Finite Group Databasebeta

← C2×#11724152 #13180641

Subgroups — C2×#11724152 #13180641

Order 1728.

522 conjugacy classes of subgroups; only the 132 normal subgroups are shown.

normal subgroups only

This group has 522 conjugacy classes of subgroups — too many to draw. Only its 132 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×#88946C211yesmaximal, characteristic
2C2×#88945C211yesmaximal, characteristic
3C2×#88945C211yesmaximal, characteristic
4C2×C3×#3777C211yesmaximal, characteristic
5C2×Dic3⋊#157C211yesmaximal, characteristic
6C2×Dic3⋊#157C211yesmaximal, characteristic
7C2×Dic3×#157C211yesmaximal, characteristic
8#11724152C218yesmaximal, direct product factor
9C2×#64243—31nomaximal
10C22×C3⋊#157C2211yescharacteristic
11C22×C3⋊#157C2211yescharacteristic
12C22×C3⋊#157C2211yescharacteristic
13C22×C3×#157C2211yescharacteristic
14C22×C3×#157C2211yescharacteristic
15C22×C3×#157C2211yescharacteristic
16C2×Dic3×C3⋊D3C2211yescharacteristic
17#88946C2214yes—
18#88945C2214yes—
19#88945C2214yes—
20C3×#3777C2214yes—
21Dic3⋊#157C2214yes—
22Dic3⋊#157C2214yes—
23Dic3×#157C2214yes—
24C2×C4⋊#157—31no—
25C2×C4⋊#157—31no—
26C2×C4×#157—31no—
27C2×#3777—31no—
28C2×#3777—31no—
29C2×#3777—31no—
30C2×#3777D311yescharacteristic
31#64243—38no—
32C22×C3×C3⋊D3C2311yescharacteristic
33C2×C3⋊#157—22no—
34C2×C3⋊#157—22no—
35C2×C3⋊#157—22no—
36C2×C3⋊#157C2312yes—
37C2×C3⋊#157C2312yes—
38C2×C3⋊#157C2312yes—
39C2×C3×#157—22no—
40C2×C3×#157—22no—
41C2×C3×#157—22no—
42C2×C3×#157C2312yes—
43C2×C3×#157C2312yes—
44C2×C3×#157C2312yes—
45C2×C3⋊(C3⋊Dic3)Q811yescharacteristic
46C2×C32×Dic3Q811yescharacteristic
47Dic3×C3⋊D3Q812yes—
48Dic3×C3⋊D3C2312yes—
49C2×#889—91nomaximal
50C22×#157—31no—
51C22×#157—31no—
52C22×#157—31no—
53C22×#157C2×D311yescharacteristic
54C22×#157C2×D311yescharacteristic
55C22×#157C2×D311yescharacteristic
56C2×C4×C3⋊D3—31no—
57C2×D3×Dic3—121no—
58C4⋊#157—34no—
59C4⋊#157—34no—
60C4×#157—34no—
61#3777—34no—
62#3777—34no—
63#3777—34no—
64#3777C2×D314yes—
65C22×C33C2×Q811yescharacteristic
66C2×C3×C3⋊D3C2×Q811yescharacteristic
67C2×C3×C3⋊D3#4112yes—
68C2×C3×C3⋊D3#4112yes—
69C2×C3×C3⋊D3C2411yescharacteristic
70C3⋊#157—22no—
71C3⋊#157—22no—
72C3⋊#157—22no—
73C3×#157—22no—
74C3×#157—22no—
75C3×#157—22no—
76C3⋊(C3⋊Dic3)C2×Q812yes—
77C32×Dic3C2×Q812yes—
78C2×C3×C4⋊C4—91no—
79C2×C4⋊Dic3—91no—
80C2×Dic3⋊C4—91no—
81C2×Dic3⋊C4—91no—
82C2×Dic3⋊C4—91no—
83C2×Dic3⋊C4—91no—
84C2×C4×Dic3—91no—
85#889—98no—
86C22×C3⋊D3C22×D311yescharacteristic
87C22×C3×D3—121no—
88C2×#157—22no—
89C2×#157—22no—
90C2×#157—22no—
91C2×#157—32no—
92C2×#157—32no—
93C2×#157—32no—
94C2×#157—62no—
95C2×#157—62no—
96C2×#157—62no—
97C2×#157C22×D312yes—
98C2×#157C22×D312yes—
99C2×#157C22×D312yes—
100C2×C32×C4—31no—
101C2×C3⋊Dic3—31no—
102C2×C3⋊Dic3—121no—
103C2×C3⋊Dic3—241no—
104C4×C3⋊D3—32no—
105C4×C3⋊D3—32no—
106C2×C3×Dic3—41no—
107D3×Dic3—122no—
108D3×Dic3—122no—
109C2×#125—271noSylow
110C2×C33#12512yescomplemented
111C2×C33C22×Q811yescharacteristic
112C3×C3⋊D3#12512yescomplemented
113C3×C3⋊D3C2×#4111yescharacteristic
114C3×C3⋊D3C2×#4111yescharacteristic
115C22×C3×C4—91no—
116C22×C3×C4—91no—
117C22×C3×C4—91no—
118C22×Dic3—91no—
119C22×Dic3—91no—
120C22×Dic3—91no—
121C22×Dic3—91no—
122C2×C4×D3—361no—
123C3×C4⋊C4—94no—
124C4⋊Dic3—94no—
125Dic3⋊C4—94no—
126Dic3⋊C4—94no—
127Dic3⋊C4—94no—
128Dic3⋊C4—94no—
129C4×Dic3—94no—
130C22×C32—41no—
131C22×C32—81no—
132C22×C32D3×Q811yescharacteristic
133C2×C3⋊D3#23512yes—
134C2×C3⋊D3#23712yes—
