Finite Group Databasebeta

← #13173377

Subgroups — #13173377

Order 1728.

376 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 376 conjugacy classes of subgroups — too many to draw. Only its 67 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#11724515C212yesmaximal, complemented
2C3×#64121C211yesmaximal, characteristic
3#11723108C212yesmaximal, complemented
4#11723095C211yesmaximal, characteristic, complemented
5#11723092C211yesmaximal, characteristic, complemented
6#10593780—31nomaximal
7C3×#3830C2212yes—
8#88993—22no—
9C2×C3⋊#622—22no—
10C4×C3⋊#157C2211yescharacteristic
11#88989C2212yes—
12#88988C2211yescharacteristic
13C33×D8C411yescharacteristic, complemented
14C3×#3754C412yescomplemented
15C3×#3746C411yescharacteristic, complemented
16C3×C8×C3⋊D3C2211yescharacteristic
17#64287—32no—
18#64121D311yescharacteristic
19C3×#912—62no—
20#63783—32no—
21#63770—31no—
22#63767—31no—
23C2×C3⋊#157—21no—
24C33×D4C2×C412yes—
25C3×#638—22no—
26C2×C3×C3⋊Dic3—22no—
27C3×C4×C3⋊D3C2311yescharacteristic
28C3×#634C2×C412yes—
29C3⋊#622—22no—
30C3⋊#622D412yes—
31C33×C8C2×C411yescharacteristic
32C3×C3⋊(C3⋊C8)C2×C411yescharacteristic
33#5197—91nomaximal
34#3830C2×D312yes—
35C3×#237—64no—
36#3792—62no—
37C2×#622—62no—
38C4×#157—31no—
39#3788—32no—
40#3787—31no—
41C32×D8—22no—
42C32×D8—42no—
43C32×D8Dic311yescharacteristic, complemented
44#3754Dic312yescomplemented
45#3746Dic311yescharacteristic, complemented
46C8×C3⋊D3C2×D311yescharacteristic
47C3×#214—64no—
48C3×Dic12—62no—
49C3×D3×C8—62no—
50C22×C33—22no—
51C2×C3×C3⋊D3#4111yescharacteristic
52C3⋊#157—41no—
53C33×C4C22×C411yescharacteristic
54C3×C3⋊Dic3—22no—
55C3×C3⋊Dic3C2×D411yescharacteristic
56C3×#135—91no—
57#948—92no—
58#912—62no—
59#837—92no—
60#819—91no—
61#815—91no—
62C2×#157—61no—
63C32×D4—24no—
64C32×D4—44no—
65C32×D4C2×Dic312yes—
66#638—22no—
67C2×C3⋊Dic3—22no—
68C4×C3⋊D3C22×D311yescharacteristic
69#634C2×Dic312yes—
70C3×#67—124no—
71C2×C3×Dic3—124no—
72C3×C4×D3—62no—
73C3×Dic6—64no—
74#622—32no—
75#622—62no—
76C32×C8—22no—
77C32×C8—42no—
78C32×C8C2×Dic311yescharacteristic
79C3⋊(C3⋊C8)C2×Dic311yescharacteristic
80C3×C3⋊C8—62no—
81#444—271noSylow
82C2×C33—42no—
83C2×C33C4×D411yescharacteristic
84C3×C3⋊D3—21no—
85C3×#41—92no—
86#237—64no—
87C3×Q16—91no—
88C3×SD16—92no—
89C3×D8—22no—
90C3×D8—42no—
91C3×D8#15711yescharacteristic, complemented
92C2×C3×C8—91no—
93#214—64no—
94C4×Dic3—91no—
95#208—92no—
96#208—182no—
97C2×C3⋊C8—91no—
98C2×C3⋊C8—182no—
99Dic12—62no—
100D3×C8—62no—
101C22×C32—22no—
102C22×C32—44no—
103C22×C32—84no—
104C2×C3⋊D3#23711yescharacteristic
105C2×C3×D3—62no—
106#157—121no—
107C32×C4—22no—
108C32×C4—42no—
