Finite Group Databasebeta

← C3×#520⋊C9 #13152178

Subgroups — C3×#520⋊C9 #13152178

Order 1728.

462 conjugacy classes of subgroups; only the 146 normal subgroups are shown.

normal subgroups only

This group has 462 conjugacy classes of subgroups — too many to draw. Only its 146 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×#41⋊C9C211yesmaximal, characteristic
2C3×C4×Q8⋊C9C212yesmaximal, complemented
3C32×#520C311yesmaximal, characteristic
4#520⋊C9C313yesmaximal, direct product factor
5C3×#41⋊C9C412yescomplemented
6C2×C3×Q8⋊C9C2211yescharacteristic
7C3×C9×C4⋊C4—41nomaximal
8C2×C32×#41C2×C311yescharacteristic
9C32×C4×Q8C2×C312yes—
10C32×C4×D4—32no—
11C32×#117—31no—
12C2×C32×C4⋊C4—31no—
13C2×#41⋊C9C2×C313yes—
14C4×Q8⋊C9C2×C316yescomplemented
15C2×C3×C4×C9—41no—
16C2×C3×C4×C9—42no—
17C3×Q8⋊C9C2×C411yescharacteristic, complemented
18C3×#520C913yescomplemented
19C3×#520C3211yescharacteristic
20C32×#41—32no—
21C32×#41C3×C412yes—
22C2×C32×Q8C22×C311yescharacteristic
23C2×C32×D4—31no—
24C22×C32×C4—31no—
25C22×C32×C4—32no—
26C32×C4⋊C4—31no—
27C32×C4⋊C4—32no—
28C32×C4⋊C4A411yescharacteristic
29C32×C22⋊C4—32no—
30C32×C42—32no—
31#41⋊C9C3×C416yescomplemented
32C2×Q8⋊C9C22×C313yes—
33C9×C4⋊C4—43no—
34C22×C3×C9—41no—
35C3×C4×C9—42no—
36C3×C4×C9—82no—
37C2×C3×#41C2×C913yes—
38C2×C3×#41C2×C3211yescharacteristic
39C3×C4×Q8C2×C916yes—
40C3×C4×Q8C2×C3212yes—
41C3×C4×D4—36no—
42C3×C4×D4—32no—
43C3×#117—33no—
44C3×#117—31no—
45C2×C3×C4⋊C4—33no—
46C2×C3×C4⋊C4—31no—
47C23×C32—31no—
48C32×Q8—31no—
49C32×Q8C2×C3×C411yescharacteristic
50C32×D4—32no—
51C32×D4—32no—
52C2×C32×C4—31no—
53C2×C32×C4—32no—
54C2×C32×C4—32no—
55C2×C32×C4—32no—
56C2×C32×C4—122no—
57C2×C32×C4C2×A412yes—
58C2×C32×C4C2×A411yescharacteristic
59C2×C4×C9—43no—
60C2×C4×C9—46no—
61Q8⋊C9C2×C3×C413yescomplemented
62#520C3×C911yescharacteristic, complemented, Sylow
63C2×C3×C9—41no—
64C2×C3×C9—41no—
65C2×C3×C9—41no—
66C3×#41—36no—
67C3×#41—32no—
68C3×#41C4×C916yescomplemented
69C3×#41C32×C412yes—
70C2×C3×Q8C22×C913yes—
71C2×C3×Q8C22×C3211yescharacteristic
72C2×C3×D4—33no—
73C2×C3×D4—31no—
74C22×C3×C4—33no—
75C22×C3×C4—31no—
76C22×C3×C4—32no—
77C22×C3×C4—36no—
78C3×C4⋊C4—36no—
79C3×C4⋊C4—32no—
80C3×C4⋊C4—33no—
81C3×C4⋊C4—31no—
82C3×C4⋊C4C22⋊C913yes—
83C3×C4⋊C4C3×A411yescharacteristic
84C3×C22⋊C4—36no—
85C3×C22⋊C4—32no—
86C3×C42—36no—
87C3×C42—32no—
88C22×C32—32no—
89C22×C32—61no—
90C22×C32—61no—
91C22×C32C22×A411yescharacteristic
92C32×C4—22no—
93C32×C4—31no—
94C32×C4—31no—
95C32×C4—62no—
96C32×C4C4×A412yes—
97C22×C9—43no—
98C4×C9—46no—
99C4×C9—86no—
100C2×#41C2×C3×C911yescharacteristic
101C4×Q8C2×C3×C912yes—
102C4×D4—32no—
103#117—31no—
104C2×C4⋊C4—31no—
105C3×C9—41noSylow
106C23×C3—31no—
107C23×C3—33no—
108C3×Q8—33no—
