Finite Group Databasebeta

← #10591607

Subgroups — #10591607

Order 576.

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

normal subgroups only

This group has 258 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×#3755C211yesmaximal, characteristic
2C3×Q8×Dic3C211yesmaximal, characteristic, complemented
3C2×C3×#216C211yesmaximal, characteristic, complemented
4#63864C211yesmaximal, characteristic
5Dic3⋊(C3⋊C8)C211yesmaximal, characteristic, complemented
6#63565C211yesmaximal, characteristic, complemented
7#63564C211yesmaximal, characteristic
8#5185—31nomaximal
9#5028—31nomaximal
10C2×C32×Q8C2211yescharacteristic
11C2×C4⋊(C3⋊D3)C2211yescharacteristic
12C2×C3×Dic6C2211yescharacteristic
13#3755—22no—
14C2×C3⋊(C3⋊C8)C2211yescharacteristic
15C3×#216—22no—
16C3×C4⋊Dic3—21no—
17C3×C4⋊Dic3C2211yescharacteristic
18C3×C4×Dic3—21no—
19C3×C4×Dic3C2211yescharacteristic
20C2×C3×C3⋊C8C2211yescharacteristic
21#3722—21no—
22#3722—21no—
23C2×C3×Q16—31no—
24C3×C4×Q8—31no—
25#947—31no—
26Q8×Dic3D311yescharacteristic, complemented
27C2×#216D311yescharacteristic, complemented
28C2×#215—31no—
29C2×#215—31no—
30C2×#215—61no—
31#875—31no—
32#835—31no—
33#821—31no—
34Dic3⋊C8—31no—
35#810—31no—
36#809—31no—
37C4⋊(C3⋊C8)—31no—
38C22×C3⋊D3—21no—
39C32×Q8—21no—
40C32×Q8D412yes—
41C2×C32×C4—21no—
42C2×C32×C4C2311yescharacteristic
43C4⋊(C3⋊D3)—21no—
44C4⋊(C3⋊D3)—21no—
45C2×C3×Dic3—21no—
46C2×C3×Dic3—21no—
47C2×C3×Dic3D411yescharacteristic
48C2×C3×Dic3D411yescharacteristic
49C3×Dic6—21no—
50C3×Dic6—21no—
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—
64C3×C4⋊C4—31no—
65C3×C4⋊C4—61no—
66C3×C42—31no—
67C3×C42—61no—
68#216—22no—
69#215—64no—
70#215—62no—
71#215—62no—
72(C2×D3)⋊C4—61no—
73(C2×D3)⋊C4—61no—
74(C2×D3)⋊C4—61no—
75(C2×D3)⋊C4—61no—
76C4⋊Dic3—21no—
77C4⋊Dic3C2×D311yescharacteristic
78C4×Dic3—21no—
79C4×Dic3C2×D311yescharacteristic
80C2×C3⋊C8—31no—
81C2×C3⋊C8—31no—
82C2×C3⋊C8—61no—
83C2×C3⋊C8C2×D311yescharacteristic
84C22×C32C2×D411yescharacteristic
85C2×C3⋊D3—41no—
86C2×C3⋊D3—41no—
87C2×C3⋊D3—41no—
88C32×C4—22no—
89C32×C4C2×D411yescharacteristic
90C32×C4#4111yescharacteristic
91C3×Dic3—21no—
92C3×Dic3—21no—
93C3×Dic3—41no—
94C3×Dic3—41no—
95C2×Q16—31no—
96C2×SD16—91no—
97#124—91no—
98C4×Q8—31no—
99C4⋊C8—91no—
100#103—91no—
101#102—91no—
102C22×D3—61no—
103C22×D3—61no—
104C22×D3—121no—
105C3×Q8—21no—
106C3×Q8—21no—
107C3×Q8—22no—
108C3×Q8—41no—
109C3×Q8—61no—
110C3×Q8—61no—
111C3×Q8D1212yes—
112C3×Q8#6712yes—
113C2×C3×C4—21no—
114C2×C3×C4—21no—
115C2×C3×C4—21no—
116C2×C3×C4—31no—
117C2×C3×C4—31no—
118C2×C3×C4—41no—
119C2×C3×C4—61no—
120C2×C3×C4—61no—
121C2×C3×C4C22×D311yescharacteristic
