Finite Group Databasebeta

← #28399802

Subgroups — #28399802

Order 14464.

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

normal subgroups only

This group has 994 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
1C22×D4×D113C211yesmaximal, characteristic, complemented
2C2×#16813649C212yesmaximal, complemented
3#20299111C211yesmaximal, characteristic, complemented
4#20298748C211yesmaximal, characteristic, complemented
5#20298747C212yesmaximal, complemented
6#20298734C211yesmaximal, characteristic, complemented
7#20298733C212yesmaximal, complemented
8#20298732C211yesmaximal, characteristic, complemented
9#20298601C212yesmaximal, complemented
10#20298560C211yesmaximal, characteristic, complemented
11#450×C113C211yesmaximal, characteristic, complemented
12C24×D113—21no—
13C2×#14268648C2211yescharacteristic, complemented
14C22×D452C2211yescharacteristic
15C22×C22⋊D113—21no—
16C2×D4×D113—28no—
17C2×D4×D113—21no—
18C2×D4×D113—21no—
19C2×D4×D113C2212yes—
20C2×D4×D113C2212yes—
21C22×C4×D113C2211yescharacteristic
22#16813649—28no—
23C2×#14268634C2212yescomplemented
24C2×D904C2212yescomplemented
25C2×#14268632C2212yescomplemented
26C2×#14268630C2212yescomplemented
27#16813607—21no—
28#16813598C2211yescharacteristic, complemented
29#16813597C2211yescharacteristic, complemented
30#16813596C2211yescharacteristic, complemented
31#16813592—21no—
32#16813584—21no—
33#16813582—21no—
34#16813580C2211yescharacteristic, complemented
35#16813544C2212yescomplemented
36#16813543C2212yescomplemented
37#16813542C2212yescomplemented
38C22⋊#11744751C2211yescharacteristic, complemented
39#16813534C2212yescomplemented
40C22×D4×C113C2211yescharacteristic
41C2×SD16×C113C2212yescomplemented
42C22⋊Q8×C113C2211yescharacteristic, complemented
43#102×C113C2212yescomplemented
44C22⋊C8×C113C2211yescharacteristic, complemented
45C23×D113—28no—
46C23×D113—21no—
47C23×D113—22no—
48C23×D113—22no—
49C23×D113—44no—
50C23×D113—44no—
51C23×D113D411yescharacteristic
52#14268648—24no—
53C2×D452—21no—
54C2×D452—24no—
55C2×D452—21no—
56C2×D452—21no—
57C2×D452C2312yes—
58C2×D452C2311yescharacteristic, complemented
59C2×C22⋊D113—28no—
60C2×C22⋊D113—21no—
61C2×C22⋊D113—22no—
62C2×C22⋊D113—22no—
63D4×D113—28no—
64D4×D113—28no—
65D4×D113—24no—
66D4×D113—24no—
67D4×D113—24no—
68D4×D113D418yescomplemented
69C22×Dic113D411yescharacteristic
70C2×C4×D113—24no—
71C2×C4×D113—21no—
72C2×C4×D113D411yescharacteristic
73C2×C4×D113D411yescharacteristic
74C2×C4×D113C2311yescharacteristic
75C2×C4×D113C2311yescharacteristic
76#14268634—24no—
77D904—24no—
78#14268632—24no—
79#14268630—24no—
80C2×#11744751C2312yescomplemented
81C4⋊Dic113C2311yescharacteristic, complemented
82#14268626—21no—
83#14268624—21no—
84#14268624—21no—
85#14268624—21no—
86#14268624—21no—
87C4×Dic113—21no—
88C24×C113—21no—
89C2×Q8×C113C2311yescharacteristic, complemented
90C2×D4×C113—24no—
91C2×D4×C113—21no—
92C2×D4×C113C2312yes—
93C22×C4×C113C2311yescharacteristic
94SD16×C113—24no—
95C2×C8×C113C2312yescomplemented
96C4⋊C4×C113—21no—
97C4⋊C4×C113C2311yescharacteristic, complemented
98C22⋊C4×C113—21no—
99C23×C113—24no—
100C23×C113—22no—
101C23×C113—42no—
102C23×C113—42no—
103C23×C113C2×D411yescharacteristic
104C22×D113—22no—
105C22×D113—22no—
106C22×D113—22no—
107C22×D113—28no—
108C22×D113—24no—
109C22×D113—22no—
110C22×D113—28no—
111C22×D113—21no—
112C22×D113—432no—
113C22×D113—44no—
114C22×D113—44no—
115C22×D113—44no—
116C22×D113—42no—
117C22×D113—42no—
118C22×D113—44no—
119C22×D113C2×D411yescharacteristic
120C22×D113C2×D411yescharacteristic
121Q8×C113—22no—
122D4×C113—24no—
123D4×C113—22no—
124D4×C113C2×D414yes—
125C2×C4×C113—22no—
126C2×C4×C113—21no—
127C2×C4×C113—21no—
128C2×C4×C113C2×D411yescharacteristic
129C2×C4×C113C2411yescharacteristic
130C22⋊D113—28no—
131C22⋊D113—28no—
132C22⋊D113—42no—
133C2×Dic113—22no—
134C2×Dic113—21no—
135C2×Dic113C2×D411yescharacteristic
136C2×Dic113C2×D411yescharacteristic
137D452—24no—
138D452—22no—
139D452—22no—
140D452—42no—
141D452—42no—
142D452C2×D414yes—
143C4×D113—22no—
144C4×D113—22no—
145C4×D113—22no—
146C4×D113—42no—
147C4×D113C2×D414yes—
148C8×C113—22no—
149#11744751—22no—
150C22×C113—21no—
151C22×C113—21no—
152C22×C113—24no—
153C22×C113—48no—
154C22×C113—42no—
155C22×C113—42no—
156C22×C113AGL(1,ℤ/8)12yescomplemented
157C22×C113C22×D411yescharacteristic
158C2×D113—28no—
159C2×D113—21no—
