Finite Group Databasebeta

← #13158866

Subgroups — #13158866

Order 1728.

764 conjugacy classes of subgroups; only the 212 normal subgroups are shown.

normal subgroups only

This group has 764 conjugacy classes of subgroups — too many to draw. Only its 212 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
1C3×#64078C211yesmaximal, characteristic
2C32×#869C211yesmaximal, characteristic
3C4×C3⋊(C3⋊(C3⋊C8))C211yesmaximal, characteristic
4#11722039C214yesmaximal
5#10592792—31nomaximal
6#10590290—34nomaximal
7C2×C3×#634C2212yes—
8C2×C32×Dic6—21no—
9C33×C42C2211yescharacteristic
10C2×C3⋊(C3⋊(C3⋊C8))C2212yes—
11C3×C4⋊(C3⋊Dic3)—21no—
12C3×C4⋊(C3⋊Dic3)—21no—
13C32×C4⋊Dic3—21no—
14C32×C4⋊Dic3C2212yes—
15C32×C4⋊Q8—31no—
16#64078D311yescharacteristic
17C3×#869—34no—
18C3×#869D314yes—
19#63637—34no—
20C4×C3⋊(C3⋊C8)—31no—
21C4×C3⋊(C3⋊C8)—34no—
22C4×C3⋊(C3⋊C8)—64no—
23#63565—316no—
24C2×C33×C4D411yescharacteristic
25C2×C33×C4D411yescharacteristic
26C2×C33×C4C2311yescharacteristic
27C2×C3×C3⋊Dic3—22no—
28C3×#634—24no—
29C2×C32×Dic3—22no—
30C32×Dic6—42no—
31C3⋊(C3⋊(C3⋊C8))—22no—
32#5093—94no—
33#5079—91no—
34C2×C32×Q8—61no—
35C2×#634C2×D312yes—
36C2×C3×Dic6—24no—
37C2×C3×Dic6—38no—
38C32×C4⋊C4—32no—
39C32×C4⋊C4—61no—
40C32×C42—24no—
41C32×C42C2×D314yes—
42C32×C42C2×D311yescharacteristic
43C4⋊(C3⋊Dic3)—21no—
44C4⋊(C3⋊Dic3)—21no—
45C2×C3⋊(C3⋊C8)—32no—
46C2×C3⋊(C3⋊C8)—38no—
47C2×C3⋊(C3⋊C8)—68no—
48C3×C4⋊Dic3—24no—
49C3×C4⋊Dic3—64no—
50C3×C4⋊Dic3—64no—
51C3×C4⋊Dic3C2×D318yes—
52C22×C33C2×D411yescharacteristic
53C33×C4SD1612yes—
54C33×C4Q1612yes—
55C33×C4#4112yes—
56C3×C3⋊Dic3—42no—
57C32×Dic3—42no—
58C3×C4⋊Q8—34no—
59C3×C4⋊Q8—91no—
60#869—34no—
61#869C3⋊D311yescharacteristic
62#835—94no—
63#810—916no—
64C4×C3⋊C8—94no—
65C4×C3⋊C8—91no—
66C4×C3⋊C8—184no—
67C32×Q8—122no—
68C2×C32×C4—24no—
69C2×C32×C4—24no—
70C2×C32×C4—24no—
71C2×C32×C4—62no—
72C2×C32×C4#6714yes—
73C2×C32×C4#6714yes—
74C2×C32×C4#6711yescharacteristic
75C2×C32×C4#6711yescharacteristic
76C2×C32×C4C22×D314yes—
77C2×C32×C4C22×D311yescharacteristic
78C2×C3⋊Dic3—22no—
79#634—24no—
80C2×C3×Dic3—28no—
81C2×C3×Dic3—68no—
82C3×Dic6—48no—
83C3×Dic6—616no—
84C3⋊(C3⋊C8)—616no—
85C3⋊(C3⋊C8)—62no—
86C3⋊(C3⋊C8)—68no—
87#487—271noSylow
88C2×C33#12411yescharacteristic
89C2×C33C2×SD1611yescharacteristic
90C2×C33C2×Q1611yescharacteristic
91C2×C3×Q8—64no—
92C2×C3×Q8—92no—
93C2×Dic6—21no—
94C2×Dic6—38no—
95C3×C4⋊C4—38no—
96C3×C4⋊C4—64no—
97C3×C4⋊C4—181no—
98C3×C4⋊C4—181no—
99C3×C42—24no—
100C3×C42D3214yes—
101C3×C42C2×C3⋊D311yescharacteristic
102C4⋊Dic3—21no—
103C4⋊Dic3—64no—
104C4⋊Dic3—64no—
105C4⋊Dic3C2×C3⋊D312yes—
106C2×C3⋊C8—98no—
107C2×C3⋊C8—92no—
108C2×C3⋊C8—188no—
109C22×C32—24no—
110C22×C32C2×#6714yes—
111C22×C32C2×#6711yescharacteristic
112C32×C4—28no—
113C32×C4—28no—
114C32×C4—28no—
