Finite Group Databasebeta

← #10591579

Subgroups — #10591579

Order 576.

262 conjugacy classes of subgroups; only the 62 normal subgroups are shown.

normal subgroups only

This group has 262 conjugacy classes of subgroups — too many to draw. Only its 62 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#64156C211yesmaximal, characteristic
2C2×C3×#216C211yesmaximal, characteristic, complemented
3#63843C211yesmaximal, characteristic, complemented
4C2×#3717C211yesmaximal, characteristic
5C3×#835C211yesmaximal, characteristic, complemented
6#63572C211yesmaximal, characteristic, complemented
7#63564C211yesmaximal, characteristic
8#5185—31nomaximal
9#4798—31nomaximal
10C2×C32×Q8C2211yescharacteristic
11C2×C4⋊(C3⋊D3)C2211yescharacteristic
12C2×C3×Dic6C2211yescharacteristic
13#3752—21no—
14#3752—21no—
15C4×C3⋊Dic3C2211yescharacteristic
16C3×#216—22no—
17C3×C4⋊Dic3C2211yescharacteristic
18C2×C3×C3⋊C8C2211yescharacteristic
19C2×C3×C3⋊C8C2211yescharacteristic
20Dic3⋊Dic3—21no—
21Dic32—21no—
22#3717—22no—
23C2×C3×Q16—31no—
24#947—31no—
25#947—31no—
26#947—61no—
27Q8×Dic3—31no—
28C2×#216D311yescharacteristic, complemented
29C2×#215—31no—
30C2×#204—31no—
31#887—31no—
32C3×#103—31no—
33#835D311yescharacteristic, complemented
34#821—31no—
35Dic3⋊C8—31no—
36Dic3⋊C8—31no—
37#809—31no—
38C22×C3⋊D3—21no—
39C32×Q8—21no—
40C32×Q8—21no—
41C2×C32×C4—21no—
42C2×C32×C4C2311yescharacteristic
43C2×C3⋊Dic3D411yescharacteristic
44C2×C3⋊Dic3D411yescharacteristic
45C4⋊(C3⋊D3)—21no—
46C4⋊(C3⋊D3)—21no—
47C2×C3×Dic3—21no—
48C2×C3×Dic3—21no—
49C3×Dic6—21no—
50C3×Dic6D412yes—
51C3×C3⋊C8—21no—
52C3×C3⋊C8—21no—
53#464—91noSylow
54C2×C3×Q8—21no—
55C2×C3×Q8—31no—
56C2×C3×Q8C2×D311yescharacteristic
57C2×C3×Q8C2×D311yescharacteristic
58C2×D12—31no—
59C2×D12—31no—
60C2×D12—61no—
61C2×Dic6C2×D311yescharacteristic
62C3×Q16—62no—
63C2×C3×C8—31no—
64C2×C3×C8—31no—
65C3×C4⋊C4—31no—
66#216—22no—
67#215—62no—
68(C2×D3)⋊C4—61no—
69(C2×D3)⋊C4—61no—
70(C2×D3)⋊C4—61no—
71(C2×D3)⋊C4—61no—
72(C2×D3)⋊C4—62no—
73(C2×D3)⋊C4—62no—
74C4⋊Dic3—61no—
75C4⋊Dic3C2×D311yescharacteristic
76Dic3⋊C4—61no—
77C4×Dic3—31no—
78C4×Dic3—31no—
79C4×Dic3—61no—
80C4×Dic3—61no—
81C4×Dic3—61no—
82C2×C3⋊C8C2×D311yescharacteristic
83C2×C3⋊C8C2×D311yescharacteristic
84#204—62no—
85C22×C32C2×D411yescharacteristic
86C2×C3⋊D3—41no—
87C2×C3⋊D3—41no—
88C2×C3⋊D3—41no—
89C32×C4—41no—
90C32×C4C2×D411yescharacteristic
91C32×C4#4111yescharacteristic
92C3⋊Dic3—21no—
93C3⋊Dic3—21no—
94C3×Dic3—22no—
95C3×Dic3—41no—
96C2×Q16—31no—
97C2×SD16—91no—
98#124—91no—
99C4×Q8—91no—
100C4⋊C8—91no—
101#103—31no—
102#102—91no—
103C22×D3—62no—
104C22×D3—61no—
105C22×D3—61no—
106C3×Q8—21no—
107C3×Q8—21no—
108C3×Q8—21no—
109C3×Q8—21no—
110C3×Q8—32no—
111C3×Q8—41no—
112C3×Q8—41no—
113C3×Q8—61no—
114C2×C3×C4—22no—
115C2×C3×C4—21no—
116C2×C3×C4—21no—
117C2×C3×C4—21no—
118C2×C3×C4—61no—
119C2×C3×C4—61no—
