Finite Group Databasebeta

← #13146378

Subgroups — #13146378

Order 1728.

252 conjugacy classes of subgroups; only the 30 normal subgroups are shown.

normal subgroups only

This group has 252 conjugacy classes of subgroups — too many to draw. Only its 30 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#64266⋊C3C211yesmaximal, characteristic
2#11720767C211yesmaximal, characteristic, complemented
3#11720740C211yesmaximal, characteristic, complemented
4#10593222—31nomaximal
5#10591676—31nomaximal
6#41⋊H(3,𝔽3)C2211yescharacteristic
7(C3⋊GL(2,𝔽3))⋊C3C411yescharacteristic, complemented
8(C3⋊#226)⋊C3C411yescharacteristic, complemented
9#88430—41nomaximal
10#64266D311yescharacteristic
11#64157D311yescharacteristic, complemented
12C3×#950—31no—
13#63815—31no—
14C3⋊(#131⋊C3)—31no—
15C3×GU(2,𝔽3)—31no—
16C3×GU(2,𝔽3)—61no—
17C3×#837—31no—
18#63577—31no—
19#63569—31no—
20#63554—31no—
21C4×(C3⋊D3)⋊C3—41no—
22Q8⋊H(3,𝔽3)C2×C411yescharacteristic
23#6180—41no—
24(C3⋊(C3⋊C8))⋊C3—41no—
25#5405—91no—
26#5197—91no—
27#4823—91no—
28C32×#41C2×D311yescharacteristic
29C3×#235—31no—
30C3×#41⋊C3—21no—
31C3×#41⋊C3C2×D311yescharacteristic
32C3⋊GL(2,𝔽3)Dic311yescharacteristic, complemented
33C3⋊#226Dic311yescharacteristic, complemented
34#3748—31no—
35C2×C3⋊(C3⋊C8)—31no—
36C3×#216—31no—
37C3×#215—31no—
38C3×#214—31no—
39C3×#213—31no—
40C3×C4×Dic3—31no—
41C3×#208—31no—
42C2×C3×C3⋊C8—31no—
43#3712—121no—
44#3711—61no—
45D3×C3⋊C8—61no—
46C2×(C3⋊D3)⋊C3—41no—
47C4×H(3,𝔽3)—41no—
48(C3⋊Dic3)⋊C3—41no—
49#985—91no—
50C3×#135—91no—
51#950—31no—
52#948—31no—
53#948—91no—
54#948—91no—
55#907—91no—
56#131⋊C3—91no—
57GU(2,𝔽3)—31no—
58GU(2,𝔽3)—61no—
59C3×#104—91no—
60#837—31no—
61#837—91no—
62#822—91no—
63#819—91no—
64#815—91no—
65#805—91no—
66#803—91no—
67C32×Q8C4×D311yescharacteristic
68C32×D4—31no—
69C2×C32×C4—31no—
70C4×C3⋊D3—41no—
71C3×#67—61no—
72C2×C3×Dic3—61no—
73C3×D12—31no—
74C3×C4×D3—61no—
75C3×Dic6—31no—
76C3×SL(2,𝔽3)—21no—
77C3×SL(2,𝔽3)C2×Dic311yescharacteristic
78C3⋊(C3⋊C8)—31no—
79C3⋊(C3⋊C8)S411yescharacteristic
80C3×C3⋊C8—31no—
81C3×C3⋊C8—31no—
82C3×C3⋊C8—61no—
83C3×C3⋊C8—121no—
84C3×C3⋊C8—121no—
85C3×C3⋊C8—241no—
86#444—271noSylow
87C2×H(3,𝔽3)—41no—
88(C3⋊D3)⋊C3—81no—
89C3×#41—31no—
90C3×#41—91no—
91C3×#41D3211yescharacteristic
92#235—31no—
93#41⋊C3—31no—
94#41⋊C3—61no—
95GL(2,𝔽3)—91no—
96#226—91no—
97C3×Q16—91no—
98C3×SD16—91no—
99C3×SD16—91no—
100C3×D8—91no—
101C3×M16—91no—
102C2×C3×C8—91no—
103C3×C42—91no—
104#216—31no—
105#215—31no—
106#214—31no—
107#213—31no—
108C4×Dic3—31no—
109#208—31no—
110#208—91no—
111#208—91no—
112#208—181no—
113#208—181no—
114C2×C3⋊C8—31no—
115C2×C3⋊C8—91no—
116C2×C3⋊C8—91no—
117C2×C3⋊C8—181no—
118C2×C3⋊C8—181no—
119ΓL(1,𝔽25)—181no—
120ΓL(1,𝔽25)—361no—
121D3×C8—181no—
122C22×C32—31no—
123C2×C3⋊D3—41no—
124C2×C3×D3—61no—
