Finite Group Databasebeta

← C2×C3×C3⋊#622 #13049587

Subgroups — C2×C3×C3⋊#622 #13049587

Order 1296.

420 conjugacy classes of subgroups; only the 58 normal subgroups are shown.

normal subgroups only

This group has 420 conjugacy classes of subgroups — too many to draw. Only its 58 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×C32×C3⋊Dic3C211yesmaximal, characteristic
2C3×C3⋊#622C212yesmaximal, direct product factor
3C2×C3⋊#622C311yesmaximal, characteristic, direct product factor
4C2×C3×#622—31nomaximal
5C22×C34C411yescharacteristic
6C32×C3⋊Dic3C411yescharacteristic
7C32×C3⋊Dic3C2211yescharacteristic
8C2×C3×C3⋊Dic3—21no—
9C2×C3×C3⋊Dic3D311yescharacteristic
10C2×C3×C3⋊Dic3C2×C311yescharacteristic
11C2×C32×Dic3—62no—
12C3⋊#622C2×C312yesdirect product factor
13C3×#622—32no—
14C2×C34C812yescomplemented
15C2×C34C2×C411yescharacteristic
16C2×#622—31no—
17C2×C3×C3⋊C8—91nomaximal
18C22×C33—21no—
19C22×C33—22no—
20C22×C33—44no—
21C22×C33—42no—
22C22×C33—42no—
23C22×C33Dic311yescharacteristic
24C22×C33C3×C411yescharacteristic
25C3×C3⋊Dic3—21no—
26C3×C3⋊Dic3—21no—
27C3×C3⋊Dic3Dic311yescharacteristic
28C3×C3⋊Dic3C3×C411yescharacteristic
29C3×C3⋊Dic3C2×D311yescharacteristic
30C3×C3⋊Dic3C22×C311yescharacteristic
31C32×Dic3—62no—
32C32×Dic3—62no—
33C34C2×C811yescharacteristic, complemented, Sylow
34C2×C32×C4—91no—
35C2×C3⋊Dic3C3×D311yescharacteristic
36C2×C3×Dic3—62no—
37C2×C3×Dic3—62no—
38C2×C3×Dic3—64no—
39#622—32no—
40C3×C3⋊C8—92no—
41C2×C33—22no—
42C2×C33—24no—
43C2×C33—21no—
44C2×C33—22no—
45C2×C33—48no—
46C2×C33—44no—
47C2×C33—44no—
48C2×C33—44no—
49C2×C33—42no—
50C2×C33—42no—
51C2×C33C3⋊C812yescomplemented
52C2×C33C3×C812yescomplemented
53C2×C33C2×Dic311yescharacteristic
54C2×C33C2×C3×C411yescharacteristic
55C2×C3×C8—271no—
56C2×C3⋊C8—91no—
57C22×C32—24no—
58C22×C32—22no—
59C22×C32—22no—
60C22×C32—44no—
61C22×C32—44no—
62C22×C32—42no—
63C22×C32—42no—
64C22×C32—48no—
65C22×C32—42no—
66C22×C32—42no—
67C22×C32—44no—
68C22×C32C3×Dic311yescharacteristic
69C22×C32#15711yescharacteristic
70C32×C4—91no—
71C32×C4—91no—
72C3⋊Dic3C3×Dic311yescharacteristic
73C3⋊Dic3C2×C3×D311yescharacteristic
74C3×Dic3—62no—
75C3×Dic3—62no—
76C3×Dic3—62no—
77C3×Dic3—62no—
78C3×Dic3—64no—
79C3×Dic3—64no—
80C33—21no—
81C33—22no—
82C33—44no—
83C33—42no—
84C33—42no—
85C33C2×C3⋊C811yescharacteristic, complemented
86C33C2×C3×C811yescharacteristic, complemented
87C2×C3×C4—91no—
88C2×C3×C4—91no—
89C2×C3×C4—181no—
90C2×Dic3—62no—
91C3×C8—272no—
92C3⋊C8—92no—
93C2×C32—24no—
94C2×C32—28no—
