Finite Group Databasebeta

← #13147334

Subgroups — #13147334

Order 1728.

347 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 347 conjugacy classes of subgroups — too many to draw. Only its 79 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
1C2×#88627C211yesmaximal, characteristic
2C2×#88626C211yesmaximal, characteristic
3#11721760C211yesmaximal, characteristic
4#11720783C212yesmaximal, complemented
5C4×#6188C212yesmaximal, complemented
6C2×C4×#272C2211yescharacteristic
7C2×#6188C2211yescharacteristic
8C2×#6188C2212yes—
9C2×#6188C2211yescharacteristic
10#88627C414yescomplemented
11#88626C414yescomplemented
12C4⋊C4×H(3,𝔽3)C411yescharacteristic
13#88542C412yes—
14C4⋊#1084C411yescharacteristic
15C4×#1084C2212yes—
16C3×#892—62no—
17C22×#272C2×C411yescharacteristic
18C2×C4×H(3,𝔽3)C2×C412yes—
19C2×C4×H(3,𝔽3)C2×C411yescharacteristic
20C2×#1084C2×C412yes—
21C2×#1084C2311yescharacteristic
22C4×#272C2×C412yes—
23C4×#272C2×C412yes—
24#6188C2×C412yes—
25#6188C2×C412yes—
26#6188C2×C414yes—
27C3×#405—91nomaximal
28C2×C3×C4×D3—62no—
29C32×C4⋊C4—22no—
30C3×(C2×D3)⋊C4—64no—
31C3×C4⋊Dic3—62no—
32C3×C4×Dic3—64no—
33C22×H(3,𝔽3)C22×C411yescharacteristic
34C2×#272—21no—
35C2×#272—21no—
36C2×#272C4212yes—
37C4×H(3,𝔽3)—22no—
38C4×H(3,𝔽3)C4212yes—
39#1084—22no—
40#1084C22×C411yescharacteristic
41#1084C22×C411yescharacteristic
42C2×C3×M16—91no—
43C22×C3×C8—91no—
44C3×#117—91no—
45#892—62no—
46C3×#97—92no—
47C3×C4×C8—92no—
48C22×C3×D3—62no—
49C2×C32×C4—22no—
50C2×C32×C4—24no—
51C2×C3×Dic3—62no—
52C2×C3×Dic3—64no—
53C3×C4×D3—64no—
54C3×C4×D3—64no—
55#405—91noSylow
56C2×H(3,𝔽3)C2×C4211yescharacteristic
57C2×H(3,𝔽3)#13111yescharacteristic
58C2×H(3,𝔽3)#13111yescharacteristic
59#272—22no—
60C22×C3×C4—91no—
61C2×C4×D3—62no—
62C3×M16—94no—
63C2×C3×C8—94no—
64C2×C3×C8—91no—
65C2×C3×C8—92no—
66C2×C3×C8—91no—
67C3×C4⋊C4—62no—
68C3×C4⋊C4—91no—
69C3×C4⋊C4#15711yescharacteristic
70C3×C22⋊C4—92no—
71C3×C42—92no—
72(C2×D3)⋊C4—64no—
73C4⋊Dic3—62no—
74C4×Dic3—64no—
75C22×C32—22no—
76C2×C3×D3—64no—
77C2×C3×D3—122no—
78C2×C3×D3—122no—
79C32×C4—24no—
80C32×C4—44no—
81C3×Dic3—62no—
82C3×Dic3—62no—
83C3×Dic3—124no—
84C2×M16—91no—
85C22×C8—91no—
86#117—91no—
87#97—92no—
88C4×C8—92no—
89H(3,𝔽3)#40511yescharacteristic, complemented, Sylow
90C23×C3—91no—
91C22×D3—62no—
92C2×C3×C4—62no—
93C2×C3×C4—64no—
94C2×C3×C4—91no—
95C2×C3×C4—92no—
96C2×C3×C4—92no—
97C2×C3×C4—92no—
98C2×C3×C4C2×#15712yes—
99C2×C3×C4C2×#15711yescharacteristic
100C2×Dic3—62no—
101C2×Dic3—64no—
102C4×D3—64no—
103C4×D3—64no—
104C3×C8—92no—
105C3×C8—92no—
106C3×C8—94no—
107C2×C32—22no—
108C2×C32—22no—
109C2×C32—22no—
110C3×D3—124no—
111C22×C4—91no—
112M16—94no—
113C2×C8—91no—
114C2×C8—92no—
115C2×C8—91no—
116C2×C8—94no—
117C4⋊C4—91no—
118C4⋊C4#108811yescharacteristic
119C22⋊C4—92no—
120C42—92no—
121C22×C3—62no—
122C22×C3—92no—
123C22×C3—181no—
124C22×C3—181no—
125C22×C3C22×#15711yescharacteristic
126C2×D3—64no—
127C2×D3—122no—
128C2×D3—122no—
129C3×C4—22no—
130C3×C4—64no—
131C3×C4—91no—
132C3×C4—91no—
133C3×C4—124no—
134C3×C4—182no—
135C3×C4C4×#15712yes—
136Dic3—62no—
137Dic3—62no—
138Dic3—124no—
139C32—22no—
140C23—91no—
141C2×C4—91no—
142C2×C4—92no—
143C2×C4—92no—
144C2×C4—92no—
145C2×C4C2×#108812yes—
146C2×C4C2×#108811yescharacteristic
147C8—92no—
148C8—92no—
149C8—94no—
150C2×C3—62no—
151C2×C3—62no—
152C2×C3—62no—
153C2×C3—182no—
154C2×C3C2×C4×#15711yescharacteristic
155C2×C3#6428711yescharacteristic
156C2×C3#6428911yescharacteristic
157D3—124no—
158C22—92no—
159C22—181no—
160C22—181no—
161C22C22×#108811yescharacteristic
162C4—22no—
163C4—91no—
164C4—91no—
165C4—182no—
166C4C4×#108812yes—
167C3—62no—
168C3#1059374811yescharacteristic
169C2—182no—
170C2C2×C4×#108811yescharacteristic
171C2#1172244911yescharacteristic
172C2#1172245111yescharacteristic
subgroupG/H
center, Frattini, socleC22×C3C22×#157
commutatorC2×H(3,𝔽3)C2×C42
FittingC4⋊C4×H(3,𝔽3)C4
prime radicalC22×#272C2×C4
cosocleC2×#1084C23
hypercenterC3×C4⋊C4#157
hypocenterH(3,𝔽3)#405
LCSG ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22×C3 ◁ C3×C4⋊C4

Complements

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

NH ≅ G/Ntype
4. #11720783C2semidirect
5. C4×#6188C2semidirect
10. #88627C4semidirect
11. #88626C4semidirect
89. H(3,𝔽3)#405semidirect

Maximal subgroups

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

subgroupindex
C2×#886272
C2×#886262
#117217602
#117207832
C4×#61882
C3×#4059