Finite Group Databasebeta

← #89506⋊C3 #13051634

Subgroups — #89506⋊C3 #13051634

Order 1344.

324 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 324 conjugacy classes of subgroups — too many to draw. Only its 72 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
1C2×#1157⋊C3C211yesmaximal, characteristic, complemented
2#6493⋊C3C211yesmaximal, characteristic, complemented
3C2×D56⋊C3C211yesmaximal, characteristic, complemented
4C2×#1132⋊C3C211yesmaximal, characteristic, complemented
5ΓL(1,𝔽8)×C22⋊C8C211yesmaximal, characteristic, complemented
6#6395⋊C3C211yesmaximal, characteristic, complemented
7#6390⋊C3C211yesmaximal, characteristic, complemented
8#89506C311yesmaximal, characteristic, complemented, Hall
9C22×C4×ΓL(1,𝔽8)C2211yescharacteristic
10C2×(C22⋊D7)⋊C3—21no—
11C2×(C22⋊D7)⋊C3—21no—
12#1157⋊C3—22no—
13#1157⋊C3—22no—
14C2×C4⋊AGL(1,𝔽7)C2211yescharacteristic
15C2×C4⋊AGL(1,𝔽7)C2211yescharacteristic
16C2×C4×AGL(1,𝔽7)—21no—
17C2×ΓL(1,𝔽8)⋊Q8C2211yescharacteristic
18C2×C8×ΓL(1,𝔽8)C2211yescharacteristic
19C2×C8×ΓL(1,𝔽8)C2211yescharacteristic
20(C2×AGL(1,𝔽7))⋊C4—21no—
21C4⋊(ΓL(1,𝔽8)⋊C4)C2211yescharacteristic
22D56⋊C3—22no—
23#1132⋊C3—22no—
24C2×#1157C2×C311yescharacteristic, complemented
25#6493C2×C311yescharacteristic, complemented
26C2×D56C2×C311yescharacteristic, complemented
27C2×#1132C2×C311yescharacteristic, complemented
28C7×C22⋊C8C2×C311yescharacteristic, complemented
29#6395C2×C311yescharacteristic, complemented
30#6390C2×C311yescharacteristic, complemented
31C3×#449—71nomaximal, Hall
32C23×ΓL(1,𝔽8)D411yescharacteristic
33C22×AGL(1,𝔽7)—21no—
34C22×AGL(1,𝔽7)—21no—
35C2×C4×ΓL(1,𝔽8)—21no—
36C2×C4×ΓL(1,𝔽8)—21no—
37C2×C4×ΓL(1,𝔽8)D411yescharacteristic
38C2×C4×ΓL(1,𝔽8)C2311yescharacteristic
39(C22⋊D7)⋊C3—24no—
40(C22⋊D7)⋊C3—42no—
41C2×ΓL(1,𝔽8)⋊C4—21no—
42C2×ΓL(1,𝔽8)⋊C4—21no—
43C4⋊AGL(1,𝔽7)—21no—
44C4⋊AGL(1,𝔽7)—21no—
45C4⋊AGL(1,𝔽7)—21no—
46C4⋊AGL(1,𝔽7)D412yescomplemented
47C4×AGL(1,𝔽7)—24no—
48ΓL(1,𝔽8)⋊Q8—21no—
49ΓL(1,𝔽8)⋊Q8D412yescomplemented
50C8×ΓL(1,𝔽8)—21no—
51C8×ΓL(1,𝔽8)—21no—
52C22×C4×C7C22×C311yescharacteristic
53C2×C22⋊D7—21no—
54C2×C22⋊D7—21no—
55#1157—22no—
56#1157—22no—
57C2×D28C22×C311yescharacteristic
58C2×D28C22×C311yescharacteristic
59C2×C4×D7—21no—
60C2×Dic14C22×C311yescharacteristic
61C2×C7×C8C22×C311yescharacteristic
62C2×C7×C8C22×C311yescharacteristic
63(C2×D7)⋊C4—21no—
64C4⋊Dic7C22×C311yescharacteristic
