Finite Group Databasebeta

← #13136709

Subgroups — #13136709

Order 1728.

279 conjugacy classes of subgroups; only the 39 normal subgroups are shown.

normal subgroups only

This group has 279 conjugacy classes of subgroups — too many to draw. Only its 39 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#11721097C211yesmaximal, characteristic
3#11721065C211yesmaximal, characteristic
4C2×C4×#272C2211yescharacteristic
5#88631—21no—
6#88627C412yescomplemented
7#88626—21no—
8C4×(C3⋊Dic3)⋊C3—21no—
9#3748⋊C3—21no—
10#88451C412yes—
11#88449—21no—
12#88435—21no—
13#88434C412yescomplemented
14#63855—61no—
15D3×#208—61no—
16C22×#272C2×C411yescharacteristic
17C2×C4×H(3,𝔽3)D411yescharacteristic
18C2×#1084D411yescharacteristic
19C4×#272C2×C411yescharacteristic
20C4×#272C2×C411yescharacteristic
21C4×#272D411yescharacteristic
22C4×#272Q811yescharacteristic
23C2×(C3⋊Dic3)⋊C3—22no—
24#6188—22no—
25(C3⋊(C3⋊C8))⋊C3—22no—
26#4559—91nomaximal
27C2×C3×C4×D3—61no—
28C2×C3×C4×D3—61no—
29C4×C3⋊Dic3—21no—
30#3748—21no—
31C3×C4×Dic3—61no—
32C3×#208—61no—
33Dic3⋊Dic3—62no—
34(C2×D3)⋊Dic3—62no—
35#3711—62no—
36D3×C3⋊C8—62no—
37C22×H(3,𝔽3)C22⋊C411yescharacteristic
38C2×#272—41no—
39C2×#272C4211yescharacteristic
40C2×#272C4⋊C411yescharacteristic
41C4×H(3,𝔽3)C22⋊C411yescharacteristic
42C4×H(3,𝔽3)C4⋊C411yescharacteristic
43#1084C22⋊C411yescharacteristic
44#1084C4⋊C411yescharacteristic
45(C3⋊Dic3)⋊C3—42no—
46C2×C3×M16—91no—
47#926—91no—
48C2×#208—91no—
49D3×M16—181no—
50#872—181no—
51C22×C3×D3—61no—
52C22×C3×D3—61no—
53C2×C32×C4—21no—
54C2×C32×C4—21no—
55C2×C3⋊Dic3—22no—
56C2×C3×Dic3—61no—
57C2×C3×Dic3—61no—
58C2×C3×Dic3—62no—
59C3×C4×D3—61no—
60C3×C4×D3—61no—
61C3×C4×D3—61no—
62C3×C4×D3—61no—
63C3×C4×D3—61no—
64C3×C4×D3—61no—
65C3×C4×D3—61no—
66C3×C4×D3—61no—
67C3⋊(C3⋊C8)—22no—
68C3×C3⋊C8—62no—
69#344—271noSylow
70C2×H(3,𝔽3)—21no—
71C2×H(3,𝔽3)#9511yescharacteristic
72#272—21no—
73#272—21no—
74C22×C3×C4—91no—
75C2×C4×D3—61no—
76C2×C4×D3—61no—
77C3×M16—92no—
78C3×M16—181no—
79C3×M16—181no—
80C2×C3×C8—181no—
81C3×C42—181no—
82#217—92no—
83(C2×D3)⋊C4—182no—
84C4⋊Dic3—181no—
85Dic3⋊C4—182no—
86C4×Dic3—61no—
87C4×Dic3—181no—
88#208—61no—
89#208—181no—
90#208—181no—
91C2×C3⋊C8—92no—
92ΓL(1,𝔽25)—182no—
93D3×C8—182no—
94C22×C32—21no—
95C22×C32—21no—
96C2×C3×D3—61no—
97C2×C3×D3—61no—
98C2×C3×D3—61no—
99C2×C3×D3—61no—
100C2×C3×D3—122no—
101C2×C3×D3—122no—
102C32×C4—21no—
103C32×C4—21no—
104C32×C4—21no—
105C32×C4—21no—
106C3⋊Dic3—42no—
107C3×Dic3—61no—
108C3×Dic3—61no—
109C3×Dic3—61no—
110C3×Dic3—61no—
111C3×Dic3—122no—
