Finite Group Databasebeta

← C22×C32⋊(C22⋊A4) #13180112

Subgroups — C22×C32⋊(C22⋊A4) #13180112

Order 1728.

5965 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 5965 conjugacy classes of subgroups — too many to draw. Only its 125 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×C32⋊(C22⋊A4)C213yesmaximal, direct product factor
2C26×C32C311yesmaximal, characteristic, complemented
3C22×C3×C22⋊A4C313yesmaximal, complemented
4C32⋊(C22⋊A4)C2211yescharacteristic, direct product factor
5C22×C22⋊H(3,𝔽3)—45nomaximal
6C25×C32—315no—
7C25×C32—35no—
8C25×C32C2×C313yescomplemented
9C2×C3×C22⋊A4C2×C319yescomplemented
10C2×C22⋊H(3,𝔽3)—415no—
11C26×C3—31no—
12C26×C3C3211yescharacteristic
13C22×C22⋊A4—33no—
14C24×C32—390no—
15C24×C32—360no—
16C24×C32—310no—
17C24×C32—310no—
18C24×C32—315no—
19C24×C32—315no—
20C24×C32—315no—
21C24×C32A415yescomplemented
22C24×C32C22×C311yescharacteristic, complemented
23C3×C22⋊A4C22×C313yescomplemented
24C22×C3×A4—415no—
25C22×H(3,𝔽3)—161no—
26C22⋊H(3,𝔽3)—45no—
27C25×C3—315no—
28C25×C3—345no—
29C25×C3—315no—
30C25×C3—35no—
31C25×C3—33no—
32C25×C3C2×C3213yes—
33C2×C22⋊A4—39no—
34C23×C32—315no—
35C23×C32—390no—
36C23×C32—390no—
37C23×C32—3120no—
38C23×C32—390no—
39C23×C32—330no—
40C23×C32—315no—
41C23×C32—35no—
42C23×C32—35no—
43C23×C32C2×A4115yescomplemented
44C2×C3×A4—445no—
45C26H(3,𝔽3)11yescharacteristic, complemented, Sylow
46C2×H(3,𝔽3)—163no—
47C24×C3—315no—
48C24×C3—345no—
49C24×C3—390no—
50C24×C3—360no—
51C24×C3—3270no—
52C24×C3—3180no—
53C24×C3—330no—
54C24×C3—330no—
55C24×C3—310no—
56C24×C3—310no—
57C24×C3—345no—
58C24×C3—345no—
59C24×C3—315no—
60C24×C3—315no—
61C24×C3—35no—
62C24×C3—31no—
63C24×C3C3×A415yes—
64C24×C3C22×C3211yescharacteristic
65C22⋊A4—33no—
66C22×A4—1215no—
67C22×C32—315no—
68C22×C32—310no—
69C22×C32—360no—
70C22×C32—390no—
71C22×C32—310no—
72C22×C32—315no—
73C22×C32—315no—
74C22×C32—163no—
75C22×C32C22×A415yescomplemented
76C22×C32C22⋊A411yescharacteristic, complemented
77C3×A4—415no—
78C25—315no—
79C25—35no—
80C25C2×H(3,𝔽3)13yescomplemented
81H(3,𝔽3)—161noSylow
82C23×C3—315no—
83C23×C3—345no—
84C23×C3—390no—
85C23×C3—390no—
86C23×C3—3270no—
87C23×C3—3270no—
88C23×C3—3360no—
89C23×C3—3120no—
90C23×C3—390no—
91C23×C3—3270no—
92C23×C3—330no—
93C23×C3—390no—
94C23×C3—315no—
95C23×C3—345no—
96C23×C3—315no—
97C23×C3—315no—
98C23×C3—35no—
99C23×C3—35no—
100C23×C3—315no—
101C23×C3C2×C3×A4115yes—
102C2×A4—1245no—
103C2×C32—315no—
104C2×C32—35no—
105C2×C32—169no—
106C2×C32C2×C22⋊A413yescomplemented
107C24—315no—
108C24—390no—
109C24—360no—
110C24—310no—
111C24—310no—
112C24—315no—
113C24—315no—
114C24C22⋊H(3,𝔽3)15yescomplemented
115C24C22×H(3,𝔽3)11yescharacteristic, complemented
116C22×C3—315no—
117C22×C3—345no—
118C22×C3—330no—
119C22×C3—3270no—
120C22×C3—330no—
121C22×C3—3180no—
122C22×C3—310no—
123C22×C3—390no—
124C22×C3—310no—
125C22×C3—360no—
126C22×C3—345no—
127C22×C3—345no—
128C22×C3—315no—
129C22×C3—315no—
130C22×C3—35no—
131C22×C3—31no—
132C22×C3—483no—
133C22×C3C22×C3×A415yes—
134C22×C3C3×C22⋊A411yescharacteristic
135A4—1215no—
136C32—163no—
137C32C22×C22⋊A411yescharacteristic, complemented
138C23—315no—
139C23—390no—
140C23—390no—
141C23—3120no—
142C23—330no—
143C23—390no—
144C23—35no—
145C23—315no—
146C23—35no—
147C23C2×C22⋊H(3,𝔽3)115yescomplemented
148C2×C3—315no—
149C2×C3—345no—
150C2×C3—35no—
151C2×C3—315no—
152C2×C3—33no—
153C2×C3—489no—
154C2×C3C2×C3×C22⋊A413yes—
155C22—315no—
156C22—310no—
157C22—390no—
158C22—310no—
159C22—360no—
160C22—315no—
161C22—315no—
162C22C22×C22⋊H(3,𝔽3)15yescomplemented
163C22C32⋊(C22⋊A4)11yescharacteristic, direct product factor
164C3—31no—
165C3—483no—
166C3C22×C3×C22⋊A411yescharacteristic
167C2—35no—
168C2—315no—
169C2C2×C32⋊(C22⋊A4)13yesdirect product factor
subgroupG/H
centerC22×C3C3×C22⋊A4
commutator, cosocleC24×C3C22×C32
FrattiniC3C22×C3×C22⋊A4
FittingC26×C32C3
socleC26×C3C32
hypercenterC22×C32C22⋊A4
hypocenterC24C22×H(3,𝔽3)
LCSG ▷ C24×C3 ▷ C24
UCSC1 ◁ C22×C3 ◁ C22×C32

