Finite Group Databasebeta

← C2×#11724341 #13181141

Subgroups — C2×#11724341 #13181141

Order 1728.

904 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 904 conjugacy classes of subgroups — too many to draw. Only its 143 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×#89040C212yesmaximal
2C2×C3×#3787C211yesmaximal, characteristic
3C2×#88943C214yesmaximal, complemented
4#11724341C218yesmaximal, direct product factor
5C2×#64228—31nomaximal
6C2×C3×C4×C3⋊D3C2211yescharacteristic
7C2×C3×#634—22no—
8C2×#6296C2214yes—
9C2×#6294—22no—
10#89040—24no—
11#89040C2218yes—
12C2×C3×#622C2212yes—
13C3×#3787C2214yes—
14#88943C22116yescomplemented
15C2×#3797—32no—
16C2×D3×Dic6—62no—
17C2×#3787D311yescharacteristic
18C2×#3776—34no—
19#64228—38no—
20C22×C3×C3⋊D3D411yescharacteristic
21C2×C33×C4D411yescharacteristic
22C2×C3×C3⋊Dic3—24no—
23C2×C3×C3⋊Dic3C2311yescharacteristic
24C3×C4×C3⋊D3D412yes—
25C3×C4×C3⋊D3C2312yes—
26C3×#634—28no—
27#6296—24no—
28#6296C2318yes—
29#6294—28no—
30C3×#622C2314yes—
31C22×Dic12—91nomaximal
32C2×C4×C3⋊D3C2×D311yescharacteristic
33C2×#634—22no—
34C2×C3×C4×D3—62no—
35C2×C3×Dic6—62no—
36C2×C3×Dic6—62no—
37C2×C3×Dic6—122no—
38C2×#627—34no—
39C2×#627—64no—
40C2×#624—62no—
41C2×D3×Dic3—64no—
42#3797—38no—
43#3797—64no—
44D3×Dic6—68no—
45D3×Dic6—68no—
46C2×#622C2×D312yes—
47#3787C2×D314yes—
48#3776—316no—
49C22×C33C2×D411yescharacteristic
50C2×C3×C3⋊D3Q1614yescomplemented
51C2×C3×C3⋊D3C2×D412yes—
52C33×C4C2×D412yes—
53C3×C3⋊Dic3—28no—
54C3×C3⋊Dic3C2×D411yescharacteristic
55C3×C3⋊Dic3C2411yescharacteristic
56C2×D3×Q8—182no—
57C22×Dic6—92no—
58C22×C3×C8—91no—
59C2×Dic12—98no—
60C2×Dic12—94no—
61C22×C3⋊D3D1211yescharacteristic
62C22×C3×D3—62no—
63C2×C32×C4—22no—
64C2×C32×C4—42no—
65C2×C32×C4D1211yescharacteristic
66C2×C3⋊Dic3—24no—
67C2×C3⋊Dic3C22×D311yescharacteristic
68C4×C3⋊D3D1212yes—
69C4×C3⋊D3C22×D312yes—
70#634—28no—
71C2×C3×Dic3—62no—
72C2×C3×Dic3—64no—
73C2×C3×Dic3—64no—
74C2×C3×Dic3—124no—
75C3×C4×D3—64no—
76C3×C4×D3—64no—
77C3×Dic6—68no—
78C3×Dic6—68no—
79C3×Dic6—128no—
80#627—38no—
81#627—68no—
82#627—68no—
83#627—64no—
84#624—68no—
85D3×Dic3—616no—
86#622C22×D314yes—
87C22×Q16—271noSylow
88C2×C33C2×Q1612yescomplemented
89C2×C33C22×D411yescharacteristic
90C3×C3⋊D3C2×Q1614yescomplemented
91C2×C3×Q8—182no—
92C22×C3×C4—91no—
93C22×Dic3—182no—
94D3×Q8—188no—
95D3×Q8—188no—
96C2×C4×D3—62no—
97C2×C4×D3—184no—
98C2×Dic6—62no—
99C2×Dic6—62no—
100C2×Dic6—98no—
101C2×Dic6—94no—
102C2×Dic6—122no—
103C2×Dic6—184no—
104C2×Dic6—184no—
105C2×C3×C8—94no—
106C2×C3×C8—92no—
107Dic12—916no—
108C22×C32—22no—
109C22×C32—42no—
110C22×C32C2×D1211yescharacteristic
111C2×C3⋊D3Dic1214yescomplemented
112C2×C3⋊D3C2×D1212yes—
113C2×C3×D3—68no—
114C2×C3×D3—64no—
115C32×C4—24no—
116C32×C4—44no—
117C32×C4C2×D1212yes—
118C3⋊Dic3—28no—
119C3⋊Dic3C2×D1211yescharacteristic
