Finite Group Databasebeta

← #13138759

Subgroups — #13138759

Order 1728.

471 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 471 conjugacy classes of subgroups — too many to draw. Only its 83 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×#3746⋊C3C212yesmaximal
2#11721099C212yesmaximal
3C2×#88433C212yesmaximal
4C8×#1084C211yesmaximal, characteristic
5#10590896—32nomaximal
6C2×#634⋊C3C2214yes—
7C2×C8×H(3,𝔽3)C2211yescharacteristic
8C4×#1084C2211yescharacteristic
9C2×#6180C2211yescharacteristic
10#3746⋊C3—24no—
11#88449—24no—
12#88433—24no—
13C2×#3746D312yes—
14C2×C3×Dic12—32no—
15#63866—34no—
16C2×#3719—34no—
17C3×C8×Dic3—32no—
18C3×C8×Dic3—61no—
19C2×C4×H(3,𝔽3)C2311yescharacteristic
20C2×#1084D412yes—
21C2×(C3⋊Dic3)⋊C3—24no—
22#634⋊C3—24no—
23#634⋊C3—24no—
24C8×H(3,𝔽3)D412yes—
25#6180D412yes—
26#5183—92no—
27#4699—91no—
28C2×#634C2×D314yes—
29C2×C3×Dic6—34no—
30C2×C32×C8—21no—
31C2×C32×C8C2×D312yes—
32#3746—24no—
33C3×C4×Dic3—32no—
34C3×C4×Dic3—61no—
35C2×C3×C3⋊C8—32no—
36C2×C3×C3⋊C8—61no—
37C3×Dic12—64no—
38Dic3⋊Dic3—68no—
39#3719—68no—
40C22×H(3,𝔽3)C2×D411yescharacteristic
41C4×H(3,𝔽3)C2×D411yescharacteristic
42C4×H(3,𝔽3)C2×D411yescharacteristic
43#1084Q1614yescomplemented
44(C3⋊Dic3)⋊C3—44no—
45C2×C3×Q16—92no—
46#944—94no—
47C2×#216—94no—
48C2×Dic12—32no—
49C2×Dic12—92no—
50C2×Dic12—92no—
51#869—92no—
52C3×C4×C8—91no—
53C8×Dic3—32no—
54C8×Dic3—61no—
55C2×C32×C4—21no—
56C2×C32×C4C22×D312yes—
57C2×C3⋊Dic3—24no—
58#634—24no—
59#634—24no—
60C2×C3×Dic3—34no—
61C2×C3×Dic3—62no—
62C2×C3×Dic3—64no—
63C3×Dic6—64no—
64C3×Dic6—64no—
65C32×C8—22no—
66C32×C8#6714yes—
67C3×C3⋊C8—34no—
68C3×C3⋊C8—62no—
69#494—271noSylow
70C2×H(3,𝔽3)#12711yescharacteristic
71C2×H(3,𝔽3)C2×Q1612yes—
72C2×C3×Q8—94no—
73C2×Dic6—34no—
74C2×Dic6—94no—
75C3×Q16—184no—
76C2×C3×C8—32no—
77C2×C3×C8—61no—
78C2×C3×C8—91no—
79C2×C3×C8D3211yescharacteristic
80C3×C42—91no—
81#216—188no—
82C4⋊Dic3—184no—
83Dic3⋊C4—188no—
84C4×Dic3—32no—
85C4×Dic3—61no—
86C2×C3⋊C8—32no—
87C2×C3⋊C8—61no—
88Dic12—64no—
89Dic12—184no—
90Dic12—184no—
91C22×C32—21no—
92C22×C32D3×D412yes—
93C32×C4—21no—
94C32×C4—21no—
95C32×C4D3×D412yes—
96C32×C4C2×#6712yes—
97C3⋊Dic3—44no—
98C3×Dic3—38no—
99C3×Dic3—64no—
100C3×Dic3—124no—
101C2×Q16—92no—
102C2×Q16—272no—
103C4⋊Q8—272no—
104C4×C8—91no—
105H(3,𝔽3)#49411yescharacteristic, complemented, Sylow
106C3×Q8—184no—
107C3×Q8—184no—
108C2×C3×C4—32no—
109C2×C3×C4—61no—
110C2×C3×C4—92no—
111C2×C3×C4—184no—
112C2×C3×C4C2×D3211yescharacteristic
113C2×Dic3—34no—
114C2×Dic3—62no—
115C2×Dic3—64no—
116C2×Dic3—184no—
117Dic6—64no—
118Dic6—64no—
119Dic6—184no—
120Dic6—184no—
121C3×C8—34no—
122C3×C8—62no—
123C3×C8—92no—
124C3×C8#62512yes—
125C3⋊C8—34no—
126C3⋊C8—62no—
127C2×C32—22no—
128C2×C32—21no—
129C2×C32D3×Q1614yes—
130C2×C32#94012yes—
131C2×Q8—94no—
132Q16—184no—
133Q16—544no—
134C2×C8—91no—
135C2×C8((C3⋊D3)⋊C3)⋊C211yescharacteristic
136C4⋊C4—544no—
137C42—91no—
138C22×C3—32no—
139C22×C3—61no—
140C22×C3#380211yescharacteristic
141C3×C4—32no—
142C3×C4—32no—
143C3×C4—61no—
144C3×C4—61no—
145C3×C4—94no—
146C3×C4—364no—
147C3×C4#380211yescharacteristic
148C3×C4C2×#62511yescharacteristic
149Dic3—38no—
150Dic3—64no—
151Dic3—124no—
152Dic3—364no—
153C32—21no—
154C32#518312yescomplemented
155Q8—184no—
156Q8—184no—
157C2×C4—92no—
158C2×C4—184no—
159C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
160C8—92no—
161C8#619812yes—
162C2×C3—34no—
163C2×C3—32no—
164C2×C3—62no—
165C2×C3—61no—
166C2×C3#6380212yes—
167C2×C3#6391411yescharacteristic
168C22#8865511yescharacteristic
169C4—94no—
170C4—364no—
171C4#8865511yescharacteristic
172C4C2×#619811yescharacteristic
173C3—32no—
174C3—61no—
175C3#1059090511yescharacteristic
176C2#1172083312yes—
177C2#1172114411yescharacteristic
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)#494
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
43. #1084Q16semidirect
105. H(3,𝔽3)#494semidirect
154. C32#5183semidirect

Maximal subgroups

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

subgroupindex
C2×#3746⋊C32
#117210992
C2×#884332
C8×#10842
#105908963