Finite Group Databasebeta

← #44562106

Subgroups — #44562106

Order 32128.

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

normal subgroups only

This group has 482 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×#22548738C211yesmaximal, characteristic, complemented
2#30632348C212yesmaximal, complemented
3#30632072C211yesmaximal, characteristic, complemented
4#30631925C211yesmaximal, characteristic, complemented
5#30631923C211yesmaximal, characteristic, complemented
6#30631911C212yesmaximal, complemented
7#30631848C212yesmaximal, complemented
8#30631826C212yesmaximal, complemented
9Q16×Dic251C211yesmaximal, characteristic, complemented
10C8×C22⋊D251C211yesmaximal, characteristic, complemented
11C22×Q16×C251C211yesmaximal, characteristic, complemented
12C2×#16860811C2212yes—
13#22548738—24no—
14C2×#16860797C2212yes—
15C2×#16860797C2212yes—
16C2×D2008C2211yescharacteristic
17C2×C8×D251C2211yescharacteristic
18#22548704—22no—
19#22548704C2212yes—
20#22548691—22no—
21#22548691C2212yes—
22#22548689—22no—
23#22548689C2212yes—
24Q8×Dic251C2212yes—
25C4×C22⋊D251—22no—
26C4×C22⋊D251C2211yescharacteristic
27#22548649C2211yescharacteristic
28Dic251⋊C8C2211yescharacteristic
29#22548639C2212yes—
30#22548639C2212yes—
31#22548637C2212yes—
32C22⋊(C251⋊C8)C2211yescharacteristic
33(C2×D251)⋊C8C2211yescharacteristic
34C8×Dic251C2211yescharacteristic
35C22×Q8×C251C2212yes—
36C2×Q16×C251—24no—
37C2×Q16×C251C2211yescharacteristic
38C2×Q16×C251C2212yes—
39C2×Q16×C251C2211yescharacteristic
40C22×C8×C251C2211yescharacteristic
41#16860811—24no—
42#16860811—42no—
43C2×D1004—22no—
44C2×D1004C2312yes—
45C2×C22⋊D251—22no—
46C2×C22⋊D251D411yescharacteristic
47C2×C4×D251—22no—
48C2×C4×D251C2311yescharacteristic
49#16860797—24no—
50#16860797—42no—
51D2008—22no—
52C2×C251⋊C8C2311yescharacteristic
53C2×C251⋊C8C2311yescharacteristic
54C8×D251—22no—
55C4⋊Dic251—22no—
56C4⋊Dic251C2312yes—
57Dic251⋊C4—22no—
58Dic251⋊C4D411yescharacteristic
59C22⋊Dic251D411yescharacteristic
60(C2×D251)⋊C4—22no—
61(C2×D251)⋊C4—22no—
62(C2×D251)⋊C4—22no—
63(C2×D251)⋊C4—22no—
64(C2×D251)⋊C4D411yescharacteristic
65C4×Dic251—22no—
66C4×Dic251C2311yescharacteristic
67C2×Q8×C251—22no—
68C2×Q8×C251—24no—
69C2×Q8×C251—42no—
70C2×Q8×C251C2312yes—
71C2×Q8×C251C2312yes—
72C22×C4×C251—22no—
73C22×C4×C251C2311yescharacteristic
74Q16×C251—24no—
75Q16×C251—22no—
76Q16×C251D414yescomplemented
77C2×C8×C251—22no—
78C2×C8×C251C2311yescharacteristic
79C2×C8×C251C2311yescharacteristic
80C22×D251—22no—
81C22×D251C2×D411yescharacteristic
82D1004—22no—
83D1004—22no—
84D1004—44no—
85C22⋊D251—22no—
86C22⋊D251—44no—
87C2×Dic251—22no—
88C2×Dic251C2×D411yescharacteristic
89C2×Dic251C2×D411yescharacteristic
90C2×Dic251C2×D411yescharacteristic
91C4×D251—21no—
92C4×D251—21no—
93C4×D251—44no—
94C251⋊C8—21no—
95C251⋊C8—21no—
96C23×C251C2×D411yescharacteristic
97Q8×C251—22no—
98Q8×C251—22no—
99Q8×C251—22no—
100Q8×C251—44no—
101Q8×C251C2×D414yes—
102C2×C4×C251—22no—
103C2×C4×C251—21no—
104C2×C4×C251—21no—
105C2×C4×C251—22no—
106C2×C4×C251—44no—
107C2×C4×C251C2×D411yescharacteristic
108C2×C4×C251C2411yescharacteristic
109C8×C251—21no—
110C8×C251C2×D412yes—
111C22×C251—21no—
112C22×C251—21no—
113C22×C251#13512yescomplemented
114C22×C251C22×D411yescharacteristic
115C2×D251—21no—
116C2×D251—21no—
117C2×D251—21no—
118C2×D251—42no—
119C2×D251—42no—
120C2×D251—42no—
121C4×C251—21no—
122C4×C251—24no—
123C4×C251—42no—
124C4×C251C22×D411yescharacteristic
125C4×C251H(5,𝔽2)11yescharacteristic
126Dic251—21no—
127Dic251—21no—
128Dic251—21no—
129Dic251—42no—
130C2×C251—22no—
131C2×C251D4211yescharacteristic
132C2×C251C2×#13511yescharacteristic
133C2×C251#57811yescharacteristic
