Finite Group Databasebeta

← #44559847

Subgroups — #44559847

Order 32128.

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

normal subgroups only

This group has 480 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×#22548744C212yesmaximal
2C4⋊Q8×D251C211yesmaximal, characteristic, complemented
3C4×#16860809C211yesmaximal, characteristic
4#30631898C211yesmaximal, characteristic, complemented
5#30631896C212yesmaximal, complemented
6#30631894C211yesmaximal, characteristic, complemented
7#30631892C211yesmaximal, characteristic, complemented
8#30631890C212yesmaximal, complemented
9#30631709C211yesmaximal, characteristic, complemented
10#30631694C212yesmaximal, complemented
11#461×C251C211yesmaximal, characteristic, complemented
12C2×#16860809C2211yescharacteristic
13C2×Q8×D251C2212yes—
14#22548744—28no—
15C2×Dic1004C2212yes—
16C2×#16860800C2212yes—
17C2×#16860798C2212yes—
18C2×#16860793C2212yes—
19#22548715C2211yescharacteristic
20#22548714—21no—
21#22548714—21no—
22#22548713C2212yes—
23C4×Dic502C2211yescharacteristic
24#22548680C2211yescharacteristic
25D4×Dic251C2211yescharacteristic
26C4×C22⋊D251—21no—
27(C22⋊D251)⋊C4—21no—
28#22548673C2211yescharacteristic
29#22548671—21no—
30C4⋊C4×D251—21no—
31C4⋊C4×D251—21no—
32C2×C4×Dic251—21no—
33C42×D251C2211yescharacteristic
34C4⋊(C251⋊C8)C2211yescharacteristic
35Dic251⋊C8C2212yes—
36#22548640C2212yes—
37#22548639C2212yes—
38#22548638C2212yes—
39C2×SD16×C251C2212yes—
40C4⋊Q8×C251C2211yescharacteristic
41C4×D4×C251C2211yescharacteristic
42C4⋊C8×C251C2211yescharacteristic
43#103×C251C2212yes—
44#16860809—22no—
45#16860809D414yes—
46C2×Dic502—22no—
47C2×Dic502C2312yes—
48C2×Dic502C2311yescharacteristic
49Q8×D251—24no—
50Q8×D251—24no—
51C2×C22⋊D251—21no—
52C22×Dic251—21no—
53C22×Dic251—21no—
54C2×C4×D251—22no—
55C2×C4×D251D411yescharacteristic
56C2×C4×D251D411yescharacteristic
57C2×C4×D251C2311yescharacteristic
58Dic1004—24no—
59#16860800—24no—
60#16860798—24no—
61#16860793—24no—
62C2×C251⋊C8C2312yes—
63C4⋊Dic251—21no—
64C4⋊Dic251—21no—
65C4⋊Dic251C2311yescharacteristic
66Dic251⋊C4—22no—
67Dic251⋊C4—22no—
68Dic251⋊C4—21no—
69Dic251⋊C4—21no—
70C22⋊Dic251—21no—
71(C2×D251)⋊C4—21no—
72C4×Dic251—21no—
73C4×Dic251—21no—
74C4×Dic251—21no—
75C4×Dic251D411yescharacteristic
76C4×Dic251D411yescharacteristic
77C4×Dic251C2311yescharacteristic
78C2×Q8×C251C2312yes—
79C2×D4×C251C2311yescharacteristic
80C22×C4×C251—21no—
81SD16×C251—24no—
82C2×C8×C251C2312yes—
83C4⋊C4×C251—21no—
84C4⋊C4×C251—21no—
85C4⋊C4×C251C2311yescharacteristic
86C22⋊C4×C251—21no—
87C42×C251C2311yescharacteristic
88C22×D251C2×D411yescharacteristic
89Dic502—21no—
90Dic502—22no—
91Dic502—22no—
92Dic502—44no—
93Dic502C2×D412yes—
94C22⋊D251—24no—
95C2×Dic251—21no—
96C2×Dic251—24no—
97C2×Dic251—21no—
98C2×Dic251—22no—
99C2×Dic251—42no—
100C2×Dic251C2×D411yescharacteristic
101C2×Dic251C2×D411yescharacteristic
102C2×Dic251C2×D411yescharacteristic
103C4×D251—22no—
104C4×D251—22no—
105C4×D251—44no—
106C4×D251C2×D412yes—
107C4×D251#4112yes—
108C251⋊C8—22no—
109C23×C251—21no—
110Q8×C251—22no—
111Q8×C251—22no—
112D4×C251—21no—
113D4×C251C2×D412yes—
114C2×C4×C251—21no—
115C2×C4×C251—22no—
116C2×C4×C251—42no—
117C2×C4×C251C2×D411yescharacteristic
118C2×C4×C251C2×D411yescharacteristic
119C2×C4×C251C2411yescharacteristic
120C8×C251—22no—
121C22×C251—22no—
122C22×C251—41no—
123C22×C251—41no—
124C22×C251C22×D411yescharacteristic
125C2×D251—21no—
126C2×D251—21no—
127C2×D251#12112yes—
128C4×C251—21no—
129C4×C251—42no—
130C4×C251—41no—
131C4×C251#13712yescomplemented
132C4×C251C22×D411yescharacteristic
133C4×C251C2×#4111yescharacteristic
134Dic251—22no—
135Dic251—21no—
136Dic251—42no—
137Dic251—41no—
138Dic251#12112yes—
139Dic251#13712yescomplemented
140C2×C251—42no—
141C2×C251C2×#12111yescharacteristic
