Finite Group Databasebeta

← #13147361

Subgroups — #13147361

Order 1728.

313 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 313 conjugacy classes of subgroups — too many to draw. Only its 59 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
2C2×C4⋊#1088C211yesmaximal, characteristic
3M16×#272C211yesmaximal, characteristic
4#11720775C212yesmaximal, complemented
5#11720774C212yesmaximal, complemented
6C22×#1088—21no—
7C2×C4×#272C2211yescharacteristic
8C22⋊#1088—21no—
9C4⋊#1088—21no—
10C4⋊#1088C2211yescharacteristic
11C4⋊#1088C2212yes—
12C4⋊#1088C2211yescharacteristic
13C4×#1088—21no—
14M16×H(3,𝔽3)C411yescharacteristic, complemented
15#88538C411yescharacteristic, complemented
16#88527C412yescomplemented
17C8×#272C2212yes—
18C3×D3×M16—62no—
19C22×#272D411yescharacteristic
20C2×#1088—21no—
21C2×#1088—21no—
22C2×#1088—22no—
23C2×#1088—42no—
24C2×C4×H(3,𝔽3)C2×C411yescharacteristic
25C2×#1084Q811yescharacteristic
26C4×#272C2×C411yescharacteristic
27C4×#272D411yescharacteristic
28C4×#272Q811yescharacteristic
29C4×#272C2311yescharacteristic
30C8×H(3,𝔽3)C2×C412yescomplemented
31#6180C2×C412yescomplemented
32C3×#428—91nomaximal
33C2×C3×C4×D3—62no—
34C32×M16—22no—
35C3×#208—62no—
36C3×ΓL(1,𝔽25)—64no—
37C3×D3×C8—64no—
38C22×H(3,𝔽3)C4⋊C411yescharacteristic
39C2×#272—21no—
40C2×#272—21no—
41C2×#272C4⋊C411yescharacteristic
42C2×#272C2×D411yescharacteristic
43#1088—41no—
44#1088—42no—
45#1088—41no—
46C4×H(3,𝔽3)C4⋊C411yescharacteristic
47C4×H(3,𝔽3)C22×C411yescharacteristic
48#1084C4⋊C411yescharacteristic
49#1084C2×Q811yescharacteristic
50C2×C3×M16—91no—
51C3×#117—91no—
52C2×C3×C4⋊C4—91no—
53D3×M16—62no—
54C3×#107—92no—
55C3×#106—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—24no—
64C3×C3⋊C8—64no—
65#428—91noSylow
66C2×H(3,𝔽3)—21no—
67C2×H(3,𝔽3)C2×C4⋊C411yescharacteristic
68#272—21no—
69#272AGL(1,ℤ/8)11yescharacteristic
70#272#13711yescharacteristic
71C22×C3×C4—91no—
72C22×C3×C4—181no—
73C2×C4×D3—62no—
74C3×M16—62no—
75C3×M16—91no—
76C3×M16—92no—
77C3×M16#15711yescharacteristic
78C2×C3×C8—92no—
79C3×C4⋊C4—91no—
80C3×C4⋊C4—92no—
81C3×C4⋊C4—91no—
82C3×C4⋊C4—181no—
83C3×C22⋊C4—181no—
84C3×C42—181no—
85#208—62no—
86ΓL(1,𝔽25)—64no—
87D3×C8—64no—
88C22×C32—22no—
89C2×C3×D3—62no—
90C2×C3×D3—62no—
91C2×C3×D3—122no—
92C2×C3×D3—122no—
93C32×C4—22no—
94C32×C4—22no—
95C3×Dic3—62no—
96C3×Dic3—62no—
97C2×M16—91no—
98#117—91no—
99C2×C4⋊C4—91no—
100#107—92no—
101#106—92no—
102H(3,𝔽3)#42811yescharacteristic, complemented, Sylow
103C23×C3—91no—
104C22×D3—62no—
105C2×C3×C4—62no—
106C2×C3×C4—91no—