134D251—41no—
135D251—82no—
136C251#328511yescharacteristic, complemented, Sylow
137#3285—2511nomaximal, Sylow
138C2×#135—2511no—
139C22×Q16D25111yescharacteristic, complemented
140#548—2512no—
141#495—2511no—
142#467—2511no—
143#466—2511no—
144#464—2512no—
145#452—2512no—
146#448—2512no—
147C4×Q16—2511no—
148C8×D4—2511no—
149C2×#41—2512no—
150C22×Q8C2×D25112yes—
151#135—5024no—
152C2×Q16—24no—
153C2×Q16C2×D25112yes—
154C2×Q16C2×D25111yescharacteristic
155C2×Q16C2×D25111yescharacteristic
156C2×SD16—2512no—
157C2×SD16—2512no—
158C2×D8—2511no—
159C22×C8—2511no—
160C22×C8C2×D25111yescharacteristic
161#124—5022no—
162#124—2512no—
163C22⋊Q8—5022no—
164C22⋊Q8—2512no—
165#121—5022no—
166#121—2512no—
167C4×Q8—2512no—
168C4×D4—5022no—
169C4×D4—2511no—
170#107—2511no—
171C4⋊C8—2511no—
172#103—2512no—
173#103—2512no—
174#102—2512no—
175C22⋊C8—2511no—
176C22⋊C8—2511no—
177C4×C8—2511no—
178#41—10042no—
179#41—5024no—
180C2×Q8—22no—
181C2×Q8—24no—
182C2×Q8—42no—
183C2×Q8C22×D25112yes—
184C2×Q8C22×D25112yes—
185C2×D4—5022no—
186C2×D4—5022no—
187C2×D4—2511no—
188C2×D4—2512no—
189C22×C4—22no—
190C22×C4—5022no—
191C22×C4—2511no—
192C22×C4C22×D25111yescharacteristic
193Q16—24no—
194Q16—22no—
195Q16C22⋊D25114yescomplemented
196SD16—10042no—
197SD16—5024no—
198D8—5022no—
199C2×C8—22no—
200C2×C8—5022no—
201C2×C8—2511no—
202C2×C8—2511no—
203C2×C8C22×D25111yescharacteristic
204C2×C8C22×D25111yescharacteristic
205C4⋊C4—5022no—
206C4⋊C4—5022no—
207C4⋊C4—2512no—
208C4⋊C4—2511no—
209C22⋊C4—5022no—
210C22⋊C4—5022no—
211C22⋊C4—5022no—
212C22⋊C4—5022no—
213C22⋊C4—2511no—
214C22⋊C4—2511no—
215C42—5022no—
216C42—2511no—
217C23—5022no—
218C23—2511no—
219C23D4×D25111yescharacteristic
220Q8—22no—
221Q8—22no—
222Q8—22no—
223Q8—44no—
224Q8C2×C22⋊D25114yes—
225D4—10044no—
226D4—10044no—
227D4—5022no—
228D4—5022no—
229D4—5022no—
230C2×C4—22no—
231C2×C4—21no—
232C2×C4—21no—
233C2×C4—22no—
234C2×C4—44no—
235C2×C4—10044no—
236C2×C4—5022no—
237C2×C4—5021no—
238C2×C4—5021no—
239C2×C4—2511no—
240C2×C4—2511no—
241C2×C4—2511no—
242C2×C4D4×D25111yescharacteristic
243C2×C4C23×D25111yescharacteristic
244C8—21no—
245C8—5021no—
246C8—5021no—
247C8C2×C22⋊D25112yes—
248C22—21no—
249C22—21no—
250C22—10042no—
251C22—10042no—
252C22—10042no—
253C22—5021no—
254C22—5021no—
255C22—5021no—
256C22#2254873812yescomplemented
257C22C2×D4×D25111yescharacteristic
258C4—21no—
259C4—24no—
260C4—42no—
261C4—10042no—
262C4—5021no—
263C4—5021no—
264C4—5021no—
265C4C22×C22⋊D25111yescharacteristic
266C4#2254876011yescharacteristic
267C2—22no—
268C2—20082no—
269C2—10041no—
270C2D4×C22⋊D25111yescharacteristic
271C2C2×#2254873811yescharacteristic
272C2#3063250011yescharacteristic
subgroupG/H
centerC22C2×D4×D251
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
FittingC22×Q16×C251C2
socleC22×C251C22×D4
hypercenterC22×Q16D251
hypocenterC251#3285
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×Q16

Complements

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

NH ≅ G/Ntype
1. C2×#22548738C2semidirect
2. #30632348C2semidirect
3. #30632072C2semidirect
4. #30631925C2semidirect
5. #30631923C2semidirect
6. #30631911C2semidirect
7. #30631848C2semidirect
8. #30631826C2semidirect
9. Q16×Dic251C2semidirect
10. C8×C22⋊D251C2semidirect
11. C22×Q16×C251C2semidirect
76. Q16×C251D4semidirect
113. C22×C251#135semidirect
136. C251#3285semidirect
139. C22×Q16D251semidirect
195. Q16C22⋊D251semidirect
256. C22#22548738semidirect

Maximal subgroups

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

subgroupindex
C2×#225487382
#306323482
#306320722
#306319252
#306319232
#306319112
#306318482
#306318262
Q16×Dic2512
C8×C22⋊D2512
C22×Q16×C2512
#3285251