Finite Group Databasebeta

← #10591346

Subgroups — #10591346

Order 576.

410 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 410 conjugacy classes of subgroups — too many to draw. Only its 114 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
1C22×C3×Dic6C211yesmaximal, characteristic
2C2×#3750C211yesmaximal, characteristic
3C2×Dic3⋊Dic3C212yesmaximal
4C3×C3⋊#95C211yesmaximal, characteristic
5#63579C212yesmaximal
6#5230—31nomaximal
7#4656—31nomaximal
8C22×C32×C4C2211yescharacteristic
9C22×C3⋊Dic3C2211yescharacteristic
10C22×C3⋊Dic3C2211yescharacteristic
11C22×C3×Dic3C2212yes—
12C22×C3×Dic3C2212yes—
13C2×C3×Dic6—22no—
14C2×C3×Dic6—24no—
15#3750—22no—
16Dic3⋊Dic3—24no—
17C22×C3×Q8—31no—
18C22×Dic6D311yescharacteristic
19C2×C4⋊Dic3—32no—
20C2×Dic3⋊C4—31no—
21C2×Dic3⋊C4—31no—
22C2×Dic3⋊C4—32no—
23C2×Dic3⋊C4—61no—
24C3×#95—31no—
25C3⋊#95—32no—
26C3⋊#95—32no—
27C3⋊#95D311yescharacteristic
28C23×C32C2311yescharacteristic
29C2×C32×C4—22no—
30C2×C32×C4D412yes—
31C2×C3⋊Dic3—21no—
32C2×C3⋊Dic3—21no—
33C2×C3⋊Dic3—21no—
34C2×C3⋊Dic3—22no—
35C2×C3⋊Dic3Q812yescomplemented
36C2×C3×Dic3—22no—
37C2×C3×Dic3—22no—
38C2×C3×Dic3—22no—
39C2×C3×Dic3—22no—
40C2×C3×Dic3D414yes—
41C2×C3×Dic3Q814yescomplemented
42C3×Dic6—28no—
43#395—91noSylow
44C2×C3×Q8—64no—
45C2×C3×Q8—62no—
46C22×C3×C4—21no—
47C22×C3×C4—32no—
48C22×C3×C4—32no—
49C22×C3×C4C2×D311yescharacteristic
50C22×C3×C4C2×D311yescharacteristic
51C22×Dic3—31no—
52C22×Dic3—31no—
53C22×Dic3—31no—
54C22×Dic3—31no—
55C22×Dic3—61no—
56C22×Dic3—61no—
57C22×Dic3C2×D312yes—
58C22×Dic3C2×D312yes—
59C2×Dic6—22no—
60C2×Dic6—24no—
61C4⋊Dic3—64no—
62Dic3⋊C4—64no—
63Dic3⋊C4—64no—
64Dic3⋊C4—62no—
65Dic3⋊C4—62no—
66C22×C32C2×D411yescharacteristic
67C22×C32C2×D412yes—
68C22×C32C2×Q812yes—
69C22×C32C2×Q811yescharacteristic
70C22×C32#4111yescharacteristic
71C32×C4—22no—
72C3⋊Dic3—22no—
73C3⋊Dic3—41no—
74C3×Dic3—24no—
75C3×Dic3—24no—
76C22×Q8—31no—
77C2×C4⋊C4—92no—
78C2×C4⋊C4—91no—
79#95—31no—
80#95—92no—
81C23×C3—21no—
82C23×C3C22×D311yescharacteristic
83C23×C3C22×D311yescharacteristic
84C3×Q8—68no—
85C2×C3×C4—22no—
86C2×C3×C4—22no—
87C2×C3×C4—22no—
88C2×C3×C4—34no—
89C2×C3×C4—34no—
90C2×C3×C4—42no—
91C2×C3×C4—62no—
92C2×C3×C4—62no—
93C2×C3×C4—62no—
94C2×C3×C4—62no—
95C2×C3×C4D1212yes—
96C2×C3×C4#6712yes—
97C2×Dic3—22no—
98C2×Dic3—22no—
99C2×Dic3—22no—
100C2×Dic3—22no—
101C2×Dic3—32no—
102C2×Dic3—32no—
103C2×Dic3—62no—
104C2×Dic3—62no—
105C2×Dic3—62no—
106C2×Dic3—61no—
107C2×Dic3—61no—
108C2×Dic3—61no—
109C2×Dic3—61no—
110C2×Dic3—62no—
111C2×Dic3—61no—
112C2×Dic3—62no—
113C2×Dic3—61no—
114C2×Dic3—62no—
