Finite Group Databasebeta

← #10591216

Subgroups — #10591216

Order 576.

406 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 406 conjugacy classes of subgroups — too many to draw. Only its 92 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×#3752C211yesmaximal, characteristic
2C2×#3724C211yesmaximal, characteristic, complemented
3C2×#3722C212yesmaximal
4C3×C3⋊#95C212yesmaximal, complemented
5#63579C211yesmaximal, characteristic, complemented
6#4675—32nomaximal
7C23×C3⋊D3C2211yescharacteristic
8C22×C32×C4C2211yescharacteristic
9C22×C3⋊Dic3C2211yescharacteristic
10C22×C3×Dic3C2212yes—
11C22×C3×Dic3C2212yes—
12#3752—22no—
13#3724—22no—
14#3722—24no—
15C2×(C2×D3)⋊C4—32no—
16C2×(C2×D3)⋊C4—32no—
17C2×(C2×D3)⋊C4—32no—
18C2×(C2×D3)⋊C4—61no—
19C2×Dic3⋊C4—32no—
20C3×#95—32no—
21C3⋊#95—32no—
22C3⋊#95D312yescomplemented
23C23×C32C2311yescharacteristic
24C22×C3⋊D3—21no—
25C22×C3⋊D3—21no—
26C22×C3⋊D3—22no—
27C22×C3⋊D3—22no—
28C22×C3⋊D3D412yes—
29C2×C32×C4—21no—
30C2×C32×C4—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
40C23×D3—32no—
41C23×D3—61no—
42C22×C3×C4—21no—
43C22×C3×C4—32no—
44C22×C3×C4—32no—
45C22×C3×C4C2×D312yes—
46C22×Dic3—32no—
47C22×Dic3—61no—
48C22×Dic3C2×D312yes—
49C22×Dic3C2×D312yes—
50(C2×D3)⋊C4—64no—
51(C2×D3)⋊C4—64no—
52(C2×D3)⋊C4—64no—
53(C2×D3)⋊C4—64no—
54Dic3⋊C4—64no—
55C22×C32C2×D411yescharacteristic
56C22×C32C2×D411yescharacteristic
57C22×C32#4111yescharacteristic
58C22×C32#4112yes—
59C22×C32#4112yes—
60C2×C3⋊D3—22no—
61C2×C3⋊D3—24no—
62C2×C3⋊D3—41no—
63C2×C3⋊D3—41no—
64C2×C3⋊D3—42no—
65C32×C4—41no—
66C3⋊Dic3—22no—
67C3×Dic3—42no—
68C3×Dic3—42no—
69C2×C4⋊C4—91no—
70C2×C22⋊C4—92no—
71C2×C22⋊C4—91no—
72#95—32no—
73#95—91no—
74C23×C3—21no—
75C23×C3C22×D312yes—
76C22×D3—34no—
77C22×D3—62no—
78C22×D3—62no—
79C22×D3—62no—
80C22×D3—62no—
81C22×D3—62no—
82C22×D3—62no—
83C22×D3—62no—
84C22×D3—62no—
85C22×D3—62no—
86C22×D3—122no—
87C22×D3—122no—
88C2×C3×C4—22no—
89C2×C3×C4—22no—
90C2×C3×C4—22no—
91C2×C3×C4—22no—
92C2×C3×C4—42no—
93C2×C3×C4—62no—
94C2×C3×C4—62no—
95C2×C3×C4—62no—
96C2×C3×C4—62no—
97C2×C3×C4—62no—
98C2×C3×C4—62no—
99C2×Dic3—22no—
100C2×Dic3—22no—
101C2×Dic3—22no—
102C2×Dic3—22no—
103C2×Dic3—22no—
104C2×Dic3—22no—
105C2×Dic3—34no—
106C2×Dic3—62no—
107C2×Dic3—62no—
108C2×Dic3—62no—
109C2×Dic3—122no—
110C2×C32#12111yescharacteristic
111C2×C32#12312yes—
112C2×C32#12311yescharacteristic
113C2×C32#12411yescharacteristic
114C2×C32#12612yes—
115C3⋊D3—42no—
116C24—91no—
117C22×C4—32no—
118C22×C4—32no—
119C22×C4—91no—
120C22×C4D3211yescharacteristic
121C4⋊C4—182no—
122C22⋊C4—182no—
123C22⋊C4—184no—
124C22×C3—21no—
125C22×C3—22no—
126C22×C3—21no—
127C22×C3—22no—
128C22×C3—21no—
129C22×C3#23512yes—
130C22×C3#23512yes—
131C22×C3#23512yes—
132C22×C3D3×D412yes—
133C22×C3D3×D412yes—
134C22×C3#23712yes—
135C22×C3#23912yes—
136C2×D3—64no—
137C2×D3—64no—
138C2×D3—64no—
139C2×D3—64no—
140C2×D3—124no—
141C2×D3—124no—
142C2×D3—122no—
143C2×D3—122no—
144C2×D3—122no—
145C2×D3—122no—
146C2×D3—122no—
147C2×D3—122no—
148C3×C4—42no—
149C3×C4—42no—
150C3×C4—122no—
151C3×C4—122no—
152Dic3—42no—
153Dic3—42no—
154Dic3—64no—
155Dic3—122no—
156C32#39711yescharacteristic, complemented, Sylow
157C23—92no—
158C23—181no—
159C23—182no—
160C23—182no—
161C23—181no—
162C23C2×D3211yescharacteristic
163C2×C4—22no—
164C2×C4—21no—
165C2×C4—62no—
166C2×C4—62no—
167C2×C4—62no—
168C2×C4—62no—
169C2×C4—62no—
170C2×C4—62no—
171C2×C4—92no—
172C2×C4—182no—
173C2×C3—22no—
174C2×C3—22no—
175C2×C3—21no—
176C2×C3—21no—
177C2×C3—21no—
178C2×C3#87612yes—
179C2×C3#88312yes—
180C2×C3#88412yes—
181C2×C3#88512yes—
182C2×C3#89412yes—
183C2×C3#89412yes—
184C2×C3#89812yes—
185D3—124no—
186D3—124no—
187C22—184no—
188C22—182no—
189C22—362no—
190C22—361no—
191C22—361no—
192C22#379811yescharacteristic
193C22#379912yes—
194C22#380211yescharacteristic
195C22#380412yes—
196C22#381111yescharacteristic
197C4—41no—
198C4—122no—
199C4—122no—
200C4—182no—
201C3—21no—
202C3#467512yescomplemented
203C2—362no—
204C2#6385412yes—
205C2#6386812yes—
206C2#6389711yescharacteristic
207C2#6391211yescharacteristic
208C2#6395311yescharacteristic
subgroupG/H
center, FrattiniC23C2×D32
commutator, socle, cosocleC23×C32C23
FittingC22×C32×C4C22
prime radicalC23×C3⋊D3C22
hypercenterC22×C4D32
hypocenterC32#397
LCSG ▷ C23×C32 ▷ C32
UCSC1 ◁ C23 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
2. C2×#3724C2semidirect
4. C3×C3⋊#95C2semidirect
5. #63579C2semidirect
22. C3⋊#95D3semidirect
156. C32#397semidirect
202. C3#4675semidirect

Maximal subgroups

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

subgroupindex
C2×#37522
C2×#37242
C2×#37222
C3×C3⋊#952
#635792
#46753