135C2×C3⋊D3D3×Q811yescharacteristic
136C2×C3⋊D3C23×D311yescharacteristic
137C2×C3×D3—121no—
138C2×C3×D3—122no—
139C2×C3×D3—122no—
140C2×C3×D3—121no—
141#157—22no—
142#157—22no—
143#157—22no—
144#157—62no—
145#157—62no—
146#157—62no—
147C32×C4—32no—
148C3⋊Dic3—32no—
149C3⋊Dic3—122no—
150C3⋊Dic3—242no—
151C3×Dic3—42no—
152#125—278no—
153C2×C4⋊C4—91no—
154C2×C4⋊C4—271no—
155C2×C4⋊C4—271no—
156C2×C4⋊C4—271no—
157C2×C4⋊C4—271no—
158C2×C4⋊C4—271no—
159C2×C42—271no—
160C33C2×#12511yescharacteristic, complemented, Sylow
161C23×C3—91no—
162C22×D3—121no—
163C2×C3×C4—92no—
164C2×C3×C4—92no—
165C2×C3×C4—92no—
166C2×C3×C4—121no—
167C2×C3×C4—182no—
168C2×C3×C4—182no—
169C2×C3×C4—182no—
170C2×Dic3—91no—
171C2×Dic3—92no—
172C2×Dic3—92no—
173C2×Dic3—92no—
174C2×Dic3—92no—
175C2×Dic3—92no—
176C2×Dic3—182no—
177C2×Dic3—182no—
178C2×Dic3—182no—
179C2×Dic3—361no—
180C2×Dic3—721no—
181C2×Dic3PSU(3,𝔽2)11yescharacteristic
182C4×D3—362no—
183C4×D3—362no—
184C2×C32—41no—
185C2×C32—42no—
186C2×C32—82no—
187C2×C32—81no—
188C2×C32#88912yescomplemented
189C2×C32C2×D3×Q811yescharacteristic
190C3⋊D3#88912yescomplemented
191C3⋊D3C2×#23511yescharacteristic
192C3⋊D3C2×#23711yescharacteristic
193C3×D3—122no—
194C3×D3—121no—
195C3×D3—121no—
196C22×C4—91no—
197C22×C4—91no—
198C22×C4—91no—
199C22×C4—271no—
200C22×C4—271no—
201C22×C4—271no—
202C22×C4—271no—
203C4⋊C4—94no—
204C4⋊C4—274no—
205C4⋊C4—274no—
206C4⋊C4—274no—
207C4⋊C4—274no—
208C4⋊C4—274no—
209C42—274no—
210C22×C3—41no—
211C22×C3—81no—
212C22×C3—91no—
213C22×C3—91no—
214C22×C3—92no—
215C22×C3—92no—
216C22×C3C2×PSU(3,𝔽2)11yescharacteristic
217C2×D3—121no—
218C2×D3—121no—
219C2×D3—122no—
220C2×D3—122no—
221C3×C4—122no—
222C3×C4—182no—
223C3×C4—182no—
224C3×C4—182no—
225Dic3—92no—
226Dic3—182no—
227Dic3—182no—
228Dic3—182no—
229Dic3—362no—
230Dic3—722no—
231Dic3C2×PSU(3,𝔽2)12yes—
232C32—41no—
233C32—81no—
234C32C2×#88911yescharacteristic, complemented
235C23—91no—
236C2×C4—31no—
237C2×C4—92no—
238C2×C4—92no—
239C2×C4—92no—
240C2×C4—182no—
241C2×C4—182no—
242C2×C4—182no—
243C2×C4—272no—
244C2×C4—272no—
245C2×C4—271no—
246C2×C4—272no—
247C2×C4—272no—
248C2×C4—272no—
249C2×C4—542no—
250C2×C4—542no—
251C2×C4—542no—
252C2×C3—42no—
253C2×C3—41no—
254C2×C3—82no—
255C2×C3—81no—
256C2×C3—91no—
257C2×C3—91no—
258C2×C3—92no—
259C2×C3#6424312yescomplemented
260C2×C3C22×PSU(3,𝔽2)11yescharacteristic
261D3—122no—
262D3—121no—
263D3—121no—
264C22—92no—
265C22—92no—
266C22—91no—
267C22—91no—
268C22D3×PSU(3,𝔽2)11yescharacteristic
269C4—32no—
270C4—182no—
271C4—182no—
272C4—182no—
273C4—272no—
274C4—542no—
275C4—542no—
276C4—542no—
277C3—41no—
278C3—81no—
279C3C2×#6424311yescharacteristic, complemented
280C2—91no—
281C2—91no—
282C2—92no—
283C2#1172415212yesdirect product factor
284C2C2×D3×PSU(3,𝔽2)11yescharacteristic
subgroupG/H
center, hypercenterC22D3×PSU(3,𝔽2)
commutator, cosocleC2×C3×C3⋊D3C24
FrattiniC2C2×D3×PSU(3,𝔽2)
Fitting, socleC22×C33C2×Q8
prime radicalC22×C3×C3⋊D3C23
hypocenterC33C2×#125
LCSG ▷ C2×C3×C3⋊D3 ▷ C33
UCSC1 ◁ C22

Complements

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

NH ≅ G/Ntype
8. #11724152C2direct (N × H)
110. C2×C33#125semidirect
112. C3×C3⋊D3#125semidirect
160. C33C2×#125semidirect
188. C2×C32#889semidirect
190. C3⋊D3#889semidirect
234. C32C2×#889semidirect
259. C2×C3#64243semidirect
279. C3C2×#64243semidirect
283. C2#11724152direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#889462
C2×#889452
C2×C3×#37772
C2×Dic3⋊#1572
C2×Dic3×#1572
#117241522
C2×#642433
C2×#8899