Finite Group Databasebeta

← #10591595

Subgroups — #10591595

Order 576.

326 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 326 conjugacy classes of subgroups — too many to draw. Only its 76 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×D3×Q8C211yesmaximal, characteristic
2C2×#3756C211yesmaximal, characteristic, complemented
3#63857C211yesmaximal, characteristic
4C2×#3719C211yesmaximal, characteristic, complemented
5C3×#835C211yesmaximal, characteristic, complemented
6#63565C211yesmaximal, characteristic, complemented
7(C2×D3)⋊(C3⋊C8)C211yesmaximal, characteristic
8#5228—31nomaximal
9#4809—31nomaximal
10C2×C32×Q8C2211yescharacteristic
11C2×#634C2211yescharacteristic
12C3×D3×Q8—22no—
13C3×D3×Q8—22no—
14C2×C3×C4×D3—21no—
15C2×C3×C4×D3C2211yescharacteristic
16C2×C3×Dic6—21no—
17C2×C3×Dic6C2211yescharacteristic
18#3756—22no—
19C2×C3⋊(C3⋊C8)C2211yescharacteristic
20C3×C4⋊Dic3C2211yescharacteristic
21C2×C3×C3⋊C8C2211yescharacteristic
22Dic3⋊Dic3—21no—
23(C2×D3)⋊Dic3—21no—
24#3719—22no—
25C22×C3×Q8—31no—
26C2×D3×Q8D311yescharacteristic
27C2×#216—31no—
28C2×#216—31no—
29C2×#216—31no—
30C2×#216—61no—
31#924—31no—
32C2×Dic12—31no—
33#897—31no—
34C3×#103—31no—
35#835—31no—
36#835D311yescharacteristic, complemented
37#830—31no—
38(C2×D3)⋊C8—31no—
39#810—31no—
40C22×C3×D3D411yescharacteristic
41C32×Q8—21no—
42C32×Q8D412yes—
43C2×C32×C4—21no—
44C2×C32×C4C2311yescharacteristic
45C2×C3⋊Dic3—21no—
46#634—21no—
47#634—21no—
48C2×C3×Dic3—21no—
49C2×C3×Dic3—21no—
50C2×C3×Dic3D411yescharacteristic
51C3×C4×D3—24no—
52C3×C4×D3—21no—
53C3×C4×D3—21no—
54C3×Dic6—24no—
55C3×Dic6—21no—
56C3×Dic6D412yes—
57C3⋊(C3⋊C8)—21no—
58C3×C3⋊C8—21no—
59#451—91noSylow
60C2×C3×Q8—21no—
61C2×C3×Q8—31no—
62C2×C3×Q8—61no—
63C2×C3×Q8—62no—
64C2×C3×Q8—62no—
65C2×C3×Q8C2×D311yescharacteristic
66C2×C3×Q8C2×D311yescharacteristic
67C22×C3×C4—31no—
68C22×C3×C4—61no—
69D3×Q8—22no—
70D3×Q8—22no—
71C2×C4×D3—21no—
72C2×C4×D3C2×D311yescharacteristic
73C2×Dic6—21no—
74C2×Dic6—31no—
75C2×Dic6—31no—
76C2×Dic6—61no—
77C2×Dic6C2×D311yescharacteristic
78C2×C3×C8—31no—
79C3×C4⋊C4—31no—
80#217—61no—
81#216—62no—
82#216—64no—
83#216—62no—
84#216—62no—
85(C2×D3)⋊C4—61no—
86C4⋊Dic3—61no—
87C4⋊Dic3C2×D311yescharacteristic
88Dic3⋊C4—61no—
89C2×C3⋊C8—31no—
90C2×C3⋊C8—31no—
91C2×C3⋊C8—61no—
92C2×C3⋊C8C2×D311yescharacteristic
93Dic12—62no—
94C22×C32C2×D411yescharacteristic
95C2×C3×D3—21no—
96C2×C3×D3—21no—
97C2×C3×D3Q1612yescomplemented
98C32×C4—22no—
99C32×C4C2×D411yescharacteristic
100C32×C4C2×D411yescharacteristic
101C3⋊Dic3—41no—
102C3×Dic3—21no—
103C3×Dic3—22no—
104C3×Dic3—41no—
105C22×Q8—31no—
106C2×Q16—91no—
107C2×Q16—91no—
108C22⋊Q8—91no—
109#103—31no—
110#103—91no—
111C22⋊C8—91no—
112C23×C3—31no—
113C22×D3#6711yescharacteristic
114C3×Q8—21no—
115C3×Q8—21no—
116C3×Q8—22no—
117C3×Q8—32no—
118C3×Q8—41no—
119C3×Q8—64no—
120C3×Q8—61no—
121C3×Q8D1212yes—
122C3×Q8#6712yes—
123C2×C3×C4—21no—
124C2×C3×C4—21no—
125C2×C3×C4—21no—
126C2×C3×C4—31no—
127C2×C3×C4—41no—
128C2×C3×C4—64no—
129C2×C3×C4—61no—
130C2×C3×C4—61no—
131C2×C3×C4—61no—
132C2×C3×C4—61no—
