Finite Group Databasebeta

← #11721223

Subgroups — #11721223

Order 864.

206 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 206 conjugacy classes of subgroups — too many to draw. Only its 51 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×#3701C211yesmaximal, characteristic, complemented
2#88635C211yesmaximal, characteristic, complemented
3#88548C211yesmaximal, characteristic, complemented
4C9×#216C211yesmaximal, characteristic, complemented
5#88425C211yesmaximal, characteristic, complemented
6#88417C211yesmaximal, characteristic, complemented
7D9×C3⋊C8C211yesmaximal, characteristic, complemented
8#63949—31nomaximal
9#63481—31nomaximal
10C3×Q8×C9C2211yescharacteristic
11C4⋊(C3⋊D9)C2211yescharacteristic
12C3×D36—21no—
13C3×D36C2211yescharacteristic
14C3×C4×D9—21no—
15C3×C4×D9C2211yescharacteristic
16C9×Dic6C2211yescharacteristic
17#6192—21no—
18(C3⋊D9)⋊C4—21no—
19C3⋊(C9⋊C8)C2211yescharacteristic
20C9×C3⋊C8C2211yescharacteristic
21C3×#239—31no—
22#3799—31no—
23#3755—31no—
24C3×#216D311yescharacteristic
25#3715—31no—
26#3714—31no—
27D3×C3⋊C8—31no—
28#3701—31no—
29#3701D311yescharacteristic, complemented
30C9×Q16—31no—
31#3675—31no—
32#3675—31no—
33#3675—61no—
34D72—31no—
35C8×D9—31no—
36C2×C3⋊D9—21no—
37C2×C3×D9—21no—
38C2×C3×D9D411yescharacteristic
39C3×C4×C9—21no—
40C3×C4×C9C2311yescharacteristic
41C9×Dic3—21no—
42C3×Dic9D411yescharacteristic
43#950—91no—
44#919—91no—
45C32×Q8C2×D311yescharacteristic
46C4⋊(C3⋊D3)—31no—
47C3×D12—31no—
48C3×D12—61no—
49C3×C4×D3—31no—
50C3×C4×D3—61no—
51C3×Dic6C2×D311yescharacteristic
52#626—61no—
53#624—61no—
54C3⋊(C3⋊C8)—31no—
55C3×C3⋊C8C2×D311yescharacteristic
56Q8×C9—21no—
57Q8×C9—31no—
58Q8×C9C2×D311yescharacteristic
59D36—21no—
60D36—31no—
61D36—61no—
62D36—61no—
63D36C2×D311yescharacteristic
64C4×D9—21no—
65C4×D9—61no—
66C4×D9C2×D311yescharacteristic
67C8×C9—31no—
68C9⋊C8—31no—
69C9⋊C8—61no—
70C2×C3×C9C2×D411yescharacteristic
71C3⋊D9—41no—
72C3×D9—21no—
73C3×D9—41no—
74C3×#41—91no—
75#239—31no—
76#239—91no—
77#235—91no—
78C3×Q16—31no—
79#216D911yescharacteristic, complemented
80#215—91no—
81#215—91no—
82#215—91no—
83#215—181no—
84#214—91no—
85#213—91no—
86C2×C3⋊C8—91no—
87D24—91no—
88D3×C8—91no—
89C2×C3⋊D3—61no—
90C2×C3×D3—31no—
91C2×C3×D3—61no—
92C32×C4—21no—
93C32×C4C22×D311yescharacteristic
94C3×Dic3—21no—
95C3×Dic3—31no—
96C2×D9—21no—
97C2×D9—61no—
98C2×D9—121no—
99C2×D9#6711yescharacteristic
100C4×C9—21no—
101C4×C9—21no—
102C4×C9—41no—
103C4×C9—61no—
104C4×C9C22×D311yescharacteristic
105Dic9#6711yescharacteristic
106#135—271noSylow
107C3×C9#13511yescharacteristic, complemented, Sylow
108C3×Q8—21no—
109C3×Q8—31no—
110C3×Q8C2×D911yescharacteristic
111C3×Q8D3211yescharacteristic
112C3×D4—91no—
113C3×D4—181no—
114C2×C3×C4—91no—
115C2×C3×C4—181no—
116#67—181no—
117D12—31no—
118D12—61no—
119D12—91no—
120D12—91no—
121D12—181no—
122D12—181no—
123C4×D3—31no—
124C4×D3—61no—
125C4×D3—181no—
126C4×D3—181no—
127Dic6C2×D911yescharacteristic
128C3×C8—31no—
129C3⋊C8—91no—
130C3⋊C8—91no—
131C3⋊C8—181no—
132C3⋊C8C2×D911yescharacteristic
133C2×C32D3×D411yescharacteristic
134C3⋊D3—121no—
135C3×D3—61no—
136C3×D3—121no—
137C2×C9—21no—
138C2×C9C2×#6711yescharacteristic
139D9—21no—
140D9—41no—
141D9—121no—
142D9—241no—
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—31no—
153C2×D3—61no—
154C2×D3—181no—
155C2×D3—181no—
156C2×D3—361no—
157C3×C4—21no—
158C3×C4—21no—
159C3×C4—21no—
160C3×C4—41no—
161C3×C4—61no—
162C3×C4—91no—
163C3×C4C22×D911yescharacteristic
164C3×C4C2×D3211yescharacteristic
165Dic3—21no—
166Dic3—31no—
167C32#91911yescharacteristic
168C9—21no—
169C9#95011yescharacteristic, complemented
170Q8—31no—
171Q8D3×D911yescharacteristic
172D4—91no—
173D4—181no—
174D4—271no—
175D4—541no—
176C2×C4—91no—
177C2×C4—181no—
178C2×C4—541no—
179C8—31no—
180C8—271no—
181C2×C3—21no—
182C2×C3—181no—
183C2×C3—361no—
184C2×C3D4×D911yescharacteristic
185C2×C3D3×#6711yescharacteristic
186D3—61no—
187D3—121no—
188D3—361no—
189D3—361no—
190D3—721no—
191C22—91no—
192C22—181no—
193C22—541no—
194C4—21no—
195C4—61no—
196C4—91no—
197C4C2×D3×D911yescharacteristic
198C3—21no—
199C3#6348111yescharacteristic, complemented
200C3#6394911yescharacteristic
201C2—181no—
202C2—361no—
203C2—1081no—
204C2D9×#6711yescharacteristic
subgroupG/H
centerC2D9×#67
commutator, cosocleC3×C4×C9C23
FrattiniC3×C4C2×D32
FittingC3×Q8×C9C22
socleC2×C32D3×D4
hypercenterQ8D3×D9
hypocenterC3×C9#135
LCSG ▷ C3×C4×C9 ▷ C2×C3×C9 ▷ C3×C9
UCSC1 ◁ C2 ◁ C4 ◁ Q8

Complements

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

NH ≅ G/Ntype
1. C3×#3701C2semidirect
2. #88635C2semidirect
3. #88548C2semidirect
4. C9×#216C2semidirect
5. #88425C2semidirect
6. #88417C2semidirect
7. D9×C3⋊C8C2semidirect
29. #3701D3semidirect
79. #216D9semidirect
107. C3×C9#135semidirect
169. C9#950semidirect
199. C3#63481semidirect

Maximal subgroups

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

subgroupindex
C3×#37012
#886352
#885482
C9×#2162
#884252
#884172
D9×C3⋊C82
#639493
#634813