890222
#888622
C3×#37562
#887922
#887902
D3×C3⋊(C3⋊C8)2
#641593
#639493