Finite Group Databasebeta

← #13158968

Subgroups — #13158968

Order 1728.

1296 conjugacy classes of subgroups; only the 188 normal subgroups are shown.

normal subgroups only

This group has 1296 conjugacy classes of subgroups — too many to draw. Only its 188 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#11723915C211yesmaximal, characteristic, complemented
2C3×#64084C211yesmaximal, characteristic, complemented
3C2×#88797C212yesmaximal, complemented
4C2×#88791C212yesmaximal, complemented
5C32×#803C211yesmaximal, characteristic, complemented
6#10592411—31nomaximal
7#10590351—34nomaximal
8C2×C4⋊(C3⋊(C3⋊D3))—22no—
9C2×C4⋊(C3⋊(C3⋊D3))C2212yes—
10C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
11C2×C3×#634C2211yescharacteristic
12C33×C42C2211yescharacteristic
13C3×#3752—22no—
14C2×C32×C3⋊C8C2212yes—
15#88797—24no—
16#88791—24no—
17C2×#3745—32no—
18C2×#3744—32no—
19#64084D311yescharacteristic, complemented
20#64083—31no—
21#64083—34no—
22#64083—64no—
23C3×#875—34no—
24C2×#3717—38no—
25C2×#3714—38no—
26C32×#97—31no—
27C3×#803D314yescomplemented
28C22×C3⋊(C3⋊D3)—22no—
29C22×C3×C3⋊D3—21no—
30C2×C33×C4D412yes—
31C2×C33×C4C2311yescharacteristic
32C4⋊(C3⋊(C3⋊D3))—22no—
33C4⋊(C3⋊(C3⋊D3))—22no—
34C4⋊(C3⋊(C3⋊D3))—44no—
35C2×C3×C3⋊Dic3—21no—
36C3×C4⋊(C3⋊D3)—22no—
37C3×#634—22no—
38C32×C3⋊C8D414yescomplemented
39#5058—91no—
40#4712—94no—
41C2×C4⋊(C3⋊D3)—32no—
42C2×C4⋊(C3⋊D3)—38no—
43C2×C4⋊(C3⋊D3)—68no—
44C2×C4⋊(C3⋊D3)—616no—
45C2×C4⋊(C3⋊D3)—68no—
46C2×C4⋊(C3⋊D3)—62no—
47C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
48C2×#634C2×D311yescharacteristic
49C2×C3×D12—34no—
50C2×C3×Dic6—34no—
51C2×C32×C8—32no—
52C32×C42—24no—
53C32×C42C2×D314yes—
54C32×C42C2×D311yescharacteristic
55#3752—22no—
56#3745—64no—
57#3744—64no—
58C3×(C2×D3)⋊C4—68no—
59C2×C3×C3⋊C8C2×D318yes—
60#3717—616no—
61#3714—616no—
62C22×C33C2×D411yescharacteristic
63C2×C3⋊(C3⋊D3)—42no—
64C2×C3⋊(C3⋊D3)—44no—
65C2×C3×C3⋊D3—41no—
66C2×C3×C3⋊D3—42no—
67C33×C4—22no—
68C33×C4C2×D412yes—
69C3×C3⋊Dic3—41no—
70C3×#124—91no—
71C2×#215—92no—
72C2×#213—92no—
73C2×D24—98no—
74C2×#204—98no—
75#875—34no—
76#874—94no—
77#874—91no—
78#874—184no—
79C3×#97—34no—
80#803C3⋊D311yescharacteristic, complemented
81C22×C3⋊D3—21no—
82C22×C3⋊D3—616no—
83C22×C3⋊D3—68no—
84C22×C3⋊D3—62no—
85C22×C3×D3—64no—
86C2×C32×C4—28no—
87C2×C32×C4—24no—
88C2×C32×C4D1218yes—
89C2×C32×C4#6712yes—
90C2×C32×C4C22×D314yes—
91C2×C32×C4C22×D311yescharacteristic
92C2×C3⋊Dic3—21no—
93C4⋊(C3⋊D3)—22no—
94C4⋊(C3⋊D3)—62no—
95C4⋊(C3⋊D3)—62no—
96C4⋊(C3⋊D3)—68no—
97C4⋊(C3⋊D3)—68no—
98C4⋊(C3⋊D3)—128no—
99C4⋊(C3⋊D3)—128no—
100C4⋊(C3⋊D3)—1232no—
101C4⋊(C3⋊D3)—1216no—
102C4⋊(C3⋊D3)—124no—
103#634—22no—
104C2×C3×Dic3—64no—
105C3×D12—68no—
106C3×Dic6—68no—
107C32×C8—34no—
108C3×C3⋊C8D12116yescomplemented
109#496—271noSylow
110C2×C33#12711yescharacteristic
111C2×C33AGL(1,ℤ/8)12yes—
112C3⋊(C3⋊D3)—82no—
113C3×C3⋊D3—81no—
114C2×C3×Q8—91no—
115C2×C3×D4—91no—
116C2×D12—34no—
117C2×D12—98no—
118C2×D12—92no—
119C2×D12—188no—
120C2×D12—1816no—
121C2×D12—188no—
122C2×D12—182no—
123C2×Dic6—34no—
124C2×C3×C8—38no—
125C3×C22⋊C4—182no—
126C3×C42—24no—
127C3×C42D3214yes—
128C3×C42C2×C3⋊D311yescharacteristic
129#215—184no—
130#213—184no—
131(C2×D3)⋊C4—68no—
132C2×C3⋊C8C2×C3⋊D312yes—
133D24—1816no—
134#204—1816no—
135C22×C32—24no—
136C22×C32C2×D1214yes—
137C22×C32C2×#6711yescharacteristic
138C2×C3⋊D3—41no—
