Finite Group Databasebeta

← #13137440

Subgroups — #13137440

Order 1728.

495 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 495 conjugacy classes of subgroups — too many to draw. Only its 103 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#11721715C211yesmaximal, characteristic
2#64078⋊C3C211yesmaximal, characteristic
3C2×#88439C214yesmaximal
4C4×(C3⋊(C3⋊C8))⋊C3C211yesmaximal, characteristic
5#10590387—31nomaximal
6#10590386—31nomaximal
7C2×#6229C2212yes—
8C2×#634⋊C3C2212yes—
9C42×H(3,𝔽3)C2211yescharacteristic
10C4⋊#1084—22no—
11C4⋊((C3⋊Dic3)⋊C3)—22no—
12C2×(C3⋊(C3⋊C8))⋊C3C2212yes—
13#88439—28no—
14#64078D311yescharacteristic
15C3×#869—31no—
16C3×#869—31no—
17C3×#869—31no—
18C3×#869—61no—
19C2×#3719—34no—
20C2×#3718—34no—
21C4×C3⋊(C3⋊C8)D311yescharacteristic
22C3×C4×C3⋊C8—31no—
23C2×C4×H(3,𝔽3)D412yes—
24C2×C4×H(3,𝔽3)C2311yescharacteristic
25C2×#1084—22no—
26#6229—24no—
27C2×(C3⋊Dic3)⋊C3—22no—
28#634⋊C3—24no—
29(C3⋊(C3⋊C8))⋊C3D414yes—
30#5092—91no—
31#5092—91no—
32#4699—91no—
33C2×#634C2×D312yes—
34C2×C3×Dic6—32no—
35C2×C3×Dic6—32no—
36C2×C3×Dic6—32no—
37C2×C3×Dic6—62no—
38C32×C42—21no—
39C32×C42C2×D311yescharacteristic
40C32×C42C2×D311yescharacteristic
41C4⋊(C3⋊Dic3)—22no—
42C2×C3⋊(C3⋊C8)C2×D312yes—
43C3×C4⋊Dic3—64no—
44C3×C4⋊Dic3—62no—
45C3×C4⋊Dic3—62no—
46C3×C4⋊Dic3—62no—
47C2×C3×C3⋊C8—32no—
48#3719—68no—
49#3718—68no—
50C22×H(3,𝔽3)C2×D411yescharacteristic
51C4×H(3,𝔽3)Q1614yescomplemented
52C4×H(3,𝔽3)C2×D412yes—
53#1084—42no—
54(C3⋊Dic3)⋊C3—42no—
55C3×C4⋊Q8—91no—
56C3×C4⋊Q8—91no—
57C2×#216—94no—
58C2×#216—94no—
59C2×Dic12—94no—
60#869—31no—
61#869—31no—
62#869—31no—
63#869—61no—
64#869—91no—
65C3×C4×C8—91no—
66C4×C3⋊C8—31no—
67C4×C3⋊C8—91no—
68C2×C32×C4—22no—
69C2×C32×C4—21no—
70C2×C32×C4D1212yes—
71C2×C32×C4#6712yes—
72C2×C32×C4C22×D311yescharacteristic
73C2×C32×C4C22×D311yescharacteristic
74C2×C3⋊Dic3—22no—
75#634—24no—
76C2×C3×Dic3—64no—
77C2×C3×Dic3—62no—
78C2×C3×Dic3—62no—
79C2×C3×Dic3—62no—
80C3×Dic6—64no—
81C3×Dic6—64no—
82C3×Dic6—64no—
83C3×Dic6—124no—
84C3⋊(C3⋊C8)D1214yes—
85C3×C3⋊C8—34no—
86#494—271noSylow
87C2×H(3,𝔽3)#12711yescharacteristic
88C2×H(3,𝔽3)C2×Q1612yes—
89C2×C3×Q8—92no—
90C2×C3×Q8—92no—
91C2×Dic6—32no—
92C2×Dic6—32no—
93C2×Dic6—32no—
94C2×Dic6—62no—
95C2×Dic6—92no—
96C2×C3×C8—92no—
97C3×C4⋊C4—182no—
98C3×C4⋊C4—182no—
99C3×C42—31no—
100C3×C42—31no—
101C3×C42—61no—
102C3×C42D3211yescharacteristic
103#216—188no—
104#216—188no—
105C4⋊Dic3—62no—
106C4⋊Dic3—62no—
107C4⋊Dic3—64no—
108C4⋊Dic3—62no—
109C4⋊Dic3—182no—
110C2×C3⋊C8—32no—
111C2×C3⋊C8—92no—
112Dic12—188no—
113C22×C32—21no—
114C22×C32C2×D1211yescharacteristic
115C22×C32C2×#6711yescharacteristic
116C32×C4—24no—
117C32×C4—22no—
118C32×C4Dic1214yescomplemented
