Finite Group Databasebeta

← #13144386

Subgroups — #13144386

Order 1728.

528 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 528 conjugacy classes of subgroups — too many to draw. Only its 109 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
1H(3,𝔽3)×#117C211yesmaximal, characteristic
2#11721824C211yesmaximal, characteristic
3#11720573C211yesmaximal, characteristic
4#11720532C211yesmaximal, characteristic, complemented
5#11720532C211yesmaximal, characteristic, complemented
6#11720530C211yesmaximal, characteristic, complemented
7#11720529C211yesmaximal, characteristic, complemented
8C3×#4996—34nomaximal
9C22×C4×H(3,𝔽3)C2211yescharacteristic
10C2×#6229C2211yescharacteristic
11C4⋊C4×H(3,𝔽3)C2211yescharacteristic
12C4⋊C4×H(3,𝔽3)C2211yescharacteristic
13C22⋊C4×H(3,𝔽3)—21no—
14C42×H(3,𝔽3)—21no—
15#88552—21no—
16C4⋊#1084C2211yescharacteristic
17#88540—21no—
18C2×#6180C2211yescharacteristic
19C2×#6180C2211yescharacteristic
20C32×#117D314yes—
21C3×#924—34no—
22C3×#830—34no—
23C3×#810—34no—
24C3×#810—34no—
25C3×#808—34no—
26C3×#807—34no—
27C23×H(3,𝔽3)D411yescharacteristic
28C2×C4×H(3,𝔽3)—21no—
29C2×C4×H(3,𝔽3)—21no—
30C2×C4×H(3,𝔽3)—21no—
31C2×C4×H(3,𝔽3)—21no—
32C2×C4×H(3,𝔽3)D411yescharacteristic
33C2×C4×H(3,𝔽3)C2311yescharacteristic
34C2×#1084—21no—
35C2×#1084—21no—
36#6229—21no—
37#6229—21no—
38#6180—21no—
39#6180—21no—
40C3×#485—91no—
41#4996—34no—
42C22×C32×C4C2×D314yes—
43C2×C3×Dic6—34no—
44C32×C4⋊C4C2×D314yes—
45C32×C4⋊C4C2×D314yes—
46C32×C22⋊C4—24no—
47C32×C42—24no—
48C3×#217—64no—
49C3×C4⋊Dic3—34no—
50C3×Dic3⋊C4—64no—
51C2×C3×C3⋊C8—34no—
52C2×C3×C3⋊C8—34no—
53C22×H(3,𝔽3)—21no—
54C22×H(3,𝔽3)—21no—
55C22×H(3,𝔽3)—21no—
56C22×H(3,𝔽3)C2×D411yescharacteristic
57C4×H(3,𝔽3)—21no—
58C4×H(3,𝔽3)—41no—
59C4×H(3,𝔽3)—41no—
60C4×H(3,𝔽3)#4111yescharacteristic
61C4×H(3,𝔽3)#4111yescharacteristic
62#1084—41no—
63#1084—41no—
64C3×C22⋊Q8—91no—
65C3×#117—34no—
66C3×#117C3⋊D311yescharacteristic
67#924—34no—
68C3×#107—91no—
69C3×#106—91no—
70C3×#103—91no—
71C3×#103—91no—
72C3×C22⋊C8—91no—
73#830—34no—
74#810—34no—
75#810—34no—
76#808—34no—
77#807—34no—
78C23×C32#6714yes—
79C2×C32×C4—24no—
80C2×C32×C4—24no—
81C2×C32×C4—24no—
82C2×C32×C4—24no—
83C2×C32×C4#6714yes—
84C2×C32×C4C22×D314yes—
85C2×C3×Dic3—64no—
86C2×C3×Dic3—64no—
87C3×Dic6—64no—
88C3×Dic6—64no—
89C3×C3⋊C8—64no—
90C3×C3⋊C8—64no—
91#485—91noSylow
92C2×H(3,𝔽3)—41no—
93C2×H(3,𝔽3)#12311yescharacteristic
94C2×H(3,𝔽3)#13511yescharacteristic
95C2×H(3,𝔽3)#13711yescharacteristic
96C2×C3×Q8—91no—
97C22×C3×C4—34no—
98C22×C3×C4C2×C3⋊D311yescharacteristic
99C2×Dic6—34no—
100C2×C3×C8—91no—
101C2×C3×C8—91no—
102C3×C4⋊C4—34no—
103C3×C4⋊C4—34no—
104C3×C4⋊C4—91no—
105C3×C4⋊C4—181no—
106C3×C4⋊C4C2×C3⋊D311yescharacteristic
107C3×C4⋊C4C2×C3⋊D311yescharacteristic
108C3×C22⋊C4—21no—
109C3×C22⋊C4—64no—
110C3×C22⋊C4—181no—
111C3×C42—21no—
112C3×C42—64no—
113#217—64no—
114C4⋊Dic3—34no—
115Dic3⋊C4—64no—
116C2×C3⋊C8—34no—
117C2×C3⋊C8—34no—
118C22×C32—24no—
119C22×C32—24no—
120C22×C32—24no—
121C22×C32C2×#6714yes—
