Finite Group Databasebeta

← #10590525

Subgroups — #10590525

Order 576.

265 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 265 conjugacy classes of subgroups — too many to draw. Only its 64 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
1#64106C211yesmaximal, characteristic
2#63871C211yesmaximal, characteristic
3C2×#3719C211yesmaximal, characteristic, complemented
4C2×#3710C211yesmaximal, characteristic
5C3×#810C211yesmaximal, characteristic, complemented
6C3×#807C211yesmaximal, characteristic, complemented
7#63567C211yesmaximal, characteristic, complemented
8#4887—31nomaximal
9#4813—31nomaximal
10C2×C4×C3⋊D3C2211yescharacteristic
11C2×#634C2211yescharacteristic
12C2×C3×Dic6C2211yescharacteristic
13C32×C4⋊C4C2211yescharacteristic
14#3752—21no—
15C4⋊(C3⋊Dic3)—21no—
16C3×C4⋊Dic3C2211yescharacteristic
17C2×C3×C3⋊C8C2211yescharacteristic
18C2×C3×C3⋊C8C2211yescharacteristic
19Dic3⋊Dic3—21no—
20#3722—21no—
21#3719—22no—
22#3710—22no—
23#946—31no—
24C2×#216—31no—
25C2×Dic12—31no—
26C2×D3×C8—31no—
27C2×D3×C8—31no—
28#897—31no—
29#897—31no—
30#897—31no—
31#897—61no—
32C3×#107—31no—
33C3×#103—31no—
34#816—31no—
35#810—31no—
36#810D311yescharacteristic, complemented
37#807D311yescharacteristic, complemented
38C22×C3⋊D3D411yescharacteristic
39C2×C32×C4—21no—
40C2×C32×C4C2311yescharacteristic
41C2×C3⋊Dic3—21no—
42C2×C3⋊Dic3D411yescharacteristic
43C4×C3⋊D3—21no—
44C4×C3⋊D3—21no—
45#634—21no—
46#634—21no—
47C2×C3×Dic3—21no—
48C2×C3×Dic3—21no—
49C3×Dic6—21no—
50C3×Dic6—21no—
51C3×C3⋊C8—21no—
52C3×C3⋊C8D412yes—
53#467—91noSylow
54C2×C3×Q8—31no—
55C2×C4×D3—31no—
56C2×C4×D3—31no—
57C2×C4×D3—61no—
58C2×Dic6—31no—
59C2×Dic6—31no—
60C2×Dic6—61no—
61C2×Dic6C2×D311yescharacteristic
62C2×C3×C8—31no—
63C2×C3×C8—31no—
64C3×C4⋊C4—21no—
65C3×C4⋊C4—31no—
66C3×C4⋊C4C2×D311yescharacteristic
67C3×C4⋊C4C2×D311yescharacteristic
68#216—62no—
69(C2×D3)⋊C4—61no—
70(C2×D3)⋊C4—61no—
71(C2×D3)⋊C4—61no—
72(C2×D3)⋊C4—61no—
73(C2×D3)⋊C4—62no—
74C4⋊Dic3—61no—
75C4⋊Dic3—61no—
76C4⋊Dic3—61no—
77C4⋊Dic3—62no—
78C4⋊Dic3C2×D311yescharacteristic
79Dic3⋊C4—61no—
80C2×C3⋊C8C2×D311yescharacteristic
81C2×C3⋊C8C2×D311yescharacteristic
82Dic12—62no—
83D3×C8—62no—
84D3×C8—62no—
85C22×C32C2×D411yescharacteristic
86C2×C3⋊D3—21no—
87C2×C3⋊D3—21no—
88C2×C3⋊D3Q1612yescomplemented
89C32×C4—41no—
90C32×C4C2×D411yescharacteristic
91C32×C4#4111yescharacteristic
92C3⋊Dic3—21no—
93C3⋊Dic3—41no—
94C3×Dic3—41no—
95C3×Dic3—41no—
96C2×Q16—91no—
97C22×C8—91no—
98C22⋊Q8—91no—
99C22⋊Q8—91no—
100#107—31no—
101#103—31no—
102#103—91no—
103C22×D3—31no—
104C22×D3—31no—
105C22×D3—61no—
106C3×Q8—61no—
107C3×Q8—61no—
108C2×C3×C4—22no—
109C2×C3×C4—21no—
110C2×C3×C4—21no—
111C2×C3×C4—21no—
112C2×C3×C4—61no—
113C2×C3×C4—61no—
114C2×C3×C4C22×D311yescharacteristic
115C2×C3×C4C22×D311yescharacteristic
116C2×Dic3—21no—
117C2×Dic3—21no—
118C2×Dic3—31no—
119C2×Dic3—31no—
120C2×Dic3—61no—
