Finite Group Databasebeta

← #13136720

Subgroups — #13136720

Order 1728.

349 conjugacy classes of subgroups; only the 45 normal subgroups are shown.

normal subgroups only

This group has 349 conjugacy classes of subgroups — too many to draw. Only its 45 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
1C2×#88631C211yesmaximal, characteristic
2#11721298C211yesmaximal, characteristic
3C2×#88447C211yesmaximal, characteristic
4C22×#1084C2211yescharacteristic
5C22×(C3⋊Dic3)⋊C3—21no—
6#88631—21no—
7#88631C412yescomplemented
8C2×#6188—21no—
9#3757⋊C3—21no—
10#88449C412yescomplemented
11#88447—21no—
12#88447—21no—
13#88447C412yescomplemented
14#63955—61no—
15C2×Dic32—61no—
16C23×H(3,𝔽3)D411yescharacteristic
17C2×#1084C2×C411yescharacteristic
18C2×#1084C2×C411yescharacteristic
19C2×#1084C2×C411yescharacteristic
20C2×#1084D411yescharacteristic
21C2×#1084D411yescharacteristic
22C2×#1084Q811yescharacteristic
23C2×(C3⋊Dic3)⋊C3—24no—
24C2×(C3⋊Dic3)⋊C3—22no—
25C2×(C3⋊Dic3)⋊C3—22no—
26#6188—22no—
27#4555—91nomaximal
28C22×C3⋊Dic3—21no—
29C22×C3×Dic3—61no—
30C22×C3×Dic3—61no—
31C22×C3×Dic3—61no—
32#3757—21no—
33C3×#217—61no—
34Dic3⋊Dic3—62no—
35Dic32—62no—
36Dic32—62no—
37Dic32—62no—
38C22×H(3,𝔽3)—22no—
39C22×H(3,𝔽3)C22⋊C411yescharacteristic
40C22×H(3,𝔽3)C22⋊C411yescharacteristic
41C22×H(3,𝔽3)C4⋊C411yescharacteristic
42#1084C4211yescharacteristic
43#1084C22⋊C411yescharacteristic
44#1084C4⋊C411yescharacteristic
45#1084C4⋊C411yescharacteristic
46(C3⋊Dic3)⋊C3—24no—
47(C3⋊Dic3)⋊C3—42no—
48C2×C3×M16—91no—
49#926—91no—
50C2×C4×Dic3—91no—
51C2×C4×Dic3—181no—
52#877—181no—
53C23×C32—21no—
54C23×C32—21no—
55C2×C3⋊Dic3—22no—
56C2×C3⋊Dic3—24no—
57C2×C3⋊Dic3—22no—
58C2×C3×Dic3—61no—
59C2×C3×Dic3—61no—
60C2×C3×Dic3—61no—
61C2×C3×Dic3—61no—
62C2×C3×Dic3—61no—
63C2×C3×Dic3—61no—
64C2×C3×Dic3—61no—
65C2×C3×Dic3—61no—
66C2×C3×Dic3—61no—
67C2×C3×Dic3—61no—
68C2×C3×Dic3—61no—
69C2×C3×Dic3—61no—
70C2×C3×Dic3—62no—
71C2×C3×Dic3—64no—
72C2×C3×Dic3—62no—
73#339—271noSylow
74C2×H(3,𝔽3)—21no—
75C2×H(3,𝔽3)—21no—
76C2×H(3,𝔽3)#9511yescharacteristic
77C2×H(3,𝔽3)#10412yes—
78C22×C3×C4—91no—
79C22×C3×C4—181no—
80C22×Dic3—61no—
81C22×Dic3—61no—
82C22×Dic3—61no—
83C22×Dic3—181no—
84C3×M16—92no—
85C3×M16—181no—
86C2×C3×C8—181no—
87C3×C22⋊C4—181no—
88#217—61no—
89#217—181no—
90C4⋊Dic3—92no—
91Dic3⋊C4—182no—
92C4×Dic3—92no—
93C4×Dic3—182no—
94C4×Dic3—181no—
95C4×Dic3—181no—
96C4×Dic3—182no—
97C4×Dic3—182no—
98C22×C32—22no—
99C22×C32—22no—
100C22×C32—21no—
101C22×C32—21no—
102C22×C32—21no—
103C22×C32—21no—
104C22×C32—21no—
105C22×C32—21no—
106C22×C32—42no—
107C3⋊Dic3—24no—
108C3⋊Dic3—42no—
109C3×Dic3—61no—
110C3×Dic3—61no—
111C3×Dic3—61no—
112C3×Dic3—61no—
113C3×Dic3—61no—
114C3×Dic3—61no—
115C3×Dic3—61no—
116C3×Dic3—61no—
117C3×Dic3—64no—
118C3×Dic3—122no—
119C2×M16—91no—
120#117—271no—
121C2×C42—271no—
