Finite Group Databasebeta

← C4⋊#88614 #13147101

Subgroups — C4⋊#88614 #13147101

Order 1728.

272 conjugacy classes of subgroups; only the 48 normal subgroups are shown.

normal subgroups only

This group has 272 conjugacy classes of subgroups — too many to draw. Only its 48 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
1C2×#88614C212yesmaximal, complemented
2C4×C9⋊SL(2,𝔽3)C211yesmaximal, characteristic, complemented
3C4⋊(C3⋊(#41⋊C3))—31nomaximal
4#10589679C311yesmaximal, characteristic, complemented
5C2×C9⋊SL(2,𝔽3)C2211yescharacteristic
6#88614—22no—
7C4⋊(3−1+2⋊C4)—41nomaximal
8C2×C3⋊(#41⋊C3)—32no—
9C3×C4×SL(2,𝔽3)D311yescharacteristic
10C2×#3701C2×C312yes—
11C4×Q8×C9C2×C311yescharacteristic
12#63457—32no—
13#63436—31no—
14C4×D36—31no—
15C4×Q8⋊C9—21no—
16C2×C4×3−1+2—41no—
17C2×3−1+2⋊C4—42no—
18C9⋊SL(2,𝔽3)D411yescharacteristic, complemented
19#5577—31no—
20C4⋊(#41⋊C3)—91no—
21C2×C3×SL(2,𝔽3)C2×D311yescharacteristic
22C3⋊(#41⋊C3)—62no—
23C3×C4⋊Dic3—121no—
24C2×Q8×C9C22×C311yescharacteristic
25#3701—22no—
26#3701—62no—
27C2×D36—32no—
28C2×D36—31no—
29C2×D36—32no—
30C2×C4×D9—32no—
31C2×Q8⋊C9—21no—
32C9×C4⋊C4—31no—
33C42×C9—31no—
34(C2×D9)⋊C4—32no—
35C4⋊Dic9A411yescharacteristic
36C22×3−1+2—41no—
37C4×3−1+2—42no—
383−1+2⋊C4—82no—
39C2×#239—32no—
40C2×#41⋊C3—92no—
41C3×C4×Q8C3×D311yescharacteristic
42#895—92no—
43#874—91no—
44C4×D12—91no—
45C4×SL(2,𝔽3)—31no—
46C2×C32×C4—41no—
47C2×C3×Dic3—122no—
48C3×SL(2,𝔽3)D1211yescharacteristic
49C22×D9—32no—
50Q8×C9—31no—
51Q8×C9C3×D411yescharacteristic
52C2×C4×C9—31no—
53C2×C4×C9—31no—
54C2×C4×C9—81no—
55C2×C4×C9C2×A411yescharacteristic
56C2×Dic9C2×A412yes—
57D36—62no—
58D36—62no—
59D36—62no—
60D36—122no—
61C4×D9—62no—
62C4×D9—62no—
63Q8⋊C9—21no—
64#550—91noSylow
65C2×3−1+2—41no—
66C2×3−1+2—41no—
67C2×3−1+2—41no—
68C2×C3×Q8C2×C3×D311yescharacteristic
69#239—62no—
70#239—182no—
71C2×D12—92no—
72C2×D12—91no—
73C2×D12—92no—
74C2×C4×D3—92no—
75#41⋊C3—182no—
76C2×SL(2,𝔽3)—31no—
77C3×C4⋊C4—31no—
78C3×C4⋊C4—361no—
79C3×C42—31no—
80(C2×D3)⋊C4—92no—
81C4⋊Dic3—31no—
82C22×C32—41no—
83C32×C4—42no—
84C3×Dic3—242no—
85C22×C9—81no—
86C22×C9C22×A411yescharacteristic
87C2×D9—62no—
88C2×D9—62no—
89C2×D9—62no—
90C4×C9—31no—
91C4×C9—31no—
92C4×C9—61no—
93C4×C9—82no—
94C4×C9#41⋊C312yescomplemented
95Dic9—22no—
96C2×#41—92no—
97#127—271no—
98#121—272no—
99C4×Q8AGL(1,ℤ/9)11yescharacteristic
