Finite Group Databasebeta

← #13138873

Subgroups — #13138873

Order 1728.

502 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 502 conjugacy classes of subgroups — too many to draw. Only its 81 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#11721136C212yesmaximal, complemented
2C2×#88432C211yesmaximal, characteristic
3#11720548C211yesmaximal, characteristic, complemented
4#63648⋊C3C212yesmaximal, complemented
5#11720314C211yesmaximal, characteristic, complemented
6#10590746—32nomaximal
7C2×C4⋊((C3⋊D3)⋊C3)C2212yes—
8C2×((C3⋊D3)⋊C3)⋊C4—22no—
9C2×C8×H(3,𝔽3)C2211yescharacteristic
10C4×#1084C2211yescharacteristic
11C2×#6180C2211yescharacteristic
12C4⋊((C3⋊Dic3)⋊C3)C2212yes—
13#88450—22no—
14#88432C414yescomplemented
15#63894—32no—
16Dic3×D12—32no—
17C2×#3714—32no—
18#63648D312yescomplemented
19C3×#821—32no—
20C3×#815—32no—
21C3×#815—61no—
22#63573—32no—
23C22×(C3⋊D3)⋊C3—22no—
24C2×C4×H(3,𝔽3)C2311yescharacteristic
25C2×#1084D412yes—
26C2×(C3⋊Dic3)⋊C3—22no—
27C4⋊((C3⋊D3)⋊C3)—22no—
28C4⋊((C3⋊D3)⋊C3)C2×C414yescomplemented
29((C3⋊D3)⋊C3)⋊C4—44no—
30C8×H(3,𝔽3)—21no—
31#6180C2×C412yescomplemented
32#4785—92no—
33#4711—91no—
34C2×C4⋊(C3⋊D3)C2×D312yes—
35C2×C3×D12—32no—
36C2×#624—62no—
37C2×D3×Dic3—62no—
38C2×C32×C8—21no—
39C2×C32×C8C2×D312yes—
40C4⋊(C3⋊Dic3)C2×D312yes—
41C3×C4⋊Dic3—32no—
42C3×C4×Dic3—32no—
43C3×C4×Dic3—61no—
44C2×C3×C3⋊C8—32no—
45C2×C3×C3⋊C8—61no—
46#3722—62no—
47(C2×D3)⋊Dic3—62no—
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)C22×C411yescharacteristic
54C4×H(3,𝔽3)#4111yescharacteristic
55#1084—22no—
56(C3⋊Dic3)⋊C3—42no—
57D4×Dic3—92no—
58C2×#213—92no—
59C2×D24—91no—
60#893—92no—
61C4×D12—92no—
62C3×#102—92no—
63C3×#97—91no—
64#821—32no—
65#821—92no—
66#817—91no—
67#815—32no—
68#815—61no—
69#808—92no—
70C22×C3⋊D3—22no—
71C22×C3×D3—62no—
72C2×C32×C4—21no—
73C2×C32×C4C22×D312yes—
74C2×C3⋊Dic3—22no—
75C4⋊(C3⋊D3)—22no—
76C4⋊(C3⋊D3)C4×D314yescomplemented
77C2×C3×Dic3—32no—
78C2×C3×Dic3—32no—
79C2×C3×Dic3—62no—
80C2×C3×Dic3—62no—
81C3×D12—34no—
82C3×D12—62no—
83#624—124no—
84D3×Dic3—124no—
85C32×C8—22no—
86C32×C8—22no—
87C3×C3⋊C8—34no—
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×Dic3—182no—
95C2×D12—32no—
96C2×D12—92no—
97C2×C4×D3—182no—
98C2×C4×D3—182no—
99C2×C3×C8—32no—
100C2×C3×C8—61no—
101C2×C3×C8—91no—
102C2×C3×C8D3211yescharacteristic
103C3×C4⋊C4—92no—
104C3×C42—91no—
105#217—182no—
106#213—98no—
107(C2×D3)⋊C4—182no—
108(C2×D3)⋊C4—182no—
109C4⋊Dic3—32no—
110C4⋊Dic3—92no—
111C4×Dic3—32no—
112C4×Dic3—61no—
113C2×C3⋊C8—32no—
114C2×C3⋊C8—61no—
115D24—94no—
116C22×C32—21no—
117C22×C32D3×D412yes—
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—21no—
125C32×C4—21no—
126C32×C4C2×C4×D312yes—
127C32×C4#23712yes—
128C3⋊Dic3—42no—
129C3×Dic3—62no—
130C3×Dic3—62no—
