Finite Group Databasebeta

← #44562179

Subgroups — #44562179

Order 32128.

494 conjugacy classes of subgroups; only the 139 normal subgroups are shown.

normal subgroups only

This group has 494 conjugacy classes of subgroups — too many to draw. Only its 139 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
1#30632341C211yesmaximal, characteristic, complemented
2D4×D1004C211yesmaximal, characteristic, complemented
3#30632034C211yesmaximal, characteristic, complemented
4#30632001C212yesmaximal, complemented
5#30631968C211yesmaximal, characteristic, complemented
6#30631923C212yesmaximal, complemented
7#30631883C211yesmaximal, characteristic, complemented
8#30631823C212yesmaximal, complemented
9C4×D2008C211yesmaximal, characteristic, complemented
10C2×#22548637C212yesmaximal, complemented
11C8×D4×C251C211yesmaximal, characteristic, complemented
12C2×#16860809—21no—
13C22×D1004C2212yes—
14C2×D4×D251—21no—
15C2×D2008C2211yescharacteristic
16C2×D2008C2212yescomplemented
17#22548715C2211yescharacteristic, complemented
18#22548712C2211yescharacteristic
19#22548699—22no—
20#22548695—21no—
21#22548690—21no—
22#22548689C2212yes—
23#22548689C2212yescomplemented
24#22548685—22no—
25C4×D1004C2211yescharacteristic
26C4×D1004C2211yescharacteristic, complemented
27C2×C4⋊Dic251C2212yescomplemented
28#22548649C2212yescomplemented
29#22548649C2211yescharacteristic, complemented
30#22548637C2212yes—
31#22548637C2211yescharacteristic
32#22548637C2212yescomplemented
33#22548637C2212yescomplemented
34#22548637C2211yescharacteristic, complemented
35#22548637C2212yescomplemented
36C22×C8×C251C2212yescomplemented
37C4×D4×C251C2211yescharacteristic
38C4⋊C8×C251C2211yescharacteristic, complemented
39C22⋊C8×C251C2212yescomplemented
40C4×C8×C251C2211yescharacteristic, complemented
41C23×D251—22no—
42#16860809—44no—
43C2×Dic502—21no—
44C2×D1004—24no—
45C2×D1004—21no—
46C2×D1004—22no—
47C2×D1004—21no—
48C2×D1004—42no—
49C2×D1004C2312yes—
50C2×D1004C2311yescharacteristic
51C2×D1004C2311yescharacteristic
52C2×C22⋊D251—22no—
53C2×C22⋊D251—22no—
54D4×D251—44no—
55C22×Dic251—22no—
56C2×C4×D251—21no—
57C2×C4×D251—21no—
58D2008—22no—
59D2008—42no—
60C4⋊Dic251—22no—
61C4⋊Dic251—21no—
62C4⋊Dic251C2311yescharacteristic
63C4⋊Dic251C2312yes—
64C4⋊Dic251C2311yescharacteristic
65(C2×D251)⋊C4—21no—
66(C2×D251)⋊C4—21no—
67(C2×D251)⋊C4—22no—
68(C2×D251)⋊C4—22no—
69C2×D4×C251D411yescharacteristic
70C22×C4×C251C2312yes—
71C2×C8×C251—42no—
72C2×C8×C251C2311yescharacteristic
73C2×C8×C251C2311yescharacteristic
74C2×C8×C251C2312yes—
75C4⋊C4×C251D411yescharacteristic
76C22⋊C4×C251D412yes—
77C42×C251C2311yescharacteristic
78C22×D251—21no—
79C22×D251—21no—
80C22×D251—22no—
81C22×D251—44no—
82C22×D251—44no—
83C22×D251—82no—
84Dic502—42no—
85D1004—21no—
86D1004—22no—
87D1004—21no—
88D1004—21no—
89D1004—22no—
90D1004—44no—
91D1004—42no—
92D1004—81no—
93D1004#4112yescomplemented
94C22⋊D251—44no—
95C22⋊D251—44no—
96C2×Dic251—21no—
97C2×Dic251—21no—
98C2×Dic251—22no—
99C2×Dic251—44no—
100C4×D251—42no—
101C4×D251—42no—
102C23×C251C2×D412yes—
103D4×C251D814yescomplemented
104C2×C4×C251—22no—
105C2×C4×C251—22no—
106C2×C4×C251C2×D411yescharacteristic
107C2×C4×C251C2×D412yes—
108C2×C4×C251C2×D411yescharacteristic
109C2×C4×C251C2411yescharacteristic
110C8×C251—21no—
111C8×C251—21no—
112C8×C251—22no—
113C22×C251—22no—
114C22×C251—22no—
115C22×C251C2×D814yes—
116C22×C251C22×D411yescharacteristic
117C2×D251—22no—
118C2×D251—41no—
119C2×D251—41no—
120C2×D251—42no—
121C2×D251—41no—