120C2×C3×C4C22×D311yescharacteristic
121C2×C3×C4C22×D311yescharacteristic
122C2×Dic3—21no—
123C2×Dic3—21no—
124C2×Dic3—31no—
125C2×Dic3—31no—
126C2×Dic3—31no—
127C2×Dic3—31no—
128C2×Dic3—61no—
129C2×Dic3—61no—
130D12—61no—
131D12—61no—
132D12—61no—
133D12—61no—
134D12—121no—
135D12—121no—
136Dic6—21no—
137Dic6#6712yes—
138C3×C8—61no—
139C3×C8—61no—
140C3⋊C8—21no—
141C3⋊C8—21no—
142C2×C32#12111yescharacteristic
143C2×C32#13511yescharacteristic
144C2×C32#13711yescharacteristic
145C3⋊D3—81no—
146C2×Q8—31no—
147C2×Q8D3211yescharacteristic
148C2×D4—91no—
149Q16—62no—
150SD16—182no—
151C2×C8—31no—
152C2×C8—31no—
153C4⋊C4—31no—
154C4⋊C4—181no—
155C22⋊C4—181no—
156C22⋊C4—181no—
157C42—91no—
158C42—181no—
159C22×C3—21no—
160C22×C3D3×D411yescharacteristic
161C22×C3D3×D411yescharacteristic
162C2×D3—122no—
163C2×D3—122no—
164C2×D3—122no—
165C2×D3—121no—
166C2×D3—121no—
167C2×D3—121no—
168C2×D3—121no—
169C2×D3—121no—
170C2×D3—121no—
171C3×C4—21no—
172C3×C4—21no—
173C3×C4—42no—
174C3×C4—41no—
175C3×C4—41no—
176C3×C4—62no—
177C3×C4—121no—
178C3×C4#23511yescharacteristic
179C3×C4D3×D411yescharacteristic
180C3×C4#23711yescharacteristic
181C3×C4C2×#6711yescharacteristic
182Dic3—22no—
183Dic3—41no—
184Dic3—61no—
185Dic3—61no—
186Dic3—61no—
187Dic3—61no—
188Dic3—121no—
189Dic3—121no—
190C32#46411yescharacteristic, complemented, Sylow
191C23—181no—
192Q8—21no—
193Q8—21no—
194Q8—32no—
195Q8—61no—
196D4—181no—
197D4—181no—
198C2×C4—21no—
199C2×C4—61no—
200C2×C4—61no—
201C2×C4—91no—
202C2×C4—91no—
203C2×C4C2×D3211yescharacteristic
204C8—61no—
205C8—61no—
206C2×C3—21no—
207C2×C3—21no—
208C2×C3—21no—
209C2×C3#88411yescharacteristic
210C2×C3#91611yescharacteristic
211C2×C3#91811yescharacteristic
212C2×C3#91811yescharacteristic
213C2×C3#91911yescharacteristic
214C2×C3#93911yescharacteristic
215D3—242no—
216D3—241no—
217D3—241no—
218C22—361no—
219C22—361no—
220C22—361no—
221C22#381111yescharacteristic
222C4—41no—
223C4—62no—
224C4—121no—
225C4—181no—
226C4—181no—
227C4#380411yescharacteristic
228C4D3×#6711yescharacteristic
229C3—21no—
230C3#479811yescharacteristic, complemented
231C3#518511yescharacteristic, complemented
232C2—721no—
233C2#6394511yescharacteristic
234C2#6395011yescharacteristic
235C2#6397111yescharacteristic
subgroupG/H
centerC22#3811
commutator, cosocleC2×C32×C4C23
FrattiniC2×C4C2×D32
FittingC2×C32×Q8C22
socleC22×C32C2×D4
prime radicalC2×C4⋊(C3⋊D3)C22
hypercenterC2×Q8D32
hypocenterC32#464
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×C3×#216C2semidirect
3. #63843C2semidirect
5. C3×#835C2semidirect
6. #63572C2semidirect
28. C2×#216D3semidirect
33. #835D3semidirect
190. C32#464semidirect
230. C3#4798semidirect
231. C3#5185semidirect

Maximal subgroups

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

subgroupindex
#641562
C2×C3×#2162
#638432
C2×#37172
C3×#8352
#635722
#635642
#51853
#47983