Finite Group Databasebeta

← #13158869

Subgroups — #13158869

Order 1728.

740 conjugacy classes of subgroups; only the 206 normal subgroups are shown.

normal subgroups only

This group has 740 conjugacy classes of subgroups — too many to draw. Only its 206 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
1C3×C4×#634C211yesmaximal, characteristic
2C4×C3⋊(C3⋊(C3⋊C8))C211yesmaximal, characteristic
3C32×C4⋊(C3⋊C8)C211yesmaximal, characteristic
4#11722051C212yesmaximal
5#11722020C212yesmaximal
6#10592530—31nomaximal
7#10590259—34nomaximal
8C2×C3×#634C411yescharacteristic
9C33×C42C2211yescharacteristic
10C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
11C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
12C3×C4⋊(C3⋊Dic3)C411yescharacteristic
13C3×#3750C412yes—
14C3×C4×C3⋊Dic3C2212yes—
15C2×C32×C3⋊C8C2212yes—
16C4×#634D311yescharacteristic
17C3×C4×Dic6—34no—
18C32×C4⋊C8—31no—
19#63642—32no—
20#63641—32no—
21C4×C3⋊(C3⋊C8)—31no—
22C4×C3⋊(C3⋊C8)—34no—
23C4×C3⋊(C3⋊C8)—64no—
24C3×C4⋊(C3⋊C8)D314yes—
25Dic3⋊(C3⋊C8)—38no—
26#63554—38no—
27C2×C33×C4C2×C411yescharacteristic
28C2×C33×C4C2×C411yescharacteristic
29C2×C33×C4C2311yescharacteristic
30C2×C3×C3⋊Dic3C2×C412yes—
31C2×C3×C3⋊Dic3C2×C412yes—
32C3×#634—22no—
33C3⋊(C3⋊(C3⋊C8))—21no—
34C3⋊(C3⋊(C3⋊C8))Q812yescomplemented
35C32×C3⋊C8—22no—
36#5024—91no—
37#4831—94no—
38C2×#634Dic311yescharacteristic
39C2×C3×Dic6—34no—
40C2×C32×C8—32no—
41C32×C42—24no—
42C32×C42C2×D314yes—
43C32×C42C2×D311yescharacteristic
44C4⋊(C3⋊Dic3)Dic311yescharacteristic
45#3750Dic312yes—
46C4×C3⋊Dic3C2×D312yes—
47C2×C3⋊(C3⋊C8)—31no—
48C2×C3⋊(C3⋊C8)—34no—
49C2×C3⋊(C3⋊C8)—31no—
50C2×C3⋊(C3⋊C8)—34no—
51C2×C3⋊(C3⋊C8)—64no—
52C2×C3⋊(C3⋊C8)—64no—
53C3×C4⋊Dic3—34no—
54C3×Dic3⋊C4—38no—
55C3×C4×Dic3—38no—
56C2×C3×C3⋊C8C2×D318yes—
57C22×C33C22×C411yescharacteristic
58C33×C4—21no—
59C33×C4M1612yes—
60C33×C4C2×Q811yescharacteristic
61C33×C4#4111yescharacteristic
62C3×C3⋊Dic3—22no—
63C3×C3⋊Dic3—22no—
64C3×C4×Q8—91no—
65C4×Dic6—34no—
66C3×C4⋊C8—34no—
67#815—98no—
68Dic3⋊C8—98no—
69C4⋊(C3⋊C8)—92no—
70C4⋊(C3⋊C8)C3⋊D311yescharacteristic
71#803—92no—
72C4×C3⋊C8—94no—
73C4×C3⋊C8—91no—
74C4×C3⋊C8—184no—
75C2×C32×C4—24no—
76C2×C32×C4—24no—
77C2×C32×C4—24no—
78C2×C32×C4C4×D314yes—
79C2×C32×C4C4×D314yes—
80C2×C32×C4C2×Dic311yescharacteristic
81C2×C32×C4C2×Dic311yescharacteristic
82C2×C32×C4C22×D314yes—
83C2×C32×C4C22×D311yescharacteristic
84C2×C3⋊Dic3C2×Dic312yes—
85C2×C3⋊Dic3C2×Dic312yes—
86#634—22no—
87C2×C3×Dic3—38no—
88C2×C3×Dic3—38no—
89C3×Dic6—68no—
90C32×C8—62no—
91C3⋊(C3⋊C8)—32no—
92C3⋊(C3⋊C8)—38no—
93C3⋊(C3⋊C8)—68no—
94C3⋊(C3⋊C8)—68no—
95C3⋊(C3⋊C8)—64no—
96C3⋊(C3⋊C8)—61no—
97C3×C3⋊C8—28no—
98#446—271noSylow
99C2×C33C4×Q811yescharacteristic
100C2×C33C2×M1611yescharacteristic
101C2×C33#13111yescharacteristic
102C2×C3×Q8—91no—
103C2×Dic6—34no—
104C2×C3×C8—38no—
105C3×C4⋊C4—92no—
106C3×C4⋊C4—91no—
107C3×C42—24no—
108C3×C42—92no—
109C3×C42D3214yes—
110C3×C42C2×C3⋊D311yescharacteristic
111C4⋊Dic3—34no—
112Dic3⋊C4—38no—
113C4×Dic3—38no—
114C2×C3⋊C8—94no—
115C2×C3⋊C8—91no—
116C2×C3⋊C8—94no—
117C2×C3⋊C8—91no—
118C2×C3⋊C8—184no—
119C2×C3⋊C8—184no—
120C2×C3⋊C8C2×C3⋊D312yes—
121C22×C32—24no—
122C22×C32C2×C4×D314yes—
