Finite Group Databasebeta

← #13144299

Subgroups — #13144299

Order 1728.

935 conjugacy classes of subgroups; only the 179 normal subgroups are shown.

normal subgroups only

This group has 935 conjugacy classes of subgroups — too many to draw. Only its 179 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×C22⋊C4×H(3,𝔽3)C211yesmaximal, characteristic
2C2×#88552C212yesmaximal
3C2×C4⋊#1084C211yesmaximal, characteristic, complemented
4#11720574C212yesmaximal, complemented
5#11720574C211yesmaximal, characteristic, complemented
6C3×#4953—34nomaximal
7C24×H(3,𝔽3)C2211yescharacteristic
8C22×C4×H(3,𝔽3)C2211yescharacteristic
9C22×C4×H(3,𝔽3)C2211yescharacteristic
10C22×#1084C2212yes—
11C22×#1084C2212yes—
12C22⋊C4×H(3,𝔽3)—22no—
13#88552—24no—
14C4⋊#1084—22no—
15C2×C32×C22⋊C4D314yes—
16C2×C3×#217—38no—
17C2×C3×C4⋊Dic3—34no—
18C3×C3⋊#95—38no—
19C3×C3⋊#95—34no—
20C23×H(3,𝔽3)—21no—
21C23×H(3,𝔽3)—21no—
22C23×H(3,𝔽3)—22no—
23C23×H(3,𝔽3)—22no—
24C23×H(3,𝔽3)D412yes—
25C23×H(3,𝔽3)C2311yescharacteristic
26C2×C4×H(3,𝔽3)—22no—
27C2×C4×H(3,𝔽3)—21no—
28C2×C4×H(3,𝔽3)—22no—
29C2×C4×H(3,𝔽3)D412yes—
30C2×#1084—22no—
31C2×#1084—22no—
32C2×#1084—22no—
33C2×#1084—22no—
34C2×#1084—22no—
35C2×#1084—22no—
36C3×#397—91no—
37#4953—34no—
38C24×C32C2×D314yes—
39C22×C32×C4C2×D314yes—
40C22×C32×C4C2×D314yes—
41C22×C3×Dic3—38no—
42C22×C3×Dic3—38no—
43C32×C22⋊C4—28no—
44C3×#217—616no—
45C3×C4⋊Dic3—68no—
46C22×H(3,𝔽3)—22no—
47C22×H(3,𝔽3)—24no—
48C22×H(3,𝔽3)—41no—
49C22×H(3,𝔽3)—41no—
50C22×H(3,𝔽3)—42no—
51C22×H(3,𝔽3)C2×D411yescharacteristic
52C22×H(3,𝔽3)C2×D411yescharacteristic
53C22×H(3,𝔽3)#4111yescharacteristic
54C22×H(3,𝔽3)#4112yes—
55C22×H(3,𝔽3)#4112yes—
56C4×H(3,𝔽3)—22no—
57C4×H(3,𝔽3)—41no—
58#1084—42no—
59#1084—42no—
60C2×C3×C4⋊C4—91no—
61C2×C3×C22⋊C4—34no—
62C2×C3×C22⋊C4—92no—
63C2×C3×C22⋊C4C3⋊D311yescharacteristic
64C2×#217—38no—
65C2×C4⋊Dic3—34no—
66C3×#95—91no—
67C3×#95—92no—
68C3⋊#95—34no—
69C3⋊#95—38no—
70C23×C32—24no—
71C23×C32—28no—
72C23×C32—28no—
73C23×C32—24no—
74C23×C32D1218yes—
75C23×C32C22×D314yes—
76C2×C32×C4—28no—
77C2×C32×C4—28no—
78C2×C32×C4—24no—
79C2×C32×C4#6718yes—
80C2×C3×Dic3—68no—
81C2×C3×Dic3—68no—
82C2×C3×Dic3—68no—
83C2×C3×Dic3—68no—
84C2×C3×Dic3—68no—
85C2×C3×Dic3—68no—
86#397—91noSylow
87C2×H(3,𝔽3)—42no—
88C2×H(3,𝔽3)#12111yescharacteristic
89C2×H(3,𝔽3)#12312yes—
90C2×H(3,𝔽3)#12311yescharacteristic
91C2×H(3,𝔽3)#12411yescharacteristic
92C2×H(3,𝔽3)#12612yes—
93C24×C3—34no—
94C24×C3C2×C3⋊D311yescharacteristic
95C22×C3×C4—34no—
96C22×C3×C4—34no—
97C22×C3×C4—92no—
98C22×C3×C4—92no—
99C22×C3×C4C2×C3⋊D311yescharacteristic
100C22×C3×C4C2×C3⋊D311yescharacteristic
101C22×Dic3—38no—
102C22×Dic3—38no—
103C3×C4⋊C4—182no—
104C3×C22⋊C4—22no—
105C3×C22⋊C4—68no—
106C3×C22⋊C4—184no—
107#217—616no—
108C4⋊Dic3—68no—
109C22×C32—216no—
110C22×C32—28no—
111C22×C32—48no—
112C22×C32—44no—
113C22×C32—44no—
114C22×C32C2×D1214yes—
115C22×C32#23514yes—
116C22×C32#23718yes—
117C22×C32#23718yes—
118C22×C32C2×#6714yes—
119C32×C4—28no—
120C32×C4—44no—
121C3×Dic3—128no—
122C3×Dic3—128no—
123C2×C4⋊C4—91no—
124C2×C22⋊C4—92no—
