Finite Group Databasebeta

← #13144520

Subgroups — #13144520

Order 1728.

561 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 561 conjugacy classes of subgroups — too many to draw. Only its 111 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)×#125C211yesmaximal, characteristic, complemented
2C4×C4⋊#272C211yesmaximal, characteristic
3#11720531C211yesmaximal, characteristic
4#11720531C211yesmaximal, characteristic
5#11720530C211yesmaximal, characteristic, complemented
6#11720529C211yesmaximal, characteristic, complemented
7C4⋊#6180C211yesmaximal, characteristic, complemented
8C3×#5066—34nomaximal
9C2×C4⋊#272C2211yescharacteristic
10C2×C4×#272—21no—
11C4⋊C4×H(3,𝔽3)—21no—
12C4⋊C4×H(3,𝔽3)—21no—
13C4⋊C4×H(3,𝔽3)C2211yescharacteristic
14C4⋊C4×H(3,𝔽3)C2211yescharacteristic
15C42×H(3,𝔽3)C2211yescharacteristic
16#88542—21no—
17C4⋊#1084C2211yescharacteristic
18C2×#6180C2211yescharacteristic
19C2×#6180C2211yescharacteristic
20C32×#125D314yescomplemented
21C3×C4×D12—34no—
22C3×#809—34no—
23C3×#809—34no—
24C3×#808—34no—
25C3×#807—34no—
26C3×C4⋊(C3⋊C8)—34no—
27C22×#272—21no—
28C2×C4×H(3,𝔽3)—21no—
29C2×C4×H(3,𝔽3)—21no—
30C2×C4×H(3,𝔽3)D411yescharacteristic
31C2×C4×H(3,𝔽3)D411yescharacteristic
32C2×C4×H(3,𝔽3)C2311yescharacteristic
33C2×#1084—21no—
34C4⋊#272—21no—
35C4⋊#272Q812yes—
36C4×#272—42no—
37#6180—21no—
38#6180—21no—
39C3×#478—91no—
40#5066—34no—
41C2×C3×D12—34no—
42C2×C3×C4×D3—64no—
43C32×C4⋊C4—24no—
44C32×C4⋊C4—24no—
45C32×C4⋊C4C2×D314yes—
46C32×C4⋊C4C2×D314yes—
47C32×C42C2×D314yes—
48C3×(C2×D3)⋊C4—64no—
49C3×C4⋊Dic3—34no—
50C2×C3×C3⋊C8—34no—
51C2×C3×C3⋊C8—34no—
52C22×H(3,𝔽3)C2×D411yescharacteristic
53C2×#272—22no—
54C2×#272—41no—
55C2×#272—41no—
56C4×H(3,𝔽3)—21no—
57C4×H(3,𝔽3)—21no—
58C4×H(3,𝔽3)—41no—
59C4×H(3,𝔽3)—41no—
60C4×H(3,𝔽3)C2×Q811yescharacteristic
61C4×H(3,𝔽3)#4111yescharacteristic
62#1084—41no—
63C3×#125—34no—
64C3×#125C3⋊D311yescharacteristic
65C3×C4×D4—91no—
66C4×D12—34no—
67C3×#107—91no—
68C3×#106—91no—
69C3×C4⋊C8—91no—
70C3×#102—91no—
71C3×#102—91no—
72#809—34no—
73#809—34no—
74#808—34no—
75#807—34no—
76C4⋊(C3⋊C8)—34no—
77C22×C3×D3—64no—
78C2×C32×C4—24no—
79C2×C32×C4—24no—
80C2×C32×C4#6714yes—
81C2×C32×C4#6714yes—
82C2×C32×C4C22×D314yes—
83C2×C3×Dic3—64no—
84C3×D12—38no—
85C3×D12—64no—
86C3×C4×D3—128no—
87C3×C3⋊C8—64no—
88C3×C3⋊C8—64no—
89#478—91noSylow
90C2×H(3,𝔽3)C22⋊Q811yescharacteristic
91C2×H(3,𝔽3)#13511yescharacteristic
92C2×H(3,𝔽3)AGL(1,ℤ/8)11yescharacteristic
93#272—42no—
94C2×C3×D4—91no—
95C22×C3×C4—181no—
96C2×D12—34no—
97C2×C4×D3—64no—
98C2×C3×C8—91no—
99C2×C3×C8—91no—
100C3×C4⋊C4—21no—
101C3×C4⋊C4—21no—
102C3×C4⋊C4—34no—
103C3×C4⋊C4—34no—
104C3×C4⋊C4—64no—
105C3×C4⋊C4—64no—
106C3×C4⋊C4—91no—
107C3×C4⋊C4C2×C3⋊D311yescharacteristic
108C3×C4⋊C4C2×C3⋊D311yescharacteristic
109C3×C22⋊C4—181no—
110C3×C42—34no—
111C3×C42C2×C3⋊D311yescharacteristic
112(C2×D3)⋊C4—64no—
113C4⋊Dic3—34no—
114C2×C3⋊C8—34no—
115C2×C3⋊C8—34no—
116C22×C32C2×#6714yes—
117C2×C3×D3—68no—
118C2×C3×D3—124no—
119C2×C3×D3—124no—
120C32×C4—24no—