65D56—22no—
66#1132—22no—
67C2×C3×#41—71no—
68C2×C3×SD16—71no—
69C2×C3×D8—71no—
70C3×#121—71no—
71C3×#103—71no—
72C3×#102—71no—
73C3×C22⋊C8—71no—
74C22×ΓL(1,𝔽8)—21no—
75C22×ΓL(1,𝔽8)—21no—
76C22×ΓL(1,𝔽8)—21no—
77C22×ΓL(1,𝔽8)C2×D411yescharacteristic
78C2×AGL(1,𝔽7)—22no—
79C2×AGL(1,𝔽7)—41no—
80C2×AGL(1,𝔽7)—41no—
81C2×AGL(1,𝔽7)—41no—
82C2×AGL(1,𝔽7)—41no—
83C2×AGL(1,𝔽7)—41no—
84C4×ΓL(1,𝔽8)—21no—
85C4×ΓL(1,𝔽8)C2×D411yescharacteristic
86C4×ΓL(1,𝔽8)C2×D411yescharacteristic
87ΓL(1,𝔽8)⋊C4—22no—
88ΓL(1,𝔽8)⋊C4—41no—
89#449—71noSylow
90C23×C7C3×D411yescharacteristic
91C22×D7—21no—
92C22×D7—21no—
93C2×C4×C7—21no—
94C2×C4×C7—21no—
95C2×C4×C7C3×D411yescharacteristic
96C2×C4×C7C23×C311yescharacteristic
97C22⋊D7—24no—
98C22⋊D7—42no—
99C2×Dic7—21no—
100C2×Dic7—21no—
101D28—21no—
102D28—21no—
103D28—21no—
104D28C3×D412yescomplemented
105C4×D7—24no—
106Dic14—21no—
107Dic14C3×D412yescomplemented
108C7×C8—21no—
109C7×C8—21no—
110C3×#41—142no—
111C3×#41—142no—
112C2×C3×Q8—71no—
113C2×C3×D4—71no—
114C2×C3×D4—71no—
115C2×C3×D4—141no—
116C2×C3×D4—141no—
117C22×C3×C4—71no—
118C22×C3×C4—141no—
119C3×SD16—142no—
120C3×D8—142no—
121C2×C3×C8—71no—
122C2×C3×C8—71no—
123C3×C4⋊C4—71no—
124C3×C22⋊C4—141no—
125C2×ΓL(1,𝔽8)—41no—
126C2×ΓL(1,𝔽8)#12011yescharacteristic
127C2×ΓL(1,𝔽8)#13511yescharacteristic
128C2×ΓL(1,𝔽8)AGL(1,ℤ/8)11yescharacteristic
129AGL(1,𝔽7)—42no—
130AGL(1,𝔽7)—81no—
131C2×#41—71no—
132C2×SD16—71no—
133C2×D8—71no—
134#121—71no—
135#103—71no—
136#102—71no—
137C22⋊C8AGL(1,𝔽7)11yescharacteristic, complemented
138C22×C7—21no—
139C22×C7—21no—
140C22×C7—21no—
141C22×C7C2×C3×D411yescharacteristic
142C2×D7—22no—
143C2×D7—41no—
144C2×D7—41no—
145C2×D7—41no—
146C2×D7—41no—
147C2×D7—41no—
148C4×C7—21no—
149C4×C7C2×C3×D411yescharacteristic
150C4×C7C2×C3×D411yescharacteristic
151Dic7—22no—
152Dic7—41no—
153C23×C3—71no—
154C23×C3—141no—
155C23×C3—141no—
156C3×Q8—72no—
157C3×Q8—141no—
158C3×D4—72no—
159C3×D4—141no—
160C3×D4—141no—
161C3×D4—141no—
162C3×D4—144no—
163C3×D4—282no—
164C2×C3×C4—71no—
165C2×C3×C4—71no—
166C2×C3×C4—141no—
167C2×C3×C4—141no—
168C2×C3×C4—141no—
169C2×C3×C4—144no—
170C2×C3×C4—141no—
171C3×C8—141no—
172C3×C8—141no—
173ΓL(1,𝔽8)#44911yescharacteristic, complemented, Hall
174#41—142no—
175#41—142no—
176C2×Q8—71no—
177C2×D4—71no—
178C2×D4—71no—
179C2×D4—141no—
180C2×D4—141no—
181C22×C4—141no—
182C22×C4C2×AGL(1,𝔽7)11yescharacteristic
