Finite Group Databasebeta

← #28400187

Subgroups — #28400187

Order 14464.

548 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 548 conjugacy classes of subgroups — too many to draw. Only its 143 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×M16×D113C211yesmaximal, characteristic, complemented
2#20299105C211yesmaximal, characteristic, complemented
3C2×#16813593C211yesmaximal, characteristic, complemented
4#20298848C212yesmaximal, complemented
5#20298841C212yesmaximal, complemented
6#20298831C212yesmaximal, complemented
7#20298824C212yesmaximal, complemented
8#20298640C211yesmaximal, characteristic, complemented
9#20298603C211yesmaximal, characteristic, complemented
10#20298599C211yesmaximal, characteristic, complemented
11C2×AGL(1,ℤ/8)×C113C211yesmaximal, characteristic, complemented
12C22×C22⋊D113—21no—
13C22×C4×D113C2211yescharacteristic
14C2×#14268631C2211yescharacteristic
15C2×#14268630C2212yescomplemented
16M16×D113—24no—
17C2×C8×D113C2212yescomplemented
18#16813603—21no—
19#16813598C2211yescharacteristic, complemented
20#16813598C2211yescharacteristic, complemented
21#16813593—21no—
22#16813593—21no—
23#16813593C2212yes—
24#16813593C2212yes—
25#16813593C2211yescharacteristic, complemented
26#16813593C2211yescharacteristic, complemented
27#16813590—21no—
28C4×C22⋊D113—21no—
29C4×C22⋊D113—21no—
30#16813581C2211yescharacteristic
31C2×C4⋊Dic113C2211yescharacteristic
32C2×C22⋊Dic113—21no—
33#16813556C2212yescomplemented
34#16813554C2212yescomplemented
35#16813546C2212yescomplemented
36#16813543C2212yescomplemented
37#16813543C2212yescomplemented
38#16813543C2212yescomplemented
39C2×#41×C113C2211yescharacteristic
40C22×D4×C113C2211yescharacteristic
41AGL(1,ℤ/8)×C113—24no—
42C2×SD16×C113C2212yescomplemented
43C2×D8×C113C2212yescomplemented
44C2×M16×C113C2211yescharacteristic
45C23×D113D411yescharacteristic
46C2×C22⋊D113—21no—
47C2×C22⋊D113—21no—
48C2×C22⋊D113—22no—
49C2×C22⋊D113—22no—
50C2×C22⋊D113—44no—
51C22×Dic113—21no—
52C22×Dic113D411yescharacteristic
53C2×C4×D113—22no—
54C2×C4×D113—22no—
55C2×C4×D113D411yescharacteristic
56C2×C4×D113D411yescharacteristic
57C2×C4×D113C2311yescharacteristic
58C2×C4×D113C2311yescharacteristic
59#14268631—22no—
60#14268630—24no—
61C2×#11744751C2312yes—
62C8×D113—24no—
63C4⋊Dic113—21no—
64C4⋊Dic113C2312yes—
65C4⋊Dic113C2311yescharacteristic
66C4⋊Dic113C2311yescharacteristic
67#14268626—21no—
68#14268626—21no—
69C22⋊Dic113—21no—
70C22⋊Dic113—22no—
71C22⋊Dic113—22no—
72C22⋊Dic113—21no—
73C22⋊Dic113—21no—
74#14268624—21no—
75#14268624—21no—
76C4×Dic113—21no—
77C24×C113—21no—
78#41×C113—22no—
79#41×C113—22no—
80C2×Q8×C113C2311yescharacteristic
81C2×D4×C113—21no—
82C2×D4×C113—22no—
83C2×D4×C113—21no—
84C2×D4×C113—22no—
85C2×D4×C113C2312yes—
86C2×D4×C113C2311yescharacteristic
87C22×C4×C113—21no—
88C22×C4×C113C2311yescharacteristic
89SD16×C113—24no—
90D8×C113—24no—
91M16×C113D414yescomplemented
92C2×C8×C113C2312yes—
93C23×C113—21no—
94C23×C113—22no—
95C23×C113—42no—
96C23×C113—42no—
97C23×C113—42no—
98C23×C113C2×D411yescharacteristic
