Finite Group Databasebeta

← #44562081

Subgroups — #44562081

Order 32128.

484 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 484 conjugacy classes of subgroups — too many to draw. Only its 131 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×#22548734C211yesmaximal, characteristic, complemented
2#30632336C212yesmaximal, complemented
3#30632065C211yesmaximal, characteristic, complemented
4#30631928C211yesmaximal, characteristic, complemented
5#30631925C211yesmaximal, characteristic, complemented
6#30631890C212yesmaximal, complemented
7#30631849C212yesmaximal, complemented
8#30631846C212yesmaximal, complemented
9D8×Dic251C211yesmaximal, characteristic, complemented
10C8×C22⋊D251C211yesmaximal, characteristic, complemented
11C2×#135×C251C211yesmaximal, characteristic, complemented
12C2×#16860809C2212yes—
13#22548734—24no—
14C2×Dic1004C2211yescharacteristic
15C2×#16860800C2212yes—
16C2×#16860798C2212yes—
17C2×C8×D251C2211yescharacteristic
18#22548713C2212yes—
19#22548698—22no—
20#22548696—22no—
21#22548694C2212yes—
22#22548691C2212yes—
23#22548687—22no—
24D4×Dic251C2212yes—
25C4×C22⋊D251C2211yescharacteristic
26C2×Dic251⋊C4—22no—
27#22548649C2211yescharacteristic
28Dic251⋊C8C2211yescharacteristic
29#22548640C2212yes—
30#22548639C2212yes—
31#22548636C2212yes—
32C22⋊(C251⋊C8)C2211yescharacteristic
33(C2×D251)⋊C8C2211yescharacteristic
34C8×Dic251C2211yescharacteristic
35C2×#41×C251C2212yes—
36#135×C251—24no—
37C2×Q16×C251C2211yescharacteristic
38C2×SD16×C251C2212yes—
39C2×D8×C251C2211yescharacteristic
40C22×C8×C251C2211yescharacteristic
41#16860809—24no—
42#16860809—42no—
43C2×Dic502C2312yes—
44C2×C22⋊D251—22no—
45C2×C22⋊D251D411yescharacteristic
46C22×Dic251—22no—
47C22×Dic251—22no—
48C2×C4×D251C2311yescharacteristic
49Dic1004—22no—
50#16860800—42no—
51#16860798—24no—
52C2×C251⋊C8C2311yescharacteristic
53C2×C251⋊C8C2311yescharacteristic
54C8×D251—22no—
55C4⋊Dic251C2312yes—
56Dic251⋊C4—22no—
57Dic251⋊C4—22no—
58Dic251⋊C4—22no—
59Dic251⋊C4—22no—
60Dic251⋊C4D411yescharacteristic
61C22⋊Dic251—22no—
62C22⋊Dic251—22no—
63C22⋊Dic251D411yescharacteristic
64(C2×D251)⋊C4—22no—
65(C2×D251)⋊C4D411yescharacteristic
66C4×Dic251C2311yescharacteristic
67#41×C251—24no—
68#41×C251—42no—
69C2×Q8×C251C2312yes—
70C2×D4×C251—22no—
71C2×D4×C251C2312yes—
72C22×C4×C251—22no—
73C22×C4×C251C2311yescharacteristic
74Q16×C251—22no—
75SD16×C251—24no—
76D8×C251D414yescomplemented
77C2×C8×C251—22no—
78C2×C8×C251C2311yescharacteristic
79C2×C8×C251C2311yescharacteristic
80C22×D251C2×D411yescharacteristic
81Dic502—22no—
82Dic502—22no—
83C22⋊D251—22no—
84C22⋊D251—44no—
85C2×Dic251—22no—
86C2×Dic251—22no—
87C2×Dic251—44no—
88C2×Dic251—44no—
89C2×Dic251C2×D411yescharacteristic
90C2×Dic251C2×D411yescharacteristic
91C2×Dic251C2×D411yescharacteristic
92C4×D251—21no—
93C4×D251—21no—
94C251⋊C8—21no—
95C251⋊C8—21no—
96C23×C251—22no—
97C23×C251C2×D411yescharacteristic
98Q8×C251—22no—
99Q8×C251—22no—
100D4×C251—22no—
101D4×C251—44no—
102D4×C251C2×D414yes—
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—24no—
113C22×C251—21no—
114C22×C251—21no—
115C22×C251—42no—
116C22×C251—42no—
117C22×C251C22×D411yescharacteristic
118C2×D251—21no—
119C2×D251—21no—
120C2×D251—21no—
121C4×C251—21no—
122C4×C251—42no—
123C4×C251C22×D411yescharacteristic
124C4×C251H(5,𝔽2)11yescharacteristic
125Dic251—21no—
126Dic251—21no—
