Finite Group Databasebeta

← #13144293

Subgroups — #13144293

Order 1728.

1111 conjugacy classes of subgroups; only the 183 normal subgroups are shown.

normal subgroups only

This group has 1111 conjugacy classes of subgroups — too many to draw. Only its 183 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×#6229C211yesmaximal, characteristic, complemented
2C2×C22⋊C4×H(3,𝔽3)C211yesmaximal, characteristic
3C2×#88552C212yesmaximal
4#11720574C211yesmaximal, characteristic, complemented
5#11720574C212yesmaximal, complemented
6C3×#4947—34nomaximal
7C24×H(3,𝔽3)C2211yescharacteristic
8C22×C4×H(3,𝔽3)C2211yescharacteristic
9C22×C4×H(3,𝔽3)C2211yescharacteristic
10C22×#1084C2212yes—
11C22×#1084C2212yes—
12C2×#6229—22no—
13C2×#6229—24no—
14C22⋊C4×H(3,𝔽3)—22no—
15#88552—24no—
16C22×C3×Dic6—34no—
17C2×C32×C22⋊C4D314yes—
18C2×C3×#217—38no—
19C3×C3⋊#95—38no—
20C3×C3⋊#95—34no—
21C23×H(3,𝔽3)—22no—
22C23×H(3,𝔽3)—21no—
23C23×H(3,𝔽3)—21no—
24C23×H(3,𝔽3)—22no—
25C23×H(3,𝔽3)Q812yescomplemented
26C23×H(3,𝔽3)C2311yescharacteristic
27C2×C4×H(3,𝔽3)—22no—
28C2×C4×H(3,𝔽3)—21no—
29C2×C4×H(3,𝔽3)—22no—
30C2×C4×H(3,𝔽3)D412yes—
31C2×#1084—22no—
32C2×#1084—22no—
33C2×#1084—22no—
34C2×#1084—22no—
35C2×#1084—22no—
36C2×#1084D414yes—
37#6229—28no—
38C3×#393—91no—
39#4947—34no—
40C24×C32C2×D314yes—
41C22×C32×C4C2×D314yes—
42C22×C32×C4C2×D314yes—
43C22×C3×Dic3—38no—
44C22×C3×Dic3—38no—
45C2×C3×Dic6—616no—
46C2×C3×Dic6—68no—
47C32×C22⋊C4—28no—
48C3×#217—616no—
49C22×H(3,𝔽3)—24no—
50C22×H(3,𝔽3)—22no—
51C22×H(3,𝔽3)—41no—
52C22×H(3,𝔽3)—42no—
53C22×H(3,𝔽3)—41no—
54C22×H(3,𝔽3)C2×D412yes—
55C22×H(3,𝔽3)C2×D411yescharacteristic
56C22×H(3,𝔽3)C2×Q811yescharacteristic
57C22×H(3,𝔽3)#4111yescharacteristic
58C22×H(3,𝔽3)#4112yes—
59C4×H(3,𝔽3)—22no—
60C4×H(3,𝔽3)—41no—
61#1084—24no—
62#1084—42no—
63C22×C3×Q8—91no—
64C22×Dic6—34no—
65C2×C3×C22⋊C4—34no—
66C2×C3×C22⋊C4—92no—
67C2×C3×C22⋊C4C3⋊D311yescharacteristic
68C2×#217—38no—
69C3×#95—92no—
70C3×#95—91no—
71C3⋊#95—38no—
72C3⋊#95—34no—
73C23×C32—24no—
74C23×C32—28no—
75C23×C32—28no—
76C23×C32—24no—
77C23×C32Dic618yescomplemented
78C23×C32C22×D314yes—
79C2×C32×C4—28no—
80C2×C32×C4—28no—
81C2×C32×C4—24no—
82C2×C32×C4#6718yes—
83C2×C3×Dic3—316no—
84C2×C3×Dic3—68no—
85C2×C3×Dic3—68no—
86C2×C3×Dic3—68no—
87C2×C3×Dic3—68no—
88C2×C3×Dic3—68no—
89C3×Dic6—632no—
90#393—91noSylow
91C2×H(3,𝔽3)—42no—
92C2×H(3,𝔽3)#12011yescharacteristic
93C2×H(3,𝔽3)C22⋊Q812yes—
94C2×H(3,𝔽3)C22⋊Q811yescharacteristic
95C2×H(3,𝔽3)#12412yes—
96C2×H(3,𝔽3)#12411yescharacteristic
97C24×C3—34no—
98C24×C3C2×C3⋊D311yescharacteristic
99C2×C3×Q8—184no—
100C2×C3×Q8—182no—
101C22×C3×C4—34no—
102C22×C3×C4—34no—
103C22×C3×C4—92no—
104C22×C3×C4—92no—
105C22×C3×C4C2×C3⋊D311yescharacteristic
106C22×C3×C4C2×C3⋊D311yescharacteristic
107C22×Dic3—38no—
108C22×Dic3—38no—
109C2×Dic6—616no—
110C2×Dic6—68no—
111C3×C22⋊C4—22no—
112C3×C22⋊C4—68no—
113C3×C22⋊C4—184no—
114#217—616no—
115C22×C32—216no—
116C22×C32—28no—
117C22×C32—48no—
118C22×C32—44no—
119C22×C32—44no—
120C22×C32C2×Dic614yes—
121C22×C32#23514yes—
122C22×C32D3×D418yes—
123C22×C32#23718yes—
124C22×C32C2×#6714yes—
125C32×C4—28no—
126C32×C4—44no—
127C3×Dic3—616no—