Complements

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

NH ≅ G/Ntype
1. C2×C32⋊(C22⋊A4)C2direct (N × H)
2. C26×C32C3semidirect
3. C22×C3×C22⋊A4C3semidirect
4. C32⋊(C22⋊A4)C22direct (N × H)
8. C25×C32C2×C3semidirect
9. C2×C3×C22⋊A4C2×C3semidirect
21. C24×C32A4semidirect
22. C24×C32C22×C3semidirect
23. C3×C22⋊A4C22×C3semidirect
43. C23×C32C2×A4semidirect
45. C26H(3,𝔽3)semidirect
75. C22×C32C22×A4semidirect
76. C22×C32C22⋊A4semidirect
80. C25C2×H(3,𝔽3)semidirect
106. C2×C32C2×C22⋊A4semidirect
114. C24C22⋊H(3,𝔽3)semidirect
115. C24C22×H(3,𝔽3)semidirect
137. C32C22×C22⋊A4semidirect
147. C23C2×C22⋊H(3,𝔽3)semidirect
162. C22C22×C22⋊H(3,𝔽3)semidirect
163. C22C32⋊(C22⋊A4)direct (N × H)
169. C2C2×C32⋊(C22⋊A4)direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C32⋊(C22⋊A4)2
C26×C323
C22×C3×C22⋊A43
C22×C22⋊H(3,𝔽3)4