Finite Group Databasebeta

← #13144373

Subgroups — #13144373

Order 1728.

935 conjugacy classes of subgroups; only the 167 normal subgroups are shown.

normal subgroups only

This group has 935 conjugacy classes of subgroups — too many to draw. Only its 167 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⋊C4×H(3,𝔽3)C211yesmaximal, characteristic, complemented
2C2×#88542C212yesmaximal
3C2×#88542C211yesmaximal, characteristic
4#11720574C212yesmaximal, complemented
5#11720574C211yesmaximal, characteristic, complemented
6C3×#4992—34nomaximal
7C23×#272C2211yescharacteristic
8C22×C4×H(3,𝔽3)C2212yes—
9C22×C4×H(3,𝔽3)C2211yescharacteristic
10C22×#1084C2212yes—
11C22×#1084C2211yescharacteristic
12C4⋊C4×H(3,𝔽3)—22no—
13#88542—24no—
14#88542—22no—
15C2×C32×C4⋊C4D314yescomplemented
16C2×C3×(C2×D3)⋊C4—38no—
17C2×C3×(C2×D3)⋊C4—34no—
18C3×C3⋊#95—38no—
19C3×C3⋊#95—34no—
20C23×H(3,𝔽3)C2311yescharacteristic
21C22×#272—21no—
22C22×#272—21no—
23C22×#272—22no—
24C22×#272—22no—
25C22×#272D412yes—
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)D412yes—
31C2×#1084—22no—
32C2×#1084—21no—
33C2×#1084—22no—
34C2×#1084—22no—
35C2×#1084—22no—
36C3×#397—91no—
37#4992—34no—
38C23×C3×D3—34no—
39C22×C32×C4C2×D318yes—
40C22×C32×C4C2×D314yes—
41C22×C3×Dic3—38no—
42C22×C3×Dic3—34no—
43C32×C4⋊C4—28no—
44C3×(C2×D3)⋊C4—616no—
45C3×(C2×D3)⋊C4—68no—
46C22×H(3,𝔽3)C2×D411yescharacteristic
47C22×H(3,𝔽3)C2×D411yescharacteristic
48C22×H(3,𝔽3)#4111yescharacteristic
49C22×H(3,𝔽3)#4112yes—
50C22×H(3,𝔽3)#4112yes—
51C2×#272—22no—
52C2×#272—24no—
53C2×#272—41no—
54C2×#272—41no—
55C2×#272—42no—
56C4×H(3,𝔽3)—22no—
57C4×H(3,𝔽3)—42no—
58#1084—42no—
59#1084—41no—
60C2×C3×C4⋊C4—34no—
61C2×C3×C4⋊C4C3⋊D311yescharacteristic
62C2×C3×C22⋊C4—92no—
63C2×C3×C22⋊C4—91no—
64C2×(C2×D3)⋊C4—34no—
65C2×(C2×D3)⋊C4—38no—
66C3×#95—92no—
67C3×#95—91no—
68C3⋊#95—34no—
69C3⋊#95—38no—
70C23×C32C22×D314yes—
71C22×C3×D3—38no—
72C22×C3×D3—68no—
73C22×C3×D3—64no—
74C22×C3×D3—68no—
75C22×C3×D3—64no—
76C2×C32×C4—28no—
77C2×C32×C4—28no—
78C2×C32×C4—28no—
79C2×C32×C4—28no—
80C2×C32×C4#6718yes—
81C2×C3×Dic3—68no—
82C2×C3×Dic3—68no—
83C2×C3×Dic3—68no—
84C2×C3×Dic3—68no—
85C2×C3×Dic3—64no—
86#397—91noSylow
87C2×H(3,𝔽3)#12111yescharacteristic
88C2×H(3,𝔽3)#12312yes—
89C2×H(3,𝔽3)#12311yescharacteristic
90C2×H(3,𝔽3)#12411yescharacteristic
91C2×H(3,𝔽3)#12612yes—
92#272—42no—
93C24×C3—91no—
94C23×D3—34no—
95C22×C3×C4—34no—
96C22×C3×C4—38no—
97C22×C3×C4—92no—
98C22×C3×C4—91no—
99C22×C3×C4C2×C3⋊D312yes—
100C22×C3×C4C2×C3⋊D311yescharacteristic
101C22×Dic3—38no—
102C22×Dic3—34no—
103C3×C4⋊C4—22no—
104C3×C4⋊C4—68no—
105C3×C22⋊C4—184no—
106C3×C22⋊C4—182no—
107(C2×D3)⋊C4—68no—
108(C2×D3)⋊C4—616no—
109C22×C32#23518yes—
110C22×C32D3×D414yes—
111C22×C32#23714yes—
112C22×C32#23918yes—
113C22×C32C2×#6714yes—
114C2×C3×D3—616no—
115C2×C3×D3—68no—
116C2×C3×D3—128no—
117C2×C3×D3—124no—
118C2×C3×D3—124no—
119C32×C4—28no—
120C32×C4—48no—
121C3×Dic3—128no—
122C3×Dic3—124no—
123C2×C4⋊C4#27211yescharacteristic, complemented
124C2×C22⋊C4—92no—
125C2×C22⋊C4—91no—
126#95—92no—
127#95—91no—
128H(3,𝔽3)#39711yescharacteristic, complemented, Sylow
