Finite Group Databasebeta

← C3×#10593125 #13164888

Subgroups — C3×#10593125 #13164888

Order 1728.

954 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 954 conjugacy classes of subgroups — too many to draw. Only its 98 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×#63703C211yesmaximal, characteristic, complemented
2C32×#585C311yesmaximal, characteristic
3#10593125C313yesmaximal, direct product factor
4C3×#41⋊C9—42nomaximal
5C3×C4×C22⋊C9—43nomaximal
6C32×2−1+4—32no—
7C32×H(5,𝔽2)—33no—
8C32×H(5,𝔽2)C2×C311yescharacteristic
9C2×C32×#41—32no—
10C2×C32×#41—33no—
11#63703C2×C313yescomplemented
12C2×C3×C22⋊C9—43no—
13C3×Q8⋊C9—42no—
14C3×#585C913yescomplemented
15C3×#585C3211yescharacteristic
16C32×#41—36no—
17C32×#41—36no—
18C32×#41—312no—
19C32×#41—32no—
20C32×#41A412yes—
21C2×C32×Q8—33no—
22C2×C32×Q8—32no—
23C2×C32×D4—36no—
24C2×C32×D4—33no—
25C2×C32×D4—33no—
26C2×C32×D4—33no—
27C22×C32×C4—33no—
28C22×C32×C4—31no—
29C22×C32×C4A413yes—
30#41⋊C9—46no—
31C4×C22⋊C9—49no—
32C3×C22⋊C9—43no—
33C3×C4×C9—161no—
34C3×2−1+4—36no—
35C3×2−1+4—32no—
36C3×H(5,𝔽2)—39no—
37C3×H(5,𝔽2)—33no—
38C3×H(5,𝔽2)C2×C913yes—
39C3×H(5,𝔽2)C2×C3211yescharacteristic
40C2×C3×#41—36no—
41C2×C3×#41—32no—
42C2×C3×#41—39no—
43C2×C3×#41—33no—
44C23×C32—31no—
45C23×C32—33no—
46C23×C32C2×A413yes—
47C32×Q8—36no—
48C32×Q8C2×A412yes—
49C32×D4—32no—
50C32×D4—36no—
51C32×D4—312no—
52C2×C32×C4—33no—
53C2×C32×C4—36no—
54C2×C32×C4—33no—
55C2×C32×C4—32no—
56C2×C32×C4—33no—
57C2×C32×C4—33no—
58C2×C22⋊C9—49no—
59Q8⋊C9—46no—
60#585C3×C911yescharacteristic, complemented, Sylow
61C2×C3×C9—161no—
62C3×#41—36no—
63C3×#41—318no—
64C3×#41—318no—
65C3×#41—36no—
66C3×#41—336no—
67C3×#41—312no—
68C3×#41—36no—
69C3×#41—32no—
70C3×#41C22⋊C916yescomplemented
71C3×#41C3×A412yes—
72C2×C3×Q8—39no—
73C2×C3×Q8—33no—
74C2×C3×Q8—36no—
75C2×C3×Q8—32no—
76C2×C3×D4—318no—
77C2×C3×D4—36no—
78C2×C3×D4—39no—
79C2×C3×D4—33no—
80C2×C3×D4—39no—
81C2×C3×D4—39no—
82C2×C3×D4—33no—
83C2×C3×D4—33no—
84C22×C3×C4—33no—
85C22×C3×C4—31no—
86C22×C3×C4—39no—
87C22×C3×C4—33no—
88C22×C3×C4C22⋊C919yescomplemented
89C22×C3×C4C3×A413yes—
90C22×C32—33no—
91C22×C32—32no—
92C22×C32—43no—
93C22×C32—121no—
94C22×C32—123no—
95C32×C4—32no—
96C32×C4—33no—
97C32×C4C22⋊A411yescharacteristic
98C22⋊C9—49no—
99C4×C9—163no—
1002−1+4—32no—
101H(5,𝔽2)—33no—
102H(5,𝔽2)C2×C3×C911yescharacteristic
103C2×#41—32no—
104C2×#41—33no—
105C3×C9—161noSylow
106C23×C3—31no—
107C23×C3—33no—
