Finite Group Databasebeta

← #89928⋊C3 #13052056

Subgroups — #89928⋊C3 #13052056

Order 1344.

396 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 396 conjugacy classes of subgroups — too many to draw. Only its 76 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×D4×AGL(1,𝔽7)C211yesmaximal, characteristic, complemented
2C2×D8×ΓL(1,𝔽8)C211yesmaximal, characteristic, complemented
3#6501⋊C3C211yesmaximal, characteristic, complemented
4C2×#1141⋊C3C211yesmaximal, characteristic, complemented
5#6406⋊C3C211yesmaximal, characteristic, complemented
6#6395⋊C3C211yesmaximal, characteristic, complemented
7(C2×AGL(1,𝔽7))⋊C8C211yesmaximal, characteristic, complemented
8#89928C311yesmaximal, characteristic, complemented, Hall
9C23×AGL(1,𝔽7)—21no—
10C2×D4×ΓL(1,𝔽8)C2211yescharacteristic
11C2×D4×ΓL(1,𝔽8)C2211yescharacteristic, complemented
12C2×(C22⋊D7)⋊C3—21no—
13C2×(C22⋊D7)⋊C3—21no—
14D4×AGL(1,𝔽7)—22no—
15D4×AGL(1,𝔽7)—22no—
16C2×C4⋊AGL(1,𝔽7)C2211yescharacteristic
17C2×C4×AGL(1,𝔽7)C2211yescharacteristic
18D8×ΓL(1,𝔽8)—22no—
19C2×C8×ΓL(1,𝔽8)C2211yescharacteristic, complemented
20(C22⋊Dic7)⋊C3—21no—
21#1141⋊C3—22no—
22C4⋊(ΓL(1,𝔽8)⋊C4)C2211yescharacteristic, complemented
23C2×ΓL(1,𝔽8)⋊C8C2211yescharacteristic, complemented
24C2×D4×D7C2×C311yescharacteristic, complemented
25C2×C7×D8C2×C311yescharacteristic, complemented
26#6501C2×C311yescharacteristic, complemented
27C2×#1141C2×C311yescharacteristic, complemented
28#6406C2×C311yescharacteristic, complemented
29#6395C2×C311yescharacteristic, complemented
30(C2×D7)⋊C8C2×C311yescharacteristic, complemented
31C3×#447—71nomaximal, Hall
32C23×ΓL(1,𝔽8)—21no—
33C23×ΓL(1,𝔽8)—21no—
34C22×AGL(1,𝔽7)—24no—
35C22×AGL(1,𝔽7)—21no—
36C22×AGL(1,𝔽7)—42no—
37C22×AGL(1,𝔽7)—42no—
38C22×AGL(1,𝔽7)D411yescharacteristic
39D4×ΓL(1,𝔽8)—21no—
40D4×ΓL(1,𝔽8)—21no—
41D4×ΓL(1,𝔽8)—21no—
42D4×ΓL(1,𝔽8)D412yescomplemented
43C2×C4×ΓL(1,𝔽8)C2311yescharacteristic
44(C22⋊D7)⋊C3—24no—
45(C22⋊D7)⋊C3—42no—
46C2×ΓL(1,𝔽8)⋊C4—21no—
47C2×ΓL(1,𝔽8)⋊C4D411yescharacteristic
48C4⋊AGL(1,𝔽7)—21no—
49C4⋊AGL(1,𝔽7)D412yescomplemented
50C4×AGL(1,𝔽7)—21no—
51C4×AGL(1,𝔽7)—21no—
52C8×ΓL(1,𝔽8)—21no—
53ΓL(1,𝔽8)⋊C8—21no—
54C23×D7—21no—
55C2×C7×D4C22×C311yescharacteristic
56C2×C7×D4C22×C311yescharacteristic, complemented
57C2×C22⋊D7—21no—
58C2×C22⋊D7—21no—
59D4×D7—22no—
60D4×D7—22no—
61C2×D28C22×C311yescharacteristic
62C2×C4×D7C22×C311yescharacteristic
63C7×D8—22no—
64C2×C7×C8C22×C311yescharacteristic, complemented
65C22⋊Dic7—21no—
66#1141—22no—
67C4⋊Dic7C22×C311yescharacteristic, complemented
68C2×C7⋊C8C22×C311yescharacteristic, complemented
69C22×C3×D4—71no—
70C2×C3×D8—71no—
71C2×C3×D8—71no—
72C3×#121—71no—
73C3×#102—71no—
74C3×#102—71no—
75C3×C22⋊C8—71no—
76C22×ΓL(1,𝔽8)—22no—
77C22×ΓL(1,𝔽8)—41no—
78C22×ΓL(1,𝔽8)—41no—
79C22×ΓL(1,𝔽8)—41no—
80C22×ΓL(1,𝔽8)—41no—
81C22×ΓL(1,𝔽8)—41no—
82C22×ΓL(1,𝔽8)C2×D411yescharacteristic
83C2×AGL(1,𝔽7)—21no—
84C2×AGL(1,𝔽7)—21no—
85C2×AGL(1,𝔽7)—22no—
86C2×AGL(1,𝔽7)—48no—
87C2×AGL(1,𝔽7)—41no—
88C2×AGL(1,𝔽7)—41no—
89C2×AGL(1,𝔽7)D812yescomplemented
90C4×ΓL(1,𝔽8)C2×D411yescharacteristic
91C4×ΓL(1,𝔽8)C2×D411yescharacteristic
92ΓL(1,𝔽8)⋊C4—21no—
93ΓL(1,𝔽8)⋊C4—41no—
94#447—71noSylow
95C23×C7—21no—
96C23×C7—21no—
97C22×D7—24no—
98C22×D7—21no—
99C22×D7—42no—
100C22×D7—42no—
101C22×D7C3×D411yescharacteristic
102C7×D4—21no—
103C7×D4—21no—
104C7×D4—21no—
105C7×D4C3×D412yescomplemented
106C2×C4×C7C23×C311yescharacteristic
107C22⋊D7—24no—
108C22⋊D7—42no—
109C2×Dic7—21no—
110C2×Dic7C3×D411yescharacteristic
111D28—21no—
112D28C3×D412yescomplemented