121C32×C4—24no—
122C32×C4—44no—
123C32×C4—44no—
124C32×C4D3×Q814yes—
125C32×C4#23914yes—
126C3×Dic3—124no—
127#125#27211yescharacteristic, complemented
128C4×D4—91no—
129#107—91no—
130#106—91no—
131C4⋊C8—91no—
132#102—91no—
133#102—91no—
134H(3,𝔽3)#47811yescharacteristic, complemented, Sylow
135C23×C3—181no—
136C22×D3—64no—
137C3×D4—92no—
138C3×D4—181no—
139C2×C3×C4—21no—
140C2×C3×C4—21no—
141C2×C3×C4—34no—
142C2×C3×C4—34no—
143C2×C3×C4—34no—
144C2×C3×C4—64no—
145C2×C3×C4—64no—
146C2×C3×C4—181no—
147C2×C3×C4—362no—
148C2×C3×C4#63811yescharacteristic
149C2×C3×C4#63811yescharacteristic
150C2×C3×C4C22×C3⋊D311yescharacteristic
151C2×Dic3—64no—
152D12—38no—
153D12—64no—
154C4×D3—128no—
155C3×C8—181no—
156C3×C8—181no—
157C3⋊C8—64no—
158C3⋊C8—64no—
159C2×C32#94614yes—
160C2×C32#94914yes—
161C2×C32#95014yes—
162C3×D3—128no—
163C2×D4—91no—
164C22×C4—181no—
165C2×C8—91no—
166C2×C8—91no—
167C4⋊C4—21no—
168C4⋊C4—21no—
169C4⋊C4—91no—
170C4⋊C4C2×#27211yescharacteristic
171C4⋊C4C2×#27211yescharacteristic
172C22⋊C4—181no—
173C42C2×#27211yescharacteristic
174C22×C3—34no—
175C22×C3—182no—
176C22×C3—361no—
177C22×C3—361no—
178C22×C3C2×#63811yescharacteristic
179C2×D3—68no—
180C2×D3—124no—
181C2×D3—124no—
182C3×C4—21no—
183C3×C4—21no—
184C3×C4—34no—
185C3×C4—34no—
186C3×C4—41no—
187C3×C4—41no—
188C3×C4—64no—
189C3×C4—64no—
190C3×C4—124no—
191C3×C4—124no—
192C3×C4—361no—
193C3×C4Q8×C3⋊D311yescharacteristic
194C3×C4#383211yescharacteristic
195Dic3—124no—
196C32#506614yescomplemented
197C23—181no—
198D4—92no—
199D4—181no—
200C2×C4—21no—
201C2×C4—21no—
202C2×C4—181no—
203C2×C4—362no—
204C2×C4#623511yescharacteristic
205C2×C4#623511yescharacteristic
206C2×C4C22×#27211yescharacteristic
207C8—181no—
208C8—181no—
209C2×C3—34no—
210C2×C3—34no—
211C2×C3—34no—
212C2×C3—362no—
213C2×C3#6415511yescharacteristic
214C2×C3#6415811yescharacteristic
215C2×C3#6415911yescharacteristic
216D3—128no—
217C22—182no—
218C22—361no—
219C22—361no—
220C22C2×#623511yescharacteristic
221C4—21no—
222C4—21no—
223C4—41no—
224C4—41no—
225C4—361no—
226C4Q8×#27211yescharacteristic
227C4#8874811yescharacteristic
228C3—34no—
229C3#1059276511yescharacteristic
230C2—362no—
231C2#1172186311yescharacteristic
232C2#1172187011yescharacteristic
233C2#1172187111yescharacteristic
subgroupG/H
center, socleC22×C3C2×#638
commutator, cosocleC2×C4×H(3,𝔽3)C23
FrattiniC2×C3×C4C22×C3⋊D3
FittingH(3,𝔽3)×#125C2
prime radicalC2×C4⋊#272C22
hypercenterC3×#125C3⋊D3
hypocenterH(3,𝔽3)#478
LCSG ▷ C2×C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22×C3 ◁ C3×C42 ◁ C3×#125

Complements

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

NH ≅ G/Ntype
1. H(3,𝔽3)×#125C2semidirect
5. #11720530C2semidirect
6. #11720529C2semidirect
7. C4⋊#6180C2semidirect
20. C32×#125D3semidirect
127. #125#272semidirect
134. H(3,𝔽3)#478semidirect
196. C32#5066semidirect

Maximal subgroups

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

subgroupindex
H(3,𝔽3)×#1252
C4×C4⋊#2722
#117205312
#117205302
#117205292
C4⋊#61802
C3×#50663