183SD16—142no—
184D8—142no—
185C2×C8C2×AGL(1,𝔽7)11yescharacteristic
186C2×C8C2×AGL(1,𝔽7)11yescharacteristic
187C4⋊C4—71no—
188C22⋊C4—141no—
189C2×C7—41no—
190C2×C7C3×#12011yescharacteristic
191C2×C7C3×#13511yescharacteristic
192C2×C7C3×AGL(1,ℤ/8)11yescharacteristic
193D7—42no—
194D7—81no—
195C22×C3—71no—
196C22×C3—141no—
197C22×C3—141no—
198C22×C3—141no—
199C22×C3—142no—
200C22×C3—281no—
201C22×C3—281no—
202C22×C3—281no—
203C22×C3—281no—
204C22×C3—281no—
205C3×C4—71no—
206C3×C4—71no—
207C3×C4—141no—
208C3×C4—142no—
209C3×C4—281no—
210C23—141no—
211C23—141no—
212C23C4⋊AGL(1,𝔽7)11yescharacteristic
213Q8—72no—
214Q8—141no—
215D4—72no—
216D4—144no—
217D4—141no—
218D4—141no—
219D4—141no—
220D4—282no—
221C2×C4—21no—
222C2×C4—21no—
223C2×C4—144no—
224C2×C4—141no—
225C2×C4—141no—
226C2×C4C4⋊AGL(1,𝔽7)11yescharacteristic
227C2×C4C22×AGL(1,𝔽7)11yescharacteristic
228C8—21no—
229C8—21no—
230C7C3×#44911yescharacteristic, complemented, Sylow
231C2×C3—71no—
232C2×C3—71no—
233C2×C3—71no—
234C2×C3—281no—
235C2×C3—282no—
236C2×C3—561no—
237C22—21no—
238C22—21no—
239C22—21no—
240C22—142no—
241C22—281no—
242C22—281no—
243C22—281no—
244C22—281no—
245C22—281no—
246C22C2×C4⋊AGL(1,𝔽7)11yescharacteristic
247C4—21no—
248C4—142no—
249C4—281no—
250C4D4×AGL(1,𝔽7)11yescharacteristic
251C4D4×AGL(1,𝔽7)11yescharacteristic
252C3—71noSylow
253C2—41no—
254C2—282no—
255C2—561no—
256C2#6445⋊C311yescharacteristic
257C2#6467⋊C311yescharacteristic
258C2#6471⋊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
hypercenterC22⋊C8AGL(1,𝔽7)
hypocenterC7C3×#449
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
1. C2×#1157⋊C3C2semidirect
2. #6493⋊C3C2semidirect
3. C2×D56⋊C3C2semidirect
4. C2×#1132⋊C3C2semidirect
5. ΓL(1,𝔽8)×C22⋊C8C2semidirect
6. #6395⋊C3C2semidirect
7. #6390⋊C3C2semidirect
8. #89506C3semidirect
24. C2×#1157C2×C3semidirect
25. #6493C2×C3semidirect
26. C2×D56C2×C3semidirect
27. C2×#1132C2×C3semidirect
28. C7×C22⋊C8C2×C3semidirect
29. #6395C2×C3semidirect
30. #6390C2×C3semidirect
46. C4⋊AGL(1,𝔽7)D4semidirect
49. ΓL(1,𝔽8)⋊Q8D4semidirect
104. D28C3×D4semidirect
107. Dic14C3×D4semidirect
137. C22⋊C8AGL(1,𝔽7)semidirect
173. ΓL(1,𝔽8)#449semidirect
230. C7C3×#449semidirect

Maximal subgroups

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

subgroupindex
C2×#1157⋊C32
#6493⋊C32
C2×D56⋊C32
C2×#1132⋊C32
ΓL(1,𝔽8)×C22⋊C82
#6395⋊C32
#6390⋊C32
#895063
C3×#4497