Finite Group Databasebeta

← #13138808

Subgroups — #13138808

Order 1728.

499 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 499 conjugacy classes of subgroups — too many to draw. Only its 79 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
1C2×#3745⋊C3C211yesmaximal, characteristic
2C2×#3746⋊C3C211yesmaximal, characteristic, complemented
3#11721113C212yesmaximal, complemented
4C2×#88431C212yesmaximal, complemented
5C8×#1084C211yesmaximal, characteristic, complemented
6#10590865—31nomaximal
7#10590864—31nomaximal
8C2×C4⋊((C3⋊D3)⋊C3)C2212yes—
9C2×#634⋊C3C2212yes—
10C2×C8×H(3,𝔽3)C2211yescharacteristic
11C4×#1084C2211yescharacteristic
12C2×#6180C2211yescharacteristic
13#3745⋊C3—22no—
14#3746⋊C3—22no—
15#88450—24no—
16#88431—24no—
17C2×#3746D311yescharacteristic, complemented
18C2×#3745D311yescharacteristic
19C2×C3×Dic12—31no—
20C2×C3×D24—31no—
21#63864—32no—
22#63860—32no—
23C2×#3717—32no—
24C2×#3716—32no—
25C3×C8×Dic3—31no—
26C3×C8×Dic3—31no—
27C3×C8×Dic3—61no—
28C22×(C3⋊D3)⋊C3—22no—
29C2×C4×H(3,𝔽3)C2311yescharacteristic
30C2×#1084D412yes—
31C2×(C3⋊Dic3)⋊C3—22no—
32C4⋊((C3⋊D3)⋊C3)—22no—
33C4⋊((C3⋊D3)⋊C3)—22no—
34#634⋊C3—22no—
35#634⋊C3—22no—
36C8×H(3,𝔽3)D412yes—
37#6180D412yes—
38#5191—91no—
39#5155—91no—
40#4696—91no—
41C2×C4⋊(C3⋊D3)C2×D312yes—
42C2×#634C2×D312yes—
43C2×C3×D12—32no—
44C2×C3×Dic6—32no—
45C2×C32×C8—21no—
46C2×C32×C8C2×D311yescharacteristic
47C2×C32×C8C2×D311yescharacteristic
48#3746—22no—
49#3745—22no—
50C3×C4×Dic3—31no—
51C3×C4×Dic3—31no—
52C3×C4×Dic3—61no—
53C2×C3×C3⋊C8—31no—
54C2×C3×C3⋊C8—31no—
55C2×C3×C3⋊C8—61no—
56C3×Dic12—62no—
57C3×D24—62no—
58#3722—64no—
59(C2×D3)⋊Dic3—64no—
60#3717—64no—
61#3716—64no—
62C22×H(3,𝔽3)C2×D411yescharacteristic
63C2×(C3⋊D3)⋊C3—42no—
64C2×(C3⋊D3)⋊C3—44no—
65C4×H(3,𝔽3)C2×D411yescharacteristic
66C4×H(3,𝔽3)C2×D411yescharacteristic
67#1084—22no—
68(C3⋊Dic3)⋊C3—42no—
69C2×C3×Q16—91no—
70C2×C3×D8—91no—
71#947—92no—
72C2×#215—92no—
73#936—92no—
74C2×#214—92no—
75C2×Dic12—31no—
76C2×Dic12—91no—
77C2×D24—31no—
78C2×D24—91no—
79C2×#204—92no—
80#875—92no—
81C3×C4×C8—91no—
82C8×Dic3—31no—
83C8×Dic3—31no—
84C8×Dic3—61no—
85C22×C3⋊D3—22no—
86C22×C3×D3—62no—
87C2×C32×C4—21no—
88C2×C32×C4C22×D311yescharacteristic
89C2×C32×C4C22×D311yescharacteristic
90C2×C3⋊Dic3—22no—
91C4⋊(C3⋊D3)—22no—
92C4⋊(C3⋊D3)—22no—
93#634—22no—
94#634—22no—
95C2×C3×Dic3—32no—
96C2×C3×Dic3—32no—
97C2×C3×Dic3—62no—
98C2×C3×Dic3—62no—
99C3×D12—62no—
100C3×D12—62no—
101C3×Dic6—62no—
102C3×Dic6—62no—
103C32×C8—22no—
104C32×C8#6712yes—
105C32×C8#6712yes—
106C3×C3⋊C8—32no—
107C3×C3⋊C8—32no—
108C3×C3⋊C8—62no—
109#495—271noSylow
110C2×H(3,𝔽3)#12711yescharacteristic
111C2×H(3,𝔽3)#13512yes—
112(C3⋊D3)⋊C3—82no—
113C2×C3×Q8—92no—
114C2×C3×D4—92no—
115C2×D12—32no—
116C2×D12—92no—
117C2×Dic6—32no—
118C2×Dic6—92no—
119C3×Q16—182no—
120C3×D8—182no—
121C2×C3×C8—31no—
122C2×C3×C8—31no—
123C2×C3×C8—61no—
124C2×C3×C8—91no—
125C2×C3×C8D3211yescharacteristic
126C3×C42—91no—
127#217—184no—
128#215—184no—
129#214—184no—
130(C2×D3)⋊C4—184no—
131(C2×D3)⋊C4—184no—
132C4×Dic3—31no—
133C4×Dic3—31no—
134C4×Dic3—61no—
135C2×C3⋊C8—31no—
136C2×C3⋊C8—31no—
137C2×C3⋊C8—61no—
138Dic12—62no—
139Dic12—182no—
140D24—62no—
141D24—182no—
142#204—184no—
143C22×C32—21no—
144C22×C32D3×D411yescharacteristic
145C22×C32D3×D411yescharacteristic
146C2×C3⋊D3—44no—
