Finite Group Databasebeta

← #10593951

Subgroups — #10593951

Order 576.

507 conjugacy classes of subgroups; only the 136 normal subgroups are shown.

normal subgroups only

This group has 507 conjugacy classes of subgroups — too many to draw. Only its 136 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
1C4×#634C211yesmaximal, characteristic, complemented
2C3×#874C211yesmaximal, characteristic
3C3×C4×Dic6C211yesmaximal, characteristic, complemented
4#63898C212yesmaximal, complemented
5Dic3×D12C212yesmaximal, complemented
6#63884C214yesmaximal, complemented
7#63863C212yesmaximal, complemented
8#63860C212yesmaximal, complemented
9#5713—31nomaximal
10#5585—31nomaximal
11C2×#634C2211yescharacteristic
12C2×C3×D12C2212yes—
13C2×C3×D12C2212yes—
14C2×C3×D12C2212yes—
15C2×C3×Dic6C2211yescharacteristic
16C2×D3×Dic3C2214yes—
17C32×C42C2211yescharacteristic
18C4⋊(C3⋊Dic3)C2211yescharacteristic
19#3750C2212yes—
20C4×C3⋊Dic3C2212yes—
21C3×C4⋊Dic3C2211yescharacteristic
22C3×Dic3⋊C4C2212yes—
23C3×C4×Dic3C2212yes—
24(C2×D3)⋊Dic3C2214yes—
25(C2×D3)⋊Dic3C2214yes—
26(C2×D3)⋊Dic3C2214yes—
27C3×#127—31no—
28C3×C4×Q8—31no—
29#947—32no—
30#936—32no—
31#936—32no—
32#935—34no—
33D4×Dic3—32no—
34D4×Dic3—32no—
35#894—34no—
36#893—32no—
37#875—32no—
38#874D311yescharacteristic
39C4×D12—32no—
40C4×Dic6—31no—
41C4×Dic6—31no—
42C4×Dic6—61no—
43C4×Dic6D311yescharacteristic, complemented
44C22×C3×D3C2314yes—
45C2×C32×C4C2311yescharacteristic
46C2×C32×C4C2311yescharacteristic
47C2×C32×C4C2311yescharacteristic
48C2×C3⋊Dic3C2312yes—
49C2×C3⋊Dic3C2312yes—
50#634—22no—
51C2×C3×Dic3C2312yes—
52C2×C3×Dic3C2312yes—
53C3×D12—24no—
54C3×D12—24no—
55C3×D12—42no—
56C3×Dic6—22no—
57D3×Dic3—44no—
58#559—91noSylow
59C2×C3×Q8—31no—
60C2×C3×D4—32no—
61C2×C3×D4—32no—
62C2×C3×D4—32no—
63C22×Dic3—34no—
64C2×D12C2×D312yes—
65C2×D12C2×D312yes—
66C2×D12C2×D312yes—
67C2×C4×D3—34no—
68C2×Dic6—31no—
69C2×Dic6—31no—
70C2×Dic6—61no—
71C2×Dic6C2×D311yescharacteristic
72C3×C4⋊C4—31no—
73C3×C4⋊C4—32no—
74C3×C42—21no—
75C3×C42—32no—
76C3×C42C2×D311yescharacteristic
77C3×C42C2×D311yescharacteristic
78#217—34no—
79#217—34no—
80#217—34no—
81(C2×D3)⋊C4—34no—
82(C2×D3)⋊C4—34no—
83(C2×D3)⋊C4—34no—
84C4⋊Dic3—31no—
85C4⋊Dic3—31no—
86C4⋊Dic3—61no—
87C4⋊Dic3C2×D311yescharacteristic
88Dic3⋊C4—32no—
89Dic3⋊C4—32no—
90Dic3⋊C4—62no—
91Dic3⋊C4C2×D312yes—
92C4×Dic3—32no—
93C4×Dic3—32no—
94C4×Dic3—62no—
95C4×Dic3C2×D312yes—
96C22×C32C2411yescharacteristic
97C2×C3×D3—24no—
98C2×C3×D3—24no—
99C2×C3×D3—24no—
100C32×C4—21no—
101C32×C4#4112yes—
102C32×C4#4112yes—
103C3⋊Dic3—22no—
104C3⋊Dic3—22no—
105C3×Dic3—22no—
106C3×Dic3—22no—
107#127—31no—
108#124—92no—
109#124—92no—
110#123—94no—
111C4×Q8—31no—
112C4×Q8—91no—
113C4×D4—92no—
114C4×D4—92no—
115C23×C3—34no—
116C22×D3C22×D314yes—
117C3×Q8—62no—
118C3×D4—64no—
119C3×D4—64no—
120C3×D4—122no—
121C2×C3×C4—21no—
122C2×C3×C4—21no—
123C2×C3×C4—21no—
124C2×C3×C4—32no—
125C2×C3×C4—32no—
126C2×C3×C4C22×D311yescharacteristic
127C2×C3×C4C22×D311yescharacteristic
