Finite Group Databasebeta

← #13138881

Subgroups — #13138881

Order 1728.

418 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 418 conjugacy classes of subgroups — too many to draw. Only its 81 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#11721084C212yesmaximal
2C2×#88433C211yesmaximal, characteristic
3#11720548C211yesmaximal, characteristic
4#63643⋊C3C212yesmaximal
5#11720314C211yesmaximal, characteristic
6#10590794—32nomaximal
7C2×#634⋊C3C2212yes—
8C2×C8×H(3,𝔽3)C2211yescharacteristic
9C4×#1084C2211yescharacteristic
10C2×#6180C2211yescharacteristic
11C4⋊((C3⋊Dic3)⋊C3)C2212yes—
12#88448—22no—
13#88447—22no—
14#88433C414yescomplemented
15#63844—32no—
16Dic3×Dic6—32no—
17C2×#3719—32no—
18#63643D312yes—
19C3×#816—32no—
20C3×#815—32no—
21C3×#815—61no—
22#63573—32no—
23C2×C4×H(3,𝔽3)C2311yescharacteristic
24C2×#1084D412yes—
25C2×(C3⋊Dic3)⋊C3—22no—
26C2×(C3⋊Dic3)⋊C3—22no—
27#634⋊C3—22no—
28#634⋊C3C2×C414yes—
29C8×H(3,𝔽3)—21no—
30#6180C2×C412yes—
31#4789—92no—
32#4716—91no—
33C2×#634C2×D312yes—
34C2×C3×Dic6—32no—
35C2×C32×C8—21no—
36C2×C32×C8C2×D312yes—
37C4⋊(C3⋊Dic3)C2×D312yes—
38C3×C4⋊Dic3—32no—
39C3×C4×Dic3—32no—
40C3×C4×Dic3—61no—
41C2×C3×C3⋊C8—32no—
42C2×C3×C3⋊C8—61no—
43#3724—62no—
44Dic3⋊Dic3—62no—
45Dic32—62no—
46Dic32—62no—
47#3719—38no—
48C22×H(3,𝔽3)C2×D411yescharacteristic
49C4×H(3,𝔽3)C22×C411yescharacteristic
50C4×H(3,𝔽3)#4111yescharacteristic
51#1084—22no—
52(C3⋊Dic3)⋊C3—24no—
53(C3⋊Dic3)⋊C3—42no—
54Q8×Dic3—92no—
55C2×#216—92no—
56C2×Dic12—91no—
57#887—92no—
58C4×Dic6—92no—
59C3×#103—92no—
60C3×#97—91no—
61#817—91no—
62#816—32no—
63#816—92no—
64#815—32no—
65#815—61no—
66#808—92no—
67C2×C32×C4—21no—
68C2×C32×C4C22×D312yes—
69C2×C3⋊Dic3—22no—
70C2×C3⋊Dic3—22no—
71#634—22no—
72#634C4×D314yes—
73C2×C3×Dic3—32no—
74C2×C3×Dic3—32no—
75C2×C3×Dic3—62no—
76C2×C3×Dic3—62no—
77C2×C3×Dic3—62no—
78C3×Dic6—34no—
79C3×Dic6—62no—
80C32×C8—22no—
81C32×C8—22no—
82C3×C3⋊C8—34no—
83C3×C3⋊C8—62no—
84#441—271noSylow
85C2×H(3,𝔽3)C4×D411yescharacteristic
86C2×H(3,𝔽3)#13712yes—
87C2×C3×Q8—92no—
88C2×Dic6—32no—
89C2×Dic6—92no—
90C2×C3×C8—32no—
91C2×C3×C8—61no—
92C2×C3×C8—91no—
93C2×C3×C8D3211yescharacteristic
94C3×C4⋊C4—92no—
95C3×C42—91no—
96#216—98no—
97C4⋊Dic3—32no—
98C4⋊Dic3—92no—
99C4⋊Dic3—182no—
100Dic3⋊C4—182no—
101Dic3⋊C4—182no—
102C4×Dic3—32no—
103C4×Dic3—61no—
104C4×Dic3—182no—
105C4×Dic3—182no—
106C4×Dic3—182no—
107C2×C3⋊C8—32no—
108C2×C3⋊C8—61no—
109Dic12—94no—
110C22×C32—21no—
111C22×C32D3×D412yes—
