Finite Group Databasebeta

← #13139975

Subgroups — #13139975

Order 1728.

475 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 475 conjugacy classes of subgroups — too many to draw. Only its 123 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
1C22×#6180C211yesmaximal, characteristic
2#64135⋊C3C212yesmaximal
3#11720315C212yesmaximal, complemented
4#11720314C212yesmaximal, complemented
5#10591122—32nomaximal
6C22×C4×H(3,𝔽3)C2211yescharacteristic
7C2×#6180C414yescomplemented
8C2×#6180C2211yescharacteristic
9C2×#6180C2211yescharacteristic
10#3757⋊C3—22no—
11C4⋊((C3⋊Dic3)⋊C3)C2214yes—
12C4×(C3⋊Dic3)⋊C3—22no—
13#64135D312yes—
14C3×#926—32no—
15C22×C3×C3⋊C8—32no—
16C22×C3×C3⋊C8—61no—
17#63575—34no—
18#63573—34no—
19C23×H(3,𝔽3)D411yescharacteristic
20C2×C4×H(3,𝔽3)C2×C412yes—
21C2×C4×H(3,𝔽3)D411yescharacteristic
22C2×C4×H(3,𝔽3)Q812yes—
23C2×C4×H(3,𝔽3)C2311yescharacteristic
24C2×(C3⋊Dic3)⋊C3—24no—
25#6180C2×C412yes—
26#6180C2×C412yes—
27#5106—91no—
28#4998—92no—
29C22×C32×C4—21no—
30C22×C32×C4C2×D312yes—
31#3757—22no—
32C4⋊(C3⋊Dic3)C2×D314yes—
33C4×C3⋊Dic3—22no—
34C3×#217—62no—
35C3×C4⋊Dic3—34no—
36C3×C4×Dic3—62no—
37C2×C3×C3⋊C8—38no—
38C2×C3×C3⋊C8—32no—
39C2×C3×C3⋊C8—32no—
40C2×C3×C3⋊C8—64no—
41C2×C3×C3⋊C8—61no—
42C2×C3×C3⋊C8—61no—
43C22×H(3,𝔽3)—22no—
44C22×H(3,𝔽3)C4⋊C412yes—
45C22×H(3,𝔽3)C2×D411yescharacteristic
46C4×H(3,𝔽3)C4⋊C412yes—
47C4×H(3,𝔽3)C22×C411yescharacteristic
48C4×H(3,𝔽3)C2×Q811yescharacteristic
49(C3⋊Dic3)⋊C3—44no—
50C22×C3×C8—91no—
51C3×#117—92no—
52#926—32no—
53#926—92no—
54C22×C3⋊C8—32no—
55C22×C3⋊C8—61no—
56#818—92no—
57#817—92no—
58#808—94no—
59#807—94no—
60C23×C32—21no—
61C23×C32#6712yes—
62C2×C32×C4—22no—
63C2×C32×C4—21no—
64C2×C32×C4—22no—
65C2×C32×C4—21no—
66C2×C32×C4Dic614yes—
67C2×C32×C4C4×D314yes—
68C2×C32×C4#6712yes—
69C2×C32×C4C22×D312yes—
70C2×C3⋊Dic3—24no—
71C2×C3×Dic3—64no—
72C3×C3⋊C8—34no—
73C3×C3⋊C8—34no—
74C3×C3⋊C8—62no—
75C3×C3⋊C8—62no—
76#427—271noSylow
77C2×H(3,𝔽3)—22no—
78C2×H(3,𝔽3)C2×C4⋊C411yescharacteristic
79C2×H(3,𝔽3)#13512yes—
80C22×C3×C4—32no—
81C22×C3×C4—61no—
82C22×C3×C4D3211yescharacteristic
83C2×C3×C8—94no—
84C2×C3×C8—91no—
85C2×C3×C8—91no—
86C3×C4⋊C4—94no—
87C3×C22⋊C4—182no—
88C3×C42—182no—
89#217—62no—
90#217—182no—
91C4⋊Dic3—34no—
92C4⋊Dic3—94no—
93C4×Dic3—62no—
94C4×Dic3—182no—
95C2×C3⋊C8—32no—
96C2×C3⋊C8—32no—
97C2×C3⋊C8—38no—
98C2×C3⋊C8—64no—
99C2×C3⋊C8—61no—
100C2×C3⋊C8—61no—
101C22×C32—24no—
102C22×C32—24no—
103C22×C32—22no—
104C22×C32—21no—
105C22×C32Dic3⋊C414yes—
106C22×C32C2×#6712yes—
107C32×C4—22no—
108C32×C4—21no—
109C32×C4—21no—
110C32×C4Dic3⋊C414yes—
111C32×C4C2×Dic612yes—
112C32×C4C2×C4×D312yes—