128C3×Dic3—128no—
129C22×Q8—91no—
130C2×C22⋊C4—92no—
131C2×C22⋊C4#27211yescharacteristic
132#95—92no—
133#95—91no—
134H(3,𝔽3)#39311yescharacteristic, complemented, Sylow
135C23×C3—21no—
136C23×C3—22no—
137C23×C3—22no—
138C23×C3—21no—
139C23×C3—38no—
140C23×C3—34no—
141C23×C3—68no—
142C23×C3—64no—
143C23×C3—64no—
144C23×C3—68no—
145C23×C3#63412yes—
146C23×C3C22×C3⋊D311yescharacteristic
147C3×Q8—188no—
148C2×C3×C4—22no—
149C2×C3×C4—22no—
150C2×C3×C4—21no—
151C2×C3×C4—38no—
152C2×C3×C4—68no—
153C2×C3×C4—64no—
154C2×C3×C4—68no—
155C2×C3×C4—94no—
156C2×C3×C4—182no—
157C2×C3×C4—182no—
158C2×C3×C4—182no—
159C2×C3×C4—182no—
160C2×C3×C4—182no—
161C2×C3×C4#63812yes—
162C2×Dic3—316no—
163C2×Dic3—68no—
164C2×Dic3—68no—
165C2×Dic3—68no—
166C2×Dic3—68no—
167C2×Dic3—68no—
168Dic6—632no—
169C2×C32—48no—
170C2×C32#87818yes—
171C2×C32#88518yes—
172C2×C32#92414yes—
173C2×C32#93614yes—
174C2×C32#93714yes—
175C24C2×#27211yescharacteristic
176C2×Q8—184no—
177C2×Q8—182no—
178C22×C4—92no—
179C22×C4—92no—
180C22×C4C2×#27211yescharacteristic
181C22×C4C2×#27211yescharacteristic
182C22⋊C4—22no—
183C22⋊C4—184no—
184C22×C3—24no—
185C22×C3—22no—
186C22×C3—34no—
187C22×C3—34no—
188C22×C3—38no—
189C22×C3—38no—
190C22×C3—34no—
191C22×C3—42no—
192C22×C3—41no—
193C22×C3—41no—
194C22×C3—616no—
195C22×C3—68no—
196C22×C3—128no—
197C22×C3—124no—
198C22×C3—124no—
199C22×C3C2×#63411yescharacteristic
200C22×C3#382811yescharacteristic
201C22×C3D4×C3⋊D312yes—
202C22×C3#383012yes—
203C22×C3C2×#63811yescharacteristic
204C3×C4—22no—
205C3×C4—41no—
206C3×C4—68no—
207C3×C4—124no—
208C3×C4—184no—
209C3×C4—362no—
210Dic3—616no—
211Dic3—128no—
212C32#494714yescomplemented
213C23—21no—
214C23—22no—
215C23—22no—
216C23—21no—
217C23#622912yescomplemented
218C23C22×#27211yescharacteristic
219Q8—188no—
220C2×C4—22no—
221C2×C4—22no—
222C2×C4—21no—
223C2×C4—94no—
224C2×C4—182no—
225C2×C4—182no—
226C2×C4—182no—
227C2×C4—182no—
228C2×C4—182no—
229C2×C4#623512yes—
230C2×C3—38no—
231C2×C3—38no—
232C2×C3—34no—
233C2×C3—34no—
234C2×C3—34no—
235C2×C3—42no—
236C2×C3—128no—
237C2×C3#6408712yes—
238C2×C3#6409412yes—
239C2×C3#6413311yescharacteristic
240C2×C3#6414511yescharacteristic
241C2×C3#6414611yescharacteristic
242C22—24no—
243C22—22no—
244C22—42no—
245C22—41no—
246C22—41no—
247C22C2×#622911yescharacteristic
248C22#8874011yescharacteristic
249C22D4×#27212yes—
250C22#8874412yes—
251C22C2×#623511yescharacteristic
252C4—22no—
253C4—41no—
254C4—184no—
255C4—362no—
256C3—34no—
257C3#1059264611yescharacteristic
258C2—42no—
259C2#1172173412yes—
260C2#1172174112yes—
261C2#1172182411yescharacteristic
262C2#1172184611yescharacteristic
263C2#1172184711yescharacteristic
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)#393
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
1. C22×#6229C2semidirect
4. #11720574C2semidirect
5. #11720574C2semidirect
25. C23×H(3,𝔽3)Q8semidirect
77. C23×C32Dic6semidirect
134. H(3,𝔽3)#393semidirect
212. C32#4947semidirect
217. C23#6229semidirect

Maximal subgroups

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

subgroupindex
C22×#62292
C2×C22⋊C4×H(3,𝔽3)2
C2×#885522
#117205742
C3×#49473