Finite Group Databasebeta

← #13138741

Subgroups — #13138741

Order 1728.

401 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 401 conjugacy classes of subgroups — too many to draw. Only its 95 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
1C4×#6196C211yesmaximal, characteristic
2C8×#1084C211yesmaximal, characteristic
3#63641⋊C3C212yesmaximal
4#11720310C212yesmaximal
5#11720304C211yesmaximal, characteristic
6#10590644—32nomaximal
7C2×#6196C411yescharacteristic
8C2×C8×H(3,𝔽3)C2211yescharacteristic
9C4×#1084C2211yescharacteristic
10C2×#6180C2211yescharacteristic
11C4×(C3⋊Dic3)⋊C3C2212yes—
12C2×(C3⋊(C3⋊C8))⋊C3C2212yes—
13#88449C411yescharacteristic
14#88448C412yes—
15C4×#627—32no—
16#63641D312yes—
17C3×Dic3⋊C8—32no—
18C3×C8×Dic3—32no—
19C3×C8×Dic3—61no—
20Dic3⋊(C3⋊C8)—32no—
21#63553—32no—
22Dic3×C3⋊C8—32no—
23C2×C4×H(3,𝔽3)C2311yescharacteristic
24C2×#1084C2×C411yescharacteristic
25C2×#1084C2×C411yescharacteristic
26C2×(C3⋊Dic3)⋊C3C2×C412yes—
27C2×(C3⋊Dic3)⋊C3C2×C412yes—
28#6196C814yescomplemented
29C8×H(3,𝔽3)—21no—
30#6180Q812yescomplemented
31(C3⋊(C3⋊C8))⋊C3—22no—
32#4830—92no—
33C8×Dic6—91no—
34C2×#627—32no—
35C2×C32×C8—21no—
36C2×C32×C8C2×D312yes—
37C4×C3⋊Dic3C2×D312yes—
38C2×C3⋊(C3⋊C8)C2×D312yes—
39C3×C4×Dic3—32no—
40C3×C4×Dic3—32no—
41C3×C4×Dic3—61no—
42C2×C3×C3⋊C8—32no—
43C2×C3×C3⋊C8—32no—
44C2×C3×C3⋊C8—61no—
45#3724—32no—
46Dic3⋊Dic3—32no—
47Dic3⋊Dic3—32no—
48C22×H(3,𝔽3)C22×C411yescharacteristic
49C4×H(3,𝔽3)C2×Q811yescharacteristic
50C4×H(3,𝔽3)#4111yescharacteristic
51#1084—21no—
52#1084C2×C812yes—
53(C3⋊Dic3)⋊C3—22no—
54(C3⋊Dic3)⋊C3C2×C814yes—
55C4×Dic6—91no—
56C4×Dic6—92no—
57C3×C4⋊C8—92no—
58C3×C4×C8—91no—
59Dic3⋊C8—32no—
60Dic3⋊C8—92no—
61Dic3⋊C8—92no—
62C8×Dic3—32no—
63C8×Dic3—61no—
64C8×Dic3—92no—
65C8×Dic3—92no—
66C4⋊(C3⋊C8)—91no—
67C4×C3⋊C8—92no—
68C2×C32×C4—21no—
69C2×C32×C4C22×D312yes—
70C2×C3⋊Dic3C4×D312yes—
71C2×C3⋊Dic3C4×D312yes—
72C2×C3×Dic3—32no—
73C2×C3×Dic3—32no—
74C2×C3×Dic3—32no—
75C2×C3×Dic3—32no—
76C2×C3×Dic3—61no—
77C2×C3×Dic3—61no—
78#627—38no—
79C32×C8—22no—
80C32×C8—22no—
81C3⋊(C3⋊C8)—22no—
82C3×C3⋊C8—34no—
83C3×C3⋊C8—62no—
84C3×C3⋊C8—62no—
85C8×Q8—271noSylow
86C2×H(3,𝔽3)C4×Q811yescharacteristic
87C2×H(3,𝔽3)C22×C811yescharacteristic
88C2×H(3,𝔽3)#13111yescharacteristic
89C2×Dic6—91no—
90C2×Dic6—92no—
91C2×C3×C8—32no—
92C2×C3×C8—61no—
93C2×C3×C8—91no—
94C2×C3×C8—92no—
95C2×C3×C8D3211yescharacteristic
96C3×C42—92no—
97C3×C42—91no—
98C4⋊Dic3—91no—
99C4⋊Dic3—92no—
100Dic3⋊C4—92no—
101Dic3⋊C4—92no—
102Dic3⋊C4—92no—
103C4×Dic3—32no—
104C4×Dic3—32no—
105C4×Dic3—61no—
106C4×Dic3—92no—
107C2×C3⋊C8—32no—
108C2×C3⋊C8—32no—
109C2×C3⋊C8—61no—
110C2×C3⋊C8—92no—
111C22×C32—21no—
112C22×C32C2×C4×D312yes—
113C32×C4—21no—
114C32×C4—21no—
115C32×C4#23712yes—
116C32×C4D3×Q812yes—
