Finite Group Databasebeta

← #13167439

Subgroups — #13167439

Order 1728.

428 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 428 conjugacy classes of subgroups — too many to draw. Only its 91 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×C3×C4×#157C211yesmaximal, characteristic
2#11723656C212yesmaximal
3#11722786C214yesmaximal, complemented
4#10593328—31nomaximal
5C22×C3×#157C412yes—
6C2×C3×C4×C3⋊D3C2211yescharacteristic
7C3×C4×#157C411yescharacteristic
8C3×C4×#157C411yescharacteristic
9C3×C4×#157C2212yes—
10C3×#3748—22no—
11#88809—22no—
12#88808C2214yes—
13C2×C4×#157D311yescharacteristic
14#63818—32no—
15D3×#208—62no—
16#63732—34no—
17C22×C3×C3⋊D3C2×C411yescharacteristic
18C2×C3×#157—24no—
19C2×C3×#157C2×C412yes—
20C2×C3×#157C2×C412yes—
21C2×C33×C4D411yescharacteristic
22C2×C3×C3⋊Dic3Q811yescharacteristic
23C3×C4×C3⋊D3C2×C411yescharacteristic
24C3×C4×C3⋊D3D411yescharacteristic
25C3×C4×C3⋊D3Q811yescharacteristic
26C3×C4×C3⋊D3C2311yescharacteristic
27C3×C3⋊(C3⋊C8)—24no—
28#4924—91nomaximal
29C22×#157Dic312yes—
30C2×C4×C3⋊D3C2×D311yescharacteristic
31C2×C3×C4×D3—62no—
32C4×#157Dic311yescharacteristic
33C4×#157Dic311yescharacteristic
34C4×#157C2×D312yes—
35#3748—22no—
36C3×#208—62no—
37C3×#208—62no—
38C3×#208—122no—
39#3712—62no—
40#3711—64no—
41#3710—34no—
42D3×C3⋊C8—64no—
43C22×C33C4⋊C411yescharacteristic
44C2×C3×C3⋊D3—22no—
45C2×C3×C3⋊D3C4⋊C411yescharacteristic
46C2×C3×C3⋊D3C22×C411yescharacteristic
47C3×#157—22no—
48C3×#157M1614yescomplemented
49C33×C4C4⋊C411yescharacteristic
50C33×C4C2×D411yescharacteristic
51C3×C3⋊Dic3C4⋊C411yescharacteristic
52C3×C3⋊Dic3C2×Q811yescharacteristic
53C2×C3×C42—91no—
54C2×#208—92no—
55D3×M16—182no—
56C4⋊(C3⋊C8)—94no—
57C22×C3⋊D3C2×Dic311yescharacteristic
58C22×C3×D3—62no—
59C2×#157—24no—
60C2×#157C2×Dic312yes—
61C2×#157C2×Dic312yes—
62C2×C32×C4—22no—
63C2×C32×C4—42no—
64C2×C32×C4D1211yescharacteristic
65C2×C3⋊Dic3Dic611yescharacteristic
66C4×C3⋊D3Dic611yescharacteristic
67C4×C3⋊D3D1211yescharacteristic
68C4×C3⋊D3C2×Dic311yescharacteristic
69C4×C3⋊D3C22×D311yescharacteristic
70C2×C3×Dic3—62no—
71C3×C4×D3—62no—
72C3×C4×D3—62no—
73C3×C4×D3—62no—
74C3×C4×D3—62no—
75C3⋊(C3⋊C8)—24no—
76C3×C3⋊C8—64no—
77C3×C3⋊C8—64no—
78C3×C3⋊C8—124no—
79#423—271noSylow
80C2×C33—21no—
81C2×C33C2×C4⋊C411yescharacteristic
82C3×C3⋊D3—21no—
83C3×C3⋊D3C2×M1612yes—
84C22×C3×C4—91no—
85C22×C3×C4—92no—
86C2×C4×D3—62no—
87C3×M16—182no—
88C3×C42—92no—
89C3×C42—91no—
90C3×C42—91no—
91#208—62no—
92#208—62no—
93#208—122no—
94#208—182no—
95C2×C3⋊C8—94no—
96ΓL(1,𝔽25)—184no—
97D3×C8—184no—
98C22×C32—22no—
99C22×C32—42no—
100C22×C32C4⋊Dic311yescharacteristic
101C2×C3⋊D3—22no—
102C2×C3⋊D3C4⋊Dic311yescharacteristic
103C2×C3⋊D3C22×Dic311yescharacteristic
104C2×C3×D3—62no—
105C2×C3×D3—62no—
106C2×C3×D3—124no—
107#157—22no—
