Finite Group Databasebeta

← #44559840

Subgroups — #44559840

Order 32128.

584 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 584 conjugacy classes of subgroups — too many to draw. Only its 133 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×#22548739C212yesmaximal, complemented
2#127×D251C211yesmaximal, characteristic, complemented
3C4×#16860809C211yesmaximal, characteristic, complemented
4#30631888C211yesmaximal, characteristic, complemented
5#30631886C212yesmaximal, complemented
6#30631884C211yesmaximal, characteristic, complemented
7#30631882C211yesmaximal, characteristic, complemented
8#30631880C212yesmaximal, complemented
9#30631709C211yesmaximal, characteristic, complemented
10#30631690C212yesmaximal, complemented
11#459×C251C211yesmaximal, characteristic, complemented
12C2×#16860809C2211yescharacteristic
13C2×D4×D251—21no—
14C2×D4×D251—21no—
15C2×D4×D251C2212yes—
16#22548739—28no—
17C2×#16860799C2212yescomplemented
18C2×#16860798C2212yescomplemented
19C2×#16860795C2212yescomplemented
20C2×#16860793C2212yes—
21#22548712C2211yescharacteristic
22#22548711—21no—
23#22548711—21no—
24#22548711C2212yes—
25C4×Dic502C2211yescharacteristic, complemented
26#22548680C2211yescharacteristic, complemented
27D4×Dic251C2211yescharacteristic, complemented
28C4×C22⋊D251—21no—
29(C22⋊D251)⋊C4—21no—
30#22548673C2211yescharacteristic
31#22548671—21no—
32C2×C4×Dic251—21no—
33C42×D251C2211yescharacteristic
34C4⋊(C251⋊C8)C2211yescharacteristic, complemented
35Dic251⋊C8C2212yescomplemented
36#22548637C2212yescomplemented
37#22548636C2212yescomplemented
38#22548635C2212yescomplemented
39C2×D8×C251C2212yescomplemented
40#127×C251C2211yescharacteristic
41C4×D4×C251C2211yescharacteristic, complemented
42C4⋊C8×C251C2211yescharacteristic, complemented
43#102×C251C2212yescomplemented
44C23×D251—22no—
45#16860809—22no—
46#16860809D414yescomplemented
47C2×Dic502C2311yescharacteristic
48C2×D1004—21no—
49C2×D1004—21no—
50C2×D1004C2312yes—
51C2×C22⋊D251—21no—
52C2×C22⋊D251—22no—
53C2×C22⋊D251—22no—
54C2×C22⋊D251—22no—
55D4×D251—24no—
56D4×D251—24no—
57D4×D251—44no—
58D4×D251—44no—
59C22×Dic251—21no—
60C22×Dic251—21no—
61C2×C4×D251D411yescharacteristic
62C2×C4×D251D411yescharacteristic
63C2×C4×D251C2311yescharacteristic
64#16860799—24no—
65#16860798—24no—
66#16860795—24no—
67#16860793—24no—
68C2×C251⋊C8C2312yes—
69C4⋊Dic251C2311yescharacteristic
70Dic251⋊C4—21no—
71Dic251⋊C4—21no—
72C22⋊Dic251—21no—
73(C2×D251)⋊C4—21no—
74C4×Dic251—21no—
75C4×Dic251—21no—
76C4×Dic251—21no—
77C4×Dic251D411yescharacteristic
78C4×Dic251D411yescharacteristic
79C4×Dic251C2311yescharacteristic
80C2×D4×C251—21no—
81C2×D4×C251—21no—
82C2×D4×C251C2312yes—
83C2×D4×C251C2311yescharacteristic
84C22×C4×C251—21no—
85D8×C251—24no—
86C2×C8×C251C2312yes—
87C4⋊C4×C251C2311yescharacteristic
88C22⋊C4×C251—21no—
89C42×C251C2311yescharacteristic
90C22×D251—22no—
91C22×D251—44no—
92C22×D251—44no—
93C22×D251—44no—
94C22×D251C2×D411yescharacteristic
95Dic502—21no—
96Dic502C2×D412yescomplemented
97D1004—22no—
98D1004—22no—
99D1004—42no—
100D1004—42no—
101C22⋊D251—24no—
102C22⋊D251—44no—
103C22⋊D251—44no—
104C22⋊D251—44no—
105C2×Dic251—21no—
106C2×Dic251—24no—
107C2×Dic251—21no—
108C2×Dic251—42no—
109C2×Dic251C2×D411yescharacteristic
110C2×Dic251C2×D411yescharacteristic
111C2×Dic251C2×D411yescharacteristic
112C4×D251—22no—
113C4×D251—22no—
114C4×D251C2×D412yes—
115C4×D251#4112yes—
116C251⋊C8—22no—
117C23×C251—21no—
118C23×C251—22no—
119D4×C251—22no—
120D4×C251—21no—
121D4×C251—22no—
122D4×C251—42no—
123D4×C251—42no—
124D4×C251C2×D412yescomplemented
125C2×C4×C251—21no—
126C2×C4×C251—42no—
127C2×C4×C251C2×D411yescharacteristic
128C2×C4×C251C2×D411yescharacteristic
129C2×C4×C251C2411yescharacteristic
130C8×C251—22no—
131C22×C251—22no—
132C22×C251—41no—
133C22×C251—41no—
134C22×C251—42no—
135C22×C251—42no—
136C22×C251—42no—
137C22×C251C22×D411yescharacteristic
