Finite Group Databasebeta

← #13143730

Subgroups — #13143730

Order 1728.

1199 conjugacy classes of subgroups; only the 181 normal subgroups are shown.

normal subgroups only

This group has 1199 conjugacy classes of subgroups — too many to draw. Only its 181 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
1C22×C4×#272C211yesmaximal, characteristic
2C2×#88542C212yesmaximal
3H(3,𝔽3)×#95C211yesmaximal, characteristic, complemented
4#11720574C211yesmaximal, characteristic, complemented
5#11720574C212yesmaximal, complemented
6C3×#4664—34nomaximal
7C23×#272C2211yescharacteristic
8C22×C4×H(3,𝔽3)C2212yes—
9C22×C4×H(3,𝔽3)C2211yescharacteristic
10C22×#1084C2212yes—
11C22×#1084C2211yescharacteristic
12C2×C4×#272—24no—
13C2×C4×#272C414yescomplemented
14#88542—24no—
15C22×C3×C4×D3—34no—
16C2×C3×(C2×D3)⋊C4—38no—
17C32×#95D314yescomplemented
18C3×C3⋊#95—38no—
19C3×C3⋊#95—34no—
20C23×H(3,𝔽3)C2311yescharacteristic
21C22×#272—22no—
22C22×#272—21no—
23C22×#272—21no—
24C22×#272C2×C412yes—
25C22×#272D414yes—
26C2×C4×H(3,𝔽3)—22no—
27C2×C4×H(3,𝔽3)—22no—
28C2×C4×H(3,𝔽3)—22no—
29C2×C4×H(3,𝔽3)—22no—
30C2×C4×H(3,𝔽3)C2×C412yes—
31C2×#1084—22no—
32C2×#1084—22no—
33C2×#1084—22no—
34C2×#1084—22no—
35C2×#1084C2×C412yes—
36C4×#272—28no—
37C3×#381—91no—
38#4664—34no—
39C23×C3×D3—34no—
40C22×C32×C4C2×D318yes—
41C22×C32×C4C2×D314yes—
42C22×C3×Dic3—34no—
43C22×C3×Dic3—38no—
44C2×C3×C4×D3—316no—
45C2×C3×C4×D3—616no—
46C3×(C2×D3)⋊C4—616no—
47C22×H(3,𝔽3)C22×C411yescharacteristic
48C22×H(3,𝔽3)C2×D412yes—
49C22×H(3,𝔽3)#4112yes—
50C22×H(3,𝔽3)#4111yescharacteristic
51C22×H(3,𝔽3)#4111yescharacteristic
52C2×#272—22no—
53C2×#272—22no—
54C2×#272—24no—
55C2×#272—24no—
56C2×#272C22⋊C414yes—
57C4×H(3,𝔽3)—22no—
58C4×H(3,𝔽3)—42no—
59#1084—22no—
60#1084—42no—
61C23×C3×C4—91no—
62C22×C4×D3—34no—
63C2×C3×C22⋊C4—92no—
64C2×(C2×D3)⋊C4—38no—
65C3×#95—34no—
66C3×#95—92no—
67C3×#95—91no—
68C3×#95C3⋊D311yescharacteristic
69C3⋊#95—38no—
70C3⋊#95—34no—
71C23×C32C22×D314yes—
72C22×C3×D3—38no—
73C22×C3×D3—316no—
74C22×C3×D3—68no—
75C22×C3×D3—64no—
76C22×C3×D3—64no—
77C2×C32×C4—28no—
78C2×C32×C4—28no—
79C2×C32×C4—28no—
80C2×C32×C4—28no—
81C2×C32×C4C4×D318yes—
82C2×C3×Dic3—38no—
83C2×C3×Dic3—68no—
84C2×C3×Dic3—68no—
85C2×C3×Dic3—68no—
86C2×C3×Dic3—68no—
87C3×C4×D3—632no—
88#381—91noSylow
89C2×H(3,𝔽3)C2×C22⋊C411yescharacteristic
90C2×H(3,𝔽3)#11711yescharacteristic
91C2×H(3,𝔽3)#11711yescharacteristic
92C2×H(3,𝔽3)#12312yes—
93C2×H(3,𝔽3)#12312yes—
94#272—24no—
95C24×C3—91no—
96C23×D3—34no—
97C22×C3×C4—34no—
98C22×C3×C4—38no—
99C22×C3×C4—91no—
100C22×C3×C4—94no—
101C22×C3×C4—92no—
102C22×C3×C4—184no—
103C22×C3×C4C2×C3⋊D312yes—
104C22×C3×C4C2×C3⋊D311yescharacteristic
105C22×Dic3—34no—
106C22×Dic3—38no—
107C2×C4×D3—316no—
108C2×C4×D3—616no—
109C3×C22⋊C4—184no—
110(C2×D3)⋊C4—616no—
111C22×C32C2×C4×D314yes—
112C22×C32#23518yes—
113C22×C32D3×D418yes—
114C22×C32#23714yes—
115C22×C32#23914yes—
116C2×C3×D3—316no—
117C2×C3×D3—68no—
118C2×C3×D3—616no—
119C2×C3×D3—68no—
120C2×C3×D3—616no—
121C32×C4—28no—
122C32×C4—48no—
123C3×Dic3—68no—
124C3×Dic3—128no—
125C23×C4—91no—
126C2×C22⋊C4—92no—
127#95—92no—
128#95—91no—
129#95#27211yescharacteristic, complemented
