Finite Group Databasebeta

← #13137391

Subgroups — #13137391

Order 1728.

502 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 502 conjugacy classes of subgroups — too many to draw. Only its 89 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
1C4×C4⋊((C3⋊D3)⋊C3)C212yesmaximal, complemented
2C2×#88432C211yesmaximal, characteristic
3#11720520C211yesmaximal, characteristic, complemented
4#11720318C212yesmaximal, complemented
5#11720314C211yesmaximal, characteristic, complemented
6#10590346—32nomaximal
7C2×C4⋊((C3⋊D3)⋊C3)C2212yes—
8C2×C4×(C3⋊D3)⋊C3—22no—
9C42×H(3,𝔽3)C2211yescharacteristic
10C2×#6180C2211yescharacteristic
11C2×#6180C2211yescharacteristic
12#3752⋊C3—22no—
13C4⋊((C3⋊Dic3)⋊C3)C2212yes—
14#88432C414yescomplemented
15C4×C4⋊(C3⋊D3)D312yescomplemented
16C3×C4×D12—32no—
17C2×#3714—32no—
18C3×#803—32no—
19C3×#803—61no—
20#63573—32no—
21#63564—32no—
22#63563—32no—
23C22×(C3⋊D3)⋊C3—22no—
24C2×C4×H(3,𝔽3)D412yes—
25C2×C4×H(3,𝔽3)C2311yescharacteristic
26C2×(C3⋊Dic3)⋊C3—22no—
27C4⋊((C3⋊D3)⋊C3)—22no—
28C4⋊((C3⋊D3)⋊C3)C2×C414yescomplemented
29C4×(C3⋊D3)⋊C3—44no—
30#6180—21no—
31#6180C2×C412yescomplemented
32#5014—92no—
33#4711—91no—
34C2×C4⋊(C3⋊D3)C2×D312yes—
35C2×C4×C3⋊D3—22no—
36C2×C3×D12—32no—
37C2×C3×C4×D3—62no—
38C32×C42—21no—
39C32×C42C2×D312yes—
40#3752—22no—
41C4⋊(C3⋊Dic3)C2×D312yes—
42C3×(C2×D3)⋊C4—62no—
43C3×C4⋊Dic3—32no—
44C2×C3×C3⋊C8—32no—
45C2×C3×C3⋊C8—32no—
46C2×C3×C3⋊C8—61no—
47C2×C3×C3⋊C8—61no—
48#3714—38no—
49C22×H(3,𝔽3)C2×D411yescharacteristic
50C2×(C3⋊D3)⋊C3—24no—
51C2×(C3⋊D3)⋊C3—42no—
52C2×(C3⋊D3)⋊C3—42no—
53C4×H(3,𝔽3)—22no—
54C4×H(3,𝔽3)C22×C411yescharacteristic
55C4×H(3,𝔽3)#4111yescharacteristic
56(C3⋊Dic3)⋊C3—42no—
57C3×C4×D4—92no—
58C2×#213—92no—
59C2×D24—91no—
60C4×D12—32no—
61C4×D12—92no—
62C3×#97—91no—
63#832—92no—
64#821—92no—
65#817—91no—
66#809—92no—
67#808—92no—
68#803—32no—
69#803—61no—
70C22×C3⋊D3—22no—
71C22×C3×D3—62no—
72C2×C32×C4—22no—
73C2×C32×C4—21no—
74C2×C32×C4#6712yes—
75C2×C32×C4#6712yes—
76C2×C32×C4C22×D312yes—
77C2×C3⋊Dic3—22no—
78C4⋊(C3⋊D3)—22no—
79C4⋊(C3⋊D3)C4×D314yescomplemented
80C4×C3⋊D3—44no—
81C2×C3×Dic3—62no—
82C3×D12—34no—
83C3×D12—62no—
84C3×C4×D3—124no—
85C3×C3⋊C8—34no—
86C3×C3⋊C8—62no—
87C3×C3⋊C8—62no—
88C3×C3⋊C8—62no—
89#442—271noSylow
90C2×H(3,𝔽3)C4×D411yescharacteristic
91C2×H(3,𝔽3)AGL(1,ℤ/8)12yes—
92(C3⋊D3)⋊C3—44no—
93C2×C3×D4—92no—
94C22×C3×C4—182no—
95C2×D12—32no—
96C2×D12—92no—
97C2×C4×D3—62no—
98C2×C4×D3—182no—
99C2×C3×C8—91no—
100C2×C3×C8—91no—
101C3×C4⋊C4—92no—
102C3×C22⋊C4—182no—
103C3×C42—32no—
104C3×C42—61no—
105C3×C42D3211yescharacteristic
106#213—98no—
107(C2×D3)⋊C4—62no—
108(C2×D3)⋊C4—182no—
109C4⋊Dic3—32no—
110C4⋊Dic3—92no—
111C2×C3⋊C8—32no—
112C2×C3⋊C8—32no—
113C2×C3⋊C8—61no—
114C2×C3⋊C8—61no—
115D24—94no—
116C22×C32—21no—
117C22×C32C2×#6712yes—
118C2×C3⋊D3—24no—
119C2×C3⋊D3—42no—
120C2×C3⋊D3—42no—
121C2×C3×D3—64no—
122C2×C3×D3—122no—
123C2×C3×D3—122no—
124C32×C4—22no—
125C32×C4—22no—
126C32×C4—21no—
127C32×C4—21no—
128C32×C4—42no—