125C2×C22⋊C4#27211yescharacteristic
126#95—91no—
127#95—92no—
128H(3,𝔽3)#39711yescharacteristic, complemented, Sylow
129C23×C3—21no—
130C23×C3—22no—
131C23×C3—22no—
132C23×C3—21no—
133C23×C3—38no—
134C23×C3—34no—
135C23×C3—64no—
136C23×C3—64no—
137C23×C3—68no—
138C23×C3—68no—
139C23×C3C4⋊(C3⋊D3)12yes—
140C23×C3C22×C3⋊D311yescharacteristic
141C2×C3×C4—22no—
142C2×C3×C4—22no—
143C2×C3×C4—21no—
144C2×C3×C4—38no—
145C2×C3×C4—68no—
146C2×C3×C4—64no—
147C2×C3×C4—68no—
148C2×C3×C4—182no—
149C2×C3×C4—182no—
150C2×C3×C4—182no—
151C2×C3×C4—182no—
152C2×C3×C4—182no—
153C2×C3×C4—182no—
154C2×C3×C4#63812yes—
155C2×Dic3—68no—
156C2×Dic3—68no—
157C2×Dic3—68no—
158C2×Dic3—68no—
159C2×Dic3—68no—
160C2×Dic3—68no—
161C2×C32—48no—
162C2×C32#87918yes—
163C2×C32#88618yes—
164C2×C32#93014yes—
165C2×C32#93514yes—
166C2×C32#93614yes—
167C24C2×#27211yescharacteristic
168C22×C4—92no—
169C22×C4—92no—
170C22×C4C2×#27211yescharacteristic
171C22×C4C2×#27211yescharacteristic
172C4⋊C4—182no—
173C22⋊C4—22no—
174C22⋊C4—184no—
175C22×C3—24no—
176C22×C3—22no—
177C22×C3—34no—
178C22×C3—34no—
179C22×C3—38no—
180C22×C3—38no—
181C22×C3—34no—
182C22×C3—42no—
183C22×C3—41no—
184C22×C3—41no—
185C22×C3—616no—
186C22×C3—68no—
187C22×C3—128no—
188C22×C3—124no—
189C22×C3—124no—
190C22×C3C2×C4⋊(C3⋊D3)11yescharacteristic
191C22×C3#382811yescharacteristic
192C22×C3#383012yes—
193C22×C3#383012yes—
194C22×C3C2×#63811yescharacteristic
195C3×C4—22no—
196C3×C4—41no—
197C3×C4—68no—
198C3×C4—124no—
199C3×C4—362no—
200C3×C4—362no—
201Dic3—128no—
202Dic3—128no—
203C32#495314yescomplemented
204C23—21no—
205C23—22no—
206C23—22no—
207C23—21no—
208C23C4⋊#27212yes—
209C23C22×#27211yescharacteristic
210C2×C4—22no—
211C2×C4—22no—
212C2×C4—21no—
213C2×C4—182no—
214C2×C4—182no—
215C2×C4—182no—
216C2×C4—182no—
217C2×C4—182no—
218C2×C4—182no—
219C2×C4#623512yes—
220C2×C3—38no—
221C2×C3—38no—
222C2×C3—34no—
223C2×C3—34no—
224C2×C3—34no—
225C2×C3—42no—
226C2×C3—128no—
227C2×C3#6408812yes—
228C2×C3#6409512yes—
229C2×C3#6413911yescharacteristic
230C2×C3#6414411yescharacteristic
231C2×C3#6414511yescharacteristic
232C22—24no—
233C22—22no—
234C22—42no—
235C22—41no—
236C22—41no—
237C22C2×C4⋊#27211yescharacteristic
238C22#8874011yescharacteristic
239C22#8874412yes—
240C22#8874412yes—
241C22C2×#623511yescharacteristic
242C4—22no—
243C4—41no—
244C4—362no—
245C4—362no—
246C3—34no—
247C3#1059265211yescharacteristic
248C2—42no—
249C2#1172173512yes—
250C2#1172174212yes—
251C2#1172183011yescharacteristic
252C2#1172184511yescharacteristic
253C2#1172184611yescharacteristic
subgroupG/H
center, Frattini, socleC23×C3C22×C3⋊D3
commutator, cosocleC23×H(3,𝔽3)C23
FittingC2×C22⋊C4×H(3,𝔽3)C2
prime radicalC24×H(3,𝔽3)C22
hypercenterC2×C3×C22⋊C4C3⋊D3
hypocenterH(3,𝔽3)#397
LCSG ▷ C23×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
3. C2×C4⋊#1084C2semidirect
4. #11720574C2semidirect
5. #11720574C2semidirect
128. H(3,𝔽3)#397semidirect
203. C32#4953semidirect

Maximal subgroups

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

subgroupindex
C2×C22⋊C4×H(3,𝔽3)2
C2×#885522
C2×C4⋊#10842
#117205742
C3×#49533