Finite Group Databasebeta

← #10591471

Subgroups — #10591471

Order 576.

286 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 286 conjugacy classes of subgroups — too many to draw. Only its 68 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×#3753C211yesmaximal, characteristic
2C3×D4×Dic3C211yesmaximal, characteristic, complemented
3C2×C3×#214C211yesmaximal, characteristic, complemented
4#63864C211yesmaximal, characteristic, complemented
5Dic3⋊(C3⋊C8)C211yesmaximal, characteristic, complemented
6#63565C211yesmaximal, characteristic, complemented
7#63564C211yesmaximal, characteristic, complemented
8#5165—31nomaximal
9#5016—31nomaximal
10C2×C32×D4C2211yescharacteristic
11C2×C4⋊(C3⋊D3)C2211yescharacteristic
12C22×C3×Dic3—21no—
13C2×C3×Dic6C2211yescharacteristic
14#3753—22no—
15C2×C3⋊(C3⋊C8)C2211yescharacteristic
16C3×#217—21no—
17C3×#214—22no—
18C3×C4⋊Dic3C2211yescharacteristic
19C3×C4×Dic3C2211yescharacteristic
20C2×C3×C3⋊C8C2211yescharacteristic
21#3722—21no—
22#3722—21no—
23C2×C3×SD16—31no—
24C3×C4×D4—31no—
25#947—31no—
26D4×Dic3D311yescharacteristic, complemented
27C2×#214D311yescharacteristic, complemented
28C2×#213—31no—
29C2×#213—31no—
30C2×#213—61no—
31#875—31no—
32#835—31no—
33#821—31no—
34Dic3⋊C8—31no—
35#810—31no—
36#809—31no—
37C4⋊(C3⋊C8)—31no—
38C23×C32—21no—
39C22×C3⋊D3—21no—
40C32×D4—21no—
41C32×D4D412yes—
42C2×C32×C4C2311yescharacteristic
43C4⋊(C3⋊D3)—21no—
44C4⋊(C3⋊D3)—21no—
45C2×C3×Dic3—21no—
46C2×C3×Dic3—21no—
47C2×C3×Dic3—42no—
48C2×C3×Dic3D411yescharacteristic
49C2×C3×Dic3D411yescharacteristic
50C3×Dic6—21no—
51C3×Dic6—21no—
52C3⋊(C3⋊C8)—21no—
53C3×C3⋊C8—21no—
54#463—91noSylow
55C2×C3×Q8—31no—
56C2×C3×D4—21no—
57C2×C3×D4C2×D311yescharacteristic
58C2×C3×D4C2×D311yescharacteristic
59C22×C3×C4—61no—
60C22×Dic3—21no—
61C2×D12—31no—
62C2×D12—31no—
63C2×D12—61no—
64C2×Dic6C2×D311yescharacteristic
65C3×SD16—62no—
66C2×C3×C8—31no—
67C3×C4⋊C4—31no—
68C3×C22⋊C4—61no—
69C3×C42—31no—
70#217—21no—
71#214—22no—
72#213—64no—
73#213—62no—
74#213—62no—
75(C2×D3)⋊C4—61no—
76(C2×D3)⋊C4—61no—
77(C2×D3)⋊C4—61no—
78(C2×D3)⋊C4—61no—
79C4⋊Dic3C2×D311yescharacteristic
80C4×Dic3C2×D311yescharacteristic
81C2×C3⋊C8—31no—
82C2×C3⋊C8—31no—
83C2×C3⋊C8—61no—
84C2×C3⋊C8C2×D311yescharacteristic
85C22×C32—22no—
86C22×C32—41no—
87C22×C32—41no—
88C22×C32C2×D411yescharacteristic
89C2×C3⋊D3—41no—
90C2×C3⋊D3—41no—
91C2×C3⋊D3—41no—
92C32×C4C2×D411yescharacteristic
93C32×C4#4111yescharacteristic
94C3×Dic3—21no—
95C3×Dic3—21no—
96C3×Dic3—41no—
97C3×Dic3—41no—
98C2×SD16—31no—
99C2×D8—91no—
100#124—91no—
101C4×D4—31no—
102C4⋊C8—91no—
103#103—91no—
104#102—91no—
105C23×C3—21no—
106C23×C3—21no—
107C23×C3—41no—
108C22×D3—61no—
109C22×D3—61no—
110C22×D3—121no—
111C3×Q8—61no—
112C3×Q8—61no—
113C3×D4—21no—
114C3×D4—21no—
115C3×D4—22no—
116C3×D4—41no—
117C3×D4D1212yes—
118C3×D4#6712yes—
119C2×C3×C4—21no—
120C2×C3×C4—31no—
121C2×C3×C4—31no—
122C2×C3×C4—61no—
123C2×C3×C4—61no—
124C2×C3×C4—122no—
125C2×C3×C4C22×D311yescharacteristic
126C2×C3×C4C22×D311yescharacteristic
127C2×Dic3—21no—