117C3⋊Dic3—22no—
118C3⋊Dic3D3×C814yes—
119C3×Dic3—34no—
120C3×Dic3—34no—
121C3×Dic3—62no—
122C3×Dic3—62no—
123C3×Dic3—62no—
124C3×Dic3—62no—
125C4×Q8—271no—
126C4⋊C8—92no—
127C4⋊C8—271no—
128C4×C8—91no—
129C4×C8—272no—
130H(3,𝔽3)C8×Q811yescharacteristic, complemented, Sylow
131C2×C3×C4—32no—
132C2×C3×C4—61no—
133C2×C3×C4—92no—
134C2×C3×C4—91no—
135C2×C3×C4—92no—
136C2×C3×C4—91no—
137C2×C3×C4C2×D3211yescharacteristic
138C2×Dic3—32no—
139C2×Dic3—32no—
140C2×Dic3—32no—
141C2×Dic3—32no—
142C2×Dic3—61no—
143C2×Dic3—61no—
144C2×Dic3—92no—
145C2×Dic3—92no—
146Dic6—94no—
147Dic6—98no—
148C3×C8—21no—
149C3×C8—62no—
150C3×C8—62no—
151C3×C8—92no—
152C3×C8—182no—
153C3⋊C8—34no—
154C3⋊C8—62no—
155C3⋊C8—62no—
156C3⋊C8—182no—
157C2×C32—21no—
158C2×C32—21no—
159C2×C32—21no—
160C2×C32#88712yes—
161C2×C32C2×D3×C812yes—
162C2×C32#90712yes—
163C2×Q8—271no—
164C2×C8—91no—
165C2×C8—92no—
166C2×C8((C3⋊D3)⋊C3)⋊C211yescharacteristic
167C4⋊C4—271no—
168C4⋊C4—272no—
169C42—91no—
170C42—92no—
171C22×C3—32no—
172C22×C3—61no—
173C22×C3C4×D3211yescharacteristic
174C3×C4—32no—
175C3×C4—32no—
176C3×C4—61no—
177C3×C4—61no—
178C3×C4—94no—
179C3×C4—92no—
180C3×C4—182no—
181C3×C4—181no—
182C3×C4#379711yescharacteristic
183C3×C4#380511yescharacteristic
184Dic3—34no—
185Dic3—34no—
186Dic3—62no—
187Dic3—62no—
188Dic3—62no—
189Dic3—62no—
190Dic3—94no—
191Dic3—182no—
192C32—21no—
193C32#483012yescomplemented
194Q8—274no—
195C2×C4—91no—
196C2×C4—92no—
197C2×C4—91no—
198C2×C4—92no—
199C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
200C8—21no—
201C8—92no—
202C8—182no—
203C2×C3—32no—
204C2×C3—32no—
205C2×C3—32no—
206C2×C3—61no—
207C2×C3—61no—
208C2×C3—61no—
209C2×C3D32×C811yescharacteristic
210C2×C3#6380511yescharacteristic
211C2×C3#6383811yescharacteristic
212C22C4×((C3⋊D3)⋊C3)⋊C211yescharacteristic
213C4—92no—
214C4—94no—
215C4—181no—
216C4—182no—
217C4#8865111yescharacteristic
218C4#8867211yescharacteristic
219C3—32no—
220C3—61no—
221C3#1059064511yescharacteristic
222C2C8×((C3⋊D3)⋊C3)⋊C211yescharacteristic
223C2#1172084211yescharacteristic
224C2#1172107811yescharacteristic
subgroupG/H
centerC2×C4C2×((C3⋊D3)⋊C3)⋊C2
commutatorC2×H(3,𝔽3)C22×C8
FrattiniC2×C3×C4C2×D32
FittingC2×C8×H(3,𝔽3)C22
socleC22×C3C4×D32
prime radicalC22×H(3,𝔽3)C22×C4
cosocleC2×C4×H(3,𝔽3)C23
hypercenterC2×C8((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)C8×Q8
LCSG ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C2×C4 ◁ C2×C8

Complements

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

NH ≅ G/Ntype
28. #6196C8semidirect
30. #6180Q8semidirect
130. H(3,𝔽3)C8×Q8semidirect
193. C32#4830semidirect

Maximal subgroups

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

subgroupindex
C4×#61962
C8×#10842
#63641⋊C32
#117203102
#117203042
#105906443