Finite Group Databasebeta

← #44559616

Subgroups — #44559616

Order 32128.

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

normal subgroups only

This group has 806 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
1C22×#16860811C211yesmaximal, characteristic, complemented
2C2×#22548742C212yesmaximal, complemented
3#121×D251C211yesmaximal, characteristic, complemented
4#30631835C212yesmaximal, complemented
5#30631831C212yesmaximal, complemented
6#30631826C211yesmaximal, characteristic, complemented
7#30631823C211yesmaximal, characteristic, complemented
8#30631820C211yesmaximal, characteristic, complemented
9#30631687C212yesmaximal, complemented
10#30631649C211yesmaximal, characteristic, complemented
11#448×C251C211yesmaximal, characteristic, complemented
12C2×#16860811—28no—
13C2×#16860811—21no—
14C2×#16860811—21no—
15C2×#16860811C2212yes—
16C2×#16860811C2212yescomplemented
17C22×D1004—21no—
18C22×D1004C2211yescharacteristic, complemented
19C2×D4×D251—21no—
20C2×D4×D251C2211yescharacteristic, complemented
21C22×C4×D251—21no—
22C22×C4×D251C2211yescharacteristic
23#22548742—28no—
24C2×#16860797C2212yescomplemented
25C2×D2008C2212yescomplemented
26C2×#16860795C2212yescomplemented
27C2×#16860793C2212yescomplemented
28#22548690C2211yescharacteristic, complemented
29#22548689C2211yescharacteristic
30#22548688—21no—
31#22548687—21no—
32#22548686C2211yescharacteristic, complemented
33C4⋊C4×D251C2211yescharacteristic, complemented
34C22⋊C4×D251—21no—
35#22548639C2212yescomplemented
36#22548637C2212yescomplemented
37#22548635C2212yescomplemented
38C22⋊(C251⋊C8)C2211yescharacteristic, complemented
39(C2×D251)⋊C8C2212yescomplemented
40C22×Q8×C251C2211yescharacteristic, complemented
41C2×SD16×C251C2212yescomplemented
42#121×C251C2211yescharacteristic
43#103×C251C2212yescomplemented
44C22⋊C8×C251C2211yescharacteristic, complemented
45C23×D251—21no—
46C23×D251—21no—
47C23×D251D411yescharacteristic
48#16860811—28no—
49#16860811—28no—
50#16860811—24no—
51#16860811—24no—
52#16860811—24no—
53#16860811D418yescomplemented
54C2×D1004—28no—
55C2×D1004—22no—
56C2×D1004—22no—
57C2×D1004—24no—
58C2×D1004—21no—
59C2×D1004—21no—
60C2×D1004C2311yescharacteristic
61C2×D1004C2312yes—
62C2×C22⋊D251—21no—
63C2×C22⋊D251—21no—
64C2×C22⋊D251—21no—
65D4×D251—24no—
66D4×D251—44no—
67C22×Dic251D411yescharacteristic
68C2×C4×D251—28no—
69C2×C4×D251—22no—
70C2×C4×D251—22no—
71C2×C4×D251—24no—
72C2×C4×D251—21no—
73C2×C4×D251D411yescharacteristic
74C2×C4×D251D411yescharacteristic
75C2×C4×D251C2311yescharacteristic
76C2×C4×D251C2311yescharacteristic
77#16860797—24no—
78D2008—24no—
79#16860795—24no—
80#16860793—24no—
81C2×C251⋊C8C2312yes—
82C4⋊Dic251C2311yescharacteristic
83Dic251⋊C4—21no—
84C22⋊Dic251—21no—
85(C2×D251)⋊C4—21no—
86(C2×D251)⋊C4—21no—
87C2×Q8×C251—24no—
88C2×Q8×C251—21no—
89C2×Q8×C251C2312yes—
90C2×D4×C251—21no—
91C2×D4×C251C2311yescharacteristic
92C22×C4×C251—21no—
93C22×C4×C251C2311yescharacteristic
94SD16×C251—24no—
95C2×C8×C251C2312yes—
96C4⋊C4×C251C2311yescharacteristic
97C22⋊C4×C251—21no—
98C22×D251—24no—
99C22×D251—22no—
100C22×D251—22no—
101C22×D251—22no—
102C22×D251—21no—
103C22×D251—41no—
104C22×D251—42no—
105C22×D251—42no—
106C22×D251—42no—
107C22×D251—42no—
108C22×D251—42no—
109C22×D251C2×D411yescharacteristic
110C22×D251C2×D411yescharacteristic
111D1004—28no—
112D1004—24no—
113D1004—28no—
114D1004—22no—
115D1004—22no—
116D1004—42no—
117D1004C2×D414yescomplemented
118C22⋊D251—42no—
119C22⋊D251—42no—
120C22⋊D251—42no—
121C2×Dic251—22no—
122C2×Dic251—21no—
123C2×Dic251C2×D411yescharacteristic
124C2×Dic251C2×D411yescharacteristic
125C4×D251—28no—
126C4×D251—22no—
127C4×D251—22no—
128C4×D251—28no—
129C4×D251—22no—
130C4×D251—42no—
131C4×D251C2×D414yes—
132C251⋊C8—22no—
133C23×C251—21no—
134C23×C251C2×D411yescharacteristic
135Q8×C251—24no—
136Q8×C251—22no—
137Q8×C251C2×D414yescomplemented
138D4×C251—22no—
139D4×C251—42no—
140C2×C4×C251—24no—
141C2×C4×C251—22no—
142C2×C4×C251—22no—
143C2×C4×C251—21no—
144C2×C4×C251C2×D411yescharacteristic
145C2×C4×C251C2411yescharacteristic
146C8×C251—22no—
147C22×C251—21no—
148C22×C251—21no—
149C22×C251—41no—
150C22×C251—41no—
151C22×C251—41no—
152C22×C251AGL(1,ℤ/8)12yescomplemented
153C22×C251C22×D411yescharacteristic
154C2×D251—21no—
