Finite Group Databasebeta

← #10591353

Subgroups — #10591353

Order 576.

346 conjugacy classes of subgroups; only the 108 normal subgroups are shown.

normal subgroups only

This group has 346 conjugacy classes of subgroups — too many to draw. Only its 108 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×C4⋊(C3⋊Dic3)C211yesmaximal, characteristic
2C2×C3×Dic3⋊C4C211yesmaximal, characteristic
3C2×Dic3⋊Dic3C212yesmaximal
4C3×C3⋊#95C211yesmaximal, characteristic
5#63579C212yesmaximal
6#4981—31nomaximal
7#4661—31nomaximal
8C22×C32×C4C2211yescharacteristic
9C22×C3⋊Dic3C2212yes—
10C22×C3×Dic3C2211yescharacteristic
11C22×C3×Dic3C2211yescharacteristic
12C22×C3×Dic3C2212yes—
13C4⋊(C3⋊Dic3)—22no—
14C3×Dic3⋊C4—22no—
15Dic3⋊Dic3—24no—
16C2×C3×C4⋊C4—31no—
17C2×C4⋊Dic3—31no—
18C2×C4⋊Dic3—31no—
19C2×C4⋊Dic3—32no—
20C2×C4⋊Dic3—61no—
21C2×Dic3⋊C4—32no—
22C2×Dic3⋊C4D311yescharacteristic
23C3×#95—31no—
24C3⋊#95—32no—
25C3⋊#95—32no—
26C3⋊#95D311yescharacteristic
27C23×C32C2311yescharacteristic
28C2×C32×C4—22no—
29C2×C32×C4D412yes—
30C2×C3⋊Dic3—22no—
31C2×C3⋊Dic3—22no—
32C2×C3⋊Dic3—22no—
33C2×C3×Dic3—22no—
34C2×C3×Dic3—21no—
35C2×C3×Dic3—21no—
36C2×C3×Dic3—22no—
37C2×C3×Dic3—21no—
38C2×C3×Dic3—22no—
39C2×C3×Dic3D412yes—
40C2×C3×Dic3Q814yes—
41#398—91noSylow
42C22×C3×C4—21no—
43C22×C3×C4—32no—
44C22×C3×C4—31no—
45C22×C3×C4—31no—
46C22×C3×C4C2×D311yescharacteristic
47C22×C3×C4C2×D311yescharacteristic
48C22×Dic3—32no—
49C22×Dic3—32no—
50C22×Dic3—62no—
51C22×Dic3C2×D311yescharacteristic
52C22×Dic3C2×D312yes—
53C22×Dic3C2×D311yescharacteristic
54C3×C4⋊C4—62no—
55C4⋊Dic3—64no—
56C4⋊Dic3—62no—
57C4⋊Dic3—62no—
58C4⋊Dic3—64no—
59Dic3⋊C4—22no—
60Dic3⋊C4—64no—
61C22×C32C2×D411yescharacteristic
62C22×C32C2×D411yescharacteristic
63C22×C32C2×Q812yes—
64C22×C32#4111yescharacteristic
65C22×C32#4112yes—
66C32×C4—22no—
67C3⋊Dic3—42no—
68C3×Dic3—22no—
69C3×Dic3—24no—
70C3×Dic3—41no—
71C2×C4⋊C4—31no—
72C2×C4⋊C4—92no—
73C2×C4⋊C4—91no—
74#95—31no—
75#95—92no—
76C23×C3—21no—
77C23×C3C22×D311yescharacteristic
78C23×C3C22×D311yescharacteristic
79C2×C3×C4—24no—
80C2×C3×C4—22no—
81C2×C3×C4—22no—
82C2×C3×C4—22no—
83C2×C3×C4—32no—
84C2×C3×C4—34no—
85C2×C3×C4—62no—
86C2×C3×C4—61no—
87C2×C3×C4—62no—
88C2×C3×C4—61no—
89C2×C3×C4—61no—
90C2×C3×C4—62no—
91C2×C3×C4D1212yes—
92C2×C3×C4#6712yes—
93C2×Dic3—21no—
94C2×Dic3—22no—
95C2×Dic3—22no—
96C2×Dic3—21no—
97C2×Dic3—22no—
98C2×Dic3—21no—
99C2×Dic3—64no—
100C2×Dic3—62no—
101C2×Dic3—62no—
102C2×Dic3—62no—
103C2×Dic3—62no—
