Finite Group Databasebeta

← #13147030

Subgroups — #13147030

Order 1728.

321 conjugacy classes of subgroups; only the 38 normal subgroups are shown.

normal subgroups only

This group has 321 conjugacy classes of subgroups — too many to draw. Only its 38 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#11722393C211yesmaximal, characteristic
2#131⋊H(3,𝔽3)C211yesmaximal, characteristic, complemented
3#11720740C211yesmaximal, characteristic, complemented
4C3×#5404—33nomaximal
5C3×#5144—31nomaximal
6#41⋊H(3,𝔽3)C2211yescharacteristic
7#88611C411yescharacteristic, complemented
8#88610C411yescharacteristic, complemented
9C8×#272—41nomaximal
10C3×#985—33no—
11C32×#131D311yescharacteristic, complemented
12C3×#950—31no—
13C3×#901—31no—
14C3×#131⋊C3D313yescomplemented
15C3×GU(2,𝔽3)—33no—
16C3×#837—31no—
17C3×#825—31no—
18C3×#822—31no—
19C3×C8×Dic3—31no—
20C4×#272—41no—
21Q8⋊H(3,𝔽3)C2×C411yescharacteristic
22C8×H(3,𝔽3)—41no—
23#6180—41no—
24#5404—33no—
25C3×#443—91no—
26#5144—31no—
27C32×#41C2×D311yescharacteristic
28C3×#235—31no—
29C3×#41⋊C3C2×D313yes—
30C3×GL(2,𝔽3)—33no—
31C3×#226—33no—
32C32×M16—31no—
33C2×C32×C8—31no—
34C3×#216—31no—
35C3×#215—31no—
36C3×#214—31no—
37C3×#213—31no—
38C3×C4×Dic3—31no—
39C3×#208—31no—
40C2×C3×C3⋊C8—31no—
41C3×ΓL(1,𝔽25)—61no—
42C3×D3×C8—61no—
43C3×D3×C8—123no—
44C2×#272—41no—
45C4×H(3,𝔽3)—41no—
46#1084—41no—
47#985—33no—
48C3×#135—91no—
49C3×#131—31no—
50C3×#131—91no—
51C3×#131C3⋊D311yescharacteristic
52#950—31no—
53#901—31no—
54#131⋊C3—33no—
55GU(2,𝔽3)—33no—
56C3×#108—91no—
57C3×#104—91no—
58C3×#104—91no—
59C3×C4×C8—91no—
60#837—31no—
61#825—31no—
62#822—31no—
63C8×Dic3—31no—
64C32×Q8C4×D311yescharacteristic
65C32×D4—31no—
66C2×C32×C4—31no—
67C3×#67—61no—
68C2×C3×Dic3—61no—
69C3×D12—31no—
70C3×C4×D3—61no—
71C3×C4×D3—123no—
72C3×Dic6—31no—
73C3×SL(2,𝔽3)C4×D313yes—
74C32×C8—31no—
75C32×C8—43no—
76C32×C8S411yescharacteristic
77C3×C3⋊C8—31no—
78C3×C3⋊C8—31no—
79C3×C3⋊C8—61no—
80C3×C3⋊C8—123no—
81#443—91noSylow
82C2×H(3,𝔽3)—41no—
83#272—42no—
84C3×#41—31no—
85C3×#41—91no—
86C3×#41C2×C3⋊D311yescharacteristic
87#235—31no—
88#41⋊C3—33no—
89GL(2,𝔽3)—33no—
90#226—33no—
91C3×Q16—91no—
92C3×SD16—91no—
93C3×SD16—91no—
94C3×D8—91no—
95C3×M16—31no—
96C3×M16—31no—
97C3×M16—61no—
98C3×M16—91no—
99C3×M16—181no—
100C2×C3×C8—31no—
101C2×C3×C8—31no—
102C2×C3×C8—61no—
103C2×C3×C8—91no—
104C2×C3×C8—181no—
105C3×C42—91no—
106#216—31no—
107#215—31no—
108#214—31no—
109#213—31no—
110C4×Dic3—31no—
111#208—31no—
112C2×C3⋊C8—31no—
113ΓL(1,𝔽25)—61no—
114D3×C8—61no—
115D3×C8—123no—
116C22×C32—31no—
117C2×C3×D3—61no—
118C2×C3×D3—123no—
119C32×C4—31no—
120C32×C4—43no—
121C32×C4C2×S411yescharacteristic