130H(3,𝔽3)#38111yescharacteristic, complemented, Sylow
131C23×C3—34no—
132C23×C3—92no—
133C23×C3—94no—
134C23×C3—182no—
135C23×C3—181no—
136C23×C3—181no—
137C23×C3C22×C3⋊D311yescharacteristic
138C22×D3—316no—
139C22×D3—38no—
140C22×D3—68no—
141C22×D3—64no—
142C22×D3—64no—
143C2×C3×C4—22no—
144C2×C3×C4—22no—
145C2×C3×C4—22no—
146C2×C3×C4—22no—
147C2×C3×C4—38no—
148C2×C3×C4—68no—
149C2×C3×C4—68no—
150C2×C3×C4—68no—
151C2×C3×C4—68no—
152C2×C3×C4—92no—
153C2×C3×C4—188no—
154C2×C3×C4—182no—
155C2×C3×C4—182no—
156C2×C3×C4—182no—
157C2×C3×C4—182no—
158C2×C3×C4C4×C3⋊D312yes—
159C2×Dic3—38no—
160C2×Dic3—68no—
161C2×Dic3—68no—
162C2×Dic3—68no—
163C2×Dic3—68no—
164C4×D3—632no—
165C2×C32#87214yes—
166C2×C32D3×C22⋊C414yes—
167C2×C32#88318yes—
168C2×C32#89214yes—
169C2×C32#89418yes—
170C3×D3—616no—
171C24—91no—
172C22×C4—91no—
173C22×C4—94no—
174C22×C4—92no—
175C22×C4—184no—
176C22×C4C2×#27212yes—
177C22×C4C2×#27211yescharacteristic
178C22⋊C4—184no—
179C22×C3—34no—
180C22×C3—38no—
181C22×C3—38no—
182C22×C3—34no—
183C22×C3—34no—
184C22×C3—94no—
185C22×C3—182no—
186C22×C3—184no—
187C22×C3—182no—
188C22×C3—184no—
189C22×C3C2×C4×C3⋊D311yescharacteristic
190C22×C3#382812yes—
191C22×C3D4×C3⋊D312yes—
192C22×C3#383011yescharacteristic
193C22×C3#383211yescharacteristic
194C2×D3—316no—
195C2×D3—616no—
196C2×D3—68no—
197C2×D3—616no—
198C2×D3—68no—
199C3×C4—22no—
200C3×C4—42no—
201C3×C4—68no—
202C3×C4—128no—
203C3×C4—182no—
204C3×C4—362no—
205Dic3—68no—
206Dic3—128no—
207C32#466414yescomplemented
208C23—92no—
209C23—94no—
210C23—182no—
211C23—181no—
212C23—181no—
213C23C22×#27211yescharacteristic
214C2×C4—22no—
215C2×C4—22no—
216C2×C4—22no—
217C2×C4—22no—
218C2×C4—92no—
219C2×C4—188no—
220C2×C4—182no—
221C2×C4—182no—
222C2×C4—182no—
223C2×C4—182no—
224C2×C4C4×#27212yes—
225C2×C3—34no—
226C2×C3—38no—
227C2×C3—34no—
228C2×C3—38no—
229C2×C3—34no—
230C2×C3—184no—
231C2×C3#6408111yescharacteristic
232C2×C3C22⋊C4×C3⋊D311yescharacteristic
233C2×C3#6409212yes—
234C2×C3#6410111yescharacteristic
235C2×C3#6410312yes—
236D3—616no—
237C22—94no—
238C22—184no—
239C22—182no—
240C22—184no—
241C22—182no—
242C22C2×C4×#27211yescharacteristic
243C22#8874012yes—
244C22D4×#27212yes—
245C22#8874411yescharacteristic
246C22#8874811yescharacteristic
247C4—22no—
248C4—42no—
249C4—182no—
250C4—362no—
251C3—34no—
252C3#1059236311yescharacteristic
253C2—184no—
254C2#1172171811yescharacteristic
255C2C22⋊C4×#27211yescharacteristic
256C2#1172173912yes—
257C2#1172176011yescharacteristic
258C2#1172176212yes—
subgroupG/H
center, Frattini, socleC23×C3C22×C3⋊D3
commutatorC22×H(3,𝔽3)C22×C4
FittingH(3,𝔽3)×#95C2
prime radicalC23×#272C22
cosocleC23×H(3,𝔽3)C23
hypercenterC3×#95C3⋊D3
hypocenterH(3,𝔽3)#381
LCSG ▷ C22×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C23×C3 ◁ C3×#95

Complements

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

NH ≅ G/Ntype
3. H(3,𝔽3)×#95C2semidirect
4. #11720574C2semidirect
5. #11720574C2semidirect
13. C2×C4×#272C4semidirect
17. C32×#95D3semidirect
129. #95#272semidirect
130. H(3,𝔽3)#381semidirect
207. C32#4664semidirect

Maximal subgroups

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

subgroupindex
C22×C4×#2722
C2×#885422
H(3,𝔽3)×#952
#117205742
C3×#46643