122H(3,𝔽3)#33911yescharacteristic, complemented, Sylow
123C23×C3—61no—
124C23×C3—61no—
125C23×C3GO+(4,𝔽2)11yescharacteristic
126C2×C3×C4—91no—
127C2×C3×C4—91no—
128C2×C3×C4—91no—
129C2×C3×C4—91no—
130C2×C3×C4—91no—
131C2×C3×C4—91no—
132C2×C3×C4—182no—
133C2×C3×C4—184no—
134C2×C3×C4—182no—
135C2×Dic3—61no—
136C2×Dic3—61no—
137C2×Dic3—61no—
138C2×Dic3—61no—
139C2×Dic3—61no—
140C2×Dic3—61no—
141C2×Dic3—61no—
142C2×Dic3—61no—
143C2×Dic3—61no—
144C2×Dic3—61no—
145C2×Dic3—61no—
146C2×Dic3—61no—
147C2×Dic3—64no—
148C2×Dic3—62no—
149C2×Dic3—62no—
150C2×Dic3—182no—
151C2×Dic3—184no—
152C2×Dic3—182no—
153C3×C8—182no—
154C2×C32—21no—
155C2×C32—21no—
156C2×C32—22no—
157C2×C32—21no—
158C2×C32—21no—
159C2×C32—22no—
160C2×C32—22no—
161C2×C32—41no—
162C2×C32—41no—
163C22×C4—91no—
164C22×C4—181no—
165M16—92no—
166M16—181no—
167C2×C8—181no—
168C4⋊C4—272no—
169C22⋊C4—181no—
170C42—272no—
171C42—541no—
172C42—541no—
173C22×C3—22no—
174C22×C3—62no—
175C22×C3—62no—
176C22×C3—61no—
177C22×C3—61no—
178C22×C3—61no—
179C22×C3—61no—
180C22×C3—61no—
181C22×C3—61no—
182C22×C3—122no—
183C22×C3D32⋊C411yescharacteristic
184C22×C3D32⋊C411yescharacteristic
185C22×C3#377311yescharacteristic
186C3×C4—91no—
187C3×C4—91no—
188C3×C4—91no—
189C3×C4—91no—
190C3×C4—184no—
191C3×C4—362no—
192Dic3—61no—
193Dic3—61no—
194Dic3—61no—
195Dic3—61no—
196Dic3—61no—
197Dic3—61no—
198Dic3—61no—
199Dic3—61no—
200Dic3—64no—
201Dic3—122no—
202Dic3—184no—
203Dic3—362no—
204C32—21no—
205C32—21no—
206C23#1088⋊C211yescharacteristic
207C2×C4—91no—
208C2×C4—91no—
209C2×C4—91no—
210C2×C4—91no—
211C2×C4—91no—
212C2×C4—91no—
213C2×C4—184no—
214C2×C4—182no—
215C2×C4—182no—
216C8—182no—
217C2×C3—21no—
218C2×C3—21no—
219C2×C3—61no—
220C2×C3—61no—
221C2×C3—62no—
222C2×C3—62no—
223C2×C3—62no—
224C2×C3—61no—
225C2×C3—61no—
226C2×C3—121no—
227C2×C3—121no—
228C2×C3#6373711yescharacteristic
229C2×C3#6374112yes—
230C22—22no—
231C22#8859011yescharacteristic
232C22(((C3⋊D3)⋊C3)⋊C2)⋊C411yescharacteristic
233C22(((C3⋊D3)⋊C3)⋊C2)⋊C411yescharacteristic
234C4—91no—
235C4—91no—
236C4—91no—
237C4—91no—
238C4—184no—
239C4—362no—
240C3—61no—
241C3—61no—
242C3#1059007411yescharacteristic
243C2—21no—
244C2—21no—
245C2#1172070612yes—
246C2#1172070711yescharacteristic
subgroupG/H
centerC22(((C3⋊D3)⋊C3)⋊C2)⋊C4
commutator#1084C42
FrattiniC23×C3GO+(4,𝔽2)
Fitting, prime radicalC23×H(3,𝔽3)D4
socleC22×C3D32⋊C4
cosocleC22×#1084C22
hypercenterC23#1088⋊C2
hypocenterH(3,𝔽3)#339
LCSG ▷ #1084 ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C23

Complements

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

NH ≅ G/Ntype
7. #88631C4semidirect
10. #88449C4semidirect
13. #88447C4semidirect
122. H(3,𝔽3)#339semidirect

Maximal subgroups

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

subgroupindex
C2×#886312
#117212982
C2×#884472
#45559