129C23×C3—34no—
130C23×C3—92no—
131C23×C3—182no—
132C23×C3—181no—
133C23×C3—182no—
134C23×C3—181no—
135C23×C3C22×C3⋊D311yescharacteristic
136C22×D3—38no—
137C22×D3—64no—
138C22×D3—68no—
139C22×D3—64no—
140C22×D3—68no—
141C2×C3×C4—22no—
142C2×C3×C4—22no—
143C2×C3×C4—22no—
144C2×C3×C4—22no—
145C2×C3×C4—38no—
146C2×C3×C4—68no—
147C2×C3×C4—68no—
148C2×C3×C4—68no—
149C2×C3×C4—68no—
150C2×C3×C4—182no—
151C2×C3×C4—182no—
152C2×C3×C4—182no—
153C2×C3×C4—182no—
154C2×C3×C4—181no—
155C2×C3×C4#63812yes—
156C2×Dic3—68no—
157C2×Dic3—64no—
158C2×Dic3—68no—
159C2×Dic3—68no—
160C2×Dic3—68no—
161C2×C32#89418yes—
162C2×C32#89818yes—
163C2×C32#92914yes—
164C2×C32#93814yes—
165C2×C32#94714yes—
166C3×D3—128no—
167C24—91no—
168C22×C4—92no—
169C22×C4—91no—
170C22×C4C2×#27212yes—
171C22×C4C2×#27211yescharacteristic
172C4⋊C4—22no—
173C22⋊C4—184no—
174C22⋊C4—182no—
175C22×C3—38no—
176C22×C3—34no—
177C22×C3—34no—
178C22×C3—38no—
179C22×C3—34no—
180C22×C3—184no—
181C22×C3—182no—
182C22×C3—362no—
183C22×C3—361no—
184C22×C3—361no—
185C22×C3#382812yes—
186C22×C3D4×C3⋊D311yescharacteristic
187C22×C3#383011yescharacteristic
188C22×C3#383212yes—
189C22×C3C2×#63811yescharacteristic
190C2×D3—616no—
191C2×D3—68no—
192C2×D3—128no—
193C2×D3—124no—
194C2×D3—124no—
195C3×C4—22no—
196C3×C4—42no—
197C3×C4—68no—
198C3×C4—128no—
199C3×C4—362no—
200C3×C4—361no—
201Dic3—128no—
202Dic3—124no—
203C32#499214yescomplemented
204C23—92no—
205C23—182no—
206C23—181no—
207C23—182no—
208C23—181no—
209C23C22×#27211yescharacteristic
210C2×C4—22no—
211C2×C4—22no—
212C2×C4—22no—
213C2×C4—22no—
214C2×C4—182no—
215C2×C4—182no—
216C2×C4—182no—
217C2×C4—182no—
218C2×C4—181no—
219C2×C4#623512yes—
220C2×C3—38no—
221C2×C3—38no—
222C2×C3—34no—
223C2×C3—34no—
224C2×C3—34no—
225C2×C3—362no—
226C2×C3#6410312yes—
227C2×C3#6410712yes—
228C2×C3#6413811yescharacteristic
229C2×C3#6414711yescharacteristic
230C2×C3#6415611yescharacteristic
231D3—128no—
232C22—184no—
233C22—182no—
234C22—362no—
235C22—361no—
236C22—361no—
237C22#8874012yes—
238C22D4×#27211yescharacteristic
239C22#8874411yescharacteristic
240C22#8874812yes—
241C22C2×#623511yescharacteristic
242C4—22no—
243C4—42no—
244C4—362no—
245C4—361no—
246C3—34no—
247C3#1059269111yescharacteristic
248C2—362no—
249C2#1172176212yes—
250C2#1172176612yes—
251C2#1172182911yescharacteristic
252C2#1172184811yescharacteristic
253C2#1172186411yescharacteristic
subgroupG/H
center, Frattini, socleC23×C3C22×C3⋊D3
commutator, cosocleC23×H(3,𝔽3)C23
FittingC2×C4⋊C4×H(3,𝔽3)C2
prime radicalC23×#272C22
hypercenterC2×C3×C4⋊C4C3⋊D3
hypocenterH(3,𝔽3)#397
LCSG ▷ C23×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C23×C3 ◁ C2×C3×C4⋊C4

Complements

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

NH ≅ G/Ntype
1. C2×C4⋊C4×H(3,𝔽3)C2semidirect
4. #11720574C2semidirect
5. #11720574C2semidirect
15. C2×C32×C4⋊C4D3semidirect
123. C2×C4⋊C4#272semidirect
128. H(3,𝔽3)#397semidirect
203. C32#4992semidirect

Maximal subgroups

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

subgroupindex
C2×C4⋊C4×H(3,𝔽3)2
C2×#885422
#117205742
C3×#49923