Finite Group Databasebeta

← C2×#63690⋊C3 #13164420

Subgroups — C2×#63690⋊C3 #13164420

Order 1728.

530 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 530 conjugacy classes of subgroups — too many to draw. Only its 71 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×#88691C211yesmaximal, characteristic
2#63690⋊C3C214yesmaximal, direct product factor
3C2×#3688⋊C3C212yesmaximal, complemented
4C2×C3×#951—31nomaximal
5C2×#63690C311yesmaximal, characteristic, complemented
6C2×3−1+2⋊Q8—41nomaximal
7#88691C2212yes—
8C2×#6204C2211yescharacteristic
9#3688⋊C3C2214yescomplemented
10C2×C32×#41D311yescharacteristic
11C22×C3×Dic6—31no—
12C3×#951—34no—
13C3×#951—34no—
14C2×C3×#216—32no—
15C2×C3×#214—32no—
16C2×C3×#208—31no—
17C2×#41⋊C9—21no—
18C2×#41⋊C9C2×C311yescharacteristic
19#63690C2×C314yescomplemented
20C2×#3688C2×C312yescomplemented
21C2×C4×3−1+2—41no—
22C2×3−1+2⋊C4—42no—
233−1+2⋊Q8—44no—
24#6204C2311yescharacteristic
25C2×C3×#137—91no—
26C2×#951—31no—
27C32×#41—32no—
28C32×#41C2×D312yes—
29C2×C32×Q8C2×D311yescharacteristic
30C2×C32×D4—31no—
31C22×C32×C4—31no—
32C22×C3×Dic3—61no—
33C2×C3×Dic6—32no—
34C2×C3×Dic6—34no—
35C2×C3×Dic6—62no—
36C2×C3×Dic6—61no—
37C3×#216—34no—
38C3×#216—34no—
39C3×#214—38no—
40C3×#208—34no—
41C2×C3×C3⋊C8—32no—
42C2×Dic18—41no—
43#41⋊C9—22no—
44#41⋊C9C22×C312yes—
45C2×Q8⋊C9—21no—
46C2×Q8⋊C9C22×C311yescharacteristic
47#3688C22×C314yescomplemented
48C22×3−1+2—41no—
49C4×3−1+2—42no—
503−1+2⋊C4—44no—
51C2×C3×#41—31no—
52C2×C3×#41C3×D311yescharacteristic
53C22×C3×Q8—91no—
54C22×Dic6—31no—
55C3×#137—94no—
56C3×#137—94no—
57C2×C3×Q16—92no—
58C2×C3×SD16—92no—
59C2×C3×M16—91no—
60#951—34no—
61#951—34no—
62C2×#216—32no—
63C2×#214—32no—
64C2×#208—31no—
65C23×C32—31no—
66C32×Q8—31no—
67C32×Q8C22×D311yescharacteristic
68C32×D4—32no—
69C32×D4—61no—
70C2×C32×C4—31no—
71C2×C32×C4—32no—
72C2×C32×C4—32no—
73C2×C32×C4S411yescharacteristic
74C2×C3×Dic3—64no—
75C2×C3×Dic3—62no—
76C3×Dic6—34no—
77C3×Dic6—62no—
78C3×Dic6—64no—
79C3×C3⋊C8—34no—
80C2×C4×C9—41no—
81C2×C4×C9—81no—
82C2×Dic9—42no—
83Dic18—44no—
84Q8⋊C9—21no—
85Q8⋊C9C23×C311yescharacteristic
86C2×#137—91noSylow
87C2×3−1+2—42no—
88C2×3−1+2—41no—
89C3×#41—32no—
90C3×#41—32no—
91C3×#41—32no—
92C3×#41—62no—
93C3×#41C2×C3×D312yes—
94C2×C3×Q8—31no—
95C2×C3×Q8—92no—
96C2×C3×Q8—94no—
97C2×C3×Q8—182no—
98C2×C3×Q8—181no—
99C2×C3×Q8C2×C3×D311yescharacteristic
100C2×C3×D4—31no—
101C2×C3×D4—31no—
102C2×C3×D4—61no—
103C22×C3×C4—31no—
104C22×C3×C4—31no—
105C22×C3×C4—61no—
106C22×C3×C4—181no—
107C22×Dic3—61no—
108C2×Dic6—32no—
109C2×Dic6—34no—
110C2×Dic6—62no—
111C2×Dic6—61no—
112C3×Q16—94no—
113C3×Q16—94no—
114C3×SD16—98no—
115C3×M16—94no—
116C2×C3×C8—92no—
117#216—34no—
118#216—34no—
119#214—38no—
120#208—34no—
121C2×C3⋊C8—32no—
122C22×C32—32no—
123C22×C32—62no—
124C22×C32C2×S411yescharacteristic
125C32×C4—31no—
126C32×C4—31no—
127C32×C4C2×S412yes—
128C3×Dic3—64no—
