Finite Group Databasebeta

← #13144469

Subgroups — #13144469

Order 1728.

869 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 869 conjugacy classes of subgroups — too many to draw. Only its 127 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
2H(3,𝔽3)×#121C211yesmaximal, characteristic
3C2×#88546C212yesmaximal, complemented
4#11720573C211yesmaximal, characteristic, complemented
5#11720532C212yesmaximal, complemented
6C3×#5040—34nomaximal
7C2×D4×H(3,𝔽3)—21no—
8C2×D4×H(3,𝔽3)C2211yescharacteristic
9C22×C4×H(3,𝔽3)C2211yescharacteristic
10C22×#1084—21no—
11C2×#6229—21no—
12C2×#6229—24no—
13C2×#6229C2212yes—
14C4⋊C4×H(3,𝔽3)C2211yescharacteristic
15C22⋊C4×H(3,𝔽3)—21no—
16#88546—24no—
17C2×#6180C2212yes—
18C22×C3×Dic6—34no—
19C32×#121D314yes—
20C2×C3×#214—38no—
21C3×#830—34no—
22C3×#810—38no—
23C23×H(3,𝔽3)—21no—
24C23×H(3,𝔽3)D411yescharacteristic
25D4×H(3,𝔽3)—22no—
26D4×H(3,𝔽3)—42no—
27C2×C4×H(3,𝔽3)—22no—
28C2×C4×H(3,𝔽3)—21no—
29C2×C4×H(3,𝔽3)D411yescharacteristic
30C2×C4×H(3,𝔽3)C2311yescharacteristic
31C2×#1084—24no—
32C2×#1084—22no—
33#6229—24no—
34#6229—22no—
35#6229D414yescomplemented
36#6180—22no—
37C3×#448—91no—
38#5040—34no—
39C2×C32×D4—24no—
40C2×C32×D4C2×D314yes—
41C22×C32×C4C2×D314yes—
42C22×C3×Dic3—64no—
43C2×C3×Dic6—38no—
44C2×C3×Dic6—616no—
45C2×C3×Dic6—64no—
46C32×C4⋊C4C2×D314yes—
47C32×C22⋊C4—24no—
48C3×#214—616no—
49C2×C3×C3⋊C8—38no—
50C22×H(3,𝔽3)—21no—
51C22×H(3,𝔽3)—21no—
52C22×H(3,𝔽3)—41no—
53C22×H(3,𝔽3)—41no—
54C22×H(3,𝔽3)—41no—
55C22×H(3,𝔽3)SD1612yescomplemented
56C22×H(3,𝔽3)C2×D411yescharacteristic
57C4×H(3,𝔽3)—21no—
58C4×H(3,𝔽3)—41no—
59C4×H(3,𝔽3)C2×D412yes—
60#1084—24no—
61C22×C3×Q8—91no—
62C22×Dic6—34no—
63C2×C3×SD16—92no—
64C3×#121—34no—
65C3×#121C3⋊D311yescharacteristic
66C2×#214—38no—
67C3×#103—92no—
68C3×C22⋊C8—91no—
69#830—34no—
70#810—38no—
71C23×C32—24no—
72C23×C32#6714yes—
73C32×D4—28no—
74C32×D4—48no—
75C2×C32×C4—28no—
76C2×C32×C4—24no—
77C2×C32×C4#6714yes—
78C2×C32×C4C22×D314yes—
79C2×C3×Dic3—616no—
80C2×C3×Dic3—68no—
81C3×Dic6—316no—
82C3×Dic6—616no—
83C3×Dic6—68no—
84C3×C3⋊C8—68no—
85#448—91noSylow
86C2×H(3,𝔽3)—22no—
87C2×H(3,𝔽3)—81no—
88C2×H(3,𝔽3)#12011yescharacteristic
89C2×H(3,𝔽3)C2×SD1611yescharacteristic
90C2×H(3,𝔽3)#13711yescharacteristic
91C2×C3×Q8—92no—
92C2×C3×Q8—184no—
93C2×C3×Q8—181no—
94C2×C3×D4—21no—
95C2×C3×D4—34no—
96C2×C3×D4—64no—
97C2×C3×D4C2×C3⋊D311yescharacteristic
98C22×C3×C4—34no—
99C22×C3×C4—181no—
100C22×C3×C4C2×C3⋊D311yescharacteristic
101C22×Dic3—64no—
102C2×Dic6—38no—
103C2×Dic6—616no—
104C2×Dic6—64no—
105C3×SD16—184no—
106C2×C3×C8—92no—
107C3×C4⋊C4—34no—
108C3×C4⋊C4C2×C3⋊D311yescharacteristic
109C3×C22⋊C4—21no—
110C3×C22⋊C4—64no—
111#214—616no—
112C2×C3⋊C8—38no—
113C22×C32—24no—
114C22×C32—24no—
115C22×C32—44no—
116C22×C32—44no—
117C22×C32—44no—
118C22×C32#21418yescomplemented
119C22×C32C2×#6714yes—
120C32×C4—24no—
121C32×C4—44no—
122C32×C4D3×D418yes—
123C3×Dic3—616no—
124C22×Q8—91no—
125C2×SD16—92no—
126#121#27211yescharacteristic
