Finite Group Databasebeta

← #13147351

Subgroups — #13147351

Order 1728.

347 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 347 conjugacy classes of subgroups — too many to draw. Only its 75 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
1#11722448C211yesmaximal, characteristic
2#11722445C211yesmaximal, characteristic
3C2×C8×#272C211yesmaximal, characteristic
4#11720773C212yesmaximal, complemented
5C8×#1088C212yesmaximal, complemented
6C2×C4×#272C2211yescharacteristic
7C22⋊#1088C412yes—
8#88631C412yescomplemented
9C4⋊#1088C412yes—
10C4×#1088C2212yes—
11#88627C412yescomplemented
12#88626C2212yes—
13C2×C8×H(3,𝔽3)C411yescharacteristic, complemented
14C2×#6180C411yescharacteristic, complemented
15C8×#272C411yescharacteristic, complemented
16C8×#272C411yescharacteristic, complemented
17C8×#272C2211yescharacteristic
18C8×#272C2211yescharacteristic
19C2×C3×D3×C8—62no—
20C22×#272C2×C411yescharacteristic
21C2×#1088C2×C414yes—
22C2×C4×H(3,𝔽3)C2×C411yescharacteristic
23C2×#1084C2×C411yescharacteristic
24C4×#272C2×C411yescharacteristic
25C4×#272C2×C411yescharacteristic
26C4×#272C2×C411yescharacteristic
27C4×#272C2311yescharacteristic
28#6188C2×C414yes—
29C8×H(3,𝔽3)C2×C411yescharacteristic
30C8×H(3,𝔽3)C2×C411yescharacteristic
31#6180C2×C411yescharacteristic
32#6180C2×C411yescharacteristic
33C3×#405—91nomaximal
34C2×C3×C4×D3—62no—
35C2×C32×C8—22no—
36C2×C3×C3⋊C8—62no—
37C3×D3×C8—62no—
38C3×D3×C8—62no—
39C3×D3×C8—62no—
40C3×D3×C8—62no—
41C22×H(3,𝔽3)C4211yescharacteristic
42C2×#272—22no—
43C2×#272C4211yescharacteristic
44C2×#272C22×C411yescharacteristic
45#1088—24no—
46C4×H(3,𝔽3)C4211yescharacteristic
47C4×H(3,𝔽3)C22×C411yescharacteristic
48#1084C4211yescharacteristic
49#1084C22×C411yescharacteristic
50C2×C3×M16—91no—
51C22×C3×C8—91no—
52C3×#117—91no—
53C2×D3×C8—62no—
54C3×#97—92no—
55C3×C4×C8—92no—
56C22×C3×D3—62no—
57C2×C32×C4—22no—
58C2×C3×Dic3—62no—
59C3×C4×D3—62no—
60C3×C4×D3—62no—
61C3×C4×D3—62no—
62C3×C4×D3—62no—
63C32×C8—22no—
64C32×C8—22no—
65C3×C3⋊C8—62no—
66C3×C3⋊C8—62no—
67#405—91noSylow
68C2×H(3,𝔽3)—21no—
69C2×H(3,𝔽3)C2×C4211yescharacteristic
70#272—21no—
71#272#13112yes—
72C22×C3×C4—91no—
73C2×C4×D3—62no—
74C3×M16—92no—
75C3×M16—92no—
76C2×C3×C8—62no—
77C2×C3×C8—92no—
78C2×C3×C8—91no—
79C2×C3×C8—91no—
80C2×C3×C8—91no—
81C2×C3×C8—91no—
82C2×C3×C8—91no—
83C2×C3×C8#15711yescharacteristic
84C3×C4⋊C4—92no—
85C3×C22⋊C4—92no—
86C3×C42—92no—
87C2×C3⋊C8—62no—
88D3×C8—62no—
89D3×C8—62no—
90D3×C8—62no—
91D3×C8—62no—
92C22×C32—22no—
93C2×C3×D3—62no—
94C2×C3×D3—62no—
95C2×C3×D3—68no—
96C32×C4—22no—
97C32×C4—22no—
98C3×Dic3—62no—
99C3×Dic3—62no—
100C2×M16—91no—
101C22×C8—91no—
