Finite Group Databasebeta

← #72229735

Subgroups — #72229735

Order 2688.

252 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 252 conjugacy classes of subgroups — too many to draw. Only its 77 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#89472⋊C3C211yesmaximal, characteristic
2#89452⋊C3C211yesmaximal, characteristic, complemented
3ΓL(1,𝔽8)×#364C211yesmaximal, characteristic, complemented
4#89313⋊C3C212yesmaximal
5#89308⋊C3C212yesmaximal, complemented
6#11727061C311yesmaximal, characteristic, complemented, Hall
7#6467⋊C3C2211yescharacteristic
8#6464⋊C3—21no—
9#6432⋊C3—21no—
10ΓL(1,𝔽8)×M32C2212yes—
11C4×C8×ΓL(1,𝔽8)C2211yescharacteristic
12#6393⋊C3C2211yescharacteristic
13#6391⋊C3—21no—
14#6391⋊C3C2212yes—
15#6379⋊C3—21no—
16#89472C2×C311yescharacteristic
17#89452C2×C311yescharacteristic, complemented
18C7×#364C2×C311yescharacteristic, complemented
19#89313C2×C312yes—
20#89308C2×C312yescomplemented
21C3×#2230—71nomaximal, Hall
22#1157⋊C3—21no—
23C2×ΓL(1,𝔽8)⋊Q8—41no—
24C2×C8×ΓL(1,𝔽8)D411yescharacteristic
25C2×C8×ΓL(1,𝔽8)C2311yescharacteristic
26C42×ΓL(1,𝔽8)D411yescharacteristic
27C4⋊(ΓL(1,𝔽8)⋊C4)—22no—
28C4⋊(ΓL(1,𝔽8)⋊C4)—41no—
29#1136⋊C3—21no—
30Dic28⋊C3Q811yescharacteristic, complemented
31D56⋊C3Q811yescharacteristic, complemented
32#1132⋊C3—21no—
33#1131⋊C3—41no—
34C8×AGL(1,𝔽7)—41no—
35C16×ΓL(1,𝔽8)—22no—
36#6467C22×C311yescharacteristic
37#6464—21no—
38#6432—21no—
39C7×M32C22×C312yes—
40C4×C7×C8C22×C311yescharacteristic
41#6393C22×C311yescharacteristic
42#6391—21no—
43#6391C22×C312yes—
44#6379—21no—
45C3×#498—71no—
46C3×#443—71no—
47C3×#365—72no—
48C3×#364—71no—
49C3×#360—72no—
50C2×C4×ΓL(1,𝔽8)—21no—
51C2×C4×ΓL(1,𝔽8)C2×D411yescharacteristic
52(C22⋊D7)⋊C3—41no—
53C2×ΓL(1,𝔽8)⋊C4—42no—
54C4⋊AGL(1,𝔽7)—21no—
55C4×AGL(1,𝔽7)—41no—
56ΓL(1,𝔽8)⋊Q8—21no—
57ΓL(1,𝔽8)⋊Q8—82no—
58C8×ΓL(1,𝔽8)SD1611yescharacteristic
59C8×ΓL(1,𝔽8)SD1611yescharacteristic
60C8×ΓL(1,𝔽8)C2×Q811yescharacteristic
61C8×ΓL(1,𝔽8)#4111yescharacteristic
62ΓL(1,𝔽8)⋊C8—41no—
63#2230—71noSylow
64#1157—21no—
65C2×Dic14—41no—
66C2×C7×C8C3×D411yescharacteristic
67C2×C7×C8C23×C311yescharacteristic
68C42×C7C3×D411yescharacteristic
69C4⋊Dic7—22no—
70C4⋊Dic7—41no—
71#1136—21no—
72Dic28C3×Q811yescharacteristic, complemented
73D56C3×Q811yescharacteristic, complemented
74#1132—21no—
75#1131—41no—
76C8×D7—41no—
77C7×C16—22no—
78C3×#135—71no—
79C3×#131—141no—
80C3×C4⋊Q8—141no—
81C3×M32—72no—
82C3×#108—71no—
83C3×#106—72no—
84C3×#106—141no—
85C3×#104—141no—
86C3×C4×C8—71no—
87C22×ΓL(1,𝔽8)AGL(1,ℤ/8)11yescharacteristic
88C2×AGL(1,𝔽7)—41no—
89C4×ΓL(1,𝔽8)—22no—
90C4×ΓL(1,𝔽8)C22⋊Q811yescharacteristic
91C4×ΓL(1,𝔽8)C2×SD1611yescharacteristic
92ΓL(1,𝔽8)⋊C4—41no—
93ΓL(1,𝔽8)⋊C4—82no—
94#498—71no—
95#443—71no—
96#365—72no—
97#364AGL(1,𝔽7)11yescharacteristic, complemented
98#360—72no—
99C2×C4×C7—21no—
100C2×C4×C7C2×C3×D411yescharacteristic
101C22⋊D7—41no—
102C2×Dic7—42no—
103D28—21no—
104C4×D7—41no—
105Dic14—21no—
106Dic14—82no—
107C7×C8C3×SD1611yescharacteristic
108C7×C8C3×SD1611yescharacteristic
