Finite Group Databasebeta

← #10590537

Subgroups — #10590537

Order 576.

217 conjugacy classes of subgroups; only the 60 normal subgroups are shown.

normal subgroups only

This group has 217 conjugacy classes of subgroups — too many to draw. Only its 60 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×#890C211yesmaximal, characteristic, complemented
2#63864C211yesmaximal, characteristic
3#63636C211yesmaximal, characteristic
4C3×#810C211yesmaximal, characteristic, complemented
5#63565C211yesmaximal, characteristic, complemented
6#63564C211yesmaximal, characteristic
7Dic3×C3⋊C8C211yesmaximal, characteristic, complemented
8#5068—31nomaximal
9#4800—31nomaximal
10C2×C4⋊(C3⋊D3)C2211yescharacteristic
11C2×C3×Dic6C2211yescharacteristic
12C32×C4⋊C4C2211yescharacteristic
13C2×C3⋊(C3⋊C8)C2211yescharacteristic
14C3×C4⋊Dic3—21no—
15C3×C4⋊Dic3C2211yescharacteristic
16C3×Dic3⋊C4—21no—
17C3×C4×Dic3C2211yescharacteristic
18C2×C3×C3⋊C8C2211yescharacteristic
19#3722—21no—
20#3722—21no—
21C3×#125—31no—
22#947—31no—
23#890D311yescharacteristic, complemented
24#875—31no—
25C3×#103—31no—
26#835—31no—
27#821—31no—
28C8×Dic3—31no—
29#810—31no—
30#810D311yescharacteristic, complemented
31#809—31no—
32#809—31no—
33#809—31no—
34#809—61no—
35C4×C3⋊C8—31no—
36C22×C3⋊D3—21no—
37C2×C32×C4—21no—
38C2×C32×C4C2311yescharacteristic
39C4⋊(C3⋊D3)—21no—
40C4⋊(C3⋊D3)—21no—
41C2×C3×Dic3—21no—
42C2×C3×Dic3—21no—
43C2×C3×Dic3D411yescharacteristic
44C2×C3×Dic3D411yescharacteristic
45C3×Dic6—21no—
46C3×Dic6—21no—
47C3⋊(C3⋊C8)—21no—
48C3×C3⋊C8—21no—
49#488—91noSylow
50C2×C3×Q8—31no—
51C2×D12—31no—
52C2×D12—31no—
53C2×D12—61no—
54C2×Dic6C2×D311yescharacteristic
55C2×C3×C8—31no—
56C3×C4⋊C4—21no—
57C3×C4⋊C4—31no—
58C3×C4⋊C4—61no—
59C3×C4⋊C4—61no—
60C3×C4⋊C4C2×D311yescharacteristic
61C3×C4⋊C4C2×D311yescharacteristic
62C3×C42—31no—
63(C2×D3)⋊C4—61no—
64(C2×D3)⋊C4—61no—
65(C2×D3)⋊C4—61no—
66(C2×D3)⋊C4—61no—
67C4⋊Dic3—21no—
68C4⋊Dic3C2×D311yescharacteristic
69Dic3⋊C4—21no—
70C4×Dic3C2×D311yescharacteristic
71C2×C3⋊C8—31no—
72C2×C3⋊C8—31no—
73C2×C3⋊C8—61no—
74C2×C3⋊C8C2×D311yescharacteristic
75C22×C32C2×D411yescharacteristic
76C2×C3⋊D3—41no—
77C2×C3⋊D3—41no—
78C2×C3⋊D3—41no—
79C32×C4—41no—
80C32×C4#4111yescharacteristic
81C32×C4#4111yescharacteristic
82C3×Dic3—21no—
83C3×Dic3—21no—
84C3×Dic3—41no—
85C3×Dic3—41no—
86#125—31no—
87#124—91no—
88#103—31no—
89#103—91no—
90#102—91no—
91#102—91no—
92C4×C8—91no—
93C22×D3—61no—
94C22×D3—61no—
95C22×D3—121no—
96C3×Q8—61no—
97C3×Q8—61no—
98C2×C3×C4—21no—
99C2×C3×C4—21no—
100C2×C3×C4—21no—
101C2×C3×C4—31no—
102C2×C3×C4—31no—
103C2×C3×C4—41no—
104C2×C3×C4—61no—
105C2×C3×C4—61no—
106C2×C3×C4C22×D311yescharacteristic
107C2×C3×C4C22×D311yescharacteristic
108C2×Dic3—21no—
