Finite Group Databasebeta

← #44559671

Subgroups — #44559671

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×#22548761C211yesmaximal, characteristic, complemented
2C2×#22548743C211yesmaximal, characteristic, complemented
3C2×#22548735C211yesmaximal, characteristic, complemented
4#30632268C211yesmaximal, characteristic, complemented
5#30631852C212yesmaximal, complemented
6#30631839C212yesmaximal, complemented
7#30631835C212yesmaximal, complemented
8#30631822C212yesmaximal, complemented
9C2×#22548635C211yesmaximal, characteristic, complemented
10#30631655C211yesmaximal, characteristic, complemented
11#417×C251C211yesmaximal, characteristic, complemented
12#22548761—24no—
13#22548761—24no—
14C2×#16860811C2212yes—
15C2×#16860810—21no—
16C2×#16860810C2211yescharacteristic
17C2×#16860810C2211yescharacteristic, complemented
18C22×D1004—21no—
19C22×D1004C2211yescharacteristic
20C2×D4×D251—22no—
21C2×D4×D251C2212yes—
22#22548743—24no—
23#22548735—24no—
24C2×#16860800C2211yescharacteristic, complemented
25C2×#16860799C2212yescomplemented
26C2×#16860798C2211yescharacteristic, complemented
27C2×#16860797C2211yescharacteristic, complemented
28C2×D2008C2212yescomplemented
29C2×#16860795C2211yescharacteristic, complemented
30C22×C251⋊C8C2211yescharacteristic, complemented
31#22548695—21no—
32#22548695C2212yescomplemented
33#22548690—21no—
34#22548690C2212yescomplemented
35#22548689—22no—
36#22548635C2212yescomplemented
37#22548635C2212yescomplemented
38(C2×D251)⋊C8C2212yescomplemented
39(C2×D251)⋊C8C2212yescomplemented
40C2×#41×C251C2211yescharacteristic
41C2×M16×C251C2211yescharacteristic, complemented
42C2×C4⋊C4×C251C2211yescharacteristic, complemented
43#103×C251C2212yescomplemented
44#102×C251C2212yescomplemented
45C23×D251—22no—
46C23×D251—21no—
47#16860811—24no—
48#16860811—24no—
49#16860810—28no—
50#16860810—22no—
51#16860810—22no—
52#16860810—22no—
53#16860810—22no—
54C2×Dic502D411yescharacteristic
55C2×Dic502C2311yescharacteristic
56C2×D1004—28no—
57C2×D1004—22no—
58C2×D1004—22no—
59C2×D1004—21no—
60C2×D1004—22no—
61C2×D1004D411yescharacteristic
62C2×D1004D414yescomplemented
63C2×D1004C2311yescharacteristic
64C2×D1004C2312yes—
65C2×C22⋊D251—22no—
66C2×C22⋊D251—21no—
67C2×C22⋊D251D412yes—
68D4×D251—216no—
69D4×D251—24no—
70D4×D251—24no—
71C2×C4×D251—22no—
72C2×C4×D251—21no—
73C2×C4×D251C2312yes—
74#16860800—22no—
75#16860799—24no—
76#16860798—22no—
77#16860797—22no—
78D2008—24no—
79#16860795—22no—
80C2×C251⋊C8—22no—
81C2×C251⋊C8C2311yescharacteristic
82C2×C251⋊C8C2311yescharacteristic
83C4⋊Dic251—22no—
84(C2×D251)⋊C4—22no—
85(C2×D251)⋊C4—22no—
86#41×C251—22no—
87#41×C251D414yescomplemented
88C2×Q8×C251C2311yescharacteristic
89C2×D4×C251—21no—
90C2×D4×C251C2311yescharacteristic
91C22×C4×C251—21no—
92C22×C4×C251—21no—
93C22×C4×C251C2311yescharacteristic
94M16×C251—22no—
95C2×C8×C251C2312yes—
96C4⋊C4×C251—21no—
97C4⋊C4×C251C2312yes—
98C22×D251—22no—
99C22×D251—28no—
100C22×D251—24no—
101C22×D251—21no—
102C22×D251—82no—
103C22×D251—82no—
104C22×D251—81no—
105C22×D251—81no—
106C22×D251C2×D412yes—
107Dic502—22no—
108Dic502—21no—
109Dic502—21no—
110D1004—28no—
111D1004—22no—
112D1004—24no—
113D1004—22no—
114D1004—21no—
115D1004—21no—
116D1004—82no—
117D1004C2×D414yes—
118C22⋊D251—28no—
119C22⋊D251—24no—
120C22⋊D251—42no—
121C2×Dic251—21no—
122C2×Dic251C2×D412yes—
123C4×D251—28no—
124C4×D251—22no—
125C4×D251—22no—
126C4×D251—42no—
127C251⋊C8—21no—
128C251⋊C8—21no—
129C23×C251—21no—
130C23×C251C2×D411yescharacteristic
131Q8×C251—21no—
132Q8×C251C2×D412yes—
133D4×C251—24no—
134D4×C251—21no—
135D4×C251C2×D412yes—
136C2×C4×C251—21no—
137C2×C4×C251—24no—
138C2×C4×C251—22no—
139C2×C4×C251—81no—
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—22no—
148C22×C251—41no—
149C22×C251—41no—
150C22×C251#12012yes—
151C22×C251C22×D411yescharacteristic
152C2×D251—22no—
153C2×D251—24no—
154C2×D251—22no—
155C2×D251—22no—
156C2×D251—42no—