104C2×Dic3—62no—
105C2×Dic3—62no—
106C2×Dic3—122no—
107C2×Dic3—122no—
108C2×Dic3Dic614yes—
109C2×Dic3D1212yes—
110C2×C32#12111yescharacteristic
111C2×C32C22⋊Q812yes—
112C2×C32#12311yescharacteristic
113C2×C32#12512yes—
114C2×C32C4⋊Q811yescharacteristic
115C22×C4—32no—
116C22×C4—31no—
117C22×C4—31no—
118C22×C4—92no—
119C22×C4D3211yescharacteristic
120C4⋊C4—62no—
121C4⋊C4—184no—
122C4⋊C4—182no—
123C22×C3—22no—
124C22×C3—21no—
125C22×C3—21no—
126C22×C3—22no—
127C22×C3—21no—
128C22×C3C2×Dic612yes—
129C22×C3C2×D1211yescharacteristic
130C22×C3C2×D1211yescharacteristic
131C22×C3#23511yescharacteristic
132C22×C3D3×D411yescharacteristic
133C22×C3#23712yes—
134C22×C3#23712yes—
135C22×C3D3×Q812yes—
136C22×C3#23911yescharacteristic
137C22×C3C2×#6711yescharacteristic
138C3×C4—24no—
139C3×C4—22no—
140C3×C4—22no—
141C3×C4—62no—
142C3×C4—64no—
143C3×C4—121no—
144Dic3—24no—
145Dic3—22no—
146Dic3—41no—
147Dic3—124no—
148Dic3—122no—
149Dic3—122no—
150C32#39811yescharacteristic, complemented, Sylow
151C23C2×D3211yescharacteristic
152C2×C4—22no—
153C2×C4—34no—
154C2×C4—32no—
155C2×C4—61no—
156C2×C4—62no—
157C2×C4—62no—
158C2×C4—61no—
159C2×C4—62no—
160C2×C4—61no—
161C2×C4—182no—
162C2×C4—182no—
163C2×C4—182no—
164C2×C4#62612yes—
165C2×C3—22no—
166C2×C3—21no—
167C2×C3—22no—
168C2×C3—21no—
169C2×C3—21no—
170C2×C3#86911yescharacteristic
171C2×C3#87812yes—
172C2×C3#88611yescharacteristic
173C2×C3#88912yes—
174C2×C3#89012yes—
175C2×C3#89511yescharacteristic
176C2×C3#89712yes—
177C2×C3#93011yescharacteristic
178C2×C3#93511yescharacteristic
179C2×C3#94411yescharacteristic
180C22D3×Dic612yes—
181C22#379911yescharacteristic
182C22D3×D1211yescharacteristic
183C22#380512yes—
184C22C2×#62611yescharacteristic
185C4—22no—
186C4—64no—
187C4—62no—
188C4—121no—
189C4—362no—
190C3—21no—
191C3#466111yescharacteristic, complemented
192C3#498111yescharacteristic, complemented
193C2#6384512yes—
194C2#6386611yescharacteristic
195C2#6389312yes—
196C2#6390911yescharacteristic
197C2#6396211yescharacteristic
subgroupG/H
center, FrattiniC23C2×D32
commutator, socle, prime radical, cosocleC23×C32C23
FittingC22×C32×C4C22
hypercenterC22×C4D32
hypocenterC32#398
LCSG ▷ C23×C32 ▷ C32
UCSC1 ◁ C23 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
150. C32#398semidirect
191. C3#4661semidirect
192. C3#4981semidirect

Maximal subgroups

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

subgroupindex
C2×C4⋊(C3⋊Dic3)2
C2×C3×Dic3⋊C42
C2×Dic3⋊Dic32
C3×C3⋊#952
#635792
#49813
#46613