109C32×C4C22×Dic311yescharacteristic
110C3⋊Dic3—22no—
111C3⋊Dic3D3×D411yescharacteristic
112C3×Dic3—62no—
113C3×Dic3—124no—
114#135—91no—
115#131—272no—
116#108—271no—
117#104—272no—
118#97—271no—
119C33#44411yescharacteristic, complemented, Sylow
120C3×Q8—92no—
121C3×D4—24no—
122C3×D4—44no—
123C3×D4—182no—
124C3×D4C2×#15712yes—
125C2×C3×C4—91no—
126C2×C3×C4—182no—
127#67—124no—
128C2×Dic3—124no—
129C2×Dic3—181no—
130C4×D3—62no—
131Dic6—64no—
132C3×C8—22no—
133C3×C8—42no—
134C3×C8—91no—
135C3×C8C2×#15711yescharacteristic
136C3⋊C8—62no—
137C3⋊C8—92no—
138C3⋊C8—182no—
139C2×C32—22no—
140C2×C32—42no—
141C2×C32—42no—
142C2×C32—84no—
143C2×C32—164no—
144C2×C32D4×Dic311yescharacteristic
145C3⋊D3—21no—
146C3×D3—122no—
147#41—92no—
148Q16—91no—
149SD16—92no—
150D8C3⋊#15711yescharacteristic, complemented
151M16—272no—
152M16—542no—
153C2×C8—91no—
154C2×C8—271no—
155C2×C8—542no—
156C42—271no—
157C22×C3—22no—
158C22×C3—44no—
159C22×C3—84no—
160C22×C3—91no—
161C2×D3—62no—
162C3×C4—22no—
163C3×C4—42no—
164C3×C4—91no—
165C3×C4—182no—
166C3×C4C22×#15711yescharacteristic
167Dic3—62no—
168Dic3—124no—
169Dic3—361no—
170C32—22no—
171C32—42no—
172C32#519711yescharacteristic, complemented
173Q8—92no—
174D4—182no—
175D4C2×C3⋊#15712yes—
176C2×C4—91no—
177C2×C4—182no—
178C2×C4—541no—
179C8—91no—
180C8—272no—
181C8—542no—
182C8C2×C3⋊#15711yescharacteristic
183C2×C3—22no—
184C2×C3—42no—
185C2×C3—42no—
186C2×C3—84no—
187C2×C3—164no—
188C2×C3—181no—
189C2×C3D4×#15711yescharacteristic
190D3—122no—
191C22—22no—
192C22—91no—
193C4—91no—
194C4—182no—
195C4—1081no—
196C4C22×C3⋊#15711yescharacteristic
197C3—22no—
198C3—42no—
199C3#1059378011yescharacteristic, complemented
200C2—42no—
201C2—181no—
202C2D4×C3⋊#15711yescharacteristic
subgroupG/H
centerC2D4×C3⋊#157
commutatorC33×C4C22×C4
FrattiniC4C22×C3⋊#157
FittingC33×D8C4
socleC2×C33C4×D4
prime radicalC3×#64121C2
cosocleC3×C4×C3⋊D3C23
hypercenterD8C3⋊#157
hypocenterC33#444
LCSG ▷ C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C2 ◁ C4 ◁ D8

Complements

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

NH ≅ G/Ntype
1. #11724515C2semidirect
3. #11723108C2semidirect
4. #11723095C2semidirect
5. #11723092C2semidirect
13. C33×D8C4semidirect
14. C3×#3754C4semidirect
15. C3×#3746C4semidirect
43. C32×D8Dic3semidirect
44. #3754Dic3semidirect
45. #3746Dic3semidirect
91. C3×D8#157semidirect
119. C33#444semidirect
150. D8C3⋊#157semidirect
172. C32#5197semidirect
199. C3#10593780semidirect

Maximal subgroups

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

subgroupindex
#117245152
C3×#641212
#117231082
#117230952
#117230922
#105937803
#51979