138C2×D251—21no—
139C2×D251—21no—
140C2×D251—42no—
141C2×D251—42no—
142C2×D251—42no—
143C2×D251—88no—
144C2×D251#12112yes—
145C4×C251—21no—
146C4×C251—41no—
147C4×C251AGL(1,ℤ/8)12yescomplemented
148C4×C251C22×D411yescharacteristic
149C4×C251C2×#4111yescharacteristic
150Dic251—22no—
151Dic251—21no—
152Dic251—41no—
153Dic251#12112yes—
154Dic251AGL(1,ℤ/8)12yescomplemented
155C2×C251—42no—
156C2×C251—82no—
157C2×C251C2×#12111yescharacteristic
158C2×C251C2×AGL(1,ℤ/8)11yescharacteristic
159C2×C251C2×AGL(1,ℤ/8)11yescharacteristic
160D251—22no—
161D251—82no—
162C251#303811yescharacteristic, complemented, Sylow
163#3038—2511nomaximal, Sylow
164C2×AGL(1,ℤ/8)—2512no—
165C2×#127—2511no—
166C4×#41—2511no—
167#460—2511no—
168#460—2511no—
169#460—2512no—
170#459—2512no—
171#459—2511no—
172#459D25111yescharacteristic, complemented
173#423—2511no—
174#418—2512no—
175C2×#41—2511no—
176C22×D4—5021no—
177C22×D4—5021no—
178C22×D4—2512no—
179AGL(1,ℤ/8)—5028no—
180C2×SD16—2512no—
181C2×SD16—2512no—
182C2×D8—2512no—
183C2×D8C2×D25112yescomplemented
184C2×M16—2512no—
185#127—5021no—
186#127—5021no—
187#127—2512no—
188#127—2511no—
189#127C2×D25111yescharacteristic
190C4×Q8—2511no—
191C4×Q8—2511no—
192C4×D4—5021no—
193C4×D4—5021no—
194C4×D4—2511no—
195C4×D4C2×D25111yescharacteristic, complemented
196#117—5021no—
197#117—2511no—
198C2×C42—5021no—
199C2×C42—2511no—
200C4⋊C8—2512no—
201C4⋊C8—2511no—
202C4⋊C8C2×D25111yescharacteristic, complemented
203#102—2512no—
204#102—2512no—
205#102—2512no—
206#102C2×D25112yescomplemented
207C24—5022no—
208#41—5022no—
209#41—2514no—
210C2×Q8—2511no—
211C2×D4—21no—
212C2×D4—21no—
213C2×D4—10044no—
214C2×D4—10044no—
215C2×D4—5021no—
216C2×D4—5024no—
217C2×D4—5024no—
218C2×D4—5022no—
219C2×D4—5022no—
220C2×D4—5022no—
221C2×D4—5021no—
222C2×D4—5021no—
223C2×D4—2512no—
224C2×D4C22×D25112yes—
225C2×D4C22×D25111yescharacteristic
226C22×C4—21no—
227C22×C4—5021no—
228C22×C4—5021no—
229C22×C4—2511no—
230C22×C4—2511no—
231C22×C4—2511no—
232SD16—5024no—
233SD16—5024no—
234D8—24no—
235D8—5024no—
236M16—5024no—
237C2×C8—2512no—
238C2×C8C22×D25112yes—
239C4⋊C4—5021no—
240C4⋊C4—5021no—
241C4⋊C4—2511no—
242C4⋊C4C22×D25111yescharacteristic
243C22⋊C4—21no—
244C22⋊C4—5021no—
245C22⋊C4—5021no—
246C42—5021no—
247C42—5021no—
248C42—5021no—
249C42—2511no—
250C42—2511no—
251C42—2511no—
252C42C22×D25111yescharacteristic
253C23—21no—
254C23—22no—
255C23—10044no—
256C23—10044no—
257C23—10044no—
258C23—5022no—
259C23—2511no—
260Q8—5021no—
261Q8—2512no—
262D4—22no—
263D4—21no—
264D4—22no—
265D4—42no—
266D4—42no—
267D4—10042no—
268D4—10044no—
269D4—10044no—
270D4—10044no—
271D4—10042no—
272D4—5024no—
273D4—5022no—
274D4—5022no—
275D4D4×D25112yescomplemented
276C2×C4—21no—
277C2×C4—42no—
278C2×C4—10042no—
279C2×C4—5022no—
280C2×C4—5021no—
281C2×C4—5021no—
282C2×C4—5024no—
283C2×C4—5022no—
284C2×C4—2511no—
285C2×C4—2511no—
286C2×C4—2511no—
287C2×C4—2512no—
288C2×C4—2512no—
289C2×C4D4×D25111yescharacteristic
290C2×C4D4×D25111yescharacteristic
291C2×C4C23×D25111yescharacteristic
292C8—22no—
293C8—5022no—
294C22—22no—
295C22—41no—
296C22—41no—
297C22—42no—
298C22—42no—
299C22—42no—
300C22—20088no—
301C22—10042no—
302C22—10042no—
303C22—10042no—
304C22—5021no—
305C22—5021no—
306C22—2512no—
307C22C2×D4×D25111yescharacteristic
308C4—21no—
309C4—41no—
310C4—10041no—
311C4—5021no—
312C4—5022no—
313C4—2512no—
314C4—2512no—
315C4#2254873912yescomplemented
316C4C2×D4×D25111yescharacteristic
317C4#41×D25111yescharacteristic
318C2—42no—
319C2—82no—
320C2—20082no—
321C2—5022no—
322C2#121×D25111yescharacteristic
323C2AGL(1,ℤ/8)×D25111yescharacteristic
324C2C2×#2254873911yescharacteristic
subgroupG/H
centerC22C2×D4×D251
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
Fitting#459×C251C2
socleC22×C251C22×D4
hypercenter#459D251
hypocenterC251#3038
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C22 ◁ C42 ◁ #459

