Finite Group Databasebeta

← #44562084

Subgroups — #44562084

Order 32128.

484 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 484 conjugacy classes of subgroups — too many to draw. Only its 123 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
2C2×#22548737C211yesmaximal, characteristic, complemented
3#30632339C212yesmaximal, complemented
4#30632065C211yesmaximal, characteristic, complemented
5#30631929C211yesmaximal, characteristic, complemented
6#30631891C212yesmaximal, complemented
7#30631852C212yesmaximal, complemented
8#30631847C212yesmaximal, complemented
9#30631769C211yesmaximal, characteristic, complemented
10C8×C22⋊D251C211yesmaximal, characteristic, complemented
11#467×C251C211yesmaximal, characteristic, complemented
12C2×#16860811C2212yes—
13C2×#16860810C2212yes—
14#22548738—24no—
15#22548737—24no—
16C2×Dic1004C2211yescharacteristic
17C2×#16860800C2212yes—
18C2×#16860799C2212yes—
19C2×#16860797C2212yes—
20C2×D2008C2211yescharacteristic
21C2×C8×D251C2211yescharacteristic
22#22548713C2212yes—
23#22548700—22no—
24#22548697—22no—
25#22548695C2212yes—
26#22548688—22no—
27#22548677C2212yes—
28C4×C22⋊D251C2211yescharacteristic
29C4⋊C4×D251—22no—
30Dic251⋊C8C2211yescharacteristic
31#22548638C2212yes—
32#22548635C2212yes—
33C22⋊(C251⋊C8)C2211yescharacteristic
34(C2×D251)⋊C8C2211yescharacteristic
35C8×Dic251C2211yescharacteristic
36C2×Q16×C251C2211yescharacteristic
37C22×C8×C251C2211yescharacteristic
38C22⋊Q8×C251C2212yes—
39#107×C251C2211yescharacteristic
40#103×C251C2212yes—
41#16860811—24no—
42#16860811—42no—
43#16860810—24no—
44#16860810—42no—
45C2×Dic502C2312yes—
46C2×D1004—22no—
47C2×D1004C2312yes—
48C2×C22⋊D251—22no—
49C2×C22⋊D251D411yescharacteristic
50C2×C4×D251—22no—
51C2×C4×D251—22no—
52C2×C4×D251—22no—
53C2×C4×D251C2311yescharacteristic
54Dic1004—22no—
55#16860800—42no—
56#16860799—24no—
57#16860797—24no—
58D2008D414yescomplemented
59C2×C251⋊C8C2311yescharacteristic
60C2×C251⋊C8C2311yescharacteristic
61C8×D251—22no—
62C4⋊Dic251—22no—
63Dic251⋊C4—22no—
64Dic251⋊C4—22no—
65Dic251⋊C4D411yescharacteristic
66C22⋊Dic251D411yescharacteristic
67(C2×D251)⋊C4—22no—
68(C2×D251)⋊C4—22no—
69(C2×D251)⋊C4D411yescharacteristic
70C4×Dic251C2311yescharacteristic
71C2×Q8×C251C2312yes—
72C22×C4×C251C2311yescharacteristic
73Q16×C251—22no—
74C2×C8×C251—22no—
75C2×C8×C251C2311yescharacteristic
76C2×C8×C251C2311yescharacteristic
77C4⋊C4×C251—22no—
78C4⋊C4×C251C2312yes—
79C22⋊C4×C251—22no—
80C22×D251—22no—
81C22×D251C2×D411yescharacteristic
82Dic502—22no—
83Dic502—22no—
84D1004—22no—
85D1004—44no—
86D1004C2×D414yes—
87C22⋊D251—22no—
88C22⋊D251—44no—
89C2×Dic251—22no—
90C2×Dic251C2×D411yescharacteristic
91C2×Dic251C2×D411yescharacteristic
92C2×Dic251C2×D411yescharacteristic
93C4×D251—21no—
94C4×D251—21no—
95C4×D251—44no—
96C4×D251—44no—
97C4×D251—44no—
98C251⋊C8—21no—
99C251⋊C8—21no—
100C23×C251C2×D411yescharacteristic
101Q8×C251—22no—
102Q8×C251—22no—
103C2×C4×C251—21no—
104C2×C4×C251—21no—
105C2×C4×C251—22no—
106C2×C4×C251—22no—
107C2×C4×C251C2×D411yescharacteristic
108C2×C4×C251C2411yescharacteristic
109C8×C251—21no—
110C8×C251C2×D412yes—
111C22×C251—21no—
112C22×C251—21no—
113C22×C251—21no—
114C22×C251C22×D411yescharacteristic
115C2×D251—21no—
116C2×D251—24no—
117C2×D251—21no—
118C2×D251—21no—
119C2×D251—42no—
120C2×D251—42no—
121C4×C251—21no—
122C4×C251—42no—
123C4×C251—42no—
124C4×C251C22×D411yescharacteristic
125C4×C251H(5,𝔽2)11yescharacteristic
126Dic251—21no—
127Dic251—21no—
128Dic251—42no—
129Dic251#13512yescomplemented
