Finite Group Databasebeta

← #13147925

Subgroups — #13147925

Order 1728.

990 conjugacy classes of subgroups; only the 129 normal subgroups are shown.

normal subgroups only

This group has 990 conjugacy classes of subgroups — too many to draw. Only its 129 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#11721720C211yesmaximal, characteristic, complemented
2#11721387C212yesmaximal, complemented
3#11721163C213yesmaximal, complemented
4#11721157C216yesmaximal, complemented
5#11721151C213yesmaximal, complemented
6#10594022—32nomaximal
7C22×((C3⋊D3)⋊C3)⋊C2C2211yescharacteristic
8C2×C4⋊#272C2213yes—
9C2×C4⋊#272C2213yescomplemented
10C2×#6200C2216yes—
11C2×#6199C2213yescomplemented
12C42×H(3,𝔽3)C2211yescharacteristic, complemented
13#3752⋊C3C2216yes—
14#3750⋊C3C2216yescomplemented
15#88451C2213yes—
16#88451C2213yescomplemented
17#64085D312yescomplemented
18C3×#876—32no—
19C3×#874—32no—
20C3×#874—61no—
21#63922—36no—
22#63886—36no—
23#63885—36no—
24#63884—36no—
25C22×#272C2311yescharacteristic
26C22×#272C2313yes—
27C22×(C3⋊D3)⋊C3C2312yes—
28C2×((C3⋊D3)⋊C3)⋊C2—43no—
29C2×C4×H(3,𝔽3)C2313yes—
30C4⋊#272—43no—
31C4⋊#272—43no—
32C2×(C3⋊Dic3)⋊C3C2316yes—
33#6200—46no—
34#6199—43no—
35#5720—91no—
36#5711—92no—
37C22×D32—32no—
38C2×C3×D12—36no—
39C2×C3×D12—36no—
40C2×C3×D12—63no—
41C2×C3×D12—63no—
42C2×#626—36no—
43C2×#625—36no—
44C2×D3×Dic3—36no—
45C32×C42—21no—
46C32×C42C2×D312yescomplemented
47#3752C2×D316yes—
48#3750C2×D316yescomplemented
49C3×(C2×D3)⋊C4—36no—
50C3×Dic3⋊C4—36no—
51(C2×D3)⋊Dic3—36no—
52(C2×D3)⋊Dic3—36no—
53C22×H(3,𝔽3)C2411yescharacteristic
54C2×#272—23no—
55C2×#272—26no—
56C2×#272—23no—
57C2×(C3⋊D3)⋊C3—26no—
58((C3⋊D3)⋊C3)⋊C2—44no—
59C4×H(3,𝔽3)—23no—
60(C3⋊Dic3)⋊C3—26no—
61C3×#127—91no—
62C3×#126—92no—
63#939—96no—
64#937—93no—
65#935—93no—
66#895—96no—
67#894—96no—
68#884—96no—
69#882—96no—
70#876—32no—
71#876—92no—
72#874—32no—
73#874—61no—
74C22×C3⋊D3C22×D312yes—
75C22×C3×D3—32no—
76C22×C3×D3—32no—
77C22×C3×D3—36no—
78C22×C3×D3—61no—
79C22×C3×D3—63no—
80C2×D32—126no—
81C2×C32×C4—23no—
82C2×C32×C4C22×D316yes—
83C2×C3⋊Dic3C22×D316yes—
84C2×C3×Dic3—36no—
85C3×D12—126no—
86C3×D12—126no—
87C3×D12—126no—
88C3×D12—126no—
89#626—126no—
90#625—126no—
91D3×Dic3—126no—
92#560—271noSylow
93C2×H(3,𝔽3)H(5,𝔽2)13yes—
94#272—41no—
95#272—43no—
96(C3⋊D3)⋊C3—42no—
97C23×D3—91no—
98C23×D3—92no—
99C2×C3×D4—93no—
100C2×C3×D4—93no—
101C2×#67—96no—
102C2×#67—96no—
103C22×Dic3—93no—
104C2×D12—36no—
105C2×D12—36no—
106C2×D12—63no—
107C2×D12—63no—
108C2×D12—96no—
109C2×C4×D3—96no—
110C3×C4⋊C4—96no—
111C3×C22⋊C4—96no—
112C3×C42—32no—
113C3×C42—61no—
114C3×C42D3211yescharacteristic
115#217—93no—
116#217—93no—
117(C2×D3)⋊C4—36no—
118(C2×D3)⋊C4—96no—
119(C2×D3)⋊C4—96no—
120(C2×D3)⋊C4—96no—
121Dic3⋊C4—36no—
122Dic3⋊C4—96no—
123C22×C32—21no—
124C22×C32C23×D312yes—
125C2×C3⋊D3—26no—
126C2×C3×D3—66no—
127C2×C3×D3—66no—
128C2×C3×D3—66no—
129C2×C3×D3—66no—
130C2×C3×D3—66no—
131C2×C3×D3—123no—
132C2×C3×D3—126no—
133C2×C3×D3—123no—
