Finite Group Databasebeta

← #13140042

Subgroups — #13140042

Order 1728.

545 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 545 conjugacy classes of subgroups — too many to draw. Only its 87 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
1C22×#6180C211yesmaximal, characteristic, complemented
2#64139⋊C3C212yesmaximal, complemented
3C2×#88432C211yesmaximal, characteristic, complemented
4#11720318C212yesmaximal, complemented
5#11720315C211yesmaximal, characteristic, complemented
6#10591164—32nomaximal
7C22×C4×H(3,𝔽3)C2211yescharacteristic
8C2×#638⋊C3—22no—
9C2×C4⋊((C3⋊D3)⋊C3)C2212yes—
10C2×#6180—22no—
11C2×#6180C2211yescharacteristic
12C2×#6180C2211yescharacteristic
13#3752⋊C3—22no—
14C4⋊((C3⋊Dic3)⋊C3)C2212yes—
15#88432—22no—
16#64139D312yescomplemented
17C3×#930—32no—
18C22×C3×C3⋊C8—32no—
19C22×C3×C3⋊C8—61no—
20C2×#3714—32no—
21#63575—32no—
22#63564—32no—
23#63563—32no—
24C23×H(3,𝔽3)D411yescharacteristic
25C22×(C3⋊D3)⋊C3—22no—
26C2×C4×H(3,𝔽3)—21no—
27C2×C4×H(3,𝔽3)—21no—
28C2×C4×H(3,𝔽3)D411yescharacteristic
29C2×C4×H(3,𝔽3)C2311yescharacteristic
30#638⋊C3—44no—
31C2×(C3⋊Dic3)⋊C3—22no—
32C4⋊((C3⋊D3)⋊C3)—22no—
33C4⋊((C3⋊D3)⋊C3)—22no—
34#6180—21no—
35#6180D412yescomplemented
36#5115—91no—
37#5038—92no—
38C22×C32×C4—21no—
39C22×C32×C4C2×D312yes—
40C2×#638—22no—
41C2×C4⋊(C3⋊D3)C2×D312yes—
42C2×C3×#67—62no—
43C2×C3×D12—32no—
44#3752—22no—
45C4⋊(C3⋊Dic3)C2×D312yes—
46C3×(C2×D3)⋊C4—62no—
47C3×C4⋊Dic3—32no—
48C2×C3×C3⋊C8—32no—
49C2×C3×C3⋊C8—32no—
50C2×C3×C3⋊C8—64no—
51C2×C3×C3⋊C8—61no—
52C2×C3×C3⋊C8—61no—
53C2×C3×C3⋊C8—64no—
54#3714—64no—
55C22×H(3,𝔽3)—21no—
56C22×H(3,𝔽3)—21no—
57C22×H(3,𝔽3)D812yescomplemented
58C22×H(3,𝔽3)C2×D411yescharacteristic
59C2×(C3⋊D3)⋊C3—42no—
60C2×(C3⋊D3)⋊C3—42no—
61C2×(C3⋊D3)⋊C3—42no—
62C4×H(3,𝔽3)—21no—
63C4×H(3,𝔽3)C2×D411yescharacteristic
64C4×H(3,𝔽3)#4111yescharacteristic
65(C3⋊Dic3)⋊C3—42no—
66C22×C3×C8—91no—
67C3×#121—92no—
68C2×#213—92no—
69#930—32no—
70#930—92no—
71C22×C3⋊C8—32no—
72C22×C3⋊C8—61no—
73C2×D24—91no—
74#832—92no—
75#821—92no—
76#818—91no—
77#809—92no—
78#807—92no—
79C23×C32—21no—
80C23×C32#6712yes—
81C22×C3⋊D3—22no—
82C22×C3×D3—62no—
83C2×C32×C4—22no—
84C2×C32×C4—22no—
85C2×C32×C4—22no—
86C2×C32×C4—22no—
87C2×C32×C4—21no—
88C2×C32×C4—21no—
89C2×C32×C4#6712yes—
90C2×C32×C4C22×D312yes—
91#638—44no—
92C2×C3⋊Dic3—22no—
93C4⋊(C3⋊D3)—22no—
94C4⋊(C3⋊D3)—22no—
95C3×#67—124no—
96C2×C3×Dic3—62no—
97C3×D12—62no—
98C3×D12—62no—
99C3×C3⋊C8—34no—
100C3×C3⋊C8—62no—
101C3×C3⋊C8—62no—
102C3×C3⋊C8—62no—
103#466—271noSylow
104C2×H(3,𝔽3)—22no—
105C2×H(3,𝔽3)#12111yescharacteristic
106C2×H(3,𝔽3)C2×D811yescharacteristic
107C2×H(3,𝔽3)#13511yescharacteristic
108(C3⋊D3)⋊C3—82no—
109C2×C3×D4—92no—
110C2×C3×D4—182no—
111C22×C3×C4—32no—
112C22×C3×C4—61no—
113C22×C3×C4D3211yescharacteristic
114C2×#67—62no—
115C2×#67—182no—
116C2×D12—32no—
117C2×D12—92no—
118C2×C3×C8—91no—
119C2×C3×C8—91no—
120C2×C3×C8—182no—
121C3×C4⋊C4—92no—
122C3×C22⋊C4—182no—
123#213—184no—
124(C2×D3)⋊C4—62no—
125(C2×D3)⋊C4—182no—
126C4⋊Dic3—32no—
127C4⋊Dic3—92no—
128C2×C3⋊C8—32no—
129C2×C3⋊C8—32no—
130C2×C3⋊C8—61no—
131C2×C3⋊C8—61no—
132C2×C3⋊C8—64no—
133C2×C3⋊C8—64no—
134D24—182no—
135C22×C32—22no—
136C22×C32—22no—
137C22×C32—22no—
138C22×C32—22no—
139C22×C32—22no—
140C22×C32—21no—
141C22×C32#21314yescomplemented
142C22×C32C2×#6712yes—
143C2×C3⋊D3—42no—
144C2×C3⋊D3—42no—
145C2×C3⋊D3—42no—
146C2×C3×D3—122no—
147C2×C3×D3—122no—
148C2×C3×D3—122no—
149C32×C4—22no—
150C32×C4—22no—
151C32×C4—21no—
152C32×C4—21no—
153C32×C4D3×D412yes—
