Finite Group Databasebeta

← #13144288

Subgroups — #13144288

Order 1728.

1287 conjugacy classes of subgroups; only the 237 normal subgroups are shown.

normal subgroups only

This group has 1287 conjugacy classes of subgroups — too many to draw. Only its 237 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
4C2×#88540C212yesmaximal, complemented
5#11720574C212yesmaximal, complemented
6C3×#4942—34nomaximal
7C24×H(3,𝔽3)C2211yescharacteristic
8C22×C4×H(3,𝔽3)C2212yes—
9C22×#1084—24no—
10C22×#1084C414yescomplemented
11C22×#1084C2212yes—
12C22×#1084C2211yescharacteristic
13C22×#1084C2211yescharacteristic
14C22⋊C4×H(3,𝔽3)—22no—
15#88552—22no—
16#88540—24no—
17C23×C3×Dic3—34no—
18C2×C32×C22⋊C4D314yes—
19C2×C3×#217—34no—
20C2×C3×Dic3⋊C4—38no—
21C3×C3⋊#95—38no—
22C23×H(3,𝔽3)—22no—
23C23×H(3,𝔽3)—22no—
24C23×H(3,𝔽3)C2×C412yes—
25C23×H(3,𝔽3)D412yes—
26C23×H(3,𝔽3)Q812yes—
27C23×H(3,𝔽3)C2311yescharacteristic
28C2×C4×H(3,𝔽3)—22no—
29C2×C4×H(3,𝔽3)—22no—
30C2×C4×H(3,𝔽3)—22no—
31C2×#1084—22no—
32C2×#1084—28no—
33C2×#1084—22no—
34C2×#1084—22no—
35C2×#1084—22no—
36C2×#1084C2×C412yes—
37C2×#1084C2×C412yes—
38C2×#1084D414yes—
39C3×#380—91no—
40#4942—34no—
41C24×C32C2×D314yes—
42C22×C32×C4C2×D318yes—
43C22×C3×Dic3—34no—
44C22×C3×Dic3—316no—
45C22×C3×Dic3—34no—
46C22×C3×Dic3—38no—
47C22×C3×Dic3—616no—
48C32×C22⋊C4—28no—
49C3×#217—68no—
50C3×Dic3⋊C4—616no—
51C22×H(3,𝔽3)—22no—
52C22×H(3,𝔽3)—22no—
53C22×H(3,𝔽3)—24no—
54C22×H(3,𝔽3)—24no—
55C22×H(3,𝔽3)C4⋊C414yescomplemented
56C22×H(3,𝔽3)C22×C411yescharacteristic
57C22×H(3,𝔽3)C2×D412yes—
58C22×H(3,𝔽3)C2×D411yescharacteristic
59C22×H(3,𝔽3)C2×Q811yescharacteristic
60C22×H(3,𝔽3)#4112yes—
61C4×H(3,𝔽3)—42no—
62#1084—22no—
63#1084—42no—
64#1084C22⋊C414yescomplemented
65C23×C3×C4—91no—
66C23×Dic3—34no—
67C2×C3×C4⋊C4—92no—
68C2×C3×C22⋊C4—34no—
69C2×C3×C22⋊C4—91no—
70C2×C3×C22⋊C4C3⋊D311yescharacteristic
71C2×#217—34no—
72C2×Dic3⋊C4—38no—
73C3×#95—92no—
74C3⋊#95—38no—
75C23×C32—28no—
76C23×C32—28no—
77C23×C32Dic618yes—
78C23×C32C4×D318yes—
79C23×C32#6718yes—
80C23×C32C22×D314yes—
81C2×C32×C4—28no—
82C2×C32×C4—28no—
83C2×C32×C4—28no—
84C2×C3×Dic3—38no—
85C2×C3×Dic3—316no—
86C2×C3×Dic3—38no—
87C2×C3×Dic3—632no—
88C2×C3×Dic3—68no—
89C2×C3×Dic3—68no—
90C2×C3×Dic3—68no—
91C2×C3×Dic3—68no—
92#380—91noSylow
93C2×H(3,𝔽3)—24no—
94C2×H(3,𝔽3)C2×C22⋊C411yescharacteristic
95C2×H(3,𝔽3)C2×C4⋊C411yescharacteristic
96C2×H(3,𝔽3)#11711yescharacteristic
97C2×H(3,𝔽3)#12112yes—
98C2×H(3,𝔽3)C22⋊Q812yes—
99C24×C3—34no—
100C24×C3C2×C3⋊D311yescharacteristic
101C22×C3×C4—38no—
102C22×C3×C4—91no—
103C22×C3×C4—94no—
104C22×C3×C4—91no—
105C22×C3×C4—92no—
106C22×C3×C4—184no—
107C22×C3×C4C2×C3⋊D312yes—
108C22×Dic3—34no—
109C22×Dic3—34no—
110C22×Dic3—316no—
111C22×Dic3—38no—
112C22×Dic3—616no—
113C3×C4⋊C4—184no—
114C3×C22⋊C4—22no—
115C3×C22⋊C4—68no—
116C3×C22⋊C4—182no—
117#217—68no—
118Dic3⋊C4—616no—
119C22×C32—28no—
120C22×C32—216no—
121C22×C32—28no—
122C22×C32—216no—
123C22×C32Dic3⋊C4116yescomplemented
124C22×C32C2×Dic614yes—
125C22×C32C2×C4×D314yes—
126C22×C32D3×D418yes—
127C22×C32#23718yes—
128C22×C32C2×#6714yes—
129C32×C4—48no—
130C3×Dic3—316no—
131C3×Dic3—68no—
132C3×Dic3—128no—
