Finite Group Databasebeta

← #90000⋊C3 #13052126

Subgroups — #90000⋊C3 #13052126

Order 1344.

296 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 296 conjugacy classes of subgroups — too many to draw. Only its 82 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×Q8×ΓL(1,𝔽8)C211yesmaximal, characteristic
2C2×#1144⋊C3C212yesmaximal, complemented
3#6487⋊C3C211yesmaximal, characteristic
4#6409⋊C3C212yesmaximal, complemented
5(C22⋊(C7⋊C8))⋊C3C211yesmaximal, characteristic
6#90000C311yesmaximal, characteristic, complemented, Hall
7C2×Q8×ΓL(1,𝔽8)—24no—
8C2×Q8×ΓL(1,𝔽8)—21no—
9C2×Q8×ΓL(1,𝔽8)C2212yes—
10C22×C4×ΓL(1,𝔽8)—21no—
11C22×C4×ΓL(1,𝔽8)C2211yescharacteristic
12C2×ΓL(1,𝔽8)⋊Q8C2211yescharacteristic
13(C22⋊Dic7)⋊C3—21no—
14#1144⋊C3—24no—
15C4⋊(ΓL(1,𝔽8)⋊C4)C2211yescharacteristic
16(ΓL(1,𝔽8)⋊C4)⋊C4—21no—
17C2×ΓL(1,𝔽8)⋊C8C2212yes—
18C22×C7×Q8C2×C311yescharacteristic
19C2×#1144C2×C312yescomplemented
20#6487C2×C311yescharacteristic
21#6409C2×C312yescomplemented
22C22⋊(C7⋊C8)C2×C311yescharacteristic
23C3×#451—71nomaximal, Hall
24C23×ΓL(1,𝔽8)D411yescharacteristic
25Q8×ΓL(1,𝔽8)—24no—
26Q8×ΓL(1,𝔽8)—22no—
27Q8×ΓL(1,𝔽8)D414yes—
28C2×C4×ΓL(1,𝔽8)—24no—
29C2×C4×ΓL(1,𝔽8)—22no—
30C2×C4×ΓL(1,𝔽8)—22no—
31C2×C4×ΓL(1,𝔽8)D411yescharacteristic
32C2×C4×ΓL(1,𝔽8)C2311yescharacteristic
33C2×ΓL(1,𝔽8)⋊C4—21no—
34C2×ΓL(1,𝔽8)⋊C4—21no—
35ΓL(1,𝔽8)⋊Q8—22no—
36ΓL(1,𝔽8)⋊C8—22no—
37C2×C7×Q8—24no—
38C2×C7×Q8—21no—
39C2×C7×Q8C22×C312yes—
40C22×C4×C7—21no—
41C22×C4×C7C22×C311yescharacteristic
42C2×Dic14C22×C311yescharacteristic
43C22⋊Dic7—21no—
44#1144—24no—
45C4⋊Dic7C22×C311yescharacteristic
46Dic7⋊C4—21no—
47C2×C7⋊C8C22×C312yes—
48C22×C3×Q8—71no—
49C2×C3×Q16—72no—
50C3×C22⋊Q8—71no—
51C3×#103—72no—
52C3×C22⋊C8—71no—
53C22×ΓL(1,𝔽8)—21no—
54C22×ΓL(1,𝔽8)—21no—
55C22×ΓL(1,𝔽8)Q1612yescomplemented
56C22×ΓL(1,𝔽8)C2×D411yescharacteristic
57C4×ΓL(1,𝔽8)—24no—
58C4×ΓL(1,𝔽8)—21no—
59C4×ΓL(1,𝔽8)C2×D412yes—
60ΓL(1,𝔽8)⋊C4—41no—
61ΓL(1,𝔽8)⋊C4—41no—
62#451—71noSylow
63C23×C7C3×D411yescharacteristic
64C7×Q8—24no—
65C7×Q8—22no—
66C7×Q8C3×D414yes—
67C2×C4×C7—24no—
68C2×C4×C7—22no—
69C2×C4×C7—22no—
70C2×C4×C7C3×D411yescharacteristic
71C2×C4×C7C23×C311yescharacteristic
72C2×Dic7—21no—
73C2×Dic7—21no—
74Dic14—22no—
75C7⋊C8—22no—
76C2×C3×Q8—71no—
77C2×C3×Q8—72no—
78C2×C3×Q8—141no—
79C2×C3×Q8—144no—
80C22×C3×C4—71no—
81C22×C3×C4—141no—
82C3×Q16—144no—
83C2×C3×C8—72no—
84C3×C4⋊C4—71no—
85C3×C4⋊C4—141no—
86C3×C22⋊C4—141no—
87C2×ΓL(1,𝔽8)—22no—
88C2×ΓL(1,𝔽8)#12011yescharacteristic
89C2×ΓL(1,𝔽8)C2×Q1611yescharacteristic
