Finite Group Databasebeta

← #10592411⋊C3 #13141535

Subgroups — #10592411⋊C3 #13141535

Order 1728.

544 conjugacy classes of subgroups; only the 112 normal subgroups are shown.

normal subgroups only

This group has 544 conjugacy classes of subgroups — too many to draw. Only its 112 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×#3745⋊C3C212yesmaximal, complemented
2C2×#3744⋊C3C212yesmaximal, complemented
3#64084⋊C3C211yesmaximal, characteristic, complemented
4#11721385C211yesmaximal, characteristic, complemented
5H(3,𝔽3)×#97C211yesmaximal, characteristic, complemented
6#10592411C311yesmaximal, characteristic, complemented
7C3×#4712—31nomaximal
8C2×C4⋊((C3⋊D3)⋊C3)—22no—
9C2×C4⋊((C3⋊D3)⋊C3)C2212yes—
10C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
11C2×#634⋊C3C2211yescharacteristic
12C2×C8×H(3,𝔽3)C2212yes—
13C42×H(3,𝔽3)C2211yescharacteristic
14#3752⋊C3—22no—
15#3745⋊C3—24no—
16#3744⋊C3—24no—
17C2×#3745C2×C312yescomplemented
18C2×#3744C2×C312yescomplemented
19#64084C2×C311yescharacteristic, complemented
20#64083C2×C311yescharacteristic, complemented
21C2×C3×D24—32no—
22C2×C3×#204—32no—
23C3×#875—31no—
24C3×#874—31no—
25C32×#97—21no—
26C32×#97D311yescharacteristic, complemented
27C32×#97C2×C311yescharacteristic, complemented
28C22×(C3⋊D3)⋊C3—22no—
29C22×(C3⋊D3)⋊C3—21no—
30C2×C4×H(3,𝔽3)D412yes—
31C2×C4×H(3,𝔽3)C2311yescharacteristic
32C2×(C3⋊Dic3)⋊C3—21no—
33C4⋊((C3⋊D3)⋊C3)—22no—
34C4⋊((C3⋊D3)⋊C3)—22no—
35C4⋊((C3⋊D3)⋊C3)—22no—
36C4⋊((C3⋊D3)⋊C3)—44no—
37#634⋊C3—22no—
38C8×H(3,𝔽3)D414yescomplemented
39C3×#496—91no—
40#4712—31no—
41#4712—91no—
42C2×C4⋊(C3⋊D3)—22no—
43C2×C4⋊(C3⋊D3)C22×C312yes—
44C2×C4⋊(C3⋊D3)C22×C311yescharacteristic
45C2×#634C22×C311yescharacteristic
46C2×C3×D12—32no—
47C2×C3×D12—31no—
48C2×C3×D12—62no—
49C2×C3×Dic6—31no—
50C2×C32×C8—22no—
51C2×C32×C8C2×D312yes—
52C2×C32×C8C22×C312yes—
53C32×C42—21no—
54C32×C42C2×D311yescharacteristic
55C32×C42C22×C311yescharacteristic
56#3752—22no—
57#3745—24no—
58#3744—24no—
59C3×(C2×D3)⋊C4—62no—
60C3×D24—64no—
61C3×#204—64no—
62C22×H(3,𝔽3)C2×D411yescharacteristic
63C2×(C3⋊D3)⋊C3—42no—
64C2×(C3⋊D3)⋊C3—41no—
65C2×(C3⋊D3)⋊C3—44no—
66C2×(C3⋊D3)⋊C3—42no—
67C4×H(3,𝔽3)—22no—
68C4×H(3,𝔽3)C2×D412yes—
69(C3⋊Dic3)⋊C3—41no—
70C2×C3×SD16—92no—
71C2×C3×D8—92no—
72C3×#127—91no—
73C3×#124—91no—
74C2×D24—32no—
75C2×D24—92no—
76C2×#204—32no—
77C2×#204—92no—
78#875—31no—
79#875—91no—
80#874—31no—
81#874—91no—
82C3×#97—31no—
83C3×#97—31no—
84C3×#97—61no—
85C3×#97C3×D311yescharacteristic
86C22×C3⋊D3—21no—
87C22×C3⋊D3—22no—
88C22×C3×D3—61no—