113C4×D7—21no—
114C4×D7—21no—
115C7×C8—21no—
116C7⋊C8—21no—
117C24×C3—141no—
118C2×C3×D4—71no—
119C2×C3×D4—71no—
120C2×C3×D4—71no—
121C2×C3×D4—141no—
122C2×C3×D4—142no—
123C2×C3×D4—142no—
124C2×C3×D4—141no—
125C22×C3×C4—71no—
126C3×D8—142no—
127C3×D8—142no—
128C2×C3×C8—71no—
129C2×C3×C8—71no—
130C3×C4⋊C4—71no—
131C3×C22⋊C4—141no—
132C2×ΓL(1,𝔽8)—42no—
133C2×ΓL(1,𝔽8)—81no—
134C2×ΓL(1,𝔽8)#12011yescharacteristic
135C2×ΓL(1,𝔽8)C2×D811yescharacteristic
136C2×ΓL(1,𝔽8)AGL(1,ℤ/8)11yescharacteristic
137AGL(1,𝔽7)—22no—
138AGL(1,𝔽7)—42no—
139C22×D4—71no—
140C2×D8—71no—
141C2×D8AGL(1,𝔽7)11yescharacteristic, complemented
142#121—71no—
143#102—71no—
144#102—71no—
145C22⋊C8—71no—
146C22×C7—22no—
147C22×C7—41no—
148C22×C7—41no—
149C22×C7—41no—
150C22×C7—41no—
151C22×C7—41no—
152C22×C7C2×C3×D411yescharacteristic
153C2×D7—21no—
154C2×D7—21no—
155C2×D7—22no—
156C2×D7—48no—
157C2×D7—41no—
158C2×D7—41no—
159C2×D7C3×D812yescomplemented
160C4×C7C2×C3×D411yescharacteristic
161C4×C7C2×C3×D411yescharacteristic
162Dic7—21no—
163Dic7—41no—
164C23×C3—71no—
165C23×C3—141no—
166C23×C3—141no—
167C23×C3—141no—
168C23×C3—144no—
169C23×C3—282no—
170C23×C3—282no—
171C3×D4—72no—
172C3×D4—72no—
173C3×D4—141no—
174C3×D4—141no—
175C3×D4—141no—
176C3×D4—141no—
177C3×D4—144no—
178C3×D4—282no—
179C2×C3×C4—71no—
180C2×C3×C4—71no—
181C2×C3×C4—141no—
182C2×C3×C4—141no—
183C2×C3×C4—141no—
184C3×C8—141no—
185C3×C8—141no—
186ΓL(1,𝔽8)#44711yescharacteristic, complemented, Hall
187C24—141no—
188C2×D4—71no—
189C2×D4—141no—
190C2×D4—142no—
191C2×D4—142no—
192C2×D4—141no—
193C2×D4C2×AGL(1,𝔽7)11yescharacteristic
194C2×D4C2×AGL(1,𝔽7)11yescharacteristic, complemented
195C22×C4—71no—
196D8—22no—
197D8—142no—
198C2×C8—71no—
199C2×C8C2×AGL(1,𝔽7)11yescharacteristic, complemented
200C4⋊C4—71no—
201C22⋊C4—141no—
202C2×C7—42no—
203C2×C7—81no—
204C2×C7C3×#12011yescharacteristic
205C2×C7C2×C3×D811yescharacteristic
206C2×C7C3×AGL(1,ℤ/8)11yescharacteristic
207D7—22no—
208D7—42no—
209C22×C3—72no—
210C22×C3—71no—
211C22×C3—142no—
212C22×C3—141no—
213C22×C3—141no—
214C22×C3—142no—
215C22×C3—281no—
216C22×C3—281no—
217C22×C3—281no—
218C22×C3—281no—
219C22×C3—281no—
220C22×C3—281no—
221C22×C3—281no—
222C22×C3—288no—
223C3×C4—71no—
224C3×C4—71no—
225C3×C4—141no—
226C3×C4—281no—
227C23—21no—
228C23—21no—
229C23—71no—
230C23—144no—
231C23—141no—
232C23—282no—
233C23—282no—
234D4—21no—
235D4—21no—
236D4—21no—
237D4—72no—
238D4—144no—
239D4—141no—
240D4—282no—
241D4(C22⋊D7)⋊C312yescomplemented
242C2×C4—71no—
243C2×C4—141no—
244C2×C4—141no—
245C2×C4—141no—
246C2×C4C22×AGL(1,𝔽7)11yescharacteristic
247C8—21no—
248C8—141no—
249C7C3×#44711yescharacteristic, complemented, Sylow
250C2×C3—71no—
251C2×C3—71no—
252C2×C3—71no—
253C2×C3—142no—
254C2×C3—282no—
255C2×C3—282no—
256C2×C3—561no—
257C22—22no—
258C22—41no—
259C22—41no—
260C22—41no—
261C22—41no—
262C22—41no—
263C22—72no—
264C22—141no—
265C22—141no—
266C22—142no—
267C22—288no—
268C22—281no—
269C22—281no—
270C22D4×AGL(1,𝔽7)11yescharacteristic
271C4—141no—
272C4—281no—
273C4D4×AGL(1,𝔽7)11yescharacteristic
274C4C2×(C22⋊D7)⋊C311yescharacteristic
275C3—71noSylow
276C2—42no—
277C2—81no—
278C2—142no—
279C2—282no—
280C2D8×AGL(1,𝔽7)11yescharacteristic
281C2#6474⋊C311yescharacteristic
282C2#6500⋊C311yescharacteristic
subgroupG/H
centerC22D4×AGL(1,𝔽7)
commutator, cosocleC2×C4×C7C23×C3
FrattiniC2×C4C22×AGL(1,𝔽7)
FittingC2×C7×D8C2×C3
socleC22×C7C2×C3×D4
hypercenterC2×D8AGL(1,𝔽7)
hypocenterC7C3×#447
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×D8

