Finite Group Databasebeta

← #10591009

Subgroups — #10591009

Order 576.

342 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 342 conjugacy classes of subgroups — too many to draw. Only its 70 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#64306C211yesmaximal, characteristic
2M16×C3⋊D3C211yesmaximal, characteristic, complemented
3#63874C211yesmaximal, characteristic, complemented
4C3×#825C212yesmaximal, complemented
5#63569C212yesmaximal, complemented
6#4821—32nomaximal
7C2×C4×C3⋊D3C2211yescharacteristic
8C3×#235C2212yes—
9#3811—21no—
10#3805—21no—
11#3802C411yescharacteristic, complemented
12C4×D32—21no—
13#3798—21no—
14#3797C411yescharacteristic, complemented
15#3796C412yescomplemented
16C32×M16C2211yescharacteristic, complemented
17#3748C2211yescharacteristic, complemented
18#3743—21no—
19C8×C3⋊D3—21no—
20C3×C4×Dic3C2212yescomplemented
21#3724—21no—
22#3722—21no—
23D3×#41—32no—
24D3×M16—32no—
25D3×M16—61no—
26#872—32no—
27C3×#104—32no—
28#837—32no—
29#825D312yescomplemented
30#805—32no—
31C22×C3⋊D3D411yescharacteristic
32C2×D32—21no—
33C2×C32×C4C2311yescharacteristic
34C2×C3⋊Dic3D411yescharacteristic
35C4×C3⋊D3C2×C411yescharacteristic
36C4×C3⋊D3C2×C411yescharacteristic
37C4×C3⋊D3D411yescharacteristic
38C4×C3⋊D3D411yescharacteristic
39C3×#67—22no—
40C2×C3×Dic3—22no—
41C3×D12C2×C412yes—
42C3×C4×D3—22no—
43C3×Dic6C2×C412yes—
44#627—21no—
45#626—22no—
46#625—21no—
47#624—21no—
48D3×Dic3—22no—
49C32×C8—21no—
50C3⋊(C3⋊C8)—21no—
51#421—91noSylow
52C3×#41—32no—
53#239—32no—
54D3×Q8—32no—
55#237—32no—
56#237—62no—
57D3×D4—32no—
58D3×D4—62no—
59#235—62no—
60#235C2×D312yes—
61C2×C4×D3—32no—
62C2×C4×D3—61no—
63C2×C4×D3—62no—
64C3×M16—21no—
65C3×M16C2×D312yescomplemented
66C3×C42—32no—
67(C2×D3)⋊C4—62no—
68Dic3⋊C4—62no—
69C4×Dic3C2×D312yescomplemented
70#208—32no—
71#208—61no—
72ΓL(1,𝔽25)—62no—
73ΓL(1,𝔽25)—62no—
74D3×C8—62no—
75D3×C8—62no—
76C22×C32C2×D411yescharacteristic
77C2×C3⋊D3—41no—
78C2×C3⋊D3C22⋊C411yescharacteristic
79C2×C3⋊D3C22⋊C411yescharacteristic
80C2×C3×D3—22no—
81D32—42no—
82C32×C4C22×C411yescharacteristic
83C32×C4C2×D411yescharacteristic
84C3⋊Dic3C22⋊C411yescharacteristic
85C3⋊Dic3C22⋊C411yescharacteristic
86C3×Dic3—22no—
87C3×Dic3—42no—
88C2×#41—91no—
89C2×M16—91no—
90#117—91no—
91#104—32no—
92#104—92no—
93C22×D3—32no—
94C22×D3—61no—
95C22×D3—62no—
96C3×Q8—32no—
97C3×D4—32no—
98C3×D4—62no—
99C2×C3×C4—21no—
100C2×C3×C4—62no—
101C2×C3×C4—62no—
102C2×C3×C4C22×D312yes—
103#67—22no—
104#67—62no—
105#67—62no—
106C2×Dic3—22no—
107C2×Dic3—32no—
108C2×Dic3—61no—
109C2×Dic3—62no—
110D12—62no—
111D12C4×D312yes—
112C4×D3—22no—
113C4×D3—32no—
114C4×D3—32no—
