Finite Group Databasebeta

← #13144290

Subgroups — #13144290

Order 1728.

1199 conjugacy classes of subgroups; only the 229 normal subgroups are shown.

normal subgroups only

This group has 1199 conjugacy classes of subgroups — too many to draw. Only its 229 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
1C23×#1084C211yesmaximal, characteristic
2C2×C22⋊C4×H(3,𝔽3)C211yesmaximal, characteristic
3C2×#88552C211yesmaximal, characteristic
4#11720574C214yesmaximal, complemented
5C3×#4943—34nomaximal
6C24×H(3,𝔽3)C2211yescharacteristic
7C22×C4×H(3,𝔽3)C2212yes—
8C22×#1084—22no—
9C22×#1084—22no—
10C22×#1084C414yescomplemented
11C22×#1084C2212yes—
12C22×#1084C2212yes—
13C22⋊C4×H(3,𝔽3)—22no—
14#88552—22no—
15C23×C3×Dic3—34no—
16C2×C32×C22⋊C4D314yes—
17C2×C3×#217—34no—
18C3×C3⋊#95—316no—
19C23×H(3,𝔽3)—24no—
20C23×H(3,𝔽3)C2×C412yes—
21C23×H(3,𝔽3)D412yes—
22C23×H(3,𝔽3)D412yes—
23C23×H(3,𝔽3)C2311yescharacteristic
24C2×C4×H(3,𝔽3)—22no—
25C2×C4×H(3,𝔽3)—24no—
26C2×#1084—28no—
27C2×#1084—22no—
28C2×#1084—22no—
29C2×#1084—22no—
30C2×#1084—24no—
31C2×#1084C2×C414yes—
32C3×#381—91no—
33#4943—34no—
34C24×C32C2×D314yes—
35C22×C32×C4C2×D318yes—
36C22×C3×Dic3—316no—
37C22×C3×Dic3—38no—
38C22×C3×Dic3—38no—
39C22×C3×Dic3—68no—
40C22×C3×Dic3—68no—
41C32×C22⋊C4—28no—
42C3×#217—68no—
43C22×H(3,𝔽3)—24no—
44C22×H(3,𝔽3)—24no—
45C22×H(3,𝔽3)—24no—
46C22×H(3,𝔽3)C22⋊C414yes—
47C22×H(3,𝔽3)C22×C411yescharacteristic
48C22×H(3,𝔽3)C2×D411yescharacteristic
49C22×H(3,𝔽3)C2×D411yescharacteristic
50C22×H(3,𝔽3)#4114yes—
51C4×H(3,𝔽3)—42no—
52#1084—24no—
53#1084—42no—
54C23×C3×C4—91no—
55C23×Dic3—34no—
56C2×C3×C22⋊C4—34no—
57C2×C3×C22⋊C4—91no—
58C2×C3×C22⋊C4C3⋊D311yescharacteristic
59C2×#217—34no—
60C3×#95—94no—
61C3⋊#95—316no—
62C23×C32—216no—
63C23×C32C4×D318yes—
64C23×C32D1218yes—
65C23×C32#6718yes—
66C23×C32C22×D314yes—
67C2×C32×C4—216no—
68C2×C32×C4—28no—
69C2×C3×Dic3—316no—
70C2×C3×Dic3—632no—
71C2×C3×Dic3—68no—
72C2×C3×Dic3—68no—
73C2×C3×Dic3—68no—
74C2×C3×Dic3—616no—
75#381—91noSylow
76C2×H(3,𝔽3)—24no—
77C2×H(3,𝔽3)C2×C22⋊C411yescharacteristic
78C2×H(3,𝔽3)#11712yes—
79C2×H(3,𝔽3)#12312yes—
80C2×H(3,𝔽3)#12312yes—
81C24×C3—34no—
82C24×C3C2×C3⋊D311yescharacteristic
83C22×C3×C4—38no—
84C22×C3×C4—94no—
85C22×C3×C4—92no—
86C22×C3×C4—92no—
87C22×C3×C4—182no—
88C22×C3×C4—182no—
89C22×C3×C4C2×C3⋊D312yes—
90C22×Dic3—38no—
91C22×Dic3—316no—
92C22×Dic3—38no—
93C22×Dic3—68no—
94C22×Dic3—68no—
95C3×C22⋊C4—22no—
96C3×C22⋊C4—68no—
97C3×C22⋊C4—182no—
98#217—68no—
99C22×C32—216no—
100C22×C32—216no—
101C22×C32—216no—
102C22×C32(C2×D3)⋊C4116yes—
103C22×C32C2×C4×D314yes—
104C22×C32C2×D1214yes—
105C22×C32#237116yes—
106C22×C32C2×#6714yes—
107C32×C4—48no—
108C3×Dic3—616no—
109C3×Dic3—128no—
110C23×C4—91no—
111C2×C22⋊C4—91no—
112C2×C22⋊C4#27211yescharacteristic
