Finite Group Databasebeta

← #10591300

Subgroups — #10591300

Order 576.

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

normal subgroups only

This group has 514 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
1C22×C3×D12C211yesmaximal, characteristic
2C2×C3×Dic3⋊C4C211yesmaximal, characteristic, complemented
3C2×(C2×D3)⋊Dic3C212yesmaximal, complemented
4C2×(C2×D3)⋊Dic3C212yesmaximal, complemented
5#63658C211yesmaximal, characteristic, complemented
6#5222—31nomaximal
7#4991—31nomaximal
8C23×C3×D3C2212yes—
9C22×C32×C4C2211yescharacteristic
10C22×C3⋊Dic3C2212yes—
11C22×C3×Dic3C2211yescharacteristic
12C22×C3×Dic3C2211yescharacteristic
13C2×C3×D12—24no—
14C2×C3×D12—22no—
15C3×Dic3⋊C4—22no—
16(C2×D3)⋊Dic3—24no—
17(C2×D3)⋊Dic3—24no—
18C22×C3×D4—31no—
19C22×D12D311yescharacteristic
20C2×C3×C4⋊C4—31no—
21C2×#217—32no—
22C2×#217—32no—
23C2×(C2×D3)⋊C4—32no—
24C2×(C2×D3)⋊C4—32no—
25C2×Dic3⋊C4D311yescharacteristic, complemented
26C3⋊#95—31no—
27C3⋊#95—31no—
28C3⋊#95—61no—
29C23×C32C2311yescharacteristic
30C22×C3×D3—22no—
31C22×C3×D3—22no—
32C22×C3×D3—22no—
33C22×C3×D3—22no—
34C22×C3×D3—22no—
35C22×C3×D3—22no—
36C22×C3×D3D414yes—
37C2×C32×C4—22no—
38C2×C32×C4D412yes—
39C2×C3⋊Dic3—22no—
40C2×C3⋊Dic3—22no—
41C2×C3⋊Dic3—22no—
42C2×C3×Dic3—21no—
43C2×C3×Dic3—21no—
44C2×C3×Dic3—22no—
45C2×C3×Dic3—21no—
46C2×C3×Dic3D412yescomplemented
47C3×D12—28no—
48#394—91noSylow
49C24×C3—32no—
50C23×D3C2×D312yes—
51C2×C3×D4—64no—
52C2×C3×D4—62no—
53C22×C3×C4—21no—
54C22×C3×C4—31no—
55C22×C3×C4—31no—
56C22×C3×C4C2×D311yescharacteristic
57C22×C3×C4C2×D311yescharacteristic
58C22×Dic3—32no—
59C22×Dic3—32no—
60C22×Dic3—62no—
61C22×Dic3C2×D311yescharacteristic
62C22×Dic3C2×D311yescharacteristic
63C2×D12—24no—
64C2×D12—22no—
65C3×C4⋊C4—62no—
66#217—64no—
67#217—64no—
68(C2×D3)⋊C4—64no—
69(C2×D3)⋊C4—64no—
70Dic3⋊C4—22no—
71C22×C32C2×D411yescharacteristic
72C22×C32C2×D412yes—
73C22×C32C2×D411yescharacteristic
74C22×C32#4111yescharacteristic
75C22×C32#4112yes—
76C2×C3×D3—24no—
77C2×C3×D3—24no—
78C2×C3×D3—24no—
79C2×C3×D3—42no—
80C2×C3×D3—42no—
81C2×C3×D3—42no—
82C2×C3×D3—42no—
83C32×C4—22no—
84C3⋊Dic3—42no—
85C3×Dic3—22no—
86C3×Dic3—41no—
87C22×D4—31no—
88C2×C4⋊C4—31no—
89C2×C22⋊C4—92no—
90C2×C22⋊C4—92no—
91#95—91no—
92C23×C3—21no—
93C23×C3—34no—
94C23×C3—62no—
95C23×C3—62no—
96C23×C3—62no—
97C23×C3—62no—
98C23×C3—62no—
99C23×C3—62no—
100C23×C3C22×D311yescharacteristic
101C23×C3C22×D311yescharacteristic
102C22×D3—22no—
103C22×D3—22no—
104C22×D3—22no—
105C22×D3—22no—
106C22×D3—22no—
107C22×D3—22no—
108C22×D3#6714yes—
109C3×D4—68no—
110C2×C3×C4—22no—
111C2×C3×C4—22no—
112C2×C3×C4—22no—
113C2×C3×C4—32no—
114C2×C3×C4—42no—
115C2×C3×C4—61no—
116C2×C3×C4—61no—
117C2×C3×C4—62no—
118C2×C3×C4—61no—
119C2×C3×C4#6712yes—
120C2×C3×C4#6712yes—
121C2×Dic3—21no—
122C2×Dic3—21no—
123C2×Dic3—22no—
124C2×Dic3—21no—
125C2×Dic3—64no—
126C2×Dic3—62no—
127C2×Dic3—62no—
128C2×Dic3—62no—
129C2×Dic3—62no—
130C2×Dic3—62no—
131C2×Dic3—62no—
132C2×Dic3—122no—
133C2×Dic3—122no—
