Finite Group Databasebeta

← #13140008

Subgroups — #13140008

Order 1728.

591 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 591 conjugacy classes of subgroups — too many to draw. Only its 123 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×C4×#1084C211yesmaximal, characteristic
2#63658⋊C3C212yesmaximal
3#11720334C212yesmaximal
4#11720334C212yesmaximal
5#10591223—32nomaximal
6C22×C4×H(3,𝔽3)C2211yescharacteristic
7C22×#1084C2211yescharacteristic
8C22×#1084C2211yescharacteristic
9C22×(C3⋊Dic3)⋊C3C2214yes—
10C4×#1084C414yescomplemented
11C2×C3×C4×Dic3—32no—
12C2×C3×C4×Dic3—61no—
13#63658D312yes—
14C3×C3⋊#95—32no—
15#63579—34no—
16#63579—34no—
17C23×H(3,𝔽3)C2311yescharacteristic
18C2×C4×H(3,𝔽3)—22no—
19C2×C4×H(3,𝔽3)C2×C412yes—
20C2×#1084—22no—
21C2×#1084—22no—
22C2×#1084C2×C412yes—
23C2×#1084C2×C412yes—
24C2×(C3⋊Dic3)⋊C3—24no—
25C2×(C3⋊Dic3)⋊C3—24no—
26C2×(C3⋊Dic3)⋊C3—24no—
27#4937—91no—
28#4651—92no—
29C22×C32×C4—21no—
30C22×C32×C4C2×D312yes—
31C22×C3⋊Dic3C2×D314yes—
32C22×C3×Dic3—32no—
33C22×C3×Dic3—32no—
34C22×C3×Dic3—34no—
35C22×C3×Dic3—61no—
36C22×C3×Dic3—61no—
37C3×C4×Dic3—38no—
38C3×C4×Dic3—64no—
39C22×H(3,𝔽3)C22×C411yescharacteristic
40C22×H(3,𝔽3)#4112yes—
41C22×H(3,𝔽3)#4112yes—
42C22×H(3,𝔽3)#4112yes—
43C4×H(3,𝔽3)—22no—
44#1084—22no—
45#1084—22no—
46(C3⋊Dic3)⋊C3—44no—
47C2×C3×C42—91no—
48C2×C4×Dic3—32no—
49C2×C4×Dic3—61no—
50C3×#95—92no—
51C3⋊#95—32no—
52C3⋊#95—92no—
53C3⋊#95—92no—
54C3⋊#95—92no—
55C3⋊#95—94no—
56C3⋊#95—94no—
57C23×C32—21no—
58C23×C32C22×D312yes—
59C2×C32×C4—24no—
60C2×C32×C4—24no—
61C2×C32×C4—22no—
62C2×C32×C4C4×D314yes—
63C2×C3⋊Dic3—24no—
64C2×C3⋊Dic3—24no—
65C2×C3⋊Dic3—24no—
66C2×C3×Dic3—34no—
67C2×C3×Dic3—34no—
68C2×C3×Dic3—62no—
69C2×C3×Dic3—62no—
70C2×C3×Dic3—64no—
71C2×C3×Dic3—64no—
72C2×C3×Dic3—64no—
73C2×C3×Dic3—62no—
74C2×C3×Dic3—62no—
75C2×C3×Dic3—64no—
76C2×C3×Dic3—64no—
77C2×C3×Dic3—64no—
78#383—271noSylow
79C2×H(3,𝔽3)#11711yescharacteristic
80C2×H(3,𝔽3)#11711yescharacteristic
81C2×H(3,𝔽3)#11711yescharacteristic
82C2×H(3,𝔽3)#12614yes—
83C22×C3×C4—32no—
84C22×C3×C4—61no—
85C22×C3×C4—91no—
86C22×C3×C4—91no—
87C22×C3×C4—94no—
88C22×C3×C4D3211yescharacteristic
89C22×Dic3—32no—
90C22×Dic3—32no—
91C22×Dic3—34no—
92C22×Dic3—61no—
93C22×Dic3—61no—
94C22×Dic3—94no—
95C3×C42—94no—
96C4×Dic3—38no—
97C4×Dic3—64no—
98C22×C32—22no—
99C22×C32—22no—
100C22×C32—22no—
101C22×C32—21no—
102C22×C32C2×C4×D312yes—
103C22×C32#23514yes—
104C22×C32#23712yes—
105C22×C32#23714yes—
106C22×C32#23912yes—
107C32×C4—24no—
108C32×C4—24no—
109C3⋊Dic3—44no—
110C3×Dic3—64no—
111C3×Dic3—64no—
112C3×Dic3—64no—
113C3×Dic3—64no—
114C3×Dic3—124no—
115C2×C42—91no—
116#95—92no—
117#95—272no—
118#95—272no—
119H(3,𝔽3)#38311yescharacteristic, complemented, Sylow
