Finite Group Databasebeta

← #44559659

Subgroups — #44559659

Order 32128.

700 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 700 conjugacy classes of subgroups — too many to draw. Only its 149 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×#22548762C211yesmaximal, characteristic
2C22×#16860800C211yesmaximal, characteristic, complemented
3#30632261C211yesmaximal, characteristic, complemented
4#30631846C214yesmaximal, complemented
5#30631841C214yesmaximal, complemented
6#30631697C211yesmaximal, characteristic, complemented
7#30631683C211yesmaximal, characteristic, complemented
8#30631655C211yesmaximal, characteristic, complemented
9C2×#137×C251C211yesmaximal, characteristic, complemented
10#22548762—24no—
11#22548762—24no—
12C2×#16860810—21no—
13C2×#16860810C2211yescharacteristic
14C2×#16860809—22no—
15C2×#16860809C2212yes—
16C22×Dic502—21no—
17C22×Dic502C2211yescharacteristic
18C2×Q8×D251C2212yes—
19C2×#16860800—24no—
20C2×#16860800C2214yes—
21C22×C251⋊C8C2211yescharacteristic
22#22548713—21no—
23#22548704—21no—
24#22548698—22no—
25#22548691C2214yes—
26#22548674C2211yescharacteristic
27#22548640C2214yes—
28#22548639C2212yes—
29#22548636C2212yes—
30(C2×D251)⋊C8C2214yes—
31C2×#41×C251C2211yescharacteristic
32C22×Q8×C251C2211yescharacteristic
33#137×C251—24no—
34C2×Q16×C251C2212yes—
35C2×SD16×C251C2212yes—
36C2×M16×C251C2211yescharacteristic
37#16860810—28no—
38#16860810—22no—
39#16860810—22no—
40#16860809—216no—
41#16860809—24no—
42#16860809—24no—
43C2×Dic502—28no—
44C2×Dic502—22no—
45C2×Dic502—21no—
46C2×Dic502—22no—
47C2×Dic502D414yes—
48C2×Dic502D411yescharacteristic
49C2×Dic502C2312yes—
50Q8×D251—24no—
51Q8×D251—24no—
52C2×D1004D411yescharacteristic
53C2×C22⋊D251—22no—
54C2×C22⋊D251D412yes—
55C22×Dic251—22no—
56C22×Dic251—21no—
57C2×C4×D251—22no—
58C2×C4×D251C2312yes—
59#16860800—28no—
60C2×C251⋊C8—22no—
61C2×C251⋊C8C2312yes—
62C4⋊Dic251C2312yes—
63Dic251⋊C4—24no—
64C22⋊Dic251—21no—
65(C2×D251)⋊C4—24no—
66C4×Dic251—21no—
67#41×C251—22no—
68#41×C251D414yes—
69C2×Q8×C251—21no—
70C2×Q8×C251—22no—
71C2×Q8×C251—22no—
72C2×Q8×C251C2312yes—
73C2×Q8×C251C2311yescharacteristic
74C2×D4×C251—21no—
75C2×D4×C251C2311yescharacteristic
76C22×C4×C251—21no—
77C22×C4×C251—21no—
78C22×C4×C251C2311yescharacteristic
79Q16×C251—24no—
80SD16×C251—24no—
81M16×C251—22no—
82C2×C8×C251C2312yes—
83C22×D251C2×D412yes—
84Dic502—28no—
85Dic502—24no—
86Dic502—22no—
87Dic502—22no—
88Dic502C2×D414yes—
89D1004—22no—
90C22⋊D251—28no—
91C22⋊D251—24no—
92C2×Dic251—28no—
93C2×Dic251—24no—
94C2×Dic251—22no—
95C2×Dic251—22no—
96C2×Dic251C2×D412yes—
97C4×D251—28no—
98C4×D251—22no—
99C4×D251—22no—
100C251⋊C8—22no—
101C23×C251—21no—
102C23×C251C2×D411yescharacteristic
103Q8×C251—21no—
104Q8×C251—22no—
105Q8×C251—22no—
106Q8×C251—42no—
107Q8×C251C2×D412yes—
108D4×C251—24no—
109D4×C251—21no—
110D4×C251C2×D412yes—
111C2×C4×C251—21no—
112C2×C4×C251—24no—
113C2×C4×C251—22no—
114C2×C4×C251—42no—
115C2×C4×C251C2×D411yescharacteristic
116C2×C4×C251C2×D412yes—
117C2×C4×C251C2×D412yes—
118C2×C4×C251C2411yescharacteristic
119C8×C251—22no—
120C22×C251—22no—
121C22×C251—22no—
122C22×C251—42no—
123C22×C251#12012yes—
124C22×C251C22×D411yescharacteristic
125C2×D251—22no—
126C2×D251—24no—
127C4×C251—22no—
128C4×C251—42no—
129C4×C251#12012yes—
130C4×C251C22×D411yescharacteristic
131C4×C251C22×D411yescharacteristic
132Dic251—24no—
133Dic251—22no—
134Dic251—42no—
135C2×C251—22no—
136C2×C251—42no—
137C2×C251C2×#12011yescharacteristic
138C2×C251#57812yes—
