Finite Group Databasebeta

← #13166232

Subgroups — #13166232

Order 1728.

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

normal subgroups only

This group has 332 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#11723333C211yesmaximal, characteristic
2#11723089C211yesmaximal, characteristic, complemented
3C3×#63765C211yesmaximal, characteristic, complemented
4#11722721C211yesmaximal, characteristic, complemented
5#11722720C211yesmaximal, characteristic, complemented
6#11722719C211yesmaximal, characteristic, complemented
7D3×#3708C211yesmaximal, characteristic, complemented
8#10593772—31nomaximal
9#89020C411yescharacteristic
10#88850C411yescharacteristic
11C3×C8×C3⋊D3C2211yescharacteristic
12C32×ΓL(1,𝔽25)C411yescharacteristic
13#88787C411yescharacteristic
14#88785C2211yescharacteristic
15D3×C3⋊(C3⋊C8)C2211yescharacteristic
16C3⋊#3708C2211yescharacteristic
17C3⋊#3708C2211yescharacteristic
18C3×#3708C2211yescharacteristic
19C3×#3708C2211yescharacteristic
20#64116—31no—
21#63804—62no—
22#63773—31no—
23C2×#3708—31no—
24#63765—31no—
25#63765D311yescharacteristic, complemented
26#63764—31no—
27#63764—31no—
28C4×C3⋊(C3⋊D3)C2×C411yescharacteristic
29C3×C4×C3⋊D3C2×C411yescharacteristic
30C32×C4×D3C2×C411yescharacteristic
31#6293C811yescharacteristic
32#6292C811yescharacteristic
33#6291C811yescharacteristic
34#6290C811yescharacteristic
35C33×C8C2×C411yescharacteristic
36C3⋊(C3⋊(C3⋊C8))C2×C411yescharacteristic
37C3×C3⋊(C3⋊C8)C2311yescharacteristic
38C32×C3⋊C8C2×C411yescharacteristic
39#4907—91nomaximal
40#3828—31no—
41#3798—62no—
42C32×M16—31no—
43C2×C3⋊(C3⋊C8)—31no—
44#3743—31no—
45#3743—31no—
46#3743—62no—
47#3743—122no—
48C8×C3⋊D3—31no—
49C8×C3⋊D3C2×D311yescharacteristic
50C3×ΓL(1,𝔽25)—22no—
51C3×D3×C8—62no—
52#3711—62no—
53#3710—62no—
54D3×C3⋊C8—62no—
55#3708—31no—
56#3708—31no—
57#3708C2×D311yescharacteristic
58#3708C2×D311yescharacteristic
59C2×C3⋊(C3⋊D3)C2×C811yescharacteristic
60C2×C3×C3⋊D3C2×C811yescharacteristic
61C2×C32×D3C2×C811yescharacteristic
62C33×C4C22×C411yescharacteristic
63C3⋊(C3⋊Dic3)C2×C811yescharacteristic
64C3×C3⋊Dic3C2×C811yescharacteristic
65C32×Dic3C2×C811yescharacteristic
66#907—182no—
67#901—91no—
68C3×M32—91no—
69C2×C3⋊C16—91no—
70#798—91no—
71#798—91no—
72D3×C16—91no—
73D3×C16—91no—
74C2×C32×C4—31no—
75#638—31no—
76#638—31no—
77C4⋊(C3⋊D3)—31no—
78C4×C3⋊D3—31no—
79C4×C3⋊D3—62no—
80C4×C3⋊D3—122no—
81C4×C3⋊D3C4×D311yescharacteristic
82#634—31no—
83C3×C4×D3—22no—
84C3×C4×D3—62no—
85#627—62no—
86#626—62no—
87#626—62no—
88#625—62no—
89C32×C8—22no—
90C32×C8—31no—
91C32×C8—42no—
92C32×C8C4×D311yescharacteristic
93C3⋊(C3⋊C8)—31no—
94C3⋊(C3⋊C8)—62no—
95C3⋊(C3⋊C8)—122no—
96C3⋊(C3⋊C8)C22×D311yescharacteristic
97C3×C3⋊C8—22no—
98C3×C3⋊C8—62no—
99#504—271noSylow
100C2×C33C22×C811yescharacteristic
101C3⋊(C3⋊D3)—21no—
102C3×C3⋊D3—21no—
103C32×D3—21no—
104#235—91no—
105#235—182no—
106C3×M16—62no—
107C2×C3×C8—91no—
108#208—91no—
109C2×C3⋊C8—182no—
110ΓL(1,𝔽25)—91no—
111ΓL(1,𝔽25)—182no—
112ΓL(1,𝔽25)—182no—
113ΓL(1,𝔽25)—362no—
114ΓL(1,𝔽25)#15711yescharacteristic
115D3×C8—62no—
116D3×C8—91no—