128C2×Dic3—21no—
129C2×Dic3—42no—
130C2×Dic3#6711yescharacteristic
131C2×Dic3#6711yescharacteristic
132D12—62no—
133D12—61no—
134D12—61no—
135D12—61no—
136D12—61no—
137D12—121no—
138Dic6—21no—
139Dic6—21no—
140C3×C8—61no—
141C3⋊C8—21no—
142C3⋊C8—62no—
143C3⋊C8—61no—
144C3⋊C8—61no—
145C2×C32—42no—
146C2×C32#12111yescharacteristic
147C2×C32#13511yescharacteristic
148C2×C32AGL(1,ℤ/8)11yescharacteristic
149C3⋊D3—81no—
150C2×Q8—31no—
151C2×D4—91no—
152C2×D4D3211yescharacteristic
153C22×C4—61no—
154SD16—62no—
155D8—182no—
156C2×C8—31no—
157C2×C8—91no—
158C4⋊C4—31no—
159C22⋊C4—61no—
160C22⋊C4—181no—
161C22⋊C4—181no—
162C42—31no—
163C22×C3—22no—
164C22×C3—22no—
165C22×C3—21no—
166C22×C3—42no—
167C22×C3—41no—
168C22×C3—41no—
169C22×C3—41no—
170C22×C3—41no—
171C22×C3—81no—
172C22×C3—81no—
173C22×C3D3×D411yescharacteristic
174C22×C3C2×#6711yescharacteristic
175C2×D3—122no—
176C2×D3—121no—
177C2×D3—121no—
178C2×D3—121no—
179C2×D3—121no—
180C2×D3—121no—
181C2×D3—121no—
182C2×D3—241no—
183C2×D3—241no—
184C3×C4—21no—
185C3×C4—21no—
186C3×C4—61no—
187C3×C4—61no—
188C3×C4—121no—
189C3×C4—121no—
190C3×C4C2×D1211yescharacteristic
191C3×C4#23511yescharacteristic
192C3×C4#23711yescharacteristic
193C3×C4C2×#6711yescharacteristic
194Dic3—21no—
195Dic3—21no—
196Dic3—41no—
197Dic3—41no—
198C32#46311yescharacteristic, complemented, Sylow
199C23—21no—
200C23—181no—
201Q8—61no—
202Q8—61no—
203D4—21no—
204D4—181no—
205D4—181no—
206D4#62612yes—
207C2×C4—31no—
208C2×C4—31no—
209C2×C4—61no—
210C2×C4—61no—
211C2×C4—122no—
212C2×C4C2×D3211yescharacteristic
213C8—61no—
214C8—181no—
215C2×C3—21no—
216C2×C3—21no—
217C2×C3—21no—
218C2×C3—42no—
219C2×C3—42no—
220C2×C3—82no—
221C2×C3#91411yescharacteristic
222C2×C3#91611yescharacteristic
223C2×C3#93011yescharacteristic
224C2×C3#93211yescharacteristic
225C2×C3#93911yescharacteristic
226C2×C3#95011yescharacteristic
227D3—242no—
228D3—241no—
229D3—241no—
230C22—22no—
231C22—41no—
232C22—41no—
233C22—361no—
234C22—361no—
235C22—361no—
236C22D3×#6711yescharacteristic
237C4—61no—
238C4—61no—
239C4—121no—
240C4—121no—
241C4#380411yescharacteristic
242C4C2×#62611yescharacteristic
243C3—21no—
244C3#501611yescharacteristic, complemented
245C3#516511yescharacteristic, complemented
246C2—42no—
247C2—721no—
248C2#6393411yescharacteristic
249C2#6393511yescharacteristic
250C2#6397811yescharacteristic
subgroupG/H
centerC22D3×#67
commutator, cosocleC2×C32×C4C23
FrattiniC2×C4C2×D32
FittingC2×C32×D4C22
socleC22×C32C2×D4
prime radicalC2×#3753C2
hypercenterC2×D4D32
hypocenterC32#463
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
2. C3×D4×Dic3C2semidirect
3. C2×C3×#214C2semidirect
4. #63864C2semidirect
5. Dic3⋊(C3⋊C8)C2semidirect
6. #63565C2semidirect
7. #63564C2semidirect
26. D4×Dic3D3semidirect
27. C2×#214D3semidirect
198. C32#463semidirect
244. C3#5016semidirect
245. C3#5165semidirect

Maximal subgroups

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

subgroupindex
C2×#37532
C3×D4×Dic32
C2×C3×#2142
#638642
Dic3⋊(C3⋊C8)2
#635652
#635642
#51653
#50163