139D251—42no—
140C251#301111yescharacteristic, complemented, Sylow
141#3011—2511nomaximal, Sylow
142C2×2−1+4—2511no—
143C2×#137D25111yescharacteristic, complemented
144C22×Q16—2511no—
145#536—2511no—
146#452—2514no—
147#451—2514no—
148#419—2511no—
149#416—2511no—
150#408—2511no—
1512−1+4—5024no—
1522−1+4—5024no—
153C2×#41—5021no—
154C2×#41—5022no—
155C2×#41—2512no—
156C2×#41—2511no—
157C2×#41C2×D25111yescharacteristic
158C22×Q8—5021no—
159C22×Q8—2512no—
160C22×Q8—2511no—
161C22×Q8C2×D25111yescharacteristic
162#137—24no—
163C2×Q16—5024no—
164C2×Q16—2514no—
165C2×Q16C2×D25112yes—
166C2×SD16C2×D25112yes—
167C2×M16C2×D25111yescharacteristic
168C22×C8—2511no—
169C4⋊Q8—5021no—
170#124—5021no—
171#123—5022no—
172C22⋊Q8—2514no—
173#117—2511no—
174#103—2512no—
175#103—2514no—
176#102—2512no—
177C22⋊C8—2514no—
178#41—22no—
179#41—5028no—
180#41—50216no—
181#41—5024no—
182#41—5024no—
183#41—5022no—
184#41—5022no—
185#41C22⋊D25114yes—
186C2×Q8—21no—
187C2×Q8—22no—
188C2×Q8—22no—
189C2×Q8—5028no—
190C2×Q8—5022no—
191C2×Q8—5024no—
192C2×Q8—5024no—
193C2×Q8—5021no—
194C2×Q8—5022no—
195C2×Q8—2514no—
196C2×Q8—2512no—
197C2×Q8—2511no—
198C2×Q8C22×D25112yes—
199C2×Q8C22×D25111yescharacteristic
200C2×D4—21no—
201C2×D4—5022no—
202C2×D4—2512no—
203C2×D4—2511no—
204C2×D4C22×D25111yescharacteristic
205C22×C4—21no—
206C22×C4—21no—
207C22×C4—5022no—
208C22×C4—5021no—
209C22×C4—5022no—
210C22×C4—2512no—
211C22×C4C22×D25111yescharacteristic
212Q16—24no—
213Q16—5028no—
214SD16—24no—
215M16—22no—
216C2×C8—5022no—
217C2×C8—2512no—
218C2×C8C22×D25112yes—
219C4⋊C4—5024no—
220C4⋊C4—2512no—
221C22⋊C4—5024no—
222C22⋊C4—5021no—
223C42—5021no—
224C23—21no—
225C23—2512no—
226C23D4×D25111yescharacteristic
227Q8—21no—
228Q8—22no—
229Q8—22no—
230Q8—42no—
231Q8—5028no—
232Q8—5022no—
233Q8—5024no—
234Q8—5022no—
235Q8—2514no—
236Q8C2×C22⋊D25112yes—
237D4—24no—
238D4—21no—
239D4—5028no—
240D4—5024no—
241D4—5022no—
242D4C2×C22⋊D25112yes—
243C2×C4—21no—
244C2×C4—24no—
245C2×C4—22no—
246C2×C4—42no—
247C2×C4—5028no—
248C2×C4—5022no—
249C2×C4—5028no—
250C2×C4—5022no—
251C2×C4—5024no—
252C2×C4—5022no—
253C2×C4—5022no—
254C2×C4—2512no—
255C2×C4D4×D25112yes—
256C2×C4D4×D25111yescharacteristic
257C2×C4C2×C22⋊D25112yes—
258C2×C4C23×D25111yescharacteristic
259C8—22no—
260C8—5022no—
261C22—22no—
262C22—22no—
263C22—42no—
264C22—5024no—
265C22—5022no—
266C22#2254868312yes—
267C22C2×D4×D25111yescharacteristic
268C4—22no—
269C4—42no—
270C4—10042no—
271C4—5024no—
272C4—5022no—
273C4#2254868312yes—
274C4C2×D4×D25111yescharacteristic
275C4C22×C22⋊D25111yescharacteristic
276C2—22no—
277C2—42no—
278C2—10042no—
279C2C2×#2254868311yescharacteristic
280C2#3063250312yes—
subgroupG/H
centerC22C2×D4×D251
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
FittingC2×#137×C251C2
socleC22×C251C22×D4
prime radicalC2×#22548762C2
hypercenterC2×#137D251
hypocenterC251#3011
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#137

Complements

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

NH ≅ G/Ntype
2. C22×#16860800C2semidirect
3. #30632261C2semidirect
4. #30631846C2semidirect
5. #30631841C2semidirect
6. #30631697C2semidirect
7. #30631683C2semidirect
8. #30631655C2semidirect
9. C2×#137×C251C2semidirect
140. C251#3011semidirect
143. C2×#137D251semidirect

Maximal subgroups

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

subgroupindex
C2×#225487622
C22×#168608002
#306322612
#306318462
#306318412
#306316972
#306316832
#306316552
C2×#137×C2512
#3011251