155C2×D251—21no—
156C2×D251—21no—
157C2×D251—24no—
158C2×D251—21no—
159C2×D251—21no—
160C2×D251—44no—
161C2×D251—48no—
162C2×D251—42no—
163C2×D251—42no—
164C2×D251—41no—
165C2×D251—41no—
166C2×D251—41no—
167C2×D251—84no—
168C2×D251#12012yes—
169C4×C251—24no—
170C4×C251—21no—
171C4×C251—41no—
172C4×C251C22×D412yes—
173Dic251—21no—
174Dic251—41no—
175Dic251#12012yes—
176C2×C251—22no—
177C2×C251—81no—
178C2×C251C2×#12011yescharacteristic
179C2×C251C2×AGL(1,ℤ/8)11yescharacteristic
180C2×C251#57511yescharacteristic
181D251—22no—
182D251—41no—
183D251—44no—
184D251—81no—
185C251#300611yescharacteristic, complemented, Sylow
186#3006—2511nomaximal, Sylow
187C22×#41—2511no—
188C2×AGL(1,ℤ/8)—2512no—
189C2×#121—2511no—
190#449—2512no—
191#449—2512no—
192#448—2511no—
193#448D25111yescharacteristic, complemented
194#447—2511no—
195#447—2511no—
196#417—2512no—
197#407—2511no—
198C2×#41—5028no—
199C2×#41—5021no—
200C2×#41—5021no—
201C2×#41—2512no—
202C2×#41—2512no—
203C22×Q8C2×D25111yescharacteristic, complemented
204C22×D4—5021no—
205C22×D4—5021no—
206C22×D4—2511no—
207C22×D4—2511no—
208C23×C4—5021no—
209C23×C4—2511no—
210AGL(1,ℤ/8)—5028no—
211C2×SD16—2512no—
212C2×SD16C2×D25112yescomplemented
213C2×D8—2512no—
214C2×D8—2512no—
215C2×M16—2512no—
216#121—5021no—
217#121—5021no—
218#121—2511no—
219#121—2511no—
220#121—2511no—
221#121C2×D25111yescharacteristic
222C2×C4⋊C4—2511no—
223C2×C22⋊C4—5021no—
224#103—2512no—
225#103C2×D25112yescomplemented
226#102—2512no—
227#102—2512no—
228C22⋊C8—2511no—
229C22⋊C8—2512no—
230C22⋊C8C2×D25111yescharacteristic, complemented
231C24—5021no—
232C24—5021no—
233C24—2511no—
234#41—5028no—
235#41—5028no—
236#41—5024no—
237#41—5024no—
238#41—5024no—
239#41—2518no—
240C2×Q8—24no—
241C2×Q8—21no—
242C2×Q8C22×D25112yes—
243C2×D4—21no—
244C2×D4—10044no—
245C2×D4—5021no—
246C2×D4—5028no—
247C2×D4—5022no—
248C2×D4—5022no—
249C2×D4—5024no—
250C2×D4—5021no—
251C2×D4—5021no—
252C2×D4—5024no—
253C2×D4—5021no—
254C2×D4—5021no—
255C2×D4—2511no—
256C2×D4—2512no—
257C2×D4C22×D25111yescharacteristic
258C22×C4—21no—
259C22×C4—5021no—
260C22×C4—5028no—
261C22×C4—5022no—
262C22×C4—5022no—
263C22×C4—5024no—
264C22×C4—2511no—
265C22×C4—2511no—
266C22×C4—2511no—
267C22×C4—2511no—
268C22×C4—2511no—
269C22×C4C22×D25111yescharacteristic
270SD16—24no—
271SD16—5024no—
272D8—5024no—
273D8—5024no—
274M16—5024no—
275C2×C8—2512no—
276C2×C8C22×D25112yes—
277C4⋊C4—5021no—
278C4⋊C4—2511no—
279C4⋊C4C22×D25111yescharacteristic
280C22⋊C4—21no—
281C22⋊C4—5021no—
282C22⋊C4—5021no—
283C22⋊C4—5021no—
284C23—21no—
285C23—10042no—
286C23—10041no—
287C23—10042no—
288C23—10042no—
289C23—10042no—
290C23—10042no—
291C23—5024no—
292C23—5022no—
293C23—5021no—
294C23—5022no—
295C23—5022no—
296C23—2511no—
297C23—2511no—
298C23D4×D25111yescharacteristic
299Q8—24no—
300Q8—22no—
301Q8D4×D25114yescomplemented
302D4—22no—
303D4—42no—
304D4—10042no—
305D4—10042no—
306D4—10042no—
307D4—10042no—
308D4—5028no—
309D4—5024no—
310D4—5028no—
311D4—5022no—
312D4—5022no—
313D4—2514no—
314C2×C4—24no—
315C2×C4—22no—
316C2×C4—22no—
317C2×C4—21no—
318C2×C4—10042no—
319C2×C4—5028no—
320C2×C4—5022no—
321C2×C4—5028no—
322C2×C4—5022no—
323C2×C4—5021no—
324C2×C4—5022no—
325C2×C4—5022no—
326C2×C4—2511no—
327C2×C4—2511no—
328C2×C4—2514no—
329C2×C4D4×D25111yescharacteristic
330C2×C4C23×D25111yescharacteristic
331C8—22no—
332C8—5022no—
333C22—21no—
334C22—21no—
335C22—41no—
336C22—41no—
337C22—41no—
338C22—20084no—
339C22—10048no—
340C22—10044no—
341C22—10041no—
342C22—10041no—
343C22—10041no—
344C22—10042no—
345C22—10042no—
346C22—5021no—
347C22—5021no—
348C22—5021no—
349C22—5024no—
350C22—5021no—
351C22—5021no—
352C22—2512no—
353C22#2254874212yescomplemented
354C22C2×D4×D25111yescharacteristic
355C4—24no—
356C4—21no—
357C4—41no—
358C4—10041no—
359C4—5021no—
360C4—2512no—
361C4C2×D4×D25112yes—
362C2—22no—
363C2—81no—
364C2—20081no—
365C2—10044no—
366C2—10041no—
367C2—5022no—
368C2#120×D25111yescharacteristic
369C2C2×#2254874211yescharacteristic
370C2#3063248611yescharacteristic
subgroupG/H
centerC22C2×D4×D251
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
Fitting#448×C251C2
socleC22×C251C22×D4
hypercenter#448D251
hypocenterC251#3006
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C22 ◁ C22×C4 ◁ #448