122C3×Dic3—62no—
123C3×Dic3—61no—
124C3×Dic3—123no—
125#135—91no—
126#131—91no—
127#131#27211yescharacteristic, complemented
128#108—91no—
129#104—91no—
130#104—91no—
131C4×C8—91no—
132H(3,𝔽3)—41noSylow
133C3×Q8—31no—
134C3×Q8—91no—
135C3×Q8C4×C3⋊D311yescharacteristic
136C3×D4—31no—
137C3×D4—31no—
138C3×D4—61no—
139C3×D4—91no—
140C3×D4—181no—
141C2×C3×C4—31no—
142C2×C3×C4—31no—
143C2×C3×C4—61no—
144C2×C3×C4—181no—
145C2×C3×C4—181no—
146#67—61no—
147C2×Dic3—61no—
148D12—31no—
149C4×D3—61no—
150C4×D3—123no—
151Dic6—31no—
152SL(2,𝔽3)—33no—
153C3×C8—31no—
154C3×C8—31no—
155C3×C8—31no—
156C3×C8—61no—
157C3×C8—91no—
158C3×C8—91no—
159C3×C8—123no—
160C3×C8—181no—
161C3×C8C3⋊S411yescharacteristic
162C3⋊C8—31no—
163C3⋊C8—31no—
164C3⋊C8—61no—
165C3⋊C8—123no—
166C2×C32—43no—
167C2×C32—61no—
168C2×C32C4×S411yescharacteristic
169C3×D3—126no—
170C3×D3—121no—
171#41—91no—
172#41C2×#27211yescharacteristic
173Q16—91no—
174SD16—91no—
175SD16—91no—
176D8—91no—
177M16—31no—
178M16—91no—
179M16—181no—
180C2×C8—31no—
181C2×C8—91no—
182C2×C8—181no—
183C42—91no—
184C22×C3—31no—
185C22×C3—31no—
186C22×C3—61no—
187C22×C3—181no—
188C2×D3—61no—
189C2×D3—123no—
190C3×C4—31no—
191C3×C4—31no—
192C3×C4—31no—
193C3×C4—61no—
194C3×C4—123no—
195C3×C4—182no—
196C3×C4—181no—
197C3×C4C2×C3⋊S411yescharacteristic
198Dic3—61no—
199Dic3—62no—
200Dic3—123no—
201C32—43no—
202C32#540411yescharacteristic, complemented
203Q8—91no—
204Q8C4×#27211yescharacteristic
205D4—31no—
206D4—91no—
207D4—181no—
208C2×C4—31no—
209C2×C4—181no—
210C2×C4—181no—
211C8—31no—
212C8—91no—
213C8—91no—
214C8—181no—
215C8#625811yescharacteristic
216C2×C3—31no—
217C2×C3—61no—
218C2×C3—61no—
219C2×C3—121no—
220C2×C3—123no—
221C2×C3—361no—
222C2×C3C4×C3⋊S411yescharacteristic
223D3—121no—
224D3—126no—
225C22—31no—
226C22—181no—
227C4—31no—
228C4—182no—
229C4—181no—
230C4C2×#625811yescharacteristic
231C3—31no—
232C3—123no—
233C3#1059363811yescharacteristic
234C2—61no—
235C2—361no—
236C2C4×#625811yescharacteristic
subgroupG/H
center, hypercenterC3×C8C3⋊S4
commutator, hypocenterQ8⋊H(3,𝔽3)C2×C4
FrattiniC3×C4C2×C3⋊S4
FittingC32×#131D3
socleC2×C3C4×C3⋊S4
prime radical#11722393C2
cosocle#41⋊H(3,𝔽3)C22
LCSG ▷ Q8⋊H(3,𝔽3)
UCSC1 ◁ C3×C8

Complements

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

NH ≅ G/Ntype
2. #131⋊H(3,𝔽3)C2semidirect
3. #11720740C2semidirect
7. #88611C4semidirect
8. #88610C4semidirect
11. C32×#131D3semidirect
14. C3×#131⋊C3D3semidirect
127. #131#272semidirect
202. C32#5404semidirect

Maximal subgroups

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

subgroupindex
#117223932
#131⋊H(3,𝔽3)2
#117207402
C3×#54043
C3×#51443
C8×#2724