95C2×C32—24no—
96C2×C32—24no—
97C2×C32—22no—
98C2×C32—22no—
99C2×C32—44no—
100C2×C32—44no—
101C2×C32—48no—
102C2×C32—48no—
103C2×C32—44no—
104C2×C32—44no—
105C2×C32—42no—
106C2×C32—42no—
107C2×C32—48no—
108C2×C32—416no—
109C2×C32—44no—
110C2×C32—44no—
111C2×C32—42no—
112C2×C32—42no—
113C2×C32—44no—
114C2×C32—48no—
115C2×C32C3×C3⋊C812yescomplemented
116C2×C32#62212yescomplemented
117C2×C32C2×C3×Dic311yescharacteristic
118C2×C32C2×#15711yescharacteristic
119C2×C8—271noSylow
120C22×C3—22no—
121C22×C3—21no—
122C22×C3—44no—
123C22×C3—42no—
124C22×C3—42no—
125C22×C3C3×#15711yescharacteristic
126C22×C3C3⋊#15711yescharacteristic
127C3×C4—91no—
128C3×C4—91no—
129C3×C4—91no—
130C3×C4—91no—
131C3×C4—181no—
132C3×C4—181no—
133Dic3—62no—
134Dic3—62no—
135C32—24no—
136C32—22no—
137C32—22no—
138C32—42no—
139C32—42no—
140C32—44no—
141C32—44no—
142C32—44no—
143C32—48no—
144C32—42no—
145C32—42no—
146C32C2×C3×C3⋊C811yescharacteristic, complemented
147C32C2×#62211yescharacteristic, complemented
148C2×C4—91no—
149C8—272no—
150C2×C3—22no—
151C2×C3—21no—
152C2×C3—24no—
153C2×C3—22no—
154C2×C3—44no—
155C2×C3—42no—
156C2×C3—42no—
157C2×C3—48no—
158C2×C3—44no—
159C2×C3—44no—
160C2×C3C3×#62212yescomplemented
161C2×C3C3⋊#62212yesdirect product factor
162C2×C3C2×C3×#15711yescharacteristic
163C2×C3C2×C3⋊#15711yescharacteristic
164C22C3×C3⋊#15711yescharacteristic
165C4—91no—
166C4—91no—
167C3—22no—
168C3—21no—
169C3—44no—
170C3—42no—
171C3—42no—
172C3C2×C3×#62211yescharacteristic, complemented
173C3C2×C3⋊#62211yescharacteristic, direct product factor
174C2C3×C3⋊#62212yesdirect product factor
175C2C2×C3×C3⋊#15711yescharacteristic
subgroupG/H
center, hypercenterC22×C3C3⋊#157
commutator, hypocenterC33C2×C3×C8
FrattiniC2C2×C3×C3⋊#157
Fitting, socle, prime radicalC22×C34C4
cosocleC3×C3⋊Dic3C22×C3
LCSG ▷ C33
UCSC1 ◁ C22×C3

Complements

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

NH ≅ G/Ntype
2. C3×C3⋊#622C2direct (N × H)
3. C2×C3⋊#622C3direct (N × H)
12. C3⋊#622C2×C3direct (N × H)
14. C2×C34C8semidirect
33. C34C2×C8semidirect
51. C2×C33C3⋊C8semidirect
52. C2×C33C3×C8semidirect
85. C33C2×C3⋊C8semidirect
86. C33C2×C3×C8semidirect
115. C2×C32C3×C3⋊C8semidirect
116. C2×C32#622semidirect
146. C32C2×C3×C3⋊C8semidirect
147. C32C2×#622semidirect
160. C2×C3C3×#622semidirect
161. C2×C3C3⋊#622direct (N × H)
172. C3C2×C3×#622semidirect
173. C3C2×C3⋊#622direct (N × H)
174. C2C3×C3⋊#622direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C32×C3⋊Dic32
C3×C3⋊#6222
C2×C3⋊#6223
C2×C3×#6223
C2×C3×C3⋊C89