Finite Group Databasebeta

← C3×#10589522 #13146898

Subgroups — C3×#10589522 #13146898

Order 1728.

335 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 335 conjugacy classes of subgroups — too many to draw. Only its 68 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
1C22×C3×Q8⋊C9C211yesmaximal, characteristic
2C3×#63421C212yesmaximal, complemented
3C3×#5225—31nomaximal
4#10589522C311yesmaximal, characteristic, direct product factor
5C2×C3×Q8⋊C9C412yescomplemented
6C2×C3×Q8⋊C9C2211yescharacteristic
7C3×#3676—41nomaximal
8C22×C32×Q8D311yescharacteristic
9C3×#926—31no—
10C2×C3×#208—31no—
11C22×Q8⋊C9—21no—
12C22×Q8⋊C9C2×C311yescharacteristic
13C3×#835—32no—
14C3×#835—32no—
15#63421C2×C312yescomplemented
16C23×C3×C9—41no—
17C2×C3×Dic9—42no—
18C3×Q8⋊C9C2×C411yescharacteristic
19C3×#419—91no—
20#5225—31no—
21C2×C32×Q8—32no—
22C2×C32×Q8—31no—
23C2×C32×Q8Dic312yes—
24C2×C32×Q8C2×D311yescharacteristic
25C22×C32×C4—31no—
26C3×#217—61no—
27C3×C4⋊Dic3—32no—
28C3×C4×Dic3—61no—
29C3×#208—62no—
30C2×C3×C3⋊C8—32no—
31C2×Q8⋊C9—22no—
32C2×Q8⋊C9—21no—
33C2×Q8⋊C9C3×C412yescomplemented
34C2×Q8⋊C9C22×C311yescharacteristic
35#3676—41no—
36C22×C3×C9—41no—
37C22×C3×C9—42no—
38C22×C3×C9—82no—
39C3×Dic9—82no—
40C22×C3×Q8—21no—
41C22×C3×Q8D911yescharacteristic
42C22×C3×Q8C3×D311yescharacteristic
43C2×C3×M16—91no—
44C3×#117—91no—
45#926—31no—
46C2×#208—31no—
47C3×#103—92no—
48C3×#103—92no—
49#835—32no—
50#835—32no—
51C23×C32S411yescharacteristic
52C32×Q8—32no—
53C32×Q8—31no—
54C32×Q8—61no—
55C32×Q8C2×Dic311yescharacteristic
56C2×C32×C4—32no—
57C2×C32×C4—32no—
58C2×C32×C4—31no—
59C2×C32×C4—31no—
60C2×C3×Dic3—62no—
61C23×C9—41no—
62C23×C9—81no—
63C3×C3⋊C8—62no—
64C2×Dic9—42no—
65Q8⋊C9—21no—
66Q8⋊C9C2×C3×C411yescharacteristic
67#419—91noSylow
68C2×C3×C9—42no—
69C2×C3×C9—41no—
70C2×C3×C9—82no—
71C2×C3×Q8—22no—
72C2×C3×Q8—21no—
73C2×C3×Q8—31no—
74C2×C3×Q8—31no—
75C2×C3×Q8—32no—
76C2×C3×Q8—32no—
77C2×C3×Q8—61no—
78C2×C3×Q8—62no—
79C2×C3×Q8Dic912yescomplemented
80C2×C3×Q8C2×D911yescharacteristic
81C2×C3×Q8C3×Dic312yes—
82C2×C3×Q8C2×C3×D311yescharacteristic
83C22×C3×C4—31no—
84C22×C3×C4—31no—
85C22×C3×C4—61no—
86C3×M16—182no—
87C2×C3×C8—92no—
88C3×C4⋊C4—92no—
89C3×C22⋊C4—181no—
90C3×C42—181no—
91#217—61no—
92C4⋊Dic3—32no—
93C4×Dic3—61no—
94#208—62no—
95C2×C3⋊C8—32no—
96C22×C32—22no—
97C22×C32SL(2,ℤ/4)12yes—
98C22×C32C2×S411yescharacteristic
99C32×C4—31no—
100C32×C4—31no—
101C32×C4—32no—
102C3×Dic3—122no—
103C22×C9—41no—
104C22×C9—42no—
105C22×C9—82no—
106C22×C9—84no—
107C22×C9—81no—
108C22×C9—82no—
109Dic9—82no—
110C22×Q8C3×D911yescharacteristic
111C2×M16—91no—
112#117—91no—
113#103—92no—
114#103—92no—
115C3×C9—41noSylow
116C23×C3—21no—
117C23×C3(C22⋊C9)⋊C211yescharacteristic
