Finite Group Databasebeta

← #44559668

Subgroups — #44559668

Order 32128.

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×#22548760C211yesmaximal, characteristic, complemented
2C2×#22548735C211yesmaximal, characteristic, complemented
3#30632265C211yesmaximal, characteristic, complemented
4#30631846C212yesmaximal, complemented
5#30631836C212yesmaximal, complemented
6#30631829C212yesmaximal, complemented
7#30631819C212yesmaximal, complemented
8#30631689C211yesmaximal, characteristic, complemented
9C2×#22548636C211yesmaximal, characteristic, complemented
10#30631655C211yesmaximal, characteristic, complemented
11C2×AGL(1,ℤ/8)×C251C211yesmaximal, characteristic, complemented
12#22548760—24no—
13#22548760—24no—
14C2×#16860810C2211yescharacteristic
15C2×#16860810C2211yescharacteristic
16C2×#16860809—21no—
17C2×#16860809C2212yes—
18C22×C22⋊D251—22no—
19C2×D4×D251—21no—
20C2×D4×D251C2212yes—
21#22548735—24no—
22C2×#16860800C2211yescharacteristic, complemented
23C2×#16860798C2211yescharacteristic, complemented
24C2×#16860797C2211yescharacteristic, complemented
25C2×#16860795C2211yescharacteristic, complemented
26C22×C251⋊C8C2211yescharacteristic, complemented
27#22548694—21no—
28#22548691C2212yescomplemented
29#22548689—21no—
30#22548687—22no—
31#22548686C2212yescomplemented
32C2×C4⋊Dic251C2211yescharacteristic, complemented
33#22548640C2212yescomplemented
34#22548637C2212yescomplemented
35#22548636C2212yescomplemented
36#22548636C2212yescomplemented
37(C2×D251)⋊C8C2212yescomplemented
38(C2×D251)⋊C8C2212yescomplemented
39C2×#41×C251C2211yescharacteristic, complemented
40C22×D4×C251C2211yescharacteristic
41AGL(1,ℤ/8)×C251—24no—
42C2×SD16×C251C2212yescomplemented
43C2×D8×C251C2212yescomplemented
44C2×M16×C251C2211yescharacteristic, complemented
45C23×D251—22no—
46#16860810—22no—
47#16860810—22no—
48#16860810—22no—
49#16860810D414yescomplemented
50#16860809—28no—
51#16860809—24no—
52#16860809—24no—
53C2×Dic502D411yescharacteristic
54C2×Dic502C2311yescharacteristic
55C2×D1004D411yescharacteristic
56C2×D1004C2311yescharacteristic
57C2×C22⋊D251—216no—
58C2×C22⋊D251—22no—
59C2×C22⋊D251—22no—
60C2×C22⋊D251—22no—
61C2×C22⋊D251—21no—
62C2×C22⋊D251D412yes—
63D4×D251—28no—
64D4×D251—24no—
65D4×D251—24no—
66C22×Dic251—22no—
67C22×Dic251—21no—
68C2×C4×D251—21no—
69C2×C4×D251C2312yes—
70#16860800—22no—
71#16860798—22no—
72#16860797—22no—
73#16860795—22no—
74C2×C251⋊C8—22no—
75C2×C251⋊C8C2311yescharacteristic
76C2×C251⋊C8C2311yescharacteristic
77C4⋊Dic251—21no—
78C4⋊Dic251C2312yes—
79Dic251⋊C4—22no—
80C22⋊Dic251—22no—
81(C2×D251)⋊C4—22no—
82C24×C251—21no—
83#41×C251—22no—
84#41×C251—22no—
85C2×Q8×C251C2311yescharacteristic
86C2×D4×C251—21no—
87C2×D4×C251—21no—
88C2×D4×C251—22no—
89C2×D4×C251D414yescomplemented
90C2×D4×C251C2311yescharacteristic
91C2×D4×C251C2312yes—
92C22×C4×C251—21no—
93C22×C4×C251C2311yescharacteristic
94SD16×C251—24no—
95D8×C251—24no—
96M16×C251—22no—
97C2×C8×C251C2312yes—
98C22×D251—21no—
99C22×D251—28no—
100C22×D251—82no—
101C22×D251—82no—
102C22×D251C2×D412yes—
103Dic502—22no—
104Dic502—21no—
105Dic502C2×D412yes—
106D1004—22no—
107D1004—21no—
108D1004C2×D412yes—
109C22⋊D251—28no—
