Finite Group Databasebeta

← #13136956

Subgroups — #13136956

Order 1728.

411 conjugacy classes of subgroups; only the 41 normal subgroups are shown.

normal subgroups only

This group has 411 conjugacy classes of subgroups — too many to draw. Only its 41 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×#88552C211yesmaximal, characteristic
2C22⋊#6188C212yesmaximal, complemented
3C24×H(3,𝔽3)C411yescharacteristic
4C22×#1084C411yescharacteristic
5C22×#1084C2211yescharacteristic
6C2×#6188—22no—
7#88552—22no—
8C2×C3×#217—62no—
9C23×H(3,𝔽3)—21no—
10C23×H(3,𝔽3)—21no—
11C23×H(3,𝔽3)—41no—
12C23×H(3,𝔽3)—41no—
13C23×H(3,𝔽3)C812yescomplemented
14C23×H(3,𝔽3)C2×C411yescharacteristic
15C2×#1084—21no—
16C2×#1084—41no—
17C2×#1084—41no—
18C2×#1084D412yes—
19#6188—42no—
20C3×#323—91nomaximal
21C24×C32—22no—
22C22×C3×Dic3—62no—
23C22×C3×Dic3—62no—
24C3×#217—68no—
25C22×H(3,𝔽3)—21no—
26C22×H(3,𝔽3)—21no—
27C22×H(3,𝔽3)—22no—
28C22×H(3,𝔽3)—42no—
29C22×H(3,𝔽3)—42no—
30C22×H(3,𝔽3)—41no—
31C22×H(3,𝔽3)—41no—
32C22×H(3,𝔽3)C22⋊C411yescharacteristic
33C22×H(3,𝔽3)C2×C811yescharacteristic
34C22×H(3,𝔽3)M1611yescharacteristic
35#1084—22no—
36#1084—41no—
37C2×C3×C22⋊C4—91no—
38C2×#217—62no—
39C3×C22⋊C8—92no—
40C23×C32—24no—
41C23×C32—24no—
42C23×C32—24no—
43C23×C32—22no—
44C23×C32—44no—
45C23×C32—44no—
46C2×C3×Dic3—64no—
47C2×C3×Dic3—64no—
48C2×C3×Dic3—124no—
49C2×C3×Dic3—124no—
50#323—91noSylow
51C2×H(3,𝔽3)—21no—
52C2×H(3,𝔽3)—21no—
53C2×H(3,𝔽3)—42no—
54C2×H(3,𝔽3)C22⋊C811yescharacteristic
55C2×H(3,𝔽3)#9911yescharacteristic
56C2×H(3,𝔽3)#10011yescharacteristic
57C24×C3—62no—
58C24×C3#15711yescharacteristic
59C22×C3×C4—91no—
60C22×C3×C4—91no—
61C22×Dic3—62no—
62C22×Dic3—62no—
63C2×C3×C8—182no—
64C3×C22⋊C4—182no—
65#217—68no—
66C22×C32—24no—
67C22×C32—24no—
68C22×C32—28no—
69C22×C32—22no—
70C22×C32—22no—
71C22×C32—22no—
72C22×C32—48no—
73C22×C32—48no—
74C22×C32—44no—
75C22×C32—44no—
76C3×Dic3—124no—
77C3×Dic3—124no—
78C2×C22⋊C4—91no—
79C22⋊C8—92no—
80H(3,𝔽3)#32311yescharacteristic, complemented, Sylow
81C23×C3—21no—
82C23×C3—21no—
83C23×C3—41no—
84C23×C3—41no—
85C23×C3—64no—
86C23×C3—64no—
87C23×C3—64no—
88C23×C3—62no—
89C23×C3—124no—
90C23×C3—124no—
91C23×C3#62212yes—
92C23×C3C2×#15711yescharacteristic
93C2×C3×C4—92no—
94C2×C3×C4—181no—
95C2×C3×C4—361no—
96C2×C3×C4—361no—
97C2×Dic3—64no—
98C2×Dic3—64no—
99C2×Dic3—124no—
100C2×Dic3—124no—
101C3×C8—362no—
102C2×C32—24no—
103C2×C32—24no—
104C2×C32—22no—
105C2×C32—22no—
106C2×C32—22no—
107C2×C32—48no—
108C24#108811yescharacteristic
109C22×C4—91no—
110C22×C4—91no—
111C2×C8—182no—
112C22⋊C4—182no—
113C22×C3—21no—
114C22×C3—21no—
115C22×C3—22no—
116C22×C3—42no—
117C22×C3—42no—
118C22×C3—41no—
119C22×C3—41no—
120C22×C3—64no—
121C22×C3—64no—
122C22×C3—68no—
123C22×C3—62no—
124C22×C3—62no—
125C22×C3—62no—
126C22×C3—128no—
127C22×C3—128no—
128C22×C3—124no—
129C22×C3—124no—
130C22×C3C2×#62211yescharacteristic
131C22×C3#379211yescharacteristic
132C22×C3C22⋊#15711yescharacteristic
133C3×C4—182no—
134C3×C4—361no—
135Dic3—124no—
136Dic3—124no—
137C32—22no—
138C23—21no—
139C23—21no—
140C23—41no—
141C23—41no—
142C23#618812yescomplemented
143C23C2×#108811yescharacteristic
144C2×C4—92no—
145C2×C4—181no—
146C2×C4—361no—
147C2×C4—361no—
148C8—362no—
149C2×C3—21no—
150C2×C3—21no—
151C2×C3—42no—
152C2×C3—64no—
153C2×C3—64no—
154C2×C3—62no—
155C2×C3—62no—
156C2×C3—62no—
157C2×C3—128no—
158C2×C3#6378611yescharacteristic
159C2×C3C22⋊#62211yescharacteristic
160C2×C3#6378811yescharacteristic
161C22—21no—
162C22—21no—
163C22—22no—
164C22—42no—
165C22—42no—
166C22—41no—
167C22—41no—
168C22C2×#618811yescharacteristic
169C22#8863111yescharacteristic
170C22C22⋊#108811yescharacteristic
171C4—182no—
172C4—361no—
173C3—62no—
174C3#1059022211yescharacteristic
175C2—21no—
176C2—21no—
177C2—42no—
178C2#1172079211yescharacteristic
179C2C22⋊#618811yescharacteristic
180C2#1172079411yescharacteristic
subgroupG/H
center, socleC22×C3C22⋊#157
commutatorC22×H(3,𝔽3)C2×C8
FrattiniC23×C3C2×#157
Fitting, prime radicalC24×H(3,𝔽3)C4
cosocleC22×#1084C22
hypercenterC24×C3#157
hypocenterH(3,𝔽3)#323
LCSG ▷ C22×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22×C3 ◁ C23×C3 ◁ C24×C3

Complements

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

NH ≅ G/Ntype
2. C22⋊#6188C2semidirect
13. C23×H(3,𝔽3)C8semidirect
80. H(3,𝔽3)#323semidirect
142. C23#6188semidirect

Maximal subgroups

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

subgroupindex
C2×#885522
C22⋊#61882
C3×#3239