125C32×C4—31no—
126C32×C4—41no—
127C32×C4—81no—
128C32×C4C2×S411yescharacteristic
129C3⋊Dic3—41no—
130C3×Dic3—61no—
131C3×Dic3—121no—
132#135—91no—
133#131—91no—
134#131—271no—
135#108—271no—
136#104—91no—
137#104—271no—
138#97—271no—
139H(3,𝔽3)—41noSylow
140C3×Q8—31no—
141C3×Q8—91no—
142C3×Q8D3×Dic311yescharacteristic
143C3×D4—31no—
144C3×D4—31no—
145C3×D4—61no—
146C3×D4—91no—
147C3×D4—181no—
148C2×C3×C4—31no—
149C2×C3×C4—31no—
150C2×C3×C4—61no—
151C2×C3×C4—181no—
152C2×C3×C4—181no—
153#67—61no—
154C2×Dic3—61no—
155D12—31no—
156C4×D3—61no—
157C4×D3—361no—
158Dic6—31no—
159SL(2,𝔽3)—31no—
160SL(2,𝔽3)—61no—
161C3×C8—91no—
162C3×C8—91no—
163C3×C8—181no—
164C3×C8—361no—
165C3⋊C8—31no—
166C3⋊C8—31no—
167C3⋊C8—31no—
168C3⋊C8—61no—
169C3⋊C8—91no—
170C3⋊C8—91no—
171C3⋊C8—91no—
172C3⋊C8—121no—
173C3⋊C8—181no—
174C3⋊C8—241no—
175C2×C32—41no—
176C2×C32—61no—
177C2×C32—81no—
178C2×C32C4×S411yescharacteristic
179C3⋊D3—81no—
180C3×D3—121no—
181#41—91no—
182#41((C3⋊D3)⋊C3)⋊C211yescharacteristic
183Q16—91no—
184SD16—91no—
185SD16—91no—
186D8—91no—
187M16—91no—
188M16—271no—
189M16—541no—
190C2×C8—91no—
191C2×C8—271no—
192C2×C8—541no—
193C42—91no—
194C22×C3—31no—
195C22×C3—31no—
196C22×C3—61no—
197C22×C3—181no—
198C2×D3—61no—
199C2×D3—361no—
200C3×C4—31no—
201C3×C4—31no—
202C3×C4—31no—
203C3×C4—61no—
204C3×C4—121no—
205C3×C4—181no—
206C3×C4—241no—
207C3×C4—361no—
208C3×C4D3×S411yescharacteristic
209Dic3—61no—
210Dic3—121no—
211Dic3—361no—
212C32—41no—
213C32—81no—
214C32#540511yescharacteristic, complemented
215Q8—91no—
216Q8((C3⋊D3)⋊C3)⋊C411yescharacteristic
217D4—31no—
218D4—91no—
219D4—181no—
220C2×C4—31no—
221C2×C4—181no—
222C2×C4—181no—
223C8—91no—
224C8—91no—
225C8—91no—
226C8—181no—
227C8—271no—
228C2×C3—31no—
229C2×C3—61no—
230C2×C3—61no—
231C2×C3—121no—
232C2×C3—121no—
233C2×C3—241no—
234C2×C3—361no—
235C2×C3Dic3×S411yescharacteristic
236D3—121no—
237D3—721no—
238C22—31no—
239C22—181no—
240C4—31no—
241C4—181no—
242C4—361no—
243C4((C3⋊S4)⋊C3)⋊C211yescharacteristic
244C3—31no—
245C3—121no—
246C3—241no—
247C3#1059322111yescharacteristic
248C2—61no—
249C2—361no—
250C2((C3⋊S4)⋊C3)⋊C411yescharacteristic
subgroupG/H
center, hypercenterC4((C3⋊S4)⋊C3)⋊C2
commutator, hypocenterQ8⋊H(3,𝔽3)C2×C4
FrattiniC3×C4D3×S4
FittingC32×#41C2×D3
socleC2×C3Dic3×S4
prime radical#64266⋊C3C2
cosocle#41⋊H(3,𝔽3)C22
LCSG ▷ Q8⋊H(3,𝔽3)
UCSC1 ◁ C4

Complements

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

NH ≅ G/Ntype
2. #11720767C2semidirect
3. #11720740C2semidirect
7. (C3⋊GL(2,𝔽3))⋊C3C4semidirect
8. (C3⋊#226)⋊C3C4semidirect
11. #64157D3semidirect
32. C3⋊GL(2,𝔽3)Dic3semidirect
33. C3⋊#226Dic3semidirect
214. C32#5405semidirect

Maximal subgroups

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

subgroupindex
#64266⋊C32
#117207672
#117207402
#105932223
#105916763
#884304