Finite Group Databasebeta

← #13178809

Subgroups — #13178809

Order 1728.

1564 conjugacy classes of subgroups; only the 167 normal subgroups are shown.

normal subgroups only

This group has 1564 conjugacy classes of subgroups — too many to draw. Only its 167 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×#64162C213yesmaximal, complemented
2#11723729C213yesmaximal, complemented
3#11723718C219yesmaximal, complemented
4#10595529—33nomaximal
5C24×C33C2211yescharacteristic, complemented
6C22×#1113C2211yescharacteristic, complemented
7C2×C3×#638C2219yes—
8C2×#6296C2213yescomplemented
9C3×#3757C2219yescomplemented
10#88814C2219yescomplemented
11#88813C2213yescomplemented
12#64162D313yescomplemented
13C3×#953—36no—
14C3×#953—63no—
15#63983—39no—
16#63959—318no—
17#63953—39no—
18C23×C33—43no—
19C23×C33C2313yes—
20C22×C3×C3⋊D3C2313yes—
21C2×#1113—43no—
22C3×#638—49no—
23C2×C3×C3⋊Dic3C2319yes—
24#6296—43no—
25#5594—93no—
26C24×C32—23no—
27C24×C32—41no—
28C24×C32C2×D313yescomplemented
29C22×D32—33no—
30C2×#638C2×D319yes—
31C2×C3×#67—318no—
32C2×C3×#67—69no—
33C2×#627—39no—
34#3757C2×D319yescomplemented
35C3×#217—318no—
36C3×#217—69no—
37#3722—39no—
38(C2×D3)⋊Dic3—318no—
39Dic32—39no—
40C22×C33—29no—
41C22×C33—44no—
42C22×C33C2411yescharacteristic
43C2×C3×C3⋊D3—29no—
44#1113—44no—
45C3×C3⋊Dic3—29no—
46C3×#120—93no—
47#953—36no—
48#953—63no—
49#937—99no—
50#936—99no—
51#885—918no—
52C23×C32—29no—
53C23×C32—43no—
54C23×C32—49no—
55C23×C32—418no—
56C23×C32—412no—
57C23×C32C22×D319yes—
58C22×C3⋊D3C22×D313yes—
59C22×C3×D3—36no—
60C22×C3×D3—63no—
61C2×D32—129no—
62#638—49no—
63C2×C3⋊Dic3C22×D319yes—
64C3×#67—1218no—
65C3×#67—1218no—
66C2×C3×Dic3—318no—
67C2×C3×Dic3—69no—
68#627—129no—
69H(3,𝔽4)—271noSylow
70C2×C33—43no—
71C2×C33H(5,𝔽2)13yes—
72C3×C3⋊D3—43no—
73C24×C3—23no—
74C24×C3—41no—
75C24×C3D3213yescomplemented
76C23×D3—93no—
77C2×C3×D4—99no—
78C2×#67—318no—
79C2×#67—69no—
80C2×Dic6—99no—
81C3×C22⋊C4—99no—
82#217—318no—
83#217—69no—
84#217—99no—
85(C2×D3)⋊C4—918no—
86C4×Dic3—99no—
87C22×C32—29no—
88C22×C32—218no—
89C22×C32—218no—
90C22×C32—23no—
91C22×C32—41no—
92C22×C32—418no—
93C22×C32—418no—
94C22×C32—424no—
95C22×C32—416no—
96C22×C32—412no—
97C22×C32C23×D313yes—
98C2×C3⋊D3—29no—
99C2×C3×D3—618no—
100C2×C3×D3—129no—
101D32—1212no—
102C3⋊Dic3—29no—
103C3×Dic3—618no—
104C3×Dic3—129no—
105#124—279no—
106#120—93no—
107#120—273no—
108C33H(3,𝔽4)11yescharacteristic, complemented, Sylow
109C23×C3—29no—
110C23×C3—49no—
111C23×C3—418no—
112C23×C3—412no—
113C23×C3—43no—
114C23×C3—93no—
115C23×C3C2×D3219yes—
116C22×D3—36no—
117C22×D3—63no—
118C22×D3—369no—
119C3×D4—369no—
120C2×C3×C4—99no—
121#67—1218no—
122#67—1218no—
123C2×Dic3—318no—
124C2×Dic3—69no—
125Dic6—369no—
126C2×C32—29no—
127C2×C32—43no—
128C2×C32—418no—
129C2×C32—412no—
130C2×C32—49no—
131C2×C32#100419yes—
132C3⋊D3—43no—
133C3×D3—126no—
134C3×D3—243no—
135C24—271no—
136C24#111311yescharacteristic, complemented
137C2×Q8—273no—
138C2×D4—99no—
139C22⋊C4—99no—
140C22⋊C4—279no—
141C42—273no—
142C22×C3—218no—
143C22×C3—218no—
144C22×C3—29no—
145C22×C3—23no—
146C22×C3—418no—
147C22×C3—418no—
148C22×C3—424no—
149C22×C3—416no—
150C22×C3—412no—
151C22×C3—41no—
152C22×C3—189no—
153C22×C3C22×D3213yes—
154C2×D3—618no—
155C2×D3—129no—
156C2×D3—3612no—
157C3×C4—189no—
158Dic3—618no—
159Dic3—129no—
160C32—23no—
161C32—41no—
162C32#559413yescomplemented
163C23—43no—
164C23—93no—
165C23—1083no—
166C23C2×#111313yes—
167Q8—1083no—
168D4—369no—
169C2×C4—99no—
170C2×C3—29no—
171C2×C3—418no—
172C2×C3—412no—
173C2×C3—49no—
174C2×C3—43no—
175C2×C3—363no—
176C2×C3#6433019yes—
177D3—126no—
178D3—243no—
179C22—29no—
180C22—44no—
181C22—189no—
182C22—1084no—
183C22C22×#111311yescharacteristic
184C4—189no—
185C3—23no—
186C3—41no—
187C3#1059552913yescomplemented
188C2—43no—
189C2—363no—
190C2#1172463313yes—
subgroupG/H
center, FrattiniC22C22×#1113
commutator, socle, cosocleC22×C33C24
FittingC24×C33C22
hypercenterC24#1113
hypocenterC33H(3,𝔽4)
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C24

Complements

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

NH ≅ G/Ntype
1. C3×#64162C2semidirect
2. #11723729C2semidirect
3. #11723718C2semidirect
5. C24×C33C22semidirect
6. C22×#1113C22semidirect
8. C2×#6296C22semidirect
9. C3×#3757C22semidirect
10. #88814C22semidirect
11. #88813C22semidirect
12. #64162D3semidirect
28. C24×C32C2×D3semidirect
34. #3757C2×D3semidirect
75. C24×C3D32semidirect
108. C33H(3,𝔽4)semidirect
136. C24#1113semidirect
162. C32#5594semidirect
187. C3#10595529semidirect

Maximal subgroups

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

subgroupindex
C3×#641622
#117237292
#117237182
#105955293