139C2×C3⋊D3—42no—
140C2×C3⋊D3—1232no—
141C2×C3⋊D3—1216no—
142C2×C3⋊D3—128no—
143C2×C3⋊D3—1216no—
144C2×C3⋊D3—122no—
145C2×C3⋊D3—124no—
146C2×C3×D3—124no—
147C2×C3×D3—128no—
148C32×C4—216no—
149C32×C4—28no—
150C32×C4—22no—
151C32×C4—28no—
152C32×C4C2×D1218yes—
153C32×C4D3×D412yes—
154C3⋊Dic3—41no—
155C3×Dic3—124no—
156C2×SD16—272no—
157C2×D8—272no—
158#127—271no—
159#124—91no—
160#97—31no—
161C33#49611yescharacteristic, complemented, Sylow
162C23×C3—181no—
163C22×D3—64no—
164C22×D3—1816no—
165C22×D3—188no—
166C22×D3—182no—
167C3×Q8—182no—
168C3×D4—182no—
169C2×C3×C4—28no—
170C2×C3×C4—24no—
171C2×C3×C4—181no—
172C2×C3×C4#62618yes—
173C2×C3×C4C4⋊(C3⋊D3)12yes—
174C2×C3×C4C2×D3214yes—
175C2×C3×C4C22×C3⋊D311yescharacteristic
176C2×Dic3—64no—
177D12—68no—
178D12—188no—
179D12—188no—
180D12—182no—
181D12—182no—
182D12—368no—
183D12—368no—
184D12—3632no—
185D12—3616no—
186D12—364no—
187Dic6—68no—
188C3×C8—316no—
189C3⋊C8C4⋊(C3⋊D3)14yescomplemented
190C2×C32—28no—
191C2×C32—24no—
192C2×C32#87414yes—
193C2×C32#90818yes—
194C2×C32#94011yescharacteristic
195C2×C32#94912yes—
196C3⋊D3—81no—
197C3⋊D3—2416no—
198C3⋊D3—248no—
199C3⋊D3—242no—
200C3×D3—244no—
201C2×Q8—91no—
202C2×D4—91no—
203C2×D4—272no—
204C2×D4—542no—
205SD16—544no—
206D8—544no—
207C2×C8—32no—
208C22⋊C4—182no—
209C42D3×C3⋊D311yescharacteristic
210C22×C3—24no—
211C22×C3—361no—
212C22×C3—362no—
213C22×C3C2×#62614yes—
214C22×C3C2×C4⋊(C3⋊D3)11yescharacteristic
215C2×D3—128no—
216C2×D3—124no—
217C2×D3—3632no—
218C2×D3—3616no—
219C2×D3—3616no—
220C2×D3—362no—
221C2×D3—364no—
222C2×D3—368no—
223C3×C4—216no—
224C3×C4—28no—
225C3×C4—22no—
226C3×C4—28no—
227C3×C4—361no—
228C3×C4D3×D1218yes—
229C3×C4C2×C4⋊(C3⋊D3)12yes—
230Dic3—124no—
231C32—24no—
232C32#471214yescomplemented
233C32#505811yescharacteristic, complemented
234C23—181no—
235C23—542no—
236Q8—182no—
237D4—182no—
238D4—542no—
239D4—542no—
240D4—1084no—
241C2×C4—181no—
242C2×C4#629212yes—
243C2×C4C2×D3×C3⋊D311yescharacteristic
244C8—34no—
245C2×C3—28no—
246C2×C3—24no—
247C2×C3—721no—
248C2×C3#6382518yes—
249C2×C3#6391114yes—
250C2×C3#6408311yescharacteristic
251C2×C3#6411712yes—
252D3—244no—
253D3—7216no—
254D3—728no—
255D3—722no—
256C22—362no—
257C22—361no—
258C22—1084no—
259C22—1082no—
260C22C2×#629211yescharacteristic
261C4—22no—
262C4—361no—
263C4D3×C4⋊(C3⋊D3)12yes—
264C3—24no—
265C3#1059035114yescomplemented
266C3#1059241111yescharacteristic, complemented
267C2—721no—
268C2—2162no—
269C2#1172337112yes—
270C2#1172351711yescharacteristic
subgroupG/H
centerC22C2×#6292
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×D3×C3⋊D3
FittingC33×C42C22
socleC22×C33C2×D4
hypercenterC42D3×C3⋊D3
hypocenterC33#496
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C42

Complements

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

NH ≅ G/Ntype
1. #11723915C2semidirect
2. C3×#64084C2semidirect
3. C2×#88797C2semidirect
4. C2×#88791C2semidirect
5. C32×#803C2semidirect
19. #64084D3semidirect
27. C3×#803D3semidirect
38. C32×C3⋊C8D4semidirect
80. #803C3⋊D3semidirect
108. C3×C3⋊C8D12semidirect
161. C33#496semidirect
189. C3⋊C8C4⋊(C3⋊D3)semidirect
232. C32#4712semidirect
233. C32#5058semidirect
265. C3#10590351semidirect
266. C3#10592411semidirect

Maximal subgroups

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

subgroupindex
#117239152
C3×#640842
C2×#887972
C2×#887912
C32×#8032
#105924113
#105903513