Finite Group Databasebeta

← #28400117

Subgroups — #28400117

Order 14464.

504 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 504 conjugacy classes of subgroups — too many to draw. Only its 141 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×#16813642C211yesmaximal, characteristic, complemented
2#20299165C211yesmaximal, characteristic, complemented
3#20299081C211yesmaximal, characteristic, complemented
4#20298818C212yesmaximal, complemented
5#20298811C212yesmaximal, complemented
6#20298796C212yesmaximal, complemented
7#20298791C212yesmaximal, complemented
8#20298626C211yesmaximal, characteristic, complemented
9#20298600C211yesmaximal, characteristic, complemented
10C2×#16813543C211yesmaximal, characteristic, complemented
11C2×AGL(1,ℤ/8)×C113C211yesmaximal, characteristic, complemented
12C2×#14268647C2211yescharacteristic
13#16813642—24no—
14C2×#14268637C2211yescharacteristic, complemented
15C2×#14268635C2211yescharacteristic, complemented
16C2×#14268634C2211yescharacteristic, complemented
17C2×#14268632C2211yescharacteristic, complemented
18C22×#11744751C2211yescharacteristic
19#16813618C2212yes—
20#16813609C2212yes—
21#16813593—21no—
22#16813593—21no—
23#16813591—22no—
24Q8×Dic113C2212yescomplemented
25D4×Dic113—22no—
26D4×Dic113C2212yescomplemented
27#16813581—21no—
28#16813581C2211yescharacteristic
29C2×C4⋊Dic113—21no—
30C2×C4⋊Dic113C2211yescharacteristic
31#16813551C2212yescomplemented
32#16813551C2212yescomplemented
33#16813547C2212yescomplemented
34#16813544C2212yescomplemented
35#16813543C2212yes—
36#16813543C2212yes—
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×Dic226C2311yescharacteristic
46C2×D452C2311yescharacteristic
47C2×C22⋊D113—21no—
48C2×C22⋊D113—22no—
49C2×C22⋊D113—22no—
50C22×Dic113—21no—
51C22×Dic113—22no—
52C2×C4×D113—21no—
53#14268637—22no—
54#14268635—22no—
55#14268634—22no—
56#14268632—22no—
57C2×#11744751—22no—
58C2×#11744751C2311yescharacteristic
59C2×#11744751C2311yescharacteristic
60C4⋊Dic113—21no—
61C4⋊Dic113—22no—
62C4⋊Dic113D411yescharacteristic
63C4⋊Dic113D411yescharacteristic
64C4⋊Dic113C2312yes—
65C22⋊Dic113—22no—
66C22⋊Dic113—22no—
67C22⋊Dic113—22no—
68C22⋊Dic113D412yes—
69C4×Dic113—22no—
70C4×Dic113C2312yes—
71C24×C113—21no—
72#41×C113—22no—
73#41×C113D414yescomplemented
74C2×Q8×C113C2311yescharacteristic
75C2×D4×C113—21no—
76C2×D4×C113—22no—
77C2×D4×C113—21no—
78C2×D4×C113—22no—
79C2×D4×C113C2312yes—
80C2×D4×C113C2311yescharacteristic
81C22×C4×C113—21no—
82C22×C4×C113C2311yescharacteristic
83SD16×C113—24no—
84D8×C113—24no—
85M16×C113—22no—
86C2×C8×C113C2312yes—
87C23×C113—21no—
88C23×C113—22no—
89C23×C113—42no—
90C23×C113—81no—
91C23×C113—81no—
92C23×C113C2×D411yescharacteristic
93C22×D113—21no—
94Q8×C113—21no—
95Q8×C113C2×D412yes—
96D4×C113—24no—
97D4×C113—22no—
98D4×C113—21no—
99D4×C113—22no—
100D4×C113—42no—
101D4×C113C2×D412yes—
102C2×C4×C113—21no—
103C2×C4×C113—24no—
104C2×C4×C113C2×D411yescharacteristic
105C2×C4×C113C2×D412yes—
106C2×C4×C113#4112yes—
107C2×C4×C113C2411yescharacteristic
108C22⋊D113—42no—
109C22⋊D113—82no—
110C22⋊D113—82no—
111C2×Dic113—22no—
112C2×Dic113—21no—
113C2×Dic113—82no—
114C2×Dic113—81no—
115C2×Dic113C2×D412yes—
116C2×Dic113C2×D412yes—
117D452—21no—
118D452—21no—
119C4×D113—42no—
120Dic226—21no—
121Dic226—21no—
122C8×C113—22no—
123#11744751—21no—
124#11744751—21no—
125C22×C113—21no—
126C22×C113—21no—
127C22×C113—22no—
128C22×C113—41no—
129C22×C113—41no—
130C22×C113—42no—
131C22×C113—42no—
132C22×C113—42no—
133C22×C113—84no—
134C22×C113#12112yes—
135C22×C113C22×D411yescharacteristic
136C2×D113—41no—
137C2×D113—41no—
138C2×D113—41no—
139C4×C113—22no—
140C4×C113#12112yes—
141C4×C113C22×D411yescharacteristic
142C4×C113C2×#4111yescharacteristic
143Dic113—22no—
144Dic113—22no—
145Dic113—41no—
146Dic113—42no—
147C2×C113—22no—
148C2×C113—42no—
149C2×C113—82no—
150C2×C113C2×#12111yescharacteristic
151C2×C113#57611yescharacteristic
152C2×C113#57711yescharacteristic
