Finite Group Databasebeta

← #10590800

Subgroups — #10590800

Order 576.

237 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 237 conjugacy classes of subgroups — too many to draw. Only its 68 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
1C2×C3×Dic12C211yesmaximal, characteristic
2#63866C211yesmaximal, characteristic
3Dic3×Dic6C211yesmaximal, characteristic
4C2×#3719C211yesmaximal, characteristic
5#63643C211yesmaximal, characteristic
6C3×Dic3⋊C8C211yesmaximal, characteristic
7#63566C211yesmaximal, characteristic
8#5182—31nomaximal
9#4849—31nomaximal
10C2×#634C2211yescharacteristic
11C2×C3×Dic6C2211yescharacteristic
12C2×C3×Dic6C2211yescharacteristic
13C2×C32×C8C2211yescharacteristic
14C4⋊(C3⋊Dic3)C2211yescharacteristic
15C3×C4×Dic3C2211yescharacteristic
16C2×C3×C3⋊C8C2211yescharacteristic
17C3×Dic12—22no—
18Dic3⋊Dic3—21no—
19Dic3⋊Dic3—21no—
20Dic3⋊Dic3—21no—
21Dic32—21no—
22#3719—22no—
23C2×C3×Q16—31no—
24Q8×Dic3—31no—
25#944—31no—
26C2×#216—31no—
27C2×Dic12—31no—
28C2×Dic12D311yescharacteristic
29#869—31no—
30C4×Dic6—31no—
31C3×C4⋊C8—31no—
32#835—31no—
33#816—31no—
34#816—31no—
35#816—31no—
36#816—61no—
37Dic3⋊C8D311yescharacteristic
38C2×C32×C4C2311yescharacteristic
39C2×C3⋊Dic3—21no—
40C2×C3⋊Dic3—21no—
41#634—21no—
42#634—21no—
43C2×C3×Dic3—21no—
44C2×C3×Dic3—21no—
45C2×C3×Dic3D411yescharacteristic
46C2×C3×Dic3D411yescharacteristic
47C3×Dic6—21no—
48C3×Dic6—21no—
49C3×Dic6—21no—
50C3×Dic6D412yes—
51C32×C8—21no—
52C3×C3⋊C8—21no—
53#462—91noSylow
54C2×C3×Q8—31no—
55C2×C3×Q8—31no—
56C2×Dic6—31no—
57C2×Dic6—31no—
58C2×Dic6—61no—
59C2×Dic6C2×D311yescharacteristic
60C2×Dic6C2×D311yescharacteristic
61C3×Q16—62no—
62C2×C3×C8—21no—
63C2×C3×C8—31no—
64C2×C3×C8C2×D311yescharacteristic
65C2×C3×C8C2×D311yescharacteristic
66C3×C42—31no—
67#216—62no—
68C4⋊Dic3—31no—
69C4⋊Dic3—31no—
70C4⋊Dic3—61no—
71C4⋊Dic3—61no—
72C4⋊Dic3—61no—
73C4⋊Dic3—61no—
74Dic3⋊C4—61no—
75Dic3⋊C4—61no—
76Dic3⋊C4—61no—
77C4×Dic3—61no—
78C4×Dic3—61no—
79C4×Dic3C2×D311yescharacteristic
80C2×C3⋊C8C2×D311yescharacteristic
81Dic12—22no—
82Dic12—62no—
83C22×C32C2×D411yescharacteristic
84C32×C4C2×D411yescharacteristic
85C32×C4#4111yescharacteristic
86C3⋊Dic3—41no—
87C3⋊Dic3—41no—
88C3×Dic3—22no—
89C3×Dic3—21no—
90C3×Dic3—41no—
91C3×Dic3Q1612yescomplemented
92C2×Q16—31no—
93C2×Q16—91no—
94C4⋊Q8—91no—
95C4×Q8—91no—
96C4⋊C8—31no—
97#103—91no—
98#103—91no—
99C3×Q8—32no—
100C3×Q8—61no—
101C3×Q8—61no—
102C3×Q8—61no—
103C2×C3×C4—21no—
104C2×C3×C4—31no—
105C2×C3×C4—31no—
106C2×C3×C4—61no—
107C2×C3×C4—61no—
108C2×C3×C4C22×D311yescharacteristic
109C2×C3×C4C22×D311yescharacteristic
110C2×Dic3—21no—
111C2×Dic3—21no—
112C2×Dic3—61no—