157C2×D251—42no—
158C2×D251—41no—
159C2×D251—41no—
160C2×D251—41no—
161C2×D251—84no—
162C2×D251—88no—
163C4×C251—22no—
164C4×C251—42no—
165C4×C251#12012yes—
166C4×C251C22×D411yescharacteristic
167C4×C251C22×D411yescharacteristic
168Dic251—22no—
169Dic251—41no—
170C2×C251—22no—
171C2×C251—42no—
172C2×C251C2×#12011yescharacteristic
173C2×C251#57611yescharacteristic
174C2×C251#57711yescharacteristic
175D251—42no—
176D251—44no—
177D251—81no—
178C251#301211yescharacteristic, complemented, Sylow
179#3012—2511nomaximal, Sylow
180C2×H(5,𝔽2)—2511no—
181C2×AGL(1,ℤ/8)—2511no—
182C2×#135—2511no—
183#537—2511no—
184#452—2512no—
185#450—2512no—
186#449—2512no—
187#447—2512no—
188#417D25111yescharacteristic, complemented
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
198C22×D4—5021no—
199C22×D4—5022no—
200C22×D4—2512no—
201C22×D4—2511no—
202AGL(1,ℤ/8)—5024no—
203#135—5024no—
204C2×Q16—2511no—
205C2×SD16—2511no—
206C2×SD16—2511no—
207C2×SD16—2512no—
208C2×D8—2511no—
209C2×D8—2512no—
210C2×M16C2×D25111yescharacteristic, complemented
211C22×C8—2511no—
212C22⋊Q8—5021no—
213C22⋊Q8—2512no—
214#121—5021no—
215#121—5022no—
216#121—2512no—
217C2×C4⋊C4C2×D25111yescharacteristic, complemented
218#103C2×D25112yescomplemented
219#102—2512no—
220#102—2512no—
221#102C2×D25112yescomplemented
222C22⋊C8—2512no—
223C22⋊C8—2512no—
224C24—5022no—
225C24—5021no—
226#41—22no—
227#41—5028no—
228#41—5022no—
229#41—5022no—
230#41—5024no—
231#41—5024no—
232#41—5022no—
233#41—5022no—
234#41D100414yescomplemented
235C2×Q8—2511no—
236C2×Q8—2511no—
237C2×Q8C22×D25111yescharacteristic
238C2×D4—21no—
239C2×D4—5022no—
240C2×D4—5028no—
241C2×D4—5022no—
242C2×D4—50216no—
243C2×D4—5024no—
244C2×D4—5021no—
245C2×D4—5024no—
246C2×D4—5022no—
247C2×D4—5021no—
248C2×D4—5022no—
249C2×D4—2512no—
250C2×D4—2511no—
251C2×D4—2514no—
252C2×D4—2511no—
253C2×D4—2512no—
254C2×D4C22×D25111yescharacteristic
255C22×C4—21no—
256C22×C4—21no—
257C22×C4—5022no—
258C22×C4—5021no—
259C22×C4—2512no—
260C22×C4C22×D25111yescharacteristic
261Q16—5022no—
262SD16—5024no—
263SD16—5022no—
264SD16—5022no—
265D8—5024no—
266D8—5022no—
267M16—22no—
268C2×C8—5022no—
269C2×C8—2511no—
270C2×C8—2511no—
271C2×C8C22×D25112yes—
272C4⋊C4—21no—
273C4⋊C4—5022no—
274C4⋊C4C22×D25112yes—
275C22⋊C4—5022no—
276C22⋊C4—5022no—
277C23—21no—
278C23—20081no—
279C23—20081no—
280C23—20082no—
281C23—20082no—
282C23—5028no—
283C23—5024no—
284C23—5021no—
285C23—5022no—
286C23—2512no—
287C23D4×D25111yescharacteristic
288Q8—21no—
289Q8—5021no—
290Q8—5021no—
291Q8—5022no—
292Q8C2×D100412yes—
293D4—24no—
294D4—21no—
295D4—20082no—
296D4—10042no—
297D4—5028no—
298D4—5028no—
299D4—5024no—
300D4—5021no—
301D4—5021no—
302D4—5022no—
303D4—5022no—
304D4—5024no—
305D4—2514no—
306D4C2×D100412yes—
307C2×C4—21no—
308C2×C4—24no—
309C2×C4—22no—
310C2×C4—81no—
311C2×C4—10042no—
312C2×C4—5028no—
313C2×C4—5021no—
314C2×C4—5022no—
315C2×C4—5022no—
316C2×C4—2512no—
317C2×C4D4×D25112yes—
318C2×C4D4×D25111yescharacteristic
319C2×C4C2×D100412yes—
320C2×C4C23×D25111yescharacteristic
321C8—22no—
322C8—5021no—
323C8—5021no—
324C22—21no—
325C22—21no—
326C22—22no—
327C22—41no—
328C22—41no—
329C22—20084no—
330C22—20088no—
331C22—10041no—
332C22—10041no—
333C22—10041no—
334C22—10042no—
335C22—10042no—
336C22—5022no—
337C22—5022no—
338C22—5022no—
339C22—5024no—
340C22#2254868512yes—
341C22C2×D4×D25111yescharacteristic
342C4—22no—
343C4—42no—
344C4—10041no—
345C4—5022no—
346C4#2254868512yes—
347C4C2×D4×D25111yescharacteristic
348C4C22×D100411yescharacteristic
349C2—22no—
350C2—42no—
351C2—20081no—
352C2—10044no—
353C2—10042no—
354C2C2×#2254868511yescharacteristic
355C2#3063249111yescharacteristic
356C2#3063249611yescharacteristic
subgroupG/H
centerC22C2×D4×D251
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
Fitting#417×C251C2
socleC22×C251C22×D4
hypercenter#417D251
hypocenterC251#3012
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C22 ◁ C22×C4 ◁ #417

