Finite Group Databasebeta

← #89623⋊C3 #13051732

Subgroups — #89623⋊C3 #13051732

Order 1344.

192 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 192 conjugacy classes of subgroups — too many to draw. Only its 74 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
1#6432⋊C3C211yesmaximal, characteristic
2C4×ΓL(1,𝔽8)⋊Q8C211yesmaximal, characteristic
3ΓL(1,𝔽8)×C4⋊C8C211yesmaximal, characteristic
4#6391⋊C3C212yesmaximal
5#6390⋊C3C212yesmaximal
6#89623C311yesmaximal, characteristic, complemented, Hall
7C2×ΓL(1,𝔽8)⋊Q8—21no—
8C2×ΓL(1,𝔽8)⋊Q8C2211yescharacteristic
9C2×C8×ΓL(1,𝔽8)C2212yes—
10C42×ΓL(1,𝔽8)C2211yescharacteristic
11C4⋊(ΓL(1,𝔽8)⋊C4)—21no—
12C4⋊(ΓL(1,𝔽8)⋊C4)C2212yes—
13C4⋊(ΓL(1,𝔽8)⋊C4)C2211yescharacteristic
14(ΓL(1,𝔽8)⋊C4)⋊C4—21no—
15C4×ΓL(1,𝔽8)⋊C4—21no—
16#6432C2×C311yescharacteristic
17C4×Dic14C2×C311yescharacteristic
18C7×C4⋊C8C2×C311yescharacteristic
19#6391C2×C312yes—
20#6390C2×C312yes—
21C3×#475—71nomaximal, Hall
22C2×C4×ΓL(1,𝔽8)D411yescharacteristic
23C2×C4×ΓL(1,𝔽8)D411yescharacteristic
24C2×C4×ΓL(1,𝔽8)C2311yescharacteristic
25C2×ΓL(1,𝔽8)⋊C4—21no—
26C2×ΓL(1,𝔽8)⋊C4—22no—
27C2×ΓL(1,𝔽8)⋊C4—21no—
28ΓL(1,𝔽8)⋊Q8—21no—
29ΓL(1,𝔽8)⋊Q8—42no—
30ΓL(1,𝔽8)⋊Q8Q812yescomplemented
31C8×ΓL(1,𝔽8)—22no—
32C2×Dic14—21no—
33C2×Dic14C22×C311yescharacteristic
34C2×C7×C8C22×C312yes—
35C42×C7C22×C311yescharacteristic
36C4⋊Dic7—21no—
37C4⋊Dic7C22×C311yescharacteristic
38C4⋊Dic7C22×C312yes—
39Dic7⋊C4—21no—
40C4×Dic7—21no—
41C3×C4⋊Q8—71no—
42C3×C4×Q8—71no—
43C3×#106—72no—
44C3×C4⋊C8—71no—
45C3×#103—72no—
46C22×ΓL(1,𝔽8)C2×D411yescharacteristic
47C4×ΓL(1,𝔽8)—21no—
48C4×ΓL(1,𝔽8)SD1612yes—
49C4×ΓL(1,𝔽8)C2×Q811yescharacteristic
50C4×ΓL(1,𝔽8)#4111yescharacteristic
51ΓL(1,𝔽8)⋊C4—22no—
52ΓL(1,𝔽8)⋊C4—42no—
53ΓL(1,𝔽8)⋊C4—41no—
54#475—71noSylow
55C2×C4×C7C3×D411yescharacteristic
56C2×C4×C7C3×D411yescharacteristic
57C2×C4×C7C23×C311yescharacteristic
58C2×Dic7—21no—
59C2×Dic7—21no—
60C2×Dic7—22no—
61Dic14—21no—
62Dic14—42no—
63Dic14C3×Q812yescomplemented
64C7×C8—22no—
65C2×C3×Q8—71no—
66C2×C3×Q8—141no—
67C2×C3×C8—72no—
68C3×C4⋊C4—72no—
69C3×C4⋊C4—71no—
70C3×C4⋊C4—141no—
71C3×C4⋊C4—141no—
72C3×C42—71no—
73C3×C42—141no—
74C2×ΓL(1,𝔽8)C22⋊Q811yescharacteristic
75C2×ΓL(1,𝔽8)C2×SD1611yescharacteristic