147C2×C3⋊D3—42no—
148C2×C3×D3—124no—
149C2×C3×D3—122no—
150C32×C4—21no—
151C32×C4—21no—
152C32×C4D3×D411yescharacteristic
153C32×C4D3×D411yescharacteristic
154C32×C4C2×#6711yescharacteristic
155C32×C4C2×#6711yescharacteristic
156C3⋊Dic3—42no—
157C3×Dic3—64no—
158C3×Dic3—62no—
159C3×Dic3—62no—
160C3×Dic3—122no—
161C2×Q16—91no—
162C2×SD16—272no—
163C2×D8—91no—
164#124—272no—
165C4×C8—91no—
166H(3,𝔽3)#49511yescharacteristic, complemented, Sylow
167C23×C3—182no—
168C22×D3—62no—
169C22×D3—182no—
170C3×Q8—182no—
171C3×Q8—182no—
172C3×D4—182no—
173C3×D4—182no—
174C2×C3×C4—31no—
175C2×C3×C4—31no—
176C2×C3×C4—61no—
177C2×C3×C4—92no—
178C2×C3×C4—182no—
179C2×C3×C4C2×D3211yescharacteristic
180C2×Dic3—32no—
181C2×Dic3—32no—
182C2×Dic3—62no—
183C2×Dic3—62no—
184C2×Dic3—182no—
185D12—62no—
186D12—62no—
187D12—182no—
188D12—182no—
189Dic6—62no—
190Dic6—62no—
191Dic6—182no—
192Dic6—182no—
193C3×C8—32no—
194C3×C8—32no—
195C3×C8—62no—
196C3×C8—92no—
197C3×C8#62512yes—
198C3⋊C8—32no—
199C3⋊C8—32no—
200C3⋊C8—62no—
201C2×C32—22no—
202C2×C32—21no—
203C2×C32#91212yes—
204C2×C32#91912yes—
205C2×C32#94011yescharacteristic
206C2×C32#94011yescharacteristic
207C3⋊D3—82no—
208C3×D3—242no—
209C2×Q8—92no—
210C2×D4—92no—
211Q16—182no—
212SD16—544no—
213D8—182no—
214C2×C8—91no—
215C2×C8((C3⋊D3)⋊C3)⋊C211yescharacteristic
216C22⋊C4—544no—
217C42—91no—
218C22×C3—31no—
219C22×C3—31no—
220C22×C3—61no—
221C22×C3—362no—
222C22×C3—364no—
223C22×C3#380211yescharacteristic
224C2×D3—124no—
225C2×D3—122no—
226C2×D3—362no—
227C2×D3—364no—
228C3×C4—31no—
229C3×C4—31no—
230C3×C4—31no—
231C3×C4—31no—
232C3×C4—61no—
233C3×C4—61no—
234C3×C4—182no—
235C3×C4—362no—
236C3×C4#380211yescharacteristic
237C3×C4C2×#62511yescharacteristic
238Dic3—62no—
239Dic3—62no—
240Dic3—64no—
241Dic3—122no—
242Dic3—362no—
243C32—21no—
244C32#515511yescharacteristic, complemented
245C32#519111yescharacteristic, complemented
246C23—182no—
247Q8—182no—
248Q8—182no—
249D4—182no—
250D4—182no—
251C2×C4—92no—
252C2×C4—182no—
253C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
254C8—92no—
255C8#619812yes—
256C2×C3—32no—
257C2×C3—32no—
258C2×C3—31no—
259C2×C3—31no—
260C2×C3—62no—
261C2×C3—61no—
262C2×C3—722no—
263C2×C3#6381012yes—
264C2×C3#6391411yescharacteristic
265D3—242no—
266D3—722no—
267C22—364no—
268C22—362no—
269C22#8865511yescharacteristic
270C4—182no—
271C4—362no—
272C4#8865511yescharacteristic
273C4C2×#619811yescharacteristic
274C3—31no—
275C3—31no—
276C3—61no—
277C3#1059086911yescharacteristic
278C2—722no—
279C2#1172084412yes—
280C2#1172114411yescharacteristic
subgroupG/H
centerC22#88655
commutator, cosocleC2×C4×H(3,𝔽3)C23
FrattiniC2×C3×C4C2×D32
FittingC2×C8×H(3,𝔽3)C22
socleC22×C3#3802
prime radicalC2×#3745⋊C3C2
hypercenterC2×C8((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#495
LCSG ▷ C2×C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×C8

Complements

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

NH ≅ G/Ntype
2. C2×#3746⋊C3C2semidirect
3. #11721113C2semidirect
4. C2×#88431C2semidirect
5. C8×#1084C2semidirect
17. C2×#3746D3semidirect
166. H(3,𝔽3)#495semidirect
244. C32#5155semidirect
245. C32#5191semidirect

Maximal subgroups

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

subgroupindex
C2×#3745⋊C32
C2×#3746⋊C32
#117211132
C2×#884312
C8×#10842
#105908653
#105908643