Finite Group Databasebeta

← C3×#63919 #11723244

Subgroups — C3×#63919 #11723244

Order 864.

370 conjugacy classes of subgroups; only the 112 normal subgroups are shown.

normal subgroups only

This group has 370 conjugacy classes of subgroups — too many to draw. Only its 112 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×#627C212yesmaximal
2C3×C4×C3⋊Dic3C211yesmaximal, characteristic
3C32×C4⋊Dic3C212yesmaximal
4C3×#3724C212yesmaximal
5C3×#888—32nomaximal
6#63919C311yesmaximal, characteristic, direct product factor
7C2×C33×C4C2211yescharacteristic
8C2×C3×C3⋊Dic3C2211yescharacteristic
9C2×C3×C3⋊Dic3C2211yescharacteristic
10C2×C32×Dic3C2214yes—
11C3×#627—24no—
12C2×C3×Dic6—34no—
13C2×#627C2×C312yes—
14C32×C4⋊C4—32no—
15C4×C3⋊Dic3C2×C311yescharacteristic
16C3×C4⋊Dic3—22no—
17C3×C4⋊Dic3D312yes—
18C3×C4⋊Dic3C2×C312yes—
19C3×Dic3⋊C4—34no—
20C3×C4×Dic3—32no—
21C3×C4×Dic3—61no—
22#3724C2×C312yes—
23C22×C33C2311yescharacteristic
24C33×C4Q812yescomplemented
25C3×C3⋊Dic3D412yes—
26C3×C3⋊Dic3Q812yescomplemented
27C32×Dic3—24no—
28C3×C4⋊Q8—91no—
29#888—32no—
30C2×C32×C4—21no—
31C2×C32×C4—22no—
32C2×C32×C4—34no—
33C2×C32×C4—41no—
34C2×C32×C4C2×D312yes—
35C2×C32×C4C22×C311yescharacteristic
36C2×C3⋊Dic3C22×C311yescharacteristic
37C2×C3⋊Dic3C22×C311yescharacteristic
38C2×C3×Dic3—24no—
39C2×C3×Dic3—32no—
40C2×C3×Dic3—32no—
41C2×C3×Dic3—61no—
42C2×C3×Dic3—61no—
43C2×C3×Dic3C2×D314yes—
44C2×C3×Dic3C22×C314yes—
45C3×Dic6—68no—
46#627—24no—
47C2×C33C2×D411yescharacteristic
48C2×C33C2×Q811yescharacteristic
49C2×C33C2×Q811yescharacteristic
50C2×C3×Q8—92no—
51C2×Dic6—34no—
52C3×C4⋊C4—32no—
53C3×C4⋊C4—32no—
54C3×C4⋊C4—62no—
55C3×C4⋊C4—92no—
56C3×C42—91no—
57C4⋊Dic3C3×D312yes—
58Dic3⋊C4—34no—
59C4×Dic3—32no—
60C4×Dic3—61no—
61C22×C32—21no—
62C22×C32—22no—
63C22×C32—41no—
64C22×C32C22×D312yes—
65C22×C32C23×C311yescharacteristic
66C32×C4—22no—
67C32×C4—24no—
68C32×C4—42no—
69C32×C4—64no—
70C32×C4Dic614yescomplemented
71C32×C4C3×Q812yescomplemented
72C3⋊Dic3C3×D412yes—
73C3⋊Dic3C3×Q812yescomplemented
74C3×Dic3—24no—
75C3×Dic3—24no—
76C3×Dic3—34no—
77C3×Dic3—34no—
78C3×Dic3—44no—
79C3×Dic3—62no—
80C3×Dic3—62no—
81C4⋊Q8—91noSylow
82C33C4⋊Q811yescharacteristic, complemented, Sylow
83C3×Q8—184no—
84C2×C3×C4—22no—
85C2×C3×C4—21no—
86C2×C3×C4—34no—
87C2×C3×C4—34no—
88C2×C3×C4—41no—
89C2×C3×C4—64no—
90C2×C3×C4—91no—
91C2×C3×C4—91no—
92C2×C3×C4D3211yescharacteristic
93C2×C3×C4C2×C3×D312yes—
94C2×Dic3—32no—
95C2×Dic3—32no—
