Finite Group Databasebeta

← #13136708

Subgroups — #13136708

Order 1728.

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

normal subgroups only

This group has 279 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
1#11722445C211yesmaximal, characteristic
2#11721065C211yesmaximal, characteristic
3C2×#88434C211yesmaximal, characteristic
4C2×C4×#272C2211yescharacteristic
5#88631C412yes—
6#88627—21no—
7#88626—21no—
8#3748⋊C3—21no—
9C2×(C3⋊(C3⋊C8))⋊C3—21no—
10#88435C412yes—
11#88434—21no—
12#88434—21no—
13#88434C412yes—
14D3×#208—61no—
15C2×D3×C3⋊C8—61no—
16C22×#272Q811yescharacteristic
17C2×C4×H(3,𝔽3)D411yescharacteristic
18C2×#1084C2×C411yescharacteristic
19C4×#272C2×C411yescharacteristic
20C4×#272C2×C411yescharacteristic
21C4×#272D411yescharacteristic
22C4×#272D411yescharacteristic
23#6188—22no—
24(C3⋊(C3⋊C8))⋊C3—22no—
25(C3⋊(C3⋊C8))⋊C3—22no—
26#4558—91nomaximal
27C2×C3×C4×D3—61no—
28C2×C3×C4×D3—61no—
29#3748—21no—
30C2×C3⋊(C3⋊C8)—21no—
31C3×#208—61no—
32C2×C3×C3⋊C8—61no—
33#3711—62no—
34D3×C3⋊C8—62no—
35D3×C3⋊C8—62no—
36D3×C3⋊C8—62no—
37C22×H(3,𝔽3)C4⋊C411yescharacteristic
38C2×#272—22no—
39C2×#272C4⋊C411yescharacteristic
40C2×#272C4⋊C411yescharacteristic
41C4×H(3,𝔽3)C22⋊C411yescharacteristic
42C4×H(3,𝔽3)C22⋊C411yescharacteristic
43#1084C4211yescharacteristic
44#1084C22⋊C411yescharacteristic
45C2×C3×M16—91no—
46C2×#208—91no—
47C22×C3⋊C8—91no—
48D3×M16—181no—
49C2×D3×C8—181no—
50C22×C3×D3—61no—
51C22×C3×D3—61no—
52C2×C32×C4—21no—
53C2×C32×C4—21no—
54C2×C3×Dic3—61no—
55C2×C3×Dic3—61no—
56C3×C4×D3—61no—
57C3×C4×D3—61no—
58C3×C4×D3—61no—
59C3×C4×D3—61no—
60C3×C4×D3—61no—
61C3×C4×D3—61no—
62C3×C4×D3—61no—
63C3×C4×D3—61no—
64C3⋊(C3⋊C8)—22no—
65C3⋊(C3⋊C8)—22no—
66C3×C3⋊C8—62no—
67C3×C3⋊C8—62no—
68#341—271noSylow
69C2×H(3,𝔽3)—21no—
70C2×H(3,𝔽3)#9511yescharacteristic
71#272—21no—
72#272#10812yes—
73C22×C3×C4—91no—
74C2×C4×D3—61no—
75C2×C4×D3—61no—
76C3×M16—92no—
77C3×M16—181no—
78C3×M16—181no—
79C2×C3×C8—181no—
80C2×C3×C8—181no—
81#208—61no—
82#208—92no—
83#208—181no—
84C2×C3⋊C8—61no—
85C2×C3⋊C8—92no—
86C2×C3⋊C8—181no—
87C2×C3⋊C8—181no—
88C2×C3⋊C8—181no—
89ΓL(1,𝔽25)—182no—
90D3×C8—182no—
91D3×C8—182no—
92D3×C8—182no—
93C22×C32—21no—
94C22×C32—21no—
95C2×C3×D3—61no—
96C2×C3×D3—61no—
97C2×C3×D3—61no—
98C2×C3×D3—61no—
99C2×C3×D3—64no—
100C2×C3×D3—122no—
101C32×C4—21no—
102C32×C4—21no—
103C32×C4—21no—
104C32×C4—21no—
105C3×Dic3—61no—
106C3×Dic3—61no—
107C3×Dic3—61no—
108C3×Dic3—61no—
109C2×M16—91no—
