Finite Group Databasebeta

← #13136936

Subgroups — #13136936

Order 1728.

411 conjugacy classes of subgroups; only the 37 normal subgroups are shown.

normal subgroups only

This group has 411 conjugacy classes of subgroups — too many to draw. Only its 37 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×#88542C211yesmaximal, characteristic
2C22⋊#6188C212yesmaximal, complemented
3C23×#272C411yescharacteristic
4C22×C4×H(3,𝔽3)C411yescharacteristic
5C22×#1084C2211yescharacteristic
6C2×#6188—22no—
7#88542—22no—
8C2×C3×(C2×D3)⋊C4—62no—
9C23×H(3,𝔽3)C2×C411yescharacteristic
10C22×#272—21no—
11C22×#272—21no—
12C22×#272—41no—
13C22×#272—41no—
14C22×#272C812yescomplemented
15C2×C4×H(3,𝔽3)—21no—
16C2×C4×H(3,𝔽3)—41no—
17C2×#1084—41no—
18C2×#1084D412yes—
19#6188—42no—
20C3×#323—91nomaximal
21C23×C3×D3—62no—
22C22×C32×C4—22no—
23C22×C3×Dic3—62no—
24C3×(C2×D3)⋊C4—68no—
25C22×H(3,𝔽3)—21no—
26C22×H(3,𝔽3)—21no—
27C22×H(3,𝔽3)C22⋊C411yescharacteristic
28C22×H(3,𝔽3)C2×C811yescharacteristic
29C22×H(3,𝔽3)M1611yescharacteristic
30C2×#272—22no—
31C2×#272—42no—
32C2×#272—41no—
33C2×#272—41no—
34C2×#272—42no—
35C4×H(3,𝔽3)—41no—
36#1084—22no—
37C2×C3×C22⋊C4—91no—
38C2×(C2×D3)⋊C4—62no—
39C3×C22⋊C8—92no—
40C23×C32—22no—
41C22×C3×D3—64no—
42C22×C3×D3—64no—
43C22×C3×D3—64no—
44C22×C3×D3—124no—
45C22×C3×D3—124no—
46C2×C32×C4—24no—
47C2×C32×C4—44no—
48C2×C3×Dic3—64no—
49C2×C3×Dic3—124no—
50#323—91noSylow
51C2×H(3,𝔽3)—21no—
52C2×H(3,𝔽3)—21no—
53C2×H(3,𝔽3)C22⋊C811yescharacteristic
54C2×H(3,𝔽3)#9911yescharacteristic
55C2×H(3,𝔽3)#10011yescharacteristic
56#272—42no—
57C24×C3—91no—
58C23×D3—62no—
59C22×C3×C4—62no—
60C22×C3×C4—91no—
61C22×C3×C4#15711yescharacteristic
62C22×Dic3—62no—
63C2×C3×C8—182no—
64C3×C22⋊C4—182no—
65(C2×D3)⋊C4—68no—
66C22×C32—24no—
67C22×C32—24no—
68C22×C32—22no—
69C22×C32—22no—
70C22×C32—22no—
71C2×C3×D3—68no—
72C2×C3×D3—128no—
73C2×C3×D3—128no—
74C2×C3×D3—124no—
75C2×C3×D3—124no—
76C32×C4—44no—
77C3×Dic3—124no—
78C2×C22⋊C4—91no—
79C22⋊C8—92no—
80H(3,𝔽3)#32311yescharacteristic, complemented, Sylow
81C23×C3—62no—
82C23×C3—92no—
83C23×C3—181no—
84C23×C3—181no—
85C23×C3—361no—
86C23×C3—361no—
87C23×C3C2×#15711yescharacteristic
88C22×D3—64no—
89C22×D3—64no—
90C22×D3—64no—
91C22×D3—124no—
92C22×D3—124no—
93C2×C3×C4—21no—
94C2×C3×C4—41no—
95C2×C3×C4—64no—
96C2×C3×C4—92no—
97C2×C3×C4—124no—
98C2×C3×C4—361no—
99C2×Dic3—64no—
100C2×Dic3—124no—
101C3×C8—362no—
102C2×C32—24no—
103C2×C32—24no—
104C2×C32—22no—
105C2×C32—22no—
106C2×C32—22no—
107C3×D3—128no—
108C24—91no—
109C22×C4—91no—
110C22×C4#108811yescharacteristic
111C2×C8—182no—
112C22⋊C4—182no—
113C22×C3—21no—
114C22×C3—21no—
115C22×C3—64no—
116C22×C3—64no—
117C22×C3—62no—
118C22×C3—62no—
119C22×C3—62no—
120C22×C3—182no—
121C22×C3—362no—
122C22×C3—362no—
123C22×C3—361no—
124C22×C3—361no—
125C22×C3#378711yescharacteristic
126C22×C3#378811yescharacteristic
127C22×C3C22⋊#15711yescharacteristic
128C2×D3—68no—
129C2×D3—128no—
130C2×D3—128no—
131C2×D3—124no—
132C2×D3—124no—
133C3×C4—41no—
134C3×C4—124no—
135C3×C4—182no—
136Dic3—124no—
137C32—22no—
138C23—92no—
139C23—181no—
140C23—181no—
141C23—361no—
142C23—361no—
143C23C2×#108811yescharacteristic
144C2×C4—21no—
145C2×C4—41no—
146C2×C4—92no—
147C2×C4—361no—
148C8—362no—
149C2×C3—21no—
150C2×C3—21no—
151C2×C3—64no—
152C2×C3—64no—
153C2×C3—62no—
154C2×C3—62no—
155C2×C3—62no—
156C2×C3—362no—
157C2×C3#6377411yescharacteristic
158C2×C3#6377811yescharacteristic
159C2×C3#6378811yescharacteristic
160D3—128no—
161C22—21no—
162C22—21no—
163C22—182no—
164C22—362no—
165C22—362no—
166C22—361no—
167C22—361no—
168C22#8862611yescharacteristic
169C22#8862711yescharacteristic
170C22C22⋊#108811yescharacteristic
171C4—41no—
172C4—182no—
173C3—62no—
174C3#1059020411yescharacteristic
175C2—21no—
176C2—21no—
177C2—362no—
178C2#1172078011yescharacteristic
179C2#1172078411yescharacteristic
180C2#1172079411yescharacteristic
subgroupG/H
center, socleC22×C3C22⋊#157
commutatorC22×H(3,𝔽3)C2×C8
FrattiniC23×C3C2×#157
FittingC22×C4×H(3,𝔽3)C4
prime radicalC23×#272C4
cosocleC22×#1084C22
hypercenterC22×C3×C4#157
hypocenterH(3,𝔽3)#323
LCSG ▷ C22×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22×C3 ◁ C23×C3 ◁ C22×C3×C4

Complements

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

NH ≅ G/Ntype
2. C22⋊#6188C2semidirect
14. C22×#272C8semidirect
80. H(3,𝔽3)#323semidirect

Maximal subgroups

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

subgroupindex
C2×#885422
C22⋊#61882
C3×#3239