Finite Group Databasebeta

← #28399877

Subgroups — #28399877

Order 14464.

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

normal subgroups only

This group has 764 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×#16813667C211yesmaximal, characteristic, complemented
2C2×#16813642C211yesmaximal, characteristic, complemented
3#20299176C211yesmaximal, characteristic, complemented
4#20298757C212yesmaximal, complemented
5#20298747C212yesmaximal, complemented
6#20298740C212yesmaximal, complemented
7#20298730C212yesmaximal, complemented
8#20298600C211yesmaximal, characteristic, complemented
9C2×#16813543C211yesmaximal, characteristic, complemented
10#20298566C211yesmaximal, characteristic, complemented
11C2×AGL(1,ℤ/8)×C113C211yesmaximal, characteristic, complemented
12#16813667—24no—
13#16813667—24no—
14C2×#14268647C2211yescharacteristic
15C2×#14268647C2211yescharacteristic
16C2×#14268646—21no—
17C2×#14268646C2212yes—
18C22×C22⋊D113—22no—
19C2×D4×D113—21no—
20C2×D4×D113C2212yes—
21#16813642—24no—
22C2×#14268637C2211yescharacteristic, complemented
23C2×#14268635C2211yescharacteristic, complemented
24C2×#14268634C2211yescharacteristic, complemented
25C2×#14268632C2211yescharacteristic, complemented
26C22×#11744751C2211yescharacteristic, complemented
27#16813601—21no—
28#16813598C2212yescomplemented
29#16813596—21no—
30#16813594—22no—
31#16813593C2212yescomplemented
32C2×C4⋊Dic113C2211yescharacteristic, complemented
33#16813547C2212yescomplemented
34#16813544C2212yescomplemented
35#16813543C2212yescomplemented
36#16813543C2212yescomplemented
37#16813534C2212yescomplemented
38#16813534C2212yescomplemented
39C2×#41×C113C2211yescharacteristic, complemented
40C22×D4×C113C2211yescharacteristic
41AGL(1,ℤ/8)×C113—24no—
42C2×SD16×C113C2212yescomplemented
43C2×D8×C113C2212yescomplemented
44C2×M16×C113C2211yescharacteristic, complemented
45C23×D113—22no—
46#14268647—22no—
47#14268647—22no—
48#14268647—22no—
49#14268647D414yescomplemented
50#14268646—28no—
51#14268646—24no—
52#14268646—24no—
53C2×Dic226D411yescharacteristic
54C2×Dic226C2311yescharacteristic
55C2×D452D411yescharacteristic
56C2×D452C2311yescharacteristic
57C2×C22⋊D113—216no—
58C2×C22⋊D113—21no—
59C2×C22⋊D113—22no—
60C2×C22⋊D113—22no—
61C2×C22⋊D113—22no—
62C2×C22⋊D113D412yes—
63D4×D113—28no—
64D4×D113—24no—
65D4×D113—24no—
66C22×Dic113—22no—
67C22×Dic113—21no—
68C2×C4×D113—21no—
69C2×C4×D113C2312yes—
70#14268637—22no—
71#14268635—22no—
72#14268634—22no—
73#14268632—22no—
74C2×#11744751—22no—
75C2×#11744751C2311yescharacteristic
76C2×#11744751C2311yescharacteristic
77C4⋊Dic113—21no—
78C4⋊Dic113C2312yes—
79#14268626—22no—
80C22⋊Dic113—22no—
81#14268624—22no—
82C24×C113—21no—
83#41×C113—22no—
84#41×C113—22no—
85C2×Q8×C113C2311yescharacteristic
86C2×D4×C113—21no—
87C2×D4×C113—21no—
88C2×D4×C113—22no—
89C2×D4×C113D414yescomplemented
90C2×D4×C113C2311yescharacteristic
91C2×D4×C113C2312yes—
92C22×C4×C113—21no—
93C22×C4×C113C2311yescharacteristic
94SD16×C113—24no—
95D8×C113—24no—
96M16×C113—22no—
97C2×C8×C113C2312yes—
98C23×C113—22no—
99C23×C113—24no—
100C23×C113—21no—
101C23×C113—81no—
102C23×C113—81no—