113#95—94no—
114H(3,𝔽3)#38111yescharacteristic, complemented, Sylow
115C23×C3—24no—
116C23×C3—38no—
117C23×C3—38no—
118C23×C3—38no—
119C23×C3—34no—
120C23×C3—616no—
121C23×C3C4×C3⋊D312yes—
122C23×C3C4⋊(C3⋊D3)12yes—
123C23×C3#63812yes—
124C23×C3C22×C3⋊D311yescharacteristic
125C2×C3×C4—24no—
126C2×C3×C4—22no—
127C2×C3×C4—68no—
128C2×C3×C4—616no—
129C2×C3×C4—94no—
130C2×C3×C4—188no—
131C2×C3×C4—182no—
132C2×C3×C4—182no—
133C2×C3×C4—182no—
134C2×C3×C4—184no—
135C2×Dic3—316no—
136C2×Dic3—68no—
137C2×Dic3—68no—
138C2×Dic3—632no—
139C2×Dic3—68no—
140C2×Dic3—616no—
141C2×C32—216no—
142C2×C32#87718yes—
143C2×C32#88618yes—
144C2×C32C2×(C2×D3)⋊C414yes—
145C2×C32#93518yes—
146C24C2×#27211yescharacteristic
147C22×C4—94no—
148C22×C4—92no—
149C22×C4—92no—
150C22×C4—182no—
151C22×C4—182no—
152C22×C4C2×#27212yes—
153C22⋊C4—22no—
154C22⋊C4—182no—
155C22×C3—24no—
156C22×C3—24no—
157C22×C3—24no—
158C22×C3—34no—
159C22×C3—34no—
160C22×C3—316no—
161C22×C3—34no—
162C22×C3—316no—
163C22×C3—616no—
164C22×C3—616no—
165C22×C3—616no—
166C22×C3#375214yes—
167C22×C3C2×C4×C3⋊D311yescharacteristic
168C22×C3C2×C4⋊(C3⋊D3)11yescharacteristic
169C22×C3#383014yes—
170C22×C3C2×#63811yescharacteristic
171C3×C4—42no—
172C3×C4—128no—
173C3×C4—184no—
174C3×C4—362no—
175Dic3—616no—
176Dic3—128no—
177C32#494314yescomplemented
178C23—24no—
179C23C4×#27212yes—
180C23C4⋊#27212yes—
181C23#623512yes—
182C23C22×#27211yescharacteristic
183C2×C4—24no—
184C2×C4—22no—
185C2×C4—94no—
186C2×C4—182no—
187C2×C4—182no—
188C2×C4—188no—
189C2×C4—182no—
190C2×C4—184no—
191C2×C3—24no—
192C2×C3—38no—
193C2×C3—38no—
194C2×C3—34no—
195C2×C3—38no—
196C2×C3—616no—
197C2×C3#6408612yes—
198C2×C3#6409512yes—
199C2×C3C2×#375211yescharacteristic
200C2×C3#6414412yes—
201C22—24no—
202C22—24no—
203C22—24no—
204C22#8854214yes—
205C22C2×C4×#27211yescharacteristic
206C22C2×C4⋊#27211yescharacteristic
207C22#8874414yes—
208C22C2×#623511yescharacteristic
209C4—42no—
210C4—184no—
211C4—362no—
212C3—34no—
213C3#1059264211yescharacteristic
214C2—24no—
215C2#1172173312yes—
216C2#1172174212yes—
217C2C2×#8854211yescharacteristic
218C2#1172184512yes—
subgroupG/H
center, Frattini, socleC23×C3C22×C3⋊D3
commutatorC22×H(3,𝔽3)C22×C4
FittingC2×C22⋊C4×H(3,𝔽3)C2
prime radicalC24×H(3,𝔽3)C22
cosocleC23×H(3,𝔽3)C23
hypercenterC2×C3×C22⋊C4C3⋊D3
hypocenterH(3,𝔽3)#381
LCSG ▷ C22×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C23×C3 ◁ C2×C3×C22⋊C4

Complements

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

NH ≅ G/Ntype
4. #11720574C2semidirect
10. C22×#1084C4semidirect
114. H(3,𝔽3)#381semidirect
177. C32#4943semidirect

Maximal subgroups

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

subgroupindex
C23×#10842
C2×C22⋊C4×H(3,𝔽3)2
C2×#885522
#117205742
C3×#49433