89C22×C3×D3—62no—
90C2×C32×C4—22no—
91C2×C32×C4—21no—
92C2×C32×C4D1212yes—
93C2×C32×C4C3×D412yes—
94C2×C32×C4C22×D311yescharacteristic
95C2×C32×C4C23×C311yescharacteristic
96C2×C3⋊Dic3—21no—
97C4⋊(C3⋊D3)—22no—
98C4⋊(C3⋊D3)—22no—
99C4⋊(C3⋊D3)—22no—
100C4⋊(C3⋊D3)—44no—
101#634—22no—
102C2×C3×Dic3—61no—
103C3×D12—62no—
104C3×D12—62no—
105C3×D12—62no—
106C3×D12—124no—
107C3×Dic6—62no—
108C32×C8—24no—
109C32×C8D1214yescomplemented
110C32×C8C3×D414yescomplemented
111#496—91noSylow
112C2×H(3,𝔽3)#12711yescharacteristic
113C2×H(3,𝔽3)AGL(1,ℤ/8)12yes—
114(C3⋊D3)⋊C3—82no—
115(C3⋊D3)⋊C3—81no—
116C2×C3×Q8—91no—
117C2×C3×D4—91no—
118C2×C3×D4—92no—
119C2×C3×D4—182no—
120C2×D12—32no—
121C2×D12—31no—
122C2×D12—62no—
123C2×D12—92no—
124C2×D12—91no—
125C2×D12—182no—
126C2×Dic6—31no—
127C2×Dic6—91no—
128C3×SD16—184no—
129C3×D8—184no—
130C2×C3×C8—32no—
131C2×C3×C8—32no—
132C2×C3×C8—62no—
133C2×C3×C8C2×C3×D312yes—
134C3×C22⋊C4—182no—
135C3×C42—31no—
136C3×C42—31no—
137C3×C42—61no—
138C3×C42C2×C3×D311yescharacteristic
139(C2×D3)⋊C4—62no—
140(C2×D3)⋊C4—182no—
141D24—64no—
142D24—184no—
143#204—64no—
144#204—184no—
145C22×C32—21no—
146C22×C32C2×D1211yescharacteristic
147C22×C32C2×C3×D411yescharacteristic
148C2×C3⋊D3—42no—
149C2×C3⋊D3—41no—
150C2×C3⋊D3—44no—
151C2×C3⋊D3—42no—
152C2×C3×D3—122no—
153C2×C3×D3—121no—
154C2×C3×D3—124no—
155C2×C3×D3—122no—
156C32×C4—22no—
157C32×C4—22no—
158C32×C4—22no—
159C32×C4—42no—
160C32×C4C2×D1212yes—
161C32×C4C2×C3×D412yes—
162C3⋊Dic3—41no—
163C3×Dic3—121no—
164C2×SD16—92no—
165C2×D8—92no—
166#127—91no—
167#124—91no—
168#97(C3⋊D3)⋊C311yescharacteristic, complemented
169H(3,𝔽3)#49611yescharacteristic, complemented, Sylow
170C23×C3—181no—
171C23×C3—182no—
172C22×D3—61no—
173C22×D3—62no—
174C22×D3—181no—
175C22×D3—182no—
176C3×Q8—182no—
177C3×D4—182no—
178C3×D4—182no—
179C3×D4—182no—
180C3×D4—364no—
181C2×C3×C4—32no—
182C2×C3×C4—32no—
183C2×C3×C4—31no—
184C2×C3×C4—31no—
185C2×C3×C4—62no—
186C2×C3×C4—61no—
187C2×C3×C4—181no—
188C2×C3×C4C3×D1212yes—
189C2×C3×C4C22×C3×D311yescharacteristic
190C2×Dic3—61no—
191C2×Dic3—181no—
192D12—62no—
193D12—62no—
194D12—62no—
195D12—124no—
196D12—182no—
197D12—182no—
198D12—182no—
199D12—364no—
200Dic6—62no—
201Dic6—182no—
202C3×C8—34no—
203C3×C8—34no—
204C3×C8—64no—
205C3×C8C3×D1214yes—
206C2×C32—22no—
207C2×C32—21no—
208C2×C32#87411yescharacteristic
209C2×C32#90812yes—
210C2×C32C3×#12711yescharacteristic
211C2×C32C3×AGL(1,ℤ/8)12yes—
212C3⋊D3—82no—
213C3⋊D3—81no—
214C3×D3—242no—
215C3×D3—241no—
216C2×Q8—91no—
217C2×D4—91no—
