Finite Group Databasebeta

← #13138762

Subgroups — #13138762

Order 1728.

355 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 355 conjugacy classes of subgroups — too many to draw. Only its 67 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
1#11721099C211yesmaximal, characteristic
2#11721095C211yesmaximal, characteristic
3#11720548C211yesmaximal, characteristic
4#63643⋊C3C212yesmaximal
5#11720313C212yesmaximal
6#10590898—32nomaximal
7C2×#634⋊C3C2212yes—
8C2×C8×H(3,𝔽3)C2211yescharacteristic
9C4×#1084C2211yescharacteristic
10C2×#6180C2211yescharacteristic
11C4⋊((C3⋊Dic3)⋊C3)C2212yes—
12#88449—22no—
13#88448—22no—
14#63869—32no—
15#63866—32no—
16#63643D312yes—
17C3×#816—32no—
18C3×#815—32no—
19C3×#815—61no—
20#63567—32no—
21#63566—32no—
22C2×C4×H(3,𝔽3)C2311yescharacteristic
23C2×#1084D412yes—
24C2×(C3⋊Dic3)⋊C3—22no—
25C2×(C3⋊Dic3)⋊C3—22no—
26#634⋊C3—22no—
27#634⋊C3—22no—
28C8×H(3,𝔽3)—21no—
29#6180—21no—
30#4797—92no—
31#4703—91no—
32C2×#634C2×D312yes—
33C2×C3×Dic6—32no—
34C2×C32×C8—21no—
35C2×C32×C8C2×D312yes—
36C4⋊(C3⋊Dic3)C2×D312yes—
37C3×C4⋊Dic3—32no—
38C3×C4×Dic3—32no—
39C3×C4×Dic3—61no—
40C2×C3×C3⋊C8—32no—
41C2×C3×C3⋊C8—61no—
42#3724—62no—
43Dic3⋊Dic3—62no—
44Dic3⋊Dic3—62no—
45Dic3⋊Dic3—62no—
46C22×H(3,𝔽3)C2×D411yescharacteristic
47C4×H(3,𝔽3)#4111yescharacteristic
48C4×H(3,𝔽3)#4111yescharacteristic
49#1084—22no—
50(C3⋊Dic3)⋊C3—42no—
51(C3⋊Dic3)⋊C3—42no—
52#944—92no—
53#890—92no—
54#870—91no—
55#869—91no—
56C3×#103—92no—
57C3×#97—91no—
58#835—92no—
59#816—32no—
60#816—92no—
61#816—92no—
62#815—32no—
63#815—61no—
64#810—92no—
65C2×C32×C4—21no—
66C2×C32×C4C22×D312yes—
67C2×C3⋊Dic3—22no—
68C2×C3⋊Dic3—22no—
69#634—22no—
70#634—22no—
71C2×C3×Dic3—32no—
72C2×C3×Dic3—32no—
73C2×C3×Dic3—62no—
74C2×C3×Dic3—62no—
75C2×C3×Dic3—62no—
76C3×Dic6—62no—
77C3×Dic6—62no—
78C32×C8—22no—
79C32×C8—22no—
80C3×C3⋊C8—62no—
81C3×C3⋊C8—62no—
82#491—271noSylow
83C2×H(3,𝔽3)#12411yescharacteristic
84C2×H(3,𝔽3)#13712yes—
85C2×C3×Q8—92no—
86C2×Dic6—32no—
87C2×Dic6—92no—
88C2×C3×C8—32no—
89C2×C3×C8—61no—
90C2×C3×C8—91no—
91C2×C3×C8D3211yescharacteristic
92C3×C4⋊C4—92no—
93C3×C42—91no—
94C4⋊Dic3—32no—
95C4⋊Dic3—92no—
96C4⋊Dic3—182no—
97C4⋊Dic3—182no—
98Dic3⋊C4—182no—
99Dic3⋊C4—182no—
100Dic3⋊C4—182no—
101Dic3⋊C4—182no—
102C4×Dic3—32no—
103C4×Dic3—61no—
104C2×C3⋊C8—32no—
105C2×C3⋊C8—61no—
106C22×C32—21no—
107C22×C32D3×D412yes—
108C32×C4—21no—
109C32×C4—21no—
110C32×C4#23512yes—
111C32×C4#23712yes—