110C22⋊D251—28no—
111C22⋊D251—24no—
112C22⋊D251—24no—
113C22⋊D251—82no—
114C2×Dic251—28no—
115C2×Dic251—21no—
116C2×Dic251—22no—
117C2×Dic251—81no—
118C2×Dic251C2×D412yes—
119C4×D251—24no—
120C4×D251—22no—
121C4×D251—22no—
122C251⋊C8—21no—
123C251⋊C8—21no—
124C23×C251—22no—
125C23×C251—24no—
126C23×C251—21no—
127C23×C251—81no—
128C23×C251—81no—
129C23×C251C2×D411yescharacteristic
130Q8×C251—21no—
131Q8×C251—21no—
132D4×C251—24no—
133D4×C251—22no—
134D4×C251—21no—
135D4×C251—21no—
136D4×C251—42no—
137D4×C251C2×D414yes—
138C2×C4×C251—21no—
139C2×C4×C251—42no—
140C2×C4×C251C2×D411yescharacteristic
141C2×C4×C251C2×D412yes—
142C2×C4×C251C2×D412yes—
143C2×C4×C251C2411yescharacteristic
144C8×C251—22no—
145C22×C251—21no—
146C22×C251—21no—
147C22×C251—24no—
148C22×C251—42no—
149C22×C251—42no—
150C22×C251—41no—
151C22×C251—41no—
152C22×C251—41no—
153C22×C251—84no—
154C22×C251#12012yes—
155C22×C251C22×D411yescharacteristic
156C2×D251—22no—
157C2×D251—22no—
158C2×D251—22no—
159C2×D251—22no—
160C2×D251—41no—
161C2×D251—41no—
162C2×D251—88no—
163C4×C251—41no—
164C4×C251#12012yes—
165C4×C251C22×D411yescharacteristic
166C4×C251C22×D411yescharacteristic
167Dic251—22no—
168Dic251—22no—
169Dic251—42no—
170C2×C251—22no—
171C2×C251—44no—
172C2×C251—81no—
173C2×C251C2×#12011yescharacteristic
174C2×C251#57611yescharacteristic
175C2×C251#57711yescharacteristic
176D251—42no—
177D251—42no—
178C251#301211yescharacteristic, complemented, Sylow
179#3012—2511nomaximal, Sylow
180C2×H(5,𝔽2)—2511no—
181C2×AGL(1,ℤ/8)D25111yescharacteristic, complemented
182C2×#135—2511no—
183#537—2511no—
184#452—2512no—
185#450—2512no—
186#449—2512no—
187#447—2512no—
188#417—2511no—
189C2×#102—2511no—
190#408—2511no—
191H(5,𝔽2)—5024no—
192H(5,𝔽2)—5024no—
193C2×#41—5021no—
194C2×#41—2512no—
195C2×#41—2511no—
196C2×#41—2511no—
197C2×#41C2×D25111yescharacteristic, complemented
198C22×D4—5021no—
199C22×D4—5022no—
200C22×D4—2512no—
201C22×D4C2×D25111yescharacteristic
202AGL(1,ℤ/8)—24no—
203#135—5024no—
204C2×Q16—2511no—
205C2×SD16—2511no—
206C2×SD16—2511no—
207C2×SD16C2×D25112yescomplemented
208C2×D8—2511no—
209C2×D8C2×D25112yescomplemented
210C2×M16C2×D25111yescharacteristic, complemented
211C22×C8—2511no—
212C22⋊Q8—5021no—
213C22⋊Q8—2512no—
214#121—5021no—
215#121—5022no—
216#121—2512no—
217C2×C4⋊C4—2511no—
218#103—2512no—
219#102—2512no—
220#102—2512no—
221#102—2512no—
222C22⋊C8—2512no—
223C22⋊C8—2512no—
224C24—21no—
225C24—5022no—
226#41—22no—
227#41—22no—
228#41—5028no—
229#41—5024no—
230#41—5024no—
231#41—5022no—
232#41—5022no—
233#41—5022no—
234#41—2514no—
235C2×Q8—2511no—
236C2×Q8—2511no—
237C2×Q8C22×D25111yescharacteristic
238C2×D4—21no—
239C2×D4—21no—
240C2×D4—22no—
241C2×D4—5028no—
242C2×D4—5024no—
243C2×D4—5022no—
244C2×D4—50216no—
245C2×D4—5022no—
246C2×D4—5021no—
247C2×D4—5024no—
248C2×D4—5022no—
249C2×D4—2512no—
250C2×D4—2511no—
251C2×D4—2511no—
252C2×D4C22⋊D25114yescomplemented
253C2×D4C22×D25111yescharacteristic
254C2×D4C22×D25112yes—
255C22×C4—21no—
256C22×C4—5021no—
257C22×C4—5022no—
258C22×C4—5021no—
259C22×C4—2512no—
260C22×C4C22×D25111yescharacteristic
261Q16—5022no—
262SD16—24no—
263SD16—5022no—
264SD16—5022no—
265D8—24no—
