Finite Group Databasebeta

← C22×C32×GL(3,𝔽2) #72218675

Subgroups — C22×C32×GL(3,𝔽2) #72218675

Order 6048.

1058 conjugacy classes of subgroups; only the 60 normal subgroups are shown.

normal subgroups only

This group has 1058 conjugacy classes of subgroups — too many to draw. Only its 60 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
1C22×C3×GL(3,𝔽2)C314yesmaximal, direct product factor
2C2×C32×GL(3,𝔽2)C213yesmaximal, direct product factor
3C32×GL(3,𝔽2)C2211yescharacteristic, direct product factor
4C2×C3×GL(3,𝔽2)C2×C3112yesdirect product factor
5C22×C32×S4—72nomaximal, Hall
6C22×C32×ΓL(1,𝔽8)—81nomaximal
7C22×GL(3,𝔽2)C3211yescharacteristic, direct product factor
8C3×GL(3,𝔽2)C22×C314yesdirect product factor
9C22×C32×A4—72no—
10C2×C32×S4—76no—
11C2×C32×S4—76no—
12C2×C32×ΓL(1,𝔽8)—83no—
13C2×GL(3,𝔽2)C2×C3213yesdirect product factor
14C22×C3×S4—78no—
15C22×C32×D4—211no—
16C22×C32×C7—81no—
17C22×C3×ΓL(1,𝔽8)—84no—
18C22×C3×ΓL(1,𝔽8)—88no—
19C22×C32×D3—281no—
20C2×C32×A4—76no—
21C32×S4—76no—
22C32×S4—72no—
23C32×ΓL(1,𝔽8)—81noHall
24GL(3,𝔽2)C22×C3211yescharacteristic, direct product factor
25C24×C32—72no—
26C22×C3×A4—78no—
27C22×C3×A4—148no—
28C2×C3×S4—724no—
29C2×C3×S4—724no—
30C2×C32×D4—213no—
31C2×C32×D4—216no—
32C2×C32×D4—213no—
33C22×C32×C4—211no—
34C2×C32×C7—83no—
35C2×C3×ΓL(1,𝔽8)—812no—
36C2×C3×ΓL(1,𝔽8)—824no—
37C22×C33—281no—
38C2×C32×D3—283no—
39C2×C32×D3—283no—
40C32×A4—72no—
41C22×S4—72no—
42C22×C3×D4—214no—
43C22×C3×C7—84no—
44C22×ΓL(1,𝔽8)—81no—
45C22×ΓL(1,𝔽8)—88no—
46C23×C32—76no—
47C23×C32—211no—
48C23×C32—216no—
49C22×C3×D3—284no—
50C2×C3×A4—724no—
51C2×C3×A4—1424no—
52C3×S4—78no—
53C3×S4—724no—
54C32×D4—211no—
55C32×D4—216no—
56C32×D4—213no—
57C32×D4—216no—
58C2×C32×C4—213no—
59C2×C32×C4—213no—
60C32×C7—81no—
61C3×ΓL(1,𝔽8)—84no—
62C3×ΓL(1,𝔽8)—88no—
63C2×C33—283no—
64C32×D3—281no—
65C32×D3—283no—
66C24×C3—78no—
67C22×A4—72no—
68C22×A4—148no—
69C2×S4—76no—
70C2×S4—76no—
71C2×C3×D4—2112no—
72C2×C3×D4—2124no—
73C2×C3×D4—2112no—
74C22×C3×C4—214no—
75C2×C3×C7—812no—
76C2×ΓL(1,𝔽8)—83no—
77C2×ΓL(1,𝔽8)—824no—
78C22×C32—72no—
79C22×C32—216no—
80C22×C32—213no—
81C22×C32—213no—
82C22×C32—284no—
83C22×C32—422no—
84C22×C32—564no—
85C22×C32GL(3,𝔽2)11yesmaximal normal, characteristic, direct product factor
86C2×C3×D3—2812no—
87C2×C3×D3—2812no—
88C3×A4—78no—
89C3×A4—148no—
90C32×C4—211no—
91C32×C4—213no—
92C22×D4—211noSylow
93C22×C7—81no—
94C33—281noSylow
95C23×C3—724no—
96C23×C3—214no—
97C23×C3—2124no—
98C22×D3—281no—
99C2×A4—76no—
100C2×A4—1424no—
101S4—72no—
102S4—76no—
103C3×D4—214no—
104C3×D4—2124no—
105C3×D4—2112no—
106C3×D4—2124no—
107C2×C3×C4—2112no—
108C2×C3×C4—2112no—
109C3×C7—84no—
110ΓL(1,𝔽8)—81no—
111ΓL(1,𝔽8)—88no—
112C2×C32—211no—
113C2×C32—213no—
114C2×C32—2812no—
115C2×C32—5612no—
116C2×C32C2×GL(3,𝔽2)13yesdirect product factor
117C3×D3—284no—
118C3×D3—2812no—
119C24—72no—
120C2×D4—213no—
121C2×D4—216no—
122C2×D4—213no—
123C22×C4—211no—
124C2×C7—83no—
125C22×C3—78no—
126C22×C3—2124no—
127C22×C3—2112no—
128C22×C3—2112no—
129C22×C3—281no—
130C22×C3—428no—
131C22×C3—564no—
132C22×C3C3×GL(3,𝔽2)14yesdirect product factor
133C2×D3—283no—
134C2×D3—283no—
135A4—72no—
136A4—148no—
137C3×C4—214no—
138C3×C4—2112no—
139C32—284no—
140C32—564no—
141C32C22×GL(3,𝔽2)11yescharacteristic, direct product factor
142C23—76no—
143C23—211no—
144C23—216no—
145D4—211no—
146D4—216no—
147D4—213no—
148D4—216no—
149C2×C4—213no—
150C2×C4—213no—
151C7—81noSylow
152C2×C3—214no—
153C2×C3—2112no—
154C2×C3—283no—
155C2×C3—5612no—
156C2×C3C2×C3×GL(3,𝔽2)112yesdirect product factor
157D3—281no—
158D3—283no—
159C22—72no—
160C22—216no—
161C22—213no—
162C22—213no—
163C22—422no—
164C22C32×GL(3,𝔽2)11yescharacteristic, direct product factor
165C4—211no—
166C4—213no—
167C3—281no—
168C3—564no—
169C3C22×C3×GL(3,𝔽2)14yesdirect product factor
170C2—211no—
171C2—213no—
172C2C2×C32×GL(3,𝔽2)13yesdirect product factor
subgroupG/H
center, Fitting, hypercenterC22×C32GL(3,𝔽2)
commutator, hypocenterGL(3,𝔽2)C22×C32
FrattiniC1G
LCSG ▷ GL(3,𝔽2)
UCSC1 ◁ C22×C32