115C2×Dic3—122no—
116C2×Dic3Dic614yescomplemented
117C2×Dic3#6714yes—
118Dic6—28no—
119C2×C32#12011yescharacteristic
120C2×C32C22⋊Q812yes—
121C2×C32C22⋊Q811yescharacteristic
122C2×C32C4⋊Q811yescharacteristic
123C2×C32C4⋊Q812yes—
124C2×Q8—64no—
125C2×Q8—62no—
126C22×C4—32no—
127C22×C4—32no—
128C22×C4—91no—
129C22×C4—91no—
130C22×C4D3211yescharacteristic
131C4⋊C4—184no—
132C4⋊C4—182no—
133C22×C3—22no—
134C22×C3—21no—
135C22×C3—21no—
136C22×C3—21no—
137C22×C3—22no—
138C22×C3C2×Dic612yes—
139C22×C3C2×D1211yescharacteristic
140C22×C3D3×D412yes—
141C22×C3#23711yescharacteristic
142C22×C3D3×Q812yes—
143C22×C3D3×Q811yescharacteristic
144C22×C3D3×Q811yescharacteristic
145C22×C3#23911yescharacteristic
146C22×C3C2×#6711yescharacteristic
147C22×C3C2×#6712yes—
148C3×C4—22no—
149C3×C4—22no—
150C3×C4—42no—
151C3×C4—64no—
152C3×C4—64no—
153Dic3—24no—
154Dic3—24no—
155Dic3—64no—
156Dic3—62no—
157Dic3—62no—
158Dic3—122no—
159Dic3—121no—
160Dic3—121no—
161C32#39511yescharacteristic, complemented, Sylow
162C23C2×D3211yescharacteristic
163Q8—68no—
164C2×C4—22no—
165C2×C4—34no—
166C2×C4—34no—
167C2×C4—62no—
168C2×C4—62no—
169C2×C4—62no—
170C2×C4—62no—
171C2×C4—92no—
172C2×C4—181no—
173C2×C4—181no—
174C2×C4—181no—
175C2×C4—182no—
176C2×C4#62612yes—
177C2×C3—22no—
178C2×C3—21no—
179C2×C3—21no—
180C2×C3—22no—
181C2×C3—21no—
182C2×C3#86911yescharacteristic
183C2×C3#87812yes—
184C2×C3#88211yescharacteristic
185C2×C3#88812yes—
186C2×C3#89711yescharacteristic
187C2×C3#94412yes—
188C2×C3#94411yescharacteristic
189C2×C3#94611yescharacteristic
190C2×C3#94612yes—
191C2×C3#95311yescharacteristic
192C22D3×Dic612yes—
193C22#379611yescharacteristic
194C22#379711yescharacteristic
195C22C2×#62611yescharacteristic
196C22D3×#6712yes—
197C4—22no—
198C4—64no—
199C4—64no—
200C4—182no—
201C4—361no—
202C3—21no—
203C3#465611yescharacteristic, complemented
204C3#523011yescharacteristic, complemented
205C2#6383912yes—
206C2#6386611yescharacteristic
207C2#6387111yescharacteristic
208C2#6388712yes—
209C2#6397711yescharacteristic
subgroupG/H
center, FrattiniC23C2×D32
commutator, socle, prime radical, cosocleC23×C32C23
FittingC22×C32×C4C22
hypercenterC22×C4D32
hypocenterC32#395
LCSG ▷ C23×C32 ▷ C32
UCSC1 ◁ C23 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
35. C2×C3⋊Dic3Q8semidirect
41. C2×C3×Dic3Q8semidirect
116. C2×Dic3Dic6semidirect
161. C32#395semidirect
203. C3#4656semidirect
204. C3#5230semidirect

Maximal subgroups

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

subgroupindex
C22×C3×Dic62
C2×#37502
C2×Dic3⋊Dic32
C3×C3⋊#952
#635792
#52303
#46563