122C32×C4—24no—
123C32×C4—44no—
124C32×C4—44no—
125C32×C4#23514yes—
126C32×C4#23514yes—
127C3×Dic3—124no—
128C3×Dic3—124no—
129C22⋊Q8—91no—
130#117#27211yescharacteristic
131#107—91no—
132#106—91no—
133#103—91no—
134#103—91no—
135C22⋊C8—91no—
136H(3,𝔽3)#48511yescharacteristic, complemented, Sylow
137C23×C3—34no—
138C23×C3#63811yescharacteristic
139C3×Q8—181no—
140C3×Q8—181no—
141C2×C3×C4—21no—
142C2×C3×C4—21no—
143C2×C3×C4—21no—
144C2×C3×C4—21no—
145C2×C3×C4—34no—
146C2×C3×C4—34no—
147C2×C3×C4—64no—
148C2×C3×C4—64no—
149C2×C3×C4—64no—
150C2×C3×C4—64no—
151C2×C3×C4—181no—
152C2×C3×C4—181no—
153C2×C3×C4#63811yescharacteristic
154C2×C3×C4C22×C3⋊D311yescharacteristic
155C2×Dic3—64no—
156C2×Dic3—64no—
157Dic6—64no—
158Dic6—64no—
159C3×C8—181no—
160C3×C8—181no—
161C3⋊C8—64no—
162C3⋊C8—64no—
163C2×C32—44no—
164C2×C32#92914yes—
165C2×C32#95014yes—
166C2×C32#95114yes—
167C2×Q8—91no—
168C22×C4C2×#27211yescharacteristic
169C2×C8—91no—
170C2×C8—91no—
171C4⋊C4—91no—
172C4⋊C4—181no—
173C4⋊C4C2×#27211yescharacteristic
174C4⋊C4C2×#27211yescharacteristic
175C22⋊C4—21no—
176C22⋊C4—181no—
177C42—21no—
178C22×C3—21no—
179C22×C3—21no—
180C22×C3—21no—
181C22×C3—34no—
182C22×C3—64no—
183C22×C3—64no—
184C22×C3—64no—
185C22×C3C2×#63811yescharacteristic
186C3×C4—21no—
187C3×C4—34no—
188C3×C4—34no—
189C3×C4—41no—
190C3×C4—41no—
191C3×C4—64no—
192C3×C4—124no—
193C3×C4—124no—
194C3×C4—361no—
195C3×C4—361no—
196C3×C4#382811yescharacteristic
197C3×C4#382811yescharacteristic
198Dic3—124no—
199Dic3—124no—
200C32#499614yescomplemented
201C23#623511yescharacteristic
202Q8—181no—
203Q8—181no—
204C2×C4—21no—
205C2×C4—21no—
206C2×C4—21no—
207C2×C4—21no—
208C2×C4—181no—
209C2×C4—181no—
210C2×C4#623511yescharacteristic
211C2×C4C22×#27211yescharacteristic
212C8—181no—
213C8—181no—
214C2×C3—34no—
215C2×C3—34no—
216C2×C3—34no—
217C2×C3—41no—
218C2×C3—124no—
219C2×C3#6413811yescharacteristic
220C2×C3#6415911yescharacteristic
221C2×C3#6416011yescharacteristic
222C22—21no—
223C22—21no—
224C22—21no—
225C22C2×#623511yescharacteristic
226C4—21no—
227C4—41no—
228C4—41no—
229C4—361no—
230C4—361no—
231C4#8874011yescharacteristic
232C4#8874011yescharacteristic
233C3—34no—
234C3#1059269511yescharacteristic
235C2—41no—
236C2#1172182911yescharacteristic
237C2#1172187111yescharacteristic
238C2#1172187211yescharacteristic
subgroupG/H
center, socleC22×C3C2×#638
commutator, cosocleC2×C4×H(3,𝔽3)C23
FrattiniC2×C3×C4C22×C3⋊D3
FittingH(3,𝔽3)×#117C2
prime radicalC23×H(3,𝔽3)D4
hypercenterC3×#117C3⋊D3
hypocenterH(3,𝔽3)#485
LCSG ▷ C2×C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22×C3 ◁ C22×C3×C4 ◁ C3×#117

Complements

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

NH ≅ G/Ntype
4. #11720532C2semidirect
5. #11720532C2semidirect
6. #11720530C2semidirect
7. #11720529C2semidirect
136. H(3,𝔽3)#485semidirect
200. C32#4996semidirect

Maximal subgroups

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

subgroupindex
H(3,𝔽3)×#1172
#117218242
#117205732
#117205322
#117205302
#117205292
C3×#49963