Finite Group Databasebeta

← #560⋊H(3,𝔽3) #13152207

Subgroups — #560⋊H(3,𝔽3) #13152207

Order 1728.

387 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 387 conjugacy classes of subgroups — too many to draw. Only its 53 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
1#127⋊H(3,𝔽3)C211yesmaximal, characteristic, complemented
2#126⋊H(3,𝔽3)C212yesmaximal, complemented
3C32×#560C311yesmaximal, characteristic, complemented
4C3×#560⋊C3C313yesmaximal, complemented
5C22×C22⋊H(3,𝔽3)—41nomaximal
6C32⋊(C42⋊C3)C2211yescharacteristic, complemented
7C32×#127C2×C311yescharacteristic, complemented
8C32×#126C2×C312yescomplemented
9C32×#123—31no—
10C32×#121—32no—
11C32×#120—31no—
12C3×#127⋊C3C2×C313yescomplemented
13C3×#126⋊C3C2×C316yescomplemented
14C2×C22⋊H(3,𝔽3)—42no—
15C2×C22⋊H(3,𝔽3)—41no—
16C3×#560—31no—
17C3×#560C3211yescharacteristic
18#560⋊C3—33no—
19C24×C32A411yescharacteristic
20C22×C3×A4—43no—
21C2×C32×D4—32no—
22C2×C32×D4—31no—
23C2×C32×D4—31no—
24C22×C32×C4—31no—
25C32×C4⋊C4—32no—
26C32×C22⋊C4—31no—
27C32×C22⋊C4—32no—
28C32×C22⋊C4—31no—
29C32×C42C22×C311yescharacteristic, complemented
30C3×C42⋊C3C22×C313yescomplemented
31C22×H(3,𝔽3)—161no—
32C22⋊H(3,𝔽3)—41no—
33C3×#127—31no—
34C3×#127C2×C3211yescharacteristic
35C3×#126—32no—
36C3×#126C2×C3212yes—
37C3×#123—33no—
38C3×#123—31no—
39C3×#121—32no—
40C3×#121—36no—
41C3×#120—33no—
42C3×#120—31no—
43#127⋊C3—33no—
44#126⋊C3—36no—
45C23×C32—31no—
46C23×C32—121no—
47C23×C32C2×A412yes—
48C23×C32C2×A411yescharacteristic
49C2×C3×A4—46no—
50C2×C3×A4—43no—
51C32×D4—122no—
52C32×D4—121no—
53C32×D4—121no—
54C2×C32×C4—31no—
55C2×C32×C4—32no—
56C2×C32×C4—121no—
57#560H(3,𝔽3)11yescharacteristic, complemented, Sylow
58C2×H(3,𝔽3)—162no—
59C2×H(3,𝔽3)—161no—
60C24×C3—31no—
61C24×C3C3×A411yescharacteristic
62C22×A4—123no—
63C2×C3×D4—31no—
64C2×C3×D4—36no—
65C2×C3×D4—33no—
66C2×C3×D4—33no—
67C2×C3×D4—32no—
68C2×C3×D4—31no—
69C22×C3×C4—33no—
70C22×C3×C4—31no—
71C3×C4⋊C4—32no—
72C3×C4⋊C4—36no—
73C3×C22⋊C4—36no—
74C3×C22⋊C4—33no—
75C3×C22⋊C4—33no—
76C3×C22⋊C4—31no—
77C3×C22⋊C4—32no—
78C3×C22⋊C4—31no—
79C3×C42—31no—
80C3×C42C22×C3211yescharacteristic
81C42⋊C3—33no—
82C22×C32—41no—
83C22×C32—62no—
84C22×C32—61no—
85C22×C32—61no—
86C22×C32—62no—
87C22×C32—121no—
88C22×C32—163no—
89C22×C32C22×A411yescharacteristic
90C3×A4—43no—
91C32×C4—62no—
92C32×C4—61no—
93#127C2×H(3,𝔽3)11yescharacteristic, complemented
94#126C2×H(3,𝔽3)12yescomplemented
95#123—31no—
96#121—32no—
97#120—31no—
98H(3,𝔽3)—161noSylow
99C23×C3—33no—
100C23×C3—31no—
101C23×C3—32no—