115C32×C4—122no—
116C32×C4#21418yes—
117C32×C4#21512yes—
118C32×C4#21618yes—
119C32×C4#21612yes—
120C32×C4#23712yes—
121C32×C4#23918yes—
122C3⋊Dic3—42no—
123C3×Dic3—48no—
124C3×Dic3—128no—
125C4⋊Q8—31no—
126C4⋊Q8—91no—
127#103—274no—
128C4×C8—271no—
129C33#48711yescharacteristic, complemented, Sylow
130C3×Q8—128no—
131C3×Q8—184no—
132C2×C3×C4—24no—
133C2×C3×C4—24no—
134C2×C3×C4—24no—
135C2×C3×C4—68no—
136C2×C3×C4—182no—
137C2×C3×C4#62514yes—
138C2×C3×C4#62514yes—
139C2×C3×C4#63811yescharacteristic
140C2×C3×C4#63811yescharacteristic
141C2×C3×C4C2×D3214yes—
142C2×C3×C4C22×C3⋊D311yescharacteristic
143C2×Dic3—22no—
144C2×Dic3—68no—
145Dic6—42no—
146Dic6—616no—
147C3⋊C8—1816no—
148C3⋊C8—188no—
149C3⋊C8—182no—
150C2×C32—24no—
151C2×C32—24no—
152C2×C32—24no—
153C2×C32C2×#21414yes—
154C2×C32#93611yescharacteristic
155C2×C32C2×#21511yescharacteristic
156C2×C32C2×#21614yes—
157C2×C32C2×#21611yescharacteristic
158C2×C32#94714yes—
159C2×Q8—61no—
160C2×Q8—92no—
161C2×C8—272no—
162C4⋊C4—32no—
163C4⋊C4—61no—
164C4⋊C4—181no—
165C4⋊C4—181no—
166C42D3×C3⋊D311yescharacteristic
167C22×C3—24no—
168C22×C3C2×#62514yes—
169C22×C3C2×#63811yescharacteristic
170C3×C4—28no—
171C3×C4—28no—
172C3×C4—28no—
173C3×C4—128no—
174C3×C4—362no—
175C3×C4#371518yes—
176C3×C4#371818yes—
177C3×C4#375412yes—
178C3×C4#375612yes—
179C3×C4#379618yes—
180C3×C4#383212yes—
181Dic3—42no—
182Dic3—128no—
183C32—24no—
184C32#507911yescharacteristic, complemented
185C32#509314yescomplemented
186Q8—122no—
187Q8—184no—
188C2×C4—62no—
189C2×C4—182no—
190C2×C4#629011yescharacteristic
191C2×C4#629011yescharacteristic
192C2×C4C2×D3×C3⋊D311yescharacteristic
193C8—542no—
194C2×C3—24no—
195C2×C3—24no—
196C2×C3—24no—
197C2×C3C2×#371514yes—
198C2×C3C2×#371814yes—
199C2×C3#6386314yes—
200C2×C3C2×#375411yescharacteristic
201C2×C3C2×#375611yescharacteristic
202C2×C3#6415611yescharacteristic
203C22C2×#629011yescharacteristic
204C4—122no—
205C4—362no—
206C4#8879212yes—
207C4#8879812yes—
208C4#8901712yes—
209C3—24no—
210C3#1059029014yescomplemented
211C3#1059279211yescharacteristic, complemented
212C2C2×#8879211yescharacteristic
213C2C2×#8879811yescharacteristic
214C2#1172343611yescharacteristic
subgroupG/H
centerC22C2×#6290
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×D3×C3⋊D3
FittingC33×C42C22
socle, prime radicalC22×C33C2×D4
hypercenterC42D3×C3⋊D3
hypocenterC33#487
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C42

Complements

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

NH ≅ G/Ntype
129. C33#487semidirect
184. C32#5079semidirect
185. C32#5093semidirect
210. C3#10590290semidirect
211. C3#10592792semidirect

Maximal subgroups

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

subgroupindex
C3×#640782
C32×#8692
C4×C3⋊(C3⋊(C3⋊C8))2
#117220392
#105927923
#105902903