117D3×C8—91no—
118D3×C8—182no—
119C3×C16—91no—
120C3×C16—91no—
121C3⋊C16—91no—
122C3⋊C16—91no—
123C22×C32—31no—
124C2×C3⋊D3—31no—
125C2×C3⋊D3—62no—
126C2×C3⋊D3—122no—
127C2×C3⋊D3D3×C811yescharacteristic
128C2×C3×D3—22no—
129C2×C3×D3—62no—
130C32×C4—22no—
131C32×C4—31no—
132C32×C4—42no—
133C32×C4C2×C4×D311yescharacteristic
134C3⋊Dic3—31no—
135C3⋊Dic3—62no—
136C3⋊Dic3—122no—
137C3⋊Dic3D3×C811yescharacteristic
138C3×Dic3—22no—
139C3×Dic3—62no—
140#131—271no—
141M32—91no—
142M32—271no—
143M32—271no—
144C2×C16—271no—
145C2×C16—271no—
146C2×C16—271no—
147C33#50411yescharacteristic, complemented, Sylow
148C2×C3×C4—62no—
149C2×C3×C4—91no—
150#67—91no—
151#67—91no—
152#67—182no—
153#67—182no—
154D12—91no—
155D12—182no—
156C4×D3—62no—
157C4×D3—91no—
158C4×D3—182no—
159C4×D3—362no—
160C4×D3C2×#15711yescharacteristic
161Dic6—91no—
162Dic6—182no—
163C3×C8—22no—
164C3×C8—42no—
165C3×C8—62no—
166C3×C8—91no—
167C3×C8C2×#15711yescharacteristic
168C3⋊C8—62no—
169C3⋊C8—91no—
170C3⋊C8—182no—
171C3⋊C8—362no—
172C3⋊C8C2×#15711yescharacteristic
173C2×C32—22no—
174C2×C32—42no—
175C2×C32—61no—
176C2×C32C2×D3×C811yescharacteristic
177C3⋊D3—21no—
178C3⋊D3—61no—
179C3⋊D3—122no—
180C3⋊D3—242no—
181C3×D3—42no—
182C3×D3—122no—
183#41—271no—
184M16—31no—
185M16—271no—
186M16—271no—
187C2×C8—91no—
188C2×C8—271no—
189C2×C8—271no—
190C16—91no—
191C16—91no—
192C16—271no—
193C16—271no—
194C22×C3—62no—
195C22×C3—91no—
196C2×D3—62no—
197C2×D3—91no—
198C2×D3—182no—
199C2×D3—362no—
200C2×D3#378711yescharacteristic
201C3×C4—22no—
202C3×C4—42no—
203C3×C4—62no—
204C3×C4—91no—
205C3×C4C22×#15711yescharacteristic
206Dic3—62no—
207Dic3—91no—
208Dic3—182no—
209Dic3—362no—
210Dic3#378711yescharacteristic
211C32—22no—
212C32—42no—
213C32#490711yescharacteristic, complemented
214Q8—271no—
215D4—271no—
216D4—271no—
217D4—271no—
218C2×C4—31no—
219C2×C4—91no—
220C2×C4—271no—
221C8—31no—
222C8—91no—
223C8—271no—
224C8D3×#15711yescharacteristic
225C2×C3—22no—
226C2×C3—42no—
227C2×C3—122no—
228C2×C3—181no—
229C2×C3C2×#378711yescharacteristic
230D3—21no—
231D3—122no—
232D3—181no—
233D3—362no—
234D3—722no—
235C22—31no—
236C22—91no—
237C22—271no—
238C4—31no—
239C4—91no—
240C4—271no—
241C4C2×D3×#15711yescharacteristic
242C3—22no—
243C3—42no—
244C3#1059377211yescharacteristic, complemented
245C2—61no—
246C2—181no—
247C2—541no—
248C2D3×#378711yescharacteristic
subgroupG/H
center, FrattiniC4C2×D3×#157
commutator, socleC2×C33C22×C8
FittingC33×C8C2×C4
prime radical#89020C4
cosocleC3×C3⋊(C3⋊C8)C23
hypercenterC8D3×#157
hypocenterC33#504
LCSG ▷ C2×C33 ▷ C33
UCSC1 ◁ C4 ◁ C8

Complements

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

NH ≅ G/Ntype
2. #11723089C2semidirect
3. C3×#63765C2semidirect
4. #11722721C2semidirect
5. #11722720C2semidirect
6. #11722719C2semidirect
7. D3×#3708C2semidirect
25. #63765D3semidirect
147. C33#504semidirect
213. C32#4907semidirect
244. C3#10593772semidirect

Maximal subgroups

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

subgroupindex
#117233332
#117230892
C3×#637652
#117227212
#117227202
#117227192
D3×#37082
#105937723
#49079