Finite Group Databasebeta

← #10590587

Subgroups — #10590587

Order 576.

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

normal subgroups only

This group has 241 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
1C3×#897C211yesmaximal, characteristic
2#63857C211yesmaximal, characteristic
3C2×#3719C211yesmaximal, characteristic, complemented
4C2×#3711C211yesmaximal, characteristic
5#63637C211yesmaximal, characteristic, complemented
6C3×#808C211yesmaximal, characteristic, complemented
7#63565C211yesmaximal, characteristic, complemented
8#5052—31nomaximal
9#4867—31nomaximal
10C2×#634C2211yescharacteristic
11C2×C3×C4×D3C2211yescharacteristic
12C2×C3×Dic6C2211yescharacteristic
13C32×C4⋊C4C2211yescharacteristic
14C2×C3⋊(C3⋊C8)C2211yescharacteristic
15C3×(C2×D3)⋊C4—21no—
16C3×C4⋊Dic3—21no—
17C3×C4⋊Dic3C2211yescharacteristic
18C2×C3×C3⋊C8C2211yescharacteristic
19Dic3⋊Dic3—21no—
20(C2×D3)⋊Dic3—21no—
21#3719—22no—
22#3711—22no—
23C3×C22⋊Q8—31no—
24C2×#216—31no—
25#924—31no—
26C2×#208—31no—
27C2×Dic12—31no—
28C2×ΓL(1,𝔽25)—31no—
29#897—31no—
30#897D311yescharacteristic
31C3×#106—31no—
32#835—31no—
33#810—31no—
34#810—31no—
35#810—31no—
36#810—61no—
37#808D311yescharacteristic, complemented
38C22×C3×D3D411yescharacteristic
39C2×C32×C4—21no—
40C2×C32×C4C2311yescharacteristic
41C2×C3⋊Dic3—21no—
42#634—21no—
43#634—21no—
44C2×C3×Dic3—21no—
45C2×C3×Dic3—21no—
46C2×C3×Dic3D411yescharacteristic
47C3×C4×D3—21no—
48C3×C4×D3—21no—
49C3×Dic6—21no—
50C3×Dic6—21no—
51C3⋊(C3⋊C8)—21no—
52C3×C3⋊C8D412yes—
53#470—91noSylow
54C2×C3×Q8—31no—
55C22×C3×C4—31no—
56C2×C4×D3C2×D311yescharacteristic
57C2×Dic6—31no—
58C2×Dic6—31no—
59C2×Dic6—61no—
60C2×Dic6C2×D311yescharacteristic
61C2×C3×C8—31no—
62C3×C4⋊C4—21no—
63C3×C4⋊C4—31no—
64C3×C4⋊C4—61no—
65C3×C4⋊C4C2×D311yescharacteristic
66C3×C4⋊C4C2×D311yescharacteristic
67C3×C22⋊C4—61no—
68#217—61no—
69#216—62no—
70(C2×D3)⋊C4—21no—
71(C2×D3)⋊C4—61no—
72C4⋊Dic3—21no—
73C4⋊Dic3—61no—
74C4⋊Dic3C2×D311yescharacteristic
75Dic3⋊C4—61no—
76#208—62no—
77C2×C3⋊C8—31no—
78C2×C3⋊C8—31no—
79C2×C3⋊C8—61no—
80C2×C3⋊C8C2×D311yescharacteristic
81Dic12—62no—
82ΓL(1,𝔽25)—62no—
83C22×C32C2×D411yescharacteristic
84C2×C3×D3—21no—
85C2×C3×D3—21no—
86C2×C3×D3—21no—
87C32×C4—41no—
88C32×C4C2×D411yescharacteristic
89C32×C4#4111yescharacteristic
90C3⋊Dic3—41no—
91C3×Dic3—21no—
92C3×Dic3—41no—
93C3×Dic3—41no—
94C2×Q16—91no—
95C2×M16—91no—
96C22⋊Q8—31no—
97C22⋊Q8—91no—
98#106—31no—
99#103—91no—
100#103—91no—
101C23×C3—31no—
102C22×D3#6711yescharacteristic
103C3×Q8—61no—
104C3×Q8—61no—
105C2×C3×C4—21no—
106C2×C3×C4—21no—
107C2×C3×C4—21no—
108C2×C3×C4—31no—
109C2×C3×C4—41no—
110C2×C3×C4—61no—
111C2×C3×C4—61no—
112C2×C3×C4—61no—
113C2×C3×C4—61no—
114C2×C3×C4C22×D311yescharacteristic