118C23×C3C3×S411yescharacteristic
119C3×Q8—21no—
120C3×Q8—32no—
121C3×Q8—31no—
122C3×Q8—32no—
123C3×Q8—31no—
124C3×Q8—62no—
125C3×Q8—61no—
126C3×Q8—61no—
127C3×Q8—61no—
128C3×Q8—62no—
129C3×Q8C2×Dic911yescharacteristic
130C3×Q8C2×C3×Dic311yescharacteristic
131C2×C3×C4—31no—
132C2×C3×C4—31no—
133C2×C3×C4—32no—
134C2×C3×C4—32no—
135C2×C3×C4—31no—
136C2×C3×C4—31no—
137C2×C3×C4—32no—
138C2×C3×C4—32no—
139C2×C3×C4—62no—
140C2×C3×C4—61no—
141C2×C3×C4—62no—
142C2×C3×C4—61no—
143C2×C3×C4—182no—
144C2×Dic3—62no—
145C3×C8—182no—
146C3⋊C8—62no—
147C2×C32—22no—
148C2×C32#98312yes—
149C2×C32C2×SL(2,ℤ/4)11yescharacteristic
150C2×C9—41no—
151C2×C9—42no—
152C2×C9—82no—
153C2×C9—84no—
154C2×C9—81no—
155C2×C9—82no—
156C2×Q8—31no—
157C2×Q8—32no—
158C2×Q8C3×Dic912yescomplemented
159C2×Q8C2×C3×D911yescharacteristic
160C22×C4—31no—
161M16—182no—
162C2×C8—92no—
163C4⋊C4—92no—
164C22⋊C4—181no—
165C42—181no—
166C22×C3—24no—
167C22×C3—22no—
168C22×C3—22no—
169C22×C3—22no—
170C22×C3—21no—
171C22×C3(C22⋊C9)⋊C412yes—
172C22×C3C2×(C22⋊C9)⋊C211yescharacteristic
173C22×C3C3×SL(2,ℤ/4)12yes—
174C22×C3C2×C3×S411yescharacteristic
175C3×C4—31no—
176C3×C4—31no—
177C3×C4—32no—
178C3×C4—31no—
179C3×C4—31no—
180C3×C4—32no—
181C3×C4—61no—
182C3×C4—61no—
183C3×C4—62no—
184C3×C4—362no—
185Dic3—122no—
186C32#542011yescharacteristic
187C9—41no—
188C9—81no—
189C23C3×(C22⋊C9)⋊C211yescharacteristic
190Q8—32no—
191Q8—31no—
192Q8—61no—
193Q8C2×C3×Dic911yescharacteristic
194C2×C4—32no—
195C2×C4—32no—
196C2×C4—31no—
197C2×C4—31no—
198C2×C4—182no—
199C8—182no—
200C2×C3—24no—
201C2×C3—22no—
202C2×C3—22no—
203C2×C3—22no—
204C2×C3—21no—
205C2×C3#6368912yes—
206C2×C3C2×(C22⋊C9)⋊C411yescharacteristic
207C2×C3C3×#98312yes—
208C2×C3C2×C3×SL(2,ℤ/4)11yescharacteristic
209C22—22no—
210C22C3×(C22⋊C9)⋊C412yes—
211C22C2×C3×(C22⋊C9)⋊C211yescharacteristic
212C4—31no—
213C4—31no—
214C4—32no—
215C4—362no—
216C3—21no—
217C3#1058952211yescharacteristic, direct product factor
218C3C3×#542011yescharacteristic
219C2—22no—
220C2C3×#6368912yes—
221C2C2×C3×(C22⋊C9)⋊C411yescharacteristic
subgroupG/H
centerC22×C3C2×(C22⋊C9)⋊C2
commutator, hypocenterQ8⋊C9C2×C3×C4
FrattiniC22×C3C2×C3×S4
FittingC22×C32×Q8D3
socleC22×C32C2×S4
prime radicalC23×C32S4
cosocleC2×Q8⋊C9C22×C3
hypercenterC23×C3(C22⋊C9)⋊C2
LCSG ▷ Q8⋊C9
UCSC1 ◁ C22×C3 ◁ C23×C3

Complements

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

NH ≅ G/Ntype
2. C3×#63421C2semidirect
4. #10589522C3direct (N × H)
5. C2×C3×Q8⋊C9C4semidirect
15. #63421C2×C3semidirect
33. C2×Q8⋊C9C3×C4semidirect
79. C2×C3×Q8Dic9semidirect
158. C2×Q8C3×Dic9semidirect
217. C3#10589522direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×Q8⋊C92
C3×#634212
C3×#52253
#105895223
C3×#36764