Finite Group Databasebeta

← #44559908

Subgroups — #44559908

Order 32128.

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

normal subgroups only

This group has 504 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
1C2×#22548735C211yesmaximal, characteristic, complemented
2#30632254C211yesmaximal, characteristic, complemented
3#30632170C211yesmaximal, characteristic, complemented
4#30631907C212yesmaximal, complemented
5#30631900C212yesmaximal, complemented
6#30631885C212yesmaximal, complemented
7#30631880C212yesmaximal, complemented
8#30631715C211yesmaximal, characteristic, complemented
9#30631689C211yesmaximal, characteristic, complemented
10C2×#22548636C211yesmaximal, characteristic, complemented
11C2×AGL(1,ℤ/8)×C251C211yesmaximal, characteristic, complemented
12C2×#16860810C2211yescharacteristic
13#22548735—24no—
14C2×#16860800C2211yescharacteristic, complemented
15C2×#16860798C2211yescharacteristic, complemented
16C2×#16860797C2211yescharacteristic, complemented
17C2×#16860795C2211yescharacteristic, complemented
18C22×C251⋊C8C2211yescharacteristic
19#22548711C2212yes—
20#22548702C2212yes—
21#22548686—21no—
22#22548686—21no—
23#22548684—22no—
24Q8×Dic251C2212yescomplemented
25D4×Dic251—22no—
26D4×Dic251C2212yescomplemented
27#22548674—21no—
28#22548674C2211yescharacteristic
29C2×C4⋊Dic251—21no—
30C2×C4⋊Dic251C2211yescharacteristic
31Dic251⋊C8C2212yescomplemented
32Dic251⋊C8C2212yescomplemented
33#22548640C2212yescomplemented
34#22548637C2212yescomplemented
35#22548636C2212yes—
36#22548636C2212yes—
37C2×#41×C251C2211yescharacteristic
38C22×D4×C251C2211yescharacteristic
39AGL(1,ℤ/8)×C251—24no—
40C2×SD16×C251C2212yes—
41C2×D8×C251C2212yes—
42C2×M16×C251C2211yescharacteristic
43#16860810—22no—
44#16860810—22no—
45C2×Dic502C2311yescharacteristic
46C2×D1004C2311yescharacteristic
47C2×C22⋊D251—22no—
48C2×C22⋊D251—22no—
49C2×C22⋊D251—21no—
50C22×Dic251—22no—
51C22×Dic251—21no—
52C2×C4×D251—21no—
53#16860800—22no—
54#16860798—22no—
55#16860797—22no—
56#16860795—22no—
57C2×C251⋊C8—22no—
58C2×C251⋊C8C2311yescharacteristic
59C2×C251⋊C8C2311yescharacteristic
60C4⋊Dic251—22no—
61C4⋊Dic251—21no—
62C4⋊Dic251D411yescharacteristic
63C4⋊Dic251D411yescharacteristic
64C4⋊Dic251C2312yes—
65C22⋊Dic251—22no—
66C22⋊Dic251—22no—
67C22⋊Dic251—22no—
68C22⋊Dic251D412yes—
69C4×Dic251—22no—
70C4×Dic251C2312yes—
71C24×C251—21no—
72#41×C251—22no—
73#41×C251D414yescomplemented
74C2×Q8×C251C2311yescharacteristic
75C2×D4×C251—21no—
76C2×D4×C251—22no—
77C2×D4×C251—21no—
78C2×D4×C251—22no—
79C2×D4×C251C2312yes—
80C2×D4×C251C2311yescharacteristic
81C22×C4×C251—21no—
82C22×C4×C251C2311yescharacteristic
83SD16×C251—24no—