119C32×C4#21614yescomplemented
120C32×C4C2×D1212yes—
121C32×C4D3×D412yes—
122C3⋊Dic3—42no—
123C3×Dic3—124no—
124C3×Dic3—122no—
125C3×Dic3—122no—
126C3×Dic3—122no—
127C2×Q16—274no—
128C4⋊Q8—91no—
129C4⋊Q8—91no—
130C4×C8—91no—
131H(3,𝔽3)#49411yescharacteristic, complemented, Sylow
132C3×Q8—184no—
133C3×Q8—184no—
134C2×C3×C4—32no—
135C2×C3×C4—32no—
136C2×C3×C4—31no—
137C2×C3×C4—31no—
138C2×C3×C4—62no—
139C2×C3×C4—61no—
140C2×C3×C4—182no—
141C2×C3×C4—182no—
142C2×C3×C4#62612yes—
143C2×C3×C4C2×D3211yescharacteristic
144C2×Dic3—62no—
145C2×Dic3—62no—
146C2×Dic3—64no—
147C2×Dic3—62no—
148C2×Dic3—182no—
149Dic6—64no—
150Dic6—64no—
151Dic6—64no—
152Dic6—124no—
153Dic6—184no—
154C3×C8—94no—
155C3⋊C8—34no—
156C3⋊C8—94no—
157C2×C32—22no—
158C2×C32—21no—
159C2×C32#87411yescharacteristic
160C2×C32C2×Dic1212yes—
161C2×C32#94011yescharacteristic
162C2×C32C2×#21612yes—
163C2×Q8—92no—
164C2×Q8—92no—
165Q16—548no—
166C2×C8—92no—
167C4⋊C4—182no—
168C4⋊C4—182no—
169C42((C3⋊D3)⋊C3)⋊C211yescharacteristic
170C22×C3—31no—
171C22×C3—31no—
172C22×C3—61no—
173C22×C3C2×#62611yescharacteristic
174C3×C4—32no—
175C3×C4—32no—
176C3×C4—34no—
177C3×C4—34no—
178C3×C4—62no—
179C3×C4—64no—
180C3×C4—362no—
181C3×C4—362no—
182C3×C4#371914yes—
183C3×C4D3×D1212yes—
184Dic3—122no—
185Dic3—122no—
186Dic3—124no—
187Dic3—122no—
188Dic3—362no—
189C32—21no—
190C32#469911yescharacteristic, complemented
191C32#509211yescharacteristic, complemented
192Q8—184no—
193Q8—184no—
194C2×C4—182no—
195C2×C4—182no—
196C2×C4#620012yes—
197C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
198C8—94no—
199C2×C3—32no—
200C2×C3—32no—
201C2×C3—31no—
202C2×C3—31no—
203C2×C3—62no—
204C2×C3—61no—
205C2×C3C2×#371912yes—
206C2×C3#6391111yescharacteristic
207C22C2×#620011yescharacteristic
208C4—362no—
209C4—362no—
210C4#8843914yescomplemented
211C4#8865412yes—
212C3—31no—
213C3—31no—
214C3—61no—
215C3#1059038611yescharacteristic
216C2C2×#8843912yes—
217C2#1172115811yescharacteristic
subgroupG/H
centerC22C2×#6200
commutator, cosocleC2×C4×H(3,𝔽3)C23
FrattiniC2×C3×C4C2×D32
FittingC42×H(3,𝔽3)C22
socleC22×C3C2×#626
prime radicalC22×H(3,𝔽3)C2×D4
hypercenterC42((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#494
LCSG ▷ C2×C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C42

Complements

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

NH ≅ G/Ntype
51. C4×H(3,𝔽3)Q16semidirect
118. C32×C4Dic12semidirect
119. C32×C4#216semidirect
131. H(3,𝔽3)#494semidirect
190. C32#4699semidirect
191. C32#5092semidirect
210. C4#88439semidirect

Maximal subgroups

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

subgroupindex
#117217152
#64078⋊C32
C2×#884392
C4×(C3⋊(C3⋊C8))⋊C32
#105903873
#105903863