Finite Group Databasebeta

← C3×Q16×C22⋊C9 #13152229

Subgroups — C3×Q16×C22⋊C9 #13152229

Order 1728.

402 conjugacy classes of subgroups; only the 112 normal subgroups are shown.

normal subgroups only

This group has 402 conjugacy classes of subgroups — too many to draw. Only its 112 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×Q8×C22⋊C9C212yesmaximal
2C3×C8×C22⋊C9C211yesmaximal, characteristic
3C22×C32×Q16C311yesmaximal, characteristic
4Q16×C22⋊C9C313yesmaximal, direct product factor
5C3×C4×C22⋊C9—22no—
6C3×C4×C22⋊C9C2211yescharacteristic
7C3×C9×Q16—41nomaximal
8C22×C32×Q8C2×C312yes—
9C2×C32×Q16—31no—
10C2×C32×Q16—31no—
11C2×C32×Q16—32no—
12C22×C32×C8C2×C311yescharacteristic
13Q8×C22⋊C9C2×C316yes—
14C8×C22⋊C9C2×C313yes—
15C2×C3×C22⋊C9D411yescharacteristic
16C3×Q8×C9—42no—
17C3×C8×C9—41no—
18C22×C3×Q16C913yescomplemented
19C22×C3×Q16C3211yescharacteristic
20C2×C32×Q8—32no—
21C2×C32×Q8—32no—
22C2×C32×Q8—62no—
23C22×C32×C4—22no—
24C22×C32×C4C22×C311yescharacteristic
25C32×Q16—32no—
26C32×Q16—32no—
27C32×Q16—31no—
28C32×Q16A411yescharacteristic
29C2×C32×C8—31no—
30C2×C32×C8—31no—
31C4×C22⋊C9—26no—
32C4×C22⋊C9C22×C313yes—
33C9×Q16—43no—
34C3×C22⋊C9Q1611yescharacteristic, direct product factor
35C3×C4×C9—41no—
36C3×C4×C9—82no—
37C22×C3×Q8C2×C916yes—
38C22×C3×Q8C2×C3212yes—
39C2×C3×Q16—33no—
40C2×C3×Q16—31no—
41C2×C3×Q16—33no—
42C2×C3×Q16—36no—
43C2×C3×Q16—31no—
44C2×C3×Q16—32no—
45C22×C3×C8C2×C913yes—
46C22×C3×C8C2×C3211yescharacteristic
47C23×C32C3×D411yescharacteristic
48C32×Q8—32no—
49C32×Q8—62no—
50C32×Q8—62no—
51C32×Q8C2×A412yes—
52C2×C32×C4—31no—
53C2×C32×C4—31no—
54C2×C32×C4—62no—
55C2×C32×C4—62no—
56C2×C22⋊C9C3×D413yes—
57C32×C8—31no—
58C32×C8C2×A411yescharacteristic
59Q8×C9—46no—
60C8×C9—43no—
61C22×Q16C3×C911yescharacteristic, complemented, Sylow
62C2×C3×C9—41no—
63C2×C3×Q8—36no—
64C2×C3×Q8—32no—
65C2×C3×Q8—36no—
66C2×C3×Q8—32no—
67C2×C3×Q8—66no—
68C2×C3×Q8—62no—
69C22×C3×C4—26no—
70C22×C3×C4—22no—
71C22×C3×C4C22×C913yes—
72C22×C3×C4C22×C3211yescharacteristic
73C3×Q16—32no—
74C3×Q16—36no—
75C3×Q16—36no—
76C3×Q16—32no—
77C3×Q16—33no—
78C3×Q16—31no—
79C3×Q16C22⋊C913yesdirect product factor
80C3×Q16C3×A411yescharacteristic
81C2×C3×C8—33no—
82C2×C3×C8—31no—
83C2×C3×C8—33no—
84C2×C3×C8—31no—
85C22×C32—31no—
86C22×C32—31no—
87C22×C32C3×Q1611yescharacteristic
88C32×C4—22no—
89C32×C4—31no—
90C32×C4—62no—
91C32×C4C22×A411yescharacteristic
92C22⋊C9C3×Q1613yesdirect product factor
93C4×C9—43no—
94C4×C9—86no—
95C22×Q8C2×C3×C912yes—
96C2×Q16—31no—
97C2×Q16—31no—
98C2×Q16—32no—