154C32×C4#23712yes—
155C3⋊Dic3—42no—
156C3×Dic3—122no—
157C2×D8—271no—
158C22×C8—91no—
159#121—92no—
160#107—271no—
161#102—272no—
162H(3,𝔽3)#46611yescharacteristic, complemented, Sylow
163C23×C3—32no—
164C23×C3—61no—
165C23×C3—182no—
166C23×C3#62511yescharacteristic
167C22×D3—62no—
168C22×D3—182no—
169C3×D4—182no—
170C3×D4—182no—
171C3×D4—364no—
172C2×C3×C4—21no—
173C2×C3×C4—21no—
174C2×C3×C4—32no—
175C2×C3×C4—32no—
176C2×C3×C4—62no—
177C2×C3×C4—62no—
178C2×C3×C4—62no—
179C2×C3×C4—62no—
180C2×C3×C4—61no—
181C2×C3×C4—61no—
182C2×C3×C4—182no—
183C2×C3×C4#62511yescharacteristic
184C2×C3×C4C2×D3211yescharacteristic
185#67—124no—
186#67—364no—
187C2×Dic3—62no—
188C2×Dic3—182no—
189D12—62no—
190D12—62no—
191D12—182no—
192D12—182no—
193C3×C8—92no—
194C3×C8—181no—
195C3⋊C8—34no—
196C3⋊C8—62no—
197C3⋊C8—62no—
198C3⋊C8—62no—
199C2×C32—24no—
200C2×C32—24no—
201C2×C32—21no—
202C2×C32—21no—
203C2×C32—21no—
204C2×C32C2×#21312yes—
205C2×C32#93912yes—
206C2×C32#95012yes—
207C3⋊D3—82no—
208C3×D3—242no—
209C2×D4—92no—
210C2×D4—182no—
211C22×C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
212D8—542no—
213C2×C8—91no—
214C2×C8—91no—
215C2×C8—182no—
216C4⋊C4—92no—
217C22⋊C4—182no—
218C22×C3—21no—
219C22×C3—21no—
220C22×C3—32no—
221C22×C3—34no—
222C22×C3—62no—
223C22×C3—62no—
224C22×C3—62no—
225C22×C3—62no—
226C22×C3—61no—
227C22×C3—62no—
228C22×C3—362no—
229C22×C3—362no—
230C22×C3—362no—
231C22×C3#371312yes—
232C22×C3C2×#62511yescharacteristic
233C2×D3—122no—
234C2×D3—122no—
235C2×D3—122no—
236C2×D3—362no—
237C2×D3—362no—
238C2×D3—362no—
239C3×C4—21no—
240C3×C4—32no—
241C3×C4—32no—
242C3×C4—62no—
243C3×C4—62no—
244C3×C4—61no—
245C3×C4—61no—
246C3×C4—362no—
247C3×C4#380511yescharacteristic
248C3×C4#381111yescharacteristic
249Dic3—122no—
250Dic3—362no—
251C32—21no—
252C32#503812yescomplemented
253C23—182no—
254C23#619811yescharacteristic
255D4—182no—
256D4—182no—
257D4—364no—
258C2×C4—21no—
259C2×C4—21no—
260C2×C4—182no—
261C2×C4#619811yescharacteristic
262C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
263C8—92no—
264C8—181no—
265C2×C3—22no—
266C2×C3—32no—
267C2×C3—32no—
268C2×C3—32no—
269C2×C3—64no—
270C2×C3—64no—
271C2×C3—61no—
272C2×C3—61no—
273C2×C3—61no—
274C2×C3—722no—
275C2×C3C2×#371311yescharacteristic
276C2×C3#6382211yescharacteristic
277C2×C3#6398011yescharacteristic
278D3—242no—
279D3—722no—
280C22—21no—
281C22—21no—
282C22—362no—
283C22—362no—
284C22—362no—
285C22#8843212yescomplemented
286C22C2×#619811yescharacteristic
287C4—21no—
288C4—362no—
289C4#8867211yescharacteristic
290C4#8867711yescharacteristic
291C3—32no—
292C3—61no—
293C3#1059116611yescharacteristic
294C2—22no—
295C2—722no—
296C2C2×#8843211yescharacteristic
297C2#1172106011yescharacteristic
298C2#1172131811yescharacteristic
subgroupG/H
centerC22C2×#6198
commutator, cosocleC2×C4×H(3,𝔽3)C23
FrattiniC2×C3×C4C2×D32
FittingC22×C4×H(3,𝔽3)C22
socleC22×C3C2×#625
hypercenterC22×C4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#466
LCSG ▷ C2×C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
1. C22×#6180C2semidirect
2. #64139⋊C3C2semidirect
3. C2×#88432C2semidirect
4. #11720318C2semidirect
5. #11720315C2semidirect
16. #64139D3semidirect
35. #6180D4semidirect
57. C22×H(3,𝔽3)D8semidirect
141. C22×C32#213semidirect
162. H(3,𝔽3)#466semidirect
252. C32#5038semidirect
285. C22#88432semidirect

Maximal subgroups

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

subgroupindex
C22×#61802
#64139⋊C32
C2×#884322
#117203182
#117203152
#105911643