Finite Group Databasebeta

← D3×#64243 #13180612

Subgroups — D3×#64243 #13180612

Order 1728.

522 conjugacy classes of subgroups; only the 122 normal subgroups are shown.

normal subgroups only

This group has 522 conjugacy classes of subgroups — too many to draw. Only its 122 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#11724362C212yesmaximal, complemented
2#11724361C211yesmaximal, characteristic, complemented
3C3×#64243C211yesmaximal, characteristic, complemented
4D3×C4⋊#157C212yesmaximal
5C4×D3×#157C211yesmaximal, characteristic
6#11724152C214yesmaximal, complemented
7D3×#3777C214yesmaximal
8C2×#64243—31nomaximal
9C2×D3×#157C2212yes—
10C2×D3×#157C2214yes—
11C4×D3×C3⋊D3C2211yescharacteristic
12C4⋊(C3⋊#157)C2212yes—
13C4×C3⋊#157C2211yescharacteristic
14C3×C4⋊#157C2212yes—
15C3×C4×#157C2211yescharacteristic
16#88946C2214yes—
17#88945C2214yes—
18#88945C2214yes—
19C3×#3777C2214yes—
20Dic3⋊#157C2214yes—
21Dic3×#157C2212yes—
22C2×C4⋊#157—32no—
23C2×C4×#157—31no—
24C2×#3777—34no—
25#64243—32no—
26#64243—31no—
27#64243—34no—
28#64243D311yescharacteristic, direct product factor
29C2×D3×C3⋊D3C2311yescharacteristic
30C2×C3⋊#157C2312yes—
31C2×C3⋊#157C2314yes—
32C2×C3×#157C2312yes—
33C2×C3×#157C2314yes—
34D3×#157—24no—
35D3×#157—28no—
36C4×C3⋊(C3⋊D3)Q811yescharacteristic
37C3×C4×C3⋊D3C2311yescharacteristic
38C32×C4×D3Q811yescharacteristic
39#6289Q811yescharacteristic
40Dic3×C3⋊D3C2311yescharacteristic
41D3×C3⋊Dic3Q811yescharacteristic
42D3×#125—91nomaximal
43C22×#157—32no—
44C22×#157—34no—
45C2×C4×C3⋊D3—31no—
46C4×D32—121no—
47C4⋊#157—34no—
48C4⋊#157—32no—
49C4⋊#157C2×D312yes—
50C4×#157—32no—
51C4×#157—31no—
52C4×#157C2×D311yescharacteristic
53#3777—34no—
54#3777—34no—
55#3777—34no—
56#3777C2×D314yes—
57C2×C3⋊(C3⋊D3)C2×Q811yescharacteristic
58C2×C3×C3⋊D3C2411yescharacteristic
59C2×C32×D3C2×Q811yescharacteristic
60D3×C3⋊D3#4114yes—
61C3⋊#157—22no—
62C3⋊#157—24no—
63C3×#157—22no—
64C3×#157—24no—
65C33×C4C2×Q811yescharacteristic
66C3⋊(C3⋊Dic3)C2×Q811yescharacteristic
67C3×C3⋊Dic3C2×Q811yescharacteristic
68C32×Dic3C2×Q811yescharacteristic
69C3×#125—91no—
70D3×C4⋊C4—94no—
71D3×C4⋊C4—92no—
72#890—92no—
73#889—94no—
74C42×D3—91no—
75#870—91no—
76C22×C3⋊D3—31no—
77C2×D32—121no—
78C2×#157—32no—
79C2×#157—34no—
80C2×#157—64no—
81C2×#157—68no—
82C2×#157C22×D312yes—
83C2×#157C22×D314yes—
84C2×C32×C4—31no—
85C2×C3⋊Dic3—31no—
86C4×C3⋊D3—31no—
87C4×C3⋊D3—31no—
88C4×C3⋊D3—31no—
89C4×C3⋊D3—121no—
90C4×C3⋊D3—241no—
91C4×C3⋊D3C22×D311yescharacteristic
92C3×C4×D3—41no—
93C3×C4×D3—121no—
94#624—121no—
95D3×Dic3—121no—
96D3×Dic3—121no—
97C2×#125—271noSylow
98C2×C33C22×Q811yescharacteristic
99C3⋊(C3⋊D3)#12512yescomplemented
100C3×C3⋊D3C2×#4112yes—
101C32×D3#12512yescomplemented
102C2×C4×D3—91no—
103C2×C4×D3—92no—
104C2×C4×D3—94no—
105C2×C4×D3—361no—
106C3×C4⋊C4—94no—
107C3×C4⋊C4—92no—
108C3×C42—91no—
109C4⋊Dic3—92no—
110C4⋊Dic3—94no—
111Dic3⋊C4—94no—
112Dic3⋊C4—94no—
113Dic3⋊C4—94no—
114C4×Dic3—92no—
115C4×Dic3—91no—
116C22×C32—31no—
117C2×C3⋊D3—31no—
118C2×C3⋊D3—34no—
119C2×C3⋊D3—121no—
120C2×C3⋊D3—241no—
121C2×C3⋊D3C23×D311yescharacteristic
122C2×C3×D3—41no—
123C2×C3×D3—121no—
124D32—124no—
125#157—22no—
126#157—24no—
