Finite Group Databasebeta

← #72259920

Subgroups — #72259920

Order 2016.

170 conjugacy classes of subgroups; only the 25 normal subgroups are shown.

normal subgroups only

This group has 170 conjugacy classes of subgroups — too many to draw. Only its 25 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
1SL(2,𝔽3)×AGL(1,𝔽7)C211yesmaximal, characteristic, complemented
2ΓL(1,𝔽8)⋊GL(2,𝔽3)C211yesmaximal, characteristic, complemented
3ΓL(1,𝔽8)×#226C211yesmaximal, characteristic, complemented
4#10621475C311yesmaximal, characteristic, complemented
5#6481⋊C3—31nomaximal
6ΓL(1,𝔽8)×SL(2,𝔽3)C2211yescharacteristic
7#4246⋊C3—41nomaximal
8D7⋊SL(2,𝔽3)—21no—
9D7×SL(2,𝔽3)C2×C311yescharacteristic, complemented
10#66659—31no—
11Q8×AGL(1,𝔽7)D311yescharacteristic, complemented
12C7⋊GL(2,𝔽3)C2×C311yescharacteristic, complemented
13C7×#226C2×C311yescharacteristic, complemented
14Q16×ΓL(1,𝔽8)—31no—
15#1144⋊C3—31no—
16#1143⋊C3—31no—
17#1132⋊C3—31no—
18#1131⋊C3—31no—
19C3×#983—71nomaximal, Hall
20C2×C3⋊AGL(1,𝔽7)—41no—
21C2×C3×AGL(1,𝔽7)—41no—
22Dic3×ΓL(1,𝔽8)—41no—
23#6481—31noHall
24C7⋊SL(2,𝔽3)—21no—
25C7×SL(2,𝔽3)C22×C311yescharacteristic
26Q8×ΓL(1,𝔽8)—31no—
27Q8×ΓL(1,𝔽8)C2×D311yescharacteristic
28#4246—41no—
29C4⋊AGL(1,𝔽7)—31no—
30C4⋊AGL(1,𝔽7)—61no—
31C4×AGL(1,𝔽7)—31no—
32C4×AGL(1,𝔽7)—61no—
33ΓL(1,𝔽8)⋊Q8—31no—
34C8×ΓL(1,𝔽8)—31no—
35ΓL(1,𝔽8)⋊C8—31no—
36C2×C3×SL(2,𝔽3)—71no—
37C3×GL(2,𝔽3)—71no—
38C3×#226—71no—
39C2×C3×ΓL(1,𝔽8)—41no—
40C3⋊AGL(1,𝔽7)—81no—
41C3×AGL(1,𝔽7)—81no—
42#1161—31no—
43Q8×D7C3×D311yescharacteristic, complemented
44C7×Q16—31no—
45#1144—31no—
46#1143—31no—
47#1132—31no—
48#1131—31no—
49#983—71no—
50C3×#137—211no—
51C2×D21—41no—
52C2×C3×D7—41no—
53C2×AGL(1,𝔽7)—61no—
54C2×AGL(1,𝔽7)—81no—
55C2×AGL(1,𝔽7)S411yescharacteristic
56C7×Dic3—41no—
57C4×ΓL(1,𝔽8)—31no—
58C4×ΓL(1,𝔽8)—61no—
59ΓL(1,𝔽8)⋊C4—31no—
60C3×#67—281no—
61C3×SL(2,𝔽3)—71no—
62C3×ΓL(1,𝔽8)—41noHall
63C7×Q8—31no—
64C7×Q8C2×C3×D311yescharacteristic
65D28—31no—
66D28—61no—
67C4×D7—31no—
68C4×D7—61no—
69Dic14—31no—
70C7×C8—31no—
71C7⋊C8—31no—
72C3×#41—211no—
73C2×C3×Q8—71no—
74C2×SL(2,𝔽3)—71no—
75C2×SL(2,𝔽3)—141no—
76GL(2,𝔽3)—71no—
77#226AGL(1,𝔽7)11yescharacteristic, complemented
78C3×Q16—211no—
79C3×Q16—211no—
80C3×SD16—211no—
81C3×SD16—211no—
82C3×M16—211no—