120C3⋊Dic3C23×D311yescharacteristic
121C3×Dic3—62no—
122C3×Dic3—62no—
123C3×Dic3—68no—
124C3×Dic3—68no—
125C3×Dic3—128no—
126C22×Q8—272no—
127C2×Q16—278no—
128C2×Q16—274no—
129C22×C8—91no—
130C33C22×Q1611yescharacteristic, complemented, Sylow
131C23×C3—91no—
132C22×D3—62no—
133C3×Q8—188no—
134C2×C3×C4—22no—
135C2×C3×C4—42no—
136C2×C3×C4—91no—
137C2×C3×C4—92no—
138C2×C3×C4—92no—
139C2×C3×C4—184no—
140C2×C3×C4GO+(4,𝔽2)11yescharacteristic
141C2×Dic3—62no—
142C2×Dic3—64no—
143C2×Dic3—64no—
144C2×Dic3—124no—
145C2×Dic3—188no—
146C4×D3—64no—
147C4×D3—64no—
148C4×D3—1816no—
149Dic6—68no—
150Dic6—68no—
151Dic6—98no—
152Dic6—128no—
153Dic6—188no—
154Dic6—184no—
155Dic6—188no—
156C3×C8—94no—
157C2×C32—22no—
158C2×C32—24no—
159C2×C32—44no—
160C2×C32—42no—
161C2×C32C2×Dic1212yescomplemented
162C2×C32C22×D1211yescharacteristic
163C3⋊D3C2×Dic1214yescomplemented
164C3×D3—68no—
165C2×Q8—182no—
166C2×Q8—274no—
167C2×Q8—278no—
168C2×Q8—544no—
169C22×C4—91no—
170C22×C4—542no—
171Q16—2716no—
172C2×C8—94no—
173C2×C8—92no—
174C22×C3—22no—
175C22×C3—42no—
176C22×C3—92no—
177C22×C3—94no—
178C22×C3C2×GO+(4,𝔽2)11yescharacteristic
179C2×D3—64no—
180C2×D3—68no—
181C3×C4—24no—
182C3×C4—44no—
183C3×C4—91no—
184C3×C4—91no—
185C3×C4—188no—
186C3×C4C2×GO+(4,𝔽2)12yes—
187Dic3—62no—
188Dic3—62no—
189Dic3—68no—
190Dic3—68no—
191Dic3—128no—
192C32—22no—
193C32—42no—
194C32C22×Dic1211yescharacteristic, complemented
195C23—91no—
196Q8—188no—
197Q8—278no—
198Q8—544no—
199C2×C4—91no—
200C2×C4—92no—
201C2×C4—92no—
202C2×C4—184no—
203C2×C4—548no—
204C2×C4#157⋊D311yescharacteristic
205C8—94no—
206C2×C3—24no—
207C2×C3—22no—
208C2×C3—44no—
209C2×C3—42no—
210C2×C3—94no—
211C2×C3#6422812yescomplemented
212C2×C3C22×GO+(4,𝔽2)11yescharacteristic
213D3—68no—
214C22—94no—
215C22—92no—
216C22C2×#157⋊D311yescharacteristic
217C4—91no—
218C4—91no—
219C4—188no—
220C4C2×#157⋊D312yes—
221C3—22no—
222C3—42no—
223C3C2×#6422811yescharacteristic, complemented
224C2—94no—
225C2#1172434112yesdirect product factor
226C2C22×#157⋊D311yescharacteristic
subgroupG/H
centerC22C2×#157⋊D3
commutator, cosocleC3×C3⋊Dic3C24
FrattiniC2C22×#157⋊D3
FittingC2×C33×C4D4
socleC22×C33C2×D4
prime radicalC22×C3×C3⋊D3D4
hypercenterC2×C4#157⋊D3
hypocenterC33C22×Q16
LCSG ▷ C3×C3⋊Dic3 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
3. C2×#88943C2semidirect
4. #11724341C2direct (N × H)
14. #88943C22semidirect
50. C2×C3×C3⋊D3Q16semidirect
88. C2×C33C2×Q16semidirect
90. C3×C3⋊D3C2×Q16semidirect
111. C2×C3⋊D3Dic12semidirect
130. C33C22×Q16semidirect
161. C2×C32C2×Dic12semidirect
163. C3⋊D3C2×Dic12semidirect
194. C32C22×Dic12semidirect
211. C2×C3#64228semidirect
223. C3C2×#64228semidirect
225. C2#11724341direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#890402
C2×C3×#37872
C2×#889432
#117243412
C2×#642283
C22×Dic129