102#117—91no—
103#97—92no—
104C4×C8—92no—
105H(3,𝔽3)#40511yescharacteristic, complemented, Sylow
106C23×C3—91no—
107C22×D3—62no—
108C2×C3×C4—62no—
109C2×C3×C4—91no—
110C2×C3×C4—91no—
111C2×C3×C4—91no—
112C2×C3×C4—91no—
113C2×C3×C4—91no—
114C2×C3×C4—94no—
115C2×C3×C4C2×#15711yescharacteristic
116C2×Dic3—62no—
117C4×D3—62no—
118C4×D3—62no—
119C4×D3—62no—
120C4×D3—62no—
121C3×C8—62no—
122C3×C8—62no—
123C3×C8—94no—
124C3×C8—91no—
125C3×C8—91no—
126C3×C8C2×#15711yescharacteristic
127C3×C8C2×#15711yescharacteristic
128C3⋊C8—62no—
129C3⋊C8—62no—
130C2×C32—24no—
131C2×C32—22no—
132C3×D3—64no—
133C3×D3—64no—
134C22×C4—91no—
135M16—92no—
136M16—92no—
137C2×C8—92no—
138C2×C8—91no—
139C2×C8—91no—
140C2×C8—91no—
141C2×C8—91no—
142C2×C8—91no—
143C2×C8#108811yescharacteristic, complemented
144C4⋊C4—92no—
145C22⋊C4—92no—
146C42—92no—
147C22×C3—62no—
148C22×C3—91no—
149C22×C3—91no—
150C22×C3—182no—
151C22×C3C4×#15711yescharacteristic
152C2×D3—62no—
153C2×D3—62no—
154C2×D3—68no—
155C3×C4—62no—
156C3×C4—62no—
157C3×C4—91no—
158C3×C4—91no—
159C3×C4—184no—
160C3×C4C4×#15711yescharacteristic
161C3×C4C22×#15711yescharacteristic
162Dic3—62no—
163Dic3—62no—
164C32—22no—
165C23—91no—
166C2×C4—91no—
167C2×C4—91no—
168C2×C4—91no—
169C2×C4—91no—
170C2×C4—91no—
171C2×C4—94no—
172C2×C4C2×#108811yescharacteristic
173C8—94no—
174C8—91no—
175C8—91no—
176C8C2×#108811yescharacteristic
177C8C2×#108811yescharacteristic
178C2×C3—21no—
179C2×C3—64no—
180C2×C3—62no—
181C2×C3—92no—
182C2×C3—181no—
183C2×C3C2×C4×#15711yescharacteristic
184D3—64no—
185D3—64no—
186C22—91no—
187C22—91no—
188C22—182no—
189C22C4×#108811yescharacteristic
190C4—91no—
191C4—91no—
192C4—184no—
193C4C4×#108811yescharacteristic
194C4C22×#108811yescharacteristic
195C3—62no—
196C3#1059376511yescharacteristic
197C2—21no—
198C2—92no—
199C2—181no—
200C2C2×C4×#108811yescharacteristic
subgroupG/H
centerC3×C8C2×#157
commutatorC2×H(3,𝔽3)C2×C42
FrattiniC3×C4C22×#157
FittingC2×C8×H(3,𝔽3)C4
socleC2×C3C2×C4×#157
prime radicalC22×#272C2×C4
cosocleC4×#272C23
hypercenterC2×C3×C8#157
hypocenterH(3,𝔽3)#405
LCSG ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C3×C8 ◁ C2×C3×C8

Complements

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

NH ≅ G/Ntype
4. #11720773C2semidirect
5. C8×#1088C2semidirect
8. #88631C4semidirect
11. #88627C4semidirect
13. C2×C8×H(3,𝔽3)C4semidirect
14. C2×#6180C4semidirect
15. C8×#272C4semidirect
16. C8×#272C4semidirect
105. H(3,𝔽3)#405semidirect
143. C2×C8#1088semidirect

Maximal subgroups

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

subgroupindex
#117224482
#117224452
C2×C8×#2722
#117207732
C8×#10882
C3×#4059