103C23×C113C2×D411yescharacteristic
104C22×D113—21no—
105C22×D113—28no—
106C22×D113—82no—
107C22×D113—82no—
108C22×D113C2×D412yes—
109Q8×C113—21no—
110Q8×C113—21no—
111D4×C113—24no—
112D4×C113—22no—
113D4×C113—21no—
114D4×C113—21no—
115D4×C113—42no—
116D4×C113C2×D414yes—
117C2×C4×C113—21no—
118C2×C4×C113—42no—
119C2×C4×C113C2×D411yescharacteristic
120C2×C4×C113C2×D412yes—
121C2×C4×C113C2×D412yes—
122C2×C4×C113C2411yescharacteristic
123C22⋊D113—28no—
124C22⋊D113—28no—
125C22⋊D113—24no—
126C22⋊D113—24no—
127C22⋊D113—82no—
128C2×Dic113—28no—
129C2×Dic113—21no—
130C2×Dic113—22no—
131C2×Dic113—81no—
132C2×Dic113C2×D412yes—
133D452—22no—
134D452—21no—
135D452C2×D412yes—
136C4×D113—24no—
137C4×D113—22no—
138C4×D113—22no—
139Dic226—22no—
140Dic226—21no—
141Dic226C2×D412yes—
142C8×C113—22no—
143#11744751—21no—
144#11744751—21no—
145C22×C113—21no—
146C22×C113—21no—
147C22×C113—24no—
148C22×C113—42no—
149C22×C113—42no—
150C22×C113—41no—
151C22×C113—41no—
152C22×C113—41no—
153C22×C113—84no—
154C22×C113#12012yes—
155C22×C113C22×D411yescharacteristic
156C2×D113—22no—
157C2×D113—22no—
158C2×D113—22no—
159C2×D113—22no—
160C2×D113—41no—
161C2×D113—41no—
162C2×D113—88no—
163C4×C113—41no—
164C4×C113#12012yes—
165C4×C113C22×D411yescharacteristic
166C4×C113C22×D411yescharacteristic
167Dic113—22no—
168Dic113—22no—
169Dic113—42no—
170C2×C113—22no—
171C2×C113—44no—
172C2×C113—81no—
173C2×C113C2×#12011yescharacteristic
174C2×C113#57611yescharacteristic
175C2×C113#57711yescharacteristic
176D113—42no—
177D113—42no—
178#3012—1131nomaximal, Sylow
179C113#301211yescharacteristic, complemented, Sylow
180C2×H(5,𝔽2)—1131no—
181C2×AGL(1,ℤ/8)D11311yescharacteristic, complemented
182C2×#135—1131no—
183#537—1131no—
184#452—1132no—
185#450—1132no—
186#449—1132no—
187#447—1132no—
188#417—1131no—
189C2×#102—1131no—
190#408—1131no—
191H(5,𝔽2)—2264no—
192H(5,𝔽2)—2264no—
193C2×#41—1131no—
194C2×#41—1132no—
195C2×#41—1131no—
196C2×#41—2261no—
197C2×#41C2×D11311yescharacteristic, complemented
198C22×D4—1132no—
199C22×D4—2261no—
200C22×D4—2262no—
201C22×D4C2×D11311yescharacteristic
202AGL(1,ℤ/8)—24no—
203#135—2264no—
204C2×Q16—1131no—
205C2×SD16—1131no—
206C2×SD16—1131no—
207C2×SD16C2×D11312yescomplemented
208C2×D8—1131no—
209C2×D8C2×D11312yescomplemented
210C2×M16C2×D11311yescharacteristic, complemented
211C22×C8—1131no—
212C22⋊Q8—1132no—
213C22⋊Q8—2261no—
214#121—1132no—
215#121—2261no—
216#121—2262no—
217C2×C4⋊C4—1131no—
218#103—1132no—
219#102—1132no—
220#102—1132no—
221#102—1132no—
222C22⋊C8—1132no—
223C22⋊C8—1132no—
224C24—21no—
225C24—2262no—
226#41—22no—
227#41—22no—
228#41—1134no—
229#41—2264no—
230#41—2268no—
231#41—2264no—
232#41—2262no—
233#41—2262no—
234#41—2262no—
235C2×Q8—1131no—
236C2×Q8—1131no—
237C2×Q8C22×D11311yescharacteristic
238C2×D4—21no—
239C2×D4—21no—
240C2×D4—22no—
241C2×D4—1132no—
242C2×D4—1131no—
243C2×D4—1131no—
244C2×D4—2262no—
245C2×D4—2264no—
246C2×D4—2268no—
247C2×D4—2262no—
248C2×D4—2262no—
249C2×D4—22616no—
250C2×D4—2261no—
251C2×D4—2264no—
252C2×D4C22⋊D11314yescomplemented
253C2×D4C22×D11311yescharacteristic
254C2×D4C22×D11312yes—
255C22×C4—21no—