100C4×D4—271no—
1013−1+2—41noSylow
102C22×D3—92no—
103C3×Q8—31no—
104C3×Q8C3×D1211yescharacteristic
105C2×C3×C4—31no—
106C2×C3×C4—31no—
107C2×C3×C4—121no—
108C2×C3×C4—362no—
109C2×C3×C4D3×A411yescharacteristic
110C2×Dic3—32no—
111D12—182no—
112D12—182no—
113D12—182no—
114D12—362no—
115C4×D3—182no—
116C4×D3—182no—
117SL(2,𝔽3)—31no—
118C2×C32—41no—
119C2×C32—41no—
120C2×C32—41no—
121C2×C9—81no—
122C2×C9—81no—
123C2×C9—81no—
124C2×C9D4×A411yescharacteristic
125C2×C9C2×#41⋊C311yescharacteristic
126C2×C9#99411yescharacteristic
127D9—122no—
128#41—182no—
129#41—542no—
130C2×Q8C2×AGL(1,ℤ/9)11yescharacteristic
131C2×D4—272no—
132C2×D4—272no—
133C2×D4—271no—
134C22×C4—272no—
135C4⋊C4—31no—
136C4⋊C4—91no—
137C22⋊C4—272no—
138C42—31no—
139C22×C3—121no—
140C22×C3C2×D3×A411yescharacteristic
141C2×D3—182no—
142C2×D3—182no—
143C2×D3—182no—
144C3×C4—31no—
145C3×C4—31no—
146C3×C4—61no—
147C3×C4—122no—
148C3×C4—722no—
149C3×C4C3⋊(#41⋊C3)12yes—
150Dic3—62no—
151C32—41no—
152C9—81no—
153C9C4⋊(#41⋊C3)11yescharacteristic, complemented
154C23—272no—
155Q8—31no—
156Q8C4⋊AGL(1,ℤ/9)11yescharacteristic
157D4—542no—
158D4—542no—
159D4—542no—
160D4—1082no—
161C2×C4—31no—
162C2×C4—31no—
163C2×C4—92no—
164C2×C4—542no—
165C2×C4—542no—
166C2×C4D9⋊A411yescharacteristic
167C2×C3—121no—
168C2×C3—121no—
169C2×C3—121no—
170C2×C3A4×D1211yescharacteristic
171C2×C3C2×C3⋊(#41⋊C3)11yescharacteristic
172C2×C3C3⋊#99411yescharacteristic
173D3—362no—
174C22—542no—
175C22—542no—
176C22—542no—
177C22C2×D9⋊A411yescharacteristic
178C4—31no—
179C4—31no—
180C4—61no—
181C4—182no—
182C4#8861412yescomplemented
183C3—121no—
184C3C4⋊(C3⋊(#41⋊C3))11yescharacteristic
185C2—1082no—
186C2C4⋊(D9⋊A4)11yescharacteristic
187C2C2×#8861411yescharacteristic
188C2#1172241111yescharacteristic
subgroupG/H
centerC22C2×D9⋊A4
commutator, cosocleC2×Q8×C9C22×C3
Frattini, socleC22×C3C2×D3×A4
FittingC4×Q8×C9C2×C3
hypercenterC2×C4D9⋊A4
hypocenterQ8×C9C3×D4
LCSG ▷ C2×Q8×C9 ▷ Q8×C9
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
1. C2×#88614C2semidirect
2. C4×C9⋊SL(2,𝔽3)C2semidirect
4. #10589679C3semidirect
18. C9⋊SL(2,𝔽3)D4semidirect
94. C4×C9#41⋊C3semidirect
153. C9C4⋊(#41⋊C3)semidirect
182. C4#88614semidirect

Maximal subgroups

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

subgroupindex
C2×#886142
C4×C9⋊SL(2,𝔽3)2
C4⋊(C3⋊(#41⋊C3))3
#105896793
C4⋊(3−1+2⋊C4)4