120C23×C3—32no—
121C23×C3—61no—
122C23×C3C2×D3211yescharacteristic
123C2×C3×C4—22no—
124C2×C3×C4—34no—
125C2×C3×C4—64no—
126C2×C3×C4—64no—
127C2×C3×C4—62no—
128C2×C3×C4—92no—
129C2×C3×C4—92no—
130C2×C3×C4—182no—
131C2×C3×C4—182no—
132C2×C3×C4—184no—
133C2×C3×C4—184no—
134C2×C3×C4—184no—
135C2×C3×C4#62412yes—
136C2×Dic3—34no—
137C2×Dic3—34no—
138C2×Dic3—62no—
139C2×Dic3—62no—
140C2×Dic3—62no—
141C2×Dic3—62no—
142C2×Dic3—64no—
143C2×Dic3—64no—
144C2×Dic3—64no—
145C2×Dic3—64no—
146C2×Dic3—64no—
147C2×Dic3—64no—
148C2×Dic3—184no—
149C2×Dic3—184no—
150C2×Dic3—184no—
151C2×C32—21no—
152C2×C32—24no—
153C2×C32—21no—
154C2×C32—21no—
155C2×C32#87212yes—
156C2×C32#87712yes—
157C2×C32#87914yes—
158C2×C32#89212yes—
159C2×C32#89814yes—
160C22×C4—91no—
161C22×C4—91no—
162C22×C4—94no—
163C22×C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
164C42—94no—
165C22×C3—32no—
166C22×C3—32no—
167C22×C3—34no—
168C22×C3—34no—
169C22×C3—32no—
170C22×C3—62no—
171C22×C3—62no—
172C22×C3—62no—
173C22×C3—61no—
174C22×C3#379612yes—
175C22×C3#379812yes—
176C22×C3#380512yes—
177C22×C3C2×#62411yescharacteristic
178C3×C4—22no—
179C3×C4—64no—
180C3×C4—64no—
181C3×C4—182no—
182C3×C4—182no—
183C3×C4—364no—
184Dic3—64no—
185Dic3—64no—
186Dic3—64no—
187Dic3—64no—
188Dic3—124no—
189Dic3—364no—
190C32—21no—
191C32#465112yescomplemented
192C23C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
193C2×C4—22no—
194C2×C4—92no—
195C2×C4—92no—
196C2×C4—182no—
197C2×C4—182no—
198C2×C4—184no—
199C2×C4—184no—
200C2×C4—184no—
201C2×C4#619912yes—
202C2×C3—32no—
203C2×C3—32no—
204C2×C3—32no—
205C2×C3—34no—
206C2×C3—34no—
207C2×C3—61no—
208C2×C3—61no—
209C2×C3—61no—
210C2×C3—64no—
211C2×C3#6384911yescharacteristic
212C2×C3#6386214yes—
213C2×C3#6387411yescharacteristic
214C2×C3#6395711yescharacteristic
215C22#8864912yes—
216C22#8865012yes—
217C22#8867212yes—
218C22C2×#619911yescharacteristic
219C4—22no—
220C4—182no—
221C4—182no—
222C4—364no—
223C3—32no—
224C3—61no—
225C3#1059122911yescharacteristic
226C2#1172108911yescharacteristic
227C2#1172109711yescharacteristic
228C2#1172111214yes—
229C2#1172129811yescharacteristic
subgroupG/H
centerC23C2×((C3⋊D3)⋊C3)⋊C2
commutatorC22×H(3,𝔽3)C22×C4
Frattini, socleC23×C3C2×D32
FittingC22×C4×H(3,𝔽3)C22
prime radical, cosocleC23×H(3,𝔽3)C23
hypercenterC22×C4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#383
LCSG ▷ C22×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C23 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
10. C4×#1084C4semidirect
119. H(3,𝔽3)#383semidirect
191. C32#4651semidirect

Maximal subgroups

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

subgroupindex
C2×C4×#10842
#63658⋊C32
#117203342
#105912233