Finite Group Databasebeta

← #28401048

Subgroups — #28401048

Order 14464.

564 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 564 conjugacy classes of subgroups — too many to draw. Only its 149 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×#16813655C211yesmaximal, characteristic, complemented
2C2×#16813642C211yesmaximal, characteristic, complemented
3C22×#14268635C211yesmaximal, characteristic, complemented
4#20299169C211yesmaximal, characteristic, complemented
5#20299165C211yesmaximal, characteristic, complemented
6#20298989C212yesmaximal, complemented
7#20298986C212yesmaximal, complemented
8#20298981C212yesmaximal, complemented
9#20298969C212yesmaximal, complemented
10#20298556C211yesmaximal, characteristic, complemented
11C2×AGL(1,ℤ/8)×C113C211yesmaximal, characteristic, complemented
12C2×#14268647C2211yescharacteristic
13C22×Dic226C2211yescharacteristic
14#16813655—24no—
15#16813642—24no—
16C2×Dic452C2212yescomplemented
17C2×#14268637C2211yescharacteristic, complemented
18C2×#14268636C2212yescomplemented
19C2×#14268635—24no—
20C2×#14268635C2212yes—
21C2×#14268635C2212yes—
22C2×#14268635C2211yescharacteristic, complemented
23C2×#14268634C2211yescharacteristic, complemented
24C2×#14268632C2211yescharacteristic, complemented
25C22×#11744751C2211yescharacteristic
26#16813620C2212yescomplemented
27#16813618C2212yes—
28#16813609C2212yes—
29#16813609C2212yescomplemented
30#16813603—22no—
31#16813601—22no—
32#16813593—22no—
33#16813591—22no—
34#16813581C2211yescharacteristic
35#16813532C2212yescomplemented
36C8×Dic113C2212yescomplemented
37C2×#41×C113C2211yescharacteristic
38C22×D4×C113C2211yescharacteristic
39AGL(1,ℤ/8)×C113—24no—
40C2×SD16×C113C2212yes—
41C2×D8×C113C2212yes—
42C2×M16×C113C2211yescharacteristic
43#14268647—22no—
44#14268647—22no—
45C2×Dic226—22no—
46C2×Dic226—22no—
47C2×Dic226—21no—
48C2×Dic226C2311yescharacteristic
49C2×Dic226C2312yes—
50C2×D452C2311yescharacteristic
51C2×C22⋊D113—21no—
52C2×C22⋊D113—24no—
53C22×Dic113—21no—
54C2×C4×D113—21no—
55Dic452—24no—
56#14268637—22no—
57#14268636—24no—
58#14268635—24no—
59#14268635—24no—
60#14268635—22no—
61#14268634—22no—
62#14268632—22no—
63C2×#11744751D414yescomplemented
64C2×#11744751C2311yescharacteristic
65C2×#11744751C2311yescharacteristic
66C4⋊Dic113D412yes—
67#14268626—24no—
68C22⋊Dic113—24no—
69C22⋊Dic113—24no—
70C22⋊Dic113D412yes—
71C4×Dic113C2312yes—
72C24×C113—21no—
73#41×C113—22no—
74#41×C113—22no—
75C2×Q8×C113C2311yescharacteristic
76C2×D4×C113—21no—
77C2×D4×C113—22no—
78C2×D4×C113—22no—
79C2×D4×C113—21no—
80C2×D4×C113C2312yes—
81C2×D4×C113C2311yescharacteristic
82C22×C4×C113—21no—
83C22×C4×C113C2311yescharacteristic
84SD16×C113—24no—
85D8×C113—24no—
86M16×C113D414yescomplemented
87C2×C8×C113C2312yes—
88C23×C113—21no—
89C23×C113—22no—
90C23×C113—42no—
91C23×C113—82no—
92C23×C113C2×D411yescharacteristic
93C22×D113—21no—
94Q8×C113—21no—
95Q8×C113—21no—
96D4×C113—21no—
97D4×C113—22no—
98D4×C113—22no—
99D4×C113—21no—
100D4×C113—42no—
101D4×C113—42no—
102C2×C4×C113—21no—
103C2×C4×C113—42no—
104C2×C4×C113C2×D411yescharacteristic
105C2×C4×C113C2×D412yes—
106C2×C4×C113C2×D412yes—
107C2×C4×C113C2411yescharacteristic
108C22⋊D113—42no—
109C22⋊D113—84no—
110C2×Dic113—21no—
111C2×Dic113—22no—
112C2×Dic113—42no—
113C2×Dic113C2×D414yes—
114D452—21no—
115D452—21no—
116C4×D113—42no—
117Dic226—22no—
118Dic226—21no—
119Dic226—22no—
120Dic226—21no—
121Dic226—42no—
122C8×C113C2×D414yes—
123#11744751C2×D412yes—
124#11744751C2×D412yes—
125C22×C113—22no—
126C22×C113—41no—
127C22×C113—42no—
128C22×C113—44no—
129C22×C113—42no—
130C22×C113—84no—
131C22×C113#12712yes—
132C22×C113C22×D411yescharacteristic
133C2×D113—41no—
134C2×D113—42no—
135C4×C113—41no—
136C4×C113#12712yes—
137C4×C113C22×D411yescharacteristic
138C4×C113C22×D411yescharacteristic
139Dic113—24no—
140Dic113—42no—
141Dic113—41no—
142C2×C113—22no—
143C2×C113—81no—
144C2×C113—82no—