122C2×C3×C4C22×D311yescharacteristic
123C2×Dic3—21no—
124C2×Dic3—21no—
125C2×Dic3#6711yescharacteristic
126C2×Dic3#6711yescharacteristic
127D12—62no—
128D12—61no—
129D12—61no—
130D12—61no—
131D12—61no—
132D12—121no—
133Dic6—21no—
134Dic6—21no—
135C3×C8—61no—
136C3⋊C8—21no—
137C3⋊C8—62no—
138C3⋊C8—61no—
139C3⋊C8—61no—
140C2×C32#12111yescharacteristic
141C2×C32#13511yescharacteristic
142C2×C32#13711yescharacteristic
143C3⋊D3—81no—
144C2×Q8—31no—
145C2×Q8D3211yescharacteristic
146C2×D4—91no—
147Q16—62no—
148SD16—182no—
149C2×C8—31no—
150C2×C8—91no—
151C4⋊C4—31no—
152C4⋊C4—61no—
153C22⋊C4—181no—
154C22⋊C4—181no—
155C42—31no—
156C42—61no—
157C22×C3—21no—
158C22×C3D3×D411yescharacteristic
159C22×C3C2×#6711yescharacteristic
160C2×D3—122no—
161C2×D3—121no—
162C2×D3—121no—
163C2×D3—121no—
164C2×D3—121no—
165C2×D3—121no—
166C2×D3—121no—
167C2×D3—241no—
168C2×D3—241no—
169C3×C4—22no—
170C3×C4—22no—
171C3×C4—21no—
172C3×C4—21no—
173C3×C4—42no—
174C3×C4—61no—
175C3×C4—61no—
176C3×C4—121no—
177C3×C4—121no—
178C3×C4C2×D1211yescharacteristic
179C3×C4#23511yescharacteristic
180C3×C4#23711yescharacteristic
181C3×C4C2×#6711yescharacteristic
182Dic3—21no—
183Dic3—21no—
184Dic3—41no—
185Dic3—41no—
186C32#46411yescharacteristic, complemented, Sylow
187C23—181no—
188Q8—21no—
189Q8—61no—
190Q8—61no—
191Q8#62612yes—
192D4—181no—
193D4—181no—
194C2×C4—21no—
195C2×C4—31no—
196C2×C4—31no—
197C2×C4—61no—
198C2×C4—61no—
199C2×C4C2×D3211yescharacteristic
200C8—61no—
201C8—181no—
202C2×C3—21no—
203C2×C3—21no—
204C2×C3—21no—
205C2×C3#91811yescharacteristic
206C2×C3#91911yescharacteristic
207C2×C3#93011yescharacteristic
208C2×C3#93911yescharacteristic
209C2×C3#94211yescharacteristic
210C2×C3#95011yescharacteristic
211D3—242no—
212D3—241no—
213D3—241no—
214C22—361no—
215C22—361no—
216C22—361no—
217C22D3×#6711yescharacteristic
218C4—22no—
219C4—61no—
220C4—61no—
221C4—121no—
222C4—121no—
223C4#380411yescharacteristic
224C4C2×#62611yescharacteristic
225C3—21no—
226C3#502811yescharacteristic, complemented
227C3#518511yescharacteristic, complemented
228C2—721no—
229C2#6394811yescharacteristic
230C2#6394911yescharacteristic
231C2#6397811yescharacteristic
subgroupG/H
centerC22D3×#67
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. C3×Q8×Dic3C2semidirect
3. C2×C3×#216C2semidirect
5. Dic3⋊(C3⋊C8)C2semidirect
6. #63565C2semidirect
26. Q8×Dic3D3semidirect
27. C2×#216D3semidirect
186. C32#464semidirect
226. C3#5028semidirect
227. C3#5185semidirect

Maximal subgroups

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

subgroupindex
C2×#37552
C3×Q8×Dic32
C2×C3×#2162
#638642
Dic3⋊(C3⋊C8)2
#635652
#635642
#51853
#50283