127#103—92no—
128C22⋊C8—91no—
129H(3,𝔽3)#44811yescharacteristic, complemented, Sylow
130C23×C3—21no—
131C23×C3—34no—
132C23×C3—64no—
133C23×C3#63811yescharacteristic
134C3×Q8—94no—
135C3×Q8—184no—
136C3×Q8—182no—
137C3×D4—22no—
138C3×D4—42no—
139C3×D4—68no—
140C3×D4—128no—
141C2×C3×C4—22no—
142C2×C3×C4—21no—
143C2×C3×C4—34no—
144C2×C3×C4—34no—
145C2×C3×C4—68no—
146C2×C3×C4—64no—
147C2×C3×C4—184no—
148C2×C3×C4—182no—
149C2×C3×C4#63811yescharacteristic
150C2×C3×C4C22×C3⋊D311yescharacteristic
151C2×Dic3—68no—
152C2×Dic3—616no—
153Dic6—316no—
154Dic6—68no—
155Dic6—616no—
156C3×C8—182no—
157C3⋊C8—68no—
158C2×C32—28no—
159C2×C32—84no—
160C2×C32C2×#21414yes—
161C2×C32#93714yes—
162C2×C32#95114yes—
163C2×Q8—92no—
164C2×Q8—184no—
165C2×Q8—181no—
166C2×D4—21no—
167C2×D4C2×#27211yescharacteristic
168C22×C4—181no—
169C22×C4C2×#27211yescharacteristic
170SD16—184no—
171C2×C8—92no—
172C4⋊C4C2×#27211yescharacteristic
173C22⋊C4—21no—
174C22×C3—21no—
175C22×C3—21no—
176C22×C3—34no—
177C22×C3—38no—
178C22×C3—41no—
179C22×C3—41no—
180C22×C3—41no—
181C22×C3—64no—
182C22×C3—64no—
183C22×C3—124no—
184C22×C3—124no—
185C22×C3—124no—
186C22×C3#375412yes—
187C22×C3C2×#63811yescharacteristic
188C3×C4—21no—
189C3×C4—38no—
190C3×C4—41no—
191C3×C4—64no—
192C3×C4—124no—
193C3×C4—184no—
194C3×C4D4×C3⋊D312yes—
195Dic3—616no—
196C32#504014yescomplemented
197C23—21no—
198C23#623511yescharacteristic
199Q8—94no—
200Q8—184no—
201Q8—182no—
202D4—22no—
203D4—42no—
204C2×C4—22no—
205C2×C4—21no—
206C2×C4—184no—
207C2×C4—182no—
208C2×C4#623511yescharacteristic
209C2×C4C22×#27211yescharacteristic
210C8—182no—
211C2×C3—22no—
212C2×C3—34no—
213C2×C3—34no—
214C2×C3—34no—
215C2×C3—68no—
216C2×C3—81no—
217C2×C3—244no—
218C2×C3C2×#375411yescharacteristic
219C2×C3#6414611yescharacteristic
220C2×C3#6416011yescharacteristic
221C22—21no—
222C22—21no—
223C22—41no—
224C22—41no—
225C22—41no—
226C22#8854612yescomplemented
227C22C2×#623511yescharacteristic
228C4—21no—
229C4—41no—
230C4—184no—
231C4D4×#27212yes—
232C3—34no—
233C3#1059273911yescharacteristic
234C2—22no—
235C2—81no—
236C2C2×#8854611yescharacteristic
237C2#1172184711yescharacteristic
238C2#1172187211yescharacteristic
subgroupG/H
center, socleC22×C3C2×#638
commutator, cosocleC2×C4×H(3,𝔽3)C23
FrattiniC2×C3×C4C22×C3⋊D3
Fitting, prime radicalH(3,𝔽3)×#121C2
hypercenterC3×#121C3⋊D3
hypocenterH(3,𝔽3)#448
LCSG ▷ C2×C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22×C3 ◁ C22×C3×C4 ◁ C3×#121

Complements

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

NH ≅ G/Ntype
1. C22×#6229C2semidirect
3. C2×#88546C2semidirect
4. #11720573C2semidirect
5. #11720532C2semidirect
35. #6229D4semidirect
55. C22×H(3,𝔽3)SD16semidirect
118. C22×C32#214semidirect
129. H(3,𝔽3)#448semidirect
196. C32#5040semidirect
226. C22#88546semidirect

Maximal subgroups

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

subgroupindex
C22×#62292
H(3,𝔽3)×#1212
C2×#885462
#117205732
#117205322
C3×#50403