Finite Group Databasebeta

← #28400052

Subgroups — #28400052

Order 14464.

584 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 584 conjugacy classes of subgroups — too many to draw. Only its 145 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×#16813647C211yesmaximal, characteristic, complemented
2C2×#16813618C211yesmaximal, characteristic, complemented
3#41×Dic113C211yesmaximal, characteristic, complemented
4#20298796C214yesmaximal, complemented
5#20298791C214yesmaximal, complemented
6#20298621C211yesmaximal, characteristic, complemented
7#20298604C211yesmaximal, characteristic, complemented
8#20298603C211yesmaximal, characteristic, complemented
9C2×AGL(1,ℤ/8)×C113C211yesmaximal, characteristic, complemented
10C22×D452C2211yescharacteristic
11C22×C22⋊D113—22no—
12#16813647—24no—
13C2×#14268634C2212yescomplemented
14C2×#14268632C2212yescomplemented
15C2×#14268631C2211yescharacteristic
16#16813618—21no—
17#16813618—21no—
18#16813618C2214yes—
19Q8×Dic113C2212yescomplemented
20D4×Dic113—22no—
21D4×Dic113C2212yescomplemented
22#16813581—21no—
23#16813581C2211yescharacteristic
24C2×C4×Dic113—21no—
25C2×C4×Dic113C2211yescharacteristic
26#16813551C2214yescomplemented
27#16813544C2214yescomplemented
28#16813543C2214yescomplemented
29C2×#41×C113C2211yescharacteristic
30C22×D4×C113C2211yescharacteristic
31AGL(1,ℤ/8)×C113—24no—
32C2×SD16×C113C2212yescomplemented
33C2×D8×C113C2212yescomplemented
34C2×M16×C113C2211yescharacteristic
35C23×D113—21no—
36C2×D452—22no—
37C2×D452—22no—
38C2×D452—21no—
39C2×D452C2312yes—
40C2×C22⋊D113—24no—
41C2×C22⋊D113—24no—
42C2×C22⋊D113—48no—
43C22×Dic113—22no—
44C22×Dic113D412yes—
45#14268634—24no—
46#14268632—24no—
47#14268631—22no—
48C2×#11744751C2312yes—
49C4⋊Dic113—22no—
50C4⋊Dic113C2312yes—
51C22⋊Dic113—21no—
52C22⋊Dic113—22no—
53C4×Dic113—21no—
54C4×Dic113—22no—
55C4×Dic113D411yescharacteristic
56C4×Dic113D411yescharacteristic
57C4×Dic113C2312yes—
58C24×C113—21no—
59#41×C113—22no—
60#41×C113D414yescomplemented
61C2×Q8×C113C2311yescharacteristic
62C2×D4×C113—21no—
63C2×D4×C113—22no—
64C2×D4×C113—21no—
65C2×D4×C113—22no—
66C2×D4×C113C2312yes—
67C2×D4×C113C2311yescharacteristic
68C22×C4×C113—21no—
69C22×C4×C113C2311yescharacteristic
70SD16×C113—24no—
71D8×C113—24no—
72M16×C113—22no—
73C2×C8×C113C2312yes—
74C23×C113—21no—
75C23×C113—22no—
76C23×C113—44no—
77C23×C113—42no—
78C23×C113C2×D411yescharacteristic
79C22×D113—22no—
80C22×D113—44no—
81C22×D113—42no—
82Q8×C113—21no—
83Q8×C113C2×D412yescomplemented
84D4×C113—24no—
85D4×C113—22no—
86D4×C113—21no—
87D4×C113—22no—
88D4×C113—42no—
89D4×C113C2×D412yescomplemented
90C2×C4×C113—21no—
91C2×C4×C113—24no—
92C2×C4×C113C2×D411yescharacteristic
93C2×C4×C113C2×D412yes—
94C2×C4×C113#4112yes—
95C2×C4×C113C2411yescharacteristic
96C22⋊D113—48no—
97C22⋊D113—48no—
98C2×Dic113—24no—
99C2×Dic113—22no—
100C2×Dic113—44no—
101C2×Dic113C2×D412yes—
102C2×Dic113C2×D412yes—
103D452—22no—
104D452—22no—
105D452—42no—
106C8×C113—22no—
107#11744751—22no—
108C22×C113—22no—
109C22×C113—22no—
110C22×C113—42no—
111C22×C113—42no—
112C22×C113—44no—
113C22×C113—84no—
114C22×C113#12112yes—
115C22×C113C22×D411yescharacteristic
116C2×D113—42no—
117C2×D113—44no—
118C2×D113—84no—
119C4×C113—22no—
120C4×C113#12112yes—
121C4×C113C22×D411yescharacteristic
122C4×C113C2×#4111yescharacteristic
123Dic113—22no—
124Dic113—42no—
125Dic113AGL(1,ℤ/8)14yescomplemented
126C2×C113—22no—
127C2×C113—42no—
128C2×C113—82no—
129C2×C113C2×#12111yescharacteristic
130C2×C113C2×AGL(1,ℤ/8)12yes—
131D113—82no—
132#3038—1131nomaximal, Sylow
133C113#303811yescharacteristic, complemented, Sylow
134C2×AGL(1,ℤ/8)—1131no—
135C2×AGL(1,ℤ/8)D11311yescharacteristic, complemented
136C2×#127—1131no—