107C2×C3×C4—91no—
108C2×C3×C4—91no—
109C2×C3×C4—91no—
110C2×C3×C4—91no—
111C2×C3×C4—181no—
112C2×C3×C4—181no—
113C2×C3×C4—182no—
114C2×C3×C4—362no—
115C2×C3×C4C2×#15711yescharacteristic
116C2×Dic3—62no—
117C4×D3—62no—
118C4×D3—62no—
119C4×D3—62no—
120C4×D3—62no—
121C3×C8—64no—
122C3×C8—92no—
123C3×C8C2×#15712yes—
124C3⋊C8—64no—
125C2×C32—22no—
126C2×C32—42no—
127C3×D3—62no—
128C3×D3—62no—
129C3×D3—122no—
130C22×C4—91no—
131C22×C4—181no—
132M16—91no—
133M16—92no—
134M16#108811yescharacteristic, complemented
135C2×C8—92no—
136C4⋊C4—91no—
137C4⋊C4—92no—
138C4⋊C4—91no—
139C4⋊C4—181no—
140C22⋊C4—181no—
141C42—181no—
142C22×C3—62no—
143C22×C3—91no—
144C22×C3—91no—
145C22×C3—181no—
146C22×C3—181no—
147C22×C3C4⋊#15711yescharacteristic
148C2×D3—62no—
149C2×D3—62no—
150C2×D3—122no—
151C2×D3—122no—
152C3×C4—62no—
153C3×C4—62no—
154C3×C4—91no—
155C3×C4—91no—
156C3×C4—361no—
157C3×C4—361no—
158C3×C4—362no—
159C3×C4C4⋊#15711yescharacteristic
160C3×C4C22×#15711yescharacteristic
161Dic3—62no—
162Dic3—62no—
163C32—22no—
164C23—91no—
165C2×C4—91no—
166C2×C4—91no—
167C2×C4—91no—
168C2×C4—91no—
169C2×C4—91no—
170C2×C4—181no—
171C2×C4—181no—
172C2×C4—182no—
173C2×C4—362no—
174C2×C4C2×#108811yescharacteristic
175C8—92no—
176C8C2×#108812yescomplemented
177C2×C3—21no—
178C2×C3—62no—
179C2×C3—91no—
180C2×C3—91no—
181C2×C3—122no—
182C2×C3—181no—
183C2×C3C2×C4⋊#15711yescharacteristic
184D3—62no—
185D3—62no—
186D3—122no—
187C22—91no—
188C22—91no—
189C22—181no—
190C22—181no—
191C22C4⋊#108811yescharacteristic
192C4—91no—
193C4—91no—
194C4—362no—
195C4—361no—
196C4—361no—
197C4C4⋊#108811yescharacteristic
198C4C22×#108811yescharacteristic
199C3—62no—
200C3#1059377411yescharacteristic
201C2—21no—
202C2—91no—
203C2—91no—
204C2—181no—
205C2C2×C4⋊#108811yescharacteristic
subgroupG/H
center, socleC2×C3C2×C4⋊#157
commutatorC4×H(3,𝔽3)C22×C4
FrattiniC3×C4C22×#157
FittingM16×H(3,𝔽3)C4
prime radicalC22×#272D4
cosocleC4×#272C23
hypercenterC3×M16#157
hypocenterH(3,𝔽3)#428
LCSG ▷ C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C2×C3 ◁ C2×C3×C4 ◁ C3×M16

Complements

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

NH ≅ G/Ntype
4. #11720775C2semidirect
5. #11720774C2semidirect
14. M16×H(3,𝔽3)C4semidirect
15. #88538C4semidirect
16. #88527C4semidirect
30. C8×H(3,𝔽3)C2×C4semidirect
31. #6180C2×C4semidirect
102. H(3,𝔽3)#428semidirect
134. M16#1088semidirect
176. C8C2×#1088semidirect

Maximal subgroups

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

subgroupindex
#117224482
C2×C4⋊#10882
M16×#2722
#117207752
#117207742
C3×#4289