Finite Group Databasebeta

← #10591324

Subgroups — #10591324

Order 576.

410 conjugacy classes of subgroups; only the 104 normal subgroups are shown.

normal subgroups only

This group has 410 conjugacy classes of subgroups — too many to draw. Only its 104 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×C3×(C2×D3)⋊C4C211yesmaximal, characteristic
2C2×C3×C4⋊Dic3C211yesmaximal, characteristic, complemented
3C2×#3724C212yesmaximal, complemented
4C2×(C2×D3)⋊Dic3C212yesmaximal
5#63658C211yesmaximal, characteristic, complemented
6#4993—31nomaximal
7#4949—31nomaximal
8C23×C3×D3C2211yescharacteristic
9C22×C32×C4C2211yescharacteristic
10C22×C3⋊Dic3C2212yes—
11C22×C3×Dic3C2212yes—
12C22×C3×Dic3C2211yescharacteristic
13C3×(C2×D3)⋊C4—22no—
14C3×C4⋊Dic3—22no—
15#3724—24no—
16(C2×D3)⋊Dic3—24no—
17C2×C3×C4⋊C4—31no—
18C2×C3×C22⋊C4—31no—
19C2×#217—32no—
20C2×(C2×D3)⋊C4—32no—
21C2×(C2×D3)⋊C4D311yescharacteristic
22C2×C4⋊Dic3D311yescharacteristic, complemented
23C2×Dic3⋊C4—32no—
24C2×Dic3⋊C4—32no—
25C3⋊#95—31no—
26C3⋊#95—31no—
27C3⋊#95—61no—
28C23×C32C2311yescharacteristic
29C22×C3×D3—22no—
30C22×C3×D3—21no—
31C22×C3×D3—21no—
32C22×C3×D3—22no—
33C22×C3×D3Q812yes—
34C2×C32×C4—22no—
35C2×C32×C4D412yes—
36C2×C3⋊Dic3—22no—
37C2×C3⋊Dic3—22no—
38C2×C3⋊Dic3D414yes—
39C2×C3×Dic3—22no—
40C2×C3×Dic3—22no—
41C2×C3×Dic3—22no—
42C2×C3×Dic3—22no—
43C2×C3×Dic3—21no—
44#399—91noSylow
45C24×C3—31no—
46C23×D3C2×D311yescharacteristic
47C22×C3×C4—21no—
48C22×C3×C4—31no—
49C22×C3×C4—32no—
50C22×C3×C4C2×D311yescharacteristic
51C22×C3×C4C2×D311yescharacteristic
52C22×Dic3—32no—
53C22×Dic3—32no—
54C22×Dic3—62no—
55C22×Dic3C2×D311yescharacteristic
56C22×Dic3C2×D312yes—
57C3×C4⋊C4—62no—
58C3×C22⋊C4—62no—
59#217—64no—
60(C2×D3)⋊C4—22no—
61(C2×D3)⋊C4—64no—
62C4⋊Dic3—22no—
63Dic3⋊C4—64no—
64Dic3⋊C4—64no—
65C22×C32C2×D412yes—
66C22×C32C2×D411yescharacteristic
67C22×C32C2×Q811yescharacteristic
68C22×C32#4112yes—
69C22×C32#4111yescharacteristic
70C2×C3×D3—24no—
71C2×C3×D3—22no—
72C2×C3×D3—42no—
73C2×C3×D3—41no—
74C2×C3×D3—41no—
75C32×C4—22no—
76C3⋊Dic3—24no—
77C3×Dic3—42no—
78C3×Dic3—41no—
79C2×C4⋊C4—31no—
80C2×C4⋊C4—92no—
81C2×C22⋊C4—31no—
82C2×C22⋊C4—92no—
83#95—91no—
84C23×C3—21no—
85C23×C3—32no—
86C23×C3—62no—
87C23×C3—62no—
88C23×C3—61no—
89C23×C3—61no—
90C23×C3C22×D311yescharacteristic
91C23×C3C22×D311yescharacteristic
92C22×D3—21no—
93C22×D3—22no—
94C22×D3—22no—
95C22×D3—21no—
96C22×D3Dic612yes—
97C2×C3×C4—22no—
98C2×C3×C4—22no—
99C2×C3×C4—22no—
100C2×C3×C4—42no—
101C2×C3×C4—62no—
102C2×C3×C4—62no—
103C2×C3×C4—62no—
104C2×C3×C4—61no—
105C2×C3×C4—62no—
106C2×C3×C4#6712yes—
107C2×C3×C4#6712yes—
108C2×Dic3—22no—
109C2×Dic3—22no—
110C2×Dic3—22no—
111C2×Dic3—22no—
112C2×Dic3—21no—
113C2×Dic3—34no—
114C2×Dic3—34no—
115C2×Dic3—64no—
116C2×Dic3—62no—
117C2×Dic3—62no—