256C22×C4—1132no—
257C22×C4—2262no—
258C22×C4—2261no—
259C22×C4—2261no—
260C22×C4C22×D11311yescharacteristic
261Q16—2262no—
262SD16—24no—
263SD16—2262no—
264SD16—2262no—
265D8—24no—
266D8—2262no—
267M16—22no—
268C2×C8—1131no—
269C2×C8—1131no—
270C2×C8—2262no—
271C2×C8C22×D11312yes—
272C4⋊C4—1132no—
273C4⋊C4—2262no—
274C4⋊C4—2261no—
275C22⋊C4—2262no—
276C22⋊C4—2262no—
277C23—22no—
278C23—24no—
279C23—21no—
280C23—81no—
281C23—81no—
282C23—1132no—
283C23—9042no—
284C23—9042no—
285C23—2268no—
286C23—2261no—
287C23D4×D11311yescharacteristic
288Q8—21no—
289Q8—21no—
290Q8—1132no—
291Q8—2262no—
292Q8—2261no—
293D4—24no—
294D4—22no—
295D4—21no—
296D4—21no—
297D4—42no—
298D4—1132no—
299D4—9042no—
300D4—2268no—
301D4—2268no—
302D4—2264no—
303D4—2264no—
304D4—2262no—
305D4—2261no—
306D4C2×C22⋊D11314yes—
307C2×C4—21no—
308C2×C4—42no—
309C2×C4—1132no—
310C2×C4—9041no—
311C2×C4—2262no—
312C2×C4—2268no—
313C2×C4—2261no—
314C2×C4—2264no—
315C2×C4—2262no—
316C2×C4—2262no—
317C2×C4D4×D11311yescharacteristic
318C2×C4D4×D11312yes—
319C2×C4C2×C22⋊D11312yes—
320C2×C4C23×D11311yescharacteristic
321C8—22no—
322C8—2261no—
323C8—2261no—
324C22—21no—
325C22—21no—
326C22—24no—
327C22—42no—
328C22—42no—
329C22—41no—
330C22—41no—
331C22—41no—
332C22—84no—
333C22—9048no—
334C22—4521no—
335C22—4521no—
336C22—2262no—
337C22—2262no—
338C22—2262no—
339C22—2262no—
340C22#1681359012yes—
341C22C2×D4×D11311yescharacteristic
342C4—41no—
343C4—4522no—
344C4—2262no—
345C4—2262no—
346C4#1681359012yes—
347C4C2×D4×D11311yescharacteristic
348C4C22×C22⋊D11311yescharacteristic
349C2—22no—
350C2—44no—
351C2—81no—
352C2—4522no—
353C2—4522no—
354C2C2×#1681359011yescharacteristic
355C2#2029940211yescharacteristic
356C2#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#3012
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×#16813667C2semidirect
2. C2×#16813642C2semidirect
3. #20299176C2semidirect
4. #20298757C2semidirect
5. #20298747C2semidirect
6. #20298740C2semidirect
7. #20298730C2semidirect
8. #20298600C2semidirect
9. C2×#16813543C2semidirect
10. #20298566C2semidirect
11. C2×AGL(1,ℤ/8)×C113C2semidirect
22. C2×#14268637C22semidirect
23. C2×#14268635C22semidirect
24. C2×#14268634C22semidirect
25. C2×#14268632C22semidirect
26. C22×#11744751C22semidirect
28. #16813598C22semidirect
31. #16813593C22semidirect
32. C2×C4⋊Dic113C22semidirect
33. #16813547C22semidirect
34. #16813544C22semidirect
35. #16813543C22semidirect
36. #16813543C22semidirect
37. #16813534C22semidirect
38. #16813534C22semidirect
39. C2×#41×C113C22semidirect
42. C2×SD16×C113C22semidirect
43. C2×D8×C113C22semidirect
44. C2×M16×C113C22semidirect
49. #14268647D4semidirect
89. C2×D4×C113D4semidirect
179. C113#3012semidirect
181. C2×AGL(1,ℤ/8)D113semidirect
197. C2×#41C2×D113semidirect
207. C2×SD16C2×D113semidirect
209. C2×D8C2×D113semidirect
210. C2×M16C2×D113semidirect
252. C2×D4C22⋊D113semidirect

Maximal subgroups

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

subgroupindex
C2×#168136672
C2×#168136422
#202991762
#202987572
#202987472
#202987402
#202987302
#202986002
C2×#168135432
#202985662
C2×AGL(1,ℤ/8)×C1132
#3012113