84D8×C251—24no—
85M16×C251—22no—
86C2×C8×C251C2312yes—
87C22×D251—21no—
88Dic502—21no—
89Dic502—21no—
90D1004—21no—
91D1004—21no—
92C22⋊D251—42no—
93C22⋊D251—82no—
94C22⋊D251—82no—
95C2×Dic251—22no—
96C2×Dic251—21no—
97C2×Dic251—82no—
98C2×Dic251—81no—
99C2×Dic251C2×D412yes—
100C2×Dic251C2×D412yes—
101C4×D251—42no—
102C251⋊C8—21no—
103C251⋊C8—21no—
104C23×C251—21no—
105C23×C251—22no—
106C23×C251—42no—
107C23×C251—81no—
108C23×C251—81no—
109C23×C251C2×D411yescharacteristic
110Q8×C251—21no—
111Q8×C251C2×D412yes—
112D4×C251—24no—
113D4×C251—22no—
114D4×C251—21no—
115D4×C251—22no—
116D4×C251—42no—
117D4×C251C2×D412yes—
118C2×C4×C251—21no—
119C2×C4×C251—24no—
120C2×C4×C251C2×D411yescharacteristic
121C2×C4×C251C2×D412yes—
122C2×C4×C251#4112yes—
123C2×C4×C251C2411yescharacteristic
124C8×C251—22no—
125C22×C251—21no—
126C22×C251—21no—
127C22×C251—22no—
128C22×C251—41no—
129C22×C251—41no—
130C22×C251—42no—
131C22×C251—42no—
132C22×C251—42no—
133C22×C251—84no—
134C22×C251#12112yes—
135C22×C251C22×D411yescharacteristic
136C2×D251—41no—
137C2×D251—41no—
138C2×D251—41no—
139C4×C251—22no—
140C4×C251#12112yes—
141C4×C251C22×D411yescharacteristic
142C4×C251C2×#4111yescharacteristic
143Dic251—22no—
144Dic251—22no—
145Dic251—41no—
146Dic251—42no—
147C2×C251—22no—
148C2×C251—42no—
149C2×C251—82no—
150C2×C251C2×#12111yescharacteristic
151C2×C251#57611yescharacteristic
152C2×C251#57711yescharacteristic
153D251—81no—
154C251#304511yescharacteristic, complemented, Sylow
155#3045—2511nomaximal, Sylow
156C2×AGL(1,ℤ/8)D25111yescharacteristic, complemented
157C2×#135—2511no—
158#535—2511no—
159#520—2511no—
160#464—2512no—
161#463—2512no—
162#460—2512no—
163#459—2512no—
164#424—2511no—
165#417—2511no—
166C2×#102—2511no—
167C2×#41—2511no—
168C2×#41C2×D25111yescharacteristic
169C22×D4C2×D25111yescharacteristic
170AGL(1,ℤ/8)—24no—
171#135—5024no—
172C2×Q16—2511no—
173C2×SD16—2511no—
174C2×SD16—2511no—
175C2×SD16C2×D25112yes—
176C2×D8—2511no—
177C2×D8C2×D25112yes—
178C2×M16C2×D25111yescharacteristic
179C22×C8—2511no—
180#127—2512no—
181#124—2512no—
182#121—5021no—
183#121—5021no—
184#120—5022no—
185C4×Q8—2512no—
186C4×D4—5022no—
187C4×D4—2512no—
188#117—5021no—
189#117—2511no—
190C2×C4⋊C4—5021no—
191C2×C4⋊C4—2511no—
192C4⋊C8—2512no—
193C4⋊C8—2512no—
194#103—2512no—
195#102—2512no—
196#102—2512no—
197#102—2512no—
198C24—21no—
199#41—22no—
200#41—5022no—
201#41—5022no—
202#41C22⋊D25114yescomplemented
203C2×Q8—2511no—
204C2×Q8C22×D25111yescharacteristic
205C2×D4—21no—
206C2×D4—21no—
207C2×D4—22no—
208C2×D4—22no—
209C2×D4—5021no—
210C2×D4—5022no—
211C2×D4—5022no—
212C2×D4—2511no—
213C2×D4C22×D25112yes—
214C2×D4C22×D25111yescharacteristic