102C23×C3—31no—
103C23×C3—121no—
104C23×C3—123no—
105C23×C3C2×C3×A412yes—
106C23×C3C2×C3×A411yescharacteristic
107C2×A4—126no—
108C2×A4—123no—
109C3×D4—122no—
110C3×D4—126no—
111C3×D4—121no—
112C3×D4—123no—
113C3×D4—121no—
114C3×D4—123no—
115C2×C3×C4—33no—
116C2×C3×C4—36no—
117C2×C3×C4—31no—
118C2×C3×C4—32no—
119C2×C3×C4—123no—
120C2×C3×C4—121no—
121C2×C32—31no—
122C2×C32—42no—
123C2×C32—41no—
124C2×C32—121no—
125C2×C32—166no—
126C2×C32—163no—
127C24C22⋊H(3,𝔽3)11yescharacteristic
128C2×D4—31no—
129C2×D4—32no—
130C2×D4—31no—
131C22×C4—31no—
132C4⋊C4—32no—
133C22⋊C4—31no—
134C22⋊C4—32no—
135C22⋊C4—31no—
136C42C22×H(3,𝔽3)11yescharacteristic, complemented
137C22×C3—31no—
138C22×C3—41no—
139C22×C3—62no—
140C22×C3—61no—
141C22×C3—63no—
142C22×C3—61no—
143C22×C3—66no—
144C22×C3—62no—
145C22×C3—66no—
146C22×C3—63no—
147C22×C3—121no—
148C22×C3—121no—
149C22×C3—123no—
150C22×C3—483no—
151C22×C3C22×C3×A411yescharacteristic
152A4—123no—
153C3×C4—66no—
154C3×C4—62no—
155C3×C4—63no—
156C3×C4—61no—
157C32—163no—
158C32#560⋊C311yescharacteristic, complemented
159C23—31no—
160C23—121no—
161C23C2×C22⋊H(3,𝔽3)12yes—
162C23C2×C22⋊H(3,𝔽3)11yescharacteristic
163D4—122no—
164D4—121no—
165D4—121no—
166C2×C4—31no—
167C2×C4—32no—
168C2×C4—121no—
169C2×C3—33no—
170C2×C3—31no—
171C2×C3—42no—
172C2×C3—41no—
173C2×C3—122no—
174C2×C3—121no—
175C2×C3—123no—
176C2×C3—121no—
177C2×C3—486no—
178C2×C3—483no—
179C22—41no—
180C22—62no—
181C22—61no—
182C22—61no—
183C22—62no—
184C22—121no—
185C22C22×C22⋊H(3,𝔽3)11yescharacteristic
186C4—62no—
187C4—61no—
188C3—31no—
189C3—483no—
190C3C3×#560⋊C311yescharacteristic
191C2—31no—
192C2—42no—
193C2—41no—
194C2—121no—
subgroupG/H
centerC3C3×#560⋊C3
commutator, cosocleC3×C42C22×C32
Frattini, socleC22×C3C22×C3×A4
FittingC32×#560C3
hypercenterC32#560⋊C3
hypocenterC42C22×H(3,𝔽3)
LCSG ▷ C3×C42 ▷ C42
UCSC1 ◁ C3 ◁ C32

Complements

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

NH ≅ G/Ntype
1. #127⋊H(3,𝔽3)C2semidirect
2. #126⋊H(3,𝔽3)C2semidirect
3. C32×#560C3semidirect
4. C3×#560⋊C3C3semidirect
6. C32⋊(C42⋊C3)C22semidirect
7. C32×#127C2×C3semidirect
8. C32×#126C2×C3semidirect
12. C3×#127⋊C3C2×C3semidirect
13. C3×#126⋊C3C2×C3semidirect
29. C32×C42C22×C3semidirect
30. C3×C42⋊C3C22×C3semidirect
57. #560H(3,𝔽3)semidirect
93. #127C2×H(3,𝔽3)semidirect
94. #126C2×H(3,𝔽3)semidirect
136. C42C22×H(3,𝔽3)semidirect
158. C32#560⋊C3semidirect

Maximal subgroups

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

subgroupindex
#127⋊H(3,𝔽3)2
#126⋊H(3,𝔽3)2
C32×#5603
C3×#560⋊C33
C22×C22⋊H(3,𝔽3)4