Finite Group Databasebeta

← #13159597

Subgroups — #13159597

Order 1728.

988 conjugacy classes of subgroups; only the 180 normal subgroups are shown.

normal subgroups only

This group has 988 conjugacy classes of subgroups — too many to draw. Only its 180 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×C8×C3⋊(C3⋊D3)C211yesmaximal, characteristic
2#11723413C212yesmaximal
3C3×#63644C211yesmaximal, characteristic, complemented
4C32×#817C211yesmaximal, characteristic, complemented
5#11722056C212yesmaximal, complemented
6#10592559—31nomaximal
7#10590712—34nomaximal
8C2×C4×C3⋊(C3⋊D3)C2211yescharacteristic
9C2×C33×C8C2211yescharacteristic
10C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
11C8×C3⋊(C3⋊D3)C414yescomplemented
12C3×C4⋊(C3⋊Dic3)C2212yes—
13C32×C4⋊Dic3C2212yes—
14#88804—22no—
15Dic3×C3⋊Dic3—22no—
16C2×C8×C3⋊D3—31no—
17C2×C8×C3⋊D3—34no—
18C2×C8×C3⋊D3—64no—
19#64101—32no—
20#63849—38no—
21C32×#106—31no—
22#63644D311yescharacteristic, complemented
23#63634—32no—
24C3×#817—34no—
25C3×#817D314yescomplemented
26#63576—38no—
27C22×C3⋊(C3⋊D3)D411yescharacteristic
28C2×C33×C4C2311yescharacteristic
29C2×C3⋊(C3⋊Dic3)D411yescharacteristic
30C4×C3⋊(C3⋊D3)C2×C412yes—
31C4×C3⋊(C3⋊D3)Q812yes—
32C2×C3×C3⋊Dic3—22no—
33C2×C32×Dic3—22no—
34C33×C8C2×C412yes—
35C3⋊(C3⋊(C3⋊C8))C2×C412yes—
36#4860—94no—
37#4860—91no—
38C2×C4×C3⋊D3—31no—
39C2×C4×C3⋊D3—34no—
40C2×C4×C3⋊D3—64no—
41C2×C32×C8—24no—
42C2×C32×C8C2×D314yes—
43C2×C32×C8C2×D311yescharacteristic
44C32×C4⋊C4—32no—
45#3752—62no—
46C4⋊(C3⋊Dic3)C2×D312yes—
47C4×C3⋊Dic3—62no—
48C2×C3⋊(C3⋊C8)—31no—
49C2×C3⋊(C3⋊C8)—34no—
50C2×C3⋊(C3⋊C8)—64no—
51C8×C3⋊D3—34no—
52C8×C3⋊D3—316no—
53C8×C3⋊D3—616no—
54C3×C4⋊Dic3—38no—
55C3×C4⋊Dic3C2×D318yes—
56#3722—68no—
57Dic32—68no—
58C22×C33C2×D411yescharacteristic
59C2×C3⋊(C3⋊D3)—22no—
60C2×C3⋊(C3⋊D3)C4⋊C412yes—
61C33×C4C22×C411yescharacteristic
62C33×C4C2×Q811yescharacteristic
63C3⋊(C3⋊Dic3)C4⋊C412yes—
64C3×C3⋊Dic3—42no—
65C32×Dic3—42no—
66C2×D3×C8—94no—
67C2×D3×C8—91no—
68C2×D3×C8—184no—
69#892—98no—
70#892—92no—
71C3×#106—34no—
72C3×#106—91no—
73#817—34no—
74#817C3⋊D311yescharacteristic, complemented
75#807—98no—
76#807—92no—
77C22×C3⋊D3—31no—
78C22×C3⋊D3—34no—
79C22×C3⋊D3—64no—
80C2×C32×C4—24no—
81C2×C32×C4—62no—
82C2×C32×C4C22×D314yes—
83C2×C32×C4C22×D311yescharacteristic
84C2×C3⋊Dic3—22no—
85C2×C3⋊Dic3—31no—
86C2×C3⋊Dic3—34no—
87C2×C3⋊Dic3—64no—
88C4×C3⋊D3—32no—
89C4×C3⋊D3—32no—
90C4×C3⋊D3—38no—
91C4×C3⋊D3—38no—
92C4×C3⋊D3—68no—
93C4×C3⋊D3—68no—
94C2×C3×Dic3—28no—
95C2×C3×Dic3—68no—
96C32×C8—28no—
97C32×C8C4×D318yes—
98C32×C8C4×D312yes—
99C3⋊(C3⋊C8)—32no—
100C3⋊(C3⋊C8)—38no—
101C3⋊(C3⋊C8)—68no—
102#427—271noSylow
103C2×C33C2×C4⋊C411yescharacteristic
104C2×C33#13512yes—
105C3⋊(C3⋊D3)—22no—
106C2×C4×D3—94no—
107C2×C4×D3—91no—
108C2×C4×D3—184no—
109C2×C3×C8—24no—
110C2×C3×C8D3214yes—
111C2×C3×C8C2×C3⋊D311yescharacteristic
112C3×C4⋊C4—38no—
113C3×C4⋊C4—92no—
114(C2×D3)⋊C4—182no—
115(C2×D3)⋊C4—188no—
116C4⋊Dic3—38no—
117C4⋊Dic3C2×C3⋊D312yes—
118C4×Dic3—182no—
119C4×Dic3—188no—
120C2×C3⋊C8—94no—
121C2×C3⋊C8—91no—
122C2×C3⋊C8—184no—
123D3×C8—916no—
124D3×C8—94no—
125D3×C8—1816no—
126C22×C32—24no—
127C22×C32D3×D414yes—
128C22×C32D3×D411yescharacteristic
129C2×C3⋊D3—32no—
130C2×C3⋊D3—38no—
