Finite Group Databasebeta

← #10590798

Subgroups — #10590798

Order 576.

235 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 235 conjugacy classes of subgroups — too many to draw. Only its 64 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#63873C211yesmaximal, characteristic
3#63843C211yesmaximal, characteristic
4C2×#3718C211yesmaximal, characteristic
5#63641C211yesmaximal, characteristic
6C3×#816C211yesmaximal, characteristic
7#63565C211yesmaximal, characteristic
8#5182—31nomaximal
9#4795—31nomaximal
10C2×C3×Dic6C2211yescharacteristic
11C2×C3×Dic6C2211yescharacteristic
12C2×C3×Dic6C2211yescharacteristic
13C2×C32×C8C2211yescharacteristic
14C4×C3⋊Dic3C2211yescharacteristic
15C2×C3⋊(C3⋊C8)C2211yescharacteristic
16C3×C4⋊Dic3C2211yescharacteristic
17C3×Dic12—22no—
18#3724—21no—
19#3724—21no—
20Dic3⋊Dic3—21no—
21Dic32—21no—
22#3718—22no—
23C2×C3×Q16—31no—
24Q8×Dic3—31no—
25#944—31no—
26#944—31no—
27C2×#216—31no—
28C2×#216—31no—
29C2×Dic12D311yescharacteristic
30#887—31no—
31C3×#103—31no—
32#835—31no—
33#816D311yescharacteristic
34Dic3⋊C8—31no—
35Dic3⋊C8—31no—
36Dic3⋊C8—61no—
37#810—31no—
38C2×C32×C4C2311yescharacteristic
39C2×C3⋊Dic3D411yescharacteristic
40C2×C3⋊Dic3D411yescharacteristic
41C2×C3×Dic3—21no—
42C2×C3×Dic3—21no—
43C2×C3×Dic3—21no—
44C2×C3×Dic3—21no—
45C3×Dic6—21no—
46C3×Dic6—21no—
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×C3×Q8—31no—
57C2×Dic6C2×D311yescharacteristic
58C2×Dic6C2×D311yescharacteristic
59C2×Dic6C2×D311yescharacteristic
60C3×Q16—62no—
61C2×C3×C8—21no—
62C2×C3×C8C2×D311yescharacteristic
63C2×C3×C8C2×D311yescharacteristic
64C3×C4⋊C4—31no—
65#216—62no—
66#216—62no—
67C4⋊Dic3—61no—
68C4⋊Dic3C2×D311yescharacteristic
69Dic3⋊C4—61no—
70Dic3⋊C4—61no—
71Dic3⋊C4—61no—
72Dic3⋊C4—61no—
73Dic3⋊C4—61no—
74C4×Dic3—31no—
75C4×Dic3—31no—
76C4×Dic3—61no—
77C4×Dic3—61no—
78C4×Dic3—61no—
79C2×C3⋊C8—31no—
80C2×C3⋊C8—31no—
81C2×C3⋊C8—61no—
82Dic12—22no—
83C22×C32C2×D411yescharacteristic
84C32×C4C2×D411yescharacteristic
85C32×C4#4111yescharacteristic
86C3⋊Dic3—21no—
87C3⋊Dic3Q1612yescomplemented
88C3×Dic3—22no—
89C3×Dic3—41no—
90C3×Dic3—41no—
91C3×Dic3—41no—
92C2×Q16—31no—
93C2×Q16—91no—
94C4⋊Q8—91no—
95C4×Q8—91no—
96C4⋊C8—91no—
97#103—31no—
98#103—91no—
99C3×Q8—32no—
100C3×Q8—61no—
101C3×Q8—61no—
102C3×Q8—61no—
103C3×Q8—61no—
104C3×Q8—61no—
105C2×C3×C4—21no—
106C2×C3×C4—61no—
107C2×C3×C4—61no—
108C2×C3×C4—61no—
109C2×C3×C4—61no—
110C2×C3×C4C22×D311yescharacteristic
111C2×C3×C4C22×D311yescharacteristic
112C2×Dic3—21no—
113C2×Dic3—21no—
114C2×Dic3—21no—
115C2×Dic3—21no—
116C2×Dic3—31no—
117C2×Dic3—31no—
118C2×Dic3—31no—
119C2×Dic3—31no—
120C2×Dic3—61no—
121C2×Dic3—61no—
122Dic6—21no—
123Dic6—21no—
124Dic6—21no—
125Dic6—21no—
126Dic6—21no—
127Dic6#6712yes—
128C3×C8—21no—
129C3×C8—21no—
130C3×C8—41no—
131C3⋊C8—61no—
132C3⋊C8—61no—
133C3⋊C8—121no—
134C2×C32#12111yescharacteristic
135C2×C32C2×Q1611yescharacteristic
136C2×C32#13711yescharacteristic
137C2×Q8—31no—
138C2×Q8—31no—
139C2×Q8—31no—
140Q16—62no—
141Q16—182no—
142C2×C8—91no—
143C2×C8D3211yescharacteristic
144C4⋊C4—31no—
145C4⋊C4—181no—
146C4⋊C4—181no—
147C4⋊C4—181no—
148C42—91no—
149C42—181no—
150C22×C3—21no—
151C22×C3D3×D411yescharacteristic
152C22×C3D3×D411yescharacteristic
153C3×C4—21no—
154C3×C4—21no—
155C3×C4—62no—
156C3×C4—121no—
157C3×C4—121no—
158C3×C4—121no—
159C3×C4#23511yescharacteristic
160C3×C4D3×D411yescharacteristic
161C3×C4#23711yescharacteristic
162C3×C4C2×#6711yescharacteristic
163Dic3—22no—
164Dic3—32no—
165Dic3—32no—
166Dic3—41no—
167Dic3—41no—
168Dic3—41no—
169Dic3—62no—
170Dic3—61no—
171Dic3—61no—
172Dic3—121no—
173C32#46211yescharacteristic, complemented, Sylow
174Q8—32no—
175Q8—61no—
176Q8—61no—
177Q8—61no—
178Q8—61no—
179Q8—61no—
180C2×C4—61no—
181C2×C4—61no—
182C2×C4—61no—
183C2×C4—61no—
184C2×C4—91no—
185C2×C4—91no—
186C2×C4C2×D3211yescharacteristic
187C8—21no—
188C8—181no—
189C2×C3—21no—
190C2×C3—21no—
191C2×C3—21no—
192C2×C3#88411yescharacteristic
193C2×C3#91511yescharacteristic
194C2×C3D3×Q1611yescharacteristic
195C2×C3D3×Q1611yescharacteristic
196C2×C3#91811yescharacteristic
197C2×C3#93911yescharacteristic
198C22#380211yescharacteristic
199C4—62no—
200C4—92no—
201C4—121no—
202C4—121no—
203C4—121no—
204C4—181no—
205C4#379511yescharacteristic
206C4D3×#6711yescharacteristic
207C3—21no—
208C3#479511yescharacteristic, complemented
209C3#518211yescharacteristic, complemented
210C2#6380211yescharacteristic
211C2#6381111yescharacteristic
212C2#6388511yescharacteristic
subgroupG/H
centerC22#3802
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
87. C3⋊Dic3Q16semidirect
173. C32#462semidirect
208. C3#4795semidirect
209. 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
#638732
#638432
C2×#37182
#636412
C3×#8162
#635652
#51823
#47953