Finite Group Databasebeta

← #13137394

Subgroups — #13137394

Order 1728.

544 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 544 conjugacy classes of subgroups — too many to draw. Only its 79 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#64084⋊C3C211yesmaximal, characteristic, complemented
2#11721385C211yesmaximal, characteristic, complemented
3C2×#88432C212yesmaximal, complemented
4C2×#88431C212yesmaximal, complemented
5#11720520C211yesmaximal, characteristic, complemented
6#10590371—31nomaximal
7#10590351—31nomaximal
8C2×C4⋊((C3⋊D3)⋊C3)—22no—
9C2×C4⋊((C3⋊D3)⋊C3)C2212yes—
10C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
11C2×#634⋊C3C2211yescharacteristic
12C42×H(3,𝔽3)C2211yescharacteristic
13C2×#6180C2212yes—
14#3752⋊C3—22no—
15#88432—24no—
16#88431—24no—
17#64084D311yescharacteristic, complemented
18#64083D311yescharacteristic, complemented
19C3×#875—31no—
20C3×#874—31no—
21C2×#3717—32no—
22C2×#3716—32no—
23C2×#3714—32no—
24C2×#3714—32no—
25C3×#803—31no—
26C3×#803—31no—
27C3×#803—61no—
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—
38#6180D414yescomplemented
39#5074—91no—
40#5058—91no—
41#4712—91no—
42C2×C4⋊(C3⋊D3)—22no—
43C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
44C2×C4⋊(C3⋊D3)C2×D312yes—
45C2×#634C2×D311yescharacteristic
46C2×C3×D12—31no—
47C2×C3×D12—32no—
48C2×C3×D12—62no—
49C2×C3×Dic6—31no—
50C32×C42—21no—
51C32×C42C2×D311yescharacteristic
52C32×C42C2×D311yescharacteristic
53#3752—22no—
54C3×(C2×D3)⋊C4—62no—
55C2×C3×C3⋊C8—32no—
56C2×C3×C3⋊C8—32no—
57C2×C3×C3⋊C8—62no—
58#3717—64no—
59#3716—64no—
60#3714—64no—
61#3714—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—
70C3×#127—91no—
71C3×#124—91no—
72C2×#215—92no—
73C2×#214—92no—
74C2×#213—92no—
75C2×#213—92no—
76C2×D24—92no—
77C2×#204—92no—
78#875—31no—
79#875—91no—
80#874—31no—
81#874—91no—
82C3×#97—91no—
83#803—31no—
84#803—31no—
85#803—61no—
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×C4#6712yes—
93C2×C32×C4#6712yes—
94C2×C32×C4C22×D311yescharacteristic
95C2×C32×C4C22×D311yescharacteristic
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—
108C3×C3⋊C8—34no—
109C3×C3⋊C8—34no—
110C3×C3⋊C8—64no—
111#496—271noSylow
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—
128C2×C3×C8—92no—
129C3×C22⋊C4—182no—
130C3×C42—31no—
131C3×C42—31no—
132C3×C42—61no—
133C3×C42D3211yescharacteristic
134#215—184no—
135#214—184no—
136#213—184no—
137#213—184no—
138(C2×D3)⋊C4—62no—
139(C2×D3)⋊C4—182no—
140C2×C3⋊C8—32no—
141C2×C3⋊C8—32no—
142C2×C3⋊C8—62no—
143D24—184no—
144#204—184no—
145C22×C32—21no—
146C22×C32C2×#6711yescharacteristic
147C22×C32C2×#6711yescharacteristic
148C2×C3⋊D3—44no—
149C2×C3⋊D3—41no—
150C2×C3⋊D3—42no—
151C2×C3⋊D3—42no—
152C2×C3×D3—122no—
153C2×C3×D3—122no—
154C2×C3×D3—121no—
155C2×C3×D3—124no—
156C32×C4—22no—
157C32×C4—22no—
158C32×C4—22no—
159C32×C4—42no—
160C32×C4D3×D412yes—
161C32×C4D3×D412yes—
162C3⋊Dic3—41no—
163C3×Dic3—121no—
164C2×SD16—272no—
165C2×D8—272no—
166#127—91no—
167#124—91no—
168#97—91no—
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×C4#62512yes—
189C2×C3×C4C2×D3211yescharacteristic
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—94no—
203C3⋊C8—34no—
204C3⋊C8—34no—
205C3⋊C8—64no—
206C2×C32—22no—
207C2×C32—21no—
208C2×C32#93212yes—
209C2×C32#94011yescharacteristic
210C2×C32#94011yescharacteristic
211C2×C32#94912yes—
212C3⋊D3—81no—
213C3⋊D3—82no—
214C3×D3—242no—
215C3×D3—241no—
216C2×Q8—91no—
217C2×D4—91no—
218C2×D4—92no—
219C2×D4—182no—
220SD16—544no—
221D8—544no—
222C2×C8—92no—
223C22⋊C4—182no—
224C42((C3⋊D3)⋊C3)⋊C211yescharacteristic
225C22×C3—31no—
226C22×C3—31no—
227C22×C3—61no—
228C22×C3—362no—
229C22×C3—361no—
230C22×C3—364no—
231C22×C3—362no—
232C22×C3C2×#62511yescharacteristic
233C2×D3—124no—
234C2×D3—122no—
235C2×D3—122no—
236C2×D3—121no—
237C2×D3—361no—
238C2×D3—362no—
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×C4#380212yes—
250Dic3—121no—
251Dic3—361no—
252C32—21no—
253C32#505811yescharacteristic, complemented
254C32#507411yescharacteristic, complemented
255C23—181no—
256C23—182no—
257Q8—182no—
258D4—182no—
259D4—182no—
260D4—182no—
261D4—364no—
262C2×C4—181no—
263C2×C4#619812yes—
264C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
265C8—94no—
266C2×C3—32no—
267C2×C3—32no—
268C2×C3—31no—
269C2×C3—31no—
270C2×C3—62no—
271C2×C3—61no—
272C2×C3—721no—
273C2×C3—722no—
274C2×C3#6382312yes—
275C2×C3#6391411yescharacteristic
276D3—242no—
277D3—241no—
278D3—722no—
279D3—721no—
280C22—364no—
281C22—362no—
282C22—361no—
283C22—362no—
284C22C2×#619811yescharacteristic
285C4—22no—
286C4—361no—
287C4#8865512yes—
288C3—31no—
289C3—31no—
290C3—61no—
291C3#1059035411yescharacteristic
292C2—722no—
293C2—721no—
294C2#1172106112yes—
295C2#1172114411yescharacteristic
subgroupG/H
centerC22C2×#6198
commutator, cosocleC2×C4×H(3,𝔽3)C23
FrattiniC2×C3×C4C2×D32
FittingC42×H(3,𝔽3)C22
socleC22×C3C2×#625
hypercenterC42((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#496
LCSG ▷ C2×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. #64084⋊C3C2semidirect
2. #11721385C2semidirect
3. C2×#88432C2semidirect
4. C2×#88431C2semidirect
5. #11720520C2semidirect
17. #64084D3semidirect
18. #64083D3semidirect
38. #6180D4semidirect
169. H(3,𝔽3)#496semidirect
253. C32#5058semidirect
254. C32#5074semidirect

Maximal subgroups

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

subgroupindex
#64084⋊C32
#117213852
C2×#884322
C2×#884312
#117205202
#105903713
#105903513