112C3⋊Dic3—42no—
113C3⋊Dic3—42no—
114C3×Dic3—62no—
115C3×Dic3—62no—
116C3×Dic3—122no—
117C3×Dic3—122no—
118C3×Dic3—122no—
119C4⋊Q8—271no—
120#125—271no—
121#103—92no—
122#103—272no—
123#97—91no—
124H(3,𝔽3)#49111yescharacteristic, complemented, Sylow
125C3×Q8—182no—
126C3×Q8—182no—
127C2×C3×C4—32no—
128C2×C3×C4—61no—
129C2×C3×C4—92no—
130C2×C3×C4—182no—
131C2×C3×C4—182no—
132C2×C3×C4C2×D3211yescharacteristic
133C2×Dic3—32no—
134C2×Dic3—32no—
135C2×Dic3—62no—
136C2×Dic3—62no—
137C2×Dic3—62no—
138C2×Dic3—182no—
139C2×Dic3—182no—
140Dic6—62no—
141Dic6—62no—
142Dic6—182no—
143Dic6—182no—
144C3×C8—21no—
145C3×C8—62no—
146C3×C8—62no—
147C3×C8—181no—
148C3⋊C8—62no—
149C3⋊C8—62no—
150C2×C32—22no—
151C2×C32—21no—
152C2×C32#88512yes—
153C2×C32#91512yes—
154C2×C32#91812yes—
155C2×Q8—92no—
156C2×C8—91no—
157C2×C8((C3⋊D3)⋊C3)⋊C211yescharacteristic
158C4⋊C4—92no—
159C4⋊C4—542no—
160C4⋊C4—542no—
161C42—91no—
162C22×C3—32no—
163C22×C3—61no—
164C22×C3#380211yescharacteristic
165C3×C4—32no—
166C3×C4—32no—
167C3×C4—61no—
168C3×C4—61no—
169C3×C4—182no—
170C3×C4—362no—
171C3×C4—362no—
172C3×C4#379811yescharacteristic
173C3×C4#380511yescharacteristic
174Dic3—62no—
175Dic3—62no—
176Dic3—122no—
177Dic3—122no—
178Dic3—122no—
179Dic3—362no—
180Dic3—362no—
181C32—21no—
182C32#479712yescomplemented
183Q8—182no—
184Q8—182no—
185C2×C4—92no—
186C2×C4—182no—
187C2×C4—182no—
188C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
189C8—21no—
190C8—181no—
191C2×C3—32no—
192C2×C3—32no—
193C2×C3—32no—
194C2×C3—62no—
195C2×C3—61no—
196C2×C3#6381112yes—
197C2×C3#6391511yescharacteristic
198C22#8865511yescharacteristic
199C4—182no—
200C4—362no—
201C4—362no—
202C4#8865011yescharacteristic
203C4#8867211yescharacteristic
204C3—32no—
205C3—61no—
206C3#1059090411yescharacteristic
207C2#1172084512yes—
208C2#1172114511yescharacteristic
subgroupG/H
centerC22#88655
commutator, cosocleC2×C4×H(3,𝔽3)C23
FrattiniC2×C3×C4C2×D32
FittingC2×C8×H(3,𝔽3)C22
socleC22×C3#3802
prime radicalC22×H(3,𝔽3)C2×D4
hypercenterC2×C8((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#491
LCSG ▷ C2×C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×C8

Complements

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

NH ≅ G/Ntype
124. H(3,𝔽3)#491semidirect
182. C32#4797semidirect

Maximal subgroups

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

subgroupindex
#117210992
#117210952
#117205482
#63643⋊C32
#117203132
#105908983