127#157—62no—
128#157—64no—
129C32×C4—31no—
130C32×C4—41no—
131C32×C4—81no—
132C32×C4D3×Q811yescharacteristic
133C3⋊Dic3—31no—
134C3⋊Dic3—121no—
135C3⋊Dic3—241no—
136C3⋊Dic3D3×Q811yescharacteristic
137C3×Dic3—41no—
138C3×Dic3—121no—
139#125—91no—
140#125—271no—
141#125—274no—
142#125—272no—
143C2×C4⋊C4—272no—
144C2×C4⋊C4—274no—
145C2×C42—271no—
146C33C2×#12511yescharacteristic, complemented, Sylow
147C22×D3—91no—
148C22×D3—361no—
149C2×C3×C4—91no—
150C2×C3×C4—92no—
151C2×C3×C4—94no—
152C2×C3×C4—121no—
153C2×Dic3—91no—
154C2×Dic3—92no—
155C2×Dic3—94no—
156C2×Dic3—361no—
157C4×D3—91no—
158C4×D3—91no—
159C4×D3—91no—
160C4×D3—121no—
161C4×D3—184no—
162C4×D3—188no—
163C4×D3—361no—
164C4×D3—361no—
165C4×D3—361no—
166C4×D3—721no—
167C4×D3PSU(3,𝔽2)11yescharacteristic
168C2×C32—32no—
169C2×C32—41no—
170C2×C32—81no—
171C2×C32C2×D3×Q811yescharacteristic
172C3⋊D3—32no—
173C3⋊D3—122no—
174C3⋊D3—242no—
175C3⋊D3D3×#4112yes—
176C3×D3—42no—
177C3×D3—122no—
178C22×C4—271no—
179C22×C4—272no—
180C22×C4—274no—
181C4⋊C4—94no—
182C4⋊C4—92no—
183C4⋊C4—274no—
184C4⋊C4—274no—
185C4⋊C4—274no—
186C4⋊C4—274no—
187C4⋊C4—272no—
188C42—91no—
189C42—271no—
190C42—272no—
191C22×C3—91no—
192C22×C3—121no—
193C2×D3—91no—
194C2×D3—94no—
195C2×D3—121no—
196C2×D3—364no—
197C2×D3—361no—
198C2×D3—721no—
199C2×D3C2×PSU(3,𝔽2)11yescharacteristic
200C3×C4—41no—
201C3×C4—81no—
202C3×C4—91no—
203C3×C4—121no—
204C3×C4—182no—
205C3×C4—184no—
206C3×C4C2×PSU(3,𝔽2)11yescharacteristic
207Dic3—91no—
208Dic3—121no—
209Dic3—182no—
210Dic3—184no—
211Dic3—361no—
212Dic3—721no—
213Dic3C2×PSU(3,𝔽2)11yescharacteristic
214C32—41no—
215C32—81no—
216C32D3×#12511yescharacteristic, complemented
217C23—271no—
218C2×C4—31no—
219C2×C4—91no—
220C2×C4—92no—
221C2×C4—94no—
222C2×C4—271no—
223C2×C4—271no—
224C2×C4—271no—
225C2×C4—271no—
226C2×C4—272no—
227C2×C4—274no—
228C2×C4—544no—
229C2×C4—548no—
230C2×C3—41no—
231C2×C3—81no—
232C2×C3—92no—
233C2×C3—122no—
234C2×C3C22×PSU(3,𝔽2)11yescharacteristic
235D3—92no—
236D3—122no—
237D3—362no—
238D3—722no—
239D3#6424312yesdirect product factor
240C22—31no—
241C22—91no—
242C22—274no—
243C22—271no—
244C4—31no—
245C4—91no—
246C4—182no—
247C4—184no—
248C4—271no—
249C4—542no—
250C4—544no—
251C4D3×PSU(3,𝔽2)11yescharacteristic
252C3—41no—
253C3—81no—
254C3C2×#6424311yescharacteristic, complemented
255C2—32no—
256C2—92no—
257C2—272no—
258C2C2×D3×PSU(3,𝔽2)11yescharacteristic
subgroupG/H
center, FrattiniC2C2×D3×PSU(3,𝔽2)
commutator, cosocleC2×C3×C3⋊D3C24
FittingC33×C4C2×Q8
socleC2×C33C22×Q8
prime radicalC2×D3×C3⋊D3C23
hypercenterC4D3×PSU(3,𝔽2)
hypocenterC33C2×#125
LCSG ▷ C2×C3×C3⋊D3 ▷ C33
UCSC1 ◁ C2 ◁ C4

Complements

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

NH ≅ G/Ntype
1. #11724362C2semidirect
2. #11724361C2semidirect
3. C3×#64243C2semidirect
6. #11724152C2semidirect
28. #64243D3direct (N × H)
99. C3⋊(C3⋊D3)#125semidirect
101. C32×D3#125semidirect
146. C33C2×#125semidirect
216. C32D3×#125semidirect
239. D3#64243direct (N × H)
254. C3C2×#64243semidirect

Maximal subgroups

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

subgroupindex
#117243622
#117243612
C3×#642432
D3×C4⋊#1572
C4×D3×#1572
#117241522
D3×#37772
C2×#642433
D3×#1259