113C3⋊Dic3—44no—
114C3×Dic3—124no—
115C22×C8—91no—
116#117—92no—
117#107—272no—
118#106—272no—
119H(3,𝔽3)#42711yescharacteristic, complemented, Sylow
120C23×C3—32no—
121C23×C3—61no—
122C23×C3#62511yescharacteristic
123C2×C3×C4—34no—
124C2×C3×C4—32no—
125C2×C3×C4—34no—
126C2×C3×C4—32no—
127C2×C3×C4—62no—
128C2×C3×C4—61no—
129C2×C3×C4—62no—
130C2×C3×C4—61no—
131C2×C3×C4—184no—
132C2×C3×C4#62412yes—
133C2×C3×C4#62511yescharacteristic
134C2×C3×C4#62712yes—
135C2×C3×C4C2×D3211yescharacteristic
136C2×Dic3—64no—
137C2×Dic3—184no—
138C3×C8—92no—
139C3×C8—92no—
140C3⋊C8—34no—
141C3⋊C8—34no—
142C3⋊C8—62no—
143C3⋊C8—62no—
144C2×C32—24no—
145C2×C32—24no—
146C2×C32—22no—
147C2×C32—21no—
148C2×C32C2×Dic3⋊C412yes—
149C2×C32#95014yes—
150C22×C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
151C2×C8—94no—
152C2×C8—91no—
153C2×C8—91no—
154C4⋊C4—94no—
155C22⋊C4—182no—
156C42—182no—
157C22×C3—22no—
158C22×C3—32no—
159C22×C3—34no—
160C22×C3—64no—
161C22×C3—64no—
162C22×C3—61no—
163C22×C3—62no—
164C22×C3#372412yes—
165C22×C3C2×#62511yescharacteristic
166C3×C4—32no—
167C3×C4—32no—
168C3×C4—34no—
169C3×C4—61no—
170C3×C4—61no—
171C3×C4—62no—
172C3×C4—364no—
173C3×C4#372412yes—
174C3×C4C2×#62411yescharacteristic
175C3×C4C2×#62711yescharacteristic
176Dic3—124no—
177Dic3—364no—
178C32—21no—
179C32#499812yescomplemented
180C23#619811yescharacteristic
181C2×C4—184no—
182C2×C4#619612yes—
183C2×C4#619811yescharacteristic
184C2×C4#619912yes—
185C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
186C8—92no—
187C8—92no—
188C2×C3—22no—
189C2×C3—32no—
190C2×C3—34no—
191C2×C3—64no—
192C2×C3—64no—
193C2×C3—61no—
194C2×C3—62no—
195C2×C3#6382212yes—
196C2×C3C2×#372411yescharacteristic
197C22—22no—
198C22#8844912yes—
199C22C2×#619811yescharacteristic
200C4—364no—
201C4#8844912yes—
202C4C2×#619611yescharacteristic
203C4C2×#619911yescharacteristic
204C3—32no—
205C3—61no—
206C3#1059113811yescharacteristic
207C2—22no—
208C2#1172106012yes—
209C2C2×#8844911yescharacteristic
subgroupG/H
centerC2×C4#6198
commutatorC4×H(3,𝔽3)C22×C4
FrattiniC2×C3×C4C2×D32
FittingC22×C4×H(3,𝔽3)C22
socleC22×C3C2×#625
prime radicalC23×H(3,𝔽3)D4
cosocleC2×C4×H(3,𝔽3)C23
hypercenterC22×C4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#427
LCSG ▷ C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C2×C4 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
3. #11720315C2semidirect
4. #11720314C2semidirect
7. C2×#6180C4semidirect
119. H(3,𝔽3)#427semidirect
179. C32#4998semidirect

Maximal subgroups

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

subgroupindex
C22×#61802
#64135⋊C32
#117203152
#117203142
#105911223