113C2×Dic3—61no—
114C2×Dic3—61no—
115C2×Dic3—61no—
116C2×Dic3—121no—
117C2×Dic3—121no—
118C2×Dic3D1211yescharacteristic
119C2×Dic3D1211yescharacteristic
120Dic6—21no—
121Dic6—21no—
122Dic6—21no—
123Dic6—62no—
124Dic6—61no—
125Dic6—61no—
126Dic6—61no—
127Dic6—61no—
128Dic6—121no—
129Dic6#6712yes—
130C3×C8—21no—
131C3×C8—21no—
132C3×C8—41no—
133C3×C8—61no—
134C3⋊C8—21no—
135C2×C32#12111yescharacteristic
136C2×C32C2×Q1611yescharacteristic
137C2×C32#13711yescharacteristic
138C2×Q8—31no—
139C2×Q8—31no—
140C2×Q8—91no—
141Q16—62no—
142Q16—182no—
143C2×C8—31no—
144C2×C8D3211yescharacteristic
145C4⋊C4—91no—
146C4⋊C4—181no—
147C4⋊C4—181no—
148C4⋊C4—181no—
149C42—31no—
150C42—181no—
151C22×C3—21no—
152C22×C3C2×D1211yescharacteristic
153C22×C3D3×D411yescharacteristic
154C3×C4—21no—
155C3×C4—21no—
156C3×C4—32no—
157C3×C4—62no—
158C3×C4—61no—
159C3×C4—121no—
160C3×C4D3×D411yescharacteristic
161C3×C4#23711yescharacteristic
162C3×C4#23911yescharacteristic
163C3×C4C2×#6711yescharacteristic
164Dic3—21no—
165Dic3—22no—
166Dic3—41no—
167Dic3—122no—
168Dic3—121no—
169Dic3—121no—
170Dic3—121no—
171Dic3—121no—
172Dic3—241no—
173Dic3Dic1212yescomplemented
174C32#46211yescharacteristic, complemented, Sylow
175Q8—32no—
176Q8—61no—
177Q8—61no—
178Q8—61no—
179Q8—181no—
180Q8—181no—
181C2×C4—31no—
182C2×C4—31no—
183C2×C4—61no—
184C2×C4—61no—
185C2×C4—181no—
186C2×C4—181no—
187C2×C4C2×D3211yescharacteristic
188C8—21no—
189C8—61no—
190C2×C3—21no—
191C2×C3—21no—
192C2×C3—21no—
193C2×C3#89511yescharacteristic
194C2×C3C2×Dic1211yescharacteristic
195C2×C3#90911yescharacteristic
196C2×C3D3×Q1611yescharacteristic
197C2×C3#91811yescharacteristic
198C2×C3#93911yescharacteristic
199C22D3×D1211yescharacteristic
200C4—32no—
201C4—61no—
202C4—62no—
203C4—121no—
204C4—361no—
205C4—361no—
206C4#379611yescharacteristic
207C4D3×#6711yescharacteristic
208C3—21no—
209C3#484911yescharacteristic, complemented
210C3#518211yescharacteristic, complemented
211C2D3×Dic1211yescharacteristic
212C2#6380111yescharacteristic
213C2#6388611yescharacteristic
subgroupG/H
centerC22D3×D12
commutator, cosocleC2×C32×C4C23
FrattiniC2×C4C2×D32
FittingC2×C32×C8C22
socle, prime radicalC22×C32C2×D4
hypercenterC2×C8D32
hypocenterC32#462
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×C8

Complements

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

NH ≅ G/Ntype
91. C3×Dic3Q16semidirect
173. Dic3Dic12semidirect
174. C32#462semidirect
209. C3#4849semidirect
210. C3#5182semidirect

Maximal subgroups

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

subgroupindex
C2×C3×Dic122
#638662
Dic3×Dic62
C2×#37192
#636432
C3×Dic3⋊C82
#635662
#51823
#48493