Finite Group Databasebeta

← #72229702

Subgroups — #72229702

Order 2688.

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

normal subgroups only

This group has 340 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#89680⋊C3C212yesmaximal, complemented
2#89472⋊C3C211yesmaximal, characteristic, complemented
3#89467⋊C3C211yesmaximal, characteristic, complemented
4ΓL(1,𝔽8)×#364C211yesmaximal, characteristic, complemented
5#89311⋊C3C212yesmaximal, complemented
6#11727059C311yesmaximal, characteristic, complemented, Hall
7#6467⋊C3C2211yescharacteristic
8C2×D56⋊C3—21no—
9C2×D56⋊C3C2212yes—
10#6464⋊C3—21no—
11#10620710—21no—
12ΓL(1,𝔽8)×M32C2212yes—
13C4×C8×ΓL(1,𝔽8)C2211yescharacteristic
14#6393⋊C3C2211yescharacteristic
15#6379⋊C3—21no—
16#6374⋊C3—22no—
17D112⋊C3—22no—
18#89680C2×C312yescomplemented
19#89472C2×C311yescharacteristic, complemented
20#89467C2×C311yescharacteristic, complemented
21C7×#364C2×C311yescharacteristic, complemented
22#89311C2×C312yescomplemented
23C3×#2213—71nomaximal, Hall
24#1157⋊C3—21no—
25C2×C4⋊AGL(1,𝔽7)—22no—
26C2×C4⋊AGL(1,𝔽7)—42no—
27C2×C8×ΓL(1,𝔽8)D411yescharacteristic
28C2×C8×ΓL(1,𝔽8)C2311yescharacteristic
29C42×ΓL(1,𝔽8)D411yescharacteristic
30#1136⋊C3—21no—
31Dic28⋊C3D411yescharacteristic, complemented
32D56⋊C3—22no—
33D56⋊C3—22no—
34D56⋊C3—41no—
35D56⋊C3—41no—
36D56⋊C3D411yescharacteristic, complemented
37#1132⋊C3—21no—
38#1131⋊C3—41no—
39C8×AGL(1,𝔽7)—41no—
40C16×ΓL(1,𝔽8)—22no—
41#6467C22×C311yescharacteristic
42C2×D56—21no—
43C2×D56C22×C312yes—
44#6464—21no—
45#6437—21no—
46C7×M32C22×C312yes—
47C4×C7×C8C22×C311yescharacteristic
48#6393C22×C311yescharacteristic
49#6379—21no—
50#6374—22no—
51D112—22no—
52C3×#509—72no—
53C3×#493—71no—
54C3×#443—71no—
55C3×#364—71no—
56C3×#361—72no—
57C22×AGL(1,𝔽7)—42no—
58C2×C4×ΓL(1,𝔽8)—21no—
59C2×C4×ΓL(1,𝔽8)C2×D411yescharacteristic
60(C22⋊D7)⋊C3—41no—
61C4⋊AGL(1,𝔽7)—21no—
62C4⋊AGL(1,𝔽7)—42no—
63C4⋊AGL(1,𝔽7)—42no—
64C4⋊AGL(1,𝔽7)—84no—
65C4×AGL(1,𝔽7)—41no—
66ΓL(1,𝔽8)⋊Q8—21no—
67C8×ΓL(1,𝔽8)D811yescharacteristic
68C8×ΓL(1,𝔽8)D811yescharacteristic
69C8×ΓL(1,𝔽8)C2×D411yescharacteristic
70C8×ΓL(1,𝔽8)#4111yescharacteristic
71ΓL(1,𝔽8)⋊C8—41no—
72#2213—71noSylow
73#1157—21no—
74C2×D28—22no—
75C2×D28—42no—
76C2×C7×C8C3×D411yescharacteristic
77C2×C7×C8C23×C311yescharacteristic
78C42×C7C3×D411yescharacteristic
79#1136—21no—
80Dic28C3×D411yescharacteristic, complemented
81D56—22no—
82D56—22no—
83D56—41no—
84D56—41no—
85D56C3×D411yescharacteristic, complemented
86#1132—21no—
87#1131—41no—
88C8×D7—41no—
89C7×C16—22no—
90C3×#135—71no—
91C2×C3×D8—72no—
92C2×C3×D8—141no—
93C3×#131—141no—
94C3×#127—141no—
95C3×SD32—142no—
96C3×D16—142no—
97C3×M32—72no—
98C3×#108—71no—
99C3×#104—141no—
100C3×C4×C8—71no—
101C22×ΓL(1,𝔽8)AGL(1,ℤ/8)11yescharacteristic
102C2×AGL(1,𝔽7)—41no—
103C2×AGL(1,𝔽7)—84no—
104C2×AGL(1,𝔽7)—82no—
105C4×ΓL(1,𝔽8)—22no—
106C4×ΓL(1,𝔽8)#12111yescharacteristic
107C4×ΓL(1,𝔽8)C2×D811yescharacteristic
108ΓL(1,𝔽8)⋊C4—41no—
109#509—72no—
110#493—71no—
111#443—71no—
112#364AGL(1,𝔽7)11yescharacteristic, complemented
113#361—72no—
114C22×D7—42no—
115C2×C4×C7—21no—
116C2×C4×C7C2×C3×D411yescharacteristic
117C22⋊D7—41no—
118D28—21no—
119D28—42no—
120D28—42no—
121D28—84no—
122C4×D7—41no—
123Dic14—21no—
124C7×C8C3×D811yescharacteristic