160C2×D113—21no—
161C2×D113—21no—
162C2×D113—21no—
163C2×D113—24no—
164C2×D113—416no—
165C2×D113—416no—
166C2×D113—48no—
167C2×D113—42no—
168C2×D113—42no—
169C2×D113—41no—
170C2×D113—41no—
171C2×D113—41no—
172C2×D113#12012yes—
173C2×D113AGL(1,ℤ/8)12yescomplemented
174C4×C113—21no—
175C4×C113—41no—
176C4×C113—41no—
177C4×C113C22×D412yes—
178Dic113—21no—
179Dic113—41no—
180Dic113#12012yes—
181C2×C113—22no—
182C2×C113—44no—
183C2×C113C2×#12011yescharacteristic
184C2×C113C2×AGL(1,ℤ/8)11yescharacteristic
185C2×C113C2×AGL(1,ℤ/8)11yescharacteristic
186D113—22no—
187D113—22no—
188D113—44no—
189D113—81no—
190#3003—1131nomaximal, Sylow
191C113#300311yescharacteristic, complemented, Sylow
192C23×D4—1131no—
193C2×AGL(1,ℤ/8)—1132no—
194#525—1131no—
195#450—1132no—
196#450—1131no—
197#450D11311yescharacteristic, complemented
198#447—1131no—
199#447—1132no—
200#447—1131no—
201#418—1132no—
202#407—1131no—
203C25—2261no—
204C2×#41—1131no—
205C22×D4—1132no—
206C22×D4—1132no—
207C22×D4—1131no—
208C22×D4—2261no—
209C22×D4—2261no—
210C22×D4—2261no—
211C22×D4—2268no—
212C22×D4C2×D11311yescharacteristic
213C23×C4—1131no—
214AGL(1,ℤ/8)—2268no—
215C2×SD16—1132no—
216C2×SD16C2×D11312yescomplemented
217C2×D8—1132no—
218C2×D8—1132no—
219C2×M16—1132no—
220#123—2261no—
221C22⋊Q8—1131no—
222C22⋊Q8C2×D11311yescharacteristic, complemented
223#121—1131no—
224#121—1131no—
225#120—2261no—
226C4×D4—2261no—
227C4×D4—2261no—
228#117—1131no—
229#102—1132no—
230#102—1132no—
231#102—1132no—
232#102C2×D11312yescomplemented
233C22⋊C8—1131no—
234C22⋊C8—1132no—
235C22⋊C8C2×D11311yescharacteristic, complemented
236C24—21no—
237C24—1131no—
238C24—4524no—
239C24—4524no—
240C24—2262no—
241C24—2262no—
242C24—2268no—
243C24—2261no—
244#41—2264no—
245C2×Q8C22×D11311yescharacteristic, complemented
246C2×D4—24no—
247C2×D4—21no—
248C2×D4—1138no—
249C2×D4—1132no—
250C2×D4—1131no—
251C2×D4—2264no—
252C2×D4—2261no—
253C2×D4—2261no—
254C2×D4—2264no—
255C2×D4—2262no—
256C2×D4—2262no—
257C2×D4—2264no—
258C2×D4—2268no—
259C2×D4—2268no—
260C2×D4—2261no—
261C2×D4—2268no—
262C2×D4—2264no—
263C2×D4—2261no—
264C2×D4C22×D11312yes—
265C22×C4—1131no—
266C22×C4—1131no—
267C22×C4—1131no—
268C22×C4—1131no—
269C22×C4—1131no—
270C22×C4—2261no—
271C22×C4—2264no—
272C22×C4C22×D11311yescharacteristic
273SD16—24no—
274SD16—2264no—
275D8—2264no—
276D8—2264no—
277M16—2264no—
278C2×C8—1132no—
279C2×C8C22×D11312yescomplemented
280C4⋊C4—21no—
281C4⋊C4—1131no—
282C4⋊C4—2261no—
283C4⋊C4C22×D11311yescharacteristic, complemented
284C22⋊C4—21no—
285C22⋊C4—2261no—
286C22⋊C4—2261no—
287C22⋊C4—2261no—
288C22⋊C4—2261no—
289C42—2261no—
290C23—24no—
291C23—22no—
292C23—42no—
293C23—42no—
294C23—1131no—
295C23—1131no—
296C23—4522no—
297C23—4524no—
298C23—45232no—
299C23—4524no—
300C23—4524no—
301C23—4524no—
302C23—4522no—
303C23—2268no—
304C23—2262no—
305C23—2262no—
306C23—2264no—
307C23—2268no—
308C23—2261no—
309C23—2262no—
310C23—2262no—
311C23D4×D11311yescharacteristic
312Q8—22no—
313D4—24no—
314D4—22no—
315D4—1134no—
316D4—4522no—
317D4—4522no—
318D4—4522no—
319D4—2264no—
320D4—2268no—
321D4—2268no—
322D4—2262no—
323D4—2262no—
324D4D4×D11314yes—
325C2×C4—22no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—1131no—
329C2×C4—1134no—
330C2×C4—1131no—
331C2×C4—4522no—
332C2×C4—2262no—
333C2×C4—2262no—
334C2×C4—2262no—
335C2×C4—2261no—
336C2×C4—2262no—
337C2×C4D4×D11311yescharacteristic
338C2×C4C23×D11311yescharacteristic
339C8—22no—
340C8—2262no—
341C22—21no—
342C22—21no—
343C22—24no—
344C22—48no—
345C22—42no—
346C22—42no—
347C22—1132no—
348C22—1132no—
349C22—4528no—
350C22—45216no—
351C22—45216no—
352C22—4521no—
353C22—4521no—
354C22—4521no—
355C22—4522no—
356C22—4522no—
357C22—2268no—
358C22—2264no—
359C22—2261no—
360C22—2261no—
361C22—2261no—
362C22—2261no—
363C22#1681364912yescomplemented
364C22C2×D4×D11311yescharacteristic
365C4—21no—
366C4—41no—
367C4—41no—
368C4—1132no—
369C4—4521no—
370C4—2261no—
371C4C2×D4×D11312yes—
372C2—22no—
373C2—44no—
374C2—9041no—
375C2—4524no—
376C2—2262no—
377C2—2262no—
378C2#120×D11311yescharacteristic
379C2AGL(1,ℤ/8)×D11311yescharacteristic
380C2C2×#1681364911yescharacteristic
subgroupG/H
centerC22C2×D4×D113
commutator, cosocleC2×C4×C113C24
FrattiniC2×C4C23×D113
Fitting#450×C113C2
socleC22×C113C22×D4
hypercenter#450D113
hypocenterC113#3003
LCSG ▷ C2×C4×C113 ▷ C2×C113 ▷ C113
UCSC1 ◁ C22 ◁ C22×C4 ◁ #450