118C2×Dic3—62no—
119C2×Dic3—62no—
120C2×Dic3—122no—
121C2×Dic3—122no—
122C2×C32C22⋊Q812yes—
123C2×C32C22⋊Q811yescharacteristic
124C2×C32#12312yes—
125C2×C32#12311yescharacteristic
126C2×C32#12711yescharacteristic
127C3×D3—42no—
128C24—31no—
129C22×C4—31no—
130C22×C4—32no—
131C22×C4—92no—
132C22×C4D3211yescharacteristic
133C4⋊C4—62no—
134C4⋊C4—184no—
135C22⋊C4—62no—
136C22⋊C4—184no—
137C22×C3—21no—
138C22×C3—21no—
139C22×C3—22no—
140C22×C3—22no—
141C22×C3—21no—
142C22×C3—64no—
143C22×C3—62no—
144C22×C3—122no—
145C22×C3—121no—
146C22×C3—121no—
147C22×C3C2×Dic611yescharacteristic
148C22×C3#23511yescharacteristic
149C22×C3#23512yes—
150C22×C3D3×D412yes—
151C22×C3D3×D412yes—
152C22×C3#23712yes—
153C22×C3D3×Q811yescharacteristic
154C22×C3#23911yescharacteristic
155C22×C3C2×#6711yescharacteristic
156C22×C3C2×#6711yescharacteristic
157C2×D3—24no—
158C2×D3—22no—
159C2×D3—42no—
160C2×D3—41no—
161C2×D3—41no—
162C3×C4—22no—
163C3×C4—22no—
164C3×C4—42no—
165C3×C4—122no—
166C3×C4—121no—
167Dic3—41no—
168Dic3—42no—
169Dic3—64no—
170Dic3—64no—
171Dic3—124no—
172C32#39911yescharacteristic, complemented, Sylow
173C23—32no—
174C23—61no—
175C23—62no—
176C23—62no—
177C23—61no—
178C23C2×D3211yescharacteristic
179C2×C4—22no—
180C2×C4—62no—
181C2×C4—62no—
182C2×C4—62no—
183C2×C4—61no—
184C2×C4—62no—
185C2×C4—94no—
186C2×C4—182no—
187C2×C4—182no—
188C2×C4#62512yes—
189C2×C3—22no—
190C2×C3—21no—
191C2×C3—22no—
192C2×C3—21no—
193C2×C3—21no—
194C2×C3—122no—
195C2×C3#87812yes—
196C2×C3#88312yes—
197C2×C3#89412yes—
198C2×C3#89612yes—
199C2×C3#92411yescharacteristic
200C2×C3#92911yescharacteristic
201C2×C3#93511yescharacteristic
202C2×C3#94011yescharacteristic
203C2×C3#94011yescharacteristic
204C2×C3#94611yescharacteristic
205D3—42no—
206C22—62no—
207C22—64no—
208C22—122no—
209C22—121no—
210C22—121no—
211C22D3×Dic611yescharacteristic
212C22#379911yescharacteristic
213C22#380212yes—
214C22#380412yes—
215C22C2×#62511yescharacteristic
216C4—22no—
217C4—122no—
218C4—121no—
219C4—184no—
220C3—21no—
221C3#494911yescharacteristic, complemented
222C3#499311yescharacteristic, complemented
223C2—122no—
224C2#6385412yes—
225C2#6385611yescharacteristic
226C2#6389912yes—
227C2#6391411yescharacteristic
228C2#6396111yescharacteristic
subgroupG/H
center, FrattiniC23C2×D32
commutator, socle, cosocleC23×C32C23
FittingC22×C32×C4C22
prime radicalC23×C3×D3C22
hypercenterC22×C4D32
hypocenterC32#399
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×C4⋊Dic3C2semidirect
3. C2×#3724C2semidirect
5. #63658C2semidirect
22. C2×C4⋊Dic3D3semidirect
172. C32#399semidirect
221. C3#4949semidirect
222. C3#4993semidirect

Maximal subgroups

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

subgroupindex
C2×C3×(C2×D3)⋊C42
C2×C3×C4⋊Dic32
C2×#37242
C2×(C2×D3)⋊Dic32
#636582
#49933
#49493