153D113—81no—
154#3045—1131nomaximal, Sylow
155C113#304511yescharacteristic, complemented, Sylow
156C2×AGL(1,ℤ/8)D11311yescharacteristic, complemented
157C2×#135—1131no—
158#535—1131no—
159#520—1131no—
160#464—1132no—
161#463—1132no—
162#460—1132no—
163#459—1132no—
164#424—1131no—
165#417—1131no—
166C2×#102—1131no—
167C2×#41—1131no—
168C2×#41C2×D11311yescharacteristic
169C22×D4C2×D11311yescharacteristic
170AGL(1,ℤ/8)—24no—
171#135—2264no—
172C2×Q16—1131no—
173C2×SD16—1131no—
174C2×SD16—1131no—
175C2×SD16C2×D11312yes—
176C2×D8—1131no—
177C2×D8C2×D11312yes—
178C2×M16C2×D11311yescharacteristic
179C22×C8—1131no—
180#127—1132no—
181#124—1132no—
182#121—2261no—
183#121—2261no—
184#120—2262no—
185C4×Q8—1132no—
186C4×D4—1132no—
187C4×D4—2262no—
188#117—1131no—
189#117—2261no—
190C2×C4⋊C4—1131no—
191C2×C4⋊C4—2261no—
192C4⋊C8—1132no—
193C4⋊C8—1132no—
194#103—1132no—
195#102—1132no—
196#102—1132no—
197#102—1132no—
198C24—21no—
199#41—22no—
200#41—2262no—
201#41—2262no—
202#41C22⋊D11314yescomplemented
203C2×Q8—1131no—
204C2×Q8C22×D11311yescharacteristic
205C2×D4—21no—
206C2×D4—21no—
207C2×D4—22no—
208C2×D4—22no—
209C2×D4—1131no—
210C2×D4—2262no—
211C2×D4—2262no—
212C2×D4—2261no—
213C2×D4C22×D11312yes—
214C2×D4C22×D11311yescharacteristic
215C22×C4—21no—
216C22×C4—2262no—
217C22×C4—2261no—
218C22×C4—2261no—
219C22×C4C22×D11311yescharacteristic
220Q16—2262no—
221SD16—24no—
222SD16—2262no—
223SD16—2262no—
224D8—24no—
225D8—2262no—
226M16—22no—
227C2×C8—1131no—
228C2×C8—1131no—
229C2×C8—2262no—
230C2×C8C22×D11312yes—
231C4⋊C4—1131no—
232C4⋊C4—1131no—
233C4⋊C4—1132no—
234C4⋊C4—2262no—
235C4⋊C4—2261no—
236C22⋊C4—1132no—
237C22⋊C4—2262no—
238C22⋊C4—2262no—
239C22⋊C4—2262no—
240C42—1132no—
241C42—2262no—
242C23—21no—
243C23—22no—
244C23—42no—
245C23—81no—
246C23—81no—
247C23—2261no—
248C23D4×D11311yescharacteristic
249Q8—21no—
250Q8—2261no—
251Q8—2261no—
252Q8C2×C22⋊D11312yes—
253D4—24no—
254D4—22no—
255D4—21no—
256D4—22no—
257D4—42no—
258D4—9042no—
259D4—9042no—
260D4—4522no—
261D4—2261no—
262D4—2261no—
263D4C2×C22⋊D11312yes—
264C2×C4—21no—
265C2×C4—24no—
266C2×C4—1132no—
267C2×C4—1132no—
268C2×C4—9041no—
269C2×C4—9042no—
270C2×C4—4522no—
271C2×C4—2262no—
272C2×C4—2261no—
273C2×C4D4×D11311yescharacteristic
274C2×C4C2×C22⋊D11312yes—
275C2×C4#1426864612yes—
276C2×C4C23×D11311yescharacteristic
277C8—22no—
278C8—2261no—
279C8—2261no—
280C22—21no—
281C22—21no—
282C22—22no—
283C22—41no—
284C22—41no—
285C22—42no—
286C22—42no—
287C22—42no—
288C22—84no—
289C22—4521no—
290C22—4521no—
291C22—4521no—
292C22#1681359412yes—
293C22C2×D4×D11311yescharacteristic
294C4—22no—
295C4—4521no—
296C4—4522no—
297C4—2262no—
298C4—2262no—
299C4#1681359412yes—
300C4C22×C22⋊D11311yescharacteristic
301C4C2×#1426864611yescharacteristic
302C2—22no—
303C2—42no—
304C2—82no—
305C2—9041no—
306C2C2×#1681359411yescharacteristic
307C2#2029940211yescharacteristic
308C2#2029940811yescharacteristic
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#3045
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×#16813642C2semidirect
2. #20299165C2semidirect
3. #20299081C2semidirect
4. #20298818C2semidirect
5. #20298811C2semidirect
6. #20298796C2semidirect
7. #20298791C2semidirect
8. #20298626C2semidirect
9. #20298600C2semidirect
10. C2×#16813543C2semidirect
11. C2×AGL(1,ℤ/8)×C113C2semidirect
14. C2×#14268637C22semidirect
15. C2×#14268635C22semidirect
16. C2×#14268634C22semidirect
17. C2×#14268632C22semidirect
24. Q8×Dic113C22semidirect
26. D4×Dic113C22semidirect
31. #16813551C22semidirect
32. #16813551C22semidirect
33. #16813547C22semidirect
34. #16813544C22semidirect
73. #41×C113D4semidirect
155. C113#3045semidirect
156. C2×AGL(1,ℤ/8)D113semidirect
202. #41C22⋊D113semidirect

Maximal subgroups

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

subgroupindex
C2×#168136422
#202991652
#202990812
#202988182
#202988112
#202987962
#202987912
#202986262
#202986002
C2×#168135432
C2×AGL(1,ℤ/8)×C1132
#3045113