218C2×D4—92no—
219C2×D4—182no—
220SD16—184no—
221D8—184no—
222C2×C8C2×(C3⋊D3)⋊C312yes—
223C22⋊C4—182no—
224C42C2×(C3⋊D3)⋊C311yescharacteristic
225C22×C3—31no—
226C22×C3—31no—
227C22×C3—61no—
228C22×C3—361no—
229C22×C3—362no—
230C22×C3—364no—
231C22×C3—362no—
232C22×C3C2×C3×D1211yescharacteristic
233C2×D3—124no—
234C2×D3—122no—
235C2×D3—122no—
236C2×D3—121no—
237C2×D3—362no—
238C2×D3—361no—
239C2×D3—364no—
240C2×D3—362no—
241C3×C4—22no—
242C3×C4—32no—
243C3×C4—32no—
244C3×C4—62no—
245C3×C4—62no—
246C3×C4—62no—
247C3×C4—122no—
248C3×C4—361no—
249C3×C4C2×C3×D1212yes—
250Dic3—121no—
251Dic3—361no—
252C32—21no—
253C32#471211yescharacteristic, complemented
254C32C3×#49611yescharacteristic, complemented
255C23—181no—
256C23—182no—
257Q8—182no—
258D4—182no—
259D4—182no—
260D4—182no—
261D4—364no—
262C2×C4—181no—
263C2×C4C4⋊((C3⋊D3)⋊C3)12yes—
264C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
265C8C4⋊((C3⋊D3)⋊C3)14yescomplemented
266C2×C3—32no—
267C2×C3—32no—
268C2×C3—31no—
269C2×C3—31no—
270C2×C3—62no—
271C2×C3—61no—
272C2×C3—722no—
273C2×C3—721no—
274C2×C3C3×#87411yescharacteristic
275C2×C3C3×#90812yes—
276D3—242no—
277D3—241no—
278D3—722no—
279D3—721no—
280C22—364no—
281C22—362no—
282C22—361no—
283C22—362no—
284C22C2×C4⋊((C3⋊D3)⋊C3)11yescharacteristic
285C4—22no—
286C4—361no—
287C4C2×C4⋊((C3⋊D3)⋊C3)12yes—
288C3—31no—
289C3—31no—
290C3—61no—
291C3C3×#471211yescharacteristic
292C2—722no—
293C2—721no—
294C2#1172138511yescharacteristic
295C2#64117⋊C312yes—
subgroupG/H
centerC22C2×C4⋊((C3⋊D3)⋊C3)
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C3×C4C22×C3×D3
FittingH(3,𝔽3)×#97C2
socleC22×C3C2×C3×D12
hypercenter#97(C3⋊D3)⋊C3
hypocenterC32C3×#496
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C42 ◁ #97

Complements

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

NH ≅ G/Ntype
1. C2×#3745⋊C3C2semidirect
2. C2×#3744⋊C3C2semidirect
3. #64084⋊C3C2semidirect
4. #11721385C2semidirect
5. H(3,𝔽3)×#97C2semidirect
6. #10592411C3semidirect
17. C2×#3745C2×C3semidirect
18. C2×#3744C2×C3semidirect
19. #64084C2×C3semidirect
20. #64083C2×C3semidirect
26. C32×#97D3semidirect
27. C32×#97C2×C3semidirect
38. C8×H(3,𝔽3)D4semidirect
109. C32×C8D12semidirect
110. C32×C8C3×D4semidirect
168. #97(C3⋊D3)⋊C3semidirect
169. H(3,𝔽3)#496semidirect
253. C32#4712semidirect
254. C32C3×#496semidirect
265. C8C4⋊((C3⋊D3)⋊C3)semidirect

Maximal subgroups

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

subgroupindex
C2×#3745⋊C32
C2×#3744⋊C32
#64084⋊C32
#117213852
H(3,𝔽3)×#972
#105924113
C3×#47123