133C23×C4—91no—
134C2×C4⋊C4—92no—
135C2×C22⋊C4—91no—
136C2×C22⋊C4#27211yescharacteristic
137#95—92no—
138H(3,𝔽3)#38011yescharacteristic, complemented, Sylow
139C23×C3—22no—
140C23×C3—22no—
141C23×C3—38no—
142C23×C3—38no—
143C23×C3—38no—
144C23×C3—34no—
145C23×C3—68no—
146C23×C3—68no—
147C23×C3#63412yes—
148C23×C3C4×C3⋊D312yes—
149C23×C3#63812yes—
150C23×C3C22×C3⋊D311yescharacteristic
151C2×C3×C4—22no—
152C2×C3×C4—22no—
153C2×C3×C4—22no—
154C2×C3×C4—68no—
155C2×C3×C4—68no—
156C2×C3×C4—68no—
157C2×C3×C4—92no—
158C2×C3×C4—94no—
159C2×C3×C4—92no—
160C2×C3×C4—188no—
161C2×C3×C4—182no—
162C2×C3×C4—182no—
163C2×C3×C4—182no—
164C2×C3×C4—182no—
165C2×Dic3—316no—
166C2×Dic3—38no—
167C2×Dic3—38no—
168C2×Dic3—632no—
169C2×Dic3—68no—
170C2×Dic3—68no—
171C2×Dic3—68no—
172C2×Dic3—68no—
173C2×C32—216no—
174C2×C32#87714yes—
175C2×C32#87818yes—
176C2×C32D3×C22⋊C414yes—
177C2×C32C2×Dic3⋊C414yes—
178C2×C32#93918yes—
179C24C2×#27211yescharacteristic
180C22×C4—91no—
181C22×C4—94no—
182C22×C4—91no—
183C22×C4—92no—
184C22×C4—184no—
185C22×C4C2×#27212yes—
186C4⋊C4—184no—
187C22⋊C4—22no—
188C22⋊C4—182no—
189C22×C3—22no—
190C22×C3—24no—
191C22×C3—22no—
192C22×C3—24no—
193C22×C3—34no—
194C22×C3—34no—
195C22×C3—38no—
196C22×C3—38no—
197C22×C3—34no—
198C22×C3—316no—
199C22×C3—68no—
200C22×C3—68no—
201C22×C3—616no—
202C22×C3—616no—
203C22×C3#375014yes—
204C22×C3C2×#63411yescharacteristic
205C22×C3C2×C4×C3⋊D311yescharacteristic
206C22×C3D4×C3⋊D312yes—
207C22×C3#383012yes—
208C22×C3C2×#63811yescharacteristic
209C3×C4—42no—
210C3×C4—94no—
211C3×C4—128no—
212C3×C4—182no—
213C3×C4—362no—
214Dic3—316no—
215Dic3—68no—
216Dic3—128no—
217C32#494214yescomplemented
218C23—22no—
219C23—22no—
220C23#622912yes—
221C23C4×#27212yes—
222C23#623512yes—
223C23C22×#27211yescharacteristic
224C2×C4—22no—
225C2×C4—22no—
226C2×C4—22no—
227C2×C4—92no—
228C2×C4—94no—
229C2×C4—92no—
230C2×C4—188no—
231C2×C4—182no—
232C2×C4—182no—
233C2×C4—182no—
234C2×C4—182no—
235C2×C3—24no—
236C2×C3—34no—
237C2×C3—38no—
238C2×C3—34no—
239C2×C3—34no—
240C2×C3—38no—
241C2×C3—616no—
242C2×C3#6408611yescharacteristic
243C2×C3#6408712yes—
244C2×C3C22⋊C4×C3⋊D311yescharacteristic
245C2×C3C2×#375011yescharacteristic
246C2×C3#6414812yes—
247C22—22no—
248C22—24no—
249C22—22no—
250C22—24no—
251C22#8854014yescomplemented
252C22C2×#622911yescharacteristic
253C22C2×C4×#27211yescharacteristic
254C22D4×#27212yes—
255C22#8874412yes—
256C22C2×#623511yescharacteristic
257C4—42no—
258C4—94no—
259C4—182no—
260C4—362no—
261C3—34no—
262C3#1059264111yescharacteristic
263C2—24no—
264C2#1172173311yescharacteristic
265C2#1172173412yes—
266C2C22⋊C4×#27211yescharacteristic
267C2C2×#8854011yescharacteristic
268C2#1172184912yes—
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)#380
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. C2×#88540C2semidirect
5. #11720574C2semidirect
10. C22×#1084C4semidirect
55. C22×H(3,𝔽3)C4⋊C4semidirect
64. #1084C22⋊C4semidirect
123. C22×C32Dic3⋊C4semidirect
138. H(3,𝔽3)#380semidirect
217. C32#4942semidirect
251. C22#88540semidirect

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
C2×#885402
#117205742
C3×#49423