108#157#20814yescomplemented
109C32×C4—22no—
110C32×C4—22no—
111C32×C4—42no—
112C32×C4—42no—
113C32×C4C4⋊Dic311yescharacteristic
114C32×C4C2×D1211yescharacteristic
115C3⋊Dic3C4⋊Dic311yescharacteristic
116C3⋊Dic3C2×Dic611yescharacteristic
117C3×Dic3—62no—
118C3×Dic3—62no—
119C2×M16—272no—
120C2×C42—91no—
121C4⋊C8—274no—
122C33#42311yescharacteristic, complemented, Sylow
123C23×C3—91no—
124C22×D3—62no—
125C2×C3×C4—22no—
126C2×C3×C4—42no—
127C2×C3×C4—92no—
128C2×C3×C4—91no—
129C2×C3×C4—91no—
130C2×C3×C4—91no—
131C2×C3×C4—91no—
132C2×C3×C4—91no—
133C2×C3×C4—92no—
134C2×C3×C4—184no—
135C2×C3×C4GO+(4,𝔽2)11yescharacteristic
136C2×Dic3—62no—
137C4×D3—62no—
138C4×D3—62no—
139C4×D3—62no—
140C4×D3—62no—
141C3×C8—184no—
142C3⋊C8—64no—
143C3⋊C8—64no—
144C3⋊C8—124no—
145C2×C32—22no—
146C2×C32—21no—
147C2×C32—42no—
148C2×C32—42no—
149C2×C32—82no—
150C2×C32C2×C4⋊Dic311yescharacteristic
151C3⋊D3—21no—
152C3⋊D3C2×#20812yes—
153C3×D3—64no—
154C3×D3—122no—
155C22×C4—91no—
156C22×C4—92no—
157M16—182no—
158M16—542no—
159C2×C8—274no—
160C42—92no—
161C42—91no—
162C42—91no—
163C22×C3—22no—
164C22×C3—42no—
165C22×C3—91no—
166C22×C3—91no—
167C22×C3—182no—
168C22×C3#377311yescharacteristic
169C2×D3—62no—
170C2×D3—62no—
171C2×D3—124no—
172C3×C4—22no—
173C3×C4—22no—
174C3×C4—42no—
175C3×C4—42no—
176C3×C4—94no—
177C3×C4—91no—
178C3×C4—91no—
179C3×C4—182no—
180C3×C4#377311yescharacteristic
181C3×C4C2×GO+(4,𝔽2)11yescharacteristic
182Dic3—62no—
183Dic3—62no—
184C32—22no—
185C32—42no—
186C32#492411yescharacteristic, complemented
187C23—91no—
188C2×C4—92no—
189C2×C4—91no—
190C2×C4—91no—
191C2×C4—91no—
192C2×C4—91no—
193C2×C4—91no—
194C2×C4—92no—
195C2×C4—184no—
196C2×C4#157⋊D311yescharacteristic
197C8—184no—
198C2×C3—21no—
199C2×C3—22no—
200C2×C3—42no—
201C2×C3—42no—
202C2×C3—82no—
203C2×C3—92no—
204C2×C3—181no—
205C2×C3C2×#377311yescharacteristic
206D3—64no—
207D3—122no—
208C22—91no—
209C22—91no—
210C22—182no—
211C22#8894011yescharacteristic
212C4—94no—
213C4—91no—
214C4—91no—
215C4—182no—
216C4#8894011yescharacteristic
217C4C2×#157⋊D311yescharacteristic
218C3—22no—
219C3—42no—
220C3#1059332811yescharacteristic, complemented
221C2—21no—
222C2—92no—
223C2—181no—
224C2C2×#8894011yescharacteristic
subgroupG/H
center, FrattiniC4C2×#157⋊D3
commutatorC2×C3×C3⋊D3C22×C4
FittingC2×C33×C4D4
socleC2×C33C2×C4⋊C4
prime radicalC22×C3×C3⋊D3C2×C4
cosocleC3×C4×C3⋊D3C23
hypercenterC2×C4#157⋊D3
hypocenterC33#423
LCSG ▷ C2×C3×C3⋊D3 ▷ C33
UCSC1 ◁ C4 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
3. #11722786C2semidirect
48. C3×#157M16semidirect
108. #157#208semidirect
122. C33#423semidirect
186. C32#4924semidirect
220. C3#10593328semidirect

Maximal subgroups

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

subgroupindex
C2×C3×C4×#1572
#117236562
#117227862
#105933283
#49249