Finite Group Databasebeta

← #11723585

Subgroups — #11723585

Order 864.

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

normal subgroups only

This group has 452 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#89018C211yesmaximal, characteristic, complemented
3#88862C211yesmaximal, characteristic, complemented
4C32×#216C211yesmaximal, characteristic, complemented
5#88793C211yesmaximal, characteristic, complemented
6#88791C211yesmaximal, characteristic, complemented
7C3⋊D3×C3⋊C8C211yesmaximal, characteristic, complemented
8#64128—31nomaximal
9#63949—34nomaximal
10C33×Q8C2211yescharacteristic
11C4⋊(C3⋊(C3⋊D3))C2211yescharacteristic
12C3×C4⋊(C3⋊D3)—21no—
13C3×C4⋊(C3⋊D3)C2211yescharacteristic
14C3×C4×C3⋊D3—21no—
15C3×C4×C3⋊D3C2211yescharacteristic
16C32×Dic6C2211yescharacteristic
17#6292—21no—
18#6289—21no—
19C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
20C32×C3⋊C8C2211yescharacteristic
21#3832—31no—
22#3832D311yescharacteristic, complemented
23C3×#239—34no—
24#3799—34no—
25C32×Q16—31no—
26#3755—31no—
27#3755—34no—
28#3755—31no—
29#3755—64no—
30#3745—31no—
31C8×C3⋊D3—31no—
32C3×#216D314yescomplemented
33#3715—34no—
34#3714—34no—
35D3×C3⋊C8—34no—
36C2×C3⋊(C3⋊D3)—21no—
37C2×C3×C3⋊D3—21no—
38C2×C3×C3⋊D3D411yescharacteristic
39C33×C4—21no—
40C33×C4C2311yescharacteristic
41C3×C3⋊Dic3D411yescharacteristic
42C32×Dic3—21no—
43#950—91no—
44#919—94no—
45C32×Q8—24no—
46C32×Q8—31no—
47C32×Q8C2×D314yes—
48C32×Q8C2×D311yescharacteristic
49C4⋊(C3⋊D3)—21no—
50C4⋊(C3⋊D3)—31no—
51C4⋊(C3⋊D3)—34no—
52C4⋊(C3⋊D3)—64no—
53C4⋊(C3⋊D3)—61no—
54C4⋊(C3⋊D3)C2×D311yescharacteristic
55C4×C3⋊D3—21no—
56C4×C3⋊D3—61no—
57C4×C3⋊D3C2×D311yescharacteristic
58C3×D12—34no—
59C3×D12—64no—
60C3×C4×D3—34no—
61C3×C4×D3—64no—
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×C33C2×D411yescharacteristic
71C3⋊(C3⋊D3)—41no—
72C3×C3⋊D3—21no—
73C3×C3⋊D3—41no—
74C3×#41—91no—
75#239—34no—
76#239—94no—
77#235—91no—
78C3×Q16—34no—
79#216C3⋊D311yescharacteristic, complemented
80#215—94no—
81#215—94no—
82#215—91no—
83#215—184no—
84#214—91no—
85#213—91no—
86C2×C3⋊C8—91no—
87D24—94no—
88D3×C8—94no—
89C2×C3⋊D3—21no—
90C2×C3⋊D3—61no—
91C2×C3⋊D3—64no—
92C2×C3⋊D3—124no—
93C2×C3⋊D3#6711yescharacteristic
94C2×C3×D3—34no—
95C2×C3×D3—64no—
96C32×C4—21no—
97C32×C4—24no—
98C32×C4—24no—
99C32×C4—44no—
100C32×C4—61no—
101C32×C4C22×D314yes—
102C32×C4C22×D311yescharacteristic
103C3⋊Dic3#6711yescharacteristic
104C3×Dic3—24no—
105C3×Dic3—34no—
106#135—271noSylow
107C33#13511yescharacteristic, complemented, Sylow
108C3×Q8—24no—
109C3×Q8—34no—
110C3×Q8D3214yes—
111C3×Q8C2×C3⋊D311yescharacteristic
112C3×D4—91no—
113C3×D4—181no—
114C2×C3×C4—91no—
115C2×C3×C4—181no—
116#67—181no—
117D12—34no—
118D12—64no—
119D12—94no—
120D12—91no—
121D12—184no—
122D12—184no—
123C4×D3—34no—
124C4×D3—64no—
125C4×D3—181no—
126C4×D3—184no—
127Dic6C2×C3⋊D311yescharacteristic
128C3×C8—34no—
129C3⋊C8—94no—
130C3⋊C8—91no—
131C3⋊C8—184no—
132C3⋊C8C2×C3⋊D311yescharacteristic
133C2×C32—24no—
134C2×C32D3×D414yes—
135C2×C32C2×#6711yescharacteristic
136C3⋊D3—21no—
137C3⋊D3—41no—
138C3⋊D3—121no—
139C3⋊D3—124no—
140C3⋊D3—244no—
141C3×D3—64no—
142C3×D3—124no—
143#41—91no—
144#41—271no—
145Q16—31no—
146SD16—271no—
147SD16—271no—
148D8—271no—
149C2×C8—271no—
150C22×C3—91no—
151C22×C3—181no—
152C2×D3—34no—
153C2×D3—64no—
154C2×D3—184no—
155C2×D3—181no—
156C2×D3—364no—
157C3×C4—24no—
158C3×C4—21no—
159C3×C4—24no—
160C3×C4—44no—
161C3×C4—64no—
162C3×C4—91no—
163C3×C4C2×D3214yes—
164C3×C4C22×C3⋊D311yescharacteristic
165Dic3—21no—
166Dic3—34no—
167C32—24no—
168C32#91914yescomplemented
169C32#95011yescharacteristic, complemented
170Q8—31no—
171Q8D3×C3⋊D311yescharacteristic
172D4—91no—
173D4—181no—
174D4—271no—
175D4—541no—
176C2×C4—91no—
177C2×C4—181no—
178C2×C4—541no—
179C8—31no—
180C8—271no—
181C2×C3—24no—
182C2×C3—181no—
183C2×C3—361no—
184C2×C3D3×#6714yes—
185C2×C3D4×C3⋊D311yescharacteristic
186D3—64no—
187D3—124no—
188D3—361no—
189D3—364no—
190D3—724no—
191C22—91no—
192C22—181no—
193C22—541no—
194C4—21no—
195C4—61no—
196C4—91no—
197C4C2×D3×C3⋊D311yescharacteristic
198C3—24no—
199C3#6394914yescomplemented
200C3#6412811yescharacteristic, complemented
201C2—181no—
202C2—361no—
203C2—1081no—
204C2C3⋊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. #89018C2semidirect
3. #88862C2semidirect
4. C32×#216C2semidirect
5. #88793C2semidirect
6. #88791C2semidirect
7. C3⋊D3×C3⋊C8C2semidirect
22. #3832D3semidirect
32. C3×#216D3semidirect
79. #216C3⋊D3semidirect
107. C33#135semidirect
168. C32#919semidirect
169. C32#950semidirect
199. C3#63949semidirect
200. C3#64128semidirect

Maximal subgroups

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

subgroupindex
C3×#38322
#890182
#888622
C32×#2162
#887932
#887912
C3⋊D3×C3⋊C82
#641283
#639493