112C32×C4—21no—
113C32×C4—21no—
114C32×C4C2×C4×D312yes—
115C32×C4#23712yes—
116C3⋊Dic3—24no—
117C3⋊Dic3—42no—
118C3×Dic3—64no—
119C3×Dic3—62no—
120C3×Dic3—62no—
121C3×Dic3—122no—
122C3×Dic3—122no—
123C2×Q16—271no—
124C4×Q8—272no—
125#106—271no—
126#103—92no—
127#97—91no—
128H(3,𝔽3)#44111yescharacteristic, complemented, Sylow
129C3×Q8—94no—
130C3×Q8—182no—
131C2×C3×C4—32no—
132C2×C3×C4—61no—
133C2×C3×C4—92no—
134C2×C3×C4—182no—
135C2×C3×C4—182no—
136C2×C3×C4C2×D3211yescharacteristic
137C2×Dic3—32no—
138C2×Dic3—32no—
139C2×Dic3—62no—
140C2×Dic3—62no—
141C2×Dic3—62no—
142C2×Dic3—182no—
143C2×Dic3—182no—
144Dic6—34no—
145Dic6—62no—
146Dic6—94no—
147Dic6—182no—
148C3×C8—21no—
149C3×C8—62no—
150C3×C8—62no—
151C3×C8—92no—
152C3⋊C8—34no—
153C3⋊C8—62no—
154C2×C32—22no—
155C2×C32—21no—
156C2×C32#67⋊C412yes—
157C2×C32#91512yes—
158C2×C32#91812yes—
159C2×Q8—92no—
160Q16—274no—
161C2×C8—91no—
162C2×C8((C3⋊D3)⋊C3)⋊C211yescharacteristic
163C4⋊C4—92no—
164C4⋊C4—542no—
165C42—91no—
166C42—542no—
167C22×C3—32no—
168C22×C3—61no—
169C22×C3#380211yescharacteristic
170C3×C4—32no—
171C3×C4—32no—
172C3×C4—61no—
173C3×C4—61no—
174C3×C4—184no—
175C3×C4—182no—
176C3×C4—362no—
177C3×C4C4×D3211yescharacteristic
178C3×C4#380511yescharacteristic
179Dic3—62no—
180Dic3—62no—
181Dic3—64no—
182Dic3—122no—
183Dic3—122no—
184Dic3—184no—
185Dic3—362no—
186C32—21no—
187C32#478912yescomplemented
188Q8—94no—
189Q8—182no—
190C2×C4—92no—
191C2×C4—182no—
192C2×C4—182no—
193C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
194C8—21no—
195C8—92no—
196C2×C3—32no—
197C2×C3—32no—
198C2×C3—32no—
199C2×C3—62no—
200C2×C3—61no—
201C2×C3#6381112yes—
202C2×C3#6390211yescharacteristic
203C22#8865511yescharacteristic
204C4—182no—
205C4—184no—
206C4—362no—
207C4C4×((C3⋊D3)⋊C3)⋊C211yescharacteristic
208C4#8867211yescharacteristic
209C3—32no—
210C3—61no—
211C3#1059079111yescharacteristic
212C2#1172084512yes—
213C2#1172113811yescharacteristic
subgroupG/H
centerC22#88655
commutatorC4×H(3,𝔽3)C22×C4
FrattiniC2×C3×C4C2×D32
FittingC2×C8×H(3,𝔽3)C22
socleC22×C3#3802
prime radicalC22×H(3,𝔽3)C2×D4
cosocleC2×C4×H(3,𝔽3)C23
hypercenterC2×C8((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#441
LCSG ▷ 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
14. #88433C4semidirect
128. H(3,𝔽3)#441semidirect
187. C32#4789semidirect

Maximal subgroups

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

subgroupindex
#117210842
C2×#884332
#117205482
#63643⋊C32
#117203142
#105907943