90C2×ΓL(1,𝔽8)#13711yescharacteristic
91C22×Q8AGL(1,𝔽7)11yescharacteristic
92C2×Q16—72no—
93C22⋊Q8—71no—
94#103—72no—
95C22⋊C8—71no—
96C22×C7—21no—
97C22×C7—21no—
98C22×C7C3×Q1612yescomplemented
99C22×C7C2×C3×D411yescharacteristic
100C4×C7—24no—
101C4×C7—21no—
102C4×C7C2×C3×D412yes—
103Dic7—41no—
104Dic7—41no—
105C23×C3—71no—
106C3×Q8—74no—
107C3×Q8—142no—
108C3×Q8—144no—
109C3×Q8—142no—
110C2×C3×C4—71no—
111C2×C3×C4—71no—
112C2×C3×C4—142no—
113C2×C3×C4—142no—
114C2×C3×C4—144no—
115C2×C3×C4—141no—
116C2×C3×C4—141no—
117C3×C8—142no—
118ΓL(1,𝔽8)#45111yescharacteristic, complemented, Hall
119C2×Q8—24no—
120C2×Q8—21no—
121C2×Q8—71no—
122C2×Q8C2×AGL(1,𝔽7)12yes—
123C22×C4—21no—
124C22×C4C2×AGL(1,𝔽7)11yescharacteristic
125Q16—144no—
126C2×C8—72no—
127C4⋊C4—71no—
128C4⋊C4—141no—
129C22⋊C4—141no—
130C2×C7—22no—
131C2×C7C3×#12011yescharacteristic
132C2×C7C2×C3×Q1611yescharacteristic
133C2×C7C3×#13711yescharacteristic
134C22×C3—71no—
135C22×C3—72no—
136C22×C3—141no—
137C22×C3—141no—
138C3×C4—72no—
139C3×C4—141no—
140C3×C4—144no—
141C3×C4—281no—
142C3×C4—281no—
143C23(C22⋊D7)⋊C311yescharacteristic
144Q8—24no—
145Q8—22no—
146Q8—142no—
147Q8(C22⋊D7)⋊C314yes—
148C2×C4—24no—
149C2×C4—22no—
150C2×C4—22no—
151C2×C4—141no—
152C2×C4—141no—
153C2×C4(C22⋊D7)⋊C311yescharacteristic
154C2×C4C22×AGL(1,𝔽7)11yescharacteristic
155C8—142no—
156C7C3×#45111yescharacteristic, complemented, Sylow
157C2×C3—71no—
158C2×C3—71no—
159C2×C3—71no—
160C2×C3—142no—
161C22—21no—
162C22—21no—
163C22#1144⋊C312yescomplemented
164C22C2×(C22⋊D7)⋊C311yescharacteristic
165C4—24no—
166C4—21no—
167C4—281no—
168C4—281no—
169C4C2×(C22⋊D7)⋊C312yes—
170C3—71noSylow
171C2—22no—
172C2#6505⋊C311yescharacteristic
173C2C2×#1144⋊C311yescharacteristic
174C2#1062078911yescharacteristic
subgroupG/H
centerC22C2×(C22⋊D7)⋊C3
commutator, cosocleC2×C4×C7C23×C3
FrattiniC2×C4C22×AGL(1,𝔽7)
FittingC22×C7×Q8C2×C3
socleC22×C7C2×C3×D4
prime radicalC23×ΓL(1,𝔽8)D4
hypercenterC22×Q8AGL(1,𝔽7)
hypocenterC7C3×#451
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×Q8

Complements

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

NH ≅ G/Ntype
2. C2×#1144⋊C3C2semidirect
4. #6409⋊C3C2semidirect
6. #90000C3semidirect
19. C2×#1144C2×C3semidirect
21. #6409C2×C3semidirect
55. C22×ΓL(1,𝔽8)Q16semidirect
98. C22×C7C3×Q16semidirect
118. ΓL(1,𝔽8)#451semidirect
156. C7C3×#451semidirect
163. C22#1144⋊C3semidirect

Maximal subgroups

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

subgroupindex
C22×Q8×ΓL(1,𝔽8)2
C2×#1144⋊C32
#6487⋊C32
#6409⋊C32
(C22⋊(C7⋊C8))⋊C32
#900003
C3×#4517