Finite Group Databasebeta

← Q16×C22⋊H(3,𝔽3) #13152231

Subgroups — Q16×C22⋊H(3,𝔽3) #13152231

Order 1728.

420 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 420 conjugacy classes of subgroups — too many to draw. Only its 70 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
1Q8×C22⋊H(3,𝔽3)C212yesmaximal
2C8×C22⋊H(3,𝔽3)C211yesmaximal, characteristic
3C22×C32×Q16C311yesmaximal, characteristic, complemented
4C3×A4×Q16C313yesmaximal, complemented
5C4×C22⋊H(3,𝔽3)—22no—
6C4×C22⋊H(3,𝔽3)C2211yescharacteristic
7Q16×H(3,𝔽3)—41nomaximal
8C22×C32×Q8C2×C312yes—
9C3×Q8×A4C2×C316yes—
10C2×C32×Q16—31no—
11C2×C32×Q16—31no—
12C2×C32×Q16—32no—
13C22×C32×C8C2×C311yescharacteristic
14C3×C8×A4C2×C313yes—
15C2×C22⋊H(3,𝔽3)D411yescharacteristic
16Q8×H(3,𝔽3)—42no—
17C8×H(3,𝔽3)—41no—
18C22×C3×Q16—31no—
19C22×C3×Q16C3211yescharacteristic
20A4×Q16—33no—
21C2×C32×Q8—32no—
22C2×C32×Q8—32no—
23C2×C32×Q8—62no—
24C22×C32×C4—22no—
25C22×C32×C4C22×C311yescharacteristic
26C3×C4×A4—26no—
27C3×C4×A4C22×C313yes—
28C32×Q16—32no—
29C32×Q16—32no—
30C32×Q16—31no—
31C32×Q16—43no—
32C32×Q16A411yescharacteristic, complemented
33C2×C32×C8—31no—
34C2×C32×C8—31no—
35C22⋊H(3,𝔽3)Q1611yescharacteristic, direct product factor
36C4×H(3,𝔽3)—41no—
37C4×H(3,𝔽3)—82no—
38C22×C3×Q8—32no—
39C22×C3×Q8C2×C3212yes—
40Q8×A4—36no—
41C2×C3×Q16—33no—
42C2×C3×Q16—33no—
43C2×C3×Q16—36no—
44C2×C3×Q16—31no—
45C2×C3×Q16—31no—
46C2×C3×Q16—32no—
47C22×C3×C8—31no—
48C22×C3×C8C2×C3211yescharacteristic
49C8×A4—33no—
50C23×C32C3×D411yescharacteristic
51C2×C3×A4C3×D413yes—
52C32×Q8—32no—
53C32×Q8—46no—
54C32×Q8—62no—
55C32×Q8—62no—
56C32×Q8C2×A412yes—
57C2×C32×C4—31no—
58C2×C32×C4—31no—
59C2×C32×C4—62no—
60C2×C32×C4—62no—
61C32×C8—31no—
62C32×C8—43no—
63C32×C8C2×A411yescharacteristic
64C22×Q16H(3,𝔽3)11yescharacteristic, complemented, Sylow
65C2×H(3,𝔽3)—41no—
66C2×C3×Q8—36no—
67C2×C3×Q8—36no—
68C2×C3×Q8—32no—
69C2×C3×Q8—32no—
70C2×C3×Q8—62no—
71C2×C3×Q8—66no—
72C22×C3×C4—22no—
73C22×C3×C4—31no—
74C22×C3×C4—62no—
75C22×C3×C4C22×C3211yescharacteristic
76C4×A4—33no—
77C4×A4—66no—
78C3×Q16—36no—
79C3×Q16—32no—
80C3×Q16—36no—
81C3×Q16—33no—
82C3×Q16—32no—
83C3×Q16—31no—
84C3×Q16—31no—
85C3×Q16—123no—
86C3×Q16C3×A411yescharacteristic
87C2×C3×C8—33no—
88C2×C3×C8—33no—
89C2×C3×C8—31no—
90C2×C3×C8—31no—
91C22×C32—31no—
92C22×C32—31no—
93C22×C32C3×Q1611yescharacteristic, complemented
94C3×A4C3×Q1613yescomplemented
95C32×C4—22no—
96C32×C4—31no—
97C32×C4—43no—
98C32×C4—62no—
99C32×C4—86no—
100C32×C4C22×A411yescharacteristic
101C22×Q8C2×H(3,𝔽3)12yes—
102C2×Q16—31no—
103C2×Q16—31no—
104C2×Q16—32no—