109C7×C8C2×C3×Q811yescharacteristic
110C7×C8C3×#4111yescharacteristic
111C7⋊C8—41no—
112C3×#41—141no—
113C2×C3×Q8—281no—
114C3×Q16—71no—
115C3×SD16—141no—
116C3×D8—71no—
117C3×M16—141no—
118C3×M16—281no—
119C2×C3×C8—71no—
120C2×C3×C8—71no—
121C2×C3×C8—281no—
122C3×C4⋊C4—142no—
123C3×C4⋊C4—281no—
124C3×C42—71no—
125C3×C16—142no—
126C2×ΓL(1,𝔽8)—21no—
127C2×ΓL(1,𝔽8)#47611yescharacteristic
128AGL(1,𝔽7)—81no—
129#135—71no—
130#131—141no—
131C4⋊Q8—141no—
132M32C2×AGL(1,𝔽7)12yes—
133#108—71no—
134#106—72no—
135#106—141no—
136#104—141no—
137C4×C8C2×AGL(1,𝔽7)11yescharacteristic
138C22×C7C3×AGL(1,ℤ/8)11yescharacteristic
139C2×D7—41no—
140C4×C7—22no—
141C4×C7C3×C22⋊Q811yescharacteristic
142C4×C7C2×C3×SD1611yescharacteristic
143Dic7—41no—
144Dic7—82no—
145C3×Q8—141no—
146C3×Q8—562no—
147C3×D4—141no—
148C3×D4—281no—
149C2×C3×C4—71no—
150C2×C3×C4—141no—
151C2×C3×C4—281no—
152C2×C3×C4—282no—
153C3×C8—71no—
154C3×C8—71no—
155C3×C8—71no—
156C3×C8—71no—
157C3×C8—281no—
158ΓL(1,𝔽8)#223011yescharacteristic, complemented, Hall
159#41—141no—
160C2×Q8—281no—
161Q16—71no—
162SD16—141no—
163D8—71no—
164M16—141no—
165M16—281no—
166C2×C8—281no—
167C2×C8C4⋊AGL(1,𝔽7)11yescharacteristic
168C2×C8C22×AGL(1,𝔽7)11yescharacteristic
169C4⋊C4—142no—
170C4⋊C4—281no—
171C42C4⋊AGL(1,𝔽7)11yescharacteristic
172C16—22no—
173C2×C7—21no—
174C2×C7C3×#47611yescharacteristic
175D7—81no—
176C22×C3—71no—
177C22×C3—281no—
178C3×C4—71no—
179C3×C4—71no—
180C3×C4—142no—
181C3×C4—281no—
182C3×C4—562no—
183Q8—141no—
184Q8—562no—
185D4—141no—
186D4—281no—
187C2×C4—21no—
188C2×C4—281no—
189C2×C4—282no—
190C2×C4C2×C4⋊AGL(1,𝔽7)11yescharacteristic
191C8—281no—
192C8#1132⋊C311yescharacteristic
193C8#1132⋊C311yescharacteristic
194C8#1159⋊C311yescharacteristic
195C8Q8×AGL(1,𝔽7)11yescharacteristic
196C7C3×#223011yescharacteristic, complemented, Sylow
197C2×C3—71no—
198C2×C3—141no—
199C2×C3—561no—
200C22—281no—
201C22#6471⋊C311yescharacteristic
202C4—22no—
203C4—281no—
204C4—562no—
205C4#6460⋊C311yescharacteristic
206C4C2×#1132⋊C311yescharacteristic
207C3—71noSylow
208C2—21no—
209C2—561no—
210C2#89614⋊C311yescharacteristic
subgroupG/H
centerC2#89614⋊C3
commutator, cosocleC2×C7×C8C23×C3
FrattiniC2×C8C22×AGL(1,𝔽7)
FittingC7×#364C2×C3
socleC2×C7C3×#476
prime radical#6467⋊C3C22
hypercenter#364AGL(1,𝔽7)
hypocenterC7C3×#2230
LCSG ▷ C2×C7×C8 ▷ C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C2 ◁ C2×C4 ◁ C4×C8 ◁ #364

Complements

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

NH ≅ G/Ntype
2. #89452⋊C3C2semidirect
3. ΓL(1,𝔽8)×#364C2semidirect
5. #89308⋊C3C2semidirect
6. #11727061C3semidirect
17. #89452C2×C3semidirect
18. C7×#364C2×C3semidirect
20. #89308C2×C3semidirect
30. Dic28⋊C3Q8semidirect
31. D56⋊C3Q8semidirect
72. Dic28C3×Q8semidirect
73. D56C3×Q8semidirect
97. #364AGL(1,𝔽7)semidirect
158. ΓL(1,𝔽8)#2230semidirect
196. C7C3×#2230semidirect

Maximal subgroups

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

subgroupindex
#89472⋊C32
#89452⋊C32
ΓL(1,𝔽8)×#3642
#89313⋊C32
#89308⋊C32
#117270613
C3×#22307