99C22×D113—22no—
100C22×D113—22no—
101C22×D113—41no—
102C22×D113C2×D411yescharacteristic
103C22×D113C2×D411yescharacteristic
104Q8×C113—21no—
105Q8×C113—21no—
106D4×C113—21no—
107D4×C113—22no—
108D4×C113—22no—
109D4×C113—21no—
110D4×C113—42no—
111D4×C113—42no—
112C2×C4×C113—21no—
113C2×C4×C113—42no—
114C2×C4×C113C2×D411yescharacteristic
115C2×C4×C113C2×D412yes—
116C2×C4×C113#4112yes—
117C2×C4×C113C2411yescharacteristic
118C22⋊D113—42no—
119C22⋊D113—44no—
120C22⋊D113—42no—
121C22⋊D113—44no—
122C2×Dic113—22no—
123C2×Dic113—21no—
124C2×Dic113—21no—
125C2×Dic113—22no—
126C2×Dic113—42no—
127C2×Dic113C2×D411yescharacteristic
128C2×Dic113C2×D411yescharacteristic
129C4×D113—22no—
130C4×D113—22no—
131C4×D113—21no—
132C4×D113—21no—
133C4×D113—21no—
134C4×D113—21no—
135C8×C113C2×D414yescomplemented
136#11744751—22no—
137C22×C113—21no—
138C22×C113—21no—
139C22×C113—41no—
140C22×C113—42no—
141C22×C113—42no—
142C22×C113—41no—
143C22×C113—41no—
144C22×C113—42no—
145C22×C113—84no—
146C22×C113#12112yes—
147C22×C113C22×D411yescharacteristic
148C2×D113—21no—
149C2×D113—21no—
150C2×D113—21no—
151C2×D113—21no—
152C2×D113—21no—
153C2×D113—44no—
154C2×D113AGL(1,ℤ/8)12yescomplemented
155C4×C113—41no—
156C4×C113#12112yes—
157C4×C113C22×D411yescharacteristic
158C4×C113C2×#4111yescharacteristic
159Dic113—21no—
160Dic113—21no—
161Dic113—42no—
162Dic113—41no—
163Dic113—41no—
164C2×C113—22no—
165C2×C113—81no—
166C2×C113—82no—
167C2×C113C2×#12111yescharacteristic
168C2×C113C2×AGL(1,ℤ/8)11yescharacteristic
169C2×C113#57511yescharacteristic
170D113—22no—
171D113—41no—
172#3055—1131nomaximal, Sylow
173C113#305511yescharacteristic, complemented, Sylow
174C2×AGL(1,ℤ/8)D11311yescharacteristic, complemented
175C22×M16—1131no—
176#525—1131no—
177C2×#121—1131no—
178#469—1132no—
179#468—1132no—
180#466—1132no—
181#465—1132no—
182#428—1131no—
183#418—1131no—
184#417—1131no—
185C2×#41C2×D11311yescharacteristic
186C22×D4—2261no—
187C22×D4C2×D11311yescharacteristic
188C23×C4—1131no—
189AGL(1,ℤ/8)—24no—
190C2×SD16C2×D11312yescomplemented
191C2×D8C2×D11312yescomplemented
192C2×M16—1131no—
193C2×M16—1132no—
194C2×M16—2264no—
195C2×M16C2×D11311yescharacteristic
196C22×C8—1132no—
197#123—2261no—
198C22⋊Q8—1131no—
199C22⋊Q8—1131no—
200#121—1132no—
201#121—1132no—
202#121—1131no—
203#121—1131no—
204#121—2261no—
205#121—2261no—
206#120—2261no—
207C4×D4—2261no—
208C4×D4—2261no—
209#117—1131no—
210C2×C4⋊C4—1131no—
211C2×C22⋊C4—2261no—
212#107—1132no—
213#106—1132no—
214#103—1132no—
215#102—1132no—
216#102—1132no—
217#102—1132no—
218C24—21no—
219C24—1131no—
220#41—22no—
221#41—22no—
222C2×Q8C22×D11311yescharacteristic
223C2×D4—21no—
224C2×D4—21no—
225C2×D4—22no—
226C2×D4—22no—
227C2×D4—4524no—
228C2×D4—2261no—
229C2×D4—2261no—
230C2×D4—2262no—
231C2×D4—2262no—
232C2×D4C22×D11311yescharacteristic
233C2×D4C22×D11312yes—
234C22×C4—21no—
235C22×C4—1131no—
236C22×C4—1131no—
237C22×C4—1131no—
238C22×C4—1131no—
239C22×C4—1131no—
240C22×C4—2261no—
241C22×C4—2262no—
242C22×C4—2262no—
243C22×C4C22×D11311yescharacteristic
244SD16—24no—