134D32—128no—
135C32×C4—26no—
136C32×C4—26no—
137C3⋊Dic3—26no—
138C3×Dic3—66no—
139#127—91no—
140#126—92no—
141#123—273no—
142#121—276no—
143#120—273no—
144H(3,𝔽3)#56011yescharacteristic, complemented, Sylow
145C23×C3—92no—
146C23×C3—91no—
147C23×C3—93no—
148C22×D3—32no—
149C22×D3—36no—
150C22×D3—32no—
151C22×D3—61no—
152C22×D3—63no—
153C22×D3—92no—
154C22×D3—363no—
155C22×D3—366no—
156C3×D4—363no—
157C3×D4—363no—
158C2×C3×C4—36no—
159C2×C3×C4—63no—
160C2×C3×C4—96no—
161C2×C3×C4C2×D3213yes—
162#67—366no—
163#67—366no—
164C2×Dic3—36no—
165C2×Dic3—96no—
166C2×Dic3—363no—
167D12—126no—
168D12—126no—
169D12—126no—
170D12—126no—
171D12—366no—
172C4×D3—366no—
173C2×C32—23no—
174C2×C32#100916yes—
175C3⋊D3—42no—
176C3×D3—122no—
177C3×D3—122no—
178C3×D3—126no—
179C3×D3—241no—
180C3×D3—243no—
181C24—271no—
182C2×D4—93no—
183C2×D4—93no—
184C2×D4—276no—
185C22×C4—273no—
186C4⋊C4—96no—
187C22⋊C4—96no—
188C22⋊C4—273no—
189C22⋊C4—273no—
190C42((C3⋊D3)⋊C3)⋊C211yescharacteristic, complemented
191C22×C3—32no—
192C22×C3—61no—
193C22×C3—186no—
194C22×C3—183no—
195C22×C3—186no—
196C22×C3—183no—
197C22×C3C22×D3211yescharacteristic
198C2×D3—66no—
199C2×D3—66no—
200C2×D3—66no—
201C2×D3—66no—
202C2×D3—66no—
203C2×D3—123no—
204C2×D3—126no—
205C2×D3—123no—
206C2×D3—186no—
207C2×D3—364no—
208C2×D3—368no—
209C3×C4—23no—
210C3×C4—66no—
211C3×C4—66no—
212C3×C4—186no—
213Dic3—66no—
214Dic3—186no—
215C32—21no—
216C32#571112yescomplemented
217C23—92no—
218C23—91no—
219C23—93no—
220C23—1083no—
221D4—363no—
222D4—363no—
223D4—1086no—
224C2×C4—96no—
225C2×C4—1083no—
226C2×C4C2×((C3⋊D3)⋊C3)⋊C213yes—
227C2×C3—36no—
228C2×C3—63no—
229C2×C3—362no—
230C2×C3—361no—
231C2×C3—363no—
232C2×C3#6430813yes—
233D3—122no—
234D3—126no—
235D3—122no—
236D3—241no—
237D3—243no—
238D3—362no—
239C22—183no—
240C22—186no—
241C22—183no—
242C22—186no—
243C22—1084no—
244C22C22×((C3⋊D3)⋊C3)⋊C211yescharacteristic
245C4—23no—
246C4—186no—
247C3—32no—
248C3—61no—
249C3#1059402311yescharacteristic
250C2—361no—
251C2—362no—
252C2—363no—
253C2#1172249313yes—
subgroupG/H
centerC22C22×((C3⋊D3)⋊C3)⋊C2
commutator, cosocleC22×H(3,𝔽3)C24
Frattini, socleC22×C3C22×D32
FittingC42×H(3,𝔽3)C22
hypercenterC42((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#560
LCSG ▷ C22×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C42

Complements

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

NH ≅ G/Ntype
1. #11721720C2semidirect
2. #11721387C2semidirect
3. #11721163C2semidirect
4. #11721157C2semidirect
5. #11721151C2semidirect
9. C2×C4⋊#272C22semidirect
11. C2×#6199C22semidirect
12. C42×H(3,𝔽3)C22semidirect
14. #3750⋊C3C22semidirect
16. #88451C22semidirect
17. #64085D3semidirect
46. C32×C42C2×D3semidirect
48. #3750C2×D3semidirect
144. H(3,𝔽3)#560semidirect
190. C42((C3⋊D3)⋊C3)⋊C2semidirect
216. C32#5711semidirect

Maximal subgroups

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

subgroupindex
#117217202
#117213872
#117211632
#117211572
#117211512
#105940223