115C2×C3×C4C22×D311yescharacteristic
116C2×Dic3—21no—
117C2×Dic3—21no—
118C2×Dic3—61no—
119C2×Dic3—61no—
120C2×Dic3—121no—
121C2×Dic3#6711yescharacteristic
122C4×D3—21no—
123C4×D3—21no—
124Dic6—21no—
125Dic6—21no—
126Dic6—62no—
127Dic6—61no—
128Dic6—61no—
129Dic6—61no—
130Dic6—61no—
131Dic6—121no—
132C3×C8—32no—
133C3⋊C8—61no—
134C3⋊C8—61no—
135C3⋊C8—121no—
136C3⋊C8D1212yes—
137C2×C32#12111yescharacteristic
138C2×C32#13711yescharacteristic
139C2×C32#13711yescharacteristic
140C3×D3—41no—
141C2×Q8—31no—
142C2×Q8—91no—
143C22×C4—31no—
144Q16—182no—
145M16—182no—
146C2×C8—31no—
147C2×C8—91no—
148C4⋊C4—31no—
149C4⋊C4—61no—
150C4⋊C4—181no—
151C4⋊C4D3211yescharacteristic
152C22⋊C4—61no—
153C22⋊C4—181no—
154C22×C3—21no—
155C22×C3—61no—
156C22×C3—61no—
157C22×C3—61no—
158C22×C3D3×D411yescharacteristic
159C22×C3C2×#6711yescharacteristic
160C2×D3—21no—
161C2×D3—21no—
162C2×D3—21no—
163C3×C4—21no—
164C3×C4—21no—
165C3×C4—41no—
166C3×C4—41no—
167C3×C4—61no—
168C3×C4—81no—
169C3×C4—121no—
170C3×C4—121no—
171C3×C4C2×D1211yescharacteristic
172C3×C4D3×D411yescharacteristic
173C3×C4#23711yescharacteristic
174C3×C4#23911yescharacteristic
175Dic3—21no—
176Dic3—41no—
177Dic3—41no—
178Dic3—122no—
179Dic3—121no—
180Dic3—121no—
181C32#47011yescharacteristic, complemented, Sylow
182C23—31no—
183Q8—61no—
184Q8—61no—
185Q8—181no—
186Q8—181no—
187C2×C4—21no—
188C2×C4—31no—
189C2×C4—61no—
190C2×C4—61no—
191C2×C4—61no—
192C2×C4—61no—
193C2×C4—181no—
194C2×C4C2×D3211yescharacteristic
195C8—32no—
196C8—181no—
197C2×C3—21no—
198C2×C3—21no—
199C2×C3—21no—
200C2×C3—121no—
201C2×C3#89511yescharacteristic
202C2×C3#91511yescharacteristic
203C2×C3#91511yescharacteristic
204C2×C3#93911yescharacteristic
205C2×C3#94211yescharacteristic
206C2×C3#95111yescharacteristic
207D3—41no—
208C22—61no—
209C22—61no—
210C22—61no—
211C22D3×#6711yescharacteristic
212C4—41no—
213C4—61no—
214C4—121no—
215C4—121no—
216C4—361no—
217C4#379611yescharacteristic
218C4D3×D1211yescharacteristic
219C3—21no—
220C3#486711yescharacteristic, complemented
221C3#505211yescharacteristic, complemented
222C2—121no—
223C2#6388611yescharacteristic
224C2#6392911yescharacteristic
225C2#6394611yescharacteristic
subgroupG/H
centerC22D3×#67
commutator, cosocleC2×C32×C4C23
FrattiniC2×C4C2×D32
FittingC32×C4⋊C4C22
socleC22×C32C2×D4
prime radicalC22×C3×D3D4
hypercenterC4⋊C4D32
hypocenterC32#470
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. #63637C2semidirect
6. C3×#808C2semidirect
7. #63565C2semidirect
37. #808D3semidirect
181. C32#470semidirect
220. C3#4867semidirect
221. C3#5052semidirect

Maximal subgroups

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

subgroupindex
C3×#8972
#638572
C2×#37192
C2×#37112
#636372
C3×#8082
#635652
#50523
#48673