Finite Group Databasebeta

← #13173376

Subgroups — #13173376

Order 1728.

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

normal subgroups only

This group has 380 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#11723096C211yesmaximal, characteristic, complemented
5C8×C3⋊#157C211yesmaximal, characteristic, complemented
6#10593779—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#63771—31no—
22C8×#157—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⋊#622D411yescharacteristic
31C3⋊#622D411yescharacteristic
32C33×C8C2×C411yescharacteristic
33C3×C3⋊(C3⋊C8)C2×C411yescharacteristic
34#5196—91nomaximal
35#3830C2×D312yes—
36C3×#237—64no—
37#3792—62no—
38C2×#622—62no—
39C4×#157—31no—
40#3788—32no—
41#3787—31no—
42C32×D8—22no—
43C32×D8—42no—
44C32×D8Dic311yescharacteristic, complemented
45#3754Dic312yescomplemented
46#3746Dic311yescharacteristic, complemented
47C8×C3⋊D3C2×D311yescharacteristic
48C3×#214—64no—
49C3×Dic12—62no—
50C3×D3×C8—62no—
51C22×C33—22no—
52C2×C3×C3⋊D3#4111yescharacteristic
53C3⋊#157—22no—
54C33×C4C22×C411yescharacteristic
55C3×C3⋊Dic3—22no—
56C3×C3⋊Dic3C2×D411yescharacteristic
57C3×#135—91no—
58#948—92no—
59#912—62no—
60#837—92no—
61#819—91no—
62C8×Dic3—91no—
63C2×#157—61no—
64C32×D4—24no—
65C32×D4—44no—
66C32×D4C2×Dic312yes—
67#638—22no—
68C2×C3⋊Dic3—22no—
69C4×C3⋊D3C22×D311yescharacteristic
70#634C2×Dic312yes—
71C3×#67—124no—
72C2×C3×Dic3—124no—
73C3×C4×D3—62no—
74C3×Dic6—64no—
75#622—31no—
76#622—31no—
77#622—62no—
78C32×C8—22no—
79C32×C8—42no—
80C32×C8C2×Dic311yescharacteristic
81C3⋊(C3⋊C8)C2×Dic311yescharacteristic
82C3×C3⋊C8—62no—
83#443—271noSylow
84C2×C33—42no—
85C2×C33C4×D411yescharacteristic
86C3×C3⋊D3—21no—
87C3×#41—92no—
88#237—64no—
89C3×Q16—91no—
90C3×SD16—92no—
91C3×D8—22no—
92C3×D8—42no—
93C3×D8#15711yescharacteristic, complemented
94C2×C3×C8—91no—
95#214—64no—
96C4×Dic3—91no—
97#208—92no—
98#208—182no—
99C2×C3⋊C8—91no—
100C2×C3⋊C8—182no—
101Dic12—62no—
102D3×C8—62no—
103C22×C32—22no—
104C22×C32—44no—
105C22×C32—84no—
106C2×C3⋊D3#23711yescharacteristic
107C2×C3×D3—62no—
108#157—62no—
109C32×C4—22no—
110C32×C4—42no—
111C32×C4C22×Dic311yescharacteristic
112C3⋊Dic3—22no—
113C3⋊Dic3D3×D411yescharacteristic
114C3×Dic3—62no—
115C3×Dic3—124no—
116#135—91no—
117#131—272no—
118#108—271no—
119#104—272no—
120C4×C8—271no—
121C33#44311yescharacteristic, complemented, Sylow
122C3×Q8—92no—
123C3×D4—24no—
124C3×D4—44no—
125C3×D4—182no—
126C3×D4C2×#15712yes—
127C2×C3×C4—91no—
128C2×C3×C4—182no—
129#67—124no—
130C2×Dic3—124no—
131C2×Dic3—181no—
132C4×D3—62no—
133Dic6—64no—
134C3×C8—22no—
135C3×C8—42no—
136C3×C8—91no—
137C3×C8C2×#15711yescharacteristic
138C3⋊C8—62no—
139C3⋊C8—91no—
140C3⋊C8—91no—
141C3⋊C8—182no—
142C2×C32—22no—
143C2×C32—42no—
144C2×C32—42no—
145C2×C32—84no—
146C2×C32—164no—
147C2×C32D4×Dic311yescharacteristic
148C3⋊D3—21no—
149C3×D3—122no—
150#41—92no—
151Q16—91no—
152SD16—92no—
153D8C3⋊#15711yescharacteristic, complemented
154M16—272no—
155M16—542no—
156C2×C8—91no—
157C2×C8—271no—
158C2×C8—542no—
159C42—271no—
160C22×C3—22no—
161C22×C3—44no—
162C22×C3—84no—
163C22×C3—91no—
164C2×D3—62no—
165C3×C4—22no—
166C3×C4—42no—
167C3×C4—91no—
168C3×C4—182no—
169C3×C4C22×#15711yescharacteristic
170Dic3—62no—
171Dic3—124no—
172Dic3—182no—
173C32—22no—
174C32—42no—
175C32#519611yescharacteristic, complemented
176Q8—92no—
177D4—182no—
178D4C2×C3⋊#15712yes—
179C2×C4—91no—
180C2×C4—182no—
181C2×C4—541no—
182C8—91no—
183C8—271no—
184C8—271no—
185C8—542no—
186C8C2×C3⋊#15711yescharacteristic
187C2×C3—22no—
188C2×C3—42no—
189C2×C3—42no—
190C2×C3—84no—
191C2×C3—164no—
192C2×C3—181no—
193C2×C3D4×#15711yescharacteristic
194D3—122no—
195C22—22no—
196C22—91no—
197C4—91no—
198C4—182no—
199C4—542no—
200C4C22×C3⋊#15711yescharacteristic
201C3—22no—
202C3—42no—
203C3#1059377911yescharacteristic, complemented
204C2—42no—
205C2—181no—
206C2D4×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#443
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. #11723096C2semidirect
5. C8×C3⋊#157C2semidirect
13. C33×D8C4semidirect
14. C3×#3754C4semidirect
15. C3×#3746C4semidirect
44. C32×D8Dic3semidirect
45. #3754Dic3semidirect
46. #3746Dic3semidirect
93. C3×D8#157semidirect
121. C33#443semidirect
153. D8C3⋊#157semidirect
175. C32#5196semidirect
203. C3#10593779semidirect

Maximal subgroups

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

subgroupindex
#117245152
C3×#641212
#117231082
#117230962
C8×C3⋊#1572
#105937793
#51969