122C2×D251—41no—
123C2×D251—42no—
124C2×D251—42no—
125C2×D251—41no—
126C2×D251—88no—
127C4×C251—22no—
128C4×C251—21no—
129C4×C251C2×D812yes—
130C4×C251C2×#4111yescharacteristic
131C4×C251H(5,𝔽2)11yescharacteristic
132Dic251—41no—
133Dic251—41no—
134Dic251—42no—
135C2×C251—24no—
136C2×C251#54611yescharacteristic
137C2×C251C22×D811yescharacteristic
138C2×C251#57711yescharacteristic
139D251—42no—
140D251—81no—
141D251—82no—
142C251#329411yescharacteristic, complemented, Sylow
143#3294—2511nomaximal, Sylow
144#547—2511no—
145D42—2511no—
146#486—2511no—
147#480—2512no—
148#474—2511no—
149#466—2512no—
150#459—2511no—
151#447—2512no—
152C4×D8—2511no—
153C8×D4D25111yescharacteristic, complemented
154C2×#102—2512no—
155C2×#41—5021no—
156C22×D4—5021no—
157C22×D4—2512no—
158C2×D8—2512no—
159C2×D8—2511no—
160C22×C8C2×D25112yescomplemented
161C4⋊Q8—2511no—
162#127—2511no—
163#123—5022no—
164C22⋊Q8—5021no—
165#121—5021no—
166#121—2512no—
167#121—2512no—
168#120—5022no—
169C4×D4—2511no—
170C4×D4—2511no—
171C4×D4C2×D25111yescharacteristic
172C2×C4⋊C4—2512no—
173#107—2511no—
174#107—2512no—
175C4⋊C8C2×D25111yescharacteristic, complemented
176#102—2511no—
177#102—2512no—
178#102—2512no—
179#102—2512no—
180#102—2511no—
181#102—2512no—
182C22⋊C8C2×D25112yescomplemented
183C4×C8C2×D25111yescharacteristic, complemented
184C24—5022no—
185#41—10044no—
186C2×Q8—5021no—
187C2×D4—10042no—
188C2×D4—10044no—
189C2×D4—5024no—
190C2×D4—5022no—
191C2×D4—5022no—
192C2×D4—5021no—
193C2×D4—5022no—
194C2×D4—5021no—
195C2×D4—2512no—
196C2×D4—2511no—
197C2×D4—2511no—
198C2×D4D100411yescharacteristic
199C22×C4—5022no—
200C22×C4—5021no—
201C22×C4—5021no—
202C22×C4C22×D25112yes—
203D8—10042no—
204D8—5022no—
205C2×C8—42no—
206C2×C8C22×D25111yescharacteristic
207C2×C8C22×D25111yescharacteristic
208C2×C8C22×D25112yes—
209C4⋊C4—5022no—
210C4⋊C4—5021no—
211C4⋊C4—2511no—
212C4⋊C4—2512no—
213C4⋊C4—2511no—
214C4⋊C4D100411yescharacteristic
215C22⋊C4—5021no—
216C22⋊C4—5021no—
217C22⋊C4—5022no—
218C22⋊C4—5022no—
219C22⋊C4D100412yes—
220C42C22×D25111yescharacteristic
221C23—20082no—
222C23—10044no—
223C23—10044no—
224C23—5021no—
225C23—5022no—
226C23—5021no—
227C23C2×D100412yes—
228Q8—10042no—
229D4—20081no—
230D4—10044no—
231D4—10044no—
232D4—10044no—
233D4—10042no—
234D4—5021no—
235D4—5022no—
236D4—5021no—
237D4—5022no—
238D4—5021no—
239D4—2512no—
240D4D200814yescomplemented
241C2×C4—22no—
242C2×C4—22no—
243C2×C4—10044no—
244C2×C4—10042no—
245C2×C4—10042no—
246C2×C4—5021no—
247C2×C4—5021no—
248C2×C4—5022no—
249C2×C4C2×D100411yescharacteristic
250C2×C4C2×D100412yes—
251C2×C4C2×D100411yescharacteristic
252C2×C4C23×D25111yescharacteristic
253C8—21no—
254C8—21no—
255C8—22no—
256C22—22no—
257C22—22no—
258C22—20088no—
259C22—10041no—
260C22—10042no—
261C22—10042no—
262C22—10041no—
263C22—10042no—
264C22—10041no—
265C22—10041no—
266C22—10041no—
267C22—5022no—
268C22C2×D200814yes—
269C22C22×D100411yescharacteristic
270C4—21no—
271C4—22no—
272C4—10041no—
273C4—10041no—
274C4—10042no—
275C4C2×D200812yes—
276C4#41×D25111yescharacteristic
277C4#2254876011yescharacteristic
278C2—24no—
279C2—20082no—
280C2—20081no—
281C2—10042no—
282C2#3063233211yescharacteristic
283C2C22×D200811yescharacteristic
284C2#3063249911yescharacteristic
subgroupG/H
centerC22C22×D1004
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
FittingC8×D4×C251C2
socleC22×C251C22×D4
hypercenterC8×D4D251
hypocenterC251#3294
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C22 ◁ C4×D4 ◁ C8×D4