Complements

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

NH ≅ G/Ntype
1. C2×D4×AGL(1,𝔽7)C2semidirect
2. C2×D8×ΓL(1,𝔽8)C2semidirect
3. #6501⋊C3C2semidirect
4. C2×#1141⋊C3C2semidirect
5. #6406⋊C3C2semidirect
6. #6395⋊C3C2semidirect
7. (C2×AGL(1,𝔽7))⋊C8C2semidirect
8. #89928C3semidirect
11. C2×D4×ΓL(1,𝔽8)C22semidirect
19. C2×C8×ΓL(1,𝔽8)C22semidirect
22. C4⋊(ΓL(1,𝔽8)⋊C4)C22semidirect
23. C2×ΓL(1,𝔽8)⋊C8C22semidirect
24. C2×D4×D7C2×C3semidirect
25. C2×C7×D8C2×C3semidirect
26. #6501C2×C3semidirect
27. C2×#1141C2×C3semidirect
28. #6406C2×C3semidirect
29. #6395C2×C3semidirect
30. (C2×D7)⋊C8C2×C3semidirect
42. D4×ΓL(1,𝔽8)D4semidirect
49. C4⋊AGL(1,𝔽7)D4semidirect
56. C2×C7×D4C22×C3semidirect
64. C2×C7×C8C22×C3semidirect
67. C4⋊Dic7C22×C3semidirect
68. C2×C7⋊C8C22×C3semidirect
89. C2×AGL(1,𝔽7)D8semidirect
105. C7×D4C3×D4semidirect
112. D28C3×D4semidirect
141. C2×D8AGL(1,𝔽7)semidirect
159. C2×D7C3×D8semidirect
186. ΓL(1,𝔽8)#447semidirect
194. C2×D4C2×AGL(1,𝔽7)semidirect
199. C2×C8C2×AGL(1,𝔽7)semidirect
241. D4(C22⋊D7)⋊C3semidirect
249. C7C3×#447semidirect

Maximal subgroups

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

subgroupindex
C2×D4×AGL(1,𝔽7)2
C2×D8×ΓL(1,𝔽8)2
#6501⋊C32
C2×#1141⋊C32
#6406⋊C32
#6395⋊C32
(C2×AGL(1,𝔽7))⋊C82
#899283
C3×#4477