127Dic251—42no—
128Dic251—42no—
129Dic251#13512yescomplemented
130C2×C251—41no—
131C2×C251—44no—
132C2×C251D4211yescharacteristic
133C2×C251C2×#13511yescharacteristic
134C2×C251#57811yescharacteristic
135D251—41no—
136C251#328311yescharacteristic, complemented, Sylow
137#3283—2511nomaximal, Sylow
138C2×#135—2511no—
139C2×#135D25111yescharacteristic, complemented
140#547—2512no—
141#494—2511no—
142#467—2511no—
143#467—2511no—
144#461—2512no—
145#452—2512no—
146#452—2512no—
147C4×D8—2511no—
148C8×D4—2511no—
149C2×#41—2512no—
150C2×#41C2×D25112yes—
151#135—24no—
152#135—5024no—
153C2×Q16—2512no—
154C2×Q16—2511no—
155C2×Q16C2×D25111yescharacteristic
156C2×SD16—2512no—
157C2×SD16C2×D25112yes—
158C2×D8C2×D25111yescharacteristic
159C22×C8—2511no—
160C22×C8C2×D25111yescharacteristic
161C4⋊Q8—2512no—
162#123—5022no—
163#123—5022no—
164C22⋊Q8—2512no—
165C22⋊Q8—2512no—
166#121—5022no—
167C4×D4—2512no—
168C4×D4—2511no—
169C2×C4⋊C4—5022no—
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—24no—
179#41—42no—
180#41—10042no—
181#41—5024no—
182C2×Q8—2512no—
183C2×Q8C22×D25112yes—
184C2×D4—22no—
185C2×D4—5022no—
186C2×D4—2511no—
187C2×D4C22×D25112yes—
188C22×C4—22no—
189C22×C4—5022no—
190C22×C4—5022no—
191C22×C4—2511no—
192C22×C4C22×D25111yescharacteristic
193Q16—22no—
194Q16—10042no—
195Q16—5022no—
196SD16—24no—
197SD16—5024no—
198D8C22⋊D25114yescomplemented
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—5022no—
208C4⋊C4—5022no—
209C4⋊C4—2512no—
210C4⋊C4—2511no—
211C22⋊C4—5022no—
212C22⋊C4—5022no—
213C22⋊C4—5022no—
214C22⋊C4—2511no—
215C22⋊C4—2511no—
216C42—2511no—
217C23—22no—
218C23—2511no—
219C23D4×D25111yescharacteristic
220Q8—22no—
221Q8—22no—
222Q8—5022no—
223Q8—5022no—
224D4—22no—
225D4—44no—
226D4—10044no—
227D4—5022no—
228D4C2×C22⋊D25114yes—
229C2×C4—21no—
230C2×C4—21no—
231C2×C4—22no—
232C2×C4—44no—
233C2×C4—10044no—
234C2×C4—10044no—
235C2×C4—5022no—
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—21no—
251C22—24no—
252C22—42no—
253C22—42no—
254C22—5021no—
255C22—5021no—
256C22—5021no—
257C22C2×D4×D25111yescharacteristic
258C4—21no—
259C4—42no—
260C4—10042no—
261C4—10042no—
262C4—5021no—
263C4—5021no—
264C4—2512no—
265C4C22×C22⋊D25111yescharacteristic
266C4#2254876011yescharacteristic
267C2—41no—
268C2—44no—
269C2—10041no—
270C2D4×C22⋊D25111yescharacteristic
271C2#135×D25111yescharacteristic
272C2#3063250111yescharacteristic
subgroupG/H
centerC22C2×D4×D251
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
FittingC2×#135×C251C2
socleC22×C251C22×D4
hypercenterC2×#135D251
hypocenterC251#3283
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#135

Complements

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

NH ≅ G/Ntype
1. C2×#22548734C2semidirect
2. #30632336C2semidirect
3. #30632065C2semidirect
4. #30631928C2semidirect
5. #30631925C2semidirect
6. #30631890C2semidirect
7. #30631849C2semidirect
8. #30631846C2semidirect
9. D8×Dic251C2semidirect
10. C8×C22⋊D251C2semidirect
11. C2×#135×C251C2semidirect
76. D8×C251D4semidirect
129. Dic251#135semidirect
136. C251#3283semidirect
139. C2×#135D251semidirect
198. D8C22⋊D251semidirect

Maximal subgroups

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

subgroupindex
C2×#225487342
#306323362
#306320652
#306319282
#306319252
#306318902
#306318492
#306318462
D8×Dic2512
C8×C22⋊D2512
C2×#135×C2512
#3283251