112C2×M16—91no—
113C2×M16—271no—
114#117—271no—
115H(3,𝔽3)#34411yescharacteristic, complemented, Sylow
116C23×C3—91no—
117C22×D3—61no—
118C22×D3—61no—
119C2×C3×C4—61no—
120C2×C3×C4—61no—
121C2×C3×C4—91no—
122C2×C3×C4—91no—
123C2×C3×C4—91no—
124C2×C3×C4—91no—
125C2×C3×C4—91no—
126C2×C3×C4—182no—
127C2×C3×C4GO+(4,𝔽2)11yescharacteristic
128C2×Dic3—61no—
129C2×Dic3—61no—
130C2×Dic3—62no—
131C2×Dic3—182no—
132C4×D3—61no—
133C4×D3—61no—
134C4×D3—61no—
135C4×D3—61no—
136C4×D3—61no—
137C4×D3—61no—
138C4×D3—61no—
139C4×D3—61no—
140C3×C8—182no—
141C3×C8—182no—
142C3⋊C8—62no—
143C3⋊C8—182no—
144C2×C32—22no—
145C2×C32—21no—
146C2×C32—21no—
147C2×C32—41no—
148C3×D3—62no—
149C3×D3—121no—
150C3×D3—121no—
151C3×D3—121no—
152C22×C4—91no—
153M16—92no—
154M16—181no—
155M16—181no—
156M16—541no—
157C2×C8—181no—
158C2×C8—272no—
159C4⋊C4—541no—
160C22⋊C4—272no—
161C42—181no—
162C22×C3—61no—
163C22×C3—61no—
164C22×C3—91no—
165C22×C3—91no—
166C22×C3—361no—
167C22×C3D32⋊C411yescharacteristic
168C2×D3—61no—
169C2×D3—61no—
170C2×D3—61no—
171C2×D3—61no—
172C2×D3—122no—
173C2×D3—122no—
174C3×C4—61no—
175C3×C4—61no—
176C3×C4—61no—
177C3×C4—61no—
178C3×C4—91no—
179C3×C4—91no—
180C3×C4—362no—
181C3×C4D32⋊C411yescharacteristic
182C3×C4#377311yescharacteristic
183Dic3—61no—
184Dic3—61no—
185Dic3—61no—
186Dic3—61no—
187Dic3—122no—
188Dic3—362no—
189C32—21no—
190C32—21no—
191C23—91no—
192C2×C4—91no—
193C2×C4—91no—
194C2×C4—91no—
195C2×C4—91no—
196C2×C4—91no—
197C2×C4—182no—
198C2×C4#1088⋊C211yescharacteristic
199C8—182no—
200C8—182no—
201C2×C3—21no—
202C2×C3—62no—
203C2×C3—61no—
204C2×C3—61no—
205C2×C3—121no—
206C2×C3—181no—
207C2×C3—181no—
208C2×C3#6373711yescharacteristic
209D3—62no—
210D3—121no—
211D3—121no—
212D3—121no—
213C22—91no—
214C22—91no—
215C22—361no—
216C22(((C3⋊D3)⋊C3)⋊C2)⋊C411yescharacteristic
217C4—91no—
218C4—91no—
219C4—362no—
220C4#8859011yescharacteristic
221C4(((C3⋊D3)⋊C3)⋊C2)⋊C411yescharacteristic
222C3—61no—
223C3—61no—
224C3#1059005411yescharacteristic
225C2—21no—
226C2—181no—
227C2—181no—
228C2#1172070711yescharacteristic
subgroupG/H
centerC4(((C3⋊D3)⋊C3)⋊C2)⋊C4
commutatorC2×#272C42
FrattiniC2×C3×C4GO+(4,𝔽2)
FittingC2×C4×H(3,𝔽3)D4
socleC2×C3#63737
prime radicalC22×#272C2×C4
cosocleC2×C4×#272C22
hypercenterC2×C4#1088⋊C2
hypocenterH(3,𝔽3)#344
LCSG ▷ C2×#272 ▷ 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
6. #88627C4semidirect
13. #88434C4semidirect
115. H(3,𝔽3)#344semidirect

Maximal subgroups

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

subgroupindex
#117224452
#117210972
#117210652
#45599