Finite Group Databasebeta

← #72229729

Subgroups — #72229729

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
1ΓL(1,𝔽8)×#498C211yesmaximal, characteristic, complemented
2#89472⋊C3C211yesmaximal, characteristic
3#89283⋊C3C212yesmaximal
4#89282⋊C3C212yesmaximal, complemented
5#89258⋊C3C211yesmaximal, characteristic, complemented
6#11729104C311yesmaximal, characteristic, complemented, Hall
7ΓL(1,𝔽8)×C4⋊Q8—21no—
8#6467⋊C3C2211yescharacteristic
9#6464⋊C3—21no—
10ΓL(1,𝔽8)×#106—21no—
11ΓL(1,𝔽8)×#106C2212yes—
12C4×C8×ΓL(1,𝔽8)C2211yescharacteristic
13#6393⋊C3C2211yescharacteristic
14#6386⋊C3C2212yes—
15#6379⋊C3—21no—
16C7×#498C2×C311yescharacteristic, complemented
17#89472C2×C311yescharacteristic
18#89283C2×C312yes—
19#89282C2×C312yescomplemented
20#89258C2×C311yescharacteristic, complemented
21C3×#2230—71nomaximal, Hall
22C2×Q8×ΓL(1,𝔽8)—41no—
23#1157⋊C3—21no—
24C2×C8×ΓL(1,𝔽8)D411yescharacteristic
25C2×C8×ΓL(1,𝔽8)C2311yescharacteristic
26C4⋊C4×ΓL(1,𝔽8)—22no—
27C4⋊C4×ΓL(1,𝔽8)—41no—
28C42×ΓL(1,𝔽8)D411yescharacteristic
29#1136⋊C3—21no—
30Dic28⋊C3Q811yescharacteristic, complemented
31D56⋊C3Q811yescharacteristic, complemented
32#1132⋊C3—21no—
33#1131⋊C3—41no—
34C8×AGL(1,𝔽7)—41no—
35ΓL(1,𝔽8)⋊C16—22no—
36C7×C4⋊Q8—21no—
37#6467C22×C311yescharacteristic
38#6464—21no—
39C7×#106—21no—
40C7×#106C22×C312yes—
41C4×C7×C8C22×C311yescharacteristic
42#6393C22×C311yescharacteristic
43#6386C22×C312yes—
44#6379—21no—
45C3×#498—71no—
46C3×#443—71no—
47C3×#365—72no—
48C3×#364—71no—
49C3×#360—72no—
50Q8×ΓL(1,𝔽8)—82no—
51C2×C4×ΓL(1,𝔽8)—21no—
52C2×C4×ΓL(1,𝔽8)—42no—
53C2×C4×ΓL(1,𝔽8)C2×D411yescharacteristic
54(C22⋊D7)⋊C3—41no—
55C4⋊AGL(1,𝔽7)—21no—
56C4×AGL(1,𝔽7)—41no—
57ΓL(1,𝔽8)⋊Q8—21no—
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
64C2×C7×Q8—41no—
65#1157—21no—
66C2×C7×C8C3×D411yescharacteristic
67C2×C7×C8C23×C311yescharacteristic
68C7×C4⋊C4—22no—
69C7×C4⋊C4—41no—
70C42×C7C3×D411yescharacteristic
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)—82no—
91C4×ΓL(1,𝔽8)C22⋊Q811yescharacteristic
92C4×ΓL(1,𝔽8)C2×SD1611yescharacteristic
93ΓL(1,𝔽8)⋊C4—41no—
94#498AGL(1,𝔽7)11yescharacteristic, complemented
95#443—71no—
96#365—72no—
97#364—71no—
98#360—72no—
99C7×Q8—82no—
100C2×C4×C7—21no—
101C2×C4×C7—42no—
102C2×C4×C7C2×C3×D411yescharacteristic
103C22⋊D7—41no—
104D28—21no—
105C4×D7—41no—
106Dic14—21no—
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—21no—
132M32—72no—
133#108—71no—
134#106—21no—
135#106C2×AGL(1,𝔽7)12yes—
136#104—141no—
137C4×C8C2×AGL(1,𝔽7)11yescharacteristic
138C22×C7C3×AGL(1,ℤ/8)11yescharacteristic
139C2×D7—41no—
140C4×C7—22no—
141C4×C7—82no—
142C4×C7C3×C22⋊Q811yescharacteristic
143C4×C7C2×C3×SD1611yescharacteristic
144Dic7—41no—
145C3×Q8—141no—
146C3×Q8—562no—
147C3×D4—141no—
148C3×D4—281no—
149C2×C3×C4—71no—
150C2×C3×C4—141no—
151C2×C3×C4—282no—
152C2×C3×C4—281no—
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—41no—
161Q16—71no—
162SD16—141no—
163D8—71no—
164M16—141no—
165M16—281no—
166C2×C8—281no—
167C2×C8(C22⋊D7)⋊C311yescharacteristic
168C2×C8C22×AGL(1,𝔽7)11yescharacteristic
169C4⋊C4—22no—
170C4⋊C4—41no—
171C42(C22⋊D7)⋊C311yescharacteristic
172C16—142no—
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—82no—
184Q8—141no—
185D4—141no—
186D4—281no—
187C2×C4—21no—
188C2×C4—42no—
189C2×C4—281no—
190C2×C4C2×(C22⋊D7)⋊C311yescharacteristic
191C8—281no—
192C8#1143⋊C311yescharacteristic
193C8#1143⋊C311yescharacteristic
194C8Q8×AGL(1,𝔽7)11yescharacteristic
195C8#6665911yescharacteristic
196C7C3×#223011yescharacteristic, complemented, Sylow
197C2×C3—71no—
198C2×C3—141no—
199C2×C3—561no—
200C22—281no—
201C22#6495⋊C311yescharacteristic
202C4—22no—
203C4—82no—
204C4—281no—
205C4C2×#1143⋊C311yescharacteristic
206C4#6509⋊C311yescharacteristic
207C3—71noSylow
208C2—21no—
209C2—561no—
210C2#89856⋊C311yescharacteristic
subgroupG/H
centerC2#89856⋊C3
commutator, cosocleC2×C7×C8C23×C3
FrattiniC2×C8C22×AGL(1,𝔽7)
FittingC7×#498C2×C3
socleC2×C7C3×#476
prime radical#6467⋊C3C22
hypercenter#498AGL(1,𝔽7)
hypocenterC7C3×#2230
LCSG ▷ C2×C7×C8 ▷ C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C2 ◁ C2×C4 ◁ C4×C8 ◁ #498

Complements

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

NH ≅ G/Ntype
1. ΓL(1,𝔽8)×#498C2semidirect
4. #89282⋊C3C2semidirect
5. #89258⋊C3C2semidirect
6. #11729104C3semidirect
16. C7×#498C2×C3semidirect
19. #89282C2×C3semidirect
20. #89258C2×C3semidirect
30. Dic28⋊C3Q8semidirect
31. D56⋊C3Q8semidirect
72. Dic28C3×Q8semidirect
73. D56C3×Q8semidirect
94. #498AGL(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
ΓL(1,𝔽8)×#4982
#89472⋊C32
#89283⋊C32
#89282⋊C32
#89258⋊C32
#117291043
C3×#22307