83C2×C3×C7—41no—
84D21—81no—
85C3×D7—81no—
86C2×ΓL(1,𝔽8)—81no—
87C2×ΓL(1,𝔽8)C2×S411yescharacteristic
88AGL(1,𝔽7)—21no—
89AGL(1,𝔽7)—82no—
90AGL(1,𝔽7)—121no—
91C22×C32—281no—
92C2×C3×D3—281no—
93C3×Dic3—281no—
94#137—211noSylow
95C2×D7—61no—
96C2×D7C3×S411yescharacteristic
97C4×C7—31no—
98C4×C7—61no—
99Dic7—31no—
100C3×Q8—71no—
101C3×Q8—211no—
102C3×Q8—211no—
103C3×D4—211no—
104C3×D4—421no—
105C2×C3×C4—211no—
106C2×C3×C4—421no—
107#67—281no—
108SL(2,𝔽3)—141no—
109SL(2,𝔽3)C2×AGL(1,𝔽7)11yescharacteristic
110C3×C8—211no—
111C3×C8—211no—
112C3×C7—41no—
113ΓL(1,𝔽8)—81no—
114ΓL(1,𝔽8)#98311yescharacteristic, complemented
115C2×C32—281no—
116C2×C32—561no—
117C3×D3—561no—
118#41—211no—
119C2×Q8—71no—
120Q16—31no—
121Q16—211no—
122SD16—211no—
123SD16—211no—
124M16—211no—
125C2×C7C2×C3×S411yescharacteristic
126D7—21no—
127D7—121no—
128C22×C3—71no—
129C22×C3—281no—
130C22×C3—421no—
131C22×C3—561no—
132C2×D3—281no—
133C3×C4—211no—
134C3×C4—211no—
135C3×C4—421no—
136Dic3—41no—
137C32—281noSylow
138Q8—31no—
139Q8—211no—
140Q8D3×AGL(1,𝔽7)11yescharacteristic
141D4—211no—
142D4—421no—
143C2×C4—211no—
144C2×C4—421no—
145C8—31no—
146C8—211no—
147C7C3×#98311yescharacteristic, complemented, Sylow
148C2×C3—41no—
149C2×C3—71no—
150C2×C3—141no—
151C2×C3—561no—
152C2×C3—562no—
153C2×C3—561no—
154C2×C3—841no—
155D3—561no—
156C22—71no—
157C22—421no—
158C4—31no—
159C4—61no—
160C4—211no—
161C3—41no—
162C3—71no—
163C3—561no—
164C2—141no—
165C2—841no—
166C2S4×AGL(1,𝔽7)11yescharacteristic
subgroupG/H
center, Frattini, hypercenterC2S4×AGL(1,𝔽7)
commutator, cosocle, hypocenterC7×SL(2,𝔽3)C22×C3
FittingC7×Q8C2×C3×D3
socleC2×C7C2×C3×S4
LCSG ▷ C7×SL(2,𝔽3)
UCSC1 ◁ C2

Complements

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

NH ≅ G/Ntype
1. SL(2,𝔽3)×AGL(1,𝔽7)C2semidirect
2. ΓL(1,𝔽8)⋊GL(2,𝔽3)C2semidirect
3. ΓL(1,𝔽8)×#226C2semidirect
4. #10621475C3semidirect
9. D7×SL(2,𝔽3)C2×C3semidirect
11. Q8×AGL(1,𝔽7)D3semidirect
12. C7⋊GL(2,𝔽3)C2×C3semidirect
13. C7×#226C2×C3semidirect
43. Q8×D7C3×D3semidirect
77. #226AGL(1,𝔽7)semidirect
114. ΓL(1,𝔽8)#983semidirect
147. C7C3×#983semidirect

Maximal subgroups

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

subgroupindex
SL(2,𝔽3)×AGL(1,𝔽7)2
ΓL(1,𝔽8)⋊GL(2,𝔽3)2
ΓL(1,𝔽8)×#2262
#106214753
#6481⋊C33
#4246⋊C34
C3×#9837