Finite Group Databasebeta

← #11721224

Subgroups — #11721224

Order 864.

226 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 226 conjugacy classes of subgroups — too many to draw. Only its 49 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
1#88746C211yesmaximal, characteristic, complemented
2#88634C211yesmaximal, characteristic, complemented
3C9×#216C211yesmaximal, characteristic, complemented
4C3×#3675C211yesmaximal, characteristic, complemented
5#88426C211yesmaximal, characteristic, complemented
6#88417C211yesmaximal, characteristic, complemented
7(C3⋊D9)⋊C8C211yesmaximal, characteristic, complemented
8#63950—31nomaximal
9#63481—31nomaximal
10C3×Q8×C9C2211yescharacteristic
11C4⋊(C3⋊D9)—21no—
12C4⋊(C3⋊D9)C2211yescharacteristic
13C4×C3⋊D9—21no—
14C4×C3⋊D9C2211yescharacteristic
15C3×D36C2211yescharacteristic
16C9×Dic6C2211yescharacteristic
17#6192—21no—
18Dic3×D9—21no—
19C9×C3⋊C8C2211yescharacteristic
20C3×C9⋊C8C2211yescharacteristic
21#3832—31no—
22#3796—31no—
23C3×#216D311yescharacteristic
24C3×#215—31no—
25#3717—31no—
26#3714—31no—
27#3710—31no—
28#3701—31no—
29#3701—31no—
30#3701—61no—
31C9×Q16—31no—
32#3675—31no—
33#3675D311yescharacteristic, complemented
34D72—31no—
35C8×D9—31no—
36C2×C3⋊D9—21no—
37C2×C3⋊D9D411yescharacteristic
38C2×C3×D9—21no—
39C3×C4×C9—21no—
40C3×C4×C9C2311yescharacteristic
41C3⋊Dic9D411yescharacteristic
42C9×Dic3—21no—
43#919—91no—
44#916—91no—
45C32×Q8C2×D311yescharacteristic
46C4⋊(C3⋊D3)—31no—
47C4⋊(C3⋊D3)—61no—
48C4×C3⋊D3—31no—
49C4×C3⋊D3—61no—
50C3×D12—31no—
51C3×Dic6C2×D311yescharacteristic
52#626—61no—
53D3×Dic3—61no—
54C3×C3⋊C8—31no—
55C3×C3⋊C8C2×D311yescharacteristic
56Q8×C9—21no—
57Q8×C9—31no—
58Q8×C9C2×D311yescharacteristic
59D36—31no—
60D36—61no—
61D36—61no—
62D36—62no—
63D36—61no—
64D36C2×D311yescharacteristic
65C4×D9—31no—
66C4×D9—61no—
67C4×D9—61no—
68C4×D9—61no—
69C4×D9—62no—
70C8×C9—31no—
71C9⋊C8C2×D311yescharacteristic
72C2×C3×C9C2×D411yescharacteristic
73C3⋊D9—21no—
74C3⋊D9—41no—
75C3×D9—41no—
76#239—91no—
77#239—91no—
78#239—91no—
79#239—181no—
80#237—91no—
81C3×Q16—31no—
82C3×SD16—91no—
83#216D911yescharacteristic, complemented
84#215—31no—
85#215—91no—
86#213—91no—
87D24—91no—
88#204—91no—
89D3×C8—91no—
90D3×C8—91no—
91C2×C3⋊D3—31no—
92C2×C3⋊D3—61no—
93C2×C3×D3—61no—
94C32×C4—21no—
95C32×C4C22×D311yescharacteristic
96C3⋊Dic3—31no—
97C3×Dic3—21no—
98C2×D9—21no—
99C2×D9—31no—
100C2×D9—61no—
101C2×D9—62no—
102C2×D9—61no—
103C4×C9—22no—
104C4×C9—21no—
105C4×C9—21no—
106C4×C9—61no—
107C4×C9C22×D311yescharacteristic
108Dic9—31no—
109Dic9—61no—
110#135—271noSylow
111C3×C9#13511yescharacteristic, complemented, Sylow
112C3×Q8—21no—
113C3×Q8—31no—
114C3×Q8C2×D911yescharacteristic
115C3×Q8D3211yescharacteristic
116C3×D4—91no—
117#67—181no—
118C2×Dic3—181no—
119D12—31no—
120D12—91no—
121D12—91no—
122D12—181no—
123D12—181no—
124D12—181no—
125D12—182no—
126D12—181no—
127C4×D3—91no—
128C4×D3—91no—
129C4×D3—181no—
130C4×D3—181no—
131C4×D3—181no—
132C4×D3—181no—
133C4×D3—182no—
134Dic6C2×D911yescharacteristic
135C3×C8—31no—
136C3×C8—91no—
137C3⋊C8—31no—
138C3⋊C8C2×D911yescharacteristic
139C2×C32D3×D411yescharacteristic
140C3⋊D3—61no—
141C3⋊D3—121no—
142C3×D3—121no—
143C2×C9—21no—
144C2×C9D3×D411yescharacteristic
145D9—41no—
146D9—61no—
147D9—121no—
148D9—122no—
149D9—121no—
150#41—271no—
151#41—271no—
152Q16—31no—
153SD16—91no—
154SD16—271no—
155D8—271no—
156C2×C8—271no—
157C22×C3—181no—
158C2×D3—61no—
159C2×D3—91no—
160C2×D3—91no—
161C2×D3—181no—
162C2×D3—182no—
163C2×D3—181no—
164C2×D3—181no—
165C3×C4—22no—
166C3×C4—21no—
167C3×C4—21no—
168C3×C4—21no—
169C3×C4—61no—
170C3×C4C22×D911yescharacteristic
171C3×C4C2×D3211yescharacteristic
172Dic3—21no—
173Dic3—91no—
174Dic3—91no—
175Dic3—181no—
176C32#91911yescharacteristic
177C9—21no—
178C9#91611yescharacteristic, complemented
179Q8—31no—
180Q8D3×D911yescharacteristic
181D4—91no—
182D4—271no—
183D4—541no—
184D4—541no—
185C2×C4—271no—
186C2×C4—541no—
187C2×C4—541no—
188C8—31no—
189C8—91no—
190C2×C3—21no—
191C2×C3—361no—
192C2×C3D4×D911yescharacteristic
193C2×C3#381111yescharacteristic
194D3—121no—
195D3—181no—
196D3—181no—
197D3—361no—
198D3—362no—
199D3—361no—
200D3—361no—
201C22—181no—
202C22—271no—
203C22—541no—
204C4—21no—
205C4—61no—
206C4—271no—
207C4C2×D3×D911yescharacteristic
208C3—21no—
209C3#6348111yescharacteristic, complemented
210C3#6395011yescharacteristic
211C2—361no—
212C2—541no—
213C2—1081no—
214C2#8866811yescharacteristic
subgroupG/H
centerC2#88668
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. #88746C2semidirect
2. #88634C2semidirect
3. C9×#216C2semidirect
4. C3×#3675C2semidirect
5. #88426C2semidirect
6. #88417C2semidirect
7. (C3⋊D9)⋊C8C2semidirect
33. #3675D3semidirect
83. #216D9semidirect
111. C3×C9#135semidirect
178. C9#916semidirect
209. C3#63481semidirect

Maximal subgroups

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

subgroupindex
#887462
#886342
C9×#2162
C3×#36752
#884262
#884172
(C3⋊D9)⋊C82
#639503
#634813