109C3×Q8—31no—
110C3×Q8C2×C4×C913yes—
111C3×Q8C2×C32×C411yescharacteristic
112C3×D4—36no—
113C3×D4—32no—
114C3×D4—32no—
115C3×D4—36no—
116C2×C3×C4—31no—
117C2×C3×C4—33no—
118C2×C3×C4—36no—
119C2×C3×C4—36no—
120C2×C3×C4—32no—
121C2×C3×C4—32no—
122C2×C3×C4—32no—
123C2×C3×C4—36no—
124C2×C3×C4—122no—
125C2×C3×C4—126no—
126C2×C3×C4C2×C22⋊C916yes—
127C2×C3×C4C2×C22⋊C913yes—
128C2×C3×C4C2×C3×A412yes—
129C2×C3×C4C2×C3×A411yescharacteristic
130C2×C32—62no—
131C2×C32C2×C4×A411yescharacteristic
132C2×C32#99411yescharacteristic
133C2×C322−1+4⋊C311yescharacteristic
134C2×C9—43no—
135C2×C9—43no—
136C2×C9—43no—
137#41—32no—
138#41C3×C4×C912yescomplemented
139C2×Q8C22×C3×C911yescharacteristic
140C2×D4—31no—
141C22×C4—31no—
142C22×C4—32no—
143C4⋊C4—32no—
144C4⋊C4—31no—
145C4⋊C4C3×C22⋊C911yescharacteristic
146C22⋊C4—32no—
147C42—32no—
148C22×C3—32no—
149C22×C3—36no—
150C22×C3—61no—
151C22×C3—61no—
152C22×C3—63no—
153C22×C3—63no—
154C22×C3C22×C22⋊C913yes—
155C22×C3C22×C3×A411yescharacteristic
156C3×C4—22no—
157C3×C4—26no—
158C3×C4—31no—
159C3×C4—31no—
160C3×C4—33no—
161C3×C4—33no—
162C3×C4—62no—
163C3×C4—66no—
164C3×C4C4×C22⋊C916yes—
165C3×C4C3×C4×A412yes—
166C32#520⋊C311yescharacteristic
167C9—43no—
168C23—31no—
169Q8—31no—
170Q8C2×C3×C4×C911yescharacteristic
171D4—32no—
172D4—32no—
173C2×C4—31no—
174C2×C4—32no—
175C2×C4—32no—
176C2×C4—32no—
177C2×C4—122no—
178C2×C4C2×C3×C22⋊C912yes—
179C2×C4C2×C3×C22⋊C911yescharacteristic
180C2×C3—66no—
181C2×C3—62no—
182C2×C3C2×C4×C22⋊C913yes—
183C2×C3#6370013yes—
184C2×C32−1+4⋊C913yes—
185C2×C3C2×C3×C4×A411yescharacteristic
186C2×C3C3×#99411yescharacteristic
187C2×C3C3×2−1+4⋊C311yescharacteristic
188C22—32no—
189C22—61no—
190C22—61no—
191C22C22×C3×C22⋊C911yescharacteristic
192C4—22no—
193C4—31no—
194C4—31no—
195C4—62no—
196C4C3×C4×C22⋊C912yes—
197C3#520⋊C913yesdirect product factor
198C3C3×#520⋊C311yescharacteristic
199C2—62no—
200C2C2×C3×C4×C22⋊C911yescharacteristic
201C2C3×#6370011yescharacteristic
202C2C3×2−1+4⋊C911yescharacteristic
subgroupG/H
center, socleC22×C32C22×A4
commutator, hypocenterQ8C2×C3×C4×C9
FrattiniC22×C3C22×C3×A4
FittingC32×#520C3
prime radicalC2×C32×#41C2×C3
cosocleC2×C3×Q8C22×C32
hypercenterC32×C4⋊C4A4
LCSG ▷ Q8
UCSC1 ◁ C22×C32 ◁ C32×C4⋊C4

Complements

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

NH ≅ G/Ntype
2. C3×C4×Q8⋊C9C2semidirect
4. #520⋊C9C3direct (N × H)
5. C3×#41⋊C9C4semidirect
14. C4×Q8⋊C9C2×C3semidirect
17. C3×Q8⋊C9C2×C4semidirect
18. C3×#520C9semidirect
31. #41⋊C9C3×C4semidirect
61. Q8⋊C9C2×C3×C4semidirect
62. #520C3×C9semidirect
68. C3×#41C4×C9semidirect
138. #41C3×C4×C9semidirect
197. C3#520⋊C9direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#41⋊C92
C3×C4×Q8⋊C92
C32×#5203
#520⋊C93
C3×C9×C4⋊C44