266D8—5022no—
267M16—22no—
268C2×C8—5022no—
269C2×C8—2511no—
270C2×C8—2511no—
271C2×C8C22×D25112yes—
272C4⋊C4—5022no—
273C4⋊C4—5021no—
274C4⋊C4—2512no—
275C22⋊C4—5022no—
276C22⋊C4—5022no—
277C23—22no—
278C23—24no—
279C23—21no—
280C23—81no—
281C23—81no—
282C23—20082no—
283C23—20082no—
284C23—5028no—
285C23—5021no—
286C23—2512no—
287C23D4×D25111yescharacteristic
288Q8—21no—
289Q8—21no—
290Q8—5021no—
291Q8—5022no—
292Q8—2512no—
293D4—22no—
294D4—21no—
295D4—21no—
296D4—24no—
297D4—42no—
298D4—20082no—
299D4—5028no—
300D4—5028no—
301D4—5024no—
302D4—5024no—
303D4—5021no—
304D4—5022no—
305D4—2512no—
306D4C2×C22⋊D25114yes—
307C2×C4—21no—
308C2×C4—42no—
309C2×C4—20081no—
310C2×C4—5022no—
311C2×C4—5028no—
312C2×C4—5021no—
313C2×C4—5022no—
314C2×C4—5022no—
315C2×C4—5024no—
316C2×C4—2512no—
317C2×C4D4×D25112yes—
318C2×C4D4×D25111yescharacteristic
319C2×C4C2×C22⋊D25112yes—
320C2×C4C23×D25111yescharacteristic
321C8—22no—
322C8—5021no—
323C8—5021no—
324C22—21no—
325C22—21no—
326C22—24no—
327C22—42no—
328C22—42no—
329C22—41no—
330C22—41no—
331C22—41no—
332C22—84no—
333C22—20088no—
334C22—10041no—
335C22—10041no—
336C22—5022no—
337C22—5022no—
338C22—5022no—
339C22—5022no—
340C22#2254868312yes—
341C22C2×D4×D25111yescharacteristic
342C4—41no—
343C4—10042no—
344C4—5022no—
345C4—5022no—
346C4#2254868312yes—
347C4C2×D4×D25111yescharacteristic
348C4C22×C22⋊D25111yescharacteristic
349C2—22no—
350C2—44no—
351C2—81no—
352C2—10042no—
353C2—10042no—
354C2C2×#2254868311yescharacteristic
355C2#3063249111yescharacteristic
356C2#3063249711yescharacteristic
subgroupG/H
centerC22C2×D4×D251
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
FittingC2×AGL(1,ℤ/8)×C251C2
socleC22×C251C22×D4
hypercenterC2×AGL(1,ℤ/8)D251
hypocenterC251#3012
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
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×#22548760C2semidirect
2. C2×#22548735C2semidirect
3. #30632265C2semidirect
4. #30631846C2semidirect
5. #30631836C2semidirect
6. #30631829C2semidirect
7. #30631819C2semidirect
8. #30631689C2semidirect
9. C2×#22548636C2semidirect
10. #30631655C2semidirect
11. C2×AGL(1,ℤ/8)×C251C2semidirect
22. C2×#16860800C22semidirect
23. C2×#16860798C22semidirect
24. C2×#16860797C22semidirect
25. C2×#16860795C22semidirect
26. C22×C251⋊C8C22semidirect
28. #22548691C22semidirect
31. #22548686C22semidirect
32. C2×C4⋊Dic251C22semidirect
33. #22548640C22semidirect
34. #22548637C22semidirect
35. #22548636C22semidirect
36. #22548636C22semidirect
37. (C2×D251)⋊C8C22semidirect
38. (C2×D251)⋊C8C22semidirect
39. C2×#41×C251C22semidirect
42. C2×SD16×C251C22semidirect
43. C2×D8×C251C22semidirect
44. C2×M16×C251C22semidirect
49. #16860810D4semidirect
89. C2×D4×C251D4semidirect
178. C251#3012semidirect
181. C2×AGL(1,ℤ/8)D251semidirect
197. C2×#41C2×D251semidirect
207. C2×SD16C2×D251semidirect
209. C2×D8C2×D251semidirect
210. C2×M16C2×D251semidirect
252. C2×D4C22⋊D251semidirect

Maximal subgroups

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

subgroupindex
C2×#225487602
C2×#225487352
#306322652
#306318462
#306318362
#306318292
#306318192
#306316892
C2×#225486362
#306316552
C2×AGL(1,ℤ/8)×C2512
#3012251