125C7×C8C3×D811yescharacteristic
126C7×C8C2×C3×D411yescharacteristic
127C7×C8C3×#4111yescharacteristic
128C7⋊C8—41no—
129C3×#41—141no—
130C2×C3×D4—142no—
131C2×C3×D4—282no—
132C3×Q16—71no—
133C3×SD16—141no—
134C3×D8—71no—
135C3×D8—142no—
136C3×D8—142no—
137C3×D8—281no—
138C3×D8—281no—
139C3×M16—141no—
140C3×M16—281no—
141C2×C3×C8—71no—
142C2×C3×C8—71no—
143C2×C3×C8—281no—
144C3×C42—71no—
145C3×C16—142no—
146C2×ΓL(1,𝔽8)—21no—
147C2×ΓL(1,𝔽8)#45911yescharacteristic
148AGL(1,𝔽7)—81no—
149AGL(1,𝔽7)—162no—
150#135—71no—
151C2×D8—72no—
152C2×D8—141no—
153#131—141no—
154#127—141no—
155SD32—142no—
156D16—142no—
157M32C2×AGL(1,𝔽7)12yes—
158#108—71no—
159#104—141no—
160C4×C8C2×AGL(1,𝔽7)11yescharacteristic
161C22×C7C3×AGL(1,ℤ/8)11yescharacteristic
162C2×D7—41no—
163C2×D7—84no—
164C2×D7—82no—
165C4×C7—22no—
166C4×C7C3×#12111yescharacteristic
167C4×C7C2×C3×D811yescharacteristic
168Dic7—41no—
169C23×C3—282no—
170C3×Q8—141no—
171C3×D4—141no—
172C3×D4—282no—
173C3×D4—282no—
174C3×D4—281no—
175C3×D4—564no—
176C2×C3×C4—71no—
177C2×C3×C4—141no—
178C2×C3×C4—281no—
179C3×C8—71no—
180C3×C8—71no—
181C3×C8—71no—
182C3×C8—71no—
183C3×C8—281no—
184ΓL(1,𝔽8)#221311yescharacteristic, complemented, Hall
185#41—141no—
186C2×D4—142no—
187C2×D4—282no—
188Q16—71no—
189SD16—141no—
190D8—71no—
191D8—142no—
192D8—142no—
193D8—281no—
194D8—281no—
195M16—141no—
196M16—281no—
197C2×C8—281no—
198C2×C8C4⋊AGL(1,𝔽7)11yescharacteristic
199C2×C8C22×AGL(1,𝔽7)11yescharacteristic
200C42C4⋊AGL(1,𝔽7)11yescharacteristic
201C16—22no—
202C2×C7—21no—
203C2×C7C3×#45911yescharacteristic
204D7—81no—
205D7—162no—
206C22×C3—71no—
207C22×C3—281no—
208C22×C3—562no—
209C22×C3—564no—
210C3×C4—71no—
211C3×C4—71no—
212C3×C4—142no—
213C3×C4—281no—
214C23—282no—
215Q8—141no—
216D4—141no—
217D4—281no—
218D4—282no—
219D4—282no—
220D4—564no—
221C2×C4—21no—
222C2×C4—281no—
223C2×C4C2×C4⋊AGL(1,𝔽7)11yescharacteristic
224C8—281no—
225C8D56⋊C311yescharacteristic
226C8D56⋊C311yescharacteristic
227C8D4×AGL(1,𝔽7)11yescharacteristic
228C8#6665911yescharacteristic
229C7C3×#221311yescharacteristic, complemented, Sylow
230C2×C3—71no—
231C2×C3—141no—
232C2×C3—561no—
233C2×C3—1122no—
234C22—281no—
235C22—564no—
236C22—562no—
237C22#6471⋊C311yescharacteristic
238C4—22no—
239C4—281no—
240C4#6458⋊C311yescharacteristic
241C4C2×D56⋊C311yescharacteristic
242C3—71noSylow
243C2—21no—
244C2—561no—
245C2—1122no—
246C2#89615⋊C311yescharacteristic
subgroupG/H
centerC2#89615⋊C3
commutator, cosocleC2×C7×C8C23×C3
FrattiniC2×C8C22×AGL(1,𝔽7)
FittingC7×#364C2×C3
socleC2×C7C3×#459
hypercenter#364AGL(1,𝔽7)
hypocenterC7C3×#2213
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
1. #89680⋊C3C2semidirect
2. #89472⋊C3C2semidirect
3. #89467⋊C3C2semidirect
4. ΓL(1,𝔽8)×#364C2semidirect
5. #89311⋊C3C2semidirect
6. #11727059C3semidirect
18. #89680C2×C3semidirect
19. #89472C2×C3semidirect
20. #89467C2×C3semidirect
21. C7×#364C2×C3semidirect
22. #89311C2×C3semidirect
31. Dic28⋊C3D4semidirect
36. D56⋊C3D4semidirect
80. Dic28C3×D4semidirect
85. D56C3×D4semidirect
112. #364AGL(1,𝔽7)semidirect
184. ΓL(1,𝔽8)#2213semidirect
229. C7C3×#2213semidirect

Maximal subgroups

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

subgroupindex
#89680⋊C32
#89472⋊C32
#89467⋊C32
ΓL(1,𝔽8)×#3642
#89311⋊C32
#117270593
C3×#22137