108C23×C3—33no—
109C23×C3—39no—
110C23×C3C2×C22⋊C919yes—
111C23×C3C2×C3×A413yes—
112C3×Q8—36no—
113C3×Q8—318no—
114C3×Q8C2×C22⋊C916yes—
115C3×Q8C2×C3×A412yes—
116C3×D4—36no—
117C3×D4—32no—
118C3×D4—36no—
119C3×D4—312no—
120C3×D4—318no—
121C3×D4—336no—
122C2×C3×C4—33no—
123C2×C3×C4—36no—
124C2×C3×C4—33no—
125C2×C3×C4—39no—
126C2×C3×C4—318no—
127C2×C3×C4—39no—
128C2×C3×C4—32no—
129C2×C3×C4—33no—
130C2×C3×C4—36no—
131C2×C3×C4—39no—
132C2×C3×C4—39no—
133C2×C3×C4—33no—
134C2×C32—63no—
135C2×C32—62no—
136C2×C32C2×C22⋊A411yescharacteristic
137C2×C9—163no—
138#41—36no—
139#41—36no—
140#41—312no—
141#41—32no—
142#41C3×C22⋊C912yescomplemented
143C2×Q8—33no—
144C2×Q8—32no—
145C2×D4—36no—
146C2×D4—33no—
147C2×D4—33no—
148C2×D4—33no—
149C22×C4—33no—
150C22×C4—31no—
151C22×C4C3×C22⋊C913yescomplemented
152C22×C3—33no—
153C22×C3—32no—
154C22×C3—39no—
155C22×C3—36no—
156C22×C3—49no—
157C22×C3—43no—
158C22×C3—123no—
159C22×C3—129no—
160C22×C3—121no—
161C22×C3—123no—
162C3×C4—32no—
163C3×C4—33no—
164C3×C4—36no—
165C3×C4—39no—
166C3×C4C22⋊(C22⋊C9)13yes—
167C3×C4C3×C22⋊A411yescharacteristic
168C32#595011yescharacteristic
169C9—163no—
170C23—31no—
171C23—33no—
172C23C2×C3×C22⋊C913yes—
173Q8—36no—
174Q8C2×C3×C22⋊C912yes—
175D4—32no—
176D4—36no—
177D4—312no—
178C2×C4—33no—
179C2×C4—36no—
180C2×C4—33no—
181C2×C4—32no—
182C2×C4—33no—
183C2×C4—33no—
184C2×C3—69no—
185C2×C3—66no—
186C2×C3—63no—
187C2×C3—62no—
188C2×C3C2×C22⋊(C22⋊C9)13yes—
189C2×C3C2×C3×C22⋊A411yescharacteristic
190C22—33no—
191C22—32no—
192C22—43no—
193C22—121no—
194C22—123no—
195C4—32no—
196C4—33no—
197C4C3×C22⋊(C22⋊C9)11yescharacteristic
198C3#1059312513yesdirect product factor
199C3C3×#595011yescharacteristic
200C2—63no—
201C2—62no—
202C2C2×C3×C22⋊(C22⋊C9)11yescharacteristic
subgroupG/H
center, hypercenterC32×C4C22⋊A4
commutator, hypocenterH(5,𝔽2)C2×C3×C9
FrattiniC2×C3C2×C3×C22⋊A4
Fitting, prime radicalC32×#585C3
socleC2×C32C2×C22⋊A4
cosocleC3×H(5,𝔽2)C2×C32
LCSG ▷ H(5,𝔽2)
UCSC1 ◁ C32×C4

Complements

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

NH ≅ G/Ntype
1. C3×#63703C2semidirect
3. #10593125C3direct (N × H)
11. #63703C2×C3semidirect
14. C3×#585C9semidirect
60. #585C3×C9semidirect
70. C3×#41C22⋊C9semidirect
88. C22×C3×C4C22⋊C9semidirect
142. #41C3×C22⋊C9semidirect
151. C22×C4C3×C22⋊C9semidirect
198. C3#10593125direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#637032
C32×#5853
#105931253
C3×#41⋊C94
C3×C4×C22⋊C94