245D8—24no—
246M16—2262no—
247M16—2264no—
248M16C22⋊D11314yescomplemented
249C2×C8—1132no—
250C2×C8—2264no—
251C2×C8C22×D11312yes—
252C4⋊C4—1131no—
253C4⋊C4—1131no—
254C4⋊C4—1132no—
255C4⋊C4—2261no—
256C4⋊C4—2261no—
257C4⋊C4—2261no—
258C22⋊C4—2261no—
259C22⋊C4—2261no—
260C22⋊C4—2262no—
261C22⋊C4—2262no—
262C22⋊C4—2261no—
263C22⋊C4—2261no—
264C22⋊C4—2261no—
265C42—2261no—
266C23—21no—
267C23—22no—
268C23—42no—
269C23—42no—
270C23—42no—
271C23—1131no—
272C23—1131no—
273C23—4521no—
274C23—2262no—
275C23—2262no—
276C23D4×D11311yescharacteristic
277Q8—21no—
278Q8—21no—
279D4—21no—
280D4—22no—
281D4—22no—
282D4—21no—
283D4—42no—
284D4—42no—
285D4—4524no—
286D4—4522no—
287D4—4524no—
288D4—4522no—
289C2×C4—21no—
290C2×C4—42no—
291C2×C4—1131no—
292C2×C4—1131no—
293C2×C4—4522no—
294C2×C4—2261no—
295C2×C4—2261no—
296C2×C4—2262no—
297C2×C4—2262no—
298C2×C4—2262no—
299C2×C4—2262no—
300C2×C4—2261no—
301C2×C4—2261no—
302C2×C4—2261no—
303C2×C4—2261no—
304C2×C4D4×D11311yescharacteristic
305C2×C4C2×C22⋊D11312yes—
306C2×C4#1426864612yes—
307C2×C4C23×D11311yescharacteristic
308C8—2262no—
309C8C2×C22⋊D11314yescomplemented
310C22—21no—
311C22—21no—
312C22—41no—
313C22—42no—
314C22—42no—
315C22—41no—
316C22—41no—
317C22—42no—
318C22—84no—
319C22—1132no—
320C22—4524no—
321C22—2261no—
322C22—2261no—
323C22—2261no—
324C22—2261no—
325C22—2261no—
326C22#1681359312yes—
327C22C2×D4×D11311yescharacteristic
328C4—41no—
329C4—4522no—
330C4—4521no—
331C4—4521no—
332C4—2261no—
333C4—2261no—
334C4#1681359312yes—
335C4C22×C22⋊D11311yescharacteristic
336C4C2×#1426864611yescharacteristic
337C2—22no—
338C2—82no—
339C2—81no—
340C2—4521no—
341C2—2262no—
342C2C2×#1681359311yescharacteristic
343C2AGL(1,ℤ/8)×D11311yescharacteristic
344C2#2029939411yescharacteristic
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#3055
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×M16×D113C2semidirect
2. #20299105C2semidirect
3. C2×#16813593C2semidirect
4. #20298848C2semidirect
5. #20298841C2semidirect
6. #20298831C2semidirect
7. #20298824C2semidirect
8. #20298640C2semidirect
9. #20298603C2semidirect
10. #20298599C2semidirect
11. C2×AGL(1,ℤ/8)×C113C2semidirect
15. C2×#14268630C22semidirect
17. C2×C8×D113C22semidirect
19. #16813598C22semidirect
20. #16813598C22semidirect
25. #16813593C22semidirect
26. #16813593C22semidirect
33. #16813556C22semidirect
34. #16813554C22semidirect
35. #16813546C22semidirect
36. #16813543C22semidirect
37. #16813543C22semidirect
38. #16813543C22semidirect
42. C2×SD16×C113C22semidirect
43. C2×D8×C113C22semidirect
91. M16×C113D4semidirect
135. C8×C113C2×D4semidirect
154. C2×D113AGL(1,ℤ/8)semidirect
173. C113#3055semidirect
174. C2×AGL(1,ℤ/8)D113semidirect
190. C2×SD16C2×D113semidirect
191. C2×D8C2×D113semidirect
248. M16C22⋊D113semidirect
309. C8C2×C22⋊D113semidirect

Maximal subgroups

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

subgroupindex
C2×M16×D1132
#202991052
C2×#168135932
#202988482
#202988412
#202988312
#202988242
#202986402
#202986032
#202985992
C2×AGL(1,ℤ/8)×C1132
#3055113