115C4×D3—32no—
116C4×D3—32no—
117C4×D3—61no—
118C4×D3—61no—
119C4×D3—61no—
120C4×D3—61no—
121C4×D3—62no—
122C4×D3—62no—
123Dic6—62no—
124Dic6C4×D312yes—
125C3×C8—22no—
126C3×C8—22no—
127C3⋊C8—62no—
128C3⋊C8—62no—
129C2×C32—21no—
130C2×C32C2×C22⋊C411yescharacteristic
131C3⋊D3—21no—
132C3⋊D3—21no—
133C3×D3—42no—
134#41—32no—
135#41—92no—
136#41—181no—
137#41—181no—
138C2×Q8—91no—
139C2×D4—91no—
140C2×D4—181no—
141C22×C4—91no—
142C22×C4—181no—
143M16—91no—
144M16—181no—
145M16D3211yescharacteristic, complemented
146C2×C8—181no—
147C4⋊C4—181no—
148C22⋊C4—181no—
149C42—32no—
150C22×C3—21no—
151C22×C3—62no—
152C22×C3D3×D412yes—
153C2×D3—22no—
154C2×D3—32no—
155C2×D3—32no—
156C2×D3—61no—
157C2×D3—61no—
158C2×D3—124no—
159C2×D3—122no—
160C2×D3—122no—
161C3×C4—21no—
162C3×C4—21no—
163C3×C4—62no—
164C3×C4—122no—
165C3×C4C2×C4×D312yes—
166C3×C4D3×D412yes—
167Dic3—22no—
168Dic3—32no—
169Dic3—32no—
170Dic3—42no—
171Dic3—61no—
172Dic3—61no—
173C32#42111yescharacteristic, complemented, Sylow
174C23—91no—
175C23—181no—
176Q8—32no—
177Q8—181no—
178D4—32no—
179D4—62no—
180D4—182no—
181D4—181no—
182C2×C4—62no—
183C2×C4—62no—
184C2×C4—91no—
185C2×C4—91no—
186C2×C4—91no—
187C2×C4—91no—
188C2×C4—91no—
189C2×C4—181no—
190C2×C4—182no—
191C2×C4C2×D3211yescharacteristic
192C8—21no—
193C8—181no—
194C2×C3—22no—
195C2×C3—21no—
196C2×C3—41no—
197C2×C3—122no—
198C2×C3D3×C22⋊C412yes—
199D3—42no—
200D3—62no—
201D3—62no—
202D3—62no—
203D3—121no—
204C22—62no—
205C22—91no—
206C22—91no—
207C22—361no—
208C22—362no—
209C22#380211yescharacteristic
210C4—62no—
211C4—91no—
212C4—91no—
213C4—122no—
214C4C4×D3211yescharacteristic
215C4#381111yescharacteristic
216C3—21no—
217C3#482112yescomplemented
218C2—21no—
219C2—122no—
220C2—181no—
221C2—181no—
222C2#6392111yescharacteristic
subgroupG/H
centerC4#3811
commutatorC32×C4C22×C4
FrattiniC2×C4C2×D32
FittingC32×M16C22
socleC2×C32C2×C22⋊C4
prime radical#64306C2
cosocleC2×C32×C4C23
hypercenterM16D32
hypocenterC32#421
LCSG ▷ C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C4 ◁ C2×C4 ◁ M16

Complements

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

NH ≅ G/Ntype
2. M16×C3⋊D3C2semidirect
3. #63874C2semidirect
4. C3×#825C2semidirect
5. #63569C2semidirect
11. #3802C4semidirect
14. #3797C4semidirect
15. #3796C4semidirect
16. C32×M16C22semidirect
17. #3748C22semidirect
20. C3×C4×Dic3C22semidirect
29. #825D3semidirect
65. C3×M16C2×D3semidirect
69. C4×Dic3C2×D3semidirect
145. M16D32semidirect
173. C32#421semidirect
217. C3#4821semidirect

Maximal subgroups

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

subgroupindex
#643062
M16×C3⋊D32
#638742
C3×#8252
#635692
#48213