137C4×#41—1131no—
138#460—1134no—
139#459—1134no—
140#423—1131no—
141#418—1131no—
142#418—1131no—
143C2×#41C2×D11311yescharacteristic
144C22×D4—1131no—
145C22×D4—2262no—
146C22×D4C2×D11311yescharacteristic
147AGL(1,ℤ/8)—24no—
148AGL(1,ℤ/8)—2264no—
149C2×SD16—1132no—
150C2×SD16C2×D11312yescomplemented
151C2×D8—1132no—
152C2×D8C2×D11312yescomplemented
153C2×M16—1131no—
154C2×M16C2×D11311yescharacteristic
155#127—1134no—
156#127—2261no—
157#127—2261no—
158C4×Q8—1132no—
159C4×D4—1132no—
160C4×D4—2262no—
161#117—1131no—
162#117—2261no—
163C2×C42—1131no—
164C2×C42—2261no—
165C4⋊C8—1134no—
166#102—1134no—
167#102—1134no—
168C24—21no—
169C24—2261no—
170#41—22no—
171#41C22⋊D11314yescomplemented
172C2×Q8C22×D11311yescharacteristic
173C2×D4—21no—
174C2×D4—21no—
175C2×D4—22no—
176C2×D4—22no—
177C2×D4—1132no—
178C2×D4—4528no—
179C2×D4—2264no—
180C2×D4—2264no—
181C2×D4—2261no—
182C2×D4—2262no—
183C2×D4—2262no—
184C2×D4C22×D11312yes—
185C2×D4C22×D11311yescharacteristic
186C22×C4—21no—
187C22×C4—1132no—
188C22×C4—2262no—
189C22×C4C22×D11311yescharacteristic
190SD16—24no—
191SD16—2264no—
192D8—24no—
193D8—2264no—
194M16—22no—
195M16—2262no—
196C2×C8—1132no—
197C2×C8C22×D11312yes—
198C4⋊C4—1132no—
199C4⋊C4—2262no—
200C22⋊C4—2262no—
201C22⋊C4—2261no—
202C42—1131no—
203C42—1131no—
204C42—1132no—
205C42—2262no—
206C42—2261no—
207C23—21no—
208C23—22no—
209C23—44no—
210C23—42no—
211C23—4524no—
212C23—4522no—
213C23—2262no—
214C23D4×D11311yescharacteristic
215Q8—21no—
216Q8C2×C22⋊D11312yescomplemented
217D4—24no—
218D4—22no—
219D4—21no—
220D4—22no—
221D4—42no—
222D4—4528no—
223D4—4528no—
224D4—4522no—
225D4—2262no—
226D4—2262no—
227D4C2×C22⋊D11312yescomplemented
228C2×C4—21no—
229C2×C4—24no—
230C2×C4—1132no—
231C2×C4—1132no—
232C2×C4—4524no—
233C2×C4—2264no—
234C2×C4—2262no—
235C2×C4D4×D11311yescharacteristic
236C2×C4C2×C22⋊D11312yes—
237C2×C4#1426864612yes—
238C2×C4C23×D11311yescharacteristic
239C8—22no—
240C8—2262no—
241C22—22no—
242C22—22no—
243C22—42no—
244C22—42no—
245C22—44no—
246C22—84no—
247C22—9044no—
248C22—4522no—
249C22—4524no—
250C22#1681359412yes—
251C22C2×D4×D11311yescharacteristic
252C4—22no—
253C4—1134no—
254C4—4522no—
255C4—2262no—
256C4#1681359412yes—
257C4C22×C22⋊D11311yescharacteristic
258C4C2×#1426864611yescharacteristic
259C2—22no—
260C2—42no—
261C2—82no—
262C2—9042no—
263C2C2×#1681359411yescharacteristic
264C2AGL(1,ℤ/8)×D11312yes—
subgroupG/H
centerC22C2×D4×D113
commutator, cosocleC2×C4×C113C24
FrattiniC2×C4C23×D113
FittingC2×AGL(1,ℤ/8)×C113C2
socleC22×C113C22×D4
hypercenterC2×AGL(1,ℤ/8)D113
hypocenterC113#3038
LCSG ▷ C2×C4×C113 ▷ C2×C113 ▷ C113
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. C2×#16813647C2semidirect
2. C2×#16813618C2semidirect
3. #41×Dic113C2semidirect
4. #20298796C2semidirect
5. #20298791C2semidirect
6. #20298621C2semidirect
7. #20298604C2semidirect
8. #20298603C2semidirect
9. C2×AGL(1,ℤ/8)×C113C2semidirect
13. C2×#14268634C22semidirect
14. C2×#14268632C22semidirect
19. Q8×Dic113C22semidirect
21. D4×Dic113C22semidirect
26. #16813551C22semidirect
27. #16813544C22semidirect
28. #16813543C22semidirect
32. C2×SD16×C113C22semidirect
33. C2×D8×C113C22semidirect
60. #41×C113D4semidirect
83. Q8×C113C2×D4semidirect
89. D4×C113C2×D4semidirect
125. Dic113AGL(1,ℤ/8)semidirect
133. C113#3038semidirect
135. C2×AGL(1,ℤ/8)D113semidirect
150. C2×SD16C2×D113semidirect
152. C2×D8C2×D113semidirect
171. #41C22⋊D113semidirect
216. Q8C2×C22⋊D113semidirect
227. D4C2×C22⋊D113semidirect

Maximal subgroups

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

subgroupindex
C2×#168136472
C2×#168136182
#41×Dic1132
#202987962
#202987912
#202986212
#202986042
#202986032
C2×AGL(1,ℤ/8)×C1132
#3038113