Complements

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

NH ≅ G/Ntype
1. C22×#16860811C2semidirect
2. C2×#22548742C2semidirect
3. #121×D251C2semidirect
4. #30631835C2semidirect
5. #30631831C2semidirect
6. #30631826C2semidirect
7. #30631823C2semidirect
8. #30631820C2semidirect
9. #30631687C2semidirect
10. #30631649C2semidirect
11. #448×C251C2semidirect
16. C2×#16860811C22semidirect
18. C22×D1004C22semidirect
20. C2×D4×D251C22semidirect
24. C2×#16860797C22semidirect
25. C2×D2008C22semidirect
26. C2×#16860795C22semidirect
27. C2×#16860793C22semidirect
28. #22548690C22semidirect
32. #22548686C22semidirect
33. C4⋊C4×D251C22semidirect
35. #22548639C22semidirect
36. #22548637C22semidirect
37. #22548635C22semidirect
38. C22⋊(C251⋊C8)C22semidirect
39. (C2×D251)⋊C8C22semidirect
40. C22×Q8×C251C22semidirect
41. C2×SD16×C251C22semidirect
43. #103×C251C22semidirect
44. C22⋊C8×C251C22semidirect
53. #16860811D4semidirect
117. D1004C2×D4semidirect
137. Q8×C251C2×D4semidirect
152. C22×C251AGL(1,ℤ/8)semidirect
185. C251#3006semidirect
193. #448D251semidirect
203. C22×Q8C2×D251semidirect
212. C2×SD16C2×D251semidirect
225. #103C2×D251semidirect
230. C22⋊C8C2×D251semidirect
301. Q8D4×D251semidirect
353. C22#22548742semidirect

Maximal subgroups

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

subgroupindex
C22×#168608112
C2×#225487422
#121×D2512
#306318352
#306318312
#306318262
#306318232
#306318202
#306316872
#306316492
#448×C2512
#3006251