Finite Group Databasebeta

← #89495⋊C3 #13051624

Subgroups — #89495⋊C3 #13051624

Order 1344.

192 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 192 conjugacy classes of subgroups — too many to draw. Only its 64 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#6489⋊C3C211yesmaximal, characteristic
2#6487⋊C3C211yesmaximal, characteristic
3ΓL(1,𝔽8)×C22⋊C8C211yesmaximal, characteristic
4#6392⋊C3C211yesmaximal, characteristic, complemented
5#6391⋊C3C211yesmaximal, characteristic, complemented
6#6390⋊C3C211yesmaximal, characteristic, complemented
7#6390⋊C3C211yesmaximal, characteristic, complemented
8#89495C311yesmaximal, characteristic, complemented, Hall
9C22×C4×ΓL(1,𝔽8)C2211yescharacteristic
10C2×ΓL(1,𝔽8)⋊Q8C2211yescharacteristic
11C2×C8×ΓL(1,𝔽8)C2211yescharacteristic
12C2×C8×ΓL(1,𝔽8)C2211yescharacteristic
13(C22⋊Dic7)⋊C3—21no—
14(C22⋊Dic7)⋊C3—21no—
15C4⋊(ΓL(1,𝔽8)⋊C4)C2211yescharacteristic
16C4⋊(ΓL(1,𝔽8)⋊C4)C2211yescharacteristic
17C4⋊(ΓL(1,𝔽8)⋊C4)C2211yescharacteristic
18(ΓL(1,𝔽8)⋊C4)⋊C4—21no—
19C4×ΓL(1,𝔽8)⋊C4—21no—
20#6489C2×C311yescharacteristic
21#6487C2×C311yescharacteristic
22C7×C22⋊C8C2×C311yescharacteristic
23#6392C2×C311yescharacteristic, complemented
24#6391C2×C311yescharacteristic, complemented
25#6390C2×C311yescharacteristic, complemented
26#6390C2×C311yescharacteristic, complemented
27C3×#485—71nomaximal, Hall
28C23×ΓL(1,𝔽8)D411yescharacteristic
29C2×C4×ΓL(1,𝔽8)—21no—
30C2×C4×ΓL(1,𝔽8)—21no—
31C2×C4×ΓL(1,𝔽8)D411yescharacteristic
32C2×C4×ΓL(1,𝔽8)C2311yescharacteristic
33C2×ΓL(1,𝔽8)⋊C4—21no—
34C2×ΓL(1,𝔽8)⋊C4—21no—
35C2×ΓL(1,𝔽8)⋊C4—21no—
36C2×ΓL(1,𝔽8)⋊C4—21no—
37ΓL(1,𝔽8)⋊Q8—21no—
38ΓL(1,𝔽8)⋊Q8—21no—
39C8×ΓL(1,𝔽8)—21no—
40C8×ΓL(1,𝔽8)—21no—
41C22×C4×C7C22×C311yescharacteristic
42C2×Dic14C22×C311yescharacteristic
43C2×C7×C8C22×C311yescharacteristic
44C2×C7×C8C22×C311yescharacteristic
45C22⋊Dic7—21no—
46C22⋊Dic7—21no—
47C4⋊Dic7C22×C311yescharacteristic
48C4⋊Dic7C22×C311yescharacteristic
49C4⋊Dic7C22×C311yescharacteristic
50Dic7⋊C4—21no—
51C4×Dic7—21no—
52C3×C22⋊Q8—71no—
53C3×#117—71no—
54C3×#107—71no—
55C3×#106—71no—
56C3×#103—71no—
57C3×#103—71no—
58C3×C22⋊C8—71no—
59C22×ΓL(1,𝔽8)—21no—
60C22×ΓL(1,𝔽8)—21no—
61C22×ΓL(1,𝔽8)—21no—
62C22×ΓL(1,𝔽8)C2×D411yescharacteristic
63C4×ΓL(1,𝔽8)—21no—
64C4×ΓL(1,𝔽8)#4111yescharacteristic
65C4×ΓL(1,𝔽8)#4111yescharacteristic
66ΓL(1,𝔽8)⋊C4—41no—
67ΓL(1,𝔽8)⋊C4—41no—
68ΓL(1,𝔽8)⋊C4—41no—
69ΓL(1,𝔽8)⋊C4—41no—
70#485—71noSylow
71C23×C7C3×D411yescharacteristic
72C2×C4×C7—21no—
73C2×C4×C7—21no—
74C2×C4×C7C3×D411yescharacteristic
75C2×C4×C7C23×C311yescharacteristic
76C2×Dic7—21no—
77C2×Dic7—21no—
78C2×Dic7—21no—
79C2×Dic7—21no—
80Dic14—21no—
81Dic14—21no—
82C7×C8—21no—
83C7×C8—21no—
84C2×C3×Q8—71no—
85C22×C3×C4—71no—
86C2×C3×C8—71no—
87C2×C3×C8—71no—
88C3×C4⋊C4—71no—
89C3×C4⋊C4—71no—
90C3×C4⋊C4—71no—
91C3×C4⋊C4—141no—
92C3×C22⋊C4—141no—
93C3×C22⋊C4—141no—