131C3×Dic3—122no—
132C3×Dic3—122no—
133C2×D8—271no—
134C4×D4—272no—
135#106—271no—
136#102—92no—
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—61no—
146C2×C3×C4—92no—
147C2×C3×C4—182no—
148C2×C3×C4C2×D3211yescharacteristic
149C2×Dic3—32no—
150C2×Dic3—32no—
151C2×Dic3—62no—
152C2×Dic3—62no—
153C2×Dic3—182no—
154C2×Dic3—364no—
155D12—34no—
156D12—62no—
157D12—94no—
158D12—182no—
159C4×D3—364no—
160C4×D3—364no—
161C3×C8—21no—
162C3×C8—62no—
163C3×C8—62no—
164C3×C8—92no—
165C3⋊C8—34no—
166C3⋊C8—62no—
167C2×C32—22no—
168C2×C32—21no—
169C2×C32#67⋊C412yes—
170C2×C32#91112yes—
171C2×C32#91412yes—
172C3⋊D3—44no—
173C3×D3—124no—
174C2×D4—92no—
175C22×C4—542no—
176D8—274no—
177C2×C8—91no—
178C2×C8((C3⋊D3)⋊C3)⋊C211yescharacteristic
179C4⋊C4—92no—
180C22⋊C4—542no—
181C42—91no—
182C22×C3—32no—
183C22×C3—61no—
184C22×C3—184no—
185C22×C3—362no—
186C22×C3—362no—
187C22×C3#380211yescharacteristic
188C2×D3—64no—
189C2×D3—122no—
190C2×D3—122no—
191C2×D3—184no—
192C2×D3—362no—
193C2×D3—362no—
194C3×C4—32no—
195C3×C4—32no—
196C3×C4—61no—
197C3×C4—61no—
198C3×C4—182no—
199C3×C4—362no—
200C3×C4C4×D3211yescharacteristic
201C3×C4#380511yescharacteristic
202Dic3—62no—
203Dic3—62no—
204Dic3—122no—
205Dic3—122no—
206Dic3—362no—
207C32—21no—
208C32#478512yescomplemented
209C23—182no—
210D4—94no—
211D4—182no—
212C2×C4—92no—
213C2×C4—182no—
214C2×C4—1084no—
215C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
216C8—21no—
217C8—92no—
218C2×C3—32no—
219C2×C3—32no—
220C2×C3—32no—
221C2×C3—62no—
222C2×C3—61no—
223C2×C3—364no—
224C2×C3#6379812yes—
225C2×C3#6390211yescharacteristic
226D3—124no—
227D3—364no—
228C22—184no—
229C22—362no—
230C22—362no—
231C22#8865511yescharacteristic
232C4—182no—
233C4—362no—
234C4C4×((C3⋊D3)⋊C3)⋊C211yescharacteristic
235C4#8867211yescharacteristic
236C3—32no—
237C3—61no—
238C3#1059074111yescharacteristic
239C2—364no—
240C2#1172083912yes—
241C2#1172113811yescharacteristic
subgroupG/H
centerC22#88655
commutatorC4×H(3,𝔽3)C22×C4
FrattiniC2×C3×C4C2×D32
FittingC2×C8×H(3,𝔽3)C22
socleC22×C3#3802
prime radicalC2×#88432C2
cosocleC2×C4×H(3,𝔽3)C23
hypercenterC2×C8((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#442
LCSG ▷ C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×C8

Complements

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

NH ≅ G/Ntype
1. #11721136C2semidirect
3. #11720548C2semidirect
4. #63648⋊C3C2semidirect
5. #11720314C2semidirect
14. #88432C4semidirect
18. #63648D3semidirect
28. C4⋊((C3⋊D3)⋊C3)C2×C4semidirect
31. #6180C2×C4semidirect
76. C4⋊(C3⋊D3)C4×D3semidirect
138. H(3,𝔽3)#442semidirect
208. C32#4785semidirect

Maximal subgroups

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

subgroupindex
#117211362
C2×#884322
#117205482
#63648⋊C32
#117203142
#105907463