131C2×C3⋊D3—68no—
132C2×C3⋊D3—616no—
133C2×C3⋊D3—68no—
134C2×C3⋊D3—62no—
135C32×C4—24no—
136C32×C4—24no—
137C32×C4—122no—
138C32×C4C2×C4×D314yes—
139C32×C4C2×C4×D311yescharacteristic
140C32×C4D3×Q814yes—
141C32×C4D3×Q811yescharacteristic
142C3⋊Dic3—32no—
143C3⋊Dic3—38no—
144C3⋊Dic3—42no—
145C3⋊Dic3—68no—
146C3×Dic3—48no—
147C3×Dic3—128no—
148C22×C8—271no—
149#117—272no—
150#107—272no—
151#106—31no—
152#106—91no—
153C33#42711yescharacteristic, complemented, Sylow
154C22×D3—94no—
155C22×D3—91no—
156C22×D3—184no—
157C2×C3×C4—24no—
158C2×C3×C4—68no—
159C2×C3×C4—182no—
160C2×C3×C4C2×D3214yes—
161C2×C3×C4C22×C3⋊D311yescharacteristic
162C2×Dic3—22no—
163C2×Dic3—68no—
164C2×Dic3—94no—
165C2×Dic3—91no—
166C2×Dic3—184no—
167C4×D3—98no—
168C4×D3—98no—
169C4×D3—92no—
170C4×D3—92no—
171C4×D3—188no—
172C4×D3—188no—
173C3×C8—28no—
174C3×C8#62418yes—
175C3×C8C4×C3⋊D312yes—
176C3⋊C8—98no—
177C3⋊C8—92no—
178C3⋊C8—188no—
179C2×C32—28no—
180C2×C32—24no—
181C2×C32D3×C4⋊C414yes—
182C2×C32D3×C4⋊C411yescharacteristic
183C2×C32#91618yes—
184C2×C32#91612yes—
185C3⋊D3—616no—
186C3⋊D3—68no—
187C3⋊D3—62no—
188C22×C4—271no—
189C2×C8—271no—
190C2×C8—274no—
191C2×C8D3×C3⋊D311yescharacteristic
192C4⋊C4—32no—
193C4⋊C4—92no—
194C22⋊C4—542no—
195C42—542no—
196C22×C3—24no—
197C22×C3#380214yes—
198C22×C3D4×C3⋊D311yescharacteristic
199C2×D3—92no—
200C2×D3—98no—
201C2×D3—188no—
202C2×D3—1816no—
203C2×D3—188no—
204C2×D3—182no—
205C3×C4—24no—
206C3×C4—24no—
207C3×C4—128no—
208C3×C4—362no—
209C3×C4#379714yes—
210C3×C4C2×#62414yes—
211C3×C4C2×C4×C3⋊D311yescharacteristic
212C3×C4Q8×C3⋊D311yescharacteristic
213Dic3—42no—
214Dic3—92no—
215Dic3—98no—
216Dic3—128no—
217Dic3—188no—
218C32—24no—
219C32#486014yescomplemented
220C32#486011yescharacteristic, complemented
221C23—271no—
222C2×C4—62no—
223C2×C4—182no—
224C2×C4—271no—
225C2×C4—272no—
226C2×C4—272no—
227C2×C4C2×D3×C3⋊D311yescharacteristic
228C8—272no—
229C8#628912yes—
230C2×C3—28no—
231C2×C3—24no—
232C2×C3#6380918yes—
233C2×C3#6390014yes—
234C2×C3C4⋊C4×C3⋊D311yescharacteristic
235C2×C3#6412512yes—
236D3—1816no—
237D3—182no—
238D3—188no—
239C22—272no—
240C22—542no—
241C22#8902611yescharacteristic
242C4—122no—
243C4—272no—
244C4—362no—
245C4#8901911yescharacteristic
246C4C2×#628911yescharacteristic
247C3—24no—
248C3#1059071214yescomplemented
249C3#1059255911yescharacteristic, complemented
250C2—542no—
251C2#1172334212yes—
252C2#1172350311yescharacteristic
subgroupG/H
centerC22#89026
commutatorC33×C4C22×C4
FrattiniC2×C4C2×D3×C3⋊D3
FittingC2×C33×C8C22
socleC22×C33C2×D4
prime radicalC22×C3⋊(C3⋊D3)D4
cosocleC2×C33×C4C23
hypercenterC2×C8D3×C3⋊D3
hypocenterC33#427
LCSG ▷ C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×C8

Complements

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

NH ≅ G/Ntype
3. C3×#63644C2semidirect
4. C32×#817C2semidirect
5. #11722056C2semidirect
11. C8×C3⋊(C3⋊D3)C4semidirect
22. #63644D3semidirect
25. C3×#817D3semidirect
74. #817C3⋊D3semidirect
153. C33#427semidirect
219. C32#4860semidirect
220. C32#4860semidirect
248. C3#10590712semidirect
249. C3#10592559semidirect

Maximal subgroups

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

subgroupindex
C2×C8×C3⋊(C3⋊D3)2
#117234132
C3×#636442
C32×#8172
#117220562
#105925593
#105907123