110C2×M16—271no—
111C22×C8—271no—
112H(3,𝔽3)#34111yescharacteristic, complemented, Sylow
113C23×C3—91no—
114C22×D3—61no—
115C22×D3—61no—
116C2×C3×C4—61no—
117C2×C3×C4—61no—
118C2×C3×C4—91no—
119C2×C3×C4—91no—
120C2×C3×C4—91no—
121C2×C3×C4—91no—
122C2×C3×C4—91no—
123C2×C3×C4GO+(4,𝔽2)11yescharacteristic
124C2×Dic3—61no—
125C2×Dic3—61no—
126C4×D3—61no—
127C4×D3—61no—
128C4×D3—61no—
129C4×D3—61no—
130C4×D3—61no—
131C4×D3—61no—
132C4×D3—61no—
133C4×D3—61no—
134C3×C8—182no—
135C3×C8—182no—
136C3×C8—182no—
137C3⋊C8—62no—
138C3⋊C8—62no—
139C3⋊C8—182no—
140C3⋊C8—182no—
141C2×C32—22no—
142C2×C32—21no—
143C2×C32—21no—
144C2×C32—41no—
145C3×D3—62no—
146C3×D3—62no—
147C3×D3—62no—
148C3×D3—121no—
149C22×C4—91no—
150M16—92no—
151M16—181no—
152M16—181no—
153M16—272no—
154C2×C8—181no—
155C2×C8—181no—
156C2×C8—272no—
157C2×C8—541no—
158C2×C8—541no—
159C22×C3—61no—
160C22×C3—61no—
161C22×C3—91no—
162C22×C3—91no—
163C22×C3—182no—
164C22×C3#377311yescharacteristic
165C2×D3—61no—
166C2×D3—61no—
167C2×D3—61no—
168C2×D3—61no—
169C2×D3—64no—
170C2×D3—122no—
171C3×C4—61no—
172C3×C4—61no—
173C3×C4—61no—
174C3×C4—61no—
175C3×C4—91no—
176C3×C4—91no—
177C3×C4D32⋊C411yescharacteristic
178C3×C4D32⋊C411yescharacteristic
179Dic3—61no—
180Dic3—61no—
181Dic3—61no—
182Dic3—61no—
183C32—21no—
184C32—21no—
185C23—91no—
186C2×C4—91no—
187C2×C4—91no—
188C2×C4—91no—
189C2×C4—91no—
190C2×C4—91no—
191C2×C4#1088⋊C211yescharacteristic
192C8—182no—
193C8—182no—
194C8—182no—
195C2×C3—21no—
196C2×C3—62no—
197C2×C3—61no—
198C2×C3—61no—
199C2×C3—92no—
200C2×C3—121no—
201C2×C3—181no—
202C2×C3#6373711yescharacteristic
203D3—62no—
204D3—62no—
205D3—62no—
206D3—121no—
207C22—91no—
208C22—91no—
209C22—182no—
210C22#8859011yescharacteristic
211C4—91no—
212C4—91no—
213C4(((C3⋊D3)⋊C3)⋊C2)⋊C411yescharacteristic
214C4(((C3⋊D3)⋊C3)⋊C2)⋊C411yescharacteristic
215C3—61no—
216C3—61no—
217C3#1059005011yescharacteristic
218C2—21no—
219C2—92no—
220C2—181no—
221C2#1172070711yescharacteristic
subgroupG/H
centerC4(((C3⋊D3)⋊C3)⋊C2)⋊C4
commutator#1084C42
FrattiniC2×C3×C4GO+(4,𝔽2)
FittingC2×C4×H(3,𝔽3)D4
socleC2×C3#63737
prime radicalC22×#272Q8
cosocleC2×C4×#272C22
hypercenterC2×C4#1088⋊C2
hypocenterH(3,𝔽3)#341
LCSG ▷ #1084 ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C4 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
112. H(3,𝔽3)#341semidirect

Maximal subgroups

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

subgroupindex
#117224452
#117210652
C2×#884342
#45589