Finite Group Databasebeta

← #10590952

Subgroups — #10590952

Order 576.

326 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 326 conjugacy classes of subgroups — too many to draw. Only its 70 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×D3×Dic6C211yesmaximal, characteristic
2C2×#3746C211yesmaximal, characteristic, complemented
3#63857C211yesmaximal, characteristic
4C2×#3719C211yesmaximal, characteristic, complemented
5C3×(C2×D3)⋊C8C211yesmaximal, characteristic
6C3×#816C211yesmaximal, characteristic, complemented
7#63567C211yesmaximal, characteristic, complemented
8#4809—31nomaximal
9#4739—31nomaximal
10C2×#634C2211yescharacteristic
11C2×#634C2211yescharacteristic
12C2×C3×C4×D3C2211yescharacteristic
13C2×C3×Dic6C2211yescharacteristic
14C2×#627—21no—
15C2×D3×Dic3—21no—
16D3×Dic6—22no—
17D3×Dic6—22no—
18C2×C32×C8C2211yescharacteristic
19#3746—22no—
20C3×C4⋊Dic3C2211yescharacteristic
21C2×C3×C3⋊C8C2211yescharacteristic
22Dic3⋊Dic3—21no—
23(C2×D3)⋊Dic3—21no—
24#3719—22no—
25C2×D3×Q8—31no—
26C22×Dic6—31no—
27C2×#216—31no—
28#924—31no—
29C2×Dic12—31no—
30C2×Dic12—31no—
31C2×Dic12—31no—
32C2×Dic12—61no—
33#897—31no—
34C3×#103—31no—
35C3×C22⋊C8—31no—
36(C2×D3)⋊C8D311yescharacteristic
37#816—31no—
38#816D311yescharacteristic, complemented
39#810—31no—
40C22×C3×D3D411yescharacteristic
41C2×C32×C4C2311yescharacteristic
42C2×C3⋊Dic3—21no—
43C2×C3⋊Dic3—21no—
44#634—21no—
45#634—21no—
46#634—21no—
47#634D412yes—
48C2×C3×Dic3—21no—
49C2×C3×Dic3—21no—
50C2×C3×Dic3D411yescharacteristic
51C3×C4×D3—21no—
52C3×C4×D3—21no—
53C3×Dic6—21no—
54C3×Dic6D412yes—
55#627—24no—
56D3×Dic3—24no—
57C32×C8—21no—
58C3×C3⋊C8—21no—
59#451—91noSylow
60C2×C3×Q8—31no—
61C22×C3×C4—31no—
62C22×Dic3—61no—
63D3×Q8—62no—
64D3×Q8—62no—
65C2×C4×D3—61no—
66C2×C4×D3C2×D311yescharacteristic
67C2×Dic6—31no—
68C2×Dic6—31no—
69C2×Dic6—31no—
70C2×Dic6—31no—
71C2×Dic6—61no—
72C2×Dic6—61no—
73C2×Dic6—62no—
74C2×Dic6—61no—
75C2×Dic6—61no—
76C2×Dic6—62no—
77C2×Dic6C2×D311yescharacteristic
78C2×C3×C8—21no—
79C2×C3×C8—31no—
80C2×C3×C8C2×D311yescharacteristic
81C2×C3×C8C2×D311yescharacteristic
82C3×C4⋊C4—31no—
83#217—61no—
84#216—62no—
85(C2×D3)⋊C4—61no—
86C4⋊Dic3—61no—
87C4⋊Dic3C2×D311yescharacteristic
88Dic3⋊C4—61no—
89C2×C3⋊C8C2×D311yescharacteristic
90Dic12—62no—
91Dic12—62no—
92Dic12—64no—
93Dic12—62no—
94C22×C32C2×D411yescharacteristic
95C2×C3×D3—21no—
96C2×C3×D3—21no—
97C2×C3×D3Q1612yescomplemented
98C32×C4C2×D411yescharacteristic
99C32×C4C2×D411yescharacteristic
100C3⋊Dic3—22no—
101C3⋊Dic3—41no—
102C3×Dic3—22no—
103C3×Dic3—21no—
104C3×Dic3—41no—
105C22×Q8—91no—
106C2×Q16—91no—
107C2×Q16—91no—
108C22⋊Q8—91no—
109#103—31no—
110#103—91no—
111C22⋊C8—31no—
112C23×C3—31no—
113C22×D3D1211yescharacteristic
114C3×Q8—32no—
115C3×Q8—61no—
116C2×C3×C4—21no—
117C2×C3×C4—31no—
118C2×C3×C4—61no—
119C2×C3×C4—61no—
120C2×C3×C4—61no—
121C2×C3×C4—61no—
122C2×C3×C4C22×D311yescharacteristic
123C2×C3×C4C22×D311yescharacteristic
124C2×Dic3—21no—
125C2×Dic3—21no—
126C2×Dic3—64no—
127C2×Dic3—61no—
128C2×Dic3—61no—
129C2×Dic3—61no—
130C2×Dic3—61no—