134C2×Dic3D1212yescomplemented
135D12—28no—
136C2×C32#12011yescharacteristic
137C2×C32#12112yes—
138C2×C32#12111yescharacteristic
139C2×C32#12312yes—
140C2×C32#12411yescharacteristic
141C3×D3—44no—
142C24—32no—
143C2×D4—64no—
144C2×D4—62no—
145C22×C4—31no—
146C22×C4—31no—
147C22×C4—92no—
148C22×C4D3211yescharacteristic
149C4⋊C4—62no—
150C22⋊C4—184no—
151C22⋊C4—184no—
152C22×C3—21no—
153C22×C3—22no—
154C22×C3—21no—
155C22×C3—21no—
156C22×C3—22no—
157C22×C3—64no—
158C22×C3—64no—
159C22×C3—64no—
160C22×C3—122no—
161C22×C3—122no—
162C22×C3—122no—
163C22×C3—122no—
164C22×C3C2×D1211yescharacteristic
165C22×C3#23511yescharacteristic
166C22×C3D3×D411yescharacteristic
167C22×C3D3×D412yes—
168C22×C3#23711yescharacteristic
169C22×C3#23712yes—
170C22×C3#23912yes—
171C22×C3C2×#6711yescharacteristic
172C22×C3C2×#6711yescharacteristic
173C22×C3C2×#6712yes—
174C2×D3—24no—
175C2×D3—24no—
176C2×D3—24no—
177C2×D3—42no—
178C2×D3—42no—
179C2×D3—42no—
180C2×D3—42no—
181C3×C4—22no—
182C3×C4—22no—
183C3×C4—42no—
184C3×C4—62no—
185C3×C4—121no—
186Dic3—22no—
187Dic3—41no—
188Dic3—124no—
189Dic3—122no—
190Dic3—122no—
191C32#39411yescharacteristic, complemented, Sylow
192C23—34no—
193C23—62no—
194C23—62no—
195C23—62no—
196C23—62no—
197C23—62no—
198C23—62no—
199C23C2×D3211yescharacteristic
200D4—68no—
201C2×C4—22no—
202C2×C4—32no—
203C2×C4—61no—
204C2×C4—61no—
205C2×C4—61no—
206C2×C4—62no—
207C2×C4—182no—
208C2×C4—182no—
209C2×C4—182no—
210C2×C4#62512yes—
211C2×C3—21no—
212C2×C3—22no—
213C2×C3—22no—
214C2×C3—21no—
215C2×C3—21no—
216C2×C3—124no—
217C2×C3#89412yes—
218C2×C3#89512yes—
219C2×C3#93011yescharacteristic
220C2×C3#93512yes—
221C2×C3#93611yescharacteristic
222C2×C3#93711yescharacteristic
223C2×C3#93811yescharacteristic
224C2×C3#93912yes—
225C2×C3#94711yescharacteristic
226C2×C3#95311yescharacteristic
227D3—44no—
228C22—64no—
229C22—64no—
230C22—64no—
231C22—122no—
232C22—122no—
233C22—122no—
234C22—122no—
235C22#379511yescharacteristic
236C22#379612yes—
237C22D3×D1211yescharacteristic
238C22C2×#62511yescharacteristic
239C22D3×#6712yes—
240C4—22no—
241C4—62no—
242C4—121no—
243C4—362no—
244C3—21no—
245C3#499111yescharacteristic, complemented
246C3#522211yescharacteristic, complemented
247C2—124no—
248C2#6386311yescharacteristic
249C2#6388412yes—
250C2#6388612yes—
251C2#6390811yescharacteristic
252C2#6397611yescharacteristic
subgroupG/H
center, FrattiniC23C2×D32
commutator, socle, cosocleC23×C32C23
FittingC22×C32×C4C22
prime radicalC22×C3×D12C2
hypercenterC22×C4D32
hypocenterC32#394
LCSG ▷ C23×C32 ▷ C32
UCSC1 ◁ C23 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
2. C2×C3×Dic3⋊C4C2semidirect
3. C2×(C2×D3)⋊Dic3C2semidirect
4. C2×(C2×D3)⋊Dic3C2semidirect
5. #63658C2semidirect
25. C2×Dic3⋊C4D3semidirect
46. C2×C3×Dic3D4semidirect
134. C2×Dic3D12semidirect
191. C32#394semidirect
245. C3#4991semidirect
246. C3#5222semidirect

Maximal subgroups

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

subgroupindex
C22×C3×D122
C2×C3×Dic3⋊C42
C2×(C2×D3)⋊Dic32
#636582
#52223
#49913