Complements

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

NH ≅ G/Ntype
1. C2×#22548739C2semidirect
2. #127×D251C2semidirect
3. C4×#16860809C2semidirect
4. #30631888C2semidirect
5. #30631886C2semidirect
6. #30631884C2semidirect
7. #30631882C2semidirect
8. #30631880C2semidirect
9. #30631709C2semidirect
10. #30631690C2semidirect
11. #459×C251C2semidirect
17. C2×#16860799C22semidirect
18. C2×#16860798C22semidirect
19. C2×#16860795C22semidirect
25. C4×Dic502C22semidirect
26. #22548680C22semidirect
27. D4×Dic251C22semidirect
34. C4⋊(C251⋊C8)C22semidirect
35. Dic251⋊C8C22semidirect
36. #22548637C22semidirect
37. #22548636C22semidirect
38. #22548635C22semidirect
39. C2×D8×C251C22semidirect
41. C4×D4×C251C22semidirect
42. C4⋊C8×C251C22semidirect
43. #102×C251C22semidirect
46. #16860809D4semidirect
96. Dic502C2×D4semidirect
124. D4×C251C2×D4semidirect
147. C4×C251AGL(1,ℤ/8)semidirect
154. Dic251AGL(1,ℤ/8)semidirect
162. C251#3038semidirect
172. #459D251semidirect
183. C2×D8C2×D251semidirect
195. C4×D4C2×D251semidirect
202. C4⋊C8C2×D251semidirect
206. #102C2×D251semidirect
275. D4D4×D251semidirect
315. C4#22548739semidirect

Maximal subgroups

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

subgroupindex
C2×#225487392
#127×D2512
C4×#168608092
#306318882
#306318862
#306318842
#306318822
#306318802
#306317092
#306316902
#459×C2512
#3038251