Complements

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

NH ≅ G/Ntype
1. C2×#22548761C2semidirect
2. C2×#22548743C2semidirect
3. C2×#22548735C2semidirect
4. #30632268C2semidirect
5. #30631852C2semidirect
6. #30631839C2semidirect
7. #30631835C2semidirect
8. #30631822C2semidirect
9. C2×#22548635C2semidirect
10. #30631655C2semidirect
11. #417×C251C2semidirect
17. C2×#16860810C22semidirect
24. C2×#16860800C22semidirect
25. C2×#16860799C22semidirect
26. C2×#16860798C22semidirect
27. C2×#16860797C22semidirect
28. C2×D2008C22semidirect
29. C2×#16860795C22semidirect
30. C22×C251⋊C8C22semidirect
32. #22548695C22semidirect
34. #22548690C22semidirect
36. #22548635C22semidirect
37. #22548635C22semidirect
38. (C2×D251)⋊C8C22semidirect
39. (C2×D251)⋊C8C22semidirect
41. C2×M16×C251C22semidirect
42. C2×C4⋊C4×C251C22semidirect
43. #103×C251C22semidirect
44. #102×C251C22semidirect
62. C2×D1004D4semidirect
87. #41×C251D4semidirect
178. C251#3012semidirect
188. #417D251semidirect
210. C2×M16C2×D251semidirect
217. C2×C4⋊C4C2×D251semidirect
218. #103C2×D251semidirect
221. #102C2×D251semidirect
234. #41D1004semidirect

Maximal subgroups

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

subgroupindex
C2×#225487612
C2×#225487432
C2×#225487352
#306322682
#306318522
#306318392
#306318352
#306318222
C2×#225486352
#306316552
#417×C2512
#3012251