105C22×C8C2×H(3,𝔽3)11yescharacteristic
106H(3,𝔽3)—41noSylow
107C23×C3—31no—
108C23×C3C32×D411yescharacteristic
109C2×A4—33no—
110C3×Q8—36no—
111C3×Q8—32no—
112C3×Q8—32no—
113C3×Q8—62no—
114C3×Q8—62no—
115C3×Q8—66no—
116C3×Q8—66no—
117C3×Q8—126no—
118C3×Q8C2×C3×A412yes—
119C2×C3×C4—33no—
120C2×C3×C4—33no—
121C2×C3×C4—31no—
122C2×C3×C4—31no—
123C2×C3×C4—62no—
124C2×C3×C4—66no—
125C2×C3×C4—66no—
126C2×C3×C4—62no—
127C3×C8—33no—
128C3×C8—31no—
129C3×C8—31no—
130C3×C8—123no—
131C3×C8C2×C3×A411yescharacteristic
132C2×C32—31no—
133C2×C32—31no—
134C2×C32—43no—
135C2×C32D4×A411yescharacteristic
136C2×Q8—32no—
137C2×Q8—32no—
138C2×Q8—62no—
139C22×C4—22no—
140C22×C4C22×H(3,𝔽3)11yescharacteristic
141Q16—32no—
142Q16—32no—
143Q16—31no—
144Q16C22⋊H(3,𝔽3)11yescharacteristic, direct product factor
145C2×C8—31no—
146C2×C8—31no—
147C22×C3—31no—
148C22×C3—33no—
149C22×C3—33no—
150C22×C3—31no—
151C22×C3—31no—
152C22×C3C32×Q1611yescharacteristic
153A4—33no—
154C3×C4—22no—
155C3×C4—33no—
156C3×C4—31no—
157C3×C4—31no—
158C3×C4—62no—
159C3×C4—66no—
160C3×C4—62no—
161C3×C4—123no—
162C3×C4—246no—
163C3×C4C22×C3×A411yescharacteristic
164C32—43no—
165C32A4×Q1611yescharacteristic, complemented
166C23D4×H(3,𝔽3)11yescharacteristic
167Q8—32no—
168Q8—62no—
169Q8—62no—
170Q8C2×C22⋊H(3,𝔽3)12yes—
171C2×C4—31no—
172C2×C4—31no—
173C2×C4—62no—
174C2×C4—62no—
175C8—31no—
176C8C2×C22⋊H(3,𝔽3)11yescharacteristic
177C2×C3—33no—
178C2×C3—33no—
179C2×C3—31no—
180C2×C3—31no—
181C2×C3—31no—
182C2×C3—123no—
183C2×C3C3×D4×A411yescharacteristic
184C22—31no—
185C22—31no—
186C22Q16×H(3,𝔽3)11yescharacteristic, complemented
187C4—22no—
188C4—31no—
189C4—62no—
190C4C22×C22⋊H(3,𝔽3)11yescharacteristic
191C3—31no—
192C3—123no—
193C3C3×A4×Q1611yescharacteristic
194C2—31no—
195C2—31no—
196C2D4×C22⋊H(3,𝔽3)11yescharacteristic
subgroupG/H
centerC2×C3C3×D4×A4
commutator, cosocleC22×C3×C4C22×C32
FrattiniC3×C4C22×C3×A4
FittingC22×C32×Q16C3
socleC23×C3C32×D4
prime radicalC2×C22⋊H(3,𝔽3)D4
hypercenterC32×Q16A4
hypocenterC22Q16×H(3,𝔽3)
LCSG ▷ C22×C3×C4 ▷ C23 ▷ C22
UCSC1 ◁ C2×C3 ◁ C32×C4 ◁ C32×Q16

Complements

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

NH ≅ G/Ntype
3. C22×C32×Q16C3semidirect
4. C3×A4×Q16C3semidirect
32. C32×Q16A4semidirect
35. C22⋊H(3,𝔽3)Q16direct (N × H)
64. C22×Q16H(3,𝔽3)semidirect
93. C22×C32C3×Q16semidirect
94. C3×A4C3×Q16semidirect
144. Q16C22⋊H(3,𝔽3)direct (N × H)
165. C32A4×Q16semidirect
186. C22Q16×H(3,𝔽3)semidirect

Maximal subgroups

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

subgroupindex
Q8×C22⋊H(3,𝔽3)2
C8×C22⋊H(3,𝔽3)2
C22×C32×Q163
C3×A4×Q163
Q16×H(3,𝔽3)4