215C22×C4—21no—
216C22×C4—5021no—
217C22×C4—5022no—
218C22×C4—5021no—
219C22×C4C22×D25111yescharacteristic
220Q16—5022no—
221SD16—24no—
222SD16—5022no—
223SD16—5022no—
224D8—24no—
225D8—5022no—
226M16—22no—
227C2×C8—5022no—
228C2×C8—2511no—
229C2×C8—2511no—
230C2×C8C22×D25112yes—
231C4⋊C4—5022no—
232C4⋊C4—5021no—
233C4⋊C4—2512no—
234C4⋊C4—2511no—
235C4⋊C4—2511no—
236C22⋊C4—5022no—
237C22⋊C4—5022no—
238C22⋊C4—5022no—
239C22⋊C4—2512no—
240C42—5022no—
241C42—2512no—
242C23—21no—
243C23—22no—
244C23—42no—
245C23—81no—
246C23—81no—
247C23—5021no—
248C23D4×D25111yescharacteristic
249Q8—21no—
250Q8—5021no—
251Q8—5021no—
252Q8C2×C22⋊D25112yes—
253D4—24no—
254D4—22no—
255D4—21no—
256D4—22no—
257D4—42no—
258D4—20082no—
259D4—20082no—
260D4—10042no—
261D4—5021no—
262D4—5021no—
263D4C2×C22⋊D25112yes—
264C2×C4—21no—
265C2×C4—24no—
266C2×C4—20081no—
267C2×C4—20082no—
268C2×C4—10042no—
269C2×C4—5022no—
270C2×C4—5021no—
271C2×C4—2512no—
272C2×C4—2512no—
273C2×C4D4×D25111yescharacteristic
274C2×C4C2×C22⋊D25112yes—
275C2×C4#1686080912yes—
276C2×C4C23×D25111yescharacteristic
277C8—22no—
278C8—5021no—
279C8—5021no—
280C22—21no—
281C22—21no—
282C22—22no—
283C22—41no—
284C22—41no—
285C22—42no—
286C22—42no—
287C22—42no—
288C22—84no—
289C22—10041no—
290C22—10041no—
291C22—10041no—
292C22#2254868712yes—
293C22C2×D4×D25111yescharacteristic
294C4—22no—
295C4—10042no—
296C4—10041no—
297C4—5022no—
298C4—5022no—
299C4#2254868712yes—
300C4C22×C22⋊D25111yescharacteristic
301C4C2×#1686080911yescharacteristic
302C2—22no—
303C2—42no—
304C2—82no—
305C2—20081no—
306C2C2×#2254868711yescharacteristic
307C2#3063249111yescharacteristic
308C2#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#3045
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×#22548735C2semidirect
2. #30632254C2semidirect
3. #30632170C2semidirect
4. #30631907C2semidirect
5. #30631900C2semidirect
6. #30631885C2semidirect
7. #30631880C2semidirect
8. #30631715C2semidirect
9. #30631689C2semidirect
10. C2×#22548636C2semidirect
11. C2×AGL(1,ℤ/8)×C251C2semidirect
14. C2×#16860800C22semidirect
15. C2×#16860798C22semidirect
16. C2×#16860797C22semidirect
17. C2×#16860795C22semidirect
24. Q8×Dic251C22semidirect
26. D4×Dic251C22semidirect
31. Dic251⋊C8C22semidirect
32. Dic251⋊C8C22semidirect
33. #22548640C22semidirect
34. #22548637C22semidirect
73. #41×C251D4semidirect
154. C251#3045semidirect
156. C2×AGL(1,ℤ/8)D251semidirect
202. #41C22⋊D251semidirect

Maximal subgroups

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

subgroupindex
C2×#225487352
#306322542
#306321702
#306319072
#306319002
#306318852
#306318802
#306317152
#306316892
C2×#225486362
C2×AGL(1,ℤ/8)×C2512
#3045251