Complements

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

NH ≅ G/Ntype
1. C22×C3×GL(3,𝔽2)C3direct (N × H)
2. C2×C32×GL(3,𝔽2)C2direct (N × H)
3. C32×GL(3,𝔽2)C22direct (N × H)
4. C2×C3×GL(3,𝔽2)C2×C3direct (N × H)
7. C22×GL(3,𝔽2)C32direct (N × H)
8. C3×GL(3,𝔽2)C22×C3direct (N × H)
13. C2×GL(3,𝔽2)C2×C32direct (N × H)
24. GL(3,𝔽2)C22×C32direct (N × H)
85. C22×C32GL(3,𝔽2)direct (N × H)
116. C2×C32C2×GL(3,𝔽2)direct (N × H)
132. C22×C3C3×GL(3,𝔽2)direct (N × H)
141. C32C22×GL(3,𝔽2)direct (N × H)
156. C2×C3C2×C3×GL(3,𝔽2)direct (N × H)
164. C22C32×GL(3,𝔽2)direct (N × H)
169. C3C22×C3×GL(3,𝔽2)direct (N × H)
172. C2C2×C32×GL(3,𝔽2)direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×GL(3,𝔽2)3
C2×C32×GL(3,𝔽2)2
C22×C32×S47
C22×C32×ΓL(1,𝔽8)8