Finite Group Databasebeta

← C2×C3×#3775 #11724293

Subgroups — C2×C3×#3775 #11724293

Order 864.

276 conjugacy classes of subgroups; only the 54 normal subgroups are shown.

normal subgroups only

This group has 276 conjugacy classes of subgroups — too many to draw. Only its 54 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×C3×#627C211yesmaximal, characteristic, complemented
2C2×C3×#625C211yesmaximal, characteristic
3C2×C3×#622C211yesmaximal, characteristic, complemented
4C3×#3775C214yesmaximal, direct product factor
5C2×#3775C311yesmaximal, characteristic, direct product factor
6C22×C32×D3—21no—
7C2×C3×C3⋊Dic3C2211yescharacteristic
8C2×C32×Dic3—21no—
9C3×#627—21no—
10C3×#627C2212yescomplemented
11C3×#625—21no—
12C3×#625C2212yes—
13C3×#622C2212yescomplemented
14C2×C3×#67—61no—
15C2×C3×Dic6—61no—
16C2×#627C2×C311yescharacteristic, complemented
17C2×#625C2×C311yescharacteristic
18C2×#622C2×C311yescharacteristic, complemented
19#3775C2×C314yesdirect product factor
20C22×C33D411yescharacteristic
21C2×C32×D3—22no—
22C2×C32×D3—42no—
23C3×C3⋊Dic3D411yescharacteristic
24C3×C3⋊Dic3C2311yescharacteristic
25C32×Dic3—22no—
26C2×C3×SD16—91nomaximal
27C23×C32—61no—
28C22×C3×D3—21no—
29C22×C3×D3—21no—
30C22×C3×D3—41no—
31C2×C32×C4—61no—
32C2×C3⋊Dic3C22×C311yescharacteristic
33C3×#67—62no—
34C3×#67—62no—
35C2×C3×Dic3—21no—
36C2×C3×Dic3—21no—
37C2×C3×Dic3—41no—
38C2×C3×Dic3—61no—
39C2×C3×Dic3—61no—
40C3×Dic6—62no—
41C3×Dic6—62no—
42#627—21no—
43#627C22×C312yescomplemented
44#625—21no—
45#625C22×C312yes—
46#622C22×C312yescomplemented
47C2×C33SD1612yescomplemented
48C2×C33C2×D411yescharacteristic
49C32×D3—42no—
50C2×C3×Q8—91no—
51C2×C3×D4—91no—
52C2×#67—61no—
53C2×Dic6—61no—
54C3×SD16—94no—
55C2×C3×C8—91no—
56C22×C32—21no—
57C22×C32—21no—
58C22×C32—41no—
59C22×C32—41no—
60C22×C32—62no—
61C22×C32—122no—
62C22×C32C3×D411yescharacteristic
63C2×C3×D3—22no—
64C2×C3×D3—22no—
65C2×C3×D3—42no—
66C2×C3×D3—42no—
67C2×C3×D3—44no—
68C2×C3×D3—42no—
69C32×C4—62no—
70C3⋊Dic3C3×D411yescharacteristic
71C3⋊Dic3C23×C311yescharacteristic
72C3×Dic3—22no—
73C3×Dic3—22no—
74C3×Dic3—42no—
75C3×Dic3—61no—
76C3×Dic3—61no—
77C3×Dic3—61no—
78C3×Dic3—61no—
79C2×SD16—91noSylow
80C33C2×SD1611yescharacteristic, complemented, Sylow
81C23×C3—61no—
82C23×C3—61no—
83C23×C3—121no—
84C22×D3—21no—
85C3×Q8—92no—
86C3×Q8—181no—
87C3×D4—92no—
88C3×D4—181no—
89C2×C3×C4—61no—
90C2×C3×C4—61no—
91C2×C3×C4—91no—
92C2×C3×C4—121no—
93#67—62no—
94#67—62no—
95C2×Dic3—21no—