129C22×C9—41no—
130C22×C9—81no—
131C4×C9—42no—
132C4×C9—82no—
133Dic9—44no—
134C2×#41AGL(1,ℤ/9)11yescharacteristic
135C22×Q8—91no—
136#137—94no—
137#137—94no—
138C2×Q16—92no—
139C2×SD16—92no—
140C2×M16—91no—
1413−1+2—41noSylow
142C23×C3—31no—
143C23×C3—31no—
144C23×C3—61no—
145C3×Q8—31no—
146C3×Q8—31no—
147C3×Q8—31no—
148C3×Q8—61no—
149C3×Q8—94no—
150C3×Q8—182no—
151C3×Q8—184no—
152C3×Q8C22×C3×D311yescharacteristic
153C3×D4—32no—
154C3×D4—32no—
155C3×D4—62no—
156C3×D4—61no—
157C3×D4—61no—
158C3×D4—62no—
159C2×C3×C4—31no—
160C2×C3×C4—32no—
161C2×C3×C4—32no—
162C2×C3×C4—31no—
163C2×C3×C4—32no—
164C2×C3×C4—32no—
165C2×C3×C4—31no—
166C2×C3×C4—62no—
167C2×C3×C4—62no—
168C2×C3×C4—61no—
169C2×C3×C4—184no—
170C2×C3×C4—182no—
171C2×C3×C4C3×S411yescharacteristic
172C2×Dic3—64no—
173C2×Dic3—62no—
174Dic6—34no—
175Dic6—62no—
176Dic6—64no—
177C3×C8—94no—
178C3⋊C8—34no—
179C2×C32—62no—
180C2×C32#98412yes—
181C2×C32C22×S411yescharacteristic
182C2×C9—41no—
183C2×C9—42no—
184C2×C9—81no—
185C2×C9—82no—
186#41—32no—
187#41C2×AGL(1,ℤ/9)12yes—
188C2×Q8—92no—
189C2×Q8—94no—
190C2×Q8—182no—
191C2×Q8—181no—
192C2×Q8C2×AGL(1,ℤ/9)11yescharacteristic
193C2×D4—31no—
194C22×C4—31no—
195C22×C4—181no—
196Q16—94no—
197Q16—94no—
198SD16—98no—
199M16—94no—
200C2×C8—92no—
201C22×C3—32no—
202C22×C3—32no—
203C22×C3—31no—
204C22×C3—62no—
205C22×C3—62no—
206C22×C3—62no—
207C22×C3—122no—
208C22×C3C2×C3×S411yescharacteristic
209C3×C4—31no—
210C3×C4—31no—
211C3×C4—31no—
212C3×C4—31no—
213C3×C4—32no—
214C3×C4—61no—
215C3×C4—61no—
216C3×C4—184no—
217C3×C4C2×C3×S412yes—
218Dic3—64no—
219C32C2×#98411yescharacteristic
220C9—41no—
221C9—81no—
222C23—31no—
223Q8—31no—
224Q8—94no—
225Q8—182no—
226Q8—184no—
227Q8C22×AGL(1,ℤ/9)11yescharacteristic
228D4—32no—
229D4—61no—
230C2×C4—31no—
231C2×C4—32no—
232C2×C4—32no—
233C2×C4—184no—
234C2×C4—182no—
235C2×C4((C22⋊C9)⋊C2)⋊C311yescharacteristic
236C8—94no—
237C2×C3—31no—
238C2×C3—32no—
239C2×C3—62no—
240C2×C3—62no—
241C2×C3—122no—
242C2×C3C3×#98412yes—
243C2×C3C22×C3×S411yescharacteristic
244C22—32no—
245C22—62no—
246C22C2×((C22⋊C9)⋊C2)⋊C311yescharacteristic
247C4—31no—
248C4—31no—
249C4—184no—
250C4C2×((C22⋊C9)⋊C2)⋊C312yes—
251C3—31no—
252C3C2×C3×#98411yescharacteristic
253C2—62no—
254C2#63690⋊C312yesdirect product factor
255C2C22×((C22⋊C9)⋊C2)⋊C311yescharacteristic
subgroupG/H
centerC22C2×((C22⋊C9)⋊C2)⋊C3
commutator, cosocle, hypocenterQ8⋊C9C23×C3
FrattiniC2×C3C22×C3×S4
Fitting, prime radicalC2×C32×#41D3
socleC22×C3C2×C3×S4
hypercenterC2×C4((C22⋊C9)⋊C2)⋊C3
LCSG ▷ Q8⋊C9
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
2. #63690⋊C3C2direct (N × H)
3. C2×#3688⋊C3C2semidirect
5. C2×#63690C3semidirect
9. #3688⋊C3C22semidirect
19. #63690C2×C3semidirect
20. C2×#3688C2×C3semidirect
47. #3688C22×C3semidirect
254. C2#63690⋊C3direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#886912
#63690⋊C32
C2×#3688⋊C32
C2×C3×#9513
C2×#636903
C2×3−1+2⋊Q84