109C2×Dic3—21no—
110C2×Dic3#6711yescharacteristic
111C2×Dic3#6711yescharacteristic
112D12—62no—
113D12—61no—
114D12—61no—
115D12—61no—
116D12—61no—
117D12—121no—
118Dic6—21no—
119Dic6—21no—
120C3×C8—61no—
121C3⋊C8—21no—
122C3⋊C8—61no—
123C3⋊C8—61no—
124C3⋊C8—121no—
125C2×C32#12411yescharacteristic
126C2×C32#13511yescharacteristic
127C2×C32#13511yescharacteristic
128C3⋊D3—81no—
129C2×Q8—31no—
130C2×D4—91no—
131C2×C8—31no—
132C2×C8—91no—
133C4⋊C4—31no—
134C4⋊C4—61no—
135C4⋊C4—61no—
136C4⋊C4D3211yescharacteristic
137C22⋊C4—181no—
138C22⋊C4—181no—
139C42—31no—
140C22×C3—21no—
141C22×C3D3×D411yescharacteristic
142C22×C3C2×#6711yescharacteristic
143C2×D3—122no—
144C2×D3—121no—
145C2×D3—121no—
146C2×D3—121no—
147C2×D3—121no—
148C2×D3—121no—
149C2×D3—121no—
150C2×D3—241no—
151C2×D3—241no—
152C3×C4—21no—
153C3×C4—21no—
154C3×C4—41no—
155C3×C4—41no—
156C3×C4—61no—
157C3×C4—61no—
158C3×C4—81no—
159C3×C4—121no—
160C3×C4—121no—
161C3×C4#23511yescharacteristic
162C3×C4#23711yescharacteristic
163C3×C4#23911yescharacteristic
164C3×C4#23911yescharacteristic
165Dic3—21no—
166Dic3—21no—
167Dic3—41no—
168Dic3—41no—
169C32#48811yescharacteristic, complemented, Sylow
170C23—181no—
171Q8—61no—
172Q8—61no—
173D4—181no—
174D4—181no—
175C2×C4—21no—
176C2×C4—31no—
177C2×C4—31no—
178C2×C4—61no—
179C2×C4—61no—
180C2×C4C2×D3211yescharacteristic
181C8—61no—
182C8—181no—
183C2×C3—21no—
184C2×C3—21no—
185C2×C3—21no—
186C2×C3#88511yescharacteristic
187C2×C3#91611yescharacteristic
188C2×C3#91911yescharacteristic
189C2×C3#94711yescharacteristic
190C2×C3#95011yescharacteristic
191C2×C3#95011yescharacteristic
192D3—242no—
193D3—241no—
194D3—241no—
195C22—361no—
196C22—361no—
197C22—361no—
198C22D3×#6711yescharacteristic
199C4—41no—
200C4—61no—
201C4—61no—
202C4—121no—
203C4—121no—
204C4#379611yescharacteristic
205C4#379911yescharacteristic
206C3—21no—
207C3#480011yescharacteristic, complemented
208C3#506811yescharacteristic, complemented
209C2—721no—
210C2#6390711yescharacteristic
211C2#6393511yescharacteristic
212C2#6394911yescharacteristic
subgroupG/H
centerC22D3×#67
commutator, cosocleC2×C32×C4C23
FrattiniC2×C4C2×D32
FittingC32×C4⋊C4C22
socleC22×C32C2×D4
prime radicalC2×C4⋊(C3⋊D3)C22
hypercenterC4⋊C4D32
hypocenterC32#488
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C2×C4 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
1. C3×#890C2semidirect
4. C3×#810C2semidirect
5. #63565C2semidirect
7. Dic3×C3⋊C8C2semidirect
23. #890D3semidirect
30. #810D3semidirect
169. C32#488semidirect
207. C3#4800semidirect
208. C3#5068semidirect

Maximal subgroups

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

subgroupindex
C3×#8902
#638642
#636362
C3×#8102
#635652
#635642
Dic3×C3⋊C82
#50683
#48003