76C2×ΓL(1,𝔽8)#13711yescharacteristic
77C4⋊Q8—71no—
78C4×Q8—71no—
79#106—72no—
80C4⋊C8AGL(1,𝔽7)11yescharacteristic
81#103—72no—
82C22×C7C2×C3×D411yescharacteristic
83C4×C7—21no—
84C4×C7C3×SD1612yes—
85C4×C7C2×C3×Q811yescharacteristic
86C4×C7C3×#4111yescharacteristic
87Dic7—22no—
88Dic7—42no—
89Dic7—41no—
90C3×Q8—72no—
91C3×Q8—141no—
92C3×Q8—282no—
93C2×C3×C4—71no—
94C2×C3×C4—71no—
95C2×C3×C4—71no—
96C2×C3×C4—141no—
97C2×C3×C4—141no—
98C2×C3×C4—142no—
99C3×C8—142no—
100ΓL(1,𝔽8)#47511yescharacteristic, complemented, Hall
101C2×Q8—71no—
102C2×Q8—141no—
103C2×C8C2×AGL(1,𝔽7)12yes—
104C4⋊C4—72no—
105C4⋊C4—71no—
106C4⋊C4—141no—
107C4⋊C4—141no—
108C42—141no—
109C42C2×AGL(1,𝔽7)11yescharacteristic
110C2×C7C3×C22⋊Q811yescharacteristic
111C2×C7C2×C3×SD1611yescharacteristic
112C2×C7C3×#13711yescharacteristic
113C22×C3—71no—
114C3×C4—71no—
115C3×C4—71no—
116C3×C4—72no—
117C3×C4—141no—
118C3×C4—142no—
119C3×C4—282no—
120C3×C4—281no—
121Q8—72no—
122Q8—141no—
123Q8—282no—
124C2×C4—141no—
125C2×C4—141no—
126C2×C4—142no—
127C2×C4C4⋊AGL(1,𝔽7)11yescharacteristic
128C2×C4C4⋊AGL(1,𝔽7)11yescharacteristic
129C2×C4C22×AGL(1,𝔽7)11yescharacteristic
130C8—22no—
131C7C3×#47511yescharacteristic, complemented, Sylow
132C2×C3—71no—
133C2×C3—71no—
134C2×C3—71no—
135C22C2×C4⋊AGL(1,𝔽7)11yescharacteristic
136C4—21no—
137C4—142no—
138C4—282no—
139C4—281no—
140C4#1132⋊C312yes—
141C4#1159⋊C311yescharacteristic
142C4Q8×AGL(1,𝔽7)11yescharacteristic
143C3—71noSylow
144C2#6460⋊C311yescharacteristic
145C2C2×#1132⋊C311yescharacteristic
146C2#6472⋊C311yescharacteristic
subgroupG/H
centerC22C2×C4⋊AGL(1,𝔽7)
commutator, cosocleC2×C4×C7C23×C3
FrattiniC2×C4C22×AGL(1,𝔽7)
FittingC7×C4⋊C8C2×C3
socleC22×C7C2×C3×D4
prime radicalC22×ΓL(1,𝔽8)C2×D4
hypercenterC4⋊C8AGL(1,𝔽7)
hypocenterC7C3×#475
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C42 ◁ C4⋊C8

Complements

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

NH ≅ G/Ntype
6. #89623C3semidirect
30. ΓL(1,𝔽8)⋊Q8Q8semidirect
63. Dic14C3×Q8semidirect
100. ΓL(1,𝔽8)#475semidirect
131. C7C3×#475semidirect

Maximal subgroups

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

subgroupindex
#6432⋊C32
C4×ΓL(1,𝔽8)⋊Q82
ΓL(1,𝔽8)×C4⋊C82
#6391⋊C32
#6390⋊C32
#896233
C3×#4757