Complements

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

NH ≅ G/Ntype
1. #30632341C2semidirect
2. D4×D1004C2semidirect
3. #30632034C2semidirect
4. #30632001C2semidirect
5. #30631968C2semidirect
6. #30631923C2semidirect
7. #30631883C2semidirect
8. #30631823C2semidirect
9. C4×D2008C2semidirect
10. C2×#22548637C2semidirect
11. C8×D4×C251C2semidirect
16. C2×D2008C22semidirect
17. #22548715C22semidirect
23. #22548689C22semidirect
26. C4×D1004C22semidirect
27. C2×C4⋊Dic251C22semidirect
28. #22548649C22semidirect
29. #22548649C22semidirect
32. #22548637C22semidirect
33. #22548637C22semidirect
34. #22548637C22semidirect
35. #22548637C22semidirect
36. C22×C8×C251C22semidirect
38. C4⋊C8×C251C22semidirect
39. C22⋊C8×C251C22semidirect
40. C4×C8×C251C22semidirect
93. D1004#41semidirect
103. D4×C251D8semidirect
142. C251#3294semidirect
153. C8×D4D251semidirect
160. C22×C8C2×D251semidirect
175. C4⋊C8C2×D251semidirect
182. C22⋊C8C2×D251semidirect
183. C4×C8C2×D251semidirect
240. D4D2008semidirect

Maximal subgroups

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

subgroupindex
#306323412
D4×D10042
#306320342
#306320012
#306319682
#306319232
#306318832
#306318232
C4×D20082
C2×#225486372
C8×D4×C2512
#3294251