Finite Group Databasebeta

← #11723425

Subgroups — #11723425

Order 864.

548 conjugacy classes of subgroups; only the 140 normal subgroups are shown.

normal subgroups only

This group has 548 conjugacy classes of subgroups — too many to draw. Only its 140 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×#634C211yesmaximal, characteristic, complemented
2C2×C32×C4×D3C211yesmaximal, characteristic
3C4⋊(C3⋊(C3⋊Dic3))C211yesmaximal, characteristic, complemented
4#88806C212yesmaximal, complemented
5(C2×D3)⋊(C3⋊Dic3)C212yesmaximal
6#64133—31nomaximal
7#63856—34nomaximal
8C22×C32×D3C2211yescharacteristic
9C2×C33×C4C2211yescharacteristic
10C2×C3⋊(C3⋊Dic3)C2212yes—
11C2×C3×C3⋊Dic3C2212yes—
12C3×#634—22no—
13C2×C32×Dic3C2211yescharacteristic
14C32×C4×D3—22no—
15C22×C32×C4—31no—
16C2×#634D311yescharacteristic, complemented
17C2×C3×C4×D3D314yes—
18C2×C3×Dic6—34no—
19#3757—32no—
20C4⋊(C3⋊Dic3)—31no—
21C4⋊(C3⋊Dic3)—34no—
22C4⋊(C3⋊Dic3)—64no—
23#3750—32no—
24#3724—38no—
25(C2×D3)⋊Dic3—38no—
26C22×C33C2311yescharacteristic
27C2×C32×D3—21no—
28C2×C32×D3—21no—
29C2×C32×D3Q812yescomplemented
30C33×C4D412yes—
31C3⋊(C3⋊Dic3)—22no—
32C3×C3⋊Dic3—22no—
33C32×Dic3—21no—
34#946—91no—
35#924—94no—
36C23×C32—31no—
37C22×C3×D3C2×D314yes—
38C2×C32×C4—24no—
39C2×C32×C4—31no—
40C2×C32×C4—62no—
41C2×C32×C4C2×D314yes—
42C2×C32×C4C2×D311yescharacteristic
43C2×C3⋊Dic3—32no—
44C2×C3⋊Dic3—38no—
45C2×C3⋊Dic3—68no—
46C2×C3⋊Dic3C2×D312yes—
47#634—22no—
48C2×C3×Dic3—38no—
49C2×C3×Dic3C2×D314yes—
50C3×C4×D3—28no—
51C3×Dic6—68no—
52C2×C33C2×D411yescharacteristic
53C2×C33C2×Q811yescharacteristic
54C2×C33#4111yescharacteristic
55C32×D3—22no—
56C2×C3×Q8—91no—
57C22×C3×C4—34no—
58C2×C4×D3C3⋊D311yescharacteristic
59C2×Dic6—34no—
60#217—98no—
61(C2×D3)⋊C4—92no—
62C4⋊Dic3—94no—
63C4⋊Dic3—91no—
64C4⋊Dic3—184no—
65Dic3⋊C4—98no—
66Dic3⋊C4—92no—
67C22×C32—24no—
68C22×C32—32no—
69C22×C32—61no—
70C22×C32—61no—
71C22×C32C22×D314yes—
72C22×C32C22×D311yescharacteristic
73C2×C3×D3—24no—
74C2×C3×D3—24no—
75C2×C3×D3Dic618yescomplemented
76C32×C4—28no—
77C32×C4—61no—
78C32×C4#6718yes—
79C32×C4#6712yes—
80C3⋊Dic3—22no—
81C3⋊Dic3—62no—
82C3⋊Dic3—68no—
83C3⋊Dic3—128no—
84C3×Dic3—24no—
85C3×Dic3—68no—
86C22⋊Q8—271noSylow
87C33C22⋊Q811yescharacteristic, complemented, Sylow
88C23×C3—34no—
89C22×D3C2×C3⋊D311yescharacteristic
90C3×Q8—182no—
91C2×C3×C4—24no—
92C2×C3×C4—34no—
93C2×C3×C4—68no—
94C2×C3×C4—92no—
95C2×C3×C4D3214yes—
96C2×C3×C4C2×C3⋊D311yescharacteristic
97C2×Dic3—38no—
98C2×Dic3—98no—
99C2×Dic3—92no—
100C2×Dic3—188no—
101C2×Dic3C2×C3⋊D311yescharacteristic
102C4×D3—22no—
103Dic6—68no—
104C2×C32—24no—
105C2×C32—24no—
106C2×C32—24no—
107C2×C32—62no—
108C2×C32C2×Dic614yes—
109C2×C32#23514yes—
110C2×C32D3×Q811yescharacteristic
111C2×C32#23911yescharacteristic
112C2×C32C2×#6714yes—
113C2×C32C2×#6711yescharacteristic
114C3×D3—28no—
115C2×Q8—91no—
116C22×C4—31no—
117C4⋊C4—272no—
118C4⋊C4—271no—
119C22⋊C4—272no—
120C22×C3—24no—
121C22×C3—38no—
122C22×C3—64no—
123C22×C3—64no—
124C22×C3C2×D3214yes—
125C22×C3C22×C3⋊D311yescharacteristic
126C2×D3—21no—
127C2×D3—21no—
128C2×D3#63412yescomplemented
129C3×C4—28no—
130C3×C4—64no—
131C3×C4—182no—
132C3×C4#62518yes—
133C3×C4#63812yes—
134Dic3—21no—
135Dic3—68no—
136Dic3—188no—
137Dic3—182no—
138Dic3—368no—
139C32—24no—
140C32#92414yescomplemented
141C32#94611yescharacteristic, complemented
142C23—31no—
143Q8—182no—
144C2×C4—31no—
145C2×C4—62no—
146C2×C4—92no—
147C2×C4—272no—
148C2×C4D3×C3⋊D311yescharacteristic
149C2×C3—24no—
150C2×C3—24no—
151C2×C3—24no—
152C2×C3—68no—
153C2×C3D3×Dic614yes—
154C2×C3#379914yes—
155C2×C3C2×#62514yes—
156C2×C3C2×#63411yescharacteristic
157C2×C3#382811yescharacteristic
158C2×C3C2×#63811yescharacteristic
159D3—22no—
160C22—32no—
161C22—61no—
162C22—61no—
163C22C2×D3×C3⋊D311yescharacteristic
164C4—61no—
165C4—182no—
166C4—542no—
167C4#629012yes—
168C3—24no—
169C3#6385614yescomplemented
170C3#6413311yescharacteristic, complemented
171C2—62no—
172C2D3×#63411yescharacteristic
173C2#8902211yescharacteristic
174C2C2×#629011yescharacteristic
subgroupG/H
center, FrattiniC22C2×D3×C3⋊D3
commutator, socle, cosocleC22×C33C23
FittingC2×C33×C4C22
prime radicalC22×C32×D3C22
hypercenterC2×C4D3×C3⋊D3
hypocenterC33C22⋊Q8
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
1. C2×C3×#634C2semidirect
3. C4⋊(C3⋊(C3⋊Dic3))C2semidirect
4. #88806C2semidirect
16. C2×#634D3semidirect
29. C2×C32×D3Q8semidirect
75. C2×C3×D3Dic6semidirect
87. C33C22⋊Q8semidirect
128. C2×D3#634semidirect
140. C32#924semidirect
141. C32#946semidirect
169. C3#63856semidirect
170. C3#64133semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#6342
C2×C32×C4×D32
C4⋊(C3⋊(C3⋊Dic3))2
#888062
(C2×D3)⋊(C3⋊Dic3)2
#641333
#638563