Complements

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

NH ≅ G/Ntype
1. C22×D4×D113C2semidirect
2. C2×#16813649C2semidirect
3. #20299111C2semidirect
4. #20298748C2semidirect
5. #20298747C2semidirect
6. #20298734C2semidirect
7. #20298733C2semidirect
8. #20298732C2semidirect
9. #20298601C2semidirect
10. #20298560C2semidirect
11. #450×C113C2semidirect
13. C2×#14268648C22semidirect
23. C2×#14268634C22semidirect
24. C2×D904C22semidirect
25. C2×#14268632C22semidirect
26. C2×#14268630C22semidirect
28. #16813598C22semidirect
29. #16813597C22semidirect
30. #16813596C22semidirect
34. #16813580C22semidirect
35. #16813544C22semidirect
36. #16813543C22semidirect
37. #16813542C22semidirect
38. C22⋊#11744751C22semidirect
39. #16813534C22semidirect
41. C2×SD16×C113C22semidirect
42. C22⋊Q8×C113C22semidirect
43. #102×C113C22semidirect
44. C22⋊C8×C113C22semidirect
58. C2×D452C23semidirect
68. D4×D113D4semidirect
80. C2×#11744751C23semidirect
81. C4⋊Dic113C23semidirect
89. C2×Q8×C113C23semidirect
95. C2×C8×C113C23semidirect
97. C4⋊C4×C113C23semidirect
156. C22×C113AGL(1,ℤ/8)semidirect
173. C2×D113AGL(1,ℤ/8)semidirect
191. C113#3003semidirect
197. #450D113semidirect
216. C2×SD16C2×D113semidirect
222. C22⋊Q8C2×D113semidirect
232. #102C2×D113semidirect
235. C22⋊C8C2×D113semidirect
245. C2×Q8C22×D113semidirect
279. C2×C8C22×D113semidirect
283. C4⋊C4C22×D113semidirect
363. C22#16813649semidirect

Maximal subgroups

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

subgroupindex
C22×D4×D1132
C2×#168136492
#202991112
#202987482
#202987472
#202987342
#202987332
#202987322
#202986012
#202985602
#450×C1132
#3003113