128C2×C3×C4C22×D311yescharacteristic
129C2×C3×C4C22×D311yescharacteristic
130C2×C3×C4C22×D311yescharacteristic
131C2×C3×C4C22×D311yescharacteristic
132C2×Dic3—32no—
133C2×Dic3—32no—
134C2×Dic3—32no—
135C2×Dic3—32no—
136C2×Dic3—62no—
137C2×Dic3—62no—
138C2×Dic3—124no—
139C2×Dic3C22×D312yes—
140C2×Dic3C22×D312yes—
141D12—24no—
142D12—24no—
143D12—42no—
144C4×D3—124no—
145Dic6—22no—
146Dic6—64no—
147Dic6—62no—
148Dic6—62no—
149C2×C32C2×#4111yescharacteristic
150C2×C32C2×#4111yescharacteristic
151C2×C32H(5,𝔽2)11yescharacteristic
152C3×D3—44no—
153C2×Q8—31no—
154C2×Q8—91no—
155C2×D4—32no—
156C2×D4—32no—
157C2×D4—32no—
158C22×C4—94no—
159C4⋊C4—31no—
160C4⋊C4—32no—
161C4⋊C4—91no—
162C4⋊C4—92no—
163C22⋊C4—94no—
164C22⋊C4—94no—
165C22⋊C4—94no—
166C42—32no—
167C42—92no—
168C42D3211yescharacteristic
169C22×C3—21no—
170C22×C3—64no—
171C22×C3—64no—
172C22×C3—64no—
173C22×C3C23×D311yescharacteristic
174C22×C3C23×D311yescharacteristic
175C2×D3—24no—
176C2×D3—24no—
177C2×D3—24no—
178C3×C4—21no—
179C3×C4—21no—
180C3×C4—22no—
181C3×C4—22no—
182C3×C4—41no—
183C3×C4—62no—
184C3×C4—62no—
185C3×C4#23512yes—
186C3×C4#23712yes—
187C3×C4#23712yes—
188C3×C4#23912yes—
189Dic3—22no—
190Dic3—22no—
191Dic3—64no—
192Dic3—62no—
193Dic3—62no—
194Dic3—62no—
195Dic3—62no—
196Dic3—122no—
197C32#55911yescharacteristic, complemented, Sylow
198C23—34no—
199Q8—62no—
200Q8—182no—
201D4—64no—
202D4—64no—
203D4—122no—
204C2×C4—32no—
205C2×C4—32no—
206C2×C4—92no—
207C2×C4—92no—
208C2×C4—364no—
209C2×C4C2×D3211yescharacteristic
210C2×C4C2×D3211yescharacteristic
211C2×C4C2×D3211yescharacteristic
212C2×C3—21no—
213C2×C3—21no—
214C2×C3—21no—
215C2×C3—124no—
216C2×C3C2×#23511yescharacteristic
217C2×C3C2×#23711yescharacteristic
218C2×C3C2×#23711yescharacteristic
219C2×C3#100411yescharacteristic
220C2×C3C2×#23911yescharacteristic
221C2×C3#100911yescharacteristic
222D3—44no—
223C22—64no—
224C22—64no—
225C22—64no—
226C22C22×D3211yescharacteristic
227C4—21no—
228C4—62no—
229C4—62no—
230C4—182no—
231C4—182no—
232C4#379512yes—
233C4#379612yes—
234C3—21no—
235C3#558511yescharacteristic, complemented
236C3#571311yescharacteristic, complemented
237C2—124no—
238C2C2×#379511yescharacteristic
239C2C2×#379611yescharacteristic
240C2#6430711yescharacteristic
subgroupG/H
center, FrattiniC22C22×D32
commutator, socle, cosocleC22×C32C24
FittingC32×C42C22
prime radicalC3×#874C2
hypercenterC42D32
hypocenterC32#559
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C42

Complements

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

NH ≅ G/Ntype
1. C4×#634C2semidirect
3. C3×C4×Dic6C2semidirect
4. #63898C2semidirect
5. Dic3×D12C2semidirect
6. #63884C2semidirect
7. #63863C2semidirect
8. #63860C2semidirect
43. C4×Dic6D3semidirect
197. C32#559semidirect
235. C3#5585semidirect
236. C3#5713semidirect

Maximal subgroups

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

subgroupindex
C4×#6342
C3×#8742
C3×C4×Dic62
#638982
Dic3×D122
#638842
#638632
#638602
#57133
#55853