Finite Group Databasebeta

← C2×C4×C3⋊#622 #13173390

Subgroups — C2×C4×C3⋊#622 #13173390

Order 1728.

648 conjugacy classes of subgroups; only the 189 normal subgroups are shown.

normal subgroups only

This group has 648 conjugacy classes of subgroups — too many to draw. Only its 189 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⋊#622C212yesmaximal
2C2×C3×C4×C3⋊Dic3C211yesmaximal, characteristic
3C4×C3⋊#622C214yesmaximal, direct product factor
4C2×C4×#622—31nomaximal
5C22×C33×C4C411yescharacteristic
6C22×C3×C3⋊Dic3C411yescharacteristic
7C22×C3×C3⋊Dic3C2211yescharacteristic
8C2×C3⋊#622C418yesdirect product factor
9C2×C3⋊#622C2214yes—
10C3×C4×C3⋊Dic3C412yes—
11C3×C4×C3⋊Dic3C2212yes—
12C22×#622—32no—
13C2×C4×C3⋊Dic3D311yescharacteristic
14C2×C3×C4×Dic3—62no—
15C4×#622—34no—
16C23×C33C2×C411yescharacteristic
17C2×C33×C4C814yescomplemented
18C2×C33×C4C2×C412yes—
19C2×C3×C3⋊Dic3C814yescomplemented
20C2×C3×C3⋊Dic3C2×C411yescharacteristic
21C2×C3×C3⋊Dic3C2×C414yes—
22C2×C3×C3⋊Dic3C2×C412yes—
23C2×C3×C3⋊Dic3C2311yescharacteristic
24C3⋊#622C2×C418yesdirect product factor
25C2×C4×C3⋊C8—91nomaximal
26C22×C32×C4—22no—
27C22×C32×C4—42no—
28C22×C32×C4Dic311yescharacteristic
29C22×C3⋊Dic3Dic311yescharacteristic
30C22×C3⋊Dic3C2×D311yescharacteristic
31C22×C3×Dic3—62no—
32C22×C3×Dic3—62no—
33C2×#622—38no—
34C2×#622—34no—
35C4×C3⋊Dic3Dic312yes—
36C4×C3⋊Dic3C2×D312yes—
37C3×C4×Dic3—64no—
38C3×C4×Dic3—64no—
39C22×C33C4212yes—
40C22×C33C2×C812yes—
41C22×C33C2×C812yes—
42C22×C33C22×C411yescharacteristic
43C33×C4C2×C814yescomplemented
44C3×C3⋊Dic3C4212yes—
45C3×C3⋊Dic3C2×C814yescomplemented
46C3×C3⋊Dic3C22×C412yes—
47C2×C3×C42—91no—
48C2×C4×Dic3—62no—
49C22×C3⋊C8—92no—
50C4×C3⋊C8—94no—
51C23×C32—22no—
52C23×C32—42no—
53C23×C32C2×Dic311yescharacteristic
54C2×C32×C4—28no—
55C2×C32×C4—24no—
56C2×C32×C4—48no—
57C2×C32×C4—44no—
58C2×C32×C4C3⋊C814yescomplemented
59C2×C32×C4C2×Dic312yes—
60C2×C3⋊Dic3C3⋊C814yescomplemented
61C2×C3⋊Dic3C4×D314yes—
62C2×C3⋊Dic3C2×Dic312yes—
63C2×C3⋊Dic3C2×Dic311yescharacteristic
64C2×C3⋊Dic3C22×D311yescharacteristic
65C2×C3×Dic3—62no—
66C2×C3×Dic3—64no—
67C2×C3×Dic3—62no—
68C2×C3×Dic3—68no—
69C2×C3×Dic3—68no—
70#622—38no—
71C2×C4×C8—271noSylow
72C2×C33C4×C814yescomplemented
73C2×C33C2×C4211yescharacteristic
74C2×C33C22×C811yescharacteristic
75C2×C33C22×C811yescharacteristic
76C22×C3×C4—22no—
77C22×C3×C4—42no—
78C22×C3×C4—91no—
79C22×C3×C4—91no—
80C22×C3×C4#15711yescharacteristic
81C22×Dic3—62no—
82C22×Dic3—62no—
83C3×C42—92no—
84C3×C42—92no—
85C4×Dic3—64no—
86C4×Dic3—64no—
87C2×C3⋊C8—94no—
88C2×C3⋊C8—98no—
89C22×C32—22no—
90C22×C32—24no—
91C22×C32—24no—
92C22×C32—24no—
93C22×C32—44no—
94C22×C32—44no—
95C22×C32—44no—
96C22×C32—42no—
97C22×C32C2×C3⋊C812yes—
98C22×C32C2×C3⋊C812yes—
99C22×C32C4×Dic312yes—
100C22×C32C22×Dic311yescharacteristic
101C32×C4—28no—
102C32×C4—48no—
103C32×C4C2×C3⋊C814yescomplemented
104C3⋊Dic3C2×C3⋊C814yescomplemented
105C3⋊Dic3C4×Dic312yes—
106C3⋊Dic3C2×C4×D312yes—
107C3×Dic3—64no—
108C3×Dic3—68no—
109C3×Dic3—64no—
110C22×C8—272no—
111C2×C42—91no—
112C4×C8—274no—
113C33C2×C4×C811yescharacteristic, complemented, Sylow