133C2×C3×C4C22×D311yescharacteristic
134C2×C3×C4C22×D311yescharacteristic
135C2×Dic3—21no—
136C2×Dic3—21no—
137C2×Dic3—61no—
138C2×Dic3—61no—
139C2×Dic3—121no—
140C2×Dic3#6711yescharacteristic
141C4×D3—24no—
142C4×D3—21no—
143C4×D3—21no—
144Dic6—24no—
145Dic6—21no—
146Dic6—62no—
147Dic6—61no—
148Dic6—61no—
149Dic6—61no—
150Dic6—61no—
151Dic6—121no—
152Dic6#6712yes—
153C3×C8—61no—
154C3⋊C8—21no—
155C3⋊C8—62no—
156C3⋊C8—61no—
157C3⋊C8—61no—
158C2×C32#12011yescharacteristic
159C2×C32C2×Q1611yescharacteristic
160C2×C32#13711yescharacteristic
161C3×D3—22no—
162C2×Q8—31no—
163C2×Q8—61no—
164C2×Q8—62no—
165C2×Q8—62no—
166C2×Q8—91no—
167C2×Q8D3211yescharacteristic
168C22×C4—31no—
169C22×C4—61no—
170Q16—182no—
171Q16—182no—
172C2×C8—31no—
173C2×C8—91no—
174C4⋊C4—31no—
175C4⋊C4—181no—
176C22⋊C4—181no—
177C22×C3—21no—
178C22×C3—32no—
179C22×C3—61no—
180C22×C3—61no—
181C22×C3D3×D411yescharacteristic
182C22×C3C2×#6711yescharacteristic
183C2×D3—21no—
184C2×D3—21no—
185C2×D3#21612yescomplemented
186C3×C4—22no—
187C3×C4—22no—
188C3×C4—21no—
189C3×C4—21no—
190C3×C4—42no—
191C3×C4—61no—
192C3×C4—62no—
193C3×C4—121no—
194C3×C4C2×D1211yescharacteristic
195C3×C4D3×D411yescharacteristic
196C3×C4C2×#6711yescharacteristic
197C3×C4C2×#6711yescharacteristic
198Dic3—21no—
199Dic3—22no—
200Dic3—41no—
201Dic3—122no—
202Dic3—121no—
203Dic3—121no—
204C32#45111yescharacteristic, complemented, Sylow
205C23—31no—
206Q8—21no—
207Q8—32no—
208Q8—64no—
209Q8—61no—
210Q8—181no—
211Q8—181no—
212Q8#62612yes—
213C2×C4—21no—
214C2×C4—31no—
215C2×C4—64no—
216C2×C4—61no—
217C2×C4—61no—
218C2×C4—61no—
219C2×C4—61no—
220C2×C4—181no—
221C2×C4C2×D3211yescharacteristic
222C8—61no—
223C8—181no—
224C2×C3—21no—
225C2×C3—21no—
226C2×C3—21no—
227C2×C3—62no—
228C2×C3#88211yescharacteristic
229C2×C3#91511yescharacteristic
230C2×C3D3×Q1611yescharacteristic
231C2×C3#94211yescharacteristic
232C2×C3C2×#21611yescharacteristic
233C2×C3#95311yescharacteristic
234D3—22no—
235C22—32no—
236C22—61no—
237C22—61no—
238C22D3×#6711yescharacteristic
239C4—22no—
240C4—61no—
241C4—62no—
242C4—121no—
243C4—361no—
244C4C2×#62611yescharacteristic
245C4D3×#6711yescharacteristic
246C3—21no—
247C3#480911yescharacteristic, complemented
248C3#522811yescharacteristic, complemented
249C2—62no—
250C2D3×#21611yescharacteristic
251C2#6394611yescharacteristic
252C2#6397711yescharacteristic
subgroupG/H
centerC22D3×#67
commutator, cosocleC2×C32×C4C23
FrattiniC2×C4C2×D32
FittingC2×C32×Q8C22
socleC22×C32C2×D4
prime radicalC22×C3×D3D4
hypercenterC2×Q8D32
hypocenterC32#451
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×Q8

Complements

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

NH ≅ G/Ntype
2. C2×#3756C2semidirect
4. C2×#3719C2semidirect
5. C3×#835C2semidirect
6. #63565C2semidirect
36. #835D3semidirect
97. C2×C3×D3Q16semidirect
185. C2×D3#216semidirect
204. C32#451semidirect
247. C3#4809semidirect
248. C3#5228semidirect

Maximal subgroups

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

subgroupindex
C2×C3×D3×Q82
C2×#37562
#638572
C2×#37192
C3×#8352
#635652
(C2×D3)⋊(C3⋊C8)2
#52283
#48093