142C2×C251C2×#13711yescharacteristic
143C2×C251C2×#13711yescharacteristic
144D251—22no—
145C251#303911yescharacteristic, complemented, Sylow
146#3039—2511nomaximal, Sylow
147C2×#137—2512no—
148C2×C4⋊Q8—2511no—
149C4×#41—2511no—
150#462—2511no—
151#462—2511no—
152#462—2512no—
153#461—2512no—
154#461—2511no—
155#461D25111yescharacteristic, complemented
156#423—2511no—
157#419—2512no—
158C2×#41—2511no—
159C22×Q8—2512no—
160#137—5028no—
161C2×Q16—2512no—
162C2×Q16—2512no—
163C2×SD16—2512no—
164C2×SD16C2×D25112yes—
165C2×M16—2512no—
166C4⋊Q8—5021no—
167C4⋊Q8—5021no—
168C4⋊Q8—2512no—
169C4⋊Q8—2511no—
170C4⋊Q8C2×D25111yescharacteristic
171C4×Q8—2511no—
172C4×Q8—2511no—
173C4×D4—5021no—
174C4×D4—5021no—
175C4×D4—2511no—
176C4×D4C2×D25111yescharacteristic
177#117—5021no—
178#117—2511no—
179C2×C4⋊C4—5021no—
180C2×C4⋊C4—5021no—
181C2×C42—5021no—
182C2×C42—2511no—
183C4⋊C8—2512no—
184C4⋊C8—2511no—
185C4⋊C8C2×D25111yescharacteristic
186#103—2512no—
187#103—2512no—
188#103—2512no—
189#103C2×D25112yes—
190#41—5022no—
191#41—2514no—
192C2×Q8—5022no—
193C2×Q8—5024no—
194C2×Q8—5024no—
195C2×Q8—2512no—
196C2×Q8—2511no—
197C2×Q8C22×D25112yes—
198C2×D4—5021no—
199C2×D4C22×D25111yescharacteristic
200C22×C4—21no—
201C22×C4—5022no—
202C22×C4—5021no—
203C22×C4—5021no—
204C22×C4—2511no—
205C22×C4—2511no—
206C22×C4—2511no—
207Q16—5024no—
208Q16—5024no—
209SD16—24no—
210SD16—5024no—
211M16—5024no—
212C2×C8—2512no—
213C2×C8C22×D25112yes—
214C4⋊C4—21no—
215C4⋊C4—21no—
216C4⋊C4—5022no—
217C4⋊C4—5022no—
218C4⋊C4—5021no—
219C4⋊C4—5021no—
220C4⋊C4—5021no—
221C4⋊C4—5021no—
222C4⋊C4—2511no—
223C4⋊C4C22×D25111yescharacteristic
224C22⋊C4—21no—
225C22⋊C4—5021no—
226C22⋊C4—5021no—
227C42—5021no—
228C42—5021no—
229C42—5021no—
230C42—2511no—
231C42—2511no—
232C42—2511no—
233C42C22×D25111yescharacteristic
234C23—21no—
235C23—2511no—
236Q8—22no—
237Q8—22no—
238Q8—10044no—
239Q8—5022no—
240Q8—5022no—
241Q8—5021no—
242Q8—2512no—
243D4—21no—
244D4—5024no—
245D4D4×D25112yes—
246C2×C4—21no—
247C2×C4—22no—
248C2×C4—42no—
249C2×C4—10044no—
250C2×C4—10042no—
251C2×C4—5022no—
252C2×C4—5021no—
253C2×C4—5022no—
254C2×C4—5021no—
255C2×C4—5024no—
256C2×C4—5022no—
257C2×C4—2511no—
258C2×C4—2511no—
259C2×C4—2511no—
260C2×C4—2512no—
261C2×C4—2512no—
262C2×C4D4×D25111yescharacteristic
263C2×C4D4×D25111yescharacteristic
264C2×C4C23×D25111yescharacteristic
265C8—22no—
266C8—5022no—
267C22—22no—
268C22—41no—
269C22—41no—
270C22—5021no—
271C22—5021no—
272C22—2512no—
273C22C2×D4×D25111yescharacteristic
274C4—21no—
275C4—41no—
276C4—42no—
277C4—10041no—
278C4—10042no—
279C4—5021no—
280C4—5022no—
281C4—2512no—
282C4—2512no—
283C4#2254874412yescomplemented
284C4C2×D4×D25111yescharacteristic
285C4#41×D25111yescharacteristic
286C2—42no—
287C2—5022no—
288C2#121×D25111yescharacteristic
289C2#137×D25111yescharacteristic
290C2C2×#2254874411yescharacteristic
subgroupG/H
centerC22C2×D4×D251
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
Fitting#461×C251C2
socleC22×C251C22×D4
prime radicalC2×#16860809C22
hypercenter#461D251
hypocenterC251#3039
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C22 ◁ C42 ◁ #461

Complements

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

NH ≅ G/Ntype
2. C4⋊Q8×D251C2semidirect
4. #30631898C2semidirect
5. #30631896C2semidirect
6. #30631894C2semidirect
7. #30631892C2semidirect
8. #30631890C2semidirect
9. #30631709C2semidirect
10. #30631694C2semidirect
11. #461×C251C2semidirect
131. C4×C251#137semidirect
139. Dic251#137semidirect
145. C251#3039semidirect
155. #461D251semidirect
283. C4#22548744semidirect

Maximal subgroups

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

subgroupindex
C2×#225487442
C4⋊Q8×D2512
C4×#168608092
#306318982
#306318962
#306318942
#306318922
#306318902
#306317092
#306316942
#461×C2512
#3039251