129C32×C4C2×C4×D312yes—
130C32×C4#23512yes—
131C3⋊Dic3—42no—
132C3×Dic3—122no—
133C2×D8—271no—
134C4×D4—92no—
135#106—271no—
136#102—272no—
137#97—91no—
138H(3,𝔽3)#44211yescharacteristic, complemented, Sylow
139C23×C3—182no—
140C22×D3—62no—
141C22×D3—182no—
142C3×D4—94no—
143C3×D4—182no—
144C2×C3×C4—32no—
145C2×C3×C4—32no—
146C2×C3×C4—32no—
147C2×C3×C4—62no—
148C2×C3×C4—61no—
149C2×C3×C4—182no—
150C2×C3×C4—364no—
151C2×C3×C4#62512yes—
152C2×C3×C4C2×D3211yescharacteristic
153C2×Dic3—62no—
154C2×Dic3—182no—
155D12—34no—
156D12—62no—
157D12—94no—
158D12—182no—
159C4×D3—124no—
160C4×D3—364no—
161C3×C8—92no—
162C3×C8—181no—
163C3⋊C8—34no—
164C3⋊C8—62no—
165C3⋊C8—62no—
166C3⋊C8—62no—
167C2×C32—22no—
168C2×C32—21no—
169C2×C32C4×#6712yes—
170C2×C32#93212yes—
171C2×C32#94912yes—
172C3⋊D3—44no—
173C3×D3—124no—
174C2×D4—92no—
175C22×C4—182no—
176D8—274no—
177C2×C8—91no—
178C2×C8—91no—
179C4⋊C4—92no—
180C22⋊C4—182no—
181C42((C3⋊D3)⋊C3)⋊C211yescharacteristic
182C22×C3—32no—
183C22×C3—61no—
184C22×C3—184no—
185C22×C3—362no—
186C22×C3—362no—
187C22×C3C2×#62511yescharacteristic
188C2×D3—64no—
189C2×D3—122no—
190C2×D3—122no—
191C2×D3—184no—
192C2×D3—362no—
193C2×D3—362no—
194C3×C4—22no—
195C3×C4—32no—
196C3×C4—32no—
197C3×C4—62no—
198C3×C4—62no—
199C3×C4—61no—
200C3×C4—61no—
201C3×C4—122no—
202C3×C4—362no—
203C3×C4#379811yescharacteristic
204C3×C4C4×D3211yescharacteristic
205Dic3—122no—
206Dic3—362no—
207C32—21no—
208C32#501412yescomplemented
209C23—182no—
210D4—94no—
211D4—182no—
212C2×C4—182no—
213C2×C4—364no—
214C2×C4#619812yes—
215C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
216C8—92no—
217C8—181no—
218C2×C3—32no—
219C2×C3—32no—
220C2×C3—32no—
221C2×C3—62no—
222C2×C3—61no—
223C2×C3—364no—
224C2×C3#6382312yes—
225C2×C3C4×#62511yescharacteristic
226D3—124no—
227D3—364no—
228C22—184no—
229C22—362no—
230C22—362no—
231C22C2×#619811yescharacteristic
232C4—22no—
233C4—362no—
234C4#8865011yescharacteristic
235C4C4×((C3⋊D3)⋊C3)⋊C211yescharacteristic
236C3—32no—
237C3—61no—
238C3#1059034411yescharacteristic
239C2—364no—
240C2#1172106112yes—
241C2C4×#619811yescharacteristic
subgroupG/H
centerC22C2×#6198
commutatorC4×H(3,𝔽3)C22×C4
FrattiniC2×C3×C4C2×D32
FittingC42×H(3,𝔽3)C22
socleC22×C3C2×#625
prime radicalC2×#88432C2
cosocleC2×C4×H(3,𝔽3)C23
hypercenterC42((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#442
LCSG ▷ C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C42

Complements

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

NH ≅ G/Ntype
1. C4×C4⋊((C3⋊D3)⋊C3)C2semidirect
3. #11720520C2semidirect
4. #11720318C2semidirect
5. #11720314C2semidirect
14. #88432C4semidirect
15. C4×C4⋊(C3⋊D3)D3semidirect
28. C4⋊((C3⋊D3)⋊C3)C2×C4semidirect
31. #6180C2×C4semidirect
79. C4⋊(C3⋊D3)C4×D3semidirect
138. H(3,𝔽3)#442semidirect
208. C32#5014semidirect

Maximal subgroups

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

subgroupindex
C4×C4⋊((C3⋊D3)⋊C3)2
C2×#884322
#117205202
#117203182
#117203142
#105903463