96C2×Dic3—61no—
97C2×Dic3—61no—
98C2×Dic3C2×C3×D314yes—
99Dic6—68no—
100C2×C32—21no—
101C2×C32—21no—
102C2×C32—21no—
103C2×C32—22no—
104C2×C32—22no—
105C2×C32—22no—
106C2×C32—41no—
107C2×C32—41no—
108C2×C32—41no—
109C2×C32C2×Dic612yes—
110C2×C32D3×D412yes—
111C2×C32D3×Q812yes—
112C2×C32C2×C3×D411yescharacteristic
113C2×C32C2×C3×Q811yescharacteristic
114C2×C32C2×C3×Q811yescharacteristic
115C2×Q8—92no—
116C4⋊C4—32no—
117C4⋊C4—92no—
118C42—91no—
119C22×C3—21no—
120C22×C3—22no—
121C22×C3—41no—
122C22×C3C2×D3211yescharacteristic
123C22×C3C22×C3×D312yes—
124C3×C4—24no—
125C3×C4—22no—
126C3×C4—42no—
127C3×C4—64no—
128C3×C4—64no—
129C3×C4—92no—
130C3×C4—92no—
131C3×C4—124no—
132C3×C4#62712yescomplemented
133C3×C4C3×Dic614yescomplemented
134Dic3—24no—
135Dic3—34no—
136Dic3—34no—
137Dic3—62no—
138Dic3—62no—
139C32—22no—
140C32—21no—
141C32—41no—
142C32#88812yescomplemented
143C32C3×C4⋊Q811yescharacteristic, complemented
144Q8—184no—
145C2×C4—34no—
146C2×C4—91no—
147C2×C4—91no—
148C2×C4C3×D3211yescharacteristic
149C2×C3—21no—
150C2×C3—22no—
151C2×C3—21no—
152C2×C3—22no—
153C2×C3—22no—
154C2×C3—21no—
155C2×C3—41no—
156C2×C3—41no—
157C2×C3—41no—
158C2×C3#379711yescharacteristic
159C2×C3#380211yescharacteristic
160C2×C3C2×#62711yescharacteristic
161C2×C3C2×C3×Dic612yes—
162C2×C3C3×D3×D412yes—
163C2×C3C3×D3×Q812yes—
164C22C2×C3×D3211yescharacteristic
165C4—64no—
166C4—92no—
167C4—92no—
168C4C3×#62712yescomplemented
169C3—22no—
170C3—21no—
171C3—41no—
172C3#6391911yescharacteristic, direct product factor
173C3C3×#88812yescomplemented
174C2C3×#379711yescharacteristic
175C2C3×#380211yescharacteristic
176C2C2×C3×#62711yescharacteristic
subgroupG/H
centerC22×C3C2×D32
commutator, cosocleC22×C32C23×C3
FrattiniC22C2×C3×D32
FittingC2×C33×C4C22
socle, prime radicalC22×C33C23
hypercenterC2×C3×C4D32
hypocenterC32C3×C4⋊Q8
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C2×C3×C4

Complements

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

NH ≅ G/Ntype
6. #63919C3direct (N × H)
24. C33×C4Q8semidirect
26. C3×C3⋊Dic3Q8semidirect
70. C32×C4Dic6semidirect
71. C32×C4C3×Q8semidirect
73. C3⋊Dic3C3×Q8semidirect
82. C33C4⋊Q8semidirect
132. C3×C4#627semidirect
133. C3×C4C3×Dic6semidirect
142. C32#888semidirect
143. C32C3×C4⋊Q8semidirect
168. C4C3×#627semidirect
172. C3#63919direct (N × H)
173. C3C3×#888semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#6272
C3×C4×C3⋊Dic32
C32×C4⋊Dic32
C3×#37242
C3×#8883
#639193