131C2×Dic3—121no—
132C2×Dic3—121no—
133C2×Dic3D1211yescharacteristic
134C4×D3—21no—
135C4×D3—21no—
136C4×D3—64no—
137Dic6—21no—
138Dic6—32no—
139Dic6—32no—
140Dic6—62no—
141Dic6—64no—
142Dic6—64no—
143Dic6—61no—
144Dic6—61no—
145Dic6—62no—
146Dic6—61no—
147Dic6—61no—
148Dic6—61no—
149Dic6—61no—
150Dic6—121no—
151Dic6—121no—
152Dic6D1212yes—
153C3×C8—22no—
154C3×C8—21no—
155C3×C8—21no—
156C3×C8—61no—
157C3⋊C8—21no—
158C2×C32#12011yescharacteristic
159C2×C32C2×Q1611yescharacteristic
160C2×C32#13711yescharacteristic
161C3×D3—22no—
162C2×Q8—31no—
163C2×Q8—91no—
164C2×Q8—91no—
165C2×Q8—181no—
166C2×Q8—182no—
167C2×Q8—182no—
168C22×C4—31no—
169C22×C4—181no—
170Q16—182no—
171Q16—182no—
172C2×C8—31no—
173C2×C8D3211yescharacteristic
174C4⋊C4—31no—
175C4⋊C4—181no—
176C22⋊C4—181no—
177C22×C3—21no—
178C22×C3—32no—
179C22×C3—61no—
180C22×C3—61no—
181C22×C3C2×D1211yescharacteristic
182C22×C3D3×D411yescharacteristic
183C2×D3—21no—
184C2×D3—21no—
185C2×D3Dic1212yescomplemented
186C3×C4—21no—
187C3×C4—21no—
188C3×C4—61no—
189C3×C4—62no—
190C3×C4—121no—
191C3×C4C2×D1211yescharacteristic
192C3×C4D3×D411yescharacteristic
193C3×C4D3×D411yescharacteristic
194C3×C4D3×D411yescharacteristic
195Dic3—22no—
196Dic3—21no—
197Dic3—41no—
198Dic3—62no—
199Dic3—62no—
200Dic3—122no—
201Dic3—122no—
202Dic3—121no—
203Dic3—121no—
204C32#45111yescharacteristic, complemented, Sylow
205C23—31no—
206Q8—32no—
207Q8—61no—
208Q8—92no—
209Q8—184no—
210Q8—181no—
211Q8—181no—
212Q8—181no—
213C2×C4—31no—
214C2×C4—61no—
215C2×C4—61no—
216C2×C4—61no—
217C2×C4—61no—
218C2×C4—184no—
219C2×C4—181no—
220C2×C4—181no—
221C2×C4C2×D3211yescharacteristic
222C8—21no—
223C8—61no—
224C2×C3—21no—
225C2×C3—21no—
226C2×C3—21no—
227C2×C3—62no—
228C2×C3#88211yescharacteristic
229C2×C3#88211yescharacteristic
230C2×C3C2×Dic1211yescharacteristic
231C2×C3#90911yescharacteristic
232C2×C3#91511yescharacteristic
233C2×C3D3×Q1611yescharacteristic
234D3—22no—
235C22—32no—
236C22—61no—
237C22—61no—
238C22D3×D1211yescharacteristic
239C4—61no—
240C4—62no—
241C4—121no—
242C4—182no—
243C4—361no—
244C4D3×D1211yescharacteristic
245C4#381111yescharacteristic
246C3—21no—
247C3#473911yescharacteristic, complemented
248C3#480911yescharacteristic, complemented
249C2—62no—
250C2D3×Dic1211yescharacteristic
251C2#6380011yescharacteristic
252C2#6392311yescharacteristic
subgroupG/H
centerC22D3×D12
commutator, cosocleC2×C32×C4C23
FrattiniC2×C4C2×D32
FittingC2×C32×C8C22
socleC22×C32C2×D4
prime radicalC22×C3×D3D4
hypercenterC2×C8D32
hypocenterC32#451
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×C8

Complements

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

NH ≅ G/Ntype
2. C2×#3746C2semidirect
4. C2×#3719C2semidirect
6. C3×#816C2semidirect
7. #63567C2semidirect
38. #816D3semidirect
97. C2×C3×D3Q16semidirect
185. C2×D3Dic12semidirect
204. C32#451semidirect
247. C3#4739semidirect
248. C3#4809semidirect

Maximal subgroups

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

subgroupindex
C2×D3×Dic62
C2×#37462
#638572
C2×#37192
C3×(C2×D3)⋊C82
C3×#8162
#635672
#48093
#47393