114C23×C3—22no—
115C23×C3—42no—
116C23×C3C2×#15711yescharacteristic
117C2×C3×C4—28no—
118C2×C3×C4—24no—
119C2×C3×C4—48no—
120C2×C3×C4—44no—
121C2×C3×C4—91no—
122C2×C3×C4—92no—
123C2×C3×C4—91no—
124C2×C3×C4—94no—
125C2×C3×C4—94no—
126C2×C3×C4#62214yescomplemented
127C2×C3×C4C2×#15712yes—
128C2×Dic3—68no—
129C2×Dic3—68no—
130C2×Dic3—64no—
131C2×Dic3—62no—
132C2×Dic3—62no—
133C3⋊C8—98no—
134C2×C32—22no—
135C2×C32—22no—
136C2×C32—28no—
137C2×C32—22no—
138C2×C32—48no—
139C2×C32—42no—
140C2×C32—42no—
141C2×C32—42no—
142C2×C32C4×C3⋊C814yescomplemented
143C2×C32C22×C3⋊C811yescharacteristic
144C2×C32C22×C3⋊C811yescharacteristic
145C2×C32C2×C4×Dic311yescharacteristic
146C22×C4—91no—
147C22×C4—91no—
148C22×C4C3⋊#15711yescharacteristic
149C2×C8—274no—
150C2×C8—278no—
151C42—92no—
152C42—92no—
153C22×C3—22no—
154C22×C3—24no—
155C22×C3—24no—
156C22×C3—24no—
157C22×C3—42no—
158C22×C3—44no—
159C22×C3—44no—
160C22×C3—44no—
161C22×C3#378712yes—
162C22×C3C4×#15712yes—
163C22×C3C2×#62212yes—
164C22×C3C22×#15711yescharacteristic
165C3×C4—28no—
166C3×C4—48no—
167C3×C4—94no—
168C3×C4—92no—
169C3×C4—92no—
170C3×C4C2×#62214yescomplemented
171Dic3—68no—
172Dic3—64no—
173Dic3—64no—
174C32—22no—
175C32—42no—
176C32C2×C4×C3⋊C811yescharacteristic, complemented
177C23C2×C3⋊#15711yescharacteristic
178C2×C4—91no—
179C2×C4—92no—
180C2×C4—91no—
181C2×C4—94no—
182C2×C4—94no—
183C2×C4C3⋊#62214yesdirect product factor
184C2×C4C2×C3⋊#15712yes—
185C8—278no—
186C2×C3—28no—
187C2×C3—22no—
188C2×C3—22no—
189C2×C3—22no—
190C2×C3—48no—
191C2×C3—42no—
192C2×C3—42no—
193C2×C3—42no—
194C2×C3C4×#62214yescomplemented
195C2×C3C2×#378711yescharacteristic
196C2×C3C2×C4×#15711yescharacteristic
197C2×C3C22×#62211yescharacteristic
198C22#8898812yes—
199C22C4×C3⋊#15712yes—
200C22C2×C3⋊#62212yes—
201C22C22×C3⋊#15711yescharacteristic
202C4—94no—
203C4—92no—
204C4—92no—
205C4C2×C3⋊#62214yesdirect product factor
206C3—22no—
207C3—42no—
208C3C2×C4×#62211yescharacteristic, complemented
209C2C4×C3⋊#62214yesdirect product factor
210C2C2×#8898811yescharacteristic
211C2C2×C4×C3⋊#15711yescharacteristic
212C2C22×C3⋊#62211yescharacteristic
subgroupG/H
center, hypercenterC22×C4C3⋊#157
commutator, hypocenterC33C2×C4×C8
FrattiniC22C22×C3⋊#157
FittingC22×C33×C4C4
socle, prime radicalC23×C33C2×C4
cosocleC2×C3×C3⋊Dic3C23
LCSG ▷ C33
UCSC1 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
3. C4×C3⋊#622C2direct (N × H)
8. C2×C3⋊#622C4direct (N × H)
17. C2×C33×C4C8semidirect
19. C2×C3×C3⋊Dic3C8semidirect
24. C3⋊#622C2×C4direct (N × H)
43. C33×C4C2×C8semidirect
45. C3×C3⋊Dic3C2×C8semidirect
58. C2×C32×C4C3⋊C8semidirect
60. C2×C3⋊Dic3C3⋊C8semidirect
72. C2×C33C4×C8semidirect
103. C32×C4C2×C3⋊C8semidirect
104. C3⋊Dic3C2×C3⋊C8semidirect
113. C33C2×C4×C8semidirect
126. C2×C3×C4#622semidirect
142. C2×C32C4×C3⋊C8semidirect
170. C3×C4C2×#622semidirect
176. C32C2×C4×C3⋊C8semidirect
183. C2×C4C3⋊#622direct (N × H)
194. C2×C3C4×#622semidirect
205. C4C2×C3⋊#622direct (N × H)
208. C3C2×C4×#622semidirect
209. C2C4×C3⋊#622direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3⋊#6222
C2×C3×C4×C3⋊Dic32
C4×C3⋊#6222
C2×C4×#6223
C2×C4×C3⋊C89