145C2×C113C2×#12711yescharacteristic
146C2×C113#57712yes—
147D113—81no—
148#3154—1131nomaximal, Sylow
149C113#315411yescharacteristic, complemented, Sylow
150C2×#137—1131no—
151C2×AGL(1,ℤ/8)D11311yescharacteristic, complemented
152C2×#135—1131no—
153C22×SD16—1131no—
154#536—1131no—
155#535—1131no—
156#497—1132no—
157#496—1132no—
158#495—1132no—
159#492—1132no—
160#405—1131no—
161C2×#41—1131no—
162C2×#41C2×D11311yescharacteristic
163C22×Q8—1131no—
164C22×D4C2×D11311yescharacteristic
165#137—2264no—
166AGL(1,ℤ/8)—24no—
167#135—2264no—
168C2×Q16—1131no—
169C2×Q16—1132no—
170C2×SD16—1131no—
171C2×SD16—1132no—
172C2×SD16—1132no—
173C2×SD16—1131no—
174C2×SD16—1132no—
175C2×SD16—2264no—
176C2×SD16C2×D11312yes—
177C2×D8—1131no—
178C2×D8C2×D11312yes—
179C2×M16C2×D11311yescharacteristic
180C22×C8—1131no—
181C4⋊Q8—1132no—
182#127—1132no—
183#124—1132no—
184#124—1132no—
185#123—2262no—
186C22⋊Q8—2262no—
187#121—2262no—
188#120—2262no—
189#117—1131no—
190#97—1132no—
191C4×C8—1132no—
192C24—21no—
193#41—22no—
194#41—22no—
195#41—2262no—
196#41—2262no—
197C2×Q8—1131no—
198C2×Q8—1132no—
199C2×Q8—2262no—
200C2×Q8—2262no—
201C2×Q8—2261no—
202C2×Q8C22×D11311yescharacteristic
203C2×D4—21no—
204C2×D4—22no—
205C2×D4—21no—
206C2×D4—22no—
207C2×D4—1131no—
208C2×D4—2264no—
209C2×D4—2261no—
210C2×D4C22×D11311yescharacteristic
211C2×D4C22×D11312yes—
212C22×C4—21no—
213C22×C4—2261no—
214C22×C4—2261no—
215C22×C4C22×D11311yescharacteristic
216Q16—2262no—
217Q16—2264no—
218SD16—24no—
219SD16—2264no—
220SD16—2264no—
221SD16—2262no—
222SD16—2262no—
223SD16—2264no—
224D8—24no—
225D8—2262no—
226M16C22⋊D11314yescomplemented
227C2×C8—1134no—
228C2×C8—1131no—
229C2×C8—1131no—
230C2×C8C22×D11312yes—
231C4⋊C4—1132no—
232C4⋊C4—2264no—
233C22⋊C4—1132no—
234C22⋊C4—2264no—
235C22⋊C4—2264no—
236C42—1132no—
237C23—21no—
238C23—22no—
239C23—42no—
240C23—82no—
241C23—2261no—
242C23D4×D11311yescharacteristic
243Q8—21no—
244Q8—21no—
245Q8—4522no—
246Q8—2262no—
247Q8—2261no—
248Q8—2262no—
249Q8—2261no—
250D4—21no—
251D4—22no—
252D4—21no—
253D4—22no—
254D4—42no—
255D4—42no—
256D4—9044no—
257D4—4522no—
258D4—2261no—
259D4—2261no—
260C2×C4—21no—
261C2×C4—42no—
262C2×C4—1134no—
263C2×C4—4522no—
264C2×C4—4522no—
265C2×C4—2261no—
266C2×C4—2262no—
267C2×C4D4×D11311yescharacteristic
268C2×C4D4×D11312yes—
269C2×C4C2×C22⋊D11312yes—
270C2×C4C23×D11311yescharacteristic
271C8—1132no—
272C8—1132no—
273C8C2×C22⋊D11314yes—
274C22—22no—
275C22—41no—
276C22—42no—
277C22—44no—
278C22—42no—
279C22—84no—
280C22—4521no—
281C22—4522no—
282C22#1681361812yes—
283C22C2×D4×D11311yescharacteristic
284C4—41no—
285C4—4521no—
286C4—4522no—
287C4—2264no—
288C4#1681361812yes—
289C4C2×D4×D11311yescharacteristic
290C4C22×C22⋊D11311yescharacteristic
291C2—22no—
292C2—82no—
293C2—81no—
294C2—9041no—
295C2C2×#1681361811yescharacteristic
296C2#2029940812yes—
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#3154
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×#16813655C2semidirect
2. C2×#16813642C2semidirect
3. C22×#14268635C2semidirect
4. #20299169C2semidirect
5. #20299165C2semidirect
6. #20298989C2semidirect
7. #20298986C2semidirect
8. #20298981C2semidirect
9. #20298969C2semidirect
10. #20298556C2semidirect
11. C2×AGL(1,ℤ/8)×C113C2semidirect
16. C2×Dic452C22semidirect
17. C2×#14268637C22semidirect
18. C2×#14268636C22semidirect
22. C2×#14268635C22semidirect
23. C2×#14268634C22semidirect
24. C2×#14268632C22semidirect
26. #16813620C22semidirect
29. #16813609C22semidirect
35. #16813532C22semidirect
36. C8×Dic113C22semidirect
63. C2×#11744751D4semidirect
86. M16×C113D4semidirect
149. C113#3154semidirect
151. C2×AGL(1,ℤ/8)D113semidirect
226. M16C22⋊D113semidirect

Maximal subgroups

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

subgroupindex
C2×#168136552
C2×#168136422
C22×#142686352
#202991692
#202991652
#202989892
#202989862
#202989812
#202989692
#202985562
C2×AGL(1,ℤ/8)×C1132
#3154113