123C22×C32C22×Dic311yescharacteristic
124C32×C4—28no—
125C32×C4—24no—
126C32×C4—21no—
127C32×C4—28no—
128C32×C4—24no—
129C32×C4—24no—
130C32×C4ΓL(1,𝔽25)18yes—
131C32×C4#20812yes—
132C32×C4#23714yes—
133C32×C4D3×Q814yes—
134C32×C4D3×Q811yescharacteristic
135C32×C4#23911yescharacteristic
136C3⋊Dic3—22no—
137C3⋊Dic3—22no—
138C3×Dic3—68no—
139C3×Dic3—68no—
140C4×Q8—91no—
141C4⋊C8—31no—
142C4⋊C8—272no—
143#97—272no—
144C4×C8—271no—
145C33#44611yescharacteristic, complemented, Sylow
146C3×Q8—182no—
147C2×C3×C4—24no—
148C2×C3×C4—24no—
149C2×C3×C4—24no—
150C2×C3×C4—92no—
151C2×C3×C4—92no—
152C2×C3×C4D3×Dic314yes—
153C2×C3×C4D3×Dic314yes—
154C2×C3×C4C4×C3⋊D311yescharacteristic
155C2×C3×C4C4×C3⋊D311yescharacteristic
156C2×C3×C4C2×D3214yes—
157C2×C3×C4C22×C3⋊D311yescharacteristic
158C2×Dic3—38no—
159C2×Dic3—38no—
160Dic6—68no—
161C3×C8—68no—
162C3⋊C8—22no—
163C3⋊C8—98no—
164C3⋊C8—92no—
165C3⋊C8—188no—
166C3⋊C8—188no—
167C3⋊C8—184no—
168C3⋊C8—181no—
169C2×C32—24no—
170C2×C32—24no—
171C2×C32—24no—
172C2×C32#88714yes—
173C2×C32C2×ΓL(1,𝔽25)14yes—
174C2×C32#90714yes—
175C2×C32C2×#20811yescharacteristic
176C2×C32Q8×Dic311yescharacteristic
177C2×C32#94811yescharacteristic
178C2×Q8—91no—
179C2×C8—32no—
180C2×C8—271no—
181C2×C8—271no—
182C4⋊C4—92no—
183C4⋊C4—91no—
184C42—92no—
185C42D3×C3⋊D311yescharacteristic
186C22×C3—24no—
187C22×C3C2×D3×Dic314yes—
188C22×C3C2×C4×C3⋊D311yescharacteristic
189C3×C4—28no—
190C3×C4—24no—
191C3×C4—21no—
192C3×C4—24no—
193C3×C4—24no—
194C3×C4—28no—
195C3×C4—182no—
196C3×C4—182no—
197C3×C4#371118yes—
198C3×C4#374312yes—
199C3×C4#379614yes—
200C3×C4#379714yes—
201C3×C4#383011yescharacteristic
202C3×C4Q8×C3⋊D311yescharacteristic
203Dic3—68no—
204Dic3—68no—
205C32—24no—
206C32#483114yescomplemented
207C32#502411yescharacteristic, complemented
208Q8—182no—
209C2×C4—92no—
210C2×C4—92no—
211C2×C4Dic3×C3⋊D311yescharacteristic
212C2×C4Dic3×C3⋊D311yescharacteristic
213C2×C4C2×D3×C3⋊D311yescharacteristic
214C8—62no—
215C8—272no—
216C8—541no—
217C2×C3—24no—
218C2×C3—24no—
219C2×C3—24no—
220C2×C3#6381514yes—
221C2×C3C2×#371114yes—
222C2×C3#6384314yes—
223C2×C3#6409611yescharacteristic
224C2×C3C2×#374311yescharacteristic
225C2×C3#6411611yescharacteristic
226C22C2×Dic3×C3⋊D311yescharacteristic
227C4—21no—
228C4—182no—
229C4—182no—
230C4#8878712yes—
231C4#8901511yescharacteristic
232C4#8901911yescharacteristic
233C3—24no—
234C3#1059025914yescomplemented
235C3#1059253011yescharacteristic, complemented
236C2#1172335411yescharacteristic
237C2C2×#8878711yescharacteristic
238C2#1172341011yescharacteristic
subgroupG/H
center, FrattiniC2×C4C2×D3×C3⋊D3
commutator, socle, prime radicalC22×C33C22×C4
FittingC33×C42C22
cosocleC2×C33×C4C23
hypercenterC42D3×C3⋊D3
hypocenterC33#446
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C2×C4 ◁ C42

Complements

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

NH ≅ G/Ntype
34. C3⋊(C3⋊(C3⋊C8))Q8semidirect
145. C33#446semidirect
206. C32#4831semidirect
207. C32#5024semidirect
234. C3#10590259semidirect
235. C3#10592530semidirect

Maximal subgroups

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

subgroupindex
C3×C4×#6342
C4×C3⋊(C3⋊(C3⋊C8))2
C32×C4⋊(C3⋊C8)2
#117220512
#117220202
#105925303
#105902593