130C2×C251—41no—
131C2×C251D4211yescharacteristic
132C2×C251C2×#13511yescharacteristic
133C2×C251#57811yescharacteristic
134D251—41no—
135D251—44no—
136C251#328311yescharacteristic, complemented, Sylow
137#3283—2511nomaximal, Sylow
138C2×#135—2511no—
139C2×#135—2511no—
140#547—2512no—
141#494—2511no—
142#467—2511no—
143#467D25111yescharacteristic, complemented
144#461—2512no—
145#452—2512no—
146#452—2512no—
147C4×D8—2511no—
148C8×D4—2511no—
149C2×#41—2512no—
150C2×#41—2512no—
151#135—5024no—
152#135—5024no—
153C2×Q16—2512no—
154C2×Q16—2511no—
155C2×Q16C2×D25111yescharacteristic
156C2×SD16—2512no—
157C2×SD16—2512no—
158C2×D8—2511no—
159C22×C8—2511no—
160C22×C8C2×D25111yescharacteristic
161C4⋊Q8—2512no—
162#123—5022no—
163#123—5022no—
164C22⋊Q8—2512no—
165C22⋊Q8C2×D25112yes—
166#121—5022no—
167C4×D4—2512no—
168C4×D4—2511no—
169C2×C4⋊C4—5022no—
170#107C2×D25111yescharacteristic
171C4⋊C8—2511no—
172#103—2512no—
173#103C2×D25112yes—
174#102—2512no—
175C22⋊C8—2511no—
176C22⋊C8—2511no—
177C4×C8—2511no—
178#41—10042no—
179#41—10042no—
180#41—5024no—
181#41—5024no—
182C2×Q8—2512no—
183C2×Q8C22×D25112yes—
184C2×D4—5022no—
185C2×D4—5022no—
186C2×D4—2511no—
187C2×D4—2512no—
188C22×C4—5022no—
189C22×C4—5022no—
190C22×C4—5022no—
191C22×C4—2511no—
192C22×C4C22×D25111yescharacteristic
193Q16—22no—
194Q16—10042no—
195Q16—5022no—
196SD16—5024no—
197SD16—5024no—
198D8—2514no—
199C2×C8—22no—
200C2×C8—5022no—
201C2×C8—2511no—
202C2×C8—2511no—
203C2×C8C22×D25111yescharacteristic
204C2×C8C22×D25111yescharacteristic
205C4⋊C4—22no—
206C4⋊C4—5022no—
207C4⋊C4—5022no—
208C4⋊C4—5022no—
209C4⋊C4—2511no—
210C4⋊C4C22×D25112yes—
211C22⋊C4—22no—
212C22⋊C4—5022no—
213C22⋊C4—5022no—
214C22⋊C4—2511no—
215C22⋊C4—2511no—
216C42—2511no—
217C23—5022no—
218C23—2511no—
219C23D4×D25111yescharacteristic
220Q8—22no—
221Q8—22no—
222Q8—5022no—
223Q8—5022no—
224D4—10044no—
225D4—10044no—
226D4—5022no—
227D4—5022no—
228D4—2514no—
229C2×C4—21no—
230C2×C4—21no—
231C2×C4—22no—
232C2×C4—22no—
233C2×C4—10044no—
234C2×C4—10044no—
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—
247C8D4×D25112yes—
248C22—21no—
249C22—21no—
250C22—21no—
251C22—10042no—
252C22—10042no—
253C22—5021no—
254C22—5021no—
255C22—5024no—
256C22—5021no—
257C22C2×D4×D25111yescharacteristic
258C4—21no—
259C4—42no—
260C4—42no—
261C4—10042no—
262C4—5021no—
263C4—5021no—
264C4—2512no—
265C4C2×D4×D25111yescharacteristic
266C4#2254876111yescharacteristic
267C2—41no—
268C2—10044no—
269C2—10041no—
270C2#3063232111yescharacteristic
271C2#135×D25111yescharacteristic
272C2#3063250011yescharacteristic
subgroupG/H
centerC22C2×D4×D251
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
Fitting#467×C251C2
socleC22×C251C22×D4
hypercenter#467D251
hypocenterC251#3283
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C22 ◁ C22×C4 ◁ #467

Complements

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

NH ≅ G/Ntype
1. C2×#22548738C2semidirect
2. C2×#22548737C2semidirect
3. #30632339C2semidirect
4. #30632065C2semidirect
5. #30631929C2semidirect
6. #30631891C2semidirect
7. #30631852C2semidirect
8. #30631847C2semidirect
9. #30631769C2semidirect
10. C8×C22⋊D251C2semidirect
11. #467×C251C2semidirect
58. D2008D4semidirect
129. Dic251#135semidirect
136. C251#3283semidirect
143. #467D251semidirect

Maximal subgroups

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

subgroupindex
C2×#225487382
C2×#225487372
#306323392
#306320652
#306319292
#306318912
#306318522
#306318472
#306317692
C8×C22⋊D2512
#467×C2512
#3283251