121C2×Dic3—62no—
122C2×Dic3—61no—
123C2×Dic3—61no—
124C4×D3—61no—
125C4×D3—61no—
126C4×D3—61no—
127C4×D3—61no—
128C4×D3—121no—
129C4×D3—121no—
130Dic6—21no—
131Dic6—21no—
132Dic6—61no—
133Dic6—61no—
134Dic6—61no—
135Dic6—61no—
136Dic6—121no—
137Dic6—121no—
138C3×C8—32no—
139C3×C8—61no—
140C3⋊C8—21no—
141C3⋊C8D1212yes—
142C2×C32#12111yescharacteristic
143C2×C32C2×Q1611yescharacteristic
144C2×C32#13511yescharacteristic
145C3⋊D3—22no—
146C2×Q8—31no—
147C2×Q8—91no—
148C22×C4—91no—
149Q16—182no—
150C2×C8—31no—
151C2×C8—31no—
152C2×C8—182no—
153C4⋊C4—31no—
154C4⋊C4—181no—
155C4⋊C4—181no—
156C4⋊C4D3211yescharacteristic
157C22⋊C4—181no—
158C22⋊C4—181no—
159C22×C3—21no—
160C22×C3D3×D411yescharacteristic
161C22×C3D3×D411yescharacteristic
162C2×D3—32no—
163C2×D3—32no—
164C2×D3—62no—
165C2×D3—61no—
166C2×D3—61no—
167C2×D3—61no—
168C2×D3—61no—
169C2×D3—121no—
170C2×D3—121no—
171C3×C4—21no—
172C3×C4—21no—
173C3×C4—42no—
174C3×C4—41no—
175C3×C4—41no—
176C3×C4—121no—
177C3×C4—121no—
178C3×C4C2×D1211yescharacteristic
179C3×C4#23511yescharacteristic
180C3×C4D3×D411yescharacteristic
181C3×C4#23911yescharacteristic
182Dic3—41no—
183Dic3—41no—
184Dic3—61no—
185Dic3—61no—
186Dic3—121no—
187Dic3—122no—
188Dic3—121no—
189Dic3—121no—
190C32#46711yescharacteristic, complemented, Sylow
191C23—91no—
192Q8—61no—
193Q8—61no—
194Q8—181no—
195Q8—181no—
196C2×C4—21no—
197C2×C4—61no—
198C2×C4—61no—
199C2×C4—91no—
200C2×C4—181no—
201C2×C4—181no—
202C2×C4—181no—
203C2×C4C2×D3211yescharacteristic
204C8—32no—
205C8—61no—
206C2×C3—21no—
207C2×C3—21no—
208C2×C3—21no—
209C2×C3#88411yescharacteristic
210C2×C3#89511yescharacteristic
211C2×C3#91211yescharacteristic
212C2×C3#91611yescharacteristic
213C2×C3D3×Q1611yescharacteristic
214C2×C3D3×Q1611yescharacteristic
215D3—62no—
216D3—62no—
217D3—122no—
218C22—92no—
219C22—181no—
220C22—181no—
221C22#381111yescharacteristic
222C4—41no—
223C4—121no—
224C4—121no—
225C4—181no—
226C4—361no—
227C4#379911yescharacteristic
228C4D3×D1211yescharacteristic
229C3—21no—
230C3#481311yescharacteristic, complemented
231C3#488711yescharacteristic, complemented
232C2—182no—
233C2#6391011yescharacteristic
234C2#6393611yescharacteristic
235C2#6394411yescharacteristic
subgroupG/H
centerC22#3811
commutator, cosocleC2×C32×C4C23
FrattiniC2×C4C2×D32
FittingC32×C4⋊C4C22
socleC22×C32C2×D4
prime radicalC22×C3⋊D3D4
hypercenterC4⋊C4D32
hypocenterC32#467
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C2×C4 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
3. C2×#3719C2semidirect
5. C3×#810C2semidirect
6. C3×#807C2semidirect
7. #63567C2semidirect
36. #810D3semidirect
37. #807D3semidirect
88. C2×C3⋊D3Q16semidirect
190. C32#467semidirect
230. C3#4813semidirect
231. C3#4887semidirect

Maximal subgroups

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

subgroupindex
#641062
#638712
C2×#37192
C2×#37102
C3×#8102
C3×#8072
#635672
#48873
#48133