Finite Group Databasebeta

← #10591694

Subgroups — #10591694

Order 576.

406 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 406 conjugacy classes of subgroups — too many to draw. Only its 98 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×#3757C211yesmaximal, characteristic
2C2×C3×#217C212yesmaximal
3C2×#3724C211yesmaximal, characteristic, complemented
4#63579C211yesmaximal, characteristic, complemented
5#63579C212yesmaximal, complemented
6#4952—32nomaximal
7C24×C32C2211yescharacteristic
8C22×C3⋊Dic3C2211yescharacteristic
9C22×C3⋊Dic3C2211yescharacteristic
10C22×C3×Dic3C2212yes—
11C22×C3×Dic3C2212yes—
12#3757—22no—
13C3×#217—24no—
14#3724—22no—
15C2×C3×C22⋊C4—32no—
16C2×#217—32no—
17C2×#217—61no—
18C2×#217D312yes—
19C2×Dic3⋊C4—32no—
20C3⋊#95—32no—
21C3⋊#95—32no—
22C3⋊#95—32no—
23C23×C32—21no—
24C23×C32—21no—
25C23×C32—22no—
26C23×C32—22no—
27C23×C32D412yes—
28C23×C32C2311yescharacteristic
29C2×C3⋊Dic3—21no—
30C2×C3⋊Dic3—22no—
31C2×C3⋊Dic3—22no—
32C2×C3⋊Dic3D412yes—
33C2×C3×Dic3—22no—
34C2×C3×Dic3—22no—
35C2×C3×Dic3—22no—
36C2×C3×Dic3—22no—
37C2×C3×Dic3—22no—
38C2×C3×Dic3—22no—
39#397—91noSylow
40C24×C3—21no—
41C24×C3C2×D312yes—
42C22×C3×C4—32no—
43C22×C3×C4—32no—
44C22×Dic3—32no—
45C22×Dic3—32no—
46C22×Dic3—61no—
47C22×Dic3—61no—
48C22×Dic3C2×D312yes—
49C22×Dic3C2×D312yes—
50C3×C22⋊C4—64no—
51#217—24no—
52#217—64no—
53#217—64no—
54Dic3⋊C4—64no—
55C22×C32—22no—
56C22×C32—24no—
57C22×C32—41no—
58C22×C32—41no—
59C22×C32—42no—
60C22×C32C2×D411yescharacteristic
61C22×C32C2×D411yescharacteristic
62C22×C32#4112yes—
63C22×C32#4112yes—
64C22×C32#4111yescharacteristic
65C3⋊Dic3—22no—
66C3⋊Dic3—41no—
67C3×Dic3—42no—
68C3×Dic3—42no—
69C2×C4⋊C4—91no—
70C2×C22⋊C4—32no—
71C2×C22⋊C4—91no—
72#95—91no—
73#95—92no—
74C23×C3—22no—
75C23×C3—22no—
76C23×C3—22no—
77C23×C3—22no—
78C23×C3—22no—
79C23×C3—22no—
80C23×C3—22no—
81C23×C3—22no—
82C23×C3—22no—
83C23×C3—21no—
84C23×C3—42no—
85C23×C3—42no—
86C23×C3#6714yes—
87C23×C3C22×D312yes—
88C2×C3×C4—62no—
89C2×C3×C4—62no—
90C2×C3×C4—62no—
91C2×C3×C4—62no—
92C2×C3×C4—62no—
93C2×C3×C4—62no—
94C2×Dic3—22no—
95C2×Dic3—22no—
96C2×Dic3—22no—
97C2×Dic3—22no—
98C2×Dic3—22no—
99C2×Dic3—22no—
100C2×Dic3—34no—
101C2×Dic3—62no—
102C2×Dic3—62no—
103C2×Dic3—62no—
104C2×Dic3—62no—
105C2×Dic3—62no—
106C2×Dic3—62no—
107C2×Dic3—62no—
108C2×Dic3—122no—
109C2×Dic3—122no—
110C2×C32—42no—
111C2×C32#12111yescharacteristic
112C2×C32#12311yescharacteristic
113C2×C32#12312yes—
114C2×C32#12411yescharacteristic
115C2×C32#12612yes—
116C24D3211yescharacteristic
117C22×C4—32no—
118C22×C4—32no—
119C22×C4—91no—
120C22×C4—91no—
121C4⋊C4—182no—
122C22⋊C4—64no—
123C22⋊C4—182no—
124C22×C3—24no—
125C22×C3—24no—
126C22×C3—24no—
127C22×C3—24no—
128C22×C3—22no—
129C22×C3—22no—
130C22×C3—21no—
131C22×C3—21no—
132C22×C3—21no—
133C22×C3—44no—
134C22×C3—42no—
135C22×C3—44no—
136C22×C3—42no—
137C22×C3—42no—
138C22×C3—42no—
139C22×C3—42no—
140C22×C3—42no—
141C22×C3#23512yes—
142C22×C3#23512yes—
143C22×C3D3×D412yes—
144C22×C3#23712yes—
145C22×C3#23712yes—
146C22×C3#23712yes—
147C22×C3C2×#6712yes—
148C3×C4—122no—
149C3×C4—122no—
150Dic3—42no—
151Dic3—42no—
152Dic3—64no—
153Dic3—122no—
154Dic3—122no—
155Dic3—122no—
156C32#39711yescharacteristic, complemented, Sylow
157C23—22no—
158C23—22no—
159C23—21no—
160C23—21no—
161C23#62512yes—
162C23C2×D3211yescharacteristic
163C2×C4—62no—
164C2×C4—62no—
165C2×C4—62no—
166C2×C4—62no—
167C2×C4—62no—
168C2×C4—62no—
169C2×C4—92no—
170C2×C4—182no—
171C2×C4—182no—
172C2×C4—181no—
173C2×C3—21no—
174C2×C3—22no—
175C2×C3—22no—
176C2×C3—21no—
177C2×C3—21no—
178C2×C3—44no—
179C2×C3—44no—
180C2×C3#87912yes—
181C2×C3#87912yes—
182C2×C3#88312yes—
183C2×C3#88512yes—
184C2×C3#92912yes—
185C2×C3#93512yes—
186C2×C3#93912yes—
187C22—24no—
188C22—22no—
189C22—41no—
190C22—42no—
191C22—41no—
192C22#380412yes—
193C22#380412yes—
194C22#380511yescharacteristic
195C22C2×#62511yescharacteristic
196C22#381111yescharacteristic
197C4—122no—
198C4—122no—
199C4—182no—
200C4—361no—
201C3—21no—
202C3#495212yescomplemented
203C2—42no—
204C2#6395311yescharacteristic
205C2#6395612yes—
206C2#6396112yes—
207C2#6397511yescharacteristic
208C2#6398011yescharacteristic
subgroupG/H
center, FrattiniC23C2×D32
commutator, socle, cosocleC23×C32C23
Fitting, prime radicalC24×C32C22
hypercenterC24D32
hypocenterC32#397
LCSG ▷ C23×C32 ▷ C32
UCSC1 ◁ C23 ◁ C24

Complements

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

NH ≅ G/Ntype
3. C2×#3724C2semidirect
4. #63579C2semidirect
5. #63579C2semidirect
156. C32#397semidirect
202. C3#4952semidirect

Maximal subgroups

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

subgroupindex
C2×#37572
C2×C3×#2172
C2×#37242
#635792
#49523