99C22×C8C2×C3×C911yescharacteristic
100C3×C9—41noSylow
101C23×C3D4×C913yes—
102C23×C3C32×D411yescharacteristic
103C3×Q8—32no—
104C3×Q8—36no—
105C3×Q8—62no—
106C3×Q8—66no—
107C3×Q8—62no—
108C3×Q8—66no—
109C3×Q8C2×C22⋊C916yes—
110C3×Q8C2×C3×A412yes—
111C2×C3×C4—31no—
112C2×C3×C4—33no—
113C2×C3×C4—33no—
114C2×C3×C4—31no—
115C2×C3×C4—62no—
116C2×C3×C4—66no—
117C2×C3×C4—62no—
118C2×C3×C4—66no—
119C3×C8—31no—
120C3×C8—33no—
121C3×C8C2×C22⋊C913yes—
122C3×C8C2×C3×A411yescharacteristic
123C2×C32—31no—
124C2×C32—31no—
125C2×C32D4×A411yescharacteristic
126C2×C9—43no—
127C2×Q8—32no—
128C2×Q8—32no—
129C2×Q8—62no—
130C22×C4—22no—
131C22×C4C22×C3×C911yescharacteristic
132Q16—32no—
133Q16—32no—
134Q16—31no—
135Q16C3×C22⋊C911yescharacteristic, direct product factor
136C2×C8—31no—
137C2×C8—31no—
138C22×C3—33no—
139C22×C3—31no—
140C22×C3—31no—
141C22×C3—33no—
142C22×C3C9×Q1613yescomplemented
143C22×C3C32×Q1611yescharacteristic
144C3×C4—22no—
145C3×C4—26no—
146C3×C4—31no—
147C3×C4—33no—
148C3×C4—62no—
149C3×C4—66no—
150C3×C4C22×C22⋊C913yes—
151C3×C4C22×C3×A411yescharacteristic
152C32A4×Q1611yescharacteristic
153C9—43no—
154C23C3×D4×C911yescharacteristic
155Q8—32no—
156Q8—62no—
157Q8—62no—
158Q8C2×C3×C22⋊C912yes—
159C2×C4—31no—
160C2×C4—31no—
161C2×C4—62no—
162C2×C4—62no—
163C8—31no—
164C8C2×C3×C22⋊C911yescharacteristic
165C2×C3—33no—
166C2×C3—33no—
167C2×C3—31no—
168C2×C3—31no—
169C2×C3D4×C22⋊C913yes—
170C2×C3C3×D4×A411yescharacteristic
171C22—31no—
172C22—31no—
173C22C3×C9×Q1611yescharacteristic, complemented
174C4—22no—
175C4—31no—
176C4—62no—
177C4C22×C3×C22⋊C911yescharacteristic
178C3Q16×C22⋊C913yesdirect product factor
179C3C3×A4×Q1611yescharacteristic
180C2—31no—
181C2—31no—
182C2C3×D4×C22⋊C911yescharacteristic
subgroupG/H
centerC2×C32D4×A4
commutatorC22×C4C22×C3×C9
FrattiniC3×C4C22×C3×A4
FittingC22×C32×Q16C3
socle, prime radicalC23×C32C3×D4
cosocleC22×C3×C4C22×C32
hypercenterC32×Q16A4
hypocenterC22C3×C9×Q16
LCSG ▷ C22×C4 ▷ C23 ▷ C22
UCSC1 ◁ C2×C32 ◁ C32×C4 ◁ C32×Q16

Complements

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

NH ≅ G/Ntype
4. Q16×C22⋊C9C3direct (N × H)
18. C22×C3×Q16C9semidirect
34. C3×C22⋊C9Q16direct (N × H)
61. C22×Q16C3×C9semidirect
79. C3×Q16C22⋊C9direct (N × H)
92. C22⋊C9C3×Q16direct (N × H)
135. Q16C3×C22⋊C9direct (N × H)
142. C22×C3C9×Q16semidirect
173. C22C3×C9×Q16semidirect
178. C3Q16×C22⋊C9direct (N × H)

Maximal subgroups

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

subgroupindex
C3×Q8×C22⋊C92
C3×C8×C22⋊C92
C22×C32×Q163
Q16×C22⋊C93
C3×C9×Q164