94C3×C42—141no—
95C2×ΓL(1,𝔽8)—41no—
96C2×ΓL(1,𝔽8)#12311yescharacteristic
97C2×ΓL(1,𝔽8)#13511yescharacteristic
98C2×ΓL(1,𝔽8)#13711yescharacteristic
99C22⋊Q8—71no—
100#117—71no—
101#107—71no—
102#106—71no—
103#103—71no—
104#103—71no—
105C22⋊C8AGL(1,𝔽7)11yescharacteristic
106C22×C7—21no—
107C22×C7—21no—
108C22×C7—21no—
109C22×C7C2×C3×D411yescharacteristic
110C4×C7—21no—
111C4×C7C3×#4111yescharacteristic
112C4×C7C3×#4111yescharacteristic
113Dic7—41no—
114Dic7—41no—
115Dic7—41no—
116Dic7—41no—
117C23×C3—71no—
118C3×Q8—141no—
119C3×Q8—141no—
120C2×C3×C4—71no—
121C2×C3×C4—71no—
122C2×C3×C4—141no—
123C2×C3×C4—141no—
124C2×C3×C4—141no—
125C2×C3×C4—141no—
126C2×C3×C4—141no—
127C2×C3×C4—141no—
128C3×C8—141no—
129C3×C8—141no—
130ΓL(1,𝔽8)#48511yescharacteristic, complemented, Hall
131C2×Q8—71no—
132C22×C4C2×AGL(1,𝔽7)11yescharacteristic
133C2×C8C2×AGL(1,𝔽7)11yescharacteristic
134C2×C8C2×AGL(1,𝔽7)11yescharacteristic
135C4⋊C4—71no—
136C4⋊C4—71no—
137C4⋊C4—71no—
138C4⋊C4—141no—
139C22⋊C4—141no—
140C22⋊C4—141no—
141C42—141no—
142C2×C7—41no—
143C2×C7C3×#12311yescharacteristic
144C2×C7C3×#13511yescharacteristic
145C2×C7C3×#13711yescharacteristic
146C22×C3—71no—
147C22×C3—141no—
148C22×C3—141no—
149C22×C3—141no—
150C3×C4—71no—
151C3×C4—71no—
152C3×C4—141no—
153C3×C4—281no—
154C3×C4—281no—
155C3×C4—281no—
156C3×C4—281no—
157C23C4⋊AGL(1,𝔽7)11yescharacteristic
158Q8—141no—
159Q8—141no—
160C2×C4—21no—
161C2×C4—21no—
162C2×C4—141no—
163C2×C4—141no—
164C2×C4—141no—
165C2×C4—141no—
166C2×C4C4⋊AGL(1,𝔽7)11yescharacteristic
167C2×C4C22×AGL(1,𝔽7)11yescharacteristic
168C8—21no—
169C8—21no—
170C7C3×#48511yescharacteristic, complemented, Sylow
171C2×C3—71no—
172C2×C3—71no—
173C2×C3—71no—
174C2×C3—281no—
175C22—21no—
176C22—21no—
177C22—21no—
178C22C2×C4⋊AGL(1,𝔽7)11yescharacteristic
179C4—21no—
180C4—281no—
181C4—281no—
182C4—281no—
183C4—281no—
184C4#1159⋊C311yescharacteristic
185C4#1159⋊C311yescharacteristic
186C3—71noSylow
187C2—41no—
188C2#6449⋊C311yescharacteristic
189C2#6467⋊C311yescharacteristic
190C2#6472⋊C311yescharacteristic
subgroupG/H
centerC22C2×C4⋊AGL(1,𝔽7)
commutator, cosocleC2×C4×C7C23×C3
FrattiniC2×C4C22×AGL(1,𝔽7)
FittingC7×C22⋊C8C2×C3
socleC22×C7C2×C3×D4
prime radicalC23×ΓL(1,𝔽8)D4
hypercenterC22⋊C8AGL(1,𝔽7)
hypocenterC7C3×#485
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22⋊C8

Complements

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

NH ≅ G/Ntype
4. #6392⋊C3C2semidirect
5. #6391⋊C3C2semidirect
6. #6390⋊C3C2semidirect
7. #6390⋊C3C2semidirect
8. #89495C3semidirect
23. #6392C2×C3semidirect
24. #6391C2×C3semidirect
25. #6390C2×C3semidirect
26. #6390C2×C3semidirect
130. ΓL(1,𝔽8)#485semidirect
170. C7C3×#485semidirect

Maximal subgroups

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

subgroupindex
#6489⋊C32
#6487⋊C32
ΓL(1,𝔽8)×C22⋊C82
#6392⋊C32
#6391⋊C32
#6390⋊C32
#894953
C3×#4857