96C2×Dic3—61no—
97C2×Dic3—61no—
98Dic6—62no—
99Dic6—62no—
100C3×C8—92no—
101C2×C32—21no—
102C2×C32—21no—
103C2×C32—22no—
104C2×C32—22no—
105C2×C32—42no—
106C2×C32—42no—
107C2×C32—41no—
108C2×C32—41no—
109C2×C32—122no—
110C2×C32C3×SD1612yescomplemented
111C2×C32C2×C3×D411yescharacteristic
112C3×D3—42no—
113C3×D3—44no—
114C3×D3—42no—
115C2×Q8—91no—
116C2×D4—91no—
117SD16—94no—
118C2×C8—91no—
119C22×C3—21no—
120C22×C3—21no—
121C22×C3—41no—
122C22×C3—41no—
123C22×C3—62no—
124C22×C3—62no—
125C22×C3—122no—
126C22×C3—122no—
127C22×C3—124no—
128C22×C3—122no—
129C22×C3GO+(4,𝔽2)11yescharacteristic
130C2×D3—22no—
131C2×D3—42no—
132C3×C4—62no—
133C3×C4—62no—
134C3×C4—91no—
135C3×C4—91no—
136C3×C4—122no—
137Dic3—22no—
138Dic3—61no—
139Dic3—61no—
140Dic3—61no—
141Dic3—61no—
142C32—21no—
143C32—21no—
144C32—41no—
145C32—41no—
146C32C2×C3×SD1611yescharacteristic, complemented
147C23—61no—
148Q8—92no—
149Q8—181no—
150D4—92no—
151D4—181no—
152C2×C4—61no—
153C2×C4—91no—
154C8—92no—
155C2×C3—21no—
156C2×C3—21no—
157C2×C3—22no—
158C2×C3—22no—
159C2×C3—41no—
160C2×C3—41no—
161C2×C3—42no—
162C2×C3—42no—
163C2×C3—122no—
164C2×C3—124no—
165C2×C3—122no—
166C2×C3#377512yesdirect product factor
167C2×C3C2×GO+(4,𝔽2)11yescharacteristic
168D3—42no—
169C22—62no—
170C22—122no—
171C22C3×GO+(4,𝔽2)11yescharacteristic
172C4—62no—
173C4—91no—
174C4—91no—
175C3—21no—
176C3—21no—
177C3—41no—
178C3—41no—
179C3C2×#377511yescharacteristic, direct product factor
180C2—122no—
181C2C3×#377512yesdirect product factor
182C2C2×C3×GO+(4,𝔽2)11yescharacteristic
subgroupG/H
center, hypercenterC22×C3GO+(4,𝔽2)
commutator, cosocleC3⋊Dic3C23×C3
FrattiniC2C2×C3×GO+(4,𝔽2)
Fitting, socleC22×C33D4
prime radicalC2×C3×#625C2
hypocenterC32C2×C3×SD16
LCSG ▷ C3⋊Dic3 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22×C3

Complements

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

NH ≅ G/Ntype
1. C2×C3×#627C2semidirect
3. C2×C3×#622C2semidirect
4. C3×#3775C2direct (N × H)
5. C2×#3775C3direct (N × H)
10. C3×#627C22semidirect
13. C3×#622C22semidirect
16. C2×#627C2×C3semidirect
18. C2×#622C2×C3semidirect
19. #3775C2×C3direct (N × H)
43. #627C22×C3semidirect
46. #622C22×C3semidirect
47. C2×C33SD16semidirect
80. C33C2×SD16semidirect
110. C2×C32C3×SD16semidirect
146. C32C2×C3×SD16semidirect
166. C2×C3#3775direct (N × H)
179. C3C2×#3775